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A study of positron annihilation in copper alloys Becker, Ernest Henry 1969

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A STUDY OF POSITRON ANNIHILATION IN COPPER ALLOYS by ERNEST HENRY BECKER B.Sc, University of Manitoba, 1965 M.Sc, University of Manitoba, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September,•1969 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t 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 s t u d y . I f u r t h e r agree tha p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y o f B r i t i s h Co lumbia V a n c o u v e r 8 , Canada Date ABSTRACT An apparatus f o r the study of positron a n n i h i l a t i o n containing some novel new features has been constructed and has been used to study disordered copper a l l o y s i n order to i n v e s t i g a t e the value of t h i s technique i n determining the ele c t r o n momentum d i s -t r i b u t i o n . The concentrated alloys. C i i A l . ^ , Cu__Zn__, Cu__Zn^_Ni-_ and Cu-Jtfi,,. have been studied and e f f e c t s a t t r i b u t e d to the d e t a i l s 60 40 of the Fermi surface of these a l l o y s observed. In p a r t i c u l a r , the [ i l l } necks i n the Fermi surfaces of these a l l o y s have been observed and the widths of these necks measured. The c u t o f f s corresponding to the Fermi radius of these a l l o y s have been measured. An anisotropy i n the a n n i h i l a t i o n of positrons with electrons i n the f i l l e d B r i l l o u i n zones has been measured and an unusual e f f e c t a t t r i b u t e d to the d-band electrons of these a l l o y s has been observed. ACKNOWLEDGEMENT I wish to express my appreciation to Dr. D.LI. Williams f o r h i s i n t e r e s t , guidance and counsel during the course of t h i s work. I wish to thank Mr. P. P e t i j e v i c h f o r many h e l p f u l discussions during the course of these experiments. The assistance of Mr. Alec Fraser i n constructing sample holders i s greatly appreciated. The research was supported by a grant from the National Research Council of Canada to Dr. D. L I . Williams. A N.R.C. Student Bursary was held by the author during the year 1968-69. V TABLE OF CONTENTS Page Chapter 1 INTRODUCTION 1 Chapter 2 ELECTRONS IN PURE METALS 4 Introduction 4 Free E l e c t r o n Gas Model 5 Peri o d i c Systems 6 The Nearly Free E l e c t r o n Model 7 C o l l e c t i v e or Many Body E f f e c t s 10 OPW and Pseudo P o t e n t i a l s 10 Band Structure C a l c u l a t i o n s 13 Chapter 3 DISORDERED ALLOYS 16 Introduction 16 Experimental Methods of Studying Concentrated A l l o y s . 1 7 1. Slow Neutrons 17 2. Photons i n A l l o y s 18 Rigid Band Model 19 Averaged Po t e n t i a l s 20 Qu a l i t a t i v e E f f e c t s Caused by Lack of P e r i o d i c i t y i n Disordered A l l o y C r y s t a l s 21 Band Structure E f f e c t s 22 Chapter 4 POSITRON ANNIHILATION IN METALS 27 Introduction ..• 27 The Sommerfeld Model 28 Elect r o n - E l e c t r o n and Electron-Positron Interactions 29 L a t t i c e Interaction E f f e c t s and Core Electrons 32 v i Page Chapter 5 EXPERIMENTAL METHODS 36 Introduction 36 Positron Annihilation 37 Wide S l i t Geometry 38 1. Introduction 38 2. Angular Correlation Results 40 3. Applications of the Wide S l i t Geometry to Real Metals 41 Point Geometry 43 1. Introduction 43 2. Angular Correlation Results 44 3. Experimental Procedure 46 4. Application of the Point Geometry to Real Metals - 47 5. A Third Compromise Experimental Arrangement 50 Chapter 6 APPARATUS 51 Introduction 51 Experimental Arrangement 52 A Description of the Electronics 55 1. The Detector Assemblies 55 2. The Detector Electronics 55 3. Apparatus for Moving the Sample 57 Experimental Procedure 63 1. Introduction 63 2. Coincidence Resolving Time 63 3. Optical Alignment of the Detectors 64 4. Samples • 64 Chapter 7 Chapter 8 Experimental Resolution 1. Introduction 2. Computer Calculated Resolution 3. C o l l i n e a r Geometry Resolution 4. Temperature Broadening of the Resolution Function 5. E f f e c t of the Detector Arrangement Correction for Decay of Source and Chance Coincidences 1. Introduction 2. Analysis of C o l l i n e a r Geometry Errors 3. Analysis of Angular c o r r e l a t i o n Method Errors Sodium-22 Apparatus 1. Introduction 2. Detector Arrangement 3. E l e c t r o n i c s 4. Alignment 5. Experimental Arrangement EXPERIMENTAL RESULTS Introduction EXPERIMENTAL ANALYSIS Introduction C U 9 0 A 1 1 0 1. Introduction 2. C o l l i n e a r geometry Results 3. Angular C o r r e l a t i o n Results 4. Discussion of the C u ^ A l - ^ Results V l l Page 65 65 66 66 67 68 69 • 69 69 70 71 71 72 72 74 74 78 78 91 91 92 92 • 92 98 101 Chapter 9 Appendix I Appendix J£ Bibliography a. Charging of Conduction Electrons b. T h e o r e t i c a l F i t to Experimental Data c. Core A n n i h i l a t i o n d . Neck Widths e. Dislocations C u 8 5 Z n 1 5 1. Introduction 2. C o l l i n e a r Geometry Results 3. Angular C o r r e l a t i o n Results 4. Discussion of Cu0_Zn..- Results c o I D C u 7 0 Z n i 5 N i 1 5 1. Introduction 2. Results and Discussion C u 6 0 N i 4 0 1. Introduction 2. Results and Discussion Summary 1. Introduction 2. Band Structure E f f e c t s Second Cutoff Further Discussion CONCLUSION v m Page 101 103 103 105 106 107 107 107 109 109 111 111 112 114 114 115 118 118 118 119 121 125 127 135 137 i x LIST OF TABLES Table Page 1 Experimental Results 99 X LIST OF.FIGURES Figure Page 1. E l e c t r o n Energy i n One Dimension: Extended Zone Scheme 9 2A. Band Structure of Copper 14 2B. A Sketch of the Fermi Surface of Copper 15 3. Smearing of k Vector 23 4A. Density i n k-Space of 'Electrons 31 4B. Density i n k-Space of Electrons at the Positron 31 5. Two Photon A n n i h i l a t i o n 39 6 . Wide S l i t Geometry 7. Region Sampled by Wide S l i t Geometry 42 8A. Point Geometry 45 8B. C y l i n d r i c a l Region Sampled by Point Geometry 45 9. Comparison of Point Geometry and Wide S l i t Geometry 49 10. Schematic Diagram of Apparatus 53 11. Lead Shielding 54 12. Schematic of E l e c t r o n i c s 56 13. Preamp and Shaper C i r c u i t 58 14. Coincidence C i r c u i t 59 15. O s c i l l a t o r C i r c u i t 60 16. Time Delay Relay C i r c u i t 61 17. Schematic of Apparatus f o r Moving Sample 62 18. Schematic of Multidetector 73 19. Preamp and Shaper C i r c u i t 75 20. Schematic of Positron Focussing System 76 21. C o l l i n e a r Geometry Results, f o r Cu n A l i n 81 Angular C o r r e l a t i o n Results f o r C U - ~ A 1 ^ Q Expanded Angular C o r r e l a t i o n Data f o r C U _ QA 1 _ _ C o l l i n e a r Geometry Results f o r Cu 0_Zn 1 [ r ob lb Angular C o r r e l a t i o n Results f o r Cu-_Zn-_ Expanded Angular C o r r e l a t i o n Curves f o r Cu„_Zn^_ C o l l i n e a r Geometry Results f o r Cu__Zn-_Ni._ Angular C o r r e l a t i o n Results f o r Cu_gZn^_Ni^_ C o l l i n e a r Geometry Results f o r Cu^-Ni^. Angular C o r r e l a t i o n Results f o r Cu.-Ni.-, 5 60 40 C o l l i n e a r Geometry Results f o r Cu_-Al-_ R e s o l u t i o n Folded i n t o [ l l l j necks R e s o l u t i o n Folded i n t o S p h e r i c a l Fermi Surface of Unit Radius F i t to Angular C o r r e l a t i o n Data of C U - Q A 1 - _ . C o l l i n e a r Geometry Results f o r C u 0 _ Z n l c ob ib C o l l i n e a r Geometry Results f o r Cu__Zn-_Ni-_ C o l l i n e a r Geometry Results f o r Cu^-Ni^-Comparison of C u _ - Z n i r N i i r and Cu.-Ni /U l b l b o(J 40 Comparison of Cu Zn , Cu Ni and Cu Zn H i _ , . o b l b o (J 4 U / U l b l b R e s u l t s Comparison o f Cu^-Al..-, C u 0 _ Z n i r and Cu_„Zn i r.Ni i r „ 90 10 ob lb /O 15 15 Results 1 X l l Figure Page 40. Cu Q_Zn, c c u t o f f 130' 41. T h e o r e t i c a l [lioj A x i s , ft-ray Copper Angular C o r r e l a t i o n Results 131 42. (Count Rate)^versus @^for Cu o cZn, c Tlio] A x i s 132 * r 1 '43. (Count Rate) versus© f or (lOOj A x i s C u 6 0 N i 4 Q ^ 133 44. (count Rate) versus 0 f o r jioo] Axis C u 9 ( ) A l 1 0 134 CHAPTER 1 INTRODUCTION Present understanding of the electronic properties of metals has reached a stage of considerable' sophistication through detailed theory and precise experimentation. Much of the i n t e r -pretation of experimental effects i n metals depends upon a detailed, knowledge of the momentum d i s t r i b u t i o n of the conduction electrons i n the metal and i n p a r t i c u l a r the precise shape of the Fermi surface which i s the constant energy surface i n momentum space marking the t r a n s i t i o n between occupied and unoccupied states. The determination of the Fermi surface 'has evolved to a high degree of accuracy through the study of phenomena such as the de Haas van Alphen e f f e c t , magneto-resistance and magnetoaeoustic effects (Cochran and Haering, 1968), a l l of which depend upon long electronic mean free paths f o r t h e i r - 2 -accuracy. The f i g u r e of merit i s the quantityCO^ "Cwhere i s the cyclotron frequency of the electrons being studied and TJ i s t h e i r r e l a x a t i o n time or l i f e t i m e between c o l l i s i o n s . The extension of t h i s type of measurement to disordered a l l o y s i s l i m i t e d by the rapid decrease of 7_T as a function of solute concentration and to date i t has been p o s s i b l e only to study c e r t a i n selected d i l u t e a l l o y s up. to a maximum./v 1% solute concentration. Since much of modern technology r e l i e s not on pure metals but on concentrated disordered a l l o y s , i t i s of considerable i n t e r e s t to develop an experimental technique y i e l d i n g information about e l e c t r o n i c states i n such systems.. The paucity of experimental information has also l i m i t e d t h e o r e t i c a l developments and i t i s there-fore of considerable fundamental i n t e r e s t to study the e f f e c t that the departure from p e r i o d i c i t y has upon the e l e c t r o n i c s t a t e s . Our understanding of such well known e m p i r i c a l laws as the Hume-Rothery r u l e s should also be improved by such a technique. This t h e s i s follows a f t e r our preliminary work i n which we i n v e s t i g a t e d the value of a high r e s o l u t i o n angular c o r r e l a t i o n p o s i t r o n a n n i h i l a t i o n probe to determine the s a l i e n t features of the copper e l e c t r o n i c momentum d i s t r i b u t i o n (Williams et a l , 1968) and represents an extension to the study of the concentrated a l l o y s C U90 A 110> G U 8 5 Z n i 5 > C u 7 0 Z n 1 5 N i 1 5 a n d W -3-In the f i r s t part of t h i s t h e s i s we introduce the t h e o r e t i c a l models used to describe p o s i t r o n a n n i h i l a t i o n i n a l l o y s . Our apparatus and i n p a r t i c u l a r , d i f f e r e n c e s between our apparatus and the apparatus commonly used i s then discussed. The experimental r e s u l t s and t h e i r a n a l y s i s conclude t h i s t h e s i s . CHAPTER 2 ELECTRONS IN PURE METALS Introduction This chapter introduces the reader to the theory of pure metals i n order to e s t a b l i s h the basic concepts, which w i l l then serve as a basi s f o r ext r a p o l a t i o n to disordered systems. I f one considers a c r y s t a l l i n e s o l i d at low temperatures, one can neglect the motion o.f the r e l a t i v e l y massive n u c l e i and a t t r i b u t e many of the properties of the m a t e r i a l to the motion of electrons moving i n the p o t e n t i a l f i e l d of the sta t i o n a r y n u c l e i . Because a t y p i c a l s o l i d contains so many electrons (of the order of 23 10 1 ) s i m p l i f i c a t i o n s are necessary f o r the system to have a t r a c t a b l e s o l u t i o n . We w i l l s t a r t from the simplest model a v a i l a b l e and then proceed to more r e a l i s t i c models. , Free E l e c t r o n Gas Model The simplest system that i s relevant f o r metals i s a cloud of non-interacting free e l e c t r o n s . The Schrodinger equation fo r an e l e c t r o n i n t h i s cloud i s . where £ i s the energy of the e l e c t r o n , m i s the mass and Ti i s Planck's constant divided by 27T. i s the wave fun c t i o n f o r the el e c t r o n . We w i l l now impose the boundary cond i t i o n that the electrons be confined to a cube of side L. The normalized s o l u t i o n s to t h i s equation are then T k ( ? ) - = e - -where 2 2 2m • k i s the wave vector corresponding to the state / ^ . ( ^ ) and A i s the p o s i t i o n vector of the e l e c t r o n . We now l e t the above electron gas i n t e r a c t through the P a u l i Exclusion P r i n c i p l e . Then at a temperature of zero degrees K e l v i n , the electrons w i l l f i l l a l l the states from the lowest energy state up to an energy £" ^  c a l l e d the Fermi energy. The locus of a l l points i n k-space at which we go from f i l l e d states to empty states i s c a l l e d the Fermi surface. For the model described here the Fermi surface would be a sphere. -6-Periodic Systems . . The next step in the development of the model i s to recognize that in a metal, the electrons are moving in a periodic potential, resulting from the crystallographic regularity of the metal. The translational symmetry of the system imposes certain conditions on the electron wave functions which are expressible in Bloch's theorem which states that; fo.r any wave function that satisfies the Schrodinger equation, there exists a vector k such that translation by a l a t t i c e vector <X i s equivalent to multiplying by the phase factor exp(i k .a) (Ziman, 1965). This can be written algebraically or equivalently f k U ) = e 1 ^ U k(/}) (2-1) where Uj (/}: ) x s a function having the symmetry of the crystal. We note that i f we make U. (n. ) a constant we have a wave function describing a free electron. For convenience wc now define a reciprocal l a t t i c e having l a t t i c e vectors £!/> x £3 x a x —\ x -2 where (a-^, a.,, a ) are the l a t t i c e vectors of the crystal and ( ^ 9 , j g ) are the reciprocal l a t t i c e vectors. Those reciprocal l a t t i c e -7-vectors are u s e f u l i n thai: the maximum independent value of k i n equation (2-1) i s l i m i t e d by the equation where-Gzjj7-^ i s a r e c i p r o c a l l a t t i c e v e ctor (Ziman, 1965). The Nearly Free E l e c t r o n Model Ele c t r o n s i n metals can i d e a l l y be put i n t o one of two cl a s s e s ; conduction electrons and core e l e c t r o n s . Conduction electrons are those electrons which are free to move throughout the metal whereas core electrons are considered to be a l l the other electrons which are bound to the i n d i v i d u a l atoms. The b a s i c assumption of the nearly free e l e c t r o n model i s that the p e r i o d i c p o t e n t i a l r e s u l t i n g from the n u c l e i and the core electrons can be treated as a weak perturbation on the free e l e c t r o n gas behavior of the conduction electrons. From standard perturbation theory, the energy £ ( k ) of the perturbed e l e c t r o n i c state i s given by , i  c w,lere e (k) - ^JL i s the energy of the unperturbed free e l e c t r o n state j and \/(/}) i s the perturbing p o t e n t i a l of the ions i n the m e t a l l i c l a t t i c e . S3.nce \/( A ) must: have the p e r i o d i c i t y of the l a t t i c e , we can write i t i n a F o u r i e r s e r i e s -8-V (n) = I- V. e « • r> G C where the G are the r e c i p r o c a l l a t t i c e vectors and \ / i s the — G expansion c o e f f i c i e n t . S u b s t i t u t i n g t h i s value of \/(A ) i n t o < ^ ] c j \ y ( / i ) | k'^ and remembering that / k \ = - W * ' * we obtain the r e s u l t that <^ k j \ / ( / I )j k ^ i s non zero only i f k _ k' + G = 0. Thus the term ^ k i s only non zero f o r G = 0, representing a constant p o t e n t i a l which merely s h i f t s a l l energy l e v e l s by \/^ . The second order term i s the term of i n t e r e s t and represents only a small c o r r e c t i o n unless cf0(]i) = £0Qs') f o r * t- h ' • 12 / / 2 / |2 which occurs when j k j = jk'j = j k + G{ . This equation defines the values of k corresponding to the boundaries of a B r i l l puin zone and i s i d e n t i c a l to the condition f o r Bragg r e f l e c t i o n of X-rays. In our case the breakdown of non-degenerate perturbation theory corresponds to Bragg r e f l e c t i o n of the electrons by the l a t t i c e and the s o l u t i o n i s r e a d i l y obtained using degenerate perturbation theory. . The e f f e c t on the energy-momentum r e l a t i o n i s to introduce energy gaps at the B r i H o u i n zone boundaries as shown i n Figure 1. C o l l e c t i v e or Many Body E f f e c t s Since the e l e c t r o n - i o n and e l e c t r o n - e l e c t r o n i n t e r a c t i o n s are through long range Coulomb forces i t might seem strange that a theory such as that described'could have any usefulness. A j u s t i f i c a t i o n of t h i s theory i s provided by the Bohm-Pines theory of plasma o s c i l l a t i o n s . In t h i s theory (Raimes 1957, 1961) a metal i s considered to be a plasma c o n s i s t i n g of a uniform d i s t r i b u t i o n of p o s i t i v e background charge i n which the electrons are embedded. This theory shows that the Coulomb forces can be divided i n t o a long-range component which only becomes important at very high e l e c t r o n energies and a short-range component of e f f e c t i v e range which i s so short that i t can often be neglected. OPW and Pseudopotentials The assumption of a nearly free e l e c t r o n gas has proven to be remarkably s u c c e s s f u l i n the study of metals, but only r e c e n t l y has a good t h e o r e t i c a l understanding of i t s v a l i d i t y been obtained using the pseudopotential theory which evolved from the orthogonalize plane wave approximation. We w i l l o u t l i n e the ideas of those t r e a t -ments to i n d i c a t e the v a l i d i t y of the nearly free e l e c t r o n model. We w i l l assume that a metal c o n s i s t s of conduction electrons and m e t a l l i c ions which contain a l l the other e l e c t r o n s . L e t \ / ( A ) be the p o t e n t i a l seen by each e l e c t r o n . The energy eigenfunctions w i l l be given by the equation l&± = (T - » V ( A ) ) t ' i = E.t : i -11-where T i s the k i n e t i c energy and i s the energy of the state ^ •• This equation holds f o r core electrons and conduction e l e c t r o n s . The orthogonalized plane wave approximation i s based on the f a c t that the conduction e l e c t r o n states must be orthogonal to the core e l e c t r o n states and that the wave fu n c t i o n resembles a plane wave i n the space between the ions. We then wish to construct a wave function which w i l l have these p r o p e r t i e s . We f o l l o w the procedure of Harrison, (1966) where the index k i n d i c a t e s conduction' electrons and the index °f i n d i c a t e s core e l e c t r o n s t a t e s . We then have the equation f o r an orthogonalized plane wave or where dTj' i s the volume element, k and A are as defined p r e v i o u s l y and the "7^ ( A . ) are the core e l e c t r o n s t a t e s . This can be re w r i t t e n i n the braket notation . CV A c a l c u l a t i o n of the quantity <^Xylc^y shows