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Observation of the transverse Stern-Gerlach effect in neutral potassium and an analysis of a charged… Enga, Eric 1969

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OBSERVATION OF THE TRANSVERSE STERN-GERLACH EFFECT IN NEUTRAL POTASSIUM AND AN ANALYSIS OF'-A. CHARGED . PARTICLE STERN-GERLACH EXPERIMENT  BY  ERIC  ENGA  B.A.Sc., U n i v e r s i t y o f B r i t i s h Columbia, 1962 M.A.Sc., U n i v e r s i t y o f B r i t i s h Columbia, 1966  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS  FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  i n the Department o f PHYSICS  We accept t h i s required  t h e s i s as conforming t o t h e  standard  THE UNIVERSITY OF BRITISH COLUMBIA December, 1969  In p r e s e n t i n g  t h i s t h e s i s in p a r t i a l  f u l f i l m e n t o f 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 I f u r t h e r agree tha  permission for extensive  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 .  permission.  Depa rtment 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  shall  not  thesis  Department o r  I t i s understood that c o p y i n g o r gain  study.  copying o f t h i s  be g r a n t e d by the Head of my  of t h i s thesis f o r f i n a n c i a l written  a v a i l a b l e f o r r e f e r e n c e and  publication  be a l l o w e d w i t h o u t  my  ABSTRACT  Two experiments a r e d e s c r i b e d . of  One i s t h e s u c c e s s f u l  observation  the r e s o n a n t d e f l e c t i o n o f a beam o f n e u t r a l potassium atoms a t a  frequency o f 7.2 Mhz, i n agreement w i t h the p r e d i c t i o n s o f t h e t h e o r y of  t h e T r a n s v e r s e S t e r n - G e r l a c h (TGS) experiment.  The o t h e r i s a  p r o p o s a l f o r a charged p a r t i c l e S t e r n - G e r l a c h experiment, which i s based on an e x t e n s i o n o f the TSG experiment t o time independent, inhomogeneous magnetic  f i e l d s h a v i n g the form  B(r,z) = B k Q  I f the f i e l d are  +  B^rle ^ 1  B ^ ( f ) i s w e l l chosen, the charged p a r t i c l e  trajectories  c o n f i n e d i n a s t a b l e beam by the r e s u l t i n g L o r e n t z f o r c e s f o r  motion g e n e r a l l y a l o n g the z a x i s .  T h i s i s , i n f a c t , the p r i n c i p l e o f  s t r o n g f o c u s i n g which i s now w i d e l y used i n a c c e l e r a t o r d e s i g n .  But  i n such a system i t i s a l s o p o s s i b l e t o s a t i s f y the c r i t e r i o n f o r a TSG  experiment, s i n c e i n a frame o f r e f e r e n c e moving w i t h a p a r t i c l e  i n t h e z d i r e c t i o n , the f i e l d "B\ (r) i s r o t a t i n g i n time.  - iii  -  TABLE OF CONTENTS Page ABSTRACT  i i  LIST OF FIGURES  iv  ACKNOWLEDGEMENTS  v i i  CHAPTER 1  Introduction  2  C l a s s i c a l Theory o f the N e u t r a l P a r t i c l e v e r s e S t e r n - G e r l a c h Experiment  3  E x p e r i m e n t a l Procedure  4  Experimental Results  5  A P r o p o s a l f o r a Charged P a r t i c l e TSG Experiment...  6  The T r a j e c t o r i e s o f an Ion w i t h Zero Moment i n a DC H e l i c a l Quadrupole  7  8  9 10  1 Trans-  14 ...  21 31  Magnetic :44  The T r a j e c t o r i e s o f an Ion w i t h Non-Zero -Magnetic Moment i n a DC H e l i c a l Quadrupole f o r Weak L o r e n t z Forces  53  The I n f l u e n c e o f the S o l e n o i d F i e l d on the T r a j e c t o r i e s , and the S t e r n - G e r l a c h E f f e c t f o r Large L o r e n t z Forces i n a DC H e l i c a l Quadrupole  67  ...  Experimental Considerations  87  C o n c l u d i n g Remarks  97  BIBLIOGRAPHY  99  APPENDIX A  The H e l i c a l Quadrupole Magnetic  B  The D e r i v a t i o n o f the T r a j e c t o r i e s o f an Ion i n a H e l i c a l Quadrupole F i e l d C o n s i s t i n g o f Orthogonal E l e c t r i c and Magnetic F i e l d s Superimposed on a Homogeneous F i e l d and on which an A d d i t i o n a l R a d i a l Force i s Introduced  C  Field  An Ion Gun Design S u i t a b l e f o r Producing Diameter, Well C o l l i m a t e d Ion Beams  101  105  Small . 112  - iv -  LIST OF FIGURES Figure 2-1  2-2  Page Schematic p l o t o f a component o f the S t e r n - G e r l a c h f o r c e v e r s u s time i n the impulse a p p r o x i m a t i o n ....  6  R e p r e s e n t a t i o n o f the r - f f i e l d i n the r o t a t i n g c o - o r d i n a t e system (x, y, z). I t has been assumed that ( f ) i s i n the x-y p l a n e and makes an angle - <f> w i t h r e s p e c t t o the x - a x i s . i s the component o f J a l o n g the e f f e c t i v e f i e l d a t t = 0  6  2-3  I l l u s t r a t i o n o f the o r i e n t a t i o n o f B-^(r) = G ( x - i y ) at a p o s i t i o n r = x+iy  3-1  Schematic scale)  3-2  diagram  o f the o v e r a l l system  11  (not t o 1  5  Drawing o f t h e d e f l e c t i o n system w i t h c o n s t r u c t i o n d e t a i l s omitted  18  3-3  E l e c t r i c a l c i r c u i t o f the quadrupole w i r e s showing d e t a i l s o f the p a r a l l e l resonant c i r c u i t used  19  4-1  (a)  (b)  P l o t o f t h e percentage i n c r e a s e i n beam i n t e n s i t y at t h e d e t e c t o r as a f u n c t i o n o f p o s i t i o n i n the x-y p l a n e due t o the f o c u s i n g e f f e c t o f the d e f l e c t i o n system i n a l a r g e , u n i f o r m - i n t e n s i t y beam Contour map o f the f o c u s e d " p i p " , as cons t r u c t e d from F i g . 4 - l a . The numbers on the contours a r e percentage i n c r e a s e s i n beam i n t e n s i t y due t o the r o t a t i n g quadrupole f i e l d (ordinate of F i g . 4-la). The d o t t e d l i n e s a r e i n t e r p o l a t e d contours between w i d e l y separated p o i n t s  26  27  4-2  Beam i n t e n s i t y p r o f i l e f o r u n d e f l e c t e d narrow beam.  28  4-3  R e l a t i v e change o f t h e beam i n t e n s i t y p a t t e r n f o r a narrow beam a t resonance. The u n d e f l e c t e d beam p r o f i l e i s shown i n F i g u r e 4-2  29  P l o t o f the beam i n t e n s i t y near t h e beam c e n t e r as a f u n c t i o n o f f i e l d B f o r f i x e d frequency and d i f f e r e n t amplitude o? the r - f f i e l d and i t s gradient  30  (a)  33  4- 4  5- 1  A h e l i c a l quadrupole wire system  -  V  -  Figure  Page (b)  5-2  5-3  An i r o n core quadrupole, showing the e l e c t r o d e s n e c e s s a r y to produce an e l e c t r i c f i e l d o r t h o g o n a l to the magnetic f i e l d (a)  (b)  7-1  A quadrupole wire system showing the e l e c t r o d e s n e c e s s a r y t o produce an e l e c t r i c f i e l d o r t h o g o n a l to the magnetic f i e l d  I l l u s t r a t i o n o f the f r e q u e n c y v e c t o r s i n the r o t a t i n g c o - o r d i n a t e system (x, y, 1). B^(r) i s i n the x,y p l a n e and makes an angle -<j> with r e s p e c t t o the x a x i s . Jg Is- the component of J a l o n g the e f f e c t i v e f i e l d An i l l u s t r a t i o n o f the v e c t o r Jg s i n 0 i n the (x, y, z) c o - o r d i n a t e system at the time t ...  ^3  36  40 40  A p l o t o f t r a j e c t o r i e s from e q u a t i o n (7-6) . The d o t t e d t r a j e c t o r i e s are those o b t a i n e d when the adiab'atic c o n d i t i o n i s s a t i s f i e d as the p a r t i c l e c r o s s e s the a x i s . The s o l i d t r a j e c t o r i e s are those o b t a i n e d when the s i g n o f Jg changes as the p a r t i c l e c r o s s e s the a x i s  63  7-2  T r a j e c t o r i e s s i m i l a r t o those o f F i g . 7-1  63  7-3  Trajectories  s i m i l a r t o those o f F i g . 7-1  64  7-4  T r a j e c t o r i e s s i m i l a r to those o f F i g . 7-1  65  7-5  An i l l u s t r a t i o n o f a system o f stops which would g i v e 100% p o l a r i z a t i o n o f the emerging beam  66  A somewhat s i m p l e r system to t h a t i l l u s t r a t e d i n F i g . 7-5 which would g i v e p a r t i a l p o l a r i z a t i o n o f the i n n e r beam  66  A c h a r t o f the r e g i o n s where the r o o t s Rj and R 2 are r e a l o r complex. The l i g h t r e g i o n s i n d i c a t e t h a t both are r e a l , and the shaded r e g i o n s i n d i c a t e t h a t one or both are complex  71  I l l u s t r a t i o n o f the dimensions and a h e l i c a l s t r i p  of a h e l i c a l  88  I l l u s t r a t i o n o f the dimensions w i r e h e l i c a l winding  of a rectangular  7- 6  8- 1  9- 1  9-2  9-3  line  P l o t o f minimum power v e r s u s wire r a d i u s f o r a r e c t a n g u l a r copper wire h e l i c a l winding at room temperature, from e q u a t i o n (9-11)  88  91  - vi -  Figure A-l  Page I l l u s t r a t i o n o f the o r i e n t a t i o n o f the B,(f) = -G(y+ix) at the p o i n t  field  r = x+iy by e q u a t i o n  102  A-2  P l o t o f the f a c t o r C(q) d e f i n e d  (a-8)..  C-l  Diagram o f the i o n gun d e s c r i b e d  C-2  T y p i c a l performance o f the i o n gun d e s c r i b e d i n appendix C. V i s the p o t e n t i a l o f the i o n i z i n g chamber  i n appendix C  104 118  119  - v i i-  ACKNOWLEDGEMENT  I t i s a p l e a s u r e t o acknowledge the v e r y g r e a t a s s i s t a n c e and enthusiasm  o f P r o f e s s o r Myer Bloom, who has guided t h i s p r o j e c t t o  i t s present stage.  S p e c i a l thanks a r e due him f o r h i s h e l p d u r i n g  the w r i t i n g o f t h i s  dissertation.  I s i n c e r e l y thank Dr. H i n Lew, who c o n t r i b u t e d much o f the d e s i g n for the n e u t r a l p a r t i c l e experiment, and who was most h e l p f u l w i t h a d v i c e on many experimental The  experimental  matters.  work r e c e i v e d e s s e n t i a l support  P h y s i c s machine shop, e s p e c i a l l y from Doug  from the  Stonebridge.  I wish t o thank P e t e r F l a i n e k f o r h i s t e c h n i c a l a s s i s t a n c e , and t h e members o f the D i v i s i o n o f Pure P h y s i c s , N a t i o n a l Research C o u n c i l , f o r t h e i r h e l p w i t h the p o r t i o n o f the experiment which was performed t h e r e . S p e c i a l thanks a r e due my w i f e , S y l v i a , f o r h e r support t h i s work. T h i s r e s e a r c h was supported Research C o u n c i l o f Canada.  by grants from t h e N a t i o n a l  during  CHAPTER 1 Introduction  In t h e c o n v e n t i o n a l S t e r n - G e r l a c h experiment a beam o f atoms having magnetic moment u i s passed geneous magnetic f i e l d  B(f).  through  a time-independent  inhomo-  For s i m p l i c i t y , suppose t h a t u = yYiJ,  where Jfi i s the a n g u l a r momentum and y i s t h e gyromagnetic r a t i o , and t h a t B(r) c o n s i s t s o f a l a r g e homogeneous p a r t B z a x i s and an inhomogeneous p a r t B^(?) with  |  q  o r i e n t e d along t h e  (?) | << | B* | .  The atoms  a c q u i r e momentum w h i l e i n t h e r e g i o n o f the inhomogeneous f i e l d  because  of  about  t h e S t e r n - G e r l a c h f o r c e F = (u . V ) B ( f ) .  the z a x i s a t t h e Larmor frequency  CJ = -yH > t h e o n l y q  ent c o n t r i b u t i o n t o F a r i s e s from J . momentum f o r t >> u important  Since J precesses  time-independ-  T h e r e f o r e , t h e n e t change i n  ^ i s proportional to J .  One o f t h e most  and best-known r e s u l t s i n modern p h y s i c s i s t h a t J_ i s found  e x p e r i m e n t a l l y t o take on o n l y the d i s c r e t e v a l u e s M = - J , - J + 1, ,+ J .  We can say t h a t t h e S t e r n - G e r l a c h experiment p r o v i d e s us with a  method f o r p r e p a r i n g a s p i n system i n any one o f t h e d i s c r e t e quantum s t a t e s M. A few year ago i t was shown t h e o r e t i c a l l y  (Bloom and Erdman 1962)  .that a more g e n e r a l form o f t h e S t e r n - G e r l a c h experiment may be d e f i n e d , u s i n g time-dependent inhomogeneous magnetic f i e l d s , i . e .  - 2 -  B(f,t) = B  This generalized  d-l)  + B ^ r ) c o s wt  Q  f o r m o f t h e S t e r n - G e r l a c h e x p e r i m e n t was named t h e  "Transverse S t e r n - G e r l a c h " experiment  (TSG) b e c a u s e  i t was p r e d i c t e d  t h a t f o r (JJ = C J , 2 J + 1 d e f l e c t e d beams s h o u l d a l s o b e o b s e r v e d , t h e q u a n t u m o  number M' = - J , - J + 1, ... + J a s s o c i a t e d w i t h e a c h o f t h e s e d e f l e c t e d beams i n t h e TSG e x p e r i m e n t b e i n g a s s o c i a t e d w i t h J^, J aj  the projection of  along;„ the x a x i s o f a c o o r d i n a t e s y s t e m r o t a t i n g w i t h a n g u l a r v e l o c i t y ;  Q  about  the z axis.  This arises  inhomogeneous m a g n e t i c rotating  fields,  at.resonance. to  field  from t h e f a c t  that, i fthe o s c i l l a t i n g  at a given position i s s p l i t  i n t o two  one o f t h e m i s s y n c h r o n o u s  with the precessing J vector  Thus, t h e c u m u l a t i v e changes  i n momentum a r e p r o p o r t i o n a l  J . z In  this  t h e s i s we d e s c r i b e t w o e x p e r i m e n t s .  One i s t h e s u c c e s s f u l  o b s e r v a t i o n o f t h e r e s o n a n t d e f l e c t i o n o f a beam o f n e u t r a l  potassium  atoms a t a f r e q u e n c y o f 7.2 M h z , i n a g r e e m e n t w i t h t h e p r e d i c t i o n s o f the  t h e o r y o f t h e TSG e x p e r i m e n t .  The r e s u l t s  of this  experiment  have  a l r e a d y been p u b l i s h e d  ( B l o o m , E n g a , Lew 1 9 6 7 ) .  for  S t e r n - G e r l a c h e x p e r i m e n t , w h i c h i s based on an  a charged p a r t i c l e  The o t h e r i s a p r o p o s a l  e x t e n s i o n o f t h e TSG e x p e r i m e n t t o t i m e - i n d e p e n d e n t , inhomogeneous f i e l d s have t h e form  B(r,z) = B k + o  If the f i e l d  B (r) i s well  B , ( f ) e iwz 1^ J  chosen, the charged p a r t i c l e  (1-2)  trajectories  - 3 -  w i l l be c o n f i n e d i n a s t a b l e beam by the r e s u l t i n g L o r e n t z f o r c e s f o r motion g e n e r a l l y a l o n g the z a x i s .  T h i s i s , i n f a c t , the p r i n c i p l e o f  s t r o n g f o c u s i n g which i s now w i d e l y used i n a c c e l e r a t o r d e s i g n .  But  i n such a system i t i s a l s o p o s s i b l e t o s a t i s f y the c r i t e r i o n f o r a TSG  experiment, s i n c e i n a frame o f r e f e r e n c e moving w i t h the p a r t i c l e  i n the z d i r e c t i o n , the f i e l d the TSG experiment,  B^(r) i s r o t a t i n g i n time.  In analogy t o  i t appears now t o be p o s s i b l e t o observe the 2J + 1  beams i n such an experiment. The c l a s s i c a l t h e o r y o f the n e u t r a l p a r t i c l e TSG experiment i s p r e s e n t e d i n Chapter 2.  The e x p e r i m e n t a l procedure and the apparatus  are d e s c r i b e d i n Chapter 3, w h i l e t h e e x p e r i m e n t a l measurements and t h e i r i n t e r p r e t a t i o n a r e g i v e n i n Chapter 4. We develop t h e e x t e n s i o n o f the TSG experiment space v a r y i n g f i e l d s  t o time  independent,  i n Chapter 5 and propose a S t e r n - G e r l a c h experiment  f o r charged p a r t i c l e s which i s developed i n Chapters 6 through 9.  In  Chapter 6 we c o n s i d e r the beam b e h a v i o r f o r the L o r e n t z f o r c e s o n l y . In Chapter 7 we i n t r o d u c e t h e S t e r n - G e r l a c h f o r c e i n the l i m i t  t h a t the  L o r e n t z f o r c e i s comparible t o , o r weaker than, t h e S t e r n - G e r l a c h f o r c e , and examine t h e beam t r a j e c t o r i e s .  The case when the L o r e n t z f o r c e i s  l a r g e r than the S t e r n - G e r l a c h f o r c e i s t r e a t e d i n Chapter 8. Chapter 9 t h e e x p e r i m e n t a l problems o f t h i s experiment  In  a r e examined and  the r e s u l t s o f some p r e l i m i n a r y e x p e r i m e n t a l work d e v e l o p i n g a s u i t a b l e i o n beam a r e p r e s e n t e d .  CHAPTER 2 C l a s s i c a l Theory o f the N e u t r a l P a r t i c l e  Transverse  S t e r n - G e r l a c h Experiment  A n e u t r a l atom o f mass m and magnetic moment IT h a v i n g  initial  p o s i t i o n r ( 0 ) and momentum p(0) i s assumed t o i n t e r a c t with t h e magnetic f i e l d  B ( r , t ) g i v e n by e q u a t i o n  (1-1) .  The p o s i t i o n and  momentum o f t h e atom a t any time t a r e g i v e n by  r(t) = f(0) +  1  p(f)dt'  (2-1)  o  p ( t ) = p(0) + f 'o  where the i n s t a n t a n e o u s  F(f)dt'  (2-2)  S t e r n - G e r l a c h f o r c e F ( t ) i s g i v e n by  F ( t ) = Y ^ ( " ( t ) • V ) B ( f ( t ) ,t)  (2-3)  w h i l e the time dependence o f J ( t ) i s governed by the equations  d J / d t = yJ x B ( f ( t ) , t )  S o l u t i o n o f equations  (1-1), (2-1)-(2-4)  J ( t ) as a f u n c t i o n o f r ( 0 ) , p ( 0 ) , t and J ( 0 ) .  (2-4)  g i v e s r ( t ) , p ( t ) , and In many c a s e s , i f t i s  - 5 made s u f f i c i e n t l y  l a r g e , t h e r e i s a c o r r e l a t i o n between r ( t ) and J ( t ) ,  so t h a t measurement o f the i n t e n s i t y d i s t r i b u t i o n i n the beam one to  t o draw c o n c l u s i o n s about t h e allowed v a l u e s s o l v e equations  (1-1),  ( 2 r l ) - ( 2 - 4 )  f o r a general  e s s e n t i a l features o f the Stern-Gerlach the "impulse approximation", for  t h e experimental  of J.  enables  It i s difficult  B(f,t).  The  experiment can be found by u s i n g  which, i n f a c t , i s a v e r y good a p p r o x i m a t i o n  arrangement used here and d e s c r i b e d i n t h e next  chapter.  