Non-adiabatic spin transitions in metastable hydrogen by Ralph Dale Hight n A thesis submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in PHYSICS Montana State University © Copyright by Ralph Dale Hight (1975) Abstract: We consider the problem of non-adiabatic passage of an oriented spin through an inhomogeneous magnetic field. Differential equations for the spin state amplitudes for an arbitrary magnetic field are derived for the two cases of spin 1/2 and spin 1. The field is shown to be completely characterized by a single parameter called the adiabaticity parameter α The adiabaticity parameter is the ratio of the field magnitude to the field rotation rate. By means of a pedagogical example, a criterion for defining regions of adiabaticity (no spin flips) and non-adiabaticity (spin flips) is developed. Using this criterion, we define transition lengths for regions where transitions occur. We solve the amplitude equations for a simple type of magnetic field and calculate the spin flip probability. The field is assumed constant everywhere except in a small region where the axial component goes linearly through zero, reversing its direction. We constructed a magnetic field that closely approximates the above field. By using the F = 1 hyperfine states in the 2S 1/2 level of atomic hydrogen, we confirm the theory by comparison with experiment. From the experimentally verified theory; we propose a novel method of producing a beam of metastable (2S 1/2) atomic hydrogen polarized completely in the F = 1, mf = 0 hyperfine state.  NOH-ADIABATIC SPIN TElANSITIONS IN METAS TAB EE HYDROGEN n i By • RALPH DALE RIGHT A t h e s i s s u bm it te d i n p a r t i a l f u l f i l lm e n t o f t h e r e q u ir em e n ts f o r t h e d eg re e o f DOCTOR OF PHILOSOPHY in PHYSICS APPROVED: C ha irm an , Exam in ing Comm ittee [a /o r D epartm en tHead G rad u a te Lean MONTANA STATE UNIVERSITY Bozeman, M ontana May, 1975 i i i ACKNOWLEDGMENTS I w ou ld l i k e t o e x p re s s my .a p p r e c ia t io n and g r a t i t u d e t o D r. R . T. R o b iscb e f o r s u g g e s t in g t h i s w ork and f o r h i s s u p p o r t d u r in g th e d a rk t im e s . I w ish t o th a n k my d e a r w if e f o r h e r u n f a i l i n g s u p p o r t and p a t i e n c e . I w o u ld .l i k e t o th a n k Mr. F re d B lan k enb e rg . f o r h i s e l e c t r o n i c w iz a rd r y and f r i e n d s h i p . To a l l t h e o th e r s b o th l a r g e and sm a l l who h e lp e d me e i t h e r v o l u n t a r i l y o r i n v o l u n t a r i l y , I th a n k y o u . - i v - TABLE OF CONTENTS' .Page A cknow ledgem ents i i i ' T ab le o f C o n te n ts i v L i s t o f T ab le s v i - L i s t o f F ig u r e s v i i C h a p te r I . I n t r o d u c t i o n I 1 . H i s to r y and I n t e r e s t ; I 2 . What i s N o n -a d ia b a t iC S p in P a ssag e 3 3- O verv iew o f t h e E xp e rim en t 10 I I . T heo ry . 20 1 . A S p in F l i p P rob lem 20 2 . A d i a b a t i c i t y P a ram e te r ' 28 3 . Two L ev e l P a s sag e ■ 37 4 . T h ree L ev e l N o n - a d ia b a t ic P a s sag e 1+3 5 . E v o lu t io n M a tr ix 4-7 6 . V e lo c i ty and B eam -s ize A v e rag in g 67 7 . Summary 73 I I I . E x p e r im e n ta l A p p a ra tu s 74 1 . O verv iew 74 2 . Beam S ou rce 77 3 . E x c i t a t i o n 79 4 . H y p e r f in e S t a t e A n a ly z e r 90 ’ 5 . D e te c to r s 99 -V - 6 . ' F lo p p e r . 7- E ro c ed u re TV. D a ta .a n d C o n c lu s io n s ■ ■ I* F lo p p e r C u rves ' . ' , . ; 2 . Time o f F l i g h t S p e c t r a ■ ' ' ' • 3" P u re , P o la r i z e d Beam ■ 4 o .C o n c lu s io n A pp end ic e s • I . T h ree l e v e l S p in F l i p P r o b a b i l i t y f o r C o n s ta n t a : T I . E eb ed a fS. Computer P rog ram f o r . th e E h ro lu tion M a tr ix . I I I . T h ree l e v e l I n i t i a l l y N on -ze ro S p in F l i p P r o b a b i l i t i e s IV . V e lo c i ty and Beam A v e ra g in g Computer F i t s V. Q uench ing of. t h e ..231. M e ta s ta b le . S t a t e in . s A tom ic Hydrogen V I. A v e ra g in g o v e r V e lo c i ty Window / • V I I . M agne tic F i e l d o f . a S e t o f O pposing S o le n o id s V I I I . . U n c e r t a i n t y . i n th e - • . P o p u la t io n - . 'F o o tn o te s ■ . • Page 111 119 . 133 ' 133 . 153 .158 165 170 186 i 138 393 /1 9 8 205 207 210-.B ib l io g r a p h y LIST OF TABLES T ab le Page I . T y p ic a l A lp h a tro n O p e ra tin g C o n d i t io n s TB T I. V a r io u s M achine P a ram e te rs 132 I I I . F lo p p e r D a ta Summary T ab le lU 2 IV . TOF S p e c t r a D a ta 159 v i i LIST OF FIGURES F ig u re Page 1 . D iagram o f a s p in p a r t i c l e e n t e r i n g a m ag n e tic f i e l d . 5 2 . S p in f l i p p r o b a b i l i t y f o r a s p in +% v e r s u s a d i a b a t i c i t y p a ram e te r f o r a . c o n s t a n t m ag n e tic f i e l d r o t a t i n g u n ifo rm ly th ro u g h l 80 d e g re e s . 9 3. Zeeman d iag ram o f t h e f i n e s t r u c t u r e and h y p e r f in e s t r u c t u r e s t a t e s i n t h e 2S^ and 2P^ l e v e l s o f a tom ic h y d ro g en . 2 2 '12 4. M o n -v e lo c ity s e l e c t e d f l o p p e r c u rv e show ing th e con - ■ v e r s io n p e r c e n ta g e o f as a f u n c t io n o f f lo p p e r c u r r e n t . ' 15 5* V e lo c i ty s e l e c t e d f lo p p e r c u r v e s . 17 6 . A s im p le r o t a t i n g m ag n e tic f i e l d . 31 7 . The in v e r s e a d i a b a t i c i t y p a ram e te r v e r s u s d i s t a n c e f o r t h e m agn e tic f i e l d i n F ig u re 6 . ' 35 8 . Two l e v e l s p in f l i p p r o b a b i l i t y f o r t h e a d i a b a t i c i t y ' p a ram e te r shown i n F ig u re 7 as a f u n c t io n o f m ag n e tic f i e l d . ' - 42 9 . Computer g e n e r a te d s o lu t i o n s (% 's ) f o r t h e m agn itu d e s o f t h e e lem en ts o f e( i r ,0 ) w i th t h e i r c u rv e f i t t e d a n a l y t i c f u n c t io n s ( s o l i d l i n e s ) . 55 10 . Computer g e n e r a te d p h a s e . 57 11 . T h ree l e v e l s p in f l i p p r o b a b i l i t y v e r s u s m ag n e tic 64 f i e l d . 12 . V e lo c i ty and beam a v e ra g e d t h r e e - l e v e l s p in f l i p p r o b a b i l i t y v e r s u s m ag n e tic f i e l d . 72 13 . A lp h a tro n d iag ram . j6 14 . Hydogen d i s s o c i a t i o n c u rv e . 8l v i i i 15 . P r e s s u r e d ependence o f th e f r a c t i o n a l d i s s o c i a t i o n and m e ta s ta b le i n t e n s i t y . 16 . A tom ic h y d ro g en 2NOH e x c i t a t i o n c u rv e . s 17 . Q uen ch ab le s i g n a l s as a f u n c t i o n o f e -g u n c o l l im a t i n g f i e l d . ■ 18 . (3^ q u ench c u rv e . 19 . or qu en ch c u r v e . 20 . Beam n o tc h c u r v e . 21 . Lamb s h i f t r e s o n a n c e c u r v e . 22 . Time o f f l i g h t e l e c t r o n i c s d ia g ram . 23 . T y p ic a l t im e o f f l i g h t s p e c trum w i th t im e m ark . 2 4 . ' V e lo c i t y s e l e c t i o n e l e c t r o n i c s d ia g ram . 25.. C ro s s - s e c t io n * o f f l o p p e r . 26 . M agn e tic f i e l d s f o r t h e f lo p p e r shown i n F ig u r e 25 . 27 . S p a t i a l v a r i a t i o n o f t h e a d i a b a t i c i t y p a r a m e te r : % 's a r e e x p e r im en t a l l y d e te rm in e d and s o l i d . l i n e s a r e t h e o r e t i c a l e x p e r im e n ta l . 28 . Maximum.' n o n - a d i a b a t i c i t y f o r t h e second c r o s s i n g p o i n t a s a f u n c t io n o f f lo p p e r c u r r e n t . 29 . ■ M agn itude and r o t a t i o n a n g le f o r t h e m agn e tic f i e l d e n v iro nm en t. 3 0 . Oven tem p e ra tu r e v e r s u s th e . p e a k p o s i t i o n i n t h e t im e o f f l i g h t sp e c trum . 86 88 93 95 98 - 101 105 107 109 113 116, 118 121 123 128 83 31 . -G a u s s ia n c u rv e f i t t o t h e t im e o f f l i g h t s p e c trum . '131 i x 32 . F loppeTr c u rv e I . 33- F loppesr c u rv e I I . ' 3 4 . F loppear • c u rv e I I I . 35- V e lo c i t y s e l e c t e d f lo p p e r c u r v e . 3 6 . O s c l l l i a t io n s from f lo p p e r c u rv e I . 37« O s c i l l a t i o n s from f lo p p e r c u rv e I I 3 8 . O s c i l l a t i o n s from f lo p p e r c u rv e I I I . 3 9 . P e r io d p3o>t fiyr t h e o s c i l l a t i o n s i n F ig u r e s 3 6 , 3 7 , and 38 . 4 0 . Time o f f l i g h t s p e c t r a f o r t h e s t a t e . 4 1 . P e r io d p l o t s o f t h e o s c i l l a t i o n s i n th e t im e o f f l i g h t s p e c t r a . -AD IB s t a t e i n i t i a l l y n o n -z e ro f lo p p e r • c u r v e s . 4 3 . T h ree l e v e l c o n s ta n t a s p in f l i p • p r o b a b i l i t y . 4 4 . C om parison o f t h e a p p ro x im a te L-Z en v e lo p e w i t h t h e com pu te r i n t e g r a t i o n . 4 5 . C om parison o f t h e damped o s c i l l a t o r y te rm a p p ro x im a t io n w i th t h e com pu te r i n t e g r a t i o n . 4 6 . F lo p p e r c r o s s - s e c t i o n . 4 7 ; F lo p p e r f i e l d c o n f i g u r a t io n . 135 137 139 l 4 i - 145 147 149 152 ' 155 157 161 169 190 192 201 20 4 ABSTRACT We c o n s id e r t h e p ro b lem o f n o n - a d ia b a t i c p a s s a g e o f an o r i e n t e d ■ s p in th ro u g h an inhomogeneous m ag n e tic f i e l d . D i f f e r e n t i a l e q u a t io n s f o r t h e s p in s t a t e am p litu d e s f o r an a r b i t r a r y m ag n e tic f i e l d a re d e r iv e d f o r t h e two c a s e s o f s p in § and s p in I . The f i e l d i s shown t o b e c o m p le te ly c h a r a c t e r i z e d by a s i n g l e p a ram e te r c a l l e d th e a d i a b a t i c i t y p a ram e te r a- The a d i a b a t i c i t y p a ram e te r i s t h e r a t i o o f t h e f i e l d m ag n itu d e t o t h e f i e l d r o t a t i o n r a t e . By means o f a p e d a g o g ic a l e x am p le , a c r i t e r i o n f o r d e f i n i n g r e g io n s o f a d i a b a t i c i t y (n o s p in f l i p s ) and n o n - a d i a b a t i c i t y ( s p i n f l i p s ) i s ' d e v e lo p e d . • U s in g t h i s c r i t e r i o n , we d e f in e t r a n s i t i o n l e n g th s f o r r e g io n s w here t r a n s i t i o n s o c c u r . We s o lv e t h e am p litu d e e q u a t io n s f o r a s im p le ty p e o f m ag n e tic f i e l d and c a l c u l a t e t h e s p in f l i p p r o b a b i l i t y . The f i e l d i s assumed c o n s ta n t ev e ryw h e re e x c e p t i n a sm a l l r e g io n w here t h e a x i a l component" go es l i n e a r l y th ro u g h z e r o , r e v e r s in g i t s d i r e c t i o n . We c o n s t r u c te d a m ag n e tic f i e l d t h a t c l o s e l y a p p ro x im a te s t h e above f i e l d . By u s in g t h e F = I h y p e r f i n e s t a t e s i n t h e 2 S i l e v e l o f a tom ic h y d ro g en , we c o n f i rm th e t h e o r y by com pa riso n w itf i e x p e r im e n t . From t h e e x p e r im e n ta l ly v e r i f i e d t h e o r y > we p ro p o se a n o v e l m ethod o f p r o d u c in g a beam o f m et a s t a b I e ( i ) a tom ic h y d ro g en p o l a r i z e d c o m p le te ly i n t h e F = I , m^ = 0 8 h y p e r f i n e s t a t e . CHAPTER I INTRODUCTION 1 .1 H is to r y and I n t e r e s t The p ro b lem o f n o n - a d ia b a t i c p a s s a g e o f an o r i e n t e d s p in th ro u g h an inhomogeneous m ag n e tic f i e l d was s t u d i e d i n th e 1 9 3 0 's f o r i t s p o t e n t i a l im p o r ta n c e i n m easu rem en ts o f t h e m agn itu d e and s ig n o f n u c le a r m ag n e tic m om ents. ^ T h e o r ie s w ere d e v e lo p e d by M a jo r a n a ^ ) and G u t t i n g e r ^ ^ , w i th e x te n s io n s by R a b i ^ ) / and S c hw in g e r^ ^ ) , and o t h e r s . E x p e r im en ts w ere, perfo rm ed , b y S t e r n ^ ^ and Segre^ Only t h e g ro s s a s p e c t s o f t h e t h e o r y w ere t r a c t a b l e due t o i t s c om p le x ity (n o com pu te rs w ere a v a i l a b l e t o s o lv e t h e n o n - l i n e a r e q u a t i o n s ) . The e x i s t i n g te c h n o lo g y s e v e r e ly l im i t e d t h e e x p e r im e n ta l i n v e s t i g a t i o n s . The s i t u a t i o n was soon rem ed ied i f n o t c l a r i f i e d b y t h e r a p id d e ­ v e lo pm en t o f m o le c u la r beam m ag n e tic r e s o n a n c e m e th o d s . The th e o r y o f t h e s e e x p e r im e n ts was e a s i e r t o h a n d le and y i e l d e d f a r more in fo rm a t io n th a n c o u ld b e e x p e c te d from , n o n - a d ia b a t i c p a s s a g e w o rk . N o n -a d ia b a t ic t r a n s i t i o n s / som etim es c a l l e d " s p in f l i p s " o r "Majo r a h a f l o p s , " w ere th e n r e l e g a t e d t o t h e s t a t u s o f a s im p le w a rn in g t o a l l a tom ic and m o le c u la r p h y s i c i s t s : n am e ly , a v o id th em , f o r th e y a r e n o t w e l l u n d e r ­ s to o d , and th e y w i l l a t b e s t d e p o la r i z e a beam o f o r i e n t e d s p in s V ery l i t t l e w o rk on M ajo rana f lo p s was done u n t i l r e c e n t l y . An e x p e r im e n t^ } w h ich u s e d M ajo rana f l o p s t o s e l e c t i v e l y p o p u l a t e : a p a r t i c u l a r s p in s t a t e , fo und u n e x p la in e d s t r u c t u r e i n t h e p r o d u c t io n c u rv e f o r th e s p in s t a t e o f i n t e r e s t . S in c e th e m a jo r a im o f t h a t - 2 - e x p e r im en t ' was n o t t h e s tu d y o f M a jo ran a f lo p s p e r s e , t h e phenomena was m e re ly n o te d and o n ly a c u r s o ry t r e a tm e n t o f t h e d a t a was done . The p ro b lem o f e x p la in in g . t h i s s t r u c t u r e was l e f t u n t i l 1970- when we b eg an th e w ork p r e s e n te d h e r e . The n o n - a d ia b a t i c p a s s a g e o f a beam o f o r i e n t e d s p in s th ro u g h an inhomogeneous m ag n e tic f i e l d i s , i n i t s e l f , o f i n t e r e s t a s an e x e r c i s e i n quantum m ech an ic s . However, t h e p ro b lem can b e r e l a t e d t o th e more g e n e r a l phenomena o f s c a t t e r i n g . In f a c t , t h e am p litu d e e q u a t io n s f o r c h a rg e exchange s c a t t e r i n g a r e i d e n t i c a l i n s t r u c t u r e ^ ^ ^ ^ ) t o th e e q u a t io n s d e s c r i b in g n o n - a d ia b a t i c p a s s a g e . - In m ost s c a t t e r i n g e x p e r i ­ m en ts , any " s t r u c t u r e " i n t h e am p l i tu d e s , w h ich i s p r e d i c t e d b y v a r io u s a p p ro x im a te s o l u t i o n s , te n d s t o b e w iped o u t by a n e c e s s a r y a v e ra g in g o v e r t h e im p ac t p a ram e te r and th e v e l o c i t y d i s t r i b u t i o n o f t h e beam . T hus, t h e v a l i d i t y o f d i f f e r e n t a p p ro x im a tio n s c an n o t b e t e s t e d as lo n g as t h e g r o s s a s p e c t s o f t h e s c a t t e r i n g t h e o r i e s a r e s im i l a r . T h is s u g g e s t s t h a t a d e f i n i t i v e s tu d y o f n o n - a d ia b a t i c p a s s a g e m igh t p ro v id e a c o n v e n ie n t and c o n t r o l l a b l e a n a lo g u e t e s t f o r a v a r i e t y o f c o l l i s i o n p ro b lem s . In f a c t , v a r io u s ap p ro x im a te t h e o r i e s d e v e lo p e d f o r s c a t t e r ­ in g p rob lem s c an b e ch ecked i n d e t a i l o n ly i f - t h e s t r e n g t h and d u r a ­ t i o n o f t h e i n t e r a c t i o n can b e c o n t r o l l e d . T h is i s p o s s i b l e , i n p r i n c i p l e , f o r a n o n - a d ia b a t i c p a s sa g e e x p e r im e n t . D u rin g t h e c o u rs e o f t h i s s tu d y , a p o s s i b l e new m ethod f o r th e p r o d u c t io n o f a c om p le te ly p o l a r i z e d beam o f n e u t r a l m e ta s ta b le - 3 - hyd rog en was d i s c o v e r e d . We p ro p o se a m ethod u s in g M ajo ran a f lo p s d i r e c t l y , w h ich i s i r o n i c i n v iew o f t h e u s u a l w a rn in g s a b o u t t h e d e ­ p o l a r i z i n g p r o p e r t i e s o f t h e M a jo ran a f l o p s , p ^ ^ th e r work a lo n g t h i s l i n e i s now b e in g a t te m p te d , and r e s u l t s w i l l b e •r e p o r te d ' e ls ew h e re . 1 .2 What i s W o n -a d ia b a t ic S p in P a ssag e? An atom i n a n o n -z e ro s p in s t a t e , m oving th ro u g h an inhomogeneous m ag n e tic f i e l d , i s a c te d on b y a t im e - v a ry in g f i e l d i n t h e a tom 's r e s t f ram e . I f t h e f i e l d v a r i a t i o n i s s u f f i c i e n t l y s low , t h e a d i a b a t i c , th e o rem p r e d i c t s t h a t t h e s p in w i l l rem a in a l ig n e d w i th t h e f i e l d and w i l l n o t b e " f l ip p e d " ( i . e . w i l l n o t make a t r a n s i t i o n t o a d i f f e r e n t s p in s t a t e ) . . I f , how ever, t h e f i e l d r o t a t e s a t a . r a t e on th e o r d e r o f t h e s p in Larmor f r e q u e n c y , th e n t h e r e a r e F o u r i e r .com ponents o f th e f i e l d n e a r t h e s p in f l i p r e s o n a n c e f r e q u e n c y . These F o u r i e r components may b e l a r g e enough t o " f l i p " t h e s p in . I f s p in f l i p s o c c u r , th e n th e m o tio n o f t h e s p in i n t h e f i e l d i s a " n o n - a d ia b a t ic p a s s a g e ." An i l l u s t r a t i o n o f " s p i n - f l i p s " may b e g iv e n b y c o n s id e r in g a s im p le tw o - l e v e l p ro b lem . ^ The d e t a i l s o f t h i s ' tw o - l e v e l p ro b lem w i l l b e g iv e n i n C h a p te r I I and we w i l l o n ly d i s c u s s t h e h i g h l i g h t s h e r e . C o n s id e r t h e e n t r y o f a p o l a r i z e d s p in g p a r t i c l e a t a v e l o c i t y v i n t o a m ag n e tic f i e l d if w h ich i s a f u n c t i o n o f p o s i t i o n as shown i n F ig u re I . The s p in i n t e r a c t s w i th t h e m agn e tic f i e l d th ro u g h i t s k■ F ig u re I : D iag ram o f a s p in p a r t i c l e e n t e r in g a m ag n e tic f i e l d . I LfN I Z <3 ( t^Q‘ti) H ^ I 'O ld A NIdS *”6— m ag n e tic moment ji, w here j2 = TC a r e t h e P a u l i s p in m a t r ic e s f o r a s p in g p a r t i c l e , and ^ i s t h e Bohr m agneton . The s p in H am il to n ia n i s th e n s im p ly ^ * I? and th e s p in wave f u n c t i o n i s \K t) = a x( t ) e x p ( - Im l t ) Qf1 + a g( t ) exp( - iu )g t) a s - H ere Of1 , g a r e t h e s p in -u p and sp in -d ow n b i s p i n o r s , r e s p e c t i v e l y . T im e -d ep end en t p e r t u r b a t i o n th e o r y t h e n .g i v e s t h e e q u a t io n s o f m o tion f o r t h e am p litu d e s ^ , ^ ( t ) . D e f in in g t h e m agn e tic f i e l d i n s p h e r i c a l p o l a r c o o r d in a te s and p e r fo rm in g th e u n i t a r y t r a n s f o rm a t io n G ( t) = §( t ) B ( t ) , t h e e q u a t i o n s . o f m o tio n i n m a t r ix n o t a t i o n may b e s e p a r a te d i n t o an e ig e n - v a lu e te rm and a c o u p lin g te rm , a s ( I ) 1 0 . O - e x p ( - i0) d f i /d t = -.(iu ) ( t ) / 2) B - d 6( t ) / d t / 2 g . O - I e x p ( i0) O The Larmor f r e q u e n c y , uu ( t ) , i s d e f in e d as w = ( 2 | ^ / f t ) H ( t ) and d 9 /d t ' i s t h e m ag n e tic f i e l d t u r n i n g r a t e . In g e n e r a l , E q u a tio n ( I ) c an n o t b e s o lv e d e x p l i c i t l y . However, i t i s in ' a c o n v e n ie n t fo rm t o d i s c u s s t h e a s ym p to t ic b e h a v io r when e i t h e r u> o r d e / d t i s much l a r g e r t h a n t h e o t h e r . When to » d 9 / d t , we have w eak c o u p l in g and do n o t r e a l l y e x p e c t any s p in s t o b e a p p r e c ia b ly f l i p p e d . Qn th e o t h e r .h an d , when d 9/ d t » to, we e x p e c t a l a r g e number o f sp in s- t o . b e - f l i p p e d . I t w i l l b e c o n v e n ie n t a t t h i s p o i n t t o d e f in e t h e r a t i o o f to and d 9 / d t ' a s t h e " a d i a b a t i c i t y p a r a m e te r " : o ( t ) = u / ( d 9 / d t ) . When -o' » I , t h e - e ig e n -v a lu e m a t r ix d om in a te s ( to » d 9 /d t ) . The a p p ro x i ­ - 7 - m ate s o lu t i o n s f o r t h e a( t ) am p litu d e s w i th th e i n i t i a l c o n d i t io n s U1(O ) = I , u3(o ) = 0 , dQ/dt ~ 0 ; and 6( Bq ) = 0 a r e s im p ly : | a i ( t ) l 2 = I ; and | a 3( t ) | 2 = 0- T h is i s known .as a d i a b a t i c p a s s a g e ; and i n f a c t , (% » I i s one o f th e c r i t e r i a u se d t o d e s ig n b e n d in g m agne ts f o r S te rn -G e r la c h ty p e e x p e r im e n ts . When a « I , t h e non ­ d i a g o n a l m a tr ix - d om in a te s and f o r U1 (O) = I , a 3(o ) = 0 , we h ave th e f i n a l s t a t e am p litu d e s a1 ( t ) = I and a s ( t ) = 0 . However, now th e . m ag n e tic f i e l d h a s changed d i r e c t i o n s w h i le t h e s p in h a s n o t ; h e n c e , t h e r e h a s beam a s p in f l i p . The s p in d id n o t f o l lo w th e m ag n e tic f i e l d r o t a t i o n . F o r a more- s p e c i f i c ex am p le , c o n s id e r t h e c a s e o f a c o n s ta n t m agn itu d e m agn e tic f i e l d r o t a t i n g u n i fo rm ly from 6 = 0 t o 0 = rr. T h a t i s , a i s a c o n s ta n t and t h e am p litu d e e q u a t io n s may b e s o lv e d e x a c t l y . The s p in f l i p p r o b a b i l i t y from C h ap te r I I i s P (rr ,a ) = s i n 2 [ n ( l + or2) s / 2] / ( l + a 2) . ( 2) As shown i n I ilg u re 2 , f o r a » I , t h e p r o b a b i l i t y o f a s p i n - f l i p o c c u r r in g i s o f o r d e r u n i t y w h i le f o r a » I , P (rr ,a ) i s c lo s e t o z e r o . I f a m ag n e tic f i e l d i s t o b e d e s ig n e d w here l e s s th a n 1% 'o f th e s p in s a r e f l i p p e d , a minimum v a lu e f o r a may b e c a l c u l a t e d b y P( r r ,a ) . « O .Q l ( a ^ 1 0 ) . F o r m ost l a b o r a to r y m ag n e tic f i e l d s t h e p a r a m e t r ic c o n s t a n t , a i s n o t a " c o n s ta n t" b u t i s a f u n c t i o n o f t o r 0 . We can. how eve r, e s t im a t e the- a p p ro x im a te w id th o f any n o n - a d i a b a t i c r e g io n . The c a l c u l a t i o n o f any s p in f l i p p r o b a b i l i t y may b e b ro k e n i n t o r e g io n s o f a d i a b a t i c p a s s a g e , w here t h e s p in s im p ly F ig u re 2 : S p in f l i p p r o b a b i l i t y f o r a s p in ^ v e r s u s a d i a b a t i c i t y p a ram e te r f o r a c o n s ta n t m ag n e tic f i e l d , r o t a t i n g u n i fo rm ly th ro u g h 180 d e g r e e s . Sp in F lip P ro ba bi lit y P( 7r, a) * P ( - T f , a . ) -Si n2[y(l + a2) /2 ] /( l + a ) Adiabaf ic i ty Parameter (a) -1 0 - f o l lo w s t h e f i e l d , and n o n - a d ia h a t i c r e g i o n s , w here t h e am p litu d e e q u a t io n s h ave t o b e s o lv e d e i t h e r a p p ro x im a te ly o r n u m e r ic a l ly . 1 -3 O verv iew o f t h e E xp e rim en t T h is t h e s i s r e p o r t s th e r e s u l t s o f a m o d if ie d v e r s i o n o f th e e x p e r im e n t b r i e f l y m en tio n ed i n S e c t io n 1 .1 . T ha t e x p e r im e n t , by R o b isco e e t . a l . u s e d a p o l a r i z e d beam o f n e u t r a l m e ta s ta b le ( 2S i) h y d ro g en a tom s . F ig u re 3 shows th e Zeeman d ia g ram o f t h e a tom ic s ■ s t a t e s u s e d i n t h i s e x p e r im e n t . The f i n e s t r u c t u r e s t a t e s a r e l a b e l e d u s in g t h e n om en c la tu re o f W. E . Lamb^ , a s ff(.2Si: m = '+ !) , P(2Si: m = ~ i) , e( SPi : m = +^), s 0 S d s u and f ( 2P^: mT = -•§) . The h y p e r f in e s t r u c t u r e i s in c lu d e d f o r r e a s o n s •s d t h a t w i l l become a p p a re n t l a t e r . We g e n e r a te a beam o f p u re 2S(o?) ■ s t a t e atom s and th e n p a s s t h e beam th ro u g h a " f l o p p e r . " The f lo p p e r u s e d b y R ob isco e was a s o le n o id c o - a x i a l w i th t h e beam . The f l o p p e r i s u s e d t o "buck o u t" a r e s i d u a l a x i a l m ag n e tic f i e l d g e n e r a te d dow nstream b y a H e lm ho ltz c o i l whose p u rp o se i s d e s c r ib e d i n C h a p te r I I I . As t h e c u r r e n t i n t h e f l o p p e r i s i n c r e a s e d , t h e a x i a l m ag n e tic f i e l d i s d r iv e n th ro u g h " z e ro " and r e ­ v e r s e s d i r e c t i o n . The m e ta s ta b le a tom t r a v e l s th ro u g h t h e f l o p p e r ' e x p e r i e n c in g an a x i a l m ag n e tic f i e l d t h a t r a p i d l y r o t a t e s b y ISO0(Tr) d e g r e e s . The a. s t a t e l o s e s i t s a x is o f q u a n t i z a t i o n as i t p a s s e s th ro u g h t h i s " z e ro f i e l d " r e g io n , and a r e d i s t r i b u t i o n o f t h e p o p u la t io n o f t h e J = § s u b s t a t e s (m.j = ± -g) o c c u r s . A t f i r s t g l a n c e , a bou t e q u a l ' f i n a l s t a t e p o p u la t io n s or 's and p ' s w ould b e e x p e c te d . 