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Fermi radii of lithium by positron annihilation. 1971

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THE FERMI RADII OF LITHIUM BY POSITRON ANNIHILATION by JOHN JOSEPH PACIGA B.Sc.', U n i v e r s i t y of Guelph, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE . i n the Department of P h y s i c s We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF A p r i l , BRITISH 1971 COLUMBIA In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of P h y s i c s The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada A p r i l 15, 1971 ACKNOWLEDGEMENTS The author i s inde b t e d to Dr. D. L I . Wi l l i a m s f o r su g g e s t i n g the t o p i c of t h i s study and p r o v i d i n g s u p e r v i s i o n throughout i t s d u r a t i o n . A number of i l l u m i n a t i n g d i s c u s s i o n s and suggestions p r o v i d e d by Jim McLarnon and the l a t e Peter P e t i j e v i c h i s g r a t e f u l l y acknowledged. The former i s a l s o to be thanked f o r t e c h n i c a l a s s i s t a n c e with the e l e c t r o n i c s . A N a t i o n a l Research C o u n c i l S c h o l a r s h i p p r o v i d e d f i n a n c i a l a s s i s t a n c e f o r the author d u r i n g the e a r l y stages of t h i s work. A b s t r a c t A p o s i t r o n a n n i h i l a t i o n experiment i n v o l v i n g c o l l i n e a r p o i n t geometry i s used to make a d i r e c t comparison of the ^110 a n < ^ ^100 Fermi r a d i i i n a s i n g l e c r y s t a l of l i t h i u m . I t i s found t h a t i s g r e a t e r than ^-^00 ^ v ^.6 ± 1>2%, i n agreement w i t h theory and a phenomenological i n t e r p r e t a - t i o n of an e a r l i e r long s l i t experiment. The h i g h e r momentum components of the p o s i t r o n wavefunction are c a l c u l a t e d by a d i r e c t method and found to be n e g l i g i b l e . On the other hand, a l e s s s t r a i g h t f o r w a r d estimate based on a f l a t t e n e d S e i t z p o t e n t i a l shows t h a t the h i g h e r momentum components of the e l e c t r o n wavefunction s i g n i f i c a n t l y reduce the e x p e r i m e n t a l l y observed a n i s o t r o p y . Hence, the d i f f e r e n c e of 5.6% should be regarded as an upper l i m i t on the true d i s t o r t i o n of the Fermi s u r f a c e of l i t h i u m . -v- TABLE OF CONTENTS Page L i s t of Tables . v i i L i s t o f F i g u r e s v i i i Chapter I: INTRODUCTION 1 Chapter I I : SUMMARY OF THEORY AND EXPERIMENT 4 A B a s i c P r i n c i p l e of Angular C o r r e l a t i o n Ex- periments Using P o s i t r o n A n n i h i l a t i o n 4 B Momentum D i s t r i b u t i o n s i n Angular C o r r e l a - - t i o n Experiments 8 C Exp e r i m e n t a l Geometries 11 ( i ) Long S l i t Geometry 11 ( i i ) P o i n t Geometry 14 D Band S t r u c t u r e C a l c u l a t i o n s and the Fermi S u r f a c e of L i t h i u m 17 ( i ) N e a r l y Free E l e c t r o n s 17 ( i i ) O r t h o g o n a l i z e d Plane Waves 21 ( i i i ) Band C a l c u l a t i o n s i n L i t h i u m 24 E E x p e r i m e n t a l R e s u l t s of Donaghy § Stewart 29 F Other Methods Used to Study L i t h i u m 32 Chapter I I I : SAMPLE PREPARATION $ EXPERIMENTAL APPARATUS 35 A Sample P r e p a r a t i o n 35 ( i ) C r y s t a l Growth 35 ( i i ) C r y s t a l O r i e n t a t i o n 38 ( i i i ) C u t t i n g and E t c h i n g 42 B The P o s i t r o n A n n i h i l a t i o n F a c i l i t i e s 43 - v i - Page Chapter IV: ANALYSIS OF DATA 53 A E x p e r i m e n t a l R e s u l t s 53 B Higher Momentum Components of the P o s i t r o n Wavefunction 60 C Higher Momentum Components of the E l e c t r o n Wavefunction 63 D Enhancement and A n n i h i l a t i o n s w i t h Core E l e c t r o n s 71 Chapter V: CONCLUSIONS 73 Appendix I 76 B i b l i o g r a p h y 79 - V l l - LIST OF TABLES Table 1 Band c a l c u l a t i o n s i n l i t h i u m Table 2 E x p e r i m e n t a l r e s u l t s Table 3 F o u r i e r c o e f f i c i e n t s of the c r y s t a l p o t e n t i a l i n l i t h i u m Table 4 C o e f f i c i e n t s of the h i g h e r momentum components of the e l e c t r o n wavefunction Page 27 59 64 68 - v i i i - LIST OF FIGURES F i g u r e Page 1 Angular r e l a t i o n between 2 a n n i h i l a t i o n photons 5 2 Long s l i t geometry 12 3 P o i n t geometry 15 4 S p l i t t i n g of energy l e v e l s i n extended zone scheme 2 2 5 F i r s t B r i l l o u i n zone of L i t h i u m 25 6 C r u c i b l e used f o r growing L i c r y s t a l s 36 7 Laue photographs of the L i c r y s t a l 41 8 Block diagram of experimental apparatus 44 9 Pre- amp and shaper c i r c u i t 46 10 C o i n c i d e n c e c i r c u i t 47 11 Apparatus i n v i c i n i t y of sample 50. 12 E x p e r i m e n t a l r e s u l t s 56 13 Cumulative experimental r e s u l t s 57 14 C r o s s - s e c t i o n s through the f i r s t B r i l - l o u i n zone of l i t h i u m 66 CHAPTER I INTRODUCTION A f t e r DeBenedetti and h i s c o l l e a g u e s p u b l i s h e d the f i r s t measurements of s u f f i c i e n t angular r e s o l u t i o n on gold i n 1950, the a p p l i c a t i o n of p o s i t r o n a n n i h i l a t i o n techniques to study the Fermi s u r f a c e of metals expanded s i g n i f i c a n t l y . T h i s f l o u r i s h of a c t i v i t y was p a r t l y a r e s u l t of the l i m i t a t i o n s of the more t r a d i t i o n a l t o o l s of Fermiology. Techniques such as the anomalous s k i n e f f e c t , c y c l o t r o n resonance, magnetoresis- tance, or the de Haas-van Alphen e f f e c t r e q u i r e a long mean f r e e path of the e l e c t r o n i n the metal, and consequently i t i s d e s i r a b l e to work at l i q u i d helium temperatures. However, there are two cases i n which low temperature experiments are i n e f f e c t i v e . The f i r s t case i s t h a t of d i s o r d e r e d a l l o y s f o r which the mean f r e e path of the e l e c t r o n i s . s h o r t even at low temperatures. Secondly, a number of m e t a l l i c c r y s t a l s undergo s t r u c t u r e t r a n s f o r m a t i o n s at low temperatures. In both of these i n s t a n c e s , p o s i t r o n a n n i h i l a t i o n s t u d i e s have a d e f i n i t e advantage as the mean f r e e path does not enter i n t o the an- a l y s i s .' L i t h i u m i s i n the second category mentioned above'since i t undergoes a M a r t e n s i t i c t r a n s f o r m a t i o n at 78°K from the body-centred c u b i c s t r u c t u r e to the hexagonal c l o s e - p a c k e d s t r u c t u r e . This occurrence i s r a t h e r u n f o r t u n a t e because -2- l i t h i u m has the s i m p l e s t e l e c t r o n i c s t r u c t u r e of any metal and c a l c u l a t i o n s of i t s band s t r u c t u r e are abundant. However, no r e l i a b l e experimental evidence of i t s Fermi surface s t r u c t u r e e x i s t e d u n t i l the p o s i t r o n a n n i h i l a t i o n experiments conducted by Donaghy and Stewart i n 1 9 6 4 . 2 , 3 , 4 The present study of the Fermi surface of l i t h i u m was undertaken f o r s e v e r a l reasons. The r e s u l t s of Donaghy and Stewart agreed w e l l w i t h a p r e d i c t e d a s p h e r i c i t y of about 51 i n the Fermi surface of l i t h i u m , but i t was necessary to r e l y on a phenomenological model to i n t e r p r e t t h e i r r e s u l t s . The present work uses a d i f f e r e n t experimental geometry which allows a more d i r e c t comparison of the r a d i i of the Fermi surface i n v a r i o u s c r y s t a l l o g r a p h i c d i r e c t i o n s . In a d d i t i o n , i t was f e l t t hat an independent measurement was d e s i r a b l e both because the a s p h e r i c i t y i s not a p p r e c i a b l y greater than the t y p i c a l experimental accuracy of p o s i t r o n a n n i h i l a t i o n e x p e r i - ments (approximately 1 % ) , and a l s o because the r e s u l t s of Donaghy and Stewart were not i n agreement w i t h those of c e r t a i n x-ray experiments. In t h i s t h e s i s i t w i l l be assumed that the reader i s f a m i l i a r w i t h the p r o p e r t i e s and n o t a t i o n of r e c i p r o c a l space as discussed i n most s o l i d s t a t e textbooks. (The f i r s t . c h a p t e r of Ziman's book'' i s adequate f o r t h i s purpose.) A l s o , the l i t e r a t u r e on p o s i t r o n a n n i h i l a t i o n i n metals i s so extensive that no attempt w i l l be made to review i t e x h a u s t i v e l y . For example, work on other metals w i l l l a r g e l y be ig n o r e d and no d e t a i l s of the p h y s i c s of the a n n i h i l a t i o n process w i l l be given. Numerous review a r t i c l e s and l i s t s of r e f e r e n c e s on these t o p i c s e x i s t elsewhere.^'^'^'^ The m a t e r i a l necessary f o r an understanding of the theory and experiments r e l e v a n t to t h i s study i s co n t a i n e d i n Chapter I I . Chapter I I I d e s c r i b e s the method used.to grow l i t h i u m s i n g l e c r y s t a l s and i t a l s o g i v e s the d e t a i l s of the apparatus used i n the a c t u a l p o s i t r o n a n n i h i l a t i o n experiment. The data i s p r e s e n t e d i n Chapter IV and d i s c u s s e d i n r e l a t i o n to the e x i s t i n g t h e o r e t i c a l and experimental work. P a r t i c u l a r emphasis i s gi v e n to an examination of the hig h e r momentum Chapter V p r o v i d e s a summary of the c o n c l u s i o n s which can be drawn from t h i s study. CHAPTER II SUMMARY OF THEORY AND EXPERIMENT A B a s i c P r i n c i p l e of Angular C o r r e l a t i o n Experiments u s i n g P o s i t r o n A n n i h i l a t i o n : When a p o s i t r o n and an e l e c t r o n a n n i h i l a t e i n a metal, two 0.511 MeV gamma rays are emitt e d at e x a c t l y 180° to each other i n the c e n t r e of mass frame of r e f e r e n c e . In the l a b o r a t o r y frame, however, the gamma rays d e v i a t e from a n t i c o l l i n e a r i t y because of the momentum of the a n n i h i l a t i n g p a r t i c l e s . In an o v e r s i m p l i f i e d i n t e r p r e t a t i o n , i t i s the momentum of the e l e c t r o n b e f o r e a n n i h i l a t i o n which causes the d e v i a t i o n from 180° i f the n o p i t . r o n i s assumed to be at r p s t , The apparent s i m p l i c i t y of measuring the angular c o r r e l a - t i o n of a n n i h i l a t i o n r a d i a t i o n to o b t a i n i n f o r m a t i o n about the momentum d i s t r i b u t i o n of e l e c t r o n s i n metals i s immediately e v i d e n t from the equ a t i o n r e l a t i n g the t r a n s v e r s e momentum of an e l e c t r o n , P , and the angle 0 by which the two a n n i h i l a t i o n photons, and d e v i a t e from 1 8 0 ° . A p p l y i n g the law of c o n s e r v a t i o n of momentum to f i g u r e 1 gives P^ = 2mc cosa where the angle a i s d e f i n e d i n the diagram and mc i s the momentum a s s o c i a t e d w i t h each a n n i h i l a t i o n photon (the r e s t mass of an e l e c t r o n m u l t i p l i e d by the v e l o c i t y of l i g h t ) . From the diagram, cosa = cos(90° - 8/2) = s i n ( 9 / 2 ) , hence -5- ' F i g u r e 1: A n g u l a r r e l a t i o n b e t w e e n t w o a n n i h i l a t i o n p h o t o n s . P t = 2mc sin(6/2) Since the deviation from a n t i c o l l i n e a r i t y i s small (G is t y p i c a l l y less than 20 m i l l i r a d i a n s ) , this gives P t = mc6 (1) From this equation, measuring 8 should give d i r e c t i n f o r - mation about the momentum of an electron before a n n i h i l a t i o n , but i n order to extract such information from an experiment several s i m p l i f y i n g assumptions are usually made. Perhaps the most s i g n i f i c a n t of these has already been mentioned; namely, that the positron i s thermalized or has an energy of approx- imately 0.025 eV. Consequently, i t can be assigned zero momen- tum so that only the electron i s responsible for the deviation of the. photons from 180°. Early c a l c u l a t i o n s x u showed that a -12 positron thermalized i n 3 * 10 second, a time much shorter than the measured a n n i h i l a t i o n t i m e s ^ which are t y p i c a l l y of the order of 2 x 10 second. Hence, for any experiment performed at room temperature i t i s quite safe to assume that the positron i s completely thermalized before a n n i h i l a t i o n . 