UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

The fermi surface of copper by positron annihilation Petijevich, Peter 1966

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1967_A6_7 P58.pdf [ 3.59MB ]
Metadata
JSON: 831-1.0103712.json
JSON-LD: 831-1.0103712-ld.json
RDF/XML (Pretty): 831-1.0103712-rdf.xml
RDF/JSON: 831-1.0103712-rdf.json
Turtle: 831-1.0103712-turtle.txt
N-Triples: 831-1.0103712-rdf-ntriples.txt
Original Record: 831-1.0103712-source.json
Full Text
831-1.0103712-fulltext.txt
Citation
831-1.0103712.ris

Full Text

THE FERMI SURFACE OF COPPER BY POSITRON ANNIHILATION by PETER PETI JEVICH B . A . S c . , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Depar tment o f PHYSICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA Sep tember , 1966 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I furth e r agree that per-mission for extensive copying of t h i s t h e s i s for s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives,, I t i s understood that copying or p u b l i -c a t i o n of t h i s t h e s i s for f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of FH/SICS  The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. D a t e -Sep/ I91G ABSTRACT A study o f th© Form I surface o f copper at room tcmperswre has been made by moans o f e p o s i t r o n a n n i h i l a t i o n technique. A p o s i t r o n a c t i v a copper s i n g l e c r y s t a l was placed midway between two ••point" s c i n t i l l a t i o n countsrs operated In tie;© coincidence. Th® co-incidence count rat e was measured f o r various c r y s t a l o r i e n t a t i o n s and the count r a t e Interpreted as a measure o f tha dlemeter o f tho Fermi surface. Tho experiment y i e l d s a Forrol s u r f s c o that Is Spherics! In k-ipaco except f o r p r o t r u s i o n s In tho { 1 1 1 ^ d i r e c t i o n s which aro estimated to subtend an angle of about 20° at £ w 0. Wit h i n exportirtental e r r o r tho r e s u l t s are c o n s i s t e n t w i t h those obtained by othor methods near 0° K. ACKNOWLEDGEMENTS The a u t h o r w i s h e s t o e x p r e s s h i s g r a t i t u d e t o Dr . G. Jones f o r h i s h e l p f u l a d v i c e and k i n d s u p e r v i s i o n t h r o u g h o u t the d u r a t i o n o f t h i s wo rk . Thanks a r e a l s o due t o D r . D. L l . W i l l i a m s f o r h i s many h e l p f u l d i s c u s s i o n s and s u g g e s t i o n s . The a u t h o r w i s h e s t o t h a n k D r . E. T e g h t s o o n i a n and Mr . N. R. R i s e b r o u g h o f the Depar tment o f M e t a l l u r g y f o r t h e i r a s s i s t a n c e w i t h the x - r a y d i f f r a c t i o n s t u d y o f the c r y s t a l used in the p r e s e n t wo rk . Thanks a r e a l s o e x t e n d e d t o P r o f e s s o r S. H, de Jong o f the D e p a r t -ment o f C i v i l E n g i n e e r i n g f o r mak ing a v a i l a b l e the e n g i n e e r ' s t r a n s i t used in t h i s w o r k . The a u t h o r a l s o w i s h e s t o t hank Mr . A . F r a s e r f o r h i s a s s i s t a n c e w i t h t h e d e s i g n o f the c r y s t a l r o t a t i o n a s s e m b l y . TABLE OF CONTENTS Page Chapter I INTRODUCTION 1 Chapter II MOTION OF ELECTRONS IN METALS 7 A. I n t r o d u c t i o n 7 B. The O n e - E l e c t r o n A p p r o x i m a t i o n 8 C. The F r e e - E l e c t r o n Model 10 D. The C r y s t a l L a t t i c e 12 E. The R e c i p r o c a l L a t t i c e 17 F. M o t i o n o f an E l e c t r o n i n a C r y s t a l L a t t i c e 18 1. P e r t u r b a t i o n Theory f o r Weak P e r i o d i c P o t e n t i a l s 20 2 . E f f e c t o f E l e c t r o n C o r r e l a t i o n s on the Fermi S u r f a c e 22 3. J u s t i f i c a t i o n o f the O n e - E l e c t r o n A p p r o x i m a t i o n 22 k. Energy Band C a l c u l a t i o n s by use o f t h e O n e - E l e c t r o n Model 23 5. The Fermi S u r f a c e o f Copper: Theory and Experiment 24 Ch a p t e r I I I ANNIHILATION OF POSITRONS 27 A. I n t r o d u c t i o n 27 B. Free A n n i h i l a t i o n o f P o s i t r o n s 27 C. P o s i t r o n A n n i h i l a t i o n from a Bound S t a t e 29 1. Bound E l e c t r o n - P o s i t r o n Systems 29 2 . P o s i t r o n i u m 30 3 . P o s i t r o n i u m A n n i h i l a t i o n 30 D. A n g u l a r C o r r e l a t i o n o f Two-Photon A n n i h i l a t i o n o f P o s i t r o n s 3 i Page E. A n g u l a r C o r r e l a t i o n G e o m e t r i e s 33 1. I n t r o d u c t i o n 33 2 ( a ) . Wide S l i t Geometry 33 2 ( b ) . D e t e r m i n a t i o n o f Fermi S u r f a c e s by Wide S l i t Geometry 35 3 ( a ) . P o i n t Geometry 37 3 ( b ) . D e t e r m i n a t i o n o f Fermi S u r f a c e s by use o f P o i n t Geometry 38 h. Col l i n e a r P o i n t Geometry kO F. A n n i h i l a t i o n o f P o s i t r o n s i n E l e c t r o n Gases ^0 1. E f f e c t o f E l e c t r o n - E l e c t r o n I n t e r a c t i o n s kO 2. E f f e c t o f E l e c t r o n - P o s i t r o n I n t e r a c t i o n s k] 3. E f f e c t o f E l e c t r o n - E l e c t r o n and E l e c t r o n - P o s i t r o n I n t e r a c t i o n s k\ h. Compar i son w i t h E x p e r i m e n t k2 G. A n n i h i l a t i o n o f P o s i t r o n s in Real M e t a l s kk 1. I n t r o d u c t i o n kk 2 . E f f e c t o f the P e r i o d i c C r y s t a l L a t t i c e P o t e n t i a l kk 3. E f f e c t o f Core A n n i h i l a t i o n h$ H. L i f e t i m e s o f P o s i t r o n s in M e t a l s k$ C h a p t e r IV EXPERIMENTAL ARRANGEMENT hi A . M e t a l C r y s t a l and P o s i t r o n Source kS B. O r i e n t a t i o n o f C r y s t a l 50 C. S p a t i a l S t a b i l i t y o f Notch and P i n Assemb l y 52 D. C r y s t a l H o l d e r 52 E. H o l d e r Suppor t 53 F. Gamma C o u n t e r s 55 G. E l e c t r o n i c s 57 H. S t a b i l i t y o f E l e c t r o n i c s 57 Page C h a p t e r V RESULTS AND CONCLUSIONS 61 A . I n t r o d u c t i o n 61 B. E x p e r i m e n t 62 C. R e s u l t s 63 D. I n t e r p r e t a t i o n o f t h e R e s u l t s 65 1. E f f e c t s o f F i n i t e Source and D e t e c t o r S i z e 65 2. R e s o l u t i o n F u n c t i o n f o r F i n i t e D e t e c t o r s and P o i n t C r y s t a l 66 3. R e s o l u t i o n F u n c t i o n f o r F i n i t e D e t e c t o r s and F i n i t e C r y s t a l 67 E. I n t e r p r e t a t i o n o f t h e R e s u l t s 69 F. A c c u r a c y A t t a i n a b l e w i t h the Method 70 1. R e s o l u t i o n and C o u n t i n g Ra te 70 2. S t a b i l i t y 71 G. D i s c u s s i o n 72 H. C o n c l u s i o n s 73 APPENDICES . A . . SOLUTION OF ABEL'S INTEGRAL EQUATION 75 B. EXPECTED ANGULAR CORRELATION CURVE WIDTH 77 C. EFFECT OF CORE ANNIHILATION 80 B i b l i o g r a p h y 8 l v i i i Table LIST OF TABLES Page Comparison of Theoretical and Experimental Fermi Surface Dimensions 26 L IST OF FIGURES F i gure Page 1. B o d y - c e n t e r e d C u b i c S t r u c t u r e ]h 2. F a c e - c e n t e r e d C u b i c S t r u c t u r e 15 3. Hexagona l C l o s e - p a c k e d S t r u c t u r e 16 h. B r i l l o u i n Zone f o r F . C . C . L a t t i c e 19 5. Copper Fermi S u r f a c e D e t a i l s 25 6. Two-Photon A n n i h i l a t i o n 32 7. W i d e - s l i t Geometry 32 8. W i d e - s l i t Geometry R e s u l t s 35 9. R e g i o n Sampled by W i d e - s l i t Geometry 35 10. P o i n t Geometry 38 11. Reg ion Sampled by P o i n t Geometry 38 12. Col l i n e a r P o i n t Geometry 38 13. A n n i h i l a t i o n i n E l e c t r o n Gases hi 14. P o s i t r o n A n n i h i l a t i o n Ra tes h3 15. E x p e r i m e n t a l A r rangement 4 8 16. Notch and P i n A r rangement 51 17. C r y s t a l H o l d e r Sh 18. Remote C o n t r o l 5h 19. P r e a m p l i f i e r and Shaper C i r c u i t 58 20. C o i n c i d e n c e C i r c u i t 59 21. E x p e r i m e n t a l R e s u l t s Sh 22. Fermi S u r f a c e Neck D e t a i l s 65 CHAPTER I INTRODUCTION One o f the fundamenta l p rob l ems in the t h e o r y o f t he s o l i d s t a t e i s t o f i n d a s o l u t i o n o f the S c h r o d i n g e r e q u a t i o n f o r the many-body sys tem d e -f i n e d by a c r y s t a l l i n e s o l i d . S i n c e a d i r e c t s o l u t i o n o f t h i s p r o b l e m i s no t p o s s i b l e c o n s i d e r a b l e e f f o r t has been d i r e c t e d toward the p r o d u c t i o n o f mode l s w h i c h a p p r o x i m a t e the many-body s y s t e m . By the use o f t h e s e mode ls i t i s t hen p o s s i b l e to make p r e d i c t i o n s about the p r o p e r t i e s o f t he c r y s t a l l i n e s ys tem t h a t can be compared w i t h t h o s e d e r i v e d f rom e x p e r i m e n t . It i s thus d e s i r a b l e t o be a b l e t o make r e l i a b l e measurements o f t h e s e p r o p e r t i e s so t h a t t hey can be compared w i t h the p r e d i c t i o n s o f t h e v a r i o u s t h e o r e t i c a l mode l s and a l s o because the d a t a may be used t o s u g g e s t newer and b e t t e r m o d e l s . A s p e c i a l c a s e o f t h i s p r o b l e m t h a t i s p a r t i c u l a r l y i n t e r e s t i n g o c c u r s when the c r y s t a l l i n e s o l i d i s a m e t a l . Work on t h i s a s p e c t o f t he p r o b l e m can be c o n s i d e r e d t o have s t a r t e d w i t h the f r e e - e l e c t r o n a p p r o x i m a t i o n 2 o f S o m m e r f e l d (1928) i n w h i c h i t wa s a s s u m e d t h a t t h e e l e c t r o n s o f t h e s o l i d do n o t i n t e r a c t w i t h e a c h o t h e r o r w i t h t h e i o n c o r e s o f t h e c r y s t a l l a t t i c e . D e s p i t e i t s o v e r s i m p l i f i c a t i o n o f t h e a c t u a l p h y s i c a l s i t u a t i o n , t h e t h e o r y u s u a l l y g i v e s g ood q u a l i t a t i v e r e s u l t s . H o w e v e r , i n o r d e r t h a t s a t i s f a c t o r y q u a n t i t a t i v e r e s u l t s be o b t a i n e d , i t i s u s u a l l y n e c e s s a r y t o i n c l u d e t h e e f f e c t o f t h e i o n c o r e s . The e x t e n s i v e t h e o r e t i c a l w o r k on t h i s a s p e c t o f t h e p r o b l e m i s e s s e n t i a l l y a n e x t e n s i o n o f t h e s t u d i e s o f H a r t r e e (1928), B l o c h (1928), a n d F o c k (1930). I n c l u s i o n o f t h e e f f e c t o f t h e i o n c o r e s g r e a t l y c o m p l i c a t e s t h e p r o b l e m s o t h a t a c c u r a t e c a l c u l a t i o n s a r e d i f f i c u l t t o p e r f o r m , b u t i t u s u a l l y l e a d s t o r e s u l t s t h a t a r e i n b e t t e r a c c o r d w i t h t h e l a r g e b o d y o f a v a i l a b l e e x p e r i m e n t a l d a t a t h a n d o e s t h e s i m p l e S o m m e r f e l d m o d e l . H o w e v e r , i t i s r e a s o n a b l e t o a s s u m e t h a t a g r e e m e n t w i t h e x p e r i m e n t w o u l d be f u r t h e r i m p r o v e d i f i t w e r e p o s s i b l e t o i n c l u d e t h e e f f e c t o f e l e c -t r o n - e l e c t r o n i n t e r a c t i o n s . C o n s i d e r a b l e p r o g r e s s on t h i s a s p e c t o f t h e p r o b -l e m h a s r e s u l t e d f r o m t h e e a r l y s t u d i e s o f Bohm a n d P i n e s ( P i n e s , 1955; P i n e s , 1963). M o s t c a l c u l a t i o n s o f e n e r g y b a n d s i n m e t a l s a t p r e s e n t a r e s t i l l b a s e d o n t h e i n d e p e n d e n t p a r t i c l e m o d e l ( C h a p t e r I I ) , h o w e v e r , b e c a u s e o f t h e d i f f i -c u l t y o f a d e q u a t e l y i n c l u d i n g t h e e f f e c t s o f e l e c t r o n - e l e c t r o n i n t e r a c t i o n ( R e i t z , 1955; C a l l a w a y , 1958). S i n c e t h e w o r k o f W i g n e r a n d S e i t z (1933) o n s o d i u m , w e l l o v e r a h u n d r e d b a n d c a l c u l a t i o n s f o r m e t a l s h a v e b e e n p e r f o r m e d . In r e c e n t y e a r s t h e a d v e n t o f h i g h - s p e e d c o m p u t i n g m a c h i n e s h a s made p o s s i b l e a n i n c r e a s i n g n u m b e r o f e x t e n s i v e c a l c u l a t i o n s t h a t a g r e e w e l l w i t h o n e a n o t h e r a n d w i t h e x p e r i m e n t . H o w e v e r , w i t h a f e w e x c e p t i o n s , g o o d a c c o r d w i t h e x p e r i m e n t i s l i m i t e d t o m e t a l s o f t h e a l k a l i g r o u p o r t h e a l k a l i n e e a r t h g r o u p ( C a l l a w a y , 1958). 3 An e x c e p t i o n t h a t i s o f p a r t i c u l a r i n t e r e s t i s c o p p e r . Here t he agreement o f r e c e n t e x t e n s i v e band c a l c u l a t i o n s w i t h e x p e r i m e n t i s good (Sega l 1, 1962; B u r d i c k , 1963). The e x p e r i m e n t a l s i t u a t i o n i s a l s o r a t h e r u n i q u e s i n c e t h e e l e c t r o n i c p r o p e r t i e s o f coppe r have been more t h o r o u g h l y i n v e s t i g a t e d than t h o s e o f any o t h e r me ta l (Sega 11, 1962). For e x a m p l e , t he Fermi s u r f a c e w h i c h d e s c r i b e s the h i g h e s t o c c u p i e d e l e c t r o n e n e r g y l e v e l a t a b s o l u t e z e r o , has been s t u d i e d by a t l e a s t f i v e ma jo r e x p e r i m e n t a l t e c h -n i q u e s ( H a r r i s o n and Webb, I 9 6 0 ) . These i n c l u d e the methods o f anomalous s k i n e f f e c t , m a g n e t o r e s i s t a n c e , m a g n e t o a c o u s t i c e f f e c t , c y c l o t r o n r e s o n a n c e , and the de Haas-van A l p h e n e f f e c t . A l l f i v e methods g i v e r e s u l t s t h a t a r e in a c c o r d w i t h t he model o f t he Fermi s u r f a c e f o r c o p p e r p r o p o s e d by P i p p a r d ( 1 9 5 7 ) . In t h i s model t he Fermi s u r f a c e c o n s i s t s o f a s p h e r i c a l c e n t r a l p a r t ( the " b e l l y " ) t o g e t h e r w i t h e i g h t p r o t r u s i o n s ( " n e c k s " ) a l o n g the f i l l } d i r e c t i o n s . As d i s c u s s e d in C h a p t e r I I , t h i s model f o r the Fermi s u r f a c e o f coppe r i s i n good a c c o r d w i t h t he s u r f a c e p r e d i c t e d by r e c e n t band c a l c u l a -t i o n s ( S e g a l l , 1962; B u r d i c k , 1963). The e x p e r i m e n t a l t e c h n i q u e s m e n t i o n e d above a r e a l l l i m i t e d t o use on f a i r l y p u r e me ta l spec imens a t v e r y low t e m p e r a t u r e s ( ~ 4 K) due to the r e q u i r e m e n t o f l o n g e l e c t r o n i c mean f r e e p a t h ( H a r r i s o n and Webb, I 9 6 0 ) . T h i s r e q u i r e m e n t i s r a t h e r r e s t r i c t i v e . For e x a m p l e , i t makes i m p o s s i b l e a s a t i s -f a c t o r y s t u d y o f a l l o y s . The r e q u i r e m e n t o f low t e m p e r a t u r e s c o m p l i c a t e s the s t u d y o f m e t a l s such as sod ium and p o t a s s i u m in w h i c h low t e m p e r a t u r e phase t r a n s i t i o n s o c c u r . The r e s t r i c t i o n t o low t e m p e r a t u r e s a l s o e l i m i n a t e s the p o s s i b i l i t y o f e x a m i n i n g a d e q u a t e l y t h e t e m p e r a t u r e dependence o f t he " s h a r p -n e s s " o f the Fermi s u r f a c e b o u n d a r y . F i n a l l y , i t i s t o be n o t e d t h a t in t h e s e methods one g e n e r a l l y examines o n l y the e l e c t r o n s t a t e s t h a t r e s i d e nea r t h e Fermi s u r f a c e . I t w o u l d be o f c o n s i d e r a b l e i n t e r e s t t o be a b l e t o p robe the k e n t i r e v a l e n c e band . A t e c h n i q u e t h a t does no t s u f f e r f rom the above l i m i t a t i o n s i n -v o l v e s a s t u d y o f the a n n i h i l a t i o n o f p o s i t r o n s in m e t a l s . In t h i s method the meta l t o be s t u d i e d i s bombarded w i t h p o s i t r o n s . Upon c o l l i s i o n w i t h an e l e c t r o n , t h e r e i s a chance t h a t the p a i r may a n n i h i l a t e each o t h e r . In n e a r l y e v e r y such a n n i h i l a t i o n two n e a r l y a n t i p a r a l l e l pho tons r e s u l t . The s m a l l d e v i a t i o n f rom s t r i c t a n t i p a r a 1 1 e l i s m ( ~1 0 r a d i a n s ) i s r e l a t e d t o the c e n t r e - o f - m a s s momentum o f t h e a n n i h i l a t i n g e l e c t r o n - p o s i t r o n p a i r by t h e s i m p l e r e l a t i o n p^ = mce where p^ i s t he component o f p a i r momentum p e r p e n d i c u l a r t o t h e n e a r l y a n t i p a r a l l e l pho ton p a i r , m the e l e c t r o n i c r e s t m a s s , c the speed o f l i g h t , and a t h e a n g u l a r d e v i a t i o n f rom 180°. The e n e r g y o f the pho tons e m a n a t i n g f rom the sample i s abou t 0.51 Mev ( a n n i h i -l a t i o n r a d i a t i o n ) so t h a t gamma-ray a t t e n t u a t i o n and s c a t t e r i n g by the sample w i l l be n e g l i g i b l e i f t h e sample i s r e a s o n a b l y s m a l l . S i n c e c a l c u l a t i o n s by L e e - W h i t i n g (1955) i n d i c a t e t h a t t h e p o s i t r o n i s e s s e n t i a l l y a t r e s t ( therma-1 i z e d ) b e f o r e i t a n n i h i l a t e s , t he c e n t r e - o f - m a s s momentum o f the p a i r i s e s s e n t i a l l y t h a t o f t he e l e c t r o n . The a n g u l a r c o r r e l a t i o n o f the gamma-ray p a i r s e m i t t e d in p o s i t r o n a n n i h i l a t i o n in s o l i d s has been o b s e r v e d by many w o r k e r s . The e a r l y work o f K l e m p e r e r (193*0 a n d t h a t o f Montgomery and B e r i n g e r (19^2) e s t a b l i s h e d t h e t ime c o i n c i d e n c e and nea r c o l l i n e a r i t y o f t he two gamma-rays e m i t t e d in p o s i t r o n a n n i h i l a t i o n . The f i r s t d e t a i l e d a n g u l a r d i s t r i b u t i o n was o b t a i n e d by De B e n e d e t t i e t a l . ( 1 9 5 0 ) . S i n c e then a n g u l a r c o r r e l a t i o n s t u d i e s o f many e l e m e n t s and compounds have been made. The s u b j e c t has been r e v i ewed by W a l l a c e ( i 9 6 0 ) . Most o f t he work on Fermi s u r f a c e s by t h e p o s i t r o n a n n i h i l a t i o n 5 t e c h n i q u e has used the " w i d e - s l i t " method ( C h a p t e r I I I ) . On t h e b a s i s o f t he f r e e - e l e c t r o n model i t i s e a s y t o show t h a t t h i s w i d e - s l i t method samples a s l i c e t h r o u g h the Fermi s p h e r e (Chap te r I I I ) . By the use o f t h i s t e c h n i q u e i t has been p o s s i b l e t o o b t a i n the momentum d i s t r i b u t i o n o f the e l e c t r o n s w i t h i n a me ta l a t t e m p e r a t u r e s r a n g i n g f rom K (Stump, 1955) t o beyond the m e l t i n g p o i n t ( G u s t a f s o n e t a l . , 1963)> p r o v i d i n g a r a t h e r d i r e c t v e r i -f i c a t i o n o f F e r m i - D i r a c s t a t i s t i c s . However , due t o f i n i t e r e s o l u t i o n and o t h e r c o m p l i c a t i n g f a c t o r s ( c o r e a n n i h i l a t i o n and o t h e r h i g h e r momentum e f f e c t s a r i s i n g f rom the p r e s e n c e o f the c r y s t a l l a t t i c e ) w h i c h a re d i s -c u s s e d in C h a p t e r I I I , t he method has no t been u s e f u l f o r y i e l d i n g q u a n t i -t a t i v e i n f o r m a t i o n about t h e d e t a i l e d shapes o f Fermi s u r f a c e s . A " p o i n t g e o m e t r y " method w h i c h samp les a c y l i n d r i c a l vo lume in momentum space has been used by a few w o r k e r s (Co lomb ino e t a l . , 1963; F u j i -w a r a , 1965). A l t h o u g h t h i s method p e r m i t s improved r e s o l u t i o n compared t o the w ide s l i t m e t h o d , the r e s u l t s a r e a g a i n o b s c u r e d by c o r e a n n i h i l a t i o n and o t h e r l a t t i c e e f f e c t s mak ing i t d i f f i c u l t t o use the method f o r an a c c u r a t e s t u d y o f Fermi s u r f a c e t o p o l o g y . The work o f t h i s t h e s i s d e s c r i b e s a deve lopment o f the " p o i n t geome t r y " t e c h n i q u e w h i c h o f f e r s a d v a n t a g e s f o r the s t u d y o f Fermi s u r f a c e s . W i t h t h i s method ( co l l i n e a r p o i n t geomet r y " ) i t i s p o s s i b l e t o examine the Fermi s u r f a c e more q u a n t i t a t i v e l y than appea r s p o s s i b l e by the " w i d e - s l i t " o r " p o i n t g e o m e t r y " methods because c o r e a n n i h i l a t i o n and h i g h e r momentum e f f e c t s a r i s i n g f rom the p r e s e n c e o f the c r y s t a l l a t t i c e p l a y a r e l a t i v e l y l e s s i m p o r t a n t r o l e in the new t e c h n i q u e . The p r i n c i p l e s o f t h i s new " c o l l i n e a r p o i n t geome t r y " t e c h n i q u e a r e d i s c u s s e d i n Chap t e r I I 1 o f t h e p r e s e n t w o r k . T h i s c h a p t e r i s f o l l o w e d 6 by C h a p t e r IV in w h i c h the e x p e r i m e n t a l a r r angement i s d i s c u s s e d in some d e t a i l . F i n a l l y , in C h a p t e r V some e x p e r i m e n t a l r e s u l t s o b t a i n e d f rom an a p p l i c a t i o n o f t h i s new method to a s t u d y o f the Fermi s u r f a c e o f coppe r a t room t e m p e r a t u r e a r e p r e s e n t e d . These r e s u l t s a r e used t o c o n s t r u c t a Fermi s u r f a c e f o r c o p p e r w h i c h i s t hen compared w i t h t h e coppe r Fermi s u r f a c e o b t a i n e d a t v e r y low t e m p e r a t u r e s by the more p r e c i s e c o n v e n t i o n a l t e c h -n i q u e s m e n t i o n e d above . The d i s c u s s i o n c l o s e s w i t h a s t a t e m e n t o f t h e c o n -c l u s i o n s t h a t have been drawn f rom t h e p r e s e n t s t u d y . 