Impulse Approximation r ( t ) = r ( 0 ) = r = c o n s t a n t  i n the r e g i o n  B (f)^0 1  In t h i s a p p r o x i m a t i o n t h e change i n momentum Ap due t o the S t e r n Gerlach  f o r c e i s c a l c u l a t e d , assuming t h a t t h e displacement  i n t h e r e g i o n o f t h e inhomogeneous f i e l d  i s negligible.  o f t h e atom  T h i s change i n  momentum can be measured by a l l o w i n g t h e atoms t o undergo f r e e for  flight  a time T between t h e inhomogeneous f i e l d r e g i o n and the d e t e c t o r ,  so t h a t t h e displacement  due t o t h e S t e r n - G e r l a c h  force i s related to  the momentum change by  Ar  =  (T/m) Ap  ( ? - ^  Now, i f one c a l c u l a t e s any component F ^ ( t ) o f t h e S t e r n - G e r l a c h f o r c e , u s i n g equations  (1-1),  ( 2 - 3 ) and ( 2 - 4 ) one o b t a i n s a p l o t  such  as t h a t shown i n F i g . 2-1 i n which F ^ ( t ) i s made up o f a constant p l u s o t h e r terms which o s c i l l a t e s i n u s o i d a l l y with time. t h a t a r e long compared w i t h t h e p e r i o d o f o s c i l l a t i o n T Ap^(t) i s p r o p o r t i o n a l t o t h e time-averaged v a l u e constant  term, i . e .  q  term  For times of F^(t),  o f F^, which i s t h e  -  6 -  L.  >  "time F i g u r e 2-1.  Schematic  p l o t o f a component o f the S t e r n - G e r l a c h f o r c e  v e r s u s time i n the impulse  F i g u r e 2-2.  approximation.  Representation of the r f f i e l d system  (x,7,z).  i n the r o t a t i n g c o o r d i n a t e  I t has been assumed t h a t B ^ ( r ) i s i n the  x,y p l a n e and makes an angle -<j> with r e s p e c t t o the x - a x i s . i s t h e component o f J along t h e e f f e c t i v e f i e l d  a t t = 0.  - 7 -  (Ap(t))  t  >  >  »  t  t<F(t)>  (2-6)  o  The  component o f J which i s q u a n t i z e d i n each o f t h e 2J + 1  d e f l e c t e d beams i s found by examining t h e dependence o f <F(t)>  on J .  For t h e c o n v e n t i o n a l S t e r n - G e r l a c h experiment, OJ = 0 and i t i s obvious from equations  (1-1), (2-3), and (2-4) t h a t  3.B, (r) (Ap(t))  so t h a t J  i s quantized.  z  t  >  y  t  ,  t ^ J  (2-7)  z  F o r w ^ 0, t h i s i s no longer t r u e , as w i l l be  seen below. We f i r s t w r i t e down e q u a t i o n  (2-4) i n t h e impulse  approximation  a f t e r making t h e t r a n s f o r m a t i o n  J  J> x  Jy> and J  z  ±  = J  +i/iit ± iJ = J e y ±  x  (2-8) •  a r e the components o f J i n a c o o r d i n a t e system  rotating  w i t h a n g u l a r v e l o c i t y At about the z a x i s .  dJ /dt +  dJ /dt z  where  = ;i(AojJ  +  + l / 2 w J ) ±i(w Cos o j t J  = -l/4i(J a) +  1 +  1  z  lz  - j'_a) ) - l / 4 i ( J c j _ e + 1  +  1  +  - l/2w |  2 l u t  +  u  1  - J*_  w 1 +  e"  )  2 l W t  (2-9)  )  (2-10)  - 8-  =  co  = - B (f)  1  It  ,  AOJ  (JJ-CU  Y  o  1  a) = - Y B o 'o  3  ai  1 ±  (2-11) ^ 1  = u  l x  ±  ico  (2-12)  l y  i s w e l l known (Winter 1955) t h a t t h e term i n v o l v i n g a)^ cos cot z  g i v e s r i s e t o resonance e f f e c t s at io = w /2n, w h i l e t h e terms i n u, + ^ o 1±  l a ) t  e  g i v e resonances at u / ( 2 n + l ) , where n i s an i n t e g e r .  These m u l t i p l e  o  quantum e f f e c t s a r e important when t h e i n e q u a l i t y fied.  << B  q  i s not s a t i s -  As we s h a l l demonstrate i n Chapter 4, a l a r g e resonance i s indeed  observed i n t h e TSG experiment near CJ = CO /2 f o r l a r g e v a l u e s q  Replacement o f t h e L i n e a r O s c i l l a t i n g We now assume t h a t t h e v a l u e  F i e l d by a R o t a t i n g  of B ^ .  Field  o f B ^ i s s u f f i c i e n t l y small  that the  i n f l u e n c e o f t h e terms which a r e e x p l i c i t l y time dependent i n equations (2-9)  and (2-10) i s n e g l i g i b l e , so t h a t t h e s e terms may be dropped.  T h i s corresponds t o assuming t h a t field and  B ( r ) cos tot by ( 1 / 2 ) B ( r ) e  = 0 and t o r e p l a c i n g t h e magnetic 1 ( 0 t  ,  i . e . , one o f amplitude  r o t a t i n g w i t h a n g u l a r v e l o c i t y co about t h e z a x i s .  linear oscillating  field  i n t h e same sense as t h e s p i n B-^ (see, f o r example,  i s e f f e c t i v e near resonance f o r s m a l l  Ramsey 1963, p. 146). vector J = (J ,Jy,J_) x  Of c o u r s e , a  i s decomposed i n t o two o p p o s i t e l y r o t a t i n g  f i e l d s , but o n l y t h e one t h a t r o t a t e s precession  (1/2)B(r)  I t i s w e l l known t h a t under t h e s e c o n d i t i o n s the p r e c e s s e s about t h e e f f e c t i v e f i e l d  a t an a n g u l a r  frequency  u  = [(Aco)  2  + 1/4  i ^  2  ]  1  /  2  (2-13)  9 -  a? is oriented at an angle © w i t h respect to the z axis, as illustrated in Fig. 2-2 for an r - f . field oriented in the x-y plane in the laboratory frame at an angle the position r .  -<}> with respect to the x axis at  Representing this by a complex number,  V  r  )  =  B  lx  ( f )  +  i B  ly  C ? )  '  ( 2 _ 1 4  )  = B^rje"^  the angle  6 i s g i v e n by  — - — = - T o  t a n 6 =. 1/2  We s p e c i f y t h e i n i t i a l  ~  c o n d i t i o n s f o r J i n terms o f a c o o r d i n a t e j  system f i x e d i n t h e r o t a t i n g frame.  i s t h e component  along t h e e f f e c t i v e f i e l d , which has p o l a r angles Fig.  2-2.  angles  (1/2TT  these  initial  and  (2-15)  a r e the components  o f J at t = 0  (6, -cb) as shown i n  along t h e axes having  polar  + 0, -<£) and (1/2TT, 1/2'TT - <j>) , r e s p e c t i v e l y .  In terms o f  c o n d i t i o n s , t h e s o l u t i o n s t o equations  ( 2 9 ) , and  (2-8),  r  (2-10) a r e as f o l l o w s :  J (t) +  = J _ * ( t ) = {l/2(l+cos6) ( J + i J ) e x p [ i ( c o + u ) t ] + l / 2 ( l - c o s 0 ) 1  2  e  ( J ^ - i ^ ^ x p t i f i o - i o ^ t J + J g S i n O e x p f i w t ) } exp(-i<j>),  J  z  (2-16)  (t)= - l / 2 s i n 0 [ ( J + i J ) e x p ( i w t ) + ( J - i J ) e x p ( - i c o t ) ] + J c o s 6 1  2  e  1  2  e  3  (2-17)  -  Using equations  10  and  (2-3)  -  (1-1),  the Stern-Gerlach force may  be  written  9B  F(t)  (r) - 3 ^ — + JJt)  (?)  3B  = v*[J (t) ~ ± r -  * J (t)  x  y  8B  (?)  Jcos.t (2-18)  Special Case In the experiment to be described i n the next two chapters the o s c i l l a t i n g f i e l d was produced by a four-wire system, as shown i n Fig.  2 - 3 . Near the center of symmetry of the four wires  (r << R ) , the  f i e l d i s well approximated by  B^r)  = G(x-iy) = Gr*  (2-19)  where G i s the gradient of the f i e l d having dimensions gauss/cm and i s given by  G  =  0.8  —  (2-20)  R  o where I i s the current i n each wire i n amperes and R  q  i s the distance i n  centimeters from the center of symmetry to the center of each wire.  The  orientation of t h i s f i e l d f o r a p o s i t i o n r = x+iy i s shown i n F i g . 2 - 3 . Using eqs.  (2-18)  and  (2-19),  F(t)  i t i s seen that  = YfiGJ.(t)costot  (2-21)  - 11 -  - 12 -  From equations  (2-16) and (2-21), we see t h a t F ( t ) c o n s i s t s o f terms which  o s c i l l a t e a t f r e q u e n c i e s OJ , 2to, and 2oo ± UJ . time-independent  In a d d i t i o n , t h e r e i s one  term which g i v e s t h e time-averaged  <F(t)>  = 1/2 yfiGJ  f o r c e t o be  sine e *  (2-22)  1  3  T h i s e x p r e s s i o n c o n t a i n s the f o l l o w i n g i n f o r m a t i o n . (a)  The time-averaged  f o r c e i s p r o p o r t i o n a l t o the v a l u e o f t h e compon-  ent o f J a l o n g the e f f e c t i v e f i e l d  i n the r o t a t i n g frame.  It i s this  component o f J which i s q u a n t i z e d i n the g e n e r a l i z e d S t e r n - G e r l a c h experiment on n e u t r a l atoms. is (b)  independent o f time  t h a t t h i s component o f J  i n the inhomogeneous f i e l d  region.  The change o f momentum Ap i s i n the r a d i a l d i r e c t i o n i n the x-y  plane.  Since  rings.  The same e f f e c t  i s q u a n t i z e d , a c i r c u l a r beam i s decomposed i n t o 2J + 1 i s obtained f o r the conventional  experiment f o r t h i s geometry (c)  I t may be noted  (Beenewitz and Paul  The dependence o f Ap on frequency  Stern-Gerlach  1954).  i s c o n t a i n e d i n the f a c t o r  sine = 2[(Aco) T h e r e f o r e , when t h e r o t a t i n g f i e l d  2  2 + W l  /4]  1 / 2  approximation  i n momentum i s maximum a t t h e Larmor frequency of for  Aoo.  I t may be noted  i s v a l i d , the change  and i s an even f u n c t i o n  t h a t the c o n v e n t i o n a l S t e r n - G e r l a c h  experiment  t h i s geometry ( B = 0 = OJ) i s a l s o d e s c r i b e d by the t h e o r y . Q  In a d d i t i o n , one can say from the argument p r e s e n t e d equation  (2-21) t o e q u a t i o n  i n going  (2-22) t h a t , i n o r d e r t o e s t a b l i s h  •  from  - 13 -  q u a n t i z a t i o n o f J along the e f f e c t i v e f i e l d ,  the time spent by the atoms  i n the inhomogeneous f i e l d r e g i o n must be much g r e a t e r than t h e p e r i o d of  p r e c e s s i o n i n the e f f e c t i v e f i e l d ,  i . e . , t >>  ^.  F i n a l l y , t h e r e s u l t s g i v e n here may be compared w i t h the quantum mechanical  c a l c u l a t i o n o f Bloom and Erdman (1962) f o r J = 1/2.  potassium atom h a v i n g the average v e l o c i t y i n a t y p i c a l beam has a de B r o g l i e wavelength  much l e s s than 10  system used i n t h i s experiment  _g  A  experiment  cm, w h i l e t h e c o l l i m a t i o n  (and o t h e r t y p i c a l beam  experiments)  _3 l o c a l i z e s the atoms t o not l e s s than 5 x 10  cm.  Thus, quantum mechanics  i s o n l y needed t o d e s c r i b e the angular momentum p r o p e r t i e s o f atoms. When the s p i n f u n c t i o n s a s s o c i a t e d w i t h d i f f e r e n t ( i n momentum space) i n the quantum mechanical Erdman 1962, e q u a t i o n s  l o c a l i z e d wave packets  calculation  (Bloom and  (28)-(36)) a r e examined f o r a r b i t r a r y w, i t i s  seen t h a t they correspond t o s p i n s q u a n t i z e d a l o n g t h e e f f e c t i v e For  J = 1/2, t h e r e a r e o n l y two such s p i n s t a t e s . However, a t the low  f i e l d s a t which the p r e s e n t experiment  was done, the n u c l e a r s p i n , I =  3/2 i s s t r o n g l y coupled t o the e l e c t r o n s p i n J t o form a t o t a l momentum F = 1,2.  i n weak e x t e r n a l f i e l d s , these s p l i t  components w i t h f i v e d i s t i n c t 1, 0, -1, -2, where y of  field.  the  2 S.. ,  ground  O  e f f e c t i v e magnetic  i n t o e i g h t Zeeman  moments M g y /4,M  i s the Bohr magneton and g  s t a t e o f potassium.  angular  J  r  J O  = 2, r  i s the Lande g f a c t o r  CHAPTER 3 E x p e r i m e n t a l Procedure  The main aim o f t h i s experiment was  t o d e t e c t the p r e d i c t e d  trans-  v e r s e S t e r n - G e r l a c h resonance and t o check the t h e o r y o f c h a p t e r 2.  Since  measurement o f the resonance frequency t o an a c c u r a c y o f a few p e r c e n t was  sufficient  f o r t h i s purpose, i t was p o s s i b l e t o keep the d e s i g n o f the  equipment r e l a t i v e l y s i m p l e .  A schematic diagram o f the o v e r a l l  system  i s shown i n F i g . 3-1. A p o t a s s i u m beam was produce and t o d e t e c t .  chosen s i n c e potassium atoms are easy both t o  The beam was produced by h e a t i n g potassium metal  t o 300°C i n an i r o n oven, w i t h a h o l e o f diameter 0.005 cm t o form the beam.  T h i s temperature corresponds t o a vapor p r e s s u r e f o r potassium  o f about 1 T o r r .  A single f i l l i n g  o f the oven was  f o r 60 hours o f continuous o p e r a t i o n .  s u f f i c i e n t to l a s t  Micrometer adjustments were  p r o v i d e d t o p o s i t i o n the oven i n the p l a n e t r a n s v e r s e t o the beam. The c o l l i m a t i o n was  done w i t h a h o l e p l a c e d 31.4  cm from the oven.  Any one o f t h r e e d i f f e r e n t s i z e d h o l e s o f diameter 0.17 0.005 cm,  cm,  0.0125  cm,  r e s p e c t i v e l y , c o u l d be brought i n t o p o s i t i o n by admustment o f  a s i n g l e micrometer. The c e n t e r o f the d e f l e c t i o n system, which was was p l a c e d 15.6 was  10 cm i n l e n g t h ,  cm from the c o l l i m a t i n g h o l e , and the d e t e c t o r opening  196 cm from the c o l l i m a t i n g h o l e .  The  long f r e e - f l i g h t r e g i o n o f  136 Detector aperture  Free Flujkt Region.  Solenoid Magnet  Quadrupole Were  Electron Muliiplier  Deflector Tootked  Pre- Amp nal  F i g u r e 3-1.  Chopper Pittances in Cffn-hmetfers Hot t« Sc«l« Schematic diagram o f the o v e r a l l system  (not t o s c a l e ) .  about  180 cm which t h i s p r o v i d e d had the advantage  o f increasing the  s i z e o f t h e d e f l e c t i o n s and c o r r e s p o n d i n g l y r e d u c i n g the t o l e r a n c e s n e c e s s a r y f o r the d e t e c t o r p o s i t i o n i n g and opening. The d e t e c t o r was a tungsten h o t - w i r e i o n i z e r .  The ions were  c o l l e c t e d on an e l e c t r o n m u l t i p l i e r a f t e r crude mass a n a l y s i s . spectrometer was r e a l l y not n e c e s s a r y but was used because i n t e g r a l p a r t o f the i o n i z e r which was a l r e a d y on hand.  The mass  i t was an  The d e t e c t o r  opening was r e c t a n g u l a r and was formed by f o u r k n i f e - e d g e d jaws which were each p o s i t i o n e d by a micrometer.  The h o t wire was 0.018 cm i n  diameter and was p l a c e d 25 cm from the d e t e c t o r opening. The beam was chopped a t 30 c.p.s. w i t h a toothed wheel d r i v e n by a synchronous  motor.  The output o f the e l e c t r o n m u l t i p l i e r was a m p l i f i e d  w i t h a narrow-band 30 c.p.s. p r e a m p l i f i e r , w i t h a g a i n o f 100, and then fed  i n t o a commercial  lock-in amplifier  (Princeton Applied  Research  Model J B 4 ) . With t h e K oven a t about 300°C a beam c o l l i m a t e d w i t h t h e 0.005 cm _9 diameter h o l e gave a maximum output c u r r e n t o f 10 multiplier.  A from t h e e l e c t r o n  With an e s t i m a t e d g a i n o f 10^ f o r t h e m u l t i p l i e r ,  this  meant t h a t the i n c i d e n t K beam corresponded, a f t e r i o n i z a t i o n , t o an -14 ion  c u r r e n t o f about  Commercial  10  A ( n e g l e c t i n g l o s s e s i n the mass s p e c t r o m e t e r ) .  t u n g s t e n w i r e , even t h e s o - c a l l e d undoped k i n d , c o n t a i n s l a r g e  amounts o f K i m p u r i t y and an i n i t i a l the  trial  showed t h a t the n o i s e from  w i r e would swamp any expected resonance s i g n a l w i t h o b s e r v a t i o n time  c o n s t a n t s o f a few seconds.  Hence i t was n e c e s s a r y t o go t o a potassium-  f r e e wire grown from W(C0)^, a c c o r d i n g t o the r e c i p e - d e s c r i b e d by Greene (1961).  With such a wire and t h e maximum u s a b l e oven temperature  o f 300°C  - 17 -  we  o b t a i n e d a s i g n a l - t o - n o i s e r a t i o f o r the d i r e c t beam o f 30 t o 1 w i t h  a 3-second time c o n s t a n t on the l o c k - i n The vacuum system was stainless steel.  b u i l t p a r t l y o f b r a s s and p a r t l y of nonmagnetic  Three o i l d i f f u s i o n pumps w i t h l i q u i d n i t r o g e n c o o l e d  b a f f l e s were used,  one  f r e e - f l i g h t region. by v a l v e s .  amplifier.  i n each o f the two  The  end chambers and one  i n the  t h r e e r e g i o n s c o u l d be i s o l a t e d from each o t h e r  A p r e s s u r e o f 3 x 10  _7  T o r r was  o b t a i n e d , except  i n the oven  -7 r e g i o n , where the p r e s s u r e was  8 x 10  Torr.  The d e f l e c t i o n system c o n s i s t e d o f a s o l e n o i d and a w i r e system, as i l l u s t r a t e d  i n F i g . 3-2.  quadrupole  Four push rods at each end  the s o l e n o i d allowed a t h r e e d i m e n s i o n a l p o s i t i o n i n g o f the assembly w i t h i n the vacuum The  s o l e n o i d had  1 p a r t i n 600  entire  envelope.  overwound ends and gave a measured homogeneity o f  over the 10-cm  center region.  The method used t o match the v e r y low r e s i s t a n c e o f the wires t o the nominal quadrupole 3-3.  The  of  50-ohm output o f the r - f . source was  quadrupole  t o make the  wires p a r t o f a p a r a l l e l resonant c i r c u i t , as shown i n F i g . c i r c u i t used had an impedance o f 120  quency o f 7.22  fre-  Mhz.  The maximum c u r r e n t used corresponds  ohms at a resonant  i n the wires was  11.3  t o a d r i v i n g v o l t a g e a c r o s s the c i r c u i t  a t o t a l power d i s s i p a t i o n o f 35 W i n the c i r c u i t .  A  (peak).  This  o f 65 V r.m.s., and To absorb  this  heat,  the wires were cemented w i t h epoxy i n t o a c c u r a t e l y machined b r a s s b l o c k s , and the s i l v e r e d mica c a p a c i t o r was c o n t a c t w i t h the r e s t o f the system. water-cooled  copper  g l u e d to a copper p l a t e i n thermal T h i s assembly was  tube on which the s o l e n o i d was  a l l mounted i n a  wound.  Solenoid cjuadrvpale  w i r e  tf  -toning co i l  2  7 10 cm  20 cm  If c o p p e r e<fOall<j diq.  F i g u r e 3-2.  Drawing omitted.  u M v e S (*M€>ftW<?)  s p a c e d ow a  circle  o f t h e d e f l e c t i o n system w i t h c o n s t r u c t i o n  details  2 m*r\  *  18 fcWG copper w t r c s e a c U <j(i/ed -to b r a s s  [O.S c r * "tolal lervej-fk backing  blocks  o lnpo"t  rf  -tuning coi I  tao orm5  # 8 A W G copper  ai 7.2  2-tonus, IcmdiV*,. KO i  \ Silver wuca, leads  F i g u r e 3-3.  O.tcw IOA^  E l e c t r i c a l c i r c u i t o f t h e quadrupole w i r e s showing d e t a i l s o f t h e p a r a l l e l r e s o n a n t c i r c u i t used.  - 20 -  The  brass backing  b l o c k s produce image c u r r e n t s which a c t as a second  quadrupole system with twice the ^ r e l a t i v e s p a c i n g o f the w i r e s . image c u r r e n t s reduce the f i e l d The maximum wire  c u r r e n t s used  o f approximately  680  g r a d i e n t due (11.3  A  These  to the wires by about  25%.  (peak)) corresponded to a g r a d i e n t  gauss/cm, as c a l c u l a t e d by u s i n g e q u a t i o n  (21), t a k i n g  the image c u r r e n t s i n t o account. To  f a c i l i t a t e the alignment o f the quadrupole wires  the beam, two  holes  0.073 cm  i n diameter, one  with r e s p e c t  at each end  to  o f the s o l e n o i d ,  were f i x e d r e l a t i v e l y a c c u r a t e l y on the a x i s o f the quadrupole system. These h o l e s c o u l d then be  l o c a t e d and  the d e f l e c t i o n system  centered,  u s i n g the beam i n t e n s i t y as the i n d i c a t o r .  A crystal controlled oscillator kilowatt  ( H e a t h k i t SB-400) d r i v i n g a  l i n e a r a m p l i f i e r (Heathkit SB-200) was  one-  used f o r the r - f . power  source. The  d e f l e c t i o n of an atom having  i n i t i a l velocity v parallel  to the  -2 z a x i s i s Cv  , where the constant  resonance, put The  sin 6=1)  and  C i s obtained  equation  from e q u a t i o n  (19.3) o f Kusch and  average d e f l e c t i o n i s found by f i r s t  showing t h a t  (v  u s i n g the v e l o c i t y d i s t r i b u t i o n f u n c t i o n of the beam.  _2  (2-22)(at  Hughes  (1959).  ) = m/(2kT),  For our  system T =  39 573°K.  For  K i n a small f i e l d  B  i t i s a p p r o p r i a t e t o c a l c u l a t e the  d e f l e c t i o n o f an atom i n a s p i n s t a t e Mp weights o f the s t a t e s M  = 2, 1, 0,  -1,  = 1, s i n c e the  statistical  -2 are 1, 2, 2, 2,  1, r e s p e c t i v e l y .  r  In order to compare w i t h experiment, we  g i v e here the r e s u l t  f o r the average d e f l e c t i o n f o r a g r a d i e n t o f 430  gauss/cm.  o f 0.011  cm  CHAPTER 4 Experimental  The  Results  t h e o r e t i c a l p r e d i c t i o n s which have been v e r i f i e d a r e c o n t a i n e d  i n eq. (2-22). immediately  The s i g n i f i c a n c e o f t h i s e q u a t i o n i s d i s c u s s e d  f o l l o w i n g i t i n Chapter  2.  To v e r i f y t h e e x i s t e n c e o f t h e r a d i a l f o r c e f i e l d ,  a large  diameter  beam c o l l i m a t e d o n l y by t h e 0.073-cm h o l e i n the d e f l e c t i o n system was produced. of  T h i s beam was o f e s s e n t i a l l y u n i f o r m  0.3 cm a t the d e t e c t o r .  S i n c e the expected  i n t e n s i t y over a diameter most p r o b a b l e  deflection  was l e s s than 0.03 cm, no l a r g e change would occur i n the i n t e n s i t y p r o f i l e except  near t h e p o i n t c o r r e s p o n d i n g  t o the o r i g i n o f t h e r a d i a l  force f i e l d .  Here an i n c r e a s e i n i n t e n s i t y should occur w i t h a r a d i u s  approximately  equal t o the average d e f l e c t i o n , r e s u l t i n g i n a s o r t o f  "pip"  a t the c e n t e r o f a u n i f o r m  field.  T h i s i s due t o those atoms  which a r e f o r c e d inwards and converge on the c e n t e r a r e a . effect  i s a consequence o f any d e f l e c t i o n system t h a t produces a  constant r a d i a l f o r c e f i e l d ,  and i s i d e n t i c a l , f o r example, with t h a t  o b t a i n e d i n a c o n v e n t i o n a l quadrupole f o c u s i n g experiment and  This focusing  Paul  1954) .  (Bennewitz  T h i s " p i p " o r focused spot was e x p l o r e d i n c o n s i d e r a b l e  d e t a i l by moving the d e t e c t o r a p e r t u r e , p r e v i o u s l y d e s c r i b e d , over t h e area o f the p i p . The  The a p e r t u r e dimensions were s e t a t 0.010 x 0.010 cm.  r e s u l t s a r e shown i n F i g . 4 - l a . In F i g . 4 - l a each curve r e p r e s e n t s a s t e p - b y - s t e p  scan o f the beam  - 22 -  field  i n the v e r t i c a l d i r e c t i o n .  The d i f f e r e n t curves correspond t o  d i f f e r e n t h o r i z o n t a l s e t t i n g s o f the aperture. was o b t a i n e d by measuring field  off (strictly  quadrupole on. of  field  the f i g u r e .  the beam i n t e n s i t y f i r s t with the d e f l e c t i n g  s p e a k i n g , w i t h the s o l e n o i d f i e l d  was o f f resonance)  The percentage  Each p o i n t on the curves  o f f so t h a t the  and then w i t h the d e f l e c t i n g  i n c r e a s e i n i n t e n s i t y i s p l o t t e d a l o n g the o r d i n a t e  From t h i s f a m i l y o f curves a rough  contour map o f equal  i n t e n s i t y l i n e s may be c o n s t r u c t e d , as shown i n F i g . 4 - l b . for  field  Neglecting  the moment the secondary peaks i n F i g s . 4 - l a and 4 - l b , i t may be  noted, f i r s t , t h a t t h e f o c u s i n g e f f e c t symmetric; about  i s approximately  cylindrically  s e c o n d l y , t h a t i t occurs c o m p l e t e l y w i t h i n a diameter o f  0.1 cm, with no o t h e r o b s e r v a b l e f l u c t u a t i o n s w i t h i n t h e 0.3-cm  diameter  circle,  over which t h e whole beam extended;  and t h i r d l y ,  that  the h a l f width a t h a l f maximum o f the f o c u s e d spot f o r a g r a d i e n t o f 430  gauss/cm i s about  0.025 cm, which i s i n agreement with the o r d e r  of magnitude c a l c u l a t e d i n a p r e v i o u s s e c t i o n . The  c e n t e r o f the f o c u s e d spot i s assumed t o be the p o s i t i o n o f the  symmetry a x i s o f the quadrupole The  w i r e s i n the p l a n e o f t h e d e t e c t o r .  secondary peaks shown i n F i g s . 4 - l a and 4 - l b p r o b a b l y  from some a b e r r a t i o n i n the f o c u s i n g system.  They almost  arise  certainly'do  not a r i s e from any s p a t i a l r e s o l u t i o n o f the beam i n t o i t s v a r i o u s 'components s i n c e the beam i s broad compared t o the d e f l e c t i n g power available. The  f o c u s i n g e f f e c t was then s t u d i e d on a v e r y f i n e beam.  With a  c o l l i m a t o r h o l e 0.005 cm i n diamter, the u n d e f l e c t e d beam p r o f i l e i s shown i n F i g . 4-2.  - 23 -  The of  f u l l width  a t h a l f i n t e n s i t y i s about 0.030 cm.  t u r n i n g on the quadrupole f i e l d  The  i s as shown i n F i g . 4-3,  effect  the  h a v i n g been o b t a i n e d and p l o t t e d i n e x a c t l y the same manner as in  Fig. 4-la.  One  the v e r t i c a l = -20  those  can see t h a t the curves are r o u g h l y symmetrical x (0.001 cm)  p o s i t i o n and  f o c u s i n g a c t i o n a t v e r t i c a l = -20 (0.001 cm).  results  x  that there i s a strong  (0.001 cm)  and h o r i z o n t a l = -15  T h i s p o s i t i o n c o i n c i d e s with the c e n t e r o f the  spot r e p r e s e n t e d i n F i g s . 4 - l a and 4 - l b .  about  x  focused  In o t h e r words, i t i s the  p o s i t i o n o f the symmetry a x i s o f the quadrupole wires a t the d e t e c t o r . S i n c e the zeros o f the axes i n F i g . 4-3  correspond  u n d e f l e c t e d beam, the quadrupole wires and a l i g n e d to about 0.02  cm  i n 180  cm  t o the c e n t e r o f the  the u n d e f l e c t e d beam were  (the f r e e - f l i g h t  length) or 1 p a r t i n  io . 4  As shown i n F i g . 4-3,  t h e r e i s a d e p l e t i o n o f atoms a t about .0.-025 cm  from the d e f l e c t o r a x i s (as measured i n the plane o f the d e t e c t o r ) . Father out i n the wings o f the beam t h e r e i s a p o s i t i v e change i n i n t e n s i t y a g a i n , i n d i c a t i n g t h a t some atoms have been thrown outwards from the c e n t e r o f the beam. These are defocused  atoms, i . e . , atoms  which, i n the r o t a t i n g frame, have e f f e c t i v e magnetic moments i n the x d i r e c t i o n o f o p p o s i t e s i g n t o those o f the focused atoms. i s so, then we  have r e s o l v e d the beam i n t o two  components.  our low d e f l e c t i n g power, coupled w i t h the Maxwellian  I f that Apparently,  distribution  v e l o c i t i e s i n the beam, has p r e c l u d e d our o b s e r v a t i o n o f a l l f i v e the expected  components.  of of  I t s h o u l d be p o i n t e d out t h a t , because o f the  p r e l i m i n a r y nature o f the p r e s e n t experiment, no attempt has been made to  a c h i e v e the u l t i m a t e i n r e s o l u t i o n or l i n e shape. Fig.  4-4  shows the f i e l d  dependence o f the resonance.  The  - 24 -  d e t e c t o r p o s i t i o n and opening were h e l d f i x e d with r e s p e c t t o the beam, i n a p o s i t i o n t h a t had a good s i g n a l - t o - n o i s e r a t i o . of  t h e quadrupole  field  The s t r e n g t h  B^, a t c o n s t a n t g r a d i e n t , a c t i n g on t h e beam  was v a r i e d by moving the d e f l e c t i o n system  and h o l d i n g t h e beam f i x e d ,  i n o t h e r words, by sending the beam down t h e d e f l e c t i o n system a t d i f f e r e n t d i s t a n c e s from the zero a x i s . by changing The  The f i e l d  g r a d i e n t was changed  the r - f . v o l t a g e d r i v e a c r o s s t h e system.  curves were o b t a i n e d by h o l d i n g t h e frequency f i x e d a t 7.22 Mhz  and changing  the s o l e n o i d f i e l d  (B, ) i n d i s c r e t e s t e p s , and r e a d i n g the  output meter o f t h e l o c k - i n a m p l i f i e r . was p e r i o d i c a l l y checked  The u n d e f l e c t e d beam i n t e n s i t y  t o ensure t h a t the system was n o t d r i f t i n g .  These curves a r e i n agreement w i t h eq. (2-22) .  The resonance  maximum occurs f o r B.. = a) / Y = 10.4 gauss, the f i e l d f o r a Larmor o o ' • p r e c e s s i o n frequency o f 7.2 Mhz, and the l i n e width f o r each c u r v e , g i v e n by the p l o t o f sin/; 6 v e r s u s B , i s a p p r o x i m a t e l y equal t o B^q  attempt  has been made, however, t o make a d e t a i l e d t h e o r e t i c a l f i t o f  the observed  l i n e shape.  The double quantum resonance Fig.  NO  which appears  i n two o f the curves o f  4-4 i s d i s c u s s e d t h e o r e t i c a l l y i n chapter 2.  I t occurs f o r l a r g e  B^ (10 t o 16 gauss) as p r e d i c t e d , but i n t h i s experiment o n l y occurs i n t h e r e g i o n o f t h e quadrupole  f i e l d where the h i g h e r 2  o r d e r terms o f t h e expansion B^ = G-^(xi - y j ) + G^(x becoming important.  a l a r g e B^  2 - y ) i + ... a r e  The e f f e c t s o f these h i g h e r - o r d e r terms have been  n e g l e c t e d i n the t h e o r y , as have been the components o f B^ p a r a l l e l t o B , which occur near the ends o f the w i r e s . o' The  shift  o f t h e double quantum resonance  which i s apparent  towards lower  fields  i n the two curves i s p r o b a b l y a B l o c h - S i e g e r t s h i f t  - 25 (Ramsey 1963, p. 222), which results from the use of a l i n e a r l y oscillating field B  instead of a true rotating  field.  4 Total  unifor-rw  Gradient = Detect  intensify  haim LuUiW  at  delector  -  =  0-3.C  *4-50 gauss / c w  opening  0.01 on* * O-OI c**\  i - ; -tio «>  i  ON  CD  Si ct  -W Vertical F i g u r e 4-la.  Plot  -*o position  o f <Jet*e+or  o f the percentage i n c r e a s e  position  O  -ao  .001 c m ) ( a r b i t r a r y  +«K> ^ero  +60  position)  i n beam i n t e n s i t y a t the d e t e c t o r  i n the x,y p l a n e due t o the f o c u s i n g  l a r g e , u n i f o r m - i n t e n s i t y beam.  -v20  as a f u n c t i o n o f  e f f e c t o f the d e f l e c t i o n system  in a  - 27 -  rlori^orftal Figure 4-lb.  post*. ( . 0 0 1 cm)  Contour map of the focused "pip", as constructed from Fig. 4 - l a .  The numbers on the contours are percentage  increases i n beam intensity due to the rotating quadrupole f i e l d  (ordinate of F i g . 4 - l a ) .  The dotted lines  are interpolated contours between widely separated points.  - 28 -  C arbt"tror^ -$era position) Figure 4-2.  Beam i n t e n s i t y p r o f i l e for undeflected narrow beam.  - 29 -  Vertical position of detector center of Figure 4-3.  relative "to  deflected beam X . 001 cm  Relative change of the beam intensity pattern for a narrow beam at resonance. in Fig. 4-2.  The undeflected beam p r o f i l e i s shown  I 0  I 0.2.  I 0.«f  I  1 O.fc  I O-fc  Solenoid F i g u r e 4-4.  I 1-2.  L_i «.0  current  C  a w  1 1-4  1  1 1-8  P ) s  P l o t o f the beam i n t e n s i t y near the beam c e n t e r as a f u n c t i o n o f the f i e l d B  f o r f i x e d frequency and d i f f e r e n t amplitude o f the r f f i e l d  and i t s g r a d i e n t .  1 SO  CHAPTER 5 A P r o p o s a l f o r a Charged  P a r t i c l e TSG Experiment  The major o b s t a c l e t o p e r f o r m i n g a c o n v e n t i o n a l S t e r n - G e r l a c h experiment on charged p a r t i c l e s  i s that the Lorentz f o r c e a s s o c i a t e d  w i t h a charged p a r t i c l e moving i n a magnetic f i e l d  i s f a r l a r g e r than  the  i n h e r e n t i n any i o n i c  Stern-Gerlach force.  beam would  The spread o f v e l o c i t i e s  g i v e r i s e t o an u n c e r t a i n t y i n t h e L o r e n t z f o r c e which  undoubtedly mask t h e e f f e c t  would  o f the Stern-Gerlach force.  When t h e n e u t r a l p a r t i c l e TSG experiment was proposed i t seemed likely  t h a t i t would be a d a p t a b l e t o charged p a r t i c l e s ,  s i n c e i t would  be p o s s i b l e t o send a charged p a r t i c l e beam p a r a l l e l t o t h e l a r g e homogeneous magnetic  field  plane perpendicular to t h i s magnetic f i e l d .  and t o observe resonant d i s p l a c e m e n t s i n t h e f i e l d due t o an o s c i l l a t i n g inhomogeneous  Some o f t h e d i f f i c u l t i e s  a s s o c i a t e d w i t h one s p e c i a l  type o f charged p a r t i c l e TSG experiment have a l r e a d y been d i s c u s s e d (Bloom and Erdman 1962; R a s t a l l  1962; Byrne  1963).  In a d a p t i n g t h e n e u t r a l p a r t i c l e TSG experiment t o charged our p r e l i m i n a r y thoughts were t h a t i n t h i s geometry the  particles  t h e amplitude o f  i o n beam o s c i l l a t i o n s produced by t h e o s c i l l a t i n g L o r e n t z f o r c e  would be i n v e r s e l y p r o p o r t i o n a l t o t h e f r e q u e n c y o f t h e o s c i l l a t i n g magnetic f i e l d ,  so t h a t by u s i n g a h i g h enough frequency t h e s e o s c i l l a t i o n s  would not s e r i o u s l y i m p a i r t h e s p a t i a l r e s o l u t i o n o f t h e beam o r t h e  - 32 Stern-Gerlach e f f e c t .  In g e n e r a l terms these c o n c l u s i o n s were c o r r e c t ,  a l t h o u g h a t the time i t was not r e a l i z e d broad c a t e g o r y o f s t r o n g f o c u s i n g  that t h i s  system i s one o f a  systems.  P r e l i m i n a r y work was begun on the e x t e n s i o n o f the ac quadrupole system, which was used i n the n e u t r a l p a r t i c l e particles.  Much o f the work a t t h i s  TSG experiment, t o charged  stage went i n t o p r o d u c i n g an i o n  beam, but s e v e r a l p r o t o t y p e ac quadrupoles were a l s o b u i l t . o u t s e t , i t was r e a l i z e d  At the  t h a t two major problems would have t o be over-  come t o u t i l i z e the ac quadrupole f o r a TSG experiment w i t h i o n i c The f i r s t problem was t o reduce the e l e c t r i c drop a l o n g the w i r e s . the  f i e l d produced by the v o l t a g e  T h i s e l e c t r i c f i e l d was s u f f i c i e n t l y  strong, f o r  low energy beams which must be used, t o produce v e r y l a r g e  t i o n s i n the beam.  I t was r e a l i z e d  beams.  much l a t e r t h a t t h i s  oscilla-  electric  field  c o u l d be u t i l i z e d t o good advantage t o c a n c e l the magnetic L o r e n t z f o r c e oscillations,  but t h i s  without s i g n i f i c a n t l y  d i d not e l i m i n a t e the problem o f r e d u c i n g i t r e d u c i n g the o s c i l l a t i n g magnetic  field.  The second problem was to c o o l the quadrupole w i r e s well that very high f i e l d  sufficiently  g r a d i e n t s c o u l d be produced.  Most o f the work which went i n t o the ac quadrupole was c e n t e r e d on t h e s e two problems; however, t h e a u x i l l i a r y problems o f g e n e r a t i n g h i g h power RF (the d e s i g n s were based on an RF power input o f 4000 w a t t s ) , grounding and s h i e l d i n g  i t , and matching the impedances,  also  took c o n s i d e r a b l e time. In t h e middle o f t h i s development  i t was r e a l i z e d  t h a t a much  s i m p l e r system c o u l d be b u i l t u s i n g time independent c u r r e n t s i n a h e l i c a l quadrupole, such as i s i l l u s t r a t e d i n F i g . 5 - l a . T h i s i d e a l e d  - 33 -  Figure 5-la.  A h e l i c a l quadrupole wire system.  F i g u r e 5-lb.  A quadrupole wire system showing the e l e c t r o d e s to produce an e l e c t r i c field.  field  necessary  o r t h o g o n a l t o the magnetic  - 34 to the extension of the TSG experiment to space varying f i e l d s as mentioned i n Chapter 1, and f i n a l l y we r e a l i z e d that we had been working a l l along within the general concept of strong focusing systems. Several months l a t e r we added an e l e c t r i c f i e l d orthogonal to the magnetic f i e l d i n the quadrupole, as i l l u s t r a t e d i n F i g . 