4■ F ig u re 3 : Zeeman d iag ram o f t h e ■f i n e s t r u c t u r e and h y p e r f in e s t r u c t u r e s t a t e s i n t h e 2 S i and l e v e l s o f atom ic, s s h y d ro g en . v a. states Fine Structure -- Hyperfine-------- B states e states f states PTirr= O Magnetic Field (gauss) -1 3 - R o b is c o e , e t . a l , , m easu red th e (3 s t a t e p o p u la t io n r e l a t i v e t o th e t o t a l 28jL s t a t e p o p u la t io n as. a f u n c t i o n o f th e c u r r e n t i n t h e s f l o p p e r . I t was e x p e c te d t h a t a s t h e f lo p p e r c u r r e n t i n c r e a s e d , t h e p s t a t e p o p u la t io n w ou ld s a t u r a t e a t a c o n s ta n t v a lu e . However, as shown i n F ig u r e 4 , t h e (3 s t a t e p o p u la t io n r e a c h e d a maximum, and th e n s lo w ly d e c r e a s e d - - w i th a damped o s c i l l a t o r y s t r u c t u r e . R ob isco e n o te d t h a t no p s t a t e s w ere p o p u la te d u n t i l t h e f i e l d was d r iv e n th ro u g h z e ro ( r e v e r s e d d i r e c t i o n ) . T h is p a r t i c u l a r |3 s t a t e p r o d u c t io n cu rv e ( F ig u r e 4) in c lu d e s c o n t r i b u t i o n s from beam atom s a t v a r io u s v e l o c i t i e s ( t h e a tom ic beam • i s p ro d u ce d b y e f f u s i o n from an oven and h en ce h as a v e l o c i t y d i s t r i ­ b u t io n ) . To a v o id v e l o c i t y a v e ra g in g e f f e c t s , a p r o d u c t io n c u rv e f o r a s e l e c t e d sm a l l r a n g e o f v e l o c i t i e s was m easu red by a t im e - o f - f l i g h t . t e c h n iq u e ; r e s u l t s s im i l a r t o t h o s e we m easu red w i th a f lo p p e r l i k e R o b is c o e 1s i s shown i n F ig u re 5 . The s t r u c t u r e i s now much more p r o ­ n oun ced . R o b is c o e , e t . a l . , a l s o o b s e rv e d an o s c i l l a t o r y s t r u c t u r e i n t h e v e l o c i t y d i s t r i b u t i o n o f t h e (3 s t a t e s . I f we ig n o r e h y p e r f in e s p l i t t i n g , we may a n a ly z e t h i s p ro b lem b y c o n s id e r in g a two l e v e l s p in § sy s tem ; t h i s i s s u f f i c i e n t t o d e s c r ib e t h e h y d ro g en f i n e - s t r u c t u r e . S u p p o r t f o r t h i s a p p ro a ch i s found i n a t h e o r e t i c a l p a p e r b y J . .H e in r ic h s / ^ H e in r ic h s c a l c u l a t e d th e p r o ­ b a b i l i t y f o r an e l e c t r o n i c t r a n s i t i o n i n an a tom ic c o l l i s i o n . The th e o r y H e in r ic h s u s e d was an e x te n s io n o f th e L andau -Z ener th e o r y L andau -Z en e r th e o r y c o n s id e r s t h e p s e u d o -c ro s s in g o f two e l e c t r o n i c ( 18, 19) F ig u re 4 : Non v e l o c i t y s e le c te d - - f lo p p e r cu rv e co nv e rs io n , p e r c e n ta g e o f P g ' s ;as f lo p p e r c u r r e n t . show ing, th e , f u n c t i o n o f - - 0.37 > . FIG.4 t i I F ig u re 5 .: V e lo c i ty s e l e c t e d f lo p p e r c u rv e . T l Y I Bp Fraction Conversion PO i I I I r n i7 n -1 8 - e n e rg y l e v e l s and d e v e lo p s ' am p litu d e e q u a t io n s i d e n t i c a l t o th e two l e v e l n o n - a d i a b a t i c p a s s a g e am p litu d e e q u a t io n s d e r iv e d i n C h ap te r I I . H e in r i c h s ' c a l c u l a t e d p r o b a b i l i t y f o r an e l e c t r o n i c t r a n s i t i o n has t h e same s t r u c t u r a l fo rm a s t h a t fo und i n th e |3 s t a t e p r o d u c t io n c u r v e . . i However, i t i s d i f f i c u l t t o b e l i e v e t h a t th e h y p e r f in e s p l i t t i n g i n t h e 2 S i e n e rg y l e v e l i s i g n o r a b l e . The d e c o u p lin g m ag n e tic f i e l d s ' ■ s t r e n g t h f o r t h e h y p e r f in e s t a t e s i s on th e o r d e r o f 50 g a u s s , much l a r g e r th a n t h e f i e l d s i n th e f lo p p e r (w h ich a r e r a r e l y l a r g e r th a n 10 g a u s s ) . H ence , t h e t h e o r y o f M a jo ran a f lo p s s h o u ld b e a p p l i e d t o t h e f o u r h y p e r f in e l e v e l s ' a s w e l l a s t h e two f i n e s t r u c t u r e l e v e l s f o r c om p a r iso n . The f o u r l e v e l th e o r y w i l l q u ic k ly r e d u c e t o a t h r e e l e v e l , S = I , t h e o r y b y n o t in g t h a t t h e lo w e s t h y p e r f in e l e v e l , F = O , m^ = O (d e n o te d i s s e p a r a te d b y I 78 MHz from th e t h r e e o th e r h y p e r f in e s t a t e s (F = I ) a t z e ro m ag n e tic f i e l d . T h is e n e rg y l e v e l s e p a r a t i o n i s s u f f i c i e n t t o i s o l a t e t h e from th e a s t a t e s so t h e IS s t a t e is . n o t g e n e r a te d b y n o n - a d ia b a t i c p a s s a g e o f t h e a s t a t e s . We s h a l l u s e t h e f i r s t few s e c t i o n s o f C h ap te r I I t o d e f in e and d e v e lo p more c l e a r l y t h e th e o r y o f n o n - a d ia b a t i c p a s s a g e . T h is w i l l b e a c com p lish e d by s o lv in g a two l e v e l sy s tem f o r a " c o n s ta n t" a d i a b a t i c i t y p a ram e te r of. We w i l l show t h a t th e p ro b lem i s com­ p l e t e l y c h a r a c t e r i z e d by th e s i n g l e p a ram e te r a . T h is p e d a g o g ic a l ^ exam ple w i l l be u s e d t o d e f in e n o n - a d ia b a t i c r e g io n s more p r e c i s e l y th a n by th e c o n d i t io n ■ a « I . We n e x t assume a s im p le m ag n e tic f i e l d whose a x i a l component r e v e r s e s d i r e c t i o n l i n e a r l y and th e n s o lv e th e am p litu d e e q u a t io n s f o r t h e two l e v e l and t h r e e l e v e l c a s e s r e s p e c ­ t i v e l y . The c h a p te r w i l l end w i th a d i s c u s s io n o f t h e e f f e c t s o f im p ac t p a ram e te r and v e l o c i t y a v e r a g in g . I n C h a p te r I I I , we w i l l d e s c r ib e t h e e x p e r im e n ta l a p p a ra tu s n e c e s s a r y t o p e r fo rm t h e e x p e r im e n ts . The e x p e r im e n ta l m ag n e tic f i e l d w i l l b e com pared t o t h e i d e a l i z e d f i e l d u s e d i n C h a p te r I I b y com paring t h e assum ed fo rm o f t h e a d i a b a t i c i t y p a ram e te r t o t h e e x p e r im e n ta l l y m easu red a- The r e s u l t s o f t h e e x p e r im en t w i l l th e n be g iv e n i n t h e f i r s t few s e c t i o n s o f C h a p te r IV w i th t h e o r e t i c a l f i t s . A t t h e end o f C h a p te r IV we w i l l p ro p o se a m ethod o f p ro d u c in g ■ a p o l a r i z e d beam o f m e ta s ta b I e h y d ro g en atom s u s in g M ajo rana -19" t r a n s i t i o n s . CHAPTER TI THEORY 2 .1 A S p in F l i p P rob lem The two l e v e l s p in f l i p p ro b lem t h a t we w i l l c o n s id e r i n t h i s s e c t i o n i s f a s h io n e d a f t e r a p a p e r b y R . T. R o b i s c o e / ' ^ We u s e h i s fo rm a lism b e c a u s e we w i l l e x te n d i t t o a t h r e e l e v e l s y s tem l a t e r . We w i l l sp end some t im e d i s c u s s in g th e a d i a b a t i c i t y p a ram e te r and u s e i t t o d e f i n e n o n - a d i a b a t i c r e g i o n s . C o n s id e r a s p in g p a r t i c l e moving w i th v e l o c i t y v th ro u g h an inhom ogeneous m ag n e tic f i e l d H a s shown i n F ig u r e I . The s p in g e n - e r a t e s a m ag n e tic moment p, = g p ^ c , w he re g i s t h e Lande g - f a c t o r , ' Hg i s t h e Bohr m agne ton , and 6 a r e t h e P a u l i s p in m a t r ic e s f o r . s p i n S = g-. The i n t e r a c t i o n o f t h e m ag n e tic moment w i th t h e e x t e r n a l m ag n e tic f i e l d is- d e s c r ib e d b y th e H am il to n ia n JC = jt * li. The tim e e v o lu t io n o f th e s p in may b e c a l c u l a t e d from .t im e -d e p e n d e n t p e r t u r ­ b a t i o n th e o r y f o r a g iv e n s e t o f i n i t i a l c o n d i t i o n s . A ssum ing th e s p in s t a t e v e c to r i s . \ |/( t) = Sa ( t ) e x p ( - i E1t / + a g( t ) exp( - iE gt / fi) , w here Of1 a r e ■ th e s p in - u p and sp in -d ow n b i s p i n o r s r e s p e c t i v e l y , S c h r o d in g e r ' s e q u a t io n y i e l d s t h e m a t r ix e q u a t io n 1 lH i t ) ) = -(in -g /fc) . Hx - U i1A \ a s ( t ) ) Vx+ iiv - t / ^ a 8C t Y (I) W ith t h e m ag n e tic f i e l d i n s p h e r i c a l p o l a r c o o r d i n a t e s , - 2 1 - ^ ( t ) = H ( t ) ( s i n 9 co s 0 , s i n 9 s i n 0 , cos 9 ) , E q u a tio n ( I ) becomes diG/dt = -i(cu(t)/2)3CE, K ( 2) co s 9 s i n 9 e x p ( - i 0 ) 1s i n 9 e x p ( i0 ) -co s 9 w here a( t ) i s t h e Larmor f re q u e n c y 2( |j,B/ ti)H( t ) , and r e p r e s e n t s t h e column v e c to r i n E q u a t io n ( I ) . To s tu d y t h e c a s e s o f weak ( a d i a b a t i c ) and s t r o n g ( n o n - a d ia b a t i c ) c o u p l in g , E q u a t io n (2 ) i s d i a g o n a l iz e d a t e a ch i n s t a n t o f t im e by a change i n t h e b a s i s v e c t o r . The r e p r e s e n t a t i o n i s changed from G t o fi. v i a t h e u n i t a r y t r a n s f o rm a t io n s §•: G = S B . The B e q u a tio n s a r e d B /d t = - ( ia ) /2 ) I O O - I 1 / 0 ~exp( - i 0 ) \ B - [ ( d 9 /d t ) /2 j I B \ e x p ( i0 ) O (3 ) where co s ( 9 /2 ) ' - s i n ( 9 /2 ) e x p ( - i0 ) s i n ( 9 /2 ) e x p ( i 0 ) cos ( 9 /2 ) (4) The am p litu d e e q u a t io n i s now d iv id e d i n t o an e ig e n v a lu e m a t r ix ( f i r s t te rm on th e f i g h t i n E q u a t io n .3.)., and a c o u p lin g m a t r ix ( s e c o n d te rm on th e r i g h t s id e o f E q u a t io n 3 ) . The cases , o f weak and s t r o n g c o u p lin g may th e n b e d i s c u s s e d by com paring t h e r e l a t i v e s i z e s o f t h e c o e f f i ­ c i e n t s o f t h e m a t r i c e s , nam ely w(t ) and d 9 / d t . I t i s c o n v e n ie n t a t t h i s p o i n t t o d e f in e t h e r a t i o o f (jo(t) and d 9 ( t ) / d t a s t h e " a d i a b a t i c i t y p a ram e te r" a - 2 2 - Q '(t) = o < t ) / ( d 6 ( t ) / d t ) . (5 ) I f o' » I , t h e e ig e n v a lu e s m a t r ix d om in a te s , and when or « I , t h e c o u p l in g m a t r ix i s d om in an t. F o r t h e s e two l im i t i n g c a s e s , th e am p li ­ tu d e s w i l l now be d e r iv e d and th e s o lu t i o n s d i s c u s s e d . I . or » I , A d ia b a t i c Case When or » I , t h e IB am p litu d e e q u a t io n i s a p p ro x im a te ly , 1 O 1 dB /d t = - ( i a j /2 ) B10 -ij The s o lu t i o n s a r e fo und s t r a i g h t f o rw a r d l y t o b e ( 6) = b ^ s C t 0 ) e x p |> iX ( t ) / 2 ] , (7 ) Where X(t ) i s t h e i n t e g r a t e d Larmor f r e q u e n c y , d e f in e d a s X (t) = J w(T)dT . (8 ) t O . T ra n s fo rm in g b a c k t o th e . Q r e p r e s e n t a t i o n b y G ~ S B , and assum ing t h e i n i t i a l c o n d i t io n s ^1C to ) = I and a 3 ( t o ) = 0 , t h e ex trem e a d i a b a t i c c a s e ( 8 = 0 ; 8 = Bq) y i e l d s a i ( t ) = c o s (X ( t ) / 2 ) - i co s PE s i n ( X ( t ) / 2 ) , , (9) a 3( t ) = - i s i n PE s i n ( X ( t ) / 2 ) I f - G i s a r b i t r a r i l y s e t t o z e r o , t h i s r a t h e r c om p lic a te d s o lu t i o nO • r e d u c e s t o - 2 3 “ O1 ( t ) = e x p [ - ( i / 2 ) X1 J o(T)d.T] => ^ ( t ) | 2 = I t O ( 10 ) a a ( t ) = 0 T h is shows t h a t f o r a m ag n e tic f i e l d w i th c o n s ta n t d i r e c t i o n ( o r " r e - i s o n a b ly " c o n s t a n t ) , ' t h e r e a r e no " s p in f l i p s , " no m a t t e r how d r a s t i c a l l y t h e m agn itu d e o f t h e f i e l d may c h a n g e . I I . a « I , N o n -a d ia b a t ic ' Case F o r of « I , t h e c o u p l in g m a t r ix i n E q u a tio n (3 ) d om in a te s . The B am p litu d e e q u a t io n becomes I O -ex p ( - i 0 ) dB/dt = -j(d e /d t)/2 e x p ( i0 ) O I t i s fo u n d t h a t b y c h an g in g th e in d e p e n d e n t v a r i a b l e t t o 0 , t h e c o u p le d e q u a t io n s become Ib lyZdG = e x p ( - i0 ) b s / 2 , d b3/d 0 = - e x p ( i0 ) ^ / 2 . B. ( n ) ( 12) (^ 3 ) The s o lu t i o n s a r e I 1 (G) = co s ( 0 / 2 ) , bg( 0) = -exp ( i0 ) s i n ( 0 /2 ) T ra n s fo rm in g b a c k t o t h e a1 am p l i tu d e s , and u s in g th e same i n i t i a l c o n d i t io n s on a 1 ^s ( t Q) as b e f o r e , t h e r e s u l t s a r e a 1 ( t ) = 1 , a s ( t ) = O • ( -2 4 - T h is r e s u l t shows t h a t t h e s p in rem a in s a l ig n e d w i th th e i n i t i a l mag­ n e t i c f i e l d r e g a r d l e s s o f how th e f i e l d r o t a t e s . F o r i n s t a n c e , i f t h e m ag n e tic f i e l d r o t a t e s b y l8 o ° ( rr) y th e n th e i n i t i a l s p in - u p s t a t e b e ­ comes th e f i n a l sp in -d ow n s t a t e and a com p le te " s p in f l i p " h a s o c c u r r e d To summ arize , i f t h e . a d i a b a t i c i t y p a ram e te r a i s g r e a t e r th a n o n e , th e n th e s p in w i l l rem a in a l ig n e d w i th t h e m ag n e tic f i e l d and no s p in f l i p s o c c u r ; i f a « I , some s p in s w i l l b e f l i p p e d . The dem ar- . c a t i o n l i n e b e tw een t h e a d i a b a t i c and n o n -a d ia b a t i c c a s e s i s f a i r l y am b iguou s . However, a q u a n t i t a t i v e i n d i c a t i o n o f how th e s p in f l i p p r o b a b i l i t y v a r i e s w i th t h e a d i a b a t i c i t y p a ram e te r a o v e r i t s e n t i r e r a n g e may b e g a in e d b y s o lv in g E q u a t io n (3 ) e x a c t l y ; t h i s i s p o s s ib l e f o r t h e c a s e o f or = c o n s t a n t . The c h o ic e o f a = p a r a m e t r ic c o n s ta n t ( in d e p e n d e n t o f 0) c o r ­ re sp o n d s t o a c o n s ta n t m agn itu d e m ag n e tic f i e l d r o t a t i n g u n i fo rm ly th ro u g h a g iv e n a n g le . In p r a c t i c e , t h i s may b e a c com p lish e d b y s e n d ­ in g a beam o f p o l a r i z e d s p in s th ro u g h a f i e l d r o t a t o r . ^ The mag­ n e t i c f i e l d w i l l b e ch o sen t o r o t a t e from 0 = 0 to 9 = rr a t t im e s . t = 0 ( e n t r y ) and t = + “ ( e x i t ) r e s p e c t i v e l y . The i n i t i a l co n ­ d i t i o n s w i l l b e a 1( t = 0) = I and a 3 ( t = 0) = 0 . The s o l u t i o n i s b egun b y t r a n s f o rm in g away th e d i a g o n a l m a t r ix i n E q u a t io n (3 ) by a n o th e r change i n r e p r e s e n t a t i o n , n am e ly , c h an g in g B t o C by t h e u n i t a r y t r a n s f o rm a t io n H: C = Cf B. The C am p litu d e e q u a t io n i s -2 5 - i ae 2 at - e x p [ - i ( A ( t ) T 0) ] / e x p ( - iX ( t ) / 2 ) O e x p [ i ( X (t) - 0) J\ O C i ( t ) ^ ( " b ) (15 ) (16) O e x p ( iX ( t ) / 2 ) I f t h e m ag n e tic f i e i a g r a a i e n t i s w eak enough so as n o t t o a f f e c t th e p a r t i c l e t r a j e c t o r y , t h e a z im u th a l a n g le 0 becomes an u n im p o r ta n t" c o n s ta n t p h a se t h a t w i l l a r b i t r a r i l y b e s e t t o z e r o . By c h an g in g th e in a e p e n b e n t v a r i a b l e from t t o 0, i t i s f o u n i t h a t t h e s e t o f c o u p le a a i f f e r e n t i a l e q u a t io n s f o r th e C am p li tu a e s become I c 1/ a e = exp( iX( 9 ) /2 ) c s / 2 , ac3/a 0 = - e x p ( - iX (0 ) /2 ) C1 / 2 . (17) X (t) may b e w r i t t e n as a f u n c t i o n o f 9 , a s ■fc 0 ' 9 x( t ) = J • u)(T)aT = J [a>( e ) /(a e ,/a t ) ]ae; = J a( 9)a6 = x( 8 ). t O 0O 6O The c ■ e q u a t io n s may b e a e c o u p le a , y i e i a i n g a^Cj/ae2 - i« (9 ) ac1/a 9 - C1/4 = o , a2c 3/a 9 2 + i a ( 9) a cg /ae + c3/4 = o . ( 18) ( ] 9 ) The s o lu t io n s ' may b e fo u n a by s t r a i g h t f o r w a r l m eans, r e c a l l i n g t h a t by c h o ic e o' i s a c o n s t a n t . A f t e r a p p ly in g t h e b o u n la r y c o n i i t i o n s , t h e s o lu t i o n s a r e -26- G1(B) = exp( io '0 /2 ) [ c o s ( k 0 /2 ) - i ( or/k) s in ( .k e /2 ) ] Cg(B) = - ( 1 /k ) e x p ( - io '0 /2 ) s i n ( k B /2 ) , ( 20) i w here k = ( I + as ) s . The G am p litu d e s a r e found b y a p p ly in g th e f t r a n s f o rm a t io n s § and G t o t h e C s o l u t i o n s , t h a t i s , G = S3- C. S e t t i n g 0 = TT t o f i n d th e f i n a l s t a t e am p litu d e s a f t e r com p le te p a s s a g e th ro u g h t h e m ag n e tic f i e l d , i t i s found t h a t t h e am p litu d e s a r e ] a 1 ( CCl 12 = ( l / k ) 3 s i n 2( k n /2 ) = IMCCYERl Y ( 21) Ia3(T r)I2 = c o s3 (kTr/2) + ( a/k ) 2 s i n 3(kTr/2) . The e x p r e s s io n f o r Ia1 (Tr) j2 r e p r e s e n t s t h e p r o b a b i l i t y t h a t th e s p in ends up a n t i - p a r a l l e l w i th r e s p e c t t o t h e f i n a l f i e l d . H ence , i t i s t h e s p in f l i p p r o b a b i l i t y , IMCCYGRl , g rap h ed i n F ig u re 2 . The p r o b a b i l i t y t h a t a s p in i s f l i p p e d i s , a s e x p e c te d , o f o r d e r u n i t y when cv « I and z e ro when cn » I . The i n t e r e s t i n g f e a t u r e o f P(TT,a) i s i t s o s c i l l a t o r y s t r u c t u r e , w h ich i s s im i l a r t o a two s l i t F r a u n h o fe r d i f f r a c t i o n i n t e n s i t y p a t t e r n . T h is i s due t o t h e r a t h e r s h a rp b o u n d a ry c o n d i t i o n s . By s h a r p , we mean s e t t i n g a1 = I , and a 3 = 0 a t p r e c i s e l y t = 0 . E x p e r im e n ta l ly , th e i n i t i a l c o n d i t io n s a r e n o t t h a t s h a rp and p re sum ab ly th e o s c i l l a t i o n s i n IMCCYEyl , w ould b e damped. ^^ I t i s a l s o i n t e r e s t i n g t h a t w here th e o s c i l l a t i o n s p e a k , t h e s p in f l i p p r o b a b i l i t y i s s i g n i f i c a n t l y g r e a t e r th a n zero . H ence , a s t r o n g e r c r i t e r i o n t h a t a « I o r a » I c an b e d ev e lo p ed t o d i s t i n g u i s h n o n - a d ia b a t i c and a d i a b a t i c s i t u a t i o n s . Assum ing t h a t we w ish t o d e f in e n o n - a d ia b a t i c r e g io n s as r e g io n s w here t h e s p in f l i p -2 7 - p r o b a b i l i t y i s g r e a t e r th a n 0 . 0 1 , th e n t h e v a lu e f o r a, w here IMCCYERl i s a lw ays l e s s t h a n 0 -0 1 f o r i n c r e a s i n g a, i s a b o u t 10- The maximum o f e a ch o s c i l l a t i o n o c c u rs a t 1 / ( 1 + of3) , so f o r l / ( l + o'2 ) £ 0 -0 1 , we h ave a S 3D. E ve ry m ag n e tic f i e l d r e g io n w here E• 5 10 i s c a p a b le o f in d u c in g a s p in f l i p a t a l e v e l o f 1% o r m ore . T h is c r i t e r i o n a llow s any m agn e tic f i e l d t o b e d iv id e d i n t o a d i a b a t i c and n o n - a d ia b a t i c r e g io n s b y d e f i n in g a " t r a n s i t i o n l e n g th " d e te rm in e d b y a ^ 10 . T here a r e two o t h e r c a s e s , o t h e r th a n a = p a r a m e t r ic c o n s ta n t in d e p e n d e n t o f 9 , t h a t have b e e n s o lv e d e x a c t l y . ^ ^ The c a se o f a ccQ i s t h e w e l l known L andau -Z ener p seu do l e v e l - c r o s s i n g , a p p ro x im a tio n f o r c h a rg e exchange p r o c e s s e s . ^ The s o l u t i o n f o r or cc g i s th e f u n c t i o n c a l l e d t h e L andau -Z ene r e n v e lo p e ( L-Z e n v e lo p e ) . The e n v e lo p e r i s e s from z e ro t o a maximum and th e n d e c r e a s e s t o z e ro w i th i n c r e a s i n g . (1 5 ) a . There- i s no o s c i l l a t o r y s t r u c tu r e , a t a l l . As shown b y J . H e in r ic tv ' o s c i l l a t i o n s o c c u r on t h i s e n v e lo p e i f t h e t r a n s i t i o n r e g io n i s assum ed t o b e f i n i t e ; Landau and Z en e r assum ed th e i n t e r a c t i o n e x te n d e d from t = - co t o t = + co. The L-Z e n v e lo p e rem a in s t h e a sym p to tic fo rm f o r t h e p seudo l e v e l - c r o s s i n g p ro b lem even w i th a f i n i t e t r a n s i ­ t i o n r e g io n . T h is i s im p o r ta n t s in c e we w i l l r e f e r t o a l l su ch a s ym p to t ic fo rm s a s L-Z e n v e lo p e s . The s o l u t i o n f o r or °c 0 w i l l b e d e r i v e d . i n S e c t io n 2 .3 from t h e two l e v e l n o n - a d ia b a t i c p o i n t o f v iew . The se co n d c a se t h a t h a s b e en s o lv e d e x a c t ly i s a cx cosh ( c o n s ta n t X t ) (22) - 2 8 “ T h is c a s e i s o f l i t t l e i n t e r e s t t o u s s in c e t h e c o n s t r u c t i o n o f a m ag n e tic f i e l d w i th t h i s b e h a v io r w ould b e e x trem e ly d i f f i c u l t , i f a t a l l p o s s i b l e . I t i s c l e a r l y e a s i e r t o c o n s t r u c t a " s im p le " m ag n e tic f i e l d r e g io n and th e n s o lv e t h e am p litu d e e q u a t i o n s . I t i s f o r t u n a t e t h a t t h e " s im p le " m ag n e tic f i e l d we u s e d i n t h i s e x p e r im en t i s s im i l a r t o th e p e r t u r b a t i o n u s e d b y Landau and Z e n e r . 2 .2 -A d ia b a t ic i ty P a ram e te r The a d i a b a t i c i t y p a ram e te r d e f in e d i n E q u a tio n ( 5 ) ; may b e r e ­ w r i t t e n t o r e f l e c t t h e v e l o c i t y d e p e n d en c e . I f t h e p a r t i c l e v e l o c i t y i s v = d z / d t , th e n a becomes o' = [2fj,gH /ft(d0/dz) ] / v » (2 3 ) The a d i a b a t i c i t y p a ram e te r i s much l e s s a t h ig h v e l o c i t i e s th a n a t low v e l o c i t i e s . T h is s h o u ld b e o b v io u s from an i n t u i t i v e s t a n d p o i n t . The s low e r t h e p a r t i c l e t r a v e l s , t h e more t im e i t h a s t o f o l lo w th e f i e l d r o t a t i o n . I n any e x p e r im e n t w here t h e p a r t i c l e beam c o n ta in s s p in s w i th some d i s t r i b u t i o n o f v e l o c i t i e s , i t i s th u s n e c e s s a r y t o a v e ra g e t h e s p in f l i p p r o b a b i l i t y o v e r t h e beam v e l o c i t y d i s t r i b u t i o n , - b e c a u s e a d epend s u pon th e p a r t i c l e v e l o c i t y . W ith t h i s d e f i n i t i o n f o r a , a new f u n c t i o n a p p e a r s , nam ely d 0 /d z . T h is i s j u s t t h e s p a t i a l r o ­ t a t i o n - r a t e o f th e m ag n e tic f i e l d i n s t e a d o f th e t e m p o r a l -r o t a t i o n r a t e . I t i s o b v io u s ly e a s i e r t o m easu re d 9 /d z th a n d 0 / d t . The " s im p le " m ag n e tic f i e l d we ch o se t o u s e i n t h i s 'e x p e r im en t was b a s e d i n p a r t on t h e fo rm f o r H ( t) f o r t h e c a se o f h = p a r am e t r ic c o n s t a n t . I n t h a t c a s e , t h e m agn itu d e o f t h e m ag n e tic f i e l d was a c o n s ta n t w h i le i t r o t a t e d i t s d i r e c t i o n by 180 d e g r e e s . As we m en tio n ed e a r l i e r , t h i s i s r a t h e r d i f f i c u l t t o a c com p lish e x p e r im e n ta l l y . However, we c an v i s u a l i z e a f i e l d c y l i n d r i c a l l y symm etric a b o u t th e p a r t i c l e beam . A ssum ing t h e s p in t r a j e c t o r y i s a s t r a i g h t l i n e d e ­ f i n i n g an a x i s , we can c o n s t r u c t a m ag n e tic f i e l d i n i t i a l l y aim ed a lo n g t h e a x i a l d i r e c t i o n e s s e n t i a l l y c o n s ta n t i n m ag n itu d e , f o r i n s t a n c e a s o le n o id . A no th e r s o le n o id w i th a f i e l d o r i e n t e d o p p o s i t e t o th e f i r s t s o le n o id t h e n g e n e r a te s a m ag n e tic f i e l d o p p o s i te i n d i r e c t i o n and th e a x i a l f i e l d w i l l t h e n have r o t a t e d b y 180 d e g re e s as t h e p a r t i c l e t r a v e l s th ro u g h th e f i r s t s o le n o id i n t o t h e se co nd o n e . The p ro b lem now i s t h e q u e s t io n o f j u s t how th e a x i a l f i e l d r e v e r s e s i t s d i r e c t i o n . One a p p ro x im a t io n w ou ld b e f o r th e a x i a l f i e l d t o b e e s s e n t i a l l y con ­ s t a n t up t o a p o i n t w here i t b e g in s t o d e c r e a s e i n m agn itu d e l i n e a r l y t o z e r o . The a x i a l f i e l d c o u ld th e n i n c r e a s e i n m agn itu d e l i n e a r l y b u t i n t h e o p p o s i t e d i r e c t i o n u n t i l i t r e a c h e d a p o i n t w here i t becomes a c o n s ta n t a g a in o n ly o p p o s i t e ly o r i e n t e d t o t h e i n i t i a l f i e l d . T h is ty p e o f f i e l d i s shown i n F ig u r e 6 . A ssum ing c y l i n d r i c a l symm etry , we may th e n c a l c u l a t e t h e r a d i a l component o f t h e m agn e tic f i e l d from th e d i f f e r e n t i a l fo rm o f B io t and S a v a r t 1s law , V • H =' 0 . The r a d i a l f i e l d i s a l s o p l o t t e d i n F ig u r e 6 . I t s h o u ld b e n o te d t h a t " r e a l " m ag n e tic f i e l d s a r e sm oo th ly v a ry in g w i th c o n tin u o u s d e r i v a t i v e s . W hile we can come c lo s e t o t h i s ty p e o f F ig u re 6 : A s im p le m odel f o r a 'm ag n e tic f i e l d r o t a t i n g , by 180 d e g r e e s . -3 1 - FIG.6 "32- magnet i c f i e l d i n t h e l a b o r a to r y , t h e f i n a l j u s t i f i c a t i o n f o r a ssum ing t h i s fo rm i s i n com paring or f o r t h e r e a l f i e l d t o t h e t h e o r e t i c a l a we w i l l d e r i v e . T h is i s done i n C hap te r- I I I and i t w i l l b e s e e n th e n t h a t t h e com p a riso n i s much b e t t e r th a n m ig h t b e e x p e c te d a t t h i s p o i n