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A simple model for studying the gravitationally induced electric field inside a metal Shegelski, Mark Raymond Alphonse 1982

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A SIMPLE MODEL FOR STUDYING THE GRAVITATIONALLY INDUCED ELECTRIC FIELD INSIDE A METAL by MARK RAYMOND ALPHONSE SHEGELSKI B.Sc, The University Of Calgary, 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department Of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1982 © Mark Raymond Alphonse Shegelski, 1982 In presenting t h i s thesis in 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 fr e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that copying or publication of t h i s 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 of Physics The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: 18 October 1982 ABSTRACT I f a m e t a l o b j e c t i s p l a c e d i n a g r a v i t a i o n a l f i e l d , .the n u c l e i and e l e c t r o n s i n the m e t a l w i l l s i n k . T h i s w i l l produce a new charge d i s t r i b u t i o n i n s i d e t h e m e t a l . A m o d i f i e d charge d i s t r i b u t i o n i m p l i e s a m o d i f i e d e l e c t r i c f i e l d i n t h e m e t a l i n t e r i o r . T h i s t h e s i s i n v e s t i g a t e s some p o s s i b l e p h y s i c a l p r o c e s s e s w h i c h g i v e r i s e t o t h e g r a v i t a t i o n a l l y i n d u c e d e l e c t r i c f i e l d i n s i d e a m e t a l . To t h i s end, a s i m p l e model o f a m e t a l i s c o n s t r u c t e d . C o m p r i s i n g the jriodel a r e i o n s , a r r a n g e d on a d i f f e r e n t i a l l y compressed l a t t i c e , and a gas- o f c o n d u c t i o n e l e c t r o n s . An i o n i s r e p r e s e n t e d by a n u c l e u s and an e l e c t r o n w h i c h a r e c o n f i n e d t o g e t h e r i n s i d e a h a r d , m a s s l e s s , s p h e r i c a l s h e l l . The n u c l e u s i s t r e a t e d as a p o i n t p a r t i c l e w h i l e the e l e c t r o n i s r e p r e s e n t e d by a wave f u n c t i o n . The c o n d u c t i o n e l e c t r o n c o n s t i t u e n t i s m o d e l l e d as a gas o f n o n i n t e r a c t i n g f e r m i o n s w h i c h i s s u b j e c t t o an e x t e r n a l l i n e a r p o t e n t i a l , The d e s i g n o f t h e model f a c i l i t a t e s t h e i n v e s t i g a t i o n o f two p o s s i b l e s o u r c e s o f t h e e l e c t r i c f i e l d : g r a v i t a t i o n a l l y i n d u c e d i o n i c d i p o l e moments, and t h e charge "imbalance i n t h e metal ' i ; To - f i r s t border i n - g , ,< o n l y the' f i r s t ; '/source . ^ m a t t e r s , c o n t r i b u t i n g .;. a p p r o x i m a t e l y -Mg*/qe t o the e l e c t r i c f i e l d , where M i s t h e i o n i c mass, g* i s t h e a c c e l e r a t i o n due t o g r a v i t y , and q e i s the e l e c t r o n i c c h a r ge. The n e t ' g r a v i t a t i o n a l l y i n d u c e d e l e c t r i c f i e l d i s a l s o found t o be a p p r o x i m a t e l y -Mg*/qe, i i i TABLE OF CONTENTS Abstract i i L i s t of Tables v i i L i s t of Figures v i i i Acknowledgement ix Note on Numbering of Equations, Footnotes, and References to the Literature x CHAPTER 1. INTRODUCTION 1 1.1 Statement and Origin of the Problem .. 1 1.2 Review of the Literature 3 1.3 Purpose of thi s Thesis 7 1.4 Outline of the Model 8 CHAPTER 2. THE LATTICE OF IONS 15 Plan of the Chapter 15 2.1 Setting up the Schrbdinger Equation for the Electron Inside the Impenetrable Shell 18 i v (A) S e t t i n g up the e q u a t i o n s f o r the p e r t u r b e d p o t e n t i a l problem 23 (B) S e t t i n g up the e q u a t i o n s f o r the p e r t u r b e d boundary problem 28 (C) S e t t i n g up an e q u a t i o n f o r E-| ; summary 37 2.2 S o l u t i o n t o the S c h r o d i n g e r E q u a t i o n f o r the E l e c t r o n I n s i d e the Imp e n e t r a b l e S h e l l 42 (A) The u n p e r t u r b e d problem; d e t e r m i n a t i o n of >^ and E 0 42 (B) S o l u t i o n t o the p e r t u r b e d p o t e n t i a l problem 43 (C) S o l u t i o n t o the p e r t u r b e d boundary problem 49 (D) C a l c u l a t i o n of ; summary 54 2.3 The P o s i t i o n of the Nu c l e u s 64 2.4 C a l c u l a t i o n of the D i p o l e Moment of Each Ion i n Terms of Ef, 66 2.5 C a l c u l a t i o n of Ep i n Terms of £ 70 2.6 S o l u t i o n s f o r f> and E^ 73 2.7 The Average E l e c t r i c F i e l d Due t o the L a t t i c e of Ions 74 2.8 L a t t i c e Types 78 (A) C u b i c l a t t i c e 79 V (B) Face-centered cubic l a t t i c e 79 (C) Closest packing; type 1 - 82 (D) Closest packing; type 2 82 (E) Hexagonal closest packed 83 2.9 Ewald Sum 84 2.10 Evaluation of the Ewald Sum and Calculation of E ^ p ^ s 94 CHAPTER 3 THE CONDUCTION ELECTRONS 98 3.1 Description of the Problem of a Free Fermi Gas Subject to an External Linear Potential 101 3.2 S t a t i s t i c a l Mechanics Approach- 105 (A) Review of some basic s t a t i s t i c a l mechanical ideas 108 (B) Sp e c i a l i z a t i o n to case of no external p o t e n t i a l ; review of the ideal Fermi gas 113 (C) Case of an external linear potential 118 (i) Approach due to Van den Berg and Lewis 126 ( i i ) A n a l y t i c i t y of the l o c a l number density in the fugacity 133 v i ( i i i ) S p e c i a l i z a t i o n to case T=0K 134 3.3 Perturbative Approach 135 3.4 Linearity of the Local Number Density in the Parameter « 142 3.5 Contribution of Internal Charge Density to the Ambient Internal E l e c t r i c F i e l d 144 CHAPTER 4 DETERMINATION OF E q v e 149 CHAPTER 5 CONCLUSION 155 Bibliography 159 Appendix A - Alternate Way of Deriving an Equation for °fEz 161 Appendix B - A Quick Derivation of the Local Number Density in a Fermi Gas Subject to an External Linear Potential 170 Appendix C - Low Density Fermi Gas in an External Linear Potential at Absolute Zero 171 Appendix D -"Proofs'of Equations (3-3.28) and (3-3.29) 180 v i i L IST OF TABLES I . V a l u e s of ^ f o r L a t t i c e s (A) t h r o u g h (E) 95 I I . V a l u e s of cxz f o r L a t t i c e s (A) th r o u g h (E) f o r some V a l u e s of Z h 95 I I I . V a l u e s of <*3 f o r L a t t i c e s (C) th r o u g h (E) f o r some v a l u e s of Z n 96 v i i i LIST OF FIGURES 1. Nucleus and Electron Inside the Impenetrable Shell 9 2(a). Contact Arrangement of the Shells 10 2(b). Non-contact Arrangement of the Shells 11 3. Placement of the Nucleus Inside the Impenetrable Shell 19 4. Position of the Boundary Relative to the Nucleus 29 5. Relationship between 6, rt,(9), of and R 30 6. The Regions and V) 34 7. Forces Acting on the Nucleus in the F u l l Perturbation Problem 56 8. The Spherical Region 2 75 9. The Face-centered Cubic La t t i c e 80 10. The Airy Functions Aj (v) and Bj(v) 123 11. The Local Number Density n(u) as a Function of the Parameter oc 144 A.1. Coordinates From the Center of the Shell Versus Coordinates from the Nucleus 161 C.1. k 2 as a Function of X in the L-*00 Limit for the Energy Surface £0 176 C.2. The Energy Surface £„ in L-a> Limit in t-space ... 176 ix ACKNOWLEDGEMENT I wish to thank my supervisor, Dr. W.G. Unruh, for suggesting the topic of t h i s thesis, the basic outline of the model, for a s s i s t i n g with the development of the model, and for encouraging me along the way. I espe c i a l l y appreciate Dr. Unruh's assistance, encouragement, and concern for my well-being during the f i n a l stages of t h i s work. I am indebted to Kathy Nikolaychuk and Stephanie Mundle, who typed the manuscript and were very patient with me. Kathy and Stephanie also contributed with numerous other d e t a i l s during the la s t few days of work. Todd Mundle must not be forgotten, as he too assisted greatly in several imporant respects. Indeed, without Kathy, Stephanie, and Todd helping me, I would not have finished t h i s thesis at the time that I did. Stephanie also deserves thanks for providing me with reassurance and encouragement, and espe c i a l l y for coping so well with a l l of my thesis induced idiosyncrasies. I would also l i k e to express appreciation to Dr. R. Barrie, Dr. L. Sobrino, John Hebron and Matthew Choptuik for some helpful discussions; Matthew also succeeded in introducing me to the computer. My thanks to N.S.E.R.C. for their f i n a n c i a l assistance. Many other people helped me in other ways; my thanks to a l l of you. X NOTE ON NUMBERING OF EQUATIONS, FOOTNOTES, AND REFERENCES TO THE LITERATURE Equations are numbered i n d i v i d u a l l y in each section of each chapter: equation (7.4), for example, means the fourth equation of section 7. In a given chapter, i f an equation i s referred to from another chapter, the chapter number i s included. Otherwise i t i s not. Equation (2-2.3) means the t h i r d equation in the second section of chapter 2, whereas equation (2.3) means the t h i r d equation of the second section of the present chapter. Footnotes are numbered consecutively throughout a chapter. They are denoted by curved parentheses: means the t h i r d footnote in the present chapter. The footnotes are l i s t e d at the end of the chapter. References to the l i t e r a t u r e are symbolized by square brackets: refers to the ninth item l i s t e d in the Bibliography. 1 CHAPTER 1  INTRODUCTION (1.1) STATEMENT AND ORIGIN OF THE PROBLEM If a metal object i s placed in a g r a v i t a t i o n a l f i e l d , a macroscopic e l e c t r i c f i e l d w i l l be created in the i n t e r i o r of the metal. The e l e c t r i c f i e l d r e sults because gravity causes the nuclei and electrons to reside at positions which are di f f e r e n t than their corresponding positions in zero g r a v i t a t i o n a l f i e l d . A d i f f e r e n t positioning of the nuclei and electrons implies a d i f f e r e n t charge density inside the metal. A modified internal charge density produces a modified e l e c t r i c f i e l d in the metal' i n t e r i o r . For similar reasons, gravity also induces an e l e c t r i c f i e l d exterior to the metal. The following problem w i l l be considered in th i s thesis: what i s the g r a v i t a t i o n a l l y induced e l e c t r i c f i e l d in a metal, and what are the main physical processes which give r i s e to t h i s e l e c t r i c f i e l d ? The interest in such g r a v i t a t i o n a l l y induced e l e c t r i c f i e l d s first arose because of a series of experiments by Witteborn and Fairbank designed to measure the acceleration of ions, electrons, and positrons in the earth's g r a v i t a t i o n a l f i e l d . To shi e l d against external e l e c t r o s t a t i c f i e l d s , the p a r t i c l e s were constrained to f a l l in a cavity formed by a 2 conductor. The results of the experiments seemed to indicate that an e l e c t r i c f i e l d mg/qe pervaded the cavity of the conductor, where m was the electron mass, g the acceleration due to gravity, and q e the electronic charge. Witteborn suggested that gravity had affected the conductor in such a way as to induce the e l e c t r i c f i e l d mg/qe inside the cavity. Schiff and B a r n h i l l then did the f i r s t t h e o r e t i c a l c a l c u l a t i o n of the gravity-induced e l e c t r i c f i e l d in a cavity surrounded by a metal. The result of their c a l c u l a t i o n was Ee5^" =mgVqe • However, another th e o r e t i c a l c a l c u l a t i o n was done by Dessler, Michel, Rorschach and Trammell . They obtained for E e x^ a f i e l d directed oppositely to g and with a magnitude of approximately Mg/qe, where M is the ionic mass. Besides being in opposite d i r e c t i o n s , these two e l e c t r i c f i e l d s d i f f e r in strength by approximately fiv e orders of magnitude! To confuse the issue, t h i s discrepancy has been observed experimentally as well. Fairbank, Lockhart and Witteborn"111 reported a temperature dependence of E * : they found the magnitude of l s e * ^ to vary from about mg/qe at low temperatures to about Mg/cL, at higher temperatures,with a t r a n s i t i o n at about 4 K . Schiff and Ba r n h i l l also examined the question of the gr a v i t a t i o n a l l y induced e l e c t r i c f i e l d ~E t n^ inside a metal. They claimed that E would be approximately mg/qg . Dessler et. a l . calculated E l t l and concluded that i t would be opposite to g and have a magnitude of approximately Mg/qe. This discrepancy i s rather i n t e r e s t i n g . This introduction now presents a brief description of the techniques employed in the l i t e r a t u r e to calculate E e x t and E i , v t . ( 1 . 2 ) REVIEW OF THE LITERATURE The approach Schiff and B a r n h i l l used to calculate E e x* was to introduce a c l a s s i c a l test p a r t i c l e of i n f i n i t e s i m a l charge into the cavity formed by the conductor. They showed that E e x - t could be expressed in terms of the s h i f t induced by the test charge in the location of the v e r t i c a l component of the center of the mass of the conductor. In their analysis, Schiff and B a r n h i l l ignored the s h i f t contributed by the n u c l e i . They calculated only the contribution of the electronic constituent of the metal, and obtained mg/qe for E* k^ . In their paper, Schiff and B a r n h i l l also b r i e f l y discussed the internal e l e c t r i c f i e l d , and made the following claim: It i s apparent that each electron and nuclei in the metal must be acted upon by an average e l e c t r i c f i e l d of such magnitude that i t exactly balances i t s weight.°} For an electron, t h i s f i e l d i s mg/qe. They further argued that, because the electrons occupy most of the volume of the metal and the nuclei only a small f r a c t i o n , E L i s very close to mgVqe. This same f i e l d , they stated, would be expected inside the cavity of the conductor. 4 Dessler et. a l . pointed out that Schiff and Ba r n h i l l had not taken into account the compression of the metal in the di r e c t i o n of the g r a v i t a t i o n a l f i e l d . They presented the following h e u r i s t i c discussion about . The net charge density inside the metal must be very small, for otherwise there would result a huge internal e l e c t r i c f i e l d . Using e l a j - t V i c i t y theory, i t is-easy to calculate the number density of ions as a function of height inside the metal. Charge n e u t r a l i t y then gives the number density of electrons as a function of height. Treating the electrons, l o c a l l y , as a free Fermi gas allows one to in turn compute the pressure gradient experienced by the electrons. Balancing t h i s force against the e l e c t r i c and gr a v i t a t i o n a l forces f e l t by the electrons gives E L n . The f i n a l expression obtained by this approach i s (1 ) where £f i s the Fermi energy of the electron gas, Q% is the average number density of ions in the metal, 6 i s the Poisson r a t i o of the metal and Y i t s Young's modulus. Dessler et. a l . pointed out that t y p i c a l values for metals ~*'(nr for the parameters in equation (1) give E L oppositely directed to g and with a magnitude of the order of Mg/qe. In their paper, Dessler et. a l . also calculated E , again obtaining a f i e l d opposite to g and of the order of Mg/qe. 5 Subsequent to the work of Dessler et. a l . , Herring®3 reexamined the Schiff and B a r n h i l l approach to the problem. He found that the s h i f t induced by the test charge in the mass moment of the nuclear constituent of the metal was much greater than that of the electronic constituent. Herring estimated that the cavity f i e l d would be of the order of Mg/qe and opposite to The basic physical ideas and assumptions of the Dessler et. a l . h e u r i s t i c discussion have been restated by Harrison1"5"1 and [10] Leung but in d i f f e r e n t terms. Leung, for example, used the Boltzmann transport equation to obtain equation (1) for E L r i t . Dessler et. a l . went beyond th e i r h e u r i s t i c discussion in "* ' n't attempting to determine EL . They introduced a model of a metal which they used to derive the following equation for E : V n -- («vrv»/?«' where ju. i s the l o c a l chemical p o t e n t i a l of electrons in the metal. They then assumed that depends only on the l o c a l number density of electrons n at the height z, and replaced VJJ. by ( dju/dK ) (3n/2)Z ). Assuming charge n e u t r a l i t y , they calculated dr\/bz from e l a s t i c i t y theory. Estimating d/U/dn gave them an estimate for E t n* . Again they found E t r y^ to be opposite to g and to have a magnitude of order Mg/qe. Peshkm presented an expression for E similar to equation (2): 6 where ju}°' i s the l o c a l chemical potential for free electrons. F is a correction to account for the eff e c t s not included in the free electron representation. E L may be estimated from equation (3) i f an estimate for F i s av a i l a b l e . As such, equations (2) and (3) may be regarded to be the same. One other approach has been presented in the l i t e r a t u r e for the purpose of finding Ecni . Rieger^" 1 calculated E v i a a consideration of the electron-phonon in t e r a c t i o n . He f i r s t determined a transformation between the usual phonon operators when there i s no gravity and the ones which apply when there i s . Then, he replaced the former by the l a t t e r in the energy expression associated with the electron-phonon in t e r a c t i o n . The result he obtained included a scalar part ( i . e . i t involved no phonon operators), which he interpreted as being the e l e c t r o s t a t i c potential energy experienced by the electrons due to the d i f f e r e n t i a l compression of the l a t t i c e . Leung, Papim and Rystephanick c r i t i c i z e d Rieger's analysis, partly because of his choice of normal modes for the l a t t i c e . They avoided choosing normal modes. Instead, they calculated the e l e c t r o s t a t i c potential f e l t by the electrons in terms of the deviation from p e r i o d i c i t y of the l a t t i c e s i t e s . Both Rieger and Leung et. a l . obtained E L oppositely directed to g and of magnitude roughly Mg/q£. 7 ( 1 . 