12 At low temperatures, however, both theory and experi- 13 ment indicate that the s i t u a t i o n i s quite d i f f e r e n t , and i n some cases the positron may annihilate before complete therm- a l i z a t i o n . A s i g n i f i c a n t r e s u l t of these studies has been the discovery that positrons a t t a i n a minimum possible energy at low temperature. For example, a positron in lithium reaches a minimum observed energy of approximately 0.017 eV at 200°K and i t s energy cannot be lowered by a f u r t h e r r e d u c t i o n i n temperature. This suggests t h a t the r e s o l u t i o n of any p o s i t r o n a n n i h i l a t i o n experiment has an u l t i m a t e l i m i t , s i n c e the motion of the p o s i t r o n w i l l b l u r f i n e s t r u c t u r a l d e t a i l s of the Fermi s u r f a c e . In u s i n g e q u a t i o n (1) to o b t a i n i n f o r m a t i o n about the Fermi' s u r f a c e , i t i s i m p l i e d t h a t p o s i t r o n s do a n n i h i l a t e w i t h e l e c t r o n s near the Fermi l e v e l . A n n i h i l a t i o n w i t h core e l e c - trons does, i n f a c t , c o n t r i b u t e to a broad experimental back- ground, but because the p o s i t r o n i s r e p e l l e d from the p o s i t i v e i o n core, t h i s c o n t r i b u t i o n i s small f o r metals l i k e l i t h i u m i n which the volume of the i o n core i s s m a l l . Another assumption which s i m p l i f i e d the i n t e r p r e t a t i o n of e a r l y experiments i s th a t the p r o b a b i l i t y of a n n i h i l a t i o n i s independent of the v e l o c i t y of the e l e c t r o n . That t h i s i s not 14 s t r i c t l y t r u e was made e v i d e n t by Kahana, who proposed that the a n n i h i l a t i o n r a t e f o r p o s i t r o n s has a momentum dependent enhancement f a c t o r , e ( y ) , o f form e(y) = a + b y 2 + cy1* where y= k/kp and a,b,c are p o s i t i v e constants which depend on the e l e c t r o n d e n s i t y . The enhancement f a c t o r i s a measure of the i n c r e a s e d d e n s i t y of e l e c t r o n s of momentum k at the p o s i t r o n and i t i s e v i d e n t t h a t the a n n i h i l a t i o n c r o s s - s e c t i o n i n c r e a s e s s l i g h t l y w i t h i n c r e a s i n g e l e c t r o n momentum. In r e l a t i o n to t h i s i t must be assumed t h a t the t o t a l momentum of the a n n i h i - l a t i n g p a i r i s not a f f e c t e d by p a r t i c l e s around the p a i r . Th i s i s r easonable because the e l e c t r o n - p o s i t r o n p a i r i s e l e c t r i - c a l l y n e u t r a l and appears to move q u i t e f r e e l y through the l a t t i c e . B Momentum D i s t r i b u t i o n s i n Angular C o r r e l a t i o n Experiments: The observed momentum d i s t r i b u t i o n r e s u l t i n g from the a n n i h i l a t i o n p a i r s can be expressed i n mathematical terms as f o l l o w s . In an independent p a r t i c l e approximation which n e g l e c t s many-body e f f e c t s , the p r o b a b i l i t y P k(K) th a t an e l e c t r o n w i t h wavevector k w i l l a n n i h i l a t e with a t h e r m a l i z e d p o s i t r o n and y i e l d a photon pair- w i t h momentum p_ = fiK i s 1 . 1 6 p r o p o r t i o n a l to ' / <Kk,r) <!>+(r) e " 1 - ' - dr c r y s t a l where ip(k,r) and <j>+(r) are the e l e c t r o n and p o s i t r o n wave- f u n c t i o n s at p o s i t i o n r . The observed momentum d i s t r i b u t i o n p (K) o f the photon p a i r s i s p r o p o r t i o n a l to a sum over a l l o c c u p i e d s t a t e s i n k-space, i n c l u d i n g core s t a t e s : p ( K ) = C I k$k, / i K k , r ) e " 1 ^ dr (2) c r y s t a l where C = a 3 / 4 i r 2 , cr = 1/137, and kp i s the Fermi r a d i u s . The above equation i s v a l i d f o r angular momentum 1=0, but Mi j - 17 narends has determined i t f o r the more g e n e r a l a n i s o t r o p i c case of 1*0. To o b t a i n the a c t u a l number of photon p a i r s emitted into a region of k-space, one attempts to integrate the annihilation probability p(K) over the region of interest. It is immediately evident that a precise analysis of the angular correlation curves requires a detailed knowledge of the two wavefunctions i|;(k,r) and <j>+(r). To a fi r s t approximation the positron wavefunction can be considered constant. This assumption is reasonable everywhere except in the ion core region where <j>+(r) is essentially zero, and therefore i t is particularly good in metals such as lithium in which the core occupies only 6% of the volume of the unit cell. A detailed analysis of the effect of both the positron and electron wave- functions on the angular distributions in silicon and aluminum 18 has been performed by Stroud and Ehrenreich with excellent results. These authors have determined single particle wave- functions for positrons in solids by utilizing the close rela- tionship of x-ray form factors to the Fourier coefficients of the potential seen by the positron. A more complete discussion of their method will be given in Chapter IV. For the electron wavefunction, the simplest case which can be considered is that of the free electron theory of Sommer- feld in which the atom lattice is completely ignored and the electrons are considered to be a gas of non-interacting. particles restricted only by the boundaries of the metal and the Pauli exclusion principle. The electron wavefunction is simply a plane wave -10- ^(k) = 1//V e 1 - ' - where V i s the volume of the metal. The energy of an e l e c t r o n s t a t e i s r e p r e s e n t e d by the f r e e e l e c t r o n p a r a b o l a E(k) = . — 2m At 0°K a l l s t a t e s are completely f i l l e d up to the Fermi l e v e l , beyond which the o c c u p a t i o n number drops a b r u p t l y to zero. The constant energy s u r f a c e a t which t h i s occurs d e f i n e s the Fermi sphere of r a d i u s kp n 2 k a E = P F 2m * The f r e e e l e c t r o n theory p r e d i c t s angular c o r r e l a t i o n curves remarkably w e l l , even though i t f a i l s completely i n pre- d i c t i n g p o s i t r o n l i f e t i m e s i n m e t a l s . x y The reason i s mainly t h a t i t i g n o r e s the e l e c t r o n - p o s i t r o n a t t r a c t i o n which en- hances the e l e c t r o n d e n s i t y at the p o s i t r o n and reduces i t s l i f e t i m e . T h i s theory w i l l , however, be s u f f i c i e n t to e x p l a i n the e x p e r i m e n t a l geometries used i n p o s i t r o n a n n i h i l a t i o n experiments, and a d i s c u s s i o n of the more r e f i n e d approximations to the e l e c t r o n wavefunction w i l l be d e f e r r e d to a l a t e r sec- t i o n . I t i s s u f f i c i e n t to note at t h i s p o i n t t h a t i f the f r e e e l e c t r o n wavefunction i s s u b s t i t u t e d i n t o e q u a t i o n (2) with <}>+(r) = constant, then p (K) - p(K) i s s p h e r i c a l l y symmetric, equal to a con s t a n t i n s i d e the Fermi sphere, and equal to zero o u t s i d e . ' ' -11- C E x p e r i m e n t a l Geometries: ( i ) Long s l i t geometry The e x p e r i m e n t a l arrangement used by a l l e a r l y i n v e s t - i g a t o r s to study the angular c o r r e l a t i o n of a n n i h i l a t i o n rad- i a t i o n i s termed the long s l i t geometry. The sample i s p l a c e d midway between two gamma ray d e t e c t o r s which are p o s i t i o n e d behind a s e t of h o r i z o n t a l l e a d s l i t s as shown i n f i g u r e 2 ( a ) . An e x t e r n a l source such as sodium-22 can be used to shine p o s i t r o n s onto the sample, or the sample i t s e l f can be a source of p o s i t r o n s as i n the case of neutron i r r a d i a t e d copper-64. The s l i t s on one s i d e of the apparatus can be r a i s e d to d e f i n e an angle at the sample of 6 = z/D where z i s the d i s - tance r a i s e d and D i s one h a l f the d i s t a n c e between the detec- t o r s . I t i s e v i d e n t t h a t the g e o m e t r i c a l r e s o l u t i o n of such an arrangement i s r e l a t e d to the width of the s l i t s and the d i s t a n c e D to the sample. The number of photons w i t h a w e l l d e f i n e d z-component measured wi t h the long s l i t geometry i s g i v e n by CO oo . N(K ) « / / p(K) dK dK (3) - oo - oo y where K z = mcQ/ft from e q u a t i o n ( 1 ) . The i n t e g r a l i n the x - d i r e c t i o n i s a r e s u l t of the f a c t t h a t the s l i t s are,wider i n t h i s d i r e c t i o n than the angle c o r r e s p o n d i n g to the width of the Fermi s u r f a c e , and the i n t e g r a l over the y - d i r e c t i o n a r i s e s because the Doppler s h i f t i n energy due to the momentum of the -12- Figure 2(b): Region of Fermi sphere sampled by long slit apparatus. -13- a n n i h i l a t i n g p a i r cannot be d e t e c t e d . As s t a t e d p r e v i o u s l y , the momentum d i s t r i b u t i o n of equation (2 ) i s g e n e r a l l y a n i s o t r o p i c because of many-body e f f e c t s and the presence of the l a t t i c e p o t e n t i a l ; however, f o r s i m p l i c i t y the p r e s e n t d i s c u s s i o n w i l l be r e s t r i c t e d to the f r e e e l e c t r o n case. Hence eq u a t i o n (3 ) becomes CO oo N(K ) « p(K) ' / / dK xdK - 00 - 00 ' which i s p r o p o r t i o n a l to the area of a s l i c e of the Fermi sphere at con s t a n t as shown i n f i g u r e 2 ( b ) . Using the f a c t t h a t K 2 = K 2 + K 2 + K 2 = K 2 + K 2 , x y z r z ' 00 N ( K J - 2TT p ( K ) / K d K . z o r r < Since KdK = K dK f o r K = constant, t h i s becomes r r z ' 00 N ( K ) "c 2TT p(K) / KdK Z K z But p(K) = 0 i f K z i s g r e a t e r than the Fermi r a d i u s kp, thus N(K z) ^ k 2 - K 2 f o r K z ^ k F (4) N(K ) = 0 f o r K > k p E q u a t i o n (4) i s the i n v e r t e d p a r a b o l a which c h a r a c t e r i z e s angular c o r r e l a t i o n experiments u s i n g the long s l i t geometry. -14- The s h a r p - c u t - o f f i n the curve at the Fermi r a d i u s i s u n s h i f t e d even w i t h the i n c l u s i o n of many-body e f f e c t s such as the 19 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 . In an a c t u a l experiment the f r e e e l e c t r o n p a r a b o l a i s superimposed on a much broader back- ground which a r i s e s from such e f f e c t s as chance c o i n c i d e n c e s , core a n n i h i l a t i o n s , and h i g h e r momentum components of the e l e c t r o n and p o s i t r o n wavefunctions. These h i g h e r momentum components, along w i t h the a c t u a l shape of the Fermi s u r f a c e , s i g n i f i c a n t l y a f f e c t the form o f the angular c o r r e l a t i o n curves ( i i ) P o i n t geometry and the R o t a t i n g Specimen Method In the p o i n t geometry arrangement which was i n t r o d u c e d i n 7 0 ?~\ ? 7 the 1960's, ' ' the width of the s l i t i s reduced i n the x - d i r e c t i o n i n order to d e f i n e two components of momentum i n s t e a d of j u s t one. T h i s arrangement i s i l l u s t r a t e d i n f i g u r e 3 ( a ) ; the r e g i o n o f k-space sampled by the d e t e c t o r s being the chord of f i g u r e 3(b). Assuming the f r e e e l e c t r o n case as b e f o r e , e q u a t i o n (3) becomes oo N(K ) «  P ( K ) / dK CO or, N ( K ) «= p(K) / dK r o where K^ i s the semi-chord shown i n f i g u r e 3(b). Proceeding as i n the long s l i t case, KdK = K dK f o r constant K , and changin -15- Figure 3(b): Region of Fermi sphere sampled by point geometry apparatus. -16- to the a p p r o p r i a t e l i m i t s the i n t e g r a l becomes » KdK N(K ) - p ( K ) / z K K z r » d(K 2 - K 2) or, N(K ) « p(K) / Z . Z K / K 2 - K 2 z / z I n t e g r a t i n g from K to the Fermi l e v e l k„, the r e s u l t i s N(K z) « / k 2 - K 2 . - (5) Since K z = mc0/h from equation (1), the angular c o r r e l a t i o n curve f o r the p o i n t geometry arrangement i s e l l i p t i c a l (or c i r c u l a r f o r a s u i t a b l e c h o i c e of axes). The advantage of t h i s geometry i s t h a t another component of momentum i s d e f i n e d ; , — ~ " - - J-.^-i—J....^ V- i V/ ^iJ- ̂ count r a t e because of the narrower s l i t s . I f 6=0 i t i s obvious from equation (5) that the number of counts i s d i r e c t l y p r o p o r t i o n a l to the r a d i u s of the Fermi sphere, kp. I f the Fermi s u r f a c e i s a c t u a l l y a n i s o t r o p i c , r o t a t i o n of the c r y s t a l w i t h the d e t e c t o r s at 6=0 w i l l map out a comparison of the 'diameters' of the Fermi s u r f a c e f o r d i f - f e r e n t d i r e c t i o n s . T h i s method i s termed the c o l l i n e a r p o i n t geometry or r o t a t i n g specimen method and was used independently 2 2 23 by W i l l i a m s , e_t a_l. and Sueoka to study the Fermi s u r f a c e of copper. In the c o l l i n e a r p o i n t geometry the c r y s t a l must be made -17- c y l i n d r i c a l to a v o i d a n i s o t r o p i e s a r i s i n g from gamma ray ab- s o r p t i o n . I t should a l s o be noted that the curves obtained by r o t a t i n g the c r y s t a l cannot give a value f o r the amount of a n n i h i l a t i o n w i t h core e l e c t r o n s . T h i s i n f o r m a t i o n must be o b t a i n e d from t h e o r e t i c a l c a l c u l a t i o n s or from a r e g u l a r s i d e - ways experiment (8*0) u s i n g e i t h e r the long s l i t or p o i n t geometry. D Band S t r u c t u r e C a l c u l a t i o n s § the Fermi Surface of L i t h i u m : ( i ) N e a r l y Free E l e c t r o n s In r e a l metals, the presence of the l a t t i c e p o t e n t i a l n e c e s s i t a t e s the use of a more s o p h i s t i c a t e d approximation to the e l e c t r o n wavefunction than i s g i v e n by the f r e e e l e c t r o n model. Many such approximations have been developed, but only a b r i e f i n t r o d u c t i o n to two of the s i m p l e r ones, the n e a r l y f r e e e l e c t r o n (NFE) theory and the o r t h o g a n a l i z e d plane wave (OPW) method w i l l be g i v e n here. S e v e r a l good r e f e r e n c e s ^' 2 4 2 5 2 6 ' ' are a v a i l a b l e f o r f u r t h e r d e t a i l s on these and other more r e f i n e d techniques such as p s e u d o p o t e n t i a l theory, c e l l - u l a r methods, augmented plane waves (APW's), and the Green's f u n c t i o n method. A g r e a t d e a l of understanding about the behaviour of e l e c t r o n s i n metals can be o b t a i n e d by a p p l y i n g f i r s t order p e r t u r b a t i o n theory to f r e e e l e c t r o n s . In the NFE approxima- t i o n the i n n e r or core e l e c t r o n s are c o n s i d e r e d to be t i g h t l y bound to the nucleus, and the conduction e l e c t r o n s behave as n e a r l y f r e e but s u b j e c t to Bragg r e f l e c t i o n s at the B r i l l o u i n zone boundaries because of the presence of a weak p e r i o d i c p o t e n t i a l V ( r ) . No exact s o l u t i o n e x i s t s f o r the r e l a t e d Schrodinger e q u a t i o n f * 2 ^ <Kr) = E(k) ^ ( r ) f ft2 V 2 + V(r) 2m but to f i r s t o rder p e r t u r b a t i o n theory the energy of an e l e c t r o n s t a t e can be r e p r e s e n t e d by (Ref. 5, page 70): \zk2 - , , r |<k|V(r)|k->| 2 E(k) = 2_JL- + <k|V(r)|k> + L . 