7 CHAPTER I I MOTION OF ELECTRONS IN METALS A . I n t r o d u c t i o n I f one c o n s i d e r s a c r y s t a l l i n e s o l i d a t f a i r l y low t e m p e r a t u r e s so t h a t t h e m o t i o n o f the r e l a t i v e l y m a s s i v e n u c l e i can be n e g l e c t e d , one can to a good a p p r o x i m a t i o n ( Z iman , I960) w r i t e the S c h r o d i n g e r e q u a t i o n f o r the s y s t em as N NZ MZ r- i*JfcV- r/ i fV i " i ^ ^ T F T F T 1 T - E 1 P Here i t has been assumed t h a t the l a t t i c e i s p e r f e c t and t h a t t h e c r y s t a l i s composed o f N atoms each w i t h z e l e c t r o n s . The p o s i t i o n v e c t o r o f e l e c t r o n A, i s d e n o t e d by and t h a t o f n u c l e u s j by f^ - . The r e m a i n i n g symbo ls have the u s u a l m e a n i n g ; -e i s t h e e l e c t r o n i c c h a r g e , m the e l e c t r o n i c r e s t mass , 27i"1i P l a n c k ' s c o n s t a n t and E the e n e r g y e i g e n v a l u e f o r t h e s y s t e m . The i n -t e r p r e t a t i o n o f the te rms in t h e above H a m i l t o n i a n i s w e l l - k n o w n ( Z i m a n , 1 9 6 0 ) . 8 The f i r s t t e rm r e p r e s e n t s the k i n e t i c ene rgy o f the e l e c t r o n s , t he second the p o t e n t i a l e n e r g y o f the e l e c t r o n s in t h e n u c l e a r Coulomb f i e l d , and t h e t h i r d t e rm t h e p o t e n t i a l ene rgy o f the e l e c t r o n - e l e c t r o n Coulomb i n t e r -a c t i o n . I f more than one t y p e o f atom i s p r e s e n t the H a m i l t o n i a n o f e q u a t i o n (2-1) i s e a s i l y g e n e r a l i z e d and l e a d s t o an e x p r e s s i o n t h a t i s o n l y s l i g h t l y more c o m p l i c a t e d . B. The O n e - E l e c t r o n A p p r o x i m a t i o n The above e q u a t i o n i s q u i t e gene r a l and i s in f a c t one o f the f u n -damenta l e q u a t i o n s i n t h e t h e o r y o f s o l i d s . However , s i n c e in a t y p i c a l s o l i d one has a v e r y l a r g e number o f i n t e r a c t i n g p a r t i c l e s i t i s q u i t e i m -p o s s i b l e t o s o l v e the S c h r o d i n g e r e q u a t i o n d i r e c t l y . N e v e r t h e l e s s the p r o b -lem can be made more o r l e s s t r a c t a b l e i f one t r e a t s the e l e c t r o n s as s t a -t i s t i c a l l y i ndependen t ( the o n e - e l e c t r o n a p p r o x i m a t i o n ) . The wave f u n c t i o n f o r the s y s t em t h e n becomes ( A n d e r s o n , 1963) ^(7, •••*,.) 5 Vfovjft (2-2) where t h e (^>p a r e o n e - e l e c t r o n f u n c t i o n s ( i n c l u d i n g s p i n ) . I f , in a d -d i t i o n to the o n e - e l e c t r o n a p p r o x i m a t i o n , t he e l e c t r o n - e l e c t r o n i n t e r a c t i o n te rm i s r e p l a c e d by i t s a v e r a g e v a l u e one o b t a i n s t h e H a r t r e e e q u a t i o n w i t h o n e - e l e c t r o n e i g e n v a l u e ; where i = l , 2 , . . . N z . In t h e s e e q u a t i o n s the a v e r a g e i n t e r a c t i o n te rm has a s i m p l e i n t e r p r e t a t i o n i f i t i s no t ed t h a t s i n c e the cha rge d e n s i t y o f e l e c t r o n j a t p o i n t ?i i s -e l^ -Oj-)! » t n e p o t e n t i a l e n e r g y a s s o c i a t e d w i t h e l e c t r o n i and volume e l emen t d£ i s J ^ 4 Then the q u a n t i t y can be i n t e r p r e t e d as the p o t e n t i a l ene rgy o f e l e c t r o n i in the c h a r g e c l o u d o f e l e c t r o n j ( Ra imes , 1 9 6 l ) . In o r d e r t o s o l v e the H a r t r e e e q u a t i o n s one assumes a s e t o f ^ } c a l c u l a t e s (i^ ) and t hence a new s e t o f yi w h i c h a r e in t u r n used in the H a r t r e e e q u a t i o n s and so o n , u n t i l a s e l f - c o n s i s t e n t r e s u l t i s o b t a i n e d . S i n c e e l e c t r o n s a r e f e r m i o n s the f u n c t i o n IP must be a n t i s y m m e t r i c i n the e l e c t r o n c o o r d i n a t e s . However , the p r o d u c t f u n c t i o n g i v e n by e q u a t i o n (2-2) does no t obey t h i s c o n d i t i o n . However, i t i s p o s s i b l e t o s a t i s f y the a n t i s y m m e t r y r e q u i r e m e n t by f o r m i n g a s u i t a b l e l i n e a r c o m b i n a t i o n o f p r o d u c t f u n c t i o n s ; t h e S l a t e r d e t e r m i n a n t , g i v e n b y ; xs*) yJi*»J w h i c h i s a n t i s y m m e t r i c as r e q u i r e d . I f f o r two e l e c t r o n s one has = ( s p a t i a l and s p i n c o o r d i n a t e s i d e n t i c a l ) t he d e t e r m i n a n t v a n i s h e s , in a c c o r d w i t h t h e P a u l i e x c l u s i o n p r i n c i p l e . U s i n g t h e d e t e r m i n a n t a 1 f u n c t i o n i t i s p o s s i b l e t o c o n s t r u c t a b e t t e r s e t o f o n e - e l e c t r o n e q u a t i o n s ( A n d e r s o n , 1 9 6 3 ) . These a r e the H a r t r e e -Fock e q u a t i o n s ; ^ ^ 10 where <r d e n o t e s the s p i n l a b e l . A l t h o u g h t h e s e e q u a t i o n s a r e known as the H a r t r e e - F o c k e q u a t i o n s , t hey a r e a c t u a l l y a s p e c i a l c a s e o f a more g e n e r a l method ( M e s s i a h , 1962) t h a t i s o f c o n s i d e r a b l e i m p o r t a n c e in a t o m i c ( H a r t r e e , 1 9 5 7 ) , n u c l e a r (Brown, 1 9 6 4 ) , and s o l i d s t a t e p r o b l e m s ( A n d e r s o n , 1 9 6 3 ) . However , in what f o l l o w s , t h e d i s c u s s i o n o f the method w i l l be l i m i t e d t o i t s use i n t h e t h e o r y o f m e t a l s . Due t o t h e i r mutua l Coulomb r e p u l s i o n , t h e e l e c t r o n s i n a s o l i d w i l l t end t o a v o i d each o t h e r (Coulomb c o r r e l a t i o n ) . Thus the e l e c t r o n s w i l l no t move i n d e p e n d e n t l y o f each o t h e r as i s assumed in t h e o n e - e l e c t r o n m o d e l . In t h e H a r t r e e and H a r t r e e - F o c k methods t h e Coulomb c o r r e l a t i o n o f t h e e l e c t r o n s i s no t t aken i n t o a c c o u n t . In the H a r t r e e - F o c k m e t h o d , h o w e v e r , t he P a u l i e x c l u s i o n p r i n c i p l e does i n t r o d u c e a c o r r e l a t i o n between e l e c t r o n s o f l i k e s p i n . In g e n e r a l , i n c l u s i o n o f t h e Coulomb c o r r e l a t i o n s and s p i n c o r r e l a t i o n s w i l l r educe the e n e r g y o f a m a n y - e l e c t r o n sys tem because the c o r r e l a t i o n s a r e e f f e c t i v e in mak ing the e l e c t r o n s spend l e s s o f t h e i r t i m e nea r each o t h e r . In a t r e a t m e n t o f m e t a l s by the o n e - e l e c t r o n method i t i s o f t e n found b e t t e r t o i g n o r e a l l c o r r e l a t i o n s than t o i n c l u d e o n l y some o f them (Ra imes , 1 9 6 l ) . Thus t h e H a r t r e e method f r e q u e n t l y y i e l d s b e t t e r r e -s u l t s than does the u n m o d i f i e d H a r t r e e - F o c k method . D e s p i t e i t s l i m i t a t i o n s , the H a r t r e e method has o f t e n been used in t h e t h e o r y o f m e t a l s ( R e i t z , 1 9 5 5 ) . In p a r t i c u l a r , i t can be used to d e r i v e t h e p r o p e r t i e s o f a f r e e - e l e c t r o n g a s . T h i s i s p a r t i c u l a r l y u s e f u l s i n c e a f r e e - e l e c t r o n gas can be r e g a r d e d as a c r u d e model o f a m e t a l . C. The F r e e - E l e c t r o n Model I f one r e p l a c e s the c h a r g e d i s t r i b u t i o n o f t h e ion c o r e s by a 11 uniform distribution of positive charge, and the charge distribution of the valence electrons by a uniform distribution of negative charge so that the net charge is zero, the Hartree equation simplifies to in, (2-4) For a cube of side L, containing N electrons it is seen that a solution of (2-4) is ^ (2-5) provided that 6 ( * ) = | £ (2-6) Application of the usual periodic boundary conditions (Ziman, I960; Raimes, 1961): ^(x+L.y.z) =^(x,y+L,z) = ,z+L) then yields t=27r(n] T+n2 j+n3 k) (2-7) where the nt- are positive or negative integers or zero and T, j*, £ are unit vectors along the cube edges. From (2-7) it is seen that the integers n,. representing "orbital" states (Raimes, 196l) define a lattice in k-space. Since each cube of side 2 7T will contain one such orbital state, the number of orbital states per L 3 unit volume of k-space is L Thus in a volume element dk* of T<-space there 3 , ^F*-are L .dk orbital states. Generalizing slightly, it is seen that for a metal 8 TT' of volume v there are 2vdk electron states (spin degeneracy included) in a volume element dk of k-space. It is evident from (2-6) that the surfaces of constant energy are spheres. Thus, as a consequence of Fermi-Dirac statistics, the occupied 12 r e g i o n o f k-space a t t h e a b s o l u t e z e r o o f t e m p e r a t u r e w i l l be a u n i f o r m l y dense s p h e r e . D e n o t i n g t h e r a d i u s o f t h i s s p h e r e ( t he Fermi sphe re ) by and o b s e r v i n g t h a t t h e r e a r e N e l e c t r o n s t a t e s w i t h i n the Fermi s p h e r e t h e f o l l o w i n g c o n d i t i o n i s o b t a i n e d ; (2v )4?rk^=N (8T ' )3 F o r , k p = and the Fermi ene rgy i s (37TN) ( V ) 2-*i 2-m I V I D. The C r y s t a l L a t t i c e In a c r y s t a l l i n e s o l i d t h e a t o m i c n u c l e i fo rm a p e r i o d i c a r r a y known as the c r y s t a l l a t t i c e . Due t o the p e r i o d i c i t y o f t h i s a r r a y i t i s p o s s i b l e t o g e n e r a t e the e n t i r e s e t o f l a t t i c e p o i n t s by use o f the c o n c e p t o f a u n i t c e l l . ( Z iman , 1964) I f the u n i t c e l l can be chosen to be a p a r a l -l e l e p i p e d c o n t a i n i n g one a t o m , the c r y s t a l l a t t i c e i s s a i d t o be a B r a v a i s l a t t i c e . On t h e o t h e r h a n d , i f a u n i t c e l l w i t h more t h a n one atom i s r e -q u i r e d t h e c r y s t a l l a t t i c e i s s a i d t o be a B r a v a i s l a t t i c e w i t h a b a s i s s i n c e t h e p o s i t i o n s o f t h e v a r i o u s atoms in the u n i t c e l l must be s p e c i f i e d . I t i s e a s i l y seen t h a t i f t he c r y s t a l l a t t i c e i s c o n s i d e r e d t o be made up o f u n i t c e l l s , each c o n t a i n i n g one atom ( B r a v a i s l a t t i c e ) , any l a t t i c e p o i n t can be r eached f rom any o t h e r by a t r a n s l a t i o n t h r o u g h a v e c t o r o f t he form T = \]a] + 1 2 ? 2 + 1 ^ (2-8) where a ^ , a ^ , and a r e v e c t o r s d e f i n e d by the edges o f the u n i t c e l l , and 13 l j , 1^ » and 1^ a r e i n t e g e r s . E q u a t i o n (2-8) i s a l s o v a l i d f o r a c r y s t a l l a t t i c e made up o f u n i t c e l l s c o n t a i n i n g more than one atom each ( B r a v a i s l a t t i c e w i t h a b a s i s ) . However, in t h i s c a s e the two atoms so l i n k e d must r e s i d e a t e q u i v a l e n t s i t e s w i t h i n t h e i r r e s p e c t i v e u n i t c e l l s . Most m e t a l s c r y s t a l l i z e in one o f t h r e e s t r u c t u r e s . These t h r e e a r e t he b o d y - c e n t e r e d c u b i c , the f a c e - c e n t e r e d c u b i c and t h e hexagonal c l o s e -packed s t r u c t u r e s , diagrams o f which a r e shown in F i g u r e s 1, 2, and 3» r e -s p e c t i v e l y . N e a r l y a l l o f the common m e t a l s have one o f t h e s e s t r u c t u r e s . For example, l i t h i u m , sodium, and p o t a s s i u m a r e b o d y - c e n t e r e d c u b i c whereas c o p p e r , s i l v e r and g o l d a r e f a c e - c e n t e r e d c u b i c . Examples o f t h e hexagonal c l o s e - p a c k e d s t r u c t u r e a r e b e r y l l i u m , magnesium and z i n c . Figure 1: Body-Centered.Cubic St r u c t u r e Figure 2: Face-Centered Cubic St r u c t u r e 16 k^^^ ; — j H • >i y / a i a, Figure 3: Hexagonal Close-Packed S t r u c t u r e 17 E. The R e c i p r o c a l L a t t i c e In t h e t h e o r y o f m e t a l s i t i s c o n v e n i e n t t o i n t r o d u c e a l a t t i c e known as the r e c i p r o c a l l a t t i c e . ( Z iman , 1964) T h i s l a t t i c e can be d e f i n e d by means o f the r e c i p r o c a l l a t t i c e v e c t o r G * = m 1 K ] + m 2b* 2 + mj&3 (2-9) where b*^  = 2K ^ x a ^ , b^ = 2 7 r x a^ , D^-2/ra*^ x a^ and m^  , m^, and a^- x a* l y a ^ x a" a y a*2 x a*^  may be p o s i t i v e o r n e g a t i v e i n t e g e r s o r z e r o . The u n i t c e l l , o r z o n e , in the r e c i p r o c a l l a t t i c e i s then o b t a i n e d ( K i t t e l , 1956) by f i n d i n g the s h o r t e s t n o n - z e r o r e c i p r o c a l l a t t i c e v e c t o r s s a t i s f y i n g the Bragg c o n d i t i o n (i< + CT = k 2 (2-10) where T< i s a g e n e r a l v e c t o r i n the r e c i p r o c a l s p a c e . From t h i s e q u a t i o n i t can be seen t h a t 2k-C i - - G so t h a t each zone boundary i s normal t o a r e c i p -G r o c a l l a t t i c e v e c t o r a t i t s m i d p o i n t . The c e l l t hus formed i s known as t h e f i r s t B r i l l o u i n zone o r the r educed z o n e . In p a r t i c u l a r , f o r a f a c e - c e n t e r e d c u b i c l a t t i c e one may t a k e the p r i m i t i v e t r a n s l a t i o n v e c t o r s t o be 3, = a(t + j) 2 a \ = a t f + ft) 2 3 = a(f* + ft) 5 2 where a i s t he l e n g t h o f a cube edge and T , j , and ft a r e u n i t v e c t o r s a l o n g 18 the cube e d g e s . From the above e q u a t i o n s one then e a s i l y o b t a i n s G = 27t (m 1 -rr^+m^) T + (n^+n^-m^j + (-rr^+ir^+m^) K j Thus the s h o r t e s t n o n - z e r o r e c i p r o c a l l a t t i c e v e c t o r s a r e the e i g h t v e c t o r s 2 * (+?+J+K) and t h e n e x t s h o r t e s t a r e the s i x v e c t o r s 2r (+27) ; _27r(+2j) ; a a a 2.7[(+2k). The i n t e r s e c t i o n o f t h e p l a n e s normal t o t h e s e v e c t o r s (a t t h e i r m i d p o i n t s ) d e f i n e s the f i r s t B r i l l o u i n zone shown as t h e t r u n c a t e d o c t a h e d r o n in F i g u r e k. F. M o t i o n o f an E l e c t r o n i n a C r y s t a l L a t t i c e I f t he f i r s t two te rms in t h e H a m i l t o n i a n o f (2-1) a r e r e t a i n e d and t h e a s s u m p t i o n made t h a t t h e i on c o r e s a r e c l o s e d s h e l l s o f e l e c t r o n s r i g i d l y a t t a c h e d t o t h e i r n u c l e i so t h a t one has a s ys tem o f n o n - i n t e r a c t i n g e l e c t r o n s mov ing in a l a t t i c e o f ion c o r e s , e q u a t i o n (2-1) r educes to [-i" v*w>Ji"fT (2.,o where V ( r ) i s t he p e r i o d i c p o t e n t i a l due t o the l a t t i c e . Now c o n s i d e r a s i m p l e B r a v a i s l a t t i c e w i t h d i m e n s i o n s N 2 a 2 > and N^a^ a l o n g a*j , a* 2 , and a*^  r e s p e c t i v e l y ; where N^ , N 2 > and N^ a r e i n t e g e r s . The p o t e n t i a l ene rgy o f an e l e c t r o n i n t h e l a t t i c e w i l l have t h e p e r i o d i c i t y o f t he l a t t i c e ; V(r") = + T) (2-12) I t i s w e l l known (He ine , I 9 6 0 ; T i n k h a m , 1964) t h a t s o l u t i o n s o f (2-11) s u b j e c t i !<• r t o the c o n d i t i o n s (2-12) a r e the B l o c h f u n c t i o n s ^( i* ) = e u j*(?) where U j » ( r ) has the p e r i o d i c i t y o f t h e l a t t i c e ; U £ ( r ) = u^ (r+f) . A p p l i c a t i o n o f t h e u s u a l p e r i o d i c boundary c o n d i t i o n s ' ^ ^ ) = ^(r+N^a*^) = ^fc^^ 2^ 2) = 1^T» (r+N_a* ) g i v e s k*-a = 2 r n , , fLa, = 2 r n „ , Toa* = 2rr i whence k 5 5 N j N N 3 £ = r u b * . + r u b * - + J2,tT_ N N 1 2 3 Figure k: B r i l l o u i n Zone f o r Face-Centered Cubic L a t t i c e where > r^, and n^ are i n t e g e r s . From t h i s e x p r e s s i o n for k i t i s seen t h a t , j u s t as in the f r e e e l e c t r o n c a s e , t h e r e w i l l be vdiL o r b i t a l s t a t e s 8T-> itS-f in a vo lume e l e m e n t dk o f k-space . Use o f (2-8) and (2-9) g i v e s e = 1 s i n c e a.