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Experimental and theoretical studies of the behaviour of an H-ion beam during injection and acceleration… Root, Laurence Wilbur 1974

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EXPERIMENTAL AND THEORETICAL STUDIES OF THE BEHAVIOUR OF AN H" ION "BEAM DURING INJECTION AND ACCELERATION IN THE TRIUMF CENTRAL REGION MODEL CYCLOTRON by  LAURENCE WILBUR ROOT B . S c . , Oregon State U n i v e r s i t y , 1968 M . S c , U n i v e r s i t y of B r i t i s h C o l u m b i a , 1972  A t h e s i s submitted the  in p a r t i a l  requirements  for  fulfilment  of  the degree of  Doctor of P h i l o s o p h y  in the  Department of Physics  We accept t h i s required  t h e s i s as conforming to  standard  THE UNIVERSITY OF BRITISH COLUMBIA April,  1974  the  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r  an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e  and  study.  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may by h i s r e p r e s e n t a t i v e s .  be granted by  permission.  ?^  Department of  Y$ ^  5  The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada  Date  n  ^  $  Department or  I t i s understood t h a t c o p y i n g or p u b l i c a t i o n  of t h i s t h e s i s f o r f i n a n c i a l g a i n written  the Head of my  fiW  s h a l l not be  allowed' without  my  ABSTRACT A comparison is made between the experimental behaviour of injection  the H" beam in the TRIUMF c e n t r a l  process and the  first  In o r d e r to o p t i m i z e emittance of central  the  electrostatic  the e l e c t r o s t a t i c  r a d i o - f r e q u e n c y a c c e l e r a t i n g gap. by experimental  l i n e e x i t must be matched to  i n j e c t i o n elements:  observations.  It  detail.  the phase space  To t h i s e n d , a t h e o r e t i c a l  inflector,  The a x i a l  s i x a c c e l e r a t e d turns are s t u d i e d in  injection  p r o p e r t i e s o f the  theoretical  region c y c l o t r o n .  the c y c l o t r o n performance  region acceptances.  ion o p t i c a l spiral  the beam at  and  the  study was made of  the magnet b o r e ,  the  the  d e f l e c t o r and the  first  In many cases these r e s u l t s were confirmed was a l s o shown t h e o r e t i c a l l y  that by a  s u i t a b l e c h o i c e of the a c c e l e r a t i n g gap, under optimum c o n d i t i o n s , 10% of the  i n j e c t e d beam can be d i r e c t e d w i t h i n the  in the v e r t i c a l injection 2.5  acceptances.  The e f f e c t s  l i n e were a l s o measured.  radial  of a chopper and buncher in  A minimum p u l s e l e n g t h o f  nsec was o b t a i n e d with a bunching f a c t o r o f To a c c e l e r a t e a beam to f u l l  acceptance and 30% w i t h the  approximately  3-0.  radius, vertical  s t e e r i n g had at  first  to be p r o v i d e d by means o f a s y m m e t r i c a l l y - p o w e r e d t r i m c o i l s and e l e c t r o static  deflection plates  f o r each t u r n .  be c o n s i s t e n t with the e f f e c t s alignments measured l a t e r .  The s t e e r i n g r e q u i r e d  of magnetic f i e l d  The s i z e and shape of  asymmetries and dee m i s the v e r t i c a l tune v  were found to be c o n s i s t e n t w i t h t h e o r y .  The v e r t i c a l  to be 0.17  T h i s agreed with the  ± 0.03  v a l u e o f 0.17-  f o r 20 deg phase i o n s .  The t r a n s i t i o n  i s known to  z  beam envelopes was  estimated  predicted  phase which s e p a r a t e s the v e r t i c a l l y - f o c u s e d  and defocused phases was e s t i m a t e d  to be -3  ± 3 d e g , w h i l e the  predicted  v a l u e was 0 deg. The r a d i a l  beam d i a g n o s t i c t e c h n i q u e s used f o r d e t e r m i n i n g  c e n t r i n g and isochronous o p e r a t i n g c o n d i t i o n s are d i s c u s s e d . i i  proper  With these  techniques 0.15  it  was p o s s i b l e to c e n t r e a 30 deg phase i n t e r v a l  i n . , which was the approximate  s i m p l i f i e d treatment of explain q u a l i t a t i v e l y The e f f e c t s calculated.  radia 1 -1ongitudinal  in our measurements.  o f space charge on the  first  beam.  s i x a c c e l e r a t e d turns  For a beam o c c u p y i n g a phase width of 30 d e g , these  The experimental  f o r average a c c e l e r a t e d c u r r e n t s  o b s e r v a t i o n s made on h i g h - c u r r e n t  are  effects below  beams are  d e s c r i b e d . ; p r i o r t o the shutdown of the c y c l o t r o n beams o f up to average c u r r e n t were a c c e l e r a t e d .  A  c o u p l i n g is g i v e n and used to  the b e h a v i o u r o f a small e m i t t a n c e  are p r e d i c t e d to be n e g l i g i b l e 100 u A .  uncertainty  to w i t h i n  1^0 u A  TABLE OF CONTENTS Page 1.  INTRODUCTION 1.1 1.2 1.3 1.4 1.5 1.6  2.  3.6 3-7  Introduction Median Plane Components The Ion Source The Beam Line The Magnet The RF System The Vacuum System Median Plane Diagnostic Probes INJECTION SYSTEM  Introduction Optical Properties Gf the Magnet Bore The Spiral Inflector The Deflector Experimental Performance of the InflectorDeflector System The Injection Gap Measurement of the Injection Gap Focal Powers  PROPERTIES OF THE 4.1 4.2 4.3 4.4 4.5 4.6  5.  CENTRAL REGION MODEL CYCLOTRON  PROPERTIES OF THE 3.1 3-2 3.3 3.4 3-5  4.  Fundamentals of Beam Transport Advantages of External Sources Neutral Beam Injection Median Plane Ion Injection Axial Injection Systems The TRIUMF Injection System  DESCRIPTION OF THE 2.1 2.2 2.3 2.4' 2.5 2.6 2.7 2.8  3.  1  INJECTED BEAM  Introduction The Central Region Acceptance Matching Calculations Experimental Observations on the Injected Beam Properties of the Chopped Beam Properties of the Bunched Beam"  VERTICAL STUDIES IN THE  5.4 5-5 5.6  13 13 13 13 14 15 16 17 18 20 20 20 22 25 28 32 36 39 39 39 40 43 45 52 57  CRC  5. 1 Introduct ion 5.2 V e r t i c a l Beam Losses Due 5.3EliminatingB  '  4 7 8 9 10 11  to Dee Misalignment  57 57 61  r  Confirmation of the Dee Misalignment Theory Estimating the Transition Phase Properties•of the Vertical Beam P r o f i l e s iv  64 65 67  Page 6.  7.  RADIAL STUDIES IN THE CRC  70  6.1 6.2 6.3 6.^ 6.5 6.6  70 70 71 75 78  Int roduct ion CRG Operating Conditions Radial Beam Diagnostic Techniques Results Concerning Isochronism and Centring Radial-Longitudinal Coupling: A Simply Theory Radial-Longitudina1 Coupling: Comparison with Experiment  83  ACCELERATION OF HIGH-CURRENT BEAMS  85  7-1 7-2  85  7-3 7-4 7.5  Introduct ion Approximations for the Space Charge E l e c t r i c Field Vertical Space Charge Calculations Radial-Longitudinal Space Charge Effects High-Current Measurements in the CRC  86 89 92 Sk  References  100  Figures  103  Appendices A.  Optical Properties of the Spiral  B.  First-Order Properties of the Horizontal Deflector  218  C.  Evaluation of the E l e c t r i c Field Due to a Rectangular Parallelepiped of Uniform Charge Density  226  v  Inflector  208  LIST O F TABLES Page "I II III IV  V  VI  VII  Magnet  Bore T r a n s f e r M a t r i x  21  CRC I n f l e c t o r T r a n s f e r M a t r i x  25  Deflector  21  Transfer Matrix  V e r t i c a l Behaviour o f a Beam P a s s i n g Through the I n f l e c t o r - D e f l e c t o r System f o r V a r i o u s Parameters  30  Radial Behaviour o f a Beam P a s s i n g Through the I n f l e c t o r - D e f l e c t o r System f o r V a r i o u s Parameters  31  Comparison Between Measured and C a l c u l a t e d Bunching Factors  54  Changes in Beam Spot S i z e Due to Energy Spread of Buncher  56  LIST OF FIGURES Page 2.1  Photograph of TRIUMF c e n t r a l  2 . 2  Median plane view of  2 . 3  region c y c l o t r o n  103  cyclotron  1  0  4  Photograph of CRC i n f 1 e c t o r - d e f 1 e c t o r assembly  1  0  5  2.4  Schematic drawing o f  the CRC i n j e c t i o n  1  0  6  2 . 5  B  radii  in.  1  0  7  2 . 6  Average v e r t i c a l  radius  1  0  8  2 . 7  Magnetic c o n t r i b u t i o n to v| v e r s u s r a d i u s  2 . 8  C o n t r i b u t i o n to B  2.9  z  v e r s u s azimuth at  of  line  1 0 in., 2 0  magnetic f i e l d  i n . and 3 0  as a f u n c t i o n of  | Q Q  from a s y m m e t r i c a l l y - e x c i t e d t r i m c o i l s  r  C o n t r i b u t i o n s to B  z  1  Diagram o f  1  CRC showing i n s t a l l a t i o n  of  rf  1  1  1  2  c a v i t y and  p o s i t i o n of d i a g n o s t i c probes  1  2 . 1 1  View of an r f  2 . 1 2  Cut-away view o f c e n t r a l  3-1  V e r t i c a l magnetic f i e l d a l o n g c e n t r a l magnet a x i s as a f u n c t i o n of d i s t a n c e from median plane  r e s o n a t o r segment  from a v e r t i c a l  3 . 2  0  produced by s y m m e t r i c a l l y - e x c i t e d  t ri m c o i 1 s  2 . 1 0  1  113  s e c t i o n of CRC r e s o n a t o r s as viewed  plane along the dee gap c e n t r e  ]\h  line  Behaviour of beam d i v e r g i n g from a s i n g l e p o i n t as  it  Behaviour of  parallel  beam as  it  "i.k  x-y  p r o j e c t i o n of  spiral  inflector  3 . 5  x-z  p r o j e c t i o n of  spiral  inflector  3 . 6  y - z p r o j e c t i o n of  spiral  inflector  3 . 7  C o - o r d i n a t e system f o r  inflector  3 . 8  Inflector  trajectories  f o r an i n i t i a l  h = 0.1  3.9  Inflector  trajectories  f o r an i n i t i a l  P  3 . 1 0  Inflector  trajectories  f o r an i n i t i a l  u = 0.1  3..11  Inflector  trajectories  f o r an i n i t i a l  P  vii  1  5  1  1  6  1  1  7  1  1  8  passes  through the magnet bore 3 - 3  1  passes through magnet bore  119 1  optics  2  0  121  n  u  in.  = 0 . 0 1 rad  = 0 . 0 1  in. rad  1  2  2 123  12^ 125  Page 3-12  Inflector t r a j e c t o r i e s  3-13  Beam at i n f l e c t o r exit obtained by starting with a ±10 mrad divergent beam at z ='44 in.  127  Beam at i n f l e c t o r exit obtained by s t a r t i n g with a ±0.1 in. p a r a l l e l beam at z = 44 i n .  128  Beam at deflector exit obtained by s t a r t i n g from a ±10 mrad divergent beam at z = 44 in.  129  3.14  3.15 3-16  Beam at deflector ±0.1  for an i n i t i a l  P  v  = 0.01.  126  exit obtained by starting with a 130  i n . p a r a l l e l beam at z = 44 i n .  3.17  Ion centring for different deflector  3.18  Contour plot of equipotentials for 1.5 x 1.5 i n . symmetric i nj'ect ion gap Injection gap v e r t i c a l focal power versus phase of the ion at the gap centre  3.19 3.20  Injection  gap  voltages  131  133  radial focal power versus phase of the 134  ion at the gap centre 3.21  Contour plots of equipotentials for CRC  3.22  Graph of injection gap e l e c t r i c f i e l d s versus distance from centre of gap Graph of radial injection gap focal powers versus v e r t i c a l injection gap focal powers  3.23  1 32  injection gap  4.1  Calculated CRC acceptances  4.2  CRC acceptances at low energy side of 1.5 symmetric injection gap  4.3  Typical  4.4  Radial overlaps for gap geometries  4.5  V e r t i c a l overlaps for gap geometries not used in CRC  ^ £  Photographs of s c i n t i l l a t o r probe at 90 deg and 270 showing unaccelerated beam  matching solution  135  I36 137 138  for CRC  in. x 1.5 in.  injection gap not used  139 140  in CRC  141 142 deg 143  4.7  Inflector entrance p r o f i l e s  1^2)  4.8  Beam at 2-3/4  145  4.9  Chopper phase width and energy spread as a function of chopper voltage Energy spread versus d r i f t distance for a beam being acted upon by longitudinal space charge forces  4.10  turns for different chopper phase settings  viii  146 147  Page 4.11  Phase i n t e r v a l  versus d r i f t  a c t e d upon by l o n g i t u d i n a l 4.12 '4.13 4.14  4.15  4.16  4.17  d i s t a n c e f o r a beam being space charge f o r c e s  O s c i l l o s c o p e photographs of U n a c c e l e r a t e d beam f o r  the chopped beam s i g n a l  1-3/4  turns  150 phase at  buncher 151  illustrating  effect  of 152  Bunched phase versus i n i t i a l 220 i n . d r i f t to chopper  phase at  buncher f o r a 153  Bunching f a c t o r as a f u n c t i o n o f buncher v o l t a g e v a r i o u s chopper c o n d i t i o n s as a f u n c t i o n o f  149  v a r i o u s chopping and bunching  conditions Bunched phase as a f u n c t i o n of i n i t i a l f o r a 500 i n . d r i f t to i n j e c t i o n gap Phase probe s i g n a l s at buncher  148  5.1  f(R)  5.2  E f f e c t of dee misalignment c o r r e c t i o n p l a t e s s c i n t i l l a t o r at 1-1/4 turns  for 154  radius  155  156  5.3  T r a n s m i s s i o n curves d u r i n g  5.4  Correction plate  5.5  Measured dee misalignments  159  5.6  Median plane B  160  5.7  Scintillator first  f  initial  as viewed on  CRC experiments  v o l t a g e s b e f o r e shimming out B  157 158  f  components  p i c t u r e s of  the beam as i t  a c c e l e r a t e d to f u l l  appeared when  radius  161  5.8  Measured B  r  before  5.9  Measured B  p  after  5.10  Correction plate  5.11  6 ( A P ) versus r a d i u s f o r dee misalignment  5.12  Calculated  5.13  T r a n s m i s s i o n versus turn number curves f o r  5.14  i n j e c t i o n phases Per cent t r a n s m i s s i o n as a f u n c t i o n of the maximum phase c o n t a i n e d in a beam whose t o t a l phase width is 21 ± 3 deg  168  Photographs of s c i n t i l l a t o r  169  5.15  shimming  162  shimming  163  voltages a f t e r  shimming out  Z  B  r  164  experiment  165 166  as a f u n c t i o n of energy various  probe at 90 deg and 270  ix  deg  167  5.16  5.17  Comparison between observed and calculated beam envelopes for a phase of approximately 0 deg  Page 170  Comparison between observed and calculated beam envelopes for a phase of approximately 15 deg  171  Comparison between observed and calculated beam envelopes for a phase of approximately 20 deg  172  Comparison between ideal v e r t i c a l acceptance e l l i p s e and the e l l i p s e required to obtain an improved f i t to measured envelopes  173  Graph of turn number at which a minimum is observed as a function of the phase of the minimum  174  6.1  Typical radial turn patterns obtained using median plane diagnostic probes  175  6.2  5 deg, 20 deg and 30 deg phase t r a j e c t o r i e s plotted on radius versus dee voltage curves  176  Effect of beam centring as seen on radius versus dee voltage plots  177  6.4  Effect of non-isochronous operating conditions on beam radii  178  6.5  Well-centred isochronized t r a j e c t o r i e s in CRC  179  6.6  Phase probe measurements f o r d i f f e r e n t magnet potentiometer settings  l8o  Schematic diagram of geometry orbits  181  5.18  5-19  5.20  6.3  6.7 6.8  for off-centred cyclotron  A n a l y t i c a l l y calculated radial beam widths along 90 deg and 270 deg azimuths  182  A n a l y t i c a l l y calculated radial beam widths along 0 deg azimuth  I83  A n a l y t i c a l l y calculated beam size versus turn number along the 180 deg azimuth  184  6.11  Phase h i s t o r i e s calculated using simple a n a l y t i c theory  185  6.12  a)  6.9 6.10  b)  6.13  a) b)  6.14  a) b)  Phase h i s t o r i e s f o r y = -0.3 in. starting from an i n i t i a l phase of 5 deg Phase h i s t o r i e s for y = -0.3 i n . starting from an i n i t i a l phase of 29 deg c  186  c  Phase phase Phase phase  h i s t o r i e s f o r y = 0.0 s t a r t i n g from an i n i t i a l of 5 deg h i s t o r i e s f o r y = 0.0 s t a r t i n g from an i n i t i a l of 29 deg  186  c  187  c  Radial beam widths at 90 deg and 270 deg f o r y Radial beam widths at 0 deg f o r y = -0.3 i n . c  c  = -0.3 in.  187 188 188  Page 6.15  6.16  6.17  7-1  7-2  7.3  7.4  Radial  beam w i d t h s beam w i d t h s  a) b)  Radial Radial  beam w i d t h s a t beam wi d t h s a t  a)  Radial  b)  Radial  beam w i d t h s beam w i d t h s  Shape o f interval  ?  Effect fields  7.6  of  conducting  for  = -0.15  c  y  c  = -0.15  in.  in.  MeV beam o c c u p y i n g a  190 190  0.15  in.  191 191  phase 192  distributions  on  showing  with  various  transverse  space  charge  electric  spread  showing  phase o f  Accelerated  7-12  Typical current  charge  on  the  effect  of  space  charge  showing  the  effect  of  space  charge  ions  due  deg  197  to  longitudinal  current  measured  ion  due  to  effects  198  longitudinal  199  (total  (total  space charge  beam c e n t r o i d  phase  space e l l i p s e s  3 deg  7.11  space  196  phase space e l l i p s e s  phase  of  ions  beam e n v e l o p e s  27  effects  195  beam e n v e l o p e s  deg p h a s e  phase o f  phase  as  a  at  turn  #6  spread at  spread  function  a s s u m i n g an  0 deg  turn  #6  of  to  30  to  30  chopper  injection  deg)  a s s u m i n g an  0 deg  source emittance  200 injection  deg)  201  voltage  versus  ion  202  source 203  7.13  Scintillator  7-14  C o m p a r i s o n b e t w e e n beam ion s o u r c e c u r r e n t s  radii  P h o t o g r a p h o f beam s p o t at 6-3/4 turns  produced  7.16  y  deg  ions  Radial displacement of space charge e f f e c t s  7-15  for  270  at  beam e n v e l o p e s  7-8  Radial  deg  boundaries  Energy  7-10  deg and  193  deg p h a s e  Vertical  Radial  90  at  charge  7.7  7.9  189  1  Vertical  on 30  for  0  deg p h a s e  15  189  at  19*  Vertical  on  at  t h e c e n t r o i d o f a 1.3 b e t w e e n 0 and 30 d e g  E versus y dimens ions  0 7.5  Radial  a) b)  photograph  comparing  high  obtained  and with  low  current  high  and  beams  204  low 205  by  100  pA on  tantalum  block 206  T r a n s m i s s i o n v e r s u s t u r n number s o u r c e c u r r e n t a t 400 pA  x1  with  chopper o f f  and  ion 207  ACKNOWLEDGEMENTS  I would l i k e to thank Dr. M.K. Craddock f o r many helpful  suggestions and for supervising my studies while  at the University of B r i t i s h Columbia.  In addition, I  would l i k e to thank Dr. E.W. Blackmore and Dr. G. Dutto for help and guidance throughout this work.  Finally I  would l i k e to thank Miss Ada Strathdee for her patience and perseverance in typing this Financial  thesis.  support from TRIUMF during the course of  this work is g r a t e f u l l y  acknowledged.  FOREWORD The work Dr.  E.W.  in t h i s  Blackmore and Dr.  done in d i r e c t  specifically  of  the data  The experimental  is responsible Design of  the  2)  C a l c u l a t i o n of inflector,  spiral  except  the a u t h o r .  the  ion o p t i c a l and the  p r o p e r t i e s of  Thus,  injection  to the c e n t r a l  the magnet  the  the phase space emittance  5)  I n t e r p r e t a t i o n of  the v e r t i c a l the  of  region acceptances.  I n t e r p r e t a t i o n of chopper and buncher measurements.  the v a l u e s of  bore,  gap.  A)  estimate  where  inflector.  s t u d i e s on the matching o f  the beam l i n e  c a l c u l a t i o n s and  for:  deflector  Theoretical  thesis are,  r e s p o n s i b i l i t y of  s u p e r v i s i o n of  work d e s c r i b e d was  The t h e o r e t i c a l  i n c l u d e d in t h i s  n o t e d , s o l e l y the  1)  3)  G. D u t t o .  c o l l a b o r a t i o n w i t h them.  interpretations  author  t h e s i s was done under the g e n e r a l  beam envelopes and the data used to  transition  phase and the v e r t i c a l  tune  v .  z 6)  Interpretation of  7)  The s i m p l i f i e d  the  radial  theoretical  turn  pattern  treatment o f  data.  the  radia 1 -1ongitudina1  c o u p l i n g phenomena and the comparisons between  theory and  8)  The space charge c a l c u l a t i o n s on the  turns.  9)  The  interpretations  of  The f o l l o w i n g were the author 1)  and Dr.  E.W.  2)  the h i g h - c u r r e n t result  Blackmore and Dr.  G.  six  measurements.  of a d i r e c t  c o l l a b o r a t i o n between  the  Dutto:  The s t u d i e s on the dee misalignment described  first  experiment.  correction plate  voltages  in S e c t i o n 5-  The measurement of  the anomalous B  xiii  f  component d e s c r i b e d in S e c t i o n  5.3-  3)  The shimming that removed the anomalous B  h)  The adjustment of the machine parameters  r  component.  to obtain  isochronism and  centring. 5)  The measurements which produced  the data used in this thesis.  The construction of the CRC and the development of the CRC hardware was exclusively the responsibi1ity of Dr. E.W.  Blackmore.  The design of  the central region dee geometry and d e f l e c t i o n plates was, except where s p e c i f i c a l l y noted, the sole work of Dr. G. Dutto.  In addition, the author  does not take any r e s p o n s i b i l i t y for the design or optimization of the injection line upstream of the magnet bore.  xiv  1.  INTRODUCTION  The cyclotron has gone through three stages of evolution.  The f i r s t  stage encompassed the development of the n o n - r e l a t i v i s t i c isochronous cyclotron by E.O. Lawrence and his c o l l a b o r a t o r s . of accelerating high-current  These machines were capable  1  beams to n o n - r e l a t i v i s t i c energies.  stage saw the development of the r e l a t i v i s t i c synchrocyclotron. machines can accelerate beams to r e l a t i v i s t i c energies;  The second These  2  however, they have a  low duty cycle and the currents produced are much smaller than those in an isochronous cyclotron.  With the discovery of the sector-focusing  principle,  3  the cyclotron went through a third stage of evolution, and i t became possible to construct  isochronous, high duty cycle, high-current  cyclotrons  operating  at r e l a t i v i s t i c enrrgies. The development of the sector-focused construction of large 'meson factory  1  cyclotron has made possible the  cyclotrons.  These machines are  designed to produce beams of protons in the 200-600 MeV energy range with currents of the order of 100 uA, and one of their functions w i l l intense secondary beams of pions. are being constructed. and  Currently two cyclotron  'meson factories'  These are the SIN cyclotron in Zurich,  the TRIUMF cyclotron in Vancouver, Canada.  be to produce  Switzerland, * 1  5  The TRIUMF cyclotron is based on a six-sector magnet and is designed to accelerate 100 uA of H" ions to 500 MeV.  The use of H" ions makes possible  the extraction of at least two beams of variable energy from 200 to 500 MeV with nearly 100% e f f i c i e n c y . The quality of the accelerated beam in the cyclotron emittance, energy resolution and microscopic  its  duty factor) is to a large  extent determined in the f i r s t few turns at the centre. are c r i t i c a l  (i.e.  Since these turns  in determining high-energy beam q u a l i t y , a f u l l - s c a l e working  model of the central region of the TRIUMF cyclotron was b u i l t .  This  - 2'central region cyclotron  1  (CRC) was designed to accelerate 100 uA of H" ions  over s i x turns to an energy of 2.5 MeV. This thesis describes the experimental  investigation of the behaviour  of the beam in the CRC and attempts to explain these results t h e o r e t i c a l l y . In some cases, theoretical investigations were required to f i l l  gaps in  existing work; in p a r t i c u l a r , the ion optical properties of the s p i r a l elect r o s t a t i c i n f l e c t o r and the f i r s t  radio-frequency  accelerating gap were  calculated, and their design chosen, by the author. This chapter b r i e f l y describes the problems which were investigated and reviews the various possible injection systems. components are described.  In Chapter 2 the CRC  In Chapter 3 the ion optical properties and design  of the elements of the injection system closest to the cyclotron are considered.  The ions enter the CRC by t r a v e l l i n g a x i a l l y through the magnet  bore and are bent into the median plane and centred f o r injection by means of two e l e c t r o s t a t i c bending electrodes.  The ions then enter the cyclotron  central  region through a radio-frequency  gap').  Theoretical and experimental  accelerating gap (the 'injection  studies of the effects of these elements  on the beam behaviour are reported. In Chapter k some of the physical properties of the injected beam are investigated.  In order to minimize the energy spread and v e r t i c a l height of  the beam at extraction, the ion optical properties of the injected beam must be carefully matched to the central region acceptances. examined both t h e o r e t i c a l l y and experimentally. and bunching the incoming beam is studied.  This problem is  Next, the effect of chopping  As i t s name implies, the chopper  chops the beam into pulses of a few nanoseconds duration by means of deflection across a s l i t at a multiple of the cyclotron frequency.  The buncher is  e s s e n t i a l l y a two-gap linear accelerator which produces bunches of beam in which the peak current is higher than the continuous ion source current.  - 3 These features are desirable, because the cyclotron is only capable of accelerating pulses of beam which arrive at the dee gaps within certain r f phase l i m i t s .  The experimental  performance of the chopper and buncher are  compared with the predictions of theory. In Chapter 5 the v e r t i c a l behaviour of the beam during acceleration in the CRC is examined.  When the CRC was f i r s t placed in operation the beam  was lost v e r t i c a l l y after about three turns.  These losses were caused by  e l e c t r i c f i e l d distortions along the dee gaps due to v e r t i c a l dee misalignments, and by the presence of a radial magnetic f i e l d component on the median plane of the cyclotron.  A description of how the dee misalignments were  compensated by placing v e r t i c a l e l e c t r o s t a t i c correction plates after the dee gap crossings and how the anomalous radial magnetic f i e l d was removed by shimming the magnet is given.  After this had been completed, an experiment  was performed to determine whether or not the v e r t i c a l  impulses due to dee  misalignments could be predicted t h e o r e t i c a l l y from the measured size of the misalignments.  A comparison is made between the results of this experiment  and the predictions of theory. In the central region of the cyclotron the dee gaps act as electros t a t i c lenses which can focus or defocus an ion depending upon the value of the r f phase at which the ion crosses the gap. By making transmission measurements, the value of the transition phase which separates ly focused phases from the v e r t i c a l l y defocused  the v e r t i c a l -  phases was measured; and by  observing the shape of the v e r t i c a l beam envelopes the strength of the focusing  forces was estimated.  theory.  These values are compared with the predictions of  In addition, the shape of the v e r t i c a l beam envelopes is compared  with theoretical predictions. In Chapter 6 the radial motion of the beam is studied.  If a high-  quality beam is to be accelerated, the ions must be properly centred, and the  -  k -  value of the magnetic f i e l d must be adjusted so that the ion rotation frequency matches the frequency of the accelerating voltage.  It is demon-  strated how radial measurements on the accelerated beam may be used to determine the proper centring conditions and the proper setting for the magnet current. When the centre of curvature of an ion is displaced from the dee gap centre l i n e , the rf phase at which i t crosses the dee gaps w i l l be affected. Since the energy gain at a dee gap is dependent on the gap-crossing phase, such a centring error w i l l a l t e r the energy d i s t r i b u t i o n within the beam, thus producing radial-1ongitudina1 coupling.  A simplified treatment of this  coupling is given, and is used to explain the observed radial beam widths. Chapter 7 deals with the high-current performance of the CRC.  Here  the effects of v e r t i c a l , radial and longitudinal space charge forces during the f i r s t s i x accelerated turns are calculated.  The f i n a l portion of the  chapter deals with the performance of the CRC when accelerating average currents in the 10-100 uA range. We begin by discussing some of the fundamental concepts used in studying beam transport systems. 1.1  Fundamentals of Beam Transport A beam consists of a group of many particles with neighbouring t r a -  jectories.  We can therefore represent the beam as a volume in an abstract  six-dimensional  'phase space' whose co-ordinates are positions and momenta.  As the beam travels through real space under the influence of an e l e c t r i c and/or magnetic f i e l d the beam volume w i l l  travel through phase space.  In  general the shape of the volume w i l l change as a function of time, but the six-dimensional volume i t s e l f is usually governed by L i o u v i l i e ' s  theorem.  6  This theorem states that under the action of forces which can be derived from a Hamiltonian, the motion of a group of p a r t i c l e s is such that the local  - 5 density of representative points in the appropriate phase space remains everywhere constant.  This is equivalent to requiring that the volume occu-  pied by a beam in phase space remains constant.  L i o u v i l i e ' s theorem is  applicable whenever a group of ions moves under the influence of external e l e c t r i c and/or magnetic f i e l d s provided no ions are lost from the beam. Although the most general phase space is a six-dimensional whose co-ordinates are position  hypervolume  and momenta components in three different  directions, when there is no coupling between different d i r e c t i o n s , we need only consider the behaviour of two-dimensional projections of the hypervolume. Under these circumstances,  L i o u v i l i e ' s theorem requires that the area of the  two-dimensional projections must remain constant. occurrence  This is a quite common  in beam transport systems, and the area of the two-dimensional  projection is known as the beam emittance.*  For the sake of convenience,  these projections are usually assumed to be either e l l i p s e s or parallelograms. In this thesis, we shall work exclusively with e l l i p s e s . The determination broken into two parts.  of beam behaviour in a transport system can be First  is the determination  of the 'central trajectory':  the path of some representative p a r t i c l e in the group, usually one at the centroid of the phase space volume.  