A SIMPLE MODEL FOR STUDYING THE GRAVITATIONALLY ELECTRIC FIELD INSIDE A METAL INDUCED by MARK RAYMOND ALPHONSE SHEGELSKI B.Sc, The U n i v e r s i t y 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 P h y s i c s We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA October © 1982 Mark Raymond Alphonse S h e g e l s k i , 1982 In presenting this thesis in partial fulfilment of requirements f o r an advanced degree at the U n i v e r s i t y of Columbia, I available for permission agree her the Library shall reference and study. I for purposes may or that extensive be granted by representatives. p u b l i c a t i o n of t h i s t h e s i s allowed without my Department of written Date: 18 October 1982 further British it freely agree that copying of t h i s t h e s i s f o r s c h o l a r l y the Head of my It for is Department or understood financial permission. Physics The U n i v e r s i t y of B r i t i s h 2075 Wesbrook Place Vancouver, Canada V6T 1W5 make the Columbia gain that by his copying or shall not be ABSTRACT If a metal nuclei object i s placed i n a gravitaional field, and e l e c t r o n s i n t h e m e t a l w i l l a new c h a r g e distribution distribution inside implies sink. This w i l l the metal. a modified electric produce A modified field .the charge i n the metal interior. 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 which give rise to the inside a metal. gravitationally physical induced To t h i s e n d , a s i m p l e processes electric model o f a field metal i s constructed. Comprising t h e jriodel a r e i o n s , a r r a n g e d o n a d i f f e r e n t i a l l y compressed l a t t i c e , is represented a n d a gas- o f c o n d u c t i o n e l e c t r o n s . by a n u c l e u s a n d an e l e c t r o n w h i c h together i n s i d e a hard, massless, is by spherical shell. An ion are confined The nucleus 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 t h e e l e c t r o n i s r e p r e s e n t e d a wave f u n c t i o n . modelled The conduction electron as a gas o f n o n i n t e r a c t i n g fermions which an e x t e r n a l l i n e a r possible sources of the e l e c t r i c the ; To - f i r s t border contributing investigation field: i n d u c e d i o n i c d i p o l e moments, a n d t h e c h a r g e metal'i i s subject to potential, The d e s i g n o f t h e m o d e l f a c i l i t a t e s two constituent i s of gravitationally "imbalance i n the i n - g , ,< o n l y t h e ' f i r s t ; '/source . ^ m a t t e r s , .;. a p p r o x i m a t e l y -Mg*/q e to the electric field, 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 induced e l e c t r i c electronic field charge. The net 'gravitationally i s a l s o f o u n d t o be a p p r o x i m a t e l y -Mg*/q , e iii TABLE OF CONTENTS Abstract i i L i s t of T a b l e s List vii of F i g u r e s viii Acknowledgement ix Note on Numbering References CHAPTER 1. of Equations, Footnotes, and t o the L i t e r a t u r e x INTRODUCTION 1 1.1 Statement and O r i g i n of the Problem 1.2 Review of the L i t e r a t u r e 3 1.3 Purpose of t h i s T h e s i s 7 1.4 O u t l i n e of the Model 8 CHAPTER 2. THE LATTICE OF IONS Plan of the Chapter 2.1 .. 1 15 15 S e t t i n g up the Schrbdinger Equation for the E l e c t r o n I n s i d e the Impenetrable S h e l l 18 iv (A) Setting up t h e e q u a t i o n s f o r the perturbed p o t e n t i a l problem (B) Setting up t h e e q u a t i o n s f o r the p e r t u r b e d boundary problem (C) S e t t i n g up a n e q u a t i o n Solution for 37 to the Schrodinger the Electron Impenetrable (A) Equation Inside the Shell The u n p e r t u r b e d 42 problem; d e t e r m i n a t i o n o f ^> a n d E (B) Solution 0 Solution 43 to the perturbed boundary problem (D) Calculation of 49 ; summary 54 2.3 The P o s i t i o n o f t h e N u c l e u s 2.4 Calculation 64 o f t h e D i p o l e Moment of E a c h I o n i n Terms o f Ef, 2.5 Calculation 2.6 Solutions 2.7 The A v e r a g e E l e c t r i c o f Ep i n Terms o f £ f o r f> a n d E^ Field to the L a t t i c e of Ions 2.8 L a t t i c e Types (A) Cubic 42 to the perturbed p o t e n t i a l problem (C) 28 f o r E-| ; summary 2.2 23 lattice 66 70 73 Due 74 78 79 V (B) F a c e - c e n t e r e d cubic (C) C l o s e s t packing; type 1 - 82 (D) C l o s e s t packing; type 2 82 (E) Hexagonal c l o s e s t packed 83 2.9 Ewald Sum 2.10 Evaluation lattice 84 of the Ewald Sum and C a l c u l a t i o n of E ^ p ^ s CHAPTER 3 THE CONDUCTION ELECTRONS 3.1 79 94 98 D e s c r i p t i o n of the Problem of a Free Fermi Gas Subject t o an External Linear 3.2 Statistical (A) Potential Mechanics Approach- Review of some b a s i c 105 statistical mechanical ideas (B) 101 108 S p e c i a l i z a t i o n t o case of no e x t e r n a l p o t e n t i a l ; review of the i d e a l Fermi gas (C) Case of an e x t e r n a l linear potential (i) 118 Approach due to Van den Berg and Lewis (ii) 113 126 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 fugacity i n the 133 vi ( i i i ) S p e c i a l i z a t i o n to case T=0K 3.3 P e r t u r b a t i v e Approach 3.4 L i n e a r i t y of the L o c a l Number Density 3.5 134 135 i n the Parameter « 142 C o n t r i b u t i o n of I n t e r n a l Charge Density t o the Ambient Internal E l e c t r i c CHAPTER 4 DETERMINATION OF E CHAPTER 5 CONCLUSION Field 144 149 q v e 155 Bibliography 159 Appendix A - A l t e r n a t e Way of D e r i v i n g an Equation for ° E 161 f z Appendix B - A Quick D e r i v a t i o n of the L o c a l Number D e n s i t y i n a Fermi Gas Subject 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 Appendix C - Low Density 170 Fermi Gas i n an E x t e r n a l L i n e a r P o t e n t i a l at Absolute Appendix D - " P r o o f s ' o f and (3-3.29) Equations Zero 171 (3-3.28) 180 vii L I S T OF TABLES I. Values of ^ II. V a l u e s o f cx z forLattices (A) t h r o u g h forLattices (A) t h r o u g h ( E ) f o r some V a l u e s o f Z III. V a l u e s o f <* 3 n 95 95 h for Lattices f o r some v a l u e s o f Z (E) (C) t h r o u g h ( E ) 96 vi i i LIST OF FIGURES 1. Nucleus and E l e c t r o n I n s i d e the Impenetrable S h e l l 2(a). Contact Arrangement 2(b). Non-contact Arrangement 3. Placement 9 of the S h e l l s 10 of the S h e l l s 11 of the Nucleus I n s i d e the 19 Impenetrable S h e l l 4. P o s i t i o n of the Boundary R e l a t i v e to the Nucleus 29 5. R e l a t i o n s h i p between 6, rt,(9), of and R 30 6. The Regions 34 7. Forces A c t i n g on the Nucleus i n and V) the F u l l P e r t u r b a t i o n Problem 56 8. The S p h e r i c a l Region 2 75 9. The Face-centered Cubic L a t t i c e 80 10. The A i r y F u n c t i o n s Aj (v) and B j ( v ) 11. The L o c a l Number D e n s i t y n(u) as a 123 F u n c t i o n of the Parameter oc A.1. 144 C o o r d i n a t e s From the Center of the S h e l l Versus C o o r d i n a t e s from the Nucleus C.1. k as a F u n c t i o n of X i n the L-* 00 2 f o r the Energy Surface £ C.2. 161 Limit 176 0 The Energy Surface £„ i n L-a> L i m i t i n t-space ... 176 ix ACKNOWLEDGEMENT I wish t o suggesting thank my the t o p i c supervisor, Dr. W.G. Unruh, f o r of t h i s t h e s i s , the b a s i c o u t l i n e of the model, f o r a s s i s t i n g with the development of the model, and f o r encouraging me along the way. I e s p e c i a l l y a p p r e c i a t e Dr. Unruh's a s s i s t a n c e , encouragement, and concern f o r my w e l l - b e i n g d u r i n g the f i n a l stages of t h i s work. I am indebted to Kathy Nikolaychuk who typed the also last f o r g o t t e n , as respects. Stephanie Mundle, the manuscript and were very p a t i e n t with me. and Stephanie during and he contributed few days of too a s s i s t e d Indeed, with without numerous work. details Todd Mundle must not be greatly Kathy, other Kathy in several imporant Stephanie, and Todd h e l p i n g me, I would not have f i n i s h e d t h i s t h e s i s at the time that I did. Stephanie also deserves thanks for providing reassurance and encouragement, and e s p e c i a l l y with a l l of my t h e s i s induced I would a l s o l i k e Barrie, Dr. L. to me with f o r coping so w e l l idiosyncrasies. express appreciation to Dr. R. Sobrino, John Hebron and Matthew Choptuik f o r some h e l p f u l d i s c u s s i o n s ; Matthew a l s o succeeded in introducing me t o the computer. My thanks t o N.S.E.R.C. for their financial assistance. Many other people helped me i n other ways; my thanks t o a l l of you. X NOTE ON NUMBERING OF EQUATIONS, FOOTNOTES, AND REFERENCES TO THE LITERATURE Equations a r e numbered i n d i v i d u a l l y c h a p t e r : equation of s e c t i o n 7. i n each s e c t i o n of each (7.4), f o r example, means the f o u r t h In a given chapter, i f an equation i s referred to from another c h a p t e r , the chapter number i s i n c l u d e d . it i s not. second Equation s e c t i o n of chapter 2, whereas equation denoted by s e c t i o n of the present curved parentheses: footnote i n the present c h a p t e r . i n the (2.3) means the chapter. Footnotes are numbered c o n s e c u t i v e l y throughout are Otherwise (2-2.3) means the t h i r d equation t h i r d equation of the second They equation a chapter. means the t h i r d The f o o t n o t e s are listed at the end of the c h a p t e r . References brackets: Bibliography. to the refers literature to the ninth are symbolized item listed by square in the 1 CHAPTER 1 INTRODUCTION (1.1) STATEMENT AND ORIGIN OF THE PROBLEM If a metal o b j e c t i s p l a c e d i n macroscopic the metal. the electric field The e l e c t r i c nuclei different and than gravitational their field. gravitational field, results to because reside at corresponding gravity causes p o s i t i o n s which are positions in m o d i f i e d i n t e r n a l charge d e n s i t y produces field i n the metal' i n t e r i o r . induces an e l e c t r i c The following zero A d i f f e r e n t p o s i t i o n i n g of the n u c l e i and e l e c t r o n s i m p l i e s a d i f f e r e n t charge d e n s i t y i n s i d e the A a w i l l be c r e a t e d i n the i n t e r i o r of field electrons a For s i m i l a r metal. a modified e l e c t r i c reasons, g r a v i t y also f i e l d e x t e r i o r to the metal. problem what i s the g r a v i t a t i o n a l l y w i l l be c o n s i d e r e d i n t h i s induced e l e c t r i c field in thesis: a metal, and what are the main p h y s i c a l processes which give r i s e t o t h i s electric field? The fields interest first Witteborn arose To shield such because and Fairbank ions, e l e c t r o n s , and field. in gravitationally of a series induced of electric experiments by designed t o measure the a c c e l e r a t i o n of positrons against in the earth's gravitational external electrostatic p a r t i c l e s were c o n s t r a i n e d t o f a l l in a cavity f i e l d s , the formed by a 2 conductor. that an The results electric of the experiments field mg/q pervaded e seemed t o i n d i c a t e the c a v i t y of the conductor, where m was the e l e c t r o n mass, g the a c c e l e r a t i o n due to g r a v i t y , and q e the e l e c t r o n i c charge. that g r a v i t y had a f f e c t e d the conductor induce the e l e c t r i c Schiff and f i e l d mg/q e Barnhill was surrounded E ^" =mgVq • then by a metal. e such of E ^ first theoretical electric field in a theoretical e in i s the calculation calculation . ionic was They obtained mass. o p p o s i t e d i r e c t i o n s , these two e l e c t r i c Besides fields differ s t r e n g t h by approximately f i v e o r d e r s of magnitude! To confuse the i s s u e , t h i s d i s c r e p a n c y experimentally reported a magnitude to as t o a f i e l d d i r e c t e d o p p o s i t e l y t o g and with a magnitude approximately Mg/q , where M being in e x way d i d the done by D e s s l e r , M i c h e l , Rorschach and Trammell for a The r e s u l t of t h e i r However, another e5 in suggested i n s i d e the c a v i t y . c a l c u l a t i o n of the g r a v i t y - i n d u c e d cavity Witteborn as well. temperature of ls *^ e Fairbank, dependence has Lockhart of and E * : t o vary from about mg/q e been they observed Witteborn" found 111 the at low temperatures about Mg/cL, a t higher temperatures,with a t r a n s i t i o n a t about 4 K. S c h i f f and B a r n h i l l gravitationally induced They claimed that E et. to al. calculated E also electric would be l t l examined field the q u e s t i o n of the ~ E ^ i n s i d e a metal. tn approximately and concluded that mg/qg . Dessler i t would be o p p o s i t e g and have a magnitude of approximately Mg/q . e This discrepancy i s rather i n t e r e s t i n g . This introduction now presents the l i t e r a t u r e to c a l c u l a t e (1.2) REVIEW OF The to a brief description THE E of the and e x t E i,vt techniques employed i n . LITERATURE approach S c h i f f and Barnhill used to c a l c u l a t e i n t r o d u c e a c l a s s i c a l t e s t p a r t i c l e of into the c o u l d be charge cavity the mass of Barnhill the only the metal, and In the location ignored calculated of the the of the the contributed contribution their of the obtained mg/q e electric field, by the v e r t i c a l component of the In t h e i r paper, S c h i f f internal induced conductor. shift charge They showed that shift analysis, by the electronic was e x infinitesimal conductor. expressed i n terms of the in of the formed by E * E e x - t test center Schiff nuclei. and They constituent f o r E* ^ . k and Barnhill also and made the b r i e f l y discussed following claim: It i s apparent that each e l e c t r o n and n u c l e i in the metal must be acted upon by an average e l e c t r i c field of such magnitude that i t e x a c t l y balances i t s weight.