-b*. = 27r£- . Thus i t f o l l o w s t h a t the v e c t o r £ i s no t u n i q u e l y de-1 J J^ i(E + u) • r - f f f - F t e r m i n e d s i n c e ^ ( ^ ) = e Ce ^ ] ' s a ^ s o ° f B l o c h f o r m . For t h i s k r eason the v e c t o r \Z i s o f t e n r e s t r i c t e d to t he f i r s t B r i l l o u i n zone ( r educed zone scheme) . In t h i s scheme the ene rgy i s a many-va lued f u n c t i o n o f R. By t r a n s l a t i n g r e g i o n s o f t h e f i r s t B r i l l o u i n zone t h r o u g h r e c i p -r o c a l l a t t i c e v e c t o r s G, one can fo rm p o l y h e d r a t h a t s y m m e t r i c a l l y s u r r o u n d the f i r s t z o n e . The f a c e s o f t h e s e p o l y h e d r a a r e a g a i n g i v e n by c o n d i t i o n ( 2 - 1 0 ) . The vo lume between t h e f i r s t zone and the n e x t p o l y h e d r o n d e f i n e s t h e second B r i l l o u i n z o n e . H i g h e r zones a r e d e f i n e d in a s i m i l a r manner . By use o f t h i s e x t e n d e d zone scheme the ene rgy £ (k) can be w r i t t e n as a s i n g l e -v a l u e d f u n c t i o n o f fc f o r a l l k*. The e x t e n d e d zone scheme then p e r m i t s easy c o m p a r i s o n o f an ene rgy s p e c t r u m £ (k) w i t h the p a r a b o l i c f r e e e l e c t r o n s p e c -t rum S°{k) = f i 2 k 2 . 2m 1. P e r t u r b a t i o n Theo ry f o r Weak P e r i o d i c P o t e n t i a l s I f t he p e r i o d i c p o t e n t i a l V("r) o f t h e c r y s t a l l a t t i c e i s assumed s m a l l , s t a n d a r d p e r t u r b a t i o n t h e o r y ( A n d e r s o n , 1963; Z i m a n , 1964) y i e l d s f o r t h e e l e c t r o n e n e r g y €($ - a\£) +<lc|vtf)| E > + £ K E lv(?) 1 E ^ l 2 where £ (k) = fi k i s t he e n e r g y o f t he u n p e r t u r b e d f r e e e l e c t r o n s t a t e | k> . However , s i n c e V (?) has t h e p e r i o d i c i t y o f t he l a t t i c e i t may be w r i t t e n as a F o u r i e r s e r i e s ( Z iman , 1964) 21 < i G - r V ( r ) = ^ V_ e so t h a t the m a t r i x e l emen t <k* | V ( r ) | i s n o n z e r o o n l y i f £ - £ + G = 0 . I f t h i s c o n d i t i o n i s s a t i s f i e d , t he m a t r i x e l emen t i s s i m p l y C U | V ( r ) ) 0= and the e x p r e s s i o n f o r the e n e r g y r educes t o S ( i c ) = €(£) + vn / v J 2 (2-13) From t h i s e q u a t i o n i t f o l l o w s t h a t t h e e n e r g y s u r f a c e s € = € (£) w i l l in g e n e r a l no t be s p h e r i c a l . In p a r t i c u l a r , the Fermi s u r f a c e € (£) = € f where £p i s t h e h i g h e s t o c c u p i e d e n e r g y l e v e l a t t he a b s o l u t e z e r o o f t em-p e r a t u r e , need no t be s p h e r i c a l , in c o n t r a s t t o t h e p e r f e c t l y f r e e e l e c t r o n 2, 2 c a se f o r wh i ch € = fi k . 2m From the p e r t u r b a t i o n e x p a n s i o n (2—13) i t can be seen t h a t the method f a i l s i f a d e g e n e r a c y $<) = € o c c u r s . T h i s d e g e n e r a c y con-2 2 d i t i o n i s e q u i v a l e n t t o the B ragg c o n d i t i o n "k = (k* + "5) o f ( 2 - 1 0 ) . In o t h e r w o r d s , t he p e r t u r b a t i o n e x p a n s i o n (2-13) i s not v a l i d when k l i e s nea r a B r i l l o u i n zone b o u n d a r y . However , n e a r a zone boundary i t i s s t i l l p o s s i b l e ( Z i m a n , 1964) t o expand the e l e c t r o n wave f u n c t i o n i n a s e r i e s o f t h e fo rm i k • r i k - r i G•r Tl f (?) - e u R(P) = e ft a ^ e ~ ~ ) (2-14) When k* l i e s n e a r the zone boundary d e f i n e d by T< = (k*+(j) i t i s p o s s i b l e , as a f i r s t a p p r o x i m a t i o n , t o i g n o r e a l l c o e f f i c i e n t s a^+Q e x c e p t a^ and a ^ + ^ in o r d e r t o o b t a i n an e x p r e s s i o n f o r t h e e n e r g y . The ene rgy i s then g i v e n by £(K) = V 0 + 1/2 (€°(K)+ £(£+£) + 1/2 where the minus s i g n r e f e r s to s t a t e s " i n s i d e " the zone boundary ( Ik* - Gl^G*/ ) IGI 2 22 and where the p l u s s i g n r e f e r s to s t a t e s " o u t s i d e " the zone boundary ( Ik*• Gl >IG1 ) . From (2-15) i t i s a g a i n seen t h a t in g e n e r a l t he Fermi s u r f a c e IGJ 2 w i l l be n o n - s p h e r i c a l . In a d d i t i o n , t h e ene rgy s p e c t r u m £= t (k") w i l l g e n e r -a l l y p o s s e s s d i s c o n t i n u i t i e s a t the zone b o u n d a r i e s . A d e t a i l e d d i s c u s s i o n o f t h e s e d i s c o n t i n u i t i e s may be found in many t e x t b o o k s , f o r e x a m p l e , K i t t e l (1955), A n d e r s o n ( 1 9 6 3 ) , o r Ziman (1964). 2. E f f e c t o f E l e c t r o n C o r r e l a t i o n s on the Fermi S u r f a c e S i n c e t h e c o n c e p t o f a w e l l - d e f i n e d Fermi s u r f a c e i s based upon the o n e - e l e c t r o n a p p r o x i m a t i o n w h i c h l a r g e l y n e g l e c t s e l e c t r o n - e l e c t r o n e f f e c t s , i t i s o f c o n s i d e r a b l e i n t e r e s t t o examine the e f f e c t s o f e l e c t r o n - e l e c t r o n i n t e r a c t i o n on t h e Fermi s u r f a c e . Such c o n s i d e r a t i o n s ( L u t t i n g e r , I960; C o r n w e l1, 1964) show t h a t a s y s t em o f , i n t e r a c t i n g e l e c t r o n s in a c r y s t a l does p o s s e s s a " s h a r p " Fermi s u r f a c e ( L u t t i n g e r , I960) a l t h o u g h in g e n e r a l i t w i l l d i f f e r in shape (but not in symmetry) f rom the Fermi s u r f a c e o f a s ys tem o f n o n - i n t e r a c t i n g e l e c t r o n s . ( C o r n w e l1, 1964) Such an a n a l y s i s a l s o shows t h a t the Fermi s u r f a c e f o r i n t e r a c t i n g e l e c t r o n s has t h e same symmetry as the Fermi s u r f a c e f o r e l e c t r o n s i n t e r a c t i n g t h r o u g h a H a r t r e e - F o c k s e l f - c o n s i s t e n t f i e l d ( C o r n w e l1, 1964). 3. J u s t i f i c a t i o n o f the O n e - e l e c t r o n A p p r o x i m a t i o n A j u s t i f i c a t i o n o f the o n e - e l e c t r o n a p p r o x i m a t i o n f o r a s ys tem o f i n t e r a c t i n g e l e c t r o n s i s p r o v i d e d by the Bohm-Pines t h e o r y o f p l asma o s c i l -l a t i o n s . In t h i s t h e o r y (Ra imes , 1957; 196l) a me ta l i s r e g a r d e d as a p lasma . composed o f a u n i f o r m d i s t r i b u t i o n o f p o s i t i v e backg round c h a r g e i n w h i c h the e l e c t r o n s a r e embedded. T h i s t h e o r y shows t h a t p lasma o s c i l l a t i o n s may be a s s o c i a t e d w i t h a quantum o f ene rgy (p lasmon) hw > f. and t h a t p lasmons in 23 a me ta l a t o r d i n a r y t e m p e r a t u r e s w i l l n o r m a l l y be in t h e i r ground s t a t e and c o n s e q u e n t l y may be o f t e n i g n o r e d . The t h e o r y a l s o shows t h a t t h e l o n g - r a n g e p a r t o f t he e f f e c t i v e Coulomb i n t e r a c t i o n i s a s s o c i a t e d w i t h the p l a s m o n s . o The r e m a i n i n g Coulomb i n t e r a c t i o n has an e f f e c t i v e range o f "'I A w h i c h i s so s h o r t t h a t o f t e n i t t oo may be n e g l e c t e d . The e x p e r i m e n t a l work on p lasma o s c i l l a t i o n s has been d i s c u s s e d by P i n e s ( 1955 , 1 9 5 6 ) . 4. Ene rgy Band C a l c u l a t i o n s by use o f the O n e - e l e c t r o n Model Energy band c a l c u l a t i o n s a r e in p r i n c i p l e based on the n u m e r i c a l s o l u t i o n o f the H a r t r e e - F o c k e q u a t i o n s ( C a l l a w a y , 1 9 5 8 ) . As t h e s e c a l c u l a -t i o n s a r e d i f f i c u l t v a r i o u s a p p r o a c h e s and a p p r o x i m a t i o n s have been emp loyed . S i n c e t h e f i r s t c a l c u l a t i o n f o r sod ium by W igner and S e i t z ( 1 9 3 3 ) , w e l l o v e r a hundred e n e r g y band c a l c u l a t i o n s f o r m e t a l s a l o n e have been p u b l i s h e d . Many o f t h e s e c a l c u l a t i o n s have been f o r t h e same m e t a l . For e x a m p l e , the c a l -c u l a t i o n f o r sod ium has been made more than t en t i m e s by use o f v a r i o u s a p -p r o a c h e s ( S l a t e r , 1 9 6 3 ) . The agreement o f ene rgy band c a l c u l a t i o n s w i t h e x p e r i m e n t i s o f t e n no t s a t i s f a c t o r y . Fo r the a l k a l i m e t a l s the agreement w i t h the a v a i l a b l e e x p e r i m e n t a l d a t a i s f a i r l y s a t i s f a c t o r y ( C a l l a w a y , 1 9 5 8 ) . On the o t h e r h a n d , w i t h the e x c e p t i o n o f c o p p e r , few d e t a i l e d c a l c u l a t i o n s f o r the n o b l e m e t a l s have been p u b l i s h e d . For c o p p e r , however , a t l e a s t a dozen c a l c u l a -t i o n s have been p e r f o r m e d ( S l a t e r , 1 9 6 3 ) . Agreement o f t h e r e c e n t work by Segal 1 (1962) and B u r d i c k (1963) w i t h e x p e r i m e n t i s good . For the few d i -v a l e n t m e t a l s f o r w h i c h d e t a i l e d band c a l c u l a t i o n s e x i s t , i t a p p e a r s t h a t o n l y f o r b e r y l l i u m i s t h e r e s a t i s f a c t o r y agreement between t h e o r y and e x -p e r i m e n t ( C a l l a w a y , 1 9 5 8 ) . R e l a t i v e l y few c a l c u l a t i o n s have been p e r f o r m e d f o r t r i v a l e n t , q u a d r i v a l e n t o r p e n t a v a l e n t m e t a l s . However, t he c a l c u l a t i o n s 2k f o r a luminum appea r t o be in q u a l i t a t i v e a c c o r d w i t h e x p e r i m e n t ( C a l l a w a y , 1 9 5 8 ) . Fo r t h e t r a n s i t i o n e l e m e n t s t h e c a l c u l a t i o n s a r e v e r y d i f f i c u l t and t h e r e s u l t s t end t o be r a t h e r q u a l i t a t i v e ( C a l l a w a y , 1 9 5 8 ) . 5. The Fermi S u r f a c e o f Copper ; Theory and E x p e r i m e n t The e n e r g y band c a l c u l a t i o n s o f Segal 1 (1962) and B u r d i c k (1963) f o r coppe r show q u a n t i t a t i v e agreement w i t h e x p e r i m e n t . Fo r e x a m p l e , Segal 1 has c a l c u l a t e d t h e r a d i u s o f t h e " n e c k s " w h i c h l i e in the { i l l } d i r e c t i o n s as w e l l as the a v e r a g e " b e l l y . " r a d i u s o f the Fermi s u r f a c e . For the neck 8 -1 r a d i u s Segal 1 o b t a i n s a v a l u e o f 0 . 2 8 t 0 .03 x 10 cm w h i c h compares w e l l 8 -1 w i t h the e x p e r i m e n t a l v a l u e o f 0 . 2 6 x 10 cm ( Joseph and T h o r s e n , 1 9 6 4 ) . Fo r 8 -1 a v e r a g e " b e l l y " r a d i u s he o b t a i n s 1.33 - 0.01 x 10 cm w h i c h i s in good 8 -1 a c c o r d w i t h the e x p e r i m e n t a l v a l u e o f abou t 1.32 x 10 cm o b t a i n e d f rom the work o f Morse ( i 9 6 0 ) . The d e t a i l e d r e s u l t s o f B u r d i c k and Segal 1 a r e com-p a r e d w i t h e x p e r i m e n t in T a b l e I. The symbo ls used in T a b l e I a r e d e f i n e d in F i g u r e 5- A l l d i m e n s i o n s a r e e x p r e s s e d in te rms o f the f r e e - e l e c t r o n Fermi 8 -1 s p h e r e r a d i u s k^ . = 1.365 x 10 cm . In the t a b l e the number (1) under k r r e f e r s t o t h e neck r a d i u s as measured a l o n g a l i n e p a s s i n g t h r o u g h the c e n t e r o f t he hexagona l zone f a c e and the m i d p o i n t o f one e d g e , whereas (2) under k r e f e r s t o the neck r a d i u s as measured a l o n g a l i n e p a s s i n g t h r o u g h the c e n t e r o f t he hexagona l B r i l l o u i n zone f a c e and one c o r n e r . The numbers ( l ) t o (k) a p p e a r i n g under k ^ g and k ^ ^ r e f e r t o the d i f f e r e n t v a l u e s o b t a i n e d when the d i r e c t i o n o f the sound waves used i n the m a g n e t o a c o u s t i c measurements i s changed . F u r t h e r d e t a i l s may be found i n the pape r by Bohm and E a s t e r l i n g ( 1 9 6 2 ) . 25 Figure 5: Copper Fermi< Surface D e t a i l s TABLE I COMPARISON OF THEORETICAL AND EXPERIMENTAL FERMI SURFACE DIMENSIONS FOR COPPER (AFTER BOHM AND EASTERLING, 1962) FERMI SURFACE DIMENSION BOHM-EASTERLING (a) ROAF (b) '100 '110 (1). (2) ( 0 (2) (3) (4) " (2) /k *100' 110 ;ioo M 10 V ( 0 " (2) /k 'lOO 7 110 1.036 t 0.021 1 .076 1.100 t 0.060 0.957 t 0.01 1 0.943 . 0.956 t 0.011 0.959 i 0.032 0.956 t 0.01 1 0.852 t 0.009 0.815 t 0.021 0.284 t 0.042 0.195 t 0.01 1 0.191 t 0.01 1 0.200 1.11 1.14 SEGALL (c) SEGALL (d) BURDICK (e) 1 .04 f 0.015 1.02 t 0.015 1.05 ± 0.02 0.94 ± 0.015 0.94 t 0.015 0.97 t 0.02 0.87 0.015 0.87 t 0.015 0.84 t 0.02 0.76 * 0.015 0.81 t 0.015 0.77 t 0.015 0.29 0.02 0.24 t 0.02 0.31 t 0.04 0.14 0.02 0.21 t 0.02 0.15 t 0.04 0.17 t 0.02 1.10 1 .09 1.09 (a) M a g n e t o a c o u s t i c e f f e c t (b) Dimensions deduced by Roaf (1962) from Shoenberg's de Haas-van A l p h e n d a t a (1962), P i p p a r d ' s (1957), and Morton's (i960) anomalous s k i n e f f e c t measurements. (c) Band t h e o r y c a l c u l a t i o n (Chodorow p o t e n t i a l ) (d) Band t h e o r y c a l c u l a t i o n (-^-dependent p o t e n t i a l ) (e) Band t h e o r y c a l c u l a t i o n (Chodorow p o t e n t i a l ) 27 CHAPTER I I I ANNIHILATION OF POSITRONS A . I n t r o d u c t i o n In r e c e n t y e a r s c o n s i d e r a b l e t h e o r e t i c a l and e x p e r i m e n t a l work has been done in the f i e l d o f p o s i t r o n a n n i h i l a t i o n . T h i s c h a p t e r o u t l i n e s some o f the t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s r e l e v a n t t o the s t u d y o f Fermi s u r f a c e s by p o s i t r o n a n n i h i l a t i o n . B. F ree A n n i h i l a t i o n o f P o s i t r o n s F r ee p o s i t r o n - e l e c t r o n a n n i h i l a t i o n may p r o c e e d by 1, 2, 3 o r more p h o t o n s , as l o n g as l i n e a r momentum, a n g u l a r momentum, e n e r g y and o t h e r s e -l e c t i o n r u l e s a r e s a t i s f i e d ( Jauch and R o h r l i c h , 1955). In t h i s r e s p e c t , one-pho ton a n n i h i l a t i o n can o n l y o c c u r in the p r e s e n c e o f an e x t e r n a l s ys tem a b l e t o t a k e up the r e c o i l momentum. The one-pho ton p r o c e s s i s r a t h e r u n -l i k e l y and in a t y p i c a l s o l i d o v e r 95% o f the p o s i t r o n s a n n i h i l a t e by two 28 o r more p h o t o n s . Fo r p o s i t r o n - e l e c t r o n p a i r s , a s i d e f rom s e l e c t i o n r u l e c o n s i d e r a -t i o n s , t he p r o b a b i l i t y f o r a n n i h i l a t i o n i n t o n+1 pho tons i s s m a l l e r than t h a t f o r a n n i h i l a t i o n i n t o n pho tons by a f a c t o r o f o r d e r < * where w i s the f i n e s t r u c t u r e c o n s t a n t and n 2. Thus l i t t l e e r r o r w i l l be i n c u r r e d i f p r o c e s s e s f o r w h i c h n > 3 a r e i g n o r e d . In f a c t d e t a i l e d c o n s i d e r a t i o n s i n -d i c a t e t h a t t h e r a t i o o f t he s p i n - a v e r a g e d c r o s s s e c t i o n s f o r t h e two-photon and t h r e e - p h o t o n p r o c e s s e s i s ^%-3t = 372 (Berko and H e r e f o r d , 1 9 5 6 ) . T h i s r a t i o i s i n r e a s o n a b l e agreement w i t h the e x p e r i m e n t a l r e s u l t o f Basson (1954) f o r a l u m i n i u m , v i z . ^/rjg = 406 ± 5 0 . T h u s , the a n n i h i l a t i o n o f p o s i -t r o n s w i t h c o n d u c t i o n e l e c t r o n s in a meta l o c c u r s p r e d o m i n a n t l y by t h e e m i s -s i o n o f two pho tons (Graham and S t e w a r t , 1 9 5 4 ) . The c r o s s s e c t i o n f o r two-photon a n n i h i l a t i o n o f a f r e e p o s i t r o n w i t h a f r e e e l e c t r o n was f i r s t o b t a i n e d by D i r a c (1930) by means o f a " p l a n e wave" c a l c u l a t i o n ( i m p l y i n g no Coulomb d i s t o r t i o n ) . In the n o n - r e l a t i v i s t i c l i m i t , t h i s r e s u l t f o r the s p i n - a v e r a g e d c r o s s s e c t i o n p e r e l e c t r o n reduces to 2 where r0 = i s t he c l a s s i c a l e l e c t r o n r a d i u s and v the r e l a t i v e v e l o c i t y mc o f t h e p o s i t r o n and e l e c t r o n . A l t h o u g h t h i s c r o s s s e c t i o n d i v e r g e s as t h e r e l a t i v e v e l o c i t y v a p p r o a c h e s z e r o , t h e a n n i h i l a t i o n r a t e ( i . e . p r o b a b i 1 i t y p e r u n i t t ime o f a n n i h i l a t i o n ) o f a p o s i t r o n in an e l e c t r o n gas i s i n d e p e n d -e n t o f v e l o c i t y , b e i n g g i v e n by R = Nvoj" = *r0 cN ( 3 - 0 where N i s t h e e l e c t r o n d e n s i t y . Because o f t he s u c c e s s o f t he e l e c t r o n gas model o f t he c o n d u c t i o n e l e c t r o n s in m e t a l s , i t m i g h t be e x p e c t e d t h a t t he mean l i f e t i m e s o f p o s i -t r o n s i n m e t a l s wou ld be a p p r o x i m a t e l y t h o s e e x p e c t e d on the b a s i s o f (3-1) -9 y i e l d i n g v a l u e s o f T = J_ ~ 10 s e c and i n v e r s e l y p r o p o r t i o n a l t o the c o n -R d u c t i o n e l e c t r o n d e n s i t y . The o b s e r v e d l i f e t i m e s , howeve r , a r e s u r p r i s i n g l y -10 c o n s t a n t ( f ~ 2 x 10 sec ) d e s p i t e t he l a r g e v a r i a t i o n i n e l e c t r o n d e n s i t y f rom one meta l t o a n o t h e r ( W a l l a c e , I960) . Much o f t h i s d i s a g r e e m e n t p r o b a -b l y a r i s e s f rom the n e g l e c t o f t he p o s i t r o n - e l e c t r o n i n t e r a c t i o n i n v o l v e d i n d e r i v i n g (3-0 and w i l l be examined in the d i s c u s s i o n o f l i f e t i m e s be low ( S e c t i o n H) . C. P o s i t r o n A n n i h i l a t i o n f rom a Bound S t a t e 1. Bound E l e c t r o n - P o s i t r o n Systems The p o s s i b l e e x i s t e n c e o f a bound p o s i t r o n - e l e c t r o n sys tem ( e + e) a n a l o g o u s to the hyd rogen atom was s u g g e s t e d by M o h o r o v i c i c 7 (1934) s h o r t l y a f t e r t h e e x p e r i m e n t a l d i s c o v e r y o f t h e p o s i t r o n ( A n d e r s o n , 1932; B l a c k e t t and O c c h i a l i n i , 1933). In a d d i t i o n t o t h i s t w o - e l e c t r o n sys tem known as p o s i t r o n i u m ( R u a r k , 1945) s e v e r a l o t h e r " p b l y e l e c t r o n " sys tems (e e e , e e e , and e e e e) were c a l c u l a t e d t o p o s s e s s s t a b l e bound s t a t e s (Whee le r , 1946), the t h r e e - e l e c t r o n sys tems b e i n g s t a b l e by 0.20 ev ( H y l l e r a a s , 1947) and the f o u r - e l e c t r o n s y s t em by 0.11 ev ( H y l l e r a a s and O r e , 1947). However , t h e s e t h r e e and f o u r - e l e c t r o n sys tems a r e u n l i k e l y t o be o b s e r v e d bo th because o f t h e s m a l l p r o b a b i l i t y o f t h e i r b e i n g f o r m e d , and b e c a u s e . o f t h e i r r eady b reak up by c o l l i s i o n s ( D e u t s c h , 1953). The l i f e t i m e a g a i n s t a n n i h i l a t i o n o f t h e s e t h r e e and f o u r - e l e c t r o n s ys t ems i s e s t i m a t e d t o be ~ 1 0 " ^ s e c . In a d d i t i o n to t h e s e p o l y e l e c t r o n s y s t e m s , d y n a m i c a l l y s t a b l e bound s t a t e s o f p o s i t r o n s w i t h m o l e c u l e s o r i ons a r e a l s o p o s s i b l e . For 30 e x a m p l e , c a l c u l a t i o n s i n d i c a t e t h a t p o s i t i o n h y d r i d e e H s h o u l d be s t a b l e by about 0 .23 ev (Neamtan e t a l . , 1962) and p o s i t r o n c h l o r i d e by about 4 . 