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High resolution positron annihilation study in the -phase region of the copper gallium and copper germanium… McLarnon, James Gordon 1973

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A HIGH RESOLUTION POSITRON ANNIHILATION STUDY IN THE a-PHASE REGION OF THE COPPER GALLIUM AND COPPER GERMANIUM SYTEMS by  JAMES GORDON McLARNON B.Sc, University of Alberta, 1963 M.Sc,  University of Alberta, 1965  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Physics We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA July, 1973  In presenting  this thesis i n p a r t i a l fulfilment of the requirements for  an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference  and study.  I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may by his representatives.  be granted by the Head of my Department or  It i s understood that copying or publication  of this thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission.  Department The University of B r i t i s h Columbia Vancouver 8 , Canada  ABSTRACT A. high resolution positron annihilation angular distribution study has been made of a sequence of ct-phase alloys, of copper gallium I and copper germanium up to electron per atom ratios of about n =  1.25.  While the primary study has concerned the variation of the radius of the neck features of the Fermi surface with electron concentration, data has also been obtained for the <110>  Fermi cut-off for a l l alloys  and i n addition a <100> cut-off has been obtained for the highest concentration copper gallium alloy. The two concentrated alloy systems show different behavior which i s i n agreement with low concentration studies done on CuGa and CuGe by Coleridge and Templeton.  The CuGa neck radius i f found to increase at  a more rapid rate than predicted by r i g i d band theory, particularly at the highest concentration studied, whereas the CuGe behavior i s below the r i g i d band predictions. General consistency i s obtained between the neck variation and the <110>  cut-off change.  For the most concentrated  CuGa a l l o y the results allow us to sketch a tentative Fermi surface and also provide convincing proof that the Fermi surface i s most unlikely to contact the (200) Brillouin zone boundary even at the limit of the a-phase.  This refutes the explanation given by Hume-Rothery and Roaf to  account f o r the occurrence of the Hume-Rothery rules. The present results are compared with the other existing experiment a l data for these alloys, in particular the optical absorption data of Montgomery and Pells, and i t is concluded that, while i t i s not possible to make any definitive theoretical statements from the present data, the two different measurements provide complementary details of the alloy band  Ill  structure which would serve as an excellent test of any theory. The recent calculations of Das and Joshi using the coherent potential model has had reasonable success i n explaining the optical data for the a-phase of the copper zinc system and i t i s hoped that the present work w i l l stimulate the application of such theory to the CuGa and CuGe systems.  To my mother and father  TABLE OF CONTENTS Page List of Tables  viii  List of Figures  ix  CHAPTER I:  INTRODUCTION  CHAPTER I I :  1  ELECTRONS IN METALS  6  A.  Nearly Free Electron Theory  B_.  Orthogonalized Plane Wave Theory  11  C.  Augmented Plane Wave Theory  13  CHAPTER I I I :  ELECTRONS IN ALLOYS  6  19  A.  Introduction  19  B.  Rigid Band Model  20  C.  Tight Binding Method  23  D.  Treatment of Alloy Potentials  25  (i) (ii) (iii) E_.  Average Potential  25  T Matrix  25  Coherent Potential  26  Other Methods Applicable (i) (ii) (iii)  to Disordered Alloys  29  Electronic Specific Heat  29  Optical Measurements  30  Inelastic Neutron Scattering  32  -viPage  CHAPTER IV:  THE POSITRON ANNIHILATION EXPERIMENT  33  A.  Introduction  33  IS.  Angular Distribution of Gamma Radiation  33  CI.  Detector Geometries  35  Wide S l i t Geometry  35  P o i n t Geometry  39  Crossed S l i t Geometry  41  Present Geometry  41  I).  Thermalization of the Positron  42  E-.  Enhancement  42  F_.  Previous Studies  43  (i) (ii)  (iii) (iv)  CHAPTER V:  POSITRON ANNIHILATION FACILITIES  .' 49  A.  Introduction  49  B.  Samples Studied  49  C_.  Detector Arrangement  52  D_.  Electronics  54  (i) (ii)  Pre-Amp D i s c r i m i n a t o r  54  Coincidence Box  57  E_.  R e s o l u t i o n E f f e c t s and Detector Considerations  58  F.  Chance Coincidence  60  (5.  Neutron Damage  60  -viiPage CHAPTER VI:  EXPERIMENTAL ANALYSIS  62  A.  Theoretical Resolution  66  B_.  Copper Analysis  72  C_.  Copper Alloy Analysis  72  (i) (ii) D.  Smooth Curve Behavior  73  Difference Curve Behavior  73  Core Effects  CHAPTER VII:  INTERPRETATION AND CONCLUSIONS  92 96  A.  Introduction  96  B_.  Rigid Band Theory  96  C_.  Alloy Fermi Surface  105  p_.  Discussion  106  E_.  Optical Studies  112  F.  Electronic Specific Heat Studies  114  G_.  Hume-Rothery Rules  115  H_.  Summary  116  Appendix  118  Bibliography  120  -viiiLIST OF TABLES Page Table 1  Electron per Atom Ratio for Several Alloys  21  Table 2  Previous Measurements of Copper Alloy Neck Radii  48  Table 3  Neck Radius Results - Difference Curve and Direct Analysis  90  Table 4  Density of States in Copper for Various Energies  98  Table 5  Rigid Band Energies for Different Electron per Atom Ratios  98  Rigid Band Prediction and Experimental Data  98  Table 6  —ixLIST OF FIGURES Figure  Page  1  Energy Gaps i n k Space  2  Copper Fermi Surface  3  Muffin Tin Potential  4  Copper Band Structure  18  5  Density of States Versus n (Rigid Band Theory)  21  6  Optical Transitions i n Copper  31  7  Momentum Conservation i n Positron Annihilation  34  8'A'  Wide S l i t Geometry  37  Sampling Region for Wide S l i t Geometry  37  Point Geometry  38  Sampling Region for Point Geometry  38  Enhancement Effect on Electron Density  44  Electron-Electron Effects on Electron Density  44  11  Block Diagram of Apparatus  50  12  Arrangement of One Set of Detectors  53  13  Pre-Amp and Shaper Circuit  55  14  Coincidence Circuit  15  Resolution Function Folded into Copper Fermi Surface  16  Resolution Function Folded into Copper Fermi Surface  ' B' 9'A' 'B* 10'A' 'B*  - Core Included  10 ' '  .  10 14  56 64  65  Figure  Page  17'A',*B','C'  Copper Angular Distribution Data  18  Folded Copper Data and Neck Radius  19  Copper <110>  Cut-Off  69 70 •'  71  20-25  Reck Radius Data for Alloys  74-79  26-31  <110>  80-85  Cut-Off Data for Alloys  32  CuGa (12.7) <100> Cut-Off  86  33  Neck Radius Results  87  34  <110>  Cut-Off Results  88  35'A'  Difference Curves for CuGe  89  'B*  Difference Curves for CuGa  89  36  Previous Copper Aluminum Data with Present Trends for CuGa, CuGe  37  94  Previous Copper Zinc Data with Present Trends for CuGa, CuGe  95  38  Energy Band Diagram for Neck Radius and <110>  39  Neck Radius Results and Rigid Band Theory  100  40  <110>  101  41  Energy Band Diagram for'<100> Cut-Off  102  42  CuGa Fermi Surface Contours  107  43  CuGe Fermi Surface Contours  108  44  Optical Transitions in Copper  111  45  Experimental Resolution Function  119  Cut-Off Results and Rigid Band Theory  Cut-Off  99  -1-  CHAPTER I INTRODUCTION The positron annihilation experiment has become an important tool in the study of the momentum distribution of electrons i n solids. The momentum distribution of the conduction electrons can be related to the Fermi surface and thus i s of considerable importance i n the study of metal physics. In particular, the technique of positron annihilation has been successfully applied to concentrated alloy systems which are inaccessible to experimental methods requiring that the electronic states be sufficiently long lived before collisions with impurity atoms occur. To a great extent previous positron annihilation experiments have been carried out on known Fermi surfaces to substantiate the r e l i a b i l i t y of the method.  From these experiments, the limitations and a b i l i t i e s of  the technique have been evaluated, leading to a further degree of sophistication i n the application.  The a b i l i t y to ascertain specific  features of the Fermi surface of metals, such as the contact with Brillouin zone boundaries, has been enhanced through the use of d i f f e r ent detector geometries.  These geometries, which greatly determine the-  resolution of the system, refer to the apertures preceding the detectors which count the gamma radiation resulting from an electron-positron annihilation. By applying the conservation of momentum law to the annihilation i t can readily be shown that the transverse momentum of the electronpositron pair i s related to the deviat ion of the gamma pair from 180°. If the angular deviation i s denoted by 6 and the transverse momentum by P  then P = mc6. If one set of detectors i s moved so that the angle 9  -2-  is varied, then an angular distribution can be plotted for P . Further, i f the positron momentum i s assumed small compared to the electron momentum, the angular distribution w i l l directly reflect the behaviour of the transverse momentum of the conduction electrons.  Thus the posi-  tron annihilation technique i s able to ascertain the momentum distribution of the conduction electrons in k space and information regarding the Fermi surface of the material can be garnered. Generally speaking, most positron annihilation experiments make use of radio-active sources such as sodium-22 for the positron. In this case additional apparatus i n the form of a magnet and stable power supply are needed to focus the positrons onto the sample.  This type  of experiment places a great dependence on the surface of the sample to be studied since the penetration depth of the positron i s small. A copper allo'y can be studied in a different and more effective manner.  If the abundant isotope of copper, atomic weight 63, i s placed  in a neutron flux then a reaction takes place which produces a radioactive isotope of copper with atomic weight 64.  This isotope possesses  a half l i f e of 12.7 hours and undergoes radio-active decay with the emission of a positron.  The resultant annihilation of the positron  with an electron depends on the volume rather than the surface of the sample since positrons are  produced throughout the sample.  The source  of neutrons in theNpresent experiment i s the Atomic Energy reactor at Chalk River, Ontario.  The samples are irradiated for a period of three  days and subsequently possess about 600 milli-curies of positron activity.  -3-  The essence of the present experiment i s the a p p l i c a t i o n of one p a r t i c u l a r geometry t o the study of copper g a l l i u m and copper germanium a l l o y systems.  This experiment was motivated by previous study of  copper z i n c ( W i l l i a m s , 1968) s i n c e g a l l i u m and germanium f o l l o w z i n c i n the  p e r i o d i c t a b l e w i t h the a d d i t i o n of one and two 4p e l e c t r o n s  respectively.  A l l o y s of 3.3, 6.0 and 8.2 atomic percent germanium and  4 . 9 , 8.7 and 12.7 atomic percent g a l l i u m were studied.  The Fermi sur-  face of copper has been p r e v i o u s l y determined (Pippard, 1958) and i s found to c o n s i s t of neck r e g i o n s i n the <111> 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 where the Fermi surface has made contact w i t h the B r i l l o u i n zone boundary. In the present experiment the neck was used as a monitor i n the study of the d i f f e r e n t copper a l l o y s to a s c e r t a i n the e f f e c t s of adding a d i f f e r ent number of conduction e l e c t r o n s to the i n t r i n s i c copper.  The r e g i o n  corresponding t o the l o c a t i o n of the Fermi surface boundary i n the <110> d i r e c t i o n was a l s o studied i n order t o p l o t Fermi s u r f a c e contours f o r the a l l o y s and observe the c o n t r i b u t i o n of core e l e c t r o n s t a t e s t o the a n n i h i l a t i o n process.  A s i n g l e run to determine the <100> c u t - o f f f o r  the highest e l e c t r o n per atom r a t i o a l l o y was a l s o undertaken. As a background f o r the present experiment the n e a r l y f r e e e l e c t r o n theory, o r t h o g o n a l i z e d plane wave theory, and the augmented plane wave theory a r e discussed under the general heading of e l e c t r o n s i n metals. In p a r t i c u l a r , the treatment of the augmented plane wave theory w i l l lead to c a l c u l a t i o n s f o r the"band s t r u c t u r e of copper*.  This chapter i s a l s o  used to d e r i v e or e x p l a i n such fundamental concepts as momentum space, Fermi s u r f a c e , and B r i l l o u i n zone boundary so that the l a t e r r e f e r e n c e may be made to them.  -4-  The transition from the study of pure metals to the behaviour of concentrated, disordered, alloys i s exceedingly d i f f i c u l t .  As well as  the inapplicability of the usual high powered experimental techniques to concentrated alloys, the loss of the inherent periodicity of the metal severely complicates the theoretical calculations.  A simple  approach i n this study i s to assume that the band structure of the host is not altered i n the alloying process and that the introduction of impurity electrons can be accounted for by simply scaling the Fermi energy according to the electron per atom ratio.  This rigid band model  would be expected to possess some validity for dilute alloys, however Stern (1969) has indicated that the rigid band model may be applicable to concentrated noble metal alloys.  This applicability i s due to the  large separation between the conduction bands for the noble metals and is related to the shielding effect of conduction electrons about solute atoms. The only detailed theoretical model which has had success i n accounting for the properties of disordered alloys i s the coherent potential method of Soven (1966).  The complexity of the problem i s  evident from the nature of the potential which i s energy dependent and further, has an imaginary part to account for the damped wave functions characteristic of disordered systems.  The coherent potential model has  been applied to the copper zinc system by Soven (1966) and Das and Joshi (1972) with marked success i n describing optical data for that alloy.  It i s hoped that the present measurements w i l l stimulate a  similar analysis of the copper gallium and copper germanium systems.  -5-  Various conclusions can be drawn from the present data.  For  example the measurements of the alloy neck radii and Fermi surface cut-offs can be related to changes in the pure copper band structure with alloying.  It should be stressed however, that the application  of present measurements to band structure considerations i s severely limited by such effects as the strong d band-conduction band interaction i n copper.  The experimental data can, however, be used to plot  Fermi surface contours for the alloys which w i l l be extremely useful in subsequent band calculations as applied to concentrated alloys.  As  well, the present work i s relevant to the Hume-Rothery rules and leads to a definitive statement concerning the a-6 phase transition in the copper gallium system.  