° } For an electron, because the the nuclei This cavity same of this field i s mg/q . e They f u r t h e r e l e c t r o n s occupy most of the only a small f r a c t i o n , E field, the they conductor. stated, L argued t h a t , volume of the i s very c l o s e to would metal and mgVq . be expected i n s i d e e the 4 Dessler not et. a l . taken into direction of pointed account the out that S c h i f f and the compression gravitational field. f o l l o w i n g h e u r i s t i c d i s c u s s i o n about The for net charge d e n s i t y otherwise there Using elaj-tVicity Barnhill had of the metal i n the They presented the . i n s i d e the metal must be very small, would r e s u l t a huge i n t e r n a l e l e c t r i c field. theory, i t i s - e a s y t o c a l c u l a t e the number density of ions as a function Charge n e u t r a l i t y then g i v e s the number d e n s i t y of e l e c t r o n s as a f u n c t i o n of h e i g h t . free Fermi gas of height T r e a t i n g the allows one to inside electrons, in experienced by the electrons. against the gravitational electrons gives E L n . and The f i n a l metal. locally, as a turn compute the pressure gradient electric the Balancing forces expression this felt obtained force by the by this approach i s (1 ) where £f i s the Fermi average number d e n s i t y energy of the e l e c t r o n gas, Q% i s the i n the metal, 6 of ions 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 f o r metals ~*'(nr for the parameters i n equation (1) give E to g and with a magnitude of the order In again their paper, Dessler obtaining a f i e l d opposite et. L oppositely directed of Mg/q . e a l . also calculated E to g and of the order , of Mg/q . e 5 Subsequent to reexamined the the S c h i f f and found that the s h i f t moment work of Dessler et. al., B a r n h i l l approach to the problem. induced by the test charge in of the n u c l e a r c o n s t i t u e n t of the metal was than t h a t of the e l e c t r o n i c c o n s t i t u e n t . b a s i c p h y s i c a l ideas and a l . heuristic discussion have the He much g r e a t e r and e 3 mass H e r r i n g estimated the c a v i t y f i e l d would be of the order of Mg/q The Herring® that opposite to assumptions of the D e s s l e r e t . been r e s t a t e d by Harrison " " 1 5 and 1 [10] Leung but in d i f f e r e n t terms. Boltzmann t r a n s p o r t equation Dessler attempting et. ju. is the to o b t a i n equation . L n local number density dr\/bz them an estimate to g and . Lrit a model of for E a : p o t e n t i a l of e l e c t r o n s i n the depends only Assuming from e l a s t i c i t y for E * . t n charge theory. Again they on z, and the r e p l a c e d VJJ. neutrality, Estimating found E t r y local d/U/dn ^ to be they gave opposite to have a magnitude of order Mg/q . e Peshkm equation introduced of e l e c t r o n s n at the height ) (3n/2)Z ). calculated They chemical They then assumed t h a t ( dju/dK (1) f o r E the -- («vrv»/?«' metal. by used used to d e r i v e the f o l l o w i n g equation V where example, a l . went beyond t h e i r h e u r i s t i c d i s c u s s i o n i n "* ' n't to determine E metal which they Leung, f o r (2): presented an expression for E similar to 6 where ju}°' i s the l o c a l chemical p o t e n t i a l f o r f r e e e l e c t r o n s . i s a c o r r e c t i o n to account free electron equation the representation. (3) i f an equations f o r the e f f e c t s not i n c l u d e d estimate E may L for F be i n the estimated i s available. F from As such, (2) and (3) may be regarded t o be the same. One other approach has been presented i n the l i t e r a t u r e f o r purpose calculated E v i a of c o n s i d e r a t i o n of determined a finding E . cni the Rieger^" electron-phonon 1 interaction. He a first t r a n s f o r m a t i o n between the usual phonon o p e r a t o r s when there i s no g r a v i t y and the ones which apply when there i s . Then, he r e p l a c e d the former by the latter e x p r e s s i o n a s s o c i a t e d with the electron-phonon result he phonon operators), which he p o t e n t i a l energy to the d i f f e r e n t i a l compression analysis, lattice. calculated the energy interaction. The obtained i n c l u d e d a s c a l a r p a r t ( i . e . i t i n v o l v e d no electrostatic Leung, in Papim partly and interpreted as being the experienced by the e l e c t r o n s due of the l a t t i c e . Rystephanick criticized Rieger's because of h i s c h o i c e of normal modes f o r the They avoided choosing normal modes. the e l e c t r o s t a t i c p o t e n t i a l f e l t Instead, by the e l e c t r o n s i n terms of the d e v i a t i o n from p e r i o d i c i t y of the l a t t i c e Both Rieger and Leung et. they a l . obtained d i r e c t e d to g and of magnitude roughly Mg/q . £ E L sites. oppositely 7 (1.3) PURPOSE OF THIS THESIS The what l i t e r a t u r e p r o v i d e s one with a p a r t i a l understanding of physical Dessler et. the processes are important i n t h i s problem. a l . h e u r i s t i c d i s c u s s i o n , f o r example, d i f f e r e n t i a l compression The points out of the l a t t i c e due to g r a v i t y , and the p r e s s u r e g r a d i e n t experienced by the e l e c t r o n s as a r e s u l t . In the l i t e r a t u r e , one a l s o f i n d s e x p r e s s i o n s f o r E However, understanding one of does what not obtain the p r i n c i p a l does one l e a r n how the metal responds so as t o support from itself, or how the . literature an sources of E^ * a r e . Nor n t o the g r a v i t a t i o n a l this response field generates the does not f e e l q u i t e convinced that E ^ has sources of E ^ . tn Moreover, in one f a c t been found. al. should be The h e u r i s t i c d i s c u s s i o n taken as an of et. an unequivocal d e t e r m i n a t i o n of E . and not T h i s i s why D e s s l e r cn obvious (2) — that assumptions v)tt the r e s u l t i s any of better D e s s l e r e t . a l . made their model the compression of -- But i t i s not known than E " , or that the c aboutju are v a l i d . Rieger and Leung e t . a l . both assumed that E to field a l . c o n s t r u c t e d a model with which t o c a l c u l a t e E ^. in equation et. i n d i c a t i o n of some of the b a s i c p h y s i c a l responses of a metal to a g r a v i t a t i o n a l as Dessler the l a t t i c e . c n Finally, was p r i m a r i l y due Again, i t i s not obvious that t h i s i s n e c e s s a r i l y so. In of order t o c o n t r i b u t e t o the understanding of t h 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 field in a problem metal, this 8 thesis presents sources of E a . simple model which i n c l u d e s some p o s s i b l e The r e l a t e d c h a l l e n g e of not be c o n s i d e r e d here. finding E e shall x (Attempts have been made by H u t s o n ^ and •DO Hanni t o provide a t h e o r e t i c a l e x p l a n a t i o n of the temperature dependance of E The section r e p o r t e d by Fairbank, Lockhart and Witteborn.) b a s i c f e a t u r e s of the model are d e s c r i b e d i n of t h i s i n t r o d u c t i o n . two b a s i c purposes i n mind. electric field inside a physical insight electric f i e l d s i n metals. i n t o the the next The model has been designed with One i s to metal. calculate The problem of other the average t o p r o v i d e some gravitationally induced (1.4) OUTLINE OF THE MODEL In t h i s model, a metal s h a l l be regarded as comprised of an i o n i c c o n s t i t u e n t and an e l e c t r o n i c c o n s t i t u e n t . The a ions are represented by the f o l l o w i n g composite classical point particle s i n g l e quantum mechanical of mass M and charge impenetrable The p o i n t p a r t i c l e , t r e a t e d c l a s s i c a l l y , the ion. The Z^q , and a e e l e c t r o n , a r e enclosed together of a massless, undeformable, of entity: electron, shell (see f i g u r e 1). r e p r e s e n t s the represented inside nucleus by a wave f u n c t i o n , models the t i g h t l y bound e l e c t r o n c l o u d surrounding the nucleus. The net charge of the i o n i s ( Z f , - l ) q e and i t s mass i s M+m. The e l e c t r o n i c c o n s t i t u e n t of the metal i s modelled by a 9 Figure 1• Nucleus and E l e c t r o n I n s i d e the Impenetrable noninteracting Shell e l e c t r o n gas which w i l l be spread throughout the i n s i d e of the metal. The s h e l l s are arranged, locally, in a lattice 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. order to simulate a f a c t of an o b j e c t p l a c e d i n height. T h i s d i f f e r e n t i a l packing r e v e a l e d by e l a s t i c a gravitational The d i f f e r e n t i a l packing may i s done i n theory: the d e n s i t y field decreases with be achieved i n one of two ways. One height way and i s to have the r a d i i to situate the another, as i n f i g u r e 2 ( a ) . shells such that of these s h e l l s s h e l l s i n a c t u a l c o n t a c t with The other way is with one to arrange the they do not touch one another. In t h i s way, the number of s h e l l s per u n i t volume may without increase diminish with height r e q u i r i n g the r a d i u s of the s h e l l s to change (see f i g u r e 2(b)). In either case, the change i s very g r a d u a l : i n a t y p i c a l 10 Ral\i of Mis Figure Figure 2(a). 