6 5 ev ( S imons , 1 9 5 3 ) . A g a i n , as in the c a se o f the t h r e e and f o u r - e l e c t r o n s y s t e m s , l i t t l e e x p e r i m e n t a l work has been done on t h e s e " compounds " (Green and L e e , 1 9 6 4 ) . 2 . . Pos i t r o n ium P o s i t r o n i u m b e a r s a r e semb lance to the hydrogen atom in t h a t i t i s a l s o composed o f a p a i r o f o p p o s i t e l y c h a r g e d p a r t i c l e s . S i n c e the reduced mass o f p o s i t r o n i u m i s h a l f t h a t o f the hydrogen a tom, the p o s i t r o n i u m e n e r -gy l e v e l s w i l l be h a l f t h o s e o f hydrogen and i t s Bohr r a d i u s t w i c e as l a r g e . Thus t h e p o s i t r o n i u m i o n i z a t i o n e n e r g y i s 6 . 8 ev and the p o s i t r o n Bohr r a d i u s i s 1.06 A . A d i s c u s s i o n o f the f i n e s t r u c t u r e o f p o s i t r o n i u m may be found i n , f o r e x a m p l e , Deu t sch ( 1 9 5 3 ) . 3 . P o s i t r o n i u m A n n i h i l a t i o n P o s i t r o n i u m decays by two and t h r e e quantum a n n i h i l a t i o n ; the ' S Q •> (para-) s t a t e d e c a y i n g by two-photon a n n i h i l a t i o n and the S^  ( o r tho- ) s t a t e 3 by t h r e e p h o t o n s , two-photon decay o f the S^  s t a t e b e i n g f o r b i d d e n by the s e l e c t i o n r u l e s . The l i f e t i m e f o r two-photon a n n i h i l a t i o n i s g i v e n by = 1.25 x 10 n sec and t h a t f o r t h r e e - p h o t o n a n n i h i l a t i o n b y ^ = 1.4 x -7 3 10 n s e c . Thus f o r a n n i h i l a t i o n f r o m the ground s t a t e (n = 1) the l i f e -t i m e s become T ^ l . 2 5 x 1 0 1 0 s e c and T = 1.4 x 10^ s e c . I f the p o s i t r o n i u m 1 3 i s fo rmed in any e x c i t e d s t a t e ( o t h e r than S q s t a t e s o r the 2 S^  s t a t e ) i t s h o u l d r a d i a t e o p t i c a l l y t o the ground s t a t e b e f o r e a n n i h i l a t i n g ( D e u t s c h , 1 9 5 3 ) . S i n c e p o s i t r o n i u m f o r m a t i o n does not seem l i k e l y in m e t a l s (Wa11 ace 1960 ; Kanazawa e t a l . , 1965; see a l s o , f o r e x a m p l e , t he r e c e n t work on l i f e t i m e s by Kugel e t a l . , 1966, p o s i t r o n i u m w i l l no t be d i s c u s s e d f u r t h e i n the p r e s e n t work . The r a t h e r e x t e n s i v e s u b j e c t o f p o s i t r o n i u m f o r m a t i o n and decay in s o l i d s , l i q u i d s , and gases has been r e v i ewed by W a l l a c e ( i 9 6 0 ) . P o s i t r o n i u m c h e m i s t r y has been d i s c u s s e d by Green and Lee (1964). D. A n g u l a r C o r r e l a t i o n o f Two-Photon A n n i h i l a t i o n o f P o s i t r o n s Energy and momentum a r e c o n s e r v e d i f t h e two pho tons (each o f e n e r gy mc ) a r e e m i t t e d in o p p o s i t e d i r e c t i o n s i n the c e n t e r - o f - m a s s f rame ( H e i t l e r , 1954). I f t he c e n t e r o f mass f rame i s in m o t i o n r e l a t i v e to the l a b o r a t o r y r e f e r e n c e f r a m e , the a n g l e between the photon d i r e c t i o n s may d e -p a r t f rom 180°. The d e p a r t u r e f rom c o l l i n e a r i t y o f the two pho tons can be found by t r a n s f e r r i n g the c o l l i n e a r two-photon sys tem f rom the c e n t e r o f mass f rame t o the l a b o r a t o r y f rame by means o f an a p p r o p r i a t e L o r e n t z t r a n s -f o r m a t i o n . The c a l c u l a t i o n i s s i m i l a r t o t h a t used t o c a l c u l a t e the a b e r -r a t i o n o f l i g h t in r e l a t i v i s t i c o p t i c s ( Becker and S a u t e r , 1964). I f the r e l a t i v e e l e c t r o n v e l o c i t y v i s v e r y s m a l l compared t o the v e l o c i t y o f l i g h t ( v / c ^ l O ) i t i s p o s s i b l e t o d e r i v e a s i m p l e e x p r e s s i o n f o r the d e v i a t i o n o f t h e pho ton d i r e c t i o n s f rom a n t i p a r a 1 1 e l i sm. In t h i s c a se the k i n e t i c e n e r g y o f the p a i r i s v e r y s m a l l compared to the e n e r g y c a r r i e d o f f by the two pho tons so t h a t each pho ton w i l l have an ene rgy v e r y 2 n e a r l y equa l t o mc . Thus each pho ton has momentum mc, and s i n c e i s s m a l l i t i s seen f rom F i g u r e 6 t h a t t h e t r a n s v e r s e component o f p a i r momentum i s g i v e n by (3-2) T h i s i s t he fundamenta l angle-momentum r e l a t i o n used in two-photon a n g u l a r c o r r e l a t i o n s t u d i e s . 32 33 E. A n g u l a r C o r r e l a t i o n G e o m e t r i e s 1. I n t r o d u c t ion The re a r e e s s e n t i a l l y t h r e e ways by wh i ch t h e Fermi s u r f a c e has been s t u d i e d by use o f a n g u l a r c o r r e l a t i o n o f a n n i h i l a t i o n r a d i a t i o n . Most a n g u l a r c o r r e l a t i o n e x p e r i m e n t s have used the " w i d e s l i t " geometry d e s c r i b e d be low ( S t e w a r t , 1957) a l t h o u g h some r e c e n t work has been done by use o f a " p o i n t " geometry (Co lomb ino e t a l . , 1963, 1964' F u j i w a r a 1965). The work d e s c r i b e d i n t h i s t h e s i s u t i l i z e s a t h i r d method ( " c o l l i n e a r p o i n t geomet r y " ) in w h i c h the a n g l e does not v a r y . These methods a r e o u t l i n e d be l ow . 2 (a) Wide S l i t Geometry There a r e s e v e r a l v a r i a t i o n s o f the w ide s l i t method (S tewar t ,1957; Berko and P l a s k e t t , 1958) but i t i s s u f f i c i e n t t o c o n s i d e r S t e w a r t ' s a r r a n g e -ment s i n c e t h e methods a r e e q u i v a l e n t . In t h i s a r r a n g e m e n t , shown in F i g u r e 7, t he gamma-ray p a i r s a r e c o u n t e d i n t ime c o i n c i d e n c e by means o f d e t e c t o r s mounted b e h i n d two s l i t s t h a t can be t aken t o l i e in the h o r i z o n t a l p l a n e . The r a d i o a c t i v e sample c o n t a i n i n g a n n i h i l a t i n g e l e c t r o n - p o s i t r o n p a i r s i s a l l o w e d t o move in a l i n e p e r p e n d i c u l a r t o the p l a n e o f , and midway be tween , t h e s e two s l i t s . By mov ing the s o u r c e v e r t i c a l l y , pho tons e m i t t e d i n t ime c o i n c i d e n c e a t v a r i o u s a n g l e s may be o b s e r v e d . Some t y p i c a l d i m e n s i o n s ( S t e w a r t , 1957) f o r t h e d i s t a n c e s shown in the f i g u r e a r e D = 100 i n . X = 1.5 i n . h = 0.050 i n . F u r t h e r e x p e r i m e n t a l d e t a i l s can be found in S t e w a r t ' s p a p e r . From F i g u r e 7 and by (3-2) i t i s seen t h a t , a p p r o x i m a t e l y , p z = mc9 = 2mcz/D (3-3) where z i s t h e v e r t i c a l d i s p l a c e m e n t . Thus i n t h i s method t h e t r a n s v e r s e component o f p a i r momentum p^  i s d i r e c t l y p r o p o r t i o n a l t o the d i s p l a c e m e n t z . I f now t h i s a p p r o a c h i s c o n s i d e r e d in te rms o f t he f r e e - e l e c t r o n model i t i s seen t h a t f o r each s e t t i n g z o f t he d i s p l a c e m e n t , one samp les a s l i c e t h r o u g h t h e Fermi vo lume as shown in F i g u r e 9. The s l i c e t h i c k n e s s , o f c o u r s e , i s d e t e r m i n e d by t h e f i n i t e r e s o l u t i o n o f t h e e x p e r i m e n t a l a r r a n g e -ment . F o l l o w i n g the a n a l y s i s o f S t ewa r t (1957) i t i s seen t h a t t he c o -i n c i d e n c e coun t r a t e w i l l be g i v e n by n(PJ c* j3(p-) dpx dp — oo _ OO where ^ ( p ) i s t he d e n s i t y o f momenta o f t he a n n i h i l a t i n g p a i r s . I f ^ ( p ) i s assumed i s o t r o p i c so t h a t ^ (p ) = J>(p) , t he c o u n t r a t e becomes n ( p z ) <* jfip) 2KPf d P f_ = 27T C (p) pdp 2 2 2 2 2 2 where p = P r + P z = P x + Py + Pz • T h u s , d i f f e r e n t i a t i o n o f t h i s e x p r e s s i o n f o r n(p^) g i v e s f(px ) <X - 1 dn (p , ) P. d P * so t h a t by (3-3) i t f o l l o w s t h a t P ( p ) <* -1 d n ( z ) . (3-4) ' z dz For t he i d e a l f r e e - e l e c t r o n ca se in w h i c h i t i s assumed t h a t -36 (a) The momentum o f the p o s i t r o n i s z e r o ( i . e . t he momenta o f t h e t h e r m a l i z e d p o s i t r o n s a r e n e g l e c t e d compared t o the e l e c t r o n momenta. (b) The e l e c t r o n momentum d i s t r i b u t i o n i s i s o t r o p i c and t aken t o be o f u n i f o r m d e n s i t y j> up t o the maximum, o r Fermi momentum p and z e r o f o r p> p . F r one has n (p „ ) j ° Pdp o r n ( p j " ( p j - > * ) (3-5) w h i c h i s t he e q u a t i o n o f an i n v e r t e d p a r a b o l a . As m i g h t be e x p e c t e d , t he e x p e r i m e n t a l c o u n t i n g r a t e c u r v e s d e v i a t e f rom an e x a c t i n v e r t e d p a r a b o l a . i n the c a se o f t h e a l k a l i m e t a l s the d e -v i a t i o n i s s m a l l . T h i s c o n f i r m s the s t a t e m e n t t h a t t h e a l k a l i m e t a l s e x h i b i t v e r y n e a r l y f r e e e l e c t r o n b e h a v i o u r . But as one p a s s e s t h r o u g h the a l k a l i n e e a r t h s and b e y o n d , i n c r e a s i n g l y l a r g e r " t a i l s " a r e s u p e r i m p o s e d on the i n -v e r t e d p a r a b o l a . Fo r c o m p a r i s o n , t h e r e s u l t s f o r sod ium and c o p p e r ( S t e w a r t , 1957) a r e shown in F i g u r e 8. In t h i s f i g u r e ©> d e n o t e s the a n g l e c o r r e s p o n d i n g t o the Fermi momentum pp . Be rko and P l a s k e t t a t t r i b u t e most o f t he t a i l o f t he c o p p e r c u r v e t o a n n i h i l a t i o n o f p o s i t r o n s w i t h the e l e c t r o n s o f the i on c o r e s ( co r e a n n i h i l a t i o n ) . They have e s t i m a t e d t h e c o r e c o n t r i b u t i o n f o r c o p p e r and f i n d t h a t t h e i r e s t i m a t e i s in r e a s o n a b l e agreement w i t h e x p e r i m e n t . "However , i t s h o u l d be no ted t h a t t h e r e a r e d i f f i c u l t i e s a s s o c i a t e d w i t h t h e s e c o r e a n n i h i l a t i o n c a l c u l a t i o n s . These d i f f i c u l t i e s w i l l be d i s c u s s e d be low . 37 2 (b) D e t e r m i n a t i o n o f Fermi S u r f a c e s by Wide S l i t Geometry I f a p a r t i c u l a r d i r e c t i o n o f a meta l s i n g l e c r y s t a l i s a l i g n e d w i t h the z a x i s in the w ide s l i t m e t h o d , i t i s found t h a t the shape o f the a n g u l a r c o r r e l a t i o n c u r v e may depend upon the c h o i c e o f c r y s t a l o r i e n t a t i o n ( Be rko and P l a s k e t t , 1 9 5 8 ) . A l t h o u g h t h i s a n i s o t r o p y has been i n t e r p r e t e d as b e i n g due to a n o n s p h e r i c a l Fermi s u r f a c e i t i s u s u a l l y d i f f i c u l t t o make q u a n t i -t a t i v e s t a t e m e n t s about the t o p o l o g y o f the Fermi s u r f a c e on the b a s i s o f such measu remen ts . A ma jo r s o u r c e o f d i f f i c u l t y i s the f a c t t h a t the method samples a s l i c e t h r o u g h momentum s p a c e , g i v i n g a r e s u l t t h a t i s an a v e r a g e o v e r a c o n s i d e r a b l e r e g i o n . In a d d i t i o n , t h i s a v e r a g i n g makes i t d i f f i c u l t t o o b s e r v e the sha rp " b r e a k " in the a n g u l a r c o r r e l a t i o n c u r v e w h i c h s h o u l d (Majumdar , 1 9 6 5 ) o c c u r a t a n g l e s c o r r e s p o n d i n g t o the Fermi momentum. A l s o , r e c e n t work w i t h l i t h i u m by Donaghy e t a l . ( 1 9 6 5 ) s u g g e s t s t h a t the h i g h e r momentum components o f t he e l e c t r o n wave f u n c t i o n may m o d i f y the shape o f the a n g u l a r c o r r e l a t i o n c u r v e s u f f i c i e n t l y so t h a t the o b s e r v e d " b r e a k s " in the a n g u l a r c o r r e l a t i o n c u r v e no l o n g e r c o r r e s p o n d t o r a d i i ( S t ewa r t e t a l . , 1 9 6 2 ) o f the Fermi s u r f a c e . D e s p i t e t h e s e l i m i t a t i o n s , the method has been used t o s t u d y the Fermi s u r f a c e o f a number o f m e t a l s ; i n s e v e r a l c a ses c o n -f i r m i n g q u a l i t a t i v e l y the r e s u l t s o b t a i n e d by the more p r e c i s e c o n v e n t i o n a l m e t h o d s . M e t a l s in the form o f s i n g l e c r y s t a l s t h a t have been examined by t h i s method i n c l u d e sod ium (Donaghy e t a l . , 1 9 6 5 ) , b e r y l l i u m ( S t ewa r t e t a l . , 1 9 6 2 ; B e r k o , 1 9 6 2 ) , magnesium ( B e r k o , 1 9 6 2 ) , a luminum and c o p p e r (Berko and P l a s k e t t , 1 9 5 8 ) as w e l l as ho lm ium, e r b i u m , and y t t r i u m ( W i l l i a m s e t a l . , 1 9 6 6 ) . 3 (a) P o i n t Geometry In t h i s method two " p o i n t " d e t e c t o r s a r e used in p l a c e o f t he w ide s l i t a r r angement d i s c u s s e d a b o v e . W i th t h i s a r r angement the r e g i o n o f momen-turn space t h a t i s sampled i s an i n f i n i t e c y l i n d e r whose d i a m e t e r i s d e t e r -mined by the d e t e c t o r geomet r y . The a r r angement i s i l l u s t r a t e d in F i g u r e 10. U s i n g a n o t a t i o n s i m i l a r t o t h a t used in the d i s c u s s i o n o f the w ide s l i t c a s e , i t i s seen t h a t f o r c o u n t i n g r a t e a t p one e x p e c t s ( F i g u r e 11) — ( 3 " 6 ) The s o l u t i o n o f t h i s i n t e g r a l e q u a t i o n ( A p p e n d i x I) i s oo rz I f , as f o r t h e w i d e s l i t c a s e , ^ (p ) i s t a k e n t o be c o n s t a n t i n s i d e the Fermi s u r f a c e and z e r o o u t s i d e , i t i s seen f rom (3-6) t h a t the c o u n t i n g r a t e i s g i v e n by n ( p z ) « \ p* - P* (3-9) 2 2 so t h a t a p l o t o f n v s . p^ i s 1 i n e a r . 3 (b) D e t e r m i n a t i o n o f Fermi S u r f a c e s by Use o f P o i n t Geometry The use o f p o i n t geometry wou ld be e x p e c t e d t o y i e l d a n g u l a r c o r -r e l a t i o n c u r v e s c o n t a i n i n g more s t r u c t u r e t han t h o s e c h a r a c t e r i z i n g the w ide s l i t geometry s i n c e the method samples a s m a l l e r r e g i o n in momentum s p a c e . A p r e l i m i n a r y s t u d y o f the Fermi s u r f a c e o f coppe r has been made by t h i s method ( F u j i w a r a , 1965). A l t h o u g h t h e r e s u l t s d e f i n i t e l y show more d e t a i l t han do t h o s e o f Be rko and P l a s k e t t (1958), the c o p i o u s p r e s e n c e o f h i g h e r momentum components a g a i n makes q u a n t i t a t i v e i n t e r p r e t a t i o n i n terms o f the Fermi s u r f a c e d i f f i c u l t . 39 I Figure 10; Point Geometry >Figure. 11:, Region Sampled by Point Geometry i Figure 12: C o l l i n e a r Point Geometry 40 4. Col l i n e a r P o i n t Geometry In t h i s method the r a d i o a c t i v e me ta l s i n g l e c r y s t a l and the two " p o i n t " d e t e c t o r s rema in c o l l i n e a r and the c o i n c i d e n c e coun t r a t e o b t a i n e d as a f u n c t i o n o f c r y s t a l o r i e n t a t i o n ( F i g u r e 1 2 ) . On the b a s i s o f t h e f r e e -e l e c t r o n model t h i s method a l s o samples a c y l i n d r i c a l vo lume o f momentum s p a c e . T h i s volume e l emen t r o t a t e s about the o r i g i n o f momentum space as t h e c r y s t a l o r i e n t a t i o n v a r i e s so t h a t one wou ld e x p e c t v a r i a t i o n s in c o u n t -ing r a t e w i t h c r y s t a l o r i e n t a t i o n to be i n t e r p r e t a b l e in te rms o f r a d i i t o t h e Fermi s u r f a c e . Tha t i s , f rom ( 3 - 9 ) , i f 8 (and hence p^) i s z e r o , then n o( p^  . I f i t i s assumed t h a t c o n t r i b u t i o n s to the c o u n t i n g r a t e f rom c o r e a n n i h i l a t i o n s a r e i s o t r o p i c and i f i t i s assumed t h a t t h e n e t e f f e c t o f a l l o t h e r h i g h e r momentum e f f e c t s i s r o u g h l y i s o t r o p i c , t h e n v a r i a t i o n s i n t he c o i n c i d e n c e c o u n t i n g r a t e a r e a d i r e c t measure o f f l u c t u a t i o n s o f t he Fermi s u r f a c e r a d i u s . F. A n n i h i l a t i o n o f P o s i t r o n s in E l e c t r o n Gases When p o s i t r o n s a n n i h i l a t e in a dense gas o f i n t e r a c t i n g e l e c t r o n s the momentum d i s t r i b u t i o n o f t he pho ton p a i r s s h o u l d d i f f e r f rom the s i m p l e F e r m i - D i r a c momentum d i s t r i b u t i o n c h a r a c t e r i s t i c o f n o n - i n t e r a c t i n g e l e c t r o n s and shown as c u r v e a o f F i g u r e 13- T h i s d e v i a t i o n a r i s e s f rom the e f f e c t o f e l e c t r o n - e l e c t r o n and e l e c t r o n - p o s i t r o n c o r r e l a t i o n s ( D a n i e l and Vosko,1960; Kahana , I960; 1963). S e v e r a l a t t e m p t s have been made t o c a l c u l a t e t he e f f e c t o f t h e s e v a r i o u s c o r r e l a t i o n s on the momentum d i s t r i b u t i o n o f t he gamma-ray p a i r s (Hatano e t a l . , 1965)-1. E f f e c t o f E l e c t r o n - e l e c t r o n I n t e r a c t i o n s I f t he e l e c t r o n - p o s i t r o n c o r r e l a t i o n s a r e i g n o r e d and o n l y t he e l e c t r o n - e l e c t r o n c o r r e l a t i o n s t aken i n t o c o n s i d e r a t i o n , the momentum d i s -t r i b u t i o n o f t h e gamma-ray p a i r s w i l l be i d e n t i c a l t o t h a t o f t h e i n t e r -a c t i n g e l e c t r o n s (Hatano e t a l . , 1965)- D a n i e l and Vosko have c a l c u l a t e d t h e momentum d i s t r i b u t i o n o f such a sys tem o f i n t e r a c t i n g e l e c t r o n s , and f o r an e l e c t r o n i c d e n s i t y c o r r e s p o n d i n g t o t h a t i n sod ium they o b t a i n t h e r e s u l t s shown in c u r v e b o f F i g u r e 13. I t i s t o be no ted t h a t i n t h i s c a se t h e p o s i t r o n a c t s as an i d e a l p r o b e s i n c e e l e c t r o n - p o s i t r o n c o r r e l a t i o n s have been i g n o r e d . 