A comparison of the experimental work with  other methods applicable to noble metal alloys serves as a useful summary of the current knowledge of the copper alloy system.  -6-  CHAPTER II ELECTRONS IN METALS  A.  Nearly Free Electron Theory The present chapter includes a study of some of the common  methods for studying the band structure of pure metals.  A survey of  the various methods w i l l help to elucidate the band structure of copper and serve as an introduction to the study of copper alloy systems. The simplest approach to study the behavior of conduction electrons i n metals i s to assume that the electrons are a gas of noninteracting particles moving in a constant potential.  If the wave  function of the conduction electrons i s denoted by ^ the Schrodinger Equation is written  The solution to this equation for the wavefunction t[i and energy e can easily be found by imposing the boundary conditions that the electrons occupy a cube of side L.  The solutions are;  2-2  I  b  £  ^  *  2-3  The electron wave functions are plane waves with index k, the wave number.  The energies f i l l up k space  such that at zero degrees Kelvin  -7-  a l l states below  =  are occupied.  The energy  i s known as the  Fermi energy and the region enclosed by the boundary comprises occupied k space. The free electron theory explains many general features of the electronic behavior i n solids quite well but often f a i l s to explain specific features such as the deviation of the Fermi surface from spheri c i t y for many metals.  The reason for this i s that the assumptions  underlying the theory do not correspond with the r e a l i s t i c situation. In r e a l i t y the electron w i l l move i n a potential due to the ions and the other electrons i n the crystal.  This potential V(r_) w i l l be periodic i n  nature because of the regularity of the crystal lattice structure and consequently may be expanded i n terms of reciprocal lattice vectors G_; c  Mir) - I V  «  &  G-t  e "  2  _  4  If the wave functions of the system are labelled by ^ ( j ) then  where the periodic function U^Cr) i s invariant under a lattice translation T so  ".  .  2-6  The electron wave functions, which are plane waves i n the free electron theory are modified by the periodic function U^Cr)  a n  d are termed Bloch  waves. Provided that V(r) i s weak the free electron energy EgCk) given by equation 2-3 i s modified by perturbation theory so that the nearly  -8free electron energy It*)*  i s given by f . l * ) W J } l ^ l i > • l  k  '  ( J  ,,. ~, £  2-7  *tk'  The f i r s t order term <k|V|k> simply represents a shift in the energy levels of the electrons as can be verified by making the substitution for V(r) as given in 2-4.  To consider the second order term in more  detail  < * W l & V - fir?**' ? 1  c  '  2-8  The matrix element becomes, on using equation 2-4 for V(r)  2-9  The integral i s a delta function which i s non-zero only i f r  2-10  Thus the only contribution to the second order energy term i s derived from the mixing of unperturbed states which differ by a reciprocal lattice vector.  In the case where  2  from which  m  L~> •-•  -•  E Q Q O  = EgCk -  G )  then,  2  -U  -9-  2-12  E q u a t i o n 2-12 d e f i n e s t h e B r i l l o u i n zone boundary, t h e p o i n t a t w h i c h Bragg r e f l e c t i o n o f e l e c t r o n s by t h e l a t t i c e atoms o c c u r .  The Bragg  r e f l e c t i o n g i v e s r i s e t o energy gaps i n k space as shown i n two dimens i o n s i n f i g u r e 1.  The e f f e c t o f t h e l a t t i c e on t h e e l e c t r o n wave  f u n c t i o n s i s t o i n t r o d u c e h i g h e r momentum components i n t o t h e wave function;  2-13  where t h e a  r  a r e g i v e n by  2-14  I t s h o u l d be mentioned f o r c o m p l e t e n e s s t h a t h i g h e r momentum components c a n a l s o c o n t r i b u t e t o t h e p o s i t r o n wave f u n c t i o n due t o t h e e x c l u s i o n of t h e p o s i t r o n from t h e core regions of the s o l i d . The band s t r u c t u r e o f t h e n o b l e m e t a l s has been found t o c o n s i s t of 5 narrow d bands l o c a t e d s e v e r a l e l e c t r o n v o l t s below t h e F e r m i energy w h i c h h y b r i d i z e w i t h t h e c o n d u c t i o n bands. r e s u l t s i n an extremely d i s t o r t e d Fermi surface  This i n t e r a c t i o n  (as compared w i t h a  f r e e e l e c t r o n s p h e r e ) such t h a t t h e s u r f a c e i s i n c o n t a c t w i t h t h e <111> zone boundary.  Thus one may speak o f t h e " n e c k s " o f copper w h i c h a r e  -10-  Figure 2 :  Copper Fermi Surface  -li-  the regions of contact as shown in figure the copper Fermi surface  2.  The neck feature of  i s used as a monitor in the study on the  effects of alloying. T h e nearly free electron model must be modified when applied to the  noble metal alloys due to the presence of the d-bands.  Pseudo-  potential theory has been found to be applicable to the study of some d i — and tri-valent metals by Heine and Weaire (1966) and is mentioned at several points i n this chapter.  The following sections, which  discuss the O.P.W. and A.P.W. theory, w i l l lead to the band structure of copper. ]}.  Orthogonal ized Plane Wave Theory In the orthogonal plane wave (O.P.W.) treatment the conduction  electron states are made orthogonal to the core electron states.  The  conduction electron behavior i s represented by plane waves in the inter-ionic regions of the crystal and the core electrons are restricted to individual ions. U t i l i z i n g the notation and method of Harrison (1966) an orthogonalized plane wave x  t  ^  s  related to a coi-e electron state \p by a  2-15  where dr' i s a volume element.  The expression can be  conveniently  written i n ket notation  2-16  -12-  which satisfies the orthogonality requirement since ^Xjjc-*  =  0*  *f  the projection operator P = l|a><a| (the operator which projects a functions onto the core states) i s introduced then  2-17  The conduction band states  can be expanded in terms of recipK.  rocal l a t t i c e vectors as a linear combination of O.P.W.s  2-18  If the expanded  of equation 2-18 are inserted into the  Schrodinger equation and a l l terms involving the projection operator are installed in the l e f t hand side of the equation one can obtain  T ^ * " h ' h * i  Khere;  2  pseudofunction  "PI;  • 5$ flc'ft' '  pseudopotentlal  W  V«l * |  l  fc'  £* C  _  19  >  «*>«» = «W * % ^  and the relation between the true and pseudo wave function i s  KV-d-/^ ^  2-20  The net result of this analysis i s the introduction of a local pseudo-potential operator given by (E^_ - H)P which has the effect of  P  -13-  counteracting the strong attractive potential  due to the ions.  In essence the conduction electron plane waves are only slightly modified i n the v i c i n i t y of the ions. The application of the O.P.W. theory to the transition metals i s somewhat ambiguous since a separation of the s ,p and 3d electrons into conduction and core states respectively, i s not always possible.  The pseudo functions however have been used  with marked success by Stroud and Ehrenreich (1968) to describe the positron annihilation angular distribution spectrums of simpler metals such as aluminum and silicon..  A third method has been found to be  especially useful i n leading to the band structure of copper and this method i s now discussed.  C_. Augmented Plane Wave A necessary requirement for any method to yield tractable solutions for the conduction electrons i n a periodic potential i s that the rapid convergence be obtained when the wave function i s expanded i n terms of the basis vectors. The augmented plane wave method (A.P.W.) does not separate core and conduction electron states as in the O.P.W. technique but instead matches exact solutions of the Schrodinger equation for the distinct regions of k space.  The potential i n the A.P.W. method has two compo-  nents; namely in the region about each ion core the potential i s spherically symmetric (termed muffin tin) and in the region outside the core the potential i s constant, figure 3.  The Schrodinger equation can  be solved exactly inside the muffin t i n region by spherical harmonics and outside by plane waves.  A brief description of the A.P.W. method  as discussed by Ziman (1969) i s now given.  -14-  Figure 3:  Muffin Tin Potential  -15-  The s o l u t i o n o f t h e S c h r o d i n g e r  equation f o r the m u f f i n t i n  r e g i o n i s g i v e n by  2-21 where:  ^ <w e x p a n s i o n  coefficients  Rg  s o l u t i o n of r a d i a l equation i n muffin t i n region  ^ ^  s p h e r i c a l harmonic  The s o l u t i o n o u t s i d e t h e m u f f i n t i n r e g i o n i s s i m p l y an of p l a n e  waves  k  k  2-22  •  I t c a n be shown t h a t t h e c p n d i t i o n s imposed on t h e e x p a n s i o n C  expansion  coefficients  by m a t c h i n g t h e s o l u t i o n s a t t h e r a d i u s o f t h e m u f f i n t i n p o t e n t i a l  are  Sectc) I f the C  ~"  2  a r e p l a c e d i n t h e e x p r e s s i o n f o r <{>(r) t h e r e s u l t a n t s o l u t i o n  is termed an augmented p l a n e wave ^ C r )  where  The Z d e n o t e s t h e c e n t r e o f t h e m u f f i n t i n . s p h e r e . d e s c r i b i n g the conduction e l e c t r o n s ^ C l )  *  s  a  The wave f u n c t i o n  c o m p o s i t e o f t h e augmented  -16-  plane waves  YU>  *J  fy-/*'  2-25  where the a expansion coefficients can be found from variational procedures. The definitive work on the band structure of copper has been done by Burdick (1963) using the A.P.W. treatment, and Segall (1962) using a Green's function approach.  The band structure i s shown in figure A.  The bands arising from the Is, 2s, 2p, 3s, and 3p levels of the atom l i e well below the conduction band and are consequently f u l l y occupied. The 3d band i s also f u l l and since the 3d states are localized, the band i t s e l f is quite narrow with a consequent high density of states. It has been found (Segall 1962)  that the electrons in the <;111> neck  regions of the copper Fermi surface exhibit p-like behavior (electron density greater in the inter-ionic regions) whereas the electrons in the <100> belly regions are s-like i n character (density greater at the ion sites).  The two distinct characters would be expected to affect  the introduction of impurity electrons into the copper system i n different ways and this is discussed further in the following chapter. The band structure calculations are completely consistent with the experimental data garnered by-Halse (1969), Shoenberg (1962) and Joseph et a l l (1966).  The most significant feature of figure 2 is that  the Fermi surface has made contact with the Brillouin zone boundary in the <111>  crystallographic direction resulting in the neck regions of  copper.  It i s this particular feature of the Fermi surface which w i l l  -17-  be used as a monitor in the study of the effects of alloying copper with different  impurities.  The present chapter has attempted  to provide, i n a cursory  manner, a theoretical basis for the behavior of the conduction electrons i n copper.  As well, a foundation for the study of copper alloys  in the following chapter, has been l a i d .  Some of the concepts intro-  duced i n this chapter, such as Brillouin zones, w i l l also be referred to i n succeeding chapters.  -18-  -19-  CHAPTER III ELECTRONS IN ALLOYS A.  Introduction The eigenfunctions of the conduction electrons i n pure metals are  Bloch functions arising from the translational invariance of the l a t tice.  The Bloch states have well defined energies given by  3-1  and f i l l up the Fermi surface i n k space.  If impurity atoms are now  randomly inserted into the pure metal, the concept of Bloch states becomes somewhat tenuous since the periodicity of the lattice i s destroyed.  The following questions pertaining to the study of these dis-  ordered alloys now arise; I.  The v a l i d i t y , i n a quantitative sense, of the application of the theory of pure metals to disordered alloys.  II.  I f such v a l i d i t y does exist, the dependence of the alloy band structure on the concentration and nature of the impurity atoms.  III.  The modification of pure metal theories to conform to reality i n the study of disordered alloys.  IV.  The experimental approach to justify the relevant theories pertaining to disordered alloys. The following discussion w i l l attempt to answer the aforementioned  questions and also to relate the role of positron annihilation to the study of disordered alloys.  -20-  .B.  Rigid Band Model A simplistic approach in this study i s to assume the only effects  of alloying a solute impurity with a valence greater than that of the solvent host i s to scale the Fermi energy in accord with the number of conduction electrons.  This assumption i s the basis of the rigid band  model which also holds that the constant energy surfaces and the density of states curve remain unchanged i n the disordered alloy. Quantitative support for the rigid band model is shown i n table 1 where the electron per atom ratio n at the alpha phase boundary i s given for several alloys including the two studied in the present experiment.  The result that the different alloys possess similar electron  per atom ratios lends credence to the rigid band model as shown i n figure 5 where the density of states N(e) i s plotted versus n using rigid band theory (Ziman 1960).  The maximum in N(e) occurs near n = 1.36  which i s the point at which a free electron sphere has expanded s u f f i ciently to encounter the Brillouin zone boundary.  The expected behavior  in the density of states and also the electronic specific heat  where  C '^<k*Wt)T  3-1  e  is to decrease once the zone boundary has been reached. With regard to the latter prediction concerning the behavior of the density of states the rigid band model conflicts with the band structure results for copper discussed i n the previous chapter.  It has been  found that the pure copper Fermi surface, that i s n = 1, i s sufficiently distorted that contact has already been made with the zone boundary.  -21-  Table 1:  ALLOY  n (cC phase boundary)  CuZn  1.384  CuAl  1.408  CuGa  1.406  CuGe  1.360  CuSi  1.420  CuSn  1.270  Electron per Atom ratios (n) for several alloys  n  Figure 5:  Density of States versus n ( Rigid Band theory )  -22-  Furthermore, much experimental evidence has accumulated to show that for the noble metal alloys the electronic specific heat, hence the density of states, actually increases with n which contradicts the rigid band theory.  Other experimental evidence (Stern 1970) also indi-  cates failures in the rigid band theory as applied to concentrated alloy systems.  Rather than totally discard the rigid band model i t i s  worthwhile to consider some modifications of the model. In the modified theory the eigenstate cf^ of the rigid band model transforms to a state in the alloy system which i s not an eigenstate due to the lack of periodicity i n the alloy l a t t i c e .  Similarly an  energy transformation takes place  Fry* *fc*> *id  3-2  where the real part of the energy E*(k) corresponds to the true eigenstate of the alloy system and the imaginary part T(k) denotes the l i f e time of the state.  