3 ) PURPOSE OF THIS THESIS The l i t e r a t u r e provides one with a p a r t i a l understanding of what physical processes are important in t h i s problem. The Dessler et. a l . h e u r i s t i c discussion, for example, points out the d i f f e r e n t i a l compression of the l a t t i c e due to gravity, and the pressure gradient experienced by the electrons as a r e s u l t . In the l i t e r a t u r e , one also finds expressions for E . However, one does not obtain from the l i t e r a t u r e an understanding of what the p r i n c i p a l sources of E^ n* are. Nor does one learn how the metal responds to the gr a v i t a t i o n a l f i e l d so as to support i t s e l f , or how thi s response generates the sources of E t n^ . Moreover, one does not f e e l quite convinced that E ^ has in fact been found. The he u r i s t i c discussion of Dessler et. a l . should be taken as an indication of some of the basic physical responses of a metal to a gr a v i t a t i o n a l f i e l d and not as an unequivocal determination of E . This i s why Dessler et. a l . constructed a model with which to calculate Ecn^. But in equation (2) — the result of their model -- i t i s not obvious that v)tt is any better known than E c " , or that the assumptions Dessler et. a l . made aboutju are v a l i d . F i n a l l y , Rieger and Leung et. a l . both assumed that E c n was primarily due to the compression of the l a t t i c e . Again, i t i s not obvious that t h i s i s necessarily so. In order to contribute to the understanding of t h i s problem of the g r a v i t a t i o n a l l y induced e l e c t r i c f i e l d in a metal, t h i s 8 thesis presents a simple model which includes some possible sources of E . The related challenge of finding E e x s h a l l not be considered here. (Attempts have been made by Hutson^ and •DO Hanni to provide a t h e o r e t i c a l explanation of the temperature dependance of E reported by Fairbank, Lockhart and Witteborn.) The basic features of the model are described in the next section of thi s introduction. The model has been designed with two basic purposes in mind. One i s to calculate the average e l e c t r i c f i e l d inside a metal. The other to provide some physical insight into the problem of g r a v i t a t i o n a l l y induced e l e c t r i c f i e l d s in metals. (1.4) OUTLINE OF THE MODEL In th i s model, a metal s h a l l be regarded as comprised of an ionic constituent and an electronic constituent. The ions are represented by the following composite e n t i t y : a c l a s s i c a l point p a r t i c l e of mass M and charge Z^q e, and a single quantum mechanical electron, are enclosed together inside of a massless, undeformable, impenetrable s h e l l (see figure 1). The point p a r t i c l e , treated c l a s s i c a l l y , represents the nucleus of the ion. The electron, represented by a wave function, models the t i g h t l y bound electron cloud surrounding the nucleus. The net charge of the ion i s (Zf,-l)q e and i t s mass i s M+m. The electronic constituent of the metal i s modelled by a 9 Figure 1• Nucleus and Electron Inside the Impenetrable Shell noninteracting electron gas which w i l l be spread throughout the inside of the metal. The s h e l l s are arranged, l o c a l l y , in a l a t t i c e pattern. However, they w i l l be more t i g h t l y packed near the bottom of the metal than near the top. This d i f f e r e n t i a l packing i s done in order to simulate a fact revealed by e l a s t i c theory: the density of an object placed in a g r a v i t a t i o n a l f i e l d decreases with height. The d i f f e r e n t i a l packing may be achieved in one of two ways. One way i s to have the r a d i i of these s h e l l s increase with height and to situate the s h e l l s in actual contact with one another, as in figure 2(a). The other way i s to arrange the sh e l l s such that they do not touch one another. In t h i s way, the number of s h e l l s per unit volume may diminish with height without requiring the radius of the she l l s to change (see figure 2(b)). In either case, the change i s very gradual: in a t y p i c a l 10 Ral\i of Mis Figure 2(a). Contact Arrangement of the Shells Figure 2(b). Non-contact Arrangement of the Shells metal bar 1 meter high, the density near the bottom of the bar is about one part in 10 times greater than near the top . As such the arrangement of the s h e l l s , l o c a l l y , may be regarded to constitute a l a t t i c e pattern. The reader may protest that, because of gravity, the she l l s 11 may not be arranged as in figure 2(b). Because of t h i s possible objection, the following comment i s made. The model has been constructed to take into consideration some of the key physical phenomena which give r i s e to the g r a v i t a t i o n a l l y induced e l e c t r i c f i e l d inside a metal, hereupon denoted by E^ v e . However, the model does not purport to be r e a l i s t i c in a l l respects. In p a r t i c u l a r , for the purposes of ca l c u l a t i n g E Q v e and providing physical insight into the problem of gravity induced e l e c t r i c f i e l d s , i t i s not important to dis t i n g u i s h between figures 2(a) and 2(b). What i s important i s how the nucleus and electron within each s h e l l respond to gravity: the function of the model i s to give an idea of how a metal w i l l be affected by a g r a v i t a t i o n a l f i e l d . Consider what w i l l happen inside of each s h e l l because of gravity. When there i s no gravity, the nucleus w i l l be at the center of the s h e l l and the ground state wave function of the electron w i l l be spherically symmetric. Gravity w i l l displace both the nucleus and the electron downwards; in the case of the electron, t h i s means that more of the electron w i l l be in the lower half of the s h e l l than in the upper h a l f . The nucleus, being much heavier than the electron, w i l l sink more than the electron. In consequence, the electron w i l l hold the nucleus up against gravity. This r e d i s t r i b u t i o n of charge inside the s h e l l w i l l endow the ion with a g r a v i t a t i o n a l l y induced dipole moment. It s h a l l be shown in the next chapter that these ionic dipole moments are 12 a key source of E a v e . The effect of gravity on the electronic constituent w i l l be taken into account by regarding the noninteracting electron gas to be subject to an external linear p o t e n t i a l . A c a l c u l a t i o n s h a l l be made to determine the number density of those electrons as a function of height. The charge density as a function of height may be found by comparing the number densities of s h e l l s and electrons. This internal charge density represents another source of E a v e. . Gravity induces an e l e c t r i c f i e l d in the metal i n t e r i o r . The microscopic version of this f i e l d — not E Q v e ! — plays a key role in p o l a r i z i n g the ions. As such, t h i s microscopic f i e l d must be taken into account in deriving the induced dipole moment. The microscopic f i e l d pervading a s h e l l w i l l be assumed from the outset to be uniform through the s h e l l and have a nonzero component only in the v e r t i c a l d i r e c t i o n . The macroscopic version of the gravity induced e l e c t r i c f i e l d , E 0y« t w i l l be regarded to influence the electronic constituent of the metal. Thus, the linear potential to which the electron gas i s subject s h a l l be taken as due to both -» -> gravity and E Qve • This assumes that E^ve does not depend on height. Several simplifying features have been incorporated into the model. The impenetrable s h e l l s were introduced so as to render tractable the solving of Schrbdinger 1s equation for the electron. For the same reason, the e l e c t r i c f i e l d pervading the s h e l l i s assumed to be uniform. This assumption w i l l be quite 13 good i f the shel l s are arranged as in figure 2(b), and spaced far apart. The interactions between the i n d i v i d u a l conduction electrons as well as those between the electrons and the l a t t i c e , have been taken into account in an approximate manner, via the lin e a r potential the electron gas experiences. In other words, i t has been assumed that the detailed arrangement of the sh e l l s may be ignored. Conceptually, t h i s i s much the same as the free electron approximation, which i s made for metals in zero g r a v i t a t i o n a l f i e l d and with zero i n t e r n a l net charge -» density. F i n a l l y , E Q V c has been assumed to be independent of height. Actually, E a v e varies with height, but only s l i g h t l y . This claim w:i 11 be v e r i f i e d . -* The sources of E Q v e are the ionic dipoles, which contribute -» an average e l e c t r i c f i e l d E ^ o l a s , and the internal charge density, giving r i s e to a f i e l d E c ^ . The f i e l d due to a l l other sources — such as charge external to the metal, the surface charge density, and the surface dipole density s h a l l be included together by a term E c , £ . The equation for E ove i s n Chapter 2, Ejl/>t)/es w i l l be calculated. The result w i l l be that E d l ^ o U s i s opposite to g and of the order of Mg/qe. E C K w i l l be calculated in Chapter 3. E C \ , w i l l turn out to be second order i n g. (4) 14 H . I . Schiff and M.V. B a r n h i l l , Phys. Rev. 151, 1067 (1966). ZA.J. Dessler et. a l . , Phys. Rev. 168, 738 (1968). 15 CHAPTER 2  THE LATTICE OF IONS PLAN OF THE CHAPTER The ultimate objective of th i s chapter i s to calculate E d i p o i e s ' t n e average e l e c t r i c f i e l d due to the dipole moments of the ions. In order to calculate E<i\p0\es , i t i s necessary to know the g r a v i t a t i o n a l l y induced dipole moment p of each ion. p may be calculated i f the d i s t r i b u t i o n of charge inside the s h e l l i s known. As such, the position of the nucleus inside the s h e l l and the electron wave function must be found. The location of the nucleus w i l l be determined by requiring that there be zero net force on the nucleus. The electron wave function w i l l be obtained by solving Schrodinger's equation. The problem of determining the wave function for the electron has three important c h a r a c t e r i s t i c s : (1) the wave function must vanish on the s h e l l and is i d e n t i c a l l y zero outside of the s h e l l , (2) the si n g u l a r i t y of the Coulomb potential experienced by the electron is at the s i t e of the nucleus, which i s not the center of the s h e l l , and- (3) in addition to the Coulomb pot e n t i a l , the electron also experiences a potential which i s linear in the v e r t i c a l d i r e c t i o n . This l i n e a r potential i s a combination of the gr a v i t a t i o n a l potential and the potential due to the e l e c t r i c f i e l d which 16 pervades the s h e l l . Thus, the e l e c t r i c f i e l d in the s h e l l w i l l be taken to be the sum of three terms: E c^, Eext , and the e l e c t r i c f i e l d in the s h e l l due to the dipole moments of a l l of the other ions. As noted in chapter 1, the e l e c t r i c f i e l d pervading the s h e l l w i l l not, in fact, be uniform. However, the f i e l d w i l l be approximated to be uniform. This -simplification i s made so that the Schrbdinger equation for the electron can be readily solved. The farther apart the shel l s are spaced, the better t h i s approximation w i l l be. Let ~Ef>(r) denote the e l e c t r i c f i e l d at the point r inside a s h e l l that i s due to the dipole moments of a l l of the other ions. In the SchrcJdinger equation, Ep(r) w i l l be replaced by a uniform f i e l d . A reasonable choice for t h i s representative uniform f i e l d i s i s the average value of Ep(r) inside the s h e l l . E l e c t r o s t a t i c theory t e l l s us that t h i s average i s the value of Ep(r) at the center ? c of the s h e l l , EpCr c). As w i l l be shown in t h i s chapter, the equilibrium position of the nucleus, r n , i s a very small distance from the center of the s h e l l . As such, Ef>(rc) w i l l d i f f e r from E p ( r M ) by a tiny amount. E p ( r n ) has a simple interpretation. Imagine replacing each ion by a point dipole p and situating the dipoles at the s i t e s of the nuc l e i . One then has a l a t t i c e of dipoles. Ep(r*„) i s the- e l e c t r i c f i e l d at the s i t e of one of those dipoles due to a l l of the other dipoles in the l a t t i c e . As such, E| 3(? h) w i l l hereupon be denoted by Ej,. TWs, t n e e l e c t r i c f i e l d in a s h e l l created by the dipole 17 moments of a l l of the other ions w i l l be taken to be uniform and equal to Ef,. It i s E p which w i l l enter into the Schrbdinger equation for the electron. For an appropriate choice of -* l a t t i c e , Ep w i l l be in the v e r t i c a l d i r e c t i o n . The reader w i l l notice that the electron wave function is going to depend on E*p and oc, where <* is the distance between the center of the s h e l l and the nucleus. This means that p w i l l depend on E p and °(. It i s easy to see that t w i l l depend on Ep. Three forces act on the nucleus: gravity, the e l e c t r o s t a t i c force due to the electron, and the e l e c t r o s t a t i c force due to a l l charge exterior to the s h e l l . The second of these forces — • depends on the electron wave function, or <* and Ep. The l a t t e r depends on Ef>. Requiring these forces to cancel gives an equation for oc in terms of Ep. That w may be expressed in terms of E/> implies that p can be written in terms of Ep. Since Ej, can be calculated in terms of p — for a given choice of l a t t i c e -- p can be determined. 'Having found p, E dholes may be calculated. Summarizing, the plan of t h i s chapter i s as follows: 1. Set up the SchrSdinger equation for the electron inside the s h e l l , with the nucleus displaced a distance oc from the center and' a uniform e l e c t r i c f i e l d Ec/, + ESxt + E^ , pervading the s h e l l . 2. Solve for the wavefunction and energy of the electron as a function of °t and Ep. 3. Solve for the position of the nucleus by requiring that there be zero net force acting on the nucleus. 18 4. Find the dipole moment p in terms of Ep . 5. Choose a l a t t i c e and find E|> in terms of p. 6. Solve the two equations combining E^ and p to determine p in terms of g, Ec/,, Eev.£ and Z n . 7. Calculate t d,-|, o l e i. . These steps are now carried out in d e t a i l . (2.1) SETTING UP THE SCHRODINGER EQUATION FOR THE ELECTRON  INSIDE THE IMPENETRABLE SHELL Let x, Y., and z, be unit, mutually perpendicular, dimensionless vectors in the x-, y-, and z-directions, respectively. A cartesian coordinate system i s employed; the space i s Euclidean. Choose the z-direction such that the gra v i t a t i o n a l f i e l d g=-g£ with g>0. Down s h a l l mean in the dir e c t i o n of g and up in the opposite d i r e c t i o n . Place the nucleus at a distance <x below the center of the s h e l l , i . e . : at ( 0 , 0 , - Of ), so ^ >0 means the nucleus i s indeed below the center of the s h e l l (see figure 3 ) . Let a be the value of <* for which there i s zero net force on the nucleus. If <*^ a, an external, imaginary force i s required to hold the nucleus in place. Choose the o r i g i n of the coordinate system to coincide with the nucleus. The e l e c t r i c f i e l d inside the s h e l l , due to a l l charges exterior to the s h e l l , i s 19 °* Shell £\e.c-tron shell Figure 3. Placement of the Nucleus Inside the Impenetrable Shell ~ (1.1) where E x ^ Edk 4 t e x t ) (1.2) and Ey, , E c^ , Eex± have a l l been defined previously The Hamiltonian for the electron i s , (1,3A) n 4 ; ouistJle (1,3B) 20 where V-V/ZTJ, h i s Planck's constant, k=i/4ne,D, £0 i s the e l e c t r i c p e r m i t t i v i t y of free space, r i s the distance of the point (x,y,z) from the nucleus, r=Jt 2-+y z - t-z 2', z c i s the elevation from the center of the s h e l l , and £=m3 +?€ES . ( K 4 ) Schrodinger's (time-independent) equation for the electron is or O > M l ( 1.6) Equation (1.6) w i l l be solved by using perturbation theory. oc and e are treated as very small parameters; that i s , <X i s small compared to R, and € i s small compared to kZ^o^VR2". as such, ^ and E are expanded in Taylor series about <tf=0, £=0: (1.7) 21 ^l0 + *\*<K*Xz+<>f\* - ( L B ) Note that only the ground state eigenfunction and ground state energy eigenvalue are sought. The reason for t h i s i s that, in most metals at or below room temperature, the energy gap between the ground state of the ion and the f i r s t excited state i s large compared to k9T , where k B i s the Boltzmann constant. Consequently, the proportion of ions not in the ground state i s a very small number. The ^j, 4\ and X\ are a l l functions of "r, where "r i s position inside the s h e l l , as measured from the nucleus. The " E ^ , c£{ and 8\ are a l l constants. Since &< and e are very small, i t i s s u f f i c i e n t for the purposes of thi s model to determine ^ and E to lowest order in oc and € . Accordingly, the approximate solution for 0 w i l l be taken to, be ^ = ^ + 0 ^ | . (1.9) As w i l l be proven, "E, and 'fc^  are both zero, and so the approximate solution for E w i l l be taken as E-E.+<r rVe 4V«*S,- " ' , 0 ) The problem reduces to finding , ^, ^ in equation (1.9), E Q , Ez , £2, <f| in equation (1.10), and to showing that 22 both E| and E ( are zero. A l l of these quantities, except for E\ , may be found by breaking the problem down into three seperate problems. One of these i s "the unperturbed problem". As the name suggests, <*=0 and £ =0 in t h i s case; the nucleus i s at the center of the s h e l l , and there i s no external p o t e n t i a l . fa and Ep are the solutions for ^  and E, respectively, when <* = 0 and e = 0, and are referred to as the unperturbed eigenfunction and the unperturbed eigenvalue. The other two reduced problems are "the perturbed potential problem", in which ^=0 but e^O, and "the perturbed boundary problem", wherein e = 0 but <*j^ 0. It i s clear that the <f>^ and the fEj may be found by solving the perturbed potential problem. S i m i l a r l y , the and the E,- are determined by solving the perturbed boundary problem. Before proceeding to determine the quantities in equations (1.9) and (1.10), two points should be made regarding equations (1.7) and (1.8). F i r s t , i t i s not necessary that these be Taylor series; equations (1.9) and (1.10) w i l l s t i l l be excellent approximations to and E provided that equations (1.7) and (1.8) are at least asymptotic expansions. Second, # and € are not dimensionless. It is possible to define dimensionless parameters ctr«/R and €/(k<le/&2), but i t i s ce r t a i n l y not mandatory. These parameters, oc and €, emerge naturally in solving for ^, 4V c £ 2 and £|, as sh a l l be seen shortly. The next step i s to set up the perturbed potential problem 23 and obtain expressions for the 9^  and the eE.;. Then, the perturbed boundary problem s h a l l be posed, and expressions for the ^ and the found. (A) Setting up the Equations for the Perturbed Potential Problem The perturbed potential problem is so named because i t d i f f e r s from the unperturbed problem only in that the electron experiences the small linear potential £ z c . With the nucleus at the centre of the s h e l l , £*=(), z=zc and Schro'dinger' s equation becomes ( 1 . 