t . From F ig u r e 6, we s e e t h a t f o r j z | ^ Lq , t h e f i e l d i s p u r e ly a x i a l and n o n - r o t a t i n g . T h is i s t h e a d i a b a t i c r e g io n w here th e s p in s im p ly rem a in s a l i g n e d o r a n t i - a l i g n e d w i th th e f i e l d . H ence , a i s i n f i n i t e b e c a u s e d 8/ d t i s z e r o . How a t • L q , a s t h e p a r t i c l e t r a v e l s f rom l e f t t o r i g h t , t h e m ag n e tic f i e l d b e g in s t o change r a p i d l y . T h a t i s , H = H z / z and H = - H r / 2 z w here r i s t h e r a d i a l z o ' o T o ' o d i s t a n c e o f f t h e z a x i s . By d e f i n in g z = v t and u s in g F ig u re I , we h ave f o r t h e Larmor f r e q u e n c y and f i e l d r o t a t i o n r a t e w (t) = g( May , q/ ft) p / s i n 6 ( t ) , d 9 /d t = ( v /p z o ) s i n 2 0 ( t ) , (2k) where p = r / 2 z Q. The a d i a b a t i c i t y p a ram e te r i s Q)/ g iv e n th e i n i t i a l am p litu d e s a t 9 = 0 ( t = - ro) . T here a r e two a p p ro a ch e s t o ■ f i n d i n g G( CCl . The f i r s t m ethod i s d i r e c t i n t e g r a t i o n o f t h e C am p litu d e e q u a t io n s f o r a s e t o f i n i t i a l c o n d i t i o n s , s e t t i n g 9 = rr, a and t r a n s f o rm in g b a c k t o th e - G am p l i tu d e . The second ap p ro a ch i s S -m a t r ix t h e o r y , som etim es c a l l e d t h e e v o lu t io n m a t r ix o r p r o p a g a to r t h e o r y , w h ich d i r e c t l y g iv e s C(Tr) from C( 0 ) . E ach o f t h e s e m ethods w i l l b e d i s c u s s e d , b e g in n in g w i th t h e f i r s t m e thod . From E q u a tio n ( 3 8 ) , t h e c o u p le d e q u a t io n s a re d c3/d 0 = - e x p [- iX ( 9) ] C2/ J 2 . T hese may b e d e c o u p le d , w i th th e r e s u l t I 3C1JdO3 - 3 ia ( ' 0) d 2c 1/ d 0 2 + [ I - 2a3( 9) - i d a ( 0 ) /d 9 ] Gc1JdO - ~ la ( 9 ) C1 = 0 , G3C3JdO3 - [ d ln a ( 0) J d 0 ]G2C3J d 2 0 + [ I + o;2( 9) I d c 3JdO - - [d ln o f ( 9 )JdO jc 3 = 0 , (40 ) G3C3JdO3 + S ia (G )G 2C3JdG2 + [ I - 2a;2( 0) + Ida(G )JdO ] Gc3JdO + + ia;( 9) c 3 = 0 . The o n ly c a s e w here E q u a tio n s (4-0) h ave b e e n s o lv e d a n a l y t i c a l l y i s f o r a = c o n s t a n t . The s o lu t i o n s f o r a = c o n s ta n t f o r v a r io u s i n i t i a l Gc1JdG = e x p [ iX (9 ) ] C3J aJ 2 , Gc3JdO = [ex p [iX ( 9) ] C3 - e x p [- iX ( 9) ] C1J J vTiT" (39) -4 6 - c o n d i t io n s a r e g iv e n i n A ppendix I f o r com pa riso n w i th t h e two l e v e l c a s e . F o r a (9 ) oc s i n ^ 8, th e E q u a tio n s (4o) c a n n o t b e s o lv e d a n a l y t i c a l l y . The e q u a t io n s can b e n u m e r ic a l ly i n t e g r a t e d , g iv e n i n i t i a l c o n d i t i o n s , f o r e a ch v a lu e o f a . T h is h as b e e n d on e ; how- e v e r , due t o th e- e x tr em e ly s low co n v e rg en c e o f t h e com pu te r s o l u t i o n , t h i s a p p ro a c h was ab andoned . The r e a s o n f o r t h e s low co n v e rg en ce o f . t h e com pu te r s o l u t i o n i s t h e p r e s e n c e o f t h e te rm e x p [ iX (9 ) ] . F o r 8 c lo s e t o z e r o , t h e s t a r t i n g p o in t o f t h e n um e r ic a l i n t e g r a t i o n , \ ( 9) i s a p p ro x im a te ly a (0 ) &8. Cv(O) i s e x tr em e ly l a r g e a t t h i s p o in t and t h e te rm exp [iX ( 9) ] o s c i l l a t e s v e r y r a p i d l y . T h e r e f o r e , v e ry sm a l l in c rem e n ts AG ' a r e n eeded f o r t h e n um e r ic a l i n t e g r a t i o n a t t h e s t a r t ( and a l s o a t t h e c o n c lu s io n , b e c a u s e X(Tr) i s a l s o i n f i n i t e ) . T h is le a v e s - t h e S-m a t r ix a p p ro a ch t o s o lv in g t h e am p litu d e e q u a t i o n s . S in c e t h e e q u a t io n s c a n n o t b e s o lv e d a n a l y t i c a l l y , i t i s d o u b t f u l t h a t S -m a t r ix th e o r y w i l l l e a d t o a n a l y t i c s o l u t i o n s . How­ e v e r p a s t u s e s o f t h i s m ethod have l e d t o more t r a c t a b l e com puter s o lu t i o n s t h a n th e s t r a i g h t f o rw a r d n u m e r ic a l i n t e g r a t i o n o f th e d i f - • f e r e n t i a l e q u a t i o n s . Due t o t h e v a r i e t y o f m ethods d e v e lo p e d u s in g t h e e v o lu t io n m a t r ix , h e lp from an e s t a b l i s h e d t h e o r i s t i n t h i s f i e l d was s o l i c i t e d . W.R. T ho rson a t th e U n iv e r s i t y o f A lb e r t a g r a c io u s ly c o n s e n te d t o a p p ly h i s com pu te r m e th o d s ^ ^ t o s o lv in g , t h i s t h r e e l e v e l p ro b lem . - The g e n e r a l a p p ro a ch o f S -m a t r ix t h e o r y i s -g iven i n m ost quan tum m ech an ic s te x tb o o k s / h en ce th e th e o r y and th e p r o ­ p e r t i e s . o f t h e e v o lu t io n m a t r ix w i l l b e q u o ted w i th o u t p r o o f . -4 7 - 2 .5 E v o lu t io n M a tr ix The f i n a l s t a t e a m p l i tu d e s ' C(Tr) c an b e r e l a t e d t o t h e i n i t i a l c o n d i t io n s C(O) b y t h e e v o lu t io n m a t r ix G(TTjO) . The r e l a t i o n s h i p i s C(Tr) = PMqOl C (O ). ( 4 l ) Some o f t h e w e l l known p r o p e r t i e s o f t h e S -m a t r ix 6 a r e ^^ ^ 1 . G(8j 6 ) o b e y s j column by co lum n, t h e same d i f f e r e n t i a l e q u a t io n a s t h e C am p litu d e s do ; 2 . G(E^jG) = ^ ( 6, 80) 3. G(GjGo) = 8(6 ,# ) e (* j8o ) j 4 . Gt G = G Gt = I , 5 . d e t |G (8 j6 o ) | = +1 g D PM PE hPq) = I f o r a l l G and 8 . A n o th e r p r o p e r t y o f G i s t h a t i t may b e r e p r e - ( 2 7 )s e n te d b y a u n i t a r y t h r e e d im e n s io n a l r o t a t i o n g ro u p x ' b e c a u s e th e e v o lu t io n m a t r ix j u s t p e rfo rm s a r o t a t i o n w i t h in t h e b a s i s s e t . . . H ence , t h e fo rm u s in g E u le r a n g le s (or, p,Y) may b e u se d t o r e p r e s e n t th e p r o p a g a to r j as I + co s p) exp [ - ! (o r + y ] /2 s i n p exp ( -± i) / J 2 ^ l - cos p) exp [ i ( 01 - y ] / 2 6( column I ) (42) “48~ / - s i n (3 e x p ( - iY ) /v r 2 - S( column 2) co s p % s i n p exp( ±c()/J~2 (42) / ( I - cos P) e x p [ - i ( o ' -■ F l V i A b £( column 3) = - s i n P exp [ iY l /V 2 \ ( I + co s p) exp [ i ( a + p) ] / 2 I f , from p r o p e r t y I , t h e d i f f e r e n t i a l e q u a t io n s f o r t h e .E u le r a n g le s a r e f o u n d , t h e same p ro b lem a r i s e s w i th t h e n um e r ic a l i n t e g r a t i o n o f t h e s e e q u a t io n s as a ro s e w i th E q u a tio n s (4-0) . However, h a l f th e p r o ­ b lem , t h e end o f t h e i n t e g r a t i o n a s 9 rr, may b e e l im in a te d b y ex ­ p l o i t i n g t h e symmetry o f Cv(B), n am e ly : cv( 9) = EMCd • s lD E q u a t io n ( 4 l ) may b e w r i t t e n as The e lem en ts o f 6MCCi 2 ,0 ) may b e • r e l a t e d t o t h e e lem en ts o f £( CCY CCi 2) b y lo o k in g a t th e symmetry i n t h e am p litu d e e q u a t i o n s . Each column o f 6 ( 9 ,0 )■ m ust obey th e same e q u a t io n s a s C; t h e r e f o r e C(TT) = g(Tr,Tr/2) e ( n /2 ,0 ) C (O ). ( t e ) / V 0’ 6o) Z 0 exp [iX ( 9) J \ = ( 2 s ) I -e x p [ - iX ( 0) ] 0 exp (iX ( 9) 3 X \ 0 - -e x p [ - iX ( 9) ] ’ J 0 ] EL Eg ( 8 ,8 ) d8 J \E 3 j ( 0 ,G o ) (4 4 ) -49 - H ere i s th e i e n t r y from th e to p i n £ ( column j ) . S e t t i n g 0 = - 9 , t h e i n t e g r a t e d Larmor f re q u e n c y becomes -6 / , 9 / / X(-9) = “ J E o( - 9) d9 = -J cx( 9)d 9 -X e ) O vO and t h e s ig n r e v e r s e d e q u a t io n s a r e H|mIOJH I 0 - e x p [ - i \ ( 9) 3 ' 0 ■ 1 d ES j ( -G ,0o) exp [iX ( 9) ] 0 -e x p [ - iX ( 0) ] d9 L o\ exp [iX ( 9) ] 0 IlE= j(-6’e=)J (4 5 ) . f E i j ( - 9 ; 0o)^ E s j f - 8 ' G^) . V 0- 0O )/ • By i n s p e c t i o n , th e f o l lo w in g c o r re sp o n d en c e may b e m ade, ■ (46 ) j : e M _ 9> ®0) Ei s ( - 0^9o) Ei s ( - 0^9oA i : N, M0^0o) : N, M0^0o) : NxM 0J 0o )^ ESl( - 0, 0O) W - 0^0o). E3 3 ( - 0, 0O) M E33( 9^0o) ESg( 9> 9Ol : NxM0^0o) \ Esa( "9^0o) Es s ( _0 ^0O). E33 ( - 0^0o) / a1M9 7 0^0o) 1M97 0^0o) En ( 0; 0o)/ T h is r e l a t i o n s h i p a llow s t h e e v a lu a t i o n o f S(TtiO ). S in c e 8( Q , -9 ) = eJ( - 9 , 9q ) , £( TT,rr/2) may b e fo und i n te rm s o f t h e e lem en ts o f e( xCi A Y0) and v i c e v e r s a . Upon w r i t i n g o u t £( CCY0) i n te rm s o f th e e lem en ts o f NMCCi 2 , 0 ) , t h e e v o lu t io n m a t r ix i s fo u nd to have th e f o l lo w in g sym m e tr ie s , b: x x MCq[l : x , M Cq 01) ■ ■ Ei s( CCYs l b G(TTi O) E31(TTjO) -Eas(TfiO) Ei s 755 i0 ) \ Esi(HiO) Eai 755i0 ) E11 ( Tt,Q)J (4 7 ) -5 0 - From th e sym m etrie s i n t h e t h r e e d im e n s io n a l r o t a t i o n g ro u p and E q u a tio n ( 4 2 ) , / , UMCCY0) = El a * ( r r ,0 ) and E13(TTjO) = -E 31(TTj O ). The f i n a l sym m etric fo rm f o r E(H j O) i s th e n g iv e n as S(Hj O) f cta o n ly E11MeO[ l O Eg3(H j O )j E13(H j O) and E13(H j O) n eed t o be E( yPr me ( m ust a l s o obey th e am p litu d e e q u a t i o n s , t h e n um e r ic a l i n t e ­ g r a t i o n s w i l l end a t 0 = u / 2 . F o r 6 z i 2' from z e r o ? EMR9) becomes f r e q u e n c ie s w h ich a llo w s r a p i d co n v e rg en ce o f th e n um e r ic a l i n t e g r a t i o n . T here i s s t i l l t h e p ro b lem o f i n i t i a t i n g t h e i n t e g r a t i o n a t 0 = Oj b u t T ho rson h a s d e v e lo p e d a t e c h n iq u e t o c irc um v en t t h i s h a l f o f th e p ro b lem . S ( H / 2 J b ) . may b e r e p r e s e n t e d b y t h e r o t a t i o n m a t r ix g iv e n b y . E q u a t io n - ( 4 2 ) . U s in g P r o p e r ty I f o r t h e e v o lu t io n m a t r i x J th e c a l c u l a t e d t o c om p le te ly d e te rm in e S(Hj O ). D e f in in g t h e te rm s o f NMz i AO0) a s E ^ j j t h e r e q u i r e d e lem en ts o f S(Hj O) a r e (4 9 ) sm a lle r , and t h e o s c i l l a t i o n s i n t h e te rm exp [iX ( 0) ] h ave sm a lle r - - 5 1 - d i f f e r e n t i a l e q u a t io n s f o r th e E u le r a n g le s a re^ dcv/dB = s i n [ a '+ X( 9) ] cos p / s i n p } d p /d 8 = - cosCcv + X (0)] , (50 ) d y /d e = - sin[(% + X(6) ] / s i n p . The i n i t i a l c o n d i t io n s a r e : a t 6 = 0 , E ^ (O jO ) = 6^ . , w i th d a /d 0 and d y /d 0 f i n i t e a t 0 = 0 . The r e s u l t s a r e |3 = 0 , of = rr - \ ( 8) , ■ and y - X( 8) - rr, a t 8 = 0 . D elo s and T h o r s o n ^ ^ h av e d e v e lo p ed a t e c lm iq u e t o g e n e r a te a s e t o f i n i t i a l c o n d i t io n s a t 6 K WE w 0 from th e o r i g i n a l - i n i t i a l c o n d i t io n s a t 0 = 0 . S t a r t i n g w i th i n i t i a l c o n d i t io n s a t 8 > 0 , t h e E q u a tio n s ( 50 ) may e a s i l y b e n u m e r ic a l ly i n t e g r a t e d , p ro v id e d 8^ i s s u f f i c i e n t l y g r e a t e r th a n z e ro f o r e x p [ iX (8) ] t o b e " s lo w ly o s c i l l a t i n g . " E s s e n t i a l l y , th e m ethod o f D e lo s and T ho rson i s t o assume t h a t e a ch s o l u t i o n ( e a ch column v e c to r i n 6) i s a s y m p to t i c a l l y d om ina ted . b y one l a r g e and " s low ly " v a ry in g c o e f f i c i e n t w h i le t h e o th e r s a r e " d r iv e n " b y t h i s c o e f f i c i e n t and a r e sm a l l . T h is a s sum p tio n h a s b een i n v e s t i g a t e d i n a p a p e r b y D elos and Thorson^ and h a s b e e n shown t o y i e l d c o r r e c t r e s u l t s . E ach column v e c to r h a s a d i f f e r e n t s o lu t i o n d e p en d in g upon w h e th e r i t s i n i t i a l c o n d i t io n h a s t h e t o p , m id d le , o r- b o t tom e n t r y n o n -z e ro a t 6 = 0 ( E^ .( 0 , 0.) = 6^ ) . As an exam ple 3 t h e i t e r a t i o n e q u a t io n s f o r th e colum n E ^ . . w i l l b e d e r iv e d . ■ D en o tin g th e column b y B1 , e g , and e 3 , t h e i n i t i a l c o n d i t io n s a r e -B1(O) = I , e 3(o ) = C a (O) = 0 . T ho rson w r i t e s -5 2 - e 3 = 021e1e5c p [- iX ( t ) ] ^ and e 3 = 033e aex p [- iA ( t ) ] ^ f o r t h e " sm a ll" d r iv e n te rm s . By u s in g E q u a tio n ( 38) , d e1/ d t = Q e .x p [ i \ ( t ) ] e a w here Q =(‘d 9 /d t ) j 2 and B1 becomes e i ( t ) = e ° exp [ p 03J Qd53 • (51) 0 -C O S im i la r m a n ip u la t io n s f o r e g and e3 y i e l d t h e f o l lo w in g i t e r a t i o n e q u a t io n s f o r 03a and 033 fe l = -l(Q /uj)E l + 031 - 0S103@ I “ (i/w )d .02 l/c lt, 032 = [ - 0 ( 1 + 03103s) - d 03a/ d t ] / [ - 2 i u ) + ( d 0 a i / a t ) / 0 9 i ] , (52 ) where w = Larmor f r e q u e n c y . The i d e a i s now t o s o lv e t h e s e e q u a t io n s i t e r a t i v e l y j t h a t i s - 031 ~ -iQ /'w => d 0 3 ! /d t = [ - i (d Q /d t) /( J 0 + iQ( d(ju/dt)/uj3 ] , (53) 033 ~ -iQ /u)=> d033/ d t = [ - i d Q /d t ) / 2 W 4 2 i Q( da)/dt)/cju2 ] , and t h e n t o s u b s t i t u t e t h e d e r i v a t i v e s , e t c , i n t o t h e r i g h t hand s id e o f E q u a tio n s ( 51 ) and (5 2 ) and r e p e a t . F o r 0 / uu s u f f i c i e n t l y sm a l l ( 0 sm a l l) t h e i t e r a t i o n w i l l g e n e r a te an a d eq u a te s e t o f s t a r t i n g v a lu e s f o r t h e E q u a t io n ( 5 0 ) . The n e x t h ig h e r i t e r a t i o n may b e u sed t o c h e ck th e p r e c i s i o n o f t h e s t a r t i n g v a lu e s a t th e 9 c h o sen . T he■ o t h e r two c a s e s a r e : S1(O) = e 3 (o ) = 0 , e g(o ) = I , and : C3i(O) = S3(O) = 0 , S3(O) = I . E ro c ed u re s i d e n t i c a l t o t h e above a r e u s e d t o g e n e r a te s t a r t i n g v a lu e s a t 9 . From h e r e , t h e s o lu t i o n s f o r £( Cxi 2 ,0 ) a r e g e n e r a te d b y n um e r ic a l i n t e g r a t i o n from 9 t o r r /2 , and . - 5 3 - t h e e lem en ts o f kMCCY0) a r e c a l c u l a t e d from E q u a tio n (4 9 ) ( s e e A ppend ix I I f o r com pu te r p ro g ram ) . Computer g rap h s o f th e e lem en ts o f £( TTjO) > F ig u r e 9 , s u g g e s t a n a l y t i c f u n c t i o n s . T ho rson fo und ( to s i x s i g n i f i c a n t f ig u r e s ) t h a t : E13MCCY0) = exp( - m o p2/ 4 ) , and E13 i s r e a l . The o t h e r e lem en ts w e re fo u n d t o "be h e l a t e d t o s im i l a r f u n c t i o n s . I n te rm s o f E13MDCCY0 ) t h e o th e r e lem e n ts , a g a in t o s i x f i g u r e s , a r e | = 1 - E13(TTjO) ) Ka(TT,0)| = (ZE13Cn,0)(l - Eis(TTfO)P , (5%) Egg(TTjO) = [ I - ZESg(TTjQ)]. Es s (TTjO) i s r e a l , w h i le t h e a rgum en ts o f E11(TTjO) and E13(TTjO) a r e r e l a t e d b y A rg fE11( TTjO)] = Z X A rg fE l g (T rjO )]. The a rgum en t o f E11(TTjO) was fo und t o b e sm oo th ly v a ry in g from . 558 6 a t ^ == 0 t o . 0 a t a = Co, a s shown i n F ig u re 10 . U n f o r tu n a t e ly , A rg fE11(TTjO) ] i s n o t an e a s i l y r e c o g n iz a b le f u n c t i o n b u t c an b e a p p ro x im a te d by - ( a;0p2 + 1 .2 ) w i t h in one p e r c e n t . The f a c t t h a t t h e e lem en ts o f S(TfjO) c an b e f i t t e d so c l o s e l y b y s im p le a n a l y t i c f u n c t io n s was q u i t e s u r p r i s i n g and s u g g e s t s t h a t t h e r e may e x i s t a n a l y t i c s o lu t i o n s f o r t h e o r i g i n a l am p litu d e e q u a t io n s ( E q u a tio n 4o). The e v o lu t io n m a t r ix f o r t h e C am p litu d e s i s now g iv e n by / ( I - E 13) O ^ [ZE13C I-E 13) ( 55) ) s ■ 3 S(TTjO)= - fZ E 13( l . E 130 ] 5e ^ [ I -Z E ^ 3F [ZE13( I - E 13) ]& e - \ E13 - [Z E 13( I - E 13) ( I - E 13) G - I * / F ig u re 9 : Computer g e n e r a te d s o lu t i o n s ( p o in t s ) f o r th e m agn itu d e o f t h e e lem en ts o f £ (rr,0 ) "With t h e i r c u rv e f i t t e d a n a l y t i c f u n c t io n s ( s o l i d l i n e s ) . M at ri x El em en t - 5 6 - \ F ig u re 10 : Computer g e n e r a te d p h a s e : A rg tE 1 S( aaOx[ lV Y -57' FIG.IO - 5 8 - where 0 = AxgfE11 ( rr^O) ] . To o b ta in t h e PMCCl a m p l i tu d e s , th e 8 4« and J t r a n s f o rm a t io n s a r e a p p l i e d . R e c a l l in g t h a t G (t ) = 8 ( t )G ( t ) G ( t ) and C(Tr) = C( 2) £( r r /2 ,0 ) C (O ), G( CCl becomes (5 6 ) G(Tr) = 8( AA( 3^ CCbCCi 2) 2) C( r r /2 ,0 ) 3 ( CCi A Y s lS^(O)G(O) = P(ThO)G(O). + From E q u a t io n ( 36) , §( rr) and 8 (0 ) a r e I 0 0 I' I l 0 • 0 ' 0 bAA( S 0 - I 0 , §1’( 0) = 0 I 0 I 0 0 j \° 0 I i From E q u a t io n (3 7 ) and X( 6l K h. a ( e ' ) d 8 ' , t h e 3 m a t r ic e s a r e \ lBr 3(iT,Tr/2) 3 (n /2 ,0 ) where X e x p ( - IX tt) 0 0 exp( iX0 ) 0 0 0 I 0 0 I 0 0 0 exp (iX ^ ) 0 0 exp( - i X j f 7 cy(8 ' ) d 8 ' and X = T , a ( 8 ' ) d 8 ' . U s in g ' th e aAAP r 0 j TT/ 2 e v o lu t io n m a t r ix i n t h e fo rm o f E q u a tio n ( 4 8 ) , we f i n d t h e p ro p a g a to r P(ThO) t o be F(ThO) e i ( Xn-+X0 ) E i iV ( e iX° ~E13 -E 1Z e ' 1 '10 (5 7 ) E1X e"i( Vr"\)) B ia e - i^ r r . Ei s • _5 9 ” The sum and d i f f e r e n c e o f \ and g r e a t l y r e d u c e i f a{ 8) i s X becomes -X and rr osymm etric a b o u t 6 = AAP r ( u• 9) = a( AAm v 6) ] (X + X )= 0 w h ile (X - X ) = 2X • The f i n a l s t a t e am p litu d e s a r e ' TT o / v TT O/ TT. t h e n r e l a t e d t o th e i n i t i a l s t a t e am p litu d e s by E 3e '13 -iX n -jI F -E= ^iXrr E1;l* a2iX" % A iX " (5 8 ) / a I(TT)X a3( AA( E u e - I ^ T T , ^ ,3 TT Tr/2 . where ' X = V of GzIdQ = V off 6 #)d 0 . Keep i n m ind how ever, AA J Tr/2 Jo th a t i f 0/ 0) i s n o t p e r fe c t ly , symm etric about 6 = AAP2 th e n ( X + X ) i s no lo n g e r ze ro and (X - X ) i s n o t e x a c t ly 2X • As a .c h e c k on o u r e v o lu t io n m a t r ix , we e x p e c t t h e f i n a l s t a t e am p li tu d e s t o b e i d e n t i c a l t o t h e i n i t i a l s t a t e am p litu d e s i f th e m ag n e tic f i e l d becomes e x trem e ly a d i a b a t i c . T h is may b e a c com p lish ed b y l e t t i n g <% -> «>. When fc E •w / x vMCCY0) -> O and from E q u a tio n (54 ) we f i n d (58a) I S1(TT)X O O e x p [ i ( 2X^-0) ]\ Za1(O) \ a S(Tr) I= O -1 O I aa(0) ^a3 ( TT) j ^ x p [ - i ( 2X ^-0 )] O ° I ^ a 3(O) J Tha t i s , IaI(Tr) I2 = | a 3( 0 ) I2 , I a 3 (T f) I 2 = |a£( 0 ) | 2 and |a3 (TT) J2 = Ia1(O)I2. R e c a l l t h a t a ^ n ) i s a l ig n e d w i th t h e f i n a l f i e l d and h en ce c o rr e s p o n d s t o t h e am p litu d e a n t i a l i g n e d w i th th e . i n i t i a l f i e l d - [ i . e . a 3( o ) ) i f t h e r e i s a d i a b a t i c p a s s a g e . The same ~6o- a p p l i e s f o r t h e a3 ( rr) am p li tu d e . T h is can b e v e ry c o n fu s in g so we w i l l t r y t o e l u c i d a t e t h e i d e a a l i t t l e m ore . I f a s p in a l ig n e d w i th th e i n i t i a l f i e l d f o l lo w s th e f i e l d r o t a t i o n , th e n i t w i l l become a n t i ­ a l ig n e d w i th t h e i n i t i a l f i e l d and h en ce * Ia1(O) | 2 -» | a 3 ( rr) | 2 ^ i f t h e f i e l d i s a d i a b a t i c . A t th e o th e r e x trem e , i f t h e m ag n e tic f i e l d i s n o n - a d i a b a t i c th e n a c om p le te i n te r c h a n g e o f s p in - u p and sp in -dow n s h o u ld o c c u r . T h is may b e a c com p lish e d b y l e t t i n g a a p p ro a ch z e ro . I f or -> 0 , t h e n E11(TTjO) I and we have U s in g a rgum en ts s im i l a r t o a b ov e , we s e e t h a t i n f a c t t h a t t h e s p in s rem a in a l i g n e d w i th t h e i n i t i a l f i e l d and h ence a r e f l i p p e d b e c au se t h e f i n a l f i e l d d i r e c t i o n i s o p p o s i t e t o t h e i n i t i a l f i e l d . The im p l ic a t i o n s o f t h i s p r o p e r t y w i l l b e made . c l e a r e r i n t h e l a s t c h a p te r w here we d i s c u s s how we m ig h t p ro d u ce a p u re p o l a r i z e d beam o f m e ta ­ s t a b l e a tom ic h y d ro g en . R e c a l l i n g th e d i s c u s s io n on i n i t i a l c o n d i t io n s b e f o r e s e c t i o n t h r e e o f t h i s c h a p t e r , we d e te rm in e d t h a t t h e p r o b a b i l i t y am p litu d e o f i n t e r e s t i s t h e 2S(a;; m^ = +1) am p l i tu d e . From E q u a t io n s (54 ) and ( 58 ) we f i n d S1(Ti) = Ea3S1(O) - El s Xe x p ( i \ n) a s (0 ) + E11^ e x p f2 iA ^ )a 3( 0 ) . (59) ( 5 8 b ) -61- I f o n ly one s t a t e i s i n i t i a l l y n o n - z e ro , th e n , s in c e th e e lem en ts E „ a r e sm oo th ly v a ry in g n o n -o s d i l a t o r y f u n c t i o n s , we can n o t have any o s c i l l a t o r y s t r u c t u r e i n | a ^ O ) | T h e r e f o r e , a t l e a s t two s t a t e s a r e i n i t i a l l y n o n -z e ro to form an i n t e r f e r e n c e te rm be tw een th e complex e x p o n e n t i a l s . I t i s a ls o a p p a re n t t h a t i f th e i n i t i a l s t a t e s a r e p r e p a r e d i n c o h e r e n t l y , th e n e c e s s a r y p h a se a v e ra g in g w i l l e l im in a te th e i n t e r f e r e n c e te rm and h en ce no o s c i l l a t i o n i n I a ^ ( CCl I ^ . We t h e r e f o r e assume t h a t th e i n i t i a l s t a t e s a r e p r e p a r e d c o h e r e n t ly and from e a r l i e r rem ark s assume a ^ (0) = 0 and a^CO) = a ^ (0 ) = I / / 2 . I n A ppendix I I I , we w i l l g iv e , th e r e s u l t s f o r th e more g e n e r a l c a se o f a l l t h r e e i n i t i a l s t a t e s p o p u la te d . The s p in f l i p p r o b a b i l i t y , u s in g E q u a t io n (54) and th e . i n i t i a l c o n d i t io n s a ^ (0 ) = 0 and a^CO) = a ^ (0) = 1 / /2 , i s U 1 (Tr)I2 = [ I - S - 7raOp ] / 2- 2 e - 1Taop / 4 [ l - e -1ra° p / 2 ] 3 /z [ cos ( X ^ / a ) ] , (60 ) w i th (j) = A rg[E11 (TrsO) ] = -(o t^p2 + 1 .2 ) 3 ^2 , and = / ^ 2 a ( 8) d 8 . B e fo re any com pa riso n s can be m ade, A m ust b e e v a lu a te d . U n fo r tu n a te - 2 3 I y , a ( 8) = aQp / s i n 0 l e a d s to an in d e te rm in a n t s form f o r A^. T ha t i s 9 A 9 co s 0 1+COS0 A . = a p f* s i n 0d0 = —r p Lim [ ----- + I n —-----------------— ] = <». ^ 0 ^O s i n 0 s i n 0O O O We may r e w r i t e A^ i n te rm s o f th e s p a t i a l v a r i a b l e z ( 61)■ xTT = i -/-O= w (z )d z - (uB/v h ) / ° j H ( z ) | d z , T h is w i l l a l s o b e i n d e te rm in a n t b e c a u se o f th e low e r l im i t How­ e v e r , th e n o n - a d i a b a t i c r e g io n does n o t p h y s i c a l ly e x te n d to i n f i n i t y , and by u s in g th e c r i t e r i o n d ev e lo p ed i n S e c t io n 2 .1 f o r d e f in in g th e t r a n s i t i o n r e g io n , th e low e r l im i t i s r e p la c e d by v I q . A now becomes cCC K f ° z IH (z) Idz = (yB/v fi) pE [ |H (z ) |d z . (62) O The m agn itu d e o f th e t o t a l m ag n e tic f i e l d i s |H ( z ) [ and i s d e f in e d i n S e c t io n 2 .1 as I1H (Z)I = [E2z + , (65 ) w i th H^ = e E z /R and H^ = •e Ep (p = r /2 R ) . The i n t e g r a t i o n may now b e p e r fo rm e d , w i th th e r e s u l t . . \ = ao 5 ( z o ,R ,p ) , , (64) w i th £ (z R ,p ) = Z i z h p 2R2)1"2 + (pR )2l n { [ z + (z ^+ p2R2) l5] /p R } ] /2 R 2 . (65)O O O O -O The f i n a l t h r e e - l e v e l s p in f l i p p r o b a b i l i t y , | a^(rr) | ^ o f E q u a tio n (60 ) i s g rap h ed i n F ig u re 11 f o r th e same s e t o f Zq , p , E v a lu e s as t h e tw o - le v e l p r o b a b i l i t y g iv en i n F ig u re 8 . The p rom in en t d i f f e r e n c e b e tw een th e tw o - le v e l and t h r e e - l e v e l a p p ro x im a t io n s . i s th e a sym p to tic -6 3 - le v e l s p in f l i p p r o b a b i l i t y ve rsus m agne tic f i e l d .F ig u re 11: Three Sp in F lip P ro ba bi iil y V = I.Ox i0 cm sec R = I c m FiG'11 —65- l im i t as e E (and h ence aj) i n c r e a s e s . The t h r e e - l e v e l s p in f l i p p r o b a b i l i t y s a t u r a t e s a t 50% p o p u la t io n a s e E i n c r e a s e s w h i le th e tw o - le v e l p r o b a b i l i t y d e c re a s e s to z e ro . They b o th h av e s im i l a r - damped o s c i l l a t o r y s t r u c t u r e s , su p e rim po sed on th e a s ym p to t ic e n v e lo p e s . These a s ym p to t ic e n v e lo p e s a r e the, f i r s t te rm s on th e r i g h t hand s id e o f E q u a t io n s (31) and ( 6 0 ) . F o r Landau and Z en e r, th e e n v e lo p e s a r e r e f e r r e d to as L-Z e n v e lo p e s . I t i s c u r io u s t h a t f o r th e t h r e e - l e v e l c a s e , th e L-Z e n v e lo p e and th e damping en v e lo p e i n th e o s c i l l a t o r y te rm a r e d ep en d en t o n ly on th e m agn itu d e o f aQ and n o t on th e e x p l i c i t form f o r a ( 6) . Only th e o s c i l l a t o r y te rm c o n ta in s in f o rm a t io n a b o u t th e f u n c t i o n a l d ependence o f a ( 8) on 0 ( o r OH Z). o r t ) th ro u g h a (0 )d 0 . D ig r e s s in g f o r a moment, we r e c a l l t h a t n o n - a d i a b a t i c p a s sa g e i s a n a lo g o u s to c h a rg e ex ch ange . F o r t h i s ty p e o f s c a t t e r i n g , th e L andau -Z en e r th e o ry f i t s a s u r p r i s i n g number o f c ro s s s e c t i o n s . We s u s p e c t t h a t th e r e a s o n f o r su ch su c c e s s i s t h a t th e fo rm o f th e i n t e r a c t i o n g iv e s r i s e o n ly to v e ry sm a ll o s c i l l a t o r y s t r u c t u r e s w h i le th e g ro s s f e a tu r e s o f t h e c ro s s s e c t io n s a r e s im p ly p r o p o r t i o n a l to .th e s t r e n g t h o f th e i n t e r a c t i o n . Comparing th e o r i g i n a l d a t a , F ig u re 5,. w i th th e two t h e o r i e s , , we m ig h t s u s p e c t t h a t th e tw o - le v e l th e o ry would f i t f a i r l y w e l l . In v iew o f th e h y p e r f in e s t r u c t u r e , t h i s i s q u i t e s u r p r i s i n g . I t was n o t u n t i l we found t h a t t h e r e w ere two c o u p le d n o n - a d i a b a t i c r e g io n s i n th e o r i g i n a l f lo p p e r t h a t we u n d e rs ta n d w hat was h a p p e n in g . The -6 6 - two c o u p le d t r a n s i t i o n r e g io n s w ere v e ry s im i l a r i n sh ap e and -m agn itu d e . H ence , a s t h e c u r r e n t i n th e f lo p p e r was i n c r e a s e d , a b e a t i n g e f f e c t was g e n e r a te d t h a t had a v e ry s h o r t p e r io d o s c i l l a t i o n sup e rim po sed upon a lo n g p e r io d o s c i l l a t i o n . The m ag n e tic f i e l d and h en ce # was n e v e r ru n much h ig h e r th a n enough t o com p le te h a l f a c y c le o f th e l a r g e r o s c i l l a t i o n . T h e r e f o r e , th e two l e v e l th e o r y w ould n o t f i t t h e d a t a ,no m a t te r w hat p a ram e te r s we u s e d . We w ere n o t a b le t o c om p le te ly e l im in a te one o f th e c r o s s in g p o i n t s . However, we w ere a b le t o make one o f th em rem a in p u re n o n - a d ia b a t i c r e g a r d l e s s o f w hat t h e se co nd r e g io n d i d . W ith new b o u n d a ry c o n d i t io n s im posed by th e f i r s t n o n - a d ia b a t i c r e g io n , we w ere a b le t o o b t a in d a t a t h a t was r e a s o n a b ly f i t t e d b y th e t h r e e - l e v e l t h e o r y . The f lo p p e r m ag n e tic f i e l d s a r e g iv e n and d i s c u s s e d i n C h a p te r I I I . The c o r r e s p o n d in g s p in f l i p c u rv e s , w i th t h e o r e t i c a l c u rv e f i t s a r e g iv e n and d i s c u s s e d i n C h a p te r IV . F o r r e a s o n s g iv e n i n C h a p te r I I I we w ere n o t a b le t o s e l e c t a sm a l l enough v e l o c i t y r a n g e f o r th e p a r t i c l e beam to e l im in a te a v e l o c i t y a v e r a g e . A ls o , b e c a u s e o f t h e f i n i t e beam s i z e we found we h ad t o a v e ra g e th e th e o r y o v e r th e beam c ro s s s e c t i o n . T h e r e f o r e , b e f o r e c o n c lu d in g t h i s c h a p te r , we w i l l do t h e a v e ra g in g , o v e r th e v e l o c i t y d i s t r i b u t i o n and th e beam s i z e . S in c e we a l r e a d y know t h a t t h e t h r e e - l e v e l th e o r y f i t s t h e d a t a , we w i l l c o n s id e r o n ly th e -67- t h r e e - l e v e l s p in f l i p p r o b a b i l i t y . . 