Second is the determination of the motion  of the other p a r t i c l e s relative to the central trajectory, i.e. the change in shape of the phase space volume as a function of time. If the p a r t i c l e s in the beam carry a charge q and have a mass m, their equation of motion is  m7 = q (~r  x  If +  E)  (1 .1)  where E is the e l e c t r i c f i e l d , B is the magnetic f i e l d , r is the position vector and the dots  indicate d i f f e r e n t i a t i o n with respect to time.  This  - Some authors define the beam emittance to be the phase space area divided by IT.  equation  solved for the i n i t i a l  conditions of the centroid of the phase  space volume yields the central trajectory r ( t ) . c  For the other t r a j e c t o r i e s  we write "r(t) = ? ( t ) + Ar(t) . c  Using this expression we can expand Eq. 1.1 in terms of Ar. Usually, a l l of the t r a j e c t o r i e s within a beam w i l l  l i e close enough to the central trajectory  that we need only keep f i r s t - o r d e r terms in Ar. This is the linear approximation, and we may use i t to write our solutions for Ar in the matrix Ax AP  7  Ax AP>  X  Ay  = R(t')  APy  Ay AP  V  Az AP  form  Az AP,  7  t=t'  t=0  where Ax, Ay and Az are the three components of Ar and AP , APy and AP are X  Z  the three components of Ar. R(t) is known as a transfer matrix, and i t is a s i x by six matrix whose components are time dependent.  R(t) operates on  a column vector consisting of components of Ar and Ar evaluated at time t=0 and generates a column vector containing the components of Ar and Ar at some different time t=t'.  The transfer matrix may be calculated by solving  the linearized equations of motion for s i x linearly-independent sets of initial  conditions. One method for tracing a phase space volume through a beam transport  system is to start with a large number of points on the surface of the phase space volume and follow their progress  through the beam transport system.  This obviously requires a great deal of computational have been developed. ' 8  9  labour, and shortcuts  The TRANSPORT matrix formalism  described by K.L. Brown  - 7 assumes that the phase space volume is contained ellipsoid.  Such an e l l i p s o i d may  in a six-dimensional  hyper-  e a s i l y be tracked through a series of  linear beam transport elements using techniques  from elementary linear  algebra. A somewhat s i m i l a r method has been described by K.G. case of a two-dimensional phase space.  This transformation  Steffen  for the  9  is used for the  v e r t i c a l space charge calculations in Chapter 7The techniques  described  in this section w i l l be used extensively  throughout this thesis to study the behaviour of beams in the CRC. beginning  these studies, we shall b r i e f l y review some of the work being done  on cyclotron injection s y s t e m s . ' 10  1.2  Before  11  Advantages of External Sources The f i r s t cyclotron sources were placed within the cyclotron. Recently  emphasis has shifted to the development of external sources located outside of the machine. 1)  The main advantages of an external source are l i s t e d below:  If high currents are accelerated, the ion source must be at high power levels. an external  2)  This high power is more easily supplied to  source.  With an external source d i f f e r e n t i a l pumping may  be used between  the source and the central region of the cyclotron. f i e s the problem of maintaining 3)  operated  This simpli-  a good central region vacuum.  The current from most ion sources is limited.  If an external  source is used, a buncher can be placed between the ion source and the central region to increase the peak injection current. k)  The desire to accelerate heavy ions has produced a great deal of interest  in external sources.  These sources usually require large  amounts of arc power, and many produce large amounts of s o l i d , l i q u i d , or gaseous contaminates which can produce corrosion and/or contaminate the vacuum within the cyclotron.  These problems can  be  - 8minimized i f an external source is used. 5)  If polarized beams are to be accelerated, a polarized source must be used.  These sources are too large to be mounted internally.  In  addition, polarized sources must be shielded from stray magnetic f i e l d s , and this is easier i f the source is located away from the cyclotron magnet. 6)  Ion source maintenance is s i m p l i f i e d i f an external source is used.  The main disadvantage of an external source is that i t requires an elaborate  injection system.  several types of external 1. 3  Neutral  We shall now examine the current status of  injection systems.  Beam Injection  One of the main problems in designing an injection system f o r use with an external source is the problem of guiding the ions through the magnetic f i e l d of the cyclotron to the central region.  One solution is to inject  neutral p a r t i c l e s which may be ionized by means of a stripping f o i l or an e l e c t r i c arc once they have reached the centre. The  f i r s t published proposal  Keller at CERN, cyclotron.  12,13  but this proposal  for neutral hydrogen injection was made by was never implemented on a f u l l - s i z e d  More recently, neutral thermal ion injection has been used at  Saclay to accelerate a beam of 0.5 nA of polarized protons to an energy of 22 MeV. > 11+  15  The currents were at least a factor of ten below the currents  which could be obtained ionizer.  using an external source combined with an external  A thermal ion neutral beam injection system has also been used with  the Lyon 28 MeV synchrocyclotron; low.  however, the injected current  is again  quite  16  Neutral  beam injection systems have also been designed using  velocity p a r t i c l e s .  Neutral  beam injection is currently being  the U-120 cyclotron at the Nuclear Research Institute in Rez,  higher  installed on Czechoslovakia.  17  - 9T h i s system is designed to neutral  inject  a 40 keV beam of deuterium or hydogen.  ion i n j e c t i o n scheme u s i n g 30 keV hydrogen atoms has a l s o been  s t i g a t e d by PI is at  Dubna.  1.  Ion  Median Plane  In median plane  A  inve-  1 8  Injection i n j e c t i o n systems a beam o f  c y c l o t r o n by t r a v e l l i n g  along a more or l e s s r a d i a l  ions i s  injected  into  path from the outer  the  edge of  the c y c l o t r o n . Once the beam has reached the c e n t r e o f the c y c l o t r o n , s t e e r i n g i n f l e c t o r s are used to c e n t r e the beam f o r One p a r t i c u l a r l y  simple form of  method used by Lebedev I n s t i t u t e ions the  loop inward by t r a v e l l i n g  injection.  radial  in M o s c o w .  19  injection  is the  In t h i s  injection  along a h i l l - v a l l e y  interface,  incoming beam is a c h i e v e d by means o f the h i l l - v a l l e y  t h i s method approximately 20% of the small  system the  and f o c u s i n g o f  gradient.  Using  i n j e c t e d beam has been a c c e l e r a t e d  in a  300 keV c y c l o t r o n . An a l t e r n a t i v e median plane  j e c t i o n o f p o l a r i z e d protons at f i e l d were compensated o f the  incoming i o n s .  injection.path  injection  Saclay.  2 0 - 2 2  system has been used f o r Here the e f f e c t s  focusing.  injection  and l i t t l e o r no e x t e r n a l  is p a r t i c u l a r l y  a p p l i c a b l e to open s e c t o r  15 MeV,  i n t o one of the f i e l d - f r e e  deflection  Median plane  is  injection  is planned f o r  In such a  valley  r e q u i r e d to compensate f o r  cyclo-  regions,  the  the 580 MeV SIN  cyclo-  the 200 MeV Indiana U n i v e r s i t y c y c l o t r o n , both of which have open  sector magnets. with small  path  E l e c t r o s t a t i c quadrupoles were i n s t a l l e d a l o n g the  c y c l o t r o n the beam can be i n j e c t e d  tron and f o r  in-  o f the magnetic  in which the magnetic f i e l d between s e c t o r s is almost z e r o .  magnetic f i e l d .  the  by p l a c i n g e l e c t r o s t a t i c d e f l e c t o r s along the  to p r o v i d e  Median plane trons  trochoidal  2 3  >  injector  Both o f these c y c l o t r o n s are operated  2tt  c y c l o t r o n s , and the  respectively.  in c o n j u n c t i o n  i n j e c t i o n energies w i l l  be 70 MeV and  - 10 1.5  Axial  Injection Systems  In axial  injection systems the beam is injected through an axial hole  in the magnet yoke.  Once the beam has reached the median plane, i t is bent  through 90 deg by means of an i n f l e c t o r .  This method was  Powell for use on the Birmingham radial ridge c y c l o t r o n . groups have worked on the problem of axial  injection.  f i r s t developed by 2 5  Since then many  The hardware for these  injection systems d i f f e r s in the type of focusing elements and i n f l e c t o r used. The focusing elements between the ion source and the i n f l e c t o r have a l l been either e l e c t r o s t a t i c quadrupoles, magnetostatic quadrupoles, einzel lenses, or solenoids.  Most of the axial  injection systems which are currently  in operation have been designed to work with r e l a t i v e l y low injection energies in the 10 to 15 keV range, and as a result the space charge forces along the injection line have been very important.  With an injection energy of 15 keV  the Cyclotron Corporation has accelerated 120 uA of H" ions using an line composed of e l e c t r o s t a t i c d o u b l e t s .  26  injection  More recently the TRIUMF group has  accelerated 160 uA of H" ions using an injection energy of 278 keV and an injection line composed of e l e c t r o s t a t i c quadrupoles.  Some aspects of this  work w i l l be discussed in this thesis. A variety of i n f l e c t o r designs has been used to bend the a x i a l l y injected ions onto the median plane.  Powell f i r s t proposed using an electron  mirror consisting of two electrodes inclined at an angle of approximately k5 deg to the incoming beam.  25  The incoming beam entered the electrodes  through an aperture and was bent through an angle of 90 deg by the action of the e l e c t r i c f i e l d .  This device has the disadvantage that the electrode  potentials must be of the same order of magnitude as the potential used to accelerate the ions to their injection  energy.  The Grenoble group has designed a s p i r a l i n f l e c t o r consisting of a pair of spiral-shaped electrodes designed to produce the required v e r t i c a l  -  deflection  -  in the presence of the central magnetic f i e l d of the c y c l o t r o n . ' 2 7  This device can be operated the electrodes are d i f f i c u l t described how problem.  11  2  at lower potentials than the electron mirror, but to machine.  In a previous  thesis the author has  a numerically-controlled m i l l i n g procedure was  applied to this  29  MUller has designed a hyperbolic i n f l e c t o r whose surfaces are hyperbolas of revolution which may  e a s i l y be constructed on a lathe.  The  dimensions of the hyperbolic i n f l e c t o r are uniquely determined by the type of ion being  injected, the injection energy and the central magnetic f i e l d of the  cyclotron.  As a result i t is not compatible with some central region  geometries.  Muller has also discussed a parabolic i n f l e c t o r which can be  constructed by bending sheet metal plate.  Like the hyperbolic  inflector,  the dimensions of the parabolic i n f l e c t o r are uniquely determined by the injection parameters. The electron optical properties of the electron mirror, the hyperbolic i n f l e c t o r and the parabolic i n f l e c t o r have been analyzed papers. ' 30  Recently  3 1  the most general J. Pabot. 1.6  in previous  the author has calculated the optical properties of  type of Grenoble spiral  These results are given  i n f l e c t o r studied by J. Belmont and  in Appendix A.  The TRIUMF Injection System The TRIUMF cyclotron is designed to accelerate an average current of  100 yA of H"  ions to an energy of 500 MeV.  The microscopic  duty factor w i l l  be approximately 10%, and hence the ion source must supply peak currents in the milliampere  range.  This requires an external  ion source, since the high  gas flow required by the H" ion source would severely degrade the central region vacuum i f the source were to be mounted internally and operated without the aid of d i f f e r e n t i a l pumping.  In addition, i t is anticipated that TRIUMF  w i l l be used to accelerate smaller currents of polarized ions, and as noted  - 12 e a r l i e r this requires an external source. Axial Neutral  injection is the most practical form of injection for TRIUMF.  injection systems would be incapable of producing the high peak  currents required, and median plane injection would be d i f f i c u l t magnetic f i e l d  is appreciable even in the valleys of the TRIUMF magnet.  The TRIUMF central injection system. 3-0  region parameters are ideally suited for an axial  The r e l a t i v e l y low central magnetic f i e l d of  kG, coupled with a high-energy gain of approximately  a high injection energy of approximately an i n f l e c t o r .  since the  300 keV,  approximately  400 keV per turn and  leaves adequate space for  Lack of space is frequently a problem in more compact  central  regions where the radii of the f i r s t few turns is small. A high injection energy of 300 keV  is being used in TRIUMF in order to  minimize space charge problems along the injection l i n e . energy was 1)  A higher  injection  not used for the following reasons: Higher potentials would be required on the i n f l e c t o r and i t appeared that proper positioning of the i n f l e c t o r would be more d i f f i cult.  2)  The f i r s t turn must clear the injection gap.  This sets a minimum  radius gain for the f i r s t turn and hence an upper limit on the injection energy. 3)  Increasing the injection energy decreases  the turn spacing on  the f i r s t few turns where the injection energy is an appreciable fraction of the total energy. f i r s t few turns more An i n f l e c t o r similar for use in TRIUMF.  29  This could make diagnostics on the  difficult.  in p r i n c i p l e to the Grenoble design was  The electron mirror was  potentials would be too high.  constructed  rejected because the operating  For the TRIUMF parameters the hyperbolic and  parabolic i n f l e c t o r designs were too large to f i t conveniently into the TRIUMF central  region.  - 13 2. 2.1  DESCRIPTION OF THE  CENTRAL REGION MODEL CYCLOTRON  Introduct ion The central region model cyclotron (CRC)  of the f i r s t 30 in. of the TRIUMF c y c l o t r o n ; H"  kG.  It is designed to accelerate  32,33  ions to an energy of approximately 2.5 MeV  approximately 3.0  is a f u l l - s c a l e working model  in a mean magnetic f i e l d  A photograph of the CRC  of  is shown in Fig. 2.1, and in  this section we shall b r i e f l y describe i t s components. 2.2  Median Plane Components Fig.  H~  2.2  shows a schematic diagram of the median plane of the CRC.  The  ions are produced in an external source and enter the cyclotron by  t r a v e l l i n g along the central magnet axis.  The  ions are bent onto the median  plane by means of a spiral e l e c t r o s t a t i c i n f l e c t o r .  After leaving the  i n f l e c t o r the ions enter a horizontal radial steering deflector which provides the extra radial deflection required to centre the trajectories for acceleration.  A photograph of the inf1ector-def1ector assembly is shown in Fig. 2.3. After leaving the deflector the ions cross the injection gap where they  are accelerated by a nominal peak rf voltage of 100 kV. quently accelerated in a two dee  The  rf system which is operated  of 200 kV and at a nominal frequency of 23.1  MHz,  ions are subseat a peak voltage  which is the f i f t h harmonic  of the ion rotation frequency. To compensate for v e r t i c a l  impulses due to dee misalignments, e l e c t r o -  s t a t i c correction plates were i n s t a l l e d for the f i r s t five dee gap crossings. A description of the operation of these plates is given  in Section 5.  The centre post in the TRIUMF cyclotron consists of a stainless steel post which mechanically mechanical support 2.3  The  supports the centre of the machine.  is not required and no centre post was  In the CRC  used.  Ion Source  An Ehlers type external  ion source purchased from the Cyclotron  this  - 14 Corporation, Berkeley, C a l i f o r n i a was used in the CRC.  This source is  3l+  designed to produce a current of 2.0 mA of 12 keV ions with an emittance of 64 mm.mrad in both directions, and at the time of purchase  i t was the only  available type of H" source capable of producing the currents required for TRIUMF. The Ehlers ion source is followed by a 90 cm acceleration tube which accelerates the beam to approximately 300 keV.  The acceleration tube con-  tains 36 re-entrant electrodes having an aperture of 7-5 cm.  Voltage was  supplied to the tube by means of a Cockroft-Wa1 ton generator which is capable of supplying a voltage of 288 ± 12 kV at a current of 10 mA and a regulation of  1 part in \0 . k  Vacuum pressure is maintained  in the acceleration tube and ion source  by means of a 10 in. o i l diffusion pump connected to the ion source chamber and by means of a 6 in. o i l diffusion pump connected to a box located between the ion source and the acceleration tube. Emittance-defining s l i t s and an einzel acceleration tube and the ion source.  lens were placed between the  The emittance-defining s l i t s were used  to decrease the emittance of the injected beam, and the einzel  lens was used  to.match the optics of the ion source to the optics of the acceleration tube. 2.4  The Beam L i n e  3 5  The beam line between the acceleration tube and the i n f l e c t o r is approximately 22 m long and is shown schematically in Fig. 2.4.  Focusing in  this line is provided by 50 e l e c t r o s t a t i c quadrupoles each of which has an aperture of 5 cm and a length of 10 cm.  Over the long d r i f t  sections the  quadrupoles are arranged in identical  'repeat sections' with spacings of  approximately 30 cm.  in the periodic section are operated at  The quadrupoles  voltages of approximately ±3-0 kV.  Some of the remaining matching  are operated at voltages as high as ±10 kV.  quadrupoles  - 15 The beam line also contains three right angle bends. made dispersionless by using a combination with quadrupoles.  These bends are  of two k$ deg bends in conjunction  The bend voltages are ±30 kV, the electrode spacing is  3.8 cm and the radius of curvature is 38 cm. In addition to the quadrupoles  and bends, the beam line also contains  an rf chopper which may be used to r e s t r i c t the phase interval of the injected beam and a buncher which may be used to increase the peak current contained in the interval accepted by the cyclotron. in more detail  These devices w i l l be discussed  in Section k.  For high current measurements i t was desirable to limit the duty cycle of  the injected beam in order to avoid overheating the diagnostic probes and  the CRC components.  This low duty cycle was obtained by placing a mechanical  chopper wheel in the beam line.  This wheel was operated at a frequency of  30 rev/sec, and i t produced a beam with a duty cycle of approximately  10%.  Beam diagnostic devices along the injection line consisted of several vibrating wire scanners placed at key positions.  These devices consisted of  wire probes which were swept across the beam, and they provided a CRT display of  the beam p r o f i l e in two transverse directions.  be monitored  In addition, currents could  on several sets of beam scrapers placed before each quadrupole  or bend electrode and on several removable beam stops placed at key points along the 1i ne. The beam line elements were enclosed in a 15 cm diam stainless steel tube.  A pressure of approximately  5  x  IO" Torr was maintained along the beam 7  line by means of s i x titanium sublimation pumps. 2.5  The Magnet The CRC magnet is a s i x pole magnet designed to simulate the TRIUMF  magnetic f i e l d out to a.radius of kO i n . , and i t produced a central magnetic f i e l d of about 3.0 kG across a gap of 20.8  in.  The magnet produces a  -  16 -  s i g n i f i c a n t amount of f l u t t e r toward the outer r a d i i , and Fig. 2.5 shows a graph of the measured magnetic f i e l d versus azimuth at radii of 10 i n . , 20 in. Fig. 2.6  and 30 in. and Fig. 2.7  shows the average magnetic f i e l d at a function of radius  shows the magnetic contribution to v  2  as a function of radius.  The main magnet c o i l s contain 80 turns, and they are powered from a 50 V 2700 A power supply which is regulated to 1 part in 10^.  The magnet  also contains five trim c o i l s mounted at radii of 14 i n . , 20.5  i n . , 27 i n . ,  33.5  in. and kO in.  plane, and they may  These c o i l s are mounted symmetrically about the median either be excited symmetrically to trim up B__ or they  be excited asymmetrically to provide a radial B vertically. '  r  component to steer the beam  Measured trim c o i l contributions to B  of operation are shown in Figs. 2.8  and 2.9.  may  r  and B  z  for the two modes  The magnet also contained one  set of s i x harmonic c o i l s ; however, these were never used. 2.6  The RF System  36  The CRC contains a two dee accelerating system composed of tuned quarter wave length cavities which are operated at a nominal frequency of 23.1  MHz.  another. Fig. 2.10.  Each dee  is constructed by placing two sets of cavities above one  A drawing of the rf cavities mounted in the CRC A h in. gap  ion t r a j e c t o r i e s . the vacuum tank.  is shown in  is l e f t between the cavities to provide room for the  The ground arms of the c a v i t i e s are connected  d i r e c t l y to  The cavity hot arms are cantilevered and v e r t i c a l  alignment  is achieved by adjusting the position of the l e v e l l i n g arms. The rf cavities are constructed in sections, one of which is shown in Fig. 2.11.  Each section is 32  in. wide, and these sections are joined  together to form the complete resonators.  A l l sections in TRIUMF w i l l  be  identical except for the centre sections which have been modified to accommodate the centre post. are used.  In the CRC only the eight centre resonator sections  The outer edges of these cavities which would normally be joined  - 17 to additional resonator segments have been terminated with flux guides. In the central region the dee gap and v e r t i c a l directions.  is tapered  in both the horizontal  To provide this tapering, s p e c i a l l y formed quadrant  plates have been attached to the dee gap end of the central region segments.  resonator  This tapering is v i s i b l e in Fig. 2 . 1 2 , which shows a schematic  view of the rf structures as viewed from along the dee gap centre l i n e .  The  beam scrapers used to intercept any beam which is t r a v e l l i n g too high or too low are also shown. Rf energy is supplied to the resonators by means of a coupling loop located near the root end of the resonators.  Tuning panels actuated by means  of bellows are placed in each resonator root piece in order to provide fine tuning of the resonator c a v i t i e s . The tuning panels are controlled by a feedback arrangement which allows the system to run at a fixed The tetrode.  resonators are driven by a single Eimac 4 C W 2 5 0 0 0 0  gap.  The Vacuum System The  resonators are enclosed  internal dimensions are 82  in a stainless steel vacuum chamber whose  in. x 409  well as the upper magnet sectors may jacking system. of two  kW power  This system is capable of producing a  kV peak-to-peak voltage across the dee  2.7  200  This tetrode is connected to the coupling loop by a tuned quarter  wave length of transmission l i n e . 200  frequency.  in. x 17  in.  The  l i d of this chamber as  be l i f t e d using a permanent 1y-instal1ed  The vacuum seal between the tank and the tank l i d consists  rectangular Viton gaskets with an internal pump out.  Four windows  have been placed in the sides and ends of the tank so that the inside of the cyclotron can be viewed when the l i d is lowered. I n i t i a l l y the vacuum tank is pumped down to approximately using mechanical pumps and a blower.  5  x  10~^  Torr  Four 1 2 in. o i l d i f f u s i o n pumps are  then used to reduce the pressure to approximately  5  x  10"  7  Torr.  The pump-  - 18 down time is less than 12 2.8  hours.  Median Plane Diagnostic Probes Three sets of probes were i n s t a l l e d in the CRC along the 0 deg, 90  and 270  deg azimuths defined in Fig. 2.2  and 2.10. 0.2  of a single v e r t i c a l flag whose width was  deg  The 0 deg probe consisted  in.  making shadow measurements onto the 90 deg and 270  This probe was deg probes.  used for  Due to the  high e l e c t r i c f i e l d along the dee gap centre l i n e , no attempt was  made to take  current readings d i r e c t l y from this probe. The 90 deg probe consists of three current pickup plates. most plate is 0.2  in. wide by 1.6  in. high and mounted  on i t is a 1 in. diam NE901 lithium glass s c i n t i l l a t o r which was the beam o p t i c a l l y . was  inner-  in. high and serves as a d i f f e r e n t i a l  The second plate is ).h in. wide by 1.6  current probe.  The  The third plate is h in. wide by 1.6  used to view  in. high, and i t  used for measuring integrated beam currents. The 270  deg probe is identical with the 90 deg probe except that the  integrated current plate is replaced with a phase probe.  The phase probe is  an rf shielded probe which can be used in conjunction with a fast oscilloscope to measure the time structure of the beam. The two current-measuring  probes at 90 deg and 270  deg were attached  to cantilevered I-beams which were driven r a d i a l l y by a Slo Syn SS250 stepping motor.  The stepping motor c o n t r o l l e r contained an internal pulse generator  whose output was  used to drive a d i g i t a l  readout  for monitoring the probe  position and to drive synchronously a six-channel Rikadenki  chart recorder to  produce current versus radius plots. The 0 deg probe was  driven by a brass screw which was  rf by a s p e c i a l l y shaped housing.  It was  shielded from the  also driven by a Slo Syn stepping  motor operating in conjunction with a pulse generator. The probe heads were a l l constructed of tantalum and copper.  No special  - 19 provisions were made for cooling the probes, and they were not capable of withstanding of time.  average currents  in excess of approximately 10 yA f o r any length  A l l probe plates had 0.2 in. wide lips at the top and bottom to  pick up secondary electrons which move along the magnetic f i e l d these l i p s , no bias voltages were required.  lines.  With  -  3 .  3 . 1  2  -  0  PROPERTIES OF THE INJECTION SYSTEM  I nt roduct ion To obtain high quality accelerated beams, i t is necessary  the properties of the injected beam.  to understand  This requires an understanding  beam transport devices which link the ion source to the cyclotron.  of the Most of  the CRC injection line is composed of e l e c t r o s t a t i c quadrupoles and e l e c t r o s t a t i c bends whose properties are well understood. ' The properties of the 5  magnet bore lens, the spiral  9  i n f l e c t o r , the horizontal steering deflector and  the injection gap are less well established.  In this section we shall  investigate these elements. 3-2  Optical Properties of the Magnet Bore The  ions enter the CRC by t r a v e l l i n g through a hole in the magnet bore  which acts as a lens.  In this section we shall calculate i t s properties.  We shall assume that the magnetic f i e l d around the central magnet axis is r a d i a l l y symmetric, in which case there w i l l only be radial and v e r t i c a l magnetic f i e l d components present.  For points close to the magnet axis we can  write the magnetic f i e l d B(r,z) to f i r s t order in the form r  9B (0,z) Z  „  r where B (0,z) is the vertical magnetic f i e l d along the central axis, k is the z  v e r t i c a l unit vector, r is a radial unit vector, and r is the radial distance from the magnet axis.  A vector potential A, which may be used to obtain the  above magnetic f i e l d , is given by  where 6 is a unit vector pointing in the theta direction with respect to a c y l i n d r i c a l co-ordinate system where z-axis lies along the central magnet axis. An appropriate Lagrangian is  21  L = j m(r where (r, 9,  2  + z  + r 0 ) + - ^ - ^ B (0,Z)  2  2  2  z  z) are the c y l i n d r i c a l co-ordinates of an ion moving along the  magnet bore and the dots indicate d i f f e r e n t i a t i o n with respect to time.  From  Lagrange's equations, we obtain the equations of motion mr  - mre qB  qrB 9  2  0  z  (3-D  K  z  (3.2)  -^7=0  2m  where K is a constant whose value depends on the i n i t i a l conditions. At the time these calculations were being performed  the magnetic f i e l d  had been measured out to a distance of 38 in. above the median plane. of  the magnetic f i e l d d i s t r i b u t i o n  is shown in Fig. 3-1.  The magnetic f i e l d  extends beyond 30 i n . , but at the time this calculation was additional data was  not available.  being done  As a rough approximation, the data shown  in Fig. 3-1 was extrapolated l i n e a r l y , and the magnetic f i e l d was fall  A plot  assumed to  to zero at z = 44 in. Eqs. 3-1 and 3-2 were solved numerically, and the calculated magnet  bore transfer matrix is given in Table I . Table I . Ax AP  X  =  Ay AP  y  z =  13  MAGNET BORE TRANSFER MATRIX  0.82  2 7  -0.37  -1. 2  Ax  -0.092  0 71  -0.012  -0. 5  AP  X  0.37  1  2  0.82  2. 7  Ay  0.012  0  5  -0.092  0. 71  APy z = 44 in.  in. J  Units for Ax Units for AP  X  V  J  and Ay are 0.1 in. and APy are 10 mrad  Ax and Ay are Cartesian co-ordinates which are measured in directions dicular to the magnet axis. equations AP of motion.  X  = dx/dz and AP  AP y  X  perpen-  and APy are divergences defined by the  = dy/dz where the z-axis l i e s along the direction  - 22 Fig.  3-2 shows what happens to a set of ions which i n i t i a l l y diverge  from a point on the magnet axis kk in. above the median plane.  