t h a t the are orthogonal to the core states as required. We now define f o r convenience the p r o j e c t i o n operator P = We then have . • . \ = " d - ^ > -Nov; expand the conduction e l e c t r o n state i n terms.of a l i n e a r combina-tion of X, . -12-6 where t h i s i s an expansion over r e c i p r o c a l l a t t i c e vectors. Put t h i s wave f u n c t i o n i n t o the Schrodinger equation and we get ^ a & ( k ) H(l-P) / k = E k ^ V - } ( 1 _ P ) + - / > ( 2 _ 2 ) G G where H i s the Iksiiiltonian f o r the system and i s the energy of the state | k ^ . I f we take a l l terms i n v o l v i n g the p r o j e c t i o n operators to one side of the equation we get • • where T J X A/and (j)^  are defined by the previous equation. Harrison, (1966) c a l l s \/J the pseudonotential and (f)^ the pseudowave f u n c t i o n . We see that d) k = £. v^/-1 G> G ** and W = V ( A ) (E k-E^. ) /or><V/ (2-3) We see now how i t i s p o s s i b l e f o r the conduction electrons i n a metal to move i n a nearly free manner i n s p i t e of the long range forces which create the p o t e n t i a l \ / ( A ) . ... The e f f e c t i v e p o t e n t i a l (2-3) i n t h i s representation involves a second term which could conceivably cancel the Coulomb p o t e n t i a l of the i o n s . This i s , of course, only a p l a u s i b i l i t y argument. We have not shown that such c a n c e l l a t i o n a c t u a l l y occurs but the experimental evidence supports t h i s idea. -13-Band Structure C a l c u l a t i o n s The orthogonalized plane waves we have described lend themselves to numerical methods f o r the computation of the i n equation (2-2). The various approaches to t h i s problem of c a l c u l a t i n g Ej are based i n p r i n c i p l e on the s o l u t i o n of the Hartree-Fock equations (Callaway, 1958). Recent c a l c u l a t i o n s of versus k f o r copper have been done by S e g a l l , (1962) and Burdick, (1963). Their agreement with each other and with experimental evidence i s good. Figure 2.a shows the r e s u l t s of these c a l c u l a t i o n s . -phe bands occupied by the 3d electrons of copper overlap with the band occupied by the 4s e l e c t r o n s . This implies that some of the d-band electrons are only l o o s e l y bound to the m e t a l l i c ions. In other words, the d i v i s i o n of electrons i n t o conduction s t a t e s and core states does not seem to be a v a l i d approach i n copper. The electrons from the other f i l l e d atomic s h e l l s i n copper w i l l be t i g h t l y bound since they would appear i n bands w e l l below those shown i n Figure 2a. The Fermi surface i n d i c a t e d by these c a l c u l a t i o n s and found experimentally i s shown i n Figure 2b. -14-COPPER FIG. 2 A B A N D S T R U C T U R E O F C O P P E R T H E S T A N D A R D G R O U P T H E O R E T I C A L N O T A T I O N I S U S E D . -15-F i g . 2B A Sketch of the Fermi surface of Copper Note; The polyhedron, represents the B r i l l o u i n zone. CHAPTER 3 DISORDERED ALLOYS Introduction Chapter 2 dealt with electrons i n pure metals, an area i n which there i s extensive t h e o r e t i c a l understanding. Numerical band structure c a l c u l a t i o n s , pseudopotential theory and the e x p e r i -mental techniques f o r measuring Fermi surfaces have been quite s u c c e s s f u l at l e a s t i n the study of the simpler metals. Beyond the study of pure metals l i e s the much l a r g e r and so f a r v i r t u a l l y unexplored f i e l d of the study of the e l e c t r o n i c states of disordered a l l o y s . Ordered a l l o y s present no p a r t i c u l a r problem since t h e i r band structure can be c a l c u l a t e d by the orthodox techniques. The l o s s of t r a n s l a t i o n a l invariance which occurs when impurity atoms arc in s e r t e d at random in t o some of the l a t t i c e s i t e s of -17-a s i n g l e c r y s t a l i s the primary reason that the t h e o r e t i c a l models we applied to pure metals are no longer v a l i d i n the d i s c u s s i o n of disordered a l l o y s . As discussed i n Chapter 1, the r e s u l t i n g high r e s i d u a l r e s i s t i v i t y and short e l e c t r o n i c mean free path of concentrated disordered a l l o y s makes i t impossible to use such pre-c i s i o n techniques as the de Haas van Alphen experiments to determine t h e i r e l e c t r o n i c structure and we are forced to use other l e s s accurate methods i n order to obtain t h i s information. Experimental Methods of Studying Concentrated A l l o y s We w i l l now review b r i e f l y some of the techniques which are capable of y i e l d i n g information about the band structure and Fermi surface of concentrated disordered a l l o y s i n s p i t e of the short mean free path of the conduction electrons i n these m a t e r i a l s . .1. Slow Neutrons The i n e l a s t i c s c a t t e r i n g of slow neutrons by phonons 'is a w e l l e s t a b l i s h e d method f o r determining the d i s p e r s i o n curves (energy versus wave number q) of the various phonon modes i n a metal. I f we consider a metal as a l a t t i c e of ions immersed i n a sea of conduction electrons, a phonon corresponds to a p e r i o d i c displacement of these ions and produces a p e r i o d i c e l e c t r i c f i e l d which i s then r a p i d l y screened out by the conduction e l e c t r o n s . This type of phonon-electron i n t e r a c t i o n has a strong dependence on the shape of the Fermi surface. Kolm, (1959) pointed out that f o r c e r t a i n values of n corresnonding to ^ Lk n . the screening a b i l i t y of the -18-conduction electrons w i l l change abruptly causing Kohn anomalies or kinks i n the phonon d i s p e r s i o n curves. These Kohn anomalies are d i f f i c u l t to observe experimentally, t h e i r prominence depends on the (J shape of the Fermi surface and on the strength of the e l e c t r o n -phonon coupling, but some materials such as l e a d - t e l l u r i u m a l l o y s have been studied (Ng and Brockhouse, 1967). 2 . Photons i n A l l o y s This subject covers a wide area of research ranging from X-ray studies to photoemission studies using o p t i c a l frequencies. Small angle Compton s c a t t e r i n g of X-rays has proven u s e f u l i n metals, p a r t i c u l a r l y f o r metals having low atomic numbers ( P h i l i p s and Weiss, 1968). The main disadvantage of t h i s method i s that s c a t t e r i n g with core electrons tends to obscure the r e s u l t s . Other techniques such as the study of the K-absorption edges of X-rays f a l l i n g on metals (Yeh and Azaro'f'f, 1967) have been applied to a l l o y s . Conversely, f a s t electrons f a l l i n g on a metal surface may e j e c t core e l e c t r o n s . The X-rays r e s u l t i n g from conduc-t i o n electrons f a l l i n g i n t o the e n e r g e t i c a l l y w e l l defined core states can be studied and the energy of the conduction band electrons deter-mined. This type of data shows considerable' s t r u c t u r e , but the a n a l y s i s i s d i f f i c u l t without p r i o r knowledge of the band structure of the a l l o y being studied. Photon experiments i n the o p t i c a l range are j u s t as ' d i f f i c u l t to analyse. In p a r t i c u l a r there i s a strong tendency -19-f o r more than one e l e c t r o n to be involved i n photon-electron i n t e r -actions with intermediate states between the i n i t i a l and f i n a l s t a t e s . Experimentally the required, high p u r i t y and f i n i s h of the metal-surfaces i s d i f f i c u l t to a t t a i n . Experiments such as photo-emission studies and o p t i c a l r e f l e c t i v i t y experiments are some of the common techniques used. One of the more s u c c e s s f u l examples of o p t i c a l r e f l e c t i v i t y experiments, i n v o l v i n g the p o l a r - r e f l e c t i o n Faraday e f f e c t has been c a r r i e d out i n g o l d - s i l v e r a l l o y s by Mc A l i s t e r and Stern, (1965). R i g i d Band Model We wi l l , now discuss some of the t h e o r e t i c a l concepts ap p l i c a b l e to disordered a l l o y s . The most common model used p a r t i c u l a r l y f o r d i l u t e a l l o y s i s the r i g i d band model. In t h i s model the concepts of a pure metal such as Fermi surface and BrJlleuin zone are applied to disordered a l l o y s even though t h e i r meaning becomes rather vague f o r concentrated a l l o y s . The a l l o y i s assumed to have the same e l e c t r o n i c band structure as the pure host metal. The only e f f e c t of a l l o y i n g i s a change i n the Fermi l e v e l on the band structure diagram caused by a change i n the number of conduction electrons and by the volume e f f e c t associated with a change i n the l a t t i c e para-meters. Stern, (1967, 1968) discusses the conditions required f o r the r i g i d band model to be v a l i d . B a s i c a l l y , the condit i o n required -20-i s that the r e d i s t r i b u t i o n of e l e c t r o n i c charge due to a l l o y i n g must not be too l a r g e . Much of h i s d i s c u s s i o n applies only to d i l u t e a l l o y s and t h i s i s the region where the r i g i d band model has given f a i r l y good agreement with the experimental r e s u l t s ( C h o l l e t and Templeton, 1968). Averaged P o t e n t i a l s The l a c k of a p e r i o d i c p o t e n t i a l i n disordered a l l o y c r y s t a l s implies that the conduction electrons of the a l l o y w i l l be scattered o f f the m e t a l l i c ions i n the l a t t i c e s i t e s . The p o t e n t i a l s of the two m e t a l l i c ions w i l l i n general be quite d i f f e r e n t , however, the amount of conduction e l e c t r o n s c a t t e r i n g o f f these ions i s i n many cases quite s i m i l a r allowing an averaged s c a t t e r i n g matrix to be used to describe the i n t e r a c t i o n of the electrons with the ions. This averaging of the e f f e c t of the two constituents of the a l l o y then reduces the problem to one having t r a n s l a t i o n a l symmetry throughout the c r y s t a l and band structure c a l c u l a t i o n s then become po s s i b l e . This type of a n a l y s i s has been c a r r i e d out f o r disordered alpha brass by Amar et a l , (1967) and by Soven, (1966). Their r e s u l t s are i n q u a n t i t a t i v e agreement with each other, but they are not very conclusive except f o r Seven's c a l c u l a t i o n that the conduction electrons are. w e l l l o c a l i z e d i n momentum space, that i s the spread i n k i s l e s s than a few percent of k„ -21-Stern, (1966) has studied g o l d - s i l v e r a l l o y s using an average p o t e n t i a l as the basis, f o r h i s c a l c u l a t i o n s . Since the p o t e n t i a l s of the gold and s i l v e r ions are quite simil£ir, h i s r e s u l t s are i n good agreement with those experimentally obtained using the polar r e f l e c t i o n Faraday e f f e c t (McAlister and Stern, 1965). I t i s of i n t e r e s t to note that the d-band core electrons i n the system were found to be more strongly a f f e c t e d by the a l l o y i n g and were not expected to be c a l c u l a b l e using an averaged p o t e n t i a l of t h i s nature. Q u a l i t a t i v e E f f e c t s Caused by Lack of P e r i o d i c i t y i n Disordered A l l o y C r y s t a l s We w i l l now consider a disordered a l l o y containing two d i f f e r e n t atoms. The conduction electrons w i l l s c a t t e r o f f the '• impurity atoms and give r i s e to a charging e f f e c t around the i n d i v i d u a l atoms (Stern, 1966) because of the l a c k of c r y s t a l sym-metry i n the system. This p i l e up of charge w i l l act i n such a way as to n e u t r a l i z e the charge d i f f e r e n c e s between the ions. The amount of charging that takes place depends on the d i f f e r e n c e s i n the s c a t t e r i n g matrix of the host and of the impurity atoms. Band structure e f f e c t s can also be important as we s h a l l see l a t e r . A feature r e l a t e d to. charging i s the f a c t that the B l o c h wave functions discussed i n Chapter 2 are now no longer eigenfunctions of the system. We must how associate a l i f e t i m e with the Bloch states - 2 2 -i n disordered a l l o y s . This w i l l cause a smearing i n the wave vector k of the Bloch states because k i s now no longer a good quantum number. Stern, (1968) has c a l c u l a t e d t h i s smearing f o r cases where the perturbation due to the impurity atoms i s small. His r e s u l t s are shown i n Figure 3. Band Structure E f f e c t s Most of the previous theory has been developed p r i m a r i l y f o r the explanation of r e s u l t s i n d i l u t e a l l o y s although the p h y s i c a l ideas presented w i l l also hold i n concentrated a l l o y s . Stern, (1969) has developed some ideas which- should also hold f o r concentrated noble metal a l l o y s . In the band structure of pure copper we f i n d the bands containing the 3d and 4s electrons e n e r g e t i c a l l y well- separated from the higher unoccupied bands. We now assume that'whatever the e f f e c t s of a l l o y i n g , even i n concentrated copper a l l o y s these bands remain e n e r g e t i c a l l y separated from the other e l e c t r o n i c bands. Since the 3d bands i n copper are f i l l e d and the 4s band i s h a l f - f i l l e d we can replace these bands by a model described as fol l o w s . We consider a pure metal with one band e n e r g e t i c a l l y w e l l separated from the r e s t and that t h i s band contains one conduction e l e c t r o n per .atom. In the t i g h t binding approximation - 2 3 -P(K) F i g . 3 Smearing of k Vector Note; The d i s t r i b u t i o n o f wave number i n an eigexistate f o r a d i l u t e a l l o y . The width of the peak i s p r o p o r t i o n a l to the f r a c t i o n a l amount of im p u r i t y -24-the wave fun c t i o n f o r the pure metal can be written where Cjjjifi- A) i s the atomic wave fun c t i o n of the pure metal constituents, A denotes the l a t t i c e s i t e s and k i s the wave vector of the e l e c t r o n i c state. We now replace one of the host atoms i n the l a t t i c e by an impurity atom which contributes two electrons per atom to the band. In the atomic state these two electrons occupy the state ^ 2 * We now wish to show the importance of whether or not (j) 2 1 S energetic-a l l y w e l l i s o l a t e d from above i n the atomic s t a t e . Because of the p o s i t i v e nuclear charge of the impurity atom we assume that both electrons remain i n the v i c i n i t y of the impurity atom i n our t i g h t binding approximation. This implies that the state 0 2 o r " ^ e Impurity atom i s f u l l since there are two electrons i n i t . Suppose we now add an a d d i t i o n a l e l e c t r o n to the system. I f 1 3 e n e r g e t i c a l l y i s o l a t e d from higher atomic states i n the impurity atom i t i s c l e a r that the wave f u n c t i o n of t h i s a d d i t i o n a l e l e c t r o n w i l l have a smaller amplitude at the impurity atom than away from i t since there.are no a v a i l a b l e states about the impurity atom. In other words we are merely .stating that we cannot put more than two electrons i n t o the atomic state (J)ir" (Jj-j ^ s no^ e n e r g e t i c a l l y i s o l a t e d from above, there i s no reason why the a d d i -t i o n a l e l e c t r o n should be a f f e c t e d by the f a c t that(j)^ i s f i l l e d and the amplitude of the e l e c t r o n i c wave function w i l l not be smaller about tiio impurity atom. In p r a c t i c e , our model i s a l i t t l e o v e r s i m p l i f l e d i n that the a d d i t i o n a l e l e c t r o n we placed i n the state ^2 3'-s n o ^ completely l o c a l i z e d about the impurity atom and there w i l l be some s c a t t e r i n g of conduction electrons o f f the•impurity atom, however, i t w i l l be much l e s s than would be expected from the simple f r e e e l e c t r o n theory. In ad d i t i o n i t i s clear, that i f i s e n e r g e t i c a l l y i s o l a t e d the conduction e l e c t r o n s c a t t e r i n g w i l l be much l e s s l o c a l i z e d about the impurity atom than f o r a (j) which i s not e n e r g e t i c a l l y i s o l a t e d . In other words, the conduction e l e c t r o n s c a t t e r i n g f o r an i s o l a t e d ( j ^ state w i l l be more P - l i k e and f o r an e n e r g e t i c a l l y degenerate (j)^ the conduction e l e c t r o n s c a t t e r i n g w i l l be more p.-like. ..,.