2 .6 V e lo c i ty and B eam -S ize A ve rag ing The t h r e e - l e v e l Sg s t a t e p r o b a b i l i t y . E q u a tio n (60) , n a t u r a l l y d iv id e s i t s e l f i n t o two p a r t s , th e L-Z en v e lo p e ( f i r s t te rm ) and th e damped o s c i l l a t o r y te rm (se co n d t e rm ) . S in c e th e te rm s a r e a d d i t i v e , e a ch may b e in d e p e n d e n t ly a v e ra g e d . The a v e rag ed L-Z e n v e lo p e may­ be u sed to cu rv e f i t t h e d a ta and th e o s c i l l a t i o n s rem oved ; Then, u s in g th e p a ram e te r s from th e a v e rag ed L-Z e n v e lo p e , th e o s c i l l a t i o n s can be s e p a r a t e l y f i t t e d s in c e th e o s c i l l a t i o n s a lo n e a r e a f u n c t io n o f th e sh ap e o f th e a d i a b a t i c i t y p a ram e te r and a r e t h e o n ly te rm s c o n ta in in g th e t r a n s i t i o n w id th I q . T h e r e f o r e , e a ch te rm w i l l be done s e p a r a t e l y . The beam -av e rag in g r e q u i r e s th e a ssum p tio n o f u n ifo rm d e n s i ty a c ro s s th e beam c r o s s - s e c t i o n . A t p r e s e n t , we h ave no way o f a c t u a l l y m ea su r in g th e d e n s i ty w i t h in th e c r o s s - s e c t i o n a l a r e a . H ow ever, how w e l l th e a v e ra g e d L-Z e n v e lo p e f i t s th e d a t a i s an i n d i c a t i o n a s to th e v a l i d i t y o f t h i s a s sum p tio n . The a v e ra g in g i s s im p ly th e i n t e g r a t i o n o v e r t h e L-Z en v e lo p e t im e s t h e beam d e n s i t y from -p = 0 to p = p = r /2 'R , th u s o max 82 0 h [ ' 0 - ~ exp(-TraoP2) ]2(pdp/p^) . (66) -6 8 - T h is i n t e g r a l can b e done and s im p ly g iv e s [ I - [ I - e x p (-Ira o p 2) } / TTCioPo ] . (67) The v e l o c i t y a v e ra g in g r e q u i r e s a d e t a i l e d know ledge o f th e a c tu a l v e lo c i t y d is t r ib u t io n o f th e p a r t i c le beam. We have shown (35}t h a t th e beam v e l o c i t y d i s t r i b u t i o n N(v ) can b e ap p ro x im a ted b y v J N(v) = No ( v / a ) n e x p ( - v 2/ a 2) . (68) H e re , ot = (2kT /m )-2, th e mean th e rm a l v e l o c i t y o f th e beam and n i s (35) a v a r i a b l e e x p o n en t' u s u a l l y a round 5 .5 . T h is a p p ro x im a tio n to t h e m easu red v e l o c i t y d i s t r i b u t i o n i s r a t h e r p o o r and th e i n t e g r a t i o n o f E q u a tio n (67 ) o v e r th e d i s t r i b u t i o n g iv en by E q u a t io n (68 ) c an no t b e done a n a l y t i c a l l y . S in c e e x p e r im e n ta l ly we m easu re th e tim e o f f l i g h t d i s t r i b u t i o n and th e n c a l c u l a t e d th e v e l o c i t y d i s t r i b u t i o n , we t r i e d c u rv e f i t t i n g th e tim e o f f l i g h t d a ta in th e h o p e .o f f in d in g an i n t e g r a b l e d i s t r i b u t i o n . S u r p r i s i n g l y , th e tim e o f f l i g h t i s f a i r l y w e l l c u rv e f i t t e d by th e G au ss ia n form aMC l = To e x p [ - ( r - T o ) 2 /AT2) I , w here aE i s th e peak tim e o f f l i g h t and GA i s r e l a t e d to th e f u l l w id th a t th e h a l f maximum p o i n t s . F ig u re 30 i n C h a p te r IV shows a t y p i c a l t im e o f f l i g h t sp e c trum w i th such a G au ss ia n f i t . The i n t e g r a t i o n o f E q u a t io n (67) (w ith aQ « 1 /v and v = . P A m where Si i s th e l e n g th th e p a r t i c l e t r a v e l s i n tim e Cl -6 9 - o v e r th e G au ss ia n d i s t r i b u t i o n a ls o can n o t b e done a n a l y t i c a l l y . However, s in c e th e G au ss ia n form i s sym m etric a b ou t Cq , th e p eak i n th e tim e o f f l i g h t sp e c trum , we th o u g h t th e i n t e g r a t i o n w ould le a d to s im p ly r e p l a c i n g v i n by & / . Computer i n t e g r a t i o n o f E q u a tio n (67 ) o v e r th e G au ss ia n d i s t r i b u t i o n showed v e ry l i t t l e d e v ia t i o n from s im p ly s e t t i n g V = H/aE i n E q u a tio n (67 ) ( s e e A ppendix IV ) . We t h e r e f o r e h av e f o r th e beam and v e l o c i t y a v e rag ed L-Z e n v e lo p e [ I - ( I - e x p (-Trao (TE l p 2) /-ITcto (TE l I AVD Mg' l ' The a d i a b a t i c i t y p a ram e te r aQ(Cq ) i s now "o^To) = V o * " (70) The damped o s c i l l a t o r y te rm i n th e | a^(-jr) | 2 p r o b a b i l i t y i s much m ore d i f f i c u l t to a v e ra g e . To f in d an e x p re s s io n w h ich w ou ld r e p r e s e n t ( to b e t t e r th a n one p e r c e n t ) th e a v e rag ed damped o s c i l l a t o r y te rm , we u se d i n t u i t i o n and th e n a c t u a l l y d id th e beam and v e l o c i t y a v e ra g in g on a com pu te r and compared o u r " d e r iv e d " e x p re s s io n w i th th e " a c tu a l " com pu te r a v e ra g e d te rm . Our i n t u i t i o n to o k th e f o l lo w in g c o u rs e . The damped o s c i l l a t o r y te rm c o n ta in s a s im p le e v e lo p e t im e s an ' o s c i l l a t o r y te rm . The e n v e lo p e i s a f u n c t io n s im i l a r to t h e L-Z e n v e lo p e and h en ce was v e l o c i t y a v e ra g ed as we d id f o r th e L-Z e n v e lo p e . The c o s in e te rm was th e n a n a l y t i c a l l y a v e ra g e d , i g n o r in g th e tim e - 7 0 ~ dependence o f th e p h a s e , u s in g th e G au ss ian d i s t r i b u t i o n . The r e ­ s u l t s w ere th e n m u l t i p l i e d t o g e t h e r t o y i e l d /2 exp(-Tra MCG l 1 b i 1|.) [ I - e x p ( -TC;E MCE l p ^ /2 ) ] 3,/2 x X exp 43v • bG■aAbASGA ( ( P r 1 r 5+ cos U 7CMaq ) - /2). . (T l) A ga in , A ppend ix IV shows th e com pu te r i n t e g r a t i o n s compared to E q u a t io n ( 7 1 ) . • U n f o r tu n a t e ly , we c o u ld n o t f in d a f u n c t io n to a p p ro x i­ m a te th e beam a v e ra g in g o f E q u a t io n ( 7 1 ) . T h e r e f o r e , E q u a t io n (71) was i n t e g r a t e d by com pu te r o v e r th e beam c r o s s - s e c t i o n to g iv e t h e o r e t i c a l c u rv e s to compare to th e o s c i l l a t i o n s removed from th e d a ta a f t e r f i t t i n g th e a v e ra g e d L-Z e n v e lo p e to th e d a ta . A ga in , a s a t e s t o f o u r f i n a l damped o s c i l l a t o r y te rm , we compare i n A ppendix IV th e com p le te beam and v e l o c i t y a v e ra g e d te rm done by com pu te r to o u r a p p ro x im a ted damped o s c i l l a t o r y te rm . To w i t h in a b ou t one p e r c e n t , t h e two compare v e ry f a v o r a b ly and th e r e s u l t i n g s a v in g s i n com puter money was o f t h e o r d e r o f .a f a c t o r o f 10 . F ig u r e 12 shows th e v e l o c i t y and beam av e rag ed t h r e e - l e v e l s p in f l i p p r o b a b i l i t y f o r th e same p a ram e te r s u se d i n F ig u r e 8 and 11 w here th e tw o - le v e l and t h r e e - l e v e l u n av e rag ed p r o b a b i l i t i e s a r e g iv en r e s p e c t i v e l y . T here i s a p ronounced damping o f th e o s c i l l a t i o n s and a d e c re a s e i n th e p e r io d o f th e o s c i l l a t i o n s due to th e a v e ra g in g o v e r th e beam and v e l o c i t y d i s t r i b u t i o n s . .iL .F ig u r e 12 : V e lo c i ty and beam a v e ra g e d t h r e e l e v e l s p in f l i p p r o b a b i l i t y v e r s u s m ag n e tic f i e l d . Sp in F lip P ro ba bi lit y V = 1.0 %IO cm/sec R = Icm AT= 40.x |0 sec H o ( g a u s s ) FIG. 12 -7 3 - 2 .7 Summary In summary, we h ave re d u c ed th e d e s c r i p t i o n o f th e n on - a d i a b a t i c p a s s a g e o f an o r i e n t e d s p in to a s i n g l e p a ram e te r c a l l e d , a p p r o p r i a t e l y , th e a d i a b a t i c i t y p a ram e te r a . The a d i a b a t i c i t y p a ram e te r i s s im p ly th e r a t i o o f th e Larm or f re q u e n c y f o r th e s p in to th e m ag n e tic f i e l d r o t a t i o n r a t e . By means o f a p e d a g o g ic a l ex am p le , we h av e shown t h a t we may r e s t r i c t o u r a t t e n t i o n to o n ly th e n o n - a d i a b a t i c r e g io n s . T hese r e g io n s a r e d e f in e d by t r a n s i t i o n l e n g th s found by th e c r i t e r i o n d e v e lo p ed i n S e c t io n 2 .1 . S p in f l i p p r o b a b i l i t i e s h av e b e en d e r iv e d f o r th e n o n - a d i a b a t i c p a s s a g e o f a s p in % and a s p in I . A v e ry s im p le m a g n e tic f i e l d c o n f i g u r a t io n i s assum ed as shown i n F ig u r e 6 . F o r e x p e r im e n ta l r e a s o n s , we- have done a beam c r o s s - s e c t i o n and v e l o c i t y d i s t r i b u t i o n a v e ra g e f o r th e s p in I c a s e . The d e t a i l s o f th e e x p e r im e n t w i l l now b e p r e s e n te d and d i s c u s s e d i n d e t a i l . CHAPTER I I I - EXPERIMENTAL APPARATUS ■ 3 . 1 O verv iew In t h i s c h a p te r we w i l l p r e s e n t t h e d e t a i l s c o n c e rn in g th e p r o ­ d u c t io n and d e t e c t i o n o f an a tom ic m e ta s ta b le hyd rogen beam. D e t a i l s o f t h e f l o p p e r , t h e d e v ic e g e n e r a t i n g th e n o n - a d ia b a t i c m ag n e tic f i e l d , w i l l b e g iv e n t o g e t h e r w i th com pa riso n s o f th e t h e o r e t i c a l .a d i a b a t i c i t y p a ram e te r t o th e e x p e r im e n ta l ly d e te rm in e d a- We w i l l c o n c lu d e t h i s c h a p te r w i th t h e p ro c e d u re s u s e d i n t a k in g d a t a and th e n u m e r ic a l c a l c u l a t i o n s r e q u i r e d t o a r r i v e a t a p r o d u c t io n c u r v e . o r s p in f l i p p r o b a b i l i t y . We c o n d u c te d t h e e x p e r im en t i n an a tom ic beam s p e c t r o m e te r , f 23) h ick -n am ed t h e a lp h a t r o n f o r t h e f i n e s t r u c t u r e m easu rem en tsv ' i t - was u se d f o r i n t h e p a s t . The a lp h a t r o n p ro d u ce s a beam o f 00 a/ s m e ta s ta b le h y d ro g en atom s b y e l e c t r o n im pac t e x c i t a t i o n o f th e rm a lly - d i s s o c i a t e d m o le c u la r h y d ro g e n . The m e ta s ta b le beam e n te r s an e x p e r i ­ m e n ta l r e g io n w here an e x t e r n a l p e r t u r b a t i o n , su ch as t h e m agn e tic f i e l d u s e d i n t h i s e x p e r im e n t , may b e a p p l i e d . The m e ta s ta b le beam _ ■ i s th e n d e t e c t e d a f t e r p a s s a g e th ro u g h a h y p e r f in e s t a t e a n a ly z e r . We- d e t e c t t h e m e ta s ta b le i n t e n s i t y e i t h e r by Auger t r a n s i t i o n s i n a s u r f a c e d e t e c t o r o r b y a B end ix C h a n n e l tro n E le c t r o n M u l t i p l i e r . T hese d e t e c t o r s a r e s e n s i t i v e o n ly t o t h e e x c i t e d s t a t e s i n hyd rogen and n o t t h e g round s t a t e . F ig u re 13 shows t h e f o u r m ain s e c t i o n s o f th e a lp h a t r o n . The two -7 5 - F ig u re 13 : A lp h a tro n d iag ram . i ] QEE QUENCH FlO E-GUN HELMHOLTZ COILOVEN DETECTOR FIG.13 - 7 7 - u p s tr e am s e c t io n s , , t h e oven and e x c i t a t i o n ch am be rs , a r e r i g i d l y m ounted w h i le th e two dow nstream s e c t i o n s , t h e s t a t e a n a ly z e r and d e t e c t i o n cham ber, a r e m ounted on a m ovable p l a t f o rm s e t i n a l ig n e d t r a c k s . T h is d e s ig n a llow s e a sy a c c e s s t o a l l cham bers w h i le m ain ­ t a i n i n g a d e q u a te 'm a c h in e a lig n m e n t . 'E a c h o f t h e t h r e e cham bers a r e 12 in c h d iam e te r 1 /8 in c h w a l l b r a s s tu b e s from 8 in c h t o 10 in c h i n l e n g t h . The cham bers a r e i n d i v i d u a l l y e v a cu a te d b y t h r e e c o ld - t r a p p e d 4 in c h o i l d i f f u s i o n pumps e x h a u s te d by a W elch d u o - s e a l fo repum p . P r e s s u r e s o f 5 -0 X KT^ t o r r a r e t y p i c a l w i th no in p u t g a s t o t h e a lp h a t r o n . P r e s s u r e s w i th an a tom ic hyd rog en beam a re l i s t e d i n T ab le I . 3 .2 Beam S ou rce We p ro d u ce a beam o f g round s t a t e ( I S jJ h y d ro g en a tom s( ^ by s e f f u s i o n ^ ^ ) from an o v en . The oven i s a, h o llow tu n g s t e n c y l in d e r I in c h i n l e n g th and 0 .1 in c h i n 'd i a m e t e r w i th a 0 .2 in c h X 0 .0 1 in c h s l i t . P r e p u r i f i e d h y d ro g en gas i s le a k e d i n t o one end o f t h e oven w i th t h e o t h e r end c a p p ed . We m easu re t h e gas p r e s s u r e i n t h e oven r e l a t i v e t o t h e oven chamber p r e s s u r e b y an o i l m an om e te r^ "^ u s in g Dow C o rn in g 704 d i f f u s i o n pump o i l . The oven i s J o u le h e a te d b y a v a r i a b l e v o l t a g e s o u rc e c a p a b le o f p ro d u c in g tem p e ra tu re s from room tem p e ra tu r e (300°K ) t o t h e m e l t in g p o i n t o f th e t u n g s t e n oven (3400°K) We m easu re t h e oven tem p e ra tu re b y a Leeds and N o r th ru p . o p t i c a l p y rom e te r c o r r e c t e d f o r t h e e m i s s iv i ty o f t u n g s t e n . T y p ic a l — 7 8- TABLE I ' T y p ic a l O p e ra tin g C o n d it io n s Oven chamber p r e s s u r e E x c i t a t i o n chamber p r e s s u r e D e te c to r chamber p r e s s u r e Oven c u r r e n t Oven tem p e ra tu re P e r c e n t d i s s o c i a t i o n 4 X 10 ^ t o r r -63 X 10 t o r r 8 X 10 ^ t o r r 250 amps a . c . 2900° K 70 # 1 4 .0 v o l t s 1000 • (j, amp 600 g au ss E l e c t r o n gun p l a t e v o l ta g e P l a t e c u r r e n t C o l l im a t in g f i e l d - 7 9 - o p e r a t in g c o n d i t io n s a r e g iv e n i n T ab le I., I f we assum e th e rm a l e q u i l ib r iu m b e tw een th e h y d rog en gas and th e oven a t t em p e ra tu re T, th e n th e m o le c u la r hyd rog en s h o u ld d i s ­ s o c i a t e i n t o a tom ic h y d rog en a t s u f f i c i e n t l y h ig h T. The f r a c t i o n a l d i s s o c i a t i o n % , d e f in e d as t h e r a t i o o f t h e p a r t i a l , p r e s s u r e o f a tom ic h y d ro g en t o t h e t o t a l p r e s s u r e F ig u re 14 shows th e f r a c t i o n a l d i s s o c i a t i o n we m easu red compared t o 15 shows a p r e d i c t e d d e c r e a s e i n % w i th i n c r e a s e i n t h e g a s p r e s s u r e . Even th o u g h t h e a b s o lu t e number o f h y d ro g en atoms i n c r e a s e s , % d e c r e a s e s due t o r e c om b in a t io n w i t h in th e oven . Our beam o f g round s t a t e h y d rog en atom s i s c o l l im a te d b y an a d ju s t a b l e w id th s l i t (4o m il w id th t y p i c a l ) b e f o r e e n t e r i n g th e e x c i t a t i o n r e g io n . 3 .3 E x c i t a t i o n We e x c i t e t h e beam o f IS i h y d ro g en atom s t o t h e m e ta s ta b le 281 s t a t e b y e l e c t r o n im pac t i n an e l e c t r o n gun / The e l e c t r o n s gun i s a s im p le d io d e t h a t p ro d u ce s a beam o f e l e c t r o n s ' p e r p e n d ic u l a r t o t h e m ach ine a x is , .w h o s e e n e rg y we c an v a ry . T h is g eom e try re d u c e s s p u r io u s s i g n a l s a t t h e d e t e c t o r due t o s t r a y o r r e c o i l e l e c t r o n s from th e e l e c t r o n g u n , b u t c om p lic a te s t h e k in em a tic s o f t h e i n e l a s t i c % = P(H)ZPtotal / (72) (32 ,33)h a s b e e n c a l c u l a t e d a s a f u n c t i o n o f t em p e ra tu re and p r e s s u r e I F ig u re lU : Hydrogen d i s s o c i a t i o n c u rv e . F R A C T IO N D IS S O C IA T IO N i.o HYDROGEN D lSSOC I A T I O N CURVE -a ^G V EN ’ WM H G & / Z Z / X i / i / / X i / f / / / I COH I X / - n 1 0 0 0 r IG.I4 /f / Z 2000 ! OVEN ( 6K ) ------- 1---------------1---------------1---------- _ u 3 0 0 0 I I - I - ] _ _ „ 1- - — . 1 1 ' I I I K I ' . I ■ j -8 2 - F ig u re 15 : P r e s s u r e d ependence o f th e f r a c t i o n a l d i s s o c i a t i o n and m e ta s ta b I e i n t e n s i t y . DI SS OC IA TI ON !- ■x - - v 'x1A = 282 0°K F* I G . I O ^ OVEN (MM OF O I L ) TOTAL SIGNAL ( —84— c o l l i s i o n . T b . e r e c o i l e f f e c t s f o r c e u s t o aim th e g round s t a t e . o beam a t a 10 a n g le w i th r e s p e c t t o t h e m ach ine a x is from above th e e l e c t r o n g un . By m ea su r in g t h e m e ta s ta b le i n t e n s i t y as a f u n c t io n o f e l e c t r o n e n e rg y , we g e n e r a te an e x c i t a t i o n c u rv e c h a r a c t e r i s t i c o f th e IS 28 e x c i t a t i o n . A t y p i c a l e x c i t a t i o n c u rv e i s shown i n F ig u re 1 6 . We m easu red a t h r e s h o ld o f 1 0 .1 v o l t s w h ich compares . f a v o ra b ly w i th t h e c a l c u l a t e d t h r e s h o ld o f 1 0 .2 v o l t s b y M o i s e iw i t s c h / The g e n e r a l sh ap e o f o u r e x c i t a t i o n c u rv e i s s im i l a r t o t h a t g iv e n by M o is e iw i ts c h and m easu red by o th e r s / 3 7 ^ 3 8 ) ^jie p e a ic o f t h e e x c i t a t i o n c u rv e o c c u rs a ro u n d '1 2 .6 v o l t s b u t v a r i e s w i th t h e age o f t h e e l e c t r o n gun and i s u s u a l l y a ro und l 4 . 0 v o l t s . The a g in g p ro c e s s b u i l d s a d i e l e c t r i c l a y e r o f "goo" on t h e anode w h ich th e n c a u se s a l a r g e f r a c t i o n o f t h e a p p l i e d v o l t a g e t o be d ropped a c r o s s i t th u s d e c r e a s in g t h e a c c e l e r a t i n g v o l ta g e f o r t h e e l e c t r o n s . . We assume t h a t t h e e x c i t a t i o n o f t h e 2S i l e v e l i n a tom ic h y d rog en p ro d u ce s a p p ro x i - s m a te ly e q u a l p o p u la t io n s o f t h e 2Si m ag n e tic s u b s t a t e s . The e l e c t r o n gun i s p la c e d i n a m ag n e tic f i e l d p a r a l l e l t o th e e l e c t r o n beam and p e r p e n d ic u l a r t o th e m ach ine a x is o r m e ta s ta b le beam . The m ag n e tic f i e l d c o l l im a te s t h e e l e c t r o n s , t h e r e b y i n c r e a s i n g th e c u r r e n t d e n s i t y i n t h e p a th o f th e a tom s . T h is c u r r e n t d en s ity - i n c r e a s e i s r e f l e c t e d b y an i n c r e a s e i n t h e m e ta s ta b le s i g n a l w i th i n c r e a s i n g s t r e n g t h o f t h e c o l l im a t i n g f i e l d as we show i n F ig u re I ? . . F ig u re 1 6 : A tom ic h y d rog en 2S^ ! e x c i ta t io n c u r v e . Q U EN CH AB LE SI GN AL 7 B S y T 5 S B S R C A T 5 9 - RG.I6 HYDROGEN EXCITATION CURVE I OVEN = 2 8 2 C° K Z ' x1A = 2.3 NN , I I I I i i I X I I I IOOON I I \ \ I I I I / / 10 .1 x ' d 5 9 5 * S 1 9 * ' d ” Z/ 'k— — k— ^ IO 15 PLATE VOLTAGE 20 07 F ig u re I ? : Q uenchab le s ig n a l s a s a f u n c t io n o f e -g u n c o l l im a t i n g f i e l d . — 5'5Bd SIGNAL 3 0 0 OLLlMATlfxJG FIELD (G) 7 0 0 FIG.17 -88 - - 8 9 - T here i s a se co n d im p o r ta n t e f f e c t c a u se d by th e m o tio n a l e l e c t r i c f i e l d g e n e ra te d , b y th e c o l l im a t i n g m ag n e tic f i e l d t h a t t h e m e ta s ta b le s e e s as i t p a s s e s th ro u g h th e e l e c t r o n gun . As we show i n F ig u re 3 ; th e a and £3 f in e , s t r u c t u r e l e v e l s d iv e rg e and a p p ro a ch th e 2P i s l e v e l r e s p e c t i v e l y w i th i n c r e a s i n g m ag n e tic f i e l d . B ecau se o f th e o r i e n t a t i o n o f th e m ag n e tic f i e l d , we a l s o coup le , th e a . and . .p s t a t e s t o t h e 2P^ s t a t e s b y a m o t io n a l e l e c t r i c f i e l d (~ 6 v o l t s / c m S' a t 600 g a u s s ) . T h is c o u p lin g i s a S t a r k c o u p lin g whose s t r e n g t h i s 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 e n e rg y l e v e l s e p a r a t i o n . H ence , b e c a u s e o f t h e s h o r t l i f e t im e o f t h e 2P s t a t e , we may p r e f e r e n t i a l l y "quench" o r d e p o p u la te t h e (3 s t a t e w h i le l e a v in g a s t a t e p o p u la te d . The 2P l i f e t im e (~ 1 .6 X 10 ^ s e c . ) i s so s h o r t t h a t a t o u r th e rm a l v e l o c i t i e s (~ 1 .0 X 10° c m / s e c . ) , any p o p u la te d 2P s t a t e d ecay s w i t h in I mm o f i t s p r o d u c t io n . H ence , t h e 2P s t a t e - n e v e r r e a c h e s ■ t h e d e t e c t o r w h ich i s s i t u a t e d some 80 cm from th e e l e c t r o n gun . The th e o r y o f q u en ch in g i s d i s c u s s e d i n more d e t a i l i n A ppend ix V. The d e g re e o f {3 q u en ch in g t h a t o c c u rs i n t h e e l e c t r o n gun i s d e p en d en t upon t h e m agn itu d e o f t h e e -g u n c o l l im a t i n g f i e l d as shown i n F ig u re 17 . The £3 c o n te n t o f th e m e ta s ta b le beam can, b e v a r i e d from 0 t o ab o u t 20 p e r c e n t b e f o r e t h e i n t e n s i t y o f t h e beam i s r e ­ duced t o t h e n o is e l e v e l . T y p ic a l o p e r a t in g c o n d i t io n s f o r t h e e l e c t r o n gun a r e g iv e n in T ab le I . A t 590 g a u s s , t h e s u r v iv in g m e ta s ta b le beam i s p o l a r i z e d - 9 0 - i n t h e a(m .j = +g) f i n e s t r u c t u r e s t a t e j t h e p s t a t e s a r e com­ p l e t e l y q u en ch ed . 