1 1 ) with the boundary condition - o ( 1 . 1 2 ) This problem w i l l be solved to the lowest n o n t r i v i a l order in the small parameter e. To t h i s end, write 24 H = H0 + ez (1.13) where H . ' - £ V < U . l+oO-, r > ^ Express j£ and E as perturbative expansions about £=0: £& + e£E, e^ eE2 + •• • ( L i e ) Substituting these expansions into the equation H^=E^ leads to the following equations: (1.17.0) 25 (1.17.1) (1.17.2) (1.17.n) The boundary condition (1.12) implies that It = 0 (1.18.0) = 0 (1.18.1) r-R (1.18.2) (1.18.n) 26 Equations (1.17.0) and (1.18.0) v e r i f y that fa and E0 are the unperturbed eigenfunction and eigenvalue. It is in terms of these two quantities that the eE^ and the fa are expressed. Using the standard techniques of perturbation theory, i t i s found that 1 1 1 r\ = l 0o u (1.19) where the integrations are over the region r<R. The normalization condition, gives r i s e to the equations r J > (1.20) 27 Equation (1.20) reduces equation (1.19) to E, = f dJr f z % „ [ dV^„ (1.21) According to the results of standard perturbation theory, equations (1.17) and (1.18) and (1.20) allow for the determination of a l l of the -^ and a l l of the €E{ , in terms of <fi0 and E Q . It is worthwhile to write out the equations which specify ^,, £E| and &E^. Before doing so, note that a simple physical argument demonstrates that ^E^O for a l l odd i. The idea i s simply that the physical energy E i s invariant under a change in sign of € . This implies that E(e) - lE(-6) = 2e eE{ + 2e3 ^E^ + ... =0, which can be true for a l l £ only i f E|=0, E=0, .... In p a r t i c u l a r , €E|=0. Consequently, the equations specifying and E z are: ( H . - £ 0 ) ^ =-z(4 (1 .22.a) O (1.22.b) r=R r (1.22.c) R 28 (1.23) For the sake of completeness, the equation for E ( given: is also E i (1.24) Once r0 has been determined, i t s h a l l be shown that equation (1.24) gives £E,=0, as required. (B) Setting up the Equations for the Perturbed Boundary Problem In t h i s problem, the nucleus i s placed a distance oc below the centre of the s h e l l , as in figure 3, and there i s no external linear p o t e n t i a l . The electron wave function i s given by the solution to L „ y = E » - 0 " 2> m r > (1.25) inside the s h e l l , and i s zero outside the s h e l l . Note that the function fi, given by equation (1.25), w i l l not i t s e l f be zero 29 exterior to the s h e l l . This difference between the electron wave function and the function ^ should be c a r e f u l l y noted by the reader. The Hamiltonian for t h i s problem d i f f e r s from that for the unperturbed problem only in that the position of the s h e l l r e l a t i v e to the nucleus i s d i f f e r e n t . If oc is small, then t h i s difference in position of the boundary i s a small one, and the problem may be regarded as a problem in which the boundary has been s l i g h t l y perturbed. Figure 4. Position of the Boundary Relative to the Nucleus An equation for the position of the boundary can be derived by examining figure 4. The boundary in the unperturbed problem is a shell- of radius R entered on the nucleus. Here i t i s a s h e l l of radius R centered on a point a distance °C above the nucleus. Let ^(0) be the distance from the nucleus to the boundary in the perturbed boundary problem, where 0 i s the polar 30 a n g l e . From f i g u r e 5, note t h a t ( r t ( G ) - cxcose) 2' + (tfsin©)2- = RL o r , s o l v i n g f o r r \ 3 ( 0 ) , E xpanding r^(0) i n a T a y l o r s e r i e s about oV=0 g i v e s (1 . 2 7 ) F i g u r e 5. R e l a t i o n s h i p between Q, r h ( 9 ) , c x and R Expanding <fi and E i n T a y l o r s e r i e s about <* = 0: (1 . 2 8 ) 31 (1.29) Again, a l l of the E<, with \ odd, are zero. The reason i s the same as in the perturbed potential problem; the physical energy E can not depend on the sign of OC. In consequence, equation (1.29) s i m p l i f i e s to (1.30) 2, 1 i~Lf Inserting equations (1.28) and (1.30) into L0^=E^ leads to the set of equations: o r 0 - L0 Yo (1.31.0) (1.31.1) (1.31.2) The boundary conditions which must be obeyed by the <J>{ are obtained by writing <fi in the form of a Taylor series about r=R: (1.32) 32 where and <P i s the azimuthal angle. $ vanishes on the s h e l l , at r=r t(9). Accordingly, inserting the form (1.28) for 0 into the right hand side of equation (1.32), and evaluating at r=r b(9), leads to ii. I, e, f) + a (a,©,?)+cse^^q?)! + « 1U«,e,<r ,) + c O I0^ /(( l,e /r; C (1.33) 2 {sin1© t'a.e.r > + | to© £ ( K A H + • • • : O where the result has been expressed as a power series in oc. Equation (1.33) can be true for arbit r a r y <X only i f the c o e f f i c i e n t of each power of oc i s zero. This means that ^ ( W ) + cose^ (R,e,<p)-o (1.34.0) (1.34.1) 33 (1.34.2) The only other condition to s a t i s f y is the normalization of the eigenfunction: USA (1.35) where the integration i s over the i n t e r i o r of the s h e l l . The l e f t hand side of equation (1.35) may be expanded in powers of OC. To see how to do so, consider figure 6. V n i s the region i n t e r i o r to a spherical surface of radius R centered on the nucleus, is the region exterior to and i n t e r i o r to the s h e l l , and i s the region i n t e r i o r to V n and exterior to the s h e l l . Note that d3r = d5r+ J V - (1.36) d r means to integrate over the region V^, where r<R. (I Equation (1.36) may be applied to the l e f t hand side of the equation (1.35). 34 Figure 6. The Regions V-^ and Vp, Using the expansion (1.28) for <p, and equation (1.36), the l e f t hand side of equation (1.35) may be converted into a power series in o( . Equation (1.35) i s true only i f the order unity term in t h i s power series i s equal to one and i f the c o e f f i c i e n t of every other power of oc i s zero. The only term of order unity in the power series i s ^ d^r^y^, , since the integrals J v d 3 r ^ and jVr^jf are of order <*J. Thus, (1 .37) The condition in which ^ enters i i (1 .38) obtained by recognizing that the c o e f f i c i e n t of cc i s zero. The 35 higher order terms give equations involving i{, </3, ••• , but since only fa and ^, w i l l be required, ultimately, there i s no need to write these equations down. Summarizing, equations (1.31.0) , (1.34.0) and (1.37) define the unperturbed eigenfunction and eigenvalue, while equations (1.31.1), (1.34.1) and (1.38) specify . Grouping these l a t t e r three equations together, I ii) - L (I) (1.39.a) (1.39.b) r= R ( 1 .39.c) Equations for the ^E^ may also be derived. Only those for 0<Z\ and °!E2 s h a l l be presented. Proceeding from the expansions (1.28) and (1.29) for ^ and E, TuJ=Ef implies that (1.40) s h a l l be used to generate an equation for E-t . Multiplying both sides by „^ , integrating over V A , and using equation (1.37) 36 Combining equation (1.25) for L 0, the relationships (1.31.0) and (1.34.0), and Green's theorem, equation (1.41) s i m p l i f i e s to where dS i s the element of area on the surface of V n and the integration i s over the surface of V n, where r=R. The boundary condition (1.39.b) allows ^ to be eliminated from equation. (1 .42 ) , giving *. . , , , . . (1 .43) Equation (1.43) s h a l l be used to v e r i f y that ^E|=0. Using the same techniques, equation (1.31.2) may be converted into the following expression for E^: <v( in a, i r (1 .44) The form of ^ on the surface of V n is described by equation (1.34.2). As such, once ^ has been determined, ^E^ may be calculated. 37 (C) Setting up an Equation for £; ; Summary An equation for £ i may be obtained by dealing with the f u l l perturbation problem. The method to use to get t h i s expression for Ei begins by equating the terms on each side of the equation (Lp +&ZC Recall that z c i s height as measured from the center of the s h e l l , while z i s height as measured from the nucleus. z c and z are therefore related by Z c - Z -oc , (1.45) Using the expansions (1.7) and (1.8) for ^  and £ along with equation (1.45), equating the c o e f f i c i e n t s o f o n each side of (L 0+ez c )^=E Ogives where 6 E i and * E , have been acknowledged as being zero. Performing the same operations on equation (1.46) as were used to derive expressions for " ^ E , and^ E j . , (1.47) The form of ~)C, on the surface of V n i s obtained by requiring)^, as given by equation (1.7), to vanish on the surface of the s h e l l . Using the same approach as that which led 38 to the boundary conditions (1.34), i t i s found that - - COS© . (1.48) This result transforms equation (1.47) into R ^ R (1.49) At t h i s stage, equations have been derived for a l l of the quantities appearing in the approximate solutions for ^ and E given by equation (1.9) and (1.10). Summarizing: (1.50.a) (1,50.b) (1 ,50.c) 39 (1.51.a) = 0 (1.51.b) (1.51.c) (1.52.a) (1.52.b) 40 0 (1.52.c) o / o (1.53.a) •I 0 (1.53.b) .it [ 2m (1.53.C) 2 = 2_m T.'z (1.53.d) 41 - - C o 5 Q<1>; (1.53.d') J (1.53.e) r (1.54) Note that the approximate solution not i d e n t i c a l l y zero on the s h e l l , but d i f f e r s from zero by terms of 2 o r d e r ^ and oi£. This i s acceptable, because the wavefunctions are required to only f i r s t order. 42 (2.2) SOLUTION TO THE SCHRODINGER EQUATION FOR THE ELECTRON  INSIDE THE IMPENETRABLE SHELL (A) The Unperturbed Problem; Determination of and of Bo The essential features of the model w i l l be independent of the choice made for R. R i s chosen for convenience to be „ n n (2.1) where *Q <2 •2) 0 - Z e i s t h e Bohr radius. For t h i s choice of R, when r<R i t is easy to see that J^ > i s e s s e n t i a l l y the hydrogenic wavefunction /Aoo , but with a di f f e r e n t normalization; 7^ z©o i s the spherically symmetric wave function belonging to the f i r s t excitation energy of the hydrogenic problem. Thus, where V- a, 3 <2-4> N i s determined from the normalization condition (1.51.a) and i s 43 N (2.5) E D i s P-7Zn * r - - ^ 4 te 0 8tf (2.6) (B) Solution to the Perturbed Potential Problem The f i r s t thing to observe is that having found *f0 , i t i s t r i v i a l to v e r i f y that equation (1.53a) g i v e s 6 E , =0. Using equation (2.6) in equation (1.51.a) and performing some simple manipulations, the following equation for ft > r e s u l t s : It i s easy to v e r i f y from equation (2.7) that ft, is of the form 44 Using equation (2.8), i t i s easy to show that condition (1.51.C ) i s s a t i s f i e d . Where t(P) must s a t i s f y ^ " o ) + ( ^ ^ ^ = p J ( 2 - p ) • ( 2 - 9 ) The boundary condition (l.5l.b) for <f>> translates into i ( D = Q • ( 2 ' 1 0 ) Putting (2.11) s i m p l i f i e s the problem to T / /cp + (|-l)r'(p) = 2 - p • <2-'2> Solving equation (2.12) for Tr' by using the well known "integrating factor method", and then integrating to get X , i t is found that P X (2.13) 45 The constants c, and c 2 must be determined. One condition to be s a t i s f i e d is t(2)=0. Another i s needed in order to determine both constants. The second condition i s that 0, must not have a 9 s i n g u l a r i t y . This requirement i s a consequence of the need for iftj1 to be integrable. Recalling the expansion (1.7) for ^ , in expressing j ^ V ^ * ^ as a power series in <* and £, i t i s seen that the c o e f f i c i e n t of the 6 1 term w i l l be R \ ' In order that the \ dV©*0, part of this be f i n i t e , i t i s necessary that </>, have no worse a s i n g u l a r i t y at (?=0 than (? . There are only two terms in (2.13) which blow up l i k e One involves c z . The two w i l l cancel i f the choice Q--72 (2.14) is made. With th i s determination of c 2,equation (2.13) may be written as x (2.15) The form (2.15) for t(P) i s very obscure. However, using P rp » power series expansions f o r e and \'1 <Ax reveals that not only the ^ 2 terms in equation (2.15) cancel, but so do the Q~' terms. As such, t(P) may be expressed as a power series: 46 (2.16) Inserting equation (2.16) into equation (2.9) allows the c j to be determined. The result i s : ^ } ' ' c > ^ 5 ? T 0 r + ^ C J P (2-,7) Where C; - C . CHXO^) J •^5 3o (2.18) The boundary condition (2.10) fixes c, ; to four decimal places C^-OM^Z (2.19) t(t?) may be spec i f i e d by either equation (2.15) or equations (2.17) and (2.18). Before using the solution for 0( to calculate € E 2 , note that t((?)<0 in the range 0<£<2, which implies ,^<0 in the upper half of the s h e l l and 4>\ >0 in the lower half. Physically, t h i s means that the electron sinks down (assuming €>0) under the influence.of the external perturbing p o t e n t i a l . 47 6 E 2 i s determined by evaluating equation (1.53.b). Using equations (2.3) and (2.8), L Z ~ -2/4 \<gl LL (2.20) It where (2 .21 ) The simplest way to solve for e E z ' i s to put X3(Z-x)-£(x)e"XJix^W(D)e P . (2.22) w(^) must s a t i s f y (2.23) Using the power series form of t ( P ) , the following solution for w(P) r e s u l t s : (2.24) where 48 V 4 = -iC, V / (2.25) Since w(0)=0, equations (2.21) and (2.22) combine to give -z (2.26) Evaluating equation (2.26) to four decimal places, c £/=-O.l082 • (2.27) From equations (2.20) and (2.5) i t is found that, to four decimal places, 49 (2.28) (C) Solution to the Perturbed Boundary Problem It i s easy to show that equation (1.53.c) says that ^E,=0. A l l that i s required i s to observe, using equation (2.3), that the r a d i a l integral in equation (1.53.c) i s f i n i t e . Since the integral of cos© over the surface of a sphere is zero, ^E, =0 follows immediately. is found almost as e a s i l y . F i r s t note by comparing equations (1.50.a) and (1.52.a) as well as equations (1.50.b) and (l.52.b) that 9/ s a t i s f i e s the same equation as >4> , but a di f f e r e n t boundary condition. In consequence, ^ i s e s s e n t i a l l y the hydrogenic wave funct ion (2.29) In s i s t i n g that V) s a t i s f y equation d.52.b) fixes N, : whence, 50 i ( H V « 9 (2.30) Equation (2.30) also s a t i s f i e s the requirement (1.52.c). Notice that 'A i s posit i v e in the part of the s h e l l above the nucleus, and negative below i t . This means that, when the s h e l l i s moved s l i g h t l y upwards, the electron gets pushed up above the nucleus a l i t t l e b i t , so more of the electron w i l l be above the nucleus than below i t . The physical reason for th i s i s b a s i c a l l y just that there i s s l i g h t l y more room available inside the s h e l l above the nucleus than below i t : the extra room l i e s in the v i c i n i t y of the s h e l l . Equation (1.53.d) is readily evaluated by making use of equation (1.53.d ) and equations (2.3) and (2.30) for ^ and \°i , respectively. The f i n a l result i s - %lFMlZjl<%Z . (2.31) There are at least two other ways to derive equation (2.31). They serve as a check and are therefore presented. One of these methods involves some key physical ideas, and i s described below. The other i s more mathematical and i s given in Appendix A. The physical method begins with the r e a l i z a t i o n that 51 F € , ( « A - - ^ - ( 2 - 3 2 ) e l OCX where F^ , («-)=Fe2. («.)z i s the e l e c t r o s t a t i c force exerted on the nucleus by the electron as a function of oc, and E i s the energy eigenvalue in the perturbed boundary problem: t ^ 0 f / E i f ^ t f ••• (2.33) T Equation (2.33) i s constructed as follows: In the perturbed potential problem i t i s clear that an external force i s required to act on the nucleus, when <*-/0, in order to keep i t in place. This force must be equal in magnitude to Fgj(°<.), but oppositely directed. If «- is increased by an i n f i n i t e s i m a l a m o u n t , , while maintaining the magnitude of the external force the same as that of Fei (°0 , then the external agent, acting in the same di r e c t i o n as the displacement of the nucleus, does an amount of work Fei (<>0<Po<. This work goes into increasing the energy of the nucleus-electron system b y S E : Fei (©Ocf«-=(£E, which i s equation (2.32). Notice that equation (2.32) indicates that ^,=0, for the force exerted on the nucleus by the electron when°<~=0 i s zero; using E=E „ + """Ei + <=<" z. + in equation (2.32) and evaluating at Qf = 0: J<K-o 1 Inserting equation (2.33) into equation (2.32): 52 re,C«) = Z«("'E2.+ ^of i%+- . (2.34). may be found by obtaining an expression for F e / (ex.), expanding i t , and finding the order o<-term. Regarding the electron in the s h e l l as a cloud of charge with charge density qe/^(r)/ , (2.35) By symmetry, the x- and y-components of F e t (°c) are zero; hence, only [ F £ , =F £, (<*) i s i n t e r e s t i n g . It i s ^ n (2.36) Using the expansion (2.37) in equation (2.36), Iw-^Zrfli/^i + zi/<£fA< ••} • (2-38) Only the order oi term on the right hand side is desired, so the 53 terms not e x p l i c i t l y w r i t t e n down may be i g n o r e d . F u r t h e r , note t h a t the f i r s t i n t e g r a l i s of o r d e r oi.3. The o r d e r c< p o r t i o n of Fe/ (<*~) i s o b t a i n e d by f i n d i n g the o r d e r one p o r t i o n of the second i n t e g r a l i n e q u a t i o n ( 2 . 3 8 ) . W r i t i n g (2.39) i t i s c l e a r t h a t the o r d e r one term here i s the f i r s t i n t e g r a l on the r i g h t hand s i d e . The o t h e r two i n t e g r a l s a r e of o r d e r T a k i n g t h i s o r d e r one term from e q u a t i o n ( 2 . 