„...../._ These Ideas are d i r e c t l y a p p l i c a b l e to noble metal a l l o y s and are i n agreement with r e s i s t i v i t y measurements. The r e s i d u a l r e s i s t a n c e of d i l u t e copper a l l o y s i s lower than that expected from a simple theory such as the Born approximation which assumes a nea r l y - f r e e electron l i k e theory. This i s now explained as being due to the f a c t that-the conduction electrons cannot s c a t t e r o f f the impurity atoms because a l l the a v a i l a b l e atomic states are already f i l l e d . Band structure c a l c u l a t i o n s i n copper i n d i c a t e that electrons i n the " b e l l y " of the copper Fermi surface, that i s i n the JlO()| d i r e c t i o n have mainly s-wave symmetry about the copper -26-corcs whereas the e l e c t r o n i c states i n the £lll] Fermi surface d i r e c -t i o n have p-type symmetry. According'to Stern, (.1969) the s-Iike " b e l l y " e l e c t r o n s w i l l be more strongly scattered by a l o c a l i z e d s h i e l d i n g charge than the p - l i k c | l l l j neck electrons and conversely f o r the n o n - l o c a l i z e d s c a t t e r i n g which occurs when the state (J}^ i s e n e r g e t i c a l l y i s o l a t e d as described i n the previous paragraphs. We therefore expect d i f f e r e n t amounts of charging from these two types of e l e c t r o n i c states depending on whether or not the state 0 i s f u l l or not. This t h e o r e t i c a l d i s c u s s i o n of a l l o y s has been highly q u a l i t a t i v e but as we s h a l l see i n Chapter 7, these ideas are d i r e c t l y a p p l i c a b l e to C u ^ A l ^ a l l o y where the state (j)^ i s not e n e r g e t i c a l l y w e l l separated from higher atomic states and to Cu Zn O O Lb where i t i s . CHAPTER 4 ' POSITRON ANNIHILATION IN METALS Introduction In t h i s chapter we w i l l discuss the i n t e r a c t i o n s between the p o s i t r o n and the metal to demonstrate the r e l a t i o n s between 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 gamma ray p a i r s and the momentum d i s t r i b u t i o n of the electrons i n the metal. Since the l i f e t i m e of a positron before a n n i h i l a t i o n i s of the order of 10 seconds and a t y p i c a l source strength i s 400 m i l l i c u r i e , there would never be more than a few positrons i n the sample at any one time. Any i n t e r a c t i o n between positrons can therefore be neglected. As was o r i g i n a l l y shown by Lee-Whiting, (1955), the positr o n s , which are emitted by beta decay (end point e n e r g y 6 Mev), r a p i d l y lose energy, f i r s t by i o n i z a t i o n and e x c i t a t i o n of the host atoms and then by i n t e r a c t i o n with the conduction electrons by creat i n g electron-hole p a i r s u n t i l they reach an energy of order KT where K i s the Boltzman constant and T i s the absolute temperature^ i n a time r e f e r r e d to as the thermalization time. A recent q u a n t i t a -t i v e c a l c u l a t i o n by Garbotte and Arora, (1967) estimates the thermaliza t i o n time as a function of e l e c t r o n density f o r an i n t e r a c t i n g e l e c t r o n gas. This c a l c u l a t i o n gives a t h e r m a l i z a t i o n time to room temperature of 5 x 10 seconds f o r sodium as compared to an experimentally determined l i f e t i m e before a n n i h i l a t i o n of-^ ^ x 10 ^ seconds. Thermalization of the positron to lower temperatures takes much longer and at l i q u i d helium temperature, thermalization of the p o s i t r o n need not be complete before a n n i h i l a t i o n (Stewart and Shand, 1966). Because a l l work c a r r i e d out by us was done at room temperature we can assume thermalization i s complete . before a n n i h i l a t i o n . We w i l l now consider a number of models of i n c r e a s i n g s o p h i s t i c a t i o n i n which the positron i s assumed to be i n i t s ground state and the thermal energy °^ the p o s i t r o n i s neglected. The Sommerfeld Model The simplest relevant model, i s the Sommerfeld model i n which the effects of the p o s i t r o n - e l e c t r o n i n t e r a c t i o n s are ignored and the electrons are assumed to be i n a non-interacting e l e c t r o n gas. The p o s i t r o n Is assumed to be i n the k= 0 state and i s therefore represented by a constant wave fu n c t i o n . -29-Pos.itron a n n i h i l a t i o n i n t h i s gas w i l l y i e l d i n most instances, • two gamma rays, given o f f at an angle of 180° to each other i n the center of mass frame of reference. Each gamma ray w i l l have an energy equal to the r e s t mass of an el e c t r o n i n the center of mass reference frame. In the lab frame, since we have neglected the momentum of the posi t r o n , the momentum of the gamma ray p a i r w i l l be the same as the momentum of the e l e c t r o n before a n n i h i l a -t i o n and an examination of the momentum spectrum of gamma ray p a i r s y i e l d s the e l e c t r o n i c momentum d i s t r i b u t i o n d i r e c t l y . E l e c t r o n - E l e c t r o n and El e c t r o n - P o s i t r o n Interactions The Sommerfeld model discussed above i s an o v e r s i m p l i f i c a -t i o n of the a c t u a l f a c t s . C a l c u l a t i o n s of the l i f e t i m e s of positrons i n an e l e c t r o n gas i n d i c a t e that 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 i s quite important i n determining the l i f e t i m e s (Kahana i n Stewart and R o e l l i g , 1967). Positron l i f e t i m e s c a l c u l a t e d using the simnple Sommerfeld model are about a f a c t o r of 8 longer than the experimental r e s u l t s f o r a simple metal such as sodium. In s p i t e of t h i s , the Sommerfeld model i s i n quite reasonable agreement with the experiment-a l l y observed angular d i s t r i b u t i o n of the a n n i h i l a t i o n gamma ray pa i r s . We w i l l now t r y to formulate a c l e a r e r p h y s i c a l p i c t u r e of a thermalized p o s i t r o n i n an el e c t r o n gas. As i s usual i n such s i t u a t i o n s the electrons w i l l move i n such a manner as to s h i e l d the charge of the positron. A l l e n e r g e t i c a l l y adjacent states i n t o which the electrons deep i n s i d e the Fermi surface could be scattered are f i l l e d . The exclusion p r i n c i p l e w i l l therefore prevent them from being strongly a f f e c t e d by the p o s i t r o n . Electrons near the Fermi surface w i l l be more strongly a f f e c t e d and w i l l move to screen the charge of the positron. C a l c u l a t i o n s by Kahana, (.1960, .1963) of the p o s i t r o n l i f e t i m e s i n metals i n d i c a t e that the e l e c t r o n density at the p o s i t r o n i s about 10 times the e l e c t r o n ' d e n s i t y i n a normal metal. This e f f e c t i v e l y enhances the a n n i h i l a t i o n p r o b a b i l i t y of the p o s i t r o n and does so i n . a manner which i s dependent on the p o s i t i o n of the e l e c t r o n i n k-space. So f a r t h i s would i n d i c a t e a r e s u l t f o r the angular d i s t r i b u t i o n of gamma ray p a i r s which i s considerably d i f f e r e n t from that predicted by the Sommerfeld model. However, t h i s i s an e l e c t r o n gas i n which the electrons also i n t e r a c t with each other. I f t h i s i s taken i n t o account the e l e c t r o n i c density does not drop sharply to zero i n k-space at the Fermi surface as was assumed i n the Sommerfeld model (Daniel and Vosko, .1960). Figure 4 shows the v a r i a t i o n of the p r o b a b i l i t y of p o s i t r o n a n n i h i l a t i o n and the v a r i a t i o n of e l e c t r o n density i n k-space as a function of the e l e c t r o n wave•vector k i n an i n t e r a c t i n g electron gas. The two e f f e c t s tend to cancel each other l e a v i n g a net r e s u l t that i s quite close to the Sommerfeld model. Sodium i s known to have a Fermi surface which i s c l o s e l y s p h e r i c a l and the -31-0 Kp ITig. 4 Schematic Sketches of; (a) Density i n k-Space of E l e c t r o n s F o l l o w i n g Daniel and Vosko Compared \yith Simple Free E l e c t r o n Theory, (b) Density i n k Space of E l e c t r o n s at the p o s i t r o n or of A n n i h i l a t i o n photon P a i r s from Kahana. e l e c t r o n wavefunctions are close to being f r e e - e l e c t r o n - l i k e . Results f o r 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 }f -rays observed by Stewart, (11.964) i n t h i s metal are i n close agreement with the Sommerfeld model. The Sommerfeld model was observed to give almost as good a f i t to the data as the more so p h i s t i c a t e d theory discussed above. L a t t i c e I n t e r a c t i o n E f f e c t s and Gore El e c t r o n s As we have now seen, the theory of p o s i t r o n a n n i h i l a t i o n f o r an i n t e r a c t i n g e l e c t r o n gas has been w e l l v e r i f i e d f o r sodium which i s a metal that i s c l o s e l y f r e e - e l e c t r o n l i k e i n character.-In the more general case, however, the e f f e c t s of the l a t t i c e poten-t i a l upon both the e l e c t r o n and positron wave functions must be considered. We have also so f a r neglected 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 with the t i g h t l y bound core e l e c t r o n s . In a p e r i o d i c l a t t i c e the wavefunction des c r i b i n g a pos i t r o n can be wr i t t e n as a plane wave expansion over the r e c i p r o c a l l a t t i c e vectors of the c r y s t a l /> r\ ( 4-1) where k i s the wave vector of the pos i t r o n . This w i l l introduce higher momentum components i n t o the wavefunction of the pos i t r o n which we have previously considered to be a s i n g l e plane wave. Another approach to t h i s i s to note that the r e p u l s i o n of the pos i t r o n from the p o s i t i v e ions w i l l create holes i n the p o s i t r o n wave-functions at the l a t t i c e s i t e s . From the Heiscnberg Uncertainty p r i n c i p l e i t i s c l e a r that t h i s w i l l introduce higher momentum terms i n the positron wavefunction. As previously mentioned, the l a t t i c e p o t e n t i a l w i l l also a f f e c t the conduction electrons i n the metal. A p h y s i c a l way of s t a t i n g t h i s would be to say that the more a n i s o t r o p i c the Fermi' surface i s , the more important the higher momentum components i n the el e c t r o n wavefunction become. An expansion s i m i l a r to (4-1) w i l l • describe the e l e c t r o n i c wavefunction. The importance of these • higher momentum components i s discussed i n more d e t a i l i n the f o l -lowing paragraphs. A d e t a i l e d attempt to v e r i f y our understanding of l a t t i c e e f f e c t s has been undertaken f o r two representative materials aluminum and s i l i c o n by Stroud and Ehrenreich, (.1968). They used e l e c t r o n density d i s t r i b u t i o n s obtained from X-ray d i f f r a c t i o n data to c a l c u l a t e the po s i t r o n wavefunction which was found to contain appreciable higher momentum components as discussed above. A pseudo-p o t e n t i a l method was used to c a l c u l a t e the e l e c t r o n i c wavefunctions. From these two r e s u l t s they could then c a l c u l a t e the angular d i s t r i b u t i o n i n momentum space of the gamma ray p a i r s i n the independent p a r t i c l e approximation using the formula ^ - I l [ d A ? r ' * A ^(n)ikS(<y)) (4-2) where ^ (/}) and ^  //J) are the p o s i t r o n and el e c t r o n wave fu n c t i o n s , -34-k and JI are quantum numbers d e f i n i n g the e l e c t r o n i c state and /Yl .j ^ ] S the occupation number of the state k,Jl . f ( P ) i s the p r o b a b i l i t y that an a n n i h i l a t i o n event w i l l y i e l d two photons of t o t a l momentum £• ' These c a l c u l a t i o n s are i n good agreement with the experimental data f o r aluminum and s i l i c o n i n s p i t e of the f a c t that the i n t e r a c t i o n of electrons and positrons i s not accounted f o r i n the a n n i h i l a t i o n process i n equation (4-2). These c a l c u l a t i o n s i n d i c a t e that the shape of the p o s i -t r o n wavefunction has a s i g n i f i c a n t e f f e c t on the f i n a l r e s u l t s i n s i l i c o n but i n aluminum the f i n a l r e s u l t s are not strongly dependent on the exact form of the. p o s i t r o n wavefunction i n s o f a r as a n n i h i l a t i o n with conduction electrons i s concerned. For electrons with momenta outside the Fermi surface,, the f i n a l r e s u l t s become more dependent on the form of the p o s i t r o n wavefunction. We therefore conclude that the 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 gamma ray p a i r s in momentum space w i l l be strongly dependent on the p o s i t r o n wavefunction f o r a n n i h i l a t i o n with core ele c t r o n s . The f a c t that the positron wavefunction i s r a p i d l y varying i n the v i c i n i t y of the ion cores w i l l make t h i s c a l c u l a t i o n even more d i f f i c u l t . The c a l c u l a t i o n s discussed are i n good agreement with experiment i n s p i t e of the f a c t that several-approximations are made. The pseudopotential method i s used to compute • the e l e c t r o n i c wave--35-functions. As discussed previously, l i f e t i m e measurements i n d i c a t e that the e l e c t r o n i c density at the p o s i t r o n i s 10 times the normal density. Since n e i t h e r s i l i c o n nor aluminum i s as f r e e - e l e c t r o n l i k e i n nature as sodium, the agreement of the c a l c u l a t i o n s with e x p e r i - . mental r e s u l t s encourages the assumption that features i n the Fermi surfaces of other materials w i l l be r e f l e c t e d i n the momentum d i s t r i b u t i o n of the a n n i h i l a t i o n gamma ray p a i r s . As already implied i n the above, there i s no reason why a p o s i t r o n should not a n n i h i l a t e with the core e l e c t r o n s i n a metal; For ions i n which the core electrons are t i g h t l y bound, the number of a n n i h i l a t i o n s of t h i s type i s r e l a t i v e l y small. This can be seen as p a r t l y because the volume occupied by the core, electrons i n metals such as sodium i s small compared to the t o t a l volume of the metal (/v~9% f o r sodium according to Donaghy, 1964). In addition to t h i s the r e p u l s i o n of the p o s i t r o n from the p o s i t i v e i o n cores i n which the core electrons are located w i l l reduce the a n n i h i l a t i o n p r o b a b i l i t y with these e l e c t r o n s . For a metal such as copper where the d-band electrons are n e i t h e r t i g h t l y bound nor free to move throughout the metal (See R.G. Chambers i n Cochran and Haering, 1968) the s i t u a t i o n i s much more complicated but we would i n t u i t i v e l y expect the p r o b a b i l i t y of positron a n n i h i l a t i o n with these e l e c t r o n s to be r e l a t i v e l y l a r g e r than i n the case of sodium. CHAPTER 5 EXPERIMENTAL METHODS Introduction Having now established that a reasonable t h e o r e t i c a l understanding of the posi t r o n a n n i h i l a t i o n process e x i s t s at l e a s t f o r simple metals we w i l l now consider the various experimental detector geometries which may be used to study the angular d i s t r i -bution of a n n i h i l a t i o n gamma ray p a i r s i n momentum space. In r e l a t i n g the r e s u l t s obtained by these experiments to the e l e c t r o n i c states we w i l l assume that the electrons i n a metal form a perturbed e l e c t r o n gas. The- l a t t i c e introduces an ambiguity i n t o the k-vcctor of the e l e c t r o n i c states since the state corresponding to k i s i d e n t i c a l to the state corresponding to k + G. In r e l a t i n g the momentum of the a n n i h i l a t i o n gamma ray pair to the wave vector of the an n i h i l a t e d e l e c t r o n we w i l l assume G = 0 except i n s o f a r as departures of the -37-e l e c t r o n wave function from a plane wave forces us to introduce higher momentum components i n t o equation (4-1). This i s compatible with the r e s u l t s of Stroud and Ebrenrcich, (.1969) as discussed i n Chapter 4 and with experimental evidence on metals i n general. For reasons of c l a r i t y i n the discu s s i o n of detector arrangements that follows we w i l l assume that the positrons a n n i h i l a t e i n an e l e c t r o n gas. L a t t i c e e f f e c t s wiJ.l be ignored and the momentum of the e l e c t r o n w i l l be assumed to be 17k. The Sommerfeld model w i l l be used throughout. Because our detector arrangement used i s rather unusual, we w i l l f i r s t present the most common experimental technique i n order to make the i n t r o d u c t i o n to our method more l u c i d . P ositron A n n i h i l a t i o n We now.consider a positron a n n i h i l a t i n g with an e l e c t r o n i n an e l e c t r o n gas. In the center of mass reference frame, p o s i t r o n a n n i h i l a t i o n w i l l u s u a l l y produce two gamma rays, emitted at an angle of 180° to each other. The k i n e t i c energy of the el e c t r o n i n a metal i s s u f f i c i e n t l y small i n comparison to the r e s t mass of the el e c t r o n so that each of the two gamma rays may be considered to have 2 an energy of m c or 511 keV'in the lab reference frame. In f a c t the k i n e t i c energy of the el e c t r o n contributes a Dopp.ler s h i f t of tiie order of 2 keV (Hotz et a l , 1968), but t h i s s h i f t i s too small to be detected by our N a l ( T l ) detectors. -38-The deviation of the angle between the gamma rays from 180° i n the lab reference frame can be detected, however. Let 0 be as defined i n Figure 5 where P-p i s the component of the momentum as shown i n the f i g u r e . We make the approximation that the energy 2 of each gamma ray i s mQC = ti 60 i n the usual notation, where UJ i s the angular frequency of the gamma rays. Then by conservation of momentum c 2 SIN 6/^ P T c P T = m o ° 0 ( 5 - 1 ) f o r small 0 . Since the Fermi surface c u t o f f of copper occurs at P ^-5.2 m i l l i r a d i a n s (See Table 1) equation (5-1) i s the standard r e l a t i o n used i n two photon angular c o r r e l a t i o n experiments. Wide S l i t Geometry 1. Introduction The most common method used i n v i r t u a l l y a l l experiments before 1966 i s the so c a l l e d wide s l i t geometry i n which the detectors are long ( e f f e c t i v e l y i n f i n i t e ) , narrow s l i t s . We w i l l now apply t h i s technique to a non-interacting e l e c t r o n gas. In Figure 6 we assume that the detectors are narrow s l i t s l y i n g i n the xy plane of width H and having length |_ . The separation between source and detectors i s D. The source of gamma rays i s -39-F1G. 6 W I D E S L I T G E O M E T R Y located at the o r i g i n of the axes. This system w i l l count coincidence gamma ray p a i r s given o f f i n the xy plane. The number of gamma rays detected having a net component of momentum i n the Z d i r e c t i o n i s determined by the angular r e s o l u t i o n of the apparatus. The experimental procedure then i s to move one of the detectors i n the Z d i r e c t i o n and to count the number of coincidences as a function-of the distance Z from the xy plane. This procedure measures the d i s t r i b u t i o n of the gamma ray pair s i n momentum space as a'function of the Z component of t h e i r net momentum. 2. Angular C o r r e l a t i o n Results We w i l l now c a l c u l a t e the r e s u l t s to be expected from the procedure ou t l i n e d above, f o r p o s i t r o n a n n i h i l a t i o n i n a free e l e c t r o n gas. We have already derived the r e l a t i o n (5-1) P • = m c 0 T o We now su b s t i t u t e the r e l a t i o n Q - Z/j} i n t o (5-1) where Z i s the v e r t i c a l displacement of one detector from the xy plane. We then have the r e l a t i o n P T = \ r m o c Z / D Let ^ ( P ) be the density o f - a n n i h i l a t i n g p a i r s of gamma rays i n momentum space or equ i v a l e n t l y the density of e l e c t r o n i c states of the electron gas i n momentum space since f o r our model Tik - P. The coincidence counting rate as a function of P,7 v.dll be given by oQ oft The reason we can int e g r a t e to i n f i n i t y i n the Px d i r e c t i o n i s because the s l i t s are assumed to be much wider than the width corresponding to the Fermi radius. The range of energies of the gamma ra}'s accepted i s much l a r g e r than the spread due to the Doppler s h i f t corresponding to the Fermi energy and so we can also integrate P^ to i n f i n i t y . assume that ^ (P) i s i s o t r o p i c so that ^ (P) - ^ (P). Then ce the Fermi surface i s s p h e r i c a l we can write where P 2 = P 2 + P 2 =• P 2 + P 2 + P 2 as shown i n Figure 7. Then A Z x y z to since ^*(P) = 0 f o r P greater than the Fermi momentum P^ n(P ) c < P I P dP Z \ n(P z) ( P f 2 - P z 2 ) (5-2) This i s the equation of a parabola. 3. A p p l i c a t i o n s of the Wide S l i t Geometry to Real Metals As discussed i n Chapter 4 t h i s procedure has been applied to the s p h e r i c a l Fermi surface of sodium (Donaghy, 1964) and the F i g . 