In momentum  space the magnet bore rotates the ions through an angle of approx 36 deg.  In  real xy space the bore acts approximately as a d r i f t space with an added deflection due to the magnetic Fig. radius.  field.  3-3 shows what happens to an i n i t i a l l y parallel beam of 0.1  in.  In xy space the ions are rotated through an angle of approx 28 deg.  In momentum space the ions arrive at the i n f l e c t o r entrance with a transverse momentum component of approximately 1 mrad, which is approximately opposite the direction of the i n i t i a l displacement at kk in. above the median plane. Thus we see a small focusing e f f e c t . 3.3  The Spi ral  Inflector  The TRIUMF spiral inflector.  inflector ' 2  9  is a modification of the Grenoble spiral  There are two major differences between the two designs.  First,  the TRIUMF i n f l e c t o r had to be designed to operate in a non-homogeneous magnetic f i e l d while the Grenoble i n f l e c t o r was designed to operate in a homogeneous magnetic f i e l d .  Second, the electrode spacing was kept constant  in the TRIUMF i n f l e c t o r while the electrode spacing in the Grenoble  inflector  decreased toward the i n f l e c t o r e x i t . To calculate the trajectories and electrode geometry for the TRIUMF i n f l e c t o r a computer code AXORB was developed.  29  AXORB numerically integrates  trajectories through a measured magnetic f i e l d and an a n a l y t i c a l l y electr i c field.  Before constructing the electrode surfaces i t was decided to test the  AXORB approximations.  This was done by numerically calculating the potential  d i s t r i b u t i o n within the spiral  i n f l e c t o r using the relaxation method.  Trajectories were then numerically integrated through the potential tion.  approximated  distribu-  Details of a similar calculation applied to a considerably different  i n f l e c t o r geometry may be found in reference 26.  - 23 Figs. 3-4  to 3-6  show three views of the i n f l e c t o r geometry which was  used for the test c a l c u l a t i o n . was  The v e r t i c a l height of the central  trajectory  13 in. and the nominal electrode voltages were ±27-25 kV for an  energy of 300 keV and a median plane magnetic f i e l d of 3.0  kG.  shown in the figures are ruled surfaces which were generated  injection  The electrodes  using AXORB.  The surfaces were 2 in. wide, and the spacing between electrodes was For this p a r t i c u l a r geometry the electrode t i l t = tan"  1  1 in.  6 is given by  '13 in.-z)  I 11  in.  J •  If we cut the electrode surfaces with a plane which is perpendicular to the central trajectory at some point a distance z above the median plane, then 0 is the angle which the electrode cross-sections make with respect to a horizontal  line lying in the plane.  The two constants  in the above equation  were selected so that the ions would be correctly positioned r a d i a l l y once they reached  the i n f l e c t o r e x i t .  The dashed central trajectory shown in Figs. 3-4 using AXORB.  to 3-6 was calculated  The crosses indicate points on the central trajectory which were  obtained by integrating through the relaxation potential d i s t r i b u t i o n .  The  two sets of results are in good agreement. AXORB also contains a f a c i l i t y for computing the trajectories of ions whose i n i t i a l  conditions d i f f e r s l i g h t l y from those of the central  trajectory.  To study these trajectories we define the optical co-ordinate system shown in Fig. 3-7.  The o r i g i n of this co-ordinate system moves along with an ion  t r a v e l l i n g on the central trajectory. along the unit vectors h, v, and u.  The quantities h, v, and u are measured v is the central trajectory  tangent  vector, h is a horizontal vector which is perpendicular to v, and u = h x v. The direction of h is defined so that u w i l l have a positive v e r t i c a l component,  n, v, and u may  be specified as functions of r where r is the  position vector of the o r i g i n of the h, v and u co-ordinates.  - 2k  P (t) =  r(t)  • h ( r ( t ) + Ar(t))  r(t)  • u ( r ( t ) + Ar(t))  c  h  P„(t) =  c  U  P ()=  [  t  (  t  " o°( c(t))] • 0 ( T ( t ) )  )  v  t  F  c  V  Here "r (t) is the central trajectory position vector, vector of some other trajectory, and V ion.  We have defined  is the speed of the central  q  c  P^ and P  c  trajectory. tum  P  y  trajectory  Ar(t) so that r ( t ) + Ar"(t) represents a point on the  central trajectory such that T"(t) - ["F (t) + A~r(t)] w i l l v(r~ (t) + AF(t)).  r~(t) is the position  u  be perpendicular to  are divergences with respect  to the central  represents the r e l a t i v e difference between the forward momen-  component of the displaced  trajectory and the central  trajectory.  Figs. 3.8 to 3-12 show t r a j e c t o r i e s which were started out approximately 2 in. in front of the i n f l e c t o r entrance with a single non-zero component chosen from among h, u, P^, P  y  and P .  Results are given both for calculations  which were done using AXORB and for calculations done using the numerically calculated potential.  The two results agree f a i r l y well, indicating that the  AXORB approximations are reasonably v a l i d . The  geometry of the i n f l e c t o r which was actually i n s t a l l e d in the CRC  differed s l i g h t l y from the geometry used for the test calculation.  The CRC  i n f l e c t o r was designed for an injection energy of 293 keV with a median plane magnetic f i e l d of 2.99 kG. The nominal electrode 6 was defined  by  9 = tan ^—pj—^' . — . -1  n  voltages were ±26.5 kV, and  The electrode width and spacing  remained unchanged. If we start with specimen beams shown in Fig. 3-2 and 3-3 and continue through the i n f l e c t o r , then we arrive at the i n f l e c t o r exit with the results shown in Figs. 3-13 and 3-14.  The i n i t i a l l y divergent beam arrives at the  - 25 i n f l e c t o r exit with a v e r t i c a l extent of ±0.4 in. and a radial extent of ±0.14  in.  radial  The maximum v e r t i c a l  divergence is ±48 mrad, and the maximum  divergence is ±27 mrad.  The i n i t i a l l y parallel  beam arrives at the  i n f l e c t o r exit with a v e r t i c a l extent of ±0.12 in. and a radial extent of ±0.06 in.  The maximum v e r t i c a l  divergences are ±12 mrad v e r t i c a l l y and  ±6 mrad radial1y. The i n f l e c t o r  i s also dispersive.  that a change of 1% in the value of P  y  From Table I I or Fig. 3-12 we see  produces a v e r t i c a l  displacement of  -0.31 i n . and a v e r t i c a l divergence of -22 mrad at the i n f l e c t o r Radially we arrive  exit.  at the i n f l e c t o r exit with a displacement of -0.042 i n .  and a divergence of 7-7 mrad.  In order to limit this dispersion the energy  spread introduced by the chopper and buncher should be kept as low as poss ible. Table I I .  CRC Inflector  h  0.20  -0.53  -0.65  h u  0.64  -0.14  -0.39  1.3  -0.29  1.2  0.27  P  —  Pu  -0.77  V  p  -1.2  0.0  v  0.0  Transfer Matrix  0.41  0.0  h  -1.4  0.0  0.16  1.1  0.0  h u  0.23  1.6  0.0  P  0.38  -1.5  1.0  V  0.0  1.0  p„  0.0  inf1ector ^ ex 11  p  u  V  i nf1ector exi t  Units for h, u and v are 0.1 in  3.4  Units for P  n  and P  Units for P  v  are \% of v  u  are 10 mra Q  The Deflector The deflector consists of a c y l i n d r i c a l  parallel  to the axis of the cylinder.  If r  1  capacitor with a magnetic f i e l d  and r are the inner and outer 2  radii of the deflector electrodes, then the e l e c t r i c f i e l d  inside of the  - 26 deflector w i l l be given approximately by AV E(r) =  r &n(r /r ) 2  1  where AV is the potential difference across the electrodes and E(r) is the e l e c t r i c f i e l d strength at radius r. A l l radii are measured from a polar axis running along the central axis of the c y l i n d r i c a l electrodes.  The equations  of motion may be written in the form mr = mr9  2  -E(R) - qBrG  (3-3)  — ( m r 6 ) = qBrr dt  (3.4)  2  where B is the value of the magnetic f i e l d , q is the ion charge, and m is the ion mass. Eqs. 3-3 and 3-4 have the t r i v i a l solutions r(t) = R = constant = 6(0) + e t  9(tj  0  provi ded .  0  _ qBR + /(qBR)2 + 4mRqE  O  2mR  These solutions correspond to an ion moving on a c i r c l e of constant with constant trajectory.  angular v e l o c i t y 9 . Q  The e l e c t r i c f i e l d E(R)  We shall c a l l  radius R  this trajectory the central  required to produce a central trajectory of  radius R is given by E(R)  = ^ q t  i 1 R  mag-  R  where E^ is the ion k i n e t i c energy and R g ma  ion  in magnetic f i e l d B.  is the cyclotron radius of the  - 27 To achieve proper centring the deflector was central trajectory radius of 6.5 36 deg.  designed  to have a  in. and an e f f e c t i v e azimuthal  extent of  For a 300 keV H" ion in a 3-0 kG magnetic f i e l d , this requires a  central trajectory e l e c t r i c f i e l d  intensity of 33 kV/in.  by placing potentials of approximately  16.5  This was  achieved  kV on a pair of c y l i n d r i c a l  electrodes whose inner and outer radii were 6 in. and 7 i n . , respectively. The electrodes were constructed of aluminum and each electrode was  2 in. wide.  The CRC deflector is shown in Fig. 2.2. Eqs. 3-3 and 3.4  cannot be solved a n a l y t i c a l l y for arbitrary  conditions; however, approximate solutions may techniques  described in Appendix B.  initial  be obtained using the  These techniques were used to calculate  the deflector transfer matrix shown in Table I I I . Table I I I . Ar AP  Az AP  Z  As APQ  1  f  0.38  0.0  0.0  0.0  0.18  -1.0  0.77  0.0  0.0  0.0  0.81  Ar AP  0.0  0.0  1.0  0.41  0.0  0.0  Az  0.0  0.0  0.0  1.0  0.0  0.0  AP  -0.78  •0.18  0.0  0.0  1.0  0.37  As  0.0  0.0  0.0  0.0  0.0  1.0  AP  0.78 R  Deflector Transfer Matrix  deflector exi t  r  Z  Q  def1ector exi t  Units for Ar, Az and As are 0.1 Units for AP  r  and AP  Units for AP  Q  are  Z  are  in.  10 mrad  \% of v  0  Here we have defined the variables Ar  = r(t) - R  As  = R(6(t) - 8 t) 0  AP  r  = r/v  AP  e  = (§(t)r(t) - 0 R)/v  0  o  o  - 23 z P where v  = v e r t i c a l distance o f f median plane z  = z/v  0  is the speed of the 300 kev  Q  acts as a d r i f t space in the v e r t i c a l  ion.  We have assumed that the deflector  direction.  If we start with the specimen beams shown in Figs. 3-13  and 3-14  and  trace the trajectories to the deflector e x i t , then we arrive at the results shown in Figs. 3-15  and 3-16.  In the case of the i n i t i a l l y divergent beam,  the maximum radial divergences have decreased the radial extent of the beam has  from ±27 mrad to ±8 mrad, and  increased from  ±0.15 in. to ±0.24 in. In  the case of the i n i t i a l l y parallel beam, we see that the maximum displacements and divergences are approximately  the same after the deflector as they were  before i t . Fig.  3-17  shows how  the co-ordinates of the ion's centre of curvature  vary when the deflector voltage is changed.  Here the x-y co-ordinate system  is centred on the magnet axis and the x-axis points along the centre of the dee.  The theoretical points were calculated using the results given in  Appendix B, and the measured points were obtained using the techniques described in Section 3-5-  The theory and measurements appear to be in f a i r  agreement. 3-5  Experimental  Performance of the Inflector-Deflector System  With an injection energy of 278 keV and a median plane magnetic f i e l d of 2.91  kG the i n f l e c t o r voltage required to produce a horizontal beam was  predicted to be ±25.1  kV, and the deflector voltage required to centre the  beam along the dee gap centre line was  predicted to be ±15-25 kV.  mentally the optimum voltages were found to be respectively. values may  Experi-  ±24.6 kV and ±15.0 kV,  The small differences between the experimental  and theoretical  be attributed to fringe f i e l d s which modify the e f f e c t i v e length of  - 29 the electrodes and electrode alignment errors which in some places could be as large as ±0.02 in. The transmission of the system was found to be between 90 and 100%. For peak currents of the order of 100 uA the r e l i a b i l i t y was high and no problems were encountered with sparking after the system had been properly conditioned.  When the peak currents were increased to 600 uA the performance  deteriorated, and i t was impossible to operate the i n f l e c t o r sparking.  for more than a minute without  This problem was never solved; however, i t is  anticipated that the performance of the i n f l e c t o r could be improved by caref u l l y optimizing the input beam, by additional conditioning, by cleaning the electrode surfaces again, and by improving  the vacuum.  To obtain additional information, a series of median plane measurements were performed on the unaccelerated beam. the probes described in Section 2.8.  These measurements were made using  For these studies the d i f f e r e n t i a l probe  width was reduced to 0.1 i n . , and the centre post structure was removed so that the beam could travel 3/4 turn without  encountering  obstructions.  Measurements were made by varying a l l of the operating parameters and noting how the position of the beam changed.  Parameters varied included the  magnetic f i e l d B, the injection energy E, the deflector voltage V^, the i n f l e c t o r voltage V. and the direction and position of the beam at the i n f l e c t o r entrance.  The l a t t e r measurements were made by using a set of elec-  t r o s t a t i c deflection plates located 16 in. above the i n f l e c t o r entrance to steer the incoming beam.  A l l measurements were made on a low emittance beam  of approx 0.2 IT in. mrad. The  results of the v e r t i c a l measurements are shown in Table IV. The  f i r s t two columns of the table contain the measured and calculated values of Az/Ax where Az is the v e r t i c a l displacement parameter x is changed by Ax.  of the beam at 90 deg when  The last two columns of the table contain  - 30 Table IV. VERTICAL BEHAVIOUR OF A BEAM PASSING THROUGH THE INFLECTOR-DEFLECTOR SYSTEM FOR VARIOUS PARAMETERS  Quant i ty  Measured  Calculated  Measured  Calculated  Az/Ax  Az/Ax  AP /Ax  AP /Ax  X  B  -2.0  E  -0.16  V  in./kG in./keV  1.2 in./kV  i  -1.9 -0.14  z  in./kG in./keV  -1.3  in./kG  z  -1.0  in./kG  -0.049 in./keV  -0.038 in./keV  0.38 in./kV  0.25 in./kV  0.81 in./kV  AP  n  90  in./rad  80 in./rad  27 in./rad  26 in./rad  AP  U  35 in./rad  50 in./rad  14 in./rad  17 in./rad  Az measured at 90 deg azimuth Estimated error in measured Az/Ax == 20% Estimated error in measured AP /Ax ? 30% z  measured and calculated values of AP /Ax where AP z  z  is the change in the v e r t i -  cal momentum of the beam when parameter x is changed by Ax. determined  AP was z  by measuring Az at 90 deg and 270 deg and using the relation . , Az(in.)@270 deg - Az(in.)@90 deg AP (m.) = . z  IT  It was estimated that Az could be estimated with an accuracy of ±20%. estimated error in AP  Z  is then given by  /l  The  20% - 30%.  In Table IV we see that most of the measured values agree  reasonably  well with the calculated values in view of the error estimates given above. There do appear to be discrepancies in Az/AV. ,  AP^/AV. and Az/AP  u>  and these  discrepancies have never been adequately explained. Table V contains the measured and calculated values of Ar/Ax where Ar represents a change in radius along either the 0 deg, 90 deg or 270 deg azimuth.  Many of the radial perturbations are much smaller than their v e r t i c a l  - 31 " Table V.  Quant i ty AR i n. AB [kG, Z  AR AE  in. keV AR f i•n. APh' rad  RADIAL BEHAVIOUR OF A BEAM PASSING THROUGH THE DEFLECTOR SYSTEM FOR VARIOUS PERTURBATIONS Theory 0 deq  Measu red 0 deg  Theory 90 deq  -0.9 ± 0.1  -1.7 0.012  0.02  ±  -3-9  0.02  Measured 90 deq -4.4  ± 0.4  INFLECTOR-  Theory 270 deq -2.9  Measured 270 deq -2.8  0.022 -0.02  0.052 0.059 ±0.01  ± 0.2 ±  0.02  ,  AR AP  i n. rad r. i AR i n. AV j kV *, AR i n. AV kV j  1.4  -5 ± 4  5.8  7 ± 4  -5.8  no data  -14  -20 ± 4  -6.8  0 ± 8  6.8  no data  0 ±  -0.19  U  -0.015  0.06  -0.16 ±  0.05  -0.064 -0.058 ± 0.005  -0.033 -0.030 ± 0.005  0.19  0.19  0.064 0.061  ±  0.05  ± 0.005  d  L  counterparts, and this makes accurate measurements of these quantities d i f f i cult.  The radial perturbations could be measured with an accuracy of ±0.03 in.  and this was  approximately  the same order of magnitude as many of the motions  themselves.  In such cases the radial measurements s t i l l  serve a useful  purpose because they place an upper limit on the measured value of the quantity. Within the estimated errors most of the measured values in Table V agree with the theoretical predictions. However, there are some discrepancies. The largest of these are in Ar/AB 0 deg.  z  at 0 deg, Ar/AE at 270  The measured value of Ar/AB^ at 0 deg appears to be f a i r l y accurate,  but i t is half as large as the predicted value. 270  deg and Ar/AP^ at  The measurements of Ar/AE at  deg and Ar/AP^ at 0 deg are considerably less accurate, which may  for the measured and predicted values having the opposite sign. discrepancies may  account  These last  two  not be too s i g n i f i c a n t because the absolute values of these  quantities are both quite small.  - 32 3-6  The  Injection  Gap  After leaving the deflector the ions are accelerated by one-half of the peak dee-to-dee voltage as they cross the 1 in. gap between the centre post exit and the puller electrode connected  to the dee.  acts as a lens whose radial and v e r t i c a l focal of the entrance and exit  The  injection  gap  lengths depend on the geometry  openings.  The focal powers for several injection gap geometries were calculated using numerical  techniques.  The  injection gap potentials were calculated by  solving Laplace's equation in three dimensions using the relaxation method with a grid spacing of 0.125  in.  3 7  The focal powers were then calculated by  using the computer code TRIWHEEL to numerically integrate t r a j e c t o r i e s the potential  distribution.  The f i r s t  through  3 8  injection gap geometry had symmetric 1.5  openings in both the centre post and the p u l l e r electrode.  in. x 1.5  in. square  The equipotentials,  as viewed in v e r t i c a l and radial planes cutting through the centre of the gap, are shown in Fig. 3-18.  These equipotentials are approximately  respect to a v e r t i c a l plane cutting through  symmetric with  the centre of the gap  halfway  between the entrance and e x i t . Figs. 3-19  and 3-20  show curves of focal power versus gap-crossing  phase for this symmetric geometry.  Here the 0 deg phase ion crosses the centre  of the gap at the instant the rf voltage has reached positive phase ions arrive l a t e r . -15  i t s peak, and the  The curves cover a phase range between  deg and kS deg which encompasses the phase acceptance  range of the  cyclotron. The focusing of dee gap lenses has been studied by a number of authors, and much of these analyses can be applied to the injection gap. Figs. 3.19  and 3-20  3 9 - 1 + 1  From  we see that for the symmetric geometry the focusing becomes  stronger as the phase increases.  This is mainly due to the rf voltage  - 33 variation e f f e c t .  An ion crossing the gap with a positive phase w i l l see a  3 9  higher e l e c t r i c f i e l d near the gap entrance than at the gap e x i t b e c a u s e the rf voltage decreases in the time required for the ion to cross the gap. Since the e l e c t r i c f i e l d s near the gap entrance are focusing while the f i e l d s near the gap exit are defocusing, the rf voltage variation effect produces a net  focusing e f f e c t .  Negative phases encounter the opposite situation and  are defocused. For the symmetric geometry, the transition phases between focusing and defocusing are -10 deg and -15 deg, and the focusing for 0 deg is positive. This is due to velocity gain and deflection e f f e c t s .  Since the ions are  accelerated as they cross the gap, they spend more time in the focusing f i e l d s near the gap entrance than they do in the defocusing f i e l d s near the gap e x i t , and this produces a focusing e f f e c t .  In addition, due to the focusing at the  gap entrance, the ions w i l l be farther from the central axis of the gap when they pass through the focusing f i e l d s than when they pass through the defocusing f i e l d s , and this produces additional A rough check on the symmetric  focusing.  geometry result was made by calculating  the focal power of a lens consisting of two 1.5  in. diam cylinders separated  by a distance of 1 i n . Assuming that the lens is excited with an r f voltage whose wave length is much longer than the dimensions of the lens, the e l e c t r i c f i e l d s near the axis of the cylinder can be written in the form'*  2  V  2?  Er =  where E  r  f tanh  0.33 Vr  1.32  sech  z + S/2)  1.32  - tanh  z + S/2)  is the radial f i e l d component, E  the spacing between the cylinders, r  Q  1.32  sech  z  z - S/2)  1.32  cos  z - S/2  cot  cos  cot  is the axial f i e l d component, s is  is the radius of the cylinders, z is the  - 34 axial distance from the centre of the gap, V is the peak potential between the two cylinders, and w is the rf frequency. was  The focal power of this lens  calculated by numerically integrating the equations of motion with the  above e l e c t r i c f i e l d s , and the results are plotted in Figs. 3.19  3-20.  and  They are in good agreement with the results obtained for the symmetric geomet ry. Fig.  3-21  shows equipotential curves for the asymmetric gap which was high by 1 .25 in. used throughout the CRC experiments. Its entrance is 0.75 in. wide and i t s A  exit is 2 in. high by 1.5  in. wide.  Asymmetric geometries such as this one  produce s t a t i c focusing in addition to the dynamic focusing discussed e a r l i e r . S t a t i c focusing arises when the curvature of the equipotentials is greater on one side of the gap than on the other.  The v e r t i c a l equipotentials are more  curved at the gap entrance than at the gap e x i t .  Thus the v e r t i c a l  f i e l d is  greater at the gap entrance than at the gap e x i t , and v e r t i c a l focusing is produced.  This situation is reversed for the radial equipotentials, and  radial defocusing is produced. The magnitude of these effects is shown in Fig. 3-22 plotted the s t a t i c e l e c t r i c f i e l d components 0.1  where we have  in. o f f the central axis of  the gap as a function of distance from the centre of the gap.  The net  impulse  which an ion receives due to s t a t i c e l e c t r i c defocusing is proportional to the area under these curves, and we see that the asymmetric gap  produces  considerable s t a t i c focusing while the symmetric gap produces almost none. In addition to the gap geometries discussed previously, calculations were also performed on a gap whose entrance was 2 in. wide by 1 in. high and whose exit was wide by 0.75  1 in. wide by 2 in. high; and on a gap whose entrance was 2 in.  in. high and whose exit was 2 in. wide by 0.75  in. high.  power versus phase curves for these geometries are shown in Figs. 3-19 3.20.  Focal and  As expected, the absolute magnitude of the focal power increases as the  - 35 injection gap is made more asymmetric. The radial and v e r t i c a l focal powers cannot be varied independently of one another, and i t is possible to show that * 1  1 1' — +— f  where f  r  r  f  z  h  0  9AT  ,  RT H C  is the radial focal length, f  energy gain through the gap,  is the v e r t i c a l focal length, AT is  <> j is the gap-crossing  phase, h is the rf  harmonic number, R is the magnetic radius of the ion as i t crosses and T  c  the gap,  is the ion energy at the centre of the gap. If a s t a t i c e l e c t r i c f i e l d  Eq. 3-5 the ratio of the v e r t i c a l be -1.  ,  (3-5)  This is equivalent  is applied to the gap, then according to focal power to the radial focal power must  to requiring that in Fig. 3-22 the ratio of the  area under the radial e l e c t r i c f i e l d curve to the area under the vertical e l e c t r i c f i e l d curve must be -1.  These areas were calculated and for the  asymmetric geometries i t was found that the deviation from unity was never more than ±0.04.  This  is commensurate with the size of the numerical errors in  the c a l c u l a t i o n . A comparison between the results predicted by Eq. 3-5 and the actual numerical results was made.  Using TRIWHEEL i t was found that AT(cfj) was given  to within \% accuracy by AT(<j>) = 96 keV cos (<j>) ,  (3-6)  assuming a peak rf voltage of 100 kV.  1  c  = 350 keV, we  , . i  1  — + — = 0.062 i n . f f r  Taking R = 11 in. and T  - 1  sincb .  z  In Fig. 3*23 we have plotted the TRIWHEEL calculated values o f l / f versus z  l/f  r  for phases of -15 deg, 15 deg and 45 deg.  The calculated values may be  - 36 approximated f a i r l y well with straight lines having a slope of -kS deg; however, in order to obtain good f i t s the value of <j> in Eq. 3-6 had to be increased by approximately  10 deg.  This discrepancy is due to acceleration  and deflection effects which were neglected in the derivation of Eq. 3-6. This 10 deg s h i f t is consistent with the fact that the transition phase for the symmetric gap-was approximately  -10 deg.  It is shown in the references cited above that the phase dependence of the focal power of a lens can be expressed approximately — « s i n (ij)  in the form  + ty)  where -\\> is a geometry-dependent phase angle which is approximately the transition phase.  equal to  In Figs. 3-19 and 3-20 we have obtained good f i t s to  a l l of the calculated focal powers using sin curves.  This provides an addi-  tional check on the operation of the computer programs and provides a useful means for parameterizing the focal properties of the injection gap. 3-7  Measurement of the Injection Gap Focal Powers The v e r t i c a l focal power of the injection gap was calculated by  measuring Az/AVj where Az is the change in the v e r t i c a l beam position when the i n f l e c t o r voltage is changed by AV.j.  Using the thin lens  approximation  we can write Az AV:  Az rf on at 90 deg  AV  TTP  rf o f f at 90 deg  2 f  z  AZ' A  V  i  rf o f f 'at 0 deg  (3.7)  where P is the ion momentum after the injection gap measured in cyclotron units, f  is the v e r t i c a l focal length of the lens, and the subscripts i n d i -  cate the azimuth and operating conditions under which the quantities were measured.  The measured values are  37 Az.  = AV | rf on at 90 deg  0.63 ± 0.05 in./kV  Az AV; rf o f f at 0 deg  =  0.60 ± 0.05 in./kV  Az  = AVj r f o f f at 90 deg P  1.2 ± 0.1 in./kV  =  11.7 ± 0.2 i n .  Substituting these values into Eq. 3-7 and solving f o r the focal power, we find l / f  = 0.060 ± 0.015 i n . " . 1  z  of 0.058 i n .  - 1  This agrees well with the theoretical value  f o r the case of 0 deg phase ions.  As predicted in Section 3-6, the injection gap vertical  focal power  appeared to be r e l a t i v e l y independent of phase over the region of interest. This was v e r i f i e d experimentally by noting that f o r a beam occupying a phase interval between approximately 0 deg and 30 deg Az AV:  rf on at 90 deg  did not depend s i g n i f i c a n t l y on the radius at which the measurement was made. A similar procedure  was  used to measure the radial focal powers.  This time we use the deflector to displace the beam r a d i a l l y at the injection gap entrance and measure the resulting change in radius at the 90 deg azimuth, For this case we write Ar AV^  Ar rf on AV(j at 90 deg  p r f off " f at 90 deg  Ar r  ' AV  d  rf o f f at 0 deg  where we have replaced the v e r t i c a l variables of Eq. 3.6 with their radial counterparts.  The measured values were  - 38 Ar AV  d  rf o f f at 90 deg  Ar rf o f f at 0 deg P  = -0.058 ± 0.005 in./kV  = -0.030 ± 0.005 in./kV  = 11.7 ± 0.2 in.  Ar AV7 I rf on ="°-°9 ± ° at 90 deg  0 1  i n  -/  k v  The Ar/AV quantities were obtained by f i t t i n g straight lines to the measured  mentsof r versus V^.  Solving for the focal power, we obtain  1 —  r  = -0.085 ± 0.035  in." . 1  From Fig. 3-20 we see that the theoretical value for l / f "0.04 i n .  - 1  r  at 0 deg phase is  The measured and calculated values appear to d i f f e r by s l i g h t l y  more than one standard deviation.  - 39 k. 4. 1  PROPERTIES OF THE INJECTED BEAM  Introduction To achieve optimum performance the beam emittance at the end of the  injection line must be matched to the acceptance of the central region of the cyclotron.  Failure to achieve a vertical match will lead to increased beam  heights at extraction and possibly to increased beam losses during acceleration.  Failure to achieve a radial match can lead to an increase in the energy  spread of the extracted beam.  In addition, if a chopper or buncher is placed  in the injection line, it must be adjusted so that i t does not distort the injected beam emittance and so that the phase of the injected beam matches the phase acceptance of the cyclotron.  In this section we shall discuss  these and related topics. 4.2  The Central Region Acceptance The size and shape of the central region acceptance is dependent on  the shape of the rf accelerating wave and on the shape of the central region magnetic field.  It has been shown theoretically that.by flat-topping the rf  wave with a third harmonic component and/or by adding a bump to the central region magnetic field the performance of the cyclotron can be improved. * 1  3  Since no attempt was made to operate the CRC in any of these modes, we shall confine our attention to the problem of matching the acceptances which were calculated assuming that the.cyclotron is operating in the fundamental rf mode with an isochronous magnetic field. The vertical and radial phase space acceptances as they appear on the high-energy side of the injection gap are shown in Fig. 4.1.  These  acceptances are phase dependent, and results are shown for three different phases which encompass the 0 deg to 30 deg phase acceptance of the cyclotron. The vertical acceptances were calculated by starting at 20 MeV with an  - 4o e l l i p s e which was chosen to minimize the beam height at higher radii and working backward toward the injection gap, using the ray tracing program TRIWHEEL.  