3 -4 H y p e r f in e i S t a t e A n a ly z e r The beam o f a m e ta s ta b le s n e x t p a s s e s th ro u g h th e e x p e r im en ts ,! \ r e g i o n , t o b e d i s c u s s e d i n S e c t io n 3 * 6 , w here t r a n s i t i o n s b e tw een th e h y p e r f in e s t a t e s (F = I , m^ = ± 1 ,0 ) a r e in d u c e d . The r e s u l t i n g m ix tu re o f H y p e r f in e s t a t e s , w h ich in c lu d e s th e (3^ s t a t e (F = I , IRp =■ - I ) , e n t e r s t h e h y p e r f in e s t a t e a n a ly z e r w here t h e (3^ s t a t e may b e rem oved from th e beam by p r e f e r e n t i a l q u e n c h in g . The p r e f e r e n t i a l q u en ch in g i s s im i l a r t o t h a t d e s c r ib e d i n t h e s e c t i o n on t h e e l e c t r o n g un . H e re , we a p p ly a m ag n e tic f i e l d p a r a l l e l t o t h e m e ta s ta b le beam b y means o f a H e lm ho ltz c o i l . The e l e c t r i c f i e l d f o r q u en ch in g i s a p p l i e d by a v a r i a b l e v o l t ­ age s o u rc e a c r o s s a sm a l l p a r a l l e l p l a t e c a p a c i to r a t t h e c e n te r o f t h e H e lm ho ltz c o i l . The e l e c t r i c f i e l d i s p e r p e n d ic u l a r t o t h e beam . T h is c o n f i g u r a t i o n in d u c e s a t r a n s i t i o n s w i th Am = ±1 s e l e c t i o n r u l e s . (S e e A ppend ix V ). Tlius t h e a and P s t a t e s i n t h e 28 l e v e l a r e c o u p le d t o t h e f and e h y p e r f in e s t a t e s , r e s p e c t i v e l y i n th e 2P l e v e l . F o r a g iv e n e l e c t r i c f i e l d , t h e p e rc e n ta g e q u en ch in g o f a s t a t e i s d e p en d en t upon th e e n e rg y l e y e l s e p a r a t i o n f o r e ach t r a n s i t i o n , (s.ee E q u a t io n 9 5 ) . The H e lm ho ltz c o i l i s u sed t o d r iv e th e a s t a t e s f a r t h e r away from th e 2P s t a t e s w h i le th e p s t a t e w i l l a c t u a l l y c r o s s t h e 2P s t a t e t h a t i t i s c o u p le d t o (S ee F ig u re 3 ) . S o , a t - 9 1 - a p p ro x im a te I y 605 g a u s s > a v e ry sm a l l e l e c t r i c f i e l d (2 v /cm ) w i l l quench th e s t a t e w i th o u t a p p r e c ia b ly q u en ch ing t h e a s t a t e s . By a p p ly in g a l a r g e e l e c t r i c f i e l d (1 00 v /cm ) , th e ct s t a t e s .a re a l s o quenched and h en ce th e t o t a l m e ta s ta b le i n t e n s i t y can b e fo u n d . F ig u r e s 18 and 19 show th e p e r c e n ta g e q u en ch in g f o r t h e p and a s t a t e s , r e s p e c t i v e ly ; a s a f u n c t i o n o f a p p l i e d v o l t a g e . From su ch g r a p h s , a low and h ig h v o l t a g e maybe ch o sen t o a l t e r n a t e l y d e te rm in e th e p and a s t a t e p o p u la t io n s d u r in g t h e c o u rs e o f t h e e x p e r im e n t . S in c e t h e d e t e c to r s u s e d a r e s e n s i t i v e o n ly t o th e m e ta s ta b le a tom s , t h e p o p u la t io n o f t h e Pg s t a t e i s m easu red by th e d e c r e a s e i n m e ta ­ s t a b l e i n t e n s i t y upon q u en ch in g o f t h e Pg s t a t e . The p o p u la t io n o f t h e two 0 h y p e r f in e s t a t e s may b e found b y s e l e c t i v e r a d i o f r e q u e n c y q u e n c h in g , b u t due t o th e l a c k o f r a d io f re q u e n c y e q u ipm en t, t h i s was n o t d o n e . Only t h e com bined p o p u la t io n o f t h e a h y p e r f in e s t a t e s a r e fo und b y q u en ch ing i n a s t r o n g d . c . e l e c t r i c f i e l d . The c r o s s - s e c t i o n a l s i z e and s l i g h t d iv e rg e n c e o f t h e m e ta s ta b le beam p ro d u c e s an u n a v o id a b le m o tio n a l e l e c t r i c f i e l d i n t h e s t a t e a n a ly z e r . S o , a f r a c t i o n o f t h e P 's a r e a c c i d e n t l y quenched i n t h e H e lm ho ltz c o i l . T h is f r a c t i o n i s fo u n d from a "b eam -no tch " c u rv e and m ust b e a c c o u n te d f o r t o b e a b le t o c a l c u l a t e t h e a c tu a l Pg s t a t e p o p u l a t i o n . A "b eam -no tch " c u rv e i s t h e t o t a l m e ta s ta b le s i g n a l v e r s u s t h e H e lm ho ltz c o i l m ag n e tic f i e l d s t r e n g t h . The beam I-9 2 - .F ig u re 18 : Pfi quench c u rv e „ Fr ac tio na l Q ue nc h Tnverv 2 850°K = 7.5 ma = GOSc (Quench Voltage) FIG.18 F ig u re 3-9 : a quench c u r v e « Fr ac tio na l Qu en ch oven (Quench Voltage/ IOO)FIG.19 ~9 6— i n t e n s i t y w i th P 's p r e s e n t i s m easu red a s a f u n c t io n o f th e Helm­ h o l t z f i e l d . As th e f i e l d a p p ro a ch e s t h e p -e c r o s s in g p o i n t n e a r 575 g a u s s , any m o tio n a l e l e c t r i c f i e l d b e g in s t o quench th e p s t a t e and a d e c r e a s e i n th e m e ta s ta b le i n t e n s i t y i s n o te d . As th e f i e l d goes p a s t t h e c r o s s i n g p o i n t , t h e p -e l e v e l s e p a r a t i o n becomes l a r g e r and f e w e r . p 's a r e q u en ch ed . Thus a p l o t o f t h e beam i n t e n s i t y v e r s u s t h e H e lm ho ltz c o i l f i e l d e x h ib i t s a " n o tc h " n e a r 575 g a u s s / F ig u re 20 shows a t y p i c a l b e am -no tch c u rv e w here th e p e r c e n ta g e o f P 's l o s t t o a c c i d e n t a l q u en ch in g i s a p p ro x im a te ly 18.8% . T h is p e r ­ c e n ta g e was fo und t o v a ry d ep en d in g upon oven tem p e ra tu re y e x c i t a t i o n v o l t a g e and o th e r , p a r a m e te r s , so a beam n o tc h c u rv e was t a k e n d u r in g e a ch d a t a r u n . The H e lm ho ltz c o i l m ag n e tic f i e l d p la y s a c r u c i a l r o l e i n th e h y p e r f in e s t a t e a n a ly z e ^ s in c e any d r i f t o r d e v ia t i o n from a p r e s e t f i e l d w i l l a p p r e c ia b ly a l t e r th e f r a c t i o n o f p 's quenched a t a g iv e n quench v o l t a g e . The m agn e tic f i e l d was c a l i b r a t e d u s in g a f e r r i c c h lo r i d e sam p le n u c le a r m ag n e tic re so n a n c e p r o b e . The c a l i ­ b r a t i o n o f t h e H e lm ho ltz m ag n e tic f i e l d was made W ith r e f e r e n c e t o t h e v o l t a g e a c ro s s a one ohm p r e c i s i o n r e s i s t o r i n s e r i e s w i th th e pow er s u p p ly and c o i l s . The c o n v e r s io n was found t o b e 544-95 g au ss p e r v o l t (± 0 .1 2 g a u s s / v o l t ) . The d r i f t was found t o b e 0 - 40 g au ss p e r h o u r . The power s u p p ly u sed h ad a sm a l l a . c . r i p p l e w h ich g e n e r ­ a te d an a . c . f i e l d o f 0 .0 6 g au ss a t 600 g a u s s . The s t a b i l i t y and •97 F ig u re 20: Beam n o tc h c u rv e . To p of B ea m (d iv .) 1 5 . 6 d i v . lOverr^uuw 1X 82.8 div. of z o ' T J I_____ < 5 I I __________ I_____________!_____________I____________ !___________ L_Oi 5 Te Ts i.o 1.2 FIG.2 0 Helmholtz coi! Voltage - 9 9 - a c c u ra c y th u s d e te rm in e d a r e s u f f i c i e n t f o r t h i s e x p e r im e n t . A more c r u c i a l t e s t o f t h e o p e r a t io n o f t h e H e lm ho ltz c o i l w ould b e a r e ­ m easu rem en t o f a Iamb s h i f t r e s o n a n c e / w h i c h h a s a l r e a d y b een done i n t h i s m ach in e u s in g a more s t a b l e power s u p p ly . F ig u r e 21 shows a t y p i c a l Lamb s h i f t re so n a n c e f o r t h e a lp h a t r o n u s in g t h e p r e s e n t power s u p p ly f o r th e H e lm ho ltz c o i l . The m easu red c e n te r o f r e s o n a n c e was 605-8 g au ss and compares f a i r l y w e l l w i th t h e v a lu e o f 6 0 5 .3 g au ss m easu red b y R ob isco e e t . a l / The f u l l w id th a t h a l f maximum was 6 l . l g au ss a t 4o$ quench compared t o t h e m easu red 6 3 .2 g a u s s a t 40% quench by R . T-. R o b isco e ( p r i v a t e c om m un ic a tio n ) . 3 .5 D e te c to r s A f t e r we s t a t e - a n a l y z e t h e beam,, we d e t e c t th e s u r v iv in g m e ta ­ s t a b l e s b y e i t h e r a s u r f a c e d e t e c t o r o r a w indow less e l e c t r o n m u l t i p l i e r c a l l e d a G h a n n e l tro n made by B end ix C o rp o r a t io n . The s u r f a c e d e t e c to r m ea su re s t h e sm a l l e l e c t r o n c u r r e n t p ro d u ced b y th e m e ta s ta b le s s t r i k i n g a c o ld 1 n i c k e l c a th o d e . About 6$ o f th e m e ta s ta b le s d e - e x c i t e b y an Auger t r a n s i t i o n w i th an e l e c t r o n i n th e m e ta l s u r f a c e . S in c e th e m e ta s ta b le e n e rg y ( 1 0 .2 eV f o r t h e 2 S i s t a t e i n hyd rog en ) i s s ' l a r g e r th a n t h e w ork f u n c t i o n o f t h e m e ta l s u r f a c e u s e d (~ 5eV f o r .n i c k e l ) , , t h e n an e l e c t r o n w i l l b e e j e c t e d from th e m e ta l s u r f a c e . These e j e c t e d e l e c t r o n s p ro d u ce a c u r r e n t t h a t i s p r o p o r t i o n a l t o -"LUt h e m e ta s ta b le i n t e n s i t y . The Auger c u r r e n t s a r e a b o u t 10 ‘ amps. The c u r r e n t . i s a m p l i f i e d b y a R ay th eon CK5886 e l e c t r o m e t e r tu b e t o 100 S F ig u re 21 : Laaib s h i f t re so n a n c e c u r v e , -101 s t e F y IBC9 9 -^H I- FIG.21 580 - J _ _ _ _ I_ _ _ _ I- - - 6 0 0 HELMHOLTZ 6 5 0 FLD. ( G A U S S ) -1 02 - a ro und 10 am ps. The a m p l i f i e d c u r r e n t i s m easu red w i th a R ub icon s p o t l i t e g a lv a n om e te r o f 0 .6 X 10 ^ amps/mm s e n s i t i v i t y y i e l d in g a p p ro x im a te ly 250 mm ( d iv i s i o n s ) o f d e f l e c t i o n w i th a r e s p o n s e tim e o f a b o u t 10 s e c o n d s . The d e t e c t o r d r i f t i s u s u a l l y a ro u nd I . 5 mm/min w i th a b o u t ±1 d i v i s i o n o f n o i s e . The s i g n a l t o n o i s e r a t i o i s a p p ro x im a te ly 200 . We m easu re t h e m e ta s ta b le i n t e n s i t y b y n o t in g th e d i f f e r e n c e i n t h e d e t e c t o r c u r r e n t w i th t h e quench v o l t a g e on and th e n w i th t h e quench v o l ta g e o f f a f t e r a 30 second ' d e la y tim e t o a l lo w th e g a lv a n om e te r t o come t o e q u i l ib r iu m w i th t h e new c u r r e n t . To d e te rm in e t h e v e l o c i t y d i s t r i b u t i o n o f th e m e t a s t a b l e s , we m easu re t h e t im e i t t a k e s a m e ta s ta b le t o t r a v e l t h e mean d i s t a n c e o f . 86 .6 4 cm from p r o d u c t io n t o d e t e c t i o n . S in c e t h e f l i g h t t im e o f a m e ta s ta b le a tom i s a b o u t 100 p ,secs , t h e s u r f a c e d e t e c t o r i s . much to o s low f o r an a n a ly s i s o f th e v e l o c i t y d i s t r i b u t i o n . T hus, we u s e a B end ix m odel 4503 C h a n n e l tro n w i th a re sp o n a n c e t im e o f l e s s th a n 50 n ano second s t o a n a ly z e .- th e m e ta s ta b le v e l o c i t y d i s t r i b u t i o n . The m a jo r d is a d v a n ta g e o f t h e C h a n n e ltro n . i s i t s f i n i t e l i f e t im e (10^ t o t a l c o u n ts ) compared t o an e f f e c t i v e l y i n f i n i t e l i f e t im e f o r th e s u r f a c e d e t e c t o r . The C h a n n e ltro n p ro d u c e s a n e g a t iv e 0 .1 v o l t p u l s e l e s s th a n 50 n ano second s i n d u r a t i o n f o r e a ch m e ta s ta b le s t r i k i n g th e d e t e c t o r . The p u l s e i s sh ap ed by an O r te c m odel .113 p r e a m p l i f i e r b e f o r e b e in g a m p l i f i e d b y an O r te c m odel 435 a m p l i f i e r . Tlie r e s u l t i n g p u l s e s a r e fe d i n t o a c o u n te r and a r e a ro und 2 .0 v o l t s w i th th r e e - V -1 0 3 - m ic ro se c o n d s f u l l w id th a t th e h a l f maximum p o in t s . The t im e o f f l i g h t p ro c e s s i s a c com p lish ed by p u l s i n g th e e l e c ­ t r o n gun p l a t e from 0 t o + 14 .0 v o l t s r e l a t i v e t o th e c a th o d e f o r 5 p se c s e v e ry 300 ^ s e c s w i th a H ew le tt P ack a rd m odel 2l4A p u l s e r . A p p ro x im a te ly one m e ta s ta b le p e r e -g u n p u l s e i s p ro d u c e d . To m easu re t h e t im e o f f l i g h t , we u s e a t im e t o am p litu d e c o n v e r te r (TAG). The s y s tem d ia g ram i s g iv e n i n F ig u re 22 . The TAG, w h ich i s s t a r t e d b y a t r i g g e r p u l s e from th e p u l s e r c o in c id e n t w i th th e e rg u n p u l s e , p u t s o u t p u l s e s whose am p litu d e s a r e d i r e c t l y p r o p o r t i o n a l t o t h e t im e e la p s e d b e tw een th e e -g u n p u l s e and t h e C h a n n e l tro n p u ls e r e s u l t i n g from th e m e ta s ta b le p ro d u ced a r r i v i n g a t t h e d e t e c t o r some 100 ^ s e c l a t e r . The number o f p u l s e s a t e a ch am p litu d e a re c o u n te d f o r 10 m in u te s and s t o r e d i n a TMC m odel 4olD p u l s e h e ig h t a n a ly z e r . T h is i s done w i th th e quench v o l ta g e o f f and th e n a 10 m in u te c o u n t i s s u b t r a c t e d from th e a c cum u la te d d a t a w i t h , t h e quench v o l ta g e on t o e l im in a te a l l n o n -m e ta s ta b le hyd rog en s i g n a l s . The r e s u l t i n g d a t a i s m arked by a c a l i b r a t e d tim e mark g e n e r a to r b u i l t i n t o t h e TAG. The s p e c trum i s th e n p r i n t e d o u t on an X - Y r e c o r d e r . F ig u r e 23 shows a t y p i c a l TOF s p e c t r a w i th i t s c a l i b r a t e d tim e m ark . We may s e l e c t e i t h e r t h e a o r f} s t a t e t im e o f f l i g h t b y s e l e c t i v e q u e n c h in g . T hese s p e c t r a a r e shown and d i s c u s s e d i n C h a p te r IV . F ig u re 24 shows th e v e l o c i t y s e l e c t i o n sy s tem we u s e d t o ..a ttem p t t o lo o k a t s i n g l e - v e l o c i t y a tom s. We u s e t h e v a r i a b l e d e la y t r i g g e r - lo 4 - I F ig u re 22 : Time o f f l i g h t e l e c t r o n i c s d iag ram . E-GUN C H ANNE LTR O N CRTEC 113 AMPLIFIER ORTEC 465 X -Y PULSE GENERATOR H.P. 214A FIG.2 2 -lo6- i F ig u re 23 : T y p ic a l t im e o f f l i g h t s p e c trum w i th tim e m ark . Ar bi tr ar y Co un ts IOO usee t ime mark oven Arbitrary Channel NumberFlG. 2o F ig u re 2k: V e lo c i ty s e l e c t i o n e l e c t r o n i c s d ia g ram . E - GUN CHANNELTRON PULSE DELAY TRIGGER OUTPUT OUTPUT START PULSE START PULSEPULSE PREAMP LINEAR GATE ORTEC 426 ORTEC 464 SCALAR TIMER ORTEC 431 = i / F ; GENERATOR H.P. 214A FIG.2 4 -1 0 9 - -1 10 - p u l s e from th e p u l s e g e n e r a to r t o s t a r t 'a l i n e a r g a t e . The l i n e a r g a te a llow s p u l s e s w i t h in a p r e s e t t im e i n t e r v a l t o p a s s t o a c o u n te r . The p r e s e t t im e i n t e r v a l i s a d ju s t a b l e from one t o f i v e m ic ro seco n d s i n d u r a t i o n . The d e la y t im e from th e e -g u n p u ls e t o t h e t r i g g e r i n g o f t h e l i n e a r g a te i s m easu red by a H ew le tt P ack a rd m odel 532TB f r e q u e n c y c o u n te r . To g e t a r e a s o n a b le number o f c o u n ts (~1900 ) , we u s e d a f i v e m ic ro se co n d g a te and c o u n te d a l l t h e p u l s e s a llow ed th ro u g h f o r 4o s e c o n d s . The c o u n t t im e varied}, o f course^ d ep end in g upon w here we w ere sam p lin g th e t im e o f f l i g h t s p e c trum . U n f o r tu n a t e ly , we found t h a t p u l s e s o u t s id e t h e t im e i n t e r v a l s e t b y t h e l i n e a r g a te w ere b e in g c o u n te d a l s o b e c au se o f t h e r a t h e r - lo n g d e c ay t im e s f o r t h e p u l s e s (~20 p ,s e c ) . The e f f e c t i v e tim e i n t e r v a l was fo und t o b e a ro und 20 m ic ro se c o n d s . T h is i s a s i g n i f i ­ c a n t l y l a r g e f r a c t i o n o f t h e FUHM o f th e t im e o f f l i g h t d i s t r i b u t i o n (~40 ^ s e c ) . T h e r e f o r e , due t o t h i s p ro b lem and th e l a c k o f s i g n a l i n t e n s i t y a t sm a l le r t im e i n t e r v a l s , we c o u ld n o t e l im in a te th e n e c e s s i t y f o r a 'v e l o c i t y a v e ra g e o f t h e t h e o r y . F o r t u n a t e l y , th e t im e o f f l i g h t s p e c trum i s n a rrow enough t o n o t c om p le te ly e l im in a te t h e o s c i l l a t i o n s i n th e d a t a . A ppend ix VI d i s c u s s e s t h e v e l o c i t y a v e ra g in g o f t h e t h e o r y t o show t h a t t h e d a t a t a k e n w ith ' v e lo c i ty - s e l e c t i o n i s no b e t t e r th a n th e d a t a t a k e n w i th o u t t h e v e l o c i t y s e l e c t i o n . -1 11 - 3 •6 F lo p p e r The " f l o p p e r " g e n e r a te s a m ag n e tic f i e l d o f a p p ro x im a te ly th e same c o n f i g u r a t io n as d i s c u s s e d i n S e c t io n 2 .2 o f C h a p te r I I , and p o r t r a y e d i n F ig u re 6 . I t i s a m a g n e t ic a l ly s h ie ld e d s e t o f d u a l o p p o s in g s o le n o id s as shown i n d e t a i l i n F ig u re 25 . The two s o le n o id s a r e wound i n o p p o s i t e d i r e c t i o n s on a g o ld p l a t e d b r a s s s p o o l 6 in c h e s i n l e n g th w i th a I .06 cm d iam e te r b o r e . Each s o le n o id h a s an a v e ra g e o f 6 4 l t u r n s o f t e f l o n t r i p l y c o a te d #27 gauge m agnet w i r e . The mean c o i l r a d iu s i s 0 .987 cm and e a ch c o i l h a s a c a l c u l a t e d f i e l d c o n s ta n t o f O.1 1 2 g a u s s /m a , i g n o r in g c o r r e c t i o n s f o r th e f i n i t e s i z e o f th e c o i l w in d in g s and th e O .050 in c h s p a t i a l s e p a r a t i o n o f t h e two c o i l s . The m ag n e tic f i e l d g e n e r a te d by th e f lo p p e r i s d e r iv e d and d i s c u s s e d i n A ppend ix V I I . The s p o o l i s e n c a se d b y a b r a s s c y l i n d e r w rapped w i th t h r e e l a y e r s o f C o -n e tic . and two l a y e r s o f N e t ic s h i e l d in g m a t e r i a l . The w rapped c y l i n d e r i s m ounted i n s i d e a l a r g e r c y l i n d e r o f s o f t i r o n w i th end cap s t o e l im in a te a l l e x t e r n a l m ag n e tic f i e l d s from th e f l o p p e r . The end cap s have 1 .2 7 cm d iam e te r h o le s c o - a x i a l w i th t h e • f l o p p e r t o a l lo w th e beam an e n t r a n c e a n d ■e x i t . The e n t i r e f lo p p e r was v i s u a l l y a l ig n e d w i th t h e m ach ine a x i s . Power f o r t h e f lo p p e r i s s u p p l i e d b y a 6 v o l t a u tom o tiv e l e a d - a c id b a t t e r y . The c u r r e n t th ro u g h th e f lo p p e r i s c o n t in u o u s ly v a r i a b l e from one m il l iam p ( ~ 0 .1 g a u s s ) t o 200 m i l l i am peres (~20. g a u s s ) . The c u r r e n t i s m o n ito re d b y a 5-g d i g i t D i g i t a l M u lt im e te r , The c u r r e n t was s t a b l e t o b e t t e r th a n 0 .1% . - 112- F ig u re 25 : C r o s s - s e c t i o n o f f l b p p e r . E-gun end I inch Helmholtz coiI end Brass Cyiin jer rBrass Spool r 64l turns of 26gauge wire^ -J I II i l ................................ - —V---- ____- ----- - - ----- -------- - - ---- - '/ I rr~f~r~mrTT7~/~y~T'/~7'7~z~/~~/‘/ /~7~sz y^//"SA . , ■ ^% \ ^ V / / / / / / / / / / / / / y / / / / y / / / / / / / { - - ■ ■■ •■■ ............ h ^ i... ................ ...................... -------- --------------- - — -----------------------—------- - ------ --- -—- --- U Soft Iron Cylinder FIG.25 —114— The m ag n e tic f i e l d o f t h e f lo p p e r was m easu red u s in g an F« W. B e l l m odel 64o g a u s sm e te r w i th a two a x is ( a x i a l and t r a n s v e r s e ) p r o b e . We m easu red th e c o i l c o n s ta n t t o b e 0 .1 0 7 g a u s s /m a and found t h a t .the m a g n e tic f i e l d v a r i e d l i n e a r l y w ith th e c u r r e n t . F ig u re 26 shows t h e a x i a l and t r a n s v e r s e f i e l d s a s a f u n c t i o n o f t h e d i s t a n c e down th e f l o p p e r , a lo n g a l i n e 0 -279 cm o f f th e c o i l a x i s . The a x i a l f i e l d i s f a i r l y l i n e a r a s i t goes th ro u g h z e ro w h i le t h e r a d i a l ■ - f i e l d i s ' 'r e a s o n a b ly " c o n s t a n t . Of c o u r s e , an e x p e r im e n ta l m ag n e tic f i e l d c an n e v e r b e e x a c t ly l i k e t h e f i e l d d e s c r ib e d i n C h a p te r I I ( F ig u r e 6 ) , b u t t h e v a l i d i t y o f t h a t a p p ro x im a tio n t o t h e " r e a l " f i e l d can b e e s t a b l i s h e d b y com paring th e t h e o r e t i c a l a d i a b a t i c i t y p a ram e te r t o t h e e x p e r im e n ta l Once a g a in , due t o t h e d i f f i c u l t y i n p o r t r a y in g a, t h e in v e r s e o f t h e e x p e r im e n ta l a d i a b a t i c i t y p a r a ­ m e te r w i l l b e g rap h ed and compared w i th t h e o r y . F ig u re 27 compares - It h e two a p l o t s . A b e t t e r f i t b e tw een th e e x p e r im e n ta l p o in t s and th e . d a sh ed l i n e can b e e f f e c t e d b y in c r e a s i n g th e v a lu e o f p u s e d i n t h e t h e o r y b y 7% a s shown b y t h e se co nd c u rv e f i t t i n g th e e x p e r im e n ta l p o i n t s . The f i t i s s u r p r i s i n g l y good i f an e f f e c t i v e p i s u s e d i n s t e a d o f t h e m easu red p v a lu e . The p a ram e te r p; i s j u s t t h e d i s t a n c e o f f a x i s d iv id e d b y tw ic e t h e c o i l r a d iu s and i s a m easu re o f t h e i n t e r a c t i o n s t r e n g t h . We fo und upon m ea su r in g th e m ag n e tic f i e l d from i n s i d e th e e l e c t r o n gun th ro u g h t h e f lo p p e r t h a t a se co nd z e ro a x i a l f i e l d -115- F ig u re 26: Magne tic f i e ld s f o r th e f lo p p e r shown in F ig u re 2 p . = 2 5m a CL O LO N X •7116- _l___________ I------------------1- 7 9 9 T T ” s - n r u rc Gxts ; g2 t30 j - rOfl C D O J CD —d H -1 1 7 - F ig u re 27: A d ia b a t ic i t y p a ram e te r: %'G a re th e e x p e r im e n ta lly de te rm ined p o in ts and s o l id l in e s a re th e o re t ic a l . . The x - a x i s i s t h e d i s t a n c e m easu red from th e g e o m e t r i c a l c e n t e r o f t h e f l o p p e r . x ' s - E x p e r i m e n t - - - - - T h e o r y ----- P e f i D (cm) - 81 1- —119 - c r o s s i n g p o i n t e x i s t e d b e tw een th e e -g u n and th e f l o p p e r . T h is se co nd n o n - a d i a b a t i c p o i n t was e x tr em e ly s e n s i t i v e t o t h e c u r r e n t s e t t i n g i n t h e f lo p p e r and was s im i l a r t o t h e d e s ig n e d n o n - a d ia b a t i c p o i n t a t t h e c e n t e r o f t h e ' f l o p p e r . The o r i g i n a l s t a t e p r o d u c t io n c u rv e s shown i n F ig u re s 4 and 5 w ere ta k e n w i th t h e s e two n o n - a d ia b a t i c r e g i o n s . T h e r e f o r e , we t r i e d t o e l im in a te t h e second p o i n t b e tw een t h e e -g u n and t h e f l o p p e r . T h is se co nd r e g io n i s c a u se d b y th e l a r g e H e lm ho ltz m ag n e tic f i e l d . We fo und t h a t we c o u ld n o t , w i th o u r e x p e r im e n ta l s e t u p , e l im in a te t h i s f i e l d r e v e r s a l p o i n t . We c o u ld , h ow ever, make i t e x tr em e ly n o n - a d ia b a t i c f o r m ost o f t h e m agn e tic f i e l d s p ro d u c e d i n t h e f l o p p e r . Hence t h e s t r a n g e i n i t i a l c o n d i t io n s u s e d i n t h e t h e o r y c h a p te r and th e r a t h e r c o n fu s in g e x p la n a t io n s r e q u i r e d t o i n t e r p r e t t h e r e s u l t s o f b o th th e o r y and d a t a . F ig u re 28- shows t h a t t h e m easu red n o n - a d i a b a t i c i t y o f t h i s r e g io n rem a in s r e a s o n a b ly n o n - a d i a b a t i c f o r t h e c u r r e n t s u s e d i n t h e f l o p p e r . F i n a l l y , F ig u r e 29 shows t h e f i e l d m agn itu d e and a n g le m easu red t o compare t h e e x p e r im e n ta l a d i a b a t i c i t y p a ram e te r w i th t h e t h e o r e t i c a l one f o r t h e f l o p p e r we c o n s t r u c te d . 