2. E f f e c t o f E l e c t r o n - P o s i t r o n I n t e r a c t i o n s By an a p p r o x i m a t e s o l u t i o n o f a B e t h e - G o l d s t o n e e q u a t i o n f o r a p o s i t r o n - e l e c t r o n p a i r in a sea o f i n t e r a c t i n g e l e c t r o n s , Kahana (i960;1963) o b t a i n e d the pho ton p a i r momentum d i s t r i b u t i o n g i v e n by c u r v e c o f F i g u r e 13. In t h i s c a l c u l a t i o n the two-body c o r r e l a t i o n s between the p o s i t r o n and the a n n i h i l a t i n g e l e c t r o n have been a c c u r a t e l y a c c o u n t e d f o r . However, w i t h the e x c e p t i o n o f the c o r r e l a t i o n s a s s o c i a t e d w i t h t h e s c r e e n i n g o f t h e a t t r a c t i v e e l e c t r o n - p o s i t r o n f o r c e , no o t h e r c o r r e l a t i o n s a r e c o n s i d e r e d . Hatano e t a l . (1965) have s u g g e s t e d t h a t the n e g l e c t o f t h e s e o t h e r c o r r e l a t i o n s may be j u s t i f i e d o n l y f o r h i g h e l e c t r o n d e n s i t i e s and not f o r t h e r e l a t i v e l y low e l e c t r o n d e n s i t i e s found i n r e a l m e t a l s . 3• E f f e c t o f E l e c t r o n - e l e c t r o n and E l e c t r o n - p o s i t r o n I n t e r a c t i o n s By s t a r t i n g w i t h a wave f u n c t i o n o f B i j 1 - D i n g l e - J a s t r o w t y p e , Hatano e t a l . have been a b l e to a p p r o x i m a t e l y i n c l u d e bo th e l e c t r o n - e l e c t r o n and e l e c t r o n - p o s i t r o n c o r r e l a t i o n s . T h e i r r e s u l t i s shown in c u r v e d o f F i g u r e 13. For the r e l a t i v e momentum d i s t r i b u t i o n o f the pho ton p a i r s t h e y o b t a i n 42 w(p) = 1 + 0 .13 ( p / p . ) 2 - 0 . 0 2 ( p / p f ) k + . . . w(0) f f in w h i c h the c o e f f i c i e n t s a r e i ndependen t o f e l e c t r o n d e n s i t y . T h i s i s t o be compared w i t h K a h a n a ' s r e s u l t ( f o r an e l e c t r o n d e n s i t y c o r r e s p o n d i n g r o u g h l y t o sodium) w(p) = 1 + 0 . 2 6 2 ( p / p , ) 2 + 0 . 2 3 3 ( p / p f ) k w(0) f f where p .^ = fik^. i s t h e u s u a l Fermi momentum. I t i s seen t h a t t h e r e i s a s u b -s t a n t i a l d i f f e r e n c e between t h e two r e s u l t s . 4 . Compar i son w i t h E x p e r i m e n t S i n c e the c o n d u c t i o n e l e c t r o n s i n sod ium can be c o n s i d e r e d t o r o u g h l y a p p r o x i m a t e an e l e c t r o n gas i t m i g h t be r e a s o n a b l e t o assume t h a t e x p e r i m e n t a l s t u d i e s o f sod ium c o u l d be used t o choose the " b e s t " o f t he t h e o r e t i c a l c u r v e s shown in F i g u r e 13 and t h u s g i v e some i n s i g h t i n t o the a n n i h i l a t i o n p r o c e s s . However , t h i s i s r a t h e r d i f f i c u l t t o do because o f e x p e r i m e n t a l u n c e r t a i n t i e s . For e x a m p l e , a l t h o u g h t h e c o m p u t a t i o n s by Hatano e t a l . appea r t o be in b e t t e r agreement w i t h S t e w a r t ' s e x p e r i m e n t a l r e s u l t s f o r sod ium than a r e t h o s e o f Kahana o r D a n i e l and V o s k o , S t e w a r t ' s r e s u l t s (1961) a r e e x p l a i n e d j u s t as s a t i s f a c t o r i l y by the s i m p l e Sommerfe ld model ( c u r v e a , F i g u r e 1 3 ) . On t h e o t h e r h a n d , r e c e n t work by Donaghy e t a 1 . (1965) w i t h sod ium a p p e a r s to g i v e r e s u l t s t h a t a r e in good a c c o r d w i t h Kahana ' s c u r v e shown in F i g u r e 13 . I t i s i m p o r t a n t to no t e however , t h a t the r e s u l t s o f Donaghy e t a l . i n v o l v e a c o r r e c t i o n f o r c o r e a n n i h i l a t i o n ( t o be d i s -c u s s e d below) w h i c h i s r a t h e r d i f f i c u l t t o e s t i m a t e ( C a r b o t t e , 1 9 6 6 ) . Thus on the b a s i s o f a n g u l a r c o r r e l a t i o n work a l o n e i t i s d i f f i c u l t a t p r e s e n t to make a d e f i n i t e c h o i c e between t h e v a r i o u s m o d e l s . i .5o p/p Figure 13s Momentum D i s t r i b u t i o n of A n n i h i l a t i o n ' Photons Emanating From an I n t e r a c t i n g E l e c t r o n Gas ( A f t e r Hatano et a l . , 1965) o © <a -p a Kanazawa et a l . B e l l and Jorgenson Kahana h i. 6 Figure ll(.: P o s i t r o n A n n i h i l a t i o n Rates ( A f t e r Kanazawa et a l . , 1965) G. A n n i h i l a t i o n o f P o s i t r o n s i n Rea l M e t a l s 1. I n t r o d u c t ion A q u a n t i t a t i v e t r e a t m e n t o f t he e f f e c t o f t h e c r y s t a l l a t t i c e upon the a n g u l a r c o r r e l a t i o n o f a n n i h i l a t i o n r a d i a t i o n f rom m e t a l s i s e x c e e d i n g l y d i f f i c u l t ( W a l l a c e , I960; C a r b o t t e , 1966). One o f the s o u r c e s o f d i f f i c u l t y i s t he p r e s e n c e o f h i g h e r momentum components i n bo th the p o s i t r o n and e l e c -t r o n wave f u n c t i o n s due to the p e r i o d i c p o t e n t i a l o f t he c r y s t a l l a t t i c e . A n o t h e r c o m p l i c a t i o n a r i s e s f rom the a n n i h i l a t i o n o f p o s i t r o n s w i t h c o r e e l e c t r o n s . Both o f t h e s e e f f e c t s y i e l d a c o n t r i b u t i o n t o the a n g u l a r c o r -r e l a t i o n a t v a l u e s o f & g r e a t e r than t h o s e w h i c h w o u l d a r i s e f rom a n n i h i l a -t i o n w i t h f r e e c o n d u c t i o n e l e c t r o n s a l o n e . 2. E f f e c t o f t h e P e r i o d i c C r y s t a l L a t t i c e P o t e n t i a l An e a r l y a t t e m p t t o c o n s i d e r the e f f e c t o f t he c r y s t a l l a t t i c e was made by De B e n e d e t t i e t a l . (1950). They c o n s i d e r e d a s i m p l e model in w h i c h , due t o Coulomb r e p u l s i o n , the p o s i t r o n wave f u n c t i o n i s e x c l u d e d f rom a v o l -ume v ( the " e x c l u d e d vo lume" ) a round the n u c d e u s . O u t s i d e t h i s e x c l u d e d e vo lume the p o s i t r o n wave f u n c t i o n i s t a k e n t o be a c o n s t a n t and t h e e l e c t r o n wave f u n c t i o n i s c o n s i d e r e d a s i m p l e p l a n e wave. The r e s t r i c t i o n o f the p o s i t r o n t o r e g i o n s s u r r o u n d i n g the " e x c l u d e d v o l u m e " r e g i o n s t h e r e b y i n t r o -duces h i g h e r momentum components i n t o the p o s i t r o n wave f u n c t i o n as wou ld be e x p e c t e d , f o r e x a m p l e , f rom the H e i s e n b e r g u n c e r t a i n t y p r i n c i p l e . The r e l a -t i v e i n t e n s i t y o f t h e s e h i g h e r momentum components i s r e l a t e d t o the q u a n t i t y v /v w h i c h i s t h e r a t i o o f t h e e x c l u d e d volume t o t h e volume o f a u n i t c e l l . e T y p i c a l v a l u e s o f v g / v range f rom 0.05 in b e r y l l i u m t o 0.29 in ba r i um ( L ang , 1956; Lang and De B e n e d e t t i , 1957). The r e s u l t s however g e n e r a l l y d i s a g r e e 45 w i t h e x p e r i m e n t ( Lang , 1956; Be rko and P l a s k e t t , 1 9 5 8 ) , t he d i s a g r e e m e n t p r o b a b l y b e i n g due t o c o r e a n n i h i l a t i o n and to c o n t r i b u t i o n s f rom the e l e c -t r o n p o s i t r o n i n t e r a c t i o n ( W a l l a c e , I 9 6 0 ) . 3. E f f e c t o f Core A n n i h i l a t i o n s By use o f u n c o r r e l a t e d o n e - p a r t i c l e e l e c t r o n wave f u n c t i o n s Berko and P l a s k e t t (1958) have c a l c u l a t e d the a n g u l a r d i s t r i b u t i o n o f pho tons r e -s u l t i n g f rom the a n n i h i l a t i o n o f p o s i t r o n s w i t h c o r e e l e c t r o n s . The c a l c u -l a t i o n s were done f o r coppe r and a luminum and a r e in f a i r l y good agreement w i t h e x p e r i m e n t . However , t he computed momentum d i s t r i b u t i o n i s q u i t e s e n s i -t i v e t o t h e shape o f the p o s i t r o n wave f u n c t i o n so t h a t i t i s d i f f i c u l t t o a s s e s s the v a l i d i t y o f t he me thod . A l s o , r e c e n t c a l c u l a t i o n s by C a r b o t t e (1966) i n d i c a t e t h a t t h e e l e c t r o n - p o s i t r o n i n t e r a c t i o n may be o f c o n s i d e r a b l e i m p o r t a n c e i n c o r e a n n i h i l a t i o n c a l c u l a t i o n s . The B e r k o - P l a s k e t t method has been a p p l i e d t o o t h e r m e t a l s by Rockmore and S t ewa r t (1965) and by T e r r e l l e t a l . ( 1 9 6 5 ) . H. L i f e t i m e s o f P o s i t r o n s i n M e t a l s I f t y p i c a l v a l u e s o f t h e e l e c t r o n d e n s i t y o f m e t a l s a r e s u b s t i t u t e d i n t o ( 3 - 1 ) , l i f e t i m e s o f the o r d e r o f 10"* s e c a r e o b t a i n e d . T h i s i s in d i s -agreement w i t h the e x p e r i m e n t a l l y o b s e r v e d l i f e t i m e s o f ~ 2 x l o ' ^ s e c . In a d d i t i o n , t he o b s e r v e d l i f e t i m e s a r e e s s e n t i a l l y i ndependent o f e l e c t r o n d e n s i t y , ( W a l l a c e , i 9 6 0 ) whereas (3-1) p r e d i c t s a d i r e c t dependence on t h e e l e c t r o n d e n s i t y . Thus a n n i h i l a t i o n i n m e t a l s c a n n o t be r ega rded as f r e e a n n i h i l a t i o n i n w h i c h the e f f e c t s o f t he Coulomb f o r c e s can be n e g l e c t e d . N e i t h e r can the l i f e t i m e s be i n t e r p r e t e d i n terms o f p o s i t r o n i u m f o r m a t i o n s i n c e t r i p l e t p o s i t r o n i u m wou ld have a l o n g l i f e t i m e o f about 10^ sec w h i c h i s not o b s e r v e d . On the o t h e r hand, r a p i d t r i p l e t t o s i n g l e t c o n v e r s i o n would l e a d t o a l i f e t i m e o f 5^ x 10^ s e c , f o u r t i m e s t h a t o f s i n g l e t p o s i -t r o n i u m ( W a l l a c e , I 9 6 0 ) . However, t h i s i s a l s o c o n s i d e r a b l y l a r g e r than the o b s e r v e d l i f e t i m e s o f ~ 2 x l o ' " s e c . The e f f e c t s o f e l e c t r o n - p o s i t r o n c o r r e l a t i o n s on the a n n i h i l a t i o n r a t e i n m e t a l s have been i n v e s t i g a t e d by Ferrel1 (1956), Kahana (1963), C a r b o t t e and Kahana (1965), and Kanazawa e t a l . (1965). These s t u d i e s i n -d i c a t e t h a t the p o s i t r o n l i f e t i m e i s v e r y s e n s i t i v e t o c o r r e l a t i o n e f f e c t s , the a n n i h i l a t i o n r a t e b e i n g d i r e c t l y dependent on t h e p r o b a b i l i t y t h a t the p o s i t r o n and e l e c t r o n a r e a t t h e same p l a c e . The r e c e n t work o f Kahana(l963), which a c c u r a t e l y t a k e s i n t o a c c o u n t t h e two-body c o r r e l a t i o n s between the p o s i t r o n and the a n n i h i l a t i n g e l e c t r o n , p r e d i c t s p o s i t r o n l i f e t i m e s t h a t a r e i n o r d e r o f magnitude agreement w i t h e x p e r i m e n t ( F i g u r e 14). H i s work a l s o g i v e s an a n g u l a r c o r r e l a t i o n c u r v e t h a t i s i n rough agreement w i t h the ex-p e r i m e n t a l work o f Stewart (1961). Kanazawa e t a l . (1965) have c a l c u l a t e d the a n n i h i l a t i o n r a t e f o r e l e c t r o n s o f z e r o momentum, t h e i r r e s u l t s b e i n g shown i n F i g u r e \k. I n c l u s i o n o f e f f e c t s o f the c r y s t a l l a t t i c e i n t o l i f e -t i m e c a l c u l a t i o n s i s a l s o v e r y d i f f i c u l t a l t h o u g h p r e l i m i n a r y work by Car-b o t t e (1966) has a l r e a d y p r o v i d e d some i n s i g h t i n t o the c o r e a n n i h i l a t i o n p r o c e s s . CHAPTER IV EXPERIMENTAL ARRANGEMENT The s c h e m a t i c d i a g r a m in F i g u r e 15 shows the e x p e r i m e n t a l a r r a n g e -ment f o r t h e " c o l l i n e a r p o i n t ge o m e t r y " method used in the p r e s e n t work . In o r d e r t o be d e t e c t e d , pho tons f rom the p o s i t r o n - a c t i v e s i n g l e c r y s t a l o f c o p p e r had t o pass t h r o u g h the 0 . 2 5 i n c h d i a m e t e r a p e r t u r e o f t he l e a d c o l l i -ma to r s t h a t were p l a c e d a t a d i s t a n c e o f about 12 f e e t f rom the coppe r c r y s t a l . I f a gamma-ray p a i r was d e t e c t e d by t h e two c o u n t e r s w i t h i n the c o i n c i d e n c e c i r c u i t r e s o l v i n g t i m e o f 10 u s e e , a c o i n c i d e n c e was r e c o r d e d by the s c a l e r . In t h i s way, w i t h i n the r e s o l u t i o n o f the a p p a r a t u s , o n l y pho ton p a i r s o f z e r o t r a n s v e r s e momentum were c o u n t e d . For each o r i e n t a t i o n o f the c r y s t a l t he t ime i n t e r v a l r e q u i r e d f o r r e c o r d i n g a p r e d e t e r m i n e d number o f c o u n t s was m e a s u r e d . T h i s made i t p o s s i b l e to a c c u r a t e l y c o r r e c t f o r the decay o f t h e p o s i t r o n a c t i v i t y (T^ = 1 2 . 9 h r ) and t hus o b t a i n a measure o f momentum 2 space a n i s o t r o p y as d i s c u s s e d in C h a p t e r I I I . PREAMP AND SHAPER tfaltTl) . DETECTOR ."SYSTEM LEAD COLLIMATOR [113 t Cu SINGLE CRYSTAL AXIS OP ROTATION COINC-IDENCE UNIT ' ( i n ) LEAD COLLIMATOR Kal ( T l ) DETECTOR SYSTEM FREAMP AND SHAPER SCALER Figure l£: Experimental Arrangement•(Schematic) A . Me ta l C r y s t a l and P o s i t r o n Source The coppe r c r y s t a l used in t h i s e x p e r i m e n t was c y l i n d r i c a l , 3.56 mm in d i a m e t e r by 3.84 mm h i g h , w i t h £l 1 l j d i r e c t i o n a l o n g i t s a x i s . Copper was chosen as the me ta l t o be examined by t h i s method f o r s e v e r a l r e a s o n s : 1. The t o p o l o g y o f the Fermi s u r f a c e o f coppe r i s w e l l known ( C h a p t e r l l ) thus mak ing i t r e l a t i v e l y easy t o a s c e r t a i n the u s e f u l n e s s o f t he new p o s i t r o n a n n i h i l a t i o n t e c h n i q u e used in t h i s e x p e r i m e n t . 2. Copper i s s t a b l e a t o r d i n a r y t e m p e r a t u r e s and i s r e a d i l y o b t a i n e d i n the fo rm o f v e r y pure s i n g l e c r y s t a l s so t h a t sample e n v i r o n m e n t o r sample i m p u r i t i e s need not be c o n s i d e r e d . 3. The use o f a c o n v e n t i o n a l p o s i t r o n s o u r c e was r e n d e r e d u n n e c e s s a r y s i n c e by the rma l n e u t r o n i r r a d i a t i o n the coppe r c r y s t a l i t s e l f became an adequa te p o s i t r o n s o u r c e . The p r e s e n t a p p r o a c h was advan tageous s i n c e i t p r o v i d e d a n e a r l y u n i f o r m d i s t r i b u t i o n o f p o s i t r o n s t h r o u g h o u t the c r y s t a l whereas c o n v e n t i o n a l s o u r c e g e o m e t r i e s " s h i n e " p o s i t r o n s on t h e sampl>e s u r f a c e ( Be rko and P l a s k e t t , 1958). A l s o , in c o n t r a s t t o some p o s i t r o n s o u r c e s , t he gamma r a d i a t i o n ema-n a t i n g f rom t h e c r y s t a l i s n e a r l y a l l a n n i h i l a t i o n r a d i a t i o n thus e n s u r i n g n o n i n t e r f e r e n c e f rom o t h e r r a d i a t i o n s . A n o t h e r i m p o r t a n t c o n s i d e r a t i o n i s t h a t o f economy. A c o n v e n t i o n a l l o n g - l i v e d s o u r c e o f compa rab l e s t r e n g t h wou ld have been p r o h i b i t i v e l y e x p e n s i v e . The main d i s a d v a n t a g e o f t h i s t y p e o f p o s i t r o n s o u r c e i s t he i n -c o n v e n i e n c e a s s o c i a t e d w i t h the s h o r t h a l f - l i f e . Due t o the r e l a t i v e l y low c o i n c i d e n c e c o u n t i n g r a t e s r e s u l t i n g f rom use o f " p o i n t " geometry i t was n e c e s s a r y t o have a r e l a t i v e l y h i g h i n i t i a l p o s i t r o n a c t i v i t y (»»50 mCJ ) ' A l t h o u g h the c o i n c i d e n c e c o u n t i n g r a t e a t the b e g i n n i n g o f a run was adequa te ( "1 c o u n t p e r second) t h i s r a t e f a l l s w i t h t i m e , mak ing i t d i f f i c u l t t o e s t a b l i s h many e x p e r i m e n t a l p o i n t s w i t h good s t a t i s t i c s . The use o f a l o n g -l i v e d s o u r c e wou ld have o b v i a t e d t h i s d i f f i c u l t y . The s i z e o f t he c r y s t a l was d e t e r m i n e d a f t e r c o n s i d e r a t i o n o f s e v e r a l c o n f l i c t i n g r e q u i r e m e n t s . From the p o i n t o f v i ew o f a n g u l a r r e s o l u -t i o n , t h e i d e a l c r y s t a l s h o u l d be v a n i s h i n g l y s m a l l . On the o t h e r h a n d , i t i s n e c e s s a r y t o have the c r y s t a l l a r g e r than a c e r t a i n minimum s i z e in o r d e r t h a t i t have s u f f i c i e n t p o s i t r o n a c t i v i t y upon i r r a d i a t i o n w i t h the the rma l n e u t r o n f l u x e s c u r r e n t l y a v a i l a b l e . In a d d i t i o n , i t was d e s i r a b l e t o choose the c r y s t a l l a r g e enough so t h a t few o f the p o s i t r o n s p roduced in the c r y s t a l e s c a p e . T a k i n g t h e s e c o n s i d e r a t i o n s i n t o a c c o u n t , t h e sample was chosen t o be a c y l i n d e r w i t h d i a m e t e r r o u g h l y equa l t o h e i g h t so t h a t i t s e x t e n t , as seen f rom t h e " p o i n t " d e t e c t o r s , wou ld be as s m a l l as p o s s i b l e . W i th the d i m e n s i o n s g i v e n above i t i s e asy to e s t i m a t e , by use o f r e s u l t s f rom P r i c e e t a l . (1957) and Evans ( 1 9 5 5 ) , t h a t l e s s than f i v e p e r c e n t o f t he p o s i t r o n s w i l l e s c a p e f rom the s a m p l e . B. O r i e n t a t i o n o f C r y s t a l The c o p p e r c r y s t a l used in the p r e s e n t work was a c y l i n d e r p o s -s e s s i n g a f jni l d i r e c t i o n a l o n g i t s a x i s . By means o f x - r a y d i f f r a c t i o n (Laue b a c k - r e f l e c t i o n ) t he o t h e r f i l l } d i r e c t i o n s o f the c r y s t a l were d e -t e r m i n e d r e l a t i v e t o a f i d u c i a l l i n e on the c r y s t a l b a s e . The c r y s t a l was then p l a c e d in a " n o t c h and p i n " a r r angemen t ( F i g u r e 16) w h i c h , upon r o t a t i o n , b r o u g h t s u c c e s s i v e { " i l l } d i r e c t i o n s o f t h e c r y s t a l i n t o the d i r e c t i o n d e -f i n e d by the two " p o i n t " d e t e c t o r s . T h u s , as can be seen f rom the d i s c u s s i o n ALUMINUM SUPPORTING PLATE Figure 16: Notch and P i n Assembly 52 in C h a p t e r s II and I I I , t he necks o f the Fermi s u r f a c e (wh ich f o r coppe r o c c u r i n the { i l l } d i r e c t i o n s ) can be examined by t h i s a r r a n g e m e n t . C. S p a t i a l S t a b i l i t y o f No t ch and P i n A s s e m b l y In the d e s i g n and c o n s t r u c t i o n o f the n o t c h and p i n a s s e m b l y c o n -s i d e r a b l e c a r e was t aken to m a x i m i z e the s p a t i a l s t a b i l i t y o f the a r r a n g e -ment . S t a b i l i t y measurements were made by v a r y i n g the v e r n i e r d i a l s e t t i n g and c a r e f u l l y o b s e r v i n g the c r y s t a l p o s i t i o n . The measurements i n d i c a t e d t h a t m o t i o n o f the c e n t r e o f mass o f the c r y s t a l f rom i t s mean p o s i t i o n was l e s s than 0 . 