The radial acceptances were calculated by working backward  35  from a c i r c u l a r emittance at 20 MeV. mum  These acceptances should y i e l d a maxi-  v e r t i c a l beam envelope of approx 0.4  in. at extraction and a maximum  energy spread of ±600 keV. Fig. 4.2  shows the acceptances of Fig. 4.1  tracked back to the low-energy  side of the 1.5  tion gap described in Section 3-6.  after they have been  in. * 1.5  in. symmetric  injec-  To achieve a matched condition the beam  at the deflector exit must f a l l within the overlapping area of the phasedependent acceptances.  The v e r t i c a l acceptances for different phases overlap  very well, and the overlap acceptance is e s s e n t i a l l y the same as the 0 deg acceptance.  The radial overlap region is much smaller than any of the indi-  vidual acceptances, and in this case we have defined a new overlap e l l i p s e which is contained in the overlap area. The shapes of the acceptances on the low-energy  side of the injection  gap may be varied by varying the focal power of the injection gap.  Calcula-  tions similar to those described above were performed for each of the injection gap geometries discussed in Section 3-6, the low-energy  and the overlap acceptances on  side of the injection gap are shown in Figs. 4.3  to  4.5-  Changing the focal power of the injection gap changes the orientation of the acceptances, and this makes the geometry of the injection gap an important parameter which can be varied to help achieve matching. 4.3  Matching Calculations If we assume the injected beam is mono-energetic,  matrix from a point 44 entrance is given by  then the transfer  in. above the median plane to the injection gap  -  f  Ar(in.)  -  41  (  0.2  -0.42  -0. 58  -0.23  AP (in.)  0.24  7.4  0. 52  1.2  AP (rad)  Az(in.)  1.9  72  -0. 73  -0.47  Ay(in.)  AP (In.)  1.1  47  -0. 43  1.3  AP (rad)  r  z  x  y  where the x- and y-co-ordinates are defined in Section 3-2, and AP gap.  z  Ax(i n.)  and Ar, AP  r>  Az  are displacements and relative momentum components at the injection  We note that the motion is strongly coupled, and this makes the matching  problem d i f f i c u l t . The beam at the magnet bore entrance can be represented as a volume in a four-dimensional space whose co-ordinates are Ax, AP transfer matrix in Eq. 4.1 exit. of  x>  Ay and AP^.  The  may be used to track this volume to the deflector  Here, i f matching is to be achieved, the two-dimensional projections  the hypervolume  in Ar - AP  r  and Az - AP  space must f a l l within the central  z  region acceptance e l l i p s e s . Unfortunately, l i t t l e is known about the four-dimensional hypervolume occupied by the injected beam.  Conventional two-dimensional Ax - AP  x  and  Ay - APy phase space measurements using the moving s l i t method have been made near the ion source; however, these measurements do not uniquely determine the  four-dimensional volume occupied by the beam.  In view of this fact, we  shall assume that the injected beam at the magnet bore entrance 44 in. above the  median plane is enclosed in an e l l i p s o i d whose equation is a Ax n  2  + 2a A*AP 12  x  + a 2AP 2  2 x  + a Ay 3 3  2  + 2a i AyAP 3  t  + a^i+APy = 1. 2  y  (4.2) This e l l i p s o i d w i l l be uniquely determined, provided we assume that i t s projections onto the Ax-AP  plane and the Ay-AP  v  plane coincide with the  measured and/or calculated two-dimensional phase space e l l i p s e s in these pianes. The e l l i p s o i d  in Eq. 4.2 w i l l be transformed into a new e l l i p s o i d once  - 42 the beam has reached the injection gap. e l l i p s o i d may  The c o e f f i c i e n t s of the new  be calculated in terms of the c o e f f i c i e n t s in Eq. 4.2 and the  transfer matrix in Eq. 4.1  using the TRANSPORT matrix formalism.  8  Due to the coupling between the four co-ordinates, i t appeared tical  to determine  the c o e f f i c i e n t s in Eq. 4.1  imprac-  by starting with the central  region acceptances at the injection gap and working backward to the magnet bore entrance.  As a result, matching  solutions were obtained by s t a r t i n g at  the entrance to the magnet bore and systematically varying the Ax-AP  x  and  Ay-APy projections of the injected e l l i p s o i d until a good overlap was obtained between the injection gap acceptances and the projections of the transformed e l l i p s o i d at the injection gap.  For this calculation the Ax-AP and Ay-APy x  projections at the bore entrance were assumed to have areas of 0.5T which is the nominal  in.mrad,  ion source emittance after the accelerator tube.  It was  assumed that the v e r t i c a l and radial acceptances at the injection gap have an area of 0.005TT i n .  2  This required that the radial acceptances discussed in  the previous section be scaled up by a factor of approximately Fig. 4.3 shows a typical matching the  CRC  solution which was obtained using  injection gap whose entrance is 1.25  whose exit  is 2 in. high by 1.5  in. wide.  5/3-  in. wide by 0.75  in. high and  Here we achieve a v e r t i c a l overlap  of approximately 30% and a radial overlap of approximately 70%. also shows the two-dimensional  Fig. 4.3  projections of the hypere11ipsoid at the magnet  bore entrance and at the i n f l e c t o r entrance required to obtain these solutions. Figs. 4.4 and 4.5 show typical matching  solutions which were obtained  using the acceptances produced by the other gaps described in Section 3.6. Radial overlaps of approximately 70% can be obtained with any of these gaps; however, the CRC gap appears to give superior v e r t i c a l matching.  The CRC  is superior v e r t i c a l l y because the major axis of i t s acceptance e l l i p s e approximately along the line A z * 2AP . Z  gap  lies  From the last two rows of the matrix  -  in Eq. 4.1,  43 -  we see that the Az-AP d i s t r i b u t i o n of the beam at the injection z  gap entrance w i l l  tend to l i e along this line irregardless of the shape of  the injected beam. It should be noted that the magnetic f i e l d  region between 41  in. above  the median plane and the i n f l e c t o r entrance plays an important part in the matching calculations. field  The rotational effect introduced by the magnetic  in this region allows us to obtain improved v e r t i c a l overlaps.  In the  main TRIUMF cyclotron the stray v e r t i c a l magnetic f i e l d w i l l extend over a much larger region, and a detailed study of the optical properties of the v e r t i c a l beam line should be made before attempting to match the calculated central  region acceptances. Matching between the beam line and the i n f l e c t o r was achieved using a  quadrupole t r i p l e t located d i r e c t l y above the i n f l e c t o r entrance. vidual quadrupoles Section 2.4,  The indi-  in this t r i p l e t were identical to those described in  and they were installed approximately 35 i n . , 44.5  above the median plane.  in. and 54 in.  Calculations done using the computer program  TRANSPORT indicated that this t r i p l e t could produce the required matching when operated at voltages in the 0-6  kV range.  8  The exact voltages required  depended upon the shape of the emittance at the t r i p l e t entrance, and f o r the most part these voltages were adjusted experimentally to optimize the performance of the cyclotron during operation. 4.4  Experimental Observations on the Injected Beam The two-dimensional  the ion source.  emittances of the beam were measured d i r e c t l y after  Unfortunately i t was p r a c t i c a l l y impossible to predict the  shape of the emittances at the magnet bore entrance in terms of the measured emittances due to losses along the beam l i n e .  As a result, the quadrupole  settings required to optiimize the injected beam emittance were found by t r i a l and error, and the behaviour of the accelerated beam was used as a diagnostic  -  tool.  44 -  An emittance measurement could conceivably have been made at the  magnet bore entrance; however, there was very l i t t l e  room available in the  beam line at this point f o r i n s t a l l i n g an emittance-measuring  apparatus with-  out extensively rearranging some of the beam line components. Fig. 4.6 shows the unaccelerated beam as viewed on the s c i n t i l l a t o r probes at 90 deg and 270 deg. These pictureswere taken at low-current levels, 0.25ir  and the injected ion source emittance was estimated to be The beam in Fig. 4.6(a) appears to be diverging v e r t i c a l l y .  in.mrad.  Comparing the  beam sizes at the two azimuths and assuming that the beam does not pass through a waist in t r a v e l l i n g between 90 deg and 270 deg, i t appears that the maximum value of AP  z  is greater than approximately 0.08 in. The beam half-  height at 90 deg appears to be approximately 0.3 in. These observations appear to be reasonably consistent with the predictions of Section 3-3total  The  radial width of the beam at these two azimuths appears to be between  approximately 0.1 in. and 0.2 in. Again, this appears to be reasonably consistent with the predictions of Section 3-3Fig. 4.6(b) shows a similar set of photographs which were taken with s l i g h t l y reduced voltages on the last quadrupole  in the beam line.  This beam  appears to be much less divergent than the beam in Fig. 4.8(a), and the maximum divergence appears to only be approximately 0.02  in. The fact that  it appears possible to obtain a nearly p a r a l l e l v e r t i c a l beam is not consistent with the theoretical predictions of Section 4.3-  A possible explanation  for this apparent discrepancy is that there might be correlations between the Ax and Ay co-ordinates at the entrance to the magnet bore.  If this were the  case, we would have to include additional terms in Eq. 4.2, and this would change the matching problem considerably.  So f a r , i n s u f f i c i e n t experimental  evidence is available to rule out this p o s s i b i l i t y . The beam p r o f i l e at the i n f l e c t o r entrance was measured on a p r o f i l e  - 45 monitor consisting of eight wires.  These wires had a spacing of 0.1 in. and  the diameter of the wires was 0.005 in.  Two monitors were used so that the  position and approximate size of the beam could be monitored in both the bland u-directions. The p r o f i l e currents which were measured when the photographs in Fig. 4.6(b) were taken are shown in Fig. 4.7.  In the u-direction  almost a l l of the beam f a l l s on three wires, and the estimated beam width is 0.4 ± 0.1 in. In the h-direction almost a l l of the beam f a l l s on two wires and the estimated beam width is 0.2 in. ± 0.1 i n . These spot sizes agree reasonably well with the calculated spot sizes in Fig. 4.34.5  Properties of the Chopped Beam The CRC chopper consists of two 3.0 in. deflection plates with a  spacing of 1.0 i n . , followed by a s l i t of variable width placed 15 in. downstream.  An r f voltage whose frequency  frequency  is one-half of the cyclotron rf  is applied to the plates to deflect the beam back and forth across  the s l i t to produce a chopped beam.  The phase width and relative phase of  the beam can be adjusted by varying the amplitude  and phase of the deflection  voltage, respectively. The operation of the chopper is i l l u s t r a t e d in Fig. 4.8.  Here we have  taken pictures of the beam as i t appears on the s c i n t i l l a t o r probe at 2-3/4 turns f o r relative chopper phases ranging between 40 deg and -20 deg. The chopper voltage was set at 2 kV and the s l i t width was 0.064 in. chopper at 40 deg, only positive phase ions are accelerated. is reduced  vanish.  When the phase  to 20 deg, the high v e r t i c a l envelope of the 0 deg phase ions  becomes visible near the high radius side of the beam. decreased  With the  farther, the well-focused positive t a i l At -10 deg a l l ions f a l l  As the chopper phase is  of the beam starts to  into a narrow radial band.  Here the  injected phases are centred around 0 deg and a l l of the ions are receiving nearly the same energy gain and arrive at nearly the same radius.  As the  - 46 phase is decreased to -20 deg, more negative phases are accelerated and the beam widens r a d i a l l y .  In addition, the beam starts to blow up v e r t i c a l l y ,  indicating that the negative phases are being defocused. be examined  in more detail  These effects w i l l  in later sections.  The chopper has been investigated t h e o r e t i c a l l y by J. Belmont and W. J o h o .  44  They have shown that the chopped phase interval Acf> immediately c  after the chopper s l i t  is given approximately by  4 wUd  where w is the width of the chopper s l i t , U is the potential  required to  accelerate the beam to i t s injection energy, L is the distance between the chopper and the analyzing s l i t , I is the e f f e c t i v e length of the chopper plates, V the  c  is the chopper peak-to-peak voltage, and d is the spacing between  chopper plates.  Their derivation assumes that the beam arrives at the  chopper s l i t at a waist whose width is equal to , and Ad> is defined as a w  c  full-width at half maximum.  For the CRC parameters, the above equation may  be written in the form A W  f  A  \  Ad) (rf deg) =  1  -  2  X  l  Q  3  —V (kV) c  In addition to r e s t r i c t i n g the phase width of the beam, the chopper also introduces energy spread into the beam. the  At the entrance and exit to  deflection plates there are longitudinal e l e c t r i c f i e l d components which  modify the k i n e t i c energy of the beam.  In the time required f o r the beam to  travel across the deflection plates, the magnitude of the rf voltage changes, and the ions do not see the same longitudinal f i e l d at the exit as at the entrance.  The magnitude of this effect depends upon the transverse displace-  ment of the ions as they cross the chopper plates, and is given approximately by the e q u a t i o n  44  -  E  A(j)  c  A  47 -  p  where AE is the energy spread introduced by the chopper, E is the injection energy of the beam, e is the beam emittance and Ap is the distance between the  centres of adjacent beam pulses after the chopper s l i t .  the  CRC the above equation may be written in the form AE  0.96xl0  - 1 +  In the case of  e(in.mrad) V (kV) c  —  E  w( i n.)  These equations assume that the beam enters the chopper at a waist whose width is equal to w. The energy spread introduced by the chopper produces debunching. phase width of the beam at the injection gap is given  by  4 4  2TT L d AE A*  »  q  the  (4.5)  A<f) + c  Ap • E  9  where  The  is the d r i f t distance between the chopper and the injection gap. In = 280 i n . , and the above equation may be written in the form  CRC,  A<j> (rf deg) « a  q  A<j> ( r f deg) +  85O  e(in.mrad) —-  A<f>(rf deg)  .  c  Fig.  4 . 9 shows a graph of AE/E, A<j> and A<f> as a function of V  e = 0.2  c  in.mrad and w = 0 . 0 6 4  q  in. AcJ>g does not "decrease as V  c  c  assuming that is increased  above 4 kV, and i t s minimum value appears to be about 25 deg. For  high-current beams there is also debunching due to longitudinal  space charge. by assuming  An order-of-magnitude estimate of this effect may be obtained that the chopped beam is a rectangular parallelepiped of uniform  charge density.  If we assume that the transverse dimensions AT of this  parallelepiped remain fixed as the beam travels from the chopper s l i t  to the  injection gap, then we can write the equations of motion of an ion near the  - 48 end of the parallelepiped in the form dy 2  m  dt  2  (4.6)  = q Ey (0 ,y ,0 , AT, Ay, AT)  where m is the ion mass, q is the ion charge, y is the distance of the ion from the centre of the charge d i s t r i b u t i o n and Ey is the parallelepiped e l e c t r i c f i e l d , which is calculated in Appendix C.  In order to evaluate E  we have assumed that the charge density  P  =  i  y(o)  p  (2AT)  2  v  (4.7)  y(t)  Q  remains constant throughout the para11e1epiped. Here I current at the chopper s l i t and v Eq. 4.6  Q  is the beam velocity.  was solved numerically using the Runge Kutta method.  of the energy spread and phase width versus d r i f t 4.10  is the peak beam  and 4.11.  Graphs  length are shown in Figs.  These calculations were done assuming that the chopper was  adjusted to give a 30 deg phase beam immediately after the chopper s l i t and AT was assumed to be 0.2 after t r a v e l l i n g 280 1.7  in. With a peak current of 1.2  mA we see that  in. the energy spread in the beam has increased by  keV and the phase interval has increased to 55 deg.  From Eq. 4.4  we see  that the energy spread introduced by space charge is much larger than the energy spread introduced by the chopper for currents in the mA range. More detailed analysis of this problem may be found however, these results w i l l  in reference  be s u f f i c i e n t for interpreting CRC  44.  results.  The chopper also produces an emittance-broadening e f f e c t .  This  results from the beam being swept back and forth across the chopper s l i t , and for  the CRC geometry  i t may be shown that the e f f e c t i v e emittance increases  by a factor of approximately chopper  slit.  — = 1.3  in the direction perpendicular to the  4 4  During the i n i t i a l  low-current measurements the chopper s l i t width was  - 49 set  at 0.64  in. and the beam emittance E was approximately 0.2  During these measurements the peak injection current was or  less, and space charge effects could be neglected.  in.mrad.  limited to 100 uA  Measurements with  higher currents..wi 1 1 be discussed in Chapter 6. A simple experimental estimate of A<j> may c  be obtained using the  relation average chopped beam current A(f> (rf deg) = 360 deg. x  (4.8)  c  average unchopped beam current The currents were measured on beam stop #10 buncher.  180 in. downstream from the  The experimental values shown in Fig. 4.9.  the theoretical  f a i r l y closely parallel  predictions.  An attempt was also made to estimate A$  c  using a non-intersecting  phase probe placed in the beam line d i r e c t l y after the chopper s l i t s . probe consisted of a hollow cylinder 3 in. long and 1 in. diam.  By  This measuring  the beam-induced current which flowed between the cylinder and ground i t was possible to estimate the duration of the beam pulse passing through the cylinder.  In our i n s t a l l a t i o n , the probe was connected to a Hewlett-Packard  amplifier with a 1.3 GHz bandwidth through approximately 35 f t of Phelps Dodge SLA-38-5OJ cable. type 561B  The output of the amplifier was  fed into a Tektronix,  sampling oscilloscope with a type 3S2 sampling unit.  A typical set of phase probe photographs beam pulse produces two signals.  are shown in Fig. 4.12.  The f i r s t signal  is produced when the pulse  enters the cylinder and the second when i t leaves the cylinder. a ground plane was  Each  In our case  placed d i r e c t l y after the cylinder e x i t , and the second  pulse is probably more r e l i a b l e for timing measurements. When the chopper voltage is increased from 0.8 kV to 2.0 kV the width of  the phase probe signal narrows considerably.  of  the signal  If we estimate the duration  in terms of a full-width at half maximum, then Ad> - 114 deg c  - 50 at  kV and A<J) -  0.8  voltages the  are  96  were  the  cables  bandwidth  and by t h e  target  values  to  of  the  the  the  injected  the  techniques  graphed  Ad>  the  described  along with  been  than  the  intervals  chopper  34  closer  to  the  at  slits.  to  experimentally beam f o r  the  measured  5.  points  is  of  the  kV  limit  is  connecting  p r o d u c e d by  4.12(b)  the  the  theoretical  24  the  response of  the  from  kV.  the  The  values  electronics  deg. the  maximum and  chopping c o n d i t i o n s .  in  beam on  shows s i g n a l s  deg w h i l e  from  these  Fig. this It  by  values Error  was  the  appears 20  Belmont  be o b t a i n e d  by p l a c i n g  minimum  The s i z e  measurements  4.9-  approximately  prediced  gap o r  6.0  narrowest  This  kV and 6 . 0  since  deg c o u l d p r o b a b l y  the  2.0  measurement.  limit  injection  signal  The e s t i m a t e d  values  that  of  c a n be e s t i m a t e d Chapter  or  theoretical  deg w i d e .  by m e a s u r i n g  various  4.0  these  head.  Again  about  of  the  characteristics  Acj>c a t  for  voltages  appeared 26  Fig.  deg and 24  the  It  chopper voltages  c a n be o b t a i n e d  20  probe  looking  with  theoretical  than  the  the  values  cases while  approximately  theoretical  the  which  of by  associated with  interval  narrower  the  p l a c e d on  uncertainty  was  amplifier,  in  With  respectively.  resolution  interval  calculated  in both  respectively.  estimated  phase  The  deg  chopper  were  accelerated  the  phase  of  12 d e g ,  limit  A<f)g was  phase  deg,  capacitance  of  d e g and  appeared  have  12  kV.  respectively.  26  c  which were o b t a i n e d  38  radii  A<f> was  placed after  measured were  38 d e g ,  a l s o measured  c  target  2.0  c o u l d be d e t e c t e d  A<j> was a  of  18 deg and  pulse which by  deg a t  deg and  measured v a l u e  values  set  33  C  of  are of  ±10  deg  estimated  that  the  smallest  deg w i d e . and J o h o .  either  internal  using  A<J>  bars  of  This  is  Shorter  by m o u n t i n g  slits  within  the  cyclotron. Fig. beam a t Although  90  4.13  shows s c i n t i l l a t o r  deg and 270  the  luminosity  probe  photographs  deg  for  various  of  the  beam c h a n g e s as  of  the  c h o p p i n g and b u n c h i n g the  unaccelerated conditions.  chopper voltage  is  the  -  51 -  0 [<\/ to 4.0 kV, the size and shape of the injected beam does  increased from  not appear to change by large amounts. tent with theory.  This appears to be reasonably consis-  The 4/TT emittance increase which occurs when the chopper  is turned on should produce a  /4/TT = 1.13 increase in the beam envelope,  which would be d i f f i c u l t to observe on the s c i n t i l l a t o r s . voltage of 4.0 kV, the energy spread approximately  ±0-36 keV.  With a chopper  introduced by the chopper w i l l be  Due to dispersive effects in the i n f l e c t o r - d e f l e c t o r  system, this should produce v e r t i c a l displacements  of approximately  at 90 deg and ±0.09 in. at 270 deg. The radial displacements approximately  ±0.018 in.  could be  ±0.018 in. at 90 deg and ±0.007 in. at 270 deg. Comparing  Fig. 4.13(a) with 4.13(c) we see that the chopper does appear to increase the radial size of the beam at 270 deg; however, the v e r t i c a l size of the beam at 90 deg appears to be smaller with the chopper o f f than with the chopper on. This apparent discrepancy  is probably due to the difference in luminosity of  the two beams rather than to any energy or emittance e f f e c t s .  The radial  differences between the two beams were too small to be resolved using the d i f f e r e n t i a l probes.  This indicates that the differences are less than  ±0.05 in. or so. A series of measurements were also made on the accelerated beam in order to determine the effects of the chopper.  D i f f e r e n t i a l probe measure-  ments made on the high radius side of the beam for chopper voltages  ranging  between 0.0 kV and 4.0 kV f a i l e d to detect any chopper-dependent e f f e c t s . chopper-dependent transmission effect was noted, however.  A  As the chopper  voltage was increased from 1.7 kV to 4.0 kV the measured transmission from 1/4 turn to 6-1/4  turns increased from 78% to 94%.  6.0 kV was approximately to 86%.  The transmission with  95%; however, with 9-5 kV the transmission dropped  The increase in transmission as the voltage was increased to 4.0 kV  occurs because more of the injected beam f e l l within the phase acceptance of  - 52 the cyclotron.  The drop in transmission which occurs when the voltage is  increased to 9-5 kV is possibly due to emittance-broadening  effects caused by  the dispersion which occurs when the energy spread of the injected beam is increased. 4.6  Properties of the Bunched Beam The CRC buncher consists of two 0.5 in. wide accelerating gaps which TTV  are separated from one another by a d r i f t tube whose length is where oo is the r f cyclotron frequency and v  = 6 . 6 in.  is the velocity of the beam. A  Q  voltage V coscot is applied to the d r i f t tube, and the velocity v (cp j) of an ion after i t has passed through the buncher is given approximately by . . /2(Eo + 2qV costf),) 2qV v(*j) = / Ii_ ~v + coscfri , qV « / m mv J  0  E  Q  (4-9)  0  where E  0  is the energy of the beam at the buncher entrance, q is the ion  charge, m is the ion mass and <j>j is the r f phase angle when the ion passes the second buncher gap.  If the ion d r i f t s  the buncher, then the phase angle w i l l coL <f>f = - 7 7 7 v (<j>;)  +  *i«  <f>; +  coL — v,  a distance L after passing through  have become 2qV  1 -  (4.10)  COS(j> ; ) V  0  In the CRC L = 500 in. and V is adjustable between 0 and 2.0 kV. In Fig.  4.14 we have plotted the bunched phase <j)f = — v  <))j for several different values of V. slope of the <j>f and  Around  versus the i n i t i a l  phase  o  = 210 deg we see that the  curves is less than 1, and a large number of i n i t i a l  phases are concentrated into a smaller range of bunched phases. In Fig. 4.15 we have compared the signal obtained on the phase probe at 1-3/4 turns with the buncher o f f to the signal obtained with the buncher set at approximately 1.75 kV.  These photographs were taken with the chopper  adjusted to give a phase^ interval of approximately 40 deg.  If we define the  bunching factor to be the ratio of the average accelerated current with the  - 53 " energy spread introduced by the buncher is producing phase-dependent beam losses in one or more of the dispersive elements along the injection  line.  When the chopper is turned on, i f we neglect the extra energy spread introduced by the chopper, then the energy spread of the injected beam is decreased because the phases with the highest and lowest energy w i l l be eliminated from the beam before reaching the dispersive bends in the beam  1 ine. So far our analysis has neglected the effect of longitudinal space charge debunching.  The fact that the bunching factor does not change appreci-  ably when the peak current is raised from 80 pA to 350 pA appears to be consistent with the space charge estimates given in the previous section. With the buncher set at 1.75 kV and the chopper adjusted to give A<j> = 36 deg, c  the  be approximately 25 deg,  phase width of the beam at the injection gap w i l l  provided we neglect space charge e f f e c t s .  Assuming a bunching factor of  about 2 and a peak current of 350 pA at the buncher entrance, the peak current after the chopper s l i t w i l l be the order of 700 pA.  Using Fig. 4.13 we see  that for such a beam the space charge debunching w i l l be about 10 deg, and the  high current beam w i l l  arrive at the injection gap with a phase spread of  approximately 35 deg. This beam is well within the 50 deg phase acceptance of the  central  region of the cyclotron, and since the bunching factor was  estimated in terms of average currents, the bunching factor f o r this beam w i l l be the same as the bunching factor f o r the 80 pA beam. Due to dispersion, the energy spread introduced by the buncher increases the e f f e c t i v e emittance of the injected beam.  With a buncher  voltage of 1.75 kV, a chopper voltage of 4.0 kV with a s l i t width of 0.064 i n . , the  beam w i l l occupy a phase interval of 18 deg after the chopper s l i t and  the  measured bunching factor was 1.75-  The energy spread AE introduced by  - 54 This is because as  A<t>  c  decreases, we are on the average operating along a  steeper portion of the bunched phase versus i n i t i a l Fig.  4 . l 6 , and this increases the bunching  phase curves shown in  factor.  A comparison was made between the bunching  factors which were calcu-  lated t h e o r e t i c a l l y using Eq. 4.11 and those which were measured in the CRC. The measured bunching the sixth turn.  factors were obtained by making current measurements at  The results of this comparison are shown in Table VI. Table VI.  Chopper Voltage kV  Measu red' Bunch-i ng Factor  Ca 1 cu-l-ated Bunch ing Factor  0.125  360  80  3.0 ± 0.2  4.0 ± 0.3  4.0  0.125  36  80  2.3 ± 0.2  2.2 ± 0.1  8.0  0.125  18  80  2.4 ± 0.2  2.4 ± 0.1  0.0  0.125  360  360  3.0 ± 0.2  4.0 ± 0.3  4.0  0.125  36  360  2.3 ± 0.2  2.2 ± 0.1  8.0  0.125  18  360  2.4 ± 0.2  2.4 ± 0.1  0.064  78  100  2.8 ± 0.2  2.9 ± 0.3  0.064  20  30  2.0 ± 0.2  2.3 ± 0.1  4.0  c  rf deg  Peak Current at Source uA  0  1.0  A<j>  Chopper S l i t Width i n.  COMPARISON BETWEEN MEASURED AND CALCULATED BUNCHING FACTORS  ••  was estimated using Eq. 4.5.  Measurements were done for peak ion source  currents ranging between 30 yA and 350 yA.  During these studies the buncher  voltage was measured d i r e c t l y and found to be 1.75 ± 0.3 kV.  Due to the  uncertainty in the measured buncher voltage, error estimates are given for both the estimated and calculated bunching The measured and calculated bunching measured bunching  factors. factors agree f a i r l y well.  The  factor with the chopper turned o f f is s l i g h t l y smaller than  the calculated value of F.  One possible explanation for this is that the  - 55 buncher turned on to the average accelerated current with the buncher turned off,  then i t appears from the figure that a bunching factor of between 2 and  3 is obtained under these conditions. The bunching factor depends upon the chopper voltage as well as upon the  buncher voltage.  The chopper is located approximately 280 in. from the  injection gap and 220 in. from the buncher.  Thus part of the bunching action  occurs before the beam reaches the chopper and part of the bunching action occurs after the chopper.  