7 Region Sampled by Wide S l i t Geometry -43-r e s u i t s are i n good agreement with the knovm features of the sodium Fermi surface obtained by other means. For non-spherical Fermi surfaces we would expect deviations from the parabolic shape indi c a t e d by equation (5-2). A s e r i e s of measurements along d i f f e r e n t c r y s t a l d i r e c t i o n s to determine the shape of the Fermi surface has been done s u c c e s s f u l l y f o r the simple metal l i t h i u m (Donaghy, 1964). So f a r we have discussed the a p p l i c a t i o n of equation (5-2) i n the determination of Fermi surfaces. At the point where Prj, = P the coincidence counting rate w i l l drop to zero i n the simple model so f a r discussed, thereby g i v i n g us the so c a l l e d Fermi surface c u t o f f . For the case of an i n t e r a c t i n g e l e c t r o n gas Majumdar (1965) has shown that there w i l l be a d i s c o n t i n u i t y i n the slope of the angular c o r r e l a t i o n curve at P^ = P^. This d i s c o n t i n u i t y , measured i n d i f f e r e n t c r y s t a l o r i e n t a t i o n s w i l l y i e l d the shape of the Fermi surface even through a n n i h i l a t i o n of the positrons with core electrons creates a low l e v e l slowly varying background which obscures the d i s c o n t i n u i t y i n r e a l metals, making i t d i f f i c u l t to observe. Wide s l i t geometry positron experiments i n copper are reported i n Mijnarends, (.1969) and Berko and P l a s k c t t , (1958). Point Geometry 1. Introduction The previous pages have been a review of experimental techniques that have been commonly used i n the past which were -44-u s e f u l l y applied to simple metals such as sodium and l i t h i u m but had been of l i t t l e use i n -studying the Fermi surface of copper because of the large amount of core a n n i h i l a t i o n present f o r t h i s metal. We introduced a high r e s o l u t i o n apparatus (Williams et a l , 1968) which could d i s t i n g u i s h features i n the Fermi surface more c l e a r l y . Suppose we now change the detectors previously discussed from long narrow s l i t s i n t o detectors having a c i r c u l a r cross s e c t i o n with a diameter roughly equal .to the width ff (as defined' i n Figure 6) of the narrow s l i t s . The s o l i d angle subtended by the detectors at the sample w i l l now be much smaller than that used i n the t y p i c a l wide s l i t arrangement and the coincidence counting rate of the system w i l l drop by a f a c t o r of ^ 1000 (See R.G. Chambers i n Cochran and Haering, 1968). However, we w i l l now no longer be i n t e g r a t -ing over the Px d i r e c t i o n of Figure 6. Figures 8a and 8b show a schematic diagram of our point geometry arrangement. 2. Angular C o r r e l a t i o n Results Using a notation s i m i l a r to that used f o r the wide s l i t arrangement and assuming as before a free electron gas, the coincidence counting rate i s <A O aO -45-F I G . 8 A P O I N T G E O M E T R Y F i g . 3B • . C y l i n d r i c a l Region Sampled by p o i n t Geometry -46-We assume as before that: j ( P ) i s constant i n s i d e the Fermi sphere and zero outside i t . Then n ( P z ) « - P z 2 (5-3) P h y s i c a l l y , the d i f f e r e n c e between t h i s arrangement and the wide s l i t geometry i s that instead of counting the states included i n a s l i c e through the Fermi sphere as i n the wide s l i t case we now have a c y l i n d e r . (See Figure 8b and'Figure 9b). Rephrasing t h i s we would say that as i n the wide s l i t geometry the counting rate w i l l be p r o p o r t i o n a l to the number .of electrons i n the volume element through the Fermi surface. The only d i f f e r e n c e i s that t h i s volume element has been changed from a s l i c e to a c y l i n d e r as i n d i c a t e d i n Figure 9. 3. Experimental Procedure We used two s l i g h t l y d i f f e r e n t experimental methods to study the a n n i h i l a t i o n of positrons i n metals. One method has already been described; that i s the study of 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 gamma ray p a i r s as a f u n c t i o n of P . In other words the c y l i n d e r i n Figure 9 i s moved out sideways and the. lengths of successive chords of the sphere are measured. This technique w i l l hereafter be r e f e r r e d to as the angular c o r r e l a t i o n method. -47-A v a r i a t i o n of t h i s method has been described by Williams et a l , (1968) where the angle between the detectors i s kept f i x e d , u s u a l l y at a p o s i t i o n corresponding t o © = 0 (See Figure 5 f o r a d e f i n i t i o n of © ) . The number of gamma ray p a i r s given o f f f o r t h i s value of 0 i s then measured as a f u n c t i o n of c r y s t a l o r i e n t a t i o n . In other words Q i s kept f i x e d and the c r y s t a l i s rotated about some ax i s . Since the counting rate f o r our model i s p r o p o r t i o n a l to the number of electrons found i n the c y l i n d e r through the Fermi surface i n Figure 9, the counting rate i s th e r e -fore p r o p o r t i o n a l to the diameter of the Fermi surface. In e f f e c t t h i s technique measures the diameter of the Fermi surface as a function of c r y s t a l o r i e n t a t i o n . This technique w i l l h e r e a f t e r be r e f e r r e d to as the c o l l i n e a r geometry method. 4. A p p l i c a t i o n of the Point Geometry to Real Metals Most of the d i s c u s s i o n of the wide s l i t geometry a l s o a p p l i e s to the point geometry experiments and t h i s s e c t i o n w i l l be p r i m a r i l y devoted to a d i s c u s s i o n of some of the advantages of the point geometry technique. In a manner s i m i l a r to the wide s l i t case any d e v i a t i o n i of our angular c o r r e l a t i o n s r e s u l t s from the form i n d i c a t e d by equation (5-3) would be a t t r i b u t e d to a departure of the Fermi surface from a s p h e r i c a l shape i n s o f a r as the simple theory applied above i s v a l i d . In p r a c t i c e a comparison of the angular c o r r e l a t i o n r e s u l t s for d i f f e r e n t c r y s t a l o r i e n t a t i o n s can us u a l l y be more e a s i l y r e l a t e d to features of the Fermi surface. -48-An important p r a c t i c a l consideration i n any such experiment i s the broad, slowly varying core e l e c t r o n c o n t r i b u t i o n . This c o n t r i b u t i o n w i l l be r e l a t i v e l y l a r g e r using the wide s l i t method as compared to the point geometry r e s u l t s f o r the same metal. To demonstrate t h i s we assume that the momentum d i s t r i b u t i o n of the core electrons can be represented i n k-space by a large sphere of radius / " l ^ . enclosing and concentric with the Fermi sphere of radius fl shown i n Figure 9. The core c o n t r i b u t i o n as a f r a c t i o n of the conduction e l e c t r o n c o n t r i b u t i o n w i l l be a f a c t o r of A f~L j_ l a r g e r f o r the wide s l i t geometry as compared to the point geometry method. I t should be noted that there i s considerable ambiguity i n t h i s d e f i n i t i o n of ^ however this, i s supposed only to give a crude idea of the e f f e c t considered. Another s i g n i f i c a n t advantage of the point geometry technique i s that any mixing of the e l e c t r o n i c wave functions with higher momentum states w i l l have a l e s s serious e f f e c t on our experimental r e s u l t s as compared to the wide s l i t geometry r e s u l t s . This can be seen by noting that the c y l i n d e r i n Figure 9 i s of e f f e c t i v e l y i n f i n i t e length and also that the s l i c e i n Figure 9 corresponding to the wide s l i t geometry i s of i n f i n i t e diameter. Therefore any higher momentum states f a l l i n g i n s i d e these volume elements w i l l contribute to the coincidence counting rate observed. I t i s c l e a r that more higher momentum states are l i k e l y to f a l l i n s i d e the s l i c e than the c y l i n d e r i n Figure 9. F i g . 9 Comparison of point Geometry and Wide S l i t Geometry Note; The two volume elements, shown are not for equivalent c r y s t a l orientations. -50-A d i s c u s s i o n of the e f f e c t of higher momentum components i n the e l e c t r o n i c wave functions on experiments using the wide s l i t geometry i s given by Donaghy, (1964). . 5. A Third Compromise Experimental Arrangement A study of the Fermi surface of copper and of copper aluminum a l l o y s has been done by Fujiwara et a l , (1966, 1967) using, an experimental arrangement which i s a compromise between the two extremes already discussed. Their detectors are s l i t s , but the length of these s l i t s , L, (as defined i n Figure 6) i s l e s s than the length commonly used. In f a c t , t h e i r length L corresponds to a length s l i g h t l y l e s s than the diameter of the Fermi surface of copper. Their r e s u l t s i n copper show considerably more d e t a i l than, f o r example the e a r l y work of Berko and Plaskett, (1953) who used the standard wide s l i t arrangement" on copper. Fujiwara et a l were not able to resolve some of the features of the copper Fermi surface such as the dip i n the [ l l o | d i r e c t i o n because t h i s sharp feature i s smoothed out by t h e i r r e l a t i v e l y large experimental r e s o l u t i o n . j l l l j neck i n copper and copper aluminum extends through the zone boundary, y i e l d i n g two d i s c o n t i n u i t i e s i n the angular c o r r e l a t i o n data. This i s s i m i l a r to an e f f e c t observed by lis and w i l l be Fujiwara et a l , (1967), have found evidence that the discussed f u r t h e r i n Chapter 7. CHAPTER 6 APPARATUS Introduction An e a r l y arrangement of the apparatus has been described previously ( P e t i j e v i c h , 1966). B a s i c a l l y , the apparatus consisted of sets of two N a l ( T l ) detectors wired to record coincidences between the two detectors. The p h y s i c a l scale of the apparatus was determined p r i m a r i l y by the f a c t that the copper a l l o y samples had to have a .14 2 minimum s i z e so that i r r a d i a t i o n by a f l u x of neutrons/sec/cm would create a r a d i o a c t i v e source of s u f f i c i e n t i n t e n s i t y . Because the s o l i d angle subtended by the detectors at the source was so small, the counting rate obtained from one p a i r of N a l ( T l ) detectors was not adequate f o r our purposes and the system usee! consisted of three sets of Nal('fl) coincidence detectors and t h e i r associated e l e c t r o n i c c i r c u i t s . A schematic diagram of the detector arrangement i s given i n Figure .10. In t h i s f i g u r e detector A i s aligned through the sample with detector F and these two detectors are connected through a coincidence c i r c u i t . This arrangement i s duplicated f o r detectors C and D and f o r detectors B and E. Experimental Arrangement The apparatus was shielded by lead blocks r e s t i n g on a concrete f l o o r as indic a t e d i n Figure 11 The posi t r o n a c t i v e s i n g l e c r y s t a l s were housed i n a lead c a s t l e with i n s i d e dimensions 8,r x 8" x 8". The walls of t h i s lead c a s t l e were 4 inches t h i c k . Two 2" x 8" s l i t s at opposite ends of the c a s t l e allowed the gamma rays to pass out of the lead c a s t l e along the l i n e of si g h t of the detectors. Three Nal(T.l) detectors at each end of the apparatus were mounted behind a 12" x 12 u x 4" lead block. The c o l l i m a t i n g holes i n the lead blocks were -4" i n diameter and the distance between the r a d i o a c t i v e source and the detectors was 25 f e e t . The primary purpose of the lead c a s t l e was to s h i e l d the r e s t of the room from the gamma rays. The N a l ( T l ) detectors were protected from background r a d i a t i o n and from scattered r a d i a t i o n coming around the lead blocks by l a y e r s of lead wrapped around the detectors. DETECTOR A M M DETECTOR D DETECTOR B « S O U R C E DETECTOR C DETECTOR E L E A D DETECTOR F L E A D i I C O L L I M A T O R C O L L I M A T O R FIG. 10 SCHEMATIC DIAGRAM OF A P P A R A T U S Fig I I. L E A D SHIELDING L E A D C A S T L E C O L L I M A T O R L E A D C O L L I M A T O R IN THE APPARATUS THERE ARE THREE DETECTORS BEHIND THE LEAD COLLIMATOR ON EACH END -55-A D e s c r i p t i o n of the E l e c t r o n i c s 1. The Detector Assemblies As stated before we used three sets of N a l ( T l ) coincidence, detectors i n the apparatus. The c i r c u i t r y used f o r a l l three sets was i d e n t i c a l and evolved from that used by Petij'evich, (1966). The N a l ( T l ) c r y s t a l s used were c y l i n d r i c a l i n form with each c r y s t a l o p t i c a l l y coupled to i t s own photomultiplier. Four of the c r y s t a l s were of dimension 2" x 1" diameter and the other two were 2" x I-g" diameter. The photomultipliers used consisted of four R.C.A. 6342A and. two R.C.A. 6810A tubes. The detector assemblies were enclosed i n l i g h t - t i g h t aluminum and brass housings. Each photomultiplier was magnetically shielded by means of a co-netic a l l o y s h i e l d (manufactured by-Pe r f e c t i o n Mica Co.) and o p t i c a l l y coupled to the Na l ( T l ) c r y s t a l by means of Dow Corning 20-057 coupling f l u i d . 2. The Detector E l e c t r o n i c s Power f o r the phot o m u l t i p l i e r s was supplied by an assortment of three Northeast S c i e n t i f i c Corps. Model No. RE.5001 AW1 and two Fluke model no. 405B high voltage power supplies. The two 6810A photomultipliers were connected to one power supply. Each of the other photomultipliers had.its own power supply. Figure 12 shows a schematic of the detector e l e c t r o n i c s f o r one coincidence detector. The other two sets of coincidence detectors were i d e n t i c a l l y arranged and t h e i r output fed -56-P R E A M P S H A P E R NaKTl) DETECTOR — N a l d l ) P R E A M P DETECTOR S H A P E R COINCIDENCE UNIT S C A L E R NO. 1474 T IME DELAY RELAY TO MOTOR P R I N T E R FIG. 12 . S C H E M A T I C OF E L E C T R O N I C S -57-to the same model 1474 s c a l e r . The nominal coincidence r e s o l v i n g time of each c i r c u i t was 25 nanoseconds. Each photomultiplier was connected to a p r e a m p l i f i e r and shaper with the d i s c r i m i n a t i o n l e v e l s set at a spectrum threshold of 140 keV. Each p a i r of p r e a m p l i f i e r s was connected, to a coincidence c i r c u i t which f i r e d on overlap of the timing pulses. These c i r c u i t s are shown i n Figures 13 and 14. The power needed f o r these c i r c u i t s was supplied by a Technipower model M-3.I. 5-1.5 30 v o l t power supply and some zener diodes. A Canberra model no. 1474 s c a l e r was used to record the coincidence counts. Timing was provided by an o s c i l l a t o r (see Figure 15 f o r the c i r c u i t diagram) feeding a Canberra model 1475 master s c a l e r . When the counts on the master s c a l e r reached a preset value, the model 1474 s c a l e r was p r i n t e d out on a Canberra model 1479 p r i n t e r . A pulse from one of the s c a l e r s during t h i s operation activated a r e l a y c i r c u i t (see Figure 16 f o r the c i r c u i t diagram) which turned on a motor. This.motor moved the sample to i t s next p o s i t i o n . A gating pulse kept a l l inputs to the s c a l e r s dead while the motor was turned on. 3. Apparatus f o r Moving the Sample The apparatus used to e i t h e r r o t a t e the sample about i t s a x i s , or a l t e r n a t i v e l y , to move i t sideways i s shown i n Figure 17. Turning shaft B rotates the sample. This was done f o r the c o l l i n e a r FIG. 13 PREAMP AND SHAPER CIRCUIT +. 3 0 V 3 INCHES OF H H 2 0 0 0 £ ! 0 0 K ] + l O V p n 560X1 ?220X2 J~ .OIJJF + 3 0 V 100 P F ! N ! 0 0 ! 7 J I- ! — J r OMfiC/l 2 N 3 6 4 sr o o L L ! _ J _ J O O + 2 5 V i~l2N79? 1 A fJ>!|3.3K> J 2 - 2 K 6 8 0 X ) | 6 8 0 ^ I^V ? I . 5 K ^ 0 V ° N 7 5 4 •:<h4~o * | K + 2 0 V 0.1 jJF p | — ' o T E S T 100 P F 2.2 K + 2 0 V 82X2 oMQc^ i f 0A10 •II- 2 0 0 P F T D 2 S I N , 0 0 H ^ N ^ l n i 0 0 X 2 _ L c ^ 5 ^ Ol jJF JX . O I U F 2N22I8 K J 70ipF | 6 g°^70X2 - o j :270fh-X2 JZD Q_ j _ O ^ 3 . 3 K L + i o v £ f - I N 7 5 0 SN752 + 10 V 10 K 1 INPUTS 470X2 5 . 4 K 3.3 K <T 10 I K 470X2 IN37I6 2 0 0 P F XT) J X . 2 N 9 6 4 2N797 68X2 iOJJH 100X2 2 N 9 6 4 .O ipF ^ 1 270X2 OUTPUT F I G . 1 4 C O I N C I D E N C E C I R C U I T F I G . 1 5 O S C I L L A T O R C I R C U I T I MH 2N964 - 12 V 4.3 K J ! U/ OMQCA 30 V - 2 0 V : I 8 0 K ^ 6 8 0 ^ IN37I6 J & 2N9S4 IT _IP5)JF .7 K IN 100 620 K 2N 3440 2N3440 \i 5 K 1 8 0 ^ 2N797 T 300 a J — - v 1.8 K i o I + 30 V 1NI00 .OiJJF \ K 10 K i . . IOK 2N3905>|—— IJJF ^I20K 2 N 3 9 0 3 V 8}JF 2N4400 5.6 K + 5 2 INiOO [330 KJ 2N22I8| ^18 K ODD 3 8 - 3 0 0 0 3 T I M E D E L A Y R E L A Y ^ 2N301 i L 1.2 K F I G . S 6 T I M E D E L A Y R E L A Y C I R C U I T S H A F T A SHAFT B 2ZZZ3 S A M P L E HOLDER TO FIT i INTO END OF SHAFT. i ro F I G . 1 7 S C H E M A T I C O F A P P A R A T U S F O R M O V I N G S A M P L E . -63-point geometry experiments. •' Turning shaft A causes the sample to move sideways. This was used to. obtain our angular c o r r e l a t i o n r e s u l t s . Shafts A and B were turned through predetermined angles by means of a motor connected to a 12" d i s c . A microswitch r i d i n g the rim of t h i s d i s c kept the motor turned on unless the lever of the microswitch was i n one of the notches cut i n the edge of the d i s c . The angular p o s i t i o n of these notches determined the angle through which the motor turned. Experimental Procedure 1. Introduction When the neutron i r r a d i a t e d samples a r r i v e d from Chalk River, they were immediately placed i n the holder shown i n Figure 17. The e l e c t r o n i c s was then checked and the si n g l e s counting rates of the detectors recorded. The experiment was then s t a r t e d and allowed to proceed uninterrupted f o r about 40 hours at which time the counting rates were too low to be u s e f u l l y recorded. 2. Coincidence Resolving Time Our method of measuring the r e s o l v i n g time of the apparatus was to use the random source method. I f the si n g l e s counting rate i n the two detectors from two separate sources i s N-j and N then we have that the chance coincidence counting rate i s given by CHANCE ^ 1 2 where "Cis the coincidence r e s o l v i n g time of the apparatus. By measuring N , N and N we can obtain "C . This value of the UJ IiiN\~>iii J . / r e s o l v i n g time was checked by using known time delays i n the c i r c u i t . The two methods agreed with each other and y i e l d e d a nominal value of 25 nanoseconds f o r the coincidence r e s o l v i n g time of the c i r c u i t s . 3. O p t i c a l Alignment of .the Detectors The detectors were o p t i c a l l y aligned to an accuracy of bet -ter than 0.5 mm over a distance of 15 meters using an engineer's t r a n s i t and l e v e l . Any misalignment would mean that the c y l i n d e r through the Fermi surface i n Figure 9 would not pass through the center of the Fermi sphere. A misalignment of 0.5 mm corresponds -to a P of 1% of the radius of the.Fermi sphere f o r copper. Li In a d d i t i o n to t h i s there were some problems i n ensuring that the c o l l i m a t i n g holes i n the 4 inch t h i c k c o l l i m a t i n g blocks were p a r a l l e l . t o the axis of the apparatus. The holes wvre estimated to be p a r a l l e l to the axis within \ degree. This would have a s i g n i f i c a n t e f f e c t on the shape of the experimental r e s o l u t i o n f u n c t i o n , but would not contribute s i g n i f i c a n t l y to the misalignment discussed i n the previous paragraph. 