3 9 ) , comparison of e q u a t i o n s (2.39) and (2.38) shows t h a t (2.40) E v a l u a t i n g the i n t e g r a l i n e q u a t i o n (2.40) reproduces the e x p r e s s i o n f o r °^ E z found e a r l i e r : 54 To four decimal places (2.41 ) (D) Calculation of ; Summary Using the expressions obtained for <^ a , , and <fi, , equation d.53.e) for £, i s easy to evaluate. The result obtained i s (2.42) The numerical value of t'(2) i s best found by working from equation (2.15). Denoting (2.43) d i f f e r e n t i a t i o n and evaluation at P=2 gives (2.44) An equation for c ( which i s instrumental in simplifying equation 55 (2.44) may be arrived at by evaluating equation (2.15) atp=2. Recalling that t(2)=0, the required expression for c, i s C,= - 8 - 1 2 J / - ( f A(2>) Equation (2.46) reduces equation (2.42) to (2.45) Combining equation (2.43) and (2.44), i t follows that T^a)^7-7V • (2.46) To four decimal places, r, = -o.5z?7. (2-46) E, i s dimensionless because ©te has units of energy. A second method of c a l c u l a t i n g £ ( i s now presented. This method i s in essence the same as the second method used to determine Ez and i t brings forth some key physical ideas. In the f u l l perturbation problem, when °^^a an external force is required to act on the nucleus to keep i t in place; see figure 7. Since t h i s external force w i l l be in the z-direction, 56 write fe^«> f ' < 2 - 4 9 ) /v/«clens Shell Figure 7. Forces Acting on the Nucleus in the F u l l Perturbation Problem Si m i l a r l y , as in the treatment used e a r l i e r to get ^E^, write f e l ( « ) - - F e l ( « ) Z (2.50) —> where, again, rei Fe, (<*.) i s the e l e c t r o s t a t i c force exerted on the nucleus by the electron. There are two other forces experienced by the nucleus: i t s weight, -Mgz, and the force applied to i t by the e l e c t r i c f i e l d E 5=E S z, which i s q eZ„E s z. Demanding that these four forces cancel gives 57 (2.51 ) Imagine the nucleus to be lowered by an amount ^ « — so <* increases by — with the four forces just c a n c e l l i n g , as in equation (2.51). Then the external agent does an amount of work c^W—Fgxt £«. on the system of nucleus, electron and the f i e l d s E s £ and -gz. The system gains potential energy q eZ nEs^°S loses potential energy Mgcfoc, and has the energy eigenvalue E change by C T E : 6 W = - F t t t ^ = < j e Z n E s ^ - M j S « + J £ - (2.52) Note that the SB term includes the change in the kinetic energy of the electron, the change in the e l e c t r o s t a t i c potential energy of the nucleus-electron configuration, as well as the change in the potential energy of the electron in the f i e l d s E 5z and _gz. Recall that t h i s l a t t e r potential energy was included in the Hamiltonian via the term e.zc ~&{z~<<) . Compiling equations (2.51) and (2.52): (2.53) F e ( («*.) i s given by equation (2.36). The expansion for E gives (2.54) 58 £,may be obtained by finding the part of F e^ which i s proportional to £ . Working from equation (2.36), substituting in the expansion for ^ , expanding in powers of and e , and extracting the £-proportional part, (2.55) Performing the integration indicated, rz (2.56) Write (l-x){.(x)eid\ - v(^ ) e" P (2.57) v(£) must s a t i s f y \/(p)-vrp)c(z-pK(p) , or, using equation (2.17) for t(PJ, . 1 - 6 .vJ (2.58) power series solution, beginning with a term in (? , w i l l 59 s a t i s f y equation (2.58): CD V(p) = £ v ^ J (2.59) r2-Substitution of equation (2.59) into equation (2.58) leads to the solving of the v-. The result i s J V i = C, v3 = 0 0 (2.60) J whence equation (2.59) becomes Si- 2 ^ V j - ' - C j - ' , j » 2 Evaluating equation (2.61) at (? = 2: to four decimal places: (2.61 ) <x> (2.62) vci) = -1.3373 . (2.63) 60 Inserting equation (2.63) and j^tH^^p = \ia)£x , — true because v(0)=0 — into ( 2 . 6 1 ) , to four decimal places, £, = - o . 57^ c ^, (2.64) which i s the same as equation ( 2 . 4 8 ) . Note that Eiis negative. The physical reason for t h i s i s captured in equations (2.55) and ( 2 . 5 6 ) . Imagine the nucleus to be at the center of the s h e l l , as in the perturbed potential problem. Then, according to equations (2.53) and ( 2 . 5 4 ) , Fcl «) = 4 • • • (2.65) -ioC-o Assuming e>0, the electron w i l l have sunk down a l i t t l e inside the s h e l l , and w i l l be p u l l i n g the nucleus down. This means that F&t ( c * ) ^ 0 w i l l be negative see equation ( 2 . 4 9 ) . Equation (2.65) says that, in consequence, must be negative. (If £<0, the electron has been pulled up, F e| (<x) 7 >0 and £ , « ) . ) With having been found, the problem has been completely solved to f i r s t order correction in ocand £.. The eigenfunction is (2.66) 61 l o w e s t n o n t r i v i a l o r d e r , where (2.68) y^, j^- 2 ^(p)e ^ 2 C o s £ ) ; (2.69) /V- ~ ^ , (2.70) <*o ^ - f ^ . > (2.71) 62 and t(P) i s given by either 3 1 C ; r z 3o or with Cr-OA£ZZ . The energy eigenvalue i s Z ' t. ^ 2 to lowest n o n t r i v i a l order, where 63 l_p _ . 8tf (2.78) (2.79) ^ 2 ° 2 O ^C2-p)-t(n)e~^p^-alo82 (2.80) -0.1713 a " (2.81 ) (2.82) 64 ( (2.83) IM tti\U ft7V)-1*-0.5721 -3e* Having found these quantities, the task remaining i s to use them to calculate the g r a v i t a t i o n a l l y induced e l e c t r i c f i e l d inside a metal. (2.3) THE POSITION OF THE NUCLEUS The gr a v i t a t i o n a l f i e l d , w i l l displace the nucleus from the center-of the s h e l l . The electron inside the impenetrable s h e l l responds to r e s i s t t h i s displacement, exerting an upwards p u l l on the nucleus. The nucleus also experiences a force due to the e l e c t r i c f i e l d E^. The position of the nucleus, a distance a down from the center of the s h e l l , i s determined by the cancelling of these forces. Mathematically, (3.1 ) Using equations (2.53) and (2.54) 65 Combining these two equations, the solution for a, to lowest order, i s 2% The value of a i s dependent on the f i e l d E^, as discussed e a r l i e r . To show thi s dependence more c l e a r l y , substitute mg+qeEg f o r e , and E P + Ej for E$ — equations (1.4) and (1.1), respectively into equation (3.2), to obtain (M-^S,)9-(Z ,+&He£i J Z . ^ f e r ( 3 3 ) Equation (3.3) shows how a depends on Ef,. Having found a, the next few steps are to determine the dipole moment p of each of the model ions, arrange the ions in a chosen l a t t i c e pattern, and calculate Ej, in terms of p. Since p w i l l depend on a, and a in turn on E p , i t w i l l be possible to determine both p and Ej,. Notice that a/a D is indeed very small, as was claimed early in t h i s t h e s i s . This i s easy to see by using equation (2.82) in equation (3.3) and taking Ep to be of the order Mg/qe. The — 18 result i s a/a 0 ^10 66 (2.4) CALCULATION OF THE DIPOLE MOMENT OF EACH ION IN TERMS OF An important question at t h i s stage of development of the model for the ions i s : what i s the e l e c t r i c f i e l d in a s h e l l which i s due to the nuclei and electrons in a l l of the other shells? To find the answer, i t i s necessary to f i r s t answer a clos e l y related question: what i s the e l e c t r i c f i e l d due to an ion far away from that ion? By an ion i s meant here a nucleus and an electron bounded by an impenetrable s h e l l . Recalling the discussion at the beginning of t h i s chapter, the e l e c t r i c f i e l d at the s i t e of an ion, due to a l l of the others, w i l l be given to a very good approximation simply in terms of the dipole moment~p of each ion. The monopole moments are taken into account by E ^ . Relative to a chosen o r i g i n , the dipole moment "p of a charge d i s t r i b u t i o n p(r) i s b = where the integration goes over a l l of the charge d i s t r i b u t i o n . Relative to that same o r i g i n , the contribution to the e l e c t r i c f i e l d at r^ , due to the dipole moment of the charge d i s t r i b u t i o n , i s \_\fo)~ jn - K (4.2) 67 assuming that r^ , l i e s outside the charge d i s t r i b u t i o n . Equation (4.1) s h a l l be used to calculate the dipole moment of each ion. The o r i g i n s h a l l be chosen to coincide with the s i t e of the nucleus. The charge density p(r) i s given by (4.3) whence (4.4) It i s clear that the physical features of the problem are invariant under rotations around the z-axis. That i s , the problem has a x i a l symmetry, and as such, p> must be in the z-d i r e c t i o n . With t h i s r e a l i z a t i o n , equation (4.4) s i m p l i f i e s to (4.5) where ^'Vif" z^cn'i - <4-6> Using the familiar expansion for <fi, fi0 + a ^ + €<fa + ••• (note that«=a now, because the nucleus i s held up without the imagined force F extK equat ion (4.6) may be expanded out: 68 r b e To the lowest order, (4.7) The integrals here are f a m i l i a r . From e a r l i e r work, r R (4.8) and r (4.9) Equation (4.9) i s an old r e s u l t . Equation (4.8) follows from equation (1.47) and 2r"l J 69 which i t s e l f follows from equations (1.48) and (2.46). Making the replacements (4.7) and (4.9) in (4.7), (4.10) The insertion of equation (3.4) into equation e x p l i c i t l y in terms of Ef.: (4.10) gives p (4.11) where Ex-a 2% J (4.12) and (4.13) As desired, p has now been expressed in terms of E 70 (2.5) CALCULATION OF E * IN TERMS OF p The task of this section i s to calculate the e l e c t r i c f i e l d inside a s h e l l due to the nuclei and electrons in a l l of the other s h e l l s . As noted at the very beginning of t h i s chapter, th i s e l e c t r i c f i e l d may be taken to be uniform inside the s h e l l . The basic reason behind t h i s approximation s h a l l now be repeated. Denote by E l 0 r , s (r) the e l e c t r i c f i e l d at the position r inside of a s h e l l , which i s due to a l l of the other model ions, that i s , a l l of the nuclei and electrons in the other s h e l l s . E Lons (r^ w i l l vary a l i t t l e as r varies inside the s h e l l . To —> —» include t h i s s p a t i a l v a r i a t i o n of EionS(r) in the Schrbdinger equation for the electron inside a sh e l l would have rendered the problem a very d i f f i c u l t one. For example, EZons (r) is not i n i t i a l l y known, and so the form of the potential to enter in the Schr3dinger equation would not have been known. This dilemma was circumvented from the outset by replacing E ^ s (r) by a uniform f i e l d in the z- d i r e c t i o n . Since the average of E tons over the i n t e r i o r of the s h e l l i s just the value of _=, , — > — > E Lor tS (r) at the center of the s h e l l , Elans ( r c ) , i t was decided to replace E iofys (r) in the Schro'dinger equation by E i . o ^ ( r t ) , with the idea that the model would be constructed so that E \ o r t 5 ( r t ) would be in the z-direction. But notice that E ujrtj ( r t ) w i l l d i f f e r only s l i g h t l y from ~ E \ 0 * s ( r n ), the value of —> Ei0rxs(r) at the s i t e of the nucleus, because the nucleus s i t s a 71 very small distance a below the center of the s h e l l . As such, i t is just as good to replace E <^„s (r) by E i ^ s (r„). In the introductory passage to t h i s chapter, i t was noted that Ellens (r*) i s equal to E|,. Recall what Ep i s : in a l a t t i c e of e l e c t r i c dipoles, Ep: i s the e l e c t r i c f i e l d at the s i t e of one — ? dipole due to a l l of the others. As such E Lon3 s h a l l be -> replaced by E ^ . To summarize, the e l e c t r i c f i e l d inside of a s h e l l due to the nuclei and electrons inside a l l of the other s h e l l s i s approximated, in this work, to be uniform and equal to Ep. The s h e l l s w i l l be arranged so that E ^  i s in the z-— = > d i r e c t i o n . An expression for E p in terms of p w i l l now be found. Consider the sh e l l s to be arranged in a l a t t i c e . Then the nuclei are arranged in the same l a t t i c e . To calculate Ep, imagine replacing each s h e l l by a dipole with moment p. These dipoles are arranged, therefore, in the o r i g i n a l l a t t i c e of the s h e l l s . The e l e c t r i c f i e l d at the s i t e of one of these dipoles due to a l l of the others w i l l be a sum of terms l i k e the right hand side of equation (4.2). Choose a coordinate system with i t s or i g i n at the s i t e of the dipole in question. Denote the primitive translation vectors of the l a t t i c e a ( , a z , a 3 . Then the dipoles are located at positions r- =n, a*. +nJ_a*z.+n,a>3, where 7T i s abbreviated notation for the three integers n,, nt, n 3. From equation (4.2), the e l e c t r i c f i e l d at the o r i g i n due to the dipole at r-+ i s 72 r7? V- 5 where i t has been assumed that not a l l of n ,n ,n are zero. The net e l e c t r i c f i e l d at the o r i g i n , Ep, i s obtained by summing the right hand side of equation (5.1) over a l l l a t t i c e s i t e s , excluding the one at the o r i g i n E p must be in the z-direction. This can be achieved by a suitable choice of l a t t i c e . Recall that p i s in the £-d i r e c t i o n . Therefore, according to equation (5.2), i f for every dipole located at (x,y,z), there i s one at (x,y,-z), the x- and 5> y- components of Ep w i l l vanish, where (5.4) and z-g is the z-component of rV. Denoting by d the l a t t i c e constant, equation (5.4) may be written as *~~~~Jl (5.5) 73 where the dimensionless sum S i s 2 „ 1 3?' 3 z / - r * (5.6) Equation (5.5) i s the r e l a t i o n which gives E in terms of p. (2.6) SOLUTIONS FOR p AND E Equations (4.11) and (5.5) may be combined to solve for both p and Ep. The results are: (6.1 ) (6.2) Using equations (4.12) and (4.13), equation (6.1) may be written out in f u l l as 74 (6.3) Equation (6.3) i s a very important r e s u l t , for, as w i l l be seen in the next chapter, i t enters d i r e c t l y into the expression for the average e l e c t r i c f i e l d r e s u l t i n g from the l a t t i c e of ions. • (2.7) THE AVERAGE ELECTRIC FIELD DUE TO THE LATTICE OF IONS Far away from a model ion, the e l e c t r i c f i e l d due to the ion may be taken as that produced by an e l e c t r i c dipole placed at the s i t e of the nucleus (again, the monopole moment i s being ignored). In order to calculate the average e l e c t r i c f i e l d due to the l a t t i c e of ions, however, i t i s necessary to consider the e l e c t r i c f i e l d due to an ion for positions very close to that ion, and even for positions inside of the ion! Consequently, i t is not at a l l obvious that the average f i e l d created by the ion w i l l be given in terms of just i t s dipole moment. As such, a proof i s now presented that t h i s i s indeed so. To begin with, i t i s necessary to decide how to go about ca l c u l a t i n g the average f i e l d due to the l a t t i c e of dipoles. Keeping in mind that one of the key purposes of the model i s to express ideas, the following method i s adopted: Let denote the volume of a primitive c e l l of the l a t t i c e . Consider a spherical region with volume Vt , centered 75 on the middle of one of the s h e l l s , as in figure 8. C a l l t h i s region Figure 8. The Spherical Region 2 The average e l e c t r i c f i e l d due to the l a t t i c e of ions s h a l l be taken to be the average e l e c t r i c f i e l d inside of £ that is due to the ions. The e l e c t r i c f i e l d at the point r^ in the region, due to the nucleus and electron inside the s h e l l which l i e s i n £ j , i s where the contribution due to the monopole moment i s included e x p l i c i t l y . A few simple steps show that the average of t h i s in the region S i s 76 (7.2) The integral over r* in equation (7.2) may be recognized as the e l e c t r i c f i e l d at a point r inside of a sphere of radius R£:(3U/HTI) with a uniform charge density <{e /yc . By spherical symmetry, th i s e l e c t r i c f i e l d must point r a d i a l l y outward and depend only on r: Uc \ r - r \ l Using Gauss' law, \ 3 [ ( r ) : k - — r and so (7.3) Recalling equation (4.4), the expression for "p*, and r e a l i z i n g that i£=1/n 5, where n 5 i s the number of shel l s per unit volume, equation (7.3) becomes 77 Equation (7.4) gives the average e l e c t r i c f i e l d , due to the nucleus and electron in a s h e l l , in a sphere of volume 7j2=l/ns that i s concentric with the s h e l l . It i s easy to see that equation (7.4) i s in fact a general result for any charge d i s t r i b u t i o n whose dipole moment i s p^ where p"* i s given by equation (4.1). A l l that i s required is to retrace the steps from equation (7.1) to equation (7.4) for a charge d i s t r i b u t i o n with density ^ ( r ) . The e l e c t r i c f i e l d given by equation (7.4) i s not quite the average f i e l d due to the l a t t i c e of dipoles; i t includes only the contribution by the ion inside the region. Required s t i l l i s the contribution by the ions, or dipoles, exterior to the region, as well as the so c a l l e d boundary-dipole contribution. It i s a well known result of e l e c t r o s t a t i c s that the average value of the e l e c t r i c f i e l d over a spherical region, due to changes exterior to the region, i s equal to the e l e c t r i c f i e l d due to those exterior charges at the center of the spherical region. Accordingly, the average f i e l d in the spherical region 2, due to the dipoles exterior to i t , i s the value of the f i e l d they create at the center of the sphere, and t h i s i s just Ep. The boundary-dipole contribution Eb.d. i s the contribution of an e f f e c t i v e surface charge density 6 \> to the ambient e l e c t r i c f i e l d . The cause of is the termination of the dipole density d i s t r i b u t i o n P(r*) on the surface of the metal. 6^, i s given by 78 — - > where r t l o c a t e s a p o i n t on the metal surface and r? i s the u n i t outward normal. Ignoring c o n t r i b u t i o n s from the s i d e s of the metal, E L J i s given by where r?s i s the average number d e n s i t y of s h e l l s in the E k < t E a w ^ and E p i n v o l v e n s , and all coi{rlk«tc In c o n c l u s i o n , the average f i e l d due to the l a t t i c e of d i p o l e s i s j u s t If^*,^ +Ep + Ti^ . , <»r ;(7.5) In order to evaluate equation (7.5), l a t t i c e s must be chosen, and S must be determined. (2.8) LATTICE TYPES R e c a l l the requirement that each l a t t i c e must meet:" f o r every l a t t i c e s i t e at ( x , y , z ) , there must be one at ( x , y , - z ) . Some l a t t i c e s are now des c r i b e d which meet t h i s requirement. 79 (A) Cubic L a t t i c e The primitive translation vectors in t h i s case are dx, d£, dz. The cubic l a t t i c e i s a very special case, for the f i e l d E^, given by equation (5.2), i s zero, regardless of the orientation of p. To see t h i s , consider the x-component of Ept where r?f=(n, x + i \ £ + _z)d, and the sum excludes n,=n2.