1 0 i n c h e s a l o n g the d i r e c t i o n ( v e r t i c a l ) o f t h e d e t e c t o r s and l e s s than 0 . 0 0 4 i n ches h o r i z o n t a l l y . T h i s r e s i d u a l m e c h a n i c a l i n s t a b i l i t y c auses an u n c e r t a i n t y o f < ~ 0 . 2 p e r c e n t i n the c o i n c i d e n c e c o u n t - r a t e . As w i l l be seen f rom the r e s u l t s p r e s e n t e d in C h a p t e r V, t h i s u n c e r t a i n t y i s s m a l l compared t o the u n c e r t a i n t y engendered by c o u n t i n g s t a t i s t i c s o r e l e c -t r o n i c d r i f t . D. C r y s t a l H o l d e r In the f i r s t run the coppe r c r y s t a l was a t t a c h e d by means o f epoxy a d h e s i v e t o a s m a l l c y l i n d r i c a l h o l d e r made o f h i g h p u r i t y a l um inum. H igh p u r i t y a luminum was used f o r the h o l d e r s i n c e a luminum has a s m a l l n e u t r o n a b s o r p t i o n c r o s s s e c t i o n and in a d d i t i o n , t he a c t i v i t y i nduced in i r r a d i a t e d a luminum has a v e r y s h o r t ha l f-1 i f e ( 2 . 30 m i n . ) . The c y l i n d e r composed o f c r y s t a l and h o l d e r was d e s i g n e d t o f i t i n t o a tube c o n t a i n i n g a n o t c h w h i c h engaged a s m a l l p i n on the h o l d e r ( F i g u r e 1 6 ) . T h i s method was used f o r t h e f i r s t e x p e r i m e n t a l r u n , but c o n t r a r y t o e x p e c t a t i o n ( L i v i n g s t o n e , 1963) the epoxy s u f f e r e d c o n s i d e r a b l e r a d i a t i o n damage so t h a t upon a r r i v a l t he c r y s t a l was found s e p a r a t e d f rom the h o l d e r . 53 T h i s made i t n e c e s s a r y t o remount the coppe r c r y s t a l on a new c y l i n d e r by use o f f r e s h epoxy b e f o r e a run c o u l d be made. In t h e second run the c r y s t a l was f o r c e f i t t e d ( b e f o r e i r r a d i a t i o n ) i n t o t h e end o f a t h i n - w a l l e d a luminum h o l d e r as i n d i c a t e d i n F i g u r e 17. T h i s h o l d e r had abou t the same g r o s s d i m e n s i o n s (4 .72 mm d i a m e t e r by 39-4 mm long ) as d i d the f i r s t e x c e p t t h a t i t had a h o l l o w c e n t r a l r e g i o n t o f u r t h e r r educe s c a t t e r i n g o f the a n n i h i l a t i o n r a d i a t i o n . W i t h t h i s a r r angemen t t h e r e i s l i t t l e a t t e n u a t i o n o f the gamma r a d i a t i o n by the t h i n c o v e r i n g (0 .033 i n c h e s ) o f a l um inum. E. H o l d e r Suppo r t A s i m p l i f i e d s c h e m a t i c d i a g r a m o f the h o l d e r s u p p o r t i s shown in F i g u r e 16. The c r y s t a l h o l d e r f i t s i n t o the s u p p o r t i n g tube w h i c h c o n t a i n s a n o t c h t h a t engages the p i n on t h e c r y s t a l h o l d e r . T h i s tube was a t t a c h e d t o a v e r n i e r d i a l (Armaco DV4) t h a t was s e c u r e l y b o l t e d t o a heavy a luminum p l a t e . T h i s heavy p l a t e i n t u r n was b o l t e d t o an a luminum beam t h a t was c a n t i l e v e r e d a t i t s o t h e r e n d . The s u p p o r t i n g tube was s t a b i l i z e d by means o f a s y s t em o f d i s c s and rods w h i c h , f o r c l a r i t y , have been o m i t t e d f rom the f i g u r e . The use o f t h e s e s t a b i l i z i n g e l e m e n t s made i t p o s s i b l e t o r educe the r e s i d u a l m e c h a n i c a l i n s t a b i l i t y o f t h e c r y s t a l t o the n e g l i g i b l e amount men-t i o n e d in t h e s t a b i l i t y d i s c u s s i o n above . To the end o f the v e r n i e r s h a f t was a t t a c h e d a s i m p l e p u l l e y wheel w h i c h c o u l d accommodate a s m a l l d i a m e t e r c o r d . By use o f the c o r d and a n o t h e r p u l l e y wheel a t t a c h e d t o a l o n g , s u p p o r t e d rod i t was p o s s i b l e to e a s i l y c o n -t r o l t he v e r n i e r d i a l s e t t i n g f rom a d i s t a n c e o f about t en f e e t f rom the r a d i o a c t i v e c o p p e r c r y s t a l . ( F i g u r e 18) The v e r n i e r d i a l was c o n v e n i e n t l y COPPER CRYSTAL \ \ \ \ \ N N ^ALUMINUM HOUSING \^,~-LOCATING PIN Figure; 1 7 : C r y s t a l ~ aridir-Holder 5k Cu CRYSTAL AND HOLDER \ ..SUPPORTING TUBE-11 VERNIER DIAL PULLEY ~^+y-"-7_ _ > z_ PULLEY CORD -ANGULAR CONTROL ROD TELESCOPE Figure 18: Remote Co n t r o l 5 5 read f rom t h i s d i s t a n c e by means o f a t e l e s c o p e mounted on a t r i p o d . W i t h t h i s a r r angemen t the r a d i a t i o n h a z a r d t o the o p e r a t o r was found t o be s m a l l ( l e s s than 5 mrem p e r 8 h o u r s ) . F. Gamma C o u n t e r s The " p o i n t " d e t e c t o r s c o n s i s t e d o f two Harshaw N A l ( T l ) c r y s t a l s mounted on RCA 6 3 4 2 p h o t o m u l t i p l i e r t u b e s . These c r y s t a l s were c y l i n d e r s one i n c h in d i a m e t e r by two i n c h e s l o n g . The 6 3 4 2 p h o t o m u l t i p l i e r tube was chosen because i t was e c o n o m i c a l and r e a d i l y a v a i l a b l e . The 634-2 a l s o has a s m a l l t r a n s i t t ime s p r e a d ( ^ 4 n s e c ) , t hus f a c i l i t a t i n g the s h o r t c o -i n c i d e n c e r e s o l u t i o n t i m e . On the o t h e r hand , the d e t e c t o r c r y s t a l s were chosen as a compromise between economy and e f f i c i e n c y . For e x a m p l e , t he measured e f f i c i e n c y o f t he i n d i v i d u a l d e t e c t o r s was about 0 . 4 5 , t o d o u b l e t h i s wou ld p r o b a b l y r e q u i r e a c r y s t a l a t l e a s t f o u r i n c h e s in d i a m e t e r and f o u r i n c h e s in h e i g h t . The d e t e c t o r a s s e m b l i e s were e n c l o s e d in l i g h t - t i g h t c y l i n d r i c a l a luminum h o u s i n g s , each p h o t o m u l t i p l i e r tube b e i n g m a g n e t i c a l l y s h i e l d e d by means o f a " C o - n e t i c N e t i c " a l l o y s h i e l d , (manu fa c tu r ed by P e r f e c t i o n M i c a Co . ) In a d d i t i o n each a luminum h o u s i n g was wrapped w i t h about 0.2 i n ches o f l e a d f o i l f o r r a d i a t i o n s h i e l d i n g p u r p o s e s . Each d e t e c t o r was r i g i d l y b o l t e d t o a l a r g e ( 6 x 6 x 3 in) l e a d b l o c k c o n t a i n i n g a c y l i n d r i c a l a p e r t u r e 0 . 2 5 i n c h e s in d i a m e t e r and 3 i n c h e s l o n g . The s o u r c e t o d e t e c t o r d i s t a n c e was t w e l v e f e e t . The c h o i c e o f s o u r c e t o d e t e c t o r d i s t a n c e and c o l l i m a t o r a p e r t u r e s i z e was based on s e v e r a l c o n s i d e r a t i o n s . These w i l l how be b r i e f l y d i s -c u s s e d . 56 The most i m p o r t a n t c o n s i d e r a t i o n was t h a t o f r e s o l u t i o n . From T a b l e I i t can be seen t h a t the d i a m e t e r o f the Fermi s u r f a c e " n e c k s " i s 2.rt%]0 mc. Hence, i n o r d e r t o o b s e r v e a r e a s o n a b l e c o i n c i d e n c e c o u n t i n g r a t e change (about h a l f t h a t e x p e c t e d f o r a p o i n t s o u r c e and p o i n t d e t e c t o r s ) a t c r y s t a l o r i e n t a t i o n s a s s o c i a t e d w i t h the n e c k s , a r e s o l u t i o n f u n c t i o n ( C h a p t e r V) w i t h f u l l w i d t h a t h a l f maximum o f ^]/2^r0 ' S r e c l u ' r e c ' . T h i s c h o i c e o f w i l l be f u r t h e r d i s c u s s e d below. In t h e p r e s e n t work, a c h o i c e o f D = 12 f e e t f o r the s o u r c e t o d e t e c t o r d i s t a n c e was p a r t i c u l a r l y c o n v e n i e n t s i n c e t h i s was the v e r t i c a l d i s t a n c e between s u c c e s s i v e f l o o r s o f the tower w h i c h housed the e x p e r i m e n t a l arrangement. S i n c e the c o n d i t i o n 2D>c^'> where f i s the c o i n c i d e n c e c i r c u i t r e s o l v i n g t i m e , i s amply s a t i s f i e d i t i s seen t h a t the cosmic ray c o n t r i b u t i o n t o t h e c o i n c i d e n c e count r a t e s h o u l d be n e g l i g i b l e . Indeed, an e x p e r i m e n t a l check (C h a p t e r V) showed t h a t the t o t a l background co-i n c i d e n c e r a t e ( i . e . the r a t e w i t h the r a d i o a c t i v e copper c r y s t a l a b s e n t ) was a l s o v e r y s m a l l . For D = 12 f e e t a c h o i c e o f d = 0 . 25 inches f o r t h e c o l l i m a t o r a p e r t u r e d i a m e t e r then g i v e s (Chapter V) a r e s o l u t i o n f u n c t i o n h a l f w i d t h o f W, ,?kr as r e q u i r e d . I /2 o The c h o i c e o f ^ -\/2^v0 w a s a compromise between c o i n c i d e n c e count r a t e and s e n s i t i v i t y t o the Fermi s u r f a c e t o p o l o g y . For example, r e d u c t i o n o f the r e s o l u t i o n f u n c t i o n w i d t h from W. ,„~ kr t o W, ..~ 2 r would reduce 1/2 o 1/2 o the c o i n c i d e n c e count r a t e by a f a c t o r ^ 1 6 (from about 60 c o u n t s per minute (maximum) t o about k c o u n t s per minut e (maximum).) Problems a s s o c i a t e d w i t h t h e s p e c i f i c a c t i v i t y o f the s o u r c e must a l s o be taken i n t o c o n s i d e r a t i o n . These problems a r e f u r t h e r d i s c u s s e d i n Chapter V. 57 G. E l e c t r o n i cs The s c h e m a t i c d i a g r a m o f F i g u r e 15 i n d i c a t e s t h e e l e c t r o n i c s used i n the e x p e r i m e n t . A p h o t o m u l t i p l i e r c u r r e n t p u l s e a r i s i n g f rom a gamma-ray i m p i n g i n g on one o f the N a I ( T l ) c r y s t a l s i s a m p l i f i e d , d i s c r i m i n a t e d , and s h a p e d , g i v i n g as o u t p u t a p o s i t i v e p u l s e about 25 nsec w i d e . Such p u l s e s were f ed i n t o a c o i n c i d e n c e u n i t w i t h r e s o l v i n g t ime s e t a t ~ 1 0 n s e c . The r e s u l t i n g c o i n c i d e n c e s were then r e c o r d e d by a E 11 OA decade s c a l e r . The E 110A was m a n u f a c t u r e d by O x f o r d E n g i n e e r i n g C o r p . ( t h i s f i r m i s no l o n g e r in e x i s t e n c e ) . The c i r c u i t d i a g r a m s o f the p r e a m p l i f i e r and s h a p i n g c i r c u i t and o f the c o i n c i d e n c e c i r c u i t a r e shown in F i g u r e s 19 and 20 r e s p e c t i v e l y . A f t e r d i f f e r e n t i a t i o n and p r e a m p 1 i f i c a t i o n , t h e n e g a t i v e c u r r e n t p u l s e f rom the p h o t o m u l t i p l i e r c o l l e c t o r was c l i p p e d by a s h o r t e d d e l a y l i n e , g i v i n g a b i p o l a r z e r o - c r o s s i n g s i g n a l . T h i s s i g n a l , when o f s u f f i c i e n t a m p l i t u d e , t r i g g e r e d a S c h m i t t - t y p e , z e r o - c r o s s o v e r d i s c r i m i n a t o r c i r c u i t . The o u t p u t f rom the d i s c r i m i n a t o r was then shaped t o g i v e a 25 n s e c o u t p u t p u l s e w h i c h went i n t o the c o i n c i d e n c e c i r c u i t shown in F i g u r e 20. In t h i s c i r c u i t a d j u s t -ment o f t h e c o i n c i d e n c e s e n s i t i v i t y c o n t r o l v a r i e s the i n p u t t h r e s h o l d l e v e l . For the p r e s e n t work t h i s c o n t r o l was a d j u s t e d t o g i v e a s t a n d i n g c u r r e n t o f about 1 ma t h r o u g h the t u n n e l d i o d e . T h i s c u r r e n t was s m a l l enough t o e n s u r e t h a t an o u t p u t p u l s e (- 10 v o l t s ) o c c u r r e d o n l y when two 25 n s e c p u l s e s f rom the p u l s e s h a p e r s a r r i v e d s i m u l t a n e o u s l y ( i e . w i t h i n the r e -s o l v i n g t ime o f 10 n sec ) a t t he c o i n c i d e n c e c i r c u i t i n p u t . H. S t a b i l i t y o f E l e c t r o n i c s S i n c e t h e c o u n t - r a t e v a r i a t i o n e x p e c t e d f rom the p r e s e n t method i s f a i r l y s m a l l ( about s i x p e r c e n t ) i t i s i m p o r t a n t t h a t d r i f t s in the c o i n c i d e n c e u\ TEST INPUT Figure 1 9 * preamp and Shaper C i r c u i t 1N752 +10 V SHORTED STUB (T>IL FEET OF 100JICOAXIAL CABLE) 10K COINCIDENCE SENSITIVITY 2.7K INPUTS (5 - A A A A y — -6 8 A TD-3 L V _ 10 TURNS ON FOT . ill CORE 3.3K 3.3K 2N1195 0.1 fit OUTPUT 3 o •H o (D O c © •H o C •rt! o 1 o 2N797 o CM | © 60 c o u n t i n g r a t e be m i n i m i z e d . S e v e r a l p r e c a u t i o n s were t aken t o e n s u r e t h a t l a r g e d r i f t s in the c o i n c i d e n c e coun t r a t e d i d not o c c u r . One such p r e -c a u t i o n i n v o l v e d m i n i m i z i n g the f l u c t u a t i o n s in t h e amb ien t t e m p e r a t u r e o f t he tower w h i c h housed t h e a p p a r a t u s . T h i s was e a s i l y done by o p e n i n g d o o r s i n t h e e x p e r i m e n t a l a r e a so f r e e c i r c u l a t i o n o f a i r f rom the main b u i l d i n g t o the towe r was p o s s i b l e . The ambien t t e m p e r a t u r e was m o n i t o r e d t h r o u g h o u t the c o u r s e o f t h e two e x p e r i m e n t s , t he maximum f l u c t u a t i o n s f rom the mean amb ien t t e m p e r a t u r e b e i n g about t 0.8 °C. D u r i n g bo th r u n s , c e r t a i n p o i n t s o f t h e e x p e r i m e n t a l c u r v e s ( C h a p t e r V) were r e p e a t e d t o e n s u r e t h a t l a r g e d r i f t s in the s e t -up d i d no t o c c u r . To w i t h i n the s t a t i s t i c a l u n c e r t a i n t y , 22 the p o i n t s were found t o be r e p r o d u c i b l e . T e s t s w i t h a Na s o u r c e i n d i c a t e d t h a t the c o i n c i d e n c e c o u n t i n g r a t e was s t a b l e to b e t t e r than t 0.6 p e r c e n t o v e r a 2k hour p e r i o d . 61 CHAPTER' V RESULTS AND CONCLUSIONS A. Int r o d u c t i on The e x p e r i m e n t a l a r r angement i s as shown in F i g u r e 15 o f C h a p t e r IV. The c r y s t a l s u p p o r t was i n c l i n e d a t an a n g l e , as shown, so t h a t the [ i l l ] a x i s was the a x i s o f r o t a t i o n and so t h a t upon r o t a t i o n , { i n } d i r e c -t i o n s wou ld s u c c e s s i v e l y p o i n t i n the d i r e c t i o n o f the d e t e c t o r s . These 111 j d i r e c t i o n s o f the coppe r c r y s t a l had p r e v i o u s l y been d e t e r m i n e d by s t a n d a r d x - r a y t e c h n i q u e s a t t he Depar tment o f M e t a l l u r g y x - r a y f a c i l i t i e s . As p o i n t e d ou t in C h a p t e r I I I , one wou ld e x p e c t the c o i n c i d e n c e coun t r a t e t o r i s e a t c r y s t a l o r i e n t a t i o n s c o r r e s p o n d i n g t o the " n e c k s " o f t h e Fermi s u r f a c e , w h i c h f o r c o p p e r o c c u r i n the { i l l } d i r e c t i o n s . A l s o , as a consequence o f the t h r e e - f o l d symmetry about a { ' l l j a x i s one wou ld e x p e c t peaks in t h e c o i n c i d e n c e c o u n t r a t e a t i n t e r v a l s o f 120°. From the a v a i l a b l e e x p e r i m e n t a l d a t a on t h e Fermi s u r f a c e o f coppe r ( Tab l e I) one wou ld e x p e c t f o r a p o i n t c r y s t a l and p o i n t d e t e c t o r s an e f f e c t o f about 13 p e r c e n t on t h e b a s i s o f t he a s s u m p t i o n s o f C h a p t e r IV. T h i s e s t i m a t e does not i n c l u d e h i g h e r momentum e f f e c t s such as c o r e a n n i h i l a t i o n . B. Expe r iment For t h e two runs d e s c r i b e d b e l o w , the d i s c r i m i n a t o r t h r e s h o l d s e t t i n g s o f the p u l s e s h a p e r s were about 0 . 6 v o l t s . T h i s v o l t a g e c o r r e s p o n d t o a gamma-ray e n e r g y o f about 0 .3 Mev so t h a t each gamma-ray e x p e n d i n g more than 0 .3 Mev o f e n e r g y i n one o f the N a l ( T l ) c r y s t a l s gave r i s e t o a 25 n s e c o u t p u t p u l s e f rom the a s s o c i a t e d s h a p i n g c i r c u i t . The s o u r c e s t r e n g t h a t t h e b e g i n n i n g o f each run was about 50 mCi o f p o s i t r o n a c t i v i t y , t he d u r a t i o n o f e a ch run b e i n g about f o u r h a l f - l i v e s o r about 50 h o u r s . For 50 mCi o f p o s i t r o n a c t i v i t y the random c o i n c i d e n c e r a t e w i l l be n e g l i g i b l e ( ^ 0 . 1 p e r c e n t o f t h e t r u e c o i n c i d e n c e r a t e ) . No a t t e m p t was made t o measure t h i s r a t e s i n c e the h a l f - l i f e o f the s o u r c e was so s h o r t ( 1 2 . 9 h o u r s ) . The backg round c o i n c i d e n c e r a t e ( i . e . the r a t e w i t h the r a d i o a c t i v e c o p p e r c r y s t a l a b s e n t ) was measured and found t o be about two c o u n t s p e r h o u r . T h i s r a t e i s n e g l i g i b l e compared t o the t r u e c o i n c i -dence r a t e , e x c e p t nea r t h e end o f the runs where i t c o n t r i b u t e d about 0 .4 p e r c e n t t o the t r u e c o i n c i d e n c e r a t e . The a x i s d e f i n e d by the c e n t e r o f mass o f the coppe r c r y s t a l and the c e n t e r s o f t he " p o i n t " d e t e c t o r f a c e s was v e r t i c a l so t h a t a l i g n m e n t o f t he c r y s t a l and d e t e c t o r s c o u l d be a c h i e v e d by means o f a s i m p l e plumb bob a r r a n g e m e n t . The e r r o r in t h i s a l i g n m e n t was e s t i m a t e d t o be l e s s than 1 mm T h i s r e s i d u a l m i s a l i g n m e n t a f f e c t s the c o i n c i d e n c e coun t r a t e by l e s s than 0.1 p e r c e n t . 