4. Samples The copper a l l o y s i n g l e c r y s t a l s studied were obtained from M a t e r i a l s Research Corp., Orangeburg, New York. Their p u r i t y was 99.99% and the c r y s t a l o r i e n t a t i o n s were- accurate to 1°. These -65-e r y s t a l s were of c y l i n d r i c a l shape, with a diameter of 4.5 mm and a length of 4 mm. The samples were mounted i n pure aluminum holders and then neutron i r r a d i a t e d i n an Atomic Energy of Canada Ltd. reactor at Chalk River, Ontario f o r approximately 40 hours at a f l u x i n t e n s i t y of 10"^ neutrons/sec/cm 2. These c r y s t a l s were disordered a l l o y s . They were X-rayed before and a f t e r the neutron i r r a d i a t i o n s and no evidence of any damage could be found. There was some evidence that the n i c k e l a l l o y c r y s t a l s changed i n s i z e during the neutron i r r a d i a t i o n s , but on a r r i v a l i n Vancouver they were back to t h e i r normal s i z e . This i s discussed more f u l l y i n Chapter 7. Experiments with these samples began approximately 36 hours a f t e r they were taken from the r e a c t o r . At t h i s time they had a copper-64 a c t i v i t y of two c u r i e s which corresponds to a p o s i t r o n a c t i v i t y of about 400 m i l l i -c u r i e . Experimental Resolution 1. Introduction Since the samples had been neutron i r r a d i a t e d to produce po s i t r o n a c t i v e copper-64, the positrons could be considered to be uniformly d i s t r i b u t e d throughout the sample. The simplest way of obtaining an estimate of the experimental r e s o l u t i o n i s to c a l c u l a t e -66-tbe maximum angle by which a gamma ray p a i r can deviate from 0 - 0 i n Figure 5 and s t i l l be detected as a coincidence between tv.'o of the N a l ( T l ) detectors f o r the detectors i n a set p o s i t i o n and h.ead on to each other. This corresponded to an angle of 1 m i l l i r a d i a n . 2. Computer Calculated Resolution These ideas have been improved upon by P e t i j e v i c h , (1968). The c o l l i m a t i n g holes and the sample were broken up i n t o small elements and a computer was used to numerically sum over a l l combinations of these elements to obtain the a c t u a l shape of the r e s o l u t i o n f u n c t i o n . For our apparatus, t h i s corresponded c l o s e l y to a gaussian whose f u l l width at one-half maximum height was 1 m i l l i r a d i a n . The r e s o l u t i o n f u n c t i o n was found to be of c i r c u l a r cross section perpendicular to the axis of the system. 3. C o l l i n e a r Geometry Resolution For the c o l l i n e a r geometry the relevant experimental r e s o l u t i o n i s given by the r e s o l u t i o n width at the Fermi surface and w i l l vary depending on the p a r t i c u l a r Fermi surface being studied. where the c h a r a c t e r i s t i c width of the experimental r e s o l u t i o n i s - For a free e l e c t r o n gas with Fermi energy £ ^, making use of equation (5-1) we have the r e s u l t (6-1) 0 = © , (Figure 5), c i s the v e l o c i t y of l i g h t and m i s the mass -67-of the e l e c t r o n . (p i s the angle subtended by the experimental r e s o l u t i o n function at the o r i g i n of the Fermi sphere i n k-space. To apply t h i s to copper we make the approximation that the Fermi surface of copper i s a sphere with a Fermi energy £ p = 7 ev. I f we su b s t i t u t e t h i s value of £ i n t o equation (6-1) we f i n d that our experimental r e s o l u t i o n f u n c t i o n has a FWHM of 13° subtended at the o r i g i n of the Fermi sphere i n k-space. Since the contact area of the [ i l l ] necks i n the copper Fermi surface subtends an angle of 20° at the o r i g i n of the Fermi sphere, we could reasonably expect to observe an e f f e c t i n the po s i t r o n a n n i h i l a t i o n which would be clue to the [ i l l ] necks. This reveals the main f a u l t to be found with the conventional wide s l i t "a.rrangernents. The wide s l i t performs , a complete average over the x d i r e c t i o n of Figure 6, smearing out e f f e c t s due to features such as the [ i l l ] necks. 4. Temperature Broadening of the Resolution Function So fax* i n the dis c u s s i o n of t h i s thesis i t has been assumed that the p o s i t r o n i s at r e s t before a n n i h i l a t i o n . Thermal e f f e c t s due to the f a c t that f o r our experiments both the p o s i t r o n and the electrons i n the metal were at room temperature have been ignored. This thermal energy can be thought of as e f f e c t i v e l y i n c r e a s i n g the width of the experimental r e s o l u t i o n . The e l e c t r o n s , since they obey Fermi-Dirac s t a t i s t i c s , w i l l have a n e g l i g i b l e e f f e c t on the r e s o l u t i o n (A.T. Stewart i n Stewart and R o e l l i g , 1967). The positrons, however, w i l l have a spread of energies which obey - 6 8 -Maxwell-Eoltzmann s t a t i s t i c s . At room temperature, the 1-D p o s i t r o n momentum d i s t r i b u t i o n w i l l therefore be gaussian with a FWHM corresponding to 0 - 0 . 5 m i l l i r a d i a n s (Doriagby, 1964) where <3 i s defined i n Figure 5. I f we f o l d t h i s i n t o the o p t i c a l r e s o l u t i o n of the apparatus we obtain a f i n a l r e s o l u t i o n due to both e f f e c t s corresponding to a gaussian of FWMM of 1.1 m i l l l r a d i a n s . 5. E f f e c t of the Detector Arrangement As Figure 10 i n d i c a t e s , the three d i f f e r e n t sets of coincidence detectors cannot occupy exactly the same p o s i t i o n i n space. The maximum angular spread between any two sets of N a l ( T l ) detectors was 2°. As discussed e a r l i e r , the angle subtended at the o r i g i n of the Fermi sphere by the experimental r e s o l u t i o n f u n c t i o n was 13°. The r e s o l u t i o n function was therefore not appreciably broadened by neglecting the f a c t that the three sets of detectors were not in exactly the same p o s i t i o n i n space. A geometrical way of viewing t h i s i s to think of the three sets of detectors as cr e a t i n g 3 d i f f e r e n t c y l i n d e r s through the Fermi surface i n Figure 9. The axes of the three c y l i n d e r s would be t i l t e d at 2° to each other and would intersect-each other at the o r i g i n of the axes, but the diameters of the c y l i n d e r s would be such that the i n t e r s e c t i o n of the c y l i n d e r s with the Fermi surface would subtend an angle of 13° at the o r i g i n . -69-Correction f o r Decay of Source and Chance Coincidences 1. Introduction The h a l f - l i f e of copper-64 was taken to be 12.7 + .1 hours. In order to reduce the e r r o r s associated with d r i f t s i n the e l e c t r o n i c s and c o r r e c t i o n s f o r the decay of the copper-64, the angular s e t t i n g s of the apparatus were cha nged frequently and the apparatus was recycled many times i n the course of a run. This was made more important by the f a c t t h a t the imp u r i t i e s alloyed i n t o the copper c r y s t a l s also . yi e l d e d small amounts of r a d i a t i o n with a h a l f - l i f e d i f f e r e n t from that of pure copper. 2. A n a l y s i s of C o l l i n e a r Geometry E r r o r s For the c o l l i n e a r geometry experiments, the angular p o s i t i o n of the sample was determined by a microswitch r i d i n g the rim of a 12" d i s c i n which notches had been cut at 10° i n t e r -v a l s . The coincidence counts were read out every 4 minutes and the sample rotated by 10 degrees. Since a l l c r y s t a l s used had at l e a s t 90° c r y s t a l symmetry and since the c r y s t a l was continuously advanced i n 10° steps i n one d i r e c t i o n , the time required to complete one set of readings was 36 minutes. The readings corresponding to c r y s t a l l o g r a p h i c a l l y equivalent d i r e c t i o n s were added together a f t e r f i r s t c o r r e c t i n g f o r the r a d i o a c t i v e decay of the copper-64 .and f o r the chance coincidences (8% of the t o t a l at the beginning of an experiment). Because these c o r r e c t i o n s had to be accurate only over a 36 minute i n t e r v a l , they were e a s i l y c a r r i e d out. The cor r e c t i o n s f o r the chance coincidences were accurate to 5%. Since they comprised -70-only cS% of the t o t a l t h i s would imply a 0.4% uncertainty i n the t o t a l number of counts. However, al]. points i n the 36 minute time i n t e r v a l receive almost the same c o r r e c t i o n . For t h i s reason the errors i n c o r r e c t i n g f o r the decay of the source would a l t e r the shape of the c o l l i n e a r geometry curves by l e s s than at the begin-ning -of the experiment. • 3. A n a l y s i s of Angular C o r r e l a t i o n Method E r r o r s In t h i s technique, V was var i e d by turning the screw (shaft A) i n Figure 17 one r e v o l u t i o n at the end of 4 minute i n t e r -v a l s . This corresponded to a sideways movement of the sample over a distance of .1/24 inch, or a change i n Q as defined i n Figure 5 by an amount of 0.273 m i l l i r a d i a n s . At the maximum O the d i r e c t i o n of r o t a t i o n of the motor d r i v i n g shaft A was reversed by manually moving a switch. For a maximum © of 9 m i l l i r a d i a n s , the time required to obtain one pass over the angular c o r r e l a t i o n curve was 136 minutes. The e l e c t r o n i c s was quite stable over t h i s time period (0.2%) but the co r r e c t i o n s f o r the decay of the source and the chance coincidences became more important. At 6 = 9 m i l l i r a d i ans the chance coincidences accounted f o r 40% of the t o t a l coincidence counts recorded at the beginning of an experiment when the i n t e n s i t y of the r a d i o a c t i v e source was at i t s maximum. Er r o r s i n the c o r r e c t i o n f o r h a l f - l i f e and f o r chance coincidence create an uncertainty of £ 3% i n the heights of the curve at 0 = 9 m i l l i r a d i a n s r e l a t i v e to the c e n t r a l maximum where chance coincidences account"for only 5% of the t o t a l number of coincidence counts at maximum source strength. This was s i m i l a r i n magnitude to the s t a t i s t i c a l error i n our experimental points. A checl on t h i s was obtained by comparing the data taken when the i n t e n s i t y of the r a d i o a c t i v e source was high, to data taken when the source had died-down. The agreement was within 5%. We were l a r g e l y i n t e r e s t e d i n comparing d i f f e r e n t a l l o y s to each other and a systematic e r r o r i n c o r r e c t i n g f o r chance coincidences would be approximately the same f o r a l l of these a l l o y s . Sodium-2'2 Apparatus 1. Introduction There are several disadvantages to using neutron i r r a d i a t e d copper c r y s t a l s as samples f o r positron studies. The most serious of these i s that studies are l i m i t e d to copper and i t s a l l o y s . Other problems such as the p o s s i b i l i t y of neutron damage to the c r y s t a l s and the r e l a t i v e l y short h a l f - l i f e of copper-64 are also present. Because of t h i s we purchased a 30 m i l l i c u r i e sodium-22 p o s i t r o n source from New England Nuclear Corp. A source of t h i s low i n t e n s i t y was not adequate f o r use with the three sets of N a l ( T l ) coincidence detectors discussed p r e v i o u s l y . In order to compensate f o r the low a c t i v i t y of the source a detector was designed which would combine high e f f i c i e n c y -72-i n c o l l e c t i n g counts at a lower cost than that of simply i n c r e a s i n g the number1 of detectors of the type already discussed. 2. Detector Arrangement Seven N a l ( T l ) c r y s t a l s (2" x 1" diameter; c y l i n d r i c a l i n form) were li n k e d by means of a l u c i t e l i g h t pipe to a s i n g l e photomultiplier. Figure 18 shows a schematic diagram of t h i s . The distance between the centers of the Na l ( T l ) c r y s t a l s i s 4.5 inches. The whole assembly was mounted behind a lead block 16" x 16" x 4" i n which \ diameter c o l l i m a t i n g holes had been d r i l l e d at the corners and center of a hexagon i n s c r i b e d on the 16" x 16" facie of the lead block. The distance between the centers of the c o l l i m a t i n g holes i s 4.5 inches. These detectors were used at a distance of 25 f e e t from the sample. 3. E l e c t r o n i c s The low e f f i c i e n c y of t h i s arrangement f o r l i g h t c o l l e c t i o n from the N a l ( T l ) c r y s t a l s made the use of low noise R.C.A. 8575 photomultipliers necessary. Due to poor s t a t i s t i c s i n the number of photons s t r i k i n g the photocathode of the photomultiplier, the pulses out of the photomultiplier were poorly shaped. These i r r e g u l a r photomultiplier pulses were i l l s u ited for the pr e a m p l i f i e r and shaper used with our other detectors. A simple c i r c u i t which produces timing pulses by f i r i n g on the r i s i n g edge of the photomultiplier F IG . 18 S C H E M A T I C O F U L T I D E T E C T O R pulses was used. Figure 19 gives the c i r c u i t diagram t o r t h i s . The coincidence c i r c u i t which i n d i c a t e d overlap of the timing pulses i s described i n Figure 13. This system had a coincidence r e s o l v i n g time of 50 nanoseconds. 4. Alignment This apparatus was o p t i c a l l y aligned i n a manner s i m i l a r to our other detectors. The only d i f f e r e n c e i s that here cross coincidences between unpaired c o l l i m a t i n g holes can occur and w i l l . be counted as genuine events. These w i l l occur at an angle Q - 15 m i l l i r a d i a n s (see Figure 5 f o r a d e f i n i t i o n of O ). A study of the angular c o r r e l a t i o n data i n d i c a t e s that at t h i s angle the coincidence counting r a t e i s w e l l down from the O - 0 p o s i t i o n (to about 6%). Since t h i s apparatus i s designed f o r use at low counting rates and mainly f o r c o l l i n e a r geometry experiments the neglect of cross coincidences i s v a l i d . Most samples studied w i l l also have t h e i r core a n n i h i l a t i o n examined and the number of cross coincidences can be checked f o r each p a r t i c u l a r case. 5. Experimental Arrangement An electromagnet was used to focus the positrons from the sodium-22 source onto the sample being studied. Figure 20 shows a schematic diagram of t h i s arrangement. The sample and source are along the axis of the magnetic f i e l d . The positrons w i l l s p i r a l along t h i s axis u n t i l they s t r i k e the sample. The magnetic f i e l d .POLE FACE ACTIVE AREA UJ SODIUM-22 POLE FACE ROD SAMPLE I l F I G . 2 0 S C H E M A T I C ' O F P O S I T R O N F O C U S S I N G S Y S T E M - 7 7 -used was 10 ki l o g a u s s . We estimate that over 80% of the positrons leaving the source in the forward d i r e c t i o n s t r i k e the sample. CHAPTER 7 Experimental Results Introduction At the outset of t h i s work, the only 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 using the point geometry technique to the study of metals was the preliminary study of copper by P e t i j e v i c h , (1966). Subsequently Fujiwara et a l (1966) applied t h e i r technique which i s discussed i n Chapter 5 to copper and our group r e f i n e d t h e i r measurements (Williams et a l , 1968). These studies represented the f i r s t d i r e c t observations of any d e t a i l s of the Fermi surface of copper using positron a n n i h i l a t i o n . The work i n t h i s thesis i s concerned with an extension of t h i s technique to the study of concentrated copper a l l o y s . These are of i n t e r e s t because the conventional resonance experiments such as the de Haas van Alphen experiments cannot be used i n concentrated a l l o y s . Our r e s u l t s on pure copper w i l l not be discussed i n t h i s thesis except in s o f a r as -79-required to explain the r e s u l t s i n a l l o y s . The a l l o y s discussed i n t h i s thesis are C u ^ A l , CUg<_Zn^ ,_, Cu-^Zn, ,-Ni, r and C u ^ ^ N i ^ . Eoth c o l l i n e a r geometry and angular c o r r e l a -70 15 15 60 40 • 1 • t i o n measurements were c a r r i e d out with the point detection system described e a r l i e r . The a l l o y s were chosen to cover a wide area of p o s s i b i l i t i e s i n the hope that t h i s survey would i n d i c a t e areas of i n t e r e s t which could be examined i n more d e t a i l i n future studies. The actual number of experiments was l i m i t e d by the time required to carry out an experiment and more severely by the cost $300 per experiment). A l l experiments discussed here were c a r r i e d out at room tempera-ture. In the following pages a l l 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 are symmetry di r e c t i o n s not actual c r y s t a l d i r e c t i o n s . For example the Qlioj , {l - i o j 3 £-HoJ , > etc., d i r e c t i o n s w i l l a l l be r e f e r r e d to as [ l io ] d i r e c t i o n s The theory of positron a n n i h i l a t i o n i n metals i s not i n a very complete state and, while our data has a d e f i n i t e dependence on c r y s t a l o r i e n t a t i o n , the i n t e r p r e t a t i o n of our r e s u l t s i n terms of the Fermi surfaces of these a l l o y s i s rather d i f f i c u l t . The f i g u r e s i n t h i s chapter contain the experimental data obtained i n the manner described i n previous chapters. The c o l l i n e a r geometry r e s u l t s have the corresponding c r y s t a l l o -graphic d i r e c t i o n s indicated on the f i g u r e s . These were obtained by x-rayin the c r y s t a l s and also by noting the c r y s t a l symmetry of the experimental r e s u l t s . The errors indicated are one standard d e v i a t i o n s t a t i s t i c a l errors -80-i n the number of counts. The angular c o r r e l a t i o n r e s u l t s also have a d e f i n i t e dependence on c r y s t a l o r i e n t a t i o n . D i f f e r e n t c r y s t a l o r i e n t a t i o n s of the same a l l o y have been normalized to each other at the 0 = 0 p o s i t i o n making use of the c o l l i n e a r geometry data. The c r y s t a l o r i e n t a t i o n f o r the angular c o r r e l a t i o n data i s s p e c i f i e d by the term "a x i s " , that i s f o r example, [lOo] axis implies that the c r y s t a l was moved i n the [lOo] d i r e c t i o n . The "if-ray d i r e c t i o n r e f e r s to the d i r e c t i o n of the coincidence gamma rays at the @ = 0 p o s i t i o n of the c r y s t a l . In Figures 23 and 26 an expanded presentation of the data on both sides of the © = 0 axis i s given. The data on each side of the @ = 0 axis has been independently normalized and the two sets of data adjusted v i s u a l l y to meet each other at the Q = 0 p o s i t i o n . -81-6 10 20 30~~40 GO 6 0 7 0 SO SO C R Y S T A L ORIENTATION ( D E G R E E S ) . F i g . 21 C o l l i n e a r Geometry R e s u l t s f o r Cu A l - ^ .0 .9 u .7 § b I-z o °.5 Ul o U J g A-o 2 O o .3 •2H .0 • °X° ^ » ° x X. / • • A v <» o • X X x x » X X X o X X • X X X i X o X X x • X x x • • . x x X X X V - 1 . — r 0 I 2 3 4 5 6 7 8 9 © MILLIRADIANS P i g . 22 ANGULAR CORRELATION RESULTS- FOR C u 9 0 A I | 0 [ l O Q l A X I S S T A T I S T I C A L E R R O R A T © - 0 - 0 . 