=n3=0. Summing over the cross terms n, n t and n,n3 w i l l give zero. Also, by symmetry, and in consequence, the entire sum vanishes i d e n t i c a l l y . For a cubic l a t t i c e , Ep=0 and S i s zero in equation (7.1). (B) Face-centered Cubic L a t t i c e Consider a face-centered cubic structure, oriented as in figure 9, such that the primitive translation vectors are 80 Figure 9. The Face-centered Cubic L a t t i c e come in pairs with the same x- and y- coordinates, but z-coordinates d i f f e r i n g in sign. It turns out that Ep for thi s l a t t i c e i s zero. To see t h i s , break up the sum over l a t t i c e s i t e s into two parts. Consider f i r s t the sum over the l a t t i c e s i t e s located at 81 where n, , nz, n 3 assume a l l interger values except n,=nz=n3=0. This gives Ep for a cubic l a t t i c e with l a t t i c e constant J ? d , and is therefore zero. Now consider the sum over the rest of the l a t t i c e s i t e s . Each of these s i t e s f a l l s into one of the following three sets: In each case, n i , n7, n 3 assume a l l integer values, including n^=n2=n3=0. It i s easy to show that the net contribution to E p by the dipoles at these l a t t i c e s i t e s i s also zero: one need only be aware that the n 1, n 2, n 3 in the sums are simply dummy variables. 82 (C) Closest Packing; Type 1 In t h i s l a t t i c e , the she l l s are arranged such that a layer of closest packed s h e l l s , in the x-y plane repeats in the z-d i r e c t i o n . This i s experessed precisely by the primitive translation vectors (8.2) (D) Closest Packing; Type 2 This case d i f f e r s from the previous one only in that a^ i s d i f ferent: a , = A X (8.3) This l a t t i c e may be thought of as hexagonal closest packing with 83 every second layer missing. (E) Hexagonal Closest Packed This i s not a l a t t i c e , but i s a l a t t i c e with a two point basis. Even so, i t meets the requirement of having a s h e l l at (x,y,-z) for one at (x,y,z). The basis vectors are "b~t = 0 and Having outlined the types of s h e l l arrangements to be considered, evaluation of S for each type s h a l l now be discussed. Before, leaving this section, the values of ns for each arrangement are presented. n s i s given by 1/^ and 7A is given by a"5^ • (a\xaa ) . The values of n 5 are: (A) Cubic L a t t i c e : n sd =1 (B) Face-centered Cubic L a t t i c e : n5d (C) Closest Packing; Type 1: n sd 3 = z//f 1 (8.4) (D) Closest Packing; Type 2 n 3d 3= f//^ (E) Hexagonal Closest Packed: n sd 3 = /2? (2.9) EWALD SUM 84 D<f) The obvious way to evaluate S for a given arrangement of shel l s i s to insert rp?=n, a, + n 2 a 2 + n,a 5 into equation (5.6). This i s not a very succesful procedure, however, because the resultant sum i s very slowly converging, and many terms need to be added up together to obtain reasonably good accuracy in the f i n a l r e s u l t . It i s not p r a c t i c a l to proceed in t h i s manner. A more rapidly converging expression for S is required. It i s the objective of t h i s section to provide such an expression. The e l e c t r o s t a t i c potential $ (r) at ~v due to a dipole p at r 1 i s h?)=t^£)v (9.,) i where i t has been assumed in writing equation (9.1) that " r / r * . The e l e c t r i c f i e l d E(r) at r due to the dipole at r' i s obtained from $ ( r ) by It i s easy to v e r i f y that 85 L \ l r - r l (9.3) S i n c e p=pz, p - V = p ^ z . Only the z component of E'CT) i s of i n t e r e s t , so c o n s i d e r o n l y [Id - t £ ( ^ ) i c . (9.4) Ep i s o b t a i n e d by summing e q u a t i o n (9.4) over a l l d i p o l e s except f o r the one a t the o r i g i n . Ep i s : r =o (9.5) — 7 the r ' i n e q u a t i o n (9.5) a r e the l a t t i c e s i t e s , the b^ a r e the b a s i s s i t e s a s s o c i a t e d w i t h each l a t t i c e p o i n t , and the prime means t o omit the term a t ^'=0, bi=0. Def i n e (9.6) Note t h a t S'(0)=S/d 3. The problem of d e t e r m i n i n g E P i s tantamount t o the problem of e v a l u a t i n g S' (r) a t r-O". No d e l t a f u n c t i o n s a r e hidden i n 86 equation (9.6), since the domain of rounder study excludes a l l — * —> of the s i t e s r',b;.. S p e c i f i c a l l y , equation (9.6) i s of interest —> only for r near the o r i g i n . Use to write S f ( v ) a s 1 = z p oo (9.7) (9.8) (It i s easy to v e r i f y that the operator tydz1 may be taken inside the integral sign.) The next step i s to break the integration up into two parts: from zero to G, and from G to i n f i n i t y , where G i s any f i n i t e positive number. The expression for S(r) then becomes SCf)»L % l a P + 0 f . e 1 ap (9-9) At t h i s point, the term i s added and subtracted from equation (9.9). The advantage of doing t h i s i s that the f i r s t sum w i l l then be periodic, with the 87 p e r i o d i c i t y of the l a t t i c e , a property which w i l l be exploited shortly. It i s easy to show that 2 4C 3 (9.10) Now define (9.11) where the term for r'=0, b =0 i s included. F( ) has the p e r i o d i c i t y of the l a t t i c e . Using equations (9.10) and (9.11), equation (9.9) may be written as: Now evaluate the middle term on the right-hand side of equation (9.12): bz -ifV-qiy rCO roo Doing the f i r s t integral here by parts s i m p l i f i e s the right-hand 88 side to Doing the remaining integral by parts now gives allows that t h i s be expressed as (9.14) 89 Only F(r) s t i l l needs to be found. As noted previously, F(r) has the p e r i o d i c i t y of the l a t t i c e . As such, F(~r) has an expansion of the form (9.15) — > where the g are the reciprocal l a t t i c e vectors. The F f are given by (9.16) where Vc i s the volume of a primitive c e l l and the integration i s c arried out over a primitive c e l l . Note that e L^ has the p e r i o d i c i t y of the l a t t i c e , by d e f i n i t i o n of reciprocal l a t t i c e vectors, and so equation (9.16) may be written as - 1 -~> (9.17) Mc where \C =NiTt and the integration goes over N primitive c e l l s . Equation (9.17) s h a l l be used in the l i m i t N-*-, as i s now 90 demonstrated. Inserting equation (9.11) into equation (9.17), and simplifying: r .1 £. Now, l e t N ^ - i n such a manner as to cover a l l of r-space. Then the r* integral i s the same for every "r', reducing the expression above to (9.18) where the r ? integral i s over a l l of r-space now. It i s not d i f f i c u l t to show that, for g^O, and in turn that thereby reducing equation (9.18) to (9.19) 91 The integral over (9 i s ea s i l y evaluated, with the result This s i m p l i f i e s equation (9.19) to: h---^3l/M zy-^ (9.20) Equation (9.20) applies for g^O. To f i n i s h i t i s necessary to fin d F~o. Equation (9.18) w i l l not be used to evaluate F^. Instead, (3) use w i l l be made of the following information : i f the dipoles at the s i t e s r in a l a t t i c e vary in di r e c t i o n according to pCr')=pe L^ , where p and q are constants, then the g^o" term of the rec i p r o c a l l a t t i c e sum makes the following contribution to —> -> E(r) The f i r s t term here i s the macroscopic e l e c t r i c f i e l d E(f) ai r due to a macroscopic p o l a r i z a t i o n wave P{r)=+? e^'^ , so the contribution i s simply 92 — > — > As q goes to o, the second term vanishes. The f i r s t term becomes the average e l e c t r i c f i e l d due to a uniform po l a r i z a t i o n d i s t r i b u t i o n P= ^jr , and i s -^^"pk. Thus, F ^ s ' - ^ f o r a l a t t i c e . Taking into account the basis s i t e s , (9.21 ) where Nj^ i s the number of basis s i t e s . Combining equations (9.20), (9.21), (9.5) and (9.4), equation (9.12) becomes where only the real part of equation (9.20) matters ph y s i c a l l y . — > Recall that Ep i s given in terms of S'(0) by As a check of equation (9.22), i t has already been seen that Ep i s zero for a cubic l a t t i c e : does equation (9.22) give 93 the value 0 for a cubic l a t t i c e ? The primitive l a t t i c e vectors are dx, d£, dz: the reci p r o c a l l a t t i c e i s also cubic, with l a t t i c e constant 2"-/d. Notice that, in consequence of t h i s , the sum over the ~q and the f i r s t term in the sum over the r 1 are very s i m i l a r . S'(0) i s independant of G, and the convenient thing to do i s to choose G, so the power of e i s the same in both sums. To achieve t h i s , take G= /JV/d. With th i s choice of G, the sum over the ~g and the f i r s t term of the sum over the r' cancel one another, since Vc=d . Moreover, the terms iC, and -4^/3^ add to zero, leaving only the other terms in the r' sum. This also vanishes, due to the 3 z , 2 - r ' 2 , for the same reason that equation (5.4) vanishes when evaluated for a cubic l a t t i c e . The expression (9.22) therefore passes the t e s t . The presence of the exponentials and the complimentary error function in equation (9.22) render i t a much more quickly converging expression than the expression (5.6) for S. Equation (5.6) would require the summation of many more terms than equation (9.22) to obtain the same accuracy in the resultant p a r t i a l sum. 94 (2.10) EVALUATION OF THE EWALD SUM AND CALCULATION OF E dlonles Equation (9.22) has been evaluated for the three l a t t e r s h e l l arrangements, (C) to (E), described in chapter 8. In each case, the formula i s p a r t i c u l a r i z e d to the arrangement being studied. A generalized computer program was written, and the formula evaluated for each pattern of s h e l l s . The evaluation in cases (C), (D) and (E) was very s i m i l a r . In each case, the expression was analyzed for several values of G, to ensure that the f i n a l answer would be independent of G. This in fact turned out to be so. As a further check, equation (9.22) was also evaluated for l a t t i c e (B). As required, the result was. e f f e c t i v e l y zero, independent of G. The values of S obtained in cases (C) to (E) were as follows: (C) Closest Packing; Type 1: (D) Closest Packing; Type 2: (E) Hexagonal Closest Packed: S=-0.9095420544 S=-5.105810840 (10.1) S=2.968683281 The number of s i g n i f i c a n t figures shown indicates the extent to which the result did not vary with G. E d / p o k s ' given by equation (7.5), w i l l now be calculated for each l a t t i c e and for several values of Z* in each case. To begin with, using equation (6.3) for p and the expressions for ^ E T , and £ E L F i t i s found that 95 ^ (MfZ* -loMooiV+lJ+o.wxlS (10*2> Write Then, using equations (8.4) for n s d S and (10.1) for S, the following values are found for oc,, <xZl and c< 3 : L a t t i c e : (A) (B) (C) (D) (E) v f l u e \L7o ZZ.il 20-13 /CfO 2JDX6 or c * ( : Table I : Value of oc, for l a t t i c e types (A) through (E). oc, does not depend on Z„. Value of 1 2 3 4 L a t t i c e : (A) (B) (C) (D) (E) J2.W It.11 IS.SC I3n ff.93 Z72.6 3/8.S UC.7 U3/ I3C5 ¥Z7Z ¥313 5Z% Table I I : Values of for l a t t i c e s (A) through (E) for some values of 2^. 96 <X3 i s zero for a l l Z„ for both l a t t i c e s (A) and (B). For l a t t i c e s (C) through (E): Value of L a t t i c e : Z ft : (C) (D) (E) 1 0.6989 3.923 -2.281 2 14.30 80.28 -46.68 3 73.24 411.2 -239.1 4 231 .9 1302 -757.0 Table I I I : Values of <=K3for l a t t i c e s (C) through (E) for some values of Z^ . The objective of Chapter 2 has been f u l f i l l e d . It i s indeed easy to see that £/(kc^/R2, ) i s small. Take R to be one of the order of a e , the Bohr radius, and E P to be of order Mg/q Then £/(!<<£//) i s of the order 10"16 97 I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals,  Series, and Products, 4th ed., Alan Jeffrey (Academic Press, New York, 1965), p. 307. 3 J.M. Ziman, P r i n c i p l e s of the Theory of Solids, 2nd ed. (Cambridge University Press, Cambridge, 1972) pp. 41-42. 98 CHAPTER 3  THE CONDUCTION ELECTRONS PRELIMINARY DISCUSSION In the last chapter, a model of the ions, in a metal was presented. This model was used to calculate the contribution to — > E MC of the g r a v i t a t i o n a l l y induced ionic dipole moments. In th i s chapter, a model of the conduction electrons i s constructed. The objective of t h i s chapter i s to use the model to calculate the number density of the electrons as a function of height. The reason t h i s c a l c u l a t i o n i s interesting i s because i t leads to an investigation of the charge d i s t r i b u t i o n in the i n t e r i o r of the metal. Charge ne u t r a l i t y in the metal i n t e r i o r i s not assumed in t h i s t h e s i s . Instead, the charge density inside a metal i s calculated. The charge density i s determined by comparing the number densities of ions and electrons as functions of height. The density of ions may be found using e l a s t i c i t y theory; finding the electron density is the goal of th i s chapter. Once the charge density i s known, i t s contribution E ch to ~~Eave may be calculated. The basic physical idea underlying the model i s that the electron constituent must be d i s t r i b u t e d in the metal in such a manner that the electron gas i s held up against gravity. B a s i c a l l y , two physical processes are involved in countering the 99 force of gravity on the electrons. One i s the e l e c t r o s t a t i c force experienced by the electrons. The other i s the fermion nature of the electrons, for i f there i s a nonuniformity in the electron number densisty, there w i l l be a gradient in the l o c a l pressure in the gas. The manner in which these physical concepts are incorporated into the model i s now described. The electronic constituent of a metal i s treated as a system of noninteracting fermions that i s subject to an external linear p o t e n t i a l . It i s assumed that t h i s linear potential i s in the v e r t i c a l d i r e c t i o n . The linear potential takes into account the electron-electron interactions, the interactions between the electrons and the l a t t i c e of ions, the weight of the electrons, and the externally imposed e l e c t r i c f i e l d Eex.^. Electron-electron interactions and e l e c t r o n - l a t t i c e interactions are taken into account by averaging them out. S p e c f i c a l l y , each electron i s regarded to be subject to an average potential created by the rest of the —> electrons, the l a t t i c e of ions, gravity, and the f i e l d E e x t . The slope of t h i s p o t e n t i a l i s mg+q&(E ave + E e x£ ) , where E a^e i s posi t i v e of points in the d i r e c t i o n opposite to g, and s i m i l a r l y for E e xt« It i s the E a i / e term in t h i s slope that takes into account the electron-electron and e l e c t r o n - l a t t i c e interactions. Notice the assumption that E ave i s a uniform f i e l d . Once again, t h i s i s a s i m p l i f i c a t i o n which i s made for convenience. . As wall be seen later in th i s thesis, E M in fact depends on height. However, this dependence is extremely small. Taking 100 E ave as uniform, therefore, i s an assumption that leads to results with which the assumption i s consistent: the assumption — > and the results are consistent to the order to which E a v / a i s calculated. The physical reason for t h i s agreement i s that the charge imbalance in t h i s model, although not i d e n t i c a l l y zero, is very small. This point s h a l l be further c l a r i f i e d at the appropriate stage in t h i s thesis. There i s a very simple way to obtain an expression for the number density of electrons as a function of height. In t h i s approach, i t is assumed that the electron gas behaves, l o c a l l y , as i f i t were a free gas. The c a l c u l a t i o n i s done in Appendix B. A more mathematically rigorous treatment, however, leads to a better understanding and appreciation of the problem. In addition, some interesting results are obtained along the path to the solution. In the next section of t h i s chapter, the problem i s spe c i f i e d in more d e t a i l . Subsequent sections deal with the solving of the problem and the ensuing implications in terms of g r a v i t a t i o n a l l y induced e l e c t r i c f i e l d s in metals. 101 (3.1) DESCRIPTION OF THE PROBLEM OF A FREE FERMI GAS SUBJECT TO  AN EXTERNAL LINEAR POTENTIAL The system to be studied consists of N i d e n t i c a l , noninteracting fermions, confined to a cube of edge length L, and subject to an external linear p o t e n t i a l . Because the p a r t i c l e s do not interact, the Hamiltonian of the N p a r t i c l e composite i s simply the sum of N single p a r t i c l e Hamiltonians: A A. A M A A A. H N l ( I.£)=SH ; ( f ; > t ) • (...) The caret denotes an operator, while the t i l d e i s short for a c o l l e c t i o n of N operators. The subscripts N and L serve as reminders that there are N fermions and they are confined to a cube of edge length L. p^ and r^ . are, respectively, the momentum and position operators of the i fermion. The single p a r t i c l e Hamiltonian i s : A A where m i s the mass of each fermion and ^ ( r ^ ) i s the external potential experienced by each. Because of equation (1.1) the eigenfunctions and energy A eigenvalues of H w, L can be expressed in terms of the single p a r t i c l e eigenfunctions and energy eigenvalues. The l a t t e r are 102 given by where k labels the eigenfunction and the corresponding eigenvalue. It i s c r u c i a l to mark well the meaning of k. k i s a multilabel which includes the spin l a b e l . Later in t h i s — > section, a label k s h a l l be used. The label k i s short for the composite of labels k and the spin l a b e l . The H° in equation (1.3) may be thought of as the position representation of equation (1.2). For the case of a linear potential and absolutely confining walls, *(c?)