The eigenstate for the alloy system w i l l be smeared  out  in k space where the width of the peak at half maximum i s given by  T.  The width depends directly on the impurity concentration. The  smearing of the eigenstate at the Fermi surface can, i n theory, be determined by the positron annihilation method since the momentum distribution of the conduction electrons i n k space i s measured.  In this event, the  region of k space corresponding to k > k^ would 'indicate the sharpness of the Fermi cut-off and any smearing of the k states would be evident. In general however, the positron annihilation technique i s unable to measure the smearing of the k states except for the simplest dilute  -23-  alloys such as lithium-magnesium (Stewart 1964).  The reason for this  i s that the momentum distribution of the gamma radiation i s affected by higher momentum components and also for the transition metal alloys the core states contribute an unknown amount.  Both effects contribute  to the smearing i n k space so that any definitive conclusions are d i f f i c u l t to draw.  C.  Tight Binding Method Another approach has been developed by Stern (1969) for the noble  metal alloys.  This approach requires that the conduction bands be ener-  getically well separated from the unfilled bands, a requirement which i s met by the copper alloys studied i n the present experiment. In this method the wave functions of the alloys are Wannier functions derived from a tight binding approximation.  If the wave function  of the pure metal i s denoted by ^ C r ) and the atomic wave function for the host copper i s denoted by  (r-R ) then  where the R denote the location of the host atoms i n the l a t t i c e . For —a the host atom copper, the conduction band w i l l be only half f i l l e d since the band i s capable of holding two electrons per atom corresponding to the two spin states of the electron. If an impurity atom with two valence electrons i s added at a l a t t i c e s i t e s then the wave function for the alloy  (r) can be written  -24-  4*  i  r  -  _  3  4  where d> = 6, for n f s n 1 <j> = <fo for n = s n 2 An(m) = expansion coefficients If the two conduction electrons belonging to the impurity atom remain isolated about the atom then the impurity state §^ can be said to be fully occupied and additional electrons added to the system would be effectively scattered from the impurity site.  The net result  i s that the amplitude of an additional electron state would be decreased i n the v i c i n i t y of the impurity contrary to the free electron behavior which would tend to deposit additional charge.at the impurity for shielding purposes.  If the impurity state is not fully occupied  then the scattering of conduction electrons w i l l be s-type i n nature, that i s spherically symmetric.  On the other hand, p-type scattering  of the conduction electrons w i l l occur i f the impurity states are fully occupied.  Evidence that the conduction electrons i n the <100> crystal-  lographic directions exhibit s-type scattering and the electrons i n the <111> crystallographic neck directions display p-type behavior has been obtained (Segall 1962). Support for the localization of charge theory has been obtained from resistivity measurements on noble metals (Leonard 1967).  These  measurements indicate that the resistance of the'.noble metals i s lower than predicted by theory (Born approximation) for scattering centres which are characterized by potentials that are localized and weak. The  -25-  explanation in terms of charge localization i s that the impurity atom states are f i l l e d and hence are not effective scattering centres for the conduction electrons,  D_.  Treatment of Alloy Potentials Three methods pertaining to the study of the effective potentials  seen by the conduction electrons have been applied to disordered alloy systems.  These methods are now discussed briefly with no attempt to  indicate their v a l i d i t y i n terms of experimental data.  It should be  mentioned however, that only the most sophisticated method of the three, that i s the coherent potential technique, i s expected to adequately relate to concentrated alloys. (i)  Average Potential In this treatment (Sommers 1966) the alloy potential i s simply the  average of the potentials of the two constituents weighted by their concentrations.  The average potential at each lattice site i s thus  V - o%\lt t c-.  3  _  5  This method has been found to be of use only for extremely dilute alloys with similar constituents and thus has very limited applicability. ( i i ) -T Matrix  v  This method (Beeby 1964) i s slightly more sophisticated than the average potential method.  Here an average scattering matrix, the T  matrix, i s used where the two distinct scattered waves corresponding to interactions with the two constituents of the alloy are averaged.  Thus  -26-  rather than an average potential one considers the average of the scattering properties of the two atoms.  The atomic potentials chosen  are usually muffin t i n in nature (see A.P.W. theory i n Chapter 2). The determination of the crystal potential for either constituent is often d i f f i c u l t and is even more uncertain in the' case of an alloy since the environment of l a t t i c e sites can vary.  As well Soven (1967)  concludes that the method i s totally inadequate for transition metals since i t introduces spurious band gaps in the energy spectrum and f a i l s to reproduce the electronic behavior near band edges.  Thus for alloys  and non-simple metals this method has been superceded by a third approach. (iii)  '  •  Coherent Potential The coherent potential model has been described by Soven (1966,  1967).  It i s defined as an effective potential which when placed upon  the lattice sites of an alloy w i l l reproduce the properties of that a l l o y . S u c h a potential must be energy dependent since no single potential would be expected to account for the alloy properties over a wide range of energies and in addition must be complex to provide for the damped wave functions which are characteristic of disordered alloy systems.  Thus the model simulates the conduction electron behavior i n  an alloy by introducing a medium in which a coherent potential i s placed on individual l a t t i c e sites and modifies the dynamical properties of the particle. The coherent potential method i s based on a Green's function formalism and for simplicity i s discussed for a one-dimensional system. should be noted however that the model i s expected to be perfectly adequate in describing the three dimensional system (1967).  It  The Green's function G(x,x') i s defined for a one-dimensional alloy (Soven 1967)  3-6  where H  is the Hamiltonian of the system 6 i s the Dirac delta function.  The free electron Green's function w i l l satisfy the i n i t i a l equation with the potential set to zero .  3  _  7  and the G(xx') can be expanded in terms of the Gg(xx') i n an i n f i n i t e series  G  Co * C*  + £ v 6 v& «• a  0  3-8  where  If the crystal potential i s labelled V^(x) where i refers to the constituent and the a to the l a t t i c e site then the total crystal potent i a l is  .  1  t*>  3-10  -28-  Thus  «t  r  3-11  The physical basis of the preceding equation is that the electrons move through the empty space of the crystal and are scattered by the atomic potentials.  The scattering processes can be accounted for by the  T-matrix approach discussed previously.  The average scattering i s given  by 3-12  where t^ and  describe the scattering for a given site and the c^ and  C £ are the concentrations of the alloy constituents.  The scattering  matrix is related to the crystal potential V by  3-13  and i f equations 3-11 and 3-13 are combined then  «tf  tC  'The Green's function i s now described  '  3-14  i n terms of repeated  scattering interactions where the electron i s scattered at a specific site then moves on to another site.  This expression for G i s only  approximate but can be readily summed.  In general the model has been  -29-  found to predict the position and shape of energy band edges for concentrated disordered alloys.  Specifically the coherent potential model has  been applied to a-brass by Soven (1966) and Das and Joshi (1972) and i s discussed further in a later section of the thesis.  _E. (i)  Other Methods Applicable to Disordered Alloys Electronic Specific Heat Since knowledge of the electronic specific heat relates directly  to the density of states through equation 3-1 then a measurement of the coefficient y versus the impurity concentration can be used to infer the behavior of the density of states.  Much experimental evidence has been  accumulated by Mizutani (1972) to show that the electronic specific heat of noble metal alloys increases with an increasing electron per atom ratio i n contradiction of the theory underlying the rigid band model. It i s noteworthy that this experimental evidence includes the alloy systems discussed i n the present experiment. Attempts have been made to explain these results on the enhancement of the electron-phonon interaction as the impurity concentration increases.  In particular the lead alloy system (alloys of PbBi and  PbTA) has been studied by Clune (1970) where the electron-phonon factor i s well-known.  The experimenters find that i f the effects of the  electron-phonon interaction are accounted for there i s quantitative agreement between the increase of the specific heat coefficient y and theory.  A definitive check on the relevance of the electron-phonon  interaction as applied to noble metal alloy systems may be garnered  -30-  from a study of a superconducting t r a n s i t i o n i n these a l l o y s .  Such a  t r a n s i t i o n would be expected i f the i n t e r a c t i o n i s a r a p i d l y i n c r e a s i n g f u n c t i o n o f the e l e c t r o n per atom r a t i o , however copper a l l o y s have been found to remain non-superconducting down to 0.05°K (Clune 1970). The electron-phonon i n t e r a c t i o n i s an a l t e r n a t i v e to Stern's l o c a l i z a t i o n hypothesis and much experimental data i s needed i n order to c l a r i f y the r e l e v a n c e of the two c o n t r i b u t i o n s . (ii)  O p t i c a l Measurements F i g u r e 6 i n d i c a t e s a simple p i c t u r e of the band s t r u c t u r e of  copper^  A narrow, f i l l e d d-band i s overlapped by a wide s-band c o n t a i n -  i n g one e l e c t r o n .  Two o p t i c a l band t r a n s i t i o n s are shown; that from the  d-band t o the Fermi l e v e l and that from the Fermi l e v e l to a h i g h e r , unoccupied conduction band.  A study of such t r a n s i t i o n s w i t h a change  i n i m p u r i t y c o n c e n t r a t i o n s would be expected to y i e l d i n f o r m a t i o n on the band s t r u c t u r e of the a l l o y .  For example the r i g i d band model would  hold t h a t the o n l y e f f e c t s of a l l o y i n g would be a s h i f t i n  to the  r i g h t hence the t r a n s i t i o n from the d band t o the Fermi ( l a b e l l e d as 1 i n f i g u r e 6) l e v e l would be expected t o s h i f t to lower wavelengths and the t r a n s i t i o n from the Fermi l e v e l t o the conduction band ( l a b e l l e d as 2 i n f i g u r e 6) t o s h i f t t o higher wavelengths.  Such simple behavior has  not been s u b s t a n t i a t e d . The prime d i f f i c u l t y a r i s e s "since non-direct t r a n s i t i o n s are a l s o p o s s i b l e whereby Ic i s not conserved.  Unless the-'experimenter i s able  to r e s o l v e the d i r e c t and n o n - d i r e c t t r a n s i t i o n s then d e f i n i t i v e c o n c l u sions are d i f f i c u l t . by B i o n d i  O p t i c a l s p e c t r a have been obtained f o r CuGa and CuGe  e t a l (1959) and P e l l s e t a l (1970) which r e l a t e to the changes  i n the copper band s t r u c t u r e w i t h a l l o y i n g and thus are r e l e v a n t to the present work.  Figure  6:  Optical Transitions in Copper  -32-  (iii)  I n e l a s t i c Neutron  Scattering  I n t h i s method t h e d i s p e r s i o n c u r v e s f o r phonons e v i n c e s k i n k s f o r wave numbers q c o r r e s p o n d i n g t o t w i c e t h e F e r m i s u r f a c e wave v e c t o r  i k^. of  These s o - c a l l e d Kohn anomalies o c c u r because t h e s c r e e n i n g e f f e c t t h e c o n d u c t i o n e l e c t r o n s changes f o r t h e s p e c i f i c v a l u e o f q.  The  s c r e e n i n g r e f e r s to the a b i l i t y of the conduction e l e c t r o n s t o screen the p e r i o d i c e l e c t r i c f i e l d of the l a t t i c e i o n s . anomalies  Observation of the  o f q = 2k^ thus y i e l d s i n f o r m a t i o n on t h e shape o f t h e F e r m i  surface. N e u t r o n d i f f r a c t i o n s t u d i e s have been done on l e a d by Brockhouse e t a l (1962) and aluminum by Stedman and N i l s s o n (1965).  The method i s  p r e d i c t e d t o b e o f o n l y l i m i t e d u s e by Shoenberg (1969) due t o t h e weak n a t u r e o f t h e k i n k s and t h e d i f f i c u l t i e s i n h e r e n t i n t h e method. The Kohn anomaly c a n a l s o be s t u d i e d by x - r a y d i f f u s e s c a t t e r i n g experiments.  Such a t e c h n i q u e has been a p p l i e d t o c o n c e n t r a t e d  aluminum a l l o y s b y S c a t t e r g o o d e t a l . (1970).  copper  -33-  CHAPTER IV THE POSITRON ANNIHILATION EXPERIMENT A.  Introduction In this chapter several facets of positron annihilation i n solids  are discussed.  The theoretical angular distribution of the gamma radia-  tion, i n conjunction with the different geometries, i s covered i n detail to provide a sufficient background for the experimental technique used in the present experiment.  Other concepts such as the thermalization  and lifetime of the positron and the possible enhancement of electrons in the v i c i n i t y of the positron are also included.  ]}. Angular Distribution of Gamma Radiation The methods by which the momentum distribution of conduction electrons i n metals and alloys has been studied by positron annihilation relate t o the particular geometry of the apertures placed in front of the detectors. As w i l l be shown, the geometry directly affects the measurement of the angular distribution of the gamma radiation hence determines the resolution of the experiment.  It should be noted that  the neck of pure copper subtends an angle of about 20 degrees at the origin o f k space thus the geometry used in the experiment should subtend an angle somewhat less than this. To analyze the different geometries i t i s necessary to apply the conservation of momentum to the annihilation of a positron with an electron. pair;  From figure 7 one has for the transverse momentum of the gamma  -34-  a = 6/2 + 3 6 + 3 = 90° a + 8/2 = 90°  Figure 7: Momentum Conservation in Positron Annihilation  -35-  p. : 2 m C i i f t ( | )  4-1  4-2  The momenta of the annihilation gamma pair i s equal to the sum of the momentum of the electron and positron. As w i l l be shown subsequently in this chapter, the positron i s thermalized at room temperature hence P  can be defined as the momentum of the electron i n the transverse  direction.  