9 % " ° ' 0 0 0 1 Y- R A Y D I R E C T I O N A A A D I I ' J / - R A Y D I R E C T I O N - 3 3 -1.0-j © ® @ _ © ® ® ® ® ® ® „ ® © _ .91 ® U . 8 H @ "' " .. © fjlOj A X I S S / l t o ® n z> o 6 ° ' ® ^ ® ® ^ ® ^ ,» ^ ® © y © ® ~ - @ QooD AXIS Q ® u © Qoq] X - R A Y 8 . .8- ® - 4 - 3 - 2 - 1 0 I 2 3 4 ! © (M ILL IRADIANS) F i g . , 23 EXPANDED ANGULAR CORRELATION DATA FOR C U 9 0 A I 1 0 S T A T I S T I C A L E R R O R A T 0 = 0 - ).3t% i— 1 IV> ? H- COINCIDENCE' COUNTING R A T E 5? • C D C D C D O O O O ->j • co C D o — ro O J , . L ! i 1 ! ! ! ffi O i f § O j o -co J £ co o ~ O o y j -— ! • O . o , O T • ®--€) r O O | : 5 4 — | ' , i • , © O i c — : S 4 ^ • \v ! • • • • « o H O j j @ H a? O -U-, 2 : £ — C J • ° j f ©-w _ c T . J - 1 @~ m g o j H 1 ® -C D | 60 j @-o o x x x x • x x -x °x x« « X X" X • X x* x° X* X ... < X X' X X" x ° . X X* X° A \ A A « X v " 1 1 1 1 1 1 ' 1 1 I 0 I 2 3 4 5 6 7 8 . 9 <f) ' (MILL IRADIANS) F i g 2 5 - , ^ ANGULAR CORRELATION R E S U L T S FOR C U 8 5 Z M | 5 N / Y v r i o o ' | A X I S S T A T I S T I C A L . E R R O R A T © = 0-0.0% A X ,< | q j y - R A Y D I R E C T I O N 0 0 „ [MOD A X I S 0"3 X - R A Y D I R E C T I O N -86-l.0i ' © © © @ @ © ^ „ ® © ® © • 9H .8 LU tr. % .7 o <J .6 Suo-U J Q o S 9. 8 .8 .7^ © © 0 © © © © @ @ ® @ © - . -• • ^ — ^ [ J IOJAXIS-© 0 QlQy-RAY DIRECTION © © @ © © ® © @ © © ® . "— ' < 0 @ JFooJ AX IS © © © © [TlO] X-RAY DIRECTION ^ © © © © © ® © - 5 - 4 - 3 - 2 - 1 0 I 2 3 4 5 . 0 (MIL LIRADIANS) ' ( - F i g . 26 E X P A N D E D A N G U L A R CORRELATION C U R V E S FOR C u 8 5 Z N | 5 S T A T I S T I C A L E R R O R A T © - 0-- 8 7 -104-103-UJ < I02H CD j Z I— 2: z> o o 101-1 •<0 1 (o) a loo-LU 9 o :£= Q Q O J O CROSS SECTION 1 T O Ql0~] OS-'S) 10 20~To~lo" 50~~60 70 SO SO CRYSTAL ORIENTATION ( D E G R E E S ) ; i1,ig-» , 2 7 C o l l i n e a r Geometry Re s u l t s l o r C u 7 0 Z n 1 5 l s i 1 5 ^ ion xVx . 8 i 2 : i — 8 .5H LU o •z. U J Q 4 o ' 8 .2 a X X X oX »x ,x oX X x x x x [[16] AXIS X - R A Y 12° F R O M nif] D I R E C T I O N 2 3 " F R O M U&J D I R E C T I O N 000 Dop] AXIS X - R A Y 14° F R O M [J06] D I R E C T I O N X X o SX °X X oX X • X V © X 0 X v ° *x x ' x« x X _j j j , . , 1 i r— r 0 I 2 3 4 5 6 7 8 9 0 ( M I L U R A D I A N S ) •' F i S ' i / 2 8 A N G U L A R C O R R E L A T I O N R E S U L T S F O R C U 7 0 Z N | 5 N ! | 5 S T A T I S T I C A L E R R O R A T @ « 0 ~ 1 . 2 % -89-(JicD i [3iO ttoo] T T CO) UJ CD 2f. 101-8 , 0 0 ' 99-o LLI C I o *~ o ° 9 8 9 /-f 0 CROSS SECTION 1 TO &I0] 10 20 3 0 '10 50 6 0 70 8 0 9 0 CRYSTAL ORIENTATION (DEGREES) Pig..29 C o l l i n e a r Geometry Results f o r C u ^ N i ^ -90-1.0 1® © @ © @ © ® ® ® ® © @ ^ 8 1 ® ® .7" U J £ .6-® ® © ® x TO [Tod] A X I S [f 1 o] / - RAY DIRECTION »- .5-o o o z: UJ Q o . 3 -o o @ ® ® ® ® ® ® © ® ® ® ® ® ©<s © ® J 1 , , ! , 1 1 1— O l 2 3 4 5 6 7 8 © ( M I L L I R A D I A N S ) ' F i g . 30 ANGULAR CORRELATION RESULTS FOR C u 6 0 N i 4 0 S T A T I S T I C A L E R R O R A T & - 0 - 1 . 4 % ( CHAPTER 8 Experimental Analysis The analysis of the results in chapter 7 is made d i f f i c u l t by the fact that neither the theory of positron annihilation in metals nor the theory for disordered alloys is in a very complete state, furthermore, as has been d i s c u s s e d e a r l i e r , the r e d u c t i o n "in s o l i d .angle imposes l i m i t a t i o n s on the allowed r e s o l u t i o n f o r a reasonable -cou n t i n g r a t e . A c c o r d i n g l y the r e s o l u t i o n f u n c t i o n has a s i g n i f i c a n t e f f e c t on the data and the u n f o l d i n g process -necessary to r e v e a l d e t a i l s due to' the Fermi s u r f a c e s o f the a l l o y s i n v o l v e s some u n c e r t a i n t y . Throughout this analysis we w i l l assume the simplest theory possible, that i s the simple Sommerfeld model discussed in previous chapters. Because of this and because our method of analysis is very crude the - 9 2 -conclusions reached i n t h i s chapter w i l l be of a t e n t a t i v e nature only. In order to f a c i l i t a t e t h i s analysis some of the data shown i n chapter 7 w i l l be shown again i n t h i s chapter. C u 9 0 A 1 1 0 1. Introduction Atomic aluminum has three r e l a t i v e l y loosely bound electrons i n 2 2 6 a 3s 3p s h e l l i n addition to a closed s h e l l of 2s 2p electrons. According to the r i g i d band model we would then expect there to be 1.2 conduction electrons per atom i n t h i s a l l o y . This corresponds to an increase i n the volume enclosed by the Fermi surface by a f a c t o r of 1.2 or an increase i n the average diameter of the Fermi sphere of 6%. 2. C o l l i n e a r Geometry Results A cross section perpendicular to the [ l i o ] d i r e c t i o n has been obtained using our c o l l i n e a r geometry and i s shown i n Figure 31. A comparison with the r e s u l t s f or pure copper i s also shown. The de Haas van Alphen data indicates that the Fermi surface of pure copper makes contact with the B r i l l o u i n zone boundary i n the {"ill] d i r e c t i o n and i t therefore seems reasonable to assume that the Fermi surface of C u ^ A l ^ w i l l reach the same point i n k-space i n t h i s d i r e c t i o n . For t h i s reason the two curves i n Figure 31 are normalized to the same value i n the £lH^ d i r e c t i o n . The t h e o r e t i c a l curve expected f o r pure copper assuming an i s o t r o p i c 50% p o s i t r o n a n n i h i l a t i o n with the electrons i n the f i l l e d B r i l l o u i n zones, using the known Fermi surface of copper and the simple -93-C o l l i n e a r Geometry R e s u l t s f o r Cu A l — THEORETICA - 9 4 -free electron theory discussed in Chapter 4 is also shown on this figure. An analysis of this data is made d i f f i c u l t in that the anisotropy i n the positron annihilation i s less than 6% and the s t a t i s t i c a l error i n each point in Figure 31 i s about 1%. The f i r s t comment that can be made i s that the copper and C u g Q A l 1 0 results are very similar. The only difference between the two results which is beyond s t a t i s t i c a l error occurs in the region between the J^ 31 jQ and jlOu] directions. The unknown shape of t h e f l l l j neck in the Fermi surfaces of these alloys makes i t d i f f i c u l t to assign a neck width to our collinear geometry results. It is possible to draw a f a i r l y well defined curve through our results i f we assume that the top of the fill} neck i s symmetrical about the (ill] direction, from group theoretical results we expect the j l l l j neck to f a l l off rapidly just as i t leaves the Brillouin zone boundary. The results of Figure 32A which are based on the previously discussed Sommerfeld model suggest that our resolution function does not alter the width of the neck observed significantly, however, the appli-cation of this theory neglects a l l higher momentum components in the electronic wave functions. T h i s i s d i s c u s s e d f u r t h e r i n Appendix ~JJ" , In an attempt to relate our results to the Fermi surface neck width we define a £ as the width of our experimental peaks at the height at which the points start to f a l l off from the top of the peak. To be more precise we define ^ as the width of the experimental peak at a height of 1 1/2 units (on our scale) below the top of the peak (the location of which was visually estimated) for the Cu Q nAl and Cu Z n K results and 1 -95-unit below the top of the peak f o r the C u ^ Z n ^ N i . ^ and C u ^ N i ^ r e s u l t s . In a s i m i l a r manner the angular c o r r e l a t i o n data with respect to the £lllj necks s u f f e r s from e s s e n t i a l l y the same problem. As i s i m p l i c i t i n the analysis we again expect that i t i s a quantity s i m i l a r to £ that we are measuring. The £ of the £lllj[ peak i n the C u ^ A l ^ data i n Figure 21 i s 38° t 6°. The experimental r e s u l t s f o r pure copper shown i n Figure 31 have large error bars associated with the experimental points and no attempt "was made to obtain a ^ from them1. According to Zornberg and Mueller, (1966) the j l l l | necks i n the copper Fermi surface subtend an angle of 20° at the center of the Fermi sphere. The errors f o r the ^ i n t h i s and the following data were d i f f i c u l t to determine because of the large experimental errors and because of the lack of t h e o r e t i c a l knowledge of the shape of the curves to be expected. We have chosen the crude method of drawing d i f f e r e n t p l a u s i b l e curves through the data points and estimating an error from the spread i n these curves. This i s a rather subjective process and makes these conclusions rather t e n t a t i v e i n nature. The electrons i n the f i l l e d d-bands i n copper and i n the a l l o y s also contribute to the d i s t r i b u t i o n of a n n i h i l a t i n g gamma ray p a i r s observed. One of the d i f f i c u l t i e s associated with these experiments i s that we have no way of knowing what t h i s d-band cont r i b u t i o n i s . We would, however, expect the peak widths obtained to be correct because the sides of the necks should correspond to r a p i d l y varying features of our r e s u l t s . We 6 0 0" 20~ 4 0 ORIENTATION OF CRYSTAL (DEGREES) Fig. 32 A-R E S O L U T I O N F O L D E D INTO 8 0 ' N E C K S . .1 .2 .3 .4 .5 .6 7 .8 .9 1.0 F i g . . R e s o l u t i o n Folded Into S p n e r i c a l Feruii surface of Unit Radius -98-now make the assumption that the d-band electrons are very l i t t l e a f f e c t e d by the a l l o y i n g . The t h e o r e t i c a l d i s c u s s i o n i n chapter 3 i n d i c a t e s that t h i s may not be a v a l i d assumption, however, i t allows us to carry out a simple analysis of the data as follows. We wish to c a l c u l a t e the average change i n the l e v e l of the Fermi surface assuming that the Jl.llJ d i r e c t i o n i s the normalization point f o r the copper and C u ^ A l ^ r e s u l t s . With these assumptions we f i n d that the average d i f f e r e n c e between the heights of the two curves i n Figure 31 i s - 2#< We now allow f o r an i s o t r o p i c c o n t r i b u t i o n from the electrons i n the f i l l e d B r i l l o u i n zones of 50% of the t o t a l . We then f i n d that the average Fermi l e v e l has gone up by 5$ i n ^gn^ i o ' ^ e c o n t r i b u t i o n from the electrons i n the f i l l e d B r i l l o u i n zones would have to be substan-t i a l l y d i f f e r e n t from that of pure copper (increased to 70% of the t o t a l ) to give us the correct f i g u r e of 6%. A s u b s t a n t i a l d i f f e r e n c e i n the aniso-tropy of the c o n t r i b u t i o n from the core electrons i s perhaps more l i k e l y . This w i l l be discussed again fur t h e r on i n t h i s t h e s i s . 3. Angular C o r r e l a t i o n Results The r e s u l t s so f a r are not very conclusive. However, i f they are combined with our angular c o r r e l a t i o n r e s u l t s , the i n t e r p r e t a t i o n becomes somewhat c l e a r e r . Figure 22 shows the angular c o r r e l a t i o n r e s u l t s f o r two d i f f e r e n t c r y s t a l orientations of C u ^ A l ^ with the two curves normalized to each other using the c o l l i n e a r geometry r e s u l t s . There are d i s c o n t i n u i t i e s i n the slopes of these curves corresponding to the l l i lz.B. AoqJz.B. flOOJCutoff [llofz.B. [lie] Cutoff 2- Cutoff C o l l i n e a r Geometry f Copper 5.81 6.72 5.57 7.115 4 .91 C u 9 0 A 1 1 0 . 5.74 6.58 6.0 ± .1 7.00 5 .9 ± .1 7.8 + .2 38° + 6 C u 8 5 Z n 1 5 5.76 6.67 5 . 4 f .1 7.06 5 .3 ± .1 7.0 + ,2 30° + 5 C u 7 0 Z n 1 5 N i 1 5 5.81 6.72 7.115 25° + 7 C u 6 0 N i 4 0 5.74 6.64 5.4 4 .1 7.03 7.5 + .2 25° + 7 The a l l o y l a t t i c e constants are taken from Pearson, 1967. A l l values are i n m i l l i r a d i a n s unless otherwise noted. A l l copper values are taken from the known Fermi surface of copper. The e f f e c t s of experimental r e s o l u t i o n are not included i n the e r r o r s . Table 1 Experimental Results -100-edge of the Fermi surface f or these p a r t i c u l a r c r y s t a l d i r e c t i o n s (Majumdar, 1965) . Determining the l o c a t i o n of the c u t o f f s i n the angular c o r r e l a t i o n data i s made d i f f i c u l t by lack of knowledge of the exact shape of the a l l o y Fermi surface and by the large amount of core a n n i h i l a t i o n . The Fermi surfaces are expected to be roughly s p h e r i c a l and i f the core a n n i h i l a t i o n could be substracted a p l o t of (count rate) versus Q (see equation 5-3) might be useful i n loc a t i n g the c u t o f f s . . Because we could not subtract the core a n n i h i l a t i o n , t h i s method of determining the cut o f f s was u n r e l i a b l e and we chose instead to draw a smooth l i n e through our data points and then estimate the cutoffs from these l i n e s . A computer c a l c u l a t i o n of the e f f e c t of our experimental r e s o l u t i o n on the data f or a sp h e r i c a l Fermi surface (Figure 32B) indicates the apparent c u t o f f on our data f o r such a Fermi surface w i l l be s h i f t e d by 0.27 m i l l i r a d i a n s . We have therefore chosen the rather crude method of estimating the cutoffs as described and then substracting 0.27 m i l l i r a d i a n s from t h i s estimate. Throughout the remainder of t h i s work the errors quoted with respect to the values of these c u t o f f s w i l l be what we consider to be the uncertainty i n estimating the cut o f f s on our data. The error i n our c o r r e c t i o n for the e f f e c t s of the experimental r e s o l u t i o n i s unknown. We note, however, the fac t that our angular c o r r e l a t i o n data i s very c l e a r l y d i f f e r e n t f o r d i f f e r e n t c r y s t a l o r i e n t a t i o n s of the same a l l o y . This indicates not only that the Fermi surfaces of these a l l o y s i s aniso-t r o p i c but also that our experimental technique i s s e n s i t i v e to these anisotropies and that the e f f e c t s of our experimental r e s o l u t i o n should not -101-be too large. In a d d i t i o n to t h i s a p l o t of (count rate) versus 0 f o r the £100"} axis C u g Q A l 1 0 data y i e l d e d a f a i r l y good s t r a i g h t l i n e and gave good agreement with the method described above. Table 1 shows the values of the Fermi surface r a d i i obtained i n t h i s manner and compares these r e s u l t s to the p o s i t i o n s of the B r i l l o u i n zone boundaries obtained using x-ray d i f f r a c t i o n data f o r the l a t t i c e constants of the a l l o y s and making use of equation (5-1]. We note that the Fermi surface extends f u r t h e r out (by 3%) i n k-space i n the £ioo] d i r e c t i o n than the [ i l l ] B r i l l o u i n zone boundary i n the £lll] d i r e c t i o n . This i s i n apparent c o n t r a d i c t i o n of our c o l l i n e a r geometry r e s u l t s where the ^ i l l j neck was 4% higher than the Fermi surface i n the jlOOj d i r e c t i o n . Insofar as our simple theory i s v a l i d the discrepancy i s beyond experimental e r r o r . Taking an average of the Fermi surface r a d i i i n Table 1 i n the (lOoJ , jlicj , and {lllj[. d i r e c t i o n s we conclude that the average Fermi surface radius i s 8% larger than that of pure copper. Since we are only considering 3 c r y s t a l d i r e c t i o n s , we cannot state that t h i s i s r e a l l y beyond the 6% increase i n radius expected from the r i g i d band model. Figure 22 also i n d i c a t e s a second d i s c o n t i n u i t y i n the slopes of the angular c o r r e l a t i o n curves at 7.8 m i l l i r a d i a n s . This second c u t o f f i s not very sharply defined and i s not as d i s t i n c t as we w i l l f i n d i t to be i n other copper a l l o y s . 4. Discussion of the C u ^ A l ^ Results a. Charging of Conduction Electrons -102-As mentioned previously, the r e l a t i v e p o s i t i o n of the top of the' f i l l ] neck i n k-space depended on whether i t was measured using the c o l l i n e a r geometry method or the angular c o r r e l a t i o n method. An explana-t i o n f o r t h i s apparent c o n t r a d i c t i o n was suggested by Stewart, (1964). His study of LiMg a l l o y s using positron a n n i h i l a t i o n concluded that f o r k 2/3 k„„.TT, ™,,. I T. A r w there was an e f f e c t i v e decrease i n the density of r ZONE BOUNDARY e l e c t r o n i c s t a t e s . The explanation of t h i s experimental r e s u l t , l i e s i n the pile-up of conduction electrons around the impurity atoms as discussed i n Chapter 3. The electrons from states close to the Fermi surface s h i e l d the excess charge of the impurity atoms giving r i s e to the so - c a l l e d charging e f f e c t . These electrons are i n a region (close to the p o s i t i v e ion) from wlvich the pos i t r o n wave function i s r e p e l l e d and t h i s e f f e c t i v e l y decreases the density of states as viewed by the po s i t r o n . This change i n the e f f e c t i v e density of e l e c t r o n i c states would strongly a f f e c t our c o l l i n e a r geometry r e s u l t s , but would have no e f f e c t on the values of the c u t o f f s obtained using our angular c o r r e l a t i o n method. We can now see how these ideas f i t i n with the experimental r e s u l t s . The £11Ij neck i n the Fermi surface makes contact with the B r i l l o u i n zone boundary. Electrons i n t h i s neck w i l l not be as free to be scattered as electrons i n the /100 ( d i r e c t i o n where the B r i l l o u i n zone boundary w i l l i s that the neck electrons have mainly p - l i k e symmetry whereas the ^ 100^ " b e l l y " electrons have s - l i k e symmetry. This would cause them to sc a t t e r d i f f e r e n t l y o f f the ion cores. As discussed i n Chapter 3, s - l i k e electrons -103-w i l l s c a t t e r more strongly o f f the aluminum ion cores than p - l i k e electrons. From these arguments we would expect the e f f e c t i v e density of states to be less i n the 1^0o] d i r e c t i o n than i n the [ i l l ] d i r e c t i o n as i s consistent with our experimental r e s u l t s which imply a 7% average d i f f e r e n c e i n the e f f e c t i v e density of states between the two d i r e c t i o n s . b. T h e o r e t i c a l F i t to Experimental Data In the angular c o r r e l a t i o n experiment with axis flOol and gamma surface. As indicated i n Figure 31 the d i f f e r e n c e i n heights between the [no] and £ioq] d i r e c t i o n s i s only 3%. This i s a measure of the d e v i a t i o n of the " b e l l y " of the Fermi surface from a s p h e r i c a l shape. In order to see how close the angular c o r r e l a t i o n data was to the form predicted by equation (5-3) we subtracted a gaussian f i t t e d to the wings of the curve and f i t a polynominal to the part of the data a t t r i b u t e d to the conduction electrons. The r e s u l t s are shown i n Figure 33. The errors involved i n t h i s procedure are large as we do not know what the core a n n i h i l a t i o n i s l i k e at small transverse momenta. Any attempt, using equation (5-3), to v e r i f y d i r e c t l y i n a manner s i m i l a r to that of Stewart, (1964), the decrease i n the e f f e c t i v e density of states seems impossible since the subtraction of the core electron c o n t r i b u t i o n i s so uncertain. c. Core A n n i h i l a t i o n A s i g n i f i c a n t feature of the data i n Figure 22 i s the anisotropy ray d i r e c t i o n moving out along the " b e l l y " of the Fermi -104-° • • . » 0 T~ 5 ~ T ~ 6 FIG. 33 FIT TO ANGULAR CORRELATION DATA GAUSSIAN SUBTRACTED 25.3 e - 0 . 