In Fig. 4.16 we have graphed the bunched phase  versus the i n i t i a l buncher phase, assuming a d r i f t  length of 220 in. With  the  aid of Figs. 4.14 and 4.16 we can estimate the bunching factor F using  the  relation  F  =  ~ mm  with  , ^  B  A  r n  J  ->  (4.11)  ( A d ) , 50 deg) c  min (A<j> , 50 deg) = 50 deg i f A<j> > 50 deg c  c  A<f> otherwise c  where A<J>  is the phase interval which passes the chopper and Acfig is the sum of  a l l of the phase intervals at the buncher which eventually contribute current to the 50 deg cyclotron phase acceptance interval. In F i g . 4.17 we have graphed the bunching factor versus the bunching voltage assuming that A<J» = 360 deg, 90 deg, 50 deg or 10 deg assuming that C  Ad)  c  is centred about 270 deg. A l l calculations assumed that the cyclotron is  capable of accelerating a phase range which is approximately 50 deg wide. With the chopper turned o f f we obtain a maximum bunching factor of approximately 4.5 with a bunching voltage of 2.0 kV.  The bunching factor decreases  as A(j> is decreased to 50 deg, because the chopper eliminates some of the c  phases which would eventually be bunched into the 50 deg phase acceptance of the gap.  cyclotron i f they were to d r i f t the additional 280 in. to the injection As A<J> is decreased to 10 deg the bunching factor increases s l i g h t l y . C  56 the buncher is then given by  AE = 3-5  keV cos  18  270 deg ±  With the a i d of the measured v a l u e s to e s t i m a t e radial  the  increase  in v e r t i c a l  T a b l e VII  values o b t a i n e d u s i n g the 90 scintillator  current  ±0.95  in T a b l e s IV and V ,  beam spot s i z e Az and the the  in S e c t i o n  deg and 270  deg probes.  Az was estimated  and Ar was estimated  Measu red  Theory  in.  0.30  in.  Az at  270 deg  0.24  ± 0.05  in.  0.48  in.  Ar at  90 deg  0. 1 ± 0.03  in.  0.1  in.  Ar at  270 deg  in.  0.04  in.  0.016  ± 0.03  measurements appear to agree well with the the v e r t i c a l  photographs  Using the d i f f e r e n t i a l  might be due to problems  l u m i n o s i t y of  a l s o be due to d i s p e r s i v e e f f e c t s system. in the  in  pre-  by a interpret-  in view of the changing beam l u m i n o s i t y .  p r o b e s , good e s t i m a t e s of  i r r e g a r d l e s s of the  theoretical  measurements appear to be too small  Part of these d i f f i c u l t i e s  dispersive effects  from  CHANGES IN BEAM SPOT SIZE DUE TO ENERGY SPREAD OF BUNCHER  ± 0.05  inflector-deflector  from  probes  0.08  ing the s c i n t i l l a t o r  obtained  in  i n j e c t e d beam is  90 deg  however,  f a c t o r of 2.  increase  Az at  The r a d i a l  is p o s s i b l e  2.8.  Quant i ty  dictions;  it  keV.  compares the c a l c u l a t e d v a l u e s with the measured  photographs shown in F i g . 4 . 1 3 ,  Table VII.  result  «  versus radius measurements made with the d i f f e r e n t i a l  described  might  1.75  spot s i z e which r e s u l t s when the energy spread o f  i n c r e a s e d by AE.  the  deg x  the  the beam.  beam s i z e could be  Part of  in beam l i n e elements  These d i s p e r s i v e e f f e c t s  inflector-deflector  than p r e d i c t e d by our simple  radial  theory.  might  the d i s c r e p a n c y before  the  combine with  the  system to produce a d i f f e r e n t  -  5. 5.1  57 ~  VERTICAL STUDIES IN THE CRC  Introduct ion The purpose of the v e r t i c a l  confirm experimentally CRC was f i r s t  beam s t u d i e s  the t h e o r e t i c a l l y  p r e d i c t e d beam b e h a v i o u r .  p l a c e d in o p e r a t i o n the e n t i r e  about two turns due to v e r t i c a l  difficulties.  chapter w i l l  When the  beam was l o s t v e r t i c a l l y  magnetic f i e l d  after  component.  The  first  deal w i t h the techniques used to overcome these  The remaining p o r t i o n s w i l l  done t o g a i n q u a n t i t a t i v e  to  impulses caused by dee misalignments and by  the presence of an anomalous r a d i a l p o r t i o n s of t h i s  in the CRC was to attempt  deal with experiments which were  i n f o r m a t i o n about the v e r t i c a l  behaviour o f  the  beam. 5.2  Vertical It  Beam Losses Due to Dee Misalignment  has been shown t h e o r e t i c a l l y  that  if  one dee is m i s a l i g n e d v e r t i c a l -  ly by an amount Ad w i t h r e s p e c t t o the o t h e r dee, then an ion c r o s s i n g the dee gap w i l l  be d e f l e c t e d v e r t i c a l l y Ad cosd) a =  R  through an angle a given  f(R)  4 4  f(R)  where <J> is the phase angle and R is the gap.  by  (5.1)  radius at which the  ion c r o s s e s  i s a geometric f a c t o r which i s o n l y a f u n c t i o n o f R and o f  a p p l i e d dee v o l t a g e .  T h i s f u n c t i o n has been e v a l u a t e d t h e o r e t i c a l l y  CRC geometry and is shown in F i g . 5 - 1 . is h i g h e r than the e x i t s i d e ,  then the  If  the  the for  the  the entrance s i d e o f the dee gap  ion w i l l  be d e f l e c t e d upward and v i c e  versa. TRIUMF is p a r t i c u l a r l y relatively vertical  long t r a n s i t  s e n s i t i v e to dee misalignments because the  time between dee-gap c r o s s i n g s a l l o w s  o s c i l l a t i o n amplitudes to b u i l d up.  by the f a c t q u i t e weak.  This s i t u a t i o n  relatively is  large  aggravated  that the magnetic f o c u s i n g near the c e n t r e o f the c y c l o t r o n is It  has been shown that a v e r t i c a l  lead to coherent o s c i l l a t i o n amplitudes of  dee misalignment o f  10 mm.  1 mm can  Such amplitudes can  - 58 lead to l a r g e v e r t i c a l The v e r t i c a l ±0.25 mm.  misalignment  Such a t o l e r a n c e  and in o r d e r to  relax this  were p l a c e d a f t e r plates  beam l o s s e s . tolerances  is d i f f i c u l t  in the CRC have been set  to meet u s i n g s e l f - s u p p o r t i n g d e e s ,  tolerance v e r t i c a l  the f i r s t  five  electrostatic  wedge shaped and have an azimuthal  These  The p l a t e s  e x t e n t o f about 22 deg.  set at 0.8  in.  ions r e c e i v e when a p o t e n t i a l  are  The v e r t i c a l  The gap between  E2-E5 and W2-W5 was e v e n t u a l l y i n c r e a s e d to 1 i n . which the  plates  and the p l a t e v o l t a g e s may be a d j u s t e d  impulses due to m i s a l i g n m e n t s .  between the p l a t e s was i n i t i a l l y  deflection  dee-gap c r o s s i n g s shown in F i g . 2.2.  act s e p a r a t e l y on each t u r n ,  to compensate f o r v e r t i c a l  at  The v e r t i c a l  gap  plates impulse A P , Z  d i f f e r e n c e AV is p l a c e d across  the  p l a t e s , is given by AV A6 A  p  z  =  {  s B where s is the p l a t e  co  s p a c i n g , A0 is the azimuthal  the c e n t r a l  magnetic f i e l d of  frequency.  Here we are measuring A P  are expressed as r a d i i 3.0  kG f o r  written  the CRC).  in the  5  1  )  r  e x t e n t of the p l a t e s ,  the c y c l o t r o n , and  of c u r v a t u r e  i s the  in c y c l o t r o n u n i t s ,  Z  in the c e n t r a l  For the CRC parameters,  ion  B is  rotation  in which momenta  magnetic f i e l d  (about  the above e q u a t i o n may be  form AV(kV)  A P ( i n . ) = 0.072 z  s ( i n.)  The most u s e f u l c r i t e r i o n was m i n i m i z a t i o n o f the v e r t i c a l  f o r o p t i m i z i n g the c o r r e c t i o n p l a t e phase-dependent e f f e c t s .  has passed through a m i s a l i g n e d dee, impulse a c c o r d i n g to E q . 5 . 1 . off  it  receives a v e r t i c a l  Due to t h i s centre.  1 + 6  It  Once the beam phase-dependent  impulse, the beam w i l l will  then  voltages  arrive  the next  dee gap v e r t i c a l l y  vertical  phase-dependent impulse due to the f o c u s i n g a c t i o n o f the dee.  at  r e c e i v e an a d d i t i o n a l This  - 59 " amplifies the vertical phase dependence within the beam.  If the correction  plate after the first dee-gap crossing is optimized, then the beam will arrive approximately centred vertically at the second gap crossing, and the phase-dependent effects will be minimized. Fig.  5.2(a) shows how the beam looks at 1-1/4 turns with no voltages on  the correction plates. The beam profile is slanted and off centre due to the phase-dependent impulses at the dee-gap crossings.  Fig. 5-2(b) shows the  beam as it appears after the voltage on plate Wl in Fig. 2.2 has been optimized.  The beam profile has straightened out, indicating that the beam  is crossing somewhere near the electrical centre of the dee gap at the first half-turn.  By adjusting plate El the beam can be centred on the median plane  as shown in Fig. 5.2(c).  We do not see any strong phase-dependent effects at  1-1/4 turns when we adjust correction plate El, because the probe at 1-1/4 turns is too close to the second dee-gap crossing for any such effects to become apparent. The next step is to look at the beam at 1-3/4 turns. adjusted to minimize the vertical phase-dependent effects. adjusted to centre the beam on the median plane. be left fixed. ing  El is now W2 is then  The voltage on plate Wl may  At 2-1/4 turns the entire process is repeated, and by continu-  in this fashion  a l l of the correction plates may be optimized.  When the CRC was f i r s t placed in operation, the peak dee voltage was set at approximately 8Q kV, and the full radius of 32 in. corresponded to about 8-1/4 turns. turn number for  Fig. 5>3 shows a graph of per cent transmission versus  four different operating conditions. These transmission  curves were obtained by measuring the current as a function of radius on the integrated current probe described in Section 2.8.  It was found that without  using either the correction plates or asymmetrically-excited trim coils it was impossible to get any beam past approximately 3"l/4 turns, corresponding  to a radius of approximately 20 i n s t a l l e d and optimized i t was 5-1/4  in.  60 When the correction plates were then  possible to accelerate a beam to approximately  turns, corresponding to a radius of approximately 26 i n . , before the  beam was  Fig. 5.4  lost v e r t i c a l l y .  shows the correct ion•pi ate voltages which  were required to get the beam to this radius. The magnitude of the v e r t i c a l dee misalignments  in the CRC were e s t i -  mated at radii of 12 in. and 32 in. by making measurements with a theodolite. The results of these measurements are shown in Fig. 5-5. 5.2,  Using Eqs. 5-1  and  the correction plate voltages needed to correct for the measured dee  misalignments vertical  were estimated assuming that the correction plates provide a  impulse which exactly cancels the v e r t i c a l  alignment of the preceding gap. Fig. 5-4.  impulse due to the mis-  The results of this calculation are shown in  The experimentally-determined correction plate voltages were  considerably larger than the expected correct ion- pi ate voltages calculated on the basis of the dee misalignment At this point, i t was ing a median plane B  r  r  =  decided to provide additional steering by produc-  component using asymmetrically-excited trim c o i l s .  magnitude of the required B — AB  theory.  AV A9 irco sR  r  The  correction was estimated using the relation  ,  (5-3)  ,  r  which was obtained by equating the v e r t i c a l to the v e r t i c a l  impulse due to a correction plate  impulse which the beam received when i t makes a half-turn in  a region where the radial magnetic f i e l d component is AB . r  Here R is the  radius of the turn and the remaining variables are defined as in Eq. 5.2. the CRC parameters,  the above equation becomes  AB (gauss) •» r  66  AV(kV) R(in.) s(in.)  For  - 61 I t was e s t i m a t e d t h a t a c o r r e c t i o n o f a p p r o x i m a t e l y 5 t o 10 G w o u l d i n F i g . 5.6.  r e q u i r e d a s shown  shown i n F i g . 5-6  profile  r a d i u s o f 32 shown  radii.  Scintillator  i n F i g . 5-7.  The c o r r e c t i o n  F i g . 5.4.  measured  i t was p o s s i b l e  the trim c o i l s  t o a c c e l e r a t e t h e beam t o a  photographs  The beam s t i l l  o f t h e beam a t e a c h  appears  plate voltages required  They a g r e e  the AB  t o produce  r  full  10%,  The t r a n s m i s s i o n u n d e r t h e s e c o n d i t i o n s was a b o u t  i n F i g . 5-3-  a r e shown  in  in.  By a d j u s t i n g  be  as  half-turn  t o be t o o l o w a t t h e o u t e r t o produce  this  beam a r e shown  i n o r d e r o f magnitude w i t h those expected  from the  misalignments.  The asymmetry magnetic  above d i s c r e p a n c y c o u l d e i t h e r i n the e l e c t r i c  field  the experiment ruled out.  field  h a v e been e x p l a i n e d by some v e r t i c a l  a t t h e dee gaps o r by t h e p r e s e n c e o f a  component on t h e m e d i a n p l a n e o f t h e c y c l o t r o n .  radial  At the time  was b e i n g c a r r i e d o u t , n e i t h e r o f t h e s e p o s s i b i l i t i e s  c o u l d be  Some p r e l i m i n a r y a t t e m p t s w e r e made t o t r y a n d d e t e c t a v o l t a g e  a s y m m e t r y a t t h e dee g a p , b u t t h e r e s u l t s o f t h e s e m e a s u r e m e n t s w e r e i n c o n clusive, without  4 7  because  i t was d i f f i c u l t  perturbing the entire  the r a d i a l  magnetic  field  r f system.  component near  T h i s m e a s u r e m e n t was more f r u i t f u l , next 5-3  t o p l a c e an r f p r o b e  t h e dee gap  An a t t e m p t was t h e n made t o m e a s u r e t h e median p l a n e o f t h e c y c l o t r o n .  and i t w i l l  be d i s c u s s e d i n d e t a i l  i n the  section. E1iminating In  B  r  o r d e r t o measure B , t h e i n f l e c t o r - d e f l e c t o r r  f r o m t h e CRC a n d a m o t o r d r i v e was m o u n t e d could  near  be r o t a t e d  in the horizontal  a three-dimensional Hall  probe  assembly  in i t s place.  was removed  An I-beam w h i c h  p l a n e was a t t a c h e d t o t h e m o t o r d r i v e , a n d  capable o f measuring  three orthogonal  field  c o m p o n e n t s s i m u l t a n e o u s l y was m o u n t e d o n t h e I-beam.  probe  radially  a l o n g t h e I-beam a n d by r o t a t i n g  magnetic  By m o v i n g t h e H a l l  t h e I-beam w i t h t h e m o t o r  drive  -  62  -  i t was possible to make magnetic f i e l d measurements at any desired azimuth and radius. If B ' is the magnetic f i e l d component actually measured on the radial p  Hal 1 probe, then  B where B  r = i  B' - B r  is the actual  r  2  si neb  radial magnetic f i e l d component, B  z  is the v e r t i c a l  magnetic f i e l d component, and <|> is the angular misalignment of the radial Hall probe with respect to the horizontal plane of the cyclotron. of the correction term probe which was  B  z  The value  sin<f> was estimated by placing a level on the Hall  read before each measurement.  A c a l i b r a t i o n curve for the  correction term was established by placing the Hall probe in a c a l i b r a t i o n magnet and recording B ' as a function of the level r  reading.  reading and of the B  This method appeared to be capable of determining B  mately 1 G a c c u r a c y . Fig.  r  z  probe  with approxi-  48  5-8 shows the measured values of B  plotted as a function of  r  azimuth for radii of 15 i n . , 20 in. and 25 in.  B  r  is negative almost every-  where and has an average value of somewhere between - 5 and - 6 G, as shown in Fig.  5-6.  This would in a large measure account for the strong v e r t i c a l  steering required to keep the beam on the median plane. The anomalous B^ component was  reduced by changing the position of the  magnet centre plug, the height of the transition pieces and the position of some of the shims on the magnet sectors. shimming is shown in Fig. 5 - 9 -  The B ~versus-6 curve a f t e r the r  The azimutha11y-averaged  B^ component shown  in Fig. 5 - 6 has been reduced to less than 2 G between radii of 10 in. and 30 in.  By exciting trim c o i l Ik asymmetrically with a current of 37 A, the  residual B  r  component could be reduced to less than 1 G.  - 63 While the shimming was  being done, the radial positions of the  dee  misalignment correction plates were modified  so that the'CRC could be operated  at a dee voltage of 92 kV instead of 85 kV.  With this increased dee  the f u l l  radius of 32  instead of 8-1/4  in. should be reached in approximately 6-1/4  With a 30 deg phase width beam, i t was to turn 6-1/4.  The correction plate voltages in Fig. 5.10.  turns  turns.  After the above modifications, the CRC was  from turn 1/4  voltage  again placed  possible to obtain 90%  A transmission  in operation. transmission 5-3-  curve is shown on Fig.  required to obtain this transmission  are shown  For comparison purposes we have also graphed the theoretical  correction plate voltages which would be required on the basis of the measured dee misalignments and the residual B Comparing Fig. 5.10  r  with Fig. 5-4  plate voltages have been considerably still  component. we see that the required correction  reduced; however, the required  do not agree with those predicted voltages.  explanations  for t h i s .  voltages  There are several possible  F i r s t , there might be additional e l e c t r i c a l  asymmetries which are independent of the geometrical  misalignments.  Second,  our estimates of the misalignments are f a i r l y crude since i t is d i f f i c u l t to make measurements on the dee structures once they are i n s t a l l e d tank.  Third, our measurement of B  r  Is subject to error.  A l l of these  factors would influence the accuracy of our predictions. shall see may  in the vacuum  In addition, as we  in the next section, in some cases the correction plate voltages  be shifted by as much as ±0.5  cyclotron performance. measured and predicted From the CRC  kV without causing a large deterioration in  This could produce some discrepancy  between our  values.  studies several conclusions were drawn regarding  correction plates required  in TRIUMF.  the  Since the required correction plate  voltages did not correspond to the predicted correction plate voltages, i t  - 64 was  decided to i n s t a l l  stal led in the CRC It was  rather than the geometrical ly-shaped plates proposed by G. Dutto.  also concluded  be s u f f i c i e n t  in TRIUMF a set of correction plates similar to those in  that two sets of f i v e correction plates would probably  in TRIUMF since the required voltages on plates W4 through E5  are fai rly sma 11. 5-4  Confirmation of the Dee Misalignment From the results given in Section 5-3  Theory i t appears d i f f i c u l t to predict  the required correction plate voltages in terms of the measured dee misalignments and B  r  components.  As a result, an experiment was  planned  to try and  verify the predictions of the dee misalignment theory. If 6(AP ) is the change in the required correction plate impulse when Z  the dee misalignments are changed from Ad to Ad + Ad', we  then from Eq.  5.1  see 6(AP ) = Ad' costj) f(R). Z  This equation was  tested experimentally by f i r s t determining  the correction  plate voltages required to maximize transmission when Ad' = 0 and determining  then  the voltages required to maximize transmission after lowering  quadrants #1 and #2 by 0.1  in. (quadrant #1  dee  is the f i r s t quadrant after the  inject ion gap). Fig.  5.11  plate impulses. result was  compares the predicted and observed Two experimental  results are shown.  v e r t i c a l correction The f i r s t  experimental  obtained by adjusting the correction plate voltages using the  procedure outlined in Section 5.2 without making any theoretical values.  reference to the  The q u a l i t a t i v e features of this experimental  appear to agree with the q u a l i t a t i v e features of the theoretical  result result.  The  biggest disagreement is at the outer radii where plate W4 appears to have too large a positive value.  This is compensated for by E5 having too large a  -  negative value. 6 5 % to a f u l l  6  5  -  The transmission for this p a r t i c u l a r run was approximately  radius of 30 in.  The second experimental  result was obtained by attempting  the voltages to their theoretical values. the transmission was approximately  80%.  to adjust  Such a solution was found, and Part of this 1 5 % improvement in  transmission as compared to the previous run may be due to improved operating conditions rather than to improved voltage settings, since an attempt to improve the injection conditions was made between the two runs.  Varying the  correction plate voltages around the values given in Fig. 5-11 did not result in improved transmission, and i t was concluded at the given settings.  that the plates were optimized  This would appear to confirm the dee misalignment  theory. 5-5  Estimating the Transition Phase E l e c t r i c a l v e r t i c a l focusing due to the lens action of the dee gaps  predominates in the central  region of the cyclotron.  e l e c t r i c focusing may be expressed 2  v_ = Z  AP  1 —  IT  The strength of the  in terms of the v e r t i c a l tune v , where z  Z  Z  is obtained by averaging over a h a l f - o r b i t , and AP  Z  is the v e r t i c a l  impulse  measured in cyclotron units which an ion receives when i t crosses a dee gap a distance z above the median plane.  A positive value of  while a negative value indicates defocusing.  indicates focusing  In Fig. 5 - 1 2 we have graphed  2  v  z  as a function of the dee-gap crossing energy for crossing phases ranging  between -20 deg and 30 deg. These  values were calculated by numerically  tracking t r a j e c t o r i e s through a dee gap potential d i s t r i b u t i o n which was obtained by numerically solving Laplace's equation  in three dimensions, as  described e a r l i e r . We see from the figure that e l e c t r i c a l  focusing is  strongest in the low-energy region where the dee gap transit time is an  - 66 appreciable fraction of an rf cycle. as the energy is raised and which occurs between 2.5  -  Electrical  focusing decreases rapidly  is negligible above a few MeV. and 3-0  MeV  increase in  is due to some e l e c t r i c f i e l d d i s t o r -  tions caused by the close proximity of the flux guides. will  The  These distortions  be absent in the f u l l - s c a l e cyclotron. At larger r a d i i , where the f l u t t e r in the magnetic f i e l d In Fig. 5-12  magnetic focusing becomes important.  we have graphed the v  values which are obtained by adding the effects of magnetic and The magnetic v  focusing. CYCLOPS  49  z  z  electric  values were calculated by running the computer code  on the measured CRC magnetic f i e l d .  An important  quantity which p a r t i a l l y determines the phase acceptance  of the cyclotron is the transition phase which separates focused phases from the vertically-defocused phases. the CRC  is higher,  t r a n s i t i o n phase should be approximately  In Fig. 5.12  0 deg.  are then focused and the negative phases are defocused  the v e r t i c a l l y we see that  The p o s i t i v e phases and eventually lost  vertically. The value of the transition phase was  estimated experimentally  adjusting the chopper to give a phase interval of approximately The centroid of this phase interval was ing approximately was  between ~7 deg and  by  21 ± 3  deg.  then moved over the phase range extend-  16 deg, and the transmission to 6 turns  measured using the integrated current probe at the 90 deg azimuth.  measured transmission as a function of turn number is shown in Fig.  The  5-13-  From the figure we see that the transmission remains above 30% as long as none of the negative phases are included in the phase interval.  The  transmission starts to drop when the lower limits of the phase interval is set  at -6 deg,  indicating that some of the negative; phases are being  When the phases are pushed s t i l l drastically.  In Fig. 5-14  lost.  further negative, the transmission drops  we have graphed the transmission to 6-1/4  turns as  - 67 a function of the maximum phase contained in the phase i n t e r v a l .  Extrapolat-  ing the data to zero transmission, we see that the t r a n s i t i o n phase is approximately ±3 deg, which agrees well with our theoretical estimate. 5.6  Properties of the Vertical Beam P r o f i l e s Fig. 5.15 shows a series of s c i n t i l l a t o r photographs  of the beam as i t  appears at the 90 deg and 270 deg azimuths with the correction plates optimized to provide a transmission of approximately 90% between turn 1/4 and turn 6-1/4. These pictures were taken with the chopper set at 2.0 kV to give an injected phase interval extending between approximately 0 deg and 40 deg. The accelerated current under these conditions was in the 2 to 3 fA range.  The  transmission between the ion source and the i n f l e c t o r entrance was t y p i c a l l y 70% with the chopper turned o f f , and the ion source emittance was estimated to be 0.25TT  in.mrad.  The v e r t i c a l beam envelopes are phase dependent due to the phasedependent focal powers of the dee gap lenses. ly v i s i b l e in the photographs. beam is a f a i r l y  This phase dependence is clear-  At any given turn the v e r t i c a l height of the  rapidly varying function of radius, which in turn is a  function of the injection phase.  The size of the v e r t i c a l envelope can be  used to estimate the shape of the injected emittance and the strength of the focusing forces within the cyclotron. By making measurements on the photographs as a function of radius was estimated.  in Fig. 5-15 the beam height  The phase corresponding to a given  radius was then estimated using the results of Chapter 6. 5.18 we have graphed  In Figs. 5-16 to  the measured beam envelope as a function of turn number  for phases of approximately 0 deg, 15 deg and 20 deg. The error bars indicate the approximate  uncertainty in the measurements.  In Figs. 5.16 to 5-18 we have also graphed  the v e r t i c a l envelopes which  would be obtained i f we injected the CRC acceptance e l l i p s e shown in Fig. 4.3  - 68 with the emittance scaled down to 0.25^ in.mrad.  These envelopes are calcu-  lated using the computer program described in Section 7>3 with the space charge forces set to zero.  From the figures we see that the measured envelopes are  larger and o s c i l l a t e more strongly than the acceptance envelopes, indicating that there is some mis-match at the injection gap entrance. It was found that by varying the shape of the injected emittance s l i g h t l y i t was possible to obtain a much better f i t to the data.  Fig. 5-19  compares the acceptance e l l i p s e with the e l l i p s e required to produce an improved f i t .  The improved e l l i p s e is s l i g h t l y more elongated than the  acceptance e l l i p s e .  From Figs. 5.16 to 5-18 we see that the beam envelopes  calculated with this e l l i p s e agree f a i r l y well with the measured envelopes for the 15 deg and 20 deg cases.  The f i t is not so good in the 0 deg case.  Here the maximum calculated envelope agrees approximately with the maximum measured envelope; however, most of the calculated values f a l l outside of the error bars on the measurements. this discrepancy.  There are three possible explanations f o r  F i r s t , we have assumed that the minimum injected phase was  0 deg. This may not be exactly true.  The e l e c t r i c a l  focusing is f a i r l y weak  in the v i c i n i t y of 0 deg phase, and a s h i f t of a degree or so can produce f a i r l y large changes in the shape of the envelope.  Second, the predictions  of our computer program are probably less r e l i a b l e in the v i c i n i t y of 0 deg. Since the e l e c t r i c a l  focusing is weak in this phase range, a small absolute  error in calculating the focal power of the dee gaps can produce a f a i r l y large change in the shape of the beam envelope. v i c i n i t y of 0 deg are f a i r l y t i g h t l y grouped  Third, the phases  radially.  in the  Thus, due to radial  emittance e f f e c t s , when we measure the v e r t i c a l beam envelopes on the high radius side of the beam we are actually measuring the maximum envelope out of a group of phases which extend over a few degrees. discrepancies in our measurement.  This could produce some  - 69 An attempt was made to locate the phases of the minima in the v e r t i c a l beam envelopes. These minima are f a i r l y well defined in many of the photographs in Fig. 5-15, and by measuring their radii  i t was possible to estimate  their phases. In Fig. 5.20 we have graphed the turn at which a minimum is observed as a function of the phase of the minimum. Error bars of +5 deg have been placed on the measured phases since this appears to be the approximate uncertainty in the measurement.  For phases in the 13 deg to 21 deg range i t  is possible to locate two sets of minima.  The second minimum at higher phases  is not seen because i t is o f f of the edge of the s c i n t i l l a t o r . The spacing between successive measured minima is approximately three turns in the phase range between 13 deg and 20 deg. Thus i t takes s i x turns to complete one v e r t i c a l o s c i l l a t i o n .  Recalling the v  z  may be interpreted  as the number of o s c i l l a t i o n s per turn, we find  v  = (1/6) = 0.028. 2  z  o  Referring to Fig. 5.12 we see that the combined e l e c t r i c and magnetic v values f a l l  between approximately 0.02  and 0.03  for phases in the 13 deg to  20 deg range and energies in the 1.0 to 2.5 MeV range.  Thus our measured  value agrees f a i r l y well with theory. In Fig. 5-20 we have also graphed the calculated minima positions assuming that the injected emittance was identical with the emittance required to f i t the measured v e r t i c a l envelopes.  We see that the calculated positions  agree reasonably well with the measured positions when one takes into account the uncertainty in the measurements.  - 70 6. 6.1  RADIAL STUDIES IN THE CRC  Introduct ion Several c r i t e r i a relating to the radial motion of the ions must be met  i f a high-quality beam is to be accelerated in the TRIUMF cyclotron.  First,  the value of the magnetic f i e l d must be matched to the radio frequency so that isochronism instantaneous  is achieved.  Second, the beam must be centred so that the  centres of curvatures of the orbits approach the geometric  centre of the machine as the beam is accelerated to high energy.  