jC. (i)  Detector Geometries Wide S l i t Geometry The wide s l i t geometry refers to an aperture preceding the detec-  tion system which i s placed i n the x-z plane such that & defines the x  length of the s l i t and  the width of the s l i t (figure 8'A'). A  specific crystallographic direction points along the y axis and this i s referred to as the axis of the system.  If one of the detecting systems  is translated i n the z direction then specific electrons with P^_ = P t z given by equation 4-2 w i l l be counted.  If p(P) i s defined as the den-  sity of electron states i n momentum space the coincidence counting rate N(P z ) w i l l be given by the expression 4-3  -36-  The integration in the x direction is to infinity since the width of the s l i t s i s made much greater than the width corresponding to the Fermi surface radius.  The integration i n the y direction i s also to  infinity since the detectors are unable to detect the Doppler shift i n energy of the annihilation radiation.  If the density of electron states  is assumed isotropic (Kahana 1967) and the Fermi surface spherical then  J J ( P ) cc f l P ) [ J d f c A P ^  4-4  5  Since the mapping area.is a disc in momentum space, figure 8'B', the integration can be carried out over the area of the disc using  In terms of the radius of the disc (in momentum units) the integral becomes  I  O  hr  Fi*Pf  Thus for nearly spherical surfaces such as encountered in simple metals such as sodium one would expect a parabolic distribution for the number of gamma pairs with a specific z component of momentum.  If the Fermi  -37-  Leacl Block  tector coincidence unit  Figure 8A  Figure 8B  Wide S l i t Geometry  Sampling Region for Wide S l i t Geometry  -38-  Figure 9 "B":  Sampling region for Point Geometry  surface i s not spherical but possesses features such as the necks previously discussed in connection with copper then the parabolic shape would be modified.  In the case of copper, a discontinuity i n the slope  of the parabola would occur in the neck region. It i s also interesting to examine the region corresponding to the location of the Fermi surface cut-off.  According to theory, no annihil-  ation pairs should be detected past this point but experimental data includes substantial background counts.  The reason for the background  counts beyond the Fermi cut-off i s two-fold; positrons can annihilate with the core electrons of copper and also higher momentum components can contribute to the background.  At best a discontinuity i n the slope  of the parabola would indicate the cut-off point. The major d i f f i c u l t y encountered with the wide s l i t geometry i s that a sample in the form of a disc i s taken i n k space and this may not yield sufficient resolution to show the neck features of copper.  It i s  for this reason that other methods have been developed in order to improve the resolution of the system. (ii)  Point Geometry To provide finer resolution in k space i t i s necessary to reduce  the limits of integration i n the x direction.  This i s accomplished i n  the point geometry method, figure 9'A', which consists of circular apertures preceding the detectors; thus the mapping of the Fermi surface i s done by cylinders as shown i n figure 9'B . 1  If the diameter of the  cylinder i s made sufficiently small the integration i n the x direction can be avoided.  In this case the coincidence counting rate i s given by  -401  to  4-7  60  and using the relations .1 .  o l . s i  j) A p ;  Pr <i  -V  4-9  the integral i n equation 4-7 becomes;  \  (P'-fj ) 1  1  4-10  A discontinuity i n this distribution w i l l occur when the cylinder passes through the copper neck region.  Since the volume element i n k  space, which i s mapped by the cylinder has been substantially reduced from that of the wide s l i t technique, the resolution w i l l be correspondingly improved.  The loss of counts encountered with this method can i n  part be alleviated by using stronger sources.  -41-  The  p o i n t geometry was  used i n e a r l i e r work i n t h i s l a b o r a t o r y  where t h e d i a m e t e r o f the c y l i n d r i c a l a p e r t u r e d e f i n e d a r e s o l u t i o n a b o u t 1 m i l l i - r a d i a n a t the s o u r c e . • was  I t was  found t h a t the  of  resolution  b a r e l y a d e q u a t e t o c l e a r l y d e f i n e the n e c k s o f CuZn a l l o y s .  (iii)  Crossed  Slit  Geometry  . A n o t h e r method has been d e v e l o p e d and used t o a e x t e n t by F u j i w a r a e t a l (1966, 1967, of c r o s s e d  1968).  considerable  In t h i s technique  s l i t s d e f i n e t h e r e s o l u t i o n f u n c t i o n where the  r e s o l u t i o n i n t h e x d i r e c t i o n (see f i g u r e 8'A') i m p r o v e d o v e r t h a t employed i n t h e w i d e s l i t  a pair  experimental  has been c o n s i d e r a b l y  geometry.  For a given  d e t e c t o r p o s i t i o n t h e number of gamma p a i r s w i t h s p e c i f i c t r a n s v e r s e momentum P  ( e q u a t i o n 4-2)  w i l l be g r e a t l y a t t e n u a t e d  c o n s i d e r a t i o n s p l a c e a l i m i t on t h e w i d t h of t h e (iv)  Present  thus  slits.  Geometry  I n the present  e x p e r i m e n t the p r e v i o u s p o i n t geometry was  i n order to improve the experimental  resolution.  ( s e e f i g u r e 9'A')  d e t e c t o r movement was subsequently time.  altered  This a l t e r a t i o n i n -  c l u d e d t h e p l a c e m e n t o f a p a i r o f s l i t s i n f r o n t o f the holes  statistical  collimating  such t h a t the r e s o l u t i o n i n the d i r e c t i o n o f narrowed by a f a c t o r o f about 1.5.  As i s d i s c u s s e d  t h e f i n a l r e s o l u t i o n i s among the s h a r p e s t i n use a t t h i s  As f o r t h e c r o s s e d s l i t geometry s t a t i s t i c a l c o n s i d e r a t i o n s p l a c e  a l i m i t on t h e s l i t d i m e n s i o n s and i a t i o n and  f i n a n c i a l c o n s i d e r a t i o n s (each  d e l i v e r y c o s t s i n the n e i g h b o r h o o d o f s i x hundred  l i m i t s t h e number o f samples s t u d i e d .  irrad-  dollars)  D_.  Thermalization  of the Positron  The determination of the positron life-time in copper has been made (Kohonen 1967)  and the result indicates a value of 2*10  seconds.  If this time is compared with the thermalization time of the positron, -12 about 3*10  seconds at room temperature, then the conclusion can be  made that the thermalization of the positron i s complete at the time of annihilation.  The conduction electrons w i l l possess energies of the  order of seven electron volts thus the thermal energy of the positron, about 0.02  electron volts, w i l l be negligible in comparison.  Several experiments have been carried out to determine the temperature effects of positron annihilation.  In particular, Shand (1967)  has studied the annihilation in sodium from 77"Kelvin to 600° Kelvin. The results of the Shand experiment show conclusively that the width of the angular distribution curve increases with increasing temperature. This smearing effect i s analogous to increasing the resolution of the measuring apparatus (see following chapter) and indeed many experimenters include the effects of positron motion in terms of an increase in their resolution.  The Shand data also indicates the positron effective  mass to be almost twice the free electron mass. JE.  Enhancement It would be expected that when a positron enters a metal there  would be an increase i n the conduction electron density about the positron due to Coulomb effects. This concept i s termed theoretical calculations (Kahana 1960)  enhancement and  indicate the electron density to  be an order of magnitude greater at the positron than elsewhere in the  -43-  metal.  The predicted Kahana enhancement has been found to be i n  excellent agreement with the measured angular distribution of sodium (Donaghy 1967).  A plot of the theoretical electron density in momen-  tum space versus k using the Kahana theory i s shown in figure lO'A'. The electrons near the Fermi surface are most affected since scattering states are readily available to them. A sketch of the electron density versus k in the absence of positrons and including the effects of electron-electron interactions i s shown in figure 10'B'  and indicates that the density of electrons  decreases at the Fermi surface.  This is due to some of the  electrons  accruing sufficient energies in order to occupy upper unfilled levels. The two effects tend to balance one another, indeed experimental evidence (Kahana 1967)  points to l i t t l e influence of the enhancement on the  angular distribution of gamma radiation.  F.  Previous Studies The f i r s t real evidence that positron annihilation could furnish  details of the copper Fermi surface was provided by the early crosseds l i t angular correlation results of Fujiwara (1965).  Subsequent work  by Fujiwara and his collaborators with improved resolution have produced much of the existing data on the copper and copper alloy systems. An alternative geometry, the point geometry, was f i r s t used by the University of British Columbia group (Williams et a l 1965,  1968) to  study the copper Fermi surface as a function of crystal orientation and concurrently  Sueoka (1967) used the crossed s l i t system to perform the  same type of study (the "rotating specimen" method).  This latter method  -44-  k  k  Figure 10 "B": Electron-Electron effects on Electron Density (Stewart 67  -45-  has been widely used by the Japanese workers.  It should also be noted  that Fujiwara and his associates pioneered the use of neutron irradiation to create positron active copper 64. Measurements of the neck radius and <100> cut-off have been made on a sequence of copper aluminum alloys consisting of 2.6, 5.7, 10.6 and 15.1 atomic percent aluminum by Fujiwara et a l (1968).  Their  results indicate an oscillatory behavior i n the neck radius dimension as i t i n i t i a l l y increases with the 2.6% alloy, decreases for the 5.7% alloy and subsequently increases to about twice the pure copper neck radius f o r the two most concentrated systems.  The most distinct neck  features occur for the 2.6% and 10.6% concentrations and are included here.  The other two alloys show l i t t l e evidence for a neck.  The neck  radius for the 2.6% aluminum sample was determined to be 1.25 m i l l i radians and the neck radius for the 10.6% sample was found to be 2.0 m.r.  The latter alloy was found to have a <100> cut-off of 6.2 m.r.  which can be compared with the present measurements. In this lab Becker et a l (1971) have determined the neck radius and <110> cut-off for C u  Zn._ and Becker (1970) has measured the same  oc  dimensions for a C u ^ ^l^o H ° y » a  ^  n t n e  former study the respective  neck and <110> r a d i i for CUg^Zn were found to be 1,6 m.r. and 5.3 m.r. and i n the latter study the corresponding values for C c.g ^~io u  1.95 m.r. and 5.9 m.r.  w  e  r  e  It should-be mentioned that the point geometry  used for these measurements employed an experimental resolution i n the <110> direction which was almost twice the resolution of the present work.  -46-  The long s l i t geometry has been used by Murray (1970) to measure the neck r a d i i in the copper alloys with impurity concentrations of 2.5 and 5 atomic percent aluminum.  The corresponding electron per atom  ratios are 1.05 and 1.10 respectively. The measured neck radius for the lower impurity concentration is 1.45 m.r. and for the higher one the neck radius was determined to be 1.60 m.r.  The authors also report-  ed that the angular correlation data for a Cu., Z n 0  /o  per atom ratio of 1.22,  00  sample, electron  22  did not indicate any distinct neck features,  presumably reflecting the type of geometry employed. Trifthauser (1969) has also studied a Cu  0  /o  Zn„_ sample and found 2.2.  a neck radius of 1.50 m.r. representing a 50% increase over the pure copper value.  This group also employed a long s l i t geometry to deter-  mine the momentum distribution. Five copper-aluminum alloys have been studied by Thompson (1971). The concentrations of aluminum are about 2.5, 1.1, 7.5, 10, and 15 atomic percent.  The respective neck r a d i i are 1.2, 1.3, 1.5, 1.4, and 1.7  and a l l quoted values are approximate as taken from a graph.  m.r.  The plot of  neck radius versus the electron per atom ratios for these alloys indicates a relatively smooth curve and i s reproduced at a later point i n this thesis. Recent positron work has been done on the copper-zinc alloys and of particular relevance to the present experiment, the copper-germanium system.  The former alloy study (Morinaga 1972) indicates a decrease in  the neck radius for a 4.1% .zinc concentration, a slight increase of about 10% in the neck radius for zinc concentrations up to 10% and then an  -47-  abrupt increase of about 40% in the neck radius in the region of 20 - 30% zinc.  The two alloys studied which are closest in impurity  concentration to the CuZn systems.mentioned above are those of zinc and 23.6%  zinc.  are about 1.15 m.r.  18.7%  The respective neck radii for these two alloys  and 1.3 m.r.  The overall data for the study of  seven alloys does not indicate that the neck radius is a monotonically increasing function of the electron-atom ratio. A study of the two copper germanium alloys with about 3% and 9% germanium has been reported in-a short note by Hasewaga (1972).  His  values for the neck radius of the respective alloys are 1.6 m.r.  and  2.0 m.r.  No other information regarding the experiment is available.  One further experiment should be mentioned although i t i s not concerned with concentrated  alloys.  Coleridge, Chollet and Templeton  (1968, 1971) have studied dilute impurities (less than 0.1%  concentra-  tion) in copper using the de Haas Van Alphen method and found the crosssection of the <111>  neck increases at approximately 6 times the rate of  change of the electron per atom ratio.  Generally they observe that the  copper gallium system seems to exhibit rigid band behavior whereas the copper germanium results differ markedly from the rigid band predictions. The authors note that Stern's hypothesis concerning the screening effect of the conduction electrons on the dilute impurities i s supported by their data. As well as the above work, several studies«of copper-nickel alloys have been reported.  These results are not included since the primary  concern in the present work is the addition of extra conduction electrons to the pure copper.  Table 2 summarizes the previously discussed data.  -48-  Group  Alloy  n  Fujiwara  CuAl-2.6  1.05 .  1.25  Murray  CuAl-2.5  1.05  1.45  Thompson  CuAl-2.5  1.05  1.20  Thompson  CuAl-5.0  1.10  1.30  Murray  CuAl-5.0  1.10  1.60  Thompson  CuAl-7.5  1.15  1.5  Thompson  CuAl-10  1.20  1.4  Becker  CuAl-10  1.20  1.95  Fujiwara  CuAl-10.6  1.21  2.0  Morinaga  CuZn-4.1  1.04  0.9  Morinaga  CuZn-5.4  1.05  1.1  Morinaga  CuZn-8.4  1.08  1.1  Morinaga  CuZn-10  1.10  1.1  Becker  CuZn-15  1.15  1.60  Morinaga  CuZn-18.7  1.19  1.2  Trifthauser  CuZn-22  1.22  1.45  Morinaga  CuZn-23.6  1.24  1.35  Morinaga  CuZn-24.6  1.25  1.45  Hasegawa  CuGe-3.0  1.09  1.60  Hasegawa  CuGe-9.0  1.27  2.0  Table 2:  Neck Radius  Previous Measurements of Copper Alloy Neck Radii  -49-  CHAPTER V ' POSITRON ANNIHILATION FACILITIES  A.  