2m — — — ^ «,2 k*k' 2m - " > "» r K -~ Tk* — M k ' k = <k|V(r)|k'> = / > * , ( r ) V ( r ) ^ ( r ) dr where the p e r i o d i c p o t e n t i a l V (r) can be expanded i n a F o u r i e r s e r i e s i n 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: V ( r ) = E V r e 1 ^ G — w i t h c o e f f i c i e n t s V G  = V / V<>) e - 1 ^ dr . (6) As u s u a l , V i s the volume of the metal. ' I t can be shown (Ref. 5, page 51) t h a t the m a t r i x elements are equal to Vg i f k - k 1 -19- equals a r e c i p r o c a l l a t t i c e v e c t o r and are zero otherwise Hence, + v + T. V G E(k) =  V 2m 9 r *a ( k 2 - (k-G) 2} *1 G*0 2m where V Q = <k|V(r)|k> The p o t e n t i a l V Q i s a c o r r e c t i o n f o r the mean p o t e n t i a l energy of an e l e c t r o n i n the l a t t i c e . T h i s r e s u l t does not d i f f e r g r e a t l y from the f r e e e l e c t r o n case i f the F o u r i e r c o e f f i c i e n t s of the p o t e n t i a l , V^, are s m a l l and i f k does not approach k-G i n v alue ( t h a t i s , i f k does not approach a B r i l l o u i n zone boundary). Consequently, the wavefunction of an e l e c t r o n i n the c o n d u c t i o n band w i l l d i f f e r only s l i g h t l y from a s i n g l e plane wave except at p o i n t s near the zone boundary. The above c o n d i t i o n f o r nonvanishing matrix elements i s i d e n t i c a l to the c o n d i t i o n f o r Bragg r e f l e c t i o n . The i n t r o - d u c t i o n of a Bragg plane o u t s i d e the f r e e e l e c t r o n sphere mixes a r e f l e c t e d component i n t o an otherwise s i n g l e plane wave, thereby l i n k i n g s t a t e s which d i f f e r 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 G_. In order to f i n d a s o l u t i o n near a zone boundary, one assumes t h a t the wavefunction may be expanded i n a s e r i e s o f form = 2 a r W e 1 ^ ) - ! ( 7 ) G - which can b e . s u b s t i t u t e d i n t o the Schrodinger e q u a t i o n . Such -20- a w avefunction i s s a i d to c o n t a i n h i g h e r momentum components because i t i n v o l v e s terms wi t h momenta g r e a t e r than k. This expansion does not imply t h a t the true wavefunctions ressemble plane waves, but the en e r g i e s o b t a i n e d by u s i n g such an expan- s i o n are p h y s i c a l l y r e a l i s t i c . I g n o r i n g a l l c o e f f i c i e n t s except a Q ( k ) and a ^ ( k ) , t h i s becomes * ( k , r ) = a Q ( k ) e ^ ' I + a _ £ ( k ) e ^ ^ ' L . One then s o l v e s the s e c u l a r equation h 2k 2 2m E(k) G E(k) = 0 to o b t a i n the f o l l o w i n g q u a d r a t i c e x p r e s s i o n f o r the energy: E(k) 2m 2ni + I 2 1,2 n zk 2m ^1 2m (k-G) * 4 | V G | 2 1/2 (8) I t i s e v i d e n t that the two degenerate, unperturbed s t a t e s which were, s e p a r 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 are now s p l i t i n energy. The p e r t u r b a t i o n has the g r e a t e s t e f f e c t near the B r i l l o u i n zone boundary k = ±1/2G at which p o i n t E(k) ft 2k2 2m ' G1 Away from the zone boundary the energy i s not too d i f f e r e n t from the f r e e e l e c t r o n p a r a b o l a i f the p e r t u r b i n g p o t e n t i a l i s s m a l l . -21- The c o n s t a n t energy s u r f a c e s are thus d i s t o r t e d from t h e i r u n p e r t u r b e d form o n l y near the B r i l l o u i n zone f a c e s where the d i s t o r t i o n a t k a r i s e s v i a a s i n g l e F o u r i e r c o e f f i c i e n t o f the l a t t i c e p o t e n t i a l between a f r e e e l e c t r o n s t a t e a t k and a n o t h e r such s t a t e a t k-G_. The f a m i l i a r r e s u l t o f e q u a t i o n (8) i s d e p i c t e d i n the extended zone scheme i n f i g u r e 4. In one d i m e n s i o n , the f r e e e l e c t r o n p a r a b o l a e x h i b i t s gaps at the B r i l l o u i n zone b o u n d a r i e s of magnitude 2|Vg|. S i n c e the t r u e c r y s t a l p o t e n t i a l i s n o t weak, the s u c c e s s of the n e a r l y f r e e e l e c t r o n model i s a t f i r s t s i g h t r a t h e r s u r p r i s i n g . I n s i g h t t o i t s s u c c e s s has been p r o v i d e d by p s e u d o p o t e n t i a l t h e o r y . E s s e n t i a l l y , the e f f e c t i v e pseudo- p o t e n t i a l seen by the v a l e n c e e l e c t r o n s i s weak because the r e q u i r e m e n t t h a t the w a v e f u n c t i o n s o f v a l e n c e e l e c t r o n s be o r t h o g o n a l t o c o r e s t a t e s has the e f f e c t o f i n t r o d u c i n g a r e - p u l s i v e p s e u d o p o t e n t i a l w h i c h p a r t i a l l y c a n c e l s the t r u e c r y s t a l p o t e n t i a l . ( i i ) O r t h o g o n a l i z e d P l a n e Waves In p r a c t i c e , the n e a r l y f r e e e l e c t r o n method can be a p p l i e d t o v e r y few m e t a l s because an e x p a n s i o n i n p l a n e waves r e q u i r e s a l a r g e number o f terms to d u p l i c a t e the sharp o s c i l l a t i o n s o f the a t o m i c w a v e f u n c t i o n s i n the n e ighbourhood of the i o n s . Much more r a p i d convergence i s o b t a i n a b l e w i t h the OPW method 27 d e v e l o p e d by H e r r i n g . -22- E(k) k — « G — *• F i g u r e 4: S p l i t t i n g of e n e r g y l e v e l s i n e x t e n d e d z o n e s c h e m e (one d i m e n s i o n ) . -23- In t h i s method the wavefunction of the valence e l e c t r o n s i s assumed to have an expansion of the form = I a k + G *k +G where the b a s i s s t a t e s xk a r e plane waves from which are sub- t r a c t e d a l i n e a r combination of core s t a t e s d>. •, : 1 ik« r v i . XM = ^ e — " ] b5 *i>S The c o e f f i c i e n t s b. are J b j = «)> j k|l//7 e 1 ^ so t h a t the c o n d u c t i o n e l e c t r o n wavefunction i s o r t h o g o n a l to the core s t a t e s , i . e . , <4>ikl V = ° ' T h i s o r t h o g o n a l i t y c o n d i t i o n may be c o n s i d e r e d a d i r e c t r e s u l t o f the P a u l i e x c l u s i o n p r i n c i p l e . The core s t a t e s themselves are h i g h l y l o c a l i z e d atomic o r b i t a l s which may be r e p r e s e n t e d i n the t i g h t b i n d i n g approximation as 4>, k(r) = ± E e 1 ^ (J).(r-R) . (J)j i s the core s t a t e w i t h quantum numbers j = (n,l,m), and the sum extends over N l a t t i c e s i t e s each se p a r a t e d by a d i r e c t l a t t i c e v e c t o r R. - The o r t h o g o n a l i z e d plane waves xk resemble plane waves i n the r e g i o n s between ions and are orthogonal to a l l core s t a t e s . T h i s s t r o n g modulation at the n u c l e i makes convergence of an -24- OPW expansion q u i t e r a p i d , and a s i n g l e OPW can give a u s e f u l f i r s t a pproximation to the band s t r u c t u r e of about 25 simple metals and semi-metals. Only two or three OPWs are needed to estimate the band s t r u c t u r e i n the very corners of the B r i l l o u i n zone. I t should be noted, however, t h a t the wavefunctions are not p r e c i s e e i g e n f u n c t i o n s of the g i v e n H a m i l t o n i a n , t h e r e f o r e one cannot o b t a i n s o l u t i o n s of a r b i t r a r y accuracy by i n c r e a s i n g 2 8 the number of terms i n the expansion. ( i i i ) . Band C a l c u l a t i o n s i n L i t h i u m S ince l i t h i u m has the s i m p l e s t e l e c t r o n i c s t r u c t u r e of any metal (two h i g h l y l o c a l i z e d 1-s core e l e c t r o n s and one n e a r l y f r e e 2-s c o n d u c t i o n e l e c t r o n ) , i t has been used as a t e s t case 29 f o r n e a r l y every type of band c a l c u l a t i o n . In f a c t , l i t h i u m 30 and sodium were the metals chosen f o r the f i r s t r e a l i s t i c band c a l c u l a t i o n s performed by Wigner and S e i t z i n the e a r l y 1930's. Since l i t h i u m c r y s t a l l i z e s i n the body-centred c u b i c s t r u c t u r e , i t s f i r s t B r i l l o u i n zone i s the r e g u l a r rhombic dodecahedron i l l u s t r a t e d i n f i g u r e 5. The f r e e e l e c t r o n sphere occupies only one h a l f the volume of the f i r s t B r i l l o u i n zone s i n c e l i t h i u m has one c o n d u c t i o n e l e c t r o n a s s o c i a t e d w i t h each atom, and because of the P a u l i p r i n c i p l e two are r e q u i r e d to f i l l a l l the s t a t e s i n the f i r s t zone. From the n e a r l y f r e e e l e c t r o n model one would expect t h a t the f r e e e l e c t r o n sphere -25- (001) (111) < Figure 5: F i rs t B r i l l o u i n Zone of L i th ium s h o w i n g Free E lec t ron Sphere. - 2.6- would be d i s t o r t e d by the presence of the Bragg r e f l e c t i o n planes ( B r i l l o u i n zone s u r f a c e s ) . D i s t o r t i o n w i l l be g r e a t e s t i n d i r e c t i o n s f o r which the sphere i s c l o s e s t to the zone f a c e s ; i n t h i s case i n the twelve <110> d i r e c t i o n s . In order to o b t a i n an i d e a of the shape of the Fermi s ur- 31 32 33 face of l i t h i u m , G l a s s e r and Callaway ' ' have c a l c u l a t e d the band s t r u c t u r e u s i n g the OPW method i n c o n j u n c t i o n with an e m p i r i c a l p o t e n t i a l c o n s t r u c t e d by S e i t z . I t was found t h a t the Fermi s u r f a c e i s c l o s e to being s p h e r i c a l but has s m a l l bulgee i n the <110> d i r e c t i o n s of the order of 5% of the r a d i u s . T h e i r value f o r the Fermi energy i s l i s t e d i n t a b l e 1 along w i t h the r e s u l t s o b t a i n e d by other i n v e s t i g a t o r s who used d i f f e r e n t methods. By f a r the most e x t e n s i v e c a l c u l a t i o n s of the energy bands 34 35 i n a l k a l i metals were c a r r i e d out by Ham ' u s i n g the quantum d e f e c t method. T h i s procedure does not r e q u i r e the e x p l i c i t c o n s t r u c t i o n of a p o t e n t i a l and a u t o m a t i c a l l y takes i n t o ac- count exchange, c o r r e l a t i o n , and r e l a t i v i s t i c e f f e c t s i n the i n t e r a c t i o n of va l e n c e e l e c t r o n s w i t h core e l e c t r o n s . Ham a l s o found t h a t the Fermi r a d i u s of l i t h i u m i s i n c r e a s e d by some S% i n the <110> d i r e c t i o n as compared wi t h the <111> and <100> d i r e c t i o n s . T h i s d i s t o r t i o n i s not n e a r l y enough to cause c o n t a c t w i t h the B r i l l o u i n zone face even when the l a t t i c e i s compressed s i g n i f i c a n t l y . The use of a procedure s i m i l a r to the augmented plane wave Table 1 INVESTIGATOR • LATTICE CONSTANT E p COMMENTS Callaway (32,33) 6.5183 a.u, 0.427 Ry, OPW method. Small bulges i n the Fermi s u r f a c e of about 5% i n <110> d i r e c t i o n s . No c o n t a c t w i t h zone f a c e . Ham (34, 35) quoted i n (28) 6. 651 6.5183 •0.433 •0. 430 Quantum d e f e c t method. Gives 5% bulges i n <110> d i r e c - t i o n s . No c o n t a c t . S c h l o s s e r § Marcus (28) 6.5183 •0.429 Method s i m i l a r to APW. Bulges out by 4.4% i n <110> d i r e c t i o n s § i s depressed by 2.1% i n <100> d i r e c t i o n s . No c o n t a c t . Capek (37) 6.614 •0.435 Model p o t e n t i a l . E l o n g a t e d i n <110> d i r e c t i o n s . In other d i r e c t i o n s a few % s m a l l e r than f r e e e l e c t r o n sphere. No c o n t a c t . Notes: (a) Atomic u n i t s are used with e=h=l, m=l/2 (see r e f . 