2(b). Contact Arrangement of the S h e l l s Non-contact Arrangement of the S h e l l s metal bar 1 meter high, the d e n s i t y near the bottom is about one part i n 10 times greater of the s h e l l s , l o c a l l y , may constitute a lattice pattern. reader may p r o t e s t t h a t , because the than near the top . such the arrangement The of be regarded bar As to of g r a v i t y , the s h e l l s 11 may not be arranged as i n f i g u r e 2 ( b ) . objection, The the f o l l o w i n g model Because of t h i s p o s s i b l e comment i s made. has been c o n s t r u c t e d t o take i n t o some of the key physical which gravitationally induced e l e c t r i c denoted by E ^ realistic ve in calculating E of g r a v i t y . phenomena However, the model a l l respects. induced e l e c t r i c rise to the i n s i d e a metal, hereupon does not purport to be In p a r t i c u l a r , f o r the purposes of and p r o v i d i n g Q v e field give consideration physical fields, i n s i g h t i n t o the problem i t i s not important to d i s t i n g u i s h between f i g u r e s 2(a) and 2 ( b ) . What is each s h e l l give important i s how the nucleus and e l e c t r o n respond t o g r a v i t y : the f u n c t i o n of the model within i s to an idea of how a metal w i l l be a f f e c t e d by a g r a v i t a t i o n a l field. Consider what w i l l happen i n s i d e of each s h e l l gravity. When will of there i s no g r a v i t y , the nucleus w i l l be at the c e n t e r of the s h e l l and the ground s t a t e wave electron because be s p h e r i c a l l y symmetric. function Gravity will of the displace both the nucleus and the e l e c t r o n downwards; i n the case of the electron, this means that more of the e l e c t r o n w i l l be i n the lower h a l f of the s h e l l than i n the upper being much electron. against be than the e l e c t r o n , w i l l The nucleus, sink more than the In consequence, the e l e c t r o n w i l l h o l d the nucleus up gravity. This the heavier half. r e d i s t r i b u t i o n of charge i n s i d e the s h e l l will ion with a g r a v i t a t i o n a l l y induced d i p o l e moment. shown i n the next chapter that endow It shall these i o n i c d i p o l e moments a r e 12 a key source of E The . a v e e f f e c t of g r a v i t y on the e l e c t r o n i c c o n s t i t u e n t w i l l taken i n t o account by to be subject regarding the n o n i n t e r a c t i n g electron to an e x t e r n a l linear potential. A calculation s h a l l be made to determine the number d e n s i t y of those as a f u n c t i o n of h e i g h t . height and may be ave This role i n t e r n a l charge d e n s i t y induces microscopic key charge d e n s i t y as a from an e l e c t r i c field v e r s i o n of t h i s f i e l d — in The the polarizing microscopic outset to of another the ions. i n the metal not E As Q v e ! interior. — such, t h i s plays a microscopic induced dipole f i e l d pervading a s h e l l w i l l be assumed be nonzero component only The function represents f i e l d must be taken i n t o account i n d e r i v i n g the moment. electrons . . Gravity The The gas found by comparing the number d e n s i t i e s of s h e l l s electrons. source of E be uniform through the s h e l l and have a i n the v e r t i c a l d i r e c t i o n . macroscopic v e r s i o n of the gravity induced electric field, E y« 0 will t be regarded to influence the e l e c t r o n i c c o n s t i t u e n t of the metal. Thus, the l i n e a r p o t e n t i a l to which the electron gas i s subject shall be taken as due to both -» g r a v i t y and -> E Q ve • T h i s assumes that E ^ v e does not depend on height. Several the model. render The For features have been i n c o r p o r a t e d impenetrable s h e l l s were tractable electron. shell simplifying the introduced as to s o l v i n g of S c h r b d i n g e r s equation f o r the 1 the same reason, the e l e c t r i c i s assumed to be uniform. so into f i e l d pervading T h i s assumption w i l l be the quite 13 good i f the far apart. s h e l l s are arranged as i n f i g u r e 2(b), and spaced The i n t e r a c t i o n s between the electrons as well as those individual between the conduction electrons and the l a t t i c e , have been taken i n t o account i n an approximate manner, v i a the l i n e a r p o t e n t i a l the e l e c t r o n gas e x p e r i e n c e s . words, i t has been assumed that the d e t a i l e d arrangement of the s h e l l s may be ignored. the In other free electron Conceptually, approximation, zero g r a v i t a t i o n a l f i e l d density. -» E Finally, height. Actually, E a v Q V and c with t h i s i s much the as which i s made f o r metals i n zero internal net charge has been assumed t o be independent of v a r i e s with h e i g h t , e same but only slightly. w:i 11 be v e r i f i e d . -* sources of E are the i o n i c d i p o l e s , which This claim The Q v e contribute -» an average density, field , E^olas giving r i s e to a f i e l d E ^ . such as charge e x t e r n a l t o density, and i n c l u d e d together the surface by a term E , £ . c and the i n t e r n a l charge The f i e l d due t o a l l other c sources — charge electric the metal, dipole the density surface s h a l l be The equation f o r E ove i s (4) n Chapter 2, Ej / l/>t) be E K C that E will d l ^ o U s es i s opposite w i l l be c a l c u l a t e d . t o g and of the be c a l c u l a t e d i n Chapter 3. second order i n g. order E \ , will C The r e s u l t w i l l turn of Mg/q . e out to be 14 H.I. (1966). Z A.J. S c h i f f and M.V. B a r n h i l l , Phys. Rev. D e s s l e r e t . a l . , Phys. Rev. 168, 738 151, 1067 (1968). 15 CHAPTER 2 THE LATTICE OF IONS PLAN OF THE CHAPTER The E d i p o i e s ultimate ' t n average e of the i o n s . objective of this electric chapter is to calculate f i e l d due to the d i p o l e In order to c a l c u l a t e moments , i t i s necessary to E<i\p \es 0 know the g r a v i t a t i o n a l l y induced d i p o l e moment p of each i o n . may be c a l c u l a t e d i s known. and the i f the d i s t r i b u t i o n of charge i n s i d e the s h e l l As such, the p o s i t i o n of the nucleus i n s i d e the s h e l l the e l e c t r o n nucleus wave f u n c t i o n will must be found. be determined net f o r c e on the nucleus. electron function problem has potential three must o u t s i d e of of The e l e c t r o n the vanish shell, experienced on by nucleus, which i s not the addition wave the shell the the wave and will function be of the (1) f o r the the wave i s i d e n t i c a l l y zero singularity electron center of there be zero function characteristics: the (2) that location equation. determining important The by r e q u i r i n g obtained by s o l v i n g Schrodinger's The p of the Coulomb i s at the s i t e of the shell, t o the Coulomb p o t e n t i a l , the e l e c t r o n also and- (3) i n experiences a p o t e n t i a l which i s l i n e a r i n the v e r t i c a l d i r e c t i o n . This potential linear potential and the i s a combination potential due of the g r a v i t a t i o n a l to the e l e c t r i c f i e l d which 16 pervades be the s h e l l . taken to electric be field Thus, the e l e c t r i c the sum of field i n the s h e l l three terms: E ^ , c will , and the Et ex i n the s h e l l due to the d i p o l e moments of a l l of the other i o n s . As noted in chapter 1, the e l e c t r i c s h e l l w i l l not, i n f a c t , be uniform. approximated to be uniform. the Schrbdinger The farther Let However, the f i e l d w i l l be This -simplification i s made so that equation f o r the e l e c t r o n can be r e a d i l y s o l v e d . apart approximation f i e l d pervading the the shells are spaced, the better this w i l l be. denote the e l e c t r i c ~Ef>(r) ions. In the SchrcJdinger equation, E p ( r ) w i l l be r e p l a c e d by field. uniform field Electrostatic at A to inside a that Ep(r) due r shell uniform is f i e l d at the p o i n t the d i p o l e moments of a l l of the other reasonable i s i s the average choice for this representative value of E p ( r ) i n s i d e the s h e l l . theory t e l l s us that t h i s average the center ? c a of the s h e l l , E p C r ) . i s the value of As w i l l be shown c i n t h i s chapter, the e q u i l i b r i u m p o s i t i o n of the nucleus, r , i s n a very small d i s t a n c e from the center of the Ef>(r ) w i l l d i f f e r c ion by p n the- e l e c t r i c M interpretation. One then has a l a t t i c e of field Imagine r e p l a c i n g each hereupon be denoted t n e dipoles. Ep(r*„) is at the s i t e of one of those d i p o l e s due t o of the other d i p o l e s i n the l a t t i c e . TWs, such, a p o i n t d i p o l e p and s i t u a t i n g the d i p o l e s at the s i t e s of the n u c l e i . all As from E ( r ) by a t i n y amount. E ( r ) has a simple p shell. As such, E| (? ) 3 h will by Ej,. electric field i n a s h e l l c r e a t e d by the d i p o l e 17 moments of a l l of the other ions w i l l be taken t o be uniform and equal to Ef,. It i s E p which w i l l equation f o r the -* lattice, Ep w i l l be i n the v e r t i c a l The electron. enter For an into the Schrbdinger appropriate choice of direction. reader w i l l n o t i c e that the e l e c t r o n wave function i s going to depend on E*p and oc, where <* i s the d i s t a n c e between the center of the depend on E shell and °(. p Three f o r c e s a c t on force all due charge to and the nucleus. T h i s means that p w i l l I t i s easy t o see that t w i l l depend on Ep. the the nucleus: gravity, electrostatic the e l e c t r o n , and the e l e c t r o s t a t i c e x t e r i o r t o the s h e l l . The second of f o r c e due t o these forces —• depends on the e l e c t r o n wave f u n c t i o n , or <* and Ep. depends on Ef>. equation of E/> Requiring these f o r oc i n terms of Ep. implies be to cancel i n terms of p — determined. gives That w may be expressed that p can be w r i t t e n i n terms of Ep. can be c a l c u l a t e d -- p can forces The l a t t e r 'Having an i n terms Since Ej, f o r a given c h o i c e of l a t t i c e found E dholes p, may be calculated. Summarizing, the p l a n of t h i s chapter 1. Set the s h e l l , center the up the SchrSdinger equation f o r the e l e c t r o n with the nucleus d i s p l a c e d and' a i s as f o l l o w s : uniform e l e c t r i c a distance f i e l d E /, + E t c Sx oc inside from the + E^, pervading shell. 2. Solve f o r the wavefunction and energy of the e l e c t r o n as a f u n c t i o n of °t and Ep. 3. Solve f o r the p o s i t i o n of the nucleus by r e q u i r i n g that there be zero net f o r c e a c t i n g on the nucleus. 18 4. F i n d the d i p o l e moment p i n terms of Ep . 5. Choose a l a t t i c e and f i n d E|> i n terms of p. 6. Solve the two equations combining E^ and p t o determine p in terms of g, E /,, E .£ and Z . c n ev C a l c u l a t e t d ,-|, o l e i . . 7. These steps are now c a r r i e d out i n d e t a i l . (2.1) SETTING UP THE SCHRODINGER EQUATION FOR THE ELECTRON INSIDE THE IMPENETRABLE SHELL Let Y., x, dimensionless vectors respectively. space and z, in unit, mutually gravitational field of g Choose Down z-directions, i s employed; the the z - d i r e c t i o n g=-g£ with g>0. and up perpendicular, the x-, y-, and A c a r t e s i a n c o o r d i n a t e system i s Euclidean. direction be such shall mean i n the opposite d i r e c t i o n . of Of ), so ^ > 0 means the nucleus the s h e l l there (see f i g u r e 3 ) . i s zero imaginary Choose i . e . : at i s indeed below the center L e t a be the value of <* f o r which net f o r c e on the nucleus. force i s required i n the Place the nucleus a t a d i s t a n c e <x below the c e n t e r of the s h e l l , (0,0,- that the to hold I f <*^a, an e x t e r n a l , the nucleus in place. the o r i g i n of the c o o r d i n a t e system to c o i n c i d e with the nucleus. The electric field i n s i d e the s h e l l , e x t e r i o r t o the s h e l l , i s due t o a l l charges 19 shell ° * Shell £\e.c-tron F i g u r e 3. Placement of the Nucleus Inside the Impenetrable Shell ~ (1.1) where Ex ^ Edk text ) 4 and Ey, , E ^ c , E± ex (1.2) have a l l been d e f i n e d p r e v i o u s l y The Hamiltonian f o r the e l e c t r o n i s , (1,3A) n 4 ; ouistJle (1,3B) 20 where V-V/ZTJ, h i s Planck's permittivity (x,y,z) of free constant, space, from the nucleus, r k=i/4ne, , is r=Jt -+y -t-z ', z 2 from the c e n t e r of the s h e l l , 2 £ i s the e l e c t r i c D 0 the d i s t a n c e of the p o i n t z c is the elevation and £=m3 +? E . € Schrodinger's ( K 4 ) S (time-independent) equation f o r the e l e c t r o n is or ( 1.6) O Equation Ml (1.6) w i l l be s o l v e d by using p e r t u r b a t i o n oc and e are t r e a t e d as very small > compared small parameters; that to R, and € i s small compared to theory. i s , <X i s kZ^o^VR ". 