Introduction The copper a l l o y t o be s t u d i e d i s p l a c e d mid-way between two  systems o f d e t e c t o r s w h i c h a r e s e p a r a t e d by 16 meters  ( f i g u r e 11).  Each system i s composed o f seven d e t e c t o r s w h i c h a r e a l i g n e d i n p a i r s t h r o u g h t h e sample.  The gamma d e t e c t o r s a r e preceeded by a 10 cm  t h i c k b l o c k o f l e a d i n t o w h i c h a r e d r i l l e d 6 mm  diameter h o l e s .  In  e s s e n c e t h i s means t h a t t h e space o f t h e sample i s mapped by seven d i f f e r e n t c y l i n d e r s each one s l i g h t l y i n c l i n e d t o a n o t h e r (by <0.2°). One s y s t e m o f d e t e c t o r s can be v a r i e d i n a sideways d i r e c t i o n c o r r e s p o n d i n g t o a t r a n s l a t i o n i n t h e <110>  direction.  P u l s e s f r o m t h e a l i g n e d p a i r s o f d e t e c t o r s a r e f e d i n t o a pre-amp d i s c r i m i n a t o r and t h e n t o a c o i n c i d e n c e box w h i c h y i e l d s an o u t p u t i f t h e i n c o m i n g p u l s e s a r e s e p a r a t e d by l e s s t h a n 25 nano-seconds.  An •  o u t p u t p u l s e f r o m t h e c o i n c i d e n c e box i n d i c a t e s a l e g i t i m a t e p o s i t r o n a n n i h i l a t i o n e v e n t and a p l o t o f t h e c o i n c i d e n c e o u t p u t v e r s u s d e t e c t o r position  IJ.  s e r v e s as t h e b a s i s f o r t h e e x p e r i m e n t a l r e s u l t s .  Samples S t u d i e d  * The seven samples  s t u d i e d were e i t h e r p u r c h a s e d from M a t e r i a l s  ** Research C o r p o r a t i o n (Cu, CuGa 4.9%,  CuGa 8.7%,  CuGa 12.7%, CuGe  6.0%)  A l l compositions r e f e r to atomic percent c o n c e n t r a t i o n of impurity i n copper. M a t e r i a l s R e s e a r c h C o r p o r a t i o n , Orangeburg, New  York.  -50-  T power supply  Pb sample  —  ^  Pb d e t e c t o r  ^  pre-amp  motor  printer  -*  master  time  timer  delay  scaler 1  —  • i .  coincidence box  Figure  11:  Block  Diagram  of Apparatus  -51-  or grown in the metallurgy department at the University of British Columbia (CuGe 3.3%,  CuGe 8.2%).  The alloys were a l l grown by the  Bridgman method and a l l samples were independently analysed to determine the impurity  concentration.  In a l l cases the bulk single crystal is cut with a spark cutter in the form of a cylinder with the axis along a <111> direction. a <110>  crystallographic  Since we desired to sweep the moveable set of detectors in  direction (to effect the most distinct neck-belly boundary  possible) i t was necessary to define this direction on the sample. This was accomplished by squaring the sides corresponding to a  <110>  direction with a spark cutter; thus the sample would s i t i n the holder such that a <111> and a <110>  axis would point in the direction of the detectors  axis would point in the direction of detector movement.  The factors influencing the sample size include the geometry of the apparatus and the radio-activity desired.  Since s l i t s of dimen-  sions 3-5 mm by 6 mm in the z and x directions were installed in front of the 6 mm holes for a l l runs the corresponding sample dimensions, on the average, were made 2.8 mm x 4.5 mm.  The length of the sample was  6 mm to insure adequate statistics for a given run.  After spark cutting  the samples were chemically polished to remove surface residue and re-x-rayed  to insure single crystal structure.  The prepared copper alloys are sent to Atomic Energy of Canada and subsequently placed in the NRU  reactor (neutron flux of l.S'lO^n/  2  cm /sec) for a period of three days.  The radiation of the normal copper  63 with neutrons yields the positron active copper 64 isotope with a half l i f e of 12.7 hours.  The radioactive sample, usually consisting of  -52-  600 milli-curies of positron activity, is sent via Air Canada and placed in the holder.  As the experiment progresses corrections must  be applied for the radio-active decay.  These corrections were mini-  mized by counting at each detector position for either four or eight minutes and re-cycling the detectors back and forth about the centre axis to allow for folding of the results. The useful counting time for a sample is about one  C_.  day.  Detector Arrangement A schematic of one pair of the detecting apparatus i s shown in  figure 12.  A l l but two of the gamma radiation detectors consist of  sodium iodide crystals coupled to RCA 6342-A photo-multiplier tubes. The other two sodium iodide scintillators were coupled to RCA photomultiplier tubes.  6810  The detectors are encased in lead sheets to  minimize the background radiation. As previously mentioned, one system of seven detectors is variable in a direction parallel to a <110>  crystallographic axis of the sample.  This was accomplished by sending a start pulse from the master timer to a motor.  This pulse, would activate the motor which would turn a threaded  axle connected to the detectors.  The stop pulse occurs when a micro-  switch riding the rim of a disc connected to the axle reaches a groove cut in the disc.  Each movement of the detectors between the start and  stop pulses corresponds to an angle of about O ^ m i l l i - r a d i a n s at the sample.  -53-  Figure 12: Arrangement of one set of detectors  -54-  .  During the course of the experiment better resolution was  obtain-  ed by cutting another groove in the disc opposite to the i n i t i a l groove. The enhanced resolution has the drawback of poorer statistics and requires highly radioactive samples. Since the seven sets of detectors cannot occupy the same position in real space the <111>  axis of the system was located such that four  sets of detectors were to one side of the axis and the other three sets to the opposite side. detectors and the <111>  The maximum angle between the extreme sets of axis of the system i s 1.8°.  The correspondence  in k space i s that each mapping cylinder is slightly inclined to each other however this w i l l have l i t t l e bearing on the results since the neck of copper intercepts the origin of k space at an angle of 20°. This point i s discussed further in the section on resolution. D_. (i)  Electronics Pre-amp Discriminator The function of this c i r c u i t , figure 13, i s to produce a well de-  fined positive pulse of 1.5 volts amplitude capable of being driven through 8 meters of cable to a coincidence box.  The negative  pulses  from the photo-multiplier tube are f i r s t inverted by the 2N964 transistor and then amplified and re-inverted by the 2N706 A transistor.  The  purpose of the delay l i n e i s to produce a bipolar pulse, the negative portion triggering a change in state of the bistable multivibrator.  The  positive portion of the bipolar pulse resets the pair to their original state.  The output from the multivibrator triggers the tunnel diode which  produces a positive pulse of 0.5 volts amplitude and 25 nano-seconds  +30V  2N964  270  1  2N706A  u  |—A W 3_ /  S80 18K  OUT  1 100K  Fig.13: P r e - a m p  and  shaper  circuit  U  -  Figure 14:  Coincidence Circuit  -57-  width.  The f i n a l stage of the circuit is an emitter follower to match  the cable impedance.  The second output was used as a monitor to observe  the output pulses. The circuit acts as an energy discriminator in that gamma radiation with energy below 150 kev w i l l not produce an output.  The discrim-  ination i s necessary to insure that random noise pulses from the photomultiplier tube w i l l not trigger the circuit. (ii)  Coincidence Box The coincidence c i r c u i t i s shown in figure 14.  I n i t i a l l y the  tunnel diode and transistors are biased off. A change in the quiescent state of these components occurs i f two pulses from the discriminators of the matched detectors overlap to produce an input pulse of three volts.  When the tunnel diode is triggered on, the voltage at the base  of the 2N797 increases turning the transistor on.  The output pulse i s  stretched by the 200 pf capacitor and is about 5 volts amplitude.  This  pulse i s fed into a scaler which i s controlled by a master timer. The nominal resolving time of the coincidence box can be adjusted by varying the bias on the tunnel diode through the 10 k pot. The measurement of the resolving time can be obtained from the random source method; two uncorrelated sources are placed near the two sets of detectors and the random coincidence rate Nc measured.  fVj = 2 Y W INlz c  where  and  4  From the relation  ,  5  _!  are the single rates, the resolving time T can be found.  The resolving times of the seven coincidence boxes employed in the experiment ranged from 18 nano-seconds to 25 nano-seconds.  -58-  E_.  Resolution Effects and Detector Considerations The resolution of the point geometry system was determined by  dividing the sample and the collimating holes into very small elements and then summing over a l l these elements with a computer.  If a l l  possible combinations of these elements are taken into consideration and the resultant angles noted one can plot the resolution function directly.  This resolution function i s described i n greater detail i n  the following chapter. The important consideration i s the angle the resolution function subtends at the origin of k space.  Designating this angle by <f> we can  proceed as follows; take the ratio of the transverse momentum P and the momentum defined by a vector drawn from the origin of k space to the Fermi surface P^. This ratio i s equal to the sine of one-half the resolution angle <j>. The transverse momentum had already been defined (Chapter IV) and i s equal to m^cQ.  The Fermi momentum i s given by  1/2 (2m £ A n  where e,. i s the Fermi energy so  5-:2  Using  = 7 e.v. and 8 = 1 milli-radian the angle $ i s equal to 13°.  Since the angle subtended at the origin of k space by the necks of V  copper i s 20° then the neck features should be defined by the apparatus. Prior to commencing the present experiment an improvement to the resolution was made since i t was known we would be dealing with impurity concentrations of sufficient magnitude (electron atom ratios of 1.25) such that the neck region might not be as* distinct as for intrinsic  -59-  copper. 6 mm  The improvement was made by placing lead s l i t s preceding  the  holes so that the width along the direction of detector movement  became 3.5 mm.  In terms of the angular resolution at the origin of k  space the angle <J> i s now about 7°.  The resolution function is no  longer cylindrical in nature but becomes a rectangular pipe mapping k space.  The resolution was found to be adequate up to the highest  impurity concentration studied in the present experiment. As previously mentioned a <111>  axis of the sample defines the  axis of the system however since seven sets of detectors at either end are used i t is impossible to position them a l l along the axis.  Figure  12 indicates one set of detector positions; four sets of detectors (defining two vertical planes) were positioned to one side of the axis and three sets of detectors ( a l l i n the same vertical plane) were located on the other side.  The distance from the <111>  of detectors is 25 cms  axis to the extreme set  and this corresponds to an angle of 1.8°.  In  essence this means that the mapping surfaces are inclined to one another by <4° in the case of the extreme sets of detectors and the third set lies somewhat in-between. It must be noted that the contribution of the positron motion has not been considered up to this point.  Since the experiments are done at  room temperature the positrons are thermalized and w i l l be characterized by a momentum distribution given by Maxwell-Boltzmann s t a t i s t i c s . Donaghy (1964) has found that the positron motion can be thought of as a contribution to the resolution function of an additional 0.05 radians.  milli-  This contribution does not appreciably alter the calculated  resolution function.  -60-  F_.  Chance Coincidence The contribution of chance coincidence to the correlation data  can be estimated using equation 5-1.  If one uses an average value for  the resolving time of 21 nano-seconds and average singles values of 1800 counts per second (as determined at the start of each run) then the calculated chance coincidence rate i s about 8 counts per minute. An average value of coincidence counts for that period of time i s ^ 200 so the chance coincidence contributes about 4%.  The cut-off data gen-  erally has about 80 counts per minute at the start, however since the cut-offs are not measured until the neck widths have been determined the singles rates are much lower, of the order of 1000 counts per second.  In this case the chance coincidence constitutes about 3%.  The  corrections for this time varying contribution to the angular correlation were easily carried out.  G_.  Neutron Damage The use of neutron radiation to create the source of positrons has  one major drawback i n that possible damage may be done to the sample. The extent of such damage i s largely unknown and most experimenters simply follow the convention of re-x-raying their sample after the neutron bombardment.  No alteration i n the x-ray pattern as compared  with the pre-neutron bombardment pattern i s considered sufficient evidence that no major damage has occurred to the sample. obviously of only limited accuracy.  This method is  Recent work howevery by Senicky  (1973), has compared the angular correlation obtained from a neutron  irradiated copper sample with that using a sodium 22 source for the positrons.  No difference between the curves i s observed for angles  less than 7 m.r. however the two graphs differ for values of .6 above 8 m.r.  In this experiment the largest angle studied was 6 m.r. so  this recent data coupled with the x-ray analysis i s considered v e r i f i cation that the effect of neutron bombardment on the angular distribution i s small.  In any event no sample in this experiment was irradiated  twice - i f a second run was desired then a second sample was cut.  CHAPTER VI EXPERIMENTAL ANALYSIS  A.  Theoretical  Resolution  The effects of the experimental resolution on the determination of the pure copper Fermi surface has been calculated by Petijevich (appendix I) for both the point geometry and the new higher resolution geometry.  This computer determined resolution i s found by dividing the  collimating s l i t s and the sample into small elements and calculating the number of coincidences possible between these elements as a function of angle.  The resultant resolution function i s closely represented by a  Gaussian with a f u l l width-half maximum (F.W.H.M.) i n the z direction of 1.13 m.r. for the point geometry and a F.W.H.M. of 0.735 m.r. for the high resolution (the F.W.H.M. in the x direction for the high resolution is 1.12 m.r.).  The resolution also includes an enhancement factor  (Kahana 1967) , however i t ignores the complicated effects associated with the contribution of higher momentum components to the electron and positron wavefunctions.  