3 .7 P ro c ed u re Now t h a t t h e equ ipm en t u se d i n o u r e x p e r im en t h a s b e e n d e s c r ib e d , we w i l l g iv e an o u t l i n e o f t h e p ro c e d u re u s e d i n o b t a i n in g th e r e q u i r e d d a t a . We b e g in b y s im p ly t u r n i n g on a l l t h e equ ipm en t and a l lo w in g e v e ry th in g t o come t o e q u i l ib r iu m . T h is u s u a l l y t a k e s 120 I F ig u re 28 : A p l o t o f th e a d i a b a t i c i t y p a ram e te r f o r t h e se co n d n o n - a d ia b a t i c r e g io n a s a f u n c t io n o f f lo p p e r c u r r e n t . \ 30 20 10 - 121 - FIG.2 8 __L IOx I (mo) I -1 2 2 - ■ I' F ig u re - E x p e r im e n ta l m agn itu d e - and' r o t a t i o n a n g le f o r t h e ■ m ag n e tic f i e l d e n v iro nm en t . t h e s p in p a r t i c l e e x p e r ie n c e s d u r in g t r a n s i t f rom t h e e -g u n th ro u g h t h e f lo p p e r H b6 'i F F ( BEAM AXIS e 'S -MAGNETIC F L D X1S - F L D . A N G L E D ( INCHES) 6 (D E G R E E S ) - 124 - ab o u t one t o two h o u r s , The f i r s t d a t a r e c o rd e d i s a com p le te r e a d ­ o u t o f a l l m ach ine p a r a m e te r s . Then p e r i o d i c a l l y we " re a d o u t" th e m ach ine t o m o n ito r t h e c o n d i t io n s u n d e r w h ich we a r e t a k i n g d a t a . A f t e r t h e i n i t i a l r e a d o u t , we r e c o r d th e d e t e c t o r d r i f t i f we a r e u s in g th e s u r f a c e d e t e c t o r t o o b 'ta in t h e n o i s e l e v e l i n h e r e n t in t h e e q u ipm en t. T here i s a l s o a n o i s e l e v e l a s s o c i a t e d w i th t h e beam i n s t a b i l i t i e s , b u t t h i s h a s r a r e l y b e e n n o t i c e a b l e . I f t h e C hanne l- t r o n i s b e in g u s e d , a d a rk c o u n t (n o beam) i s r e c o rd e d a s t h e n o is e i n h e r e n t i n t h a t d e t e c t o r s y s tem . I f t h e d e t e c to r s a r e w o rk ing p r o p e r l y , an e x c i t a t i o n c u rv e , F ig u r e 1 6 , i s r e c o rd e d t o d e te rm in e t h e optimum e l e c t r o n e x c i t a t i o n v o l t a g e f o r t h e m e ta s ta b le beam . The e x c i t a t i o n .c u rv e i s a g ra p h o f t h e q u en ch ab le s i g n a l a s a f u n c t io n o f e -g u n a c c e l e r a t i n g v o l t a g e . I f t h e m e ta s ta b le i n t e n s i t y i s l a r g e enough , u s u a l l y g r e a t e r .th an 100 d i v i s i o n s on th e g a lv a n om e te r o r 2000 c o u n ts / s e c o n d w i th t h e C h a n n e l t r o n , t h e H e lm ho ltz c o i l i s s e t t o t h e - e c r o s s i n g p o i n t (605 g a u s s ) and an a quench c u rv e i s m easu red ( F ig u r e 19) . The f lo p p e r i s th e n tu rn e d on t o some l a r g e f i e l d (~10 g a u s s ) t o p ro d u ce s * A n o th e r quench c u rv e i s r e c o rd e d f o r t h e Pgt s ( F ig u r e I B ) . From th e se - two quench c u r v e s , we s e l e c t two v o l t a g e s and d e te rm in e th e p e r c e n ta g e quench f o r e a c h . F o r i n s t a n c e , a t 7 v o l t s we quench a p p ro x im a te ly lOOfo o f t h e Pfi1 s b u t no . a s t a t e s . A t 300 v o l t s a l l o f t h e pB s t a t e s a r e quenched and a b o u t 97% o f t h e o f 's . T hese f r a c t i o n s a r e d e n o te d b y f . ^ and -1 25 - f r e s p e c t i v e l y . We m easu re a "beam -notch c u rv e t o d e te rm in e th e amount o f ^ ' s a c c i d e n t l y quenched b y t h e H e lm ho ltz f i e l d . The d i f f e r e n c e i n th e m e ta s ta b le i n t e n s i t y b e tw een z e ro and 605 g au ss i n t h e H e lm ho ltz c o i l i s t h e amount o f f ^ ' s l o s t . From th e 7*0 v o l t q u en ch ab le s i g n a l a t 605 g a u s s i n t h e H e lm ho ltz c o i l , we may d e te rm in e th e p a ram e te r Tl w h ich when m u l t i p l i e d t im e s th e a p p a re n t amount o f p ^ 's found , from t h e 7 .0 v o l t quench s i g n a l g iv e s t h e t r u e i n t e n s i t y . W ith th e H e lm ho ltz f i e l d s e t t o 605 g a u s s , we r e c o r d th e 7 .0 and 300 v o l t quench s i g n a l s a s a f u n c t i o n o f t h e c u r r e n t i n t h e f l o p p e r ; we d e n o te t h e s e s i g n a l s a s QS(7) and QS(300 ) r e s p e c t i v e l y . From t h i s d a t a , we c a l c u l a t e t h e Pg s t a t e p r o d u c t io n as a f u n c t i o n o f th e f l c p p e r c u r r e n t . The number o f pg ' s p r e s e n t i s s im p ly &B = 0^(7). (74) The number o f u ’ s , d e n o te d as A, i s t h e t o t a l 300 v o l t q u en ch ing s i g n a l m inus t h e number o f p ^ 's , A = (09 (300 ) - 13%)/^. (75) The Pg s t a t e p o p u l a t i o n , I 12 > i n te rm s o f m e a su ra b le p a r am e te r s , i s I Pg Is = ( Y f p ) Q S ( 7 ) / [ Q B ( 3 0 0 ) - ( V f p ) 0 S ( 7 ) / f o , + ( V Y % 0 8 ( 7 ) ] (76 ) We may r e d u c e th e c om p le x ity o f E q u a tio n ( 76 ) b y d e f i n in g a p a ram e te r A - 1I lV -1 26 - I i y s = f+QS(7)/[Q9(300) + _ Y fp )Q0(7)] (77) In A ppend ix V I I I , we c o n s id e r i n d e t a i l t h e u n c e r t a i n i t y i n th e s t a t e p o p u la t io n c a l c u l a t e d from th e u n c e r t a i n t i e s i n t h e m easu red p a r a m e te r s . However, we fo und th e u n c e r t a i n t y t o b e a r e l a t i v e l y c o n s ta n t m ag n itu d e o f ± 0 .025 compared .to t h e ra n g e o f |P g | 2 from 0 t o 0 .5 0 . F o r t h e d a t a t a k e n a t a g iv e n TOF w i th a f i n i t e t im e i n t e r v a l , we u s e d t h e same p ro c e d u re s as above th ro u g h th e ' t a k in g o f an e x c i t a t i o n c u r v e . A t t h i s p o i n t , we m easu re t h e m e ta s ta b le i n t e n s i t y ( c o u n t r a t e ) a s a f u n c t i o n o f d e c r e a s in g e -g u n p u l s e w id th a t a c o n s ta n t f r e q u e n c y o r r e p e t i t i o n r a t e . As t h e p u l s e w id th d e c r e a s e s , so s h o u ld t h e m e ta s ta b le i n t e n s i t y i n d i r e c t p r o p o r t i o n . W ith a 5 ■ m ic ro se c o n d p u l s e w id th a t a r e p e t i t i o n r a t e o f 250 m ic ro s e c o n d s , a TOF s p e c trum i s t a k e n o f t h e u ' s ( t h e f lo p p e r c u r r e n t i s z e r o ) . We th e n s e t t h e f lo p p e r c u r r e n t t o some l a r g e v a lu e t o p ro d u ce Pglg and t a k e a TOF s p e c trum o f t h e Pgl s * The TAC - TMC g e a r i s th e n r e p l a c e d w i th t h e "m anual" sy s tem shown i n F ig u re 24 . The p ro c e d u re from h e r e i s th e n i d e n t i c a l t o t h a t d e s c r ib e d p r e v io u s ly f o r th e s u r f a c e d e t e c t o r d a t a . To a n a ly z e t h e s u r f a c e d e t e c t o r d a t a ( t h e v e l o c i t y a v e ra g ed d a ta ) we n eed t o know th e p o s i t i o n and f u l l w id th a t h a l f maximum o f th e TOF s p e c trum . From t h e TOF s p e c t r a , we can p l o t t h e TOF p eak a s a f u n c t i o n o f oven tem p e ra tu re a s shown i n F ig u re 3 0 . From t h i s F ig u re 3 0 : Oven tem p e ra tu re v e r s u s t h e p e a k p o s i t i o n s i n t h e t im e o f f l i g h t s p e c trum . ■• V -1 28 - 2 9 0 0 - 2800 2700 FIG.30 TO. F. (/.(.sec) -1 2 9 - c u rv e , we may d le te rm ine t h e TOF p e ak f o r t h e s u r f a c e d e t e c t o r d a ta from th e oven t e m p e r a tu r e . The FWrHM o f m ost o f t h e TOF s p e c t r a was f a i r l y constan t:; a t 4o pp ec . A t y p i c a l TOF sp e c trum w i th a g a u s s i a n f i t i s shown ±m F ig u r e 31- T ab le I I l i s t s t h e f i x e d p a ram e te r s and some o f t h e t y p i c a l e x p e r im e n ta l .p a ram e te rs u s e d by th e th e o r y t o f i t t h e e x p e r im e n ta l d a ta .. Armed mm w i th t h e r e q u i r e d d a t a , we w i l l now p ro c e e d t o th e l a s t c h a p te r where t h e e x p e r im e n ta l (3^ s t a t e p r o d u c t io n ' c u rv e s and TOF d i s t r i b u t i o n s a r e g iv e n and compared t o th e t h e o r y . -1 3 0 - Ii F ig u re 3 1 : G au s s ia n c u rv e f i t t o t h e TOF7s sp e c trum . \ N or m al iz ed In te ns ity 119 Msec x's -Experiment — EXPHT-II9/23 .7)2 ) Toven- 2 7 5 5° K i.O.S-. (yusec)FIG.3 -1 32 - T ab le I I V a r io u s M achine P a ram e te r s Mean l e n g th from e l e c t r o n gun t o d e t e c t o r 86.64 cm T y p ic a l t im e - o f - f l i g h t 97 |j,sec T y p ic a l FWHM 40 'jj,sec T y p ic a l V e lo c i ty 0.893 X IO^ cm/ 1 F lo p p e r c o i l r a d iu s (R) O.987 cm F lo p p e r "bore r a d iu s ( f )x max/ 0.53 cm p = r /2RIm ax max' 0.268 F lo p p e r c o i l c o n s ta n t 0 .1 0 7 g au ss /m a CHAPTER' IV DATA AND CONCLUSIONS 4 .1 F lo p p e r Curves We w i l l p r e s e n t i n t h i s f i n a l c h a p te r th e d a ta from th e e x p e r i ­ ment d e s c r ib e d i n C h a p te rs I I and I I I . The d a ta w i l l b e a n a ly z e d u s in g th e th e o ry d e r iv e d i n C h ap te r I I . We w i l l th e n p r e s e n t o u r c o n c lu s io n s i n c lu d in g s u g g e s t io n s f o r f u r t h e r e x p e r im e n ts . F ig u r e s 32 th ro u g h 34 p r e s e n t s t h r e e s t a t e p r o d u c t io n c u rv e s . T hese a r e th e s t a t e p o p u la t io n s a s a f u n c t io n o f th e c u r r e n t i n th e f lo p p e r . The d a ta was ta k e n w i th th e s u r f a c e d e t e c to r , a n d a r e t h e r e ­ f o r e a v e ra g ed o v e r t h e beam v e l o c i t y d i s t r i b u t i o n . F ig u r e 35 shows a " f l o p p e r " cu rv e ta k e n a t te m p t in g to s e l e c t a sm a ll v e l o c i t y window to t r y to e l im in a t e th e v e l o c i t y a v e ra g in g e f f e c t s . F o r r e a s o n s d i s ­ c u s se d i n S e c t io n 3 . S 9 low i n t e n s i t i e s and too w ide a v e l o c i t y w indow , we d id n o t s e e an i n c r e a s e i n th e o s c i l l a t o r y s t r u c t u r e as p r e d i c te d by th e o ry . T h e r e f o r e , we w i l l d i r e c t o u r a t t e n t i o n to th e s p in f l i p p r o b a b i l i t y a v e ra g ed o v e r th e beam v e l o c i t y d i s t r i b u t i o n and th e beam c r o s s - s e c t i o n . The s o l i d c u rv e s i n F ig u r e s 32 th ro u g h 34 a r e th e cu rv e f i t t e d L an d au -Z en e r e n v e lo p e s from E q u a tio n (69) i n C h ap te r I I . From th e oven tem p e ra tu re f o r e a c h , f l o p p e r c u rv e , we d e te rm in e d th e p eak TOF from F ig u rq 30 . We a llow ed th e v a lu e o f p 0 , a m easu re o f th e i n t e r a c t i o n s t r e n g t h ! r e c a l l in g t h a t H+ = -HoPQ(p o = r /2 R ) ] to v a ry to o b t a in th e a s ym p to t ic f i t shown. T ab le I I I l i s t s th e f ix e d p a ra ­ m e te rs w i th th e c o r r e s p o n d in g pQ v a lu e f o r e ach c u rv e . The F ig u re 32 : F lo p p e r Curve I . I0 P op ul at io n Data -x -x -x - L~Z Envelope oven 2 900°K *O.E peak = 95psec Iflop (nna) 136 r F ig u re 3 3 : F lo p p e r Curve I I , L-Z Envelope 'oven T.O.F peak = 97jjsec (ma)—> FIG.33 F ig u re 3 4 : F lo p p e r Curve I I I , -139- F ig u re 3 5 : V e lo c i ty s e l e c t e d f lo p p e r cu rv e , v Po pu la tio n oven FIG.35 -141- "142- TABLE I I I D a ta Summary T ab le F lo p p e r C urves Po Zg(Cm) ^ ( t i s e c I 0 .5 0 ± 0 .0 1 1 .80 ± 0.05 9 5 .0 I I 0 .5 3 ± 0 .0 1 1 .8 1 ± 0 .0 5 ■ 9 7 .0 I I I 0 .5 1 ± 0 .0 1 ' 1 .7 7 ± 0 .0 5 100 .0 a v e ra g e d 0 .5 1 ± 0 .0 2 1 .7 9 ± 0 .0 5 -1 4 3 - a v e ra g e po v a lu e i s 0 .$ 1 . The sm a l l o f f s e t i n t h e L-Z e n v e lo p e s i s due t o th e p r e s e n c e o f a sm a l l r e s i d u a l m ag n e tic f i e l d i n t h e f lo p p e r a t z e ro s o le n o id c u r r e n t . The r e s i d u a l f i e l d i s a b o u t O.25 g au ss so t h e c u r r e n t r e q u i r e d t o d r iv e t h e f lo p p e r a x ia l - f i e l d i n i t i a l l y th ro u g h z e ro i s a p p ro x im a te ly 2 .3 m i l l i am p e re s . T h is r e s i d u a l f i e l d c a u se s an asymm etry i n t h e a d i a b a t i c i t y p a ram e te r a. We t h e r e f o r e , e x p e c t t h e p h a s e o f t h e o s c i l l a t i o n s t o b e s i g n i f i c a n t l y d i f f e r e n t th a n t h e t h e o r e t i c a l p h a se d i s c u s s e d i n S e c t io n 2 .5 . We s u b t r a c t t h e L-Z en v e lo p e from th e d a ta t o s e p a r a t e th e o s ­ c i l l a t i o n s from th e a s ym p to t ic c u rv e . The o s c i l l a t i o n s a r e now g iv e n i n F ig u re s 36 th ro u g h 3 8 . The t h e o r e t i c a l o s c i l l a t i o n s a re shown u s in g th e p e a k TOF and pQ v a lu e s from t h e i r r e s p e c t i v e L-Z e n v e lo p e s . We a llow ed th e t r a n s i t i o n l e n g th Lq " t o v a ry t o f i t ’ t h e p e r i o d o f th e o s c i l l a t i o n s . S in c e we d id n o t e x p e c t t h e t h e o r e t i c a l p h a se t o f i t , we u s e d t h e t h e o r e t i c a l p h a se p lu s a c o n s ta n t p h a se f a c t o r t o o b t a i n t h e " b e s t " f i t . A g a in , we l i s t t h e ' pE v a lu e s w i th t h e f i x e d p a ram e te r s i n T ab le I I I . The a v e ra g e z v a lu e o r t r a n ­ s i t i o n l e n g th i s I . 79 cm. I t i s r e a d i l y a p p a re n t t h a t t h e o s c i l l a t i o n s go o u t o f p h a se w i th t h e th e o r y f a i r l y r a p i d l y . As we h av e a l r e a d y i n d i c a t e d , t h e a sym e try i n t h e a d i a b a t i c i t y p a ram e te r a c a u se s th e i n t r o d u c t i o n o f a n o th e r f lo p p e r c u r r e n t d ep en d en t p h a se i n t o th e o s c i l l a t o r y te rm . To b e a b le t o p r e d i c t t h i s d e p en d en c e , we. would n eed t o Imow th e f l o p p e r ' s m ag n e tic f i e l d c o n f i g u r a t io n more a c c u r a t e l y -1 4 4 - F ig u re 3 6 : O s c i l l a t i o n s from f lo p p e r c u rv e I „ Po pu la tio n -145 - Oscillations from Fig.32*^ Theory ---- T = 95usec FIG.3G F ig u re 37 : O s c i l l a t i o n s from f lo p p e r c u rv e I I „ Bs Po pu la tio n -1 4 7 - Oscillations from Theory ---- = 97 usee FIG.37 F ig u re 3 8 : O s c i l l a t i o n s from f lo p p e r cu rv e I I I . -1 4 9 - Oscillafions from Fig. 34 Theory ---- .79 cm T =IO 0 usee ___ FIG.3 8 -1 50 - t h a n we a r e p r e s e n t l y c a p a b le o f m e a su r in g . F o r t h i s r e a s o n , we w i l l o n ly docum ent t h e i n i t i a l and f i n a l p h a se v a lu e s w i th o u t t r y i n g t o e x p la i n th em t h e o r e t i c a l l y . The i n i t i a l e x p e r im e n ta l p h a s e a p p e a rs t o b e n / 2 where th e o r y p r e d i c t s AAP r and th e n a p p ro a ch e s -rr w ith i n c r e a s i n g f l o p p e r c u r r e n t a s shown i n F ig u r e 39 w he re th e o r y p r e d i c t s HAAd F ig u re 39 i s s im p ly a p l o t o f t h e p o s i t i o n o f th e maximum and minimum p o s i t i o n s f o r t h e o s c i l l a t i o n s i n F ig u r e s 36 th ro u g h 3 8 . The a rgum en t o f t h e o s c i l l a t i o n s may be w r i t t e n as ACC i / i p + 0 . When i / i p = n , t h e o s c i l l a t i o n i s a t a maximum s h i f t e d b y th e p h a se 0. By p l o t t i n g th e p o s i t i o n i w here t h e maximums o c c u r v e r s u s i n t e g r a l numbers and t h e p o s i t i o n s o f t h e minimums a t h a l f i n t e g e r v a l u e s , t h e p e r io d and p h a se 'maybe d e te rm in e d , S in c e t h e p e r i o d depends upon an a v e ra g e o v e r th e v e l o c i t y d i s t r i b u t i o n and th e beam c r o s s - s e c t i o n , t h e p e r io d s w ere d e te rm in e d b y com pu te r f i t s r a t h e r t h a n g r a p h i c a l l y d e te rm in in g t h e p e r i o d . The i n i t i a l p h a se and a s ym p to t ic f i n a l p h a s e , . h ow eve r, maybe found from t h i s f i g u r e by a l t e r n a t e l y e x t r a p o l a t i n g t o z e ro c u r r e n t and th e n t o l a r g e c u r r e n t v a l u e s . S e t t i n g a s id e t h e p ro b lem w i th t h e p h a s e , t h e t h e o r e t i c a l f i t s t o t h e d a t a a r e f a i r l y good . The s u r p r i s i n g r e s u l t i s t h e r a t h e r l a r g e v a lu e s f o r b o th p ' and z . However, as p r e v io u s ly d i s c u s s e d t h e e x p e r im en t was ham pered by th e p r e s e n c e o f a second n o n - a d ia b a t i c r e g io n . Even th o u g h we t r i e d t o make t h i s r e g io n e x tr em e ly non- P e r io d p l o t f o r t h e o s c i l l a t i o n s i n F ig u r e s 3 5 , 36 , I P 3 O 4 56 2 FIG.39 N —> -1 53 - a& ia b a t ic and in d e p e n d e n t o f th e f lo p p e r ' c u r r e n t , t h e f a c t t h a t b o th P0 and % a r e a b o u t tw ic e t h e i r c a l c u l a t e d v a lu e s i n d i c a t e s t h a t we d id n o t q u i t e s u c c e e d . Our t h e o r e t i c a l v a lu e f o r p i s 0 .2 7 compared t o t h e m easu red pQ = 0 .5 I and o u r c a l c u l a t e d v a lu e o f z i s O.89 cm i n s t e a d o f t h e m easu red I . 79 cm. The f a c t t h a t t h e second n o n - a d ia b a t i c r e g io n i s a p p ro x im a te ly t h e same l e n g th a s o u r d e s ig n e d n o n -a d ia b a t i c r e g io n m igh t a c co u n t f o r t h e f a c t o r o f two i n pE b u t i s m ere s p e c u l a t i o n on o u r p a r t . S t i l l , t h e t h e o r e t i c a l f i t t o th e d a t a i s s u r p r i s i n g l y good i n s p i t e o f t h e v a lu e s f o r pQ and LE D We c o u ld c o n c lu d e a t t h i s p o i n t t h a t f u r t h e r i n v e s t i g a t i o n i s n e c e s s a r y b y c om p le te ly r e d e s ig n in g t h e m ag n e tic f i e l d s t o e l im in a te - t h e se co nd n o n - a d i a b a t i c r e g io n . T h is i s p r o b a b ly n o t w o r th w h i le i n v iew o f t h e TOF s p e c t r a . 4 .2 T im e -o f -F l ig h t (TOF) S p e c t r a A n o th e r com pa riso n b e tw een th e o r y and e x p e r im en t i s a f f o r d e d by t h e p r e s e n c e o f o s c i l l a t i o n s i n th e TOF s p e c t r a a l s o p r e d i c t e d b y th e t h e o r y . From S e c t io n 2 .6 , we s e e t h a t t h e p e r io d o f t h e o s c i l l a t i o n s i n t h e TOF s p e c t r a a r e l i n e a r i n th e t im e o f f l i g h t , and th e p e r io d s o f t h e o s c i l l a t i o n s a r e f u n c t io n s o f p and z . F ig u r e 4o shows t h r e e TOF s p e c t r a t a k e n a t d i f f e r e n t - f lo p p e r c u r r e n t s e t t i n g s . A g a in , u s in g a p e r io d p l o t s im i l a r t o F ig u re 39 > we s e e i n F ig u re 4 l t h a t th e p e r io d o f t h e o s c i l l a t i o n s i s l i n e a r i n t h e tim e o f f l i g h t and th e s lo p e s o f t h e s e l i n e s , w h ich g iv e s t h e p e r io d s compare f a v o r a b ly t o F ig u re 40 : Time o f f l i g h t s p e c t r a f o r th e p s t a t e . Channel NumberFIG.40 .-U -Ji__ u r. -156- F ig u re 4 l : P e r io d p l o t s o f t h e - 'o s c i l l a t i o n s i n th e TOF s p e c t r a . FIG.41 Integer -1 5 8 - th e c a l c u l a t e d p e r io d s u s in g th e p' and pE v a lu e s o b ta in e d from th e f l o p p e r c u rv e s . T ab le .IV com pares t h e o b se rv e d p e r io d s t o th e c a l ­ c u l a t e d p e r io d s i n te rm s o f m il l i a m p e r e s o f f lo p p e r c u r r e n t . Jh e ach c a s e , t h e ag reem en t i s f a i r l y good c o n s id e r in g t h a t t h e TOF i s known t o o n ly f i v e p e r c e n t . S o , we h ave a n o th e r t e s t o f t h e th e o r y y i e l d i n g pQ and z v a lu e s c o n s i s t e n t w i th t h e f lo p p e r c u rv e s . T h e r e f o r e , we b e l i e v e t h a t f u r t h e r i n v e s t i g a t i o n w ou ld o n ly le a d t o b e t t e r a g re em en t-b e tw e en th e o r y and e x p e r im en t and i s p ro b a b ly n o t w a r ra n te d a t p r e s e n t due t o t h e more e x c i t i n g t h e o r e t i c a l l y p r e d i c t e d p o l a r i z a t i o n e x p e r im e n t . 4 .3 P u re P o la r i z e d Beam The t h e o r y from S e c t io n 2 .6 p r e d i c t s t h a t t h e (3^ s t a t e p opu ­ l a t i o n a p p ro a ch e s t h e i n i t i a l u p p e r a;(F = + 1 , Iti^ 1 = +1) s t a t e p o p u la t io n as t h e m ag n e tic f i e l d i n c r e a s e s . So f a r , we have shown o n ly one c a s e w here t h i s i s a p p a r e n t ly t r u e . T ha t i s , w i th t h e (Sg s t a t e i n i t i a l l y z e r o , t h e f r a c t i o n a l Pg s t a t e p o p u la t io n i n f a c t a p p ro a ch e s t h e 50 p e r c e n t l e v e l w i th i n c r e a s i n g f lo p p e r c u r r e n t . S in c e , we c a n n o t a t p r e s e n t d e te rm in e t h e a s t a t e p o p u la t io n s i n d i ­ v i d u a l l y , we a llow ed t h e i n i t i a l (L s t a t e p o p u la t io n t o be non z e ro , b y d e c r e a s in g th e c o l l im a t i n g f i e l d i n . t h e e -g u n ( s e e F ig u r e 17 i n C h a p te r I I I ) , t o c h e ck t h i s p r e d i c t i o n . F ig u re 42 shows t h e (3g s t a t e a p p ro a c h in g t h e p r e d i c t e d i n i t i a l ' a s t a t e p o p u l a t io n . The d a sh ed l i n e i n F ig u re 42 i s t h e a v e ra g e d L-Z e n v e lo p e f o r a (3g -15 .9 - TABLE IV TOF S p e c t r a D a ta ' i O bserved p e r io d (ma) A c tu a l f lo p p e r C u r r e n t (ma) I . 2 4 ,0 2 7 .4 I I 23 -3 E ? .5 I I I 3 2 .5 3 0 .0 . - l6 o - i F ig u re 4 2 : s t a t e i n i t i a l l y n o n -z e ro f lo p p e r c u rv e . ,5- o O p d O n CO T0ven " 2 8 6 0 °K 0 FIG.42 flopX m a ) Data L-Z Envelope 50 —1 6 2 - p o p u la t io n i n i t i a l l y n o n -z e ro . The r e a s o n f o r t r y i n g t o check th e a sym p to tic b e h a v io r i s t h a t t h e th e o r y p r e d i c t s an in g e n io u s m ethod f o r p ro d u c in g a p u re p o la r i z e d , beam o f m e ta s ta b le a tom ic h y d ro g en . I f we s t a r t w i th a beam c o n ta in in g o n ly th e a s t a t e s , we may c om p le te ly dump th e u p p e r a s t a t e (ny, = +1) i n t o th e s t a t e , w h ich i n t u r n can b e d i f f e r e n t i a l l y q u en ch ed , l e a v in g a beam w i th o n ly t h e low e r a s t a t e (m^ = 0 ) p o p u la te d . T hu s , c u r io u s ly en ough , M a jo ran a s p in f l i p s — w h ich a r e u s u a l l y th o u g h t o f a s a n u i s a n c e , i n s o f a r as th e y d e p o la r iz e a beam o f o r i e n t e d s p in s — may . b e u s e d t o p ro d u ce a f u l l y p o l a r i z e d beam . We a re p ro c e e d in g to t e s t t h i s m ethod and t h e r e s u l t s w i l l b e p u b l i s h e d l a t e r . 4 .4 ' C o n c lu s io n I n c o n c lu s io n , we have s t u d i e d t h e o r e t i c a l l y th e p ro b lem o f n o n - a d i a b a t i c p a s s a g e o f an o r i e n t e d s p in th ro u g h an inhomogeneous m a g n e tic f i e l d f o r t h e c a s e s o f s p in % and s p in I , and 'e x p e r im e n ta l ly t h e s p in I c a s e . We have shown t h a t t h e m ag n e tic f i e l d can b e com­ p l e t e l y c h a r a c t e r i z e d by a s i n g l e p a ram e te r d e s ig n a te d as th e a d i a b a t i c i t y p a ram e te r a . T h is p a ram e te r i s s im p ly th e r a t i o o f t h e Larmor f re q u e n c y t o th e f i e l d r o t a t i o n r a t e . T ha t i s a = to /( d e / d t ) . By s o lv in g th e am p litu d e e q u a t io n s f o r a s p in % in a f i e l d d e f in e d by c o n s ta n t a , we h ave shown t h a t t h e m ag n e tic f i e l d can b e s e c t io n e d i n t o n o n - a d i a b a t i c r e g i o n s , r e g io n s w here s p in f l i p s o c c u r , and r e g io n s o f a d i a b a t i c i t y o r w here no t r a n s i t i o n s o c c u r . A c r i t e r i o n was d e v e lo p e d f o r d e f i n i n g t h e . l e n g t h o f t h e n o n - a d ia b a t i c r e g io n s and s in c e no t r a n s i t i o n s o c c u r i n t h e a d i a b a t i c r e g i o n s , t h e p ro b lem o f a s p in p a s s a g e i s re d u c e d t o lo o k in g a t o n ly t h e n o n - a d ia b a t i c r e g io n s . We d e r iv e d t h e s p in f l i p p r o b a b i l i t y f o r th e two c a s e s o f s p in |r and s p in I , u s in g a s im p le r o t a t i n g m ag n e tic f i e l d . T h is p a r t i c u l a r f i e l d i s s im p ly a c o n s ta n t a x i a l f i e l d up t o a p o i n t ( - z ) w here t h e a x i a l component o f t h e f i e l d goes l i n e a r l y th ro u g h z e ro and th e n r e v e r s e s i t s d i r e c t i o n u n t i l a p o i n t ( z ) i s r e a c h e d w here i t i s sym m etric w i th r e s p e c t t o t h e z e ro f i e l d c r o s s in g p o i n t . From t h i s p o i n t o n , t h e f i e l d i s a c o n s t a n t , r e v e r s e d i n d i r e c t i o n from th e i n i t i a l l y c o n s ta n t f i e l d . T h i s . f i e l d a rra n g em en t was ch o sen e x p e r i ­ m e n ta l ly f o r i t s s im p l i c i t y i n f u n c t i o n a l fo rm and e a s e o f c o n s t r u c t i o n . An e x p e r im en t was p e rfo rm ed u s in g t h e F = I h y p e r f in e s t a t e o f t h e 2 S i l e v e l i n a tom ic h y d ro g en . The e x p e r im e n ta l f i e l d was shown t o 2 y i e l d a p p ro x im a te ly t h e same a d i a b a t i c i t y p a ram e te r a s t h e t h e o r e t i c a l o n e . The r e s u l t i n g e x p e r im en ta l , d a t a a g re e d q u a l i t a t i v e l y w i th t h e s p in I t h e o r y . Q u a n t i t a t i v e l y , t h e two v a r i a b l e p a ram e te r s Po = ( r / 2 R ) , a m easu re o f t h e i n t e r a c t i o n s t r e n g t h , and . Lq , th e t r a n s i t i o n l e n g th w ere tw ic e as l a r g e as e x p e c te d . We p ro p o se t h a t t h i s d i s c r e p a n c y i s due t o t h e p r e s e n c e o f a second n o n - a d ia b a t i c r e g io n . F i n a l l y , we w ou ld l i k e t o add t h a t o u r p ro p o se d a n a lo g u e f o r -1 64 - c e r t a i n s c a t t e r i n g p rob lem s i s f a i r l y com p le te inasm uch as we can c o n t r o l th e s t r e n g t h and d u r a t io n o f t h e i n t e r a c t i o n t o a c e r t a i n e x t e n t . We h ave p ro p o se d a new m ethod f o r p ro d u c in g a p u re p o l a r i z e d beam o f m e ta s ta b le h y d rog en a tom s . T h is m ethod i s b e in g i n v e s t i g a t e d . APPENDIX I THREE ISZEL S P H FL IP PROBABILITY FOR CONSTANT a The amplifbade e q u a t io n s f o r t h e s p in I sy s tem f o r c o n s ta n t a may be s o lv e d a n a ly t i c a l l y by a ssum ing th e fo rm c^( 9) cc exp(a.