2 as such, ^ and E are expanded i n T a y l o r s e r i e s about <tf=0, £=0: (1.7) 21 ^l z Note that only the ground s t a t e state energy eigenvalue are between the sought. and ground The reason f o r t h i s i s temperature, the energy ground s t a t e of the i o n and the f i r s t s t a t e i s l a r g e compared constant. (LB) eigenfunction t h a t , i n most metals at or below room gap - + *\*<K*X +<>f\* 0 to k T 9 Consequently, the , where k i B proportion s the excited Boltzmann of ions not i n the of "r, where ground s t a t e i s a very small number. ^j, 4\ and X\ a r e The position "E^, inside the a l l functions shell, "r i s as measured from the nucleus. The £{ and 8\ are a l l c o n s t a n t s . c Since &< and e are very s m a l l , i t is sufficient f o r the purposes of t h i s model t o determine ^ and E to lowest order i n oc and € . Accordingly, the approximate solution f o r 0 w i l l be taken to, be ^= "E, ^ + 0 ^ | and . As w i l l be proven, 'fc^ are approximate s o l u t i o n f o r E w i l l be taken as (1.9) both zero, and so the E-E.+<r Ve V«*S,r The problem (1.9), E , Q E , z reduces to "' 4 finding , ^, ^ i n equation £ , <f| i n equation ( 1 . 1 0 ) , and to showing 2 ,0) that 22 both E| and E\ , may be E are z e r o . ( found by A l l of these q u a n t i t i e s , except f o r breaking the problem down into three seperate problems. One of these is "the unperturbed problem". suggests, <*=0 and £ =0 i n t h i s case; the nucleus of the s h e l l , As the name i s at the center fa and there i s no e x t e r n a l p o t e n t i a l . and E are p the s o l u t i o n s f o r ^ and E, r e s p e c t i v e l y , when <* = 0 and e = 0, are eigenfunction and the other two reduced problems are "the p e r t u r b e d potential referred unperturbed The to as the unperturbed eigenvalue. problem", i n which ^=0 but problem", f Ej and wherein e^O, e = 0 but <*j^0. may be found by s o l v i n g the Similarly, the and the and "the p e r t u r b e d boundary I t i s c l e a r that the <f>^ and the perturbed potential E,- are determined problem. by s o l v i n g the p e r t u r b e d boundary problem. Before proceeding t o determine (1.9) First, Taylor equations series; i t i s not e x c e l l e n t approximations (1.7) and € are not equations that these be (1.10) will still be provided that equations (1.9) and to and dimensionless. parameters c e r t a i n l y not mandatory. naturally necessary E (1.8) are at l e a s t asymptotic dimensionless seen in and (1.10), two p o i n t s should be made r e g a r d i n g equations (1.7) and (1.8). and the q u a n t i t i e s in solving It ctr«/R These f o r ^, and expansions. i s possible €/(k<le/& ), 4 V c £ 2 oc to define but 2 parameters, Second, # and €, i t is emerge and £|, as s h a l l be shortly. The next step i s to s e t up the perturbed p o t e n t i a l problem 23 and obtain expressions for the 9^ and the E.;. e perturbed boundary problem s h a l l be posed, and the ^ and the differs perturbed potential from the unperturbed problem is of the so named because i t problem only i n that c centre for f o r the Perturbed P o t e n t i a l Problem experiences the small l i n e a r p o t e n t i a l £ z . the expressions found. (A) S e t t i n g up the Equations The Then, the the electron With the nucleus at s h e l l , £*=(), z=z and Schro'dinger' s equation c becomes (1.11) with the boundary c o n d i t i o n - o (1.12) T h i s problem w i l l be s o l v e d to the lowest n o n t r i v i a l in the s m a l l parameter e. To t h i s end, write order 24 H = H + ez (1.13) 0 where H.'-£V<U. l+oO-, r>^ Express j£ and E as p e r t u r b a t i v e expansions about £ = 0 : £ + eE, e^E + •• • £ & Substituting these expansions the f o l l o w i n g equations: e 2 i n t o the equation H^=E^ (Lie) leads to (1.17.0) 25 (1.17.1) (1.17.2) (1.17.n) The boundary c o n d i t i o n (1.12) i m p l i e s that It = 0 (1.18.0) = 0 (1.18.1) r-R (1.18.2) (1.18.n) 26 Equations unperturbed these that fa and (1.17.0) and (1.18.0) v e r i f y eigenfunction and eigenvalue. Using the standard techniques are 0 the I t i s i n terms of two q u a n t i t i e s that the E ^ and the fa are e E expressed. of p e r t u r b a t i o n theory, i tis found that 1 1 1 r\ = l 0 o (1.19) u where the i n t e g r a t i o n s are over the region r<R. The n o r m a l i z a t i o n c o n d i t i o n , gives r i s e to the r J equations > (1.20) 27 Equation (1.20) reduces According E, = f f the results to equations equation z dr J (1.17) and 0 % „ [ of standard (1.18) and (1.21) dV^„ perturbation (1.20) allow ^- and a l l of the E{ , i n d e t e r m i n a t i o n of a l l of the <fi (1.19) to € theory, for the terms of E . and Q It ^,, E | is and £ argument worthwhile to w r i t e out the equations which s p e c i f y & E^. Before doing so, note t h a t a demonstrates that ^E^O simple f o r a l l odd i. physical The idea i s simply that the p h y s i c a l energy E i s i n v a r i a n t under a change i n T h i s i m p l i e s that E(e) - E(-6) = 2e E sign of € . ... =0, l which can be t r u e f o r a l l £ only i f In p a r t i c u l a r , E|=0. € and E z Consequently, + e { E|=0, the equations 2e ^E^ 3 E=0, + .... specifying are: (H.-£ )^ =-z(4 0 O (1 .22.a) (1.22.b) r=R r (1.22.c) R 28 (1.23) For the sake of completeness, the equation for E is ( also given: Ei Once r 0 has been (1.24) determined, i t s h a l l be shown that equation (1.24) g i v e s E,=0, as r e q u i r e d . £ (B) S e t t i n g up the Equations the f o r the Perturbed Boundary Problem In t h i s problem, the nucleus i s placed a distance centre in of the shell, external linear potential. as figure oc below 3, and there i s no The e l e c t r o n wave f u n c t i o n i s given by the s o l u t i o n to L„y=E »-0 inside " 2>m > r the s h e l l , and i s zero o u t s i d e the s h e l l . f u n c t i o n fi, given by equation (1.25) (1.25), w i l l not Note that the itself be zero 29 exterior to the wave f u n c t i o n and the shell. T h i s d i f f e r e n c e between the e l e c t r o n the f u n c t i o n ^ should be carefully noted reader. The Hamiltonian f o r t h i s problem d i f f e r s from that f o r the unperturbed problem only i n relative to the nucleus that the is different. position of the problem may be regarded been s l i g h t l y F i g u r e 4. and as a problem i n which the boundary the has P o s i t i o n of the Boundary R e l a t i v e to the Nucleus for the p o s i t i o n of the boundary can be by examining f i g u r e 4. shell- of The Let boundary in the unperturbed r a d i u s R entered s h e l l of r a d i u s R centered nucleus. this perturbed. An equation a shell If oc is s m a l l , then d i f f e r e n c e i n p o s i t i o n of the boundary i s a small one, is by ^(0) be boundary in the perturbed on the nucleus. on a p o i n t a the distance derived problem Here i t i s a °C above d i s t a n c e from the nucleus the to the boundary problem, where 0 i s the p o l a r 30 angle. From f i g u r e 5, n o t e (r (G) t that - c x c o s e ) ' + (tfsin©) - = 2 2 R L or, s o l v i n g for r\ (0), 3 Expanding r^(0) i n a T a y l o r Figure Expanding 5. s e r i e s a b o u t oV=0 g i v e s R e l a t i o n s h i p b e t w e e n Q, <fi and E i n T a y l o r r (9),cx h and (1 .27) (1 .28) R s e r i e s a b o u t <* = 0: 31 (1.29) Again, a l l of the same E<, with \ odd, are zero. The reason is the as i n the perturbed p o t e n t i a l problem; the p h y s i c a l energy E can not depend on the s i g n of OC. In consequence, equation (1.29) s i m p l i f i e s to (1.30) 2, Inserting equations 1 i~Lf (1.28) and (1.30) i n t o L ^=E^ 0 leads to the set of e q u a t i o n s : o r 0 - L 0 Yo (1.31.0) (1.31.1) (1.31.2) The boundary c o n d i t i o n s which must be obeyed by the <J> { obtained by w r i t i n g <fi i n the form of a T a y l o r s e r i e s about r=R: (1.32) are 32 where and <P i s the azimuthal r=r (9). angle. $ vanishes r i g h t hand s i d e of equation the shell, the form (1.28) f o r 0 Accordingly, inserting t on (1.32), and e v a l u a t i n g at at i n t o the r=r (9), b leads to ii. I, e, f) + a (a,©,?)+cse^^q?)! + « U«,e,<r ) + 1 , c 0^ (( ,e r; / OI l / C (1.33) {sin © t'a.e.r > + | to© £ ( K A H 1 2 + •• • : O where the Equation result (1.33) can has be been true expressed for as a power arbitrary c o e f f i c i e n t of each power of oc i s z e r o . series <X only i n oc. if the T h i s means that (1.34.0) ^ ( W ) + cose^(R,e,<p)-o (1.34.1) 33 (1.34.2) The only other c o n d i t i o n to s a t i s f y the i s the n o r m a l i z a t i o n of eigenfunction: (1.35) USA where the i n t e g r a t i o n i s over the i n t e r i o r of the s h e l l . The left powers of OC. hand s i d e of equation (1.35) may be expanded i n To see how t o do so, c o n s i d e r f i g u r e 6. r e g i o n i n t e r i o r to a s p h e r i c a l s u r f a c e of r a d i u s R the nucleus, i s the region e x t e r i o r to shell, and i s the r e g i o n i n t e r i o r shell. Note that dr= 3 d r means to integrate d r+ 5 over to V V n centered on and i n t e r i o r to the n and e x t e r i o r t o the (1.36) JV- the i s the region V^, where r<R. (I Equation (1.36) equation (1.35). may be applied to the l e f t hand s i d e of the 34 F i g u r e 6. Using the expansion left The Regions V-^ and Vp, (1.28) f o r <p, and equation hand s i d e of equation s e r i e s i n o( . Equation (1.35) may be converted (1.35) i s true only i f the (1.36), the i n t o a power order unity term i n t h i s power s e r i e s i s equal t o one and i f the c o e f f i c i e n t of every other power of oc i s z e r o . in J v the d 3 r ^ power and series jVr^jf The only term of order u n i t y i s ^ d^r^y^, are of order , since <*. J the integrals Thus, (1 .37) The c o n d i t i o n i n which ^ enters i i (1 .38) obtained by r e c o g n i z i n g that the c o e f f i c i e n t of cc i s zero. The 35 higher order terms give s i n c e only fa and ^, w i l l equations involving i{, </, ••• , but 3 be r e q u i r e d , u l t i m a t e l y , there is no need t o w r i t e these equations down. Summarizing, define the equations equations unperturbed (1.31.1), these l a t t e r (1.31.0) , eigenfunction (1.34.1) (1.34.0) and and (1.37) eigenvalue, and (1.38) s p e c i f y . while Grouping three equations t o g e t h e r , I ii) - L (I) (1.39.a) (1.39.b) r= R ( 1 .39.c) Equations and ° E ! 2 f o r the ^E^ may shall a l s o be d e r i v e d . Only those for Z\ 0< be presented. Proceeding from the expansions (1.28) and (1.29) f o r ^ and E, TuJ=Ef i m p l i e s t h a t (1.40) shall be used M u l t i p l y i n g both s i d e s by equation (1.37) to generate an equation ^„ , i n t e g r a t i n g over V , A for and E-t . using 36 Combining equation (1.34.0), where (1.25) f o r L , the r e l a t i o n s h i p s 0 and Green's theorem, equation dS is integration condition the (1.31.0) and (1.41) s i m p l i f i e s to element of area on the s u r f a c e of V i s over the s u r f a c e of V , where r=R. The boundary to be e l i m i n a t e d from equation. n (1.39.b) allows ^ and the n (1 .42 ) , g i v i n g *. Equation Using converted . , ,, . (1 .43) (1.43) s h a l l be used to v e r i f y that ^E|=0. the same techniques, equation (1.31.2) may be i n t o the f o l l o w i n g e x p r e s s i o n f o r E^: in a, <v( The . form (1.34.2). calculated. of ^ on the s u r f a c e of V As such, once ^ has been (1 i r n .44) i s d e s c r i b e d by equation determined, ^E^ may be 37 (C) S e t t i n g up an Equation f o r £ i may be o b t a i n e d by d e a l i n g with the f u l l An equation perturbation for Ei f o r £; ; Summary problem. The method to use to get t h i s e x p r e s s i o n begins by equating the equation (L +&Z p terms Recall C that on z c each side of the i s height as measured from the c e n t e r of the s h e l l , while z i s height as measured the nucleus. z c and z a r e t h e r e f o r e r e l a t e d by Z - Z -oc , (1.45) c Using the expansions equation 0 c where E i 6 (1.7) and (1.8) f o r ^ and £ along with (1.45), equating the c o e f f i c i e n t s o f o n ( L + e z )^=E from each s i d e of Ogives and * E , have been acknowledged as being z e r o . Performing the same o p e r a t i o n s on equation used to d e r i v e e x p r e s s i o n s f o r " ^ E , (1.46) as were and^Ej., (1.47) The form of ~)C, on the surface equation of (1.7), V to n i s obtained by r e q u i r i n g ) ^ , as given by vanish on the s u r f a c e of the s h e l l . Using the same approach as that which l e d 38 to the boundary c o n d i t i o n s (1.34), i t i s found that - - COS© T h i s r e s u l t transforms equation (1.48) . (1.47) i n t o (1.49) R At this ^R stage, equations have been d e r i v e d f o r a l l of the q u a n t i t i e s appearing given by equation i n the approximate s o l u t i o n s f o r ^ (1.9) and (1.10). and E 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 (1.52.c) 0 o / o •I 0 .it [ 2m 2 = 2_ m T.'z (1.53.a) (1.53.b) (1.53.C) (1.53.d) 41 (1.53.d') - - C o 5 Q<1>; J (1.53.e) (1.54) r Note that the approximate solution 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 not from zero by terms of 2 order^ and oi£. T h i s i s a c c e p t a b l e , because the are r e q u i r e d to only f i r s t order. wavefunctions 42 (2.2) SOLUTION TO THE SCHRODINGER EQUATION FOR THE ELECTRON INSIDE THE IMPENETRABLE SHELL (A) The Unperturbed Problem; Determination of The essential and of Bo f e a t u r e s of the model w i l l be independent of the c h o i c e made f o r R. R i s chosen „ n f o r convenience t o be (2.1) n where *Q 0 isthe Bohr - < e •) 2 2 Z radius. For t h i s c h o i c e of R, when r<R i t i s easy to see that J^> i s essentially different function the hydrogenic normalization; belonging hydrogenic problem. to wavefunction /Aoo , but with 7^z©o i s the s p h e r i c a l l y symmetric the first excitation energy of a wave the Thus, where V- a, 3 N i s determined from the n o r m a l i z a t i o n c o n d i t i o n <-> 2 4 (1.51.a) and i s 43 N E D (2.5) is r - - ^ P-7 n 4 Z te * 8tf 0 (B) S o l u t i o n to the Perturbed P o t e n t i a l The trivial first t h i n g to observe to v e r i f y that equation Using equation some simple (2.6) (2.6) i s t h a t having found *f , i t i s 0 (1.53a) g i v e s E , =0. 