26, pg. 55). The u n i t of l e n g t h i s the Bohr r a d i u s of hydrogen (0.52917 A ) , and the u n i t of energy i s the Rydberg (13.6049 eV). (b) The Fermi energy i n a s p h e r i c a l approximation i s -0.422 Ry. The zone boundary i n the <110> d i r e c t i o n i s at -0.412 Ry. and c o n t a c t occurs i f the Fermi energy i s g r e a t e r than t h i s v a l u e . - 2 8 - (APW) method l e a d s to r e s u l t s w h i c h are i n s u b s t a n t i a l agree- ment w i t h the p r e v i o u s a u t h o r s . U s i n g a f l a t t e n e d S e i t z p o t e n t i a l , the c a l c u l a t i o n s o f S c h l o s s e r and M a r c u s 2 ^ i n d i c a t e a s l i g h t l y l a r g e r outward b u l g e of about 1% i n the <110> d i r e c t i o n s . T h i s method d i f f e r s from the c e l l u l a r method, Green's f u n c t i o n method, and the OPW method i n t h a t a c o m p o s i t e r e p r e s e n t a t i o n o f the w a v e f u n c t i o n i s used i n s t e a d o f an ex- p a n s i o n i n a s i n g l e s e t of w a v e f u n c t i o n s o b t a i n e d by o r t h o g o n - a l i s i n g t o c o r e s t a t e f u n c t i o n s . I n the o r i g i n a l APW method, f o r example, an e x p a n s i o n i n p l a n e waves i s used i n the o u t e r p a r t o f the c e l l and an e x p a n s i o n i n s p h e r i c a l waves i s used i n the i n n e r p a r t . 3 6 A n t o n c i k has employed a p s e u d o p o t e n t i a l method and a c h i e v e d agreement w i t h G l a s s e r and C a l l a w a y by u s i n g the same 37 l a t t i c e c o n s t a n t and p o t e n t i a l . Capek a l s o r e p o r t s agree- ment (see t a b l e 1). He bases h i s c a l c u l a t i o n s on a model p o t e n t i a l i n the form of s p h e r i c a l ^-dependent w e l l s . T h i s use o f n o n - l o c a l model p o t e n t i a l s i s s i m i l a r t o , b u t not q u i t e the same as the p s e u d o p o t e n t i a l method. The t e c h n i q u e was 3 8 used e a r l i e r by Heine and Abarenkov who a l s o found t h a t the Fermi s u r f a c e of l i t h i u m behaved as mentioned p r e v i o u s l y . I t i s e v i d e n t t h a t t h e r e i s s u b s t a n t i a l agreement.among the d i f f e r e n t t h e o r e t i c a l c a l c u l a t i o n s f o r l i t h i u m . A q u i t e r e m a r k a b l e f a c t w h i c h appears from t h e s e c a l c u l a t i o n s i s t h a t the band s t r u c t u r e does not appear t o be v e r y s e n s i t i v e t o -29- u n c e r t a i n t i e s i n the p o t e n t i a l , s i n c e s i g n i f i c a n t d i f f e r e n c e s e x i s t i n the c h o i c e s of the v a r i o u s authors. D e s p i t e t h i s agreement, there are s e v e r a l suggestions that the theory developed may not be accurate i n a l l r e s p e c t s . Some e a r l y work, 3^ , 4 <^ f o r example, i n d i c a t e d a c o n s i d e r a b l e amount of c o n t a c t w i t h the zone f a c e s . A l s o , Cornwall and Wohlfarth 4"^ have developed an energy band i n t e r p o l a t i o n scheme based on G l a s s e r and Callaway's r e s u l t s , and they suggest t h a t although the Fermi s u r f a c e does not appear to touch the zone boundary, the d i f f e r e n c e i s onl y 0.025 Ry. which i s w i t h i n the accuracy of ±0.05 Ry. s t a t e d by G l a s s e r and Callaway. On the other hand, i t has been found t h a t the Fermi s u r f a c e s o f the other a l k a l i metals are c o n s i d e r a b l y l e s s d i s t o r t e d than p r e d i c t e d by Ham's c a l c u l a t i o n s . 3 * * ' 4 2 S e v e r a l other r e l e v a n t t h e o r e t i c a l papers w i l l be d i s c u s s e d a f t e r an examination of the experimental work which has been performed on l i t h i u m s i n g l e c r y s t a l s . E E x p e r i m e n t a l R e s u l t s o f Donaghy and Stewart: The most d e f i n i t i v e e x p e r i m e n t a l d e t e r m i n a t i o n of the Fermi s u r f a c e of l i t h i u m to date was c a r r i e d out by Donaghy 2 3 4 and Stewart. ' ' A long s l i t apparatus was used to study the angular 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 r a d i a t i o n from c r y s t a l s o r i e n t e d i n the <110>, <111>, and <100> d i r e c t i o n s . The r e - s u l t i n g e x p e r i m e n t a l curves were s i g n i f i c a n t l y d i f f e r e n t f o r -30- the three o r i e n t a t i o n s , and by c o n s t r u c t i n g a phenomenological model wi t h s e v e r a l a d j u s t a b l e parameters, Donaghy and Stewart were able to determine that the Fermi s u r f a c e of l i t h i u m i s a n i s o t r o p i c , w i t h the r a d i u s k-^Q g r e a t e r than k-^Q by about 5% as p r e d i c t e d t h e o r e t i c a l l y . I t was a l s o determined that the Fermi s u r f a c e does not c o n t a c t the zone boundary i n the <110> d i r e c t i o n u n l e s s i t does so by an u n r e a l i s t i c a l l y narrow neck. The model which these authors chose was a sphere w i t h twelve bumps superimposed towards the zone faces i n the <110> d i r e c t i o n s . The c a l c u l a t e d curves agreed q u a n t i t a t i v e l y w i t h the measured momentum d i s t r i b u t i o n s i n the r e g i o n 0<6<2.5 m i l l i r a d i a n s but d i f f e r e d f o r 0>2.5 m i l l i r a d i a n s . The d i s - crepancy f o r 0>2.5 m i l l i r a d i a n s was a t t r i b u t e d to the f a c t t h a t some of the photon p a i r s a r i s e from Umklapp processes and t h e r e f o r e i n v o l v e h i g h e r momentum components o f the e l e c t r o n wavefunction. The observed d i s t r i b u t i o n i s p r o p o r t i o n a l to the areas o f s l i c e s through the Fermi s u r f a c e i f £ = hk, but not i f h i g h e r momentum components p_ = h(k+G_) are i n v o l v e d . These p a r t i c u l a r e xperimental r e s u l t s have been d i s c u s s e d 43 from a t h e o r e t i c a l v i ewpoint by M e l n g a i l i s and DeBenedetti. A p p l y i n g an a b b r e v i a t e d form of the OPW method used by Callaway, they found t h a t the n u m e r i c a l l y c a l c u l a t e d angular c o r r e l a t i o n curves showed a n i s o t r o p i e s i n agreement with the statements of Donaghy and Stewart. I t was necessary to i n c l u d e both the e f f e c t of the shape of the Fermi s u r f a c e and the e f f e c t of the -31- departure of the e l e c t r o n wavefunction from a s i n g l e plane wave: the a n i s o t r o p i e s c o u l d not be e x p l a i n e d by the shape alone. In a d d i t i o n , these authors computed the momentum d i s - t r i b u t i o n r e s u l t i n g from a n n i h i l a t i o n w i t h the two core e l e c - trons i n l i t h i u m . For the long s l i t geometry, a broad, f l a t curve r e s u l t s which i s about one order of magnitude s m a l l e r than the condu c t i o n e l e c t r o n c o n t r i b u t i o n . B e t t e r q u a n t i t a - t i v e agreement wi t h the experimental curves c o u l d be ob t a i n e d by i n c l u d i n g Kahana's enhancement f a c t o r which allows f o r 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 . Another p o s s i b l e i n t e r p r e t a t i o n of the r e s u l t s of Donaghy 44 and Stewart has been suggested by Stachowiak. I f the theory 17 which Mijnarends d e r i v e d to determine a n i s o t r o p i c momentum d i s t r i b u t i o n s i n p o s i t r o n a n n i h i l a t i o n experiments i s a p p l i e d to Donaghy and Stewart's r e s u l t s , the evidence suggests a p o s s i b i l i t y of c o n t a c t w i t h the B r i l l o u i n zone face i n the <110> d i r e c t i o n . T h i s i s i n f e r r e d from the f a c t t h a t the d e n s i t y of s t a t e s curve d e r i v e d from Mijnarends theory does not drop to zero at the zone boundary. However, i t appears t h a t more experimental o r i e n t a t i o n s are necessary to d e f i n e t h i s r e s u l t more a c c u r a t e l y , because one must determine an ex- pansion i n Kubic harmonics and the r e s u l t s of Donaghy and Stewart only allow the d e t e r m i n a t i o n of three terms i n t h i s expansion. -32- F Other Methods used to study L i t h i u m : As s t a t e d i n Chapter I, the u s u a l low temperature tech- niques such as the de Haas-van Alphen e f f e c t cannot be used to study l i t h i u m because i t has a phase t r a n s i t i o n at 78°K. However, s e v e r a l experiments u t i l i z i n g x-rays can be performed at room temperature and have an advantage over p o s i t r o n a n n i h i - l a t i o n i n th a t no assumptions need be made about p o s i t r o n wavefunctions, t h e r m a l i z a t i o n times, or 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 . In p a r t i c u l a r , Compton s c a t t e r i n g has been used to study 45 46 47 l i t h i u m ' ' s i n c e the t h e o r e t i c a l Compton p r o f i l e shows a d i s c o n t i n u i t y at the Fermi s u r f a c e s i m i l a r to the one found i n p o s i t r o n a n n i h i l a t i o n experiments. In a study which a l s o i n - 46 eluded s e v e r a l other substances, P h i l l i p s and Weiss found th a t t h e i r r e s u l t s f o r p o l y c r y s t a l l i n e l i t h i u m y i e l d e d v a l u e s f o r the Fermi momentum of 0.593 ± 0.015 (a.u.) f o r l i t h i u m and 0.509 ± 0.02 (a.u.) ^ f o r sodium, whereas the f r e e e l e c t r o n v alues are 0.588 and 0.481 r e s p e c t i v e l y . In c o n t r a s t , p o s i t r o n a n n i h i l a t i o n experiments y i e l d momentum d i s t r i b u t i o n s which are c l o s e r to f r e e e l e c t r o n theory. The d i s c r e p a n c y f o r sodium 48 i s e s p e c i a l l y p u z z l i n g s i n c e de Haas-van Alphen experiments i n d i c a t e t h a t the Fermi s u r f a c e of sodium i s s p h e r i c a l to a p r e c i s i o n g r e a t e r than one p a r t i n one thousand. A l s o , the r e s u l t s of P h i l l i p s and Weiss f o r a s i n g l e c r y s t a l of l i t h i u m do not agree w i t h Donaghy and Stewart's p o s i t r o n r e s u l t s , s i n c e -33- no d e v i a t i o n from s p h e r i c a l symmetry was found i n the <110>, <100>, and <111> d i r e c t i o n s w i t h i n t h e i r s t a t e d accuracy of 3%. Another method of o b t a i n i n g i n f o r m a t i o n about the Fermi s u r f a c e i s through the s o f t x-ray emission, spectrum. I f vac- a n c i e s are c r e a t e d i n the K - s h e l l of l i t h i u m by f a s t e l e c t r o n s , these v a c a n c i e s are f i l l e d by e l e c t r o n s from the co n d u c t i o n band. The energy of the s o f t x-ray emitted i s nto = E(k) - E c where the energy of the core s t a t e s , E , i s w e l l d e f i n e d and E(k) i s the energy of an e l e c t r o n i n s t a t e k. U n f o r t u n a t e l y , 49 t h i s p i c t u r e i s o v e r s i m p l i f i e d , and r e s u l t s f o r l i t h i u m are mainly q u a l i t a t i v e s i n c e a c t u a l experiments are d i f f i c u l t to . . - . 50 , . . ,. i n t e r p r e t r e i i a D i y . i t i s rouna m a t tne snape or tne spectrum i s p u z z l i n g i n the case of l i t h i u m , but i s reasonably w e l l understood i n sodium and potassium. Ham ( r e f . 35, page 2539) suggests that the spectrum may i n d i c a t e t hat the Fermi s u r f a c e of l i t h i u m c o n t a c t s the B r i l l o u i n zone f a c e , and t h i s d i s c r e p a n c y between p o s i t r o n a n n i h i l a t i o n experiments and the x-ray r e s u l t s i s s t i l l u n r e s o l v e d . I t i s obvious t h a t both of the methods d i s c u s s e d above are u s e f u l only f o r l i g h t metals. In the f i r s t case both core and c o n d u c t i o n e l e c t r o n s s c a t t e r e q u a l l y , and i n the second case the K - s h e l l can be f i l l e d from any l e v e l above i t , t h e r e - f o r e i n t e r p r e t a t i o n becomes d i f f i c u l t i f s e v e r a l higher l e v e l s are occupied. Even more important i s the f a c t t h at good - 3 4 - r e s o l u t i o n i s d i f f i c u l t to o b t a i n i n - x - r a y experiments. In Compton s c a t t e r i n g , f o r example, there i s a c h a r a c t e r i s t i c broadening of the Compton p r o f i l e which has been a t t r i b u t e d to the p o s s i b i l i t y that the experimental curve contains more high 4 7 momentum than the t h e o r e t i c a l curve; however, t h i s broadening may a c t u a l l y be caused by the f a c t that a weaker band from x-ray Raman s c a t t e r i n g overlaps w i t h the Compton p r o f i l e . CHAPTER I I I SAMPLE PREPARATION AND EXPERIMENTAL APPARATUS A Sample P r e p a r a t i o n : ( i ) C r y s t a l growth Although attempts to grow s i n g l e c r y s t a l s of l i t h i u m were 52 made as e a r l y as 1917 when H u l l was determining the s t r u c t u r e s of v a r i o u s elements by x-ray d i f f r a c t i o n , i t was not u n t i l 1958 53 t h a t c r y s t a l s of reasonable s i z e were grown by Nash and Smith. 