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Electron motion in a pre-accelerator cavity Vermeulen, Fred Eric 1962

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E L E C T R O N MOTION I N A P R E - A C C E L E R A T O B C A V I T Y  by FRED E R I C B.Sc,  VERMEULEN  University  A T H E S I S SUBMITTED  In  required  FOR  THE DEGREE  OF  OF A P P L I E D S C I E N C E  the Department of  Electrical  ¥e a c c e p t  1960  I N P A R T I A L F U L F I L M E N T OF  THE REQUIREMENTS MASTER  of Alberta,  this  Engineering  thesis  as conforming  to the  standard  Members o f t h e D e p a r t m e n t of E l e c t r i c a l The  University  Engineering  of B r i t i s h  September,  1962  Columbia  In presenting  this thesis i n p a r t i a l fulfilment of  the r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t t h e 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 s t u d y . for extensive  I f u r t h e r agree t h a t p e r m i s s i o n  c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be  g r a n t e d by t h e Head o f my Department o r by h i s  representatives.  I t i s understood t h a t copying o r p u b l i c a t i o n o f 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 s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n  Department o f  &.&CTX-'CAjL  ^ / V ^ / V ^ M / ^  The U n i v e r s i t y o f B r i t i s h Columbia, Vancouver 8, Canada. Date  Q<T77/g.  permission.  II  ABSTRACT  Relativistic  electron motion  uniform, uni-directional electric information an  obtained  i s applied  e l e c t r o n beam w h i c h  into  a high  accelerator particular graphical The  field  the main  i s injected uniformly  Numerical  equations  of motion  and used  transit  a t low energy  angle p r e -  c a l c u l a t i o n s a r e made  o f an e l e c t r o n  to obtain  main a c c e l e r a t o r . form  and i s designed  the pre-accelerator joint  The  for a  form.  particular  with  i s examined.  c a v i t y , and t h e r e s u l t s a r e p r e s e n t e d i n  wave a r e d e r i v e d  graphical  field  varying,  t o study the dynamics of  i n t e n s i t y , long  cavity.  i n a time  operation  accelerator.  This  i n a  numerical data  travelling  data  on a  i s presented i n  t o be u s e d  r e s u l t s to supply  of the pre-accelerator  i n  conjunction  information and the  on  VIII  ACKNOWLEDGEMENT  The Dr.  author wishes  G.B. W a l k e r ,  course  of this  The and  The for 1961,  for his help  colleagues author  i sgratefully  i s indebted  and f o r a d d i t i o n a l Grant  Department.  and encouragement d u r i n g t h e  by d i s c u s s i o n s  t h e award o f a B u r s a r y  Council  professor,  study.  a i d afforded  other  t o thank the supervising  with  M r . C.R. J a m e s  acknowledged.  to the National  Research  i n 1960 and a S t u d e n t s h i p  support  (BT-68) g r a n t e d  from  a National  to the E l e c t r i c a l  Council i n  Research Engineering  Ill  T A B L E OF  CONTENTS Page  List  of I l l u s t r a t i o n s  V  Acknowledgement  VIII  1.  Introduction.  1  2.  Electron Motion  i n the Pre-Accelerator....  2.1  The E q u a t i o n s  2.2  General  2.3  of Motion.  9  2.2.1  Maximum E n e r g y  2.2.2  Paths i n Energy-Phase and i n P o s i t i o n - P h a s e  Transfer.......  «  o  o  .  .  .  .  .  «  Beam C u r r e n t  2.5  Specific Results  2.6  Beam C h a r g e D e n s i t y  2.7  P a r t i c u l a r Deductions  2.8  Beam C h a r g e D e n s i t y Suggestions f o r an Improved  2.8.2  Space Space...12  «  2.4  2.8.1  o  .  «  .  .  «  «  «•o•«..15  20  .  24 ....33 Regarding 37 Injection  A Suggested I n j e c t i o n V e l o c i t y F u n c "fc i o n o » » o o 9 * 9 » o « o o o * 6 a o * o 0 o A Suggested Modulation  Electron Motion  9  of Electron.  Trajectories..o..«  3.  4  Results........  Classification  4  4 0  Device.47  i n the Main  .A.C c G 16 r3/t o r • 0 • 0 0 0 0 000 e o 0 0 0 00 « o o 0 9 0 0 0 0 00 •»• 4 9  3.1  The E q u a t i o n s  of  3.2  Electron  3.3  C o r r e l a t i o n of Electron Motion i n the P r e - A c c e l e r a t o r and A c c e l e r a t o r . . 5 9  Orbits..  M o t i o n . a . o o « . . o e o . o . 4 9  ...... «  54  IV  Page 3.4  Further the  4.  Hamiltonian  Conclusion  References  Computations  Involving  Function  61 65 66  V  LIST  OP  ILLUSTRATIONS  Figure  Page  1A  The  L i n e a r E l e c t r o n A c c e l e r a t o r System  2A  Partial  2B  Energy-Phase  2C  Position-Phase  2D  Classification Phase  2E  Profile  of the P r e - A c c e l e r a t o r C a v i t y . . . .  Space  for E =20.0xl0  13  f o r E =20.0x10^ v o l t s / m e t e r . c of T r a j e c t o r i e s by I n j e c t i o n  14  6  Space  °.  E l e c t r o n Energy Versus volts/meter  6  c  « ».  T =30.Oxl0 eV 3  Q  18  0  E l e c t r o n Energy Versus E =20.0xl0  volts/meter  6  c  U =180.0°  E l e c t r o n Energy Versus E =20.0xl0  volts/meter  6  c  2H  E l e c t r o n Energy Versus E  c  Position T =30.Oxl0 eV 3  Q  •..  0  2G  =20.0xl0  6  volts/meter  18 Position .T =30.Oxl0 eV 3  Q  Position T =30.Oxl0 eV 0 3  n  o .19  u =302.0°. 0  21  Injection  2J  UQ  and  Phase  volts/meter  6  c  T and E  |3 V e r s u s  =20.0xl0 c  UQ V e r s u s A r r i v a l  Phase  u,  21  I / I Q V e r s u s u-^  E =20.0xl0 2K  16  Position  u =103.9°. .. 2F  4  volts/meter...  c  UQ.......  E =20.0xl0  3  6  T =30.Oxl0 eV  25  T =30;0xl0 eV 0  26  3  Q  u^  volts/meter  3  n  VI  Figure  Page  2L  UQ  and  I / I Q V e r s u s u^ T _ = 2 ' . O O x l 0 e V . . . . 27 0  2M  E =20.0xl0 volts/meter c T and 3 V e r s u s E =20.0xl0  T =2.00xl0 eV....  28  T =2.00xl0 eV....  29  T =2.00xl0 eV  30  6  volts/meter  6  c  2N  volts/meter  6  c  3  Q  T a n d 8 V e r s u s u-^ E =60.0xl0  volts/meter  6  c  2P  3  Q  UQ and I / I Q V e r s u s u^ E =60.0xl0  20  3  (uQ+2ixm) V e r s u s  3  Q  (\x-^-2nn)  at  z=z-^ a n d a t z = z ^ + 1 0 c m . E =40.0xl0  volts/meter  6  c  2Q  T and 8 V e r s u s E =40.0xl0 Injection  2S  Paths of  31  T =2.00xl0 eV.... 3  32  z  34  3  Q  Q  volts/meter  6  c  2R  u  T =2.00xl0 eV  0  P h a s e UQ V e r s u s P o s i t i o n e^,  e ^ +  i n Position-Phase  Space 2T  39  An I l l u s t r a t i o n o f E l e c t r o n Near the C a v i t y Entrance E =20.0xl0  volts/meter  6  c  Bunching  T =2.00xl0 eV 3  0  41  2U  Beam C h a r g e D e n s i t y  a t u=320° T =2.00xl0 eV 0 a t u=320°  42  2V  E =20.0xl0 volts/meter c Beam E n e r g y a n d V e l o c i t y E =20.0xl0  T =2.00xl0 eV....  43  6  c  3A  Partial Cavity  6  volts/meter  Profile  3  n  3  Q  o f one M a i n  Accelerator 49  VII  Figure  Page  3B  p V e r s u s «z •^m  3C  T Versus  /v , ph  volts/meter  6  a  3D  T Versus  3E  Application  3F  (z-z ) Versus a  A  =10.0xl0 a  55  A  E =10.0xl0  E  i n a ¥ave a t v , = c / 2 . . . . ph  (Limiting  6  57  Orbits)  of P r e - A c c e l e r a t o r  58 Data  60  A  volts/meter  T =.389xl0 eV a 6  63  E L E C T R O N MOTION I N A P R E - A C C E L E R A T O R C A V I T Y  1.  A linear  electron  main components. system  energy the is  electron  bunched,  travelling  The  This  system where  system  of pre-accelerating  that  upon e n t e r i n g sufficient  travelling The i.e.  electrons spread These  the main  a large  a  amount o f e n e r g y .  the high  energy  electron  experiments.  