In addition, i t should be noted that the posit-  ron mass i s chosen to be equal to the free electron mass (experimental evidence indicates a positron mass i n the neighborhood of twice this value), however this over-estimation in broadening i s partly compensated for i n that the s l i t width adapted" for the experiment i s slightly wider than used in the theoretical calculations. The effect of folding-the resolution function into the copper Fermi surface described by Roaf (1962), along with the angular distribution data for copper, are shown i n figure 15. It is obvious from the figure  -63-  that relatively poor agreement i s achieved and the contribution of core annihilation, principally from the d band, to the angular distribution must be considered.  An indication of the difficulty involved in account-  ing for the core effects i s apparent from the work of Senicky (1973). The Senicky calculation, using a plane wave expansion in terms of reciprocal lattice vectors (or by an alternate method using a Wigner-Seitz wavefunction) for the positron wavefunction and free atomic wave functions for the core electrons, indicate a core to conduction electron counting rate of not more than 0.1 at 6 = 0 m.r. This result, which pertains to the identical orientation used in the present work, i s well below the counting rate for 9 > 9^ encountered in the copper angular distribution i n figure 15 and at odds with experimental data from other positron work on copper.  For the sake of comparison, we have assumed  that the higher of the Senicky calculated core contributions w i l l be unaffected by the present resolution function (since the core effects are relatively constant for 9 < 6 m.r) and have arbitrarily adjusted the core to conduction ratio at 6 m.r. to be 0.25 relative to the theoretical conduction electron distribution discussed earlier.  The results are  shown i n figure 16 and slightly better overall agreement i s achieved. I t is possible the inclusion of the electron higher momentum components would further improve the agreement, however the use of Kahana's enhancement factor is questionable i n view of the results of Fujiwara (1966, 1968) that an abnormally large* enhancement occurs i n th.e neck region. A consistent corroboration of this has been found in the results of Morinaga (1972).  On the other hand evidence for "negative enhancement" of the  higher momentum components has been found by Fujiwara et a l (1971). Yet  -64-  e  <  -•*«-  o  Folded Resolution  o  Q  o  Present Copper data  o o  —  no data * gathered —v \  "0 \ I  \ \  o I  0  1  I  o « I  o  \  0  «  0  Vo?  Figure 15:  210  Ji5  430  5-25  tto  Resolution Function folded into Copper Fermi Surface  m.r.  -65-  i  .  •  J05  2.io  Figure 16:  ;  \\5  42o  S.1S  iia  Resolution Function folded into Copper Fermi Surface—core included  -66-  a n o t h e r c o m p l i c a t i o n may  a r i s e f r o m the d i f f r a c t i o n o f the gamma r a y s  (Hyodo e t a l 1971, Hyodo 1973) w h i c h can. have s i g n i f i c a n t e f f e c t s upon the  shape o f the a n g u l a r d i s t r i b u t i o n c u r v e s . We  c o n c l u s i o n o f M o r i n a g a (1972) who  can o n l y echo t h e i i  s t a t e s t h a t the e x i s t e n c e o f t h e  enhancement e f f e c t combined w i t h t h e unknown c o r e c o n t r i b u t i o n c o m p l i c a t e s t h e s t u d y o f t h e F e r m i s u r f a c e and i t may  greatly  t h e r e f o r e be  d i f f i c u l t t o d e t e r m i n e q u a n t i t a t i v e l y an unknown F e r m i s u r f a c e . i n g l y we h a v e a d o p t e d the e m p i r i c a l approach o f r e l a t i n g d i s t i n c t  Accorddevia-  t i o n s f r o m a smooth a n g u l a r d i s t r i b u t i o n c u r v e t o t h e neck r a d i u s and Fermi s u r f a c e  cut-offs.  S i n c e t h e p u r e c o p p e r neck r a d i u s and <110> measured  c u t - o f f have been  a c c u r a t e l y ( H a l s e 1969) t h e n a comparison o f t h e known v a l u e s  w i t h t h e p r e s e n t d a t a s h o u l d e n a b l e us t o d e t e r m i n e t h e e f f e c t s o f o u r m e a s u r i n g a p p a r a t u s on t h e t r u e v a l u e .  D i s t i n c t d e v i a t i o n s f r o m smooth  b e h a v i o r a r e c h a r a c t e r i s t i c o f n o t o n l y t h e pure copper d a t a b u t a l s o o f the  a l l o y r e s u l t s as w e l l .  The f o l l o w i n g s e c t i o n i n c l u d e s an a n a l y s i s  b a s e d on d r a w i n g a smooth c u r v e t h r o u g h the a n g u l a r d i s t r i b u t i o n d a t a and a t t r i b u t i n g t h e d i f f e r e n c e between the measured neck r a d i u s and resolution.  T h i s method i s supplemented i n a l a t e r s e c t i o n where t h e  pure copper d a t a i s s u b t r a c t e d from the a l l o y B_.  Copper  <110>  results.  Analysis  The a n a l y s i s o f the p u r e copper sample w i l l be d e s c r i b e d i n d e t a i l s i n c e a common a n a l y t i c p r o c e d u r e was used f o r a l l t h e a l l o y s a m p l e s . The n e c k r e g i o n o f copper i n t e r s e c t s t h e B r i l l o u i n zone boundary a t r i g h t angles ( S e g a l l 1962), thus a sharp d i s c o n t i n u i t y i n the a n g u l a r c o r r e l a t i o n  -67-  graph should be evident when the moveable detector system passes through the neck region.  The finite resolution of the apparatus and thermal  motion of the positron w i l l introduce a smearing effect on the results. Accordingly, the location of the neck region on an angular distribution curve should be manifested i n a reasonably distinct drop from a central plateau region to a smooth shoulder described by a parabolic curve.  The  sample i s oriented such that a <lll>direction is aligned along the axis of the system (towards the detectors) and a <100> direction is parallel to the detector motion. Figure 17'A' i s the i n i t i a l sweep of the moveable detector assembly from the nominal centre position of 9 = 0 milli-radians to 9 = 3 milli-radians.  Figure 17'B' also includes the combined results of  sweeps 2 and 3 where sweep 2 is from 9 = 3 milli-radians (right hand side of <111> axis of the system) to 9 = -3 milli-radians (left hand side of axis) and sweep 3 is the return pass.  A l l results have been  corrected for the radioactive decay of the sample. the neck region of copper.  Both curves indicate  For the right hand side of the 9 = 0 ° posi-  tion, sweep 1 indicates a drop off in the v i c i n i t y of 1.0 m.r. and this' is verified by sweeps 2 and 3. With regard to the l e f t hand side of the 0 = 0 position, sweeps 2 and 3 indicate a discontinuity in the region about 1 m.r. indicating that the true centre of the correlation coincides with the nominal centre.  In the study of the alloys a l l but two of the  angular correlations were symmetrical  about the nominal centre.  The two  exceptions indicated a shift of 0.1 m.r. was necessary to align the true and nominal centre; the shift presumably reflecting the placement of the sample.  -68-  Th e total number of counts for both sides of the 0 = 0 ° axis i s shown in figure 17'C.  The data i s based on an even number of sweeps  for each side (that is the number of sweeps from 0 to +3 milli-radians is the same as the number of sweeps in the opposite direction) to counter the effects of possible d r i f t in the electronics. The results of figure 17'C  indicate  clearly that the true centre  of the angular distribution coincides with the nominal centre.  The l o -  cation of the true centre enables folding of the data whereby the number of counts in channels symmetric about the true centre are added. The folded data are shown in figure 18.  The neck radius of copper  is determined to be 1.20 m.r. with an uncertainty of 0.10 m.r.  This  result can be compared with previous determinations of the neck radius (Halse 1969 and Lee 1969) which yielded a value of 0.99 m.r.  The d i f f e r -  ence between the two numbers i s attributed to the effects of the experimental resolution adding about 0.20 m.r. to the present measurement. The data for the <110>  copper cut-off is shown i n figure 19.  The  discontinuity in the angular distribution occurs at 5.15 ± 0.10 m.r. is indicated by an arrow i n figure 19.  and  The accepted value for this cut-  off from the work of-both Halse and Lee i s 4.98 m.r.,  thus the cut-off  data provides strong confirmation that the effect of the experimental resolution i s to add about 0.2 m.r. to the measured cut-off. correction i s applied to a l l neck radius, <110>  This  cut-offs, and to the  single <100> cut-off measurements for the copper alloys. It should be noted that the <110> at 7.1 m.r.,  Brillouin zone boundary occurs  thus the Fermi surface of pure copper f a l l s well short of  -690  0  'A" Sweep 1 2.0  6 m.r.  1.0  2.0  Q  1.0  2.0  1.0 0  t  'B" Sweeps 2 and 3  m.:  1.0 6  6  .95  CO  $ .90 O  c_>  > •H  ni . 85 <y  .80 'C" Total Counts  Figure 17:  Copper Angular Distribution Data  6  m.r.  -70-  -71-  -72-  this boundary in contrast to the made.  <111> direction where contact has been  The significant contribution of core electron states to the  angular distribution i s obvious in the region of k > k  C^. (i)  for figure 19.  Copper Alloy Analysis Smooth Curve Behavior The analysis of the copper alloy samples follows the same format  as for the pure copper sample, that i s f i r s t defining the true centre of the angular distribution graph then folding the results about that centre.  The apparatus is constantly cycled at periods of four or eight  minutes to reduce any d r i f t effects of the electronics.  A similar correc-  tion of 0.20 m.r. is applied to the alloy data for the effects of the experimental resolution. The following copper alloy samples have been studied: 12.7% gallium and 3.3, 6.0, and 8.2% germanium. tions are expressed in atomic percent. twice and i s correct to within 0.1%.  A.9, 8.7, and  A l l impurity concentra-  Each composition has been analyzed Three graphs have been compiled for  each sample, the determination of the neck radius from the folded angular distribution, the neck width compared with the pure copper neck width, and the <110>  alloy cut-off with the corresponding pure copper value. A l l  quoted errors are based on one standard deviation s t a t i s t i c a l errors so a point representing 10,000 counts i s correct to ± 1%. the <110>  In the graphs for  cut-offs the error bars for the pure copper data have been re-  duced in order to f a c i l i t a t e the interpretation of the alloy behavior. The arrows on the graphs indicate the uncorrected neck radius or cut-off feature.  The neck radius has 1% peak statistics for a l l samples.studied  with the exception of the intermediate CuGe sample which has 1.25% istics.  stat-  -73-  The results of the direct analysis of the alloy data, corrected for the effects of experimental  resolution, are shown in figures 20 to  25 for the neck radius data and i n figures 26 to 31 for the <110> cutoffs.  A single determination of the <100> cut-off for the CuGa 12.7%  sample is shown in figure 32.  .  .  The corrected neck radii and <110> cut-offs are shown i n figure 33 and 34 respectively. It i s readily apparent from both figures that the CuGa Fermi surface i n the neck and <110> regions i s expanding more rapidly than the CuGe Fermi surface.  The neck radius for the highest  CuGa concentration studied i s over twice that of pure copper whereas the <110> cut-off i s 11% higher i n the alloy than i n copper.  The corres-  ponding values for CuGe relative to copper are 87% and 10% respectively. Two neck radius measurements by Hasegawa (1972) are also included i n figure 33. (ii)  Difference Curve Behavior One of the disadvantages of the method used for determining the  neck radius i n the alloys i s that the results tend to be highly dependent upon one or two points i n the angular distribution curve, which are of course subject to s t a t i s t i c a l fluctuations. Another approach i s to subtract a smooth copper curve from a l l the alloy results and then to require that the difference curves bear some resemblance to one another within the alloy series'. This family of curves places somewhat less emphasis upon individual points although, as w i l l be readily apparent from the data, i t is also a moot point but we would expect similar features to be related. This procedure i s applied to the CuGa data i n figure 35'A' and the CuGe data in figure 35'B'.  Shown above the figures i s the difference  1.0  -74-  1 K T  1  \  4  to  •u § 0.95 o u  >  \ x  •H 4-J nJ rH 0)  Pi  1  A  M  \  0,90  0.5  1.0  Figure 20:  1.5.  2.0  3.'0  2.5  0  m.r.  Cu(95.l)Ga(4.9) Neck Radius  » o  © © o ©  o o  o *  o  Copper  6  Alloy  0  ro  u C O  o >  0  Pi  e  o  Copper and CuGa .4.9  o  Angular Distribution  »  0  1  1.0  -75-  n \  w  1 3  0.95  O  u > •U CO  rH Pi  t  N  •i  0.90  0.5 Figure 21:  1.0  1.5  2.0  3.0  2.5  0  Cu(91.3)GaC8.7) Neck Radius e 6  e  0 0  fi  t>  O  8  °  o § ©  o Copper © Alloy  w  0  0  tt  •  6  e  e  0 0  ©  &  0  o 0  °  O  Copper and CuGa 8.7 Angular Distribution  0  m.:  0.5  1.0  Figure 22:  1.5  2.0  2.5  3.0  3.5  Cu(87.3)Ga(12.7) Neck Radius ©  0  0  © © o o  &  ©  O o  0  o  6  Q Copper 0  0  o Alloy  0  ©  Counts  o  ©  0 0  0  •rH •U  0 0 e  o o  n  Copper and CuGa 12.7 Angular Distribution  e  $  -77-  1.0 1  *  m  f.l  4->  § o u  0.95  \  cu >  V  iH +J  crj  r-H  1  cu Pi  0.90  0.5 Figure 23:  1.0  1.5  2.0  2.5  .  3.0  0  Cu(96. 7)Ge(3. 3) Neck Radius © ©  © ft  o  °  ° °  *  e  ©  6 0  o Copper  •  0  © Alloy 0  to  C  3 0 •H  O  co  o  PS  6 0  e  o  o  "  #  o  ft  Copper and CuGe 3.3 Angular Distribution  m.r.  -78-  1.0  kl CO  •u  0.95  \  c 3  o  -  _\1  >  •h 0)  Pi  0.90  \  0.5  1.0  Figure 24;  1.5  2.0  r-  3.0 G m.r,  2.5  Cu(94. 0)GeC6, 0) Neck Radius • °  0 e  o G  8 0  0  o  o  Copper  © Alloy  •  •  o  o CO  0  ©  ©  0  O  §  0  o c_> <D 4J  cd cu Pi  o  »  6  •  0  0  0 e  ©  Copper and CuGe 6.0 Angular Distribution  1.0  -790v  c 3 o 0.95 o o  M  >  •H •U rd tH CU  Pi  r  1N  0.90  0.5 Figure 25:  1.0  2.0  1.5  \  3.0.  6  m.r.  O  Copper  Cu(91.8)Ge(8.2) Neck Radius 0  6  ° o o  O o ®  6  © Alloy  0  co  4-1  e  O  0  c 3 O  cu  o  e  o  > G  cu PJ  Copper and CuGe 8.2 Angular Distribution  O  Copper  © Alloy  4. 1  3.5  4.0  Figure 26:  ^•5  5.0  5.5  Cu(95.1)Ga(4.9) <110>  6.0 cut-off  R e l a t i v e Counts o  o  o  o  CO  -E-  Ul  ON  O  O  o  oo  CO Cn]  C3  o  o  us-. 0—-I 1  c Cn  o £3 C3  Cn  -o- ,  O  o v r. c rr C  s  Cn Cn  )-a  O  > o 3  - T 8 -  o o  1.0  0.9  *1\  0.8  O  Copper  © Alloy 0.7, \  \  5  v  0.6 c 3 o  I  00 N3  CO  I  >  u 0.5\  rd rH  •Mr  Pi  f  0.4  0.3  3.5  4.0  Figure 28:  4.5  5.0  6.0  5.5  Cu(87.3)Ga(12.7) <110>  cut-off  0  m.r.  1.0  0.9  O Copper  0.8  © Alloy  *M  0.7  5  I,  co  £ o u  5  0.6 i  co  4  0.5  I  t  Pi  0.4  0,3  3.5  4.0  Figure 29:  4.5  5.0  5.5  Cu(96.7)Ge(3.3) <110>  6.0 cut-off  0  m.r.  .1.0  si  O Copper O Alloy  0.  f  9  i  0.3  3.5  4.0  Figure 30:  4.5  5.0  Cu(94.0)Ge(6.0) <110>  5.5 cut-off  6.0  e  m. r .  Relative Counts o •  to  o •  o •  Ui  / *-o-« I  1  /  o .  o .  CO  »vl  >  J  —©—•  o •  CT>  i-o-<  o  v;  -S9-  M  O O  ^  ro  o  .  VO  Figure 32:  Cu(87.3)Ga(12.7) <100> cut-off  -87-  2.2 © O © 0  2.0  Copper Gallium Copper Germanium Copper Copper Germanium (Hasegawa)  CO  c  CO •H  cd  r-<  I  •H  1.8  I CO 3 •H  cd  1.6  Pi  u J3  1.4  1.2  1.0  1.0 Figure 3 3 :  1.1 Neck Radius Results  n  1.2  -88-  1.0  Figure 34:  @  Copper Gallium  Q  Copper Germanium  ©  Copper  1.2  1.1  n  <110> Cut-off Results  ]  -89-  Zone Boundary Alloy Fermi Surface Copper Fermi Surface  Mapping Area  Scale: 1 c.m,=0.22 m.r  COPPER GERMANIUM tr  0 cms.  