Third, care  must be taken to avoid radial phase space distortions due to effects such as the radial-longitudinal coupling e f f e c t .  5 0  In this section we shall discuss  the techniques which were employed to obtain a well-centred  isochronous beam  in the CRC and try to interpret the radial effects which were observed in the CRC. 6.2  CRC Operating  Conditions  The CRC was designed  to operate at an r f frequency of 23.1  MHz with a  mean magnetic f i e l d of 3-033 kG, a peak dee voltage of 100 kV, and an injection energy of 298 keV. found that their resonant 23-1 MHz.  When the rf cavities were excited in the CRC i t was frequency was approximately  This became the operating frequency  22.17 MHz instead of  of the cyclotron.  In order to preserve the 23-1 MHz orbit geometry with the 22.17  MHz  operating frequency, to obtain isochronism the magnetic f i e l d was scaled down by the ratio of the two frequencies, and a l l energies and voltages were scaled down by the square of the ratio of the two frequencies. magnetic f i e l d was reduced to 2.911  Thus the mean  kG, the r f voltage was reduced to 92 kV,  and the injection energy was reduced to 274 keV.  These values became the  nominal CRC operating parameters. When the CRC was f i r s t placed in operation the control settings required to produce the above operating conditions had to be determined. Most  -  71  -  of these settings were determined experimentally using the beam as a t i c tool. 6.3  We shall now  discuss some of the techniques which were employed.  Radial Beam Diagnostic Techniques The  radial measurements described in this section were made using the  probes described in Section 2.8.  A typical set of radial turn patterns  obtained with these probes is shown in Fig. 6.1. d i r e c t l y from the chart recorder. all  diagnos-  These patterns were obtained  Turn pattern (a) was  three heads of the 90 deg probe together.  Patterns  (b) and  probe patterns which were measured at 90 deg and 270  (d) shows the 0 deg shadow measurement.  Pattern  case were collected on the 90 deg probe. be accurate to within  ±0.05 i n . , and  connecting  This turn pattern is useful  for estimating the transmission of the cyclotron. differential  obtained by  (c) are  deg.  The actual currents in this  The probe c a l i b r a t i o n appeared to  i t was  found that radial measurements  made on the accelerated beam could usually be reproduced with an accuracy of ±0.15  i n . , providing the cyclotron operating conditions remained stable  between measurements.  L i t t l e use was  made of the phase probe i n i t i a l l y  because average currents of the order of 10 yA were required to produce a usable signal on i t . During the early course of the CRC work, currents of this magnitude were not available for acceleration due to d i f f i c u l t i e s along the beam line from the ion source. The  radial motion in the TRIUMF central region has been studied  theoretically. ' * 4 3  voltage curves  5 1  5 2  Fig. 6.2  shows a series of radius versus applied dee  (the slanted lines) which were calculated by numerically  integrating trajectories through a magnetic f i e l d obtained by measurement and through an e l e c t r i c f i e l d which was equation  calculated by numerically solving Laplace's  in three dimensions for the central region boundary conditions.  Each of the slanted lines represents the variation of radius with dee voltage at a given turn number assuming that the ion crosses the injection gap with a  - 72 phase of 5 deg. These curves a l l represent wel1-centred t r a j e c t o r i e s , and the goal of the CRC radial studies was to try and duplicate these t r a j e c t o r i e s as closely as possible. Treating the figure now as a plot in (oblique) radius-turn number co-ordinates and ignoring the dee voltage scale, i f we plot the measured radius at each turn i t is possible to estimate the degree of centring that has been achieved.  If the experimental  beam is centred, then a l l of the  measured points should 1ie on a straight v e r t i c a l versus-dee-voltage  line.  Although the radius-  curves shown in Fig. 6.2 were constructed for a 5 deg  injection phase, these curves are also useful f o r studying the behaviour of trajectories which were injected with other phases.  On Fig. 6.2 we have  plotted two calculated trajectories which were started out with  initial  phases of 20 deg and 30 deg assuming an applied dee voltage of 92 kV. trajectories appear as f a i r l y straight v e r t i c a l voltages are approximately  These  l i n e s , and the e f f e c t i v e dee  87 kV and 80 kV, respectively. These values agree  to within a k i l o v o l t with the values which are obtained assuming a cosine law f o r the variation of the energy gain with respect to the injection phase, and this suggests that the phase range of the injected beam can be estimated using the cosine law in conjunction with the measured beam r a d i i . The behaviour of individual phases within the beam could be studied experimentally by using the chopper to r e s t r i c t the phase range of the injected beam and adjusting the phase interval of the injected beam so that one of the extreme phases f e l l at the desired phase which had been singled out for study.  Measurements on the desired phase could then be made by making  measurements on the inner or outer radius of the beam. Due to the radial-longitudinal coupling e f f e c t ,  4 9  the CRC beam quality  is p a r t i c u l a r l y sensitive to centring errors in a d i r e c t i o n perpendicular to the dee gap centre line.  Such centring errors are e a s i l y detectable using  - 73 the radius-versus-dee-voltage  curves.  Fig. 6.3  shows two sets of t r a j e c t o r i e s  which were measured during the course of the CRC experiments.  The  first  trajectory (deflector voltage ±3200 V) zig-zags between alternate half-turns. From the figure we see that the radii measured on the 270  deg probe are too  large and the radii measured on the 90 deg probe are too small.  This  indi-  cates that the centre of curvature of the ions is displaced away from the dee gap centre line towards the 270  deg probe.  by decreasing the deflector voltage.  This condition may  be corrected  In the second case shown in Fig.  6.3  the deflector voltage has been reduced to ±2750 V, and we see that much of the zig-zagging has been eliminated. Additional  information which is helpful in determining  is measurement of the beam sizes along the 0 deg, 90 deg, azimuths. with  The  radial width of a well-centred beam w i l l  increasing azimuth.  The  isochronous  condition may  u  deg  increase smoothly  deg of azimuth.  This w i l l  be  6.5-  in Section  Radial beam diagnostics may The  deg and 270  radial width of a poorly-centred beam w i l l  fluctuate up and down as we move through 360 discussed in more detail  180  proper centring  also be used to help determine  isochronism.  be written in the form  = ^  (6.1) m  where co is the angular frequency of the rf accelerating voltage, h is an integer representing the harmonic number (h = 5 for TRIUMF), B is the azimuthally-averaged magnetic f i e l d , and q/m being accelerated.  In the CRC  is the charge to mass ratio of the ion  the rf frequency was  or so by the high Q of the cavity resonators. considered  fixed to within  As a result, co had to be  fixed, and the value of the magnetic f i e l d was  i sochron i sm.  0.01%  varied to achieve  - Ik If the value of the magnetic f i e l d frequency of the orbiting  is too high, then the rotational  ion w i l l be too high, and as the ion is accelerated  it w i l l cross each successive dee gap with a s l i g h t l y more negative phase. If B is too low, the opposite w i l l occur, and the phase s h i f t w i l l be in the positive d i r e c t i o n .  The magnitude of this phase s h i f t A<|) is given by AB — B m -0.6 deg rf/(gauss-turn)AB  Acj>(rf deg/turn) = -360 h  (6.2)  where AB is the difference between the actual magnetic f i e l d and the isochronous magnetic f i e l d B.  Deviations of just a few gauss from isochronism can  produce phase s h i f t s after s i x turns which are s i g n i f i c a n t compared to the 0 deg to 30 deg phase interval which the CRC is designed to accelerate. One diagnostic technique which may be employed to help determine isochronism is to inject a beam which occupies a phase range between approximately 0 deg and some positive phase and measure the radius of the positive phase side of the beam during acceleration. Fig. G.k shows radius versus e f f e c t i v e dee voltage plots for trajectories which were calculated assuming that the magnetic f i e l d was isochronous and that the magnetic f i e l d from isochronism by ±6.0 G.  When the magnetic f i e l d  differed  is too high, the gap-  crossing phases are gradually shifted negative, and this causes an ion which is i n i t i a l l y normally  injected with a positive phase to gain more energy than i t would  i f the magnetic f i e l d were isochronized.  Thus such an ion w i l l  gradually d r i f t toward higher radii than would normally be the case. the magnetic f i e l d  is too low, we see the opposite e f f e c t .  When  The high radius  0 deg side of the beam is r e l a t i v e l y unaffected by small changes in isochronism because the energy gain varies approximately  as the cosine of the gap-crossing  phase, and the cosine curve is much f l a t t e r in the v i c i n i t y of 0 deg than i t is in the v i c i n i t y of some higher positive phase such as 20 deg or 30 deg.  - 75 Additional  information which was found useful  in determining  isochronism can be obtained from the v e r t i c a l behaviour of the beam. the v e r t i c a l shift  Since  focusing properties of the dee gaps are phase dependent, any  in the dee-gap crossing phases produces a corresponding change in the  v e r t i c a l behaviour of the beam.  If the phase s h i f t s are negative, then a l l  phases w i l l be less strongly focused.  This can lead to increased v e r t i c a l  losses and to an increased beam size over the phases which are shifted negative.  If the phase s h i f t s are positive, then the beam in some cases w i l l  be more strongly focused than usual, and the v e r t i c a l beam sizes w i l l be reduced over some phase ranges.  These effects are most easily noticeable  near the 0 deg high radius side of the beam.  Although this c r i t e r i o n is  somewhat vague, i t proved useful during the i n i t i a l  commissioning  stages of  the CRC when the magnetic f i e l d differed by as much as 10 or 20 G from i sochron i sm. 6.4  Results Concerning Isochronism and Centring Using the beam diagnostic techniques described in the previous section,  the beam was properly centred and an isochronous setting f o r the magnetic f i e l d was obtained.  Fig. 6.5  shows measured trajectories corresponding to  phases of approximately 0 deg, 20 deg and 30 deg. These t r a j e c t o r i e s l i e on r e l a t i v e l y straight v e r t i c a l tions from isochronism.  l i n e s , indicating that there are no large devia-  In the region between 2-1/4  and 5 l / 4 turns there is _  almost no zig-zagging between adjacent half-turns, and the centring appears to be good to within ±0.15 in. in this region, which is the approximate uncertainty in our measurements. During the f i r s t 2-1/4 so good.  turns the centring does not appear to be quite  These discrepancies are probably due to transverse impulses at the  dee-gap crossings.  These impulses eventually compensate for one another, and  - 76 the beam ends up centred at the higher  radii.  There also appears to be some centring error in the region beyond 5-lA  turns.  This error is probably  due to e l e c t r i c f i e l d distortions in  this region which were caused by the close proximity of the resonator guides.  This problem w i l l  flux  not exist in the main TRIUMF cyclotron.  The operating conditions required to obtain the trajectories shown in Fig.  6.5 are summarized below:  =  22.165 MHz  =  278.7 ± 0.5 kev  =  92 ± 2 kV  Deflector electrode-to-electrode voltage  =  28.0 kV  Inflector electrode-to-electrode voltage  =  kS.2 kV  Rf frequency measured on frequency synthesizer Estimated  injection energy based on radial  measurements made on the unaccelerated Rf voltage estimated  from measured beam radii  = 870  Magnet pot setting Main magnet current  =2601 A  Trim coi1 TI current  =  Trim c o i l Jk current  65 A  (symmetrically  excited)  80 A  (symmetrically  excited)  60 A  (asymmetrically excited to remove residual B components)  0  Trim c o i l T2 current Trim c o i l T3 current  beam  =  r  Trim c o i l T5 current The  r f frequency and the r f voltages agree well with the optimum values  in Section 6.2, but the injection energy is somewhat high.  given  Lowering the  injection energy would have require re-optimizing a l l of the bending voltages in the beam line, and this was not done because i t appeared possible to obtain satisfactory operation at 278.7 keV.  From Fig. 3.17 we see that the  deflector voltage appears a b i t too low. The fact that the beam can be centred with a deflector voltage of 28.0 kV rather than 30.0 kV is probably  - 77 " due to the presence of some compensating transverse impulses at one or more of  the gap crossings.  the main magnet c o i l .  The trim c o i l s TI and T3 contribute to B  For the above operating conditions, Fig. 2.6 shows  the estimated average value of B ing  z  as a function of radius.  the true isochronous magnetic f i e l d  estimated B  z  A line represent-  is also shown in the figure.  The  curve appears to be 3 to 8 G below the line representing true  isochronism. estimate of B  as well as  2  This difference is about three parts in a thousand, and our z  in terms of the coil  currents could be o f f by this amount.  When the beam line was optimized to provide higher currents, i t was possible to verify that isochronism had been obtained using the phase probe. Fig.  6.6  shows a series of photographs of the phase probe signal as i t  appeared on an osciol1oscope with a 100 MHz  bandwidth.  The f i r s t three phase probe photographs were taken at 1-3/4 2-3/4  turns and 3 3/4 _  turns with a magnet pot setting of 869.  that the signal from the phase probe is approximately  turns,  Here we  see  stationary with  respect to the rf waveform at a l l three turns, indicating that the magnetic field  is approximately  isochronous.  Pictures at 1-3/4  also taken for magnet pot settings of 865 and 873s h i f t of approximately  1.5 ± 0.5  the probe is moved from 1-3/4  and 3 3/4 _  turns were  In each case we see a  nsec in the peak position of the signal as  turns to 3 3/4 -  turns.  This corresponds  to a  phase s h i f t of approximately 6 ± 2 rf deg/turn, which is equivalent to a change in the average value of the magnetic f i e l d of 10 ± 3 G. on the magnetic f i e l d  indicated that changing  Measurements  the magnet pot setting by  4 divisions should change the average magnetic f i e l d by approximately  8.8  which agrees f a i r l y well with the results of our phase probe measurement. Techniques similar to this w i l l  probably be used extensively during the  commissioning stages of the TRIUMF cyclotron.  G,  - 78 6.5  Radial-Longitudinal  Coupling:  A Simple Theory  When an ion's centre of curvature dicular to the dee gap, different phases.  The  one side of the dee  the ion w i l l ion w i l l  is displaced in a direction perpen-  cross successive dee gaps at s l i g h t l y  in general  receive a higher energy gain on  than on the other side of the dee, and this  causes the centre of the ion's  trajectory to move along the dee gap  direction of the lower energy gap crossing. radial may  1ongitudina1-coup 1ing.  in the  This e f f e c t is known as  In this section we shall show how i t  50  be used to explain the observed beam sizes which were obtained with  various centring conditions. Consider an ion whose centre of curvature y-disp1acement in a d i r e c t i o n perpendicular shown in Fig. 6.7.  is i n i t i a l l y given a  to the dee gap centre l i n e , as  As a notational convenience we have labelled axis cross-  ings with consecutive  integers, and we shall use  integer subscripts to denote  the value of various quantities at the axis crossings. our analysis, we shall assume that the magnetic f i e l d The  ions w i l l  inside 30  then travel along c i r c u l a r arcs.  In order to simplify is constant  everywhere.  This is a good approximation  in. where the f l u t t e r never exceeds approximately 0.007.  If an ion arrives at the f i r s t gap crossing with phase d>i and with energy E , then after crossing the gap we 0  r  2  =  find  / 2m(E + qV cos<h)  .  0  (6.3)  qB  where V is the peak dee-to-dee voltage, m is the ion mass, q is the ion charge, B = ^  is the value of the magnetic f i e l d , r  radius of curvature  at point two  in Fig. 6.7,  2  is the value of the  h is the rf harmonic number,  and co is the rf frequency. From Fig. 6.7 a phase  we see that the ion w i l l arrive at the l e f t dee gap with  - 79 " 2hy <f>3 =  <h  (  (6.4)  f"2  Then the rad i us of curvature at poi nt h w i l l be g iven by  / 2m E  + qV(cos<|>i + cos<J>3)  0  (6.5)  qB Continuing  in this fashion, we see that in general 2 >  =  2(-l)  I  t 2h + l +1  (  +  h y  j + 1  c  (6.6)  2j  j=l  and the radius of curvature at the next y-axis crossing is given by n+l 2m E„ + qV J cos^j-i JLd 1  (6.7)  2n+2 qB We now look at the position of the centre of the ion's o r b i t .  If we  assume that there are no transverse kicks at the dee-gap crossings, then the position of the orbit centre in the y-direction w i l l position of the orbit centre  remain fixed.  The  in the x-direction w i l l o s c i l l a t e back and  forth across the centre of the dee as the ion is accelerated at subsequent gap crossings.  If we work in cyclotron units, the ion momentum w i l l be  measured in terms of the ion's cyclotron radius in the central magnetic field.  Using these units, we can write  R  2n+1  R  .n+l  = (-D  2n =  In the above equations  +  ^ o H ) x + 2 [ 2n-l +  (6.8)  Yc r 2n  n  0  AP  +  AP  2n-5  +  (6.9)  we have defined AP; to be the ion momentum gain in  cyclotron units which the ion receives when i t passes gap i , R; is the measured orbit radius r e l a t i v e to centre of the cyclotron at axis crossing i ,  - .80. x  c  is the i n i t i a l  x-co-ordinate of the orbit centre, and y  c  is the i n i t i a l  y-co-ordinate of the orbit centre. Since AP quantities are phase dependent, the R quantities w i l l be phase dependent.  If we now define  AR(<j>,^,y ) = R(<j>,y ) - R(*,y ), c  c  c  we immediately see  AR (*,*,y ) 2n  AR  c  =  r  ( < f > 2 n  2n+l (*^'Vc) =  'yc) "  r 2  (6.10)  n^' c> v  2  2n-l 2n-l VY  AP  2n- *2n-5 (  5  )  AP  (  }  5  In deriving the above results we have- assumed that r , y 0  phase independent.  (6.H)  " 2n- *2n-5 j c  and x  c  are a l l  This would normally be the case p r i o r to the f i r s t dee-  gap cross i ng. Since AP  6.6 and 6.7 to evaluate 2n+l = r„ 2n+2 - r„ 2n, we can use Eqs. ^ 0  the AP terms and hence the AR terms. tance beam w i l l  The radial beam spread for a zero emit-  then be given by the maximum value of AR(cf),^,y ) , which is c  obtained when <> j and if) are allowed to range over the rf phase interval occupied by the injected beam. Before comparing the measured beam widths with those predicted by the above theory, i t is useful to examine the q u a l i t a t i v e predictions of the theory.  F i g . 6.8 shows the calculated beam spreads along the 90 deg and the  270 deg azimuths. azimuths. E  Q  Figs. 6.9 and 6.10 show beam along the 0 deg and 180 deg  These calculations were done assuming B = 2.9 kG, V = 164 kV,  = 273 keV, and that the injected beam occupied a phase interval  0 deg and 30 deg.  between  From Fig. 6.8 we see that the beam sizes along the 90 deg  -  and 270 deg azimuths increase as y  81 -  increases.  c  This can be explained by  examining the phase history of the beam as shown in Fig. 6.11.  This phase  history was calculated using Eqs. 6.5 and 6.6 and assuming that the ion was injected with a 0 deg phase.  Although this graph was constructed for a  0 deg phase ion, the phases follow almost identical curves for any i n i t i a l phase between 0 deg and 30 deg, provided the o r i g i n of the graph is s h i f t e d by the appropriate amount in the d>-d i rect ion.  If we assume that the energy  AE(d)) which an ion gains at a given gap crossing is proportional to cos(cj)) where <J> is the phase at which the ion crosses the gap, then we have 6 ^AE (cf>) <* - in(<j>) t5 cp  (6.12)  s  where 6 AE(cj))j 6cf).  As y  is the change in energy gain when d) is changed by an amount  is increased, we note from Fig. 6. 11 that the phases are shifted  c  towards p o s i t i v e values. phases w i l l smaller.  In accordance with Eq. 6.7  the larger p o s i t i v e  receive less energy at the gap crossings and their radii w i l l be  On the other hand, the cyclotron radii of zero degree ions  w i l l be r e l a t i v e l y unaffected because of the sine)) factor. follows from Eq. 6.10 that this w i l l  It immediately  lead to increased beam sizes along the  90 deg and 270 deg azimuths. From Eq. 6.11 we see that the beam widths along the 0 deg and 180 deg azimuths depend only upon the momentum gains at the opposite azimuth. Fig.  6.11 it is clear that for any y  I Ad> (180  deg) I »  c  From  ^ 0  | Acf> (0 deg) | «  0  where Ac|>(l80 deg) and Ad) (0 deg) are phase s h i f t s at successive 0 deg and 180 deg azimuths caused by the centring error y . The AP terms in Eq. 6.11 c  will  depend primarily on the value of the gap crossing phase at the indicated  azimuth.  Since Acf> (0 deg)«  0 for any y , i t follows from Eq. 6.11 c  that AR  - 82 along the 180 deg azimuth w i l l be nearly independent of y . hand, A<}>(180 deg) w i l l  increase as y  that along the 0 deg azimuth AR w i l l behaviour is i l l u s t r a t e d  c  On the other  is increased; and we see from Eq. increase as y  c  6.11  is increased. This  in Figs. 6.9 and 6.1,0.  It should be noted that the above q u a l i t a t i v e arguments are only applicable when y  is r e l a t i v e l y small.  When y  becomes large and the gap  crossing phases are shifted by large amounts, the assumptions above discussions break down.  used in the  It is then necessary to consider the behaviour  of individual phases throughout the injected phase interval. The simple analytic theory outlined so far has several First,  shortcomings.  i t does not take into account any details of the magnetic  field.  Secondly, i t does not take into account the p o s s i b i l i t y of transverse kicks at the dee-gap crossings.  In addition, the energy gain at the various gap  crossings does not s t r i c t l y correspond to a V cos(<|>) function nor is the e f f e c t i v e value of V independent of radius.  Despite these shortcomings, we  shall see shortly that the simple analytic theory is capable of making f a i r l y good q u a l i t a t i v e predictions about beam sizes. If we compare the phase h i s t o r i e s predicted by our simple analytic model with the phase h i s t o r i e s computed using the computer program PINWHEEL, we obtain the graphs shown in Figs. 6.12 constructed for y„ = -0.3'in. and y_ = 0.  and 6.13.  Graphs have been  For each of these values of y_,  phase h i s t o r i e s have been constructed for i n i t i a l  phases of 5 deg and 29  deg.  In order to take into account a small amount of non-isochronism in the PINWHEEL magnetic f i e l d at the outer r a d i i , a constant phase s h i f t of approximately -1.4  deg/turn between the third and sixth turn was  included in the  analytic approximation. From the graphs we see that in general the phases agree to rather better than ±2 deg though cases have been found where the discrepancies are  - 83 as  l a r g e as k d e g .  T h e s e d i s a g r e e m e n t s c a n be a t t r i b u t e d  t o the presence o f  a t r a n s v e r s e momentum k i c k w h i c h t h e i o n r e c e i v e s a t t h e f i r s t dee-gap c r o s s i n g . for  the remaining  6.6  t r a n s v e r s e k i c k changes t h e e f f e c t i v e  by u s i n g ions  radial  Coup 1 i n g :. C o m p a r i s o n w i t h  longitudinal  c o u p l i n g was i n v e s t i g a t e d e x p e r i m e n t a l l y  in the y - d i r e c t i o n . 270 d e g a z i m u t h s u s i n g  T h e beam r a d i i  probes d e s c r i b e d  t h e 0 d e g , 90 d e g earlier.  t h e 180 d e g a z i m u t h b e c a u s e t h e r e was no  M e a s u r e m e n t s w e r e made f o r d e f l e c t o r v o l t a g e s  28.25 kV a n d 26.5 kV.  of the injected  were measured a l o n g  the d i f f e r e n t i a l  M e a s u r e m e n t s w e r e n o t made a l o n g  o f 32 k V , 30 kV,  W i t h a d e f l e c t o r v o l t a g e o f 28.25 k V , t h e a c c e l e r a t e d  t r a j e c t o r y a p p e a r e d t o be f a i r l y w e l l c e n t r e d , a n d i t was assumed y  c  = 0 f o rthis  other in  voltage.  The v a l u e o f y  d e f l e c t o r v o l t a g e s was t h e n  radius at the f i r s t  value o f y  c  c  Experiment  the d e f l e c t o r t o displace the centres of curvature  probe t h e r e .  value of y  part o f the a c c e l e r a t i o n process.  Radial-Longitudina1 The  and  This  right-hand  was f o u n d  f o r t h e measurements  c  estimated  that  made w i t h  by m e a s u r i n g t h e r e l a t i v e  quarter-turn for different  change  deflector voltages.  The  t o be -0.3 i n . , -0.15 i n . a n d 0.15 i n . f o r d e f l e c t o r  v o l t a g e s o f 32 kV, 30 kV a n d 26.5 k V , r e s p e c t i v e l y . All the  o f t h e a b o v e m e a s u r e m e n t s w e r e made u s i n g a c c e l e r a t e d c u r r e n t s i n  1~3 yA r e g i o n .  injected emittance placed width  In o r d e r  was l i m i t e d  a t the ion source. o f the injected  adjusted  to minimize  emittance  to approximately  Under these  the s i z e of the  0.2 i n . m r a d u s i n g  slits  operating conditions the total  beam was t y p i c a l l y  less  than  0.1  radial  i n . The c h o p p e r was  t o g i v e a n i n j e c t e d p h a s e i n t e r v a l w h o s e w i d t h was 27 ± 5 d e g . T h e  centroid of this  phase i n t e r v a l  was l o c a t e d a t 15 ± 3 d e g .  A c o m p a r i s o n between t h e t h e o r e t i c a l Figs.  effects,  6.14 t o 6.17.  Theoretical f i t s  PINWHEEL a n d t h e s i m p l e  analytic  m e a s u r e d beam s i z e s i s g i v e n i n  have been done, both  theory o u t l i n e d earlier.  u s i n g t h e program Reasonable f i t s t o  - 84 the experimental  data was obtained by assuming that the injected phase  interval extended between 5 deg and 29 deg. As seen from the graphs, there appears to be f a i r l y  reasonable  agree-  ment between the measured and theoretical beam sizes f o r the y x* 0, -0.15 i n . , c  and -0.3 in. cases. case.  The agreement is not quite so good in the  0.15 i n .  We see in Fig. 6.17(b) that the PINWHEEL calculated beam size appears  to agree roughly with the measured beam size, but the a n a l y t i c a l l y - c a l c u l a t e d beam size is too large.  This disagreement is probably due to the presence of  a transverse kick at the f i r s t turn.  This was mentioned e a r l i e r in connection  with the phase history comparisons. In Fig. 6.17(a) we see that the experimental big  o s c i l l a t i o n s on alternate half-turns.  beam sizes make f a i r l y  This could indicate that the  various phases are not centred with the same value of y . c  This situation  could arise i f there were strong phase-dependent transverse kicks somewhere during the acceleration process.  These o s c i l l a t i o n s are predicted by PINWHEEL  but are expected to be considerably smaller.  This type of discrepancy might be  explained by some type of geometric radial alignment error in the CRC or by the possible i n a b i l i t y of the relaxation code to correctly calculate the PINWHEEL e l e c t r i c f i e l d values  in certain irregular geometric regions such as  the region near the f i r s t turn on the injection gap side of the dee. It should be noted that the emittance of the injected beam was ignored during the above c a l c u l a t i o n s . The inclusion of the proper injected emittance into the calculation might remove some of the apparent discrepancies between the theoretical predictions and the experimental  results.  No attempt  was made to include the effect of the beam emittance because at the time of these measurements the shape of the injected radial emittance had not been determined unambiguously.  - 85 7. 7.1  ACCELERATION OF HIGH-CURRENT BEAMS  Introduction Most of the measurements described so far have been made on beams whose  average current was  10 yA or less.  There were several reasons for t h i s .  F i r s t , the i n i t i a l measurements were aimed at determining tory properties of the beam, and emittances decreased  the central trajec-  to f a c i l i t a t e these studies the beam  were reduced by placing s l i t s at the ion source e x i t . the amount of peak current available for acceleration.  probes were uncooled higher than 10 yA.  and would have overheated  This Second, the  with average beam currents  Third, in order to minimize the p o s s i b i l i t y of accidental-  ly dumping large amounts of current on uncooled  surfaces, i t was  desirable to  gain some experience with low currents before accelerating high currents. Fourth, during the i n i t i a l  stages of operation, the injection line had not  been optimized and had a poor transmission for peak currents greater than approximately  200 yA.  emittance-1imiting  Once the i n i t i a l measurements had been completed, the  s l i t s were removed and the beam line was  high-current transmission.  A mechanical chopper was  optimized for  installed in the injec-  tion line so that the duty cycle could be reduced and high peak current measurements could be made using uncooled  probes.  These modifications allowed  average currents in excess of 100 yA to be accelerated to s i x turns. As the beam current is increased, space charge effects become increasingly important.  