performs  i t s first  function,i . e .  t h e e l e c t r o n beam, t o e n s u r e the electron  beam  t o move i n s y n c h r o n i s m w i t h t h e  wave.  injection  that  energy and  i t moves i n s y n c h r o n i s m w i t h  the main a c c e l e r a t o r  velocity  intermediate  i s f e d through  additional  beam t h e n e n t e r s  f o rphysical  that  has  i tgains  the main a c c e l e r a t o r ,  injection  three  gun, the i n j e c t i o n  e l e c t r o n beam  wave, t h e r e b y g a i n i n g  i s available  consistsof  gun i s t h e source o f a low o r  through which  Upon l e a v i n g beam  are the electron  The b u n c h e d  accelerator  system  accelerator.  e l e c t r o n , beam.  injection  accelerator  These  and the main  The  INTRODUCTION  system  of bunching enter  i t s second  function,  t h e e l e c t r o n beam, t o e n s u r e  the main a c c e l e r a t o r  and d u r i n g electrons  performs  the correct  then ride  with  a narrow  that phase  fraction  of the r - f cycle.  on t h e c r e s t  of the t r a v e l l i n g  2 wave a n d t h e r e b y since  This  a maximum  t h e e l e c t r o n s move n e a r  subjected the  gain  each other,  Also,  they are  t o t h e same a c c e l e r a t i o n s a n d t h e r e f o r e  main a c c e l e r a t o r w i t h narrow  energy  prerequisite An  amount o f e n e r g y .  spread  a narrow  experiments.  s y s t e m may p e r f o r m  above f u n c t i o n s .  Various  types  i n the l i t e r a t u r e .  either or both  of i n j e c t i o n  12 described  spread.  o f t h e e l e c t r o n beam i s a  f o r many p h y s i c a l  injection  energy  leave  of the  systems a r e  3  ' '  The  injection  system which  consists  of a high  field  cavity.  This  i s analyzed  intensity,  long  i n this  transit  c a v i t y i s a component o f a l i n e a r  system which has been designed  to obtain  study  angle accelerator  information  on  4 a new t y p e The  of main  purpose  information  accelerator.  of this  which w i l l  and  the f i e l d  the  field  intensity  intensity  study  i s to provide  representative  r e l a t e t h e e l e c t r o n g u n beam i n the pre-accelerator  i n the main  accelerator.  energy  cavity to  Gun  Pre-Accelerator Ca^/// Fig.  1A  First Main Accelerator The L i n e a r  Second  Cavities  Electron Accelerator  System  4 2. E L E C T R O N MOTION I N THE P R E - A C C E L E R A T O R  2.1  The E q u a t i o n s  The  details  pertinent  of Motion  of the pre-accelerator  to this  study  a r e shown i n F i g u r e  details  a r e t h e e l e c t r o n beam e n t r a n c e  tunnel,  t h e two r e - e n t r a n t  b e t w e e n them*  Fig.  2A  cavity that are  tunnel  c a v i t y noses,  The c a v i t y h a s c y l i n d r i c a l  Partial  Profile  2A.  These  and  exit  and t h e gap  region  symmetry.  of the Pre-Accelerator  Cavity  5  The  electromagnetic ZQ^  for  : E; = 1 z  fields ;  Z ^ Z^  E  a r e assumed t o b e ;  r ^ r^  •  sin(tot)  . .. ( 2 - 1 )  c  H = 0  ...(2-2)  and f o r z - z  0  z ^ z E  ;  1  = H = 0  r^v  Q  .  ...(2-3)  where  and  E  i s the electric  field  intensity  H  i s the magnetic  field  intensity  t  i s time  <c  i s the frequency  1  i s the unit  z  , cold,  beam o f s p e c i f i e d  the  i n the z-direction.  cylindrical  and i r r o t a t i o n a l  c u r r e n t and v o l t a g e  enters  I t i s assumed t h a t t h e presence  cavity  represented the  vector  *  A uniform  z = ZQ.  of operation  does n o t a l t e r by equations  (2-l),  t h e gap a t  of this  the electromagnetic  electron  beam i n  fields  ( 2 - 2 ) , and ( 2 - 3 ) , and t h a t  m o t i o n o f a n y o n e e l e c t r o n i s n o t a f f e c t e d .by,,other-  * I m p l i c i t l y , a u n i f o r m b e a m must be e v e r y w h e r e continuous. S u c h a b e a m may b e t h o u g h t o f a s b e i n g l i k e a c o m p r e s s i b l e f l u i d t h a t h a s b e e n s u b d i v i d e d i n t o many homogeneous volume i n c r e m e n t s , each o f w h i c h has t h e e l e c t r o n i c c h a r g e t o mass r a t i o . I n t h i s way, t h e existence of i n d i v i d u a l e l e c t r o n s i s acknowledged, y e t the beam i s u n i f o r m a n d h a s a c o n t i n u o u s c u r r e n t a n d c h a r g e density a t every i n t e r i o r p o i n t . 1  6  electrons. computed obtain  T h e p a t h o f e a c h e l e c t r o n may t h e r e f o r e  independently,  the total  The  motion  o f t h e beam. o f an e l e c t r o n  p + e A )= - e grad I0 - v . J j \ *m  p ^m  i s t h e m e c h a n i c a l momentum o f  1  i n an  f i e l d is"*  dt  d  where  a n d a l l p a t h s may b e summed t o  equation of motion  electromagnetic  be  the of  electron,  i n terms  i t s v e l o c i t y v , rest  m a s s niQ,  the speed o f l i g h t  i n vacuo  and c,  given  ...(2-4)  by m 5  =  m  v °  ...(2-5)  E q u a t i o n (2-4) does n o t account f o r t h e energy t h a t t h e e l e c t r o n l o s e s b y radiation.6»' I n accordance with the c l a s s i c a l theory of r a d i a t i o n , the r a t i o of energy l o s s to e n e r g y g a i n o f a n e l e c t r o n t h a t moves a t n e a r t h e s p e e d o f l i g h t ancj, i s a c c e l e r a t e d a l o n g i t s d i r e c t i o n o f m o t i o n i s 2 E  e R  s-53 m  Q  c  where R i s t h e (assumed) c l a s s i c a l e l e c t r o n r a d i u s . This r a d i a t i o n l o s s becomes s i g n i f i c a n t i f t h e energy g a i n , -(E e R), o f t h e e l e c t r o n as i t t r a v e l s a d i s t a n c e R i s o f t h e same o r d e r a s i t s r e s t e n e r g y (mQ c^)» I n the present case t h i s l o s s i s n e g l i g i b l e , i . e * 2 E e R -S ^ 2 3 m c Q  IO" • • n  1 3  7  e  i s the electronic stand this  and  charge  f o r a negative  (The s y m b o l  quantity  Consider  will  throughout  work),  5 a n d 0. a r e t h e e l e c t r o m a g n e t i c scalar  e  potentials  an e l e c t r o n  vector  and  respectively.  that  i s injected  into  the  cavity  gap a t (r,  0,  z) =  ( 0 , 0,  T  =  T 0  t  =  t  z ) Q  ...• ( 2 - 6 )  Q  where T i s t h e e l e c t r o n The  electron v  is  obtained  energy.  injection velocity .= v • < zO  z  . ( 2 - 7 )  n  from  T  the m  =  relationship c  Q  2 - m  ~2\  ...(2-8)  c  Q  l .e  V  z0  =  1  C  -  ...(2-9) 1  0  + m  By v i r t u e  of equation  Equations  (2-1) and  (2-2),  (2-4),  Q  c  A i s a s s u m e d t o be  thus,  yield  negligible.  8  |  0  V  |=  g  e E  sin(at)  ...(2-10)  where v c  K  Also, the  since  radial  there  a r e no f o r c e s  i n t h e r and © d i r e c t i o n s ,  a n d a n g u l a r momenta a r e c o n s e r v e d .  Integration  of equation  (2-10) w i t h  respect  t o time  yields K  fl  =  +  p  e E —  ,  m  c«  \/l - 0 ^ 2  Prom t h e i n i t i a l is  conditions,  equations  ...(2-11)  (2-6) and (2-7),  K  d e t e r m i n e d as Q K  e E  n  =  +  V - Po 1  It  Q  cos(ttt)  follows  from 8  =  equation s i ntan ^  —  cos(«t ) 0  ...(2-12)  o"  m  c  (2-11)  that  (K - a c o s u )  i.e. 8 =  K - a cos u yi  + ( K - a- COB u )  where phase a n g l e and e E a •=  mQ  c c«  u = ttt  ...