54 A t about the same time, Champier developed a s l i g h t l y d i f - f e r e n t method u s i n g a slow c o o l i n g r a t e of 2 C°/hour to grow s i n g l e c r y s t a l s which were 3 m i l l i m e t e r s t h i c k and a few square c e n t i m e t e r s i n area. However, the technique of Nash and Smith with i t s f a s t e r c o o l i n g r a t e of 30 C°/hour has been p r e f e r r e d by subsequent i n v e s t i g a t o r s , ^ ' ^ ' ^ and i t i s a s l i g h t l y im- proved v e r s i o n * ^ of t h e i r method which i s used here. The c r y s t a l s were grown by a m o d i f i e d Bridgman technique u s i n g the t h r e e - s e c t i o n c r u c i b l e shown i n f i g u r e 6. The bottom s e c t i o n i s a growing cup made from a 2-1/4 i n c h l e n g t h of 1/2 I.D. seamless s t a i n l e s s s t e e l p i p e wrapped wi t h asbestos i n s u l a t i o n over which i s wound nichrome hea t e r wire i n such a way t h a t the h o t t e s t p o r t i o n of the c y l i n d e r i s at the- bottom. Cu r r e n t through the wire i s c o n t r o l l e d manually by means of a v a r i a c . The middle s e c t i o n i s a m i l d s t e e l plunger which f i t s snugly i n t o the growing cup and has a tapered hole t h a t forms a -36- H 3 - 4-1/2' -H 1-1/2" Copper Cooling Fins Mild Steel Plunger 2-1/4 1/2" LD. Stainless Steel Growing Cup A s b e s t o s — N i c h r o m e Heater Wire Mild Steel Plug Figure 6: Cross-section of Crucible Used for Growing Lithium Single C r y s t a l s . -37- 1/8 i n c h diameter n u c l e a t i o n t i p at the top. L a s t l y , f o u r copper c o o l i n g f i n s f i t onto the top of the plunger to. complete the e f f e c t of pr o d u c i n g a h i g h temperature g r a d i e n t along the l e n g t h of the c r u c i b l e . I t i s i n the d e s i g n of the plunger that t h i s method d i f f e r s from the st a n d a r d Bridgman technique i n which c r y s t a l s are grown by c o o l i n g upwards from the bottom of the melt. Since molten l i t h i u m has a hig h s u r f a c e t e n s i o n and a low d e n s i t y (0.534 gm/cm3 at 20°C and 0.498 gm/cm3 at 300°C), the f o r c e of gravi-ty w i l l not cause i t to flow i n t o the apex of a s m a l l c o n i c a l t i p at the bottom of the melt. With the open t i p design of f i g u r e 6, however, the molten l i t h i u m i s f o r c e d up the cen- t r a l channel when the plunger i s lowered i n t o the c r u c i b l e cup. Another advantage of c o o l i n g from the top of the melt i s th a t i m p u r i t i e s tend to s e t t l e downwards and spurious n u c l e a t i o n s are l i m i t e d to the s m a l l p o r t i o n of the c r y s t a l which s o l i d i f i e s l a s t . Because l i t h i u m r e a c t s w i t h a l l known m o l e c u l a r gases at hi g h temperatures, the e n t i r e growing process was c a r r i e d out i n a glove box f i l l e d w i t h argon. A s m a l l q u a n t i t y of phos- phorus pentoxide was used to absorb water vapour s i n c e l i t h i u m at temperatures above i t s m e l t i n g p o i n t (179°C) r e a c t s r a p i d l y i n the presence of moisture to form a b l a c k h y g r o s c o p i c n i t r i d e , L i j N , which i s e a s i l y d i s t i n g u i s h a b l e from the b r i g h t s i l v e r c o l o u r of the pure metal. The a c t u a l procedure was as f o l l o w s . L i t h i u m rods 12.5 m i l l i m e t e r s i n diameter and 99.9% pure were purchased from Koch-Light L a b o r a t o r i e s L i m i t e d . An i n g o t of proper s i z e f o r the furnace was cut and scraped of i t s oxide l a y e r b e f o r e b e i n g p l a c e d i n t o the growing cup. To r e t a r d chemical r e a c t i o n s and to f a c i l i t a t e removal from the c r u c i b l e a f t e r c o o l i n g , the i n g o t and a l l s u r f a c e s touching i t were coated w i t h petroleum j e l l y . The l i t h i u m was then heated i n the growing cup to approximately 320°C and s t i r r e d a f t e r 1-1/2 hours of h e a t i n g . During t h i s p e r i o d the n u c l e a t i o n plunger was a l s o preheated to the same temperature i n a heater s i m i l a r i n d e s i g n to the c r u c i b l e con- t a i n i n g the molten l i t h i u m . A f t e r the h e a t i n g p e r i o d had passed, the s u r f a c e of the melt was skimmed to remove the b l a c k n i t r i d e c r u s t . Then the plunger was lowered i n t o the c r u c i b l e cup, f o r c i n g molten l i t h i u m up the c e n t r a l cone to form a n u c l e a t i o n bead at the top. The copper c o o l i n g f i n s were added and c o o l i n g was con- t r o l l e d manually w i t h the v a r i a c at about 30 C°/hour u n t i l s o l i d i f i c a t i o n was complete. The c r y s t a l was then removed from the mould and coated with petroleum j e l l y to p r o t e c t i t from o x i d i z i n g to a white powder d u r i n g the process of o r i e n t a t i o n . ( i i ) C r y s t a l O r i e n t a t i o n Since l i t h i u m has a low e l e c t r o n d e n s i t y , Laue t r a n s m i s s i o n p a t t e r n s have u s u a l l y been used to determine c r y s t a l o r i e n t a - -39- t i o n . ' ' ' In a darkened room the Laue spots can a c t u a l l y be viewed d i r e c t l y on a f l u o r e s c e n t screen and compared wi t h st a n d a r d p a t t e r n s f o r a body-centred c u b i c s t r u c t u r e such as 59 those g i v e n by Majima and Togino. Laue t r a n s m i s s i o n photographs p r o v i d e d a r a p i d method of d e t e r m i n i n g whether or not a p a r t i c u l a r sample was a good s i n g l e c r y s t a l . The sample was mounted on a goniometer and t r a n s l a t e d so t h a t i t s o r i e n t a t i o n w i t h r e s p e c t to the x-ray beam d i d not change. Using P o l a r o i d Type 57 f i l m , exposure times of only f i v e minutes were r e q u i r e d at s e t t i n g s of 12 keV and 20 ma. The primary beam e a s i l y passed through the e n t i r e two i n c h l e n g t h of a sample, and i f the p a t t e r n s on s u c c e s s i v e photo- graphs were i n v a r i a n t , i t meant t h a t a s i n g l e c r y s t a l was p r e s e n t . 5 7 Feder was able to o b t a i n s a t i s f a c t o r y b a c k - r e f l e c t i o n photographs of l i t h i u m c r y s t a l s by u s i n g long exposure times, and s i n c e t h i s method i s more f a m i l i a r i t was used to determine the a c t u a l o r i e n t a t i o n of a sample. S e t t i n g s of 10 keV, 30 ma, and 1-1/2 hours of exposure w i t h wet process I l f o r d x-ray f i l m r e s u l t e d i n the photographs shown i n f i g u r e 7. F i g u r e 7(a) i s taken along the <001> c r y s t a l l o g r a p h i c d i r e c t i o n and f i g u r e 7(b) i s w i t h i n two degrees of the <011> d i r e c t i o n . The. M i l l e r i n d i c e s of the spots are g i v e n on the o v e r l a y . Since l i t h i u m c r y s t a l l i z e s i n the body-centred c u b i c s t r u c t u r e , the only spots which occur are ones f o r which the sum of the i n d i c e s i s even. (a) Centred along th« ( 0 0 1 ) crystallographic axis. (b) C along the (01l) I ;-aphic axis. -42- ( i i i ) C u t t i n g and E t c h i n g Once the c r y s t a l l o g r a p h i c d i r e c t i o n s were determined, the c r y s t a l was a t t a c h e d to an aluminum b l o c k with GC E l e c t r o n i c s 'Copper P r i n t ' and cut on an A g i e t r o n Spark E r o s i o n Machine. T h i s was m o d i f i e d so t h a t the c u t t i n g edge was a t r a v e l l i n g copper wire e l e c t r o d e 0.010 i n c h i n diameter. For the s o f t 6 0 a l k a l i metals, S c h i l l e r e_t a l . have de v i s e d a r o t a t i n g blade spark c u t t e r which produces a b e t t e r cut than e i t h e r a s t a t i o n - ary blade or a wire e l e c t r o d e . However, t h e i r method was not used s i n c e the s m a l l amount of s u r f a c e damage produced by the wire e l e c t r o d e was e a s i l y removed d u r i n g the e t c h i n g p r o c e s s . U s i n g the spark c u t t e r at a working p o t e n t i a l of 100 v o l t s , a p a r a ] l e l p i p e d was cut with each face p a r a l l e l to a (100) plane. The c r y s t a l was then removed from the b l o c k and p o l i s h e d by l a p p i n g the s u r f a c e a g a i n s t a p i e c e of f i n e t i s s u e which was l a i d on a g l a s s p l a t e and charged w i t h methanol. An approximately c y l i n d r i c a l form was o b t a i n e d by l a p p i n g f o u r edges of the p a r a l l e l p i p e d . T h i s process must be c a r r i e d out i n a i r , and a m i r r o r - l i k e f i n i s h i s o b t a i n a b l e by e t c h i n g f o r a few seconds i n methanol and then r i n s i n g i n xylene. Xylene a l s o serves to remove the petroleum j e l l y c o a t i n g which was n e c e s s a r y to p r o t e c t the c r y s t a l d u r i n g any long p e r i o d s of exposure to a i r . In i t s f i n a l form the sample was c y l i n d r i c a l , 3.5 m i l l i - meters i n diameter and 4.5 m i l l i m e t e r s i n l e n g t h , with the -43- <001> d i r e c t i o n w i t h i n three degrees of the c y l i n d e r a x i s . I t was glued w i t h 'Copper P r i n t ' to the end of a brass rod one i n c h i n l e n g t h and s l i g h t l y s m a l l e r i n diameter than the c r y s t a l i t s e l f . T h i s rod served as a h o l d e r i n the r o t a t i o n experiment. B The P o s i t r o n A n n i h i l a t i o n F a c i l i t i e s : The apparatus f o r t h i s study evolved from equipment used i n p r e v i o u s i n v e s t i g a t i o n s of copper and i t s a l l o y s . A d d i t i o n a l i n f o r m a t i o n to t h a t p r o v i d e d here may be found i n the graduate theses of P e t i j e v i c h and Becker. The p o i n t geometry method d e s c r i b e d i n Chapter I I , s e c t i o n C ( i i ) i s used, and p r o v i s i o n i s made f o r e i t h e r a sideways motion of the d e t e c t o r s (8*0) or a r o t a t i o n of the specimen wi t h the d e t e c t o r s f i x e d at 8=0. The sequence of events i n an experiment may be d e s c r i b e d w i t h r e f e r e n c e to f i g u r e 8 which shows the b a s i c arrangement of the apparatus except f o r d e t a i l s i n the v i c i n i t y of the source and sample. P o s i t r o n s from the sodium-22 source are f o c u s s e d onto the s u r f a c e of the sample where a n n i h i l a t i o n s w i t h e l e c t r o n s produce gamma r a y s . A d e t e c t o r arrangment con- s i s t i n g of a N a l ( T l ) s c i n t i l l a t o r c r y s t a l (Nuclear E n t e r p r i s e s Inc. type 408), and a p h o t o m u l t i p l i e r tube (RCA-6342A or RCA-6810A), i s p l a c e d behind a 4" t h i c k l e a d b l o c k 25 .feet away from the sample. The c o l l i m a t i n g hole i n the l e a d b l o c k l e a d i n g to the N a l ( T l ) c r y s t a l i s only 1/4 i n c h i n diameter, t h e r e f o r e assuming that the sample i s a p o i n t source, the -44- High V o l t a g e P S . Pre-amp S h a p e r C i r c u i t Na-22 S o u r c e A L e a d Nal(TI) C r y s t a l P h o t o m u l t i p l i e r S a m p l e S h a f t High V o l t a g e R S . Pre-amp Shaper C i r c u i t M o t o r T i m e r Time De lay Re lay P r i n t e r Sca le r C o i n c i d e n c e Unit Figure 8= B lock diagram of the exper imenta l apparatus . Detai ls in the vicinity of the sample are o m i t t e d . -45- geometric r e s o l u t i o n i s 0.8 m i l l i r a d i a n s . I t has been d e t e r - mined i ( r e f . 