5-lb, which serves the important function of n e u t r a l i z i n g the major portion of the Lorentz force, for a p a r t i c u l a r beam v e l o c i t y . When the dc h e l i c a l quadrupole i s used, rather than the ac quadrupole, the following important s i m p l i f i c a t i o n s occur.  There are no skin-  depth or induced current phenomena, so that additional electrodes f o r control of the e l e c t r i c f i e l d may be inserted into the quadrupole without attenuating the magnetic f i e l d .  Also, the f u l l cross-section  of the quadrupole wires carries the current so that the resistance and the power dissipated are both less than for the ac case. should be appreciated that since  In f a c t , i t  the magnetic f i e l d gradient and resistance  are each inversely proportional to the cross sectional area of the wires, and since the cooling capacity varies at least i n proportion to the area, one can make this system as large as necessary without having the cooling capacity lag behind the power required to maintain a given f i e l d  gradient.  F i n a l l y , i t i s much easier to produce, s h i e l d and ground, and match impedances, for dc currents than high frequency ac currents. So f a r , we have r e s t r i c t e d this discussion to a h e l i c a l quadrupole formed from current-carrying wires, since this i s a form of construction which i s simple and straight forward, and which has the important advantage that the additional electrodes which are inserted to produce the orthogonal e l e c t r i c f i e l d may be made concentric with the quadrupole  -  35  -  w i r e s , as i l l u s t r a t e d i n F i g . 5 - l b . v e r y h i g h degree o f f i e l d f i e l d may  Such a c o n s t r u c t i o n should g i v e a  orthogonality.  However, a h e l i c a l quadrupole  a l s o be produced u s i n g i r o n - c o r e d magnets, as i l l u s t r a t e d i n  F i g . 5-2.  In t h i s case the e l e c t r o d e s p r o d u c i n g the  orthogonal  e l e c t r i c f i e l d must be p l a c e d between the i r o n p o l e s , and g o n a l i t y a c h i e v e d may super-conducting  not be v e r y good.  quadrupoles  i n c r e a s e the magnetic f i e l d In Chapter  2 we  F i n a l l y , i t i s p o s s i b l e that  can be developed  which would  have g i v e n the c l a s s i c a l theory o f the t r a n s v e r s e  conclusions presented  I t i s c l e a r t h a t the  t h e r e r e g a r d i n g the S t e r n - G e r l a c h f o r c e may  over d i r e c t l y to the charged  n e c e s s a ry  t o c o n s i d e r the Lorentz f o r c e u n l e s s one wishes t o compute  p l a c e d i n the same f i e l d  configuration, w i l l  particle, i f  experience  the same S t e r n -  force.  In the f o l l o w i n g , we particular  essentially  r e p e a t much o f Chapter  2 f o r the  case o f a p a r t i c l e moving g e n e r a l l y i n the z d i r e c t i o n  a time-independent The  be  p a r t i c l e c a s e , s i n c e i t i s not  t r a j e c t o r i e s - . T h a t i s , a n e u t r a l p a r t i c l e or a charged  Gerlach  significantly  g r a d i e n t s which can p r e s e n t l y be a t t a i n e d .  S t e r n - G e r l a c h experiment f o r a n e u t r a l p a r t i c l e .  carried  the o r t h o -  h e l i c a l quadrupole  Stern-Gerlach f o r c e i s given  F  field. by:  = (u - y ) B ( f , z )  S G  in  (5-1)  or F  F  SGx  cr SGy  Kh  =  = (U V x 3x 1  +U  yh  + u |y 3y  +  U  z l l  )  B  x  + U ! - J )B z 3z y  ^  K  (5-2b) J  F i g u r e 5-2.  An i r o n core quadrupole, to  produce  field.  showing the e l e c t r o d e s n e c e s s a r y  an e l e c t r i c f i e l d  o r t h o g o n a l t o the  magnetic  - 37 -  F  =  SGz  TT—,+  (u  u  ^ x 8x  —  + u  y 3y  ——  z 3z  ;  )B  z  In t h e h e l i c a l quadrupole B ( f , z ) i s g i v e n by:  B(r,z) = B I + B } + B t ' x y • o v  1  B  = G (x s i n 2wz - y cos 2coz)  B^ =-Gg(y s i n 2toz + x cos 2UJZ)  B  Therefore,  substituting  o  = constant  equation  SGx  =  ^B^ x  ^SGy  =  ~^B^ x °  F  F  c r  SGz  I f we r e w r i t e e q u a t i o n  U  U  S  ^  (5-3) i n t o  n  2  O  -  S  2  Z  -  ~ y u  Z  +  U  c  o  (5-2) y i e l d s  s  2  y ^"  = 0  (5-3) i n the form:  S  ~ ^  n 2  z  — ^ z  - 38 -  then i t i s obvious t h a t i n a c o o r d i n a t e frame moving i n the z d i r e c t i o n at  velocity V  q  , the f i e l d  B(f,z) w i l l  c o n s i s t o f a component  r o t a t i n g w i t h the a n g u l a r v e l o c i t y 2ojy , and a s o l e n o i d f i e l d o  We  now  d e f i n e a c o o r d i n a t e frame *yi,y  (?) B. Q  which i s c e n t e r e d on the  p a r t i c l e and which r o t a t e s i n time a t the a n g u l a r frequency co about z a x i s , such t h a t the a n g u l a r v e l o c i t y o f the component B^(r) the z a x i s i n the frame x,y  the z a x i s  we have the r e s u l t  about  i s zero.  S i n c e i n g e n e r a l the p a r t i c l e may time about  the  have an a n g u l a r frequency  :;co i n T  ( a r i s i n g from the L o r e n t z f o r c e s , f o r example)  that;  co = 2cov  - co  (5-7)  T  where co^ = a n g u l a r frequency i n time o f the p a r t i c l e about  In at  the c o o r d i n a t e frame "x/y, J p r e c e s s e s about  an angular  the z a x i s  the e f f e c t i v e  field  frequency  u)  2 = [(Aco) +  2  /  e  1/2 ] ^  (5-8)  where  Aco  co-co  -YB^r)  =  -YB,  (5-9)  (5-10)  - 39 The e f f e c t i v e f i e l d i s oriented at an angle 0 with respect to the z axis (as i l l u s t r a t e d i n F i g . 5-3a) where:  l tan 0 = - -=• u  (5^11)  Aw  If, as i n Chapter 2, we take  as the component of J along the  e f f e c t i v e f i e l d at t = z = 0, and then calculate the time-average of the Stern-Gerlach  force, i t i s clear that only the components of  y i e l d a cummulative force, since only field  will  i s synchronous with the  (r) (This i s shown e x p l i c i t l y i n Chapter 2). Noting from F i g . 5-3b, that;  B  u  = yn\J sine 7  + terms o s c i l l a t i n g at u>  (5-12a)  + terms o s c i l l a t i n g at Ui  (5-12b)  B  u  = yfiJ Y  sine  ^—  IBJTOI  3  we can substitute equation (5-12) into (5-4) to y i e l d , f o r times large compared to —  : e  <F  < F  SGx  = ^ 3  >  SG/  J  =  Y  M  3  l l  S i n 9  F  G  B  S i n S  l l  r  G  B  This may be written  < F  SG  >  Y  tfJ  3  sine | G | B  -  (5-13)  Figure 5 - 3 a .  I l l u s t r a t i o n of the frequency vectors i n the rotating coordinate system and makes an angle  {%,y*,z~).  ^(F)  i s i n the x,y  plane  -<j> with respect to the ~ axis.  the component of J along the e f f e c t i v e f i e l d .  u2  Figure 5-3b.  An i l l u s t r a t i o n of the vector coordinate system at the time t .  s i n 0 i n the  (x,y,z)  is  - 41 -  with  sine  77—2  =  v72  (5  "  14)  T h e r e f o r e the time averaged S t e r n - G e r l a c h f o r c e i s r a d i a l , w i t h a magnitude t h a t depends on the s i n 6 f a c t o r noted above. e s s e n t i a l l y the same as e q u a t i o n f a c t o r o f 2 which Chapter 1/2  2 system  (2-22)  i n Chapter  2,  This r e s u l t i s except f o r a  e n t e r s because the l i n e a r o s c i l l a t i n g f i e l d  o f the  i s decomposed i n t o two r o t a t i n g components which have  the magnitude o f the o r i g i n a l  field.  We have d i s c u s s e d the d e t e c t i o n o f the n e u t r a l p a r t i c l e T r a n s v e r s e Stern-Gerlach e f f e c t  i n Chapters 3 and 4, and i n d i c a t e i n t h i s  t h a t o t h e r d e t e c t i o n t e c h n i q u e s may p a r t i c l e case. 1)  Resonant  We  be more s u i t a b l e f o r the charged  can suggest f o u r methods o f d e t e c t i o n as f o l l o w s ;  D e f l e c t i o n Method  The method used f o r the n e u t r a l p a r t i c l e case may charged p a r t i c l e s .  a l s o be used f o r  T h i s method i n v o l v e s a resonance i n the S t e r n - G e r l a c h  f o r c e v i a the s i n 6 f a c t o r consequent  Chapter  (see e q u a t i o n 5-14), and the d e t e c t i o n o f a  change i n the i n t e n s i t y d i s t r i b u t i o n o f the beam.  resonance i n the case o f the h e l i c a l quadrupole f i e l d  c o u l d be  The produced  by f i x i n g v , thus f i x i n g the apparent a n g u l a r f r e q u e n c y o f the h e l i c a l quadrupole f i e l d  i n the r e s t frame o f the p a r t i c l e s , and then t u n i n g the  p r e c e s s i o n o f the magnetic moment to t h i s frequency u s i n g the i n a l magnetic is anticipated  field  B. q  As we  longitud-  s h a l l see l a t e r , the width o f the resonance  t o be v e r y l a r g e .  The f a c t t h a t the l o n g i t u d i n a l  field  a l s o produces a d i r e c t e f f e c t on the beam i n t e n s i t y d i s t r i b u t i o n makes  the resonant d e f l e c t i o n method much more d i f f i c u l t particle 2)  than f o r t h e n e u t r a l  case.  Computational The  Method  computational method i n v o l v e s a d e t a i l e d comparison  of the  e x p e r i m e n t a l beam i n t e n s i t y d i s t r i b u t i o n as a f u n c t i o n o f the v a r i o u s experimental parameters motion  with t h a t computed from the equations o f  f o r t h e charged p a r t i c l e s .  I t would be n e c e s s a r y t o p i c k out  those f e a t u r e s o f t h e i n t e n s i t y d i s t r i b u t i o n which a r e a s s o c i a t e d w i t h the s m a l l 3)  Stern-Gerlach force.  Comparison Method In  t h i s method the i n t e n s i t y d i s t r i b u t i o n o f beams o f s i m i l a r i o n s  h a v i n g d i f f e r e n t magnetic  moments would be compared and t h e e f f e c t s  c h a r a c t e r i s t i c o f the Stern-Gerlach force exhibited.  F o r example,  4 + He i o n s i n t h e ground s t a t e would have two magnetic  states  -  ing  t o magnetic  moments o f a p p r o x i m a t e l y  moment o f t h e e l e c t r o n . for  t h e p r o j e c t i o n o f the magnetic  field  i n low magnetic  s p i n 1/2.  The magnetic  The Rabi Magnetic The Rabi method  a system  +  y  g  i s the magnetic  would have 3 v a l u e s  moment a l o n g t h e e x t e r n a l magnetic  f i e l d s because o f the e f f e c t o f the nucleus o f moments o f these s t a t e s would be a p p r o x i m a t e l y  y , o, and -y , with p r o b a b i l i t i e s , 4)  ±y^, where  On the o t h e r hand, ^He  correspond-  h  1/4, 1/2, and 1/4, r e s p e c t i v e l y .  Resonance Method  (Ramsey 1963) c o n s i s t s o f sending the i o n s  through  composed o f a " p o l a r i z e r " , a " d e p o l a r i z e r " and an " a n a l y z e r " .  The p o l a r i z e r and a n a l y z e r f o r charged p a r t i c l e S t e r n - G e r l a c h would be h e l i c a l quadrupole  systems which each produce  experiments  a certain  intensity  d i s t r i b u t i o n a t t h e output g i v e n a c e r t a i n d i s t r i b u t i o n o f i n t e n s i t i e s and  - 43 spins at the input.  I f t h e d i s t r i b u t i o n o f s p i n s a t each p o s i t i o n i n ,  the beam were changed i n the r e g i o n between t h e p o l a r i z e r and a n a l y z e r , the i n t e n s i t y d i s t r i b u t i o n a t t h e output o f t h e a n a l y z e r would a l s o be changed.  T r a n s i t i o n s between d i f f e r e n t s p i n s t a t e s can be produced by  a d e p o l a r i z e r i n which t h e s p i n s undergo magnetic resonance.  For  charged p a r t i c l e s  the d e p o l a r i z e r would c o n s i s t o f a l o n g i t u d i n a l time  independent f i e l d  and a t r a n s v e r s e r f . f i e l d , both o f these magnetic  f i e l d s being homogeneous.  When t h e u s u a l magnetic resonance c o n d i t i o n s  are s a t i s f i e d , t h e p r o b a b i l i t y o f a p a r t i c l e undergoing a t r a n s i t i o n from one s p i n s t a t e t o another i n t h e d e p o l a r i z e r i s l a r g e .  When t h e  beam passes from t h e p o l a r i z e r through t h e d e p o l a r i z e r t o t h e a n a l y z e r , it  i s n e c e s s a r y t o s a t i s f y c o n d i t i o n s o f " a d i a b a t i c passage"  1961)  i n order  to preserve  (Abragam,  the d i r e c t i o n o f q u a n t i z a t i o n o f t h e s p i n s  when t h e magnetic resonance c o n d i t i o n s  i n t h e d e p o l a r i z e r a r e not  satisfied. I t would seem t h a t o f t h e f o u r methods d e s c r i b e d method i s t h e best  one. T h i s method d i f f e r s  above the Rabi  from t h e o t h e r t h r e e i n  t h a t a change i n t h e i n t e n s i t y d i s t r i b u t i o n o f t h e beam i s a c h i e v e d by v a r y i n g a parameter  (the frequency o f the r f . f i e l d  which has no a p p r e c i a b l e  i n the d e p o l a r i z e r )  d i r e c t e f f e c t on t h e i n t e n s i t y d i s t r i b u t i o n ,  but which e f f e c t s t h e i n t e n s i t y o n l y by changing t h e p o p u l a t i o n s spin states.  of the  CHAPTER 6 The T r a j e c t o r i e s o f an Ion w i t h Zero Magnetic Moment i n a DC H e l i c a l Quadrupole  In t h e f o l l o w i n g c h a p t e r s we a r e a t t e m p t i n g t o f i n d out whether a h e l i c a l quadrupole system can be used t o s e p a r a t e s p i n s t a t e s i n a S t e r n - G e r l a c h experiment.  A b a s i c t h e o r e t i c a l requirement f o r such a  study i s t h e c a l c u l a t i o n o f r e a s o n a b l y p r e c i s e e x p r e s s i o n s f o r the particle trajectories.  The d i r e c t approach, o f f i n d i n g t h e s o l u t i o n s  to the e q u a t i o n o f motion o f an i o n w i t h a non-zero magnetic moment i n a h e l i c a l quadrupole system i s c o m p l i c a t e d because t h e e q u a t i o n i s non-linear.  I t has not, t o our knowledge, been s o l v e d .  We have, howev  developed approximate, but u s e f u l s o l u t i o n s which a r e based on the c h a r a c t e r i s t i c s o f t h e average S t e r n - G e r l a c h f o r c e g i v e n by e q u a t i o n (5-13).  In t h i s c h a p t e r we a r e o n l y c o n s i d e r i n g the t r a j e c t o r i e s o f  i o n s w i t h zero magnetic moment.  For t h i s case the e q u a t i o n o f motion  may be s o l v e d e x a c t l y i n the a x i a l r e g i o n , and we have done t h i s i n appendix B. In Chapter 7 we have developed s o l u t i o n s f o r a charged p a r t i c l e w i t h non-zero magnetic moment which a r e v a l i d  i n the l i m i t t h a t the  L o r e n t z f o r c e i s comparible t o t h e S t e r n - G e r l a c h f o r c e . it  In t h i s  limit  i s a v a l i d a p p r o x i m a t i o n t o decouple t h e r a d i a l motion from the  -  tangential motion.  45'  -  -  The r a d i a l equation of motion which r e s u l t s i s linear  and i s solved i n Chapter 7. In Chapter 8 we have developed approximate solutions based on the s i m i l a r i t y between the r a d i a l nature of the average Stern-Gerlach force given by equation (5-13) and a force of the form 0i, where $ i s a constant. We show that the c o e f f i c i e n t $ may be chosen to represent the cummulative effect of the Stern-Gerlach force over a range of z which must be s p e c i f i e d . The substitution of the force $r. for the average Stern-Gerlach force reduces the equation of motion to a linear equation, and the solutions to this  equation, evaluated at the end point of the a r b i t r a r y i n t e r v a l  over which $ represents the cummulative effect of the Stern-Gerlach force, y i e l d s f o r each i n t e r v a l a point of the trajectory. In the rest of this chapter we are only considering the motion caused by the Lorentz force with the solenoid f i e l d absent, since an understanding o f this motion i s very helpful before considering the additional effect of the Stern-Gerlach force.  We show that t h i s motion  has a very simple character i n the two extreme regions of operation. For very weak forces the motion i s almost planar with a sinusoidal amplitude.  For strong forces a magnitude of the Lorentz force i s  defined beyond which the motion i s divergent.  This may be c a l l e d the  point of i n s t a b i l i t y , since the beam "blows up" quite r a p i d l y beyond this point.  At t h i s point the t r a j e c t o r i e s are closely approximated by  diverging s p i r a l s , with the radius growing l i n e a r l y with distance along the optic axis, and the s p i r a l pitch synchronous with the h e l i c a l winding. Between the two extremes, the t r a j e c t o r i e s are bounded and periodic  -  46  -  with four c h a r a c t e r i s t i c frequencies, which reduce to two dn the h e l i c a l frame of reference defined i n appendix B.  The motion i s rather l i k e a  corkscrew twisted about the optic axis i n this region. In appendix B, we give the derivation of the equation of motion for a h e l i c a l quadrupole which includes both magnetic and e l e c t r i c f i e l d gradients  (Gg and Gg) and a solenoid f i e l d  (B ).  As we have mentioned,  q  a r a d i a l force term (§) i s also included, but for the rest of this chapter we are considering only the case for $ = 0. Except for the inclusion of $ and B , these equations are discussed q  quite extensively i n the l i t e r a t u r e .  G. Salardi et a l . (1968) studied  the device as a lens to c o l l e c t and focus charged p a r t i c l e s .  K.J.  Le Couteur (1967) studied p a r t i c l e guiding i n h e l i c a l multipole f i e l d s and the quadrupole i n p a r t i c u l a r .  He also considers the higher  order  terms which are important i n the o f f - a x i a l region, and shows that the device can be bent into a c i r c l e  and s t i l l confine a beam.  (1959) studied the device i n the axial region.  A.M.  L.C. Teng  Strashkevich et a l .  (1968) shows that at r e l a t i v i s t i c v e l o c i t i e s i t i s possible to make the device somewhat achromatic by adding the equivalent of the term G^ described i n appendix B.  N.I. Trotsyuk (1969) shows that the device  can  be used to focus atoms and molecules under certain conditions. Note that the e l e c t r i c and magnetic f i e l d s are orthogonal  to one  another, and as usual under this condition, the respective forces are colinear.  For this reason the gradient terms appear i n a single parameter  © : ©  =  — mv  9  o  —  (G  E  + v G ) o  B  (6-1)  -  47  -  In most o f t h e l i t e r a t u r e t h i s p o i n t i s not e x p l i c i t l y noted, s i n c e the concern i s u s u a l l y o n l y w i t h t h e magnetic f i e l d ,  because o f i t s  strong focusing properties f o r energetic p a r t i c l e s .  For our purposes  t h i s i s an important  p o i n t because by choosing  V G g we can make  a r b i t r a r i l y s m a l l without s a c r i f i c i n g the magnetic  ©  Q  field  g r a d i e n t , which o f course The  £  t o c a n c e l the term  produces t h e S t e r n - G e r l a c h  s o l u t i o n s i n appendix B a r e s t r i c t l y v a l i d  displacements numerical  G  from t h e o p t i c (z) a x i s .  Pearce  i n t e g r a t i o n t h a t f o r displacements  the h e l i c a l  step  l e n g t h , t h e equations  only f o r small  (1969) has shown by  l e s s than L/20, where L i s  are v a l i d .  are g r e a t e r than L/20, t h e a x i a l magnetic f i e l d l i n e a r terms appear i n t h e equations  effect.  When the displacements  becomes a p p r e c i a b l e , non-  o f motion, and t h e system becomes  more s t r o n g l y f o c u s i n g . If,  i n appendix B, we put B  Q  = 0, t h e term W reduces t o w, and i f ,  i n a d d i t i o n , $ = 0, we can w r i t e t h e t r a j e c t o r i e s , equations  (b-22), a s :  ojy I cosco z-cosu), Z I  x = x cosu„z + -=- y sinw,z + o ^ 2 o *1 ©  0  y  ttl  +  —*o  -2 [ — since, z- —  y = y COSUKZ — o -1  +  -^o  0 where  -  0)  .  —=• x sinto„z + OJ  2  o  -1 [ — sinco, z-  a  sinco.z] *  "I  cox  "  O  -2  <^  r L  I  sinco„z]  "  2  (6-2a)  COSOJ,  -1  -i z-cosco zl  -2  0  (6-2b)  2 »1  2  If  ©  48  2 co +  ©  2  co  ^2  co  -  -  ©  2TT_  =  (6-2c)  L  i s s m a l l , we can r e w r i t e these equations by expanding co^  and co„  to  « co + 1/2 ©/co  -\ „ |©/c/ «  1  (6-3)  Then a p p l y i n g simple t r i g o n o m e t r i c t r a n s f o r m a t i o n s , e q u a t i o n (6-2) takes t h e form:  2 &  x = x cosco„z + y smco z + J o —2 o -1  o  n  2cox +  .ris.  . . © z smcoz s m r—"" 2co  (_• z coscoz s i n •=—  (6-4a)  2OJX  © z  O  y = y coscoz - x smco„z -  -  *°  "  0  2coy  Q  +  for  "  smcoz s m  ©  2  "  © z coscoz s i n -=—  ©/co  2  << 1  T r a n s f o r m i n g t o the l a b o r a t o r y frame;  -:— 2  a (6-4b)  - 49 -  x = xcoswz - ysinwz y = ycosojz + x s i n u z  (6-5)  ~~  ©  x j? x cos o  z  ©  — 2oj  + y sm 'o  2O)X  z  ^ —  ©  Q  +  z  sm  2OJ  ^  (6-6) y « y cos o  If x , y  © z •=— 2co  + x sm o  © z - —  2 y - o u  +  y  sm  2OJ  are small, i . e . ;  2cjr  |rj  |  «|-^°  (6-7)  0 where r  o  1  =  fx ^ o  + y 'c  1 2'  then t h i s motion i s n e a r l y p l a n a r , and we can d e s c r i b e i t by the r a d i u s r;  (Note  1)  2_r r =  ©  Note t h a t t a k i n g the l i m i t simple f r e e f l i g h t ,  Note 1:  as  © z s i n •=—  + r  (6-8)  ^  2  as  ©  0 i n equation  (6-6) , one gets  expected:  r as used here i s . s t r i c t l y a one d i m e n s i o n a l c o o r d i n a t e r a t h e r than [ f | , s i n c e both p o s i t i v e and n e g a t i v e v a l u e s are a l l o w e d .  50  -  X  = X  -  z  + X  (6-9)  0  y =  y +.v 0  We can make the following observations.  The t r a j e c t o r i e s are  2  bounded provided ©  < to . For very small © , the motion i s modified from  simple free f l i g h t to a long wavelength sinusoidal motion, with a h a l f period given by:  z  ,  = 2m  (6-10)  As (Q> becomes larger, the motion becomes more complicated with two c h a r a c t e r i s t i c frequencies, or modes, co^ and a^.  In this region one  must be careful about the r e l a t i v e importance of the a x i a l (r = 0) and o f f - a x i a l p a r t i c l e s when making generalizations. Generally, when discussingvthe focusing properties of these systems, only the axial p a r t i c l e s are considered.  This involves the i m p l i c i t assumption that:  cor |r | « | ^ | ©  (6-11)  In many beam handling experiments t h i s i s a good assumption, but i n this experiment we are considering beams with  r r  From equation  o o  «  -4 10 m  <a 10"  3  (6-11), this means that we are only j u s t i f i e d i n  - 51 -  g e n e r a l i z i n g to a x i a l p a r t i c l e s i f  iL  We  »  IO"  1  m  (6-12)  i n t r o d u c e the u s e f u l d i m e n s i o n l e s s  parameter  "a":  a = -j  (6-13)  to  Then the c o n d i t i o n o f e q u a t i o n  (6-12) f o r a t y p i c a l  step length L = 2 x  -2 10  m,  becomes;  a <<  (6-14)  As w i l l l a t e r be shown, t h i s corresponds of  to a rather small  value  ©. F i n a l l y , we  c o n s i d e r what happens  N  when  2 = OJ J. i . e . , a = 1.  © 2  Taking  the l i m i t i n e q u a t i o n  x = x ~  0  + y °  s i n /2  tjz  =—  /2  (6-2), as  • + x °  = -cox z + y cos /2~ coz + — — o 'o "* i£  . /2 + y [ o J  r  s i n / 2 0)Z =  L  co  sin /2  ->w  ©  tl)Z  =—  , we  + —  O  r  get;  .  -|  0)  rcos/2" coz - 11 ~  z] J  for a = 1 Comparing terms, we  —  [ l - c o s / 2 coz]  note t h a t the dominant term becomes;  (6-15)  - 52 -  x » 0[r L  r + — o to  1  r y « -((DX + * )z + 0 [ r + — — o 'o o t o v  L  ]  J  (6-16) ^ ^  for (<o,x + y ) z | >> 0 [ r ^ o o o 1  For  L  +  r — to  ]  (6-17) J  the v a l u e s quoted e a r l i e r , i . e .  r r  o o  «  10  «  10  «s  2 x 10  4  m  -2 L  Condition  m  (6-18)  (6-17) becomes  z »  3 x 10 m 3  (6-19)  Under t h e s e c o n d i t i o n s i t can be seen t h a t the motion i s v e r y soon a simple s p i r a l  i n the l a b o r a t o r y frame, w i t h the s p i r a l  with ' the h e l i x and the r a d i u s growing l i n e a r l y w i t h (z) . 1  synchronous  CHAPTER 7 The T r a j e c t o r i e s o f an Ion w i t h Non-Zero Magnetic Quadrupole f o r Weak L o r e n t z  In t h i s  Moment i n a DC H e l i c a l  Forces  c h a p t e r an approximate e q u a t i o n o f motion i s developed  f o r t h e r a d i a l motion o f an i o n w i t h non-zero magnetic moment i n a dc h e l i c a l quadrupole  system which i s v a l i d when t h e L o r e n t z f o r c e i s  comparable t o , o r weaker than, the S t e r n - G e r l a c h f o r c e . t h a t the L o r e n t z f o r c e i s zero((B> exact t r a j e c t o r i e s  of a p a r t i c l e  f o r c e g i v e n by e q u a t i o n In o r d e r f o r (Q)  ->  In the l i m i t  0 ) , t h e e x p r e s s i o n s y i e l d the  e x p e r i e n c i n g t h e average S t e r n - G e r l a c h  (5-13) .  t o be s m a l l w h i l e m a i n t a i n i n g a l a r g e S t e r n -  G e r l a c h f o r c e , o r t h o g o n a l magnetic and e l e c t r i c f i e l d s must be u t i l i z e d as shown i n F i g . 5 - l b .  In t h i s way the L o r e n t z  force;  F = q (E + v x B)  can be made a r b i t r a r i l y s m a l l f o r a p a r t i c u l a r i n g a l a r g e magnetic f i e l d  gradient.  i s p r o b a b l y not d e s i r a b l e t o reduce  while maintain-  From a p r a c t i c a l p o i n t o f view i t t h e L o r e n t z f o r c e much below a few  p e r c e n t o f i t s v a l u e f o r t h e magnetic f i e l d r e q u i r e an extremely  velocity,  a l o n e , s i n c e t h i s would  monochromatic beam and a l s o a v e r y h i g h  t o l e r a n c e i n the a p p a r a t u s .  mechanical  - 54 -  In Chapter quadrupole f o r small  6 i t was shown t h a t the t r a j e c t o r i e s i n a h e l i c a l  i n the absence o f t h e S t e r n - G e r l a c h f o r c e a r e n e a r l y p l a n a r ©  and near a x i a l p a r t i c l e s .  Under these c o n d i t i o n s the  r a d i a l dependence o f t h e t r a j e c t o r i e s i s g i v e n by e q u a t i o n  (6-8);  (Note 1)  ji°\  2  2sF  0  ©_z_  sin  2co  +  ,  r  G  o \  ( 7  _  1 }  v  —  < K  J  1  CO  where r(z=0) = r J  o  •  dr,  j—(z=0) = r  dz  v  o  J  T h i s e q u a t i o n i s t h e s o l u t i o n t o the e q u a t i o n o f motion;  2  i-  H  =  m V  d r  ,_  2  (-^  ~2  o  7  dz  2  where; F„ = -mv H o 2  (^—) (r-r ) 2co o  In t h e r e g i o n i n which e q u a t i o n  J  (7-1) i s v a l i d , we can approximate  the L o r e n t z f o r c e i n the h e l i c a l quadrupole the average  (7-3)  2  by  ( e q u a t i o n 7-3) . I f  S t e r n - G e r l a c h f o r c e ( e q u a t i o n 5-13) i s added, a one  d i m e n s i o n a l e q u a t i o n o f motion i s o b t a i n e d which y i e l d s the d e s i r e d Note 1:  r as used here i s s t r i c t l y a one d i m e n s i o n a l c o o r d i n a t e r a t h e r than | r | , s i n c e both p o s i t i v e and n e g a t i v e v a l u e s a r e a l l o w e d .  -  r a d i a l dependence o f the  55  trajectories in this  ® 2  d r 2 dz 2  -  ,  + ( 2(j) )  =  r  5l 3 _ _ 2E U  G B  l +  region:  ® 2 2co (__) r o  (7-4)  where = Y^ 3 J  s  i  n  , (Note  6  2) (7-5)  ^  The  =  l ( s i g n r)^(Note  displacement r r  o  and  the  u r  It  = r  Note 3 :  +  (^  s h o u l d be  mations o n l y  Note 2:  (7-4)  s o l u t i o n to e q u a t i o n  ) 3 Z  initial  |G  | [1  3)  i s expressed i n terms o f the  radial velocity r  <Q> z - cos - 5 -  ] +  o  as  2(or - sin  initial  follows;  <0)  z  kept i n mind t h a t these s o l u t i o n s are v a l i d  (7-6)  approxi-  for;  In Chapter 5 we have shown t h a t the S t e r n - G e r l a c h f o r c e w i l l be r a d i a l l y inwards or outwards depending on the o r i e n t a t i o n o f the magnetic moment w i t h r e s p e c t to the e f f e c t i v e magnetic f i e l d i n the system. The r a d i a l sense o f t h i s f o r c e w i l l not change i f the a d i a b a t i c c o n d i t i o n i s not v i o l a t e d . However, i n t h i s one d i m e n s i o n a l f o r m u l a t i o n the a d i a b a t i c c o n d i t i o n can be v i o l a t e d i n the a x i a l r e g i o n s i n c e the e f f e c t i v e f i e l d goes to zero as the p a r t i c l e c r o s s e s the a x i s . Thus, a s s o c i a t e d with J 3 i s a p r o b a b i l i t y t h a t i t may change i n t h i s r e g i o n i f the a d i a b a t i c condition i s violated. The f a c t o r 3 must be i n s e r t e d because i n t h i s one d i m e n s i o n a l r e p r e s e n t a t i o n the s i g n o f the S t e r n - G e r l a c h f o r c e depends on the s i g n of r .  - 56 2cof  I'd « '  —  0 ' (7-7)  T  < K  1  Note t h a t i n the l i m i t o f  ©  ->  0, we  o b t a i n the exact  trajectories  f o r the average S t e r n - G e r l a c h f o r c e alone:  r = r  |G | + r z + — \ z ; o o 4E  <0>  r  = 0  (7-8) ^ '  T h i s would apply to a n e u t r a l p a r t i c l e i n a magnetic h e l i c a l quadrupole,  or t o a charged  i s complete f o r o r t h o g o n a l A focal  p a r t i c l e when the Lorentz f o r c e c a n c e l a t i o n e l e c t r i c and magnetic f i e l d s .  l e n g t h i s d e f i n e d by;  r(z ) f  From e q u a t i o n  = r  (7-9)  o  (7-6);  I  r © tan  z  Er  f  — =  -  (0)  —nTT-r  °  <<  i  2tor - o  ©  /jx  (7-10)  CO  A l s o note from e q u a t i o n  (7-6) t h a t a l l p a r t i c l e s are r e f o c u s e d  for: © 2o  z 2m;  n = 1, 2, 3  S i n c e t h i s f o c a l p o i n t i s independent o f u^, no S t e r n - G e r l a c h  (7-11)  effect  - 57 -  would be observed. From e q u a t i o n  (7-8) the f o c a l  length f o r ©  = 0is;  4Er z  This f o c a l  f = - u ^ r- o  =  (- )  0  y 12  l e n g t h i s o n l y d e f i n e d f o r u^ < 0.  (7-10) y i e l d s  the change o f f o c a l  length  Differentiating  equation  (Az^) f o r a change i n u^;  (7-13)  Or, expanding about u^ = 0;  Az  a) |G |Au ^ f-  =  o From e q u a t i o n  (7-14)  O  Er  1  (7-6) the maximum e x c u r s i o n , r  n  max  , occurs  for z = z , m  where;  o  zm  =  From e q u a t i o n  2u> .  ©  ,  -1 , tan (  * E © o  S 3l l U  G  B  <  ( 7  (7-6), i f u^ = 0 the f o c a l  2lT(±)  o  ©  I I - I --I  length i s ;  I  r-  "  1 5 )  - 58 -  From equation (7-6), i f u^ = 0 the focal length i s ; 3  u  °  =  2uf  2 IT CO  (7-16)  o «  1  Then equation (7-14) can be written:  Az-  f  i  G B  Er o  l  L  a  o  A U  5  (7-17)  4TT  D i f f e r e n t i a t i n g equation (7-6) yields the change i n radius f o r a change i n u :  Ar r(u =0) 3 v  |G |L R  AU,  2TT aEf o  tan  (7-18)  4cb  From equations (7-17) and (7-18) i t can be seen that the r e l a t i v e effect on the t r a j e c t o r i e s f o r a change i n u^ scales approximately as;  IS|L_ aE  In Chapter 9 these parameters are discussed from the experimental point of view, considering such things as the power requirements and cooling capcities needed to produce a given value of G , D  and the  D  energy range and collimation range i n which a beam might be produced.  - 59 -  This j u s t i f i e s , to some extent, the values chosen f o r the t r a j e c t o r i e s i n F i g . 7-1, but, of course, only a successful experiment i s the f i n a l justification.  C l e a r l y , one desires large [Gg|L, but small aE.  Chromatic aberation i s present i n these systems.  Both the para-  meters "E" and "a" change i f the energy changes. We have;  T  2  fZEp (G + Jm G) ^E ~B c  8TT  2  E  (7-19)  D  D i f f e r e n t i a t i n g , this becomes;  ^ " . ^ [ 3 ^ . ( 1 1 a  E  L  1  /  . 2 „ d m a  IOTT  E  . !]  ( 7  J  .  2 10 )  '  If the term i n the brackets i s zero, the chromatic effect of "a" vanishes This i s the condition for minimum chromatic aberration i n the system. [ 2 ?  It i s useful to solve f o r the product L J — from equation (7-20) N m Aa with — = 0, i n the following form; cl  16TT  ^ m  2  (L/a)(G /E)q  ^ ~  R  2 l )  This represents the condition for minimum chromatic aberration f o r given (L/a) and (Gg/E).. Note that equation (7-6) may be written i n terms of the r a t i o s i  (L/a) and (|G |/E), i . e . ; B  7 r = r  Q  + (L/ays~-  sin(a/L)irz  u  3  (|G |/E)[l-cos(a/L)T7z] + (L/a)«r B  o  (7-22)  - 60 -  F i g u r e s 7-1 t o 7-4 a r e p l o t s o f t h e t r a j e c t o r i e s o b t a i n e d equation  (7-6) f o r v a r i o u s v a l u e s o f (L/a) and (|Gg|/E).  from  Each f i g u r e  shows s i x t r a j e c t o r i e s , o b t a i n e d from the t h r e e i n i t i a l c o n d i t i o n s r  • = 0, r  o  o  =10  -5  ,10  -4  ,10  -3  , and two v a l u e s o f t h e magnetic moment •  u_ = +u , and -u . 3 o o The  c o n d i t i o n f o r minimum chromatic  a b e r r a t i o n ( e q u a t i o n 7-21) i s  i n o r d e r , f o r f i g u r e s 7-1 t o 7-4;  L j ^  This w i l l  = 23.7; 15.8; 7.9; 1.6.  i l l u s t r a t e that i t i s very d i f f i c u l t  t o s a t i s f y the  c o n d i t i o n f o r minimum chromatic a b e r r a t i o n , e s p e c i a l l y i f the product LG ) i s l a r g e , which i s the case when the S t e r n - G e r l a c h e f f e c t i s B  _2 large. He  +  For example, i n F i g . 7-1, i f we make L = 2 x 10  beam would r e q u i r e an energy  condition.  It i s s t i l l  o f about.02 eV t o s a t i s f y  As an example o f t h e chromatic From e q u a t i o n  —  a  this  h a r d e r t o s a t i s f y the c o n d i t i o n f o r t h e other  f i g u r e s , a l t h o u g h p a r t i c l e s o f l a r g e r mass than H e  F i g . 7-2.  m, then a  effect  (7-20) we o b t a i n :  - 11.5  E  +  would h e l p .  i n these examples, c o n s i d e r ,  K  (7-23) J  In t h i s case t h e dominant chromatic a b e r r a t i o n occurs f o r t h e " a " terms. For a He eV  +  beam, " t h e r m a l " e n e r g i e s correspond t o E -7.4 x 10  (300°C); f o r a 2 eV beam t h i s g i v e s ;  -2  - 61 -  P u t t i n g t h i s i n t o e q u a t i o n (7-23) we have f o r t h e parameters  used i n  F i g . 7-2;  —  a  »* 50%  T h i s p o i n t s out the f a c t t h a t the chromatic e f f e c t s due t o a thermal spread i n the beam energy w i l l  cause  large trajectory modifications.  In t h i s case they a r e as l a r g e as those caused by changing  " a " by 50%.  O b v i o u s l y , t h e chromatic e f f e c t s must be c a r e f u l l y c o n s i d e r e d i n a d e t a i l e d design.  Some v e l o c i t y s e l e c t i o n might prove u s e f u l , but i t  seems more l i k e l y t h a t i t would not be n e c e s s a r y i n many cases s i n c e such a l a r g e S t e r n - G e r l a c h e f f e c t can be o b t a i n e d . F i g u r e s 7-5 and 7-6 i l l u s t r a t e a simple system  o f s t o p s which  c o u l d be p l a c e d a t t h e e x i t o f the S t e r n - G e r l a c h p o l a r i z e r r e g i o n t o o b t a i n beams p o l a r i z e d i n e i t h e r o f t h e two senses shown s c h e m a t i c a l l y i n t h e diagrams. depending  O b v i o u s l y , o t h e r systems o f stops can be d e v i s e d ,  on the degree  o f p o l a r i z a t i o n which i s r e q u i r e d .  d e p i c t e d i n F i g . 7-5, f o r example, which i n v o l v e a hollow  The c o n d i t i o n s converging  i o n beam would g i v e 100% p o l a r i z a t i o n f o r both p o l a r i z a t i o n  senses,  whereas removing the r e s t r i c t i o n o f h a v i n g a h o l l o w beam produces p o l a r i z a t i o n f o r one sense o f p o l a r i z a t i o n but o n l y p a r t i a l  100%  polarization  f o r the o t h e r . An i n h e r e n t advantage which these systems possess  i s t h a t the i o n  beam i s guided by the combined L o r e n t z and S t e r n - G e r l a c h f o r c e s a t a l l  - 62 -  times.  