6 i n equation manipulations, Problem the (1.51.a) following and performing equation f o rft> results: It form i s easy to v e r i f y from equation (2.7) that ft, i s of the 44 Using equation (2.8), is s a t i s f i e d . Where t(P) must ^"o) The i t i s easy to show that c o n d i t i o n satisfy = p (2-p) • + ( ^ ^ ^ boundary c o n d i t i o n (1.51.C) J ( 2 - 9 ) 1 0 ) ( l . 5 l . b ) f o r <f>> t r a n s l a t e s i n t o i ( D = Q • ( 2 ' Putting (2.11) s i m p l i f i e s the problem to T cp + (|-l)r'(p) = 2 - p • // Solving "integrating equation (2.12) f o r Tr' by using < -' > 2 2 the w e l l known f a c t o r method", and then i n t e g r a t i n g t o get X , i t i s found that P X (2.13) 45 The constants to be s a t i s f i e d determine is t(2)=0. to 1 for ^ be 2 must be determined. One condition Another in order The singularity. the need f o r iftj £, c both c o n s t a n t s . not have a 9 (1.7) c, and is needed integrable. j^V^*^ Recalling that the the expansion as a power s e r i e s i n <* and i t i s seen that the c o e f f i c i e n t of the 6 In order \ 1 term w i l l be ' \ dV©*0, part of this be finite, it necessary that </>, have no worse a s i n g u l a r i t y at (?=0 than There One are only two involves c . z The two terms i n (2.13) which blow up w i l l cancel i f the written With t h i s determination is (? . like choice Q--72 i s made. must T h i s requirement i s a consequence of , i n expressing R 0, second c o n d i t i o n i s that to (2.14) of c , e q u a t i o n 2 (2.13) may be as x The form (2.15) f o r t(P) i s very obscure. P p » (2.15) However, using r power s e r i e s expansions f o r e the ^ terms. terms 2 As in equation such, t(P) may a nd \'1 <Ax r e v e a l s that not (2.15) c a n c e l , but so do the be expressed as a power s e r i e s : only Q~' 46 (2.16) I n s e r t i n g equation be determined. (2.16) i n t o equation (2.9) a l l o w s the c j to The r e s u l t i s : ^ } ' ' T r > ^ 5 ? c 0 + (2 ^ JP C - ,7) Where C; - C. CHXO^) •^5 J (2.18) 3o The boundary c o n d i t i o n (2.10) f i x e s c, ; to four decimal p l a c e s C^-OM^Z t(t?) may be specified (2.19) by e i t h e r equation (2.15) or equations (2.17) and (2.18). Before u s i n g the s o l u t i o n that for 0 ( to calculate t((?)<0 i n the range 0<£<2, which i m p l i e s h a l f of the s h e l l and 4>\ >0 i n the lower h a l f . means that the electron sinks € E , 2 note ^,<0 i n the upper Physically, this down (assuming €>0) under the i n f l u e n c e . o f the e x t e r n a l p e r t u r b i n g potential. 47 6 E 2 i s determined by e v a l u a t i n g equation equations (2.3) and (1.53.b). Using (2.8), L Z~ -2/4 \ l <g L L (2.20) It where (2.21 ) The simplest way t o s o l v e f o r E ' i s t o put e z X (Z-x)-£(x) " Jix^W(D)e P . 3 X (2.22) e w(^) must satisfy (2.23) Using w(P) the power s e r i e s form of t ( P ) , the f o l l o w i n g s o l u t i o n f o r results: (2.24) 48 where V 4 = -iC, / (2.25) V Since w(0)=0, equations (2.21) and (2.22) combine to give -z E v a l u a t i n g equation (2.26) to four decimal p l a c e s , c From equations decimal places, (2.26) (2.27) £/=-O.l082 • (2.20) and (2.5) i t i s found t h a t , to four 49 (2.28) (C) S o l u t i o n to the Perturbed Boundary Problem It All the that i s easy to show that equation i s required radial integral follows (1.53.c) says that ^E,=0. i s to observe, using equation integral i n equation of cos© over the surface (1.53.c) i s f i n i t e . of a sphere found almost as easily. equations (1.50.a) and (1.52.a) as w e l l (l.52.b) different is that Since the zero, ^E, =0 immediately. is and (2.3), that First as note by comparing equations (1.50.b) s a t i s f i e s the same equation as >4> , but a 9/ boundary c o n d i t i o n . In consequence, ^ i s essentially the hydrogenic wave funct ion (2.29) Insisting whence, that V) s a t i s f y equation d.52.b) f i x e s N, : 50 (2.30) i ( H V « 9 Equation (2.30) also s a t i s f i e s the requirement 'A i s p o s i t i v e i n the p a r t of the N o t i c e that shell (1.52.c). above the nucleus, and negative below i t . T h i s means t h a t , when the s h e l l i s moved s l i g h t l y upwards, the e l e c t r o n gets pushed up above the nucleus a little nucleus than basically the b i t , so more of the e l e c t r o n w i l l be above the below it. The physical reason for this j u s t that there i s s l i g h t l y more room a v a i l a b l e shell is inside above the nucleus than below i t : the e x t r a room lies i n the v i c i n i t y of the s h e l l . Equation equation (1.53.d) i s r e a d i l y e v a l u a t e d (1.53.d ) and equations respectively. The f i n a l (2.31). at use of and \°i , result i s . Z are making (2.3) and (2.30) f o r ^ - %lFMlZjl<% There by least two other ways (2.31) to d e r i v e equation They serve as a check and are t h e r e f o r e presented. of these methods involves some key physical d e s c r i b e d below. The other i s more mathematical ideas, and One is and i s given i n Appendix A. The p h y s i c a l method begins with the r e a l i z a t i o n that 51 F ,(«A--^ - € e l where F^, («-)=F . («.)z ( 2 - 3 2 ) OCX f o r c e exerted on the nucleus by the e l e c t r o n as a f u n c t i o n of oc, and E i s the energy e2 i s the e l e c t r o s t a t i c eigenvalue i n the perturbed t ^ 0 boundary problem: f / E i f ^ t f ••• (2.33) T Equation (2.33) i s c o n s t r u c t e d as f o l l o w s : In perturbed problem the external order potential i t i s c l e a r that an f o r c e i s r e q u i r e d to a c t on the nucleus, to keep it in place. This force magnitude to Fgj(°<.), but o p p o s i t e l y d i r e c t e d . must the external force the same in be equal i n I f «- i s increased by an i n f i n i t e s i m a l a m o u n t , , while maintaining of when <*-/0, the magnitude as that of Fei (°0 , then the e x t e r n a l agent, a c t i n g i n the same d i r e c t i o n as the displacement of the nucleus, does an amount of work Fei (<>0<Po<. T h i s work goes i n t o i n c r e a s i n g the energy of the n u c l e u s - e l e c t r o n Fei (©Ocf«-=(£E, which i s equation Notice force that equation exerted using E=E „ + system bySE: (2.32). (2.32) i n d i c a t e s that ^,=0, for the on the nucleus by the e l e c t r o n when°<~=0 i s zero; """Ei + <=<" z. + in equation e v a l u a t i n g at Qf = 0: J<K-o I n s e r t i n g equation 1 (2.33) i n t o equation (2.32): (2.32) and 52 r ,C«) = Z«("'E .+ ^of %+i e may be expanding found 2 by obtaining an . (2.34). expression for F e/ (ex.), i t , and f i n d i n g the order o<-term. Regarding with charge the e l e c t r o n i n the s h e l l as a density qe/^(r)/ cloud of charge , (2.35) By symmetry, hence, only [ F , £ the x- and y-components = F , (<*) i £ ^ s interesting. of F et (°c) are zero; It i s (2.36) n Using the expansion (2.37) in equation (2.36), Iw-^Zrfli/^i Only i/<£fA z + the order oi term on the r i g h t hand s i d e i s d e s i r e d , so the 53 terms not explicitly that the f i r s t The integral order o r d e r one w r i t t e n down may c< be ignored. Further, note i s o f o r d e r oi. . 3 p o r t i o n o f Fe/ (<*~) i s o b t a i n e d by p o r t i o n of t h e s e c o n d integral in finding equation the (2.38). Writing (2.39) it on is clear the r i g h t Taking of t h a t t h e o r d e r one hand s i d e . The t h i s o r d e r one equations ( 2 . 3 9 ) and term here o t h e r two term i s the integrals from e q u a t i o n first are (2.39), integral of order comparison ( 2 . 3 8 ) shows t h a t (2.40) Evaluating expression f o r °^E the z integral found i n equation earlier: (2.40) reproduces the 54 To four decimal p l a c e s (2.41 ) (D) C a l c u l a t i o n of ; Summary Using the e x p r e s s i o n s o b t a i n e d f o r <^ , a d.53.e) f o r £ , i s easy to e v a l u a t e . The , and <fi, , equation r e s u l t obtained i s (2.42) The numerical value of t'(2) equation (2.15). is best found by working from Denoting (2.43) d i f f e r e n t i a t i o n and e v a l u a t i o n at P=2 gives (2.44) An equation for c ( which i s i n s t r u m e n t a l i n s i m p l i f y i n g equation 55 (2.44) may be a r r i v e d at by e v a l u a t i n g equation R e c a l l i n g that t(2)=0, the r e q u i r e d expression (2.15) a t p = 2 . f o r c, i s C,= - 8 - 1 2 J / - ( f A(2>) Combining equation (2.43) and (2.44), T^a)^7-7V Equation To (2.45) i t f o l l o w s that • (2.46) reduces equation (2.46) (2.42) t o four decimal p l a c e s , r , = -o.5z?7. E, i s dimensionless because ©te has u n i t s of energy. A second method of c a l c u l a t i n g £ method is determine in E z essence - (2 the same ( i s now presented. This as the second method used t o and i t b r i n g s f o r t h some key p h y s i c a l ideas. In the f u l l p e r t u r b a t i o n problem, when °^^a an external f o r c e i s r e q u i r e d to a c t on the nucleus to keep i t i n p l a c e ; see f i g u r e 7. Since t h i s e x t e r n a l f o r c e w i l l be i n the z - d i r e c t i o n , 46) 56 write fe^«> f ' < 2 - 4 9 ) Shell /v/«clens F i g u r e 7. Forces A c t i n g on the Nucleus in the F u l l P e r t u r b a t i o n Problem S i m i l a r l y , as in the treatment used e a r l i e r to get ^E^, write (2.50) fel(«)--F («)Z e l where, again, —> Fr, (<*.) i s the e l e c t r o s t a t i c e ei nucleus by the e l e c t r o n . by the nucleus: the e l e c t r i c There are two i t s weight, -Mgz, f i e l d E5=E S these four f o r c e s c a n c e l and other the z, which i s q Z „ E e gives f o r c e exerted on the f o r c e s experienced f o r c e a p p l i e d to i t by s z. Demanding that 57 (2.51 ) Imagine the nucleus to be lowered by an amount ^ « i n c r e a s e s by — with the four f o r c e s j u s t c a n c e l l i n g , equation (2.51). — so <* as Then the e x t e r n a l agent does an amount of work c^W—Fgxt £«. on the system of n u c l e u s , e l e c t r o n and the f i e l d s and -gz. The in system gains p o t e n t i a l energy q Z E s ^ ° S e E £ s loses n p o t e n t i a l energy Mgcfoc, and has the energy e i g e n v a l u e E change by CTE: 6W=-F t^ = <j Z E ^-MjS« + J £ t t e n - s (2.52) Note that the SB term i n c l u d e s the change i n the k i n e t i c of the electron, the change in the e l e c t r o s t a t i c potential energy of the n u c l e u s - e l e c t r o n c o n f i g u r a t i o n , as change i n the p o t e n t i a l energy of the e l e c t r o n i n the f i e l d s and in _ gz. well energy R e c a l l that t h i s l a t t e r p o t e n t i a l energy was the H a m i l t o n i a n v i a the term e.z ~&{z~<<) . c as the E z 5 included Compiling equations (2.51) and (2.52): (2.53) F e ( («*.) i s given by equation (2.36). The expansion f o r E g i v e s (2.54) 58 £,may be obtained by proportional finding expanding proportional of F ^ e which i s to £ . Working from equation for ^ , the p a r t (2.36), s u b s t i t u t i n g i n the expansion i n powers of and e , and e x t r a c t i n g the £ - part, (2.55) Performing the i n t e g r a t i o n indicated, rz (2.56) Write (l-x){.