(2-13) 2  9  Integration  of equation  " 0  Z  Z  (2-13)  _ c_ ~ tt  / /  (K - q c o s u) i 1  An simple  analytic  solution  yields  du ^\  +(K - a c o s (i)  of equation  (2-14) i n terms of  known f u n c t i o n s c o u l d n o t be found*  A  series  expansion  i n powers o f (U-UQ) c a n be o b t a i n e d .  solution,  however, i s o f l i m i t e d  coefficients  of the series  form  a r e f u n c t i o n s o f UQ and must,  the integrand i n equation  f o r numerical  integration,  program f o r t h e Alwac written*  This  of UQ.  (2-13),  (2-^12),  (2-9),  (2-14) i s i n s u i t a b l e  i t was d e c i d e d  III E digital  program  This  u s e f u l n e s s because the  t h e r e f o r e , b e r e - e v a l u a t e d f o r e a c h new v a l u e Since  «..(2-14)  computer  that  should  incorporates equations  be  (2-14),  ( 2 - 8 ) a n d ( 2 - 6 ) , a n d was u s e d t o  o b t a i n many o f t h e r e s u l t s w h i c h a r e p r e s e n t e d this  a  throughout  work*  2«2  General  2«2ol At  this  represented indefinitely  Results  Maximum E n e r g y stage  i t i s of interest  by equations along  Transfer  both  to l e t the f i e l d  (2-1') a n d ( 2 — 2 ) c o n t i n u e directions  of the z-axis.  m o t i o n o f an e l e c t r o n t h a t i s r e l e a s e d anywhere  The  on t h e  10  axis  of the c a v i t y of Figure  several  2A may t h e n b e t r a c e d f o r  periods.  Consider  ••••  an e l e c t r o n t h a t  i s released  at  z=z •: 0  U=U  Q  T-T The  electron will  (T,u)  trace  out a path  and i n p o s i t i o n - p h a s e  properties  of these  of  2.1.  chapter By  equations  space  i n energy-phase (z,u).  p a t h s may b e d e d u c e d  (2-8),  (2-12) and  T = T ( S , u , u) = Q  general  from the equations  (2-13)  T ( p , u , u + 27in)  0  =  Some  space  0  O V  T  Q  U  0'  " ) u  (n=0,l,2,3...) ...(2-15) Equations even p e r i o d i c can  function  be t r a n s f e r r e d f r o m  vice-versa time  ( 2 - 1 5 ) show t h a t  i s therefore  the electron  of phase. the f i e l d limited,  energy  i s an  The a m o u n t o f e n e r g y  that  t o t h e e l e c t r o n and n o m a t t e r how l o n g t h e  of i n t e r a c t i o n . For  a specified field  maximum e n e r g y u~  exchange  = nit  strength  a n d a s p e c i f i e d 8Q, t h e  takes place i f  11  and half  i f the time of i n t e r a c t i o n  i s an odd number  of  periods, i.e. Au  To  calculate  first  =  (2m  + l ) it  t h i s maximum e x c h a n g e  the r e l a t i o n s h i p  equations  (m=0,l,2, . . . )  (2-5) and  T + m  Q  between  of energy,  T and p^, w h i c h  (2-8) i s found t o  ' 2 2 ^ P c + m  c'  consider  m  by  be 2 4\ c  0  ...(2-16)  Hence, dT It  follows  = v  dp  . . .(2-17)  m  from equations  (2-10),  (2-15) and  (2-17)  that T  ( 3 , mx,  (n+2m+l)Ti) - T  0  ( S Q , nix, m i )  (n+1)% e z  E 0)  sin  [x d|j.  ...(2-18) The v  integral  i n equation  tends to Ic[throughout  (2-18) w i l l  be  an extremum i f  the i n t e g r a t i o n .  In  this  case AT  = I =  AT | max  |T(+  1, n-rx, ( n + 2 m + l ) 7 i ) ^- T ( +  2c e E / to c'  1, n u ,  n%)  12  During found  this  time  t o be  the  electron displacement  (since v  =  is  easily  c)  z  z  2.2.2  " ol z  Paths  =  «  =  J  A u  (2m+l.)u  i n Energy-Phase  Space  and  in  Position-Phase  Space Refer in  now  to Figure  energy-phase  paths  are  space  labelled  are  paths  Figure  labelled  and  electromagnetic two  sets  of  origin  i n Figure  either  the  any  pairs with be  of the  both  path  2C  of  cases  simultaneously  and  electrons follow shown i n  Because  the  a f u n c t i o n of p o s i t i o n ,  and  Hence, any  (T,u) of  on  the or  some e l e c t r o n .  2B  These  That i s , the may  be  along Figure  t r a c e the  K  K  z-axis  in  suit  corresponding  (z,u)  on  origin,  Z , n  be  may,  taken  to  intermediate -coordinates  on  The  subsequent motion  and  Z  n 2C  and  n  two  these  z-axis  shifted  or negative: z - d i r e c t i o n to  shift  be  cycle.  space,  ....  quite general.  coordinates  will  The  Z^,  i s not  e  Figure  , ....  Z-^,  problem.  proper  t r a c e d f o r one  is arbitrary  coordinates  injection  the  are  positive  particular  several e l e c t r o n paths  i n position-phase  field  paths  where  , K^,  corresponding 2C  2B,  for  . n  can,  t h u s , be  e n e r g y and  used  position  of  to an  electron  13  Fig.  2B  Energy-Phase  Space  f o r E = 2 0 . 0 x 10  volts/meter.  14  15  as  a function  electron K  n  ,  of time  i f a t l e a s t one p o i n t  t r a j e c t o r y c a n be l o c a t e d  on one o f t h e c u r v e s  However, t h e r e a l u s e f u l n e s s  providing  a simple  which w i l l  2.3  physical  assist further  Classification  The  electron  now b e s t u d i e d length  picture  motion  Trajectories.  This  of Figure  2A  will  c a v i t y has a gap  and i s operated  at a  frequency  i  w i l l be c a r r i e d ,3 f o r a beam i n j e c t i o n e n e r g y TQ =30.0 x 10 eV  out  cycles/sec.  Investigations  ( e l e c t r o n v o l t s ) a t a maximum f i e l d volts/meter, 40.0  x 10 The  6  and f o r T =2.00 x 1 0 0 and 60.0 x 1 0  preceding  7  four  distinct  types  t r a j e c t o r i e s are Trajectories are  i s p o s i t i v e and below i f v z • z  F o r each t r a j e c t o r y t h e i n j e c t i o n phase UQ  reference  follows  that  6  t h e e x i t phase u^ o r UQ' a r e i n d i c a t e d .  phase  c  volts/meter.  2 E , 2 F , 2G a n d 2H. i f v  E = 2 0 . 0 x 10  eV a t E =20.0 x 1 0 , c  Representative  drawn above t h e z - a x i s  and  6  strength  parameters permit  electron motion.  negative*  3  o  shown i n F i g u r e s  is  lies i n  of the electron  i n the cavity  o f Z ^ - Z Q = 1 7 . 4 0 mm  f=2.998 x 10  of  curves  studies.  i n more d e t a i l .  9  of these  of Electron  motion  (T,u) on t h e  points  c a n e a s i l y be l o c a t e d  Further since i t  from dT „ j — = e E sin u dz c  points  of zero  t r a j e c t o r y slope  i n t h e T-z - d i a g r a m s  correspond An  these  which  Fig.  2D  such that  an  will  characteristic  injection be  reversals.  cycle  intervals  is  will  field  injection  intervals, of  to  interval  examined  in  and  can  be  subdivided  electron  injected  execute  only  of  one  the  this  into  during  that  type  interval.  corresponding  four  type  of  any  one  motion  Each of  motion  turn.  Classification  of T r a j e c t o r i e s P h a s e u,  by  Injection  17  Type  I The t r a j e c t o r y  of  those  part  shown i n F i g u r e  electrons which  are injected  of the decelerating half Of t h e s e  velocity  m e r e l y be s l o w e d  high  those  down d u r i n g  A l l of these  during  the  latter  cycle.  electrons, the e a r l i e r  t w i c e , whereas  transit.  2E i s r e p r e s e n t a t i v e  ones w i l l  injected the f i r s t  electrons leaye  reverse  somewhat  later  stage  their  of  the cavity  will  at a  energy.  Type I I The t r a j e c t o r y those part  of Figure  electrons which  enter  the c a v i t y  of the a c c e l e r a t i n g h a l f  leave  the cavity  somewhat  later  at a high  leave  2F i s r e p r e s e n t a t i v e o f  cycle.  energy.  the cavity  during  the i n i t i a l  The e a r l i e r Those  that  ones  enter  a t a low energy.  