-< C 7 } ' A - S ' ^ cJae. "A y (1.4) The eigenf unctions and eigenvalues of H*lt are expressed in terms of the f;\h and e r through the use of occupation numbers n K. Thus, the eigenstate J £ (r) belonging to the eigenvalues E^ of iW may be l a b e l l e d by the n K: ^JV^r*) i s denoted "t^^ ir). S i m i l a r l y , E i s denoted E ^ j , and i s given 103 by [{T-Z^t (1-5) The expression for "~p (r) i s more complicated. The important thing to be aware of here i s that V^^r,) may be thought of as a state of the N p a r t i c l e s in which the number of p a r t i c l e s in the single p a r t i c l e state % (r) i s n k; equation (1.5) supports this point of view. So does the requirement (1.6) The bulk properties of the Fermi gas at a temperature T may be determined by employing the rules of s t a t i s t i c a l mechanics. One piece of information obtained in t h i s way i s the average occupancy of the state % ( r ) , denoted <n K>L. More w i l l be said about <n)t>1. in the next section. For the moment, <n K >L may be thought of as the number of p a r t i c l e s e x i s t i n g in the state ^""'(r) at temperature T, although i t should be noted that t h i s i s only a coarse, i n t u i t i v e way to think of < n K > L . These average occupation numbers <nyt>L play a key role in determining the value of the l o c a l number density of fermions, denoted by n^Cr). It i s possible to proceed from the complicated expression for i ^ n ^ (r) to derive the following equat ion: 104 (1.7) Ic Equation (1.7), however, i s clear on i n t u i t i v e grounds, by thinking of <nK>u as being the number of fermions in the state It i s reasonable to expect that, deep inside the metal, the lo c a l number density of free electrons w i l l be independent of L, x and y. This fact emerges from equation (1.7) in a mathematically precise manner by taking the thermodynamic l i m i t : where -usz/L, ~r'= r^/L, n(u) i s the l i m i t of r\U)("r) and (r > /) i s the l i m i t of 1}^(r). n(u) i s a temperature dependent function, the temperature dependence entering through the < n i t > . According to equation (1.8), n(u) may be determined by finding the <n*> and the / , K ( r ? / ) , and performing the indicated summation. The result for n(n) may then be determined for the model by taking T=0; r e c a l l that the model i s for the metal in i t s ground state. This method of determining n(n) i s the one presented in the next section. l e t t i n g L->°°in such a way that the average p a r t i c l e density remains constant. I n t u i t i v e l y , therefore, i t is to be expected that, in the thermodynamic l i m i t , equation (1.7) w i l l go over to (1.8) 105 An alternate method i s presented thereafter. In thi s method, the s p e c i f i c a t i o n T=0 i s made at the beginning, which further s i m p l i f i e s the expression for n(*tt). The external potential i s treated as a perturbation, and the ^(r*) are found by using perturbative techniques. n(-u) i s evaluated to lowest order in the perturbative parameter. A comparison between the two methods i s made, and i t i s shown that both lead to the same r e s u l t . (3.2) STATISTICAL MECHANICS APPROACH Recently, a paper appeared in the l i t e r a t u r e which considers the problem of a noninteracting Bose gas subject to a weak external potential of power form. Study of thi s paper reveals that i t is very easy to extend the results of the paper to the case of a Fermi gas, provided that the fugacity does not exceed unity. The fugacity i s e , where k 8 is Boltzmann's constant, T i s the absolute temperature, and^a is the chemical p o t e n t i a l . It i s for 0<e^<l that the aforementioned extension may be made. For t h i s range of the fugacity, the results obtained are as follows: The grand canonical pressure p i s given by 106 (2.2) where (2.3) (2.4) (2.5) where c i s the slope of the external linear p o t e n t i a l , and ^ i s given in terms of the average p a r t i c l e density £ by (2.6) where 2 /V/TT-JO y ~ l e * z - M (2.7) 107 The l o c a l pressure and l o c a l number density of fermions are given by (2.8) and where IC =Z. (2.10) • " L ' The ^ in the above equations i s in fact the fugacity; = e / S / * (2.11) Equation (2.6) establishes a one-to-one correspondence between p and i~ , whence equations (2.2) and (2.6) determine a unique plot of p in terms of ^ >, for a given temperature. \ i s c a l l e d the thermal wavelength. c i s the difference in potential energy between the "bottom" of the cube, at 11=0, and the "top" , atoc = 1 . 108 N o t i c e t h a t e q u a t i o n s (2.8) and (2.9) a r e the p r e s s u r e and number d e n s i t y f o r a f r e e Fermi gas w i t h c h e m i c a l p o t e n t i a l fx-cu. The a p p e a l of the above r e s u l t s i n terms of the i n v e s t i g a t i o n a t hand i s t h i s : i f t h e s e r e s u l t s c o u l d be shown t o be t r u e f o r a l l ^  , n('u) c o u l d be o b t a i n e d f o r the f r e e e l e c t r o n s i n a metal s i m p l y by e v a l u a t i n g (2.9) i n the l i m i t (T-»0). In the remainder of t h i s s e c t i o n , an o u t l i n e i s p r e s e n t e d d e s c r i b i n g how the above r e s u l t s a r e o b t a i n e d f o r p, Q , p(tO and n ( n ) , a d i s c u s s i o n i s p r e s e n t e d t o i n d i c a t e t h a t these r e s u l t s a r e indeed v a l i d f o r a l l ^ , and the e x p r e s s i o n s a r e e v a l u a t e d i n t h e l i m i t fl-*0*. (A) Review of Some B a s i c S t a t i s t i c a l M e c h a n i c a l Ideas The work t o be done here s h a l l be done u s i n g the grand c a n o n i c a l ensemble. The grand c a n o n i c a l p a r t i t i o n f u n c t i o n i s where ft = l / k B T , T i s the t e m p e r a t u r e of the heat b a t h , V i s the volume of the system,yU i s the c h e m i c a l p o t e n t i a l of the heat b a t h and Q(/?,N,V) i s the p a r t i t i o n f u n c t i o n f o r the c a n o n i c a l 109 ensemble: QfaWhZ e p W . { 2 - 1 3 ) The sum in equation (2.13) i s over a l l occupation number sets s a t i s f y i n g equation (1.6) and E ^ ^ i s as defined previously. Combining equations (2.12) and (2.13), and using equation (1.5), Because the fermions are noninteracting, equation (2.13) may be s i m p l i f i e d . Using equation (1.6), (2.15) It i s easy to v e r i f y that the two summations in equation (2.15) may be replaced, equivalently, by summing each n k independently. This gives: 110 r S 2 Z ' i e n « r>» W £ - 0 L J .i may assume only the values 0 and 1 because of the Pauli exclusion p r i n c i p l e . Equation (2.15) has become (2.16) Equation (2.16) i s the grand canonical p a r t i t i o n function for nomnteracting fermions confined to a volume L ; the c k refer to a cube of edge length L. The grand canonical pressure i s obtained from Z via (2.17) Using equation (2.16), equation (2.17) becomes (2.18) The mean p a r t i c l e number <N>L i s given by 111 <N>L = f ^ft-^. i )z z "*° m /3z 4« or (2.19) Inserting equation (2.16) into equation (2.19) gives ^>L--2 e / 5 ( ^ - / u ) + l ' ( 2 - 2 0 > in terms of which the average p a r t i c l e number density i s Q z I M ^ , . (2.21) 1 L p i s not a funtion of L. In the thermodynamic l i m i t , as L goes to i n f i n i t y , so does <N>L , in such a way that (p stays constant. The ensemble average occupation numbers are given by or 1 12 Use of equation (2.16) once again gives * ' L " e/SC^-/*) + |' (2.23) A comparison of equations (2.20) and (2.23) shows that < M > L - T < n k \ • ( 2 - 2 4 ) 1 k The thermodynamic l i m i t consists of l e t t i n g Ir* 0 0 but such that remains constant. By equations (2.20) and (2.21), t h i s gives an i m p l i c i t equation totju , o r ^ = e ^ , in terms of :' i y — — ; — (2.25) p denotes the l i m i t of pM{^^) 113 Equations (2.25) and (2.26) together give p in terms of p , at a given temperature T. The l i m i t of equation (2.23), with given by equation (2.25), gives the <nK> which enter into equation (1.8). (B) Specia l i z a t i o n to Case of no External Potential; Review of  the Ideal Fermi Gas^ 2 ) When there i s no external p o t e n t i a l , the single p a r t i c l e eigenvalues are "pV2m, where ^ =2n^h*/L andln i s a vector whose components are integers. As such, in the l i m i t L-*-°, the sums over k may be replaced by integrals over p, as in 1 where s i s the spin of the fermion (s=xfor electrons). In this case, the expression for p, equation (2.26), becomes _(25+l)<frr / - I C O A J and the equation for p, equation (2.25), gives fe,S*l)fn-(2.27) (2.28) 114 where i s the fugacity, given by equation (2.12). Equations (2.27) and (2.28) may be written as and P= ~ J f j / ^ S ) ' (2.30) with A given by equation (2.3) r ^ M f ) by equation (2.4) and J \ ( ^ ) by equation (2.7). I f 0<^ *<l , these integral expressions may be written as power series expansions. It i s easy to do t h i s . The results are: •z— :5lZ~ , V ^ N ' (2.31) d'1 u and - ( l ! L _ r (2.32) V • it, Notice that (2.33) 115 This i s in fact true for a l l ^ , as equations (2.4) and (2.7) show. Also notice the source of the symbol to denote the integral expressions (2.4) and (2.7): the subscript i s the power of 1/j in the sum when 0<^<1. Equations (2.31) and (2.32) are useful for expanding equations (2.29) and (2.30) in the l i m i t £>A«i. For, from equations (2.30) and (2.32), may be expressed in powers of >^A , with the result 5 Zjtl V H z ^ l / (2.34) Use of equations (2.31) and (2.34) in equation (2.29) then gives h.C/|+4 p £ t > (2.35, Observe that ( > A 3 « i corresponds physically to the high temperature and/or low density l i m i t of the Fermi gas. As required, therefore, equation (2.35) reproduces, to leading order, the c l a s s i c a l ideal gas law. This result is due to the 3 fact that ^A<<1 means that the average p a r t i c l e separation i s much larger than the thermal wavelength, so quantum ef f e c t s are small. The corrections in equation (2.35) to the c l a s s i c a l ideal gas law are due to precisely those quantum e f f e c t s . 1 16 Equation (2.34) may also be inserted into equation (2.23), with the result that (2.36) to leading order. Equation (2.36) i s just the Maxwell-Boltzmann d i s t r i b u t i o n function. The other extreme for which equations (2.29) and (2.30) may be approximated i s the low temperature and/or high density l i m i t , £ A > > 1 . In th i s case, i t i s necessary to fi n d an expansion f or-p3 /2 (^ ) as Such an expansion may be obtained by st a r t i n g with equation (2.7). One obtains, as 36 - r r Z , „ 1 k^m^rw -foe?-) (2.37) S i m i l a r i l y , Equations (2.30) and (2.37) give, to lowest order, 117 or, using equation (2.3) for A , and equation (2.11) for ^ , / ^ 2 ~ U * T / ^ P ^ f A - * o ° - (2.39) Expanding in powers of kBT/£F , equations (2.37) and (2.30) give (2.40) <n^> i s given by ^i>~' eP>(^/^ju-) \ ' (2.41) with," given by equation (2.40). Thus, < n?/ 1 ^  ,r w v { *. \+0 • (2.42) The expansion for p i s obtained using equations (2.29), (2.39), (2.39), and (2.40): (2.43) These results are very d i f f e r e n t from the c l a s s i c a l r e s u l t . 118 This i s because the l i m i t >^A5>>1 corresponds to a thermal wavelength which i s large compared to average p a r t i c l e separation, and so quantum e f f e c t s are very important. It i s clear from equations (2.4) and (2.7) thatj^/i. ( p and J^(^) are monotonic increasing functions. As such, p i s a monotonic increasing function of ^ , for a given temperature. Moreover, the relationships are smooth, and as a result the ideal Fermi gas exhibits no phase t r a n s i t i o n s . (C) Case of an External Linear Potential Consider in more d e t a i l now the Schrodinger equation (1.3), with H given by equation (1.4), and with the eigenfunctions normalized as usual according to / |f<wf<l'r Cube (2.44) That ^ L . it) i s i n f i n i t e outside the cube means ^^ ) ( r > ) = 0 outside the cube, and further that (r) vanish on the walls of the cube. It i s straightforward to see that the eigenfunctions are given by (2.45) 119 where (2.46) and the eigenvalues £ ^ are given by (2.47) The label k denotes the t r i p l e ( k x , k y , k z ) , with »VtT \ , (VTT (2.48) where n y = 1 , 2, 3 ,... , n y = 1,2,3,... normalization condition J^K^(z) must s a t i s f y the (2.49) as well as the boundary conditions 120 1<<0 = £ ( Q = 0 . (2.50) The problem s p e c i f i e d by e q u a t i o n s ( 2 . 4 6 ) , (2.49) and (2.50) may be s o l v e d e x a c t l y f o r J K i (z) and EKz. P u t t i n g (2.51 ) and (2.52) c o n v e r t s the problem t o dZ _ Zw\c\} U -cL (2.53) d -1 (2.54) ft 121 (2.55) E q u a t i o n (2.53) r e v e a l s t h a t the i m p o r t a n t d i m e n s i o n l e s s parameter i n the problem i s 4 = - ^ ^ . (2.56) I f &L i s s m a l l , a u s e f u l approach i s p e r t u r b a t i o n t h e o r y , even though an e x a c t s o l u t i o n i s p o s s i b l e . T h i s approach i s undertaken i n the next s e c t i o n . The extreme where 0 0 i s v e r y l a r g e s h a l l be examined below. D e n o t i n g \ ~ — - •) (2.57) K2 cL and p u t t i n g (2.58) 122 v=.<x , y3(u- y k j ; expresses the problem as (2.59) (2.60) V Z (2.61 ) O • (2.62) The exact solutions to equation (2.60) are the two Airy functions Aj_(v) and B^(v) depicted graphically in figures 10(a) and 10 ( b ) ^ Hence, K Q^fi^W+t^g.CU) - (2.63) 123 (a) \ 1 • /» .it / / •a - / -Jo - / .% /1 \1 / 1 \* '/ • \ 1 1 / /.Ic 'A - .1 1 1 I s /-"? Wf /-7 V* As -4 \-3 -Z / i • '-•4 - -J, F i g u r e 10. The A i r y F u n c t i o n s A 7 ( v ) and By (v) a* z, b k 2 a n d }f*2 must be c h o s e n t o s a t i s f y e q u a t i o n s (2.61) and ( 2 . 6 2 ) ; t h u s : (2.64) 124 (2.65) and n' 2^  (2.66) Equations (2.64) and (2.65) combine to read (2.67) or By figure 10(a), i t is clear that equation (2.68) fixes a discrete set of values <&z , where labels those values. Equation (2.67) then gives the r a t i o ^ / t f c z for that value of ^ l c z . From there, equation (2.66) may be solved for one of the two constants, whence the other i s also known. In this way the problem i s exactly soluble. Unfortunately, the solution i s highly i m p l i c i t . In fact, 125 to the author's knowledge, expressions are not available in the l i t e r a t u r e which give the / K 2 ' s s a t i s f y i n g equation (2.68). What s h a l l be done, therefore, i s to study the problem sp e c i f i e d by equations (2.60) - (2.62) for the cases°<«1 and c*>>1. The extreme c<<<i w i l l be dealt with by using perturbation theory on equations (2.53) - (2.55). The problem posed by equations (2.60) - (2.62) for°<>>1 has been studied by the author. Some interesting r e s u l t s emerged which relate to the case of a low density Fermi gas at absolute zero in a gr a v i t a t i o n a l f i e l d . These results are presented in Appendix C. For the purposes of the present chapter, the key thing to notice is the emergence of the dimensionless parameter^. In analyzing the expressions for Q and p, equations (2.25) and (2.26), L i s to be taken to increase without bound. What does th i s imply for<=<? Is o< to increase without bound as well? What about c? Should c remain constant as L increases? Or i s i t better to hold something else fixed? Van den Berg and Lewis chose to hold cL fixed as L-»—. In that case c^- blows up with L l i k e L . Their reason for holding cL fixed was to ensure that the ef f e c t of the external potential would not be so extreme as to destroy the thermodynamic behavior of the system. They desired t h i s because they were investigating the modification of Bose-Einstein condensation due to the external p o t e n t i a l . This approach to the Fermi gas problem — holding cL fixed as l r > o a — s h a l l be considered next. The steps to be outlined 126 follow those employed by Van den Berg and Lewis, (i) Approach due to Van den Berg and Lewis The physical system considered by Van den Berg and Lewis in thei r paper i s a noninteracting Bose gas subject to an external potential of power form, c (-r)^ , w h e r e £ > 0 . (In th i s thesis, the potent i a l i s l i n e a r , s o , £ = 1 . ) They derive expressions for p, (> , pOu) and n (u) in terms of in the l i m i t L->°" with held constant. employed in th i s paper may be e a s i l y extended to the Fermi gas when the temperature and density are such t h a t ^ < 1 . For a complete understanding of these techniques, the reader should consult the paper i t s e l f . A brief sketch of the key ideas, however, is presented here, along with the p a r t i c u l a r form they take for the case of a noninteracting Fermi gas in an external li n e a r p o t e n t i a l . F i r s t , define mentioned previously, the mathematical techniques ( 2 . 6 9 ) and ( 2 . 7 0 ) 127 where the £ ^ are given by equation (2.47) and &i ) i s the smallest eigenvalue. In terms of 2f(-0 and ")7£) the occupation numbers (2.23) may be written as * c ; ' ( 2 - 7 1 ) L e"? + f CD j^(^) i s to be determined via the conditions (2.24) and (2.71); i . e . : P= -rr• = ~fr|>^ ' (2'72) The f i r s t step i s to prove that equation (2.72) leads to equation (2.6) in the l i m i t L—•>°". The second step is' to show that equation (2.18), which can be written as gives equation (2.2) for IT*-". Then, expressions (2.8) and (2.9) for p(-u) and n.(-u) have to be derived. The s t a r t i n g point of the above proofs i s to rewrite and plL\ p may be expressed as 128 or p=-(2s*IJ2 ((-ir(^L)) 5L(h, , (2.74) where S,^) = k^ e n > l fe > ( 2 . 7 5 ) It has been assumed that J (L)<1 (which implies ^<1) in order to employ X 0° A V I with x i d e n t i f i e d a s J ( L ) e " ^ . S i m i l a r i l y , using o 3 i vv . „ _ , 1^  with the i d e n t i f i c a t i o n of x as ^ ( L ) e 1 ? , p ^ may be written as ^ - ^ ( ( " 0 ( 2 . 7 6 ) 129 Define G(n) via Combining equations (2.6), (2.32) and (2.77), one result to be proven i s that, in the l i m i t L->-=, From equations (2.2), (2.31) and (2.77), another result to prove i s (2.79) Proving equations (2.78) and (2.79) i s by no means a t r i v i a l task. However, the equations Jb and 130 S, Cn) = GiC*) (2.81 ) c e r t a i n l y suggest that equations (2.78) and (2.79) w i l l result in the thermodynamic l i m i t . Equations (2.80) and (2.81) are not s u f f i c i e n t in themselves to show that equations (2.78) and The asymptotic forms of (L) and S (n) as L"y<^ are required. The mathematical manipulations required to prove equations (2.78) and (2.79) are somewhat lengthy, but the basic idea involved i s captured in equations (2.80) and (2.81). Combining equations (1.7) and (2.71), where the x- and y-dependences are omitted: they w i l l disappear as L-*°° . S i m i l a r l y , the equation for p4)(ai) i s (2.79) result from equations (2.74) and (2.76), respectively. 02.82) (2.83) Equation (2.82) may be cast into the form (2.84) 131 where From equations (2.84) and (2.85), by using a simple extension of one of the lemmas stated in the Van den Berg and Lewis paper, i t is easy to see that equation' (2.9) r e s u l t s , where n ( l A ) - | ^ ( % ) . (2.86) In a similar fashion, equation (2.8) can be shown to result by taking the l i m i t L->°° in equation (2.76). This completes the sketch of the proofs for equations (2.2), (2.6), (2.8) and (2.9). As noted previously, the proofs are v a l i d only for 0<^<1. The reason for thi s i s simply that a l l of the proofs in the Van den Berg and Lewis paper use power series expansions for the functions in question: in order to extend those proofs to the case of a Fermi gas, the power series forms for J s / J J ) a n d ^ ( J ) must be v a l i d , and thi s i s so only i f 0<^ <1 . To summarize, i f ^ i s in the range [0,1], the following equations may be e a s i l y proven by extending the results of the Van den Berg and Lewis paper: A ( 2 . 