(x)e d\ i v(£) must (2.57) - v(^) e " P satisfy \/(p)-vrp)c(z-pK(p) , or, using equation (2.17) f o r t ( P J , v.J (2.58) .1-6 power s e r i e s s o l u t i o n , beginning with a term i n (? , will 59 s a t i s f y equation (2.58): CD V(p) = £ v ^ (2.59) J r2 Substitution of equation the s o l v i n g of the v-. (2.59) i n t o equation (2.58) leads t o The r e s u l t i s J V i = C, v= 0 3 (2.60) 0 Si- whence equation 2 ^ V j - ' - C j - ' , j»2 J (2.59) becomes (2.61 ) E v a l u a t i n g equation (2.61) a t (? = 2: <x> to four decimal (2.62) places: vci) = -1.3373 . (2.63) 60 I n s e r t i n g equation ( 2 . 6 3 ) and j^tH^^p — t r u e because v ( 0 ) = 0 — = \ia)£ , x i n t o ( 2 . 6 1 ) , t o four decimal places, £, = - o . 5 7 ^ ^ , c which i s the same as equation Eiis Note that captured be at negative. the center The p h y s i c a l reason f o r t h i s Then, a c c o r d i n g c l Assuming e > 0 , shell, F &t Equation (If £ < 0 , (c*)^ 0 will will potential t o e q u a t i o n s ( 2 . 5 3 ) and ( 2 . 5 4 ) , = 4 •• • (2.65) -ioC-o the e l e c t r o n w i l l have sunk down a and is Imagine the nucleus to of the s h e l l , as i n the perturbed Fc«) that (2.48). i n equations ( 2 . 5 5 ) and ( 2 . 5 6 ) . problem. the (2.64) be p u l l i n g be the nucleus down. negative see pulled up, must be F | (<x) 7 e inside T h i s means equation ( 2 . 6 5 ) says t h a t , i n consequence, the e l e c t r o n has been little (2.49). negative. >0 and £,«).) With having been found, the problem has been completely solved to f i r s t order c o r r e c t i o n i n ocand £.. The eigenfunction is (2.66) 61 lowest n o n t r i v i a l o r d e r , where (2.68) y^, /V- j^- ^(p)e ^ C o s £ ) ; 2 2 ~ <*o ^ ^ - , f ^ . > (2.69) (2.70) (2.71) 62 and t ( P ) i s given by either 3 C; 1 z r 3o or with C -OA£ZZ . r The energy eigenvalue i s Z to lowest ' t. n o n t r i v i a l order, where ^ 2 63 (2.78) _ l_p . 8tf (2.79) °2 ^2 O ^C2-p)-t(n)e~^p^-alo82 -0.1713 a " (2.80) (2.81 ) (2.82) 64 ( (2.83) IM tti\U 3e* ft7V)-1*-0.5721 Having found these q u a n t i t i e s , the task remaining them to calculate the - i s to use g r a v i t a t i o n a l l y induced e l e c t r i c field i n s i d e a metal. (2.3) THE POSITION OF THE NUCLEUS The g r a v i t a t i o n a l f i e l d , c e n t e r - o f the s h e l l . w i l l displace The e l e c t r o n the nucleus from the i n s i d e the impenetrable responds to r e s i s t t h i s displacement, e x e r t i n g an upwards shell pull on the n u c l e u s . The nucleus a l s o experiences a f o r c e due t o the electric field E^. the center down from The p o s i t i o n of the nucleus, a d i s t a n c e a of c a n c e l l i n g of these f o r c e s . the shell, i s determined by the Mathematically, (3.1 ) Using equations (2.53) and (2.54) 65 Combining these two equations, the s o l u t i o n f o r a, to lowest order, i s 2% The value of a i s dependent on the f i e l d earlier. mg+q E e To show fore, g respectively this and E P dependence + E j f o r E$ — i n t o equation more Having shows how found a, clearly, substitute (1.4) and JZ.^fe will the next few ( steps are to determine p 3 ) i t will be the Since p possible to both p and Ej,. N o t i c e that a/a D i s indeed very s m a l l , as was claimed in t h i s t h e s i s . T h i s i s easy t o see by using equation equation and result 3 the ions i n a p a t t e r n , and c a l c u l a t e Ej, i n terms of p. depend on a, and a i n turn on E , determine r a depends on Ef,. d i p o l e moment p of each of the model ions, arrange chosen l a t t i c e (1.1), (3.2), t o o b t a i n 9 (3.3) discussed equations (M-^S,) -(Z,+&He£i Equation E^, as (3.3) i s a/a 0 ^10 — 18 taking Ep early (2.82) i n to be of the order Mg/qe. The 66 (2.4) CALCULATION OF THE DIPOLE MOMENT OF EACH ION IN TERMS OF An important q u e s t i o n at t h i s stage of development model for the which i s due shells? To ions i s : what i s the e l e c t r i c to the n u c l e i and e l e c t r o n s i n far a l l of away from that what i s the e l e c t r i c ion? the field others, w i l l terms are field at the answer a due to an shell. R e c a l l i n g the d i s c u s s i o n at the beginning of t h i s electric other By an ion i s meant here a nucleus and an e l e c t r o n bounded by an impenetrable the the in a s h e l l f i n d the answer, i t i s necessary t o f i r s t closely related question: ion field of s i t e of an i o n , due be given to a very good to a l l of the approximation of the d i p o l e moment~p of each i o n . chapter, simply in The monopole moments taken i n t o account by E ^ . R e l a t i v e to a chosen origin, 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 i n t e g r a t i o n goes over a l l of the charge R e l a t i v e to that same o r i g i n , the c o n t r i b u t i o n to field at r^ , due to the dipole moment distribution. the of the electric charge distribution, is \_\fo)~ jn - K (4.2) 67 assuming that r^, l i e s o u t s i d e the charge d i s t r i b u t i o n . Equation (4.1) s h a l l be used t o c a l c u l a t e the d i p o l e moment of each The origin shall be chosen ion. t o c o i n c i d e w i t h the s i t e of the The charge d e n s i t y p(r) i s given by nucleus. (4.3) whence (4.4) It i s c l e a r that the p h y s i c a l f e a t u r e s of the invariant under rotations around the z-axis. problem has a x i a l symmetry, and as such, p> must direction. problem are That i s , the be in the z- 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 t o (4.5) where ^'Vif" Using the familiar ••• (note t h a t « = a the imagined f o r c e ^ ' z cn expansion f o r - i <fi, - <4 fi 0 now, because t h e nucleus i s h e l d F xtK e equat ion + a ^ up + 6> €<fa + without (4.6) may be expanded out: 68 r b e To the lowest order, (4.7) The i n t e g r a l s here are f a m i l i a r . From e a r l i e r work, r (4.8) R and r Equation equation (4.9) is (4.9) an o l d r e s u l t . (1.47) and 2r"l J Equation (4.8) f o l l o w s from 69 which i t s e l f f o l l o w s from equations (1.48) and (2.46). Making the replacements (4.7) and (4.9) i n (4.7), (4.10) The i n s e r t i o n of equation explicitly (3.4) i n t o equation (4.10) gives p i n terms of Ef.: (4.11) where Ex-a 2% J (4.12) and (4.13) As d e s i r e d , p has now been expressed i n terms of E 70 (2.5) CALCULATION OF E * IN TERMS OF p The task of t h i s s e c t i o n i s t o c a l c u l a t e the e l e c t r i c inside a shell other s h e l l s . basic t o the n u c l e i and e l e c t r o n s i n a l l of the As noted at the very beginning of this electric The due f i e l d may be taken t o be uniform reason field behind this this chapter, i n s i d e the s h e l l . approximation shall now be repeated. Denote by inside E l 0 r, s ( r ) the e l e c t r i c field at i s , a l l of the n u c l e i and e l e c t r o n s i n E (r^ will vary i n c l u d e t h i s s p a t i a l v a r i a t i o n of equation f o r the e l e c t r o n problem a initially very uniform in Ei (r) onS one. equation field would not have (r) at to r e p l a c e E i i n the z- d i r e c t i o n . with E\ o r t 5 the (r ) t (r) i n the Schro'dinger idea that would be the in model the would to enter known. in This E ^ s (r) Since the average of just the value of — > the center of the s h e l l , Elans ofys To Schrbdinger replacing —> LortS the been from the outset by _=,, E shells. For example, EZons ( r ) i s not over the i n t e r i o r of the s h e l l i s tons E other —» known, and so the form of the p o t e n t i a l Schr3dinger a the i n s i d e a s h e l l would have rendered the difficult dilemma was circumvented by r a l i t t l e as r v a r i e s i n s i d e the s h e l l . —> the position of a s h e l l , which i s due t o a l l of the other model ions, that Lons the (r ), i t was decided c equation by E i.o^(r ), t be c o n s t r u c t e d so that z-direction. But notice that E ujrtj ( t ) at w i l lthe d i fs fi et re of onlythe s l nucleus, i g h t l y from ~ E \ * the ( r ), the value Ei rxs(r) because nucleus s i t s of a r 0 0 —> s n 71 very small distance a below the c e n t e r of the s h e l l . As such, i t i s j u s t as good to r e p l a c e E <^„ (r) by E i ^ s (r„). s In the i n t r o d u c t o r y passage t o t h i s c h a p t e r , i t was that Ellens ( r * ) i s equal t o E|,. of e l e c t r i c R e c a l l what Ep d i p o l e s , Ep: i s the e l e c t r i c is: noted in a lattice f i e l d at the s i t e of one — ? dipole replaced due t o a l l of -> by E ^ . the others. To summarize, the e l e c t r i c the nuclei and electrons field inside As such E s h a l l be Lon3 i n s i d e of a s h e l l due a l l of the other to shells i s approximated, i n t h i s work, t o be uniform and equal to Ep. The s h e l l s w i l l be arranged so that E^ is in the z- —=> direction. An expression for E p in terms of p w i l l now be found. Consider the s h e l l s to be arranged nuclei are arranged in the imagine r e p l a c i n g each s h e l l dipoles same in a l a t t i c e . lattice. the To c a l c u l a t e Ep, by a d i p o l e with moment are arranged, t h e r e f o r e , Then i n the o r i g i n a l p. These l a t t i c e of the shells. The e l e c t r i c to f i e l d at the s i t e of one of these d i p o l e s a l l of the others will s i d e of equation (4.2). origin site at the be a sum of terms l i k e the r i g h t hand Choose a of the primitive t r a n s l a t i o n vectors the coordinate dipole system in question. Denote the of the l a t t i c e a , a , ( with i t s z a . where 7T > J equation (4.2), notation f o r the three the e l e c t r i c d i p o l e a t r-+ i s f i e l d at z i n t e g e r s n,, n , t the origin Then 3 d i p o l e s a r e l o c a t e d at p o s i t i o n s r- =n, a*. +n _a* .+n,a 3, is abbreviated due n . due 3 From to the 72 r where it has 7? been The net e l e c t r i c V- 5 assumed that not a l l of n ,n ,n are zero. f i e l d a t the o r i g i n , Ep, i s obtained by summing the r i g h t hand s i d e of equation (5.1) over a l l lattice sites, e x c l u d i n g the one at the o r i g i n E p must s u i t a b l e choice direction. dipole be i n the z - d i r e c t i o n . of lattice. Recall T h i s can be achieved by a that T h e r e f o r e , a c c o r d i n g t o equation p is in the £- (5.2), i f f o r every l o c a t e d at ( x , y , z ) , there i s one a t ( x , y , - z ) , the x- and 5> y- components of Ep w i l l vanish, where and (5.4) z-g i s the z-component of rV. Denoting by d the l a t t i c e constant, equation (5.4) may be w r i t t e n as *~~~~Jl (5.5) 73 where the dimensionless sum S i s 2 „ 3?' 