Type I I I The t r a j e c t o r y Figure It  i n Figure  2G i s s i m i l a r  2E i n t h a t t h e e l e c t r o n r e v e r s e s  differs  to that i n  velocity  i n s o f a r as t h e e l e c t r o n r e a c h e s h i g h  twice. velocities  between r e v e r s a l s . This 10  3  very  type  eV a n d E  c  o f m o t i o n was = 20.0 x 10  narrow i n j e c t i o n  6  phase  observed  only  volts/meter. interval  f o r TQ = 3 0 . 0 x  I t occurs  and t h e r e f o r e  over only  a  18  T(eV) 210000 -  180 000  120000  60000  1  0  1  Z  1  !  1  1  6  1  8  I  10  12  11  16 zlmm)  TfeH  8  10  IZ  14  16  zlmm)  Top:  E  c  = 20.0xl0  u  Q  = 103.9°  Fig.2E Bottom: E  volts/meter T  6  Electron = 20.0xl0  Energy 6  =  30.0xl0 eV 3  Versus  volts/meter T  Position n  =  30.0xl0 eV 3  \j  C  UQ  Q  =  180.0  Fig.2F  0  Electron  Energy  Versus  Position  19  TkV) 120000  60000 -  60000  TfeV) IZOOOO  60000  -i  1  <f  4  1  1  £  8  1  :—I  10  1  IZ  H  r-  16  z (mm)  4 a.  Tops E u  c  = 20.0xl0  Q  =  volts/meter  T  Q  =  30.0xl0 eV 3  300.5°  Fig.2G B o t t o m s Eu  6  u:  Q  Electron = 20.0xl0 =  6  Energy  Versus  volts/meter  T  Position Q  =  30.0xl0 eV 3  302.0°  Fig.2H  Electron  Energy  Versus  Position  20  very The  few ( ^ 1 $ ) of t h e i n j e c t e d e l e c t r o n s final  neglected  energy  of these  electrons  are involved.  i s low and they a r e  hereafter.  Type I V The  t r a j e c t o r y of Figure  electrons which the  enter  the cavity during  accelerating half cycle  decelerating therefore  2H i s r e p r e s e n t a t i v e  half cycle.  and the i n i t i a l  These  do n o t c o n t r i b u t e  electrons  Beam Having  now w i s h primary  examined a few c h a r a c t e r i s t i c  interest i s the current  current.  electrons Consider  forming  energy  i n Figure  21.  a n d shows t h e i n j e c t i o n  function  of a r r i v a l  z = z^i  The c u r v e  Of  and v e l o c i t y o f  This  i s limited  to entrance  to motion of type  I and type  the v e l o c i t y  curve  extends  phase UQ as a  p h a s e u a t some f i x e d  enough such t h a t  we  the current.  the curve  one p e r i o d  trajectories,  as a f u n c t i o n o f phase  the  positive.  a r e r e j e c t e d and  a p i c t u r e o f t h e beam a s a w h o l e .  p o s i t i o n , and t h e instantaneous  large  of the  Current  to obtain  corresponding  part of  interest.  and  over  part  to the c a v i t y output  T h e y a r e o f no f u r t h e r s p e c i f i c  2.4  the l a s t  of those  position phases I I , and z^ i s  of a l l electrons i s  21  Fig.21  I n j e c t i o n Phase UQ V e r s u s  Let  of Figure  21 i t i s s e e n t h a t a l l t h o s e  enter the cavity  by  z =  of  these  Phase  t h e k n o w n i n j e c t i o n beam c u r r e n t be I Q .  inspection which  Arrival  d u r i n g the phase  d u r i n g t h e phase  increment  increment  du.  electrons to the instantaneous  The  u  By electrons du,Q p a s s  contribution  c u r r e n t a t z^ i s  therefore dujI(u  The  0 )  u)  = I  s i g n o f (du /.du) U z on w h e t h e r  reverse  order  .. . ( 2 - 1 9 )  du"  may b e p o s i t i v e  n  depends  0  o r n e g a t i v e and  k  t h e e l e c t r o n s go p a s t  i n which  t h e y were  z ^ i n t h e same o r  injected.  Had t h e c u r v e  22  in  Figure in  to I(u  21 b e e n  the negative  0 >  to include electrons going  z-direction,  include the sign  i t w o u l d t h e n be  analytic  reconsidering  necessary  o f 8(UQ U ) i n t h e e x p r e s s i o n f o r  e x p r e s s i o n f o r I ( U Q U ) may b e d e r i v e d b y  equations  z - z0  ~  (2-13) and  (2-14),  8 (u ^u)d[x  (6  Q  find du  0  , set  du u+du  u. c_ ft  8 (u ^)da 0  <6  0  0  =  t h e above  z  k "  z  0  0  i tfollows that u  u 8(u ^|i)du. = 0  u0  B(u +du ^u)du.  u +du  u0  From  past  u). An  To  extended  / u0  8 (u +du ^u)du 0  Q  0  23  u +du 0  u+du  0  8 (u +du 0  If  d u a n d duQ  neglected,  are small,  t h e a b o v e may  3P(u  0 t  iu  ti)  and i f second o r d e r be w r i t t e n  0  B(u +du 0  Hence,  ti)dji  quantities are  as  duQ  0  du P(u +du  0  u )  0 >  0 >  +  0  u)  i n the l i m i t ,  +  p(u +du 0  0 >  u +du ) 0  du 2  8(u +du ^u+du) 0  0  as du and d u  du 0  approach  n  P( o, u  du  0  0  u  zero,  )  .(2-20)  58(u ^)d(i Q  du,  Now,  substitute  perform  equation  the indicated  e x p r e s s i o n f o r duQ/du I(UQ u ) . yields  Multiply  the general  (2-13) i n t o  equation  differentiation. i n equation  Use t h e r e s u l t i n g  (2-19) t o o b t a i n  I ( U Q U) by the sign result,  (2-20) and  o f P'(UQ u ) .  This  24  I(u  0 f  u)-  = u / a sin u  n  l+(K-a  d|o, c o s |x)  2  3/2  (2-21) l(u  u)  n  i s the current  a t p o s i t i o n z a n d p h a s e u due  the  electrons  i n j e c t e d a t p h a s e UQ.  one  injection  phase  instantaneous desirable will the  be  current  electrons  contributions  at a different v e l o c i t y . sum  Specific The  more  The  than  to the  a t p h a s e u a n d p o s i t i o n z«  to keep these  arithmetic  2.5  contributes  In general,  to  separate total  It i s  as  each  current  i s  of the i n d i v i d u a l c o n t r i b u t i o n s .  Results  c a v i t y output  current,  electron  energy, v e l o c i t y  and i n j e c t i o n phase have been p l o t t e d f o r v a r i o u s parameters 20. to  as a f u n c t i o n  In c e r t a i n regions the instantaneous  contribution These the  they  z>z,.  The  two  output  phase  i n Figures  injection  phases  current.  The  r e s u l t s a r e f o r z = z^. must d r i f t  lesser  enter  the main a c c e l e r a t o r  kinematic  line.  For p r a c t i c a l  through a short  reasons  field-free  structure  e f f e c t o f any g i v e n  2J to  contribute  i s d i s t i n g u i s h e d by use of a d o t t e d  electrons  before  of e x i t  drift  region  at length  can  25  470  450  430  410  390  370  350  330  310  i90  270 u  f  E  c  = 20.0x10° v o l t s / m e t e r  Fig.2J  u  n  and  l / l  n  Versus  T  Q  u.  =  30.0xl0 eV J  (Oeyrees)  26  Tie/)  350000  460  4f0  420  400  380  360  340  3Z0  300  280  U, (Degrees)  E  c  = 20.0xl0  Fig.2K  6  volts/meter  T and 8 versus  u  1  T  Q  =  30.0xl0 eV 3  27  28  160  E c  110  120  100  = 20.0x10° v o l t s / m e t e r  Pig.2M  T and 8 V e r s u s  u.  380  T  Q  =  360  310  2.00xl0 eV 3  3^0  u, fOejrees)  29  30  TUV) 1000000  300000  1.0  800000 -  .9  700000  .8  600000 -  -  .7  500000 -  - .5  100000  300000  200000  .3  100000  2  0  410  E  c  T  1  1  110  400  380  = 60i0xl0  Fig.20  6  —r  310  360  volts/meter  T and 8 V e r s u s  0  -  u.  300  320  280 u,  T  0  n  =  2.00xl0 eV 3  (degrees)  u, - ZTTH  — i  1  300  280  1  260  :  1  1  1  240  220  200  1  1  180  ISO a  c  = 40i.0xl0  Pig.2P  6  volts/meter  (uQ+2n;m) V e r s u s  T  Q  =  (Degrees)  2.00xl0 eV 3  (u ~2un) a t z = Z j and 1  ' \  —  at  /40  *i?irm  0  E  fDegrees)  z = z,+10cm  —  —  —  —  —  E  c  = 40.0x10  Fig.2Q  T and  volts/meter B Versus  u  T  n  Q  =  2.00xl0 eV J  33  be  c a l c u l a t e d from the  This  has  known v e l o c i t y of  b e e n d o n e f o r one  shown i n F i g u r e  2P.  and  a function  in  v e l o c i t y as Figure  z^z^. of  2Q.  The  arrival  Figure  The  These  above phase  will  be  the  particular  s e c t i o n of  the  Beam C h a r g e Figure has  phase u  2R  = u^.  