0 0 0 6 9 X 2 FITTEO EQUATION 7 8 O (MILLIRADIANS) 9 — i 10 C u 9 0 A ! I O Ooo] ^ 5 8 6 8 - 0.18 XI0" 4 X ? "4 0.94 X 10"" X 4 The equations r e f e r to a s c a l e i n which a s p h e r i c a l Fermi surface would have a zero c o e f f i c i e n t -105 -i n the positron a n n i h i l a t i o n with electrons at higher transverse momenta. This anisotropy varies from a 20% e f f e c t at 0 = 7.1 m i l l i r a d i a n s to a 43% e f f e c t at Q= 9.0 m i l l i r a d i a n s . Anisotropics i n the a n n i h i l a t i o n of positrons with core electrons i n pure copper have been observed by Sueoka, (1969) and Mijnarends, (1969). In p a r t i c u l a r Sueoka, (1969) found an anisotropy i n the core a n n i h i l a t i o n f o r P„„„.,T i n the 11001 1 r bRMl . «- J d i r e c t i o n i n pure copper. Our r e s u l t s f o r Cu^Al^g show a s i m i l a r e f f e c t i n d i c a t i n g presumably that the d-band electrons i n C u ^ A l ^ and i n copper have s i m i l a r e l e c t r o n i c configurations. Since only 10% of the atoms are not copper atoms, t h i s r e s u l t i s not too s u r p r i s i n g . d. Neck Widths One of our angular c o r r e l a t i o n experiments ( £llo] axis and Q-Tl^ gamma ray) should give us an i n d i c a t i o n of the £lllj necks i n CUggAl^g. Figure 23 shows the angular c o r r e l a t i o n curves i n an expanded scale f o r small (3 with the data on both sides of the 0 = 0 axis shown. We a t t r i b u t e the d i s c o n t i n u i t i e s i n the shape of t h i s curve to the side of the £^3L 1 lj neck. I f we assume that the sides of the ^ l l l j neck are i n f i n i t e l y steep, our experimental r e s o l u t i o n would have the e f f e c t of moving the apparent d i s c o n t i n u i t y i n our data out by 0.7 m i l l i r a d i a n s . Alowing f o r t h i s we conclude that the width of the [ill} neck i s 3.9 m i l l i r a d i a n s . I f we assume a c u t o f f for the neck along the [ i l l ] axis of 5.74 m i l l i r a d i a n s , t h i s corresponds to a neck subtending an angle of 40° ± 4° at the o r i g i n of the Fermi sphere. Our value for of the peak from the c o l l i n e a r . geometry r e s u l t s was 38° t 6°. A s i m i l a r close up i n Figure 23 of the -106-angular c o r r e l a t i o n data taken with /VlOOj axis and Jl00j| gamma ray d i r e c t i o n does not give any information about the necks. This analysis assumes that the admixture of higher momentum components into the e l e c t r o n i c wave functions as defined i n equation (4-1) i s not too large. The admixture of e l e c t r o n i c states across the [ i l l ] zone boundary could have a s i g n i f i c a n t e f f e c t on the c o l l i n e a r geometry r e s u l t s ( i t would decrease the apparent neck width) but would not a f f e c t the value of the neck width obtained from our angular c o r r e l a t i o n r e s u l t s . e. Dislocations The l i t e r a t u r e suggests that a large number of d i s l o c a t i o n s i n a c r y s t a l w i l l cause a s i g n i f i c a n t f r a c t i o n (/^20%) of the positrons to ann i h i l a t e with electrons having a small momentum (Dekhtyar, 1968). Such an a f f e c t would appear experimentally as a large number of counts at © ^ 3 m i l l i r a d i a n s i n our angular c o r r e l a t i o n curves. D i s l o c a t i o n s , created by the neutron bombardment of our samples are not evident i n our experimental r e s u l t s . No excessive counting rates at small <3 are present i n the data i n Figure 23. This type of d i s l o c a t i o n would also a f f e c t the apparent s i z e of the [ i l l } necks i n the angular c o r r e l a t i o n r e s u l t s but would not do so for the c o l l i n e a r geometry r e s u l t s . The agreement of our data on the ( i l l ] necks as obtained by the two d i f f e r e n t methods lead us to conclude that t h i s type of d i s l o c a t i o n a n n i h i l a t i o n , i f i t occurs, accounts f o r less than 5% of the positrons. -107-C u 8 5 Z n ! 5 1. Introduction Atomic zinc has two loosely bound 4s electrons and a closed 3 d ^ s h e l l below i t . In a manner s i m i l a r to the analysis of C-UggAl^Q we would expect there to be 1.15 conduction electrons per atom i n t h i s a l l o y . This corresponds to a 4.7% increase i n the average diameter of the Fermi surface over that of copper. Both copper and zinc have f i l l e d 3d bands and we would expect the theory of Stern, (1969) discussed i n Chapter 3 to be v a l i d f o r t h i s a l l o y . 2. C o l l i n e a r Geometry Results Figure 34 shows the c o l l i n e a r geometry r e s u l t s f o r a plane perpendicular to the j l i o j d i r e c t i o n i n CUg^Zn^,^. The £ of the £lllj[ peak i s 30° t 5°. The maximum anisotropy i n the CUg^Zn^^ data i s 7 1/2%. Another d i f f e r e n c e l i e s between the [31l] and jlOo] d i r e c t i o n s . This region i s almost f l a t i n Cu Q-Zn , showing very ob 15 l i t t l e structure compared to the copper and C U Q Q A I ^ Q r e s u l t s . I f we assume as before that there i s no change i n the c o n t r i b u t i o n i n the p o s i t r o n a n n i h i l a t i o n with the core electrons we f i n d that the average diameter of the Fermi sphere has decreased from that of copper. The data i n Figure 34 drops below that of copper i n the p-ioj d i r e c t i o n implying that e i t h e r the core a n n i h i l a -t i o n i n Cu Q Z n 1 c i s d i f f e r e n t from that of copper or that there, are ob l b less conduction electrons i n s i d e the Fermi surface. -108-• ® @ C U G 5 Z n ! 5 •Fig. 34 " ' . ' X X COPPER C o l l i n e a r Geometry Results I'or Cu Zn,, THEORETICAL COPPER -109-3. Angular C o r r e l a t i o n Results The r e s u l t s of the angular c o r r e l a t i o n experiments i n CUg^Zn^ normalized to each other from our c o l l i n e a r geometry r e s u l t s are shown i n Figure 25. There appear to be two d i s c o n t i n u i t i e s i n the slopes of each of the curves. The values of 0 , corrected f o r experimental r e s o l u t i o n , f o r which these c u t o f f s appear are l i s t e d i n Table 1. In t h i s case the c o l l i n e a r geometry and the angular c o r r e l a t i o n method r e s u l t s are consistent with respect to the r e l a t i v e heights of the Fermi surface i n k-space i n the j l l l " ] and £lOo] d i r e c t i o n , assuming that the f i l l ] neck ends at the B r i l l o u i n zone boundary. This i s i n agreement with Stern's (1969) theory discussed i n Chapter 3 as we s h a l l see l a t e r . 4. Discussion of Cu c Zn.. c Results ob l b As Table 1 shows, the average of the Fermi surface c u t o f f s i n the { i l l ] , / l i o j and (j-Ooj d i r e c t i o n s i s quite close to that of pure copper. This implies that the number of conduction electrons i n CUg^Zn^^ i s very s i m i l a r to that of pure copper. This w i l l be discussed at the end of t h i s Chapter and i n the a p p e n d i x . We mentioned previously that there was some evidence f o r a second c u t o f f i n the Cu-.Al..^ angular c o r r e l a t i o n at ^ = 7.8 m i l l i -l u radians. This second c u t o f f i s reasonably d i s t i n c t i n the Cu Zn data ob lb and occurs at 7.0 I .2 for both c r y s t a l o r i e n t a t i o n s . This i s beyond the [lOu] B r i l l o u i n zone boundary and within the j l i o ] zone boundary. Since i t l i e s beyond the Fermi surface and since the smearing of the k-vector i s - 1 1 0 -o n l y / ^ 2 % of k„ f o r alpha brass (Soven, (1966)), the cause of t h i s e f f e c t r must be of atomic (core e l e c t r o n ) o r i g i n . The anisotropy i n the a n n i h i l a t i o n f o r Figure 25 i s 11% at @ =8 m i T l i r a d i a n s . Because of the unexpected behavior of our angular c o r r e l a t i o n r e s u l t s with respect to the second c u t o f f , we were unable to make a . reasonable estimate as to the amount of core a n n i h i l a t i o n at s m a l l Q . A gaussian f i t t e d to the region between the two cutoffs i n a manner s i m i l a r to our analysis f o r Cu Al y i e l d s a core a n n i h i l a t i o n of /'v'80% of the y u i o t o t a l a n n i h i l a t i o n at Q = 0. This i s c l e a r l y too large and we conclude t h i s procedure i s not v a l i d f o r CUg^-Zn^. Figure 26 shows a part of the angular c o r r e l a t i o n data of an expanded s c a l e . Again as f o r C u ^ A l ^ , the d i s c o n t i n u i t i e s i n d i c a t e the width of the [ i l l ] neck i n the CUg^Zn^^ Fermi surface. The neck width obtained from the [no] axis c r y s t a l i s 3.2 m i l l i r a d i a n s where we have corrected for the e f f e c t s of our experimental r e s o l u t i o n as we did f o r the C U Q Q A I J Q data. This corresponds to an angle of 32° t 4° subtended by the neck at the center of the Fermi sphere. This i s consistent with the c o l l i n e a r geometry r e s u l t s which gave a jT of 30° f o r the [ i l l ] peak. The [l00*] axis curve i n Figure 26 also gives us a value f o r the width of the [ i l l ] neck since the plane containing the [no] and [lOo] axes also contains the [ l l l j d i r e c t i o n i n a face centered cubic c r y s t a l . Knowing that the angle between the [lio] and [ill] axes i s 35° we c a l c u l a t e a [ i l l ] neck width of 34° t tj making use of the dip at Q = 1.8 m i l l i r a d i a n s i n the curve i n Figure 23. This i s i n good agreement -111-with our other r e s u l t s . In t h i s c a l c u l a t i o n our experimental r e s o l u t i o n should not a f f e c t the r e s u l t s since the feature i n the curve was a dip whose angular p o s i t i o n would not be s h i f t e d by the r e s o l u t i o n of our apparatus. At © = 2.7 m i l l i r a d i a n s there i s a drop i n the curve i n Figure 26 corresponding to the other side of the [ i l l ] neck. Because t h i s drop i s not as sharply defined as the other d i s c o n t i n u i t i e s , no attempt was made to get the [ i l l ] neck width from i t . A rough extrapolation of the [ l io ] axis curve i n Figure 26 indicates that the [ i l l ] neck r i s e s 10% above the re s t of the curve. This agrees with our c o l l i n e a r geometry r e s u l t s and indicates i n a manner discussed i n the Cu A l i n t e r p r e t a t i o n , that no c r y s t a l damage due to the yu l u neutron bombardment i s present. C u 7 0 Z n 1 5 N i 1 5 1. Introduction As described i n the experimental section of t h i s t h e s i s , a l l our samples were enclosed i n pure aluminum holders and then neutron i r r a d i a t e d i n the AECL reactor at Chalk River. We discovered i n the course of our experiments that the a l l o y s containing n i c k e l seemed to shrink i n s i z e during neutron bombardment, thereby causing the samples to f a l l out of t h e i r holders. Even a f t e r a l t e r i n g the holders so that the c r y s t a l s could not f a l l out, there was s t i l l a tendency f or the samples to rotate about t h e i r axis of symmetry during the neutron i r r a d i a t i o n s . For t h i s reason the two Cu^Zn-^Ni^j. angular c o r r e l a t i o n experiments were performed with - 1 1 2 -the gamma ray d i r e c t i o n a long r a t h e r unusual c r y s t a l l o g r a p h i c o r i e n t a t i o n s . X - rays o f the c r y s t a l s a f t e r the exper iments had been performed showed no ev idence o f damage and the c r y s t a l s were back to t h e i r normal s i z e . However, we t h i n k tha t the on l y way these c r y s t a l s c o u l d f a l l out o f t h e i r h o l d e r s would be f o r them to change i n s i z e d u r i n g the neut ron i r r a d i a t i o n s . We had no problems o f t h i s na tu re w i t h a l l o y s c o n t a i n i n g no n i c k e l . Atomic z i n c has two 4s e l e c t r o n s and a f i l l e d 3d s h e l l below i t . N i c k e l has a f i l l e d 3d s h e l l and no 4s e l e c t r o n s . Acco rd ing to the r i g i d band model we t h e r e f o r e have one conduc t ion e l e c t r o n per atom i n t h i s a l l o y . The l a t t i c e spac ing o f t h i s a l l o y i s ve ry c l o s e to t ha t o f copper . T h i s would l ead us to expect a Fermi s u r f a c e f o r t h i s a l l o y which i s s i m i l a r to tha t o f pure copper . 2. R e s u l t s and D i s c u s s i o n F i g u r e 35 shows the r e s u l t s o f a c o l l i n e a r geometry exper iment i n the p lane p e r p e n d i c u l a r to the f l i o ] a x i s . The r e s u l t s f o r copper a re as u s u a l p l o t t e d on the same graph w i t h the tops o f the j j l l ] peaks no rma l i zed to each o t h e r . The j l l l j peak i n C i i g j - Z n ^ N i ^ has a £ o f 25° + 7 ° . The angu la r c o r r e l a t i o n r e s u l t s are shown i n F i g u r e 28. As u s u a l the two curves i n t h i s f i g u r e are no rma l i zed to each o the r u s i n g our c o l l i n e a r geometry r e s u l t s . The c u t o f f s a re r a t h e r smeared i n compar ison to our o the r r e s u l t s . With the assumpt ion t ha t the f i l l ] neck makes con tac t w i t h the B r i l l o u i n zone boundary , a -113-104-103-Ld jg 102. CD z; ZD O u 101-y IOO z: UJ 9 o — 99 o ^ o 98 97 CROSS SECTION! TO QlQ} ~0 10 2 0 3 0 ^0~^0™~60~7o~^ .90 CRYSTAL ORIENTATION ( D E G R E E S ) F i g . 35 . C o l l i n e a r Geometry R e s u l t s 1'or C u 7 0 Z n 1 5 K i 1 5 O O O C U 7 0 Z N I 5 N , I 5 ' -X X X C O P P E R - THEORETICAL COPPER - 1 1 4 -r o u g h e s t i m a t e o f t h e c u t o f f s i n d i c a t e s ' t ha t the average d iameter o f the Fermi s u r f a c e has i n c r e a s e d ^  10% over t ha t o f copper . The smear ing o f the c u t o f f s i s presumably due to a smear ing o f the wave v e c t o r k. I t seems reasonab le to assume tha t an a l l o y c o n t a i n -i n g two d i f f e r e n t impu r i t y atoms would e x h i b i t more smear ing o f the wave v e c t o r k than the o the r a l l o y s we examined. The smear ing o f the c u t o f f s makesthe d e t e r m i n a t i o n o f the second c u t o f f p a r t i c u l a r l y d i f f i c u l t - . F i t t i n g a gauss ian to the d a t a between 6 and 8 m i l l i r a d i a n s i n F i gu re 28 would imply a core a n n i h i l a -t i o n o f 70% at © = 0 which i s s i m i l a r t o the CUg<_Zn^ and C u ^ N i ^ r e s u l t s i m p l y i n g tha t a second c u t o f f e x i s t s and t ha t an e f f e c t s i m i l a r to t ha t i n these two a l l o y s i s o c c u r r i n g . A t 0 = 6.6 m i l l i r a d i a n s the a n i s o t r o p y i n the a n n i h i l a t i o n between the two c r y s t a l o r i e n t a t i o n s i s 18%. I t decreases to 10% at 0= 9 .0 m i l l i r a d i a n s . There i s no s t r u c t u r e we can r e l a t e to necks i n the Fermi s u r f a c e . C u 6 0 N i 4 0 ' 1. I n t r o d u c t i o n As mentioned p r e v i o u s l y , a tomic n i c k e l has a c l o s e d 3d s h e l l and no 4s e l e c t r o n s . Acco rd ing to the r i g i d band model we would expect a Fermi s u r f a c e whose volume con ta ined 0.6 o f the e l e c t r o n s found i n s i d e the copper Fermi s u r f a c e . Because o f t h i s s h r i n k i n g o f the Fermi sphere we would expect the Fermi s u r f a c e o f C u , n N i . n t o become more i s o t r o p i c . I t ( -115-would seem l i k e l y t ha t the £l 1 l j necks would v a n i s h a n d t h a t there would be no con tac t o f the Fermi s u r f a c e w i t h the [ i l l ] B r i l l o u i n zone boundary. 2 . R e s u l t s and D i s c u s s i o n F i g u r e 36 shows the c o l l i n e a r geometry r e s u l t s i n a p lane p e r p e n d i c u l a r to the [l io] d i r e c t i o n . The r e s u l t s are very s i m i l a r t o the C u 7 0 Z n 1 5 N i 1 5 da ta except t ha t the a n i s o t r o p y i n the C u ^ N i ^ da ta i s s l i g h t l y reduced . . A compar ison o f the Cu^ N i ^ and C u ^ Z n ^ N i ^ r e s u l t s i s shown i n F i gu re 37. The { i l l ] peak £ f o r C u 6 Q N i 4 0 i s 25° i 7 . Without our angu la r c o r r e l a t i o n r e s u l t shown i n F i gu re 30 , the Cu N'* da*a would be p a r t i c u l a r l y d i f f i c u l t to a n a l y s e . F i gu re 30 shows two c u t o f f s , whose v a l u e s c o r r e c t e d f o r exper imen ta l r e s o l u t i o n are l i s t e d in Tab le 1. For a Fermi s u r f a c e c o n t a i n i n g on l y 60% of the e l e c t r o n s found i n the copper conduc t ion band, we would expect a Fermi s u r f a c e w i th a d iameter 18% l e s s than tha t found i n copper . I n s t e a d , i n the [loJJ d i r e c t i o n we find a c u t o f f on l y 2% l e s s than t ha t f o r pure copper . Th i s i s c o n s i s t e n t with our o ther r e s u l t s and w i l l be d i s c u s s e d l a t e r i n the a p p e n d i x . In s p i t e o f the r a t h e r s m a l l (3%) a n i s o t r o p y found i n the c o l l i n e a r geometry da ta i n F i gu re 36 , there i s some s t r u c t u r e a t t r i b u t a b l e to the [ i l l ] neck i n the angu la r c o r r e l a t i o n d a t a . In a procedure s i m i l a r to t ha t used f o r j jOu] a x i s curve f o r C u g 5 Z n 1 5 we have a d i p i n the da ta at 0 = 2.7 m i l l i r a d i a n s . Th i s i s not w e l l enough d e f i n e d f o r us to o b t a i n a neck width from i t , b u t i t s existe.nce together with the" other data suggests contact with t h e B r i l l o u i n zone boundary. -116-Ifio] aooj CROSS SECTION ± TO [IlO] F i g . 36 0 10 20 S ^ c J ^ 5 C ~ 6 0 70. 80 90 . CRYSTAL ORIENTATION (DEGREES) . . " © @ © C u g Q N i ^ X X X C O P P E R THEORETICAL COPPER C o l l i n e a r Geometry R e s u l t s f o r cu- Ni bO 40 -117-® © C u 7 0 Z n 1 5 N i 1 5 X * C u 6 0 N i 4 0 UJ < t r CD i— ZD O O 102 i 1 0 1 -100-9 9 -9 8 -9 7 ' (110) y (III) rl ® X X<?> X (311) ( 1 0 0 ) R E S U L T S C R O S S S E C T I O N J_ TO (110) 0° 10° 20° 30° 40° 50° 6 0 ° 70° 80° 9CT CRYSTAL ORIENTATION . F i g . . 37/ COMPARISON OF C u 7 0 N i l 5 Z n l 5 and C u 6 0 N i 4 0 -118-One other feature of this data is a bump in the angular correlation data at C3 = 8.0 milliradians. This bump rises 15% over a straight line extrapolation of the higher transverse momentum part of the curve and is presumably due to some feature of the core electrons. Summary 1. Introduction In the previous pages many of the features of the data have been discussed. We w i l l now try to formulate a qualitative overall picture of the results. 2. Band Structure Effects It is evident from Table 1 that the Fermi surface r a d i i of Cu^nNi^Q and Cu^Zn^Ni^i- seem larger than we would have expected from the r i g i d band model. For Cu^Al.