^Q) f o r t h e amplitind 'es and s u b s t i t u t i n g i n t o E q u a tio n ( 3 8 ) . The c h a r a c ­ t e r i s t i c e q u a tio n s f o r t h e a^ a r e U13 - Sicva12 + ( I - Sor2)U1 - io? = 0 , Ug3 i ( I + a 2) a g = 0 ^ (7 8 ) a 3'3 Hr Sicva32 + ( I - 2o'2 ) a 3 + ict = 0 . The s o lu t i o n s t o E q u a t io n ( 78 ) a r e s t r a i g h t f o rw a r d ly fo u n d t o b e S1 = icy, ± ik + icv ; a B = 0 , ± ik ; (79) a 3 = -icy, ± ik - icv ; w here k = Hf- cvs . The g e n e r a l s o lu t i o n s f o r t h e C^( 9) a r e th e n some l i n e a r com b in a tio n o f e x p o n e n t i a l s w i th a rgum en ts c o n s i s t i n g o f t h e s e t o f a ^ s ; g iv e n i n E q u a tio n ( 79) . A ssum ing th e i n i t i a l con ­ d i t i o n s C1 ( 9 = 0) = P , c g( 9 = 0) = q and c 3( 9 = 0) = r , we g e t - i 6 6 - C^(G) = ( exp( iaG ) / 2k2) 5? ( l+ ( Ecy3-H )cos. .k0 - 2 ia k s i n k6) + q( J f 2 k s i n kG4-i^/ 2 a co s k8 - J 2' a) + r ( l - co s k6 ) ] , Cg(G) = [$>(- s in ( k G ) / / 2 k + i(V 2 o;/k3) s i n 2 gkG) + ( 80 ) q( c o s (kG ) / k 2+ n 2/ k 3) + r ( s i n kG)/J~2 k + 1 ( ^ /2 a / k 3) S in 3^kQ) ] , C3(G) = ( e x p ( - ia 8 ) /2 1 c 3) [p ( § s i n s glcQ) + q ( - ^ / 2 k s i n k8+2.i^/ 2 a s i n 3gk8) + r ( 2 k 3 cos kG + 2 s in 3^kQ + 21ak s i n kG) ] . L e t t i n g th e i n i t i a l c o n d i t io n s b e com plex w i th a r b i t r a r y p h a s e s , we may w r i t e o u t t h e i n d i v i d u a l p r o b a b i l i t i e s f o r th e m ost g e n e r a l i n i t i a l c o n d i t i o n s . However, th e am p litu d e s a r e v e ry com plex and i t w ould r e q u i r e - s e v e r a l p ag e s t o s im p ly w r i t e them down. T h e r e f o r e , we w i l l b e a l i t t l e more r e s t r i c t i v e on t h e i n i t i a l c o n d i t io n s and j u s t q u o te t h e r e s u l t s . The i n t e r e s t e d r e a d e r may d e r iv e t h e g e n e r a l am p litu d e s from E q u a t io n (8 0 ) w i th enough p a t i e n c e . The f i r s t s im p l i f i c a t i o n w i l l b e p h a se a v e ra g in g o f t h e p ro b a ­ b i l i t i e s Ic^(G) j 2 . T h is w i l l e l im in a t e a l l c ro s s te rm s b e tw een th e i n i t i a l a m p l i tu d e s . F o r p = q = r = l / J 3 , t h e s p in f l i p p ro b a ­ b i l i t y I c a ( Q) j 3 e v a lu a te d a t Q = rr, i s Ic 1(Tr) j 2 = ^CI + ( 2 s in -gkTr) 3( I - s i n 3 ^krr)/ k 3 ] . (81 ) T h is i s g rap h ed i n F ig u re (4 3 ) a lo n g w i th th e -p h a s e , a v e ra g e d s p in f l i p p r o b a b i l i t y f o r p = q = I f fJ 2 and r = 0 g iv e n by Ic1 (Tr) j 2 = | [ 1 - ( S in ( ^ r r ) Z k ) 4 ] . (82) - 1 6 7 - As c an b e s e e n from F ig u re ( 43)> t h e p r o b a b i l i t y s im p ly o s c i l l a t e s a b o u t § f o r a l l t h r e e s t a t e s i n i t i a l l y o c c u p ie d . The o s c i l l a t i o n s q u ic k ly damp o u t and t h e u n i n t e r e s t i n g r e s u l t i s t h a t t h e s t a t e s rem a in a p p ro x im a te ly e q u a l ly p o p u la te d . F o r t h e r = 0 c a s e , th e s p in f l i p p r o b a b i l i t y r i s e s t o 50 p e r c e n t and rem a in s t h e r e . T here a r e sm a l l I o y f r e q u e n c y o s c i l l a t i o n s b u t t h e y a r e n e g l i g i b l y sm a l l i n am p l i tu d e . As w i l l b e s e e n l a t e r i n .C h ap te r I I , t h i s b e h a v io r o c c u rs f o r more r e a l i s t i c m ag n e tic f i e l d s th a n c o n s ta n t a . T h is a s ym p to t ic b e h a v io r i s s im p ly a c om p le te s p in f l i p . The i n i t i a l l y s p in up s t a t e iry = +1 i s c om p le te ly dumped i n t o t h e s p in down s t a t e nip = - I . I-168- F ig u re 43 : T h ree l e v e l c o n s t a n t ' a s p in f l i p p r o b a b i l i t y . I -169" FI G. 43 O O O Append ix I l • L e b e d a 's Computer P rog ram f o r th e E v o lu t io n M a tr ix R O B i S C C E MODEL P R O B L E M ' W I T H I HE TA AS I HE U O t P V A R I A B L E • L M T A R Y R E P R E S E N T A T I O N OF THE P R O P A GA T O R I M P L I C I T R E A L VK ( A=- H' 0 = 2 ) R E A L C C E C C O M M C N / C O X / C O E U ( 3 ) C C K M:C N / S C L E NX / F K ' V ' R T 2 # P I * R I u 2 ' CNE ' TWL ' | \ C F CL CCMMOK X £ C L N S / B k. T A ' AL F A A i O AM: A j C O S B J S I NB C C M M O N / B L R p / P T i I 6 ) , W T ( I 6 I * X I ' N P T C CMMON / E G N / C C H 4 H ) , Y ( 5 6 ) ' C l 6 ) ' U ( 6 ) , O S T E P - E R R U P T j ER R L C ' E RKO S ' TMAX ' T M I N ji N ' M ' Nl JN A X ' J C L U M f ' b b ' A 8 1 S ' 5 8 1 S ( NPT=I f i R E A D ( 5 ' 7 5 ) ( P T ( 1 ) , I = 1 ' N P T ) R E A D ( 5 ' 7 5 ) ( WT( I ) , 1 = 1 , NPT I CA L L B C O Q t N 7 5 F O RMAT ( 3 G 2 4 & 1 6 I C N E 5sI « CDCU T W O = S » 0 0 0 0 R T a s D S G R T ( T w a ) P I w 3 . 1 4 1 5 9 2 6 5 3 5 8 9 8 7 F X 0 2 « P I / T W 0 M = 4 ERRUp = = I » 0 0 ^ 0 5 E R X = E R R U P Z l O oODO E R R L Q = I * CD"C8 ERRCS s I » CU * 0 8 C D = 2 ' 4 C D C 1 C Mr h, K•N K f 8 ■ [ p 8 q f M A l K v 7 4 8 s U p [ 8 f M v l K + 5 K N 8 [ r p 8 ; C ( 4 ' =SaSOGlZDU C ( I S = I » 0 C U 0 Z D D D ( 2 ) = = 5 = CDOC Z DO O ( 3 S = I = S O O l Z O O D ( 4 S = S « OCOOZDD NUMAX= S OOO W R I T E ( 6 , 1 0 0 1 ) 1 0 0 1 F O R M A T ( ' I T 4 0 ' ' R C B I S C O E MOUEU S P I N F L I P P R O B L E M ' S 4 7 9 READ ( 5 > 2 0 0 1 ) FKM I N , F KMA X ' E K S T t P ' 1 1 C 2 0 0 1 F O RMA T ( 3 G 2 4 ' 1 6 ' 1 3 ) I F ( FKM I N ' L T ' O , U D O O ) GO TO 4 3 ■ FK = FKM I N C l i e , L T ' 0 C A L C U L A T E R R B A B l L I T t s BUT DONT R EAD I N I T I A L -1 7 1 - C COND I T I ON S C I I C - - C D O M R EAD I M T CON' D DONT CALC P R O b A B I L C I I C i Q f . Q . : S 2 368 f S H T P R OBAB ' I F U I C « I E » O ) GU TO 4 5 1 R EAD ( 5 * 3 5 M C U t U ( J ) M - ' I M ) 3 5 F O R M A T ( 3 F 2 4 » 1 6 ) L R I T E t 6 . » 3 6 ) I C O t O ( J M J e l M ) 3 6 F C R F A T t ' C I N I T I A L COND I T I ONS C l « ' , 6 1 6 , 8 , ' CS ■ ' 1 , 6 1 6 , 8 , ' CO = ' , 6 1 6 * 8 1 4 5 1 CCN I I NLE 4 3 1 L R I T E ( 6 , 1 0 0 2 ) F K 1 0 0 2 F ORMA T ! ' CSTRENGTH PARAMETER* FK » ' , 6 2 4 * 1 6 1 A N G I = " 1 5 D 0 V D E X P ( . 4 9 3 D 0 * 0 L 0 G ( F K ) ) XI = OS I N I A N G I ) CALL U PRAKS t ANG I ) L R I T E ( 6 ; 7 4 ) B E T A * A L F A A , GAMA , JCOUNT 74 FCRf 1ATt ' O 1 j. • BETA » ' , Q 1 6 * 8 , ' )—P B O " i s G s O / i " I OANNA = 6 1 6 * 8 , 2 X , 1 4 / ) CAL L C G E F C ( I I C ) F K= FK+ FKS TEP I F ( F K * L E * F KMAX ) GO To 4 3 1 GC TC 4 7 9 4 3 CONTI NUE END ' F O R , I S U P R A M S , U p R A M S S U B ROL T I N E U P RAK S ( ANGO) I M P L I C I T R E A L # 8 ( A - H , 0 » Z ) C 0 N M C N / E Q N / C C ( 4 8 ) , Y ( 5 6 ) , C ( 6 ) , U ( 6 l O [ N a / I Y 9 n n q I Y / n n ’ [ I , E R R C S , T K A X T K I N , N , M , N UMA X , JCOUNT COKMC N / S C L N S / B E T A , A L F A A , GAMA , C B , S B C G MM C N / S C L E N X / F K , V , R T 2 , P I , P I O H , O N E , TWO , NOFCL C 0 M M 0 N / B L R p / P U 1 6 ) , W T ( 1 6 ) . , X i , N P T EXTERNAL UN l TAR G = C N E / R T 2 C NUMERICAL I NTEGRAT I ON OF THE P H I ' S NOFCL=O c x = s i N C z : C S Y * S ; , , c B f ' y T w ( x ) = x **2 7 3 9 CALL P H I ( X I , P R , P H I X , I ERRR ) . I F ( I ERRR • E Q i C ) GO TO 7 3 8 ANeO=ANQCZTWO X I = D s I N ( A N Q O ) GC IQ 7 3 9 7 3 8 CONTI NUE SUM I = O * 0 0 0 0 N F T - N P T M CO I I P T = 1 , N P T u I a T3M P T b I Z=Pl (MPT) - 1 7 2 -X=ZVXI oTG- 2 "JsPo0k8 jX" JsvV ’ V ( C A L L P k I ( X , P H I R f P H I I , I E R R R ) TRM = P H I V W T ( MP i ) V D J A C / Z V M 2 S U M l = S L K H - T H M I F (DABS (TRM: /SUM- I ) * LT»ERRUp ) GU TO 91 I CONT I NUE 9 1 S U H = XI VSUMI F h I I - P H I X N = 4 R H O 2 1 s P R v P R + P H 1 1 v P H 1 1 Z l = O K E Z D S Q R T I ONE + R H O g l + R H 0 2 1 V K H 0 2 1 / iM O D O O ) 8 . z G K f , * . C m. z G , x l o M r h K p rd f n z G a f g i = Z l V R H O e i Z T W O ANGS l = C A T A N S ( P H I I , PR ) Y ( 4 I = A N G S l Y ( 3 I = G V S U K I TSTART=ANQQ a " r 6 K f e 8 [ [ a " S c K I r f N CSTLP= i j O ' + esTSTART ) / 2 0 0 = ODOO DIMENSION ALS) DC 1 5 4 1 I - I V J 1 5 4 1 A ( I ) = I * 0 0 - 0 2 3 7 CALL A K B C E ( T S T A R T f U N I T A R f I E R f A ) I F f B E R = G T = D WR I T E f G f 7 0 ) I ER 7C FORMAT( i CUPRAMS ERROR G ' f I U I F f I E R = EG = 4 ) GO TO 3 7 C BACKSTEP C S T L P = ( T K A X a T S T A R T ) / 4 o ODQO CO 2 ? 1 = 5 * 8 27 CALL RLNGE(TSTARTfI fUNITAR) I N D X a N V ( K + 3 I Z S = Y f I N D X + 1 ) 2 3 = 7 ( I N D X + 2 I Z l = U S QR T I 0 N E - 2 2 V Z 2 - Z 3 V Z 3 ) A N G S l = Y < I N D X + 4 I CB=Zl tZ l -ONE EB=OSGRTIOhE-CWvCB) B E T A = D A C O S f Z l Q Z D O h E I ALFAA=ANGS l GAKA = = A L F A A t i Y ( I N D X 1 3 ) RETURN END ' F Q R H S LJNI T A R I U N l T AR S U B R O U T I N E U M T A R f T V O J ) = M EK—M pM v Z1 )— x [ 7 ) . , B j . 4 - ' “ 173~ . C O M r O N / E G N / C O ( 4 8 ) , Y ( 5 6 ) , C ( 6 ) , U ( . 6 ) , O W T E P , E R R U P ' E R R L O I i E R R O S i TKAX* T M l N i N i h O N U n AX * J C U U N T C O M M O N / S O L E N X / F K i V i R T 2 i P l i P J -Oi di O N E , Tvi O i N Q F C L 6 " , 6 d r " m I ) G = G N E / F U 2 Z S = Y I N K + 1 ) ZS=YINK +H ) A R G = C N E « Z 2 v Z 2 - Z 3 * Z 3 r 9 M 3 n P , ’ a 4 O 0 OD 0 0 ) GO TO 9 3 5 0 Z l a U S Q R T I A . * l C S = D C C S I Y ( NK + 4 ) ) SS = D g I N I Y I NK + 4 i ) CG( NK + 1 ) = Z * C S * ( Z 3 - Z l - I v O S T ER C C ( N K 4 2 ) = " Q * Z 2 V C S * 0 S T E P C C ( N K + 3 ) = Q v Z 2 / Z l v S s * 0 S T E P A L F = P K Z D S I N I T C 0 ( N K + 4 ) = ( A L F + U * I Z 1 / Z 2 + Z 3 / Z 2 " Z 2 / Z 1 ) * S S ) V 0 S T E P . R E T U R N • 9 3 5 5 CO 1 6 3 I = I i N 1 6 3 C O ( N K t I ) = I o O D S O R E T U R N END ' F O R * I S C C E F C i C O E F C S UBROUT I NE C O E F C ( I l C ) DOUBLE P R E C I S I O N D B i D A i D G i D C B i D S B CCMMONZ SOLN S ZDB i DA i DG i DCD i D S B COMPLEX COEF I 3 ) i U P ( S i 3 ) i U ( 3 * 3 > i S ( 3 i 3 ) * C O ( 3 ) T G Q Q G % L T G Z L 2 T : G x v l D I M E N S I O N RROB ( S ) C N E = 1 * 0 TWO=SaO RTS = S G R T I 2 , O I A = DA • G = DO C C l I = I i 3 I CO I I ) = CMP L X I C O E Z I I ) i Q a O ) S B = D S B CB=DCB Z s ■ ( O N E t C B ) Z T W O R E = Z v C C S ( A + O ) F I M = = Z v S I N ( A t G ) L ( I i l ) = C P L X ( R E i F l M ) RE = S BZRTSVCOS I G ) F I M = = S B Z R T S V S I N ( G ) L ( S i l ) = C M P L X ( R E i F I M ) p K M 8% / K f W l p a ^ [ R E = Z v C O S ( A=G-) F I M = Z V S I N ( A = G ) - 1 7 4 - U 3 * l I =ChzP L X ( R t ^ F l M ) L ( 2 ' 2 I = C K P L X ( C D , 0 , 0 I P E a laS f i Z R T a ^ CQ S ( A ) F I K feS QZRT S ^ S IN ( A ) L ( I , 2 I = C K P L X ( R b f F I M ) L ( 3 , 2 I = = CON J G ( U ( I , 2 ) ) L ( I ' 3 I = C C N J G I U I 3 , I ) ) L ( 2 ' 3 I = = C O N J G ( U ( S f D ) L ( 3 J1 3 I = CCNJG ( U ( l f l ) ) CC 2 1 = 1 , 3 CC 2 J 52I , 3 ' 2 L P ( I f J ) = U ( J f X ) L P ( I f S ) , B qnM r Y A l L P ( 2 , l ) = " U p ( 2 , l ) L P ( 2 , 3 I = = U p ( 2 , 3 ) U P ( 3 , S I = = U p ( 3 , 2 ) CALL C P XKL T ( U P , U , S , 3 ) CC 4 1 = 1 , 3 ' CC 19 J = 1 , 3 A K P - C A t i S ( S ( I , J ) ) FAZ = A T AN S I A I K AO t S ( I f J ) I f R E A L ( S I I f J ) I ) k R I I E ( 6 , 1 5 I I , J , S ( I , J ) , A M P , FAZ 1 5 F C R K A T ( * S ( 1 f 1 1 , 1 , I f 1 1 , I I • f ' NE ® ' , 6 1 6 , 8 , ' I f G1 6 a 3 f • AKP - ' , 0 1 6 , 8 , ' FAZE ® ' , 6 1 6 * 8 I I S CONTI NUE 4 C O N T I N U E • I F N I C ' E Q ' O I R E T U R N C A L L ' C P X K L T I S f C O f C O E F f I I CO 9 1 = 1 , 3 P R O B I I I s O « C CO 6 J = I f S 6 P R C B ( I I = PROB I I ) + ( G A B S ( S ( I f J I ) * C O E O ( J ) 1 * * 2 5 C O N T I N U E WR I T E I f i f 5 1 I P R O B I I 1 , 1 = 1 , 3 1 5 1 F O R M A T ( 1 C AVERAGE P R O B A B L I T l I E S , , 3 G 1 6 « S ) CO 7 1 = 1 , 3 7 P R O B ( I ) = C A B S ( C O E F ( I ) ) V V S WR I T E ( 6 , 9 I ( P R O D ( I ) , 1 = 1 , 3 I 9' FORMAT ( ' P R O B A B I L I T I E S ’ , 3 G 1 6 * 8 Z / ) RETURN D ( : % 2 ' F O R , I N BCOGENf BCOGEN SUBROUT I NE BCOGEN I MP L I C I T R E A L V S ( A = H f O = Z ) C 0 N M C N / B S / B C C ( 2 1 f S I ) B I NOKI I , J I = F A C I I I I / F A C T ( J ) Z F A C T ( I = J ) CO ' I 1 1 = 1 , 2 1 I = I l = I IM » -1 7 5 -CO 2 J J = I f I I J = J J ^ l 2 E C O l J J f I I I = B I N O M I I , J ) 1 C G M I h L E R E T U R N : % 2 1 F C R M N F ACT f F ACT DOUBLE P R E C I S I O N F U N C T I O N F A d ( I ) ) S T C = i 7 G 2 G I F l ( I . E Q ' 0 ) . C R * ( I * E Q ' 1 ) ) RE l URN I M P L I C I T R E A L v B ( C f D f H ) D I M E N S I O N H ( S O ) DATA H / I ' O D O f 2 » O D O f 6 " O D O , ? 4 . U D O , 1 2 0 * 0 0 0 , 1 7 2 0 * 0 D 0 f 5 0 4 0 * O U O f 4 O 3 2 O * 0 D 0 f 3 6 2 S 8 0 . 0 O 0 f 2 3 S 9 l 6 8 C 0 » O D 0 M 7 S 0 O l 6 0 O » 0 D 6 # 6 ? 2 7 O g 0 g » C D 2 , 3 S 7 l 7 8 2 9 1 S » 0 D 2 f l 3 C 7 6 7 A 3 6 8 « O D 3 f S O 9 2 2 7 8 9 8 B B . 0 D 3 f W f [ m h O 6 r l VAl ( N D J t i > > d H C J ) U 3 f q a M 6 q I l K 3 [ q a M 6 q I l > N q u i 8 . : C ; . % : % 2 9 f n O r 6 8 / n v O 8 / n v S U B R O U T I N E D E R 3 ( A O U T j A l O 3 g j A3 J M N j D ) I M P L I C I T n / 3 ’ d W r 3 • e O [ • p l ppEFEMpJ8 “[8 “pj7 6 G c [ M - D I M E N S I O N A OU T i I ) O A l I I ) O A S d ) J A 3 ( I ) DO I N U P - ' l j N I N M = N U P t l S U M S = C e C C O O CO E J = I j N l S U M E = O o O D Q a CO 3 L = I j J 3 S UME = S U M e t B C O d j J d A 2 ( J=> L t I d - A S ( L ) 2 S U M 3 « S U M 3 + B C 0 ( l c J M) V A l ( N l » J + l » V S U M S I A C U T ( N U P ) = A CU T ( N U P ) t s U M 3 / D RETURN END 9 P n O I N p H I j PM I N q W n [ q a r 6 / I e r M c O I n O I r j I E R ) I M P L I C I T n / 3 ’ d W M 3d e O [ • p l f [ u u [ 6 i N [ R ’ / 6 c i 9 " O o O n a N O I c h I r s A O [ 6 / O a ’ [ O N O F C L C O M M O N / B S / B C C i E I O N r l T T Q DQ E % i : . G C i : . Z 8 r u / 6 N r [ 6 3 M / [ l D IMENS I ON A R ( 4 0 0 ) j A U 4 0 0 ) > B E T R i 4 0 0 ) J BET I ( 4 C 0 ) l D N R ( 2 C ) j D N I ( S O I J D R ( E O ) J D I ( E U ) J E X l ( 2 0 ) J E X S ( Z C ) 2GAMR(4C0I fGAMI I 4 0 0 ) ' TNR(SO) ' TN I (PO) D I M E N S I O N D ( S O ) D I M E N S I O N X X ( S O ) IER = I X = S I N T h E T A f A * S l N v » 3 / ( R T 2 * h K ) GaONEZRTS F K 2 = F K * R T 2 M A X " 1 3 DC 6 7 I « l f l 4 6 7 A ( I ) « 0 9 0 0 0 0 X X ( I ) = X X X ( 2 I = D S G R T I O N E * X * X ) DC 3 S 1 = 3 ^ 1 4 3 9 XX( I I = = X X I 1 - 2 ) A ( I ) = X v v 3 CAL L D E R 3 ( A ( S ) f X X f XJCf XX f I S f O N t ) CO 4 5 I - I f l J A ( I ) - A ( I ) Z F K S 4 5 C O N T I N U E Z = O 4 ODOO f 6 / 6 B • f 6 / R CO 3 0 I = I f 2 6 0 A R ( I ) = Z ' A I ( I ) = Z BETR( I ) =Z BETI l I I=Z GAMR( I )=Z GAMl ( I ) -Z 3 0 C O N T I N U E CO 3 g I - I f g O f % n ( I ) = Z C M ( I ) = Z C R ( I ) = Z C I ( I ) = Z D ( I ) - Z E X l ( I ) = Z E X S ( I ) = Z TNR( I ) = Z T N I ( I ) = Z 3 2 CONT I NUE DG 3 1 K = I f M A X A I ( K ) = = A ( K ) B E T I ( K ) H - s A ( K ) Z T W O GAMI ( K ) 2=!aA ( K ) Z Q 3 1 CONTI NUE P R = A R ( I ) P I = A I ( I ) CO 2 7 M= S f MAX -1 7 7 - -1 7 8 -Iv I l fsMAX = M I = 2 0 * ( M " 2 ) + 1 H - 2 O v ( M » I ) I-1 C R f l J = G M E faTWC^M AR ( I ) V - BETRI I > A' .BE' I I ( I ) ) C l ( I ) « " T k O v ( A R ( E ) V R E T I ( I J + A I ( I ) V B E T H ( I ) ) D ( I J a D R I l ) v v E + D I ( I ) v At 2 ■ CNR I I ) = B E T R I I I V I A R l I J v A R ( I ) “ A I ( I ) * A I < I ) J - T W O V B E T K I ) I V A R ( I ) V A l I I ) + GAMH ( I ) * A R ( I + 1 I =UAMI I D V A I ( 1 + 1 ) C M ( I ) = B E T I I I ) v I AR I I ) v AR I I ) t aA l I I ) v A l i I ) I 4 - TWCVBETR I I ) I v A R I I ) VA I ( I J ^ G A h R I D d S r I D l I + C A M ( D VAR I D D TNR I I J = C N R I I J v D R 11 ) + C M I I J V U I I I ) T N D l J = D N I I I J VDR I I J - =DNR I I J V-UI I I J A R ( I l ) = T N R ( I ) Z U ( I ) A I ( I l ) = T N I ( I ) Z D ( I ) P R = P R + AR I I l J P I = P D A I I I l J I F ( D A B E ( A R i I l ) Z P R ) = G E e E R X s O R = D A B S I A I ( I I I Z P I > 6 G E o E R X ) I GO TO 4 3 1 I E R - O R E T U R N 4 3 1 I F ! M EG * MAX ) WR I TE ( 6 d 0 2 4 ) P R ^ AR ( I I D P D A I { I D 1 0 2 4 FORMAT ! ' M = MAX; P R * AR # P D A I ' ; 4 G 1 6 . 9 > I F ( M * E 0 # MA X ) GO TO 7 9 C C A L C ALFA PRIME CA L L C E R 3 ( D NR i 2 D S E T R I I d A R d D AR f I D f d D C N R 5 C A L L D E R 3 ( D N R i a D B E T R I I D A I d D A K I D N i a O§ f6 / u l C A L L D E R 3 I D N R I 2 D B E T I ( I D A R d D A K I D N l D - s C e B O D O C ) C C M C A L L D E R 3 ( D M ( 2 ) * G E T l ( I ) * A R d ) ; A R ( I ) * M l * 0 N E ) C A L L D E R 3 ( D N K S ) * B E T K I ) ; A H r l y 3 " r l m u U U = f % / u l f 3 ’ ’ 8 / n v M s 6 r M A l = W / a n M r l = S n M r 8 3 " r 8 u U U = f 7 5 s 8 f [ l CO I O C N L P = I j M l l M = N y p + ! CO 1 0 1 J - I j N l - BXf= B C O t J j N - I ) J D J + I M J I = N 1 43 J + I A R X - A R ( J I ) A I X = A K J I ) P n c B P 3 u n r h l J I ) G I X = G A M I N l J I ) CNR t N I ) =fCNR I N I ) + B X v ( G R X VA R X fsG I X V A I X ) . ■ C M I M ) =ffC N K M ) + B X v ( P n c d 3 r c > P ’ c d 3nc w B R X mB E T R I J D l ) B I X - B E T K J D l ) ARX = A R ( M J I ) AI X = AI I M J I ) C R ( N I I = D R ( N I D T W Q t d X v ( A R X v B K X t i A l x V f i l X ) D I ( N l ) = D l ( N l ) * T W O » B X v ( A R X # B l X + A l x * B H X ) ' 1 0 1 C O N T I N U E 1 0 0. C O N T I N U E C D CO 1 0 6 N U P a l f M l l NI = E c K G, CO 1 0 7 J = I f N l g a o “ p j 7 l c JM - J I = U - H M J I = M a J + ! C N R X = C N R ( N l J I ) C M X = C M ( M J I ) D R X - D R ( J ) C I X = D I ( J ) C R X X s D R ( J l J I ) C I X X a D I I M J I ) TNR(M I = T N R ( NI I+ B X * ( DNRX*DRX + UNI XtD I X) T N I ( N I I = T N I ( M ) + B X * ( D N l X » D R X " D N R X * D l X ) C ( N l ) = D I N l ) + B X * ( D R X X * D R X + D l X X * O I X ) 1 0 7 C O N T I N U E 1 0 6 C O N T I N U E CA L L D C E R V ( E X l ( S ) f D f M l l f O N E ) E X H l ) = O N E y D ( I I C A L L D E R E I A R ( 1 1 + 1 I f T N R f E X l f M l I f O N E ) C A L L D E R S l A H H + 1 ) f T N H E X l f M4 I f O N E ) CO 7 8 K = I fMAX. T N R ( K ) = Z . T M ( K ) = Z 7 8 CONTI NUE TNR ( I I = B E T R ( I ) » D R ( I I + B E T I ( D v D I ( I ) T N H I J = B E T I ( I ) V D R ( I I = B E T R ( I I v U I ( I ) E E T R ( I l I = T N R ( I ) Z D ( I ) E E T K I l I = T N I I I I Z D I I ) C E E T A D E R I V E DO 102 N U P = I f M l l M = NUP - H DC 1 0 3 J = I f M l B X = B C O ( J f N l ) J I = J t I M J l = M = J + I E R X e B E T R ( N l J I ) B I X = B E T I ( N l J I ) D R X = D R ( J ) C I X = DI I J ) 7 NR I N I ) M N R ( M > + BXv . l B RXVDRX + B I X v D I X I T N H M I a T N I ( M ) + B X v ( B I X v D R X a B R X v D I X ) 1 0 3 C O N T I N U E 1 0 2 C O N T I N U E -1 79 - ~ l8 0 - CALL O E R S f B E T R I I l + ! ) ^ T N R ^ E X l O u l l i O N t ) C A L L D E R 2 ( B E T I I I l + l ) i I N I i E X l i M l i i O N t ) DG 81 Ke l iMAX T N R ( K ) a Z T N I ( K ) = Z 8 1 C O N T I N U E C CAL C GAM T N R ( I >=GAMR ( I ) VDR ( U >+ G A M l ( I J v U I f I ) T M ( I ) =GAM ( I ) VDR ( I I - ^GAMR ( I ) V U I f I ) G A M H ( I l ) = I N R ( I ) Z D ( I ) G AM l ( I l ) = T M ( 1 ) Z D ( I ) CO 1 0 4 N U P = I i M l l M = N U P v i CO I OS J = I g N l G X = G C O ( J i N l ) J I = J v S M J I = M = J + ! . G R X ffiG A M R ( N l J I ) G I X = G AM I T N I ( N l ) = T N I ( N I ) + B X V ( G i x v D R X tstG R X v D I X ) 1 0 5 C O N T I N U E 1 0 4 C O N T I N U E f S H H T : . : x S * Q . M x x > 2 M N M E x i i Q x b G % /R l T S H H C E R E ( GAMI l 1 1 + 1 I i T N I i E X l i M l l , O N t I C CLEAR WORK S P A C E CO 7 7 K= S i MAX T N R ( K ) = Z T N I ( K ) = Z C R ( K ) = Z C I ( K ) = Z E X l ( K ) = Z E X S ( K ) = Z C N R ( K ) ffiZ C N I ( K ) = Z 7 7 D ( K ) = Z 2 7 C O N T I N U E 7 9 C O N T I N U E R E T U R N END ' F O R i I N CHXKLT i CPXMLT S U B R O U T I N E CPXKLT(AiBiCiNCOL) COMPLEX A O i S I i B O g N C O L O C ( M N C O L ) DC I J = I i N C O L CO I I « l i 3 C f I i J ) = C K P L X ( 0 ' 0 i 0 . 0 ) o o o o n n o o f s o o r t o o o o o o o o o o o o o -1 8 1 - CO 2 K « l * - 3 2 C ( I ' J ) = C ( . I , J ) + A ( ' I , K ) V B ( K f J ) I CONTI NUE RETURN : % 2 ' F C R i I N AM&OE f AMBDE -!• THE ANS k ERS F OR T HE N EQUAT I ONS APPEAR I N THE L A S T N v ( M « l ) + l O A = v i j j j O 6 LOCAT I ONS OF y t WHERE K I S T H E ' C R D E R CF THE J N T E G 1 + + + + + I F AKE DE RUNS OVER THE S P E C I F I E D END P O I N T , THEN CN R E T U R N TC T HE C A L L I N G PROGRAM, P UT I N C S T E p = ZKAX = Z S T ART i I F J N T E G WAS I N THE P O S I T I V E S D I R E C T I O N , T HEN CALL Rl J NGE ( 2 S I A R T , K + i , F ) s THE ANSWERS W I L L T H E N APPEAR I N N V ( M ) + 1 ' 2 , 3 * , , , N ' N q W n [ q a r 6 / n q % P / M n 7 " " " c Y 9 l I F , X E O K , R ( EGNUM E N I T U G R R U S I M P L I C I T R E A L v B ( A - H f O - Z ) E R R O R R E T U R N S I E R R ~ i NO E R R OR I E R R = 2 J C O U N T TOO L ARGE I E R R a 4 C A N ' T S T A R T I E R R « 6 C A N ' T H A L VE E NOUGH I E R R 8 7 O S T E P TOO SMAL L I E R R ■ I N ' NLRK V E R S I O N I C A N . BE S T O P P E D EX I E R NA L Y BY S E T T I N G N U M A X t L T i J C O U N T ' D I M E N S I O N M O D I F I C A T I O N S N K A X a K A X 4 NO a OF E Q N S a e 4 S KMAXt s KAX ORDE R = 4 CO ( NKAXVMMAX) , T ( NMAXv ( MMAX + 1 ) I f Y S ( N MA X VMKA X ) , C S ( MMAXvNKAX) C , W ( N K A X ) C K ( N M A X ) , Al NMAX I f COEFP I MKAX f COEFC ( KKAX ) C C M K O N X E G N / C O I 4 8 ) , Y I 5 6 ) , CQEMPT 6 ) , C 0 E F C ( 6 ) , C S T E P # F . U R , E L . I , C E R R f R K AX , R K I N f N , M f N UMAX , J CQ UN T D IMENS I ON Y S ( 4 S ) , C S ( 4 8 ) , W ( S I , C K ( 8 ) , A ( 8 ) ■ _ K D E X a K K KX I ERI T = - I GG TO 3 E N T RY A K G D E ( R f F f I E R R f A ) J C O U N T = O I N D E X « O I E R R = 3 Z = R NK 65 N V M N K l s NM ® N C O a X E D M S g o n o r> o o r> o o o o o o o o - I 82- f \ M2 « M v I toN U x % 2 : Z j= x % 2 : Z > r I K p I S A V E ■ C I F < I N D E X s GT s 4 I GO TO ( 3 3 ' 2 B , ' 3 2 , 3 l ) j I E R R KDEX a I , 2 KDEX a KDEX + I J C O U N T ■ J C O U N T + I I F I K D E X * G T , M ) GO TO 8 3 K a N $ ( K D E x a 2 ) DUNNY a 0 S T E P / 2 « 0 D 0 0 CS = 1 . 4 1 4 2 1 3 5 6 2 3 7 3 0 9 5 Gl “ I » CDOo « 1 * 0 0 0 0 / 0 2 G 2 = = E i C D O O + 3 « 0 0 0 0 / 0 2 2 = KL ) KL * I " X E D K ' R I F L L AC 0 4 OT OG ) 0 1 - » Q E , L K t F I 8 8 UT OG ) OS - « Q E » K L ( F I I R U N Q a i CALL F I R * KDEX = I , 2 ) CO 4 I * I , N C K ( I ) «5 C O ( K + I ) 4 Y ( K 4 I Y N ) = Y ( K - I - I ) + ■ CK ( I ) / 2 , CDOC R a R + DUMMY K = K 4 N 3 = KL J K L ^ X t D K i R t F LLAC 8 8 OT OG > 0 2 - » Q E . K L f F I 0 4 OT OG I d = , Q E ' KL ( F I CALL F ( R. JKDEX1 3 ) CO 5 I a 1 , N U I » « ( 2 * 0 0 0 0 = 0 2 ) + CO ( K + I ) + G g v C K U ) 5. -Y ( K + I ) = I ( K + I ) + G I V ( C O ( K + I ) » C K ( I ) ) A - s KL ) K L ' X t D K , R ( F L LAC 0 4 QT OG > 0 1 “ , Q E ' K U F I 8 8 OT OG > 0 2 =, , Q E , KL ( F I f 3 ’ ’ 9 M n x " 8 / c i 4 ) G l s U 6 ODOO + I , , 0 0 0 0 / 0 2 DC 6 I = I i N CK l D ■ CC1< K + I ) 6 Y ( K + I ) a Y I K + I ) + G l V t C K I D y W ( D ) C l ® 2 » O D O O + OS G2 “ « 2 « CDOO =- 3 « 0 0 0 0 / 0 2 . R a R + DUMMY S « KL ) K L ' X t D K i R ( P L LAC 0 4 OT OG > 0 1 ® , Q E ' K L ( F I n o o n n o o n n o n n o o n C 8 8 OT OG J 0 £ - . G E * K L i F I T S H H ) x . b ­ 2 : Z b , wR CO 7 I K I f N k ( I ) ■ G l t f C K U ) + Qg t f Wi I ) 7 V ( K + I ) a Y I K + I ) + C 0 ( K U I / 6 » 0 0 0 0 - M I ) / 3 » 0 D 0 0 I F ( I E R R ) 3 3 * 3 3 * 2 8 CO 9 I = I ; \ S Y S ( I ) = Y ( I ) JCOUNT=JCCUNT-I THE F C L L C W I KG MODIFICATION I S TO C I R C UMV E N T ' T H E P O S S I B I L I T Y OF D O U B L I N G ' 1 S I E P SIZE WHEN TOO NEAR THE MAX VAL UE OF T HE RANGE 1 0 „ W ( I ) - R + A a O D O O t f Q S T E P I C W ( I ) = R t f S i O D O Q t f C S T E P FOR MCS T E F F I C I E N T D O U B L I N G AND E S P E C I A L L Y I F THE D I M E N S I O N S ARE C HANGED W I T H T HE O R D E R # S E T T H E I S A V E C U T O F F AT M + I I F ( I SAVE«GT ' 5 *AND*W(1 ) 'LT*RMAX,AND, I W ( I ) , G T ' R M I N ) GO TO 2 5 J C O U N T - J C C U N T tf I I = KL ) K L # M ' R ( F LLAC 0 4 OT OG ) O i * . Q E . K L ( F I 8 8 OT OQ ) OS - . G E « L K ( F I C A L L F ( R j M j l ) 1 1 R a R + C S T E P D O ' 1 3 I - I # N "8/c K r B 6 D DUMMY a C = ODOO OG 1 2 K = V M KDEX ■ KDEX + N ! ■ 1 2 DUMMY e DUMMY + C O E F p ( K ) tf CO ( KUEX ). 1 3 Y ( N M t f I ) ® Y ( N M l t f I ) tf DUMMY CO 1 5 I - I * N KDEX a I = N CO 1 5 K = I * M KCEX a K D E x tf N I-F ( K u L T * Mi) CO( KDEX ) «* CO ( KDEX tf N ) 1 5 Y ( KDEX ) » Y < KDEX tf N ) 6 ® KL ) Kl . *M,R(F LLAC 0 4 OT OG ) 0 1 " s Q t « K L ( F I 8 8 OT OG » 0 2 = . Q E ' K L ( F I CALL F(R#Mj 6 ) I F ( U C C L N T i G T i N UMAX ) GO TO 3 2 ' CO I ? I = 1 # N DUMMY O ' ODOO ■ • KCEX » I “ N —183~ -18U- CO 1 6 K = I , M . KOEX a KDEX + N . ■ 1 6 C L PMY = DUMMY + C O E F C ( K ) * C 0 ( K U E X ) 1 7 K ( I ) = Y 3 3 U O 2 1 C S TEP a C S T E P / H o O D O O I F ( D A B S ( C S T E P i a L E « O E R R l GO T O 3 0 I F ( I E R R « L T » 2 ) GO TO 2 3 CO 2 2 I ffl 1U N 2 2 Y ( I ) » Y S ( I ) J C O U N T ffl U COUNT =M + 1 I F U R L K G ' E Q . I I J C O U N T = J C O U N T = I I F ( U C C L N T . E G * I ) J C O U N T ■ O GO TO I 2 3 CO 24 I = U M 2 4 Y ( I ) « Y I NM 2 + I ) u C C U N T ■= J C O UN T = I I NDEX = O I ERR = 4 Z = R = G S T E P ^ g e O D O O GO TO i 2 5 CS TEP 13 C S T E P * 2 , 0 D 0 0 CO 2 7 I » l A NM Y S ( I ) » Y ( I ) I F ( I = N ) 2 6 , 2 6 , 2 7 2 6 Y ( I ) o Y ( NMi l + I I 2 7 C S ( I ) = C O ( I ) Z = R I NDEX * 3 -1 8 5 - I ERR a 2 GO TO a 2 8 CO 2 9 I = I i NM Y ( I ) c Y S ( I ) 2 5 C O ( I ) ■ C S ( I ) I E R R « i C C C . C CAUL F ( R i MUO ) GC TQ 1 1 SC I ERR = 5 . 