be  at  z =  that  the  or  the  and  is  of  phase  energy  are  shown  for a l l values expressed 10cm  in by  of  terms use  discover  of  a length  are, dz.  p o s i t i o n which  inspection  has  u^)  be  bound  structure.  an  of  the  k  i s the  Q  zO —  electron,  reached  the  at  those phase  distributed within  same e l e c t r o n s  It follows  / x • q(u ,u ) = q  w h e r e q(uQ  i f a  i t i s seen that  at phase u^  (v^dUQ)/©  Q  of  Density  These  cavity.  way  main a c c e l e r a t o r  w h i c h have been i n j e c t e d d u r i n g  length  the  preceding  c a v i t y o u t p u t beam w i l l  wave i n t h e  shows t h e  By  i n c r e m e n t duQ  into  can  b e e n i n j e c t e d a t p h a s e u_  electrons  in  z^  valid  data i s s u i t a b l e to  travelling  2.6  axial  injection  are  shown l a t e r  presenting  that  of  curves  z =  parameters  corresponding plots  information at  of  electrons.  2P.  It  to  set  the  an  were  contained  beam p r i o r t o  injection  that du dz  0  « ««(2—22) u.  beam c h a r g e d e n s i t y  w i t h i n the  cavity  34 a t p o s i t i o n z a n d phase u ^ due t o t h e e l e c t r o n s  injected  a t phase UQ  i n t h e beam  and where  before  injection.  charge  per unit  Fig*2R  The  i s the charge  The u n i t s  density  o f <I(UQ u ^ ) a n d  are  length.  I n j e c t i o n P h a s e UQ V e r s u s  s i g n of (duQ/dz)^  Position z  i s negative i f the electrons k a r e i n t h e same o r d e r a s t h e y w e r e p r i o r t o i n j e c t i o n . E l e c t r o n i c charge d e n s i t y i s always a negative quantity. i n equation (2(du /dz) I t is,therefore,necessary t o use 0  35  Equation for  a s p e c i f i e d beam i n j e c t i o n  electric  intensity  determine An by  (2-22) and a p l o t such as t h a t  2B  e n e r g y a n d a maximum  i n the cavity are s u f f i c i e n t to  <I(UQ u^.) b y t a k i n g  analytic expression  considering  of Figure  equations  slope  measurements.  f o r q ( u ^ u^) c a n be  (2-13) and  derived  (2-14),  u z  ~ 0  p(u ^u)du  = m  z  0  u0 To  find du dz  dz =  0  , write u,  ^  p(u +du ^u)du 0  u +du 0  Using  an argument  follows  -  0  0  Q  similar  to that  i n s e c t i o n 2.4, i t  that u,  u +du Q  38(u dz =  ^ (6  Q  u)du 3u7  0  du  c_ (6  p(u +du ^a)dn Q  u and  B (u ^u)du  0  0  36  3P(u ,|i)du 0  0  du  " a P o' o o  az = ^  (u  3u  0  u  )du  u0 From e q u a t i o n  (2-22) and t h e above,  a(u ,u ) 0  =  k  36(u ,u)du Q  B (u ,u )0  0  0  3u  0  u0 Nov,  substitute equation  perform by  (2-13) i n t o t h e above  the indicated differentiation  expression,  and g e n e r a l i z e  setting u^ = u to obtain  q(u ,u) 0  = u  a  sin  UQ  du  1 +(K-a c o s u ) '  3/2  u0 ...(2-23) In general,  more t h a n one i n j e c t i o n phase  to  the charge d e n s i t y  of  these i n d i v i d u a l contributions  UQ  a t p o s i t i o n z and phase  contributes u.  equals the t o t a l  T h e sum charge  density. For  a n y one component o f t o t a l  charge d e n s i t y  and t h e  37  corresponding that  component  Q  which  states  may b e  2.7  =  c  q(u ,u) Q  namely  0(u ,u) n  =  charge  density  charge  velocity  P a r t i c u l a r D e d u c t i o n s R e g a r d i n g Beam C h a r g e Several  either  conclusions  equation  c a n be drawn b y  (2-21) o r e q u a t i o n  d e n o m i n a t o r o f t h e R.H.S. q(u ,u) , Q  seen  that  current  D  current,it  e q u a t i o n s ( 2 - 2 1 ) and (2-23) a r e c o n s i s t e n t , l(u ,u)  the  of t o t a l  inspecting  (2-23).  of equation  Density  Designate (2-23) by  where u a  D  [q(u ,u)]  =  Q  B  0  sin  UQ  du.  +  1 +(K-a cos u)'  3/2  u0 ...(2-24)  Case  I UQ  It  follows  (n an  im  from equation D  Hence,  =  q(nu,u)  by equation  means t h a t  (2-24)  that  = SQ  (2-23)  q(nit,u) = q  This  integer)  Q  the section  o f t h e beam w h i c h i s i n j e c t e d  38  at  field  reversal  throughout  i s of constant  charge  density  i t s motion i n t h e c a v i t y gap.  This  r e s u l t may b e a r r i v e d  Inspection  at otherwise.  of the energy-phase p l o t  ( F i g u r e 2B) shows  that \  dT ^du on e a c h p a t h e  - l ' 0 e  a  n  d  e  ' u=nn K^.  +l'  (TQ,ix-duQ) ,  along  maintain their  Z-^f  v  i  g  n  respective injection  and (TQ^+du^)., w i l l p a t h be  o f e^ i n p o s i t i o n - p h a s e  To a f i r s t  in  a  Let this  be p r e c i s e l y  the  n  I t follows that three electrons  (TQ.-IX)  same p a t h . path  = 0  cavity.  Z^.  order  .  The  space  I twill  injection velocities  move a l o n g t h e . corresponding  ( F i g u r e 2C) w i l l  start  approximation  at  until  space w i l l  be a l o n g  respectively.  +  o f e ^ a n d e -^ +  curves  The r e s p e c t i v e  coordinates  o f e_^ and e ^ w i l l  (O.n+duQ).  The p a t h s  Figure  +  parallel to and n e g a t i v e starting  b e (O.u-duQ) a n d  ©Q a n d e ^ a r e s k e t c h e d i n +  2S.  Figure relative respect  of  will  e ^ has entered  d i s p l a c e d b y ( C B Q dug)/**) i n t h e p o s i t i v e  z-directions  then  (0+n),  e _ ^ a n d eQ  I t f o l l o w s t h a t the paths  position-phase  coordinates  2S s h o w s t h a t e _ ^ , e g , e ^ m a i n t a i n +  positions  as seen by an o b s e r v e r  to the cavity.  q(mx,u) = q . Q  This  their  at rest  i s i n agreement  with  with  3 9  Fig.2S  Paths  o f e ,,  e , n  e , i n Position-Phase  Space.  Case I I Electrons i.e.  s i n UQ <  monotonically  are i n j e c t e d 0.  Holding  into  an a c c e l e r a t i n g f i e l d ,  UQ f i x e d ,  i n c r e a s e w i t h u, hence  D  q(uQ,u)  will  q_(u.Q,u) m o n o t o n i c a l l y  decreases.  Case I I I Electrons i.e.  s i n u > 0. n  are injected Holding  u  n  into fixed,  a retarding q(u ,u) at n  field, first  40  increases  with u, passes  monotonically  through  density  e^,  e^ a n d e  and  U Q + <1UQ.  position This and  c  with  respective  The i n f i n i t y e  c  instantaneous  2.8  several  Case  beam e n e r g y  Let 8Q(UQ), and  phase  Figure  Figure  A Suggested  curves,  the required the crossover  the crossover  i n  similar to  charge  2V s h o w s t h e  I n j e c t i o n System Function  I I I o f s e c t i o n 2.7 r a i s e s t h e q u e s t i o n s  a t some f i x e d  o f UQ  2U s h o w s a n  Injection Velocity  that  e^.  and v e l o c i t y .  f o r an Improved  distribution  same t i m e  i s illustrated  p l o t o f e l e c t r o n p o s i t i o n a n d beam  e l e c t r o n s be i n j e c t e d d u r i n g velocity  type  constant  2R, a r e s h o w n .  Suggestions 2.8.1  p h a s e s UQ - duQ, UQ  edge o f t h e beam.  of this  density within the cavity. corresponding  electrons  e ^ , a n d e^ i n t u r n p a s s e s  at the leading  2T w h e r e  three  continually increasing values  Electron bunching  i n Figure  injection  infinite  t h e n a r i s e s a t t h e phase and  passes  involves  propagates  that  o f t h e phenomenon o f  i s p o s s i b l e i f one c o n s i d e r s  at which  infinity  Figure  and then  decreases.  A physical explanation charge  infinity,  part  they  can  o f one c y c l e w i t h  a l l cross  over  such  at the  p o i n t w i t h i n t h e cavity?"