8 7 ) 132 i (2.88) (2.89) (2.90) S i n c e e q u a t i o n s (2.87) t o (2.90) r e s u l t i n t h e l i m i t L->- i f cL i s h e l d f i x e d , the p h y s i c a l i n t e r p r e t a t i o n of t h e s e e q u a t i o n s i s as f o l l o w s : the e x t e r n a l p o t e n t i a l i s so weak t h a t , l o c a l l y , t he gas behaves as i f i t were a f r e e Fermi gas w i t h c h e m i c a l p o t e n t i a l / " -cu.; and p a r e s i m p l y t h e average p a r t i c l e d e n s i t y and average p r e s s u r e , r e s p e c t i v e l y . For a more d i r e c t r o u t e t o t h i s r e s u l t , see Appendix B. The o b v i o u s r e q u i r e m e n t a t t h i s s t age i s t o show t h a t e q u a t i o n (2.89) i s v a l i d f o r a l l ^ . Reasons why t h i s s h o u l d be e x p e c t e d a r e g i v e n n e x t . Then, e q u a t i o n s (2.87) t o (2.90) are p a r t i c u l a r i z e d t o temperature T=0 K. 133 ( i i ) A n a l y t i c i t y of the L o c a l Number D e n s i t y i n the  F u g a c i t y A p h y s i c a l reason s h a l l now be p r e s e n t e d which i n d i c a t e s t h a t niu.), as g i v e n by e q u a t i o n s (2.86) and ( 2 . 8 2 ) , i s e q u a l t o the f u n c t i o n f o r a l l ^>0. E q u a t i o n (2.89) i s v a l i d f o r 0<^<1. n(u;j[) i s an a n a l y t i c f u n c t i o n of J . I f n(u) i s a l s o a n a l y t i c i n j , i t w i l l f o l l o w , from the t h e o r y of a n a l y t i c f u n c t i o n s , t h a t n(u) i s e q u a l t o n (u,J ) f o r a l l £ . There i s a c o n v i n c i n g p h y s i c a l argument which i n d i c a t e s t h i s must be so. Assume t h a t n(&) i s not a n a l y t i c i n ^  . Then t h e r e must be a phase t r a n s i t i o n i n t h e gas f o r some v a l u e of ^ . But the gas i s c o m p r i s e d of n o n i n t e r a c t i n g f e r m i o n s . As such, the Fermi s t a t i s t i c s which the p a r t i c l e s obey g i v e r i s e t o a r e p u l s i o n between the i n d i v i d u a l f e r m i o n s , whereas a phase t r a n s i t i o n can o n l y r e s u l t i f t h e r e i s an a t t r a c t i v e f o r c e between the i n d i v i d u a l p a r t i c l e s . Thus, t h e r e can be no phase t r a n s i t i o n , and n(u-) must be a n a l y t i c i n 2j . T h i s means t h a t e q u a t i o n (2.89) i s v a l i d f o r a l l ^ . S i m i l a r r e a s o n i n g can be a p p l i e d t o e q u a t i o n s ( 2 . 8 7 ) , (2.88) and (2.90) t o i n d i c a t e t h a t a l l f o u r e x p r e s s i o n s a r e v a l i d f o r a l l t- . 134 A m a t h e m a t i c a l p r o o f of the a n a l y t i c i t y of n(u) i s not p r e s e n t e d i n t h i s t h e s i s . ( i i i ) S p e c i a l i z a t i o n t o Case T=0 K i o n s U s i n g the a s y m p t o t i c forms f o r j ° 3 / v ( J ) a n d J ^ ( J ) equat ( 2 . 3 7 ) and ( 2 . 3 8 ) , r e s p e c t i v e l y — i t i s easy t o show t h a t , i n the l i m i t , when <£> >cT ^ equat i o n s ( 2 . 8 7 ) t o ( 2 . 9 0 ) become, f o r s=^z, 7 ( 2 - 9 2 ) ( 2 .93 ) where 135 (2.96) (3.3) PERTURBATIVE APPROACH In th i s section, ^ s h a l l be taken to be a small number, and perturbation theory w i l l be used to derive an expression for n(oi) correct to the f i r s t order in°^. The gas s h a l l be taken to be at absolute zero (and i t i s to be assumed from the outset that the number of p a r t i c i e s in the system i s very large, i . e . , that the system is large enough to be regarded as macroscopic). The perturbation problem i s specified by the following equat ions: ( 3 . 1 ) (3.2) (3.3) (3.4) 136 [_^Ef\,^KZEf\... (3 .5 , Since the system i s large, ft * (r) s h a l l be taken to be periodic in the x and y dir e c t i o n s , with p e r i o d i c i t y L. j/1 ^  * j s t o vanish on the planes z=0 and z=L. This choice of boundary conditions i s made for convenience. Using standard perturbation theory techniques, one finds that (3.6) (3.7) where (3.9) 137 (3.10) and also F l " 2 (3.11) ' (f)^ljDe i ( z ) ' (3.12) where (3.13) Normalizing the eigenfunction determines A^ (3.14) z 8 ( L k J By virtu e of equation (3.11), each energy l e v e l i s elevated 138 by the same amount, -%c, independent of k. In consequence, to t h i s order in the perturbation c a l c u l a t i o n , the Fermi surface for the gas i s spherical, just as for a free gas. The Fermi wave number k F may therefore be immediately expressed in terms of the average p a r t i c l e density ^: k F = ( 3 i r ^ . ( 3 . - 5 ) (the spin s of the fermions has been taken to be £ ) . To f i r s t order in c:, the l o c a l number density of fermions n(ti) is k: IcClcr (?) Z *2c tf% ii & , (3.16) — > The summation in equation (3.16) i s over a l l k' with magnitude k less than k F. The factor 2 preceding the summation sign arises because the fermions are spin i: p a r t i c l e s . Equation (3.16) i s e a s i l y s i m p l i f i e d to where 139 I _ AcTT ( 3 . 1 8 ) ~ ~~L~ Using the well known formulae ( 3 . 2 0 ) 5 ' i n Z X = ^ ( l - C o s 2 x ) ( 3 . 2 1 ) i t i s straightforward to check that, to dominant order in the large number np-, the f i r s t term on the right hand side of equation ( 3 . 1 7 ) is simply (•> . Combining t h i s result and equations ( 3 . 1 3 ) and ( 3 . 1 4 ) , equation ( 3 . 1 7 ) reduces to + . ( 3 . 2 2 ) 140 Combining equations ( 3 . 1 9 ) , (3.21) and the relations z**A (3-24) ic (3.23) 0 0 1 1 X S - 5 W l o c = — - , 0 < X < Z T F (3.25) =1 < Z k=, ^ 6 Z H 1 ° ^ ^ ' (3.26, i t follows that the terms of order np2" in the curly parentheses in equation (3.22) cancel out. As such, i t i s necessary to go to order n F. The following expressions supplement those already c i t e d : 0*0 I ~ — a ; Hp ( 7 ° (3.27) y i 1 Z_ ^ — •»p k=nF Ic2" nF 141 ^ - A F k W W V " ' (3.28) 2 s h b c 5 $ ^ x K n F ^ V f , p (3.29) O<X<2TT Using these three equations, equation (3.22) becomes, to l e a d i n g order, (3.30) Equation (3.27) i s proven by showing t h a t Z L i k ' i s bound dx/x z and from above by dx/(x-1) . Equations (3.28) and (3.29) are proven i n Appendix D. Equation (3.30) gives the f i r s t order expression f o r the l o c a l number d e n s i t y . The mathematical c r i t e r i o n f o r which the .preceding d e r i v a t i o n i s v a l i d is<*<<1, or L«(h2"/2mc)= J-* . I f c i s of the order of mg, JLo i s of the order of 10 3 meters! This would seem to suggest t h a t , unless c turns out to be extremely s m a l l , equation (3.30) w i l l be of no use i n the present work. 142 This, in fact, turns out to be not so, and as s h a l l be seen, equation (3.30) w i l l play a key role in the model metal. (3.4) LINEARITY OF THE LOCAL NUMBER DENSITY IN THE PARAMETER cV Equations (2.92) to (2.95) are v a l i d for < * » \ and c y<f F. Consider the extreme & «£f . Expanding equations (2.92) and (2.94) in terms of the small parameter £ / c > , and retaining terms only up to f i r s t power in C / E p , the following expressions r e s u l t : (4 .1 ) (4.2) Using equation (4.2) to eliminate £ F from equation (4.1), or, equivalently: (4.3) 143 Compare equations ( 3 . 3 0 ) and ( 4 . 3 ) . From equations ( 3 . 1 5 ) and ( 3 . 1 8 ) i t i s seen that the two expressions for n (IL) are i d e n t i c a l . Yet equation ( 3 . 3 0 ) was derived assuming <=>(« 1 , whereas equation ( 4 . 3 ) was derived assuming °<>> 1 andc<<£ F. The condition C « t p translates, using equation ( 4 . 2 ) , into * < < ( 3 n 2 N ) . This means that equation ( 4 . 3 ) i s the asymptotic form for nOu) both when c*r i s small, and when i s large, provided c*r i s not too large. The obvious inference from t h i s result i s that equation ( 4 . 3 ) i s also v a l i d for a l l <=< in between these extremes, or, simply, as long as °<~«(3v7-N) . The reasoning behind this assertion i s as follows. One can imagine increasing <=*• by holding L fixed and increasing c. As long as c i s not too large -- i . e . : as long as ^ < < ( 3 - n 7 ' N ) 5 --, the external potential may be thought of as a perturbation. As c increases, n(-^) w i l l change smoothly. Physically, one does not expect n(u) to fluctuate with c. That i s , i f n(-u) i s linear in oC when oL i s small, and also when i s large, with the same slope in both regimes, i t i s reasonable to assume that n(-u) is also l i n e a r , with the same slope, in between the two extremes. See figure 11 to help c l a r i f y t h i s concept. In conclusion, equation ( 4 . 3 ) i s assumed to be v a l i d for a l l small compared to N 3. 1 44 h(u) I 44o( 44 N2/3 Figure 11. The Local Number Density n(u) as a Function of the Parameters (3.5) CONTRIBUTION OF INTERNAL CHARGE DENSITY TO THE AMBIENT INTERNAL ELECTRIC FIELD A nonzero internal charge density w i l l contribute to the average internal e l e c t r i c f i e l d . This contribution s h a l l now be calculated. The charge density at height u, denoted q(u), i s given in terms of the number density of electrons, n e(u), and the number density of ions, n-j-(u), by the following equation: ne(u) is given in turn by equation (4.3), while n x(u) may be found v i a macroscopic e l a s t i c i t y theory, which produces the following equation for an iso t r o p i c body subject to a uniform g r a v i t a t i o n a l f i e l d : 145 o X W - . h 2 6 p 4 M a < 5 ' 2 > i s Poisson's r a t i o , Y i s Young's modulus and M<^  i s the atomic mass for the material in question. px i s the number density of ions when there is no external f i e l d , while <jnx i s n I(u)-p x. If rijXu) i s chosen to be (\ at u=J-, then i t follows that nT(u) = pI+^p/M3LU^> ( 5 .3 . where Ma has been replaced by M, the ionic mass. Denote by 6(u) the charge per unit area at height u in the large L l i m i t . Since d>(u)=Lq(u)du, 6(*)~f\yg(i-u) du. (5.4) where (5.5) Equation (5.4) follows from equations (4.3), (5.1), (5.3) and the assumption of o v e r a l l charge n e u t r a l i t y : (5.6) 146 where >^e i s the average number density of free electrons. Using equation ( 5 . 4 ) , the contribution by the internal charge density to the average internal e l e c t r i c f i e l d i s e a s i l y found to be Note that E c ^ depends on height. It was assumed at the outset that Eavc i s independent of height. The question arises as to how to reconcile t h i s apparent c o n f l i c t . Recall that the objective of the model i s to develop ideas, not to produce a precise c a l c u l a t i o n of Eave . S p e c i f i c a l l y , i t i s of interest in t h i s chapter to see i f the internal charge imbalance in a metal contributes a f i r s t order term to E a v- e. Since the calculations in th i s model have been made to only f i r s t order in g, E f t ( / e w i l l be independent of height to f i r s t order in g i f Ec/j i s an order g quantity. It w i l l be shown in the next chapter that Ec^ does, in fact, vanish to f i r s t order in g. There are two possible ways to proceed. One i s to use equation ( 5 . 7 ) as i t i s , and to show that A turns out to be of order g . The other i s to use the maximum value of E c ^ ( n ) , which occurs at U = i , and to show that A i s zero to f i r s t order in g. Both approaches lead to the same r e s u l t . To be more s p e c i f i c , replace E c ^ ( u ) by E c i , ( | ) . Thus, 147 (5.8) U s i n g e q u a t i o n ( 5 . 8 ) , a s e l f - c o n s i s t e n t s o l u t i o n f o r Eave. s h a l l be found i n the next c h a p t e r . The term " s e l f - c o n s i s t e n t " i s used because Eave depends on Ec^, but E ^ depends on E a v e . The l a t t e r p o i n t f o l l o w s because A i n e q u a t i o n (5.8) depends on c, the s l o p e of the e x t e r n a l l i n e a r p o t e n t i a l e x p e r i e n c e d by the c o n d u c t i o n e l e c t r o n s , which i n t u r n i s g i v e n by I t i s c e r t a i n l y p o s s i b l e t o c a l c u l a t e what n(u) would have t o be i n o r d e r t o g i v e r i s e t o the E a v e ( u ) which i n t u r n g i v e s r i s e t o n ( u ) . T h i s problem, however, i s not r e l e v a n t t o t h i s t h e s i s . The reason i s because one does not expect the e l e c t r i c f i e l d i n a met a l t o v a r y much i n the m e t a l i n t e r i o r . As such, E a V e has been assumed t o be a p p r o x i m a t e l y u n i f o r m i n s i d e the model m e t a l . Once A has been shown t o be of o r d e r , t h i s a ssumption w i l l have been v e r i f i e d . 'M. Van den Berg and J.T. L e w i s , Comm. Math. Phys. 81, 475-494 (1981). (5.9) 148 ^See, for example, K. Huang, S t a t i s t i c a l Mechanics (John Wiley and Sons, Inc., New York") 1963), pp. 224-230. •^Equation (2.45) a c t u a l l y only gives the part of the eigenstate which can be expressed in the position representation. This may be l a b e l l e d by "k". The label "k" denotes both "It" and the spin label of the eigenstate. a 'M. Abramowitz and I.A. Stegun, eds. Handbook of Mathematical Functions (U.S. Government Printing O f f i c e , Washington, 1964), p. 446. 5At T=0 K, JX i s equal to the Fermi Energy, £^. r ° S . Gradsteyn and I.M. Ryzhik, 4th ed., ed. Alan Jeffrey, Table of Integrals, Series, and Products (Academic Press, New York, 1965), p.30. 7 I b i d . , p.31. 149 CHAPTER 4 DETERMINATION OF E»./a The basic result of chapter 2 was the construction of a model of the ions and the c a l c u l a t i o n of E d i ^ e \ e S • I n chapter 3, a modelling of the free electrons produced a result for E,.^. Using these findings, and the equation (1) E « = E A l ? . i „ + L , + t " . • ' Eavc w i l l now be determined. Write -/? Mi 4.A ' C . x / > ' r . (2) where /}x and ^  are found by comparing equation (2) to equation (2-10.3): A. =- . (4) Substituting equation (2) into equation (1): 150 where ( 6 ) Now write ^Mh - A i s (7) where, from equations (3-5.5) and (3-5.8): Inserting equation (7) into equation (5), and solving for E : r , A M3/i + AEact +AM-AAA w i»i -2, /3jytfsQe/9 i s a number. For L of the order of 10 meters and Pi of the order of 10 /(metres) , a t y p i c a l value for metals, /^ 3/^ 5 Q e/ g i s of the order of 10 . By equations ( 4 ) and ( 6 ) , and (9) (10) 151 tables II and III , i t i s clear that jh^^s%/9 1* and so equation (11) may be replaced by [ _ AM^te y ? i £ « r f ; 4 - « > L I V (12) It w i l l turn out that the term in the numerator involving i s large compared to the other terms. In consequence, the expression for E c t v e which i s consistent to f i r s t order in the small parameter ( /BzySy^ q e/g)' i s Into equation (13) put r r c° } where (14) ( 15 ) The resultant equation for i s Ar - ~3 , _J r _ i r w ( , 6 ) or, using /4y$ 5q e/g - 10 / 6 , r 2 - U (17) 152 Rewrite equation (15) as r ( o J - _ £ y ^ < 1 8 ) Equations (17) and (18) show that 6Ea<J^ i s small compared to (o) . E&ve . From tables I, II and I I I , i t i s clear that /3\y5z w i l l not be very large, and w i l l c e r t a i n l y not be of order 10^. From equations (9) and (10), using t y p i c a l values for 6 and Y^ ( s p e c i f i c a l l y , those for copper), i t follows that /fy/M/35 i s an order one number. Unless E e * t i s very large, A E u ^ « E a y / c , and R ~ F, To v e r i f y the consistency of t h i s r e s u l t , put equation (13) into equation (7) to obtain and put equations (6) and (19) into equation (2) to get p J J H 3 i p V r (fl) . <20) Equations (19) and (20) combine with equation (1) to say that where It has been assumed in thi s model that the dependence of E 4 V 4 on u i s very weak. The consistency of t h i s assumption with 153 the above results w i l l now be demonstrated. F i r s t , using equations (14), (15) and (16), note that d i f f e r s from E^vt-mg/qg by terms that are of the order of 10 Co) . C*} of E a v g . Replacing E a . v e by E« v<-mg/q e in equation (7), and (o) using equation (15) for E a v f c , gives Ec^=0. In other words, equations (14), (15), (16) and (7) imply that E cv^ ~ 1 0_Vt E a v e . Equivalently, replacing E a v / € by E4v^'-mg/qe in equation (5.5) gives A=0 ( r e c a l l that c=mg+qeEav,e ), which means that A i s of order 10-'S equation (5.7) then says that E c ^ ( M . ) X 10~'^  E a v e . Since Ecv, i s very small, equation (19) indicates that r ~ A - L p ( o ) (23) t e x t - f x <je + A C a v < ' Combining equations (1), (20) and (23) along with the fact that E CU i s n e g l i g i b l e reveals that E a v a ~ E „ ^ , as stated e a r l i e r in t h i s chapter. (o) . . . . Because E ci,«E a v, e , the assumption that E a v e i s uniform in the metal i n t e r i o r i s consistent with the results of thi s chapter. Equation (21) for E a v / e i s the same result as produced by the h e u r i s t i c argument due to Dessler et. a l . This result, however, has been arrived at here in a d i f f e r e n t manner than i t was by Dessler et. a l . Moreover, the present approach also determines how much of E a v € i s made up by the charge imbalance inside the metal, and how much by the ionic dipoles. It i s not surprising though, that the answer obtained here for E ^ e is the same as in the h e u r i s t i c treatment of Dessler e t . a l . , for the basic physics i s very much the same in each case: the l a t t i c e 154 compression i s calculated using e l a s t i c i t y theory, and the electrons are treated, b a s i c a l l y , as i f they were free l o c a l l y . Moreover, although the l o c a l number density of electrons d i f f e r s from the l o c a l number density of ions in the model presented here, the difference between the two is very small. In consequence, to dominant order, E a v e as calculated in the model is the same as E a v t as calculated by the h e u r i s t i c Dessler et. al. approach. Obtaining a f i n a l answer for E a v e was only one of the objectives of the model, as was stressed in the introduction to thi s work. In the next part of th i s thesis, the model i s compared to the l i t e r a t u r e so as to demonstrate what the model has contributed towards understanding the problem of the gr a v i t a t i o n a l l y induced e l e c t r i c f i e l d . 'Note: (1 - 2 6 ) / y =1/3K where K is the "modulus of compresibility". K is of order 101° Joules/metre 3 for most metals. 