62, page 66) t h a t the a c t u a l r e s o l u t i o n f u n c t i o n f o r t h i s apparatus i s approximately Gaussian, w i t h a f u l l width at h a l f maximum of one m i l l i r a d i a n . When a gamma ray s t r i k e s a N a l ( T l ) c r y s t a l , the l i g h t produced causes the p h o t o m u l t i p l i e r tube to emit a p u l s e . T h i s p u l s e i s f e d i n t o a c i r c u i t which a m p l i f i e s i t to 1.5 v o l t s and a l s o shapes i t i n t o a form s u i t a b l e f o r the f a s t c o i n c i d e n c e c i r c u i t . A d e t a i l e d schematic diagram of the p r e - a m p l i f i e r and shaper c i r c u i t i s giv e n i n f i g u r e 9. This c i r c u i t a l s o serves as an energy d i s c r i m i n a t o r so t h a t a l l p u l s e s below 140 keV are e l i m i n a t e d . P u l s e s from a p a i r of d e t e c t o r s which are a l i g n e d c o l l i n e a r - l y w i t h the c r y s t a l at 6=0 are f e d i n t o the f a s t c o i n c i d e n c e c i r c u i t o f f i g u r e 10. I f two p u l s e s from the p a i r of d e t e c t o r s a r r i v e a t the in p u t s w i t h i n 25 nanoseconds, t h e i r combined v o l t a g e of 3 v o l t s i s s u f f i c i e n t to f i r e the tu n n e l diode. T h i s , i n t u r n , switches on the three t r a n s i s t o r s i n the c i r c u i t and a s i n g l e c o i n c i d e n c e count i s r e g i s t e r e d on a Canberra I n d u s t r i e s model 1473 s c a l e r . The r e s o l v i n g time, T, of each c o i n c i d e n c e c i r c u i t i s measured u s i n g a random source method. I f N-̂  and ̂  are the counts produced i n two d e t e c t o r s by u n c o r r e l a t e d sources, the number of chance c o i n c i d e n c e s r e c o r d e d i s g i v e n by N c h a n c e = 2 t N 1 N 2 • +30V .01 uf 1N752 820 .01 Uf '1N100 2N9G4 1N617 2N964 ^^1N3714 1 0 0 p f 2.2K 150 100pf VVW • L 2N706A_ 47K — "~ 150 WW > 820 10K OUT 1 0 + 3OV Connections Fig. 9: Pre-amp and shaper circuit, 1N750 ® S1N752 •2.2K If 10uf 1N754 10uf Delay Line 3" of HH2000 2 0 pf •1.8K 3.3K 2N797 • 02|jf 1N100 4.7K .01 uf jr—AAMr 2.7 K 2N964 1N100 2N706A 680 - 1 8 K 27pf •AA/W-o 100K IN +30V 100 100 470 - v V v \ A 1 F igure 10: Co i n c i dence c i r c u i t . -48- By a d j u s t i n g the potentiometer which c o n t r o l s the b i a s on the tu n n e l diode i n f i g u r e 10, the r e s o l v i n g time, T, can be ad j u s t e d to approximately 25 nanoseconds. To o b t a i n the max- imum c o i n c i d e n c e count r a t e , the c a b l e lengths from the d e t e c t o r s to the c o i n c i d e n c e c i r c u i t s must a l s o be op t i m i z e d so t h a t the time r e q u i r e d f o r a p u l s e to reach the c o i n c i d e n c e c i r c u i t i s the same f o r each d e t e c t o r i n a p a i r . For an experiment i n which the c r y s t a l i s r o t a t e d , the sample h o l d e r f i t s i n t o the end of a hollow s h a f t which leads to the r o t a t i o n motor. In order to stop the r o t a t i o n only i n p a r t i c u l a r d i r e c t i o n s , a 12 i n c h diameter brass d i s c which has notches on the ci r c u m f e r e n c e ( i n t h i s case at 45° i n t e r v a l s ) i s mounted on the s h a f t of the motor. When the motor i s turned from one p o s i t i o n to the next, a m i c r o s w i t c h r i d i n g the circum- f e r e n c e of the wheel immediately stops the r o t a t i o n when the succeeding notch i s reached. To prevent such f a c t o r s as source decay and d r i f t s i n the e l e c t r o n i c s from i n t r o d u c i n g a n i s o t r o p i c s i n the experimental r e s u l t , i t i s d e s i r a b l e to count i n one p a r t i c u l a r o r i e n t a t i o n only f o r a r e l a t i v e l y s h o r t time compared with the time r e q u i r e d f o r the e n t i r e experiment. A Canberra model 1492 timer was used to c o n t r o l t h i s i n t e r v a l . A f t e r counts had been accumul- ated f o r twenty minutes i n one p o s i t i o n , a pulse from the timer a c t i v a t e d a Canberra model 1489 tape p r i n t e r which p r i n t e d out the number of c o i n c i d e n c e counts r e g i s t e r e d on each s c a l e r . -49- At the same time, the motor rotated the specimen to the next p o s i t i o n on the wheel. While the motor turned and the number of counts was being printed, a time delay relay inactivated the scalers so that no fal s e counts.could be registered. The scalers then began counting from zero i n the new p o s i t i o n and the above procedure was repeated automatically u n t i l a suf- f i c i e n t number of counts for good s t a t i s t i c s was accumulated. Figure 11 gives a more detailed view of the apparatus i n the v i c i n i t y of the sample. Since approximately one h a l f of the positrons s t r i k i n g the sample are absorbed within 0.04" (ref. 2, page 81), i t was necessary to enclose the sample i n a helium atmosphere to minimize oxidation of the surface. A l t e r n a t i v e l y , the chamber could have been evacuated, but this would have l i t t l e advantage because the penetration depth of positrons i n helium i s large. This was confirmed by the very low background count obtained when the sample was removed. The helium atmosphere was contained i n an a i r t i g h t brass box of dimensions 3" * 6" x 6" placed between the pole faces of the magnet used to focus the positrons onto the sample. Approxi- mately every eight hours, the box was flushed with helium , through a valve and outlet (not shown) located on the top and bottom of the box respectively. The a n n i h i l a t i o n photons ; escaped to the detectors through clear l u c i t e windows on the . sides of the box. The positrons were emitted from a sodium-22 source pur- - 5 0 - Shaft Leading to Rotation Motor ' J Figure 11: Top view of details of the apparatus around the s a m p l e . -51- chased from New England Nuclear C o r p o r a t i o n . I t s i n i t i a l p o s i t r o n a c t i v i t y at the time of purchase was 30 m i l l i c u r i e s , but because the h a l f - l i f e of sodium-22 i s 2.6 y e a r s , the s t r e n g t h at the time of t h i s experiment was about 10 m i l l i - c u r i e s . A s t r o n g magnetic f i e l d was used to focus the p o s i - t r o n s onto the s u r f a c e of the sample, and l e a d b l o c k s were arranged as shown i n f i g u r e 11 so t h a t the d e t e c t o r s c o u l d not r e g i s t e r a n n i h i l a t i o n s from the source, sample h o l d e r , or s i d e s of the box. In order to prevent the time r e q u i r e d to perform an ex- periment from becoming p r o h i b i t i v e , i t was necessary to have s i x separate p a i r s of d e t e c t o r s accumulating counts. Each p a i r j s connected to a separate c o i n c i d e n c e c i r c u i t so t h a t no c r o s s c o r r e l a t i o n s can occur, and the outputs from two c o i n - cidence c i r c u i t s are f e d i n t o one of three separate s c a l e r s . O b v i o u s l y , s i n c e each p a i r of d e t e c t o r s cannot occupy the same p o s i t i o n i n space, the c y l i n d e r s . t h r o u g h k-space [see f i g . 3(b)] w i l l not be p a r a l l e l to each o t h e r . However, s i n c e the sample to d e t e c t o r d i s t a n c e i s l a r g e , no d e t e c t o r i s more than 2° from the true d i r e c t i o n of i n t e r e s t at 6=0. The r e s o l u t i o n f u n c t i o n of 0.8 m i l l i r a d i a n subtends a much l a r g e r angle at the Fermi r a d i u s (approximately 21.5° i n the case of l i t h i u m ) , hence the e r r o r i n t r o d u c e d i s not s i g n i f i c a n t . At the s t a r t of an experiment, the d e t e c t o r s and sample are a l i g n e d i n the 0=0 p o s i t i o n so t h a t a s i n g l e p a i r of -52- d e t e c t o r s passes c o l l i n e a r l y through the s u r f a c e of the sample. This alignment i s performed o p t i c a l l y u s i n g a surveyor's l e v e l to measure the v e r t i c a l h e i g h t of the d e t e c t o r s , and a t r a n s i t to measure the h o r i z o n t a l positions.- The accuracy of t h i s alignment i s approximately ±0.05 m i l l i m e t r e s over the e n t i r e d i s t a n c e of" 20 metres between the d e t e c t o r s . -53- CHAPTER IV. ANALYSIS OF DATA A E x p e r i m e n t a l R e s u l t s : Of the s e v e r a l p o s s i b l e r o t a t i o n s of i n t e r e s t i n a body- c e n t r e d c u b i c c r y s t a l , the most s t a t i s t i c a l l y f a v o u r a b l e one f o r comparing the <110> and <100> d i r e c t i o n s i s a b a s a l plane r o t a t i o n i n which the a x i s of r o t a t i o n i s along the <001> c r y s t a l l o g r a p h i c d i r e c t i o n . I f t h i s a x i s i s p e r p e n d i c u l a r to the l i n e j o i n i n g the d e t e c t o r s , then a s i n g l e r o t a t i o n w i l l expose the <100>, <110>, <010>, <110>, <100>, <110>, <010>, and <110> d i r e c t i o n s r e s p e c t i v e l y . Since each of these d i r e c - t i o n s i s se p a r a t e d by 45°, i t was.necessary to c o n s t r u c t the r o t a t i o n c o n t r o l wheel w i t h e i g h t notches along the circum- f e r e n c e . These notches were numbered 1 to 8, and the c r y s t a l was o r i e n t e d so t h a t an odd p o s i t i o n corresponded to a <110> d i r e c t i o n and an even p o s i t i o n corresponded to a <100> d i r e c - t i o n . In a d d i t i o n to the f a c t t h a t each of the above p r i n c i p a l d i r e c t i o n s occurs an e q u i v a l e n t of f o u r times i n 360°, the b a s a l plane r o t a t i o n has the advantage t h a t the d i r e c t i o n s e x a c t l y between the p r i n c i p a l d i r e c t i o n s are a l l e q u i v a l e n t . These m i d p o s i t i o n s do not correspond to d i r e c t i o n s w i t h low M i l l e r i n d i c e s , but they l i e w i t h i n a degree of the <520> d i r e c t i o n . I f the c r y s t a l i s r o t a t e d by 22.5° w i t h r e s p e c t to -54- th e wheel, i t i s apparent t h a t the c o i n c i d e n c e count r a t e i n each p o s i t i o n s hould be the same. T h i s serves as a u s e f u l check that any a n i s o t r o p y observed i n comparing the <110> and <100> d i r e c t i o n s i s genuine. Even w i t h s i x p a i r s of d e t e c t o r s , the t o t a l count r a t e was only about 4.5 counts each minute, and to o b t a i n good s t a t i s t i c s i t was necessary to count f o r a t o t a l of 16 days. Two t h i r d s of t h i s time was devoted to sampling the <110> and <100> d i r e c t i o n s , and one t h i r d was devoted to the m i d p o s i t i o n . As s t a t e d p r e v i o u s l y , the wheel remained at one p a r t i c u l a r notch f o r only twenty minutes b e f o r e being r o t a t e d to the next p o s i t i o n . To ensure t h a t the t o t a l counts f o r the m i d p o s i t i o n c o u l d be n o r m a l i z e d to the other two d i r e c t i o n s , c o u n t i n g was c a r r i e d out i n the p r i n c i p a l d i r e c t i o n s f o r two days and then the sample was r o t a t e d by 22.5° to the m i d p o s i t i o n f o r one day. Th i s process was repeated u n t i l the experiment was completed. F i n a l l y , a background count was taken f o r a p e r i o d of 4.7 days. T h i s was accomplished by removing the sample and h o l d e r but l e a v i n g the r e s t of the exp e r i m e n t a l apparatus e x a c t l y as b e f o r e . The background count which i s measured i n t h i s way a r i s e s from a n n i h i l a t i o n s i n the helium and the s i d e s o f the box used to c o n t a i n i t . There are a l s o a s m a l l number of chance c o i n c i d e n c e s from power l i n e f l u c t u a t i o n s , cosmic r a y s , and other d e t e c t o r n o i s e . The number of counts r e c o r d e d amounted to approximately 2% of the t o t a l c o i n c i d e n c e counts. -55- F i g u r e 12 d i s p l a y s the experimental r e s u l t s a f t e r the background has been s u b t r a c t e d . No c o r r e c t i o n has been a p p l i e d f o r the decay of the sodium-22 source s i n c e i t s h a l f - l i f e i s 2.6 years and the slow decay w i l l be averaged out by the r o t a t i o n . Each graph i n f i g u r e 12 r e p r e s e n t s the output from a s i n g l e s c a l e r or two p a i r s of d e t e c t o r s . The graphs on the l e f t cor- respond to the p r i n c i p a l d i r e c t i o n s , w i t h the <110> d i r e c t i o n s i n the odd numbered p o s i t i o n s and the <100> d i r e c t i o n s i n the even numbered p o s i t i o n s . The r i g h t - h a n d graph of each p a i r corresponds to the m i d p o s i t i o n r e s u l t s f o r the same d e t e c t o r s as on the l e f t . The e r r o r bar on each experimental p o i n t r e p r e s e n t s ( v ^ " c + , where N c i s the t o t a l number of c o i n - cidence counts r e g i s t e r e d and N h i s the c o r r e s p o n d i n g number of background counts. An examination of the graphs on the l e f t shows t h a t the <110> p o i n t s are a l l h i g h e r than the <100> p o i n t s except f o r p o s i t i o n s 4 and 5 i n the middle and bottom rows. In c o n t r a s t , the m i d p o s i t i o n graphs do not appear to e x h i b i t any d e f i n i t e o rder. These trends are i l l u s t r a t e d more c l e a r l y i n f i g u r e 13 which i s an accumulation of the r e s u l t s i n f i g u r e 12. The top graph i n d i c a t e s a s i g n i f i c a n t i n c r e a s e i n the number of counts f o r the <110> d i r e c t i o n s , whereas i n the bottom graph the e f f e c t i s markedly reduced. I f the odd p o s i t i o n s i n the bottom graph are c o n s i d e r e d s e p a r a t e l y from the even p o s i t i o n s , the a c t u a l -56- 1300 1250 - 1200 - 1150 - 1100 I t T 0 I J L I J L 3 4 5 6 7 8 1450 1400 1350 1300 I 1 I I I J I 1 I I I I L 8 1650 - 1 6 0 0 - 1 5 5 0 - 1500 - 1450 - I T J 1 I l_ 3 4 5 6 7 8 Figure 12= Exper imenta l resul ts . Coincidence counts versus wheel posit ion. On the left-hand graphs, odd posi t ions are (110) d i rect ions and even posit ions are <100) d i rect ions. The r ight-hand graphs are the midposi t ion resul ts for the same d e t e c t o r s as on the l e f t . Each graph represents the counts f rom two pairs of d e t e c t o r s . -57- 3 4 5 6 7 8 Wheel Position 3 4 5 6 Wheel Position Figure 13: Cumulative results from fig. 12. The top graph gives the (110) and (lOO) directions (odd and even positions respectively). The bot- tom graph gives the midposition result. -58- n u m e r i c a l v a l u e s are 16,679 ± 147 and 16,468 ± 146 r e s p e c t i v e l y . Although the odd p o s i t i o n count i s 1.3% h i g h e r than the even p o s i t i o n count, the d i f f e r e n c e i s w i t h i n the s t a t i s t i c a l e r r o r as would be expected f o r e q u i v a l e n t points.. The data of f i g u r e s 12 and 13 can be reduced f u r t h e r to giv e the number of counts r e g i s t e r e d i n each of the three d i f - f e r e n t d i r e c t i o n s sampled by t h i s experiment. The nu m e r i c a l r e s u l t s are g i v e n i n t a b l e 2; the f o u r rows of the t a b l e c o r - responding to the fou r p a i r s o f graphs given i n f i g u r e s 12 and 13. The m i d p o s i t i o n i s a u t o m a t i c a l l y normalized to the other two d i r e c t i o n s because of the manner i n which the experimental data was accumulated. Comoarine the <110> and <100> d i r e c t i o n s of each row of t a b l e 2, i t i s found t h a t the <110> d i r e c t i o n i s r e s p e c t i v e l y 5.9%, 4.4%, 5.6%, and 5.3% g r e a t e r than the <100> d i r e c t i o n . The net d i f f e r e n c e of 5.3% i s o b v i o u s l y s i g n i f i c a n t compared w i t h the e r r o r s of ±0.6% on each.of the two p o i n t s . T h i s r e s u l t agrees w e l l w i t h the one o b t a i n e d from the phenomeno- l o g i c a l model of Donaghy and Stewart. The m i d p o s i t i o n count i n t a b l e 2 occurs midway between the p r i n c i p a l d i r e c t i o n counts i n the f i r s t two rows, but i s h i g h e r than the <110> count i n the t h i r d row. The net r e s u l t i s t h a t the m i d p o s i t i o n t o t a l i s 4% h i g h e r than the <100> t o t a l . Ac- c o r d i n g to the r e s u l t s o f Donaghy and Stewart ( r e f . 