*"^  injection velocity  f u n c t i o n be  p o s i t i o n w i t h i n t h e c a v i t y be  phase be u . * x  z^,  a  41  U  0  (Degrees) 230  220  210  200  130  180  170  160  150  14-0  130  E  c  = 2CU0xl0  Fig.2T  An  6  volts/meter  Illustration  Near the C a v i t y  T  Q  =  2.00xl0 eV  of E l e c t r o n Entrance  3  Bunching  43  z (mm)  E'  c  = 20.0x10° v o l t s / m e t e r  Fig.2V  T  Q  = 2.00x10 eV  Beam E n e r g y a n d V e l o c i t y  a t u = 320°  Use  equation  (2-14)  to  write u  u : x  £ ft  P(p +dp ,u +du ,n) 0  u +  u0  du  0  = z From t h e above  i t follows  u  -  x  0  0  0  da  Q  z„ 0  that  u  B(B ,u ,uO 0  du  0  =  / B(B +dp ,u +du ,u) 0  u0  0  0  0  du  u0 u +du 0  0  P(-P +dP fU +du ,|i) 0  0  0  du  0  u0 Hence, n e g l e c t i n g  second  order  quantities  u 3B(3 ,u ,a)d8 0  B(8 ,u ,u ) 0  0  0  du  0  90(0 ,u ,u)du o  o  Q  30  Thus,  0  0  3u  0  o  45  u 3P(P ,u ,u) du 0  0  3u  A3 0 du  0  u0  0  u 3  P(P .u ,u) 0  du  0  38  0  u0 Substituting the  indicated  non-linear  equation  (2-13) i n t o  t h e above  differentiation yields  differential  a first  and performing order  equation,  u a  sin  UQ  du  1 +(K-a cos u)' dp  0 du,  3/2  u0 u  du 1-6  3/2 0  l + (K-a cos u)'  3/2  u0 ....(2-25) w h i c h may b e s i m p l i f i e d t o  46  dp =(1-P  fl  du  2N3/2 )  '0  2  + a sin u 0  0  0 du 1  + (K-a  cos u)'  3/2  • *•(2—26) In  using  PQ(UQ), for  one f i r s t  a value  o f UQ.  crossover position,  corresponding  phase u .  Since  injection u  thus, been reduced The  (2-26) t o c o n s t r u c t a f u n c t i o n  chooses  PQ a t some v a l u e  electron the  equation  remains  This  It  and then  process  a value  of fixing the  one i n j e c t i o n v e l o c i t y a n d phase determines constant,  to a relationship  the arrival  equation  (2-26) h a s ,  b e t w e e n PQ a n d U Q .  e v a l u a t i o n o f dPQy/duQ.  i s o f course  important  PQ(UQ)  that  PQ(UQ)  must be p o s i t i v e  c a n be r e a l i z e d  Consider  and r e a l ,  t h e consequences  P n ( u ) a n d u,n. a t w h i c h 0 0 0 r i  u: x  t o compute a f u n c t i o n  i npractice.  o f P Q ( U Q ) a t PQ^" a n d UQ"*" , t h e s e  at  x  f u n c t i o n P Q ( U Q ) c a n now b e c o n s t r u c t e d b y s u c c e s s i v e  numerical  r  forz  T h i s means  and PQ(UQ) ^  of starting being  a t a uniform  (2-25) w i t h  equation  1.  the  computation  the values of  the electrons f o r crossover  w o u l d b e c o n t r i b u t e d i f t h e beam w e r e  injected  velocity  PQ^•  that  at z  x  t o be  Comparison o f equation  ( 2 - 2 4 ) shows t h a t f o r t h e s e  values  47  d p ^ / d u Q = 0. be r e f e r r e d  2.82  This  A Suggested Modulation  to find  beam p r i o r  to  a m e t h o d t o c o m p u t e P Q ( U Q ) , we  a velocity  modulating  a large  design  driftspace.  b e d e n o t e d PQ  BQ(UQ) d u r i n g  the  that  BQ  before  fraction  decreases with  This  of view  that  which  pre-accelerator  entrance  i s placed  beyond the crossover  fraction  point  of  function  respectively increases  d p ^ / d u ^ c a n be  i s small.  or  of the cycle  zero  which  a t UQ"*" .  slowly with  this  Prom a p r a c t i c a l  be e a s i e r t o r e a l i z e  ' Q) that varies U  of the cycle during  that  function,  (UQ) t o  the v a r i a t i o n of PQ(UQ) d u r i n g  of the cycle i t may  offers  of the e l e c t r o n s .  i s recalled that  infers  fraction  PQ  most  UQ d u r i n g  a  computed.  t o m a t c h BQ  a r r i v e at the  that  by  combination  and d e n s i t y m o d u l a t e d beam, a  U  It  gap f o l l o w e d  (UQ), i s e a s i l y  or s l i g h t l y  ' Q) i s obtained  contains  This  I f the pre-accelerator  velocity  the e l e c t r o n  and i t s output v e l o c i t y  be a t t e m p t e d  most o f t h e e l e c t r o n s  slightly  modulating  flexibility  m u s t now  entrance.  to modulate  now  injection.  density  It  Device  a suitable device  Consider  which w i l l  shortly  to«  Having devised wish  c h a r a c t e r i s t i c , o f BQ(UQ) w i l l  a velocity  UQ, r a t h e r  point  function  than a  function  48  P  0  (U )  that varies  Q  desirable, 1 near u  n  •  rapidly with  therefore,  u . Q  to approximate  I t may PQCUQ)  be B  Y  ^0  ^ 0^ U  49  3.  3.1  E L E C T R O N MOTION I N THE M A I N A C C E L E R A T O R  The E q u a t i o n s  Cy  lindrical Metal Guide  Partial  Fig.3A The cavities  of Motion  Profile  o f one M a i n A c c e l e r a t o r C a v i t y  electromagnetic f i e l d  i n the main  accelerator  c a n be c o n s i d e r e d as a s u p e r p o s i t i o n o f  harmonics.  Since the cavities  are operated  space  i n a  resonant  4 condition  , these  magnitude  and equal b u t o p p o s i t e phase v e l o c i t i e s .  analysis  harmonics  exist  considers e l e c t r o n motion  fundamental  forward  travelling  i n pairs  of equal  i n the f i e l d  harmonic,  which  This  of the f o r small r  50  is  assumed t o be E = 1 E s i n to 2 •z a ph  - t  ...(3-1)  V  H.= where The  0  t h e phase v e l o c i t y  remaining harmonics  perturbations. It the  axis  this  the f i e l d  view  to r - f  will  be  and t h a t  i s attached  to this  along  r e p r e s e n t e d by equations(3-1)  of other  o f a n y one e l e c t r o n i s electrons.  to the t r a v e l l i n g  coordinate  wave r e p r e s e n t e d  The e l e c t r o n  position  frame i s  p  h  t  ...(3-3)  o f e q u a t i o n ( 3 - 2 ) , A i s assumed t o be  Equations  neglected.  i t s presence  s t a g e i t i s c o n v e n i e n t t o s e t up a  z' = z - v In  the electron  cavities  e q u a t i o n s (3-1) and ( 3 - 2 ) .  referred  = c  Furthermore, the motion  ffame which by  n  t h e e l e c t r o n beam t r a v e l s  assumed t o be i n d e p e n d e n t At  p  These r - f p e r t u r b a t i o n s  of the accelerator  (3-2).  v  subject  i s assumed t h a t  does n o t a l t e r and  ...(3-2)  (2-4) and (3-1) thus  negligible.  yield  The e n e r g y e x c h a n g e d u r i n g t w o s u c c e s s i v e p e r t u r b a t i o n s o f i p p p o s i t e s i g n a n d d u e t o t h e same s p a c e harmonic w i l l average t o zero i f the e l e c t r o n m a i n t a i n s a c o n s t a n t v e l o c i t y d u r i n g t h e t i m e 'fif i n t e r a c t i o n . This i s the case i f the e l e c t r o n t r a v e l s near the speed of l i g h t , because t h e n a d d i t i o n o r r e m o v a l o f even l a r g e amounts o f energy r e s u l t i n a n e g l i g i b l e change i n t h e e l e c t r o n ' s velocity. In t h e p r e s e n t a n a l y s i s i t i s assumed t h a t t h e e f f e c t of t h e n e g l e c t e d space h a r m o n i c s on t h e m o t i o n o f i n t e r mediate energy e l e c t r o n s i s also s e l f - c a n c e l l i n g . This i s an a p p r o x i m a t i o n .  51  dp = eE  TT—  dt  A further equation  p •^m  V  i s o b t a i n e d by d i f f e r e n t i a t i o n  (3-3) w i t h r e s p e c t to  equation  ...(3-4)  \ ph  relationship  dz _ dt ~ Using  -t  s i n to  a  d_z dt  time,  _ V  dz ( 2 - 5 ) , TT dt  ...(3-5)  ph c  of  a  be w r i t t e n  n  i n terms  of  as dz dt m m  2 ^ Pm 0 T c  2  ...(3-6)  V  +  Substitution  of equation  (3-3) i n t o  equation  (3-4),  substitution  of equation  (3-6)  equation  (3-5)  into  and results  in dp  m  dt  / =  eE  dz dt  and  elimination  <\ ...