3 z / - r * Equation 1 (5.6) (5.5) i s the r e l a t i o n which g i v e s E (2.6) SOLUTIONS FOR p AND Equations both p and Ep. (4.11) i n terms of p. E and (5.5) may be combined to s o l v e f o r The r e s u l t s a r e : (6.1 ) (6.2) Using equations out in f u l l as (4.12) and (4.13), equation (6.1) may be w r i t t e n 74 (6.3) Equation in (6.3) i s a very important the next chapter, the average electric r e s u l t , f o r , as w i l l be i t enters d i r e c t l y f i e l d resulting seen i n t o the e x p r e s s i o n f o r from the l a t t i c e of i o n s . • (2.7) THE AVERAGE ELECTRIC FIELD DUE TO THE LATTICE OF IONS Far away from a model i o n , the e l e c t r i c ion at may be taken as that produced the s i t e of the nucleus ignored). field by an e l e c t r i c due to the d i p o l e placed (again, the monopole moment i s In order t o c a l c u l a t e the average electric being f i e l d due to the l a t t i c e of i o n s , however, i t i s necessary t o c o n s i d e r the electric ion, is f i e l d due to an i o n f o r p o s i t i o n s very and even f o r p o s i t i o n s i n s i d e of the i o n ! not at a l l obvious that the average w i l l be given i n terms of j u s t proof begin with, Keeping express Let lattice. f i e l d c r e a t e d by the ion f i e l d due t o ideas, the f o l l o w i n g method Consider the volume As such, a to decide how to go about the lattice i n mind that one of the key purposes denote that i s indeed so. i t i s necessary c a l c u l a t i n g the average to Consequently, i t i t s d i p o l e moment. i s now presented that t h i s To close of of dipoles. of the model i s to i s adopted: a primitive cell a s p h e r i c a l region with volume V t of the , centered 75 on the middle of one of the s h e l l s , as i n f i g u r e 8. Call this f i e l d due to the l a t t i c e of ions shall region Figure 8. The average e l e c t r i c be taken to The S p h e r i c a l Region 2 be the average e l e c t r i c field i n s i d e of £ that i s due to the i o n s . The e l e c t r i c the r^ i n the region, due to nucleus and e l e c t r o n i n s i d e the s h e l l which l i e s i n £ j , i s where the explicitly. the f i e l d at the p o i n t contribution due to the monopole moment i s i n c l u d e d A few simple steps show that the average of t h i s i n region S i s 76 (7.2) The i n t e g r a l over r * i n e q u a t i o n the e l e c t r i c (7.2) may f i e l d at a p o i n t r i n s i d e of R :(3U/HTI) with £ a uniform symmetry, t h i s e l e c t r i c charge a be r e c o g n i z e d as sphere d e n s i t y <{e /y . c f i e l d must p o i n t radially of By radius spherical outward and depend only on r : Uc \r-r\ l Using Gauss' law, [ ( r ) : k - — \ 3 r and so (7.3) Recalling realizing equation (4.4), that i £ = 1 / n , where n u n i t volume, equation 5 5 the i s the (7.3) becomes expression number of f o r "p*, and shells per 77 Equation nucleus that (7.4) g i v e s the average e l e c t r i c and electron in a shell, field, due to the i n a sphere of volume 7j2=l/ns i s c o n c e n t r i c with the s h e l l . It i s easy to see that equation result f o r any charge the steps general d i s t r i b u t i o n whose d i p o l e moment i s p^ where p"* i s given by equation retrace (7.4) i s i n f a c t a (4.1). A l l that from equation i s required (7.1) t o equation i s to (7.4) f o r a charge d i s t r i b u t i o n with d e n s i t y ^ ( r ) . The electric f i e l d given by equation (7.4) i s not q u i t e the average f i e l d due t o the l a t t i c e of d i p o l e s ; the contribution by the ion i n s i d e the r e g i o n . i s the c o n t r i b u t i o n by the i o n s , or region, It as of the electric electrostatics field changes e x t e r i o r t o the r e g i o n , due to region. 2, due those exterior Accordingly, of an electric dipole only Required still exterior to the that i s equal t o the field. density 6^, i s given by field charges at the center of the s p h e r i c a l i n the s p h e r i c a l region of the f i e l d of the sphere, and t h i s i s j u s t Ep. boundary-dipole c o n t r i b u t i o n Eb.d. i s surface average electric t o the d i p o l e s e x t e r i o r t o i t , i s the value effective the over a s p h e r i c a l r e g i o n , due to the average f i e l d they c r e a t e at the center The dipoles, includes w e l l as the so c a l l e d boundary-dipole c o n t r i b u t i o n . i s a w e l l known r e s u l t of value it charge The cause of distribution d e n s i t y 6 \> is the the to contribution the ambient termination of the P(r*) on the s u r f a c e of the metal. 78 —-> where r t l o c a t e s a p o i n t on t h e m e t a l s u r f a c e and r? i s t h e outward normal. metal, E L J where s k < c o n t r i b u t i o n s from t h e s i d e s of t h e i s g i v e n by r? i s t h e E Ignoring unit average E t a w number d e n s i t y ^ and E p If^*,^ +Ep + shells involve In c o n c l u s i o n , t h e average f i e l d dipoles i s just of Ti^. , <» due in n , and all s to the the coi{rlk«tc lattice of ;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 d e t e r m i n e d . (2.8) LATTICE TYPES R e c a l l t h e r e q u i r e m e n t t h a t each every lattice site lattice must meet:" f o r a t ( x , y , z ) , t h e r e must be one a t ( x , y , - z ) . Some l a t t i c e s a r e now d e s c r i b e d which meet t h i s r e q u i r e m e n t . 79 (A) Cubic The dz. Lattice primitive translation The c u b i c l a t t i c e v e c t o r s i n t h i s case are dx, d£, i s a very s p e c i a l case, f o r the f i e l d E^, given by equation (5.2), i s zero, r e g a r d l e s s of the o r i e n t a t i o n of p. To see t h i s , c o n s i d e r the x-component of Ept where r?f=(n, x + i \ £ + Summing over _z)d, and the the c r o s s terms n, n sum t excludes and n,n 3 n,=n .=n =0. 2 3 w i l l give zero. A l s o , by symmetry, and i n consequence, the e n t i r e sum vanishes i d e n t i c a l l y . For a cubic l a t t i c e , E =0 and S i s zero i n equation p (B) Face-centered Consider figure Cubic (7.1). Lattice a face-centered cubic structure, 9, such that the p r i m i t i v e oriented t r a n s l a t i o n v e c t o r s are as i n 80 F i g u r e 9. The Face-centered Cubic come i n p a i r s with the coordinates d i f f e r i n g It turns out t h i s , break up the Consider first same x- and y- Lattice coordinates, but z- in sign. that sum Ep over for this lattice lattice sites i s zero. into two the sum over the l a t t i c e s i t e s l o c a t e d at To see parts. 81 where n, , n, n z 3 assume a l l i n t e r g e r values except T h i s g i v e s Ep f o r a c u b i c l a t t i c e i s t h e r e f o r e zero. lattice sites. Now c o n s i d e r Each of with l a t t i c e the sum over these sites n,=n =n =0. constant the falls rest z 3 J ? d , and of the i n t o one of the f o l l o w i n g three s e t s : I n each case, n i , n , n 7 n^=n =n =0. 2 3 be variables. assume a l l integer values, including I t i s easy t o show that the net c o n t r i b u t i o n t o E p by the d i p o l e s at these only 3 lattice sites aware that the n , n , n 1 2 3 i s also zero: one need i n the sums are simply dummy 82 (C) C l o s e s t Packing; Type 1 In t h i s l a t t i c e , of closest direction. packed This the s h e l l s are arranged such that a shells, is layer i n the x-y plane repeats i n the z- experessed precisely by the primitive translation vectors (8.2) (D) C l o s e s t Packing; Type 2 This case d i f f e r s from the p r e v i o u s one only i n that a^ i s dif ferent: a, = A X (8.3) T h i s l a t t i c e may be thought of as hexagonal c l o s e s t packing with 83 every second layer missing. (E) Hexagonal C l o s e s t Packed T h i s i s not a l a t t i c e , but basis. Even so, i t meets the requirement (x,y,-z) f o r one at ( x , y , z ) . Having considered, i s a l a t t i c e with outlined the evaluation The b a s i s types of S of for a point of having a s h e l l at vectors shell each two "b~t = 0 are and arrangements to be type shall now be discussed. Before, l e a v i n g this arrangement are presented. n by a"^ • (a\xaa ) . 5 section, s the i s given by The values of n 5 v a l u e s of n 1/^ and 7A s f o r each is given are: (A) Cubic L a t t i c e : n d =1 (B) Face-centered Cubic L a t t i c e : (C) C l o s e s t Packing; Type 1: n d = //f (D) C l o s e s t Packing; Type 2 n d = //^ (E) Hexagonal C l o s e s t Packed: n d = /2? s nd 5 3 z 1 s 3 f 3 3 s (8.4) 84 (2.9) EWALD The shells SUM D<f) obvious i s to way to e v a l u a t e S f o r a given arrangement of insert r =n, a, p? i n t o equation however, (5.6). because + n a 2 2 + T h i s i s not a the r e s u l t a n t sum n,a 5 very succesful procedure, i s very s l o w l y converging, and many terms need to be added up together t o o b t a i n reasonably good accuracy proceed in i n the f i n a l this for S i s r e q u i r e d . manner. It i s result. It is not practical to A more r a p i d l y converging expression the objective of this section to provide such an e x p r e s s i o n . The r 1 e l e c t r o s t a t i c p o t e n t i a l $ (r) at ~v due t o a d i p o l e p at is h?)=t^£)v i where i t has been assumed i n w r i t i n g equation The e l e c t r i c f i e l d E ( r ) at r due from $ ( r ) by It i s easy to v e r i f y that (9.,) (9.1) that "r/r*. to the d i p o l e a t r ' i s obtained 85 L Since p=pz, interest, p-V=p^z. so c o n s i d e r for i s obtained Only the z component o f E ' C T ) i s o f only [Id Ep (9.3) \lr-rl - £(^)ic. (9.4) t by summing e q u a t i o n t h e one a t t h e o r i g i n . E p (9.4) over a l l d i p o l e s except is: (9.5) r =o — 7 the r ' i n equation (9.5) a r e the l a t t i c e s i t e s , the b^ are the basis sites associated w i t h each l a t t i c e p o i n t , and t h e prime means t o o m i t t h e t e r m a t ^'=0, bi=0. Def i n e (9.6) Note that The of S'(0)=S/d . 3 problem of d e t e r m i n i n g evaluating S' (r) at E P i s tantamount r-O". No d e l t a to the problem functions a r e hidden i n 86 equation of (9.6), s i n c e the domain of rounder —* —> the s i t e s r',b;.. S p e c i f i c a l l y , equation study excludes a l l (9.6) i s of i n t e r e s t —> only f o r r near the o r i g i n . Use 1 z = p oo (9.7) to w r i t e S ( v ) a s f (9.8) (It is easy to i n s i d e the i n t e g r a l The parts: next from verify tydz may be taken 1 sign.) step i s to zero t h a t the operator break the integration to G, and from G to i n f i n i t y , f i n i t e p o s i t i v e number. SCf)»L % up into two where G i s any The e x p r e s s i o n f o r S ( r ) then becomes l a +0 P f.e 1 ap (9-9) At t h i s p o i n t , the term i s added and s u b t r a c t e d from equation (9.9). doing t h i s i s that the f i r s t then be p e r i o d i c , with the sum w i l l The advantage of 87 periodicity of the l a t t i c e , a p r o p e r t y which w i l l be e x p l o i t e d shortly. It i s easy to show t h a t 4C 3 2 (9.10) Now d e f i n e (9.11) where the term periodicity equation Now equation for r'=0, of the l a t t i c e . is included. Using equations F( ) has the (9.10) and (9.11), (9.9) may be w r i t t e n a s : e v a l u a t e the middle term on the right-hand side of (9.12): rCO b -ifV-qiy z Doing b =0 the f i r s t integral roo here by p a r t s s i m p l i f i e s the right-hand 88 s i d e to Doing the remaining allows i n t e g r a l by p a r t s now that t h i s be expressed gives as (9.14) 89 Only F(r) still needs to be found. 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 noted previously, 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 p r i m i t i v e c e l l and is carried out over a p r i m i t i v e c e l l . p e r i o d i c i t y of the l a t t i c e , vectors, the integration Note that e ^ L by d e f i n i t i o n of r e c i p r o c a l and so equation (9.16) may be w r i t t e n -1 has the lattice as -~> (9.17) Mc where \C =NiT Equation t and the i n t e g r a t i o n goes over N p r i m i t i v e (9.17) s h a l l be used in the limit N-*-, as cells. is now 90 demonstrated. and Inserting equation (9.11) i n t o equation simplifying: .1 £. r Now, l e t N ^ - i n such a manner as t o cover a l l of r-space. the (9.17), r* i n t e g r a l Then i s the same f o r every "r', reducing the e x p r e s s i o n above t o (9.18) where the r i n t e g r a l ? It and i s over a l l of r-space now. i s not d i f f i c u l t t o show t h a t , f o r g^O, i n t u r n that thereby reducing equation (9.18) t o (9.19) 91 The i n t e g r a l over (9 i s e a s i l y e v a l u a t e d , with the r e s u l t T h i s s i m p l i f i e s equation (9.19) t o : h---^3l/M zy-^ Equation (9.20) a p p l i e s f o r g^O. (9.20) To f i n i s h i t i s necessary to f i n d F~o. Equation use w i l l at be made of the f o l l o w i n g i n f o r m a t i o n : the s i t e s pCr')=pe ^ E(r) The to reciprocal first a r Instead, i f the d i p o l e s i n a l a t t i c e vary i n d i r e c t i o n a c c o r d i n g t o , where p and q a r e c o n s t a n t s , then the g^o" term L —> -> the (9.18) w i l l not be used t o e v a l u a t e F^. (3) l a t t i c e sum makes the f o l l o w i n g c o n t r i b u t i o n t o term here macroscopic contribution of i s the macroscopic polarization i s simply wave electric P{r)=+? field E(f) ai r due e^'^ , so the 92 — > —> As q goes to o, the