^ the Fermi surface diameter is about what the r i g i d band model predicts whereas the "Fermi surface radius in Cu Zn ob l b is too small. We consider that the CUg^Zn^^ and C u ^ A l ^ results are consistent with the theoretical predictions of Stern, (1969) discussed in Chapter 3. The extra electron contributed by the zinc atom wi l l remain localized about the zinc impurity, which implies that the number of conduction electrons is v i r t u a l l y unchanged from that of copper. C u ^ A l ^ does not exhibit the band structure effects discussed in Chapter 3 and the aluminum contributes -119-alraost as many conduction electrons to the Fermi sea as predicted by the r i g i d band model. These ideas are borne out by the anisotropy i n the e f f e c t i v e density of states observed f o r C u ^ A l ^ and the absence of t h i s e f f e c t i n CUg^Zn^^. As discussed i n Chapter 3 the p - l i k e [ i l l ] neck electrons w i l l be more strongly scattered i n Cu^^Zn^^ than the s - l i k e £l0uj " b e l l y " electrons and the exact converse w i l l occur i n C u ^ A l ^ . The contact of the Fermi surface with the [ i l l ] zone boundary w i l l oppose t h i s s c a t t e r i n g , however, as discussed f o r C u n n A l i n . The r e s u l t , as discussed f o r Cu o rZn 1 [-yU 1U ob l b appears to be an equal amount of charging by the j j o o ] and [ i l l ] electrons i n contrast to the s i t u a t i o n discussed i n the C u ^ A l ^ se c t i o n . The s i t u a t i o n for Cu, nNi._ and C u _ _ Z n 1 r N i i r i s not quite as 60 40 70 15 15 n c l e a r cut. The bands discussed i n Chapter 3 are not f u l l . Stern, (1969) predicts that noble metal a l l o y s i n which the bands are not f u l l w i l l have more l o c a l i z e d conduction electron s h i e l d i n g than f o r noble metal a l l o y s containing zinc. The large number of conduction electrons indicated by the r a d i i of the Fermi surfaces f o r C u ^ N i ^ and f o r Cu^^Zn^^Ni^ imply that some of the d-band states may be empty. These a l l o y s are more concentrated than CUgrZn^[. and C u ^ A l ^ g and i t appears that large changes i n the band structure have occurred p l a c i n g the d-bands e n e r g e t i c a l l y above the conduction band. Second Cutoff A consistent feature of our angular c o r r e l a t i o n data i s the presence of two discontinuities i n the slopes of the curves. This second -120-c u t o f f seems to be independent of c r y s t a l o r i e n t a t i o n and B r i l l o u i n zone boundaries. Since higher momentum components i n the conduction electron wave functions should be dependent on these quantities we t e n t a t i v e l y conclude that t h i s feature of our data i s associated with the core electron a n n i h i l a t i o n . Charging of the ions, due to s c a t t e r i n g of conduction electrons o f f the impurity atoms was suggested experimentally by Stewart, (1964) and discussed t h e o r e t i c a l l y by Stern, (1966) . The p r o b a b i l i t y of p o s i t r o n a n n i h i l a t i o n with such electrons i s complicated by the f a c t that they are i n a region frcm which the p o s i t r o n i s repulsed. In any event the scattered conduction elements do not have any momenta larger than P„ .. However, J ° Fermi s c a t t e r i n g of these electrons o f f n i c k e l or zinc w i l l presumably be d - l i k e and a mixing of these states with the other d-states seems l i k e l y . The conduction electrons should also screen the nucleus, allowing the p o s i t r o n wave function to penetrate further into the ion core. The aluminum a l l o y would not exhibit t h i s e f f e c t as strongly since conduction electron s c a t t e r -ing o f f t h i s atom w i l l be s - l i k e , allowing less mixing i n with higher momentum states. Therefore, p o s i t r o n a n n i h i l a t i o n with these higher momentum states w i l l not be increased as much for aluminum a l l o y s as for a l l o y s containing zinc or n i c k e l . These speculations are of a q u a l i t a t i v e nature only but the data i s consistent with these ideas. -121-Further Discussion . '• ' Figures 38 and 39 show a comparison of angular c o r r e l a t i o n curves for s i m i l a r c r y s t a l o r i e n t a t i o n s of d i f f e r e n t a l l o y s . We have normalized the peaks of the curves to each other although the d i f f e r e n t amounts of charging of the impurity atom may render t h i s type of normalization meaning-l e s s . The C u 7 Q Z n 1 5 N i 1 5 data i s included i n these figures because our c o l l i n e a r geometry indicated that t h i s Fermi surface i s close to i s o t r o p i c . Therefore the f a c t that the gamma ray d i r e c t i o n s were along d i f f e r e n t c r y s t a l o r i e n t a t i o n s from those of the other a l l o y s i s not important. Figure 38 indicates that along the flOcTj a x i s , the C u ^ N i ^ curve bulges out above the others. The c o n t r i b u t i o n from the electrons at 0^8.2 m i l l i r a d i a n s i s s i m i l a r f or a l l the a l l o y s except C u ^ N i ^ . We should perhaps use t h i s p o r t i o n of the a n n i h i l a t i o n as normalization point f o r the d i f f e r e n t a l l o y s . The C u ^ N i ^ curve i n Figure 38 would then f a l l considerably below the other curves. We could then a t t r i b u t e t h i s apparently low density of e l e c t r o n i c states to a large charging e f f e c t due to the large number of n i c k e l ions present i n t h i s a l l o y s . However, i t i s also p o s s i b l e that the a n n i h i l a t i o n of positrons with core electrons i s very d i f f e r e n t f o r C u ^ N i ^ than from the other a l l o y s since there are such a large number of impurity atoms present. In our analysis we have not considered any e f f e c t s due to a renormalization and a smearing of the electron wave vector caused by a lack of c r y s t a l symmetry i n random a l l o y s as postulated by Stern, (1966). This smearing of the electron wave vectors would smear out the Fermi - 1 2 2 -1.0 .9-.8 h-< CC CD -6 1-O 5. LU O U J . 9 4 o o o • 3H .2 0 ®° ® s t a t i s t i c a l e r r o r at e=0 ®X • •. X X X CUggZn^ Ooo] AXIS _ y-RAY DIRECTION -Qiol ®X 9 • . . C U g ^ i ^ Q00J" AXIS ®X © X-RAY DIRECTION- [Ho] ®X « ® ® ® C ^ Z n ^ N i ^ QOQlAXlS ® ODD X-RAY DIRECTION ® X o ® X o ® X® • x@ • X • ®x« • x® x§>, a ° » • 1 1 1 1 1 r~ 1 1 r~ 0 . 1 2 3 4 5 6 7 8 9 0 ( M I L L I R A D I A N S ) FIG. 38 C O M P A R I S O N O F C U Q 5 Z N | 5 , C U 6 0 N I 4 Q , A N D C U ^ N J 5 N i | 5 RESULTS. CO d-P ct-H-CO ct-H* O P H fl> »-< O »-J P o >. 0 -n 1 s T) > CO o o o c (X) o o o c 00 (Jl N > O o c 3 N Cji cn XJ m co c CO o OH CD r . £ <-o > •M1 COINCIDENCE COUNTING RATE J > | '•£> cn CT) ^| CO O X X© ® X X o ® X X _ » © ® ®e x» © x © x ® X o ® . ® ® ...® © Xs, x> X o 5R x« x ° ® ® ® © X © X X _ Q ® X ° .® © e x ® * x © 0 X o o 2 8 > o c 00 N - ' O Co x 5 i TO 5 I 5 © © I i -124-s u r f a c e c u t o f f s and shou ld e x p l a i n why the Cu-^Zn., , - N i . ^ c u t o f f s are so washed ou t . On the o ther hand the Cu ,_N i . „ c u t o f f s see inaoored is t inc t o U 4 U so t ha t the s i t u a t i o n i s r e a l l y not ve ry c l e a r on t h i s p o i n t . In c o n c l u s i o n we would s t a t e tha t the l a rge amount o f core a n n i h i l a t i o n has made t h i s a n a l y s i s r a t h e r d i f f i c u l t . However, the c l e a r dependence of our r e s u l t s on c r y s t a l o r i e n t a t i o n i n d i c a t e s tha t the p o s i t r o n a n n i h i l a t i o n techn ique used shows c o n s i d e r a b l e promise f o r the s tudy o f d i s o r d e r e d a l l o y s i n the f u t u r e . CHAPTER 9 Conclusion This has been a preliminary i n v e s t i g a t i o n of a few selected copper a l l o y s . A few t e n t a t i v e conclusions have been reached and are discussed i n Chapter 8.' In t h i s type of general survey, i t i s to be expected that many questions have remained unanswered. One of the l i m i t i n g f a c t o rs i n t h i s project was the lack of money f o r neutron i r r a d i a t i o n s of our samples, which cost about $300 each. This was one of the primary reasons why more c r y s t a l orientations of each of our samples were not studied. In s p i t e of the preliminary nature of t h i s work some points of i n t e r e s t have been discovered and i t i s c l e a r that the method shows considerable promise. Anisotropies i n the a n n i h i l a t i o n of higher momentum electrons have been discovered i n the a l l o y s Cu A l . . , Cu Zn and 90 10 85 15 C u ^ Z n - ^ N i ^ . Variations i n the e f f e c t i v e density of e l e c t r o n i c states seem present i n C u q n A l i n and agree with the t h e o r e t i c a l predictions of -126-Stern, (1969). The cutoffs found by us and l i s t e d i n Table 1 give indications of the number of conduction electrons i n the various a l l o y systems. Possible explanations for the second c u t o f f observed i n the a l l o y s have been discussed i n Chapter 8. Since we now know what sort of data to expect from a l l o y systems, we are i n a p o s i t i o n to improve our apparatus. As discussed i n Chapter 6, a sodium-22 positron source and a magnet f o r focussing the positrons have recently been added to our apparatus. The number of Nal(Tl) detectors i s being increased and equipment f o r moving the equipment at one end sideways i s under construction. Angular c o r r e l a t i o n data on pure copper i s needed to enable us to compare t h i s data to that of the a l l o y s already studied. Several d i f f e r e n t c r y s t a l o r i e n t a t i o n s of some of the a l l o y s already studied are planned. In the s l i g h t l y more di s t a n t future copper a l l o y s such as copper gallium i n which, according to the r i g i d band model many electrons per impurity atom are added to the conduction band should y i e l d i n t e r e s t i n g r e s u l t s . Systems i n which core a n n i h i l a t i o n i s small, such as l i t h i u m and berylium, are well suited f o r study using the sodium-22 source. We expect the anisotropics here to be larger and these materials should lend themselves to numerical c a l c u l a t i o n s such as those done by Stroud and Ehrenreich, (1968). Appendix A c c o r d i n g to Majumdar (1965) t h e r e s h o u l d e x i s t a d i s c o n t i n u i t y i n the s l o p e s o f our a n g u l a r c o r r e l a t i o n data at the p o s i t i o n c o r r e s p o n d i n g to the edge o f the Fermi s u r f a c e . In p r a c t i s e the 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 o f t h i s p o i n t i s made d i f f i c u l t by the e f f e c t s o f e x p e r i m e n t a l r e s o l u t i o n and core a n n i h i l a t i o n . We w i l l now c o n s i d e r i n d e t a i l the d a t a o b t a i n e d for the [110] a x i s CUg^Zn^^ data to i l l u s t r a t e the methods used and the type o f d a t a o b t a i n e d f o r C u 8 5 Z n i 5 a n c ^ Cu60N"*"40* F i g u r e 40 shows the data o b t a i n e d f o r the [HO] a x i s C u n r Z n , r on an expanded s c a l e . The s t a t i s t i c a l e r r o r s are about O D 15 the s i z e of the e x p e r i m e n t a l p o i n t s . The s t r a i g h t l i n e through the p o i n t s r e p r e s e n t s a v i s u a l f i t to the data. There appear to be two d i s c o n t i n u i t i e s i n the s l o p e of the data a t A and C. Assuming t h a t the f i r s t d i s c o n t i n u i t y a t A i s due to the edge of the Fermi s u r f a c e , then the c o n t r i b u t i o n c a u s i n g the second d i s c o n t i n u i t y a t C i s not known. The o r i g i n o f the c o n t r i b u t i o n between A and C does have some b e a r i n g on the l o c a t i o n of the c u t -o f f s determined by us. I f the r e g i o n AC was due to our e x p e r i m e n t a l r e s o l u t i o n then the Fermi s u r f a c e c u t - o f f would occur a t B. I f the r e g i o n AC r e p r e s e n t s some s l o w l y v a r y i n g background c o n t r i b u t i o n not due t o the c o n d u c t i o n e l e c t r o n s i n the f i r s t B r i l l o u i n zone, then the cut«-off would o c c u r a t A. C a l c u l a t i n g the e f f e c t s due to e x p e r i m e n t a l r e s o l u t i o n i s made d i f f i c u l t by the unknown shape o f the a l l o y Fermi s u r f a c e . F i g u r e 41 p r e s e n t s an a n a l y s i s due to p'etijevich,(196 8) i n which the e f f e c t s o f e x p e r i m e n t a l r e s o l u t i o n are accounted f o r i n pure copper -128-f o r the [110] axis [ i l l ] Y~ r aY c r y s t a l o r i e n t a t i o n . This a n a l y s i s assumes the simple Sommerfeld theory and neglects core e l e c t r o n a n n i h i l a t i o n . The region AC i n Figure 40 occurred over a distance o f about 1.5 m i l l r a d i a n s . Figure 41 does not have any region s i m i l a r to AC. This would imply that, i f our simple theory i s v a l i d , the region AC i s not due to the conduction electrons in' • the f i r s t B r i l l o u i n zone and that the Fermi surface c u t - o f f corresponds to the p o s i t i o n A since the theory previously discussed i n d i c a t e d that the smearing of the Fermi surface i n alpha brass was small (^2% of k^). In.addition to t h i s we note that according to equation (5-3) a p l o t of (counting r a t e ) 2 versus 9 2 should y i e l d a s t r a i g h t l i n e , the i n t e r c e p t of which corresponds to the edge of the Fermi surface f o r a s p h e r i c a l Fermi surface. I f we add a small constant background of height K to equation (5-3) and then square both sides we obtain [ n ( p z ) ] 2 - r^2 - pj + 2K/p^ - p2* + K 2 We w i l l neglect the 2K/JP5 - p| term since i t i s small. The above analysis again assumes the simple Sommerfeld theory to be v a l i d . In a d d i t i o n the a l l o y Fermi surfaces we are studying are not s p h e r i c a l and the c o n t r i b u t i o n from the core electrons i s not w e l l known. However, a p l o t of (count r a t e ) 2 versus 6 2 i n Figure 42 y i e l d s a f a i r l y good s t r a i g h t l i n e . Such a p l o t i s not as strongly dependent on experimental r e s o l u t i o n as our other attempts to obtain the Fermi surface c u t - o f f although i t s u f f e r s from the above named disadvantages. The s t r a i g h t l i n e -129-th rough the points represents a v i s u a l f i t to the data and indicates a Fermi surface c u t - o f f in.good agreement with A i n Figure 40.. I t i s not i n good agreement with B. As previously discussed, a gaussian f i t to the region between the two cut-offs for Cu o rZn,, and Cu, (.Ni„ n i n d i c a t e d 85 15 60 40 core a n n i h i l a t i o n s of about 80% at the 0 = 0 p o s i t i o n of our angular c o r r e l a t i o n data. Recent angular c o r r e l a t i o n data obtained by us for pure copper did not appear to have a second cu t - o f f and a gaussian f i t to the t a i l of the curve in d i c a t e d a core a n n i h i l a t i o n of about 40% at 9 = 0. This leads us to the conclusion that the Fermi surface c u t - o f f corresponds to point A i n Figure 40 and that the AC contribution i s due to some unknown background e f f e c t . The cutoffs obtained by the two methods are i n good agreement with each other for CUg^Zn^, but the plot of (count rate) versus gPVor this a l l o y i s somewhat ambiguous i n view of the marked anisotropy indicated by Figure 24. The Cu^Ni^Q res u l t s (Figure 29) and the jiooj axis C U ^ Q A I - ^ Q r e s u l t s did not appear to be so anisotropic and such a plot might be more useful here. Figures 43 and 44 show such a plot. The indicated cutoffs are i n good agreement with Table 1. -130-.5 » \ I \ .4 Figure 40 C u 8 5 z n 1 5 Cutoff © . 3 bO c! •H ri o o o rt .2 -cf •H O « •H O O V \ v x c .1 5 6 7 8 © . ( m i l l i r a d i a n s ) -131-2 4 6 ^ (milliradians) Figure 42 ^ ^ (Count Rate) versus 0 for Cu^j-Zn^ { l l o ] Axis 1 X X K X X X, x. f X X X X X X ^ \ xv Corresponding Cutoff 5.4 m i l l i r a d . X v ( Background x Level' \ X x X 20 iQ (milliradians) F i g u r e 43 (Count Rate,) v e r s u s © f o r [lOOJ A x i s Cu^Ni^g X 1 N 1 . Nx C o r r e s p o n d i n g C u t o f f 5.5 N i N m i l l i r a d . Background %\ <? L e v e l v. 9. * 2 0 » 0 ( m i l l i r a d i a n s ) F i g u r e 44 (Count R a t e v e r s u s 0 f o r [ l O o j A x i s C U g 0 A l 1 0 N 1, K \ \ \ X \ \ * ^ C o r r e s p o n d i n g v C u t o f f 5 . 9 Background T,eveSL Appendix 77T The c a l c u l a t i o n s on which the c u r v e s shown i n F i g u r e 32A are; based n e g l e c t e d a l l h i g h e r momentum components i n the e l e c t r o n i c wave f u n c t i o n s . T h i s assumption i s o f d o u b t f u l v a l i d i t y , p a r t i c u l a r l y f o r those e l e c t r o n i c s t a t e s near the B r i l l o u i n zone boundary. I n t h i s appendix we w i l l d i s c u s s these h i g h e r momentum components i n g r e a t e r d e t a i l . F o r copper and copper based a l l o y s i n which t h e r e i s a neck i n the [ i l l } d i r e c t i o n making c o n t a c t w i t h the 3 r i l l o u i n zone boundary, the dominant term i n the e x p a n s i o n over r e c i p r o c a l l a t t i c e v e c t o r s o f the neck e l e c t r o n wave f u n c t i o n s w i l l be components c o r r e s p o n d i n g to the [ i l l ] r e c i p r o c a l l a t t i c e v e c t o r s . F u j i w a r a and Sueoka, (1967) d i s c u s s the a p p l i c a t i o n o f the n e a r l y - f r e e e l e c t r o n model to the case o f a Fermi sphere making c o n t a c t w i t h a B r i l l o u i n zone boundary. The r e s u l t o f such an a p p l i c a t i o n i n which the e x p a n s i o n (equation(4-1) ) i s stopped a f t e r i n c l u d i n g one r e c i p r o c a l l a t t i c e v e c t o r ( t h e [ i l l ) r e c i p r o c a l l a t t i c e v e c t o r f o r copper) i n d i c a t e s t h a t some o f the e l e c t r o n s w i l l occupy s t a t e s i n the n e x t B r i l l o u i n zone as shown B r i l l o u i n Zone Face Fermi S u r f a c e f x 1 - 1 3 6 -w h e r e ^ f ^ j i s the e l e c t r o n i c d e n s i t y such that a density of 2 i n d i c a t e s f u l l occupation of s t a t e s and ^rvn 114/ v/here G i s a r e c i p r o c a l l a t t i c e vector perpendicular to the zone face and JV^ | i s the G-th F o u r i e r component of the p o t e n t i a l energy of an e l e c t r o n . I f X/j^ -jwere known f o r the copper a l l o y s t h i s theory could be used to give an i n d i c a t i o n of the occupation of higher momentum s t a t e s . I t should be mentioned, however^ that the p o s i t r o n a n n i h i l a t i o n r e s u l t s of Fujiwara and Sueoka, ( 1 9 6 7 ) d i d not agree with t h i s theory f o r pure copper i n which the magnitude of V^/jjis known. In any event i t i s c l e a r that t h i s e f f e c t w i l l tend to round out the top of the necks shown i n Figure 3 2 A ? out i t w i l l not reduce the apparent s i z e of the necks, since as long as the c y l i n d e r shown i n Figure 9 i s p o i n t i n g along the [ i l l ] d i r e c t i o n , a t r a n s l a t i o n o f e l e c t r o n s along t h i s c y l i n d e r i n momentum space w i l l not a f f e c t the counting r a t e . I t would re q u i r e other higher momentum components corresponding to t r a n s l a t i o n through d i f f e r e n t r e c i p r o c a l l a t t i c e vectors to change t h i s counting r a t e . -137-BIBLIOGRAPHY Amar, H., Johnson, K.H., and Sommers, C.B., (1967), Phys. Rev. 153, 655. 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