(3-7)  \ ph/ V  •m 2 m  By  sin  a  wz  0  A +  ,..(3-8) Pm  2  X  "  P  h  ^2  c  of the d i f f e r e n t i a l s  dt from  ( 3 - 8 ) , an e x p r e s s i o n i s o b t a i n e d w h i c h  integrated  directly  to y i e l d  the  solution  equations can  (3-7)  be  to the  present  problem. A  second  approach to the present  by u s i n g H a m i l t o n i a n mechanics.  problem  This second  i s possible  approach i s of  52  interest  because Hamiltonian  e x t e n s i v e l y by in  the  authors  who  mechanics  i s used  d i s c u s s a c c e l e r a t o r problems  literature ' ' ' . 1  Hamilton's  3  8  9  canonical equations  _ f£i  are  - ill  ,  ' ^1 - 35 w h e r e H,  (_ ) 3  (q ,p ,t) i  i  a n d w h e r e q.  a n d p.  relationship  which follows d i r e c t l y  are c a n o n i c a l coordinates.  A further  from equations  (3-9)  (3-10) i s M dt  The  10  the Hamiltonian, i s H = H  and  }  _ ~  3H at  (  }  •••»v-> i-w  c a n o n i c a l coordinates which are s e l e c t e d f o r the  present  problem  are  = z' •Pj_ = P Since  ...(3-12) ...(3-13)  m  t h e m o t i o n i s i n one  canonical  c o o r d i n a t e s have  Equations  (3-7),  (3-9),  t o be  <6Z  = - eE  sxn  V  ph  o n l y , no f u r t h e r  specified.  (3-12) and  d E  —, 9z  direction  (3-13)  yield ...(3-14)  53  Integration  of equation eE H  Equations  (3-14)  v a ph. ^ < *  cos  r  1  (3-8), (3-10),  m p  0 m  0  I wz — ph  +h (p ,t) 1  ...(3-15)  m  v  (3-12) and  21 m  gives  (3-13)  -  2  V  yield  ...(3-16)  ph  ~  +  c Integration  of equation  s u p p l i e d by  equation  u H =  Any  ; i/ p » -^m  h (t) 2  affecting  may  be  i d e n t i c a l l y zero  contain  explicitly.  It  be  a constant  By  arbitrary.  i n order equation  of the  1  wz  + h (t) 2  The  function  It will  be  t h a t H does (3-11) H  not  will,  motion.  follows that 1  H  ' \  Hamiltonian  the c a n o n i c a l equations.  c h o s e n t o be  therefore,  in  added t o the  i s ,therefore, completely  time  of the i n f o r m a t i o n  eE v , a ph — - — c o s .«  , -p v. , + m ph  A  f u n c t i o n of time  without  (3-15) r e s u l t s  2 _,_ 2 4 c + m c 0  2  (3-16) and use  p  2  c  2  + m  -Mil  c ' -p v . + 0 ph 2  4  n  (o  cos  wz V  1  \  ph  ...(3-17) This  i s i d e n t i c a l to the Hamiltonian  f u n c t i o n used  by  g  Slater. In  summary, a f i r s t  o b t a i n e d by d i r e c t  integral  integration  of motion has  of the c a n o n i c a l  been equations  54  t with  respect  to  z  and  p  .  •in  The  canonical  integrated  with  coordinates. a constant  the  present  useful  form  3.2  respect  The  not  equations  the  thus  m o t i o n and,  problem.  I t can,  equation  Electron  equally well  some o t h e r  Hamiltonian  of  of  to  can  set  of  obtained  canonical i s , in  hence, not  h o w e v e r , be  (3-17) by  be  general,  suitable for  changed  a canonical  into  the  transformation.  Orbits  t Electron space. are  orbits  I f the  initial  substituted into  value  of H  are  lines  of  coordinates  equation  i s determined  and  constant (P ?z  the  in p  ) of  m  (3-17),  H  an  then the  -  z  electron appropriate  electron orbit  can  be  computed. A v ^ (8).  =  f a m i l y of  c/2  has  the  in orbits  p o s i t i o n X,  momentum o f  velocity Electrons  c/2..  i n a wave a t p h a s e  been reproduced  Electrons  equilibrium  orbits  an  i n Figure  labelled  the  i n orbits  labelled  B  a velocity  r e s p e c t i v e l y to  the  travelling  wave.  These  labelled  equilibrium  on  and  electrons.  unbound  of which  e l e c t r o n which t r a v e l s at  These e l e c t r o n s are  Points  from  orbit  D,  T  and  bound to C have  a  represents the  the  too  stable  phase wave.  high  and  remain i n synchronism  e l e c t r o n s are are  which  reference  A move a r o u n d  ordinate  low  unbound.  3B  velocity  s a i d to  p o s i t i o n s of  separates  the  too  with  be  unstable orbits  of  bound  55  Pm  Fig.3B  p  Versus m  Before with  v  p  n  W  *ph  proceeding to  a Wave a t v  f r o m momentum p  permit us t o r e l a t e l a t e r  ph  electron motion  = c, i ti s d e s i r a b l e  description  data.  in  z  V  m  t o change to kinetic  c_ 2  i n a wave  the basis of energy  T,  This  results to the pre-accelerator  will  56  Using (equation  t h e r e l a t i o n s h i p between  ( 2 - 1 6 ) ) a n d r e p l a c i n g toz / c b y A, a F  is  (A,T) =  constructed.  Q  For  large F  and  values  (A,T) ~  m  e  c  E  +  Q  Q  cos A  ...(3-18)  o f T, T+ m c  2  0  / ' - T 1 +  2 m c - -T7 K —  ecE to— - c o s A ^  n  +  hence,  By v i r t u e  of equation  ecE . - j —  cos. A  ...(3-19)  (3-19) t h e f o l l o w i n g d e f i n i t i o n  made., A  A Using been f  that  I 5 9~\ \l T + 2T m c  l i m F (A,T) = T-^co  is  function  H(z\p )  I t readily follows  o ( A , T ) = T+ m c * -  F  p^ and T  "I = cos e< a \ equation  computed  = 2.998x10^  solid  a  (3-18),  representative  f o r a frequency cycles/sec.  l i n e s i n Figure  3C.  of operation  These  A = A . low  Bound  energy without  approach f i x e d phase  Unbound o r b i t s a r e s t i l l  energy e l e c t r o n s .  .  The p o s i t i o n o f s t a b l e  U shaped o r b i t s , g a i n  asymptotically  '  have  o r b i t s a r e shown a s  e q u i l i b r i u m X h a s now r i s e n t o i n f i n i t y . following  orbits  positions  electrons, l i m i t and given  by  present, but only f o r  The l i m i t i n g  bound  orbit  57  Electron  Orbits  Lines I n d i c a t i n g the D u r a t i o n Travel ( i n Degrees) Along the Orbits. E  = lO.OxlO  Fig.3C  6  volts/meter  T Versus  A  of Electron  58  Tfel/J  -m  -MO  -100  -SO  -20  20  SO  100  HO A  E  a  i s indicated i n volts/meter  Fig.3D  T Versus  A (Limiting  Orbits)  180 (Degrees)  59 asymptotically A  a  = -• 1 8 0 ° .  approaches a f i x e d phase p o s i t i o n a t  A family  of these  latter  computed, a n d i s shown i n F i g u r e  3.3  information  Curves  8 to a r r i v a l  v e l o c i t y curve  calculated,and arrival This The  relative there  between  electron  energy  from  curve  are p r o v i s i o n s  i s , now,  abscissae  arrival  described  are  I t i s indicated  the  shown i n F i g u r e the r a t i o  of electrons  pre-accelerator In  3E-.  i s ( Q2 U  a similar fashion  bound  are function  entrance. on F i g u r e  shift  cavities.  phase u^, T  i n the previous i n Figure  to those  injected  versus paragraph 3E into  ~ oi^/360°« u  i t i s possible  3C.  i s arbitrary  a phase  and a c c e l e r a t o r  phase UQ v e r s u s  Using  the  angles  superimposed  for introducing  the pre-accelerator  2.  i s r e p l o t t e d as a  u^ and t h e s u p e r p o s i t i o n  that  of the  chapter  a t the main a c c e l e r a t o r curve  wave i n t h e  normalized  and t h e d r i f t l e n g t h b e t w e e n  p o s i t i o n i n g o f t h e two  Injection  T and  phase u^ a t t h e e x i t  the energy  energy  the  illustrated.  and main a c c e l e r a t o r , d r i f t  phase  that  to the t r a v e l l i n g  c a v i t y are selected  pre-accelerator  as  c a n be  relating  pre-accelerator  of  Pre-Accelerator  has been p r e s e n t e d  of i n j e c t e d electrons  main a c c e l e r a t o r  velocity  i n the  Accelerator  Sufficient binding  been  3D.  