63 The h i g h p u r i t y c o p p e r s i n g l e c r y s t a l used in the p r e s e n t work was o b t a i n e d f rom M e t a l s R e s e a r c h L t d . o f C a m b r i d g e , Eng l and and was t he rma l n e u t r o n i r r a d i a t e d a t the C h a l k R i v e r f a c i l i t i e s o f A t o m i c Energy o f Canada L t d . The f i l l ] d i r e c t i o n s o f the c r y s t a l were d e t e r m i n e d by s t a n d a r d x - r a y t e c h n i q u e s a t the Depar tment o f M e t a l l u r g y x - r a y f a c i l i t i e s b e f o r e the n e u t r o n i r r a d i a t i o n s were p e r f o r m e d . C. R e s u l t s To t e s t t he c o l l i n e a r p o i n t geometry method two runs were made. The r e s u l t s a r e shown in F i g u r e 2 1 . In the f i r s t r u n , an e x p l o r a t o r y o n e , most p o i n t s were t a k e n w i t h o n l y 2 .5 p e r c e n t s t a t i s t i c s , mak ing i t n e c e s s a r y t o p a i r p o i n t s i n o r d e r t o r educe the e r r o r (by ~1 / ) . However , in the second run a s m a l l e r a n g u l a r range was c o v e r e d , mak ing p o s s i b l e b e t t e r s t a t i s t i c s . The r e s u l t s o f t he second run a r e a l s o shown in F i g u r e 2 1 . In the r e s u l t s f o r t h e second run most p o i n t s r e p r e s e n t about 64-00 c o u n t s thus g i v i n g 1.25 p e r c e n t s t a t i s t i c s , an improvement by a f a c t o r o f two o v e r the f i r s t r u n . In t h i s second run the r e l a t i v e o r i e n t a t i o n o f c r y s t a l and h o l d e r d i f f e r e d f rom t h a t o f t he f i r s t run by about 60 ° . The peak in the c o u n t i n g r a t e c u r v e was a l s o s h i f t e d by about 6 0 ° , as e x p e c t e d . It i s seen t h a t the r e s u l t s p r e s e n t e d a r e c o n s i s t e n t w i t h an e f f e c t o f about s i x p e r c e n t . A t t h e l e f t hand s i d e o f t h e c u r v e f o r the f i r s t run can be seen some e v i d e n c e o f the e x p e c t e d 120° p e r i o d i c i t y . The c u r v e s shown in F i g u r e 21 have been c o r r e c t e d f o r decay o f the s o u r c e ^\/2 ~ ^.9 hou r s ) and f o r backg round c o i n c i d e n c e s . A l l r e a d i n g s were t aken a t room t e m p e r a t u r e . I f one assumes a u n i f o r m l y dense Fermi vo lume and assumes t h a t h i g h e r momentum e f f e c t s can be n e g l e c t e d , then t h e peaks i n the c u r v e s shown above s h o u l d be a measure o f t he d i a m e t e r o f t he Fermi volume in the i?9 65 { i l l } d i r e c t i o n s . The w i d t h o f t h e s e peaks s h o u l d be a measure o f the d i a m e t e r o f the " n e c k s " t h a t o c c u r i n the £l 1 1^ d i r e c t i o n s . The d i s -c u s s i o n below i n d i c a t e s t h a t t h i s s i m p l e i n t e r p r e t a t i o n o f the r e s u l t s i s c o n s i s t e n t w i t h the known d i m e n s i o n s o f t h e copper Fermi s u r f a c e . D. I n t e r p r e t a t i o n o f t h e R e s u l t s 1. E f f e c t s o f F i n i t e Source and D e t e c t o r S i z e W i t h the p r e s e n t e x p e r i m e n t a l c o n f i g u r a t i o n t h e change i n c o i n c i -dence c o u n t i n g r a t e due t o the p r e s e n c e o f the " n e c k s " o f the Fermi s u r f a c e w i l l be s u b s t a n t i a l l y reduced from t h e e x p e c t e d 13 p e r c e n t because o f t h e f i n i t e c r y s t a l and d e t e c t o r s i z e . In o r d e r t o e s t i m a t e the e x p e c t e d s i z e o f t h i s e f f e c t , the e x p e r i m e n t a l Fermi s u r f a c e as g i v e n i n T a b l e I can be r o u g h l y a p p r o x i m a t e d by a sphere ("the b e l l y " ) o f r a d i u s p Q upon which a r e mounted e i g h t " n e c k s " w h i c h a r e t r u n c a t e d cones. The r e l e v a n t d i m e n s i o n s o f such a cone a r e i n d i c a t e d in F i g u r e 22. The f i n i t e d e t e c t o r and c r y s t a l s i z e can be a p p r o x i m a t e l y a c c o u n t e d f o r by use o f a r e s o l u t i o n o r " d e n s i t y " f u n c t i o n . F i g u r e 22; Fermi S u r f a c e Neck D e t a i l s 66 By the use o f t h i s d e n s i t y f u n c t i o n the "Mass" o r e f f e c t i v e volume o f a p p r o p r i a t e r e g i o n s o f momentum space can be found and an e s t i m a t e made o f the e x p e c t e d c o i n c i d e n c e c o u n t i n g r a t e change. T h i s i s done by p e r f o r m i n g an e l e m e n t a r y volume i n t e g r a t i o n o v e r a c y l i n d e r c o n t a i n i n g a neck a t each end and then comparing the r e s u l t w i t h t h a t o b t a i n e d from a s i m i l a r i n t e g r a t i o n performed in a d i r e c t i o n in wh i c h t h e r e a r e no n e c k s . L e t t i n g t h e e f f e c t i v e volume o f a c y l i n d e r i n a d i r e c t i o n i n w h i c h t h e r e a r e no necks be 2m^  and t h a t o f a p a i r o f necks be 2m^, i t i s seen t h a t a measure o f the r e l a t i v e count r a t e v a r i a t i o n i s g i v e n by the r a t i o m 2 where m l m1 = | r P(r) d t d z da and m. w I i o • 0 b^+Aa+oc (5-0 iff I n2 = J r P ( r ) dz d r de + 2T|f j(r)[h +(b-fH*n«Q d r and _P(r) i s the r e s o l u t i o n f u n c t i o n . 2. R e s o l u t i o n F u n c t i o n f o r F i n i t e D e t e c t o r s and P o i n t C r y s t a l For the case o f c i r c u l a r d e t e c t o r s o f d i a m e t e r d and a " p o i n t " c r y s t a l s o u r c e t h e r e s o l u t i o n f u n c t i o n i s P ( r ) 1 - Dr r < d d D (5-2) / ( r ) = 0 r > f where D i s the s o u r c e t o d e t e c t o r d i s t a n c e . T h i s e x p r e s s i o n f o r j°(r) was o b t a i n e d as a g e n e r a l i z a t i o n o f a n u m e r i c a l example; an a n a l y t i c d e r i v a -t i o n appears t o be q u i t e d i f f i c u l t . For the p r e s e n t e x p e r i m e n t a l arrangement 67 -3 one has d = 1.70 x 10 . I f one now chooses ( C h a p t e r I) D -3 p = 5 .10 mc x 10 J o b = 1.00 mc x 10 , (5-3) h = p i u - P Q = 0 . 70 x 10 mc k e q u a t i o n s (5-1) and (5-2) y i e l d m^ = 0 . 1 2 . Here p^ i s t a k e n t o be the a ve r age b e l l y r a d i u s ( C h a p t e r I ) . I t i s i n t e r e s t i n g t o n o t e t h a t i f the necks a r e t a k e n t o be c y l i n d e r s o f r a d i u s b i n s t e a d o f t r u n c a t e d c o n e s , the e f f e c t d r o p s t o - 0 . 0 8 7 , a s u b s t a n t i a l change , m, 3 . R e s o l u t i o n F u n c t i o n f o r F i n i t e D e t e c t o r s and F i n i t e C r y s t a l S i n c e the coppe r c r y s t a l used in the p r e s e n t work was a c y l i n d e r whose a x i s was i n c l i n e d a t an a n g l e o f 70 .5 ° w i t h the l i n e d e f i n e d by the d e t e c t o r s , i t i s d i f f i c u l t t o g i v e a s i m p l e a n a l y t i c e x p r e s s i o n f o r t h e r e s o l u t i o n o r d e n s i t y f u n c t i o n . A r e a s o n a b l e f i r s t c h o i c e o f r e s o l u t i o n f u n c t i o n wou ld appear t o be the f o l l o w i n g ; fir) « e V 2 - 0 / (5-4) where £ i s an e f f e c t i v e s o u r c e d i m e n s i o n (as " s e e n " by a d e t e c t o r ) . T h i s f u n c t i o n i s s i m i l a r t o t h e r e s o l u t i o n f u n c t i o n used by Lang (1956) f o r c o r -r e c t i o n o f a n g u l a r c o r r e l a t i o n r e s u l t s (wide s l i t geometry ) in w h i c h the r a t i o o f s o u r c e w i d t h t o d e t e c t o r s l i t w i d t h was 0 . 5 . I f , f o r t h e p r e s e n t a r r angement one assumes an e f f e c t i v e s o u r c e s i z e o f £ k mm one f i n d s — r_2 f ( r ) <V e 2 68 U s i n g t h i s r e s o l u t i o n f u n c t i o n and the p a r a m e t e r s (5-3) one o b t a i n s = 0 .082 w h i c h i s somewhat s m a l l e r than the p r e v i o u s v a l u e 0 .12 o b t a i n e d a s suming n e g l i g i b l e s o u r c e s i z e . I n c l u d i n g , now, f o r v a r i a t i o n s i n b p e r m i t t e d by the p r e s e n t l o w - t e m p e r a t u r e d a t a ( Tab l e I ) , t he r e s u l t a n t u n c e r t a i n t y in i s 0 . 0 0 4 . The dependence o f the c a l c u l a t e d v a l u e s o f ™2 on o t h e r p a r a m e t e r s such as the v a l u e chosen f o r <tf i s i l l u s t r a t e d by r e p e a t i n g the c a l c u l a t i o n f o r «< = T . The c a l c u l a t i o n y i e l d s m_ = 0 .072 3 'Z m l + 0 . 0 0 4 . Thus the p r e s e n t method s h o u l d be q u i t e s e n s i t i v e t o the d e t a i l e d shape o f the Fermi s u r f a c e . I t s h o u l d be no ted t h a t t h e e x p e r i m e n t a l r e s u l t s a r e c o n s i s t e n t w i t h a c o n s i d e r a b l e number o f r e s o l u t i o n f u n c t i o n s . For example the r e s o -2 2 - r -_r l u t i o n f u n c t i o n s . j*(r) e 2 and ^ ( r ) qf e4 g i v e v a l u e s o f o f 0 .082 and 0 . 0 5 0 r e s p e c t i v e l y b o t h o f w h i c h a re c o n s i s t e n t w i t h the r e s u l t s shown in F i g u r e 2 1 . The f u l l w i d t h a t h a l f maximum o f the c o r r e s p o n d i n g a n g u l a r c o r r e l a t i o n c u r v e s i s 3 2 ° and 45° r e s p e c t i v e l y ( A p p e n d i x B) however ; so t h a t the second r e s o l u t i o n f u n c t i o n may g i v e a b e t t e r f i t t o the d a t a shown in F i g u r e 2 1 . D e s p i t e t h i s d i f f i c u l t y , i t i s seen t h a t the e x p e r i m e n t a l r e s u l t s a re c o n s i s t e n t w i t h a model o f t he Fermi s u r f a c e in w h i c h " n e c k s " o c c u r in the - [ l l l j d i r e c t i o n s , s u b t e n d i n g an a n g l e o f "X ~ 20° a t k* = 0 (see T a b l e I ) . T h i s a n g l e i s d e f i n e d by the ( c i r c u l a r ) r e g i o n o f c o n t a c t o f the Fermi s u r f a c e w i t h the B r i l l o u i n zone b o u n d a r y : ~\ = 2 t a n / . k j . j where k^ k m i s the neck r a d i u s a t the zone boundary and k ( ) ( i s t he d i s t a n c e o f t h i s zone boundary f rom k = 0 . T h u s , t a k i n g i n t o c o n s i d e r a t i o n the u n c e r t a i n t i e s in t h e a v a i l a b l e Fermi s u r f a c e d i m e n s i o n s ( T a b l e I) and in the r e s o l u t i o n f u n c t i o n one i s l e d t o e x p e c t an e f f e c t o f about f i v e t o e i g h t p e r c e n t in the p r e s e n t work i f t he Fermi vo lume i s assumed t o be u n i f o r m l y dense and i f h i g h e r momentum e f f e c t s can be n e g l e c t e d . A l t h o u g h t h i s i s c o n s i s t e n t w i t h the r e s u l t s shown in F i g u r e 21, i t i s e v i d e n t t h a t a more a c c u r a t e knowledge o f the r e s o l u t i o n o f f u n c t i o n i s e s s e n t i a l i f the method i s t o b e ^ q u a n t i t a t i v e v a l u e . I t i s b e l i e v e d t h a t a more c o n v e n i e n t c h o i c e o f c r y s t a l o r i e n t a t i o n s h o u l d p e r m i t a good e s t i m a t e o f t he r e s o l u t i o n f u n c t i o n , t hus o b v i a t i n g t h i s s o u r c e o f d i f f i c u l t y . E. I n t e r p r e t a t i o n o f t h e R e s u l t s The above i n t e r p r e t a t i o n o f the r e s u l t s n e g l e c t e d c o m p l i c a t i o n s due t o h i g h e r momentum e f f e c t s and ion c o r e a n n i h i l a t i o n c o n t r i b u t i o n s . However , t he h i g h e r momentum components a s s o c i a t e d w i t h the c o n t a c t o f the Fermi s u r f a c e w i t h a hexagona l zone f a c e w i l l no t c ause d i f f i c u l t y s i n c e t h e a f f e c t e d momentum s t a t e s w i l l u s u a l l y be m e r e l y t r a n s l a t e d by a r e c i p r o c a l l a t t i c e v e c t o r t h a t l i e s e s s e n t i a l l y a l o n g the a x i s o f t he s a m p l i n g c y l i n d e r . Fo r o t h e r d i r e c t i o n s normal t o a zone b o u n d a r y , a s i m i l a r c o n s i d e r a t i o n a p p l i e s However , i f the c y l i n d e r a x i s i s not a p p r o x i m a t e l y p e r p e n d i c u l a r t o a zone f a c e s m a l l l o s s e s in the c o u n t i n g r a t e w i l l o c c u r . S i n c e in c o p p e r the non-hexagona l zone f a c e s a r e r e l a t i v e l y d i s t a n t f rom the c e n t e r o f the z o n e , i t appea r s r e a s o n a b l e to assume t h a t t h i s c o n t r i b u t i o n w i l l be f a i r l y s m a l l and w i l l p r o b a b l y be masked by the much l a r g e r c o r e a n n i h i l a t i o n c o n t r i b u t i o n . By use o f the r e s u l t s on c o r e a n n i h i l a t i o n g i v e n by Berko and P l a s k e t t (1958) an e s t i m a t e o f t h e c o r e a n n i h i l a t i o n c o n t r i b u t i o n can be made. T h i s e s t i m a t e i s made by a p p r o x i m a t i n g the c o r e a n n i h i l a t i o n a n g u l a r c o r r e -l a t i o n c u r v e by a p a r a b o l a and c o n s i d e r i n g t h i s p a r a b o l a t o c o r r e s p o n d t o a 70 Fermi s u r f a c e w i t h P Q = 1 * mc x 10 . By p r o p e r l y w e i g h i n g the c o n -t r i b u t i o n o f t h e " i o n c o r e Fermi s p h e r e " w i t h t h a t o f t he ( c o n c e n t r i c ) Fermi s p h e r e a s s o c i a t e d w i t h the c o n d u c t i o n e l e c t r o n s , i t can be r o u g h t l y e s t i m a t e d ( A p p e n d i x C) t h a t c o r e a n n i h i l a t i o n s h o u l d l ower the e x p e c t e d e f f e c t o f 8.2 p e r c e n t t o about 6 p e r c e n t . T h i s i s no t i n c o n s i s t e n t w i t h t h e p r e s e n t r e s u l t s . A more d e t a i l e d c a l c u l a t i o n does no t appea r to be j u s t i f i e d by the s t a t i s t i c s o r r e s o l u t i o n o f t h e p r e s e n t w o r k . In a d d i t i o n , r e c e n t c a l c u l a -t i o n s by C a r b o t t e (1966) i n d i c a t e t h a t c o r e a n n i h i l a t i o n c a l c u l a t i o n s may be c o n s i d e r a b l y more d i f f i c u l t than e a r l i e r b e l i e v e d . F. A c c u r a c y A t t a i n a b l e w i t h the Method I t i s p e r h a p s o f i n t e r e s t t o c o n s i d e r the u l t i m a t e a c c u r a c y a t t a i n a b l e w i t h the p r e s e n t m e t h o d , in o r d e r t h a t i t may be compared w i t h the o t h e r methods used f o r Fermi s u r f a c e d e t e r m i n a t i o n . V a r i o u s a s p e c t s o f t h i s t o p i c a r e d i s c u s s e d b e l o w . 1 . R e s o l u t i o n and C o u n t i n g Ra te An i m p o r t a n t l i m i t i n g f a c t o r in the p r e s e n t method i s t h e low c o u n t i n g r a t e . In o r d e r t o improve t h e r e s o l u t i o n by a f a c t o r o f two ( i . e . t o r educe the h a l f - w i d t h o f the r e s o l u t i o n f u n c t i o n by a f a c t o r o f two) i t i s n e c e s s a r y t o r educe the d e t e c t o r s o l i d a n g l e by a f a c t o r o f f o u r . T h i s r e d u c t i o n in d e t e c t o r s o l i d a n g l e r educes the c o i n c i d e n c e c o u n t i n g r a t e by a f a c t o r ' v J l 6 . T h i s e s t i m a t e i s f o r a " p o i n t " c r y s t a l . For a f i n i t e c r y s t a l the d e c r e a s e wou ld be even more s e v e r e . T h u s , in o r d e r t o improve the r e s o -l u t i o n by a f a c t o r o f two , w i t h the r e s t o f t he e x p e r i m e n t a l a r rangement u n -c h a n g e d , t h e s o u r c e s t r e n g t h used in the p r e s e n t work wou ld need to be i n -c r e a s e d f rom ~ 50 mCi t o about a C u r i e . P rob l ems a s s o c i a t e d w i t h the h a n d l i n g o f such s o u r c e s wou ld become much more s e v e r e t h a n t h o s e e n c o u n t e r e d so f a r . 71 However , a more i m p o r t a n t l i m i t a t i o n a r i s e s f rom c o n s i d e r a t i o n s o f s p e c i f i c a c t i v i t y . W i t h p r e s e n t l y a v a i l a b l e t he rma l n e u t r o n f l u x e s a t AECL , C h a l k R i v e r , i t wou ld be d i f f i c u l t t o o b t a i n a p o s i t r o n a c t i v i t y o f much more than a C u r i e in the p r e s e n t c r y s t a l . O t h e r p o s s i b l e improvements i n v o l v e an i n c r e a s e d d e t e c t i o n e f -f i c i e n c y . T h i s c o u l d be a c c o m p l i s h e d by l o w e r i n g the d i s c r i m i n a t i o n l e v e l s o f t h e p u l s e s h a p e r s and by u s i n g l a r g e r N a l ( T l ) c r y s t a l s . I t may then be p o s s i b l e t o improve the i n d i v i d u a l d e t e c t o r e f f i c i e n c y o f 0 .45 by a f a c t o r "2 so t h a t the c o i n c i d e n c e coun t r a t e c o u l d be improved by a f a c t o r 4 . I t i s t h u s e v i d e n t f rom t h i s d i s c u s s i o n t h a t the r e s o l u t i o n ( u s i n g the p r e s e n t c r y s t a l as a s o u r c e ) c anno t be improved by more than a f a c t o r <-»3 i f r e a s o n a b l e ( l e s s t han 40 f e e t ) s o u r c e t o d e t e c t o r d i s t a n c e s a r e to be u s e d . An a p p r o -p r i a t e l o n g - l i v e d s o u r c e , o f c o u r s e , wou ld be p r o h i b i t i v e l y e x p e n s i v e , a t p r e s e n t . 2 . S t a b i 1 i t y A n o t h e r i m p o r t a n t l i m i t i n g f a c t o r i s t he m e c h a n i c a l and e l e c t r i c a l s t a b i l i t y o f t he a r r a n g e m e n t . The sys tem c o u l d be made s u f f i c i e n t l y s t a b l e m e c h a n i c a l l y so t h a t the u n c e r t a i n t y i n the c o i n c i d e n c e c o u n t i n g r a t e c o u l d be r educed t o l e s s than 0.1 p e r c e n t . However , t he u n c e r t a i n t y in the c o -i n c i d e n c e c o u n t i n g r a t e due t o the i n s t a b i l i t y o f the e l e c t r o n c i s c anno t be r educed be low ^ 0 . 1 p e r c e n t w i t h o u t g r e a t l y i n c r e a s e d c o m p l e x i t y o f i n s t r u -m e n t a t i o n . Thus , f o r c o i n c i d e n c e coun t r a t e s in the 1 t o 10 sec "^ r ange , c o u n t i n g s t a t i s t i c s wou ld appea r t o s e t the p r a c t i c a l l i m i t on the p r e c i s i o n a t t a i n a b l e . A l l o w i n g , a t m o s t , a few hours p e r p o i n t , t o t a l c o u n t s o f no more than a few t e n s o f t housands c o u l d be o b t a i n e d l e a d i n g t o s t a t i s t i c a l u n c e r t a i n t i e s ~1 p e r c e n t p e r p o i n t . G. Di s c u s s ion By d e c r e a s i n g the s o l i d a n g l e a s s o c i a t e d w i t h a d e t e c t o r by a f a c t o r o f f o u r and i n c r e a s i n g the d e t e c t o r e f f i c i e n c y by a f a c t o r ^ 2 and u s i n g t h e maximum a v a i l a b l e t he rma l n e u t r o n f l u x e s i t s h o u l d be p o s s i b l e t o improve the r e s o l u t i o n o f the p r e s e n t a r r angement by a f a c t o r o f two w h i l e m a i n t a i n i n g s t a t i s t i c s o f »*» 1 p e r c e n t . The e l e c t r o n i c d r i f t c o u l d be r e -duced t o r-0.1 p e r c e n t by use o f r e c y c l i n g p r o c e d u r e s . In a d d i t i o n , a more c o n v e n i e n t c h o i c e o f c r y s t a l o r i e n t a t i o n s h o u l d p e r m i t a good e s t i m a t e o f the r e s o l u t i o n f u n c t i o n . T h u s , in the absence o f c o r e a n n i h i l a t i o n e f f e c t s and o t h e r h i g h e r momentum e f f e c t s , t he p r e s e n t t e c h n i q u e wou ld be c o m p e t i t i v e w i t h the o t h e r methods ( T a b l e I) e x c e p t pe rhaps f o r the de Haas van A l p h e n e f f e c t w i t h w h i c h , f o r e x a m p l e , t he Fermi s u r f a c e o f p o t a s s i u m was found to be s p h e r i c a l t o ^ 0 . 1 p e r c e n t ( S h o e n b e r g , 1965). I f c a r e f u l measu remen t s , f o r v a r i o u s c r y s t a l o r i e n t a t i o n s , a r e made w i t h such improved r e s o l u t i o n i t may be p o s s i b l e t o s e p a r a t e the c o -i n c i d e n c e c o u n t i n g r a t e c o n t r i b u t i o n due t o a n n i h i l a t i o n o f p o s i t r o n s w i t h e l e c t r o n s o f the ion c o r e s f rom the c o i n c i d e n c e c o u n t i n g r a t e due t o a n -n i h i l a t i o n o f p o s i t r o n s w i t h v a l e n c e e l e c t r o n s . I f such a s e p a r a t i o n p r o v e s p o s s i b l e , i t may p e r m i t measurements o f Fermi s u r f a c e d i a m e t e r s t o an a c c u r a c y o f ~ 1 p e r c e n t in c e r t a i n a l l o y s composed o f such t r a c t a b l e m e t a l s i f c e r t a i n a s s u m p t i o n s can be made abou t t h e e f f e c t o f a l l o y i n g on c o r e a n n i h i l a t i o n o f the c o n s t i t u e n t m e t a l s . It m i g h t a l s o be p o s s i b l e t o i n v e s t i g a t e h i g h e r momentum e f f e c t s a s s o c i a t e d w i t h t h e l o c a l i z a t i o n o f the p o s i t r o n in the p e r i o d i c c r y s t a l p o t e n t i a l o r i t may be p o s s i b l e t o examine h i g h e r momentum e f f e c t s a s s o c i a t e d w i t h p r o x i m i t y o f t he B r i l l o u i n zone b o u n d a r i e s . 