3 1 I E R R a I ERR + I 3 2 I ERR = I E R R + I 3 3 R E T U R N O a KL ) K L ; M ' R ( F L LAC 0 4 OT OG 1 0 1 * o G E 8 K L f F I SS OT OG ) O2 - , Q E , K L ( F I C C END NRU l ER R R E I 4 O l K R R E l 0 4 APPEHDIX I I I THREE LEVEL ^ INITIALLY NON ZERO SPIN FLIP PROBABILITY" The s p in f l i p p r o b a b i l i t y may b e fo und f o r a r b i t r a r y i n i t i a l c o n d i t io n s b y s im p ly w r i t i n g a 3( n) from th e m a t r ix E q u a tio n ( 5 8 ) . F o r o u r e x p e r im e n t , t h e two u p p e r a s t a t e s a r e assum ed t o b e e q u a l ly p o p u la te d . By d e f i n in g th e i n i t i a l (3^ s t a t e p o p u la t io n t o be some f r a c t i o n X o f e i t h e r one o f t h e a s t a t e s , X |of|2 = ](3g |2 , we have from E q u a t io n ( 59 ) t h e s p in f l i p am p litu d e U1(Tr) = Ei ; i e x p ( - i ( 2 X TT- 0 ) ) a 1(o )+ E 1g e x p ( - i ( XTT- |- 0 ) ) a a( o ) + 7 r 'E 13a ;L( o ) . ( 83 ) We h ave U1(O) = a 0(o ) and d e f i n in g A t o be U1(O) we h ave f o r th e s p in f l i p p r o b a b i l i t y Ia1(H)I2 = A[ { l - ( I-X )exp (-TO0P2) > 2 { (E n El s +VX E1gE13)cos(X ^-& 0) + E11E13 c o s (2 ( X^ . - g 0 )) ] ] . The d o u b le a n g le fo rm u la a llow s u s t o r e w r i t e t h e c o s ( 2 ( X - g 0 ) ) a s Ia1 (TT) I 2 = A[ { l-(l-X )exp(-TTQ '0 p2 )+2A/X e x p ( -TT os IAp 1 l X ( l - e x p ( -TTCV0P3/ 2 ) ) }+2( PE1 3( I -E 13) ) s {[( I - E 13) + ^ E13] c o s ( X^-§0) (8 2 ) + ^ T ( ZE13( I-E13) ) * COS3 ( X^-lW) . The f i r s t t e rm on t h e r i g h t hand s id e o f E q u a tio n (8 2 ) i n { } i s w ha t we ch oo se t o c a l l t h e L-Z e n v e lo p e f o r t h e ^ i n i t i a l l y non z e ro c a s e . We c o u ld s u b s t i t u t e E13 = exp( -Trcv0P2 ) ■ i n t o t h e o s c i l l a ­ t i o n e n v e lo p e s and w r i t e o u t th e f u l l b low n e x p r e s s io n . However, —187~ s in c e we o n ly w an t t o o b s e rv e t h e a s ym p to tic l im i t o f t h i s c a se and compare i t t o t h e e x p e r im e n t , we w i l l o n ly c o n c e rn o u r s e lv e s f u r t h e r w i th t h e L-Z e n v e lo p e . U s in g t h e same p ro c e d u re and a rgum en ts g iv e n i n S e c t io n 2 .6 , we c a l c u l a t e t h e ' v e l o c i t y and beam a tre rag ed L-Z en v e lo p e t o be + (4^T /'m ^ (^ )p^ )( l-exp (-T r^ (^ )p^ /2 )) ] . (84) We may now n o te t h a t t h e l im i t o f |a ^ ( rr) | 2^ as ■hEMa0 ) in c r e a s e s i s A o r t h e i n i t i a l a s t a t e p o p u l a t io n . E x p e r im e n ta l d a t a e x h ib i t i n g t h i s phenomena i s g iv e n i n F ig u re (42). APPENDIX PV VELOCITY" AND BEAM AVERAGING COMPUTER FITS In -S e c t io n 2 .6 we c la im th a t th e a ve rag in g o f th e L-Z enve lope o ve r th e t im e o f f l i g h t spec trum can be app rox im a ted b y s im p ly r e ­ p la c in g T i n a b y th e peak v a lu e o f th e TOF spec trum . F ig u re (4 4 ) compares th e L-Z enve lope w i th j u t o a n um e r ic a l in te g r a t io n o f th e beam averaged ' L-Z enve lope as summing th e Gaussian fo rm f o r th e TOF spec trum . The d if fe r e n c e between th e two a re n e g l ig ib le . F o r t h e damped o s c i l l a t o r y te rm , we c la im ed t h a t we c an s t i l l make t h e v e l o c i t y a v e ra g in g a p p ro x im a tio n ( i . e . r e p l a c i n g C b y Cq ) and j u s t p e r fo rm a n u m e r ic a l beam a v e ra g e . F ig u re (4 5 ) com pares th e a p p ro x im a t io n t o b e com p le te n um e r ic a l v e l o c i t y and beam a v e ra g ed damped o s c i l l a t o r y te rm . A g a in , t h e d i f f e r e n c e s a r e n e g l i g a b le . B oth F ig u r e s (4 4 ) and (45 ) a r e o n ly one exam p le . The com pa rison s w ere a l s o made f o r g r o s s l y d i f f e r e n t p a ram e te r s and w ere found t o f i t v e r y c l o s e l y . The m ain r e a s o n f o r t h e a p p ro x im a tio n s was t h e exp en se o f t h e n u m e r ic a l i n t e g r a t i o n s done by t h e com pu te r. -1 8 9 - I / F ig u r e 44 : Com parison o f th e ap p ro x im a te L-Z e n v e lo p e w i th th e com pu te r i n t e g r a t i o n . v 5J 3 O -Q O v_ CL O FIG.44 x's-Computer Averaged L'Z Envelope ----L-Z Approximation to lop ,___I 1 5 -190- F ig u re 4$ : Comparison o f th e damped o s c i l la t o r y , te rm a p p ro x im a tio n w ith th e computer in t e g r a t io n . XS X / / i Computer Averaged Approximation APPENDIX V QUENCHING OF THE ES1 METASTABiEE STATE IN ATOMIC HYDROGEN s Quenching is th e p rocess o f de -p o p u la t in g a r e l a t i v e l y lo n g l iv e d e x c ite d s ta te ( E S 5 ~ l / 8 s) b y open ing a decay channe l to s s th e g round s ta te th ro u g h a c lo s e ly in g s h o r t l i v e d e x c ite d s ta te ( EP1 ^ T ~ 1 .6 X 10 ^ s e e ) . The decay ch anne l i s opened b y S ta rk s ' P , m ix in g between th e two s ta te s due to an e x te r n a l ly a p p lie d e le c t r i c f i e l d . We w i l l d e s c r ib e th e p rocess o f quench ing b y two s ta te tim e dependent p e r tu r b a t io n th e o ry w ith phenom eno log ica l decay c o n s ta n ts . C ons ide r two s ta te s se pa ra ted i n ene rgy b y ftw w i t h decay con­ s ta n ts vg , Yp and coup led to g e th e r b y a s t a t i c e le c t r i c f i e l d The e le c t r i c f i e l d p roduces a r e a l m a t r ix e lement V = ' r | # g ( r ) ) . ftu) {_ # s ( r ) , & g ( t ) , Yg = 8 se c~ . _ ^ % p ( t ) , Yp = 6 -25 XlO ° s e c The 0g( r ) and A ( r ) a re th e s p a t ia l wave fu n c t io n s f o r th e ES and SP s ta te r e s p e c t iv e ly . The e qu a tio n s o f t im e dependent p e r tu r ­ b a t io n th e o ry in c lu d in g decay a re th e n if td a g( t ) / d t = V e x p ( - im t)a p ( t ) - i ( TiYg/ 2) ag( t ) if td a ( t ) / d t = V e x p ( iu ) t )a ( t ) - i ( ^Y0/ 2) a f t ) (85) % en V = o , th e s ta te s s e p a ra te ly decay w i th th e c o r r e c t fo rm -19 |&g(t)12 = | a g ( 0 ) | 2 exp ( -Yg t ) , | a p ( t ) | 3 = I a ^ ( O ) e x p ( - ^ t ) . (86) S o lu t io n s f o r a ( t ) and a ( t ) a re found b y u n co u p lin g th e s p E qua tion s (85) and ass'uming th e fo rm a ( t ) = B x p ( ^ t ) . Due to th e q u a d ra t ic e xp re ss io n s found f o r H = (u)/2fl')[-(l+e-in) ± ( l - e - iQ )Q ] s , (87) where n = 2w/YpJ S = Yg/\p, % = 2v/AYp, (88) Q = [ I - % | q | a / ( l - e - 1 0 ) 2 ]* , th e assumed s o lu t io n s become ag (t) = a%(t) e x p (^ t ) + ag (t) e%p(p,_t), ap ( t ) = b a( t ) exp( +t ) + b 8( t ) exp( p _ * t ) . U s in g th e bounda ry c o n d it io n s a (0 ) = 1 and -a ( 0 ) = 0 , th e S P c o e f f ic ie n t s a re e a s i ly found t o be ( 89 ) & l ( t ) = ( n J s Y g ) / ( k - - M-+) > as ( t ) = b ’ vH + iV s ) / ( ^ „ " p,+), M t ) = - ^ a ( t ) = ( i v / f & ) / ( p f - n ^ ) . ( 9 0 ) As can be r e a d i l y seen , th e e xp re ss io n s f o r a and a a re v e rys p c om p lic a te d . A pp ro x im a tio n s a re made to s im p l i f y th e e x p re s s io n s , namely v < fiw and e « I , t o reduce !Equa tions ( 90 ) to U1 ( t ) ~ I , aa( t ) ~ 0 , b x ( t ) ~ i q / ( I + i n ) , b 0( t ) = - b x( t ) , (91) and t h e am p litu d e s a r e -1 95 - % ( t ) ~ e % p .( ^ t ) , &p(t) ~ ( i q / ( l + ifl) J t e x p ( ^ t ) - e x p C ^ t ) ] , (92) These may b e r e w r i t t e n i n t o t h e h ig h ly s u g g e s t iv e fo rm exp[ - ( F /2 -I- i ( ££,/ft)t ) ] . T hese F 's a r e m o d if ie d d e c ay r a t e s and th e ^ E 's a r e t h e l e v e l s h i f t s due t o th e p e r t u r b a t i o n . They a r e AEg = a u ( | q | 2 / ( l + n ^ ) ) = -AEp . W ith a q u e n ch .re g io n o f f i n i t e le n g th t = A /v • where v is th e v e lo c i t y o f th e atom ; and w i th Ys b e in g sm a ll compared t o th e c o r r e c t io n te rm v ( | q j 2/ ( l + k 3) ) , th e p o p u la t io n o f th e 2S i s ta te a f t e r passage th ro u g h th e quench re g io n is la g( t ) I 3 = exp { - ( Y p l l ^ l O / l X l + f i 2) ] } . (9 5 ) The 2S i and 2P i s ta te s o f hydrogen c o n ta in fo u r h y p e r f in e s s le v e ls each. By a p p ly in g a Zeeman f i e l d and choos ing th e p ro p e r e le c t r i c f i e l d o r ie n ta t io n ( E f r H ) th e two p s ta te s can approach and even c ro ss th e SPi h y p e rf in e . s ta te s th e y a re coup led to w h ile th e s two a s ta te s d iv e rg e from th e 2P i s ta te s . Th is geom etry d r iv e s a s t r a n s i t i o n s (Amj = ±1* A ^ = ± I ) > so th e a llow ed t r a n s i t io n s are aB ^ 1B ' ' ■”196— F ig u re (3 ) i n C h a p te r I shows th e Zeeman d iag ram f o r t h e h y p e r f in e s t a t e s . Wear t h e p - e c r o s s i n g p o i n t , t h e p 's a r e " e a s i l y " quenched by a v e r y sm a l l e l e c t r i c f i e l d b e c au se th e y a r e c lo s e t o t h e 2P s t a t e s . T h is sm a l l f i e l d i s i n s u f f i c i e n t t o a p p r e c ia b ly quench th e n s t a t e s due t o t h e l a r g e e n e rg y l e v e l s e p a r a t i o n . H ence , t h i s d i f f e r e n t i a l q u en ch in g a llow s t h e d e te rm in a t io n o f t h e p s t a t e p o p u l a t io n . The m a t r ix e lem en t V i s fo und t o b e J 3 e a0 |^ | ( aQ = Bohr r a d iu s ) fo r- t h e n = 2 l e v e l i n h y d ro g en and can be r e l a t e d t o t h e v o l t a g e (V) a p p l i e d i n t h e quench r e g io n b y = V /d , w here d i s t h e e l e c t r o d e s e p a r a t i o n . From h e r e , t h e p ro b lem becomes com plex due t o t h e v e l o c i t y d i s t r i b u t i o n and n o n -u n i f o rm i ty o f th e a p p l i e d e l e c t r i c f i e l d . W ith s u f f i c i e n t a n a ly s i s t h e m o tio n a l e l e c t r i c f i e l d g e n e r a te d by th e H e lm ho ltz c o i l can b e fo und from th e beam n o tc h c u rv e . F u r th e r a n a ly s i s i s n o t w a r ra n te d b y t h i s e x p e r im en t b u t , h as p r e v i o u s ly b e e n done i n g r e a t d e t a i l . ^ F o r q u en ch in g i n t h e e l e c t r o n g u n , t h e e l e c t r i c f i e l d i s r e p la c e d b y = v X l l /c and t h e a n a ly s i s i s s t r a i g h t f o rw a r d . -1 9 7 - AVERAGING OVER VELOCITY" INTERVAL APEEHDH VI U s in g a rgum en ts s im i l a r t o th o s e i n S e c t io n 2 .6 , t h e damping o f t h e o s c i l l a t o r y p o r t i o n o f th e th e o r y i s re d u c ed t o an i n t e g r a l o v e r t h e c o s in e te rm (co s k r ) = c h C 6M Cl c o s (k T )d r . ( 96 ). I n s t e a d o f t h e G au s s ia n 6MCl u s e d i n S e c t io n 2 .6 , t h e v e l o c i t y window c an b e a p p ro x im a te d b y a c o n s ta n t so th e i n t e g r a l r e d u c e s to c J t ^ TG, ( k T ) d T = c co s kT0 , ' (9 7 ) w here &r = i ( Ts -Ta ) and aE K M av > T1 ) / 2 . T liis damping te rm , - s i n k^T ) i s com pa rab le t o t h e damping te rm from th e v e l o c i t y k&T a v e ra g in g o v e r t h e G a u s s ia n , e x p ( - ( k S /2 ) 2 . The b e s t we can do w i th t h e p r e s e n t e x p e r im e n ta l s e t up i s a 20 p se c w h i le § ~ 25 p s e c ; The damping te rm s a r e t h e n a p p ro x im a te ly g iv e n as s in ( k&T) ~ ( 1 -. (k ^T ) 3/6 + k&T ) ~ ( I - k 3(6 6 f ) + . • . ) , (98) exp( - ( k 6 / 2 ) ^ ~ ( l - ( k 6) 2/ 4 + •. • «) ^ ( I - k 2( IOi-I-) . . . ) . So damping due t o t h e f i n i t e v e l o c i t y window i s on t h e o r d e r o f th e damping due t o t h e f u l l v e l o c i t y d i s t r i b u t i o n . APPENDIX V II MAGNETIC FIELD OF A SET OF OPPOSING SOLENOIDS We w i l l d e r iv e a p p ro x im a te e x p re s s io n s f o r th e m ag n e tic f i e l d s g e n e r a te d b y th e s e t o f two o p p o s i t e ly wound s o le n o id s diagrammed i n F ig u r e 4 $ . As shown i n F ig u re 27 , t h e m easu red a d i a b a t i c i t y p a ram e te r f o r t h i s f l o p p e r i s v e r y c lo s e t o t h e t h e o r e t i c a l a d i a b a t i c i t y p a r a ­ m e te r f o r t h e m odel o r i d e a l i z e d f i e l d g iv e n i n F ig u re 6 . We w i l l t r y h e r e t o c o n v in c e you t h a t t h i s was n o t an a c c id e n t . Due t o t h e c y l i n d r i c a l symmetry o f t h e f l o p p e r , we w i l l u s e th e c y l i n d r i c a l c o o r d i n a t e s , z , r , 0. and assume t h a t t h e r e i s no 0 • component o r d ependence o f th e m ag n e tic f i e l d . As any s tu d e n t o f p h y s ic s know s, t h e m ag n e tic f i e l d on t h e a x is o f a s o le n o id i s p u re a x i a l (Hz ) and i s e a s i l y c a l c u l a t e d . The f i e l d on a x is i s s im p ly a sum o f f i e l d s due t o N lo o p s o f w i r e . We th e n add on a n o th e r s o le n o id wound i n t h e o p p o s id e d i r e c t i o n and we have t o a f a i r a p p ro x i ­ m a t io n , t h e a x i a l f i e l d due t o two o p p o s i t e ly wound s o le n o id s . ■ The f i e l d due t o one lo o p o f w ir e i s Hz( z) = I a 3/2 ( + a 2) k , w here ' z i s t h e a x i a l d i s t a n c e from th e c e n te r o f t h e lo o p and a i s t h e r a d iu s o f t h e lo o p . To c a l c u l a t e th e f i e l d due t o a s o le n o id , we w r i t e t h e c u r r e n t a s d l = N I "dx/L w here N i s t h e number o f lo o p s and L th e l e n g th o f t h e s o le n o id . We now have f o r t h e a x i a l f i e l d (99) . +L . H ( z) = W I r T a 2dx " 2 1 F ( T z )= * . ^ / a -1 99 - T h is y i e l d s . N I 2 L _____z ~ L [ ( z -L )3 + a 3 ] e + z + L [(z+L ) 3^ a 2F k . (1 00 ) 2z ( z 3+ a2) 8 k The a x i a l f i e l d i s shown i n F ig u re 47 . I n t h e c e n te r o f t h e f l o p p e r , t h e a x i a l f i e l d i s a lm o s t l i n e a r a s i t p a s s e s t h r o u g h 'z e r o . E xpand ing E q u a t io n (1 00 ) f o r z < a c .L , we f i n d Hz( z ) ss (N I/L ) z / a . (101) The f i e l d i s l i n e a r n e a r t h e c e n t e r o f t h e f l o p p e r . F o r L > z > a f ^ ( z ) = ( % I / L ) = H p , ( 1 0 2 ) and th e f i e l d i s a p p ro x im a te ly c o n s ta n t th ro u g h o u t t h e r e s t o f th e f l o p p e r . E q u a t io n s (101 ) and (102 ) a r e e x a c t ly t h e a x i a l f i e l d cho sen i n F ig u r e 6 w i th a b e in g z . F o r t h e r a d i a l f i e l d , we a g a in invok e th e d i f f e r e n t i a l fo rm o f B io t and S a v a r t ' s law V • H = O . Th is s t r a i g h t f o rw a r d l y y i e l d s H ( z , r ) = -Eq p r f o r z < a < L and # ( z , r ) = 0 f o r L > z > a , w here p = r / 2 a . We have ig n o re d th e end s o f t h e f lo p p e r s in c e t h e c o l l im a t i n g f i e l d o f t h e e -g u n and th e H e lm lio ltz f i e l d o v e r l a p th e two ends r e s p e c t i v e l y and a s was hoped , t h e s e end f i e l d s w e re a d i a b a t i c so we may ig n o re th em . The e f f e c t o f t h e f i n i t e s i z e o f t h e c o i l w in d in g s and th e sm a l l s p a t i a l s e p a r a t i o n o f t h e two s o le n o id s c a n b e c a l c u l a t e d r a t h e r F ig u re 46 : F lo p p e r c r o s s - s e c t i o n . FIG.46 201 —202— s t r a i g h t f o rw a r d l y . The r e s u l t s a r e v e ry complex e x p re s s io n s w hich r e d u c e t o t h e a p p ro x im a te f i e l d s g iv e n above to f i r s t o rd e r '. The m a jo r e f f e c t o f t h e s e two c o r r e c t i o n s i s t o a l t e r th e e x p r e s s io n f o r e E s l i g h t l y , y i e l d i n g an e E w i th in one p e r c e n t o f t h e m easu red Hq . T h e r e f o r e , we w i l l n o t go i n t o t h e d e t a i l s o f su ch a c a l c u l a t i o n . The f i r s t ' g l a r i n g d e f e c t o f t h i s c a l c u l a t i o n i s t h e a ssum p tio n t h a t o f f a x i s t h e a x i a l component o f t h e m agn e tic f i e l d i s in d ep e n d e n t o f r . The e x a c t f i e l d s f o r a i n f i n i t e l y sm a l l t h i c k n e s s o f c o i l w in d in g s h a s b e en done and up t o t h e maximum d i s t a n c e t h e beam can b e o f f a x i s ( -g a ) , t h i s a p p ro x im a tio n i s s u r p r i s i n g l y good (~ 2$,). T h e r e f o r e , t h e v e ry s im p le d e r i v a t i o n done h e re i s v e ry c lo s e t o th e a c t u a l f i e l d c o n f i g u r a t io n o f t h e f l o p p e r . 'This e x p la in s t h e c lo s e f i t b e tw een th e t h e o r e t i c a l and e x p e r im e n ta l a d i a b a t i c i t y p a r a m e te r . F ig u re 1+7: F lo p p e r f i e l d c o n f i g u r a t i o n . D ( i n c h e s ) FIG.4 7 APPENDIX V I I I STATISTICAL ANALYSIS We w i l l compute t o f i r s t o r d e r , th e u n c e r t a i n ty i n t h e c a l c u l a t e d p s t a t e p o p u la t io n from th e e x p e r im e n ta l d a t a . We w i l l u s e t h e ' f o l lo w in g n o t a t i o n P = Low v o l t a g e quench s i g n a l cv = h ig h v o l t a g e quench s i g n a l Po = a c t u a l number o f p ^ 's fdE = a c t u a l number o f cv's . T = T o t a l m e ta s ta b le s i g n a l We may now d e f in e t h e ' t o t a l number o f p ^ 's and t o t a l m e ta s ta b le i n t e n s i t y as ' t = PZfp ' T = PA + fp T| a. The IB s t a t e p o p u la t io n i s th e n s im p ly = 9 o / (P o .+ *o ) = P o /? ' D e f in in g f = f J f n as was done i n C h a p te r I I I , and d e f i n in g + cvTf P Pc = P /T , Y = ( I “ 1 / y ^ ^e have r 1B VA = ( f + Pc ) / ( l + f +Pc Y) • S ince Y <: 0 , th e n I =2 Er i s a lways g re a te r th a n f + Pc . We now in c lu d e u n c e r ta in t ie s i n f + p and ^ [ ^ o r in s ta n c e ; we have f o r f , f = f + ( l ± B f+/ f + ) . Ig n o r in g a l l c ro ss te rm s c o n ta in in g th e p ro d u c ts o f th e u n c e r ta in t ie s 3f + , dpc and BY; we have to f i r s t o r d e r —206 '-' i ^ r = ^ w here Bf+ + + 3P, + ( B/+3Y + P n ^ + ^ P r ) / ( ^ + ^+Pr^+ + v ^ C Now, s in c e f + , p , y, .Bf h, Bg , and' BY a r e a l l l e s s th a n o n e , • t h e l a s t te rm i n t h e e q u a t io n f o r a i s much l e s s th a n t h e f i r s t two t e rm s . H ence , t h e u n c e r t a i n ! t y i n t h e (3^ s t a t e p o p u la t io n i s t o f i r s t o r d e r j u s t 3 f+/ f + + BPc / P c * The v a lu e s f o r f + > p and t h e i r u n c e r t a i n ! t i e s a r e r e a s o n a b ly c o n s t a n t . T ha t i s , t h e v a lu e f o r f and ^ f+ a r e p r e d e te rm in e d b e f o r e t h e f l o p p e r c u rv e i s t a k e n and a r e a t w o rs t f - 0 .8 and Bf = 0 .0 0 5 . The v a lu e f o r Pc and BPc a r e , a t t h e i r w o r s t , ■ R = o .4 and BP = 0 .0 0 5 . We t h e r e f o r e , have a lm o s t in d ep e n d e n t Hc T o f t h e a c t u a l d a t a , a ± 0 . 0 2 . FOOTNOTES 1 K e l lo g g , R aM and Z a c h a r i a s , P h y s . R e v . , , p 4 7 2 , (1936 ) . 2 T o r re y and R abi , H ays. R e v . , 5 1 , p 379A, (1 9 3 7 )• 3 M illm an and Z a c h a r i a s , ■ P h y s . .,R ev . , 5 1 , p 380A, ( I 937 ) . 4 E . M a jo ra n a , Ruova C im ., 9 , p 4 3 ; (1 9 3 2 ) . 5 P . G u t t i n g e r , Z e i t s . F . P hysi k . , 7 3 , P 169 , ( I 9 3 I ) . 6 I . I . R a b i , P h y s . R ev . , 4 9 , p 324 (1 9 3 6 ) . 7 J . S c hw in g e r, P h y s . R e v . , 5 1 ; P 6 4 8 , (1 9 3 7 )• 8 L. Motz and M. E . R o se , P h y s . R e v . , 5 0 , P 3 48 , ( I 936 ) . 9 T. E . P h ip p s and 0 . S t e r n , Z e i t s . F . Physi k , 8 0 , p 610 , (1933) 10 R. F r i s c h and E. S e g rd , Z e i t s . F . . P h y s ik , SO, p 6 l 0 , (1 9 3 3 )- 11 N. F . Ram sey, "M o le cu la r B eam s," O xfo rd 1 s t e e l., p 4 o l , ( I 963 ) 12 M a rto n , M ethods o f E x p e r im e n ta l P h y s ic s , V o l . 4 , p a r t B, p 250, Academ ic -P re s s , ( 1967) - 13 T. R eb an e , Fn .D , T h e s is , U n iv e r s i t y o f M ich ig an . 14 C . Z e n e r , "N o n -a d ia b a t ic C ro s s in g o f E nergy L e v e ls , " P ro c . R oy . S o c . , A , 137 , P 696, (1 9 3 2 ) . 15 J . H e in r i c h s , P h y s . R e v . , 176 , No. I , p l 4 l , (5 Dec. I968 ) . 16 R . T. R o b is c o e , A. J . P . , V o l. 3 9 /2 , P 146 , (F e b . 1971)« 17 W. E. Lamb and R. C. R e th e r f o r d , P h y s . R ev . , 8 1 , . N o ,2 , p 227 , (15 J a n . I96 I ) . 18 C . Z e n e r , P r o c . R oy . S o c . , A, 137 , P 696 , ( 1 9 3 2 ) . 19 L. L andau , P h y s . Z. S o w je t . , 2 , p 4 6 , (1 9 3 2 ) . 20 F ig u re 3 o f W. L ic h te n and S . S c h u l t z , P h y s . R e v . , 116 , P 1132, (1 959 )- ~~ - 208~ 21 N. E osen and C . Z e n e r , P h y s . R e v . , 4 o , p $ 0 2 , ( 1932) . 22 G ra d sh te y n and R y zh ik , T ab le o f I n t e g r a l s , S e r i e s and P r o d u c ts , Academ ic P r e s s , p lo64-106>6>7 ( I 9S5 ) . 23 D icke and W i t tk e , I n t r o d u c t i o n t o Quantum M ech an ic s , p I9 2 . 24 J .B . D e lo s and W.R. Thors o n , P h y s . R ev . AS, p 7'28, (197 2 ) . 2? C .F . .Lebeda and W. R . Thors o n , C an ad ian J o u r n a l o f P h y s ic s , 4 8 , 24 , p 2937 , ( 1970) . 26 S c h i f f , "Quantum M ech an ic s , " M cG raw -H ill, 3 rd E d . , C h ap te r 9 , p 298 , ( 1968) . 27 M. T inkham , "Group Theo ry and Quantum M ech an ic s ," McGraw- H i l l , C h a p te r 5 , p 111 , (1 9 6 4 ) . 28 R . T. R o b is c o e , "O b s e rv a t io n s o f l e v e l C ro s s in g i n H, n = 2 , ” P h y s . R e v . , 138 , Ho. 1A, p A22-A34, (5 A p r i l 1965) . 29 .The th e rm a l e n e rg ie s a t 2800°K (~ .2$ eV) a r e i n s u f f i c i e n t t o e x c i t e t h e h y d ro g en atom t o i t s f i r s t e x c i t e d s t a t e ( 1 0 .2 eV ). 30 R i e f , "F undam en ta ls o f S t a t i s t i c a l and Therm al P h y s i c s , " M cG raw -H ill, ( 1965) . 31 Due t o t h e low co n d u c tan c e o f t h e c o n n e c t in g tu b in g from th e oven t o t h e m anom eter, t h e o i l m anometer r e a d s h ig h e r th a n t h e a c t u a l p r e s s u r e i n th e o v en . The o i l m anom eter i s only- p r o p o r t i o n a l t o t h e oven p r e s s u r e and n o t an a b s o lu t e m ea su re . 32 K. F . B o n h o e f fe r , E rg e b , d . E x a k t . H a tu rv r ie s . , 6 , 201 , ( I 927 ) . 33 W ooley , S c o t t and B riclcvredde, J . R e s e a rc h H a t . B u r . S t a n d . , 4 l , P 3 7 9 , ( 1 9 4 8 ) . 34 The o t h e r e x c i t e d s t a t e s i n h y d ro g en a r e n o t m e ta s ta b le and d e c ay b e f o r e r e a c h in g t h e d e t e c t o r . 35 R . D. H ig h t and R . T. R o b is c o e , "V e lo c i ty D i s t r i b u t i o n and T em pera tu re D ependence o f a Beam o f E x c i te d A tom s," 1 P ro c e e d in g s o f t h e M ontana Academy o f S c ie n c e , p 101 , A p r i l , 1975 . 36 B. L. M o is e iw its c h and S . J . Sm ith , " E le c tr o n Im pac t E x c i t a t i o n o f A tom s," R ev . Mod. F h y s . , 4 o , H o .2 , p 238 -353 , ( A p r i l , I 968 ) . 37 W. E. Lamb, J r . and R. C. R e t h e r f o r d , " t h e Hydrogen Atom, P a r t I I , " P h y s . Rev, (15 J a n 1 9 5 1 ) . F in e S t r u c t u r e o f , S i , N o .2 , p 227 , 38 T. ¥ . S hyn , "A M easurem ent o f th e 2 8 ^ ^ - 2P^yg en e rg y S e p a r a t io n , ( AE - S ) , i n Hydrogen (n = 2 ) ," T h e s is a t th e U n iv e r s i ty o f Mic h i g a n , . p 76 . i 39 R. T. R o h is c o e , " R e c o n c i l i a t i o n o f E x p e r im e n ta l Lamb S h i f t s , P h y s . R e v . , 1 6 8 , N o . I , p U - I l , (5 A p r i l 1968 ) . Uo R. T. R o b is c o e , " C arg ese L e c tu r e s i n P h y s i c s , " Lamb S h i f t E x p e r im e n ts , p 33 . - 2 1 0 - BIB LIOGRAPHY 1 . K. F . B o n h o e f fe r ; E rg e b . d . E x a k t . Natur\»ries . ; 6 , 201 , (1927) • 2 . J . B. D elo s and W. R. B io r s o h , P h y s . R ev . A6 , 7 28 , ( 1972) . 3 . D icke and W i t tk e , I n t r o d u c t i o n t o Quantum. M ech an ic s , p I 9 2 . 4 . F . F r i s c h and E. S e g r e ' , Z e l t s . F . Ehysi k , 8 0 , 610 , (1 9 3 3 )• 5 . G ra d sh te y n and R y zh ik , T ab le o f I n t e g r a l s , S e r i e s and P r o d u c ts , A cadem ic p r e s s , p 1 064 -1066 , ( 1965) , 6 . P . G u t t i n g e r , Z e i t s . F . F a y s Ik , 7 3 , 169 , ( I 93 I ) . 7 . J . H e in r i c h s , P h y s . R e v . , 176 , No. I , l 4 l , (5 Dec. I 968 ) . 8 . R . D. E i g h t , and R . T. R ob isco e ,, " V e lo c i ty D i s t r i b u t i o n and T em pera tu re D ependence o f a Beam o f E x c i te d A tom s," P ro c e ed in g s o f t h e M ontana Academy o f S c ie n c e s , V o l . 3 4 , p H O . 9 . K e llow , R ab i and Z a c h a r i a s , P h y s . R e v . , 5 0 ,. 4 72 , ( I 936 ) . 1 0 . W. E . Lamb and R . C . R e th e r f o r d , P h y s . R ev . , 8 1 , No. 2 , 227, (15 J a n . I 95 I ) . 1 1 . W. E. Lamb, J r . , and R . ' C . R e t h e r f o r e , "F in e S t r u c t u r e o f th e Hydrogen Atom. P a r t I I , " P h y s . R ev . , 8 1 , No. 2 , 2 27 , (15 J a n . I 95 I ) • 1 2 . L. L andau , P h y s . Z. 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