4, page 395, f i g u r e 5), the m i d p o s i t i o n should not be more than 1% h i g h e r -59- Table 2 A comparison of the n u m e r i c a l v a l u e s of the c o i n c i counts f o r the 3 d i f f e r e n t d i r e c t i o n s . The f i r s t 3 row the t a b l e correspond to the 3 p a i r s of graphs i n f i g u r e and the l a s t row corresponds to the accumulated r e s u l t s i n f i g u r e 13. <110> M i d p o s i t i o n <10 1 9885 ± 114 9629 ± 113 9337 •2 11,253 ± 123 11,046 ± 122 10,782 3 12,424 ± 126 12,472 ± 127 11,762 4 33,562 ± 210 33,147 ± 202 31,881 -60- than the <100> d i r e c t i o n . The f a c t t h at the r e s u l t here i s s i g n i f i c a n t l y g r e a t e r does not c o n s t i t u t e a s e r i o u s d i s c r e p a n c y between these two experiments. There are two reasons f o r t h i s . F i r s t l y , i t can be estimated t h a t the f u l l width at h a l f max- imum of the r e s o l u t i o n f u n c t i o n i n t h i s experiment subtends an angle of 21.5° at the Fermi r a d i u s ( r e f . 62, page 66). I f t h i s r e s o l u t i o n f u n c t i o n o v e r l a p s the s i d e of the bulge i n the <110> d i r e c t i o n , the count r a t e i n the m i d p o s i t i o n w i l l be a r t i f i c i a l l y enhanced. Secondly, i t w i l l be shown l a t e r t h a t the h i g h e r momentum components of the e l e c t r o n wavefunction can have a s i g n i f i c a n t e f f e c t on the count r a t e , and t h i s e f f e c t i s not easy to determine f o r the m i d p o s i t i o n . For these reasons the u s e f u l n e s s of the m i d p o s i t i o n r e s u l t i s l i m i t e d to v e r i f y i n g t h a t the count r a t e i s i s o t r o p i c f o r e q u i v a l e n t c r y s t a l l o g r a p h i c d i r e c t i o n s . B Higher Momentum Components of the P o s i t r o n Wavefunction: A d e t a i l e d examination of the h i g h e r momentum components of the e l e c t r o n and p o s i t r o n wavefunctions i s necessary i n order to determine whether or not the observed 5.3% d i f f e r e n c e between ^110 a n < ^ ^100 a r x S e s f r o m a genuine d i s t o r t i o n of the Fermi s u r f a c e of l i t h i u m . I f these h i g h e r momentum components are l a r g e , the n e a r l y f r e e e l e c t r o n approximation i s o b v i o u s l y i n - a p p l i c a b l e . Even when they are r e l a t i v e l y s m a l l , however, the hi g h e r momentum components have an important experimental -61- s i g n i f i c a n c e s i n c e they determine the number of s t a t e s k which are s c a t t e r 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 . Such s t a t e s are not d e t e c t e d because they g e n e r a l l y l i e o u t s i d e the experimental r e s o l u t i o n f u n c t i o n . In e q u a t i o n (2) on page 8, i t was assumed that the p o s i t r o n wavefunction <J>+(r_) c o u l d be c o n s i d e r e d c o n s t a n t . The v a l i d i t y of t h i s assumption can be examined by expanding the wavefunction of a t h e r m a l i z e d (k=0) p o s i t r o n i n a s e r i e s i n v o l v i n g r e c i p r o c a l l a t t i c e v e c t o r s K: <j>+(r) = a o + E a K e 1 ^ " ! . (9) K — I f the c o e f f i c i e n t s a^ are s m a l l , t h e i r v a l u e s are g i v e n from f i r s t order p e r t u r b a t i o n theory by a K = — (10) £ n 2 K 2 2m A value f o r a must be determined from the n o r m a l i z a t i o n con-o d i t i o n | a Q | 2 + | a^. | 2 = 1. The F o u r i e r c o e f f i c i e n t s , V^, of the p o t e n t i a l seen by the p o s i t r o n may be c a l c u l a t e d by a d i r e c t 18 method developed by Stroud and E h r e n r e i c h . By u t i l i z i n g the r e l a t i o n s h i p between the e l e c t r o n charge d e n s i t y and the x-ray form f a c t o r , the F o u r i e r c o e f f i c i e n t s can be o b t a i n e d d i r e c t l y from experiment without the n e c e s s i t y of having to choose a p a r t i c u l a r form f o r the p o t e n t i a l . The d e t a i l e d a p p l i c a t i o n of t h i s procedure to l i t h i u m i s given i n Appendix I. -62- For the two s h o r t e s t r e c i p r o c a l l a t t i c e v e c t o r s ^110 = 2/2 IT/a and K^QQ = 4-rr/a, the values of the F o u r i e r coef- f i c i e n t s are V-^Q = 0.1182 Ry. and V"2QQ = 0.0713 Ry. Atomic u n i t s (see page 27) are used to s i m p l i f y the c a l c u l a t i o n s . In these u n i t s e q u a t i o n (10) becomes K and the c o e f f i c i e n t s are A K = — , : W A 1 1 0 = 0'°636 A N C * A 2 0 0 = ^ - 0 1 ^ 2 f o r a l a t t i c e c o n s t a n t of a = 6.5183 a.u. C o n s i d e r i n g the f a c t t h a t twelve (110) and s i x (200) planes c o n t r i b u t e to the s c a t t e r i n g , the p r o b a b i l i t i e s of i n t e r e s t are: l a I  2  = '0.95 1 o 1 1 2 | a 1 1 0 | 2 = 0.0486 6 | a 2 0 0 | 2 = 0.0022 I t i s obvious from these v a l u e s t h a t the f i r s t c o e f f i c i e n t i s dominant and the procedure of t e r m i n a t i n g e q u a t i o n ( 9 ) a f t e r a i s j u s t i f i e d . The above r e s u l t i s c o n s i s t e n t w i t h the r e s u l t s o f other authors ' who have chosen a form f o r the p o t e n t i a l and then c a l c u l a t e d p o s i t r o n wavefunctions i n the Wi g n e r - S e i t z a p p r o x i - mation which r e q u i r e s s p h e r i c a l symmetry and zero s l o p e at the Wig n e r - S e i t z r a d i u s . -63- C Higher Momentum Components o£ the E l e c t r o n Wavefunction: The task of e v a l u a t i n g the hi g h e r momentum components of the e l e c t r o n wavefunction i s not as s t r a i g h t f o r w a r d as i t was f o r the p o s i t r o n . In the n e a r l y f r e e e l e c t r o n approximation, the c o n d u c t i o n e l e c t r o n wavefunction can be expanded as K k , r ) = Z a (k) e i ( ^ } ' I , (12) K - where the c o e f f i c i e n t s can be obt a i n e d from f i r s t order per- t u r b a t i o n theory as b e f o r e : a K (10 = . (13) EK-k " E k The d i f f i c u l t y i s th a t no d i r e c t method i s a v a i l a b l e f o r o b t a i n i n g the F o u r i e r c o e f f i c i e n t s o f tiie p o t e n t i a l , and i f one assumes a form f o r the p o t e n t i a l then one e f f e c t i v e l y as- sumes a shape f o r the Fermi s u r f a c e . Thus, the procedure of adopting a p o t e n t i a l to determine the e f f e c t of the h i g h e r momentum components of the e l e c t r o n wavefunction i s i n c o n s i s t e n t . N e v e r t h e l e s s , s i n c e the Fermi s u r f a c e of l i t h i u m i s not g r e a t l y d i s t o r t e d and the t h e o r e t i c a l c a l c u l a t i o n s appear to be r e l a - t i v e l y i n s e n s i t i v e to s m a l l d i f f e r e n c e s i n the p o t e n t i a l , i t i s hoped t h a t such a procedure w i l l allow a reasonable estimate of the h i g h e r momentum components to be made. The f i r s t f i v e v a l u e s o f f o r a f l a t t e n e d S e i t z p o t e n t i a l are gi v e n i n t a b l e 2 8 3 as taken from the paper of S c h l o s s e r and Marcus. These -64- Table 3 F o u r i e r c o e f f i c i e n t s of l i t h i u m f o r a f l a t t e n e d S e i t z p o t e n t i a l (taken from S c h l o s s e r § M a r c u s ^ ) The l a t t i c e c o n s t a n t i s a=6.5183 a.u. K |K| -V^ 0 0.0000 a.u. 1.00221 Ry 11U 1.3632 0.16889 200 1.9279 0.09435 211 2.3611 0.06388 220 2.7264 0.05166 -65- c o e f f i c i e n t s are very s i m i l a r to those o b t a i n e d from the OPW 32 33 method of Callaway. > As i n d i c a t e d p r e v i o u s l y , the q u a n t i t y | â - j 2 determines the p r o b a b i l i t y t h a t a photon p a i r w i l l -have momentum ft(K-k). Hence, these c o e f f i c i e n t s w i l l determine the number of p a i r s t h a t are not d e t e c t e d , and i t i s necessary to compare t h e i r e f f e c t on the count r a t e along the <110> and <100> d i r e c t i o n s . The number of counts l o s t along the <100> d i r e c t i o n w i l l be c o n s i d e r e d f i r s t . F i g u r e 14(a) shows a cross s e c t i o n through the f i r s t B r i l l o u i n zone and f r e e e l e c t r o n sphere i n k-space, with the s l i c e b e i n g taken p a r a l l e l to the b a s a l plane. I t w i l l be h e l p f u l throughout t h i s d i s c u s s i o n to r e f e r to the diagram of the f i r s t B r i l l o u i n zone of l i t h i u m given on page 25. The a c t u a l width of the r e s o l u t i o n f u n c t i o n along the <100> d i r e c t i o n i s shown by the d o t t e d l i n e s i n f i g u r e 14(a). For convenience i t w i l l be assumed t h a t t h i s r e s o l u t i o n i s a c t u a l l y a l i n e through k-space, although t h i s w i l l o b v i o u s l y l i m i t the v a l i d i t y of the r e s u l t o b t a i n e d . C o n s i d e r i n g the s t a t e k and the s t a t e s c a t t e r e d by the (110) plane 'B' i n f i g u r e 14(a), i t i s p o s s i b l e to express the e n e r g i e s r e q u i r e d i n the denominator of equation (13) as and - 6 6 - Figure 14- C r o s s - s e c t i o n s through the f i rst B r i l l o u i n z o n e of l i th ium. -67- The c o - o r d i n a t e axes are taken as shown i n the diagram. Hence, *. 2 if 2 E v , - E, = f l - 2k /Kl and s i n c e k = k sin45°, i t i s p o s s i b l e to w r i t e e q u a t i o n (13) f o r the c o e f f i c i e n t s i n atomic u n i t s as •V a v = • (14) £ K 2 ( l - /2k/K) As expected, the magnitude of a^ depends on the d i s t a n c e from the zone boundary. The c o e f f i c i e n t necessary f o r the e x p l i c i t c a l c u l a t i o n i s i n t a b l e 3, and values of | a ^ | 2 f o r s e l e c t e d d i s t a n c e s from -kp to kp are g i v e n i n column A of t a b l e 4. These c o e f f i c i e n t s are e v i d e n t l y much l a r g e r than the c o e f f i c i e n t s o f the p o s i t r o n wavefunction. An average value of | a ^ | 2 f o r column A i s e a s i l y o b t a i n e d from the u s u a l r e l a t i o n k F / |a | 2 dk -k F ^ 2 _ £_ aK k F / dk k F T h i s y i e l d s J | 2 = 0.01316 f o r the s i n g l e plane 'B' i n f i g u r e 14(a). However, fo u r such planes o f the B r i l l o u i n zone come together at 'A' to form the corner, and s i n c e there are f o u r on the other s i d e of the zone i t i s necessary to m u l t i p l y |a^| -68- Table 4 F r a c t i o n of kj 1. 