73 H. C o n c l u s i ons By use o f a new p o s i t r o n a n n i h i l a t i o n t e c h n i q u e e m p l o y i n g " c o l l i n e a r p o i n t g e o m e t r y " t he Fermi s u r f a c e o f c o p p e r was f ound t o be a n i s o t r o p i c a t room t e m p e r a t u r e . The r e s u l t s o f t he p r e s e n t work a r e c o n -s i s t e n t w i t h t h e . p i c t u r e o f t h e Fermi s u r f a c e h a v i n g " n e c k s " whose c o n t a c t w i t h t h e hexagona l f a c e s o f t he f i r s t B r i l l o u i n zone s u b t e n d s an a n g l e o f abou t 20° a t t h e o r i g i n o f k*-space as i n d i c a t e d by the more a c c u r a t e work done by o t h e r w o r k e r s n e a r a b s o l u t e z e r o . The r e s u l t s o b t a i n e d in t h i s work a r e no t o f s u f f i c i e n t p r e c i s i o n t o i n d i c a t e the e x t e n t o f t he c o n t r i b u t i o n f rom c o r e a n n i h i l a t i o n s . However , w i t h b e t t e r s t a t i s t i c s and r e s o l u t i o n , a more d e t a i l e d t r e a t m e n t o f the c o n -t r i b u t i o n o f c o r e a n n i h i l a t i o n s w i l l be r e q u i r e d b e f o r e f u r t h e r d e t a i l s o f t he Fermi s u r f a c e t o p o l o g y can be a s c e r t a i n e d . S i n c e t h e s t a t i s t i c s and r e s o l u t i o n can be improved s u b s t a n t i a l l y o v e r t h o s e employed in the p r e s e n t work i t s h o u l d be p o s s i b l e to s t u d y , in d e t a i l , o v e r a much l a r g e r t e m p e r a t u r e range than i s the case f o r o t h e r m e t h o d s , the Fermi s u r f a c e o f v a r i o u s m e t a l s . In a d d i t i o n i t s h o u l d be p o s s i b l e t o a p p l y the method t o a s y s t e m a t i c s t u d y o f a l l o y s . T h i s i s o f c o n s i d e r a b l e i n t e r e s t s i n c e l i t t l e work has been done on the Fermi s u r f a c e o f a l l o y s . . In a d d i t i o n t o i t s p r o m i s e as a t o o l f o r the i n v e s t i g a t i o n o f Fermi s u r f a c e s , the method o f c o l 1 i n e a r p o i n t geometry may p r o v e t o be o f some v a l u e in the s t u d y o f c o r e e l e c t r o n a n n i h i l a t i o n . W i t h improved r e s o -l u t i o n and s t a t i s t i c s i t may be p o s s i b l e t o s e p a r a t e the c o n t r i b u t i o n t o t h e c o i n c i d e n c e c o u n t i n g r a t e due t o c o r e e l e c t r o n s f rom the c o n t r i b u t i o n a r i s i n g f rom the c o n d u c t i o n e l e c t r o n s . T h i s wou ld make p o s s i b l e an e x p e r i m e n t a l t e s t o f c o r e a n n i h i l a t i o n c a l c u l a t i o n s . In c l o s i n g i t s h o u l d pe rhaps be n o t e d t h a t v e r y l i t t l e work on Fermi s u r f a c e s has been done by means o f p o i n t geome t r y . In v i ew o f the f a c t t h a t the p r e l i m i n a r y r e s u l t s o f F u j i w a r a (1965) o b t a i n e d by use o f non-c o l l i n e a r p o i n t geometry show much more s t r u c t u r e than do t h o s e o b t a i n e d by Be rko and P l a s k e t t (1958) u s i n g the w ide s l i t m e t h o d , the n o n c o l 1 i n e a r p o i n t geometry method may p r o v e t o be a u s e f u l complement t o the c o l 1 i n e a r p o i n t geomet ry me thod . F i n a l l y , i t can be n o t e d t h a t the adven t o f l i t h i u m d r i f t e d germanium d e t e c t o r s ( D e a r n a l e y and N o r t h r o p , 1966) p o s s e s s i n g e x c e l l e n t e n e r g y r e s o l u t i o n : may make p o s s i b l e f u r t h e r improvements in the p o i n t geometry me thod . Fo r e x a m p l e , by c o m b i n i n g some D o p p l e r s h i f t d i s c r i m i n a t i o n t o g e t h e r w i t h the u s u a l a n g u l a r s e l e c t i o n , one c o u l d a c h i e v e a sys tem w i t h v e r y much more s e n s i t i v i t y t o the Fermi s u r f a c e t o p o l o g y s i n c e the r e g i o n o f momentum space sampled by t h e p r o p o s e d method c o u l d be made v e r y s m a l l . T h u s , f o r the c o l 1 i n e a r c a s e , one c o u l d sample a c y l i n d e r i n momentum space e x t e n d i n g f rom p^ = o<P0 t o p^ = |Sp Q i n s t e a d o f f rom p^ = - p^ t o P z = + p Q . Here <X and ^ a r e p o s i t i v e c o n s t a n t s and p Q i s the Fermi momentum. I t i s seen t h a t the method h o l d s p r o m i s e f o r t h e s t u d y o f h i g h e r momentum components ( p > p ) s i n c e o( and/? c o u l d be chosen t o be g r e a t e r than u n i t y . 75 APPENDIX A SOLUTION OF A B E L ' S INTEGRAL EQUATION The s o l u t i o n o f A b e l ' s i n t e g r a l e q u a t i o n S (x) = " ( y ) d y , (x - vT 0< <* < 1 , f (a) = 0 i s g i v e n ( B o c h e r , 1929) by u (x ) = s i n 7 T < * 7T d | S ( t ) d t dx J (x .- t ) - * A. (A-l) I t i s d e s i r e d t o f i n d the s o l u t i o n o f t h e r e l a t e d i n t e g r a l e q u a t i o n f ( x ) = Ok f q ( t ) d t x 2 2 L e t w = - x , v = - t and dv = - 2 t d t and c o n s i d e r r ( - x Z ) = r (w) = - f ( x ) -G(v) = q ( t ) = q( r=y) - 2 t - 2 iT^v" r q ( t ) d t ( t * X V ) t a. r G(v )dv (w - vR whe re (A-2) A p p l i c a t i o n o f ( A- l ) g i v e s G(w) =-( d ( r ( v ) d v T dw } (w - v) d_f = dy dx = -] dw dx dw 2x dx VI = - _L d 2TA dx r ( v )dv (w - v ) * s i nee 7.6 S u b s t i t u t i o n f o r v and w y i e l d s G(w) = - _ i d / C-f(t) J [ -2t dt] 2 7Tx dx ( t l - x T ) t From (A-2) one has G(w) = q NT^W = q(x) -2 ^ -2x (A-3) Combining t h i s l a s t e q u a t i o n w i t h (A-3) then g i v e s the d e s i r e d r e s u l t g(x) -1 A * dx •a r t f ( t ) d t ( t ^ - x* )* APPENDIX B EXPECTED ANGULAR CORRELATION CURVE WIDTH C e r t a i n g e o m e t r i c r e l a t i o n s needed f o r the c a l c u l a t i o n w i l l be f i r s t o u t l i n e d . C o n s i d e r the r o t a t i o n , t h r o u g h an a n g l e ^ , o f a v e c t o r "p* about the a x i s . For the two p o s i t i o n s (1 and 2) one ^has f rom the a d j a c e n t f i g u r e t h a t P, = (P« s i n f , 0 , p Q cosf ) W ( B-l) ^ •p*2 = ( p „ s i n y c o s ^ , p Q s i n y s i n ^ , p Q c o s ^ ) The p e r p e n d i c u l a r d i s t a n c e f rom p o i n t p*^  t o the l i n e d e f i n e d by p*2 can be o b t a i n e d a f t e r d e t e r m i n a t i o n o f the c o n s t a n t " * in the e q u a t i o n U c ? 2 - P, ) • P 2 = 0 (B-2) S o l v i n g f o r the p e r p e n d i c u l a r d i s t a n c e d ^ = |<*p*2 - P*j J o n e f i n d s J 2 2 1 + cos \j/ - cos <j> ( cos <j> s i n ^ + 2 cos y) In o r d e r t o c a l c u l a t e the shape (hence the h a l f w i d t h o f the a n g u l a r c o r r e l a t i o n c u r v e s e v e r a l a p p r o x i m a t i o n s a re made; 1. The Fermi s u r f a c e " n e c k s " a re assumed t o be a p p r o x i m a t e d by c y l i n d e r s o f h e i g h t h and d i a m e t e r a , = b + _h. Here b and 1* 2 h have the same mean ing as i n C h a p t e r V. 2. An e f f e c t i v e d i s t a n c e ( i n momentum space ) dj_ i s u s e d . T h i s d i s t a n c e i s the d i s t a n c e between the t i p s o f t he v e c t o r s "p*j 78' and p 2» The q u a n t i t y p Q i s assumed t o be g i v e n by p Q = p^ + h where p i s t he m a g n i t u d e o f the Fermi momentum. 2 F 3 . Core a n n i h i l a t i o n and o t h e r h i g h e r momentum e f f e c t s a r e n e g l e c t e d . W i t h the above s i m p l i f i c a t i o n s , t he p r o b l e m reduces t o the c a l -c u l a t i o n o f t he " m a s s " o r e f f e c t i v e co lume o f a c y l i n d e r o f base a^u and - r 2 h e i g h t h i f i t s d e n s i t y v a r i e s as ^ (r) <s< e " ^ r . T h i s d e n s i t y i s assumed t o be i ndependen t o f h. The ( c i r c u l a r ) base o f t he c y l i n d e r r e s t s on the p^ - Py p l a n e , i t s c e n t e r l y i n g on the p^ a x i s a d i s t a n c e dj^ f rom t h e o r i g i n . The e q u a t i o n o f the c i r c l e i s r = rY = d^_ cos© + Y aj/ - d± sin© 6>©= s i n (ft-ejjf- ) | 2 2 2 i d d± c o s e +V - dj_ s i n e 6<®= s i n ~ ( _ a ) Here d^?a^ The e f f e c t i v e volume o f the c y l i n d e r ( i . e . a " n e c k " ) i s ® A r = 5 = d' m (f) = 2K I M r ) r d r da and the a n g u l a r c o r r e l a t i o n c u r v e i s g i v e n by £{f) = siiiL R m(o) where R = rr^ i s the c o i n c i d e n c e coun t r a t e a n i s o t r o p y e x p e c t e d a t j> = Q ( e q u a t i o n (5-1) ) . A c h o i c e o f g = J_ g i v e s a c u r v e w i t h h e i g h t m„ = 0.082 and a 2 m l f u l l w i d t h a t h a l f maximum o f about 32 . For g = J_ one o b t a i n s 0.050 and h 45 r e s p e c t i v e l y . I t w o u l d t hus appea r t h a t b e t t e r agreement w i t h t he ex-79 p e r i m e n t a l c u r v e s i s o b t a i n e d i f one chooses t h e r e s o l u t i o n f u n c t i o n - _ r v |» (r) = e 3 (see a l s o Chapter V): s i n c e one then o b t a i n s = 0.062 m, and Wj ~ 40°, in f a i r a c c o r d w i t h t h e e x p e r i m e n t a l c u r v e s , 80 APPENDIX C EFFECT OF CORE ANNIHILATION I f i t i s assumed t h a t the c o r e e l e c t r o n s can be r o u g h l y d e s c r i b e d by a s o r t o f " i o n c o r e Fermi s p h e r e " w i t h r a d i u s p p and the c o n d u c t i o n e l e c t r o n s by a ( c o n c e n t r i c ) Fermi s p h e r e o f r a d i u s p one h a s , f o r w ide s l i t g e o m e t r y , f o r the r a t i o o f t he c o i n c i d e n c e c o u n t i n g r a t e c o n t r i b u t i o n s a t z = 0 N cond = h. = / 2 ? r r d r . i_ ~ f c " K N c o r e h 2 I 2 Kb r d r were £ i s t he momentum space d e n s i t y (assumed c o n s t a n t ) f o r the c o r e e l e c t r o n s . L e t t i n g m^  be the be 11 y ' m a s s " c o n t r i but ion , m 2 the neck mass c o n t r i b u t i o n ( C h a p t e r V ) , and m^ the c o n t r i b u t i o n o f the ion c o r e e l e c t r o n s , i t i s seen t h a t f o r p o i n t geometry the c o i n c i d e n c e c o u n t i n g r a t e a n i s o t r o p y i s r educed f rom m 2 t o m 2 f 1 *1 = m 2 l l + m j m, m, I I  m^ j m^  + h 2 P , m 1 V * From Berko and P l a s k e t t ' s pape r one can e s t i m a t e h 2 = 1.15 and h l p ^ 18 x 1 0 " 3 mc so t h a t ] _ = 0 . 7 5 £ 1 + m. m l Thus c o r e a n n i h i l a t i o n wou ld be e x p e c t e d t o a f f e c t t he c o i n c i d e n c e coun t r a t e a n i s o t r o p y by about 25 p e r c e n t . Bl BL I OGRAPHY ANDERSON, C.! D. , ( 1 9 3 2 ) , Phys. Rev., 4 j _ , 4 0 5 . ANDERSON, P. W., ( 1 9 6 3 ) , Concepts i n Sol i d s , Benjamin, New York. BASSON, J . K., ( 1 9 5 4 ) , Phys. Rev., 9 6 , 6 9 1 . BECKER, R., and SAUTER, F., ( 1 9 6 4 ) , E l e c t r o m a g n e t i c F i e l d s and I n t e r a c t i o n s V o l . I, B l a c k i e , London. BENEDETTI, S., DE, COWAN, C.' E.', KONNEKER, W. R. , and RP IMAKOFF, N.:, ( 1 9 5 0 ) Phys. Rev. , JTJ, 2 0 5 . BERINGER, R., and MONTGOMERY, C. G.<, ( 1 9 4 2 ) , Phys. Rev., G]_, 2 2 2 . BERKO, S.', and HEREFORD, F.' L., ( 1 9 5 6 ) , Rev. Mod. Phys., 2 8 , 2 9 9 . BERKO, S.., and PLASKETT, J. 1 S.', ( 1 9 5 8 ) , Phys. Rev., J J _ 2 , 1 8 7 7 . BERKO, S., ( 1 9 6 2 ) , Phys. Rev., 128, 2 1 6 6 . BLACKETT, P. M. S., and OCCHIANL INI, G.' P.' S., ( 1 9 3 3 ) , P r o c . Roy. Soc. (London) A l 3 9 , 6 9 9 . BLOCH, F J , ( 1 9 2 8 ) , Z. P h y s i k , £ 2 , 5 5 5 . BOCHER, M.!, ( 1 9 2 9 ) , An I n t r o d u c t i o n t o the Study o f I n t e g r a l E q u a t i o n s , Cambridge, London. BOHM, H. V., and EASTERLING, V. J . , ( 1 9 6 2 ) , Phys. Rev., 128, 1 0 2 1 . BROWN, G., ( 1 9 6 4 ) , U n i f i e d Theory o f N u c l e a r M o d e l s , N o r t h H o l l a n d , Amsterdam. BURDICK, G. A., (l96l), Phys. Rev. L e t t e r s , 7 , 1 5 6 . BURDICK, G.'A., ( 1 9 6 3 ) , Phys. Rev. , J 2 9 , 1 3 8 . CALLAWAY, J . ! , ( 1 9 5 8 ) , S o l i d S t a t e P h y s i c s , 7 , 9 9 . CARBOTTE, J . P J , and KAHANA, S. , ( 1 9 6 5 ) , Phys. Rev., J j $ 9 , A213. CARBOTTE, J . P.', ( 1 9 6 6 ) , Phys. Rev. , J 4 4 , 3 0 9 . C0L0MBIN0, P.', Fl SCELLA, B., and TROSSI, L., ( 1 9 6 3 ) , Nuovo Cimento, 2 7 , 5 8 9 . CORNWELL, J . F., ( 1 9 6 4 ) , Phys. Kondens. M a t e r i e ±, 1 6 1 . 82 DANIEL, E J , and VOSKO, S. : H J , (i960), P h y s . R e v . , 120, 2041 . DEARNALEY, . G.., and NORTHROP, D.' C J , (1966), S e m i c o n d u c t o r C o u n t e r s f o r N u c l e a r R a d i a t i o n s , E & F. N. Spon L t d . , London . DEUTSCH, MJ , (1953), P r o g r . N u c l . P h y s . , 3, 131 . DIRAC, P. A, 1 M.1, (1930), P r o c . Camb. P h i l . Soc . 2 6 , 361. DONAGHY, J J J . . ' , STEWART, A . T..,. R0CKM0RE, D j M. , and KUSMISS, J . H j , (1965), i n ; IX th I n t e r n . C o n f . Low-Temp. P h y s . , P lenum P r e s s , New Y o r k . EVANS, R. D . , (1955), The A t o m i c N u c l e u s , M c G r a w - H i l l , New Y o r k . FOCK, V . , (1930), Z. P h y s i k , 6l_> 126. FUJIWARA, K J , (1965), J . P h y s . Soc . J a p a n , 2 0 , 1533. GRAHAM, R. L . , and STEWART, A . T . , (1954), Can . J . P h y s . 3 2 , 6 7 8 . GREEN, J . , and L E E , J . , (1964), P o s i t r o n i u m C h e m i s t r y , Academic P r e s s , New Y o r k . GUSTAFSON, D. R. , MACKINTOSH, A. R J , and ZAFFARANO, D. J . 1 , (1963), P h y s . R e v . , 130, 1455. HARRISON, WJ A J , and WEBB, M. B.', e d s , ( I 9 6 0 ) , The Fermi S u r f a c e , W i l e y , New Y o r k , HARTREE, D. R., ( 1 9 2 8 ) , P r o c . Camb. P h i l . Soc . 2 4 , 8 9 . HARTREE, D j R J , ( 1 9 5 7 ) , The C a l c u l a t i o n o f A t o m i c S t r u c t u r e s , W i l e y , New Y o r k , HATAN0, A . , KANAZAWA, HJ K J , and MIZUN0, Y J , ( 1 9 6 5 ) , P r o g r e s s o f T h e o r e t i c a l P h y s i c s , ^ 4 , 8 7 5 . HEINE, V.', ( i 960 ) Group Theo ry i n Quantum M e c h a n i c s , Pe rgamon, New Y o r k . HEITLER, W j , (1954) The Quantum Theory o f R a d i a t i o n , O x f o r d U n i v e r s i t y P r e s s , HYLLERAAS, E. A . , ( 1 9 4 7 ) , P h y s . R e v . , 21, ^ 9 1 . HYLLERAAS, E j A j , and ORE, A . , ( 1 9 4 7 ) , P h y s . R e v . , 21. 4 9 3 . JAUCH, J J M J , and R0HRLICH, F. , ( 1 9 5 5 ) , The Theory o f Pho tons and E l e c t r o n s , A d d i s o n - W e s l e y , R e a d i n g , Mass . JOSEPH, A . S j , and TH0RSEN, A . C. , ( 1 9 6 4 ) , P h y s . R e v . , 134, A979. KAHANA, S . , ( I 9 6 0 ) , P h y s . R e v . , UJ, 123 . KAHANA, S j , ( 1 9 6 3 ) , P h y s . R e v . , J 2 9 , 1622. KANAZAWA, H., OHTSUKI, Y.' H.I, and YANAGAWA, S. , ( 1 9 6 5 ) , Phys. Rev., J_38 A l l 5 5 . K1TTEL, C.;, ( 1 9 5 6 ) , Sol i d S t a t e Phys i c s , W i l e y , New York. KLEMPERER, GV, (193*0, P r o c . Camb. P h i l . Soc., 3 0 , 3 4 7 . KUGEL, Hj .W.I, FUNK, E.; G., and MiHELICH, J.i W.<, ( 1 9 6 6 ) , P h y s i c s L e t t e r s , 2 0 , 3 6 4 . LANG, L. G., ( 1 9 5 6 ) , Ph. D. D i s s e r t a t i o n , C a r n e g i e I n s t , o f Tech., ( u n p u b l i s h e d ) . LANG, L.' G.', and BENEDETTI, S. , DE, ( 1 9 5 7 ) , Phys. Rev., ]08, 9 6 4 . LEE-WHITING, G. E., ( 1 9 5 5 ) , Phys. Rev., S7_, 1557. LIVINGSTONE, W. R., ( 1 9 6 3 ) , P r i v a t e Communication. MAJUMDAR, C.: K j , ( 1 9 6 5 ) , Phys. Rev., J 4 0 , A 2 2 7 . MESSIAH, A., (1 9 6 2 ) , Quantum M e c h a n i c s , V o l . I I , W i l e y , New York. M0H0R0VICIC, S., (1 9 3 4 ) , A s t r o n . N a d i r . 25_3_, 94. MORSE, R. W.', ( i 9 6 0 ) , i n ; The Fermi S u r f a c e , e d s . W. A. H a r r i s o n and M. B. Webb, W i l e y , New York. MORTON, V.! MJ, (I960), Ph. D. T h e s i s , Cambridge U n i v e r s i t y . NEAMTAN, S. MJ, DAREWYCH, G.', and 0CZK0WSKI, G. , ( 1 9 6 2 ) , Phys. Rev., 126, 193. PINES, D., ( 1 9 5 5 ) , S o l i d S t a t e P h y s i c s , 1, 3 6 7 . PINES, D.', ( 1 9 5 6 ) , Rev. Mod. Phys . , 28, 184. PINES, D.I, ( 1 9 6 3 ) , E l e m e n t a r y E x c i t a t i o n s i n S o l i d s , Benjamin, New York. PRICE, B.I T., HORTON, C j C.', and SPINNEY, K.1 T.!, (1957) , R a d i a t i o n  Sh i e l d i n g , Pergamon, New Yor k . PIPPARD, A . l B.I, ( 1 9 5 7 ) , P h i l . T r a n s . Roy. Soc. (London), A250, 325. RA1MES, S.1, (1 9 5 7 ) , R e p o r t s on P r o g r e s s i n P h y s i c s , 20, 1. RAIMES, S., ( I 9 6 0 , The Wave Mechanics o f E l e c t r o n s i n M e t a l s , N o r t h - H o l l a n d , Amsterdam. REITZ, J . R., ( 1 9 5 5 ) , S o l i d S t a t e P h y s i c s , j . , 1. ROAF, D. J . , ( 1 9 6 2 ) , P h i l . T r a n s . Roy. Soc. (liondon) , A255, 135 . ROCKMORE, D. M.', and STEWART, A. T/, i n : Wayne S t a t e U n i v e r s i t y P o s i t r o n A n n i h i l a t i o n C o n f e r e n c e , 1965 , (To be P u b l i s h e d ) . RUARK, A J E., ( 1 9 4 5 ) , Phys. Rev., 6 8 , 2 7 8 . SEGALL, B.;, ( 1 9 6 2 ) , Phys. Rev., _[25_, 109. SHOENBERG, DJ, ( 1 9 6 2 ) , P h i l . T r a n s . Roy. Soc. (London) , A2§5_, 8 5 . SHOENBERG, D j , ( 1 9 6 5 ) , i n ; IXth I n t e r n . Conf. Low-Temp. Phys., Plenum P r e s s , York. SIMONS, L., ( 1 9 5 3 ) , Phys. Rev. 9 0 , 165 . SLATER, J.' C , ( 1 9 6 3 ) , Quantum Theory o f M o l e c u l e s and S o l i d s , M c G r a w - H i l l , New York. SOMMERFELD, A., ( 1 9 2 8 ) , Z. Phys. 4 7 , 1. STEWART, A J T., ( 1 9 5 7 ) , Can. J . Phys . , 3 5 , 168. STEWART, A J T. , ( 1 9 6 1 ) , Phys. Rev., J . 2 3 , 1587. STEWART, A.: T. , SHAND, J . B j , DONAGHY, J . ; J.', and KUSMISS, J . H. , ( 1 9 6 2 ) , Phys. Rev. , J 2 8 , 1 18 . STUMP, R., ( 1 9 5 5 ) , Phys. Rev., JJ ) 0 , 1 2 5 6(A). TERRELL, J . HJ, BERKO, S., and WE ISBERG, H. L., i n ; Wayne S t a t e U n i v e r s i P o s i t r o n A n n i h i l a t i o n C o n f e r e n c e , 1965 , (To be P u b l i s h e d ) . TINKHAM, M., ( 1 9 6 4 ) , Group Theory, M c G r a w - H i l l , New York. WALLACE, P.! R. , ( i 9 6 0 ) , S o l i d S t a t e P h y s i c s , J O , 1. WHEELER, J . A.', ( 1 9 4 6 ) , Ann. N. Y. Acad. S c i . , 4 8 , 2 1 9 . WIGNER, E.', and SEITZ, F., ( 1 9 3 3 ) , Phys. Rev., 4 3 , 8 0 4 . WILLIAMS, R. W., LOUCKS, T.1 L., and MACKINTOSH, A.R., (1966), Phys. Rev. L e t t e r s , J 6 , 168 . ZIMAN, J . M j , ( i 9 6 0 ) , E l e c t r o n s and Phonons, Oxford U n i v e r s i t y P r e s s , ZIMAN, J . M.;, (1964) , P r i n c i p l e s o f the Theory o f S o l i d s , Cambridge U n i v e r s i t y P r e s s . 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0103712/manifest

Comment

Related Items