C o r r e l a t i o n of E l e c t r o n Motion and  the  o r b i t s has  t o compute t h e  T  ^'180°  Fig.3E  Application  of P r e - A c c e l e r a t o r  Data  o  61  fraction which  which w i l l  asymptotically  which  field  i t will If  not have  bound near This  electrons  orbit  i s the  orbit  hence,  electrons  of moving  energy.  i s o p e r a t e d a t a low  arriving  those  w h i c h u l t i m a t e l y r e s u l t i n a maximum e n e r g y  gain.  adjusting  accelerator orbits  on  3.4  initial  the phase cavities,  entering  Further  energy  may  second main  Computations  be  may  orbits However,  r e l a t i o n s h i p b e t w e e n t h e two electrons  the  field  at the main a c c e l e r a t o r to enter  by  sufficient  the  i n the region  a maximum a m o u n t o f  the p r e - a c c e l e r a t o r  strength,  lies  i n t e n s i t y , and, gain  be  approaches-90°,  f o r most of i t s l e n g t h  highest near  of e l e c t r o n s  main  transferred to  new  cavity.  Involving  the  Hamiltonian  Function By  use  equation  o f t h e r e l a t i o n s h i p dT  (2-17)  and  A = ^  = v dp  , equation  m  (3-4)  from i s transformed  to ^ dz  = eE  a  sin A  Upon i n t e g r a t i o n T_  z-z„  where  z  a  a  =  denotes  /  — eE  the  a  ...(3-20) sin A  entrance of the main a c c e l e r a t o r  and  62 T  a  i s the corresponding In  injection  energy,  a long accelerator A i n equation  replaced by A approximately  to y i e l d valid  a relationship  f o r large values  ( 3 - 2 0 ) may  which  of  z - z a  be  i s  >  T-T z-z  ^ eE  The p r e s e n t is,  sinA  a  ...(3-21) a  accelerator i s short.  therefore, not s u f f i c i e n t l y  necessary  to proceed  Equation  accurate  i n a different  energy  the  from  relationship /  A =  sin  11  equation  equation  2  \  V 1 - cos A (3-18),  a f u n c t i o n o f T may into  to obtain the  g a i n as a f u n c t i o n o f d i s t a n c e . '  From  and  and i ti s  manner 2  (3-2l)  an e x p r e s s i o n f o r s i n A as  be o b t a i n e d , w h i c h ,  (3-20) r e s u l t s T  when  substituted  i n  dT 2 ecE \ a  " 2 I 2 F-T-m^c +\l T^+  2>\ ^ 2  2Tm c 0  ...(3-22)  Equations  f  (3-22) and t h e  relationship  - - e t  were programmed results  (3-18),  on the Alwac  o b t a i n e d were u s e d  I I IE digital  to plot  computer.  F i g u r e s 3 C , 3D a n d  The 3F.  63  64 Figure  3F r e l a t e s  ( z - z ) and A f o r e l e c t r o n s which £L  were  injected  curves  a t t h e same i n i t i a l  may b e s h i f t e d  different observed  Any o f these  along the z-axis to serve f o r  i n j e c t i o n angles f o r curves  energy.  A.  relating  In  t h a t case  the f i e l d  of  position,  whereas  A similar  p r o p e r t y was  z and u i n t h e p r e - a c c e l e r a t o r .  was o n l y a f u n c t i o n  i n the present  of time  case  the f i e l d ,  c o o r d i n a t e frame,  i s only a  referred  t o t h e moving  function  o f A and n o t o f time.  and n o t when  65  4.  Making  c e r t a i n s i m p l i f y i n g a s s u m p t i o n s , some a s p e c t s  of  relativistic  of  electric  electron motion p a r a l l e l  i n t e n s i t y i n a time v a r y i n g ,  uni-directional A series  CONCLUSION  electric  field  have been  to the d i r e c t i o n uniform, investigated.  o f c u r v e s v a s computed w h i c h r e l a t e t h e f i e l d  intensity  i n a pre-accelerator  c a v i t y and t h e energy o f  a u n i f o r m e l e c t r o n beam a t t h e e n t r a n c e o f t h e c a v i t y t o the of  energy  a n d v e l o c i t y o f t h e e l e c t r o n beam a t t h e e x i t  the cavity. It  was shown t h a t  a n e l e c t r o n beam i n j e c t e d a t a  non-uniform v e l o c i t y w i l l velocity  satisfies  differential The  bunch i n t e n s e l y i f i t s i n j e c t i o n  a certain first  dynamics  of electrons  i n a linear  using  constructed  on t h e a s s u m p t i o n t h a t  that  a Hamiltonian function which  time.  computed r e l a t i n g  electron  of i n t e r a c t i o n harmonic.  energy,  and main a c c e l e r a t o r  f r a c t i o n of electrons  space harmonic, enter.  the f i e l d  was  distance  I t w a s s h o w n how t o c o r r e l a t e t h e c o m p u t a t i o n s  on t h e p r e - a c c e l e r a t o r the  accelerator  o f a s i n g l e f o r w a r d - t r a v e l l i n g space  Curves were and  non-linear  equation.  was s t u d i e d  is  order  t o determine  bound t o t h e f o r w a r d - t r a v e l l i n g  and t h e o r b i t s w h i c h t h e s e  electrons  66  REFERENCES  1.  C h o d o r o w , M. , e t a l , " S t a n f o r d High-Energy Linear Electron Accelerator (Mark 1 1 1 ] " . M.L. R e p o r t No . 2 5 8 ( 1 9 5 5 ) , W.W. H a n s e n L a b o r a t o r i e s o f P h y s i c s , Stanford University, Stanford, California.  2.  Dome, G. , " E l e c t r o n B u n c h i n g b y U n i f o r m S e c t i o n s of Disk-Loaded Waveguide. P a r t A; General Study". M.L. R e p o r t N o . 7 8 0 - A ( D e c e m b e r I 9 6 0 ) , W.W. Hansen Laboratories of Physics, Stanford University, Stanford, California.  3.  L i c h t e n b e r g , A . J . , "The A p p l i c a t i o n o f P h a s e S p a c e C o n c e p t s ~to t h e D e s i g n o f a n E l e c t r o n Linac Buncher". E n g . L a b . ( 1 9 6 1 ) 19 P a r k s R o a d , University of Oxford.  4.  E n g l e f i e l d , C.G., " I n v e s t i g a t i o n i n t o t h e Properties of D i e l e c t r i c a l l y L o a d e d Slow-Wave. S t r u c t u r e s ' . ' . Report No.M.L.5 ( O c t o b e r 1 9 6 1 ) , Department o f E l e c t r i c a l E n g i n e e r i n g , The U n i v e r s i t y of B r i t i s h Columbia, Vancouver, Canada.  5.  Goldstein,  H., C l a s s i c a l M e c h a n i c s . AddisonW e s l e y P u b l i s h i n g Company, I n c . . R e a d i n g , M a s s a c h u s e t t s , U.S.A. (1959).  6.  P a n o f s k y , W.K.H. a n d P h i l l i p s , M., C l a s s i c a l E l e c t r i c i t y and Magnetism. A d d i s o n - W e s l e y P u b l i s h i n g Company, Inc. R e a d i n g , M a s s a c h u s e t t s , U.S.A.  41-955).  7.  Landau,  L. and L i f s h i t z , E. The C l a s s i c a l Theory of F i e l d s . Addison-Wesley P u b l i s h i n g Company, I n c . R e a d i n g , M a s s a c h u s e t t s , U.S.A. ( 1 9 5 1 )  •  67  8.  Slater,  J.C., "The D e s i g n o f L i n e a r Accelerators". Rev.Mod.Phys. V o l . 2 0 , No.3 ( J u l y 1948).  9.  L i c h t e n b e r g , A.J., " A p p l i c a t i o n of Phase Space C o n c e p t s t o . P a r t i c l e Dynamics i n A c c e l e r a t o r s " . Ph.D. Thesis (1961), U n i v e r s i t y of Oxford.  10.  C o l e m a n , P.D., "Theory of the Rebatron-A R e l a t i v i s t i c Electron Bunching A c c e l e r a t o r f o r Use i n M e g a v o l t Electronics'.'. J.. A p p l . P h y s . V o l . 2 8 , No.9, (Sept.1957).  11.  H a i m s o n , J . , " S o m e . A s p e c t s o f E l e c t r o n Beam O p t i c s and X-Ray P r o d u c t i o n w i t h the Linfear A c c e l e r a t o r " . IRE T r a n s . Vol.NS-9, No.2, X A p r i l 1962).  

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