T h i s means t h a t p e r t u r b i n g f o r c e s which are bound t o be  a r e not n e c e s s a r i l y s e r i o u s , s i n c e a s m a l l beam displacement o p t i c a x i s w i l l be By extending  present  from  the  compensated f o r by an a d d i t i o n a l f o c u s i n g f o r c e .  the s t r o n g f o c u s i n g Lorentz  r e g i o n on both ends one  f o r c e beyond the  can a l s o p r e - f o c u s  I f v e l o c i t y s e l e c t i o n was  and p o s t - f o c u s  polarizing the beam.  r e q u i r e d t h i s c o u l d be accomplished i n a  s e c t i o n o f the h e l i c a l quadrupole p r i o r to the p o l a r i z i n g r e g i o n  by  u s i n g the v e l o c i t y dependence of the f o c a l p o i n t s . In Chapter 9 the experimental field  g r a d i e n t Gg,  and  problems o f a c h i e v i n g a h i g h magnetic  a low beam energy E, are c o n s i d e r e d .  9-3,  i f we  assume E = 2 eV,  Gg/E  = 500  gauss/cm/eV f o r a p e r t u r e s  2 cm.  I t i s much h a r d e r  i t seems v e r y easy to produce a v a l u e  high f i e l d  gradients  imposed by the L/a may it  o f a few mm  to produce Gg/E  should be p o s s i b l e i f the a p e r t u r e step l e n g t h i s used.  and  step l e n g t h L =  = 5000 gauss/cm/eV, but i t  i s as s m a l l as p o s s i b l e and  Pulsed operation i s very s e n s i b l e f o r s i n c e i t l e s s e n s the fundamental  Any  (0 < a < 1).  Of  t o have very good o r t h o g o n a l i t y between the  and magnetic f i e l d s  i f very small values  a long  producing  restriction  l i m i t e d c o o l i n g c a p a c i t y o f the system.  be produced w i t h i n the range o f " a "  i s necessary  From F i g .  o f " a " are used.  value  of  course, electric  - 63 -  Figure 7-1.  A plot of t r a j e c t o r i e s from equation (7-6).  The dotted  t r a j e c t o r i e s are those obtained when the adiabatic is s a t i s f i e d as the p a r t i c l e crosses the axis.  condition  The s o l i d  t r a j e c t o r i e s are those obtained when the sign of J3 changes as the p a r t i c l e crosses the axis.  Figure 7-2.  Trajectories  similar to those of Fig. 7-1.  1.61 IZ  .8  •r  Radius Q-  s  Optic Avcis ttie+ers 6  •  /  /  V  .61 1.2  1.6.  Figure 7-3.  Trajectories similar to those of Fig. 7-1.  /  S  Figure  7-4.  Trajectories similar to those of F i g .  7-1.  - 66 Entrance  ^rture  ^tf/  Rtrtn.  collector passes Complete l|j polarcjccl inner beam  ^ ^ D i s k collector passes Completely polarised  ow+er beam convc/gm^ unpblav-i^ecf ior\ beam Figure 7-5.  An i l l u s t r a t i o n of a system of stops which would give 100% p o l a r i z a t i o n of the emerging beam.  /  . Km/) collector passes partly  inner  potanjed  beam  Dr-sk. collector passes Completely polarised outer beam  Sold  corvuera »nj  polarised i o r \ beam Figure 7-6.  A somewhat simpler system to that i l l u s t r a t e d i n F i g . 7-5 which would give p a r t i a l p o l a r i z a t i o n of the inner beam.  CHAPTER 8 The  I n f l u e n c e o f t h e S o l e n o i d F i e l d on t h e T r a j e c t o r i e s , and t h e S t e r n -  Gerlach E f f e c t  f o r Large  Lorentz Forces i n a DC H e l i c a l  Quadrupole  In t h i s c h a p t e r we wish t o compare the i n f l u e n c e o f t h e s o l e n o i d field  B  q  on the t r a j e c t o r i e s t o t h a t o f the S t e r n - G e r l a c h f o r c e , t o  determine,  f o r one t h i n g , i f i t i s p o s s i b l e t o u t i l i z e the resonant  n a t u r e o f the s i n 9 term i n the average S t e r n - G e r l a c h f o r c e 5-13).  (Equation  We a l s o wish t o e s t i m a t e t h e S t e r n - G e r l a c h e f f e c t when the  Lorentz f o r c e i s l a r g e compared w i t h the S t e r n - G e r l a c h f o r c e i n the dc h e l i c a l quadrupole system. d o u b t f u l t h a t t h e resonant  Our c o n c l u s i o n s a r e r a t h e r n e g a t i v e .  e f f e c t o f t h e s i n 0 term can be d e t e c t e d  by simply sweeping t h e f i e l d (sin  0=1)  It i s  B  q  on and o f f the resonance c o n d i t i o n  because the t r a j e c t o r i e s a r e m o d i f i e d too much by the  r e s u l t i n g change i n t h e Lorentz f o r c e . Stern-Gerlach e f f e c t  I t i s a l s o doubtful that the  i s u s a b l e when l a r g e Lorentz f o r c e s a r e p r e s e n t  i n the dc h e l i c a l quadrupole.  We show i n t h i s chapter t h a t the S t e r n -  G e r l a c h e f f e c t i s g e n e r a l l y s m a l l , f o r t h i s case, and t h a t t o d e t e c t i t would r e q u i r e a p r o h i b i t i v e l y s t a b l e and m e c h a n i c a l l y and a v e r y monochromatic beam.  accurate  In t h e l i g h t o f Chapter  apparatus,  7, i t appears  t h a t the key t o a s u c c e s s f u l experiment i s the near c a n c e l l a t i o n o f t h e L o r e n t z f o r c e u s i n g o r t h o g o n a l magnetic and e l e c t r i c  fields.  - 68 -  For v e r y s m a l l v a l u e s o f " r " the S t e r n - G e r l a c h f o r c e w i l l the motion, s i n c e the L o r e n t z f o r c e goes t o zero on the o p t i c but  dominate axis,  i t should be remembered t h a t the S t e r n - G e r l a c h f o r c e i s a t a l l times  very small.  In equations  (8-16),  (8-17), and  (8-18) we  have  e x p r e s s i o n s f o r the r a d i u s r * which marks the t r a n s i t i o n between the r e g i o n o f space i n which the L o r e n t z f o r c e deominates and t h a t i n which the S t e r n - G e r l a c h f o r c e dominates.  I f the S t e r n - G e r l a c h f o r c e  dominates, the S t e r n - G e r l a c h e f f e c t w i l l be l a r g e , but as we w i l l the r a d i u s r * u s u a l l y i s extremely  small.  O u t s i d e o f the r a d i u s r * , i n  the Lorentz f o r c e dominant r e g i o n we have made a simple to  show,  approximation  the e q u a t i o n o f motion f o r a non-zero magnetic moment t o e s t i m a t e  the minimum S t e r n - G e r l a c h Although  effect.  t h e average S t e r n - G e r l a c h f o r c e i s o f a v e r y simple form,  b e i n g r a d i a l and o f constant magnitude, a n o n - l i n e a r e q u a t i o n  results  when i t i s i n c o r p o r a t e d i n t o the e q u a t i o n o f motion. From e q u a t i o n  (5-13) we have;  < F  SG  >  ^* 3  =  J  S  i  n  9  I S I)  7  (8-1)  we d e f i n e ;  K  = YnJ  3  sine|G | B  (8-2)  F o l l o w i n g t h e development o f the e q u a t i o n o f motion i n appendix B, the r e s u l t i n g ; equations when F <  c ; r  >  i s i n c l u d e d are  - 69 -  x - 2Wy  ~  2 -  i  lA2,_l  - (w + © - -A ) +—V") v 4 mv o mv 2r r ,o mv  x=  0  o  ?.+ 2Wx - (W - © - j(|p> -V 2 2  }  2+  mv  o  (8-3)  =0  r  The presence of the £/r term makes these equations non-linear. The basis of most of this chapter i s the simple approximation of replacing 1 / r by l/<r>, where <r> i s a constant, representing a radius averaged over the length of the trajectory of the p a r t i c l e . replacement  reduces the equations  (8-3)  This  to the linear equations  ( b r l 3 )  by making the i d e n t i f i c a t i o n :  $ = 5/<r>  If we put <r> =  r m a x  >  (8-4)  this amounts to replacing the Stern-Gerlach  force by a force everywhere smaller, and w i l l c e r t i a n l y give a minimum f o r the Stern-Gerlach e f f e c t . We cannot j u s t i f y this approximation any further.  It seems very  reasonable, and since the r e s u l t s are not very encouraging we are not inclined to carry the analysis any further at this time, since an experiment  based on the alternative presented i n Chapter 7 appears to  be very promising.  Nevertheless, t h i s kind of analysis should be quite  helpful i n conjunction with computor calculations of the t r a j e c t o r i e s . We believe that a computor used alone may not give as much insight as approximate solutions such as these. It i s helpful to do some preliminary analysis of the equations (b-13)  which we w i l l use as the basis for our approximate solutions.  - 70 These i n c l u d e the term $, and R^'  and  R^'  can be  solved exactly.  I f both the  are r e a l , the t r a j e c t o r i e s are bounded, and  are as g i v e n  i n equation  (b-22) .  the  roots  solutions  Note t h a t the term $ can change the  f o c a l l e n g t h , the maximum e x c u r s i o n ,  or cause a r o t a t i o n about the o p t i c  axis. I f one no  or both of the r o o t s become complex, the t r a j e c t o r i e s  l o n g e r bounded, s i n c e d i v e r g e n t F i g . 8-1  terms appear i n the s o l u t i o n s .  i n d i c a t e s the r e g i o n s where the r o o t s R^'  r e a l or complex.  The  l i g h t regions  the shaded r e g i o n s  or both a r e complex, c o r r e s p o n d i n g  to divergent  T h i s f i g u r e i s d e r i v e d i n the f o l l o w i n g  way.  The  B:  r o o t s may  and  R2'  i n d i c a t e t h a t both are r e a l ,  correspond to bounded s o l u t i o n s , and one  are  be w r i t t e n , from appendix  are and  i n d i c a t e that  solutions.  (8-5)  or; W(l  - ab - a  J  1 - 4b/a  )  1/2 (8-6)  where;  - 71 -  F i g u r e 8-1.  A c h a r t o f the r e g i o n s or complex.  where the r o o t s R  The l i g h t r e g i o n s  r e a l , and the shaded r e g i o n s are  complex.  1 1  and R ' 2  are r e a l  i n d i c a t e t h a t both a r e  i n d i c a t e t h a t one or both  - 72 -  b = f/  (8-8)  Q  qB  i  (G„ + v G ) [*/* L V / M  c  The i n s i d e  radical  -  2  4m  ]  ( - ) 8  9  i s imaginary f o r ;  o < a < 4b (8-10) -4b < a < o  The  line,  a = 4b  (8-11)  m a r k s t h i s b o u n d a r y i n F i g . 8-1. The e n t i r e yields;  r o o t may R  I 1  be z e r o .  Putting  = 0  a =  for  equations b <;  (8-6) e q u a l t o z e r o  1/2 (8-12)  a  R  2  • = 0  for  = ITb  b  a = - ^ r 1-b  * "  b >  1 / 2  1/2 (8-13)  a = ^  These l i n e s  b ,  -1/2  a r e d r a w n i n F i g . 8-1.  The s y m m e t r i e s i n t h i s example, t h a t t h e lower h a l f The t r a n s f o r m a t i o n ;  s y s t e m c a n be u t i l i z e d . o f F i g . 8-1  T h e y show, f o r  i s redundant.  - 73 -  "*  I X  y  (8-14)  w -> -W © leaves the equations of motion (equation b-13)  invarient.  This i s also  true, of course, for the solutions. Since the interchanging of the x and y axis corresponds rotation of /2 v  to a  about the optic axis, or a t r a n s l a t i o n of L/4 along  the optic axis; the transformations W -»» -W and  ©  -©  are  equivalent to one another, except for a rotation or t r a n s l a t i o n . Also, a h e l i c a l f i e l d has the property that the transformations V  Q  -V  q  and co  -to  are equivalent, except for possible rotations  about, or translations along, the optic axis. Since the roots R-^' and R^'  do not change for any of these  transformations, the solutions can d i f f e r only i n the i n i t i a l conditions which they represent.  For a beam with c y l i n d r i c a l symmetry no change  would be noticed for the above transformations, since a l l i n i t i a l conditions are represented.  Thus the lower h a l f of F i g . 8-1 i s  redundant since i t represents one of the above transformations. Referring to F i g . 8-1, a l i n e defined by  "a" = constant  crosses three boundaries between the bounded and unbounded solution regions, provided |a| < 2.  At each boundary, where a root or r a d i c a l  is zero, we can solve for $ from equation (8T9) , (8-11), (8-12) and  - 74 -  (8-13).  This value of $ defines a r a d i a l force which i s the t r a n s i t i o n  between a region i n which the r a d i a l Lorentz force dominates and a region i n which the $ force dominates.  Since these r e l a t i o n s are  independent of z, we can equate:  $* = £/r*  (8-15)  where the asterisk denotes that a root or r a d i c a l i s zero.  This  r e l a t i o n defines a radius, which marks the t r a n s i t i o n i n space between the region i n which the Lorentz force dominates and that i n which the Stern-Gerlach  force dominates.  roots, the Stern-Gerlach  This means that near the zero of the  force can p r e c i p i t a t e or delay the onset of  the divergent behavior within this radius. Using equation  (8-15) to define r * , and solving f o r §* from  equations (8-9), (8-11),  (8-12), and (8-13) we obtain;  (a W 2  E  e V ) mv o  2 +  for the l i n e a = 4b i n F i g . 8-1, and;  r * 1  £  o r  the line a =  = •  ^ I 1 7 2E((l-I)aW  2  —  (8-17)  l/4(^) )) 2  +  i n F i g . 8-1, and;  r 3*  (8-18) , 2E(-(1 + 4- )aW a  qB + 1/4 ( — ) ) mv o Z  - 75  f o r the l i n e a =  i n F i g . 8-1  The S o l e n o i d F i e l d B o The i n f l u e n c e o f t h e s o l e n o i d f i e l d  B on t h e S t e r n - G e r l a c h e f f e c t o  can be e s t i m a t e d from t h e e q u a t i o n s i n appendix r o o t s o f t h e e q u a t i o n o f motion i n t h e terms,  B  appears i n t h e  q  ( e q u a t i o n s b-14 and b - 1 6 ) ;  qB Tr- 2mv o  W = OJ + —  mv  B.  (8-19)  2  1/4  2  ^0,2  ( mv o  o  In terms o f t h e d i m e n s i o n l e s s parameters  —  (  8  -  2  0  )  " a " and "b" o f F i g . 8-1, t h e s e  are; q(G a  =  ®/W  + v G )  =  Z  ^  —  O  2  1  )  W mv o qB - -ii-3  2  = /®  b  =  f  ( G  t  Y  G  E  Clearly, B  q  ]  b / q o B  (8-22)  becomes t h e dominant term i n e q u a t i o n (8-22) when;  ^  ,  2  I  and, i n e q u a t i o n ( 8 - 1 9 ) , B  ,  l*/ql  q  i s t h e dominant term f o r  I  > it  qB  2m^T o  , >  (8-23)  ^  - 76 -  The Stern-Gerlach force can be approximated by replacing $ with 5/<r>, where <r> i s approximately the aperture size of the apparatus, as outlined at the beginning of t h i s chapter. For a beam of p a r t i c l e s described by;  He  +  E  ions = 2eV  yftJg  = U  L  q  (Bohr magneton)  = 2 x 10"  (8-25)  m  2  sine = 1 and putting -4 <r>  =10  m  equation (8-23) becomes; B or,  for G  D  2 q  > 9.6 x 10" |G | (MKS)  (8.26)  8  B  = 100 gauss/cm;  D  |B | >  3.1 gauss  (8.27)  2600 gauss  (8.28)  and equation (8-24) becomes;  |BJ  >  It can be seen from equation (8-27) that rather small values of B  o  w i l l dominate the Stern-Gerlach e f f e c t .  using a swept B  q  This seems to preclude v  mode f o r detecting the e f f e c t , as we w i l l further  illustrate. The reason f o r including the solenoid f i e l d B  q  i s , of course, to  maximize the Stern-Gerlach effect v i a the term s i n 9 (eqn. 5-14) . i l l u s t r a t e , we can calculate B  To  to maximize s i n 6, and also determine  -  77  -  the resonance width, f o r the case with; _2  L  =  2 x 10  G„  =  1000  E  =  2eV  +  =  beam ion  =  u  =  6 x 10  He  YftJ r  3  max 1  ^  «  L  cm  g/cm  ( g  _  2 g )  Q  -4  m  2OJV - o  These values could apply, f o r example, to the t r a j e c t o r i e s drawn i n Figs. 7-1 to 7-3. Putting Aco = 0 to maximize s i n 6, y i e l d s , from equations (5-7) and (5-9);  B  Q  =  -- 278. g ^ s s  (8.30)  The half-width at half-maximum of s i n 9 i s ;  (Aco), = /  ±/3to,  (8-31)  It i s , of course, determined by the maximum radius of the p a r t i c l e -4  trajectory, and f o r the example c i t e d ,  B (r) n  1  max  = G r B max D  T m  a  x  = 6 x 10  m ; we have;  (eqn. 5-6) n  J  (8.32)  = 60 gauss Therefore the half-width i s ;  - 78 -  (Aw). ,  ^J-=-  Comparing equations  ~  100 gauss  (8-33)  (8-30) and (8-33), we can conclude that the  resonance i s e s s e n t i a l l y centered about B =0, and that this broad ' o resonance would be d i f f i c u l t to distinguish from the trajectory changes produced by the Lorentz forces for changing B , as noted  earlier.  It i s possible to design the apparatus to enhance the resonance effect i n order to simplify detection.  This could involve chosing an  ion with a smaller y, increasing u>, increasing E, decreasing r  , or  m£ix  decreasing Gg.  These steps a l l seem to reduce the overall e f f e c t ,  so the compromise which must be made, s a c r i f i c i n g the size of the signal in order to sharpen the resonance, must be decided e n t i r e l y i n terms of signal to noise with whatever detection scheme i s proposed. We have treated the case for small consider the remaining range of ©  ©  i n Chapter 7, now we  using the approximation outlined  in the introduction to this chapter. The easiest way to do this i s to put G„ = B =0. This does not leave out much information for the E o r  p r a c t i c a l reason that the magnetic f i e l d gradient G to make © ©  large.  alone i s s u f f i c i e n t  The only reason for introducing G„ i s to reduce  , to minimize the effects of a large ©  on the Stern-Gerlach e f f e c t .  The effect of B has been considered separately i n this chapter by q  noting when i t becomes the dominant term i n the expressions f and W. The simple magnetic h e l i c a l quadrupole  B  o  =-G  E C  =  0  - 79 The term "b" has a simple interpretation  for this case.  From  equation (8-9) we have;  b =  0  (8-34)  This i s just the r a t i o Of the r a d i a l force term ($r) to the Lorentz force (qGgV r). Q  I f $ represents the Stern-Gerlach force, then one  would expect "b" to be rather small. If we take, as t y p i c a l values, the experimental parameters of equations (8-25), and put $ = £/<r>, then;  b ~  6 x 10  -5  (8-35)  The region of interest i s c l e a r l y close to b = 0 i n F i g . 8-1. Some simple estimates of the effects on the t r a j e c t o r i e s due to d i f f e r e n t values of "b" can be made. D i f f e r e n t i a t i n g equation (8-6) we get, for constant "a";  l'  - a)  /l-4b/a  (8-36) (  2 /l-4b/a  + a)  If we consider only changes about b = 0,'and f i n d the r e l a t i v e change by dividing through by the root R ' or R ' we get;  - 80 -  A  l '  R  Ab  2-a.  -R7- T T^> {  , _  -  B  n  0  (8-37) A  D  !  2  fl  Ab  2+a.  =" —  b  _  " ° n  Except f o r the regions near a = ±1, these terms are very small since Ab i s very small.  Using the approximation $ = £/<r>, and  d i f f e r e n t i a t i n g equation (8-34)  A  b  =  (  sin 8 qv <r> ' o  Yft  n  3  N  G. B  Using the parameter values of equations (8-25), and putting yft-AJ 2u , we obtain: o  b % 10  From equations (8-37) and  (8-39)  4  (8-39) i t i s obvious that the trajectory 4  changes that t h i s w i l l cause w i l l be of the order of 1 part i n 10 , unless "a" i s very near ± 1. 4 If we suppose that the focal point changes by 1 part i n 10  due to  the Stern-Gerlach e f f e c t , then the most obvious d i f f i c u l t y i n detecting i t i s the chromatic aberration of t h i s system.  If we take as a f i r s t  approximation, that the f o c a l length varies as (equation 6-10); 2Trcomv  0  f  ©  q G  B  then the v e l o c i t y of the beam and hence i t s energy must be constant to  4 better than about 1 part i n 10 .  This r e s u l t seems t y p i c a l of operation  i n the region between very small "a" (a ^ 0) and large "a" (a = ±1). That i s , a small effect is present, but to detect i t requires a nearly monochromatic beam. From equation (8-17), for a - 1/2,  yf^J^ = U , E = 2eV, B Q  Q  = 0,  we  obtain;  r * % 10 2  m  Thus, only within this very small radius would we expect a large Stern-Gerlach effect for the operating region just discussed, which -4 i s consistent with our conclusion that at a radius of 10  m, only a  very small effect i s present. Now  consider the regions very near a = ±1. I f we put a =  note that for small "b", the roots given by equation expanded.  ±1, we  (8-6) can be  Consider the case for a = 1 then; R  1  =  R'  *  I  2  w/b u/2-3b  (8-41)  2 also  6  _  (from equation  b-19)  (from equation  b-15)  2 © Put x  o  = y o J  -co  = 0  Then for b > o, the roots are r e a l , and we can simplify equation with the above terms:  (b-22)  - 82 -  x-x  •  o ,  s i n yzuz javT  0  A -  —  " — + — (coswvb: z - cos/2 wz) a  "  ~  -  x  (8-42)  y  — (cos/ET  (JJZ  a  "  - cos/2" ojz) +  -  with  —  f/2~ sin/2~ OJZ - ^ ' ^"  a  S  N  /  /F  4b << 1 a B  o  =1 =0  b > 0 Since /b" <</2, the focal length of these equations i s very near;  z  - — 2 -  £  (8-43)  If we put $ = 5/<r> to represent the Stern-Gerlach force over this focal distance, we can estimate "b" as i n equation (8-35), for " t y p i c a l " parameter values such as those i n equation (8-25) . Taking; b = 10  -4 (8-44)  L = 2 x 10 m 2  we obtain from equation (8-43)  z  - 1 meter  £  The maximum excursion of the equation (8-42) i s determined mainly by the factor;  K r m  a  x  — — /bco  (8-45)  -  83 -  I f we a g a i n put $ = £/<r> over t h e range o f one f o c a l  l e n g t h ( z , 0 , then r w i l l be somewhat r max  6  g r e a t e r than <r>.  t o r e p r e s e n t the S t e r n - G e r l a c h f o r c e  A conservative estimate i s t o put r max  6  = <r>.  r  we can say, from e q u a t i o n  (8-45), t h a t a l l p a r t i c l e s  Then  entering the  system w i t h i n t h e angle g i v e n by:  *  =  <  r  ^  >  0  (8-46)  w i l l be f o c u s e d w i t h i n t h e d i s t a n c e z^ ( e q u a t i o n 8-46). of equation  Using t h e v a l u e s  (8-44).  y  = TT x 1 0 "  (8-47)  4  T h i s e s t a b l i s h e s t h a t a cone o f p a r t i c l e s w i t h b >o w i l l be f o c u s e d w i t h i n a r e a s o n a b l e d i s t a n c e i n t h i s system. p a r t i c l e s w i t h b < o. so t h a t t h e s o l u t i o n s  The r o o t R^' i n e q u a t i o n s  (8-41) i s imaginary,  (8-42) w i l l be d i v e r g i n g . 2 - 2  r  = x  +  C o n s i d e r now the  Computing;  2  y  we have from equation (8-42) : .2 2  r  f  O  2. 2S in  •  = (-j— + y  Q  • /2oiZ  . 2 ^  O  2—  )  U)  0).  -  2coshto,/ibT'z cos/2coz) - X y Q  -cos/2coz)-  ?  0)  X y Q  +  o  ^  sin/2 coz /jbj" ~  ,2 c  o  s  h  2  rr—rr  i r l  b  l  z  +  c  o  s  *^k  o  —  sin/2o)Z  .  (coshco/|b |' z  n  (8-48) 2  i:  /i^Tcoz  2x y  sinh 2 / | b I toz  z  -  s  Q  o — Tb I v  2  (  9 *o sinh/|b |coz + — ~ ±y " - u |'b|  o 2/2y  ,.  —2  s i n h l /  ;  2  u / | b |'  l b I-  z  cos/fuz  84 where  /|bj  < <  ^  b <o a = 1 B  o  =0  For small z this is o s c i l l a t o r y at  /^UJZ,  but as z gets larger, the  o s c i l l a t i n g terms become less important and the dominant term becomes;  y r 55 1/2  — —  s i n h /fbT" O J Z  (8-49)  If we make the conservative estimate that;  r = <r>  = r aperture  and i f we put $ = ?/<r> to represent the Stern-Gerlach force over a distance equivalent to the focal length f o r p a r t i c l e s with b > o;  v  .jb \£  then equation (8-49) gives a conservative estimate of the minimum angle which a p a r t i c l e may enter the system with  i n order to be expelled  beyond the radius of the aperture i n the distance z^. Using the parameter values of the previous example, except that -4 b = 10 , we obtain from equation (8-49) y J  o  > 5 x 10"  5  This i s a very conservative estimate,  and i t indicates that most  -4 p a r t i c l e s with b = -10  would be expelled from this system out to  -4 the radius of 10  meters within a distance of one meter of f l i g h t .  magnetic h e l i c a l quadrupole i n the a = 1 region and achieve a This example has shown that i t might be possible to operate a p a r t i c l e separation from the Stern-Gerlach  force.  An obvious problem  i s to maintain the a = 1 condition with s u f f i c i e n t p r e c i s i o n . example, noting i n Fig. 8-1,  that the slope of the l i n e  a = 1 i s 1, we can estimate that "a" must be constant  For  ' = 0 at  to at least the  order of "b", i f the apparatus i s to separate the b < o p a r t i c l e s i n a -4 diverging mode, and focus the b > o p a r t i c l e s . 4 must be constant  to about 1 part i n 10  If b ^ 10  or better.  , then "a"  We have, from  equation (8-21); qG a  2 w mv o  Thus both the gradient Gg, and the beam v e l o c i t y V  q  must be  constant  4 to about 1 part i n 10  or better.  We can a r r i v e at the same conclusions from equation (8-17) which for t h i s case can be written; r 2*  B qv (l-l/a) Q  G  (8-50)  B  Putting a = 1 yields r^* = °°, which confirms the r e s u l t that a large effect can be expected for the a = 1 region, and indicates that at the point of i n s t a b i l i t y for the Lorentz force alone (a = 1), any  - 86 - • small additional r a d i a l force i s s u f f i c i e n t to dominate the motion, i n the sense that i t w i l l p r e c i p i t a t e or delay the i n s t a b i l i t y , for any r . Equation (8-50) also y i e l d s the same s t a b i l i t y c r i t e r i o n , for i f -4 we put  = 10  m, the average radius which we have considered, we can -4 solve for "a" to obtain a - 1 + 10 . Thus "a" must be constant to 4 about 1 part i n 10 or better, i f the Stern-Gerlach force i s to dominate -4 out to the radius of 10 m.  CHAPTER 9 Experimental  Considerations  In this chapter some general remarks are made concerning the design of h e l i c a l quadrupoles and some j u s t i f i c a t i o n i s given f o r the choice of the parameter values used i n the examples of Chapters 6 to 8. To begin with we w i l l determine the r e l a t i o n s between step length, aperture size, gradient and minimum power f o r a h e l i c a l quadrupole system. Let R  q  be the radius of a h e l i x ,  l e t L be the step length, and L^ the  t o t a l length as shown i n F i g . 9-1.  Then the t o t a l length of the h e l i c a l  l i n e i s L. ,„; HT'  L, HT  q  L J l T  4TTR o L  q /4 2  +  1  (9-1)  (9-2)  I f d i s the width of a h e l i c a l s t r i p , (Fig. 9-1) then D, the width measured l o n g i t u d i n a l l y i s ;  (9-3)  If four wires are to be wound then the width of each wire cannot exceed  Figure 9-1.  I l l u s t r a t i o n of the dimensions of a h e l i c a l l i n e and a helical strip.  Figure 9-2.  I l l u s t r a t i o n of the dimensions of a rectangular wire h e l i c a l winding.  - 89 -  d*  ° 2 \\ + q /4  (9-4)  R  r  x  2  For a fixed amount of power (P), the maximum magnetic f i e l d gradient (Gg) i s obtained i f the depth of the wire (B) i s 2/3 the inner radius (Rj) of the wire, f o r fixed wire width (d) (see F i g . 9-2), assuming rectangular wire i s used.  To show this we have, from appendix A;  G  D  I a — R o  T  a  /P ^—^  (9-5)  where R i s the radius to the center of the wire and R i s the wire o resistance.  Assume d i s constant, and R^ i s constant, then;  G  R  a  — (R  dG d i f f e r e n t i a t i n g , we f i n d - r ^ —  :  =-  (9-6)  B/2r  +  B  i s a maximum f o r ;  ClD  B  =  2/3Rj  (9-7)  The minimum resistance occurs when B s a t i s f i e s equation (9-7) and "d" s a t i s f i e s the equality i n equation (9-4).  For a single h e l i c a l winding  of s o l i d , rectangular, copper wire, t h i s resistance i s 4(1.7 x 10" )L 8  R . = min  ^ _, 2 o  TTR  (1 + q /4)A 2  v  (MKS) '  v  (9-8)  where A i s a " f i l l i n g f a c t o r " to allow f o r i n s u l a t i o n on the wires.  1  90For the case of wires with a f i n i t e width "d", the factor  s i n <  ^ must  <P  be included i n the expression f o r Gg, from appendix A;  G  b  =  C  (q)  ^  8  X  l  0 2  ~  7  ^  (MKS)  (9-9)  ^o where  (j> = uod  When the space between the wires i s very small,  i s at i t s  smallest value which i s ;  = .63, $  =TT/2  (D = L/4)  (9-10)  1  From equations (9-9) and (9-10), f o r conditions of minimum resistance; 2 4 )G_ R R . ° C (q)  12  (3.94 x 10 P  =  B  M  I  (MKS)  N  (9-11)  To i l l u s t r a t e , consider systems similar to those used as examples i n Chapters 7 and 8.  Typical values are;  -2  L = 2 x 10  -2  m, 4 x 10  -2  m, 6 x 10  m  Gg = 1000 g/cm L  = 1m  T  The radius R  q  may chose to minimize the power, consistent with the  aperture required f o r the beam.  F i g . 9-3 i s a plot of P . from mm for the values given above. r  equation (9-3) versus R  q  - 91 -  |0  I  1  1  1  a  3  4  *  1  6  Ro mm  Figure 9-3.  Plot of minimum power versus wire radius f o r a rectangular copper wire h e l i c a l winding at room temperature, from equation (9-11) .  - 92 -  Quite c l e a r l y , the r e l a t i o n between the aperture size, pitch length, and the power required to produce a given gradient Gg i s an important consideration i n design.  For a given system, the maximum  gradient (Gg) w i l l be determined by the cooling capacity.  The value  chosen for most of the examples i n Chapters 7 and 8 was G  = 1000 gauss/  D  D  cm.  This i s a very conservative value.  It can e a s i l y be produced as  i l l u s t r a t e d i n F i g . 9-3 for aperture sizes consistent with the beam 4 size.  It should be possible to increase Gg to at least 10  gauss/cm  i n a p r a c t i c a l system; which would apply, f o r example, to the t r a j e c t o r i e s plotted i n F i g . 7-4,  f o r a 2eV beam.  Increasing the Stern-Gerlach force ( i . e . , by increasing G ) D  has the  important additional advantage, besides increasing the Stern-Gerlach e f f e c t , that the slow ions w i l l be less subject to stray f i e l d s and " d i r t " e f f e c t s , because of the focusing effect of this force. The Ion Beam From the theory of Chapters 6 through 8 i t i s evident that the ion beam i s a very important consideration i n designing the  apparatus.  Foremost, i t i s necessary to have a beam which i s small i n cross section and well collimated, secondly, i t should be as intense, pure, and monochromatic as possible, consistent with a very low energy. a compromise must be made between these  Obviously,  requirements.  For the purpose of giving examples i n Chapters 7 and 8 we assumed a beam energy of 2eV.  With care, i t seems possible to achieve such a  beam with the required size and collimation, and an intensity measured at least i n tens of p a r t i c l e s / s e c .  It should be kept i n mind,however,  - 93 -  that the Stern-Gerlach effect i s approximately  proportional to  G /E D  D  (see, for example, equation 7-23)  so that reducing the beam energy i s  equivalent to increasing the magnetic f i e l d gradient by a l i k e amount. A large l i t e r a t u r e exists on ion sources, but most of this work has been to develop intense sources of fast ions, and t y p i c a l l y the energy spread i s of the order of a v o l t .  An intense source of slow  Argon ions has been reported by F. Hushfar et a l . (1967). 14 2 a beam i n t e n s i t y of 10  They obtain  particles/cm sec at an average energy of 2eV  using a plasma source and a novel extraction system which allows both electrons and ions to be simultaneously neutralized beam.  extracted i n a space charge  However, the energy spread appears to be large,  about a v o l t , which i s t y p i c a l of most plasma sources. Two  good sources of slow ions adaptable to a wide variety of ions,  are photoionization and electron bombardment sources.  Of the  photoionization can produce the most monochromatic beam.  two,  Weissler  et a l . (1959) using a well defined beam of u l t r a v i o l e t r a d i a t i o n obtained from a Seya-Namioka grating monochromator could produce ions at 9 eV with an energy spread of .04 eV. However, the y i e l d of ions i s much lower than that attainable with electron bombardment. It i s useful to give a b r i e f analysis of the electron bombardment source as a source of ions for this experiment. production i s given approximately  by the  R = n a(v )n v o e e e where  formula;  ° 3 cm sec x  n s  The rate of ion  (9-12)  -  n  o  -  94  = number density of atoms  a(v )  the v e l o c i t y dependent i o n i z a t i o n cross section i n square centimeters 2 n v = the number of electrons/cm /sec e e . ^ e  J  If we consider the i o n i z a t i o n of He to He ,  with a source pressure  +  -4 of 5 x 10  Torr, then;  n  o  = 1.6 x 10  13  atoms/cm  3  -17 2 a(v ) = 3.3 x 10 cm (W.E. Lamb, J r . , and M. Skinner, 6  Typical electron densities are from 1 x 10 16 with a "weak" source up to 38 x 10  16  1950)  2 electrons/cm /sec  2 electrons/cm /sec i n a "strong"  source such as that of Plumlee (1957). I f we assume n v = 5 x 10^^ electrons/cm /sec, then e e 2  R = 2.6 x 1 0  ions/cm /sec  13  (9-13)  3  We now omit a l l d e t a i l s of ion extraction and focusing, and simply assume that the ions a l l originate i n a small sphere which radiates ions at the rate R into the s o l i d angle 4TT .  If we take the  diameter of this sphere to be equal to the diameter of the beam, then we would expect a beam of diameter  .050 cm, and a divergence f  =  =  _3 10  to have an i n t e n s i t y of; -3 2 2.6 x 1 0  13  x 4/3ir (.025) x 3  ( 2  ^°—^—  X  - 540 ions/sec  (9-14)  This performance was achieved at a beam energy of 4eV i n the ion gun described i n appendix C. The rate of ion production given i n equation (9-13) i s t y p i c a l . Lipeles (1966) reports t h i s value i n an ion source similar to that of Novick and Commins (1958). and a divergence of *  q  With a beam diameter of about .4 cm  = 0.6, he obtains a beam intensity of 3 x 1 0 ^  ions/sec. at a beam energy of 10 eV. to be almost  He also measured the energy spread  .4 eV or 4% for his beam.  His beam i n t e n s i t y , compared  with that of equation (9-14) i l l u s t r a t e s the very large decrease i n t o t a l p a r t i c l e s / s e c which one must expect when the same source i s used to produce a smaller diameter and better collimated beam. A very serious problem with slow ion beams i s the rapid f a l l i n i n t e n s i t y below an energy of some 10 eV.  For example, the ion gun  described i n appendix C would produce a beam about two orders of magnitude more intense at 10 eV than at 4 eV beam energy, as i l l u s t r a t e d in F i g . C-2.  Much of this i s due to the collimating effect of the lens  system, which increases the e f f e c t i v e s o l i d angle at the ion source at higher beam energies.  However, at a beam energy of 3 eV, the ion count  was equal to the metastable background count, some two orders of magnitude below the i n t e n s i t y at 4 eV. cut  This very rapid f a l l  (beam  o f f ) i s t y p i c a l of these sources and i s apparently due to stray  fields.  Our own experience was that the cleanliness of the aperture  next to the ionizer exit aperture had a pronounced effect on the beam cut-off.  This aperture i s presumably intercepting a large flux of  electrons and ions from the ionizer and could conceivably contaminate very quickly. We have tentatively concluded that an ion beam at 2 eV  - 96 with an i n t e n s i t y of some tens of p a r t i c l e s / s e c can be produced with an electron bombardment source i n a clean vacuum system; and with the required s i z e and collimation for the examples given i n Chapter 7. However, we believe that an apparatus could be designed to polarize a much larger and more intense beam, but t h i s would require a major e f f o r t of engineering.  9  CHAPTER 10 Concluding Remarks  This thesis has added e s s e n t i a l l y two new ingredients for consideration to the conditions for which a charged p a r t i c l e generalized Stern-Gerlach experiment can be successfully performed, from those previously reported (Bloom and Erdman 1962; Rastall 1962; Byrne 1963; Bloom, Enga and Lew 1967).  These new considerations are the extension  of the Transverse Stern-Gerlach experiment to time independent, space varying inhomogeneous magnetic f i e l d s given by equation (1-2), and the introduction of an e l e c t r i c f i e l d E(r) which i s orthogonal to the magnetic f i e l d B^(f) (equation 1-2) i n the plane transverse to the general p a r t i c l e motion.  