Along the injection line space charge effects were calculated  using either the Vladimirskij-Kapchinskij equations version of TRANSPORT. ' 53  cyclotron. ' 5 5  5 6  54  or the CERN space charge  Space charge effects are also important within the  In this section we shall calculate the effect of space charge  on the f i r s t s i x accelerated turns in the CRC. interpret the high-current CRC measurements.  These results w i l l be used to  86 7-2  Approximations for the Space Charge E l e c t r i c Field The space charge force F ( r ) experienced by an ion is given by the g  equation F_ (7) = q 11 x  B_  (7) + t Cr).  (7.1)  where B ( r ) is the magnetic f i e l d due to the moving ions within the beam and g  E (r) s  is the e l e c t r i c f i e l d due to the charge d i s t r i b u t i o n of the beam. For  a n o n - r e l a t i v i s t i c beam  \~r x B ( r ) | «  | E ( r ) | , and we can neglect the  s  s  f i r s t term in Eq. 7.1 and evaluate E ( r ) using the s t a t i c approximation s  E,(r) =  f  1  P(r') "r I 2  d l 3  beam plus conducting boundaries where p ( r ' ) is the charge density. If we neglect the presence of conducting boundaries, then the above integral w i l l only extend over the charge d i s t r i b u t i o n of the beam.  One  simple charge d i s t r i b u t i o n which can be integrated a n a l y t i c a l l y is a rectangular parallelepiped of uniform charge density whose dimensions are 2x  Q  x 2y  Q  x 2 z . The z-component of E 0  is then given by  g  E (x',y',z',x ,y ,z ) = z  0  0  0  ire. -  ' X „  (z'-z) dz dx dy (x-x') which  2  + (y-y')  2  + (z-z')  2  3/2  is evaluated a n a l y t i c a l l y in Appendix C. A rectangular parallelepiped does not correspond very closely to the  shape of the accelerated beam.  Fig. 7-1 shows the approximate shape of the  centroid of a pulse extending over a 0 deg to 30 deg phase interval after i t  - 87 had been accelerated to about 1.3 MeV. lar.  The beam i s , of course, not rectangu-  In addition, the radial width and v e r t i c a l height of the beam vary with  phase along the length of the beam.  Despite these complications i t w i l l be  seen that this charge d i s t r i b u t i o n can s t i l l  be used to estimate space charge  effects. Fig. Z. 0  Z  0  q  Q  From this figure we see that E  q  y  1.2. shows E (0 ,0 , z , X ,y ,Z ) Q  for different values of X , y and q  is pretty well  Z  Q  independent of y , provided  is greater than the smallest transverse dimension of the beam.  This  suggests that the radial and v e r t i c a l e l e c t r i c f i e l d s near the surface of the beam can be usefully approximated by expressions of the form E (*)  «  E [o,0,Ax(<f>) ,Az(cJ>) ,°°,Ax(e|))  M*)  "«  E  X  z  Z  1 0,0,Az(<|)) ,Ax(<j>) ,°°,Az(<j>)  'J  where d> is the injection phase of some ion t r a v e l l i n g near the surface of the beam, Ax(<j>) is the half-radius of a beam which Az(<j>) is the half-height of the beam which calculating E  z  is injected with phase <j>, and  is injected with phase <j>.  we use the charge d i s t r i b u t i o n  P = — i — — 4Ax(cf>)Az(<j)) v where I  In  (7.2)  is the peak current being accelerated and v is the velocity of the  beam. In addition to radial and v e r t i c a l space charge forces, there are also longitudinal space charge forces due to the presence of an azimuthal e l e c t r i c f i e l d component which may be approximated by  E U)~ 6  E (o,0,R(d>-<|> )/5, Ax,'RA(|>/5, A Z ] z  c  where Ax is the half-width of the beam averaged over a l l  phases being  - 88 accelerated, Az is half the v e r t i c a l beam height averaged over a l l phases being accelerated, R is the radius of the turn where the space charge f i e l d is being calculated, Atf> is the width of the phase interval being accelerated, <> j is the phase of the ion at which we are evaluating  Eg, and <f> is the c  average of a l l of the phases which are being accelerated. calculated using Eq. 7-2 with Ar(tj)) Az(cb)  In this case, p is  replaced by ArAz.  So far we have neglected the presence of induced charge on the conducting boundaries which enclose the beam.  An order-of-magnitude estimate of the  effect of the boundaries can be made by considering an i n f i n i t e bar of uniform charge density which is bounded by two i n f i n i t e parallel conducting plates. The Green's function for this problem may be calculated using the method of images and is given  by  5 7  { cosh ^ - (x-x') + cos-p{y+y') ] V(x,x',y,y') = e log cosh -|- (x-x') - cos  j-iy-y')  where the origin of our co-ordinate system is at some point half way between the  conducting plates, I is the spacing between the plates, e is the charge  per  unit length of a line of charge at (x',y') which is i n f i n i t e in the  z-direction and the potential V is to be evaluated at point (x,y). The x-component of the e l e c t r i c f i e l d due to a bar of charge placed between the plates was calculated by evaluating the integral x  r  r  E (x,y) = - ±  Yo  V(x,x',y,y')dx' dy  x  " o x  "Yo  and s i m i l a r l y for the y-component of the f i e l d . of the charged bar are 2x Typical  0  1  Here the transverse dimensions  x 2y . 0  results are shown in Fig. 7-3 where we have evaluated the ratio  of the e l e c t r i c f i e l d which is present when the conducting boundaries are placed around the beam to the e l e c t r i c f i e l d which is present when the charged  - 89 bar is placed in free space away from any conducting spacing of the enclosing aperture  boundaries.  Since the  in the CRC varies, calculations were  performed for I = 1 i n . , 2 in. and h in.  We see that as long as the height  of the bar does not exceed 80% of the plate spacing, the conducting a l t e r the e l e c t r i c f i e l d by 20% or less.  boundaries  This is small enough that i t can be  neglected for most purposes. 7-3  Vertical Space Charge Calculations V e r t i c a l l y the cyclotron central region acts like a series of lenses  separated  from one another by regions where magnetic focusing and space charge  defocusing forces act. may  Between dee gaps, in linear approximation,  the motion  be described by the equation dz 2  de 9  with  2  2 .+ v | z  =  ,  0  (7.3)  q  9  5E  vt = \>t - — z  Z m  mco  2  N  7  - , 9z l  z  =  0  where 0 is the azimuth of the ion, z is the distance the ion is o f f the median plane, v  2  m  z  is the magnetic focusing term, E  z  is the space charge e l e c t r i c  f i e l d , q is the ion charge, m is the ion mass, and co is the ion rotation frequency.  E  was  z  section, and v  2  m  z  calculated using the techniques  was  described in the previous  estimated by interpolating r a d i a l l y in a table of values  which were calculated using the computer code CYCLOPS. Eq. 7-3 with z.  49  assumes that the space charge e l e c t r i c f i e l d varies l i n e a r l y  Such a relation would hold exactly i f the beam were an  cylinder containing a uniform charge d i s t r i b u t i o n . this relation is approximately  infinite  Numerical tests indicate  v a l i d for a rectangular parallelepiped whose  dimensions are comparable to those of the beam. 2  v and  z  is not a rapidly varying function of 6 as we move along an o r b i t  is nearly constant over an azimuth of 10 deg or so.  Through such a region,  -  90 -  the solution to Eq. 7-3 is  z(e).  f  sin v 6  cos v 0  z  z  z(0)  if v  p (e)  -v  z  f  •}  z  sine  cos v.  r  sinh v 6 z  cosh v 0  z(e)  z  v  z  7  sinh v 0  cosh v-  7  >  0  <  0  P (O) Z  z(0)  if P (e)  2  P (0) z  I  (7.4) where we have written the v e r t i c a l momentum component P of inches.  By taking 1 0 deg steps and recalculating v  z  Z  in cyclotron units  at the end of each  step for use in the next step, these equations may be used to track ions through any desired azimuth. In order to calculate E ' using the techniques described in Section 1.2 z  we must know the radial beam widths.  For the v e r t i c a l calculation i t was  assumed that the radial widths were unaltered by space charge. charge widths were calculated using the computer code COMA  58  v e r t i c a l space charge program.  The zero space and fed into the  A l l calculations assumed that the injected  radial emittance was c i r c u l a r with an area of 0 . 0 0 5 ^ i n . . 2  The v e r t i c a l beam envelopes were calculated by starting with a phase space e l l i p s e at the injection gap entrance and tracking i t through the cyclotron using a matrix t e c h n i q u e . ' 9  58  Between dee-gap crossings the  transfer matrix in Eq. ~J.k was used to obtain the transformation c o e f f i c i e n t s , and a thin lens transfer matrix was used to obtain the transformation c o e f f i cients of the dee gap lenses.  The focal power of these lenses was calculated  using the constant gradient e l e c t r i c f i e l d approximation which was borrowed from the linear motion code COMA.  58  The constant gradient approximation  - 91 assumes that there is a constant e l e c t r i c f i e l d component across the dee gap. An a n a l y t i c expression for the focusing may be found in reference 4 3 . The  results of the v e r t i c a l space charge calculations are summarized  in Figs. 7 . 4  to 7 . 6 where we have plotted the beam envelope versus turn  number for average accelerated currents of 0, 100, 300 and 500 yA.  These  calculations were a l l performed on a 0.5^ in.mrad emittance beam which was injected with a phase interval between 0 deg and 30 deg. The beam at the injection gap entrance was diverging with a maximum momentum component of 0.1 in. and an envelope of 0.1 in. From the figures we see that the beam envelopes are not s i g n i f i c a n t l y affected by space charge unless the current is raised above 100 yA.  At 300 yA  there is some detectable blow-up, and at 500 yA the beam envelope blows up by a factor of approximately plates approximately  2.  Since the beam has to pass between correction  1 in. apart, we see that a 0 deg phase beam of 300 or  500 yA could not be accelerated without  encountering  and 30 deg, where the e l e c t r i c a l v e r t i c a l  some losses.  At 15 deg  focusing is stronger, the situation  is somewhat better, and i t would probably be possible to accelerate currents near the 500 yA level without encountering  too large a loss.  The beam envelopes could probably be reduced somewhat by optimizing the shape of the injected emittance for each value of current.  No attempt was  made to do this because the average current in TRIUMF w i l l be limited to about 100 yA at 500 MeV by the amount of residual radioactivity that can be tolerated due to H" stripping losses, and at these currents i t appears that the effects of space charge are f a i r l y small.  It is conceivable that higher currents might  be accelerated, provided they are extracted at an energy lower than 500 MeV, but there do not appear to be any immediate plans for operating TRIUMF in this mode, and no attempts were made to operate the CRC at these current levels.  - 92 7.4  Radial-Longitudinal Space Charge Effects In the region between dee-gap crossings the equations of motion may  written in c y l i n d r i c a l  r = r&  be  (r,9) co-ordinates in the form qBr . q - — e + - E m m  2  qB . ^ q — rr + — m m  f  -(r 9) dt 2  r  r  E  where q is the ion charge, m is the ion mass, B is the v e r t i c a l magnetic field, E  r  is the e l e c t r i c f i e l d due to radial space charge, and E  azimuthal e l e c t r i c f i e l d due to longitudinal space charge. the magnetic f i e l d B is constant, then the orbits w i l l  Q  is the  If we assume that  be nearly c i r c u l a r and  we can expand the equations of motion about a c i r c u l a r orbit by writing  where co =  r(t) = R + A r ( t ) ,  Ar(t) << R  9(t) = co + A9(t),  A9(t) << co  R = ^  and v is the ion velocity.  Substituting these  sions back into the equations of motion, and only keeping  expres-  linear terms in the  A quanti t i e s , gives  Ar = R  A6 =  u A9 + - E m  ( q / m ) E e  R  " "  1  (7-5)  r  (7.6)  A >  At the dee-gap crossings we shall assume that the beam is instantaneously accelerated in a direction perpendicular to the dee gap. that the energy gain at the gap  We shall assume  is given by 6E(A9) = V cos ((j) - 5A0) where V  is the peak dee-to-dee voltage and <>j is the gap-crossing phase of the perfectly c i r c u l a r unperturbed  reference trajectory.  If we use unprimed quantities  to denote values before the dee gap and primed quantities to denote values  93 after the dee gap, then we find  Ar' = Ar  R' =  AG ' =  '2(E + 6E(0))  m — qB  C  RA9 T " 7  Ar' = AR r  AO ' =  where E  R'  2(E + 6E(A6))  qB  m  m  (R' - Ar)  is the energy of the perfectly-centred unperturbed reference orbit  c  before the dee-gap crossing, and E is the energy of the space charge perturbed trajectory before the dee-gap crossing. The radial-1ongitudinal space charge calculations were done by solving Eqs.  7-5 7•6 numerically -  using the Runge-Kutta method.  The e l e c t r i c  field  components were estimated using the techniques described in Section 1.2.  To  obtain continuous estimates of the radial beam size for calculating the space charge f i e l d s , the computer program was designed to track simultaneously.  nine t r a j e c t o r i e s  These trajectories were a l l injected with the same phase.  phase space the i n i t i a l  conditions for eight of the nine  trajectories were  points equally spaced around a c i r c l e whose radius was 0.007^ i n . . 2  ninth  In  trajectory was started out in the centre of the c i r c l e .  The  For the space  charge e l e c t r i c f i e l d estimates, the v e r t i c a l beam envelopes calculated in the previous section were used. For a chopped beam the longitudinal space charge effects are largest near the maximum and minimum injected phases.  The approximate energy spread  introduced into the beam by longitudinal space charge effects is shown in Fig. 7-7.  These calculations assume that the beam was  injected with a phase  - Sh interval between 0 deg and 30 deg, and results are shown for ions which were injected with phases of 3 deg and 27 deg. After s i x turns with an average current of 500 uA, the longitudinal space charge produces an energy spread of  +10 keV.  This energy spread produces a s h i f t  as shown in Fig. 7-8. this s h i f t  in the centroid of the beam  With an average current of 500 uA, after s i x turns  is less than 0.1 in. and would be d i f f i c u l t to detect experimental-  iyFigs. 7-9 and 7-10 show at  the sixth turn.  phase space plots of the beam as i t appears  Here we see the combined effects of radial and longitud-  inal space charge e f f e c t s .  In addition to displacing the centroid of the  beam, we see that the orientation of the beam is also altered. From the above calculations, i t appears that space charge effects are r e l a t i v e l y small for average currents below 100 uA. the  For average currents in  v i c i n i t y of 500 uA the space charge effects are, however, s i g n i f i c a n t , and  some adjustment of the shape of the injected emittance would probably be required in this case to obtain an optimized beam at extraction. are  Since there  no immediate plans for operating TRIUMF at these current levels, the  subject was not pursued further. 7-5  High-Current Measurements in the CRC In the limited time available f o r high-current operation of the CRC,  the  primary objective was to demonstrate that a 100 uA beam could be  accelerated successfully to f u l l energy; detailed measurements of orbit behaviour received only limited attention. The f i r s t  high-current measurements were done with an ion source  current of approximately 350 uA.  Of this approximately 120 uA reached beam  stop #10 situated d i r e c t l y after the last bend in the beam l i n e .  With this  amount of current available, i t was possible to accelerate average currents in the  10 to 50 yA range.  - 95 In order to make high-current measurements within the cyclotron, a water-cooled  tantalum target was placed at 5 3 / 4 turns. -  In Fig. 7-11 we have  graphed the measured target current as a function of the chopper voltage for two different bunching conditions.  With the chopper and buncher both turned  o f f , the accelerated current was found to be 15 yA. The transmission from beam stop #10 to the injection gap entrance was about 30%, and the accelerated phase interval could be estimated approximately as 360 deg x  15 yA 0.9 x 120 yA  = 50 deg.  This is consistent with the limits on the phase range set by radial clearance of the centre post and the v e r t i c a l defocusing experienced by negative phases. When the chopper and/or buncher are turned on the average accelerated current I may be estimated theoretically using the relation F where  A<}>  C  is the chopper phase interval given by Eq. 4.3 and F is the bunching  factor calculated in Section 4.6.  The calculated currents have been graphed  in Fig. 7-11 along with the measured currents, and we see that the two results agree fa i rly wel1. From the results of Sections 4.5 and 7-1-7-3, the effects of space charge at peak currents of 120-450 yA should be r e l a t i v e l y small; however, the increase in ion source emittance which occurs when the source current is raised to the 450 yA level  is not negligible.  A graph of the ion source  emittance versus the ion source current is shown in Fig. 7.12. In going from 45 yA to 450 yA the source emittance approximately 0.26TT in.mrad to 0.64ir in.mrad. effect of this emittance  increases from  In order to try and observe the  increase, a comparison was made between two beams  whose peak current at the source was 45 ± 10 yA and 425 ± 25 yA, respectively.  - 96 In order to avoid overheating the probes, a mechanical  chopping wheel was  placed in the beam line to reduce the duty cycle by a factor of approximately 10.  These measurements were made with the chopper set at 4 kV with a  chopper s l i t width of 0.125 probe photographs turns.  in.  Fig. 7.13 shows a series of s c i n t i l l a t o r  comparing the high-and  low-current beams out to 2-1/4  The average accelerated current in each case was approximately  1.6 ± 0.2 yA. We see from the photographs only s l i g h t l y .  that the high and low current beams d i f f e r  The envelope of the 450 yA beam appears s l i g h t l y larger than  the envelope of the 45 yA beam, but there does not appear to be any sign of the threefold emittance increase predicted by theory.  One possible explana-  tion for this is that the losses along the beam line are decreasing the emittance of the high current beam so that the actual value of the injected emittance does not actually t r i p l e in size when the beam current is raised from 45 yA to 450 yA. A series of photographs  were also taken with the buncher set at  approximately 1.75 kV peak voltage.  Although the luminosity of the beam  increases, the buncher does not appear to produce any large d i s t o r t i o n s . Fig. beams.  7-14 shows a comparison  between the radial position of the two  Again, there appears to be l i t t l e difference between the two beams.  The apparent differences on the low radius side of the beam are probably not s i g n i f i c a n t and can be attributed to i n s t a b i l i t i e s in the system and to difficulties  in interpreting the radial turn patterns.  After the above measurements had been completed, current was increased to 600 yA.  the ion source  The current on beam stop #10 under these  conditions was 230 yA, and the average accelerated current with the buncher set  at 1.75 kV and the chopper turned o f f was 100 yA.  Assuming 90% transmis-  sion between beam stop #10 and the injection gap entrance, and assuming a  - 97 50 deg cyclotron phase acceptance along with a bunching  factor of 3.0, the  calculated current is  which agrees roughly with the measured value. The 100 yA beam on the tantalum target produced spot.  This spot is shown in Fig. 7-15, and i t appears to be about 0.7 i n .  high and 1.5 in. wide. to  a brightly-glowing  the geometric  The glowing spot probably does not correspond exactly  limits of the beam.  When the chopper was set at 2.0 kV, the accelerated current dropped to approximately one-third of i t s previous value.  With a chopper s l i t width of  0.125 i n . , a buncher peak voltage of 1.75 kV and a chopper voltage of 2.0 kV, the bunching  factor should be 2.87.  Under these conditions, the beam at the  chopper s l i t should occupy a phase interval of approximately 78 deg, and the beam at the injection gap should occupy a phase interval of approximately 20 deg. Assuming that the bunching  factor f o r the unchopped beam is approxi-  mately 3-0, we should find Average accelerated current with chopper at 2 kV  2.87 = 0.96 3.0  Average accelerated current with chopper o f f which is much larger than the observed for  ratio of 0.33-  No adequate explanation  this phenomenon was ever found; however, i t appeared  dependent.  to be current  When the ion source current was reduced to 350 yA  the ratio of  currents was found to be 0.73, which is closer to the theoretical value. This suggests that the effect is tied to the fact that the ion source emittance changes as the ion source current is changed.  No real attempt was made  to solve this problem experimentally, and a l l additional high current measurements concentrated on accelerating an unchopped beam.  - 98 After the above measurements had been completed, water cooling was i n s t a l l e d around the centre post and around the tantalum continuous  target so that a  unchopped beam could be accelerated for long periods of time with-  out overheating any components.  The beam line was also re-optimized to  improve transmission at high currents. With these  improvements the beam was again accelerated.  we have graphed the transmission versus turn number. made with approximately  400 yA of ion source current.  In Fig. 7-16  This measurement was With the chopping  wheel adjusted to reduce the duty factor by 10, the measured current on beam stop #10 was  30 yA, and the measured current at 5"l/4 turns was 2.9 yA.  Assuming a 90% transmission between beam stop #10 and the deflector e x i t , the accelerated phase interval is  2.9 PA (30 yA x 0.9)  x  360 deg = 39 deg.  This is less than the 50 deg phase acceptance obtained previously. reason f o r this is that the cooling panels  The  i n s t a l l e d around the centre post  increased the radius of the post, and as a result fewer phases have s u f f i cient energy to clear i t . After the above measurements the chopping wheel was turned o f f and the ion source current was raised to 700 yA. on beam stop #10. to 5"l/4 turns.  This yielded 450 yA of current  With the buncher o f f i t was possible to accelerate 40 yA With a buncher voltage of 1.75  kV the current increased to  120 yA. It was found possible to accelerate beams in the 90 to 100 yA range for 2-1/2 hours without  encountering  any component damage; however, there  was frequent sparking along the beam line.  The sparking rate was approxi-  mately 1 spark per minute, and the sparks occurred mainly in the i n f l e c t o r -  - 99 deflector system, in two of the beam line bends and in three of the quadrupoles.  The s t a b i l i t y of the system could probably  be improved by improving  the vacuum at certain points along the l i n e , realignment of some of the beam line elements and additional optimization of the beam line operating parameters.  However, these improvements could not be attempted because the  CRC experimental  program was c u r t a i l e d at this point so that personnel  be freed to work on the main TRIUMF cyclotron.  could  -  100 -  REFERENCES 1.  E.O. Lawrence, "The Evolution of the Cyclotron", Les Prix Nobel en 1951, Imprimente Royale, P.A. Norstedt & Soner, Stockholm, 1952, 127  2.  D. Bohm and L. Foldy, Phys. Rev. 72_, 649 (1947)  3.  J.R. Richardson, "Sector-Focusing Cyclotrons" in Progress in Nuclear Techniques & Instrumentation, Vol. 1, ed. F.M. Farley (North-Holland, Amsterdam, 1965) 3  4.  J.P. Blaser, Proc. Int. Conf. on Nuclear Structure, 506 (1968)  5.  E.W. Vogt and J.J. Burgerjon, editors, "TRIUMF Proposal and Cost Estimate" (1966)  6.  A.P. Banford, The Transport of Charged Particle  7.  S. Penner, Rev. S c i . Instr. 32_, 150 (1961)  8.  K.L. Brown and S.K. Howry, "TRANSPORT/360, A Computer Program for Designing Charged Particle Beam Transport Systems", SLAC Report  1966)  Beams (E. Spon, London,  No. 91 (1970) 910.  K.G. Steffen, High Energy Beam Optics D.J. Clark, Proc. F i f t h  1971)  (Interscience, New York, 1965)  Int. Cyclotron Conf. (Butterworths, London,  583  11.  D.J. Clark, Proc. Sixth Int. Cyclotron Conf., AIP Conf. Proc. #9 (AIP, New York, 1972) 191  12.  R. Keller, L. Dick and M. Fidecaro, "Une source de protons polarises: Etat actual de la construction", CERN 60-2 (I960)  13-  L. Dick, Ph. Levy and J. Vermeulen, Proc. Int. Conf. on Sector-Focused Cyclotrons and Meson Factories, 127, CERN 63-19 0963)  14.  J. Thirion, CEA-N 621 (1966)  15.  R. Maillard, CEA-N 1032, 67 (1968)  16.  J. Kouloumdjian, L. Feuvrais, G. Hadinger and B. Pin, Nucl. Instr. & Meth. 79, 192 (1970)  17-  V. Bejsovec, P. Bern, J . Mares and Z. Trejbal, Nucl. Instr. & Meth. 87_,  229 (1970)  18.  Yu. A. PI i s , L.M. Sonoko, M.A. Joropkov, Soviet Physics. Technical Physics 12, 3, 348 (1967)  19.  V.A. Gladyshev, L.N. Katsaurov, A.N. Kuznetson, L.P. E.M. Moroz, Sov. Atom. Energ. 18, 3, 268 (1965)  Martynova,  - 101 20. 21.  R. Beurtey a n d J. Thirion, Nucl. Instr. & Meth. 33_, 338 (1965) R. Beurtey, R. Maillard and J. Thirion, IEEE Trans. Nucl. S c i . NS-13,  4, 179 (1966)  22.  R. Beurtey and J.M. Durand, Nucl. Instr. & Meth. 5_7_, 313 (1967)  23.  H.A. Willax, Proc. Sixth Int. Cyclotron Conf., AIP Conf. Proc. #9 (AIP, New York, 1972) 114  24.  M.E. Rickey, ibid. , 1  25.  A.J. Cox, D.E. Kidd, W.B. Powell, B.L. Reece and P.J. Waterton, Nucl.  26.  A.A. Fleischer, G.O. Hendry and D.K. Wells, Proc. F i f t h Int. Cyclotron Conf. (Butterworths, London, 1971) 658  27.  J.L. Belmont and J.L. Pabot, IEEE Trans. Nucl. S c i . NS-13, 191 (1966)  28.  J.L. Belmont and J.L. Pabot, Institut des Sciences Nucleaires, Laboratoire du Cyclotron, Rapport Interne No 3 (1966)  29.  L.W. Root, "Design of a n Inflector for the TRIUMF Cyclotron", M.Sc. thesis, University of B r i t i s h Columbia (1972)  30.  R.W. Miiller, Nucl. Instr. & Meth. 54.. 29 (1967)  31.  W.B. Powell and B.L. Reece, Nucl. Instr. & Meth. 3_2, 325 (1965)  32.  E.W. Blackmore, G. Dutto, M. Zach, L. Root, Proc. Sixth Int. Cyclotron Conf., AIP Conf. Proc. #9 (AIP, New York, 1972) 95  33-  E.W. Blackmore, G. Dutto, W. Joho, G.H. Mackenzie, L. Root, M. Zach, IEEE Trans. Nucl. S c i . NS-20, 3, 248 (1973)  34.  K.W. Ehlers, Nucl. Instr. & Meth. 32., 309 (1965)  35-  B.L. D u e l l i , W. Joho, V. Rodel, B.L. White, Proc. Sixth Int. Cyclotron Conf., AIP Conf. Proc. #9 (AIP, New York, 1972) 216  36.  Instr. & Meth. 18-19, 25 (1962)  K.L. Erdman, R. P o i r i e r , O.K. Fredriksson, J.F. Weldon, W.A. Grundman,  ibid. , 451 Nucl. S c i . NS-16, 766 (1969)  37.  D. Nelson, H. Kim and M. Reiser, IEEE  38.  T.E. Zinneman, "Three-dimensional E l e c t r o l y t i c Tank Measurements and Vertical Motion Studies i n the Central Region of a Cyclotron", University of Maryland Dept of Physics and Astronomy, Technical Report No. 986 (1969)  39.  M.E. Rose, Phys. Rev. (Ser. II) 53., 392 (1938)  40.  G. Dutto, M.K. Craddock, TRIUMF internal  Trans.  report TRI-DN-71-21 (1971)  - 102 41.  M. Reiser, "First-Order Theory of E l e c t r i c a l Focusing in Cyclotron-Type Two-Dimensional Lenses with Static and Time-Varying Potential", University of Maryland, Department of Physics and Astronomy, Technical Report No. 70-125 (1970)  42.  A.B. El-Kareh and J.C.J. El-Kareh, Electron Vol. 1 (Academic Press, New York, 1970) 90  43.  G. Dutto, C. Kost, G.H. Mackenzie, M.K. Craddock, Proc. Sixth Int. Cyclotron Conf., AIP Conf. Proc. #9 (AIP, New York, 1972) 351  44.  J. Belmont and W. Joho, "Chopping and Bunching on the 300 keV Injection Line", TRIUMF internal report TRI-DN-73-16 (1973)  45.  M.K. Craddock, G. Dutto, C. Kost, Proc. Sixth Int. Cyclotron Conf., AIP Conf. Proc. #9 (AIP, New York, 1972) 329  46.  G. Dutto, private communication  47-  A. Prochazka, private communication  48.  R.B. Moore and E.W. Blackmore, private communication  49-  M.M. Gordon, W.P. Johnson, T. Arnette, Bull. Am. Phys. Soc. 9_, 473 (1964)  50.  J.L. Bolduc and G.H. Mackenzie, IEEE Trans. Nucl. S c i . NS-18, No. 3,  51.  R.J. Louis, G. Dutto and M.K. Craddock, ibid.,  52.  R.J. Louis, "The Properties of Ion Orbits in the Central Region of a Cyclotron", TRIUMF Report TRI-71-1 (1971)  53-  I'M. Kapchinskij and V.V. Valdimirskij, Proc. Int. Conf. on High-Energy Accelerators and Instrumentation (CERN, Geneva, 1959) 274  54.  F.J. Sacherer and T.R. Sherwood, IEEE Trans. Nucl. S c i . NS-18, No. 3,  55-  M. Reiser, IEEE Trans. Nucl. S c i . NS-13, 171 (1966)  56.  M.M. Gordon, Proc. F i f t h  Beams  3  Lenses  and  Optics,  287 (197D  282  1066 (197D  1971) 305  Int. Cyclotron Conf. (Butterworths, London,  57-  J- Kunz, P.L. Bayley, Phys. Rev. J_7, 56 (1921)  58.  C. Kost, "Guide to COMA 2 (Cyclotron Orbits Matrix Accelerator Version 2), TRIUMF internal report TRI-DN-73"3 (1973)  59-  E.P. Lane, Metric Differential Geometry of Curves and Surfaces of Chicago Press, Chicago, 1940) 95  60.  E. Durand, Eleotrostatique  et magnetostatique  (University  (Masson, Paris, 1953) 377  Fig. 2.1.  Photograph of TRIUMF central region cyclotron.  Fig. 2.2.  Median plane view of cyclotron.  - 105 -  Fig. 2.3-  Photograph of CRC  inflector-deflector  assembly.  Fig; 2.4.  Schematic drawing of the CRC  injection  line.  Fig. 2.5-  B  z  versus azimuth at radii of 10 in., 20 in. and 30 in.  Fig. 2.6.  Average v e r t i c a l magnetic f i e l d as a function of radius.  Fig. 2.8.  Contribution to B  r  from asymmetrically-excited  trim c o i l s .  o  4^---  T3  in  T2  Coi 1 Rad us  A _ _ \  &  \ X  \ \  \  TI  14  n.  2  T2  20  n.  2  T3 27  n.  2  T4  33  n.  1  T5  40  n.  1  o in' CO CO ZDm CD  T4  0 1  M CD Al1 coi1 currents = 50 A a a'  —i 10.0  1— 15.0  RflDIUSUNJ  Fig. 2.9.  Contributions to B  z  produced by symmetrically-excited  Tu rns/Tank  trim c o i l s .  S E C T I O N  T H R O U G H  M E D I A N  P L A N E  resonator  hot arm  Fig.  2.10.  