0 0.8 0.6 0.4 "0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Column A 2 a K1 Average 0.0543 0.0315 0.0205 0.0145 0.0107 0.0083 0.0066 0 .0053 0.0044 0.0037 0.0032 0.01316 Column B 0 .0255 0.0193 0.0150 0.0121 0.0099 0.0083 0.0070 0.0060 0.0052 0.0046 0.0040 0.01015 Column A giv e s the values of |a- K| 2 f o r the s c a t t e r i n g of s t a t e k by the plane 'B' i n f i g u r e 14(a). Column B g i v e s the valu e s of | a ^ | 2 f o r the s c a t t e r i n g o f s t a t e k by the plane 'D' i n f i g u r e 1 4 ( c ) . -69- by e i g h t to o b t a i n a t o t a l c o n t r i b u t i o n of 0.105 from these p l a n e s . F i n a l l y , i t can be shown t h a t the other four planes of the B r i l l o u i n zone are of the type ' C shown i n f i g u r e 14(b). Since these are a l l p a r a l l e l to the <100> d i r e c t i o n , each w i l l have |a-j^| 2 = 0.0083 from t a b l e 4. Adding the c o n t r i b u t i o n s from a l l twelve (110) p l a n e s , the r e s u l t i s 0.138. I t i s apparent t h a t f o u r of the s i x (200) planes w i l l a l s o a f f e c t the count r a t e , although s c a t t e r i n g from these planes i s reduced because of the s m a l l e r value of V^QQ and the l a r g e r v a l u e of K^QQ. The net e f f e c t of these planes i s to reduce the count r a t e by 0.3%. Hence, the e f f e c t of s c a t t e r i n g from both the (110) and (200) planes i m p l i e s t h a t a t o t a l of 14.1% of the a n n i h i l a t i o n p a i r s w i l l not be d e t e c t e d when the apparatus i s a l i g n e d along the <100> d i r e c t i o n . Now c o n s i d e r the number of counts l o s t when the apparatus i s a l i g n e d along the <110> d i r e c t i o n . From f i g u r e 14(a) the s c a t t e r i n g from the two planes e q u i v a l e n t to 'B' w i l l be counted except f o r a few s t a t e s which l i e o u t s i d e the r e s o l u t i o n func- t i o n . The other two planes w i l l c o n t r i b u t e 0.0083 each, or a t o t a l of 1.7%. The remaining e i g h t planes are a l l of the type 'D* shown i n f i g u r e 1 4 ( c ) , where-the r e s o l u t i o n f u n c t i o n i s along the d i r e c t i o n shown. Choosing the c o - o r d i n a t e axes as i n d i c a t e d , i t i s p o s s i b l e to c a l c u l a t e an e x p r e s s i o n f o r the c o e f f i c i e n t s s i m i l a r to e q u a t i o n (14). In t h i s case, however, k = k cos60° -70- and the r e s u l t i s V K a = . (15) ^ K 2 ( l - k/K) The r e q u i r e d F o u r i e r c o e f f i c i e n t i s once again V^lO" Values of | a ^ | 2 f o r p o i n t s from -kp to +kp are g i v e n i n column B of t a b l e 4. As expected, t h e i r i n i t i a l v a lue at kp i s lower than f o r column A, but they do not d i m i n i s l i as r a p i d l y . The average value can be c a l c u l a t e d as b e f o r e and i s equal to 0.01015. For the e i g h t planes of t h i s type the c o n t r i b u t i o n i s 8.1%, hence* the t o t a l number of counts l o s t along the <110> d i r e c t i o n from s c a t t e r i n g by the (110) planes i s 9.8%. A c o n s i d e r a t i o n of the e f f e c t of the (200) planes r a i s e s t h i s v alue to 10.2%. T r- J - i - - - ! - _ _ _ _ - .r T O o o . j i / i T O . J - 1 , i i n ^ „ — J - i n n - . 1 1 U11C X U S i C S U l i U . L ' l l i t J I U X t . X ' O X U i C l l C O J I U ^ x u u ^ d i r e c t i o n s are c o r r e c t e d f o r i n the experimental data g i v e n on page 59, the r e s u l t i n g t o t a l s are 36,985 and 36,376 respec- t i v e l y . The true d i s t o r t i o n i n the Fermi s u r f a c e would then be reduced to 1.7% which i s not a p p r e c i a b l y g r e a t e r than the s t a t i s t i c a l a ccuracy. The above c a l c u l a t i o n r e v e a l s t h a t the h i g h e r momentum components of the e l e c t r o n wavefunction have a s i g n i f i c a n t e f - f e c t on the observed a n i s o t r o p y . The i n d i c a t i o n i s t h a t the 5.3% d i f f e r e n c e between the number of counts i n the <110> and <100> d i r e c t i o n s should be regarded as an upper l i m i t to the t r u e d i s t o r t i o n of the Fermi s u r f a c e . -71- A more thorough examination of the e f f e c t of the h i g h e r momentum components of the wavefunctions i s o b v i o u s l y d e s i r a b l e . A complete c a l c u l a t i o n would have to c o n s i d e r h i g h e r terms i n the e l e c t r o n and p o s i t r o n expansions, as w e l l as the e f f e c t of the broad experimental r e s o l u t i o n f u n c t i o n . In a d d i t i o n , i t would be nece s s a r y to examine the e f f e c t of the choice of the p o t e n t i a l , s i n c e the adoption of p a r t i c u l a r v a l u e s f o r the F o u r i e r c o e f f i c i e n t s i s the most s i g n i f i c a n t assumption made i n t h i s a n a l y s i s . D Enhancement and A n n i h i l a t i o n s w i t h Core E l e c t r o n s : , ... , „ „ , . . . 16 , - m e x n g a m s ana ueiseneaetti nave rouna m a t D e t t e r quan- t i t a t i v e agreement i s o b t a i n a b l e i f Xanana's enhancement f a c t o r i s i n c l u d e d i n c a l c u l a t i n g the angular c o r r e l a t i o n curves of Donaghy and Stewart from OPW wavefunctions. The enhancement f a c t o r has been c o n s i d e r e d i s o t r o p i c i n t h i s a n a l y s i s . I t s omiss i o n s h o u l d not be s e r i o u s because the Fermi s u r f a c e o f l i t h i u m i s n e a r l y s p h e r i c a l and any e f f e c t would c e r t a i n l y be weaker than the e f f e c t of the h i g h e r momentum components of the e l e c t r o n wavefunction. In a d d i t i o n , an examination of f i g u r e 8 on page 405 of M e l n g a i l i s and DeBenedetti's paper shows that enhancement does not a f f e c t the shape of the d i s t r i b u t i o n near -72- The l a s t e f f e c t which must be considered i s that of a n n i h i l a t i o n w i t h core e l e c t r o n s . From the c o n s i d e r a t i o n s of M e l n g a i l i s and DeBenedetti, t h i s c o n t r i b u t i o n i s an order of magnitude s m a l l e r than the c o n t r i b u t i o n from conduction e l e c - trons f o r the long s l i t geometry. Since the p o i n t geometry arrangement samples an even smaller p r o p o r t i o n of the core e l e c t r o n s r e l a t i v e to the conduction e l e c t r o n s i n k-space, i t can be c a l c u l a t e d that approximately 5% of the coincidence counts a r i s e from a n n i h i l a t i o n s w i t h core s t a t e s . This small i s o t r o p i c c o n t r i b u t i o n can be s u b t r a c t e d , w i t h the r e s u l t that the t o t a l a n isotropy between the p r i n c i p a l d i r e c t i o n s increases to 5 . 6 % . -73- CHAPTER V CONCLUSIONS This chapter p r e s e n t s a b r i e f summary of the r e s u l t s of t h i s experiment. The main f e a t u r e i s that the number of counts i n the <110> d i r e c t i o n i s found to be 5.3 + 1.2 p e r c e n t g r e a t e r than the number i n the <100> d i r e c t i o n . This a n i s o t r o p y i s i n - creased to 5.6% i f the c o n t r i b u t i o n from a n n i h i l a t i o n s w i t h core e l e c t r o n s i s s u b t r a c t e d from the t o t a l number of counts. Since the c o l l i n e a r p o i n t geometry i s used, the evidence that the Fermi s u r f a c e of l i t h i u m i s d i s t o r t e d i s more d i r e c t than the evidence based on the phenomenological model used to i n t e r p r e t the r e s u l t s of Donaghy and Stewart's long s l i t ex- periment. However, the two methods agree very w e l l except f o r the enhanced value of the m i d p o s i t i o n count i n t h i s experiment. T h i s d i s c r e p a n c y i s not s e r i o u s s i n c e i t i s probably due to the e f f e c t of the h i g h e r momentum components of the e l e c t r o n w avefunction and the width of the r e s o l u t i o n f u n c t i o n . An examination of the h i g h e r momentum components of the e l e c t r o n wavefunction i n d i c a t e s t h a t the observed a n i s o t r o p y of 5.6% must be c o n s i d e r e d an upper l i m i t to the true d i s t o r t i o n o f the Fermi s u r f a c e of l i t h i u m . In t h i s experiment the h i g h e r momentum components reduce the count r a t e i n the <100> d i r e c t i o n to a g r e a t e r extent than i n the <110> d i r e c t i o n , and i f t h i s e f f e c t i s c o r r e c t e d f o r , the experimental a n i s o t r o p y i s reduced -74- to 1.7%. In c o n t r a s t , the h i g h e r momentum components of the p o s i t r o n are too s m a l l to have a s i g n i f i c a n t e f f e c t on the a n i s o t r o p y . I t i s i n t e r e s t i n g to note t h a t t h e o r e t i c a l c a l c u l a t i o n s are f a i r l y c o n s i s t e n t i n p r e d i c t i n g 5% bulges towards the zone boundaries i n the <110> d i r e c t i o n s , whereas Compton x-ray experiments were unable to r e s o l v e any a n i s o t r o p y w i t h i n an accuracy o f 3%. In a d d i t i o n , experiments on other a l k a l i metals have shown t h a t the Fermi s u r f a c e s o f these metals are c o n s i d e r a b l y l e s s d i s t o r t e d than p r e d i c t e d by the e x t e n s i v e c a l c u l a t i o n s o f Ham. The p o s i t r o n a n n i h i l a t i o n s t u d i e s c o n s t i - t u t e the only e x p e r i m e n t a l evidence f o r a 5% d i s t o r t i o n i n the Fermi:".surface of l i t h i u m , y e t i t i s apparent t h a t t h i s may p a r t i a l l y be caused by the d e v i a t i o n of the e l e c t r o n wavefunc- t i o n from a s i n g l e plane wave. I t i s v i r t u a l l y c e r t a i n t h a t the d i s t o r t i o n i s not l a r g e enough to cause c o n t a c t with the f i r s t B r i l l o u i n zone boundary i n . t h e <110> d i r e c t i o n . Concerning f u t u r e work i n t h i s area, the most important requirement i s an improved method of deter m i n i n g the e f f e c t of the h i g h e r momentum components on the c o i n c i d e n c e count r a t e s , p r e f e r a b l y i n a manner which does not r e q u i r e the p r i o r s e l e c - t i o n of a c r y s t a l p o t e n t i a l . A l o g i c a l e x t e n s i o n o f the l i t h i u m experiment would be to i n v e s t i g a t e lithium-magnesium a l l o y s . I t i s known that up to 70 a t . - % o f magnesium can be added to l i t h i u m w i t h no change -75- i n the body-centred c u b i c s t r u c t u r e and only s m a l l changes i n the l a t t i c e c o n s t a n t . P o s i t r o n a n n i h i l a t i o n experiments by S t e w a r t ^ i n p o l y c r y s t a l l i n e Li-Mg samples show that i t i s a n e a r l y f r e e e l e c t r o n a l l o y . Experiments i n s i n g l e c r y s t a l s would be of 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 the f r e e e l e c t r o n sphere of l i t h i u m expands wi t h the a d d i t i o n of magnesium u n t i l c o n t a c t occurs with the B r i l l o u i n zone f a c e . A study of t h i s e f f e c t would be v a l u a b l e because the Li-Mg a l l o y s are the s i m p l e s t to t r e a t t h e o r e t i c a l l y . -76- APPENDIX I C a l c u l a t i o n o£ the Higher Momentum Components of the E l e c t r o n 1 8 Wavefunction Using the Method of Stroud and E h r e n r e i c h : In M.K.S. u n i t s , the p o t e n t i a l V ( r ) seen by a p o s i t r o n moving i n the Coulomb f i e l d of the e l e c t r o n s and n u c l e i i s give n by 1 p(r') d r f - : V ( r ) = / , A . l 4 TTE I r - r ' I o 1 — — 1 where the t o t a l charge d e n s i t y p(r) i s a sum of the n u c l e a r and e l e c t r o n i c charge d e n s i t i e s : p(r) = p + ( r ) + p"(r) . A. 2 c o e f f i c i e n t s V K = 1 / v a / V ( - ^ e 1 - ' - dL A.3 and p K = 1/v / (r) e 1 - ' - dr . A.4 The volume of a u n i t c e l l i s denoted by v . Using the f a c t 3. t h a t these F o u r i e r c o e f f i c i e n t s are nonzero only at r e c i p r o c a l l a t t i c e v e c t o r s K, equations A . l and A.2 can be transformed to 1 pK Vv- = - — A. 5 — e K 2 o + A. 6 P K = P K + P K -77- Thus one can o b t a i n the F o u r i e r c o e f f i c i e n t s of the p o t e n t i a l seen by the p o s i t r o n i f the charge d e n s i t y can be determined. T r e a t i n g the n u c l e a r p a r t of A.6 i n terms of p o i n t charges Ze at each l a t t i c e s i t e , i t i s e v i d e n t t h a t PK = — - A ' 7 — a The e l e c t r o n charge d e n s i t y p^, can be expressed i n terms of the atomic s c a t t e r i n g f a c t o r as IPKI = | F • A. 8 — a The atomic s c a t t e r i n g f a c t o r , f , i s d e f i n e d as the r a t i o of the amplitude of the wave s c a t t e r e d by the atom to t h a t s c a t t e r e d by a s i n g l e e l e c t r o n . Values of f f o r l i t h i u m were f 0 o b t a i n e d from the Metals Reference Book. Although these valu e s are t h e o r e t i c a l , i t i s found t h a t agreement w i t h exper- i m e n t a l form f a c t o r s i s g e n e r a l l y g o o d . ^ I t i s e v i d e n t t h a t the s i g n of p^ i n equation A.8 must be n e g a t i v e s i n c e -Ze/v i s simply the average e l e c t r o n charge d e n s i t y . I n c o r p o r a t i n g the r e s u l t s of A.6, A.7, and A.8 i n t o e q u a t i o n A.5 i t f o l l o w s t h a t VK e v * — o a x v However, c a l c u l a t i o n s are performed more e a s i l y i f A.9 i s expressed i n atomic u n i t s so t h a t i n rydbergs i s given by -78- For a body-centred c u b i c c r y s t a l the volume of the u n i t c e l l i s v^ = l / 2 ( a 3 ) where the l a t t i c e c o nstant 'a' i s taken as 6.5183 a.u. The two s h o r t e s t r e c i p r o c a l l a t t i c e v e c t o r s which g i v e r i s e to nonv a n i s h i n g c o e f f i c i e n t s are K110 = 2/2TT/a and K200 = 4 7 T / a • The v a l u e s of f are gi v e n f o r values of sin6/X from 0.0 to 1.1 (A i n A), t h e r e f o r e i t i s necessary to determine sin0/A f o r (110) and (200) r e f l e c t i o n s i n l i t h i u m . T h i s i s e a s i l y accomplished u s i n g the Bragg r e l a t i o n 2dsin0 = nA, and i t i s found t h a t sinB/X = 0.205 f o r a (110) r e f l e c t i o n and sin0/A = 0.290 f o r a (200) r e f l e c t i o n . Th i n t e r p o l a t e d v a l u e s of f are 1.79 and 1.54 r e s p e c t i v e l y . 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Japan 2 0
United States 2 0
City Views Downloads
Unknown 4 2
Beijing 3 0
Tokyo 2 0
Mountain View 1 0
Ashburn 1 0

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