The development of these new considerations i n  Chapters 5 through 9 indicates that the experiment i s considerably s i m p l i f i e d from that previously envisaged, and that no fundamental difficulty  remains to the successful execution of a charged p a r t i c l e  Stern-Gerlach experiment. The most f r u i t f u l application of such an experiment would probably be the precision measurement of the low-lying energy levels of a wide variety of ions and molecular ions, which are of great interest i n chemistry and astrophysics. such measurements.  Very few methods have proven useful for  We believe that the generalized Stern-Gerlach  - 98 experiment w i l l now allow the study of such ions i n the same general way that the Stern-Gerlach  experiment has been applied to the study  of atoms and molecules. Another application i s the construction of a polarized ion source for use i n nuclear physics.  Such sources as are presently i n use ionize  after atomic state s e l e c t i o n , whereas this experiment makes possible the state s e l e c t i o n after i o n i z a t i o n . This difference may prove to be important i n future designs.  A short review of polarized ion sources  now i n use i s given by Drake (1967).  -  99  -  BIBLIOGRAPHY  Abragam, A. 1961.  Principles of Nuclear Magnetism  (Oxford University-  Press, London). Bas, Von E.B. and Gaug, H. 1968.  Z. Angew, Math Phys. (Switzerland),  Vol. 18, No. 4, 557. Bennewitz, H.G. and Paul, W. 1954. Bloom, M. and Erdman, K.L. 1962.  Z. Physik 139, 489. Can. J . Phys. 40, 179.  Bloom, M., Enga, E. and Lew, H. 1967.  Can. J . Phys. 45_, 1481.  Byrne, J . 1963.  Can. J . Phys. 41_, 1571.  Drake, J r . , C.W.  1967.  Atomic and Electron Physics, V o l . 4 pp. 226-  257 (Academic Press, New York and London). Greene, E.F. 1961.  Rev. S c i . Instr. 32_, 860.  Hanszen, K.-J. and Lauer, R. 1967.  Focusing of Charged P a r t i c l e s ,  Vol. I, pp. 251-307, (Academic Press, New York and London). Hushfar, F., Rogers, J.W.  and Webb, D. 1967.  Rarefied Gas Dynamics  F i f t h Symposium, Vol. I I , pp. 1427-1442 (Academic Press, New York). Kusch, P. and Hughes, V.W.  1959.  Handbuch der Physik, Vol. 37 (Springer  Verlag), pp. 1-172. Lamb, J r . W.E.  and Skinner, M. 1950.  Le Couteur, K.J. 1967. Lipeles, M. 1966.  Phys. Rev. 78_, 539.  Plasma Physics, Vol. 9, 457.  Thesis, Columbia University (unpublished).  Lippert, W. and Pohlit, W. 1952.  Optik 9_, 456.  Lippert, W. and P o h l i t , W. 1953.  Optik 10_, 447.  Novick, R. and Commins, E.D. 1958. Pearce, R.M.  Phys. Rev. I l l ,  822.  1969. University of V i c t o r i a , Private communication.  - 100 Pierce, J.R.  1954.  Theory and Design of Electron Beams, second  edition. (De. Van Nostrand and Co., New Jersey). Plumlee, R.H. Ramsey, N.F.  1957.  Rev. S c i . Instr. 28,  1963.  830.  Molecular Beams (Oxford University Press, reprinted  from f i r s t e d i t i o n , 1955) . R a s t a l l , P. 1962. Salardi, G.,  Can. J . Phys. 40,  1271.  Zanazzi, E. and U c c e l l i , F. 1968.  Methods, 5£, Strashkevich, A.M.  152. and Trotsyuk, N.I. 1968.  Physics, Vol. 13, no. 3, Teng, L.C. 1959.  Nuclear Instr. and  Soviet Physics - Technical  384.  H e l i c a l Quadrupole Magnetic Focusing Systems,  ANAL-55 February. Trotsyuk, N.J. 1969.  Soviet Physics - Technical Physics, Vol. 13,  No. 7, 944'. Watson, G.N.  1962.  Theory of Bessel Functions, Second Ed. (Cambridge  University Press). Weissler, G.L.,  Samson, J.A.R., Ogawa, M.,  Opt. Soc. Am.  49_,  and Cook, G.R.  338.  Winter, J . 1955.  Compt. Rend. Acad. S c i . 241,  375.  Zankel, K. 1968.  Nuclear Instr. and Methods 65,  322.  1959.  J.  - 101 APPENDIX A The H e l i c a l Quadrupole Magnetic F i e l d  The magnetic f i e l d of a quadrupole wire system near the center of symmetry i s well approximated by;  B ^ f ) = -G(y + ix)  (a-1)  at the p o s i t i o n r = x + i y , where;  G = 0.8 -Kj R  gauss/cm  (a-2)  o  with, I = current i n each wire i n amperes, R  = distance i n centimeters  q  from the center of symmetry to the center of each wire. of this f i e l d i s shown i n F i g . A-1.  The orientation  This f i e l d i s similar to the f i e l d  i l l u s t r a t e d i n F i g . 2-3, except that the orientation of the axes x,y with respect to the quadrupole wires i s d i f f e r e n t . If this system i s twisted to form a h e l i c a l quadrupole, as i l l u s t r a t e d i n F i g . 9-1, the f i e l d w i l l rotate with z at twice the rate which the wires rotate with z, i . e . , the  the f i e l d i s represented i n  a x i a l region by a f i e l d of the form;  B^z)  G  B  = -G (y + i x ) e l 2  B  = C(q)G 2TT  z  (a-3)  (a-4)  (a-5)  - 102 -  Figure A - l .  I l l u s t r a t i o n of the orientation of the f i e l d -G(y  + ix) at the point r = x + i y .  =  - 103 -  L = h e l i c a l step length  (a-6)  4?rR  The factor C(q) can be calculated from the paper of Le Couteur (1967), who solves f o r the general h e l i c a l multipole case and the h e l i c a l quadrupole i n p a r t i c u l a r .  He also considers the higher order terms  3 0(r ), which are l e f t out i n the above r e l a t i o n . We give; C(q) = l/2q K (q) + l/4q K (q) 2  3  2  1  (a-8)  where K i s a modified Bessel function of the seconddkind. n In order to make the i d e n t i f i c a t i o n with Le Couteur we give the relation  (Watson 1962, Eqn. 3, pg. 79)  zK '(z) =-vK (z) - zK ,(z) V V v-1  (a-9)  The factor C(q) i s plotted i n F i g . A-2. Equation (a-3) may be written with a homogeneous solenoid f i e l d (B ) superimposed along the z axis. Bf ,z) = B i + r  B  x  + B_H  (a-lOa)  = G (x s i n 2tdz - y cos 2uz) D  X  D  (a-lOb)  —  B^ = -Gg(y s i n 2_z + x cos 2wz)  (a-lOc)  B  (a-lOd)  z  = B  Q  - 104 -  1.0  .9 .8  .7  .5 .43  r  Z .1 0  2  Figure A-2.  3  Plot of the factor C(q) defined by equation (a  - 105 -  APPENDIX B The Derivation of the Trajectories of an Ion i n a H e l i c a l Quadrupole F i e l d Consisting of Orthogonal E l e c t r i c and Magnetic Fields  Superimposed  on a Homogeneous F i e l d and on. which an Additional Radial Force i s Introduced  From appendix A we have the magnetic f i e l d of a h e l i c a l quadrupole;  B(f,z) = B i + B j + B k  (b-la)  B^ = Gg(x s i n 2ojz - y cos 2_z)  (b-lb)  By = -Gg(y s i n 2u>z + x cos 2wz)  (b-lc)  x  B  z  = B  y  z  o  where; 2TT  _=  —  L = h e l i c a l step length G  D  = magnetic gradient  D  The corresponding e l e c t r i c h e l i c a l quadrupole f i e l d which i s orthogonal to B"(r,z) i n the x,y plane can be found by putting; E(r)  . B(r) = 0  with B = 0 z  (b-2)  thus; E(r) = E I + E J + E 1 <  (b-3a)  E  = G (y s i n 2oiz + x cos 2wz)  (b-3b)  = G (x s i n 2wz - y cos 2uz)  (b-3c)  p  E y  106 E  z  = 0  (b-3d)  where; G„ = e l e c t r i c b  gradient  It i s very useful to include i n these solutions a r a d i a l force term F which we define as: r  F  r  = $r  (b-4)  $ i s a constant, and i s l e f t a r b i t r a r y i n this derivation. It i s discussed i n Chapter 8 , where this additional force term i s used. The equation of motion which we wish to integrate i s ;  j  2—  ^ dt  _ F = q/m(v x B(f) + E(f)) + - ±  (b-5)  This may be written as;  A  2  TT = q/m(E + vB - v a x 2 ^ ^ x y z dt  B ) + - x z y m  (b-6a)  ,2  =-%r = q/m(E 2 dt n  + vB y z  v J  d z | = q/m(v B dt  - v B ) + - x x x z m  (b-6b)  2  Q  x  We make the r e s t r i c t i o n s that;  - v B) x  (b-6c)  - 107 -  v  dz -j— dt  =  z  =  v  <te constant  o  (b-7b)  then; v  v  dx dt  dx dz  dz » J T - = V X dt o  = - n r = j —  x  =  y  % dt  =  v  . (b-8a) 0  K  y o  J  (b-8b)  7  v  where; x  =  y  -  (b-8c)  %  Cb-M)  i t follows that; d x dt  2  v  x  (b-8e)  2  Adt ,2 ^-4 dt  =  v y °  =  0  (b-8f)  2  (b-8g)  With these r e s t r i c t i o n s equation (b-6c) drops out.  Substitution of  equation (b-8) into (b-6) y i e l d s ;  x  =  -X-TT (E + v B - v B ) + - ^ 5 - x 2 x y o o y 2 mv mv o o v  J  J  y y  =  q 0  mv  o  2  (E + v B - v B ) + y o x x o  =• y  %  y  v  (b-9a) J  J  mv  2  o  1  K  (b-9b) J  - 108 -  We wish to transform these equations to a set of axes (x,y) which rotate at the r a t e w z , i . e . ,  x  = x cos wz + y s i n wz  (b-lOa)  y  =  (b-lOb)  y cos wz - x s i n wz  d i f f e r e n t i a t i n g with respect to z, we obtain;  x  =  x cos wz + y s i n wz + wy  (b-lla)  y  =  y cos wz - x s i n wz - wx  (b-llb)  and d i f f e r e n t i a t i n g again;  = x cos wz + y s i n wz  (b-12a)  § 2 y + 2wx - w y = y cos wz - x s i n wz  (b-12b)  x - 2wy - w x 2  Substituting for x and y from equation (b-9) , equation (b-12) can be expressed after some manipulaton i n the following form; x - 2Wy  - (W  + ©  + f)x = 0  (b-13a)  y + 2Wx  - (W  - ©  + f)y = 0  (b-13b)  2  2  where qB ;  mv o  o  109  - l/4(—y v  •fj-  mv Equation  2  mv ' o  o  v  (b-16) J  (b-13) can be combined into a single fourth order, l i n e a r ,  homogeneous equation;  T +  2(W - f)x* + (W + © 2  + f) (W - ©  2  2  Rz Solutions of the form e are assumed.  +J©  2  (b-17)  Four roots R are obtained;  R  :  = (-W + f  R  2  = (-W + f - 6 )  R  3  = -R  (b-18c)  R  4  = -R  (b-18d)  2  2  - 4W f)  + f)x = 0  2  = (-W + f + 6 )  1 / 2  2  1 / 2  (b-18a)  (b-18b)  1 / 2  1  2  where; 6 = J®  - 4W f  2  (b-19)  2  The solutions can be written as;  x =C ^ l W  2 +  Z  2  +C e 2 R  + C e" l  Z  R  2  ©  +  f-  + C e" 2  Z  R  3  W  2  2  R l  -2WR  C  1  l  61  (b-20a)  Z  4  +  © f-R +  2 2  -2WR  +  C  2  2  6 2  (b-20b) W  2  +0+£-R  :  W  2  i_c -2WR,  + © +f-R  2  e l  3  2  —  :  -2WR„  C e ^4  2  - 110 -  The constants, C, can be obtained i n terms of the i n i t i a l conditions Denoting values at z = 0 by subscript o, we obtain the following from equations  (b-10) and  (b-11);  x -o  =  y -o  =  x -o  =  y Lo  =  x J  yo x  1  o  o  y o  +  (b-21)  oiy ~ o  J  J  -  OJX  - o  The equations(b-20), together with t h e i r f i r s t  derivatives,  evaluated at z = 0 using the values from equations  (b-21), give four  equations which can be solved for the four unknown C's.  For the case  where the four roots are a l l pure imaginary so that they may be written i n the form R = iR', where the prime denotes a r e a l number, then the solutions to equation (b-17) can be written as follows, where the constants have been solved f o r i n terms of the i n i t i a l conditions given by equation (b-21) . 01.72  x  x  = —[(cosR 'z+cosR 'z)+ 1  .  2  sinR 'z 2 R '  „, .,,2 A 2W(o)W-W + © J  6  L  2  sinR„'z 2 R* 2  „„,2 ^ 2W-© 6  y sinR 'z (cosR-j^ z - c o s R ' z ) ] + — [ 0 ) ( ^—; 1  =  ,  2  Atx sinR 'z -fj-a© , 2 R  sinR 'z 1_  2'  R  1  l '  x 2  R  sinR 'z Qr 1 l '  sinR_'z sinR 'z • W , , , 2 1 )]+ y T-[COSR 'z-cosR 'z] R' " R y (b-22a) r  f  2  n  n  1  0 0  2  1  9  Z  I l l -  -  sinR 'z  - LaC  RI  <  sinR 'z  )n  R-  +  2  [cosR^ ' Z+COSR2' z  +  2 ^  ~.  „,  sinR 'z  5  ~ —  sinR.'z R^  ®-(cosR^ ' Z-COS.R2 ' z) ]  W •^•[cosR^' z-cosR ' z] 2  P L  sinR ' z 1  R~'  +  sinR 'z z  ~~RT  5  +  0  zW  2 .-^sinR 'z +  6  z  1  RT  1  sinR 'z 1  RT  1  JJ  (b-22b)  - 112 -  APPENDIX C An Ion Gun Design Suitable f o r Producing Small Diameter, Well-Collimated, Ion Beams  The ion gun described i n this appendix was developed  specifically  to produce small diameter, well-collimated ion beams using an electron bombardment ion source.  The major inovation from standard designs such  as given by Novick and Commins (1958) or Plumlee (1957) i s the incorporat i o n of the ring focus electron gun, which i s well suited to ionizing small diameter beams because of i t s geometry.  A diagram of the ion gun  i s given i n F i g . C - l . The ring focus electron gun has been used by others for various purposes but i t has not, to our knowledge, been incorporated into an ion gun suitable for adaption to this experiment. have analyzed i t s design i n some d e t a i l .  Bas and Gauz (1968)  Zankel (1968) has calculated  electrode shapes to maintain a coaxial current flow by compensating for end e f f e c t s , although our design does not include this compensation. The advantages of the ring focus design are threefold.  It i s a very  compact, simple structure, with a c y l i n d r i c a l geometry which matches the beam geometry and reduces the various f i e l d asymmetries to a minimum. It u t i l i z e s an e l e c t r o s t a t i c focusing system which i s self-focusing at a l l bombardment energies.  E l e c t r o s t a t i c focusing i n the electron gun  has the advantage that i t does not affect the ion beam as would a magnetic focusing system.  F i n a l l y , the ring focus design produces a very intense  electron beam i n the a x i a l region where the ionizing events are most useful.  - 113 The major problem with the ring focus design i s the magnetic f i e l d produced by the filament heater current which deflects the electrons so that they do not occur.  enter the a x i a l region where the i o n i z a t i o n i s to  For our purpose this problem was  overcome by using "on-off" heater  current with a filament having a large thermal mass. produced during the " o f f " period, which may cycle.  The detector may  The useful beam i s  be a large f r a c t i o n of each  be gated i n synchronism with the heater  to increase the signal to noise.  current  This "on-off" cycle has the additional  advantage that the magnetic f i e l d from the filament current lead-in wires does not affect the ion beam, so that these wires may  be placed conven-  i e n t l y to simplify construction. The lens system consists of a Pierce lens (Pierce 1954)  to i n i t i a l l y  c o l l e c t and focus the ions, three single aperture lens, and a three element unipotential lens (Lippert and Pohlit, 1952  and 1953), followed  by a collimating aperture. The Pierce lens was constants may  chosen because i t s focal length and  be c l o s e l y calculated.  unipotential lens, may  aberation  The Pierce lens, together with the  be designed to correct to f i r s t order for  chromatic aberration, and p a r t l y correct for spherical aberration. They may  also be designed as a crude monochromator by making the system  very chromatic. Referring to Fig. C-l for an explanation of the symbols, we give the following design formulae; The ion gun w i l l focus the beam at i n f i n i t y i f ;  - 114 -  where; [SCg-i,) + 3b  1  + 4h] (c-2)  4[h + b' + g - i ] x  b ' f S C g - i ^ + 4h] B  " 16h[h + b + g - i ]  =  4 d l 1  x  i  (  _  3  )  t >n  '  IF-d-  =  C  ( c  -  4 )  The v a r i a t i o n of the focal point ( z ) of the unipotential lens £  as a function of the spread i n beam energy ( A U ) i s given by; (Hanszen q  and Lauer 1967) AU A z  f  = -  c  c h ^  -T-  "  (c  where C^^ ) i s the chromatic aberration constant 00  5)  of the unipotential  lens. For achromatic operation of the ion gun we put;  Az  = -Ai  f  (c-6)  3  where A i ^ i s the v a r i a t i o n i n the image point of the lens  preceding  the unipotential lens as a function of the spread i n beam energy.  We  give the formula; .2 Ai  = (b')' -^2. [ (h-i )  .2  1  3  V  2  where;  ( 2  ± ( l ^ f ^ g - i ^  n ^ T f " ) + J-1  2  >1+S)±  2  ±  3  Cc-7)  f  f  - 4h IXzVM  2  3  ( c  " f-  =  (C  7~  2•  .  ( c  9 )  "  -  10)  u )  ^(g-ij) F  W  The minimum chromatic aberration constant f o r a unipotential lens is; C  c h  O ) = -3 I  (c-13)  The condition for achromatic operation, equation equation  (c-6), and  (c-7), y i e l d for this p a r t i c u l a r case the unipotential lens  length (£);  ^  2  (h-i ) (1+S) 'fjCg-ijr z  2  ^ (l*S)f  )  CC-14) r  2  3  In the normal operating range (S > 0) we have f , f ^ < 0, while 2  f^<  0 for a l l S.  Such values y i e l d I >  0 which indicates that achromat  operation i s possible i n the normal operating range i f the gun dimensions are chosen to give a reasonable value of "I". The ion gun which we have b u i l t has the following nominal dimensions in inches,  - 116 =  0.20  1  =  0.25  g  =  0.075  h  =  0.160  b'  =  .158  d  =  .020  =  1.7  V  d  o z  c  The unipotential lens has the nominal a  =  0.176  b  =  0.176  d  =  0.088  I  =  0.44  f  =  0.145  dimensions;  for which; z  C  ch«  From equation (c-1), this ion gun should focus at i n f i n i t y f o r ;  or  V  =  -3V  V  = (l/2)v  In p r a c t i c e , i t i s very d i f f i c u l t to b u i l d the ion gun accurately enough to expect precise agreement between experiment  and theory. For  our gun we obtained the best results by operating the unipotential lens  " somewhat below (V), with V  - 117 ~ -2V.  Note that the condition V  = (1/2^/ i s an unfavourable one because  the p o s i t i v e ions experience only a very weakly accelerating e l e c t r i c f i e l d on exit from the ionizer. Fig.  C-2 gives t y p i c a l results for a He beam with a pressure i n +  -4 the ionizer of approximately 5 x 10  Torr (He).  These results  were obtained by d i r e c t i n g the beam through a f i e l d - f r e e region 50 cm long between the ion gun exit aperture and four knife-edge jaws which could be independently moved to explore the beam i n t e n s i t y .  The ions  were collected by an electron m u l t i p l i e r (Bendix, model 306) .  The  -7  pressure i n the f i e l d free region was 5 x 10  Torr.  The ion gun was constructed of Type 304 stainless steel with alumina insulators.  - 118 -  Electrode s«. (pafetftw) Ft lament Lomjer chamber (~l00v) (v) Pusher \ Filament focys / Spherical Anode (~V) \ (~IOOv) / /(V)  Unipotential Let  ) t  Figure C - l .  Uhipo+eKtial Lew  Untpofervfial  focal point  Wid plane  L«r\s  Diagram of the ion gun described i n appendix C.  - 119 -  £ - 7 *  10*  At Average Number of Ions per sec.  Beam dioenjence at hal-f wave, intensity  (H*) +  Beam dia*we"ter at Source » .050 cm  3  4-  £  6  7  9  V c : I o n Beam £ner<j\j  (eV)  Figure C-2. Typical performance of the ion gun described i n appendix C.  V i s the potential of the i o n i z i n g chamber.  

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