Diagram of CRC showing  t a l l a t i o n of r f cavity and position of diagnostic probes.  Fig. 2. 11 .  View of an rf resonator segment.  flux  inj ection gap pu11er  resonator hot arm tips  centre post beam scrapers  resonator ground arm  Fig. 2.12.  Cut-away.view of central section of CRC resonators as viewed from a v e r t i c a l plane along the dee gap centre l i n e .  guides  i  -  115 -  Fig. 3-2.  Behaviour of beam diverging from a single point as i t passes through the magnet bore.  A YtlNJ  Fig. 3.3.  A PYIMRFOJ  A  Y UN.J  Behaviour of p a r a l l e l beam as i t passes through magnet bore.  - 118 -  Fig.  3.4.  x-y  projection  of  spiral  inflector.  - 119 -  3-5.  x-z projection of sp i ra1 i n f l e c t o r .  Fig.  3.6.  y-z projection  of  spiral  inflector.  fixed o r i g i n  v [ 7 ( t ) + Ar(t)] c  h[r* (t) + Ar(t)] c  Co-ordinate system for i n f l e c t o r optics.  - 122 -  ORBIT PLOT  U= •  Fig.  3.8.  CM  H- A V- + PU- X PH= X PV= X  Inflector  trajectories  for  an  initial  h = 0.1  in.  - 123 -  ORBIT PLOT  U- •  Fig.  3-9.  H=  A  V-  +  PU=  X  PH- X  Inflector t r a j e c t o r i e s for an  CM CD  PV-  initial  P  h  X  = 0.01  rad.  - 124 -  ORBIT PLOT  CM  geomet r i c ent ranee  1  to  —*  ///  I  geomet r i c ex i t  V  O  i  \  4  \  \  oo a  ZD  a.  S—JL 0 083  0.u<  \  CD O  V  I— Q I  - CD H= A  g.  3 . 1 0 .  Vr +  Inflector  PU- X  trajectories  PH- X  PV-  f o r an i n i t i a l  X  u = 0 . 1 in,  - 125 -  ORBIT PLOT  m  geomet r i c exi t  U= • Fig. 3-11-  H= A  V- +  PU- X  PH- X  •vr  PV= X  Inflector t r a j e c t o r i e s for an i n i t i a l  P  u  = 0.01 rad-  -  U- ED H- A  Fig. 3.12.  V- +  126  -  PU- X  PH= X  Inflector t r a j e c t o r i e s for an  PV- X  initial  P  v  =  0.01  U13N.J  Fig. 3-13-  PU H IRFDJ  Beam at i n f l e c t o r exit obtained by starting with a ±10 mrad divergent beam at z = hk i n .  PUIMRRDJ  UUNJ  Fig.  3.H.  Beam a t i n f l e c t o r a t z = 44 i n .  exit  obtained  by s t a r t i n g  with  a 0.1  in. parallel  beam  AZUNJ  g. 3-15-  A PZIHRRD)  Beam at deflector exit obtained by starting from a ±10 mrad divergent beam at z = 44 i n .  3.16.  Beam at deflector at z = hh i n .  exit obtained by starting with a ±0.1 in. p a r a l l e l beam  131 m  2.9 kG  /  23 keV  278 keV  a  -0.030 ± 0.005 in./keV  /  /  -0.058 ± 0.005 in./keV rn  a  26 kV  V CM  a  / 1  /  /  /  30 kV -0.2  - 1  -0.1  F  =  ^  /  0.1  A  +  3h keV  3.17.  Ion centring  0.2  Theory  Measurements  for different deflector  voltages.  Equipotentials in a v e r t i c a l plane cutting through the gap centre Fig. 3.18.  Contour plot of equipotentials for 1.5  Median plane radial equipotentials  x 1.5  in. symmetric injection gap.  ig. 3.19-  Injection gap v e r t i c a l focal at the gap centre.  power versus phase of the  ion  - 134 a a  ig. 3-20.  Injection gap radial focal ion at the-gap centre.  power versus phase of the  . N median piane  ion enters gap  (a)  (b)  Equipotentials in a v e r t i c a l plane cutting through the gap centre  Fig. 3.21.  Contour plots of equipotentials for CRC  Median plane radial equ i potent i als  injection  gap.  ex 11  E  and E  z  (relative  units)  03 a"  J  entrance  r  1 -f  E  r  f o r above geometry  ><  E  z  f o r above geometry  1.0  1.5  d i s t a n c e from gap c e n t r e ( i n . ) entrance  ex 11  T  T IS"  is"  I i \  I —I.  A  E  •  E  r  z  f o r above geometry f o r above geometry  !  i g . 3-22.  Graph o f i n j e c t i o n  gap e l e c t r i c f i e l d s versus d i s t a n c e from c e n t r e o f gap.  1 2.0  - 137 -  Fig. 3.23.  Graph of radial injection gap focal powers versus v e r t i c a l gap focal powers.*  injection  Fig. 4.1.  Calculated CRC  acceptances.  APRUN.)  cn  o  vertical  acceptances  J  o  i  radial  Fig. 4 . 2 .  CRC acceptances at low energy side of 1.5  acceptances  in. x 1.5 in. symmetric injection gap.  -  140  -  A PX-PYIR}  44  i n . above  median  PU-PHIR)  plane  ...  at  inflector  A PR(IN. 1  entrance  APZIIN.J computed  —*  emi t t a n c e  CD ~  computed emi t t a n c e  -0.15 l  .0  / /  CD  ^  0.15 i  AHUN.J  acceptance  CD —  acceptance  R a d i a l o v e r l a p o f 70% a t injection gap e n t r a n c e  Fig.  4.3.  Typical  i  Vertical injection  matching  «  solution  f o r CRC  o v e r l a p o f 90% gap  injection  entrance  gap.  at  A PRUN.) CM  A PRUN.)  APR UN.)  overlap = 70%  = 70%  acceptance  computed emi ttance  acceptance  0.15  A RUN.3  A RUN.)  ARUN.)  acceptance computed emi ttance  i  computed emi ttance  T IS"  ±  k ent ranee  T  -fc  is"  I"  ±  T  exi t  T A  a"  k—2."—H ent ranee  Fig. h.k.  T  exi t  ent ranee  Radial overlaps for gap geometries not used in CRC.  ex 11  _^  APZUN.) in overlap = 58% -0 o  overlap = 38%  -  computed •^emi ttance  computed emittance  0.15  jf^/  0.15  ^  A ZUNJ  A Z U N J  / /  3  l  /  ^  / computed emittance  0.15 AZUNJ -c-  acceptance  ro  acceptance  ^  a  1  5.  1  I  ?ccentance  T 1  T  T IS"  is"  ±  I"  T entrance  exi t  Fig. 4.5-  entrance  T 1 a"  .7*'  H  H exi t  pntrance  V e r t i c a l overlaps for gap geometries not used in CRC.  ex i t  - 143 -  Fig.  4.6(a)  Fig.  4.6(b)  Photographs o f the s c i n t i l l a t o r probe at 90 deg and 270 deg; u n a c c e l e r a t e d beam with l a s t quadrupole in i n f l e c t o r matching t r i p l e t o p e r a t i n g at ±6 kV.  Photographs of the s c i n t i l l a t o r probe at 90 deg and 270 deg; unacce1erated beam with l a s t quadrupole in i n f l e c t o r matching t r i p l e t o p e r a t i n g at ±5-5 kV.  0.0  RELATIVE CURRENT 0.25  0.5  0.75  I . _  J  f M  I -e-  O  CD  n  -  73  i  L  rn  o CD -I  Q) 3  n  t- -i  CD  • CD in  o  cn o »  zr  i  Q. -I  - W  -  -1  CD O  CD  O 3  o  n  - 145 -  Chopper  phase  setti  D i r e c t i on o f increasing  radius  Chopper  voltage  = 2.0  Chopper  slit  = 0.064 i n .  width  40  deg  30  deg  20  deg  kV  10 d e g  -10  deg  •20 d e g i n.  Fig.  4.8.  Beam  a t 2-3/4  turns  fordifferent  chopper phase  settings.  3.0  4.0  5.0  CHOPPER VOLTAGE(KV.) Fig.  U.S.  Chopper phase width and energy spread as a function of chopper voltage.  Peak current at chopper  3-6 mA  1.2 mA  0.6 mA  40.0  80.0  120.0  160.0  DRIFT DISTANCE(IN.) Fig. 4.10.  200.0  240.0  280.0  Energy spread versus d r i f t distance for a beam being acted upon by longitudinal space charge forces.  Peak current at chopper 3.6 A m  1 .2  mA  0.6 mA  0.0  40.0  80.0  120.0  160.0  DRIFT DISTANCE U N . ) Fig. A . 1 1 .  200.0  240.0  280.0  Phase interval versus d r i f t distance for a beam being acted upon by longitudinal space charge forces.  149 -  v  0.8  =  c  kV  V  = 2.0  c  kV  if  \  v  4  «  c  F i g . 4.12(a).  *  - *  kV  v  c  = 6  kV  O s c i l l o s c o p e photographs o f the chopper beam s i g n a l from the n o n - i n t e r c e p t i n g phase probe (4 n s e c / d i v )  1MB*  ' 1  ! m  •  *  .'  m  v  c  F i g . 4.12(b).  =  il 1  '''1  2  kV  =  6  kV  O s c i l l o s c o p e photographs o f chopped beam beam s i g n a l from a beam i n t e r c e p t i n g t a r g e t p l a c e d 8 f t downstream of the chopper s l i t s (6 n s e c / d i v )  ig.  4.13.  Unaccelerated  beam  for  various  chopping  and  bunching  conditions.  - 151 -  Fig. k.]k.  Bunched phase as a function of i n i t i a l for a 500 in. d r i f t to injection gap.  phase at buncher  ro (a)  (b)  Beam a t 1-3/4 t u r n s with buncher o f f  Fig.  4.15-  Phase  probe  Beam with  signals  a t 1-3/4  turns  illustrating  a t 1-3/4 t u r n s b u n c h e r on  effect  of  buncher.  Fig. 4.16.  Bunched phase versus i n i t i a l a 220 in. d r i f t to chopper.  phase at buncher for  -  Fig.  4.17.  \5h  -  Bunching factor as a function of buncher voltage for various chopper conditions.  Fig. 5-1.  f ( R ) as a function of radius.  (a)  El = 0 Wl = 0  (b)  El = Wl =  (c)  0 -0.7  kV  Wl  -1.0  kV  -0.7  kV  Chopper voltage = 3 !<V for al1 runs  5.2.  Effect of dee misalignment correction plates as viewed on s c i n t i l l a t o r at 1-1/4  turns.  Typical transmission curve to a f u l l radius of 32 in. after eliminating anomalous Chopper voltage = k kV. V f « 92 kV r  3.25  Fig.  5.3-  Transmission  8.25  4.25  TURN**  curves during  i n i t i a l CRC experiments.  o  cn  Experimental voltages requi red with trim coi1s o f f  oo LA  Voltages calculated from dee misalignment theory  ^  21.0 \ * /23.0 \. F4  ^ ^\  \  \^ \ ^  |  25.0  radius (in.)  Experimental voltages required with trim c o i l s energized to elimi nate B r  Fig. 5.4.  Correction plate voltages before shimming out B  Fig. 5-5-  Measured dee misalignments.  Actual trim c o i l B correction required to get beam to f u l l radius (before shimming) r  B correction estimated using Eq. 5.3 r  / / Bp a f t e r shimming with Th asymmetrically exci ted with 37 A  / / ON  o 12.5  17.5  "2TJ.0 ~ ~ radius (in.)  27.5  -Y Measured B sh i mm i ng  —^ <c tn_J i Fig.  5.6..  Median plane B  30.0  r  components.  '  r  after  Measured B shimming  r  before  ig.  5.7.  S c i n t i l l a t o r p i c t u r e s o f t h e beam as i t a p p e a r e d when f i r s t a c c e l e r a t e d to f u l l r a d i u s (arrows p o i n t in d i r e c t i o n o f i nc r e a s i ng r a d i u s ) .  Fig. 5.8.  Measured B  r  before shimming.  BR(GAUSS) -9.0  -6.0  -3.0  - £91  0.0  3.0  6.  Fig. 5-10.  Correction plate voltages after shimming out  B. r  Fig. 5.11.  6(AP ) versus rad i us for dee misalignment experiment. Z  a  a  -5.0  0.0  5.0  10.0  15.0  20.0  25.0  MAX. PHASE(DEG.) Fig. 5-14.  Per cent transmission as a function of the maximum phase contained in a beam whose total phase width is 21 ± 3 deg.  30.0  in CD  CD  Measured Ideal Ar CD CD '  0.25  1.0  Fig. 5.16.  1.75  acceptance  Emittance adjusted to better f i t measurements  2.5  TURN #  3.25  4.0  4.75  5.5  Comparison between observed and calculated beam envelopes for a phase of approximately 0 deg.  CM  D  Fig.  5-17.  Comparison between observed and calculated beam envelopes for a phase of approximately 15 deg.  TURN Fig. 5.18.  Comparison between observed and calculated beam envelopes for a phase of approximately 20 deg.  A PZUN.J a  Fig. 5-19-  Comparison between ideal vertical acceptance e l l i p s e and the e l l i p s e required to obtain an improved f i t to measured envelopes.  Fig.  5.20.  Graph of turn number at whicha minimum is observed as a function of the phase of the minimum.  - 175 -  n I  2.0  1.0+  10  Fig. 6.1.  15  20  R (in.)  25  •  •  30  Typical radial turn patterns obtained using median plane diagnostic probes, a) integrated current measured on 90 deg probe with a l l three current pick-up plates connected together b) current measured on d i f f e r e n t i a l probe at 90 deg; c) current measured on d i f f e r e n t i a l probe at 270 deg; d) shadow measurement made with 0 deg probe  31  29  27  25  23  -h  21 T3 fD  19  17  15  -h  13  70  80  e f f e c t i v e dee voltage Fig.  6.2.  100  90  (kV)  5 deg, 20 deg and 30 deg phase t r a j e c t o r i e s on radius versus dee voltage curves.  plotted  70  80  e f f e c t i v e dee voltage Fig. 6.3-  90  100  (kV)  Effect of beam centring as seen on radius versus voltage plots.  dee  effective Fig.  6.-k.  Effect  of  dee  voltage  non-isochronous  (kV) operating  conditions  on  beam  radii.  I0  J  1  1  1  80  90  100  H  70 DEE Fig.  6.5-  VOLTAGE  Well-centred  (kV)  isochronized t r a j e c t o r i e s  in CRC.  -  g. 6.6.  180  -  Phase probe measurements f o r d i f f e r e n t magnet p o t e n t i o m e t e r s e t t i ngs.  diagram of geometry for off-centred cyclotron o r b i t s .  - 182 -  -183-  Fig.  6.9.  Analytically the  0  deg  calculated  azimuth.  radial  beam w i d t h s  along  - 184  F i g . 6.10.  Analytically calculated the 180 deg a z i m u t h .  -  beam s i z e v e r s u s t u r n number a l o n g  Fig.  6.11.  Phase h i s t o r i e s  calculated  u s i n g simple a n a l y t i c  theory.  - 186 -  0.0  1.0  2.0  3.0  4.0  5.0  6.0  TURN # Fig.  6 . 1 2 ( b ) .  Phase h i s t o r i e s f o r y phase o f . 2 9 deg.  c  =  -  0  .  3  in. starting  f r o m an  initial  - 187 -  analytic theory PINWHEEL  1.0  2.0  3.0  TURN # Fig. 6.13(a).  Phase h i s t o r i e s for y of 5 deg.  4.0  5.0  = 0 starting from an i n i t i a l  c  6.0  phase  analytic theory PINWHEEL  2.0  3.0  TURN # Fig. 6.13(b).  Phase h i s t o r i e s for y of 29 deg.  = 0 starting from an i n i t i a l  phase  - 188  Fig.  6.14(b).  Radial  beam w i d t h s  at  -  0  deg  for y  c  =  -0.3  in.  189  .—A  —ii  measured PINWHEEL  — ^ —  0.25  1.25  Fig. 6 . 1 5 ( a ) .  2.25  3.25  TURN #  analytic method  4.25  5.25  Radial beam widths at 90 deg and 270 deg f o r y  measured PINWHEEL  —A—analytic  method  6.0  3.0  TURN # Fig. 6 . 1 5 ( b ) .  Radial beam widths at 0 deg for y  c  = 0.  c  = 0.  -  Fig. 6.16(b).  190  -  Radial beam widths at 0 deg for y  = -0.15 i n .  - 191 -  Fig. 6.17(b).  Radial beam widths at 0 deg for y  c  = 0.15 in.  - 192 -  Fig. 7.1.  Shape of the centroid of a 1.3 MeV beam occupying a phase interval between 0 and 30 deg.  0.15  0.0  Fig. 7.2.  E  z  versus y  0.3  0  0.45  0.6  0.75  0.9  for charge distributions with various transverse dimensions.  Fig.  7.3.  Effect of conducting boundaries on space charge e l e c t r i c  fields.  Fig.  I.k.  V e r t i c a l beam envelopes showing effects of space charge on 0 deg phase  ions.  Fig.  1.1.  Energy spread due to longitudinal  space charge e f f e c t s .  Fig.  7 .8.  Radial  displacement of beam centroid due to longitudinal space charge e f f e c t s .  AP (in.) r  Fig. 7-9.  Radial phase space e l l i p s e s at turn #6 assuming an injection phase of 27 deg (total phase spread 0 deg to 30 deg).  Fig.  7-10.  R a d i a l p h a s e s p a c e e l l i p s e s a t t u r n #6 a s s u m i n g an' i n j e c t i o n p h a s e o f 3 deg ( t o t a l p h a s e s p r e a d 0 deg t o 30 d e g ) .  theory experiment A  buncher voltage  •-}- buncher voltage  3.J  CHOPPER Fig. 7-11.  4.5  1.75 kV 0  6.0  VOLTAGE(KV.)  Accelerated current as a function of chopper voltage.  20.0  0.0  40.0  60.0 current uA  Fig.  7.12.  80.0 (X10  1  100.0  120.0  )  Typical measured ion source emittance versus ion source current.  7-13-  S c i n t i l l a t o r p h o t o g r a p h s c o m p a r i n g h i g h and low current beams. (Arrows p o i n t in d i r e c t i o n of i n c r e a s i n g r a d i u s )  Turn  70 effective  Fig.  7.14.  80 dee  Comparison source  100  90 voltage  between  currents.  #  (kV)  beam  radii  obtained  with  high  and  low i o n  - 206 -  I  Fig.  7.15.  Photograph of beam spot produced by tantalum b l o c k at 6 - 3 / 4 turns  100  uA o n  in.  - 208 -  APPENDIX A. 1.  OPTICAL PROPERTIES OF THE  SPIRAL INFLECTOR  Introduction The parametric  equations for the central trajectory of the spiral  i n f l e c t o r described by J. Belmont and J. P a b o t  x  _ A f _ 2 _  2U y =  -  +  cos(2K - l)b  kK  cos(2K + l)b1  2K - 1  Afsin(2K + l)b 21 2K + 1  are given'by  27  (Al)  2K + 1  sin(2K - l)b) 2K - 1  (A2)  z = -A sinb  (A3)  where z is measured along the central axis of the cyclotron and the x-y plane is parallel  to the median plane of the cyclotron.  It is assumed  that the e l e c t r i c force is i n i t i a l l y directed along the x-axis.  The  remaining variables are defined by mv  2 0  A =  = E l e c t r i c radius of curvature.  Here m is the ion mass  q is the ion charge, v  Q  trajectory ion,and E  is the central trajectory e l e c t r i c  0  is the v e l o c i t y of the central  f i e l d strength at the i n f l e c t o r v t b = —2— A  entrance.  = independent variable which varies over the range 0 < b < TT/2  R =  qB  = magnetic radius of curvature  in magnetic f i e l d B  A k' K = — + 2R 2 k' is defined in terms of the electrode inclination angle 0 by the relation tan0 = k' sinb.  - 209 Eqs. Al to A3 were obtained  assuming that the e l e c t r i c f i e l d  along the central trajectory is given  "E(b) = E  0  / l + k'  sin b  2  2  " E(b)  by  | u(b) cose + h(b)  where h(b) and u(b) are the unit vectors defined The optical properties of the spiral  sine  (A4)  in Section  3.3.  i n f l e c t o r have previously been  calculated for the case in which the electrode cross-sections were horizont a l , which is equivalent  to setting k' = O.O. We shall now 29  generalize  this calculation to the case in which the electrode cross-sections t i l t e d and k' ± 2.  are  0.0  Derivation of the E l e c t r i c Field  Expression  We shall assume that the equipotential surfaces  in the v i c i n i t y of  the central trajectory are ruled surfaces as described  in Section  3-3.  Let R be the position vector of a point on one of these equipotential surfaces.  Then  R(b,-,h ) = 7(b) r  +  u (b) r  / l where h (b) r  and u (b) r  + k'  2  + h  r  h (b) r  (A5)  sin b 2  are the unit vectors which are obtained  by rotating  fi(b) and u(b) through an angle e(b) using the vector v(b) as an axis of rotation. h  r  r(b) is the position vector of a point on the central trajectory,  is a displacement measured along h , and <* is a parameter which labels r  the equipotential surface. arises due  The  factor 1//1  + k'  2  sin b 2  is a term which  to the fact that the electrode-to-electrode spacing decreases  as we move toward the i n f l e c t o r e x i t .  We note that the curve which the  position vector R(b,«,0) generates as b varies over the range 0 < b < TT/2 acts  as a generator for the ruled surface which has <* as i t s parameter.'  210  -  -  Us i ng well-known results from d i f f e r e n t i a l geometry, an expression for a normal vector n to our equipotential surface is given  dR(b,oc,Q) dS  r  rT(b,u ,h ) r  r  =  + h,  I dR(b,«,0)  x  5 9  (A6)  h.  dS  dS  1  dh.  by  where S is arc length measured along the generator of the ruled surface. Defining n = ——•, we shall assume that the e l e c t r i c f i e l d at R(b,u |n| given by  (A7)  E(b,u ,h ) = | E(b,u ,h ) | n(b,u ,h ) r  r  r  r  r  ,h ) is  r  where we have set  u,. = P  / l + k'  2  sin b ' 2  An expression for |E(b ,u ,h )| may r  r  known result from e l e c t r o s t a t i c s t h a t  '  d log |E  be obtained by using the well'  5 0  ±± +  Ri  (A8)  R2  where I is measured along the direction of E.  Rj and R  are the principal  2  radii of curvature associated with the equipotential surface at which we are applying the equation.  For a ruled surface the radius of curvature  along the rulings is i n f i n i t e , and we can set R  2  by assuming that R  2  is approximately  = 0.  equal to the radius of curvature  of the projection of the central trajectory onto the v-u that  We can estimate  r  plane.  We note  - 211 -  u (b+Ab) r  ^  Fig. Al  0(b+Ab)  v(b) AAb Wo  <z  du (b) v(b) • — • Ab db r  > l  Using the parametric equations for the central trajectory, i t is easily v e r i f i e d that 1 R  2  1 + 2Kk' s i n b 2  / 1 + k'  2  sin b "  Combining this result with A8 and performing some t r i v i a l algebraic manipulation, we immediately  A|t|  = — A  arrive at  (1 + 2Kk s i n b ) u 2  (A9)  r  where we have defined A j E| to be the difference in e l e c t r i c f i e l d strength between a point at position vector r(b) + u  r  u (b) and a point at position r  vector r(b). Eqs. A 6 , A7 and A9 may now be combined to give an expression for E(b,u ,h ). r  r  In order to derive a set of d i f f e r e n t i a l equations  governing  the behaviour of the paraxial t r a j e c t o r i e s , we must calculate a value for  - 2.12 -  AE(v,h ,u ) E E(b + v/A,h ,u ) - ? ( b , 0 , 0 ) r  r  r  r  referenced to the basis vectors v(b) , u ( b ) , and h ( b ) . r  Such an expression  r  is e a s i l y derived using Taylor's theorem.  We have to f i r s t - o r d e r terms in  v/A -*/ » AE(v,h ,u_)  -»/ x E(b,h ,u ) +  r  r  r  3E(b,h ,u )  v  r r>  A  rr  3b  Using the parametric equations  •  - E(b,0,0).  (A10)  for the central trajectory along  with our previously-derived results, i t is a somewhat lengthy but straightforward v, h  r  process to evaluation Eq. A 1 0 .  Keeping only f i r s t - o r d e r terms in  and u , we arrive at r  f  AE(v,h ,u ) = A r  k'  sinb cosb v  2  r  2  r  /l  + k'2  2K cosb  + v  + u ( l + 2Kk'sin b)  S  in2b  / l + k'  sin b +  2  k cosb  2  f k ' ,s i nb cosb  /l  + k'2 s i n b 2  2  { / l + k'2 i n b 2  S  k' cosb /l  + k'  2  J  + h. 2K cosb/ 1 + k'  - v(l + 2Kk  sin b  2  2  (All)  sin b) 2  sin b  It is easily v e r i f i e d by taking derivatives with respect to h , u , and v r  that AE s a t i s f i e s both of the Maxwell f i e l d equations  r  V • (AE) = 0 and  Vx(AE) = 0 . 3.  Formulating  the D i f f e r e n t i a l  Equations *  Both the central trajectory position vector r(b) and the paraxial position vector Tp(b)  must s a t i s f y Lorentz equations  of motion:  -  mr mr  213.  -  = q E(r) + r x B(r)  D  (A12)  = q E ( r ) + r_ x B(r ) D  Subtracting Eq. A12  (A13)  D  from Eq. A l 3 and defining Ar(b) = r (b) - r(b) we p  arrive at mKr  =  q |AET(A7) +  Ar  x  Q  .  (Al  Here we have assumed that the magnetic f i e l d  is constant  throughout the  region of interest, and we have used AE(Ar) = E ( r p ) - E ( r ) .  Using Eq.  and the parametric equations for the central trajectory we can everything We  in Eq. A10  4)  A7  represent  in terms of the basis vectors fi(b), u(b), and v(b).  f i n a l l y obtain Z = -2v  + (2K + k') cosb "h + u + 2Kk'cos b u 2  1 + 2Kk'  +  -;—•  ,2  1 + k'  z  sin b  ,  2  • ?  <  k  L  sin b z  ~ s i n b  n  +  IT = (2K + k') (-cosb "u - sinb v + sinb u) + 2Kk' k sinb (1 + 2Kk' 2  )  „  2  ( 5) A1  tT  sin b), , -(k' sinb h +  + 1 + k'  u  ,  ^  U)(A16)  sin b 2  (A16) 7 = (2K + k')(h sinb + V cosb) + 2u  (Al7)  where the dots indicate d i f f e r e n t i a t i o n with have defined the dimensionless ^ u u = A k.  Numerical  respect to b, and where we  variables  ^ v ~, h v = - h = -. A A  Results  Eqs. A15  to Al7 are too complicated  for analytic solution in closed  form, and as a result they were solved numerically using the Runge-Kutta  - 214 method. Numerical results are given here for the case in which A = 13 i n . , R = 10.25 in. and k' = -1.0. The results of this calculation are shown in Figs. A2 to A6.  Here we have  expressed our results in terms of the optical co-ordinates defined in Section 3-3-  In our p a r t i c u l a r case these divergences are given by h + 2Ku cosb  p„ = -  F u  =  A  u - 2Kb sinb  u - 2Kh cosb  Since Eqs. A2-A6 are linear, a l l solutions combinations of the solutions  can be expressed as linear  given in the figures.  No trajectory  was  started out with an i n i t i a l non-zero v displacement since the only effect of such a displacement  is to produce a constant v displacement along the  entire length of the trajectory.  I  •  H= A  A3.  Inflector optics for an i n i t i a l  V=  +  PU" X  PH= X  PV~ X  Pj-, divergence.  A5.  Inflector optics for an i n i t i a l  P  u  divergence.  - 218 APPENDIX B. 1.  First-Order  FIRST-ORDER PROPERTIES OF THE HORIZONTAL DEFLECTOR  Optical  Properties  We start with the equations of motion given in Section 3-4: • mr = mr0  qER  9  d  dt  -  z  mr 6 2  _ qB  d  2  d7  qBr6  (BI)  r  (B2)  s.  We now write r(t) and 8 ( t ) 0  2  -  in the form  r(t) = R + Ar(t)  (B3)  0 (t)  (Bh)  o  = 9  where R and 0  o  0  + A6(t)  refer to the central  trajectory which is defined to be a  c i r c l e of radius R on which an ion with constant angular velocity 8 travels.  in Eqs. B3 and Bh are assumed to be small  The delta quantities  perturbations about the central  m (R + Ar) = m(R + A r ) ( 0  0  o  qB (R + Ar)(0  trajectory. + A6)2 _  Substituting  into BI , we find  qER R + Ar  + A9).  (B5)  aER Expanding the denominator of the ^  term in Taylor series, and only  +  keeping f i r s t - o r d e r terms in the delta quantities, we find after some t r i v i a l algebraic manipulations mAr =  2mR6,  qBR  A6 +  qE  •2 m 8  °  +  R "  q B 6  °  Ar.  (B6)  We have simplified this expression by using the fact that the central trajectory co-ordinates must s a t i s f y Eq. BI.  219 Starting with Eq. B2 and integrating with respect to t, we find  mr 0 -  mr (O)0(O)  = y -  2  2  (r  -  2  (B7)  r (0)) 2  Substituting Eqs. B3 and Bk into B7 we arrive at m(R + A r ) ( 9 2  0  m(R  + AO) qB 2  I  +  (R  Ar(0)) (0 2  +  Ar)  2  -  + AO(0)  O  (R  Ar(0))  +  2  (B8)  Expanding the squared terms and only keeping f i r s t - o r d e r terms in the delta quantities, we find  AO  =  fAr  -  fqB  Ar(0)  R  l  +  m  A0(0)  (B9)  Substituting this result back into Eq. B6, we arrive at '_ 2qE _ (qB) ) 2  mAr =  .  R  m  kqE (qB) Ar + . — +  I  )  2  Ar(0)  m J  R  *  2mr0 - qBr A 6 ( 0 ) where we have used the relation m9  2  =  + qB0  (BIO)  to simplify our f i n a l  o  result, Eq. B I O can be solved to obtain  Ar(t) =  Ar(0)  s i ncot  [(2qE)/mR • A r ( 0 ) + A 8 ( 0 )  + Ar(0) where ai =  (2R9  0  - (qBR)/m)  1 - costot  (BID  2qE  / — —  mR  +  fqB)  - 220 or A r ( t ) , we then obtain ^ A r ( O ) + A6(0)[2R9 -3M o  s i ncot  A*r(t) = Ar(0) coscot + (B12) Substituting  BI1 into B9, we obtain  A'e(t) = A*9(0) +  ^mR  .  A'r(O)  .  s i ncot  Ar(0)  +  A8(0)  2R9  oBR]  r  1 - coscot  fli. 20, (BI 3)  Integrating  Eq. B13 with respect to t, we find  A0(t) = A 9 ( 0 ) + A9(0)t +  2SI +  —  mR  Ar(0)  Ar(0) + A0(O) 2R0,  1 - coscot e  m  J  t - — s i ncot co  SLl _ 20 m  r  R  (BH) The optical properties Figs. BI to B3.  of a typical deflector are i l l u s t r a t e d in  These calculations were performed assuming the CRC  deflector e l e c t r i c a l parameters.  In the figures we have compared our  analytic results with results obtained by numerically Eqs. ment.  integrating  BI and B2, and we see that the two sets of results are in good agreeThe variables used in the figures are defined by AP  r  = Ar/V  As  = R A0  A  = AV /V  e  0  0  O  = (R A9 + Ar 6)/V . 0  - 221 -  2.  Position of the Trajectory's Centre of Curvature at the Deflector Exit We shall derive an expression for the location of the ion's centre of  curvature at the deflector exit when the e l e c t r i c f i e l d  along the  deflector central trajectory is changed from E to E + AE where AE is small compared to E. We shall assume that r ( t ) and 6(t) as defined in Eqs. B2 and B3 s a t i s f y Eqs. Bl and B2 when E is replaced with E + AE.  Then, to f i r s t  order we have 2mR0 - qBR  mAr =  o  q(E + AE)  AG +  m 6  o  +  R  qBe  0  Substituting Eq. B9 into B15 and using the relation m6  2  simplify our f i n a l  mR  Ar  Im J  qAE m  Here we have assumed that Ar(0) = Ar(0) =0. these i n i t i a l Ar(t)  + qB0 to +  o  (B16)  Integrating Eq. B16 with  conditions we find =  qAE mco  Ar(t)  = ^ = 3i  (Bl5)  result, we find  q(AE - 2E)  mAr =  Ar - qAE.  >2  (cosco't - 1)  (B17)  qAE =  mco  s i nco't  1 <2E - iE>  +  fail m  (B18)  In order to calculate the position of the centre of curvature of the ion trajectory at the deflector e x i t , we may make use of a simple geometric argument.  Referring to Fig. Bh, the position vector of the centre of  -  222 -  curvature is given by  OC = (R + Ar)r + R  m a g  n  = (R + Ar)r + R g (-cos<j> r - sin<j>9) ma  (R  +  Ar- R  m  a  g  ) r  +  ^ _ 1  (B19)  where r is the unit radius vector associated with the central  trajectory  ion at the deflector e x i t , § is the unit tangent vector to the central trajectory at the deflector e x i t , n is a vector which is perpendicular to the velocity vector of the trajectory whose centre of curvature we are trying to locate, R g ma  is the cyclotron radius of the ion in the CRC  magnetic f i e l d , and v is the ion v e l o c i t y .  Expression Eq. B20 in terms  of the Cartesian basis vectors i and j shown in Fig. Bk, we find  R  OC =  (R + Ar - R  m a g  ) cosO  (R + Ar - R, mag  )  sin6  +  mag  Ar  ^atL  s in6  COS0  (B20)  Ar(0) and Ar(8) may be approximated using Eqs. B17 and B18. Eq. B20 may be used for estimating the effect which changing the deflector voltage has on the i n i t i a l centring of the cyclotron o r b i t s .  -  ill  -  Fig.  B3.  Deflector optics for an AV /V = 0.01. e  0  initial  - 226 -  APPENDIX C.  EVALUATION OF THE ELECTRIC FIELD DUE TO A RECTANGULAR PARALLELEPIPED OF UNIFORM CHARGE DENSITY  We wish to evaluate the integral E (x',y',z ,x ,y ,z ) ,  y  0  0  c o z  0  r o x  r Vo  (y'-y) dy dx dz [(x-x')  The  + (y-y')  2  + (z-z') }  2  2  3 / 2  integrations with respect to y and x may be done e a s i l y using standard  forms, and we find • o z  £n  E„ =  /(x-x')  2  + (z-z')  + (y -y') + x-x'  2  2  0  x=x. £n  /(x-x')  2  + (z-z')  + (y +y') + x-x  2  2  dz.  1  0  x=-x  r  We must now evaluate four integrals, a l l of which are of the form ZA-Z  l(a ,P,z ,z') =  In / q + a  2  2  0  dq  +P  2  (z +z') 0  with P = ±x -x' and a Q  2  = P  2  + (y ±y') where the choice of a + or - sign 0  depends upon which integral is being evaluated  and q = z-z'.  integral as i t stands is not found in any of the standard integration tables.  To evaluate i t , we f i r s t  l(a ,P,z ,z') 2  0  = q in  integrate by parts to obtain  q  2  +P q=-(z +z') 0  r z -z»  q dq 2  0  [ /a +q 2  (z +z') 0  2  + PJ / a + q 2  :  227  If we now make the change of variable where  0 if q  r = / a + q , q = S(q) / r - a 2  2  2  2  =0  1 if q > 0  S(q) =  { -] i f q < 0 , then we can write q=z -z ' 0  l ( a , P , z , z ' ) = q In  /a +q  2  2  n  2  + P q=-(z +z') 0  f/a +(z -z') 2  2  0  S(q)/r ^ 2  2  dr  r + P /a +(z +z') 2  2  0  / r ^ d r r + P  Defining f ( a , P , r ) = 2  and using standard  forms, we e a s i l y  arrive at f(0,0,r)  = - |r|  f(P ,P,r) = - / r - P  2  f(a ,P,r) = - / r - a  2  2  2  2  2  + P £nf2/r -P 2  r + P Jin 2 / r - a 2  I f a  2  ? p  2  + 2r ,  2  + 2r  lal  P ^ 0  J  and a 5* 0.  Our integral may e a s i l y be evaluated we find  2  in terms of the above functions,  228 q z =  a +q  l ( a , P , z , z ' ) = q In  2  2  0  2  '  - z 0  +P q=-(z +z') 0  ,P, / a + (z -z') 2  2  0  - f  ,P,/a  + J - 2f a ,P,|a|] + f f a , P , / a 2  + f  2 3  2  ,P, / a  2  + (z -z') 0  2  2  + (z +z') 0  + (z +z')  2  2  if z ' > z > 0 0  2  0  if z  > z' > 0.  0  The f i n a l solution to our problem is then given by the expression  E (x ,y',z ,x ,y ,z ) ,  v  ,  0  0  0  = kt\E  (x-x')  + (y -y') ,x-x',z ,\z'  2  2  0  r  x=x  r  -  I (x-x') + (y +y') ,x-x',z , |z' 2  2  0  0  x=-x  r  

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