Error bars Scale: 1 cm. =0.22 m.r  COPPER GALLIUM x  C  e> e  cms • Figure 35 " " A  art(  j "B"  Difference Curves for CuGe (top) 'and CuGa (bottom)  -90-  — ' —  ALLOY  1  •  Difference Curve (mr)  Direct Analysis (m.r.)  CuGa A.9  l.AA  1.A6  CuGa 8.7  1.73  1.62  CuGa 12.7  2.20  2.15  CuGe 3.3  1.30  1.30  CuGe 6.0  1.53  1.A7  CuGe 8.2  1.72  1.86  Table 3 Neck Radius Results - Difference Curve and Direct Analysis  * corrected for resolution  -91-  region sampled.  The curves shown represent a f i t to the data where a l l  points are equally weighted.  As expected the results indicate a similar  difference behavior in the alloys for 8 < 1 m.r.  As the neck region i s  approached in the respective alloys the curves show a general rise and the grouped results do indeed resemble that of a family of curves. The error bars for the difference points are ± 10 on the scale used.  It i s  obvious that other curves could be drawn for the difference data, however the point where the peaks begin to f a l l , which are taken to represent the alloy neck radii and are denoted by arrows, are not especially sensitive to weighing an individual point more or less. The results of the analysis are tabulated (table 3) and also included are the-corrected neck radius values previously determined by 1  direct analysis.  Reasonable agreement i s evident for most samples, how-  ever the unknown effects of the resolution make a rigorous comparison difficult.  It would be expected that the difference analysis might yield  values somewhat in excess of the neck r a d i i determined from the direct analysis, and i f so the highest CuGe neck radius as found from the direct analysis may be too high.  For this particular sample the quoted neck  radius i s taken to be an average of the direct analysis and the d i f f e r ence curve behavior which is found to be 1.79 m.r. In summary, the present data has been considered from several viewpoints.  The theoretical aspects of the resolution behavior on the  measured angular distribution were given and found to be highly dependent on the core annihilation.  This analysis is further complicated by  enhancement effects and the contribution of higher momentum components. A l l these factors make a theoretical approach untenable.  Another  -92-  procedure, whereby a smooth curve was drawn through the data points, indicated sharp discontinuities which could be related to the neck radius and <110> cut-off.  These discontinuities, when compared with  the pure copper neck and <110> cut-off, indicated 0.2 m.r. should be subtracted to account for the resolution.  It i s this analysis, which i s  dependent on distinct deviations from smooth behavior i n the angular distribution results that appear best suited to determine the alloy neck and cut-off features. A reasonable confirmation of these results was evident when difference curves were plotted for the various alloys.  This  analysis, whereby the pure copper data i s subtracted from the alloy results, includes an undetermined resolution effect and yields data points with large error bars. The present results are compared with two other alloy systems, CuAl and CuZn i n figures 36 and 37 respectively.  No discussion of this com-  parison i s attempted, however i t should be noted that the CuZn system i s mentioned i n some detail i n subsequent chapters.  D.  Core Effects It i s interesting to note that an experiment of this nature might  be valuable i n determining the core effects due to the copper d band. If an impurity i s added which w i l l not contribute to the core annihilation (that i s with a well localized d band) then the reduction i n core annihilation for k < k  f  correlated with the amount of impurity added would  serve as an indication of the copper core contribution.  I f the impurity  has a valence of three or four then the additional ionic charge would also tend to reduce any positron - core annihilation.  It would be necessary to  -93-  correct for higher momentum effects which, although low in the direction, are expected to vary rapidly for k > k^.  <110>  To a certain  extent the impurities added in the present experiment would satisfy the d band c r i t e r i a although each probably accounts for a small core annihilation.  Indeed the two most concentrated alloy samples for the  systems studied indicate a 3% decrease in the counting rate at 8 = 6 m.r., however the statistics in this region do not allow for any quantitative analysis.  -94-  O  Thompson  A  Fujiwara  U  Murray  (y Becker ©  Halse  T  0 1  10  U  t2  1.3  Figure 36: Previous Copper Aluminum data with present trends for CuGa and CuGe  -95-  O Morinaga  2.2  •v* Becker 0 Trifthauser 2.©  © Halse CO  C cd  •H  rd M I  •H  6 CO  3 •H  1.6  t> CO  A! o cu  1.4  1.0  t.o Figure 37;  u  t.2.  Previous Copper Zinc Data with present trends for CuGa and CuGe  1.3  -96-  CHAPTER VII INTERPRETATION AND CONCLUSIONS A.  Introduction The interpretation of the present results i s made d i f f i c u l t due  to the inherent problems i n applying theory to concentrated, disordered alloys.  The periodicity of the pure metal i s lost i n the alloying pro-  cess and concepts such as Bloch states and Fermi surfaces cannot be rigorously applied to the alloy.  As discussed previously, the i n i t i a l  approach in the alloy study, i s the application of the general theory of pure metals to the concentrated alloy system.  A f r u i t f u l starting  point i n the analysis i s to assume that the alloy possesses the same electronic structure as the host, that i s the additional conduction electrons occupy the k states of the pure metal.  This behavior i s the  essence of the rigid band model and has been predicted (Stern 1969) to be of some value i n describing noble metal alloys.  The present chapter  includes an analysis of the data i n terms of rigid band predictions and also considers other implications of the experimental data.  15.  Rigid Band Theory Constant energy surfaces have been computed for copper (Faulkner  1967) using the Kohn-Korringa-Rostoker  (K.K.R.) method.  The potential  used i s essentially the same as used by Burdick who derived the copper t  band structure from an A.P.W. analysis.  The integrated density of  states M(e)[M(e^) = n] has been calculated for various energies above the copper Fermi level and thus i s of use i n the analysis of the present  -97-  alloys.  A tabulation of M(e) versus energy (where M(e^) = n = 1  defines the copper Fermi level) i s shown in table 4.  From this table  i t i s possible to determine the constant energy surfaces for the alloys studied i n the present experiment.  The Fermi energies for the  alloys can be corrected for the change in the copper, lattice constant by using the results of Hume-Rothery (1936) who has studied both alloy systems.  The corrected Fermi energies have been expressed in rydbergs  in table 5.  To evaluate the rigid band behavior of the neck and <110>  cut-off i t i s necessary to refer to a band structure diagram for copper as shown i n figure 4.  The energy band structure has been expanded in  the region of interest near the Fermi level i n figure 38. in this figure are the <110>  Also shown  band and the energy band corresponding to  the copper neck both of which are indicated by dashed lines above E^ to denote non-occupancy.  The rigid band Fermi levels for n = 1.1,  1.18,  1.23, and 1.25 are drawn in increasing order above the copper Fermi level.  The predicted neck radius for these n values can be found by  measuring the distance from the <111> intercept of the  zone boundary to the corresponding  Fermi level with the <lll>band. Likewise the <110> •  cut-offs can be found by measuring from T to the intercept of the <110> band with the appropriate Fermi surface.  In both cases the predicted  r i g i d band values are found by multiplying the measured distances by the corresponding pure copper value. The rigid band calculations are compared with the experimental data i n table 6 and also i n figures 39 and 40. A similar rigid band analysis can be applied to the <100> Fermi surface using the copper band structure shown i n figure 41.  The pre-  dicted Fermi energy for the highest concentration CuGa alloy i s shown  -98-  M(E)  E(Ry) 0.66168 0.66675 0.66845 0.67014 0.67860 0.68806 0.70398 0.72090  M(E)  E(Ry)  0.97542 0.99392 1.00002 1.00613 1.03655 1.06673 1.12608 1.18444  1.24141 1.29727 1.35202 1.40578 1.43261 1.45886 1.46408 1.46932  0.73782 0.75475 0.77167 0.78859 0.79705 0.80551 0.80720 0.80889  Table 4: Density of States in Copper for Various Energies  n  AE (Rydbergs)  E (Rydbergs)  1.0  -0.192  1.10  -0.161  .031  1.18  -0.141  .051  1.23  -0.131  .061  1.25 •  -0.124  .068  j  .  ! t  Table 5: Rigid Band Energies for Different Electron per Atom Ratios  Alloy  Neck Radius R.B. Theory  Neck Radius Experiment  <110> cut-off R.B. Theory  <110> cut-off Experiment  CuGa4.9  1.34±.06 mr  1.46±.l mr  5.23±.05 mr  5.101.1 mr  CuGa8.7  1.531  1.62±  "  5.40±  "  5.41±  "  CuGal2.7  1.70±  2.15±  "  5.52±  "  5.591  "  CuGe3.3  1.34±  1.30±  "  5.23±  "  5.101  "  CuGe6.0  1.53±  1.47± . "  5.40±  "  5.351  "  CuGe8.2  1.66±  1.79±  5.48±  Table 6:  "  "  Rigid Band Prediction and Experimental Data  5.48+  11  -99-  -100-  -101-  -102-  +0.2  Figure 41:  Energy Band Diagram for <100> Cut-off  -103-  and corresponds to a <100> radius of 6.20 m.r.  The empirically deter-  mined <100> cut-off was found to be 5.92 m.r. after correction for the resolution.  Thus the expansion of the alloy Fermi surface in this  direction is less than predicted by rigid band theory. discussed  This point is  subsequently.  It should be mentioned that the pure copper k values used in the rigid band analysis have been taken from Halse (1969) and Lee (1969) who have calculated identical k values (correct to three figures). values for k<100>, k<110>, and 0.256 in units of A° are 5.56,  4.98, and  are respectively 1.44,  1.29,  These and  The corresponding k values in milli-radians 0.99.  The two copper alloy systems have been studied by Coleridge and Templeton (1971) in the dilute region of impurity concentration of less than 0.1 atomic percent.  This group used the de Haas van Alphen method  to measure the frequency of neck orbits in the alloys and, combined with previous data on the cyclotron mass (Joseph et a l 1966) and thermal mass (Martin 1966), were able to calculate dA/de.  Assuming cylindrical necks,  Coleridge finds that the predicted rigid behavior for the copper alloys i s a neck radius versus n ratio of 3.1.  Experimentally the measured  ratios for the dilute CuGa and CuGe alloys were found to be 3.0 and  1.7  respectively (the extrapolated dilute alloy ratios are shown as dashed lines i n figure 39).  On this basis Coleridge concludes that the dilute  CuGa system exhibits rigid band-like behavior whereas the CuGe alloys do not. The present data indicates a CuGa neck radius per n ratio of 3.4 for n < 1.2.  It thus appears that the addition of concentrated  Ga  -104-  impurities to pure copper has not appreciably altered the alloy neck region behavior from that of the dilute system.  No measurements were  made on the <100> cut-off by the Coleridge group due to the lack of sensitivity of this Fermi surface to changes in electron concentration. The discrepancy between the highest concentration neck radius (fot n = 1.25)  and rigid band theory is very marked and represents a true  departure from a rigid band-like behavior.  This point i s strongly  corroborated by the <100> measurement which was found to be 0.28  m.r.  lower than the rigid band prediction ( i f the Fermi surface has bulged out i n the neck region then i t must be compressed in the belly region). The present data also indicated a CuGe neck radius per n ratio of about 2.7 for n < 1.2 which is somewhat greater than the dilute measurement but s t i l l less than rigid band theory.  The present data combined  with the dilute CuGe work casts considerable doubt on the findings of Hasegawa (see figure 33) who quotes a neck radius of 1.6 m.r. CuGe alloy of n = 1.1.  for a  The Hasegawa measurement indicates a neck radius  expansion well in excess of rigid band theory.  His other result of a  2.0 m.r. neck radius for a CuGe alloy with n = 1.27 i s consistent with • the present data.  The <110>  CuGe cut-off results are in agreement with  the neck radius data since the data points are depressed below the rigid band curve for n < 1.2.  In addition, the highest concentration result  does show an upward shift as noted for the neck radius behavior. In summary, the CuGa neck radius and <110>  cut-off data exhibit  r i g i d band-like behavior for n < 1.2 and an upward departure from rigid band predictions for n = 1.25.  The neck radius data extends (and agrees  -105-  with) the observations of Coleridge for the dilute system.  The neck,  radius departure from rigid band theory for n = 1.25 i s supported by the <100> cut-off having a somewhat less than rigid band expansion. The present CuGe neck radius data extends the dilute alloy observations of Coleridge that the neck region expansion i s less than predicted by rigid band theory, although the concentrated alloys show a greater expansion than that determined for the dilute alloys.  The <110> cut-off  values are i n substantial agreement with.the neck radius data for both alloy systems presumably reflecting the pulling out of the <110> Fermi surface as the neck expands.  C_. Alloy Fermi Surface The neck radius, <110> cut-off and the single <100> cut-off data can be used to plot 'the change i n Fermi surface as impurity electrons are added to the intrinsic copper.  The copper crystal i s a face centered  cubic structure and hence the reciprocal lattice i s body-centred cubic. The Brillouin zone i s shown i n figure 42 where the hexagonal faces of the zone are perpendicular to the <111> directions and the square faces are perpendicular to the <100> directions.  The contours of the Fermi surface  of pure copper can be drawn from known data on the width of the necks subtending the origin of k space and the dimensions of the <100> and <110> cut-offs relative to the Brillouin zone boundaries i n those directions.  The inner boundary i n figure 42 represents the pure copper Fermi  surface.  The primary effect of adding additional electrons to the copper  solvent i s to increase the width of the necks as found i n the present experiment.  The increase i n the neck dimension pulls the <110> Fermi  surface outwards whereas the <100> Fermi surface expands only slightly.  -1061  In figure 42 the three boundaries beyond the pure copper Fermi surface correspond to the copper gallium alloys studied in the present experiment.  The <110> Fermi surface for the alloys appear to increase  in a monotonic fashion with no evidence of bulging towards the zone boundary.  