155 CHAPTER 5 CONCLUSION In t h i s thesis, a simple model of a metal has been constructed. The model has been used to calculate the g r a v i t a t i o n a l l y induced e l e c t r i c f i e l d in terms of the sources of the f i e l d . The sources which have been considered e x p l i c i t l y are the ionic dipole moments and the charge imbalance inside the metal; a l l other sources have been grouped together into E e x ^ . A br i e f summary of t h i s model i s now presented. The ions are arranged, l o c a l l y , in a l a t t i c e pattern. The number density of ions decrease l i n e a r l y with height. Because of gravity, these ions have a dipole moment p. These dipole moments create an average e l e c t r i c f i e l d Ecj^0/es inside the metal. In the model, p and E^ip0ieS are calculated to only f i r s t order in g. Because the density of ions changes with height, the ions do not a l l have the exact same dipole moment. However, the change in dipole moment with height i s very small. S i m i l a r l y , E ^^0)es also depends very weakly on height. Nevertheless, the expressions calculated for p and E j , - ^ ^ are correct to f i r s t order in g, for a l l positions inside the metal. Taking into account the height dependence on the number density of shel l s would produce corrections of second order in g. 156 The conduction electron number density, l i k e the ionic number density, decreases l i n e a r l y with height. The slopes of the number densities d i f f e r by a tiny amount, giving r i s e to a net charge density inside the metal. This charge density produces an e l e c t r i c f i e l d E c ^ ( u ) . A self-consistent solution for Eave shows that E c^ ( u ) « E a ^ , thereby v e r i f y i n g the i n i t i a l assumption that E a v e depends only weakly on height. The solution obtained for E a e i s the same as the Dessler et. a l . h e u r i s t i c expression for E^ . In the f i n a l analysis, Edi|ooks ' E e * t ' a n d Eck a r e a 1 1 o f order Mg/q6 and directed opposite to g*, while E c^ i s n e g l i g i b l e to the order of ca l c u l a t i o n in the model. The value of the model i s not the value of E a v & that i t produces. The model represents a metal far too simply to give a value of E<xye that can be trusted. The real worth of the model i s the physical insight i t provides. Through the model, one can understand how the g r a v i t a t i o n a l l y induced e l e c t r i c f i e l d comes about. One can envisage how a metal responds to gravity, and how thi s response generates sources of E a v e . In p a r t i c u l a r , the model provides insight into how the requirement that the nuclei be held up against gravity can give r i s e to ionic dipole moments. Moreover, the model shows very c l e a r l y how the e l e c t r i c f i e l d at the s i t e of the nucleus can be exactly - M g V q , yet be something considerably d i f f e r e n t on the average. F i n a l l y , the model reveals that charge n e u t r a l i t y in the metal i n t e r i o r need not be assumed; the charge imbalance in a metal may be shown to be so 157 minute that E c ^ makes a ne g l i g i b l e contribution to E a v e • The model has demonstrated that, in addition to the compressibility of the l a t t i c e , charge imbalance and g r a v i t a t i o n a l l y induced ionic dipoles are important concepts in the problem of gravity induced e l e c t r i c f i e l d s in metals. Besides metals, the model also generates some insight into the question of gravity induced e l e c t r i c f i e l d s in d i e l e c t r i c s . For a d i e l e c t r i c , there are no conduction electrons. The model then reduces to simply a l a t t i c e of ions, with Z n=1. Note that the model s t i l l predicts Eavc of order Mg/q£, and opposite to g, whether the l a t t i c e i s compressible or not! According to the model, the key source of E a K e for a d i e l e c t r i c ~ i s the f i e l d due to the ionic dipole moments; the ef f e c t of l a t t i c e compressibility i s not important. But in a metal, both l a t t i c e compressibility and the ionic dipole moments are s i g n i f i c a n t . The model therefore suggests the following question: What r e l a t i v e importance do these two eff e c t s have in a semi-conductor? The model presented here i s a very simple model. What i s needed i s to supplant t h i s model with one which takes into account the ionic p o l a r i z a t i o n and the charge imbalance in a more refined manner. The l a t t i c e compression w i l l also need to be considered c a r e f u l l y . One would also l i k e to see thi s proposed model possess some f l e x i b i l i t y , so that i t could give some conclusions about d i e l c t r i c s , semiconductors and metals. As a f i n a l point, the model of t h i s thesis does not deal with the physics involved in the support against gravity of the 158 ion as a unit. It would be interesting to include t h i s feature into a more elegant model, and to see what sources of the gravity induced e l e c t r i c f i e l d t h i s feature would lead to. 159 BIBLIOGRAPHY Abramowitz, M., and Stegun, I.A., eds. Handbook of Mathematical Functions with Formulas, Graphs, and  Mathematical Tables. Washington: U.S. Government Pr i n t i n g O f f i c e , 1964 Dessler, A.J.; Michel, F.C.; Rorschach, H.E.; and Trammell, G.T, "Gr a v i t a t i o n a l l y Induced E l e c t r i c F i e l d s in Conductors." Physical Review 168 (A p r i l 15 1968): 737-743. Gradstehyn, I.S., and Ryzhik, I.M. Tables of Integrals,  Series, and Products, 4th ed. Edited by Alan J e f f r e y . New York: Academic Press, 1965 Hanni, R.S., and Madey, J.M.J. "Shielding by an Electron Surface Layer." Physical Review B 17 (February 15 1978): T976-1983. Harrison, W.A. "Force on an Electron near a Metal in a Gravitational F i e l d . " Physical Review 180 (A p r i l 25 1969): 1606-1607. Herring, C. "Gravitationally Induced E l e c t r i c F i e l d near a Conductor, and i t s Relation to the Surface-Stress Concept." Physical Review 171 (July 25 1968): 1361-1369. Huang, K. S t a t i s t i c a l Mechanics New York: John Wiley and Sons, Inc., 1963. Hutson, A.R. "Electrons of the Vacuum Surface of Copper Oxide and the Screening of Patch F i e l d s . " Physical Review B 17 (February 15 1978): 1934-1939. Leung, M.C; Papani, G.; and Rystephan ick, R.G. "Gravity-Induced E l e c t r i c F i e l d s in Metals." Canadian Journal of Physics 49 (1971): 2754-2767. Leung, M.C. " E l e c t r i c F i e l ds Induced by Gravitational F i e l d s in Metals." I_l Nuovo Cimento 7 (February 11 1972): 220-224. Lockhart, J.M.; Witteborn, F.C.; and Fairbank, W.M. "Evidence for a Temperature-Dependent Surface Shielding E f f e c t in Copper." Physical Review Letters 38 (May 25 1977): 1220-1223. 160 Peshkin, M. "Gravity-Induced E l e c t r i c F i e l d Near a Conductor" Annals of Physics 46 (1968): 1-11. Peshkin, M. "Gravity-Induced E l e c t r i c F i e l d Near a Conductor." Physics Letters 29A (May 5 1969): 181-182. Rieger, T.J. "G r a v i t a t i o n a l l y Induced E l e c t r i c F i e l d in Metals." Physical Review B 2 (August 15 1970): 825-828. S c h i f f , L.I., and B a r n h i l l , M.V. "Gravitation-Induced E l e c t r i c F i e l d Near a Metal." Physical Review 151 (November 25 1966): 1067-1071. Sc h i f f , L.I. "Gravitation-Induced E l e c t r i c F i e l d Near a Metal." Physical Review B 1 (June 15 1970): 4649-4654. Van den Berg, M., and Lewis, J.T. "On the Free Boson Gas in a Weak External P o t e n t i a l . " Communications in Mathematical Physics 81 (1981): 475-494. Witteborn, F.C., and Fairbank,W.M. "Experimental Comparison of the Gravitational Force on Freely F a l l i n g Electrons and Met a l l i c Electrons." Physical Review Letters 19 (October 30 1967): 1049-1052. Ziman, J.M. Pr inc iples of the Theory of Sol ids, 2nd ed. Cambridge: Cambridge University Press, 1972. 161 APPENDIX A ALTERNATE WAY OF DERIVING AN EQUATION FOR ^ E ^ Define coordinates from the center of the s h e l l , x„,ye , z a . See figure A.1. L e t ^ ( r t ) be the ground state eigenfunction for S h e l l Figure A.I. Coordinates from the Center of the Shell. Versus Coordinates from the Nucleus. the perturbed boundary problem, where "rt denotes position from the center of the s h e l l . The (ground state) energy eigenvalue is s t i l l given by equation (1.30). It is also given by (A. 1 ) where the integral i s over the region i n t e r i o r to the s h e l l , that i s , over a l l & and a l l fa (the polar and azimuthal angles, 162 as measured from the center of the shell) and over 0<ro<R, and where H(r 0 ) is the Hamiltonian expressed in coordinates from the center of the s h e l l : <LVY\ W7> > (A.2) E i i s determined by d i f f e r e n t i a t i n g equation (A.1) twice with respect to ex and determining the order one term: 1 °^ — ^4^*hlj>. (A. 3) Notice that the l i m i t s of integration in equation (A.3) do not depend on o<. This i s precisely why H(rt), and } ( r i ) were introduced. D i f f e r e n t i a t i n g the integral once (A.4) Write Inside the s h e l l , 163 41 (A.5) It follows that As a r e s u l t , the right hand side of equation (A.4) becomes (A.6) R Equation (A.6) follows because ^R. cV Jo 3^1 (I which implies that d« 0*. r r 164 D i f f e r e n t i a t i n g a second time: (A.7) Expressions for h^/lot- and £V/h«? are required. From equation (A.5), = 1<Z o * or i l l - 1,7 Q Z C£S0 ( A > 8 ) where r and 5 are measured from the nucleus. Notice at t h i s point that, since f f d 3 r 0 = ^ / d 3 r — where 3 3 d r c and d r refer to the same physical volume element — 165 equation (A.8) may be plugged into equation (A.6) to give The order oi- term on the l e f t here i s thereby recovering equation (2.40): Calculating AxU/<) SU - - l Soc1 dz,2 3 V JY-rM-? v.' i ' AY 1 vuz»-^ z i ' V and so, using equation (A.6), 166 CM (A.9) where, again, 6 and r are measured from the nucleus. Insert equations (A.8) and (A.9) into ( A . 7 ) , and equate t h i s to (A.10) It i s required to find the order one term on the right hand side of equation (A.1 0 ) . Notice that § d 3 r e in the l a t t e r two integrals may be replaced by f*fd** i f the integral i s also changed from C d 3 r 0 to d r. The integral with the delta function then becomes Sfx, W>8c?) for which the order one term i s 167 As such, the second term in equation (A.10) makes an order one contribution of (A. 1 1 ) Replacing 4}% d r c by ^^63r makes the f i n a l i n t e gral in equation (A.10) 9 for which the order one portion i s (I This vanishes on account of the B integration. The f i r s t integral in equation (A.10) also makes an order one contribution. To see t h i s , break up the integration into two parts. One part consists of a sphere with a radius of order ot, but greater than<=<. The integration within t h i s sphere w i l l o give a contribution of order higher than ^ , and may be discarded. For the rest of the integration, expand ^ as 168 o r o \o(5) -\o<^i) \ (A.12) Only $>0 and need to be known in order to obtain the order one term being sought. To obtain $ a and $, , expand pi and t// out in powers of and, of course, in terms of v0r&off0. Doing th i s gives e (A.13) and ((VZ)e-P-/zcos90 (A.14) To lowest order in oc: cos9 a (A.15) Plugging equations (A.13), (A.14) and (A.15) into the f i r s t i ntegral in equation (A.10) gives the order one term sought: 169 d ^ 2 ^ ( l ) > - - 2 ) C 2 - P ^ P " J > ^ ^ & ^ < ^ ^ U ' , 6 > -s_-z„ 4 l^j^.(p)(2f,)c-r-Combining equation (A.11) and (A.16) gives the order one term which i s required to determine ^ . Using equation (A.10), or *P - §E 7 M M 2 his? just as before. 170 APPENDIX B A QUICK DERIVATION OF THE LOCAL NUMBER DENSITY  IN A FERMI GAS SUBJECT TO AN EXTERNAL LINEAR,POTENTIAL This appendix deals with a central problem of Chapter 3 of th i s t h e sis, namely that of determining the l o c a l number density n(oi) of fermions which are subject to an external p o t e n t i a l . The approach here assumes, from the outset, that the l o c a l properties of the Fermi gas are the same as those of a free Fermi gas. As such, n(u) i s given by where a l l these symbols are the same as in Chapter 3. The l o c a l chemical potential y- i s determined by the requirement that there is no gradient in the l o c a l potential energy, i . e . , that .yu CU) A C i t y * - r; i where i s the chemical potential of the gas. Thus: This i s the same as equation (3-2.89). 171 APPENDIX C LOW DENSITY FERMI GAS IN AN EXTERNAL LINEAR POTENTIAL AT ABSOLUTE ZERO This appendix deals with some properties of a low density Fermi gas which i s subject to an external l i n e a r p o t e n t i a l . Expressions s h a l l be derived for the Fermi energy, the pressure, and the internal energy density of the gas in terms of i t s average p a r t i c l e density. The treatment s h a l l be r e s t r i c t e d to absolute zero. The s t a r t i n g point i s part c of section 3.2. Consider equation (2.65) in the case ^ 2 < 1 and in the l i m i t Using the asymptotic forms for the Airy functions Ai(v) and B^(v) as (1) ( C D (C.2) equation (2.65) implies that (C.3) 172 From equations (2.64) and (C.3) i t follows that (C.4) The asymptotic forms of A* (v) and B; (v) a s ^ - ^ a r e ^ r^nH-v)1"* A3 B«tV,~i4^ (C6) Equations (C.4), (C.5) and (C.6) demand that', to lowest order, where v„ z denotes the nz zero of A^(v), and has been replaced by E^/Z. As figure 10(a) shows, a l l of the zeros of Ai(v) are negative. From equation (C.5), the asymptotic form of the zeros is 173 ~ (C.8) The integer labels of the zeros have been chosen to increase with ascending absolute magnitude of the zero. Equation (C.8) i s an excellent approximation to the precise value of the zeros of A i ( v ) , even for those close to the o r i g i n . For the purposes of t h i s appendix, no s i g n i f i c a n t error i s incurred by using equation (C.8) in equation (C.7). This gives C l z ( 2 ^ c \ 3 ( ^ - 0 ^ ^ (c.9) L"*"zA L 8 ) ' where the d e f i n i t i o n of has been used. The l i m i t may be regarded as holding 'c fixed and l e t t i n g L-5>«'. As such, equation (C.9) gives those eigenvalues which are less than c in the l i m i t L A mathematical point should now be c l a r i f i e d . Equation (C.9) i s v a l i d only for • ( c - , 0 ) This i s because the asymptotic forms (C.I) and (C.2) may be employed in equation (2c.22) only i f the arguments of the Airy functions are large. In consequence, i f En z i s too close to c, d1* (1 - ) may not be large, and equation (C.9) w i l l not be v a l i d . Let n z denote the value of n z for which the right hand side of equation (C.9) i s equal to c, 174 ' Z 4 3-TT U i 1 ' (C. 1 1 ) Equation (C.9) w i l l be a good approximation for En-2- provided that n ^ i s s u f f i c i e n t l y less than n-^  so as to render equation (C.10) true. To determine how much less, write From equations (C.9) and (C.11), in the large L l i m i t , ^n must s a t i s f y ^ n » ^ L c x ' / ^ (c-'3> ITT in order that equation (C.10) hold. Equation (C.13) indicates by how much n z must be less than n z in order that equation (C.9) be v a l i d . In terms of l = ! k » n -12 3 equation (C.9) may be written as \2/3 (C.14) (C.15) Using equation (C.15) in equation (2.47): I 3 ] . (C.16) 175 Equation (C.16) may be used to determine the constant energy surfaces in k-space. Let ^ k x ' V . <C-,7) For the constant energy surface £„ , kz i s given in terms of X' by n + 0-3 (C.18) where C)Z - 2. HA C" 1 " u 1 The l i m i t i n g form as L-^°°of the function k x(X) spe c i f i e d by equation (C.18) i s given graphically in figure C.1. Figure C.2 shows the constant energy surface £ 0 . This information about the constant energy surfaces may be used to determine the average number density p, the internal energy density , and the pressure p in terms of the Fermi energy . The preceding results apply provided that i s less than c by enough to make equation (C.13) true. £ F i s determined in terms of N and L, in the l i m i t L , by the equation 176 Figure C. 1 . k 2 as a Function of K in the L-»°°Limit  for the Energy Surface £ Q Figure C.2. The Energy Surface £> in L - ^ 0 ° Limit in k-Space A glance at figure C.2 indicates that equation (C.20) is equivalent to z 177 where X^-a^-ia^f3 , (c.22) (C.23) Evaluating equation (C.21) gives (C.24) where ^=N/L . Equation (C.24) may be solved for £ F (C.25) In a similar fashion, the equation for the t o t a l internal energy U, r (C.26) may be s i m p l i f i e d to get it. : 178 J or 0 -The result i s (C.27) U= y e F p . (C.28) p i s calculated from U(N,V), V=LS, via »_^U(H V) ( C 2 9 ) (C.30) For comparison, the corresponding results for ix and p for the ideal Fermi gas (no external potential) are u= , (c-31) j > = f f F p . - ( c - 3 2 ) It was claimed that the results of this appendix are v a l i d only for a low density Fermi gas. This claim emerges from equation (C.25). In order that £p- <c, i t follows from equation (C.25) that 179 (C.33) where 0 = z ( L ~ £ y * ( c . 3 4 ) To get an idea of the magnitude of p„ , consider the case c=mg and L=10 meters. This gives ^>=756 cm . Compare this to the It- -3 t y p i c a l number densities in metals, of the order of 10 cm ! The densities for which the results of t h i s appendix apply are very small indeed. M. Abramowitz and I.A. Stegun, eds. Handbook of Mathematical Functions (U.S. Government Prin t i n g Of fTce, Washinton, 1964), pp. 448-449. 2 I b i d . , pp.448-449. 180 APPENDIX D "PROOFS" OF EQUATIONS (3~3.28) AND (3-3.24) C o n s i d e r the f u n c t i o n C « T K / 0 a ^ . (D.1 ) P u t t i n g u=kx, i t i s easy t o see t h a t £ o , rip") ^ * £ °° C o $ ^ ^ „ ( D - 2 ) or From e q u a t i o n ( D . 3 ) , i t f o l l o w s t h a t the a s y m p t o t i c form of S' as nF->°° i s 5CX ;nF) ^ -1 S f h C ^ } * s f x - > 0 ( D . 4 ) T h i s c l a i m may be v e r i f i e d by w r i t i n g 5 C X ; 0 P > ^ - ^ ^ ^ Y / V - f V ; n p ) ) ^  C ( D . 5 ) and showing t h a t J a ; - o(J-p) «s nF -> - ( D < 6 ) E q u a t i o n (D.4) v e r i f i e s e q u a t i o n (3-3.28) f o r s m a l l x. 181 Consider S(x;nF) for x=n. in t h i s case, (D.7) Since, 4L _ < ? _ J < r ds (d,8) o (V2,)' N 3 t ( M Z O i N l l ( r \ F ^ ' i t follows that -L < J \ ! < 1 . (D.9) Si m i l a r l y , < x 1 <: I (D.10) 2(M) ^ ^ O ( ^ F + ZT + I ) 1 v Z t V ^ l ) ' From equations (D.9) and (D.10), (D.1 1 ) This treatment for x = n- can be repeated for x=£, x=f, x-V^ etcetera. In each case, i t w i l l be found theS i s of order n p T as nF~>°° . Since $ is thi s order for a l l these d i s t i n c t points, and also for a continuous range of x when x i s small, one i s led by i n t u i t i o n to expect i t to be true for a l l x in the range 0<X<2TT. As such, equation (3-3.28) has been "proven". If equation (3-3.28) i s true, then a similar procedure shows that the function oo l c ^ F k i s of order nF~' as n p - ^ . Accordingly, write 182 F(x;M= jj-FWCx;flF) • (D-,3> U s i n g e q u a t i o n ( 3 - 3 . 2 4 ) , i t f o l l o w s t h a t I t i s s t r a i g h t f o r w a r d t o v e r i f y t h a t e q u a t i o n (D.14) i s s a t i s f i e d by tf„ * \ C O H ) (D. 15) Thus, £- z. which i s e q u a t i o n ( 3 - 3 . 2 9 ) . 

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