The measured <100> cut-off for the highest- atomic percent  concentration copper gallium alloy indicates less bulging of the Fermi surface in the <100> direction as the alloy concentration increases. This result i s due to the proportionately greater increase in neck radius compared to the slow increase in the <100> cut-off. The copper germanium results (figure 43) are quite similar to the CuGa data although the neck expansion i s not nearly as pronounced. No <100> Fermi surface cut-off was obtained for the CuGe system, accordingly no contours have been plotted i n the belly region.  On the basis of  the CuGa results i t would be expected that the <100> CuGe Fermi surface would tend to bulge out slightly from that of the CuGa Fermi surface. This reasoning follows from the larger CuGa neck expansion relative to CuGe as shown i n figure 39. D_. Discussion There is no explicit theoretical calculation of the Fermi surface for the alloy systems studied.  The only quantitative statements that can  be made relate to the rigid band model which, i n an unmodified form, i s unable to account for electronic specific heat measurements.  I t has been  suggested by Stern (1969) however, that the noble metal alloys may fortuitously exhibit a rigid band-like behavior due i n part to the large energy gap between f i l l e d and unfilled conduction bands.  -108-  Figure 43:  Copper Germanium Fermi Surface Contours  -1091  The present CuGa neck radius data indicates a slightly larger dependence upon the electron per atom ratio than predicted by the rigid band model for n < 1.2 and a progressively larger dependence at the highest concentration studied (n = 1.25).  The latter measurement is  confirmed by the less than rigid band increase in the <100> Fermi surface cut-off for that alloy.  The neck radius behavior for n < 1.2 is in  close agreement with the Coleridge and Templeton work (1971) on dilute CuGa alloys. The <100> Fermi surface measurement has special significance in terms of the Hume-Rothery rules.  It has been suggested by Hume-Rothery  and Roaf (1961) that the reason for the a-8 phase transition in the neighborhood of n = 1.40  for the noble metal alloys may be due to contact  of the <100> Fermi surface with the Brillouin zone boundary.  If we  extrapolate the measurement for the n = 1.25 CuGa alloy, there is l i t t l e likelihood of such contact at n = 1.40.  A recent communication from  Stern also indicates there is no contact with the boundary for gold alloys up to n =  1.42.  The positron annihilation method is not sensitive to the variation in the energy gap between the lower f i l l e d conduction band (point figure 44) and the, upper unoccupied band (point  , in  in figure 44) . It is  clear however that relative to the pure copper band structure, the energy shifts resulting from the addition of gallium, are such that the energies of states in the <111>  neck region are lowered with respect to those i n  the <100> belly region. The CuGe neck radius data has an i n i t i a l rate of increase with n which i s somewhat less than rigid band and, although a subsequent rise  -110-  appears to occur for n = 1.23, the data i s consistently below the CuGa results.  Likewise the <110>  cut-offs for CuGe are generally lower than  the corresponding CuGa cut-offs for a given n.  The work of Coleridge  and Templeton on dilute CuGe alloys also indicates lower than r i g i d band behavior for the neck radius increase with n.  It would be expected that  the <100> Fermi surface radius for CuGe would be correspondingly larger than the CuGa radius for a given n however this has not been measured. The problems involved i n the interpretation of the experimental data are considerable. If the sytems studied were free electron like then information on.the conduction band energy gaps could be used to determine the change in the [111] and [200] Fourier components of a local pseudo-potential.  Unfortunately, for the case of copper, the anal-  ysis i s greatly complicated by the interaction of the s,p conduction band with the d band.  It i s this hybridization which gives rise to the dis-  torted pure copper Fermi surface. Even the simplest -model for the situation requires a non-local pseudopotential (Taylor 1969) and then no simple relation exists between the energy gaps and the Fourier components. Soven (1966) has applied a coherent potential model (see chapter III) to the a phase CuZn system. (n = 1.40) the <111>  The theory indicates that at 40% zinc  energy gap (L^  f  L^ in figure 44) should be about  10% lower than the value in pure copper with both L^, and especially L^ predicted to decrease i n energy as the concentration of zinc increases. Also, the point X^,, i s s t i l l well above the Fermi level at n = 1.4 so that no contact with the zone boundary i s predicted. Another calculation using the coherent potential model has been applied by Das and Joshi (1972) to CuZn.  This theory i s applied to the  -Ill-  -112-  measurements of the variation in the energy of the as determined by Biondi and Rayne (1959).  , -*•  transition  Reasonable agreement between  the theory and data is apparent up to the highest concentration studied which was 30% zinc.  It should be noted that the Soven calculation pre-  dicts a much less L , -*• L  energy decrease than does the Das and Joshi  theory. The application of the coherent potential theory of the CuZn system to CuGa or CuGe would not be of any value since the nature of the impurity would be expected to exert a large influence on the behavior of the alloy.  The present measurements can, in part, be related to the  results of other methods applied to concentrated copper alloys.  E.  Optical Studies The study pf optical transitions i s complicated by the assignment  of the observed transitions to the band structure involved.  The distinc-  tion between direct and non-direct transitions has also not been clearly resolved and i t is expected that the proportion of non-direct transitions w i l l increase with impurity concentration (Seib and Spicer 1969).  As  mentioned previously,- a coherent potential calculation (Das and Joshi 1972)  for CuZn in the a phase shows rather good agreement with the opti-  cal transitions in the absorption spectra observed by Biondi and Rayne (1959). More recently, Pells and Montgomery (1970) have studied the absorption spectra of CuZn, CuGa, and CuGe in the a phase.  They state that  the CuZn results are i n general agreement with those of Biondi and Rayne (1959).  The results for CuGa and CuGe seem to show a decrease in the  -113-  energy of the 1 ^ -*-  transition with alloying but i n the Ge case the  structure i s very broad.  However optical measurements by Nilsson (1970)  on CuGe indicates a sharper  , •>  transition which moves to lower  energies with alloying. Another feature i n the spectra has been assigned to a d band to Fermi surface transition and the movement of this feature with concentration i s consistent with a narrowing of the d band in a l l cases.  The amount of narrowing is related to the atomic percent-  age of impurity rather than the electron per atom ratio since for a given n the d -»• E^ transition energy has increased to a greater extent i n CuZn relative to CuGa or CuGe.  The d bands of the solutes are i n a l l cases  much more tightly bound than in copper and as Heine (1966) has suggested a "Swiss cheese" model i s appropriate for the copper d band, the holes being the solute sites.  We may note that although the data is sparse,  the core contribution to the annihilation rate of the highest  concentra-  tion alloys studied presently i s rather less than for the pure copper, presumably reflecting the fact that the solute i s contributing l i t t l e to the core annihilation since its-d bands are tightly bound and l i t t l e overlap with the positron occurs. In conclusion, the combined optical and positron annihilation results would indicate that the effect of alloying is to decrease the <111> energy gap while at the same time shifting the point energies so that a net increase i n the neck radius occurs.  , to lower The two  effects are not entirely independent i n that the .present data indicates that the downward energy shift i n the optical data indicates the L^t -*• pared with CuGe.  i s larger i n CuGa than i n CuGe and shift to be larger i n CuGa com-  The energy variations with alloying depend i n a  -114-  complex manner on the conduction band-d band interaction and the change in lattice potential.  The fact that both the energy shifts seem to be  directly related to the impurity concentration rather than the electron per atom ratio is probably significant.  J_.  Electronic Specific Heat Studies For completeness the measurements of electronic specific heats in  copper alloys is discussed.  As previously mentioned the rigid band  model predicts that the density of states for copper alloys should decrease with increasing n since the copper Fermi surface has already made contact with the Brillouin zone boundary. electronic specific heat y by equation  c a n  D e  The measurement of the  related to the density of states. N(e).  3-1.  The electronic specific heat has been measured for CuZn, CuGa, and CuGe alloys (Mizutani 1972).  The coefficient y is found to increase  with impurity concentration in a l l three systems suggesting that N(e) i s an increasing function of n. with these measurements.  Thus the simple rigid band model is at odds  One possible explanation of this behavior would  be an increase in the energy gaps for a l l three alloy systems, however previous optical data and positron annihilation work on CuZn (Morinaga 1972) indicate the opposite change in the energy gap. Another explanation has been offered by Stern (1970) and this is the charging concept discussed in chapter III. the density of states N(e) i s effectively increased due to the charging of conduction electrons about impurity atoms.  This shielding effect at impurity sites is related to the  -115-  difference in valence between the host and impurity atom and mainly affects the states near the Fermi energy.  The result i s an increase of  N(e) with n which i s independent of the solvent band structure. No contrary evidence to the charging concept has been presented, however i t s t i l l remains to be shown that the increase in y can be entirely accounted for on this basis.  It has also been suggested that  an increase in the electron-phonon enhancement factor may partially account for the behavior (Clune 1970). G_.  Hume-Rothery Rules As has been discussed earlier, a model for the Hume-Rothery rules  was advanced by Hume-Rothery and Roaf (1961) who postulated that the a phase became unstable when the Fermi surface made contact with the <200> Brillouin zone boundary.  Any reasonable extrapolation of the present  results for CuGa i s in disagreement with this prediction.  On the other  hand since the CuGe results in the neck region are consistently lower than the CuGa data i t i s possible that contact may occur for CuGe - a measurement of the <100> cut-off for a high germanium concentration would be very desirable.  Given the range of behavior observed in the  neck region for the alloys studied to date:  CuZn, CuAl, CuGa, and CuGe  one would have to conclude that the detailed shape of the Fermi surface can only be a second order effect i n determining the phase boundary. Recently Stroud and Ashcroft (1971) have claimed that the HumeRothery rules are a consequence of the rapid variation of the wave-number dependent di-electric screening function in the immediate neighborhood of 2k  as a result of their pseudopotential calculations for CuZn and  -116-  CuAl using the Lindhard dielectric function. This assumption is simil a r to the original theory of Hume-Rothery, however Stroud and Ashcroft claim that the shape of the Fermi surface i s , in part, implicit i n the use of an energy dependent pseudopotential.  Further measurements on  concentrated alloys should lead to accurate energy calculations which can be related to the a -> 8 phase transition. H_. Summary This thesis has presented data upon the Fermi surface features associated with the a phase alloys copper gallium and copper germanium. Considering the problems involved in the positron annihilation technique, we feel the data are among the most clear cut yet presented, reflecting the very high resolution employed.  The observed changes i n the pure  copper Fermi surface upon the addition of concentrated impurities leads to several conclusions. It i s possible to state that on the basis of the present work the CuGa Fermi surface w i l l not make contact with the [200] Brillouin zone boundary i n the a phase region contrary to the predictions of the HumeRothery and Roaf model.  As well the results lend support to the conten-  tion of Stern that any agreement with the r i g i d band model in this general class of alloys i s fortuitous. The Stern hypothesis, which i s supported by electronic specific heat measurements, i s related to the shielding of conduction electrons about impurity atoms.  In essence  c  Stem predicts that the effects of adding the impurity atoms w i l l be minimized by shielding so that i f an impurity atom with valence greater than one i s added, then since one can have only two electrons i n a given  -117-  state, the other conduction electrons w i l l have less probability of being i n proximity to the impurity. Thus the addition of large amount of impurities to the noble metal host does not radically alter the band structure and accordingly concentrated alloys s t i l l possess distinct Fermi surface features (as noted presently) and lower than expected resistance (as noted by Stern (1969)).  Clearly i t would be of consider-  able value to obtain the Fermi radii in other orientations in the systems studied to obtain a complete picture of the Fermi surface. In view of the cubic symmetry, very few additional orientations would be necessary. The agreement of the present concentrated alloy work with previous measurements on the same dilute alloys i s encouraging and demonstrates that the Fermi surface i n highly concentrated systems can be effectively studied by positron annihilation.  To this end i t should be noted that  more experimental and theoretical work i s required in order to make the positron method a more precise tool i n this study.  Such factors as en-  hancement, higher momentum components and core annihilation, which complicate the analysis of these experiments, w i l l have to be studied in greater d e t a i l .  Such a study,to calculate the core annihilation contribu-  tion, has been discussed i n the thesis. At present, no detailed theoretical prediction for the copper gallium or the copper germanium systems exists, however the application of the coherent potential model to copper zinc (Das and Joshi 1972) i s encouraging and should stimulate (as should the present data) similar calculations for these systems.  -118-  APpENDIX The resolution function which closely approximates that used in the present work'has been calculated by Petijevich (unpublished) and i s discussed i n chapter VI.  As mentioned i n the text of the thesis,  the calculation involves a division of the collimati'ng s l i t s and sample into small elements and finding the number of coincidences between these elements as a function of angle. The resolution function i n the direction of detector movement (z direction of figure 12) i s shown by the inner curve on figure 45. The f u l l width half maximum (F.W.H.M.) for this resolution i s 0.735 milli-radians.  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