becomes the average second electric d i s t r i b u t i o n P= ^jr , and Thus, F^s'-^for term vanishes. f i e l d due The to a uniform first term polarization i s -^^"pk. a lattice. Taking i n t o account the b a s i s sites, (9.21 ) where Nj^ (9.20), where i s the number of (9.21), (9.5) and basis sites. (9.4), equation only the r e a l part of equation Combining equations (9.12) becomes (9.20) matters physically. —> R e c a l l that Ep i s given i n terms of S'(0) As a check of equation that Ep (9.22), i t by has already i s zero f o r a c u b i c l a t t i c e : does equation been seen (9.22) g i v e 93 the value 0 f o r a c u b i c The primitive reciprocal lattice? lattice lattice is vectors are dx, d£, dz: a l s o c u b i c , with l a t t i c e constant 2"-/d. N o t i c e t h a t , i n consequence of t h i s , the sum first are very s i m i l a r . term independant in the sum of G, and over the r 1 the convenient over the ~q and t h i n g to do so the power of e i s the same i n both sums. take G= JV/d. first term of the sum With t h i s c h o i c e of G, / Vc=d . Moreover, over the the terms r' the sum cancel iC, to the 3 z , 2 - r ' , f o r the same 2 vanishes when e v a l u a t e d f o r a c u b i c The The error function of the i n equation one would equation partial require the reason sum. achieve this, another, the since This a l s o vanishes, that equation (5.4) lattice. exponentials (9.22) render summation (9.22) to o b t a i n the same is and - 4 ^ / 3 ^ add to zero, of test. and the complimentary i t a much more converging e x p r e s s i o n than the e x p r e s s i o n (5.6) (5.6) S'(0) over the ~g and e x p r e s s i o n (9.22) t h e r e f o r e passes the presence the i s to choose G, To l e a v i n g only the other terms i n the r ' sum. due the many accuracy in f o r S. quickly Equation more terms than the resultant 94 (2.10) EVALUATION OF THE EWALD SUM AND CALCULATION OF E Equation (9.22) has been dlonles e v a l u a t e d f o r the three s h e l l arrangements, (C) to ( E ) , d e s c r i b e d i n chapter 8. case, the formula studied. A is particularized generalized to the similar. out to be so. As a f u r t h e r for lattice check, (B). e f f e c t i v e l y zero, independent cases The e v a l u a t i o n i n In each case, the was analyzed f o r s e v e r a l v a l u e s of G, t o ensure that the f i n a l answer would be independent evaluated being computer program was w r i t t e n , and the ( C ) , (D) and (E) was very expression In each arrangement formula e v a l u a t e d f o r each p a t t e r n of s h e l l s . cases latter As of G. of G. This in fact equation (9.22) required, the turned was result also was. The values of S o b t a i n e d i n (C) t o (E) were as f o l l o w s : (C) C l o s e s t Packing; Type 1: S=-0.9095420544 (D) C l o s e s t Packing; Type 2: S=-5.105810840 (E) Hexagonal C l o s e s t Packed: S=2.968683281 The number of significant figures shown (10.1) indicates the extent t o which the r e s u l t d i d not vary with G. E d/poks ' given by equation (7.5), w i l l now be c a l c u l a t e d for each l a t t i c e and f o r s e v e r a l values of Z* i n each case. To begin with, e x p r e s s i o n s f o r ^ E T , and using £ E L F equation (6.3) i t i s found that for p and the 95 ^ (MfZ* -loMooiV+lJ+o.wxlS (10 * 2> Write Then, using equations (8.4) f o l l o w i n g v a l u e s a r e found Lattice: v f l u e or c * : ( for n d and (10.1) f o r S, the S s f o r oc,, and c < : <x Zl 3 (A) (B) (C) (D) \L7o ZZ.il 20-13 /CfO T a b l e I : V a l u e of oc, f o r l a t t i c e types (E) 2JDX6 (A) through ( E ) . oc, does not depend on Z„. Value of 1 Lattice: (A) (B) (C) J2.W It.11 IS.SC Z72.6 2 3 ¥Z7Z 4 Table I I : (D) I3n 3/8.S UC.7 U3/ I3C5 ¥313 (E) ff.93 5Z% V a l u e s of f o r l a t t i c e s (A) through (E) f o r some v a l u e s of 2^. 96 <X 3 lattices i s zero f o r a l l Z„ f o r both l a t t i c e s (A) and (B). For (C) through ( E ) : Value of Lattice: Zft: (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 <=K for l a t t i c e s 3 for (C) through (E) some v a l u e s of Z^ . The o b j e c t i v e of Chapter 2 has been fulfilled. It i s indeed easy to see that £/(kc^/R ) i s small. Take R to be one of the order of a , the Bohr r a d i u s , and E P to be of order Mg/q Then £ / ( ! < < £ / / ) i s of the order 10" 2, e 16 97 I.S. Gradshteyn and I.M. Ryzhik, Table of I n t e g r a l s , S e r i e s , and Products, 4th ed., Alan J e f f r e y (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 S o l i d s , 2nd ed. (Cambridge U n i v e r s i t y Press, Cambridge, 1972) pp. 41-42. 98 CHAPTER 3 THE CONDUCTION ELECTRONS PRELIMINARY DISCUSSION In the l a s t chapter, a model of the ions, presented. in a metal was T h i s model was used t o c a l c u l a t e the c o n t r i b u t i o n to — > E MC of this chapter, the g r a v i t a t i o n a l l y induced i o n i c d i p o l e moments. a constructed. model of The o b j e c t i v e the height. The reason electrons this of the e l e c t r o n s calculation as is the interior of the metal. interior i s not assumed i n t h i s density inside determined by electrons as found using the goal metal comparing functions elasticity basic that distribution thesis. Instead, number of h e i g h t . the charge The charge d e n s i t y i s densities The d e n s i t y of and of ions may be theory; f i n d i n g the e l e c t r o n Once the charge d e n s i t y ions density is i s known, i t s E ch to ~~Eave may be c a l c u l a t e d . physical electron constituent manner the function Charge n e u t r a l i t y i n the metal i s calculated. of t h i s c h a p t e r . contribution The a a interesting i s because i t leads t o an i n v e s t i g a t i o n of the charge in is of t h i s chapter i s t o use the model to c a l c u l a t e the number d e n s i t y of conduction In the idea u n d e r l y i n g the model i s that the must be d i s t r i b u t e d i n the metal i n such electron gas is held up a g a i n s t B a s i c a l l y , two p h y s i c a l processes are i n v o l v e d a gravity. i n c o u n t e r i n g the 99 f o r c e of g r a v i t y on the e l e c t r o n s . force experienced by the e l e c t r o n s . nature of the e l e c t r o n s , electron One f o r i f there number d e n s i s t y , i s the e l e c t r o s t a t i c The other i s the fermion i s a nonuniformity i n the there w i l l be a gradient i n the l o c a l pressure i n the gas. The manner incorporated constituent in into which these physical concepts the model i s now d e s c r i b e d . The e l e c t r o n i c of a metal i s t r e a t e d as a system of fermions t h a t i s subject t o an e x t e r n a l are noninteracting linear potential. It i s assumed that t h i s l i n e a r p o t e n t i a l i s i n the v e r t i c a l d i r e c t i o n . The linear potential takes i n t o account the e l e c t r o n - e l e c t r o n i n t e r a c t i o n s , the i n t e r a c t i o n s between the electrons and the l a t t i c e of i o n s , the weight of the e l e c t r o n s , and the e x t e r n a l l y imposed e l e c t r i c f i e l d E .^. ex electron-lattice interactions averaging them out. be subject to Electron-electron are taken into S p e c f i c a l l y , each e l e c t r o n an average p o t e n t i a l c r e a t e d i n t e r a c t i o n s and account by i s regarded to by the r e s t of the —> electrons, The the l a t t i c e of ions, g r a v i t y , and slope the field E e x of t h i s p o t e n t i a l i s mg+q&(E ave E £ ) , where E ^ e + e x p o s i t i v e of similarly points for E t« e x takes i n t o account the i n the d i r e c t i o n It i s the E a i / e a opposite to is g, and term i n t h i s slope electron-electron t. that and e l e c t r o n - l a t t i c e interactions. Notice again, As assumption that E ave i s a uniform f i e l d . Once 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 f o r convenience. . wall height. the be seen l a t e r in this thesis, E M i n f a c t depends on However, t h i s dependence i s extremely small. Taking 100 E ave as uniform, therefore, is r e s u l t s with which the assumption an assumption that leads i s c o n s i s t e n t : the to assumption — > and the results calculated. The c o n s i s t e n t to the order imbalance i s very small. i n t h i s model, although not This point s h a l l be further identically clarified i s a very simple way to o b t a i n an expression number d e n s i t y of e l e c t r o n s as a f u n c t i o n of as is a v / a the zero, at the for the stage i n t h i s t h e s i s . There approach, to which E p h y s i c a l reason f o r t h i s agreement i s that charge appropriate are height. i t i s assumed that the e l e c t r o n gas i f i t were a f r e e gas. The In behaves, c a l c u l a t i o n i s done in this locally, Appendix B. A more mathematically r i g o r o u s a better understanding and treatment, however, leads appreciation of the problem. a d d i t i o n , some i n t e r e s t i n g r e s u l t s are obtained along the to In path to the s o l u t i o n . In specified solving the next section i n more d e t a i l . of the problem and gravitationally of this chapter, the Subsequent sections deal problem i s with the the ensuing i m p l i c a t i o n s i n terms of induced e l e c t r i c fields in metals. 101 (3.1) DESCRIPTION OF THE PROBLEM OF A FREE FERMI GAS SUBJECT TO AN EXTERNAL LINEAR POTENTIAL The system noninteracting and subject particles be studied fermions, to do composite to an confined external not consists potential. the Hamiltonian M A A A. identical, A Because Hamiltonians: A. H (I.£)=SH;(f t) • Nl of N (...) ;> The c a r e t denotes an o p e r a t o r , while the t i l d e collection operators. The is short t h a t there are N fermions and they a r e c o n f i n e d cube edge length L. p^ for a s u b s c r i p t s N and L serve as reminders of the of the N p a r t i c l e i s simply the sum of N s i n g l e p a r t i c l e A N t o a cube of edge l e n g t h L, linear interact, of and to a r^. a r e , r e s p e c t i v e l y , the momentum and p o s i t i o n o p e r a t o r s of the i fermion. The single p a r t i c l e Hamiltonian i s : A where m A i s the mass of each fermion and ^ ( r ^ ) i s the e x t e r n a l p o t e n t i a l experienced by each. Because of equation (1.1) the eigenfunctions and energy A eigenvalues of H , w L can be expressed p a r t i c l e e i g e n f u n c t i o n s and energy i n terms of the s i n g l e 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 multilabel which mark well includes the the spin meaning of k. k label. Later is a in this —> s e c t i o n , a l a b e l k s h a l l be used. The l a b e l k i s short f o r the composite of l a b e l s k and the s p i n l a b e l . The (1.3) H° i n equation may be thought (1.2). r e p r e s e n t a t i o n of equation For of as the p o s i t i o n the case of a linear p o t e n t i a l and a b s o l u t e l y c o n f i n i n g w a l l s , "A *(c?)-< The eigenf u n c t i o n s terms of the f;\h numbers n . K Thus, e i g e n v a l u e s E^ denoted "t^^ ir). of C7 } 'A-S'^ and eigenvalues and er the iW y (1.4) cJae. through eigenstate of H* the J £ lt are expressed i n use (r) of belonging t o the may be l a b e l l e d by the n : Similarly, E K i occupation ^JV^r*) is s denoted E ^ j , and i s given 103 by [ T-Z^t - { (1 5) The e x p r e s s i o n f o r "~p ( r ) i s more c o m p l i c a t e d . The be of important t h i n g t o be aware of here i s that V^^r,) may thought of as a s t a t e of the N p a r t i c l e s particles i n the s i n g l e p a r t i c l e s t a t e (1.5) supports t h i s p o i n t of view. i n which the number % ( r ) i s n ; equation k So does the requirement (1.6) The bulk p r o p e r t i e s of the Fermi gas at a temperature T may be determined by employing the r u l e s of s t a t i s t i c a l One piece of occupancy information of the s t a t e % o b t a i n e d i n t h i s way i s the average ( r ) , denoted about <n > . i n the next s e c t i o n . )t thought 1 of as the number mechanics. <n >L. K More w i l l For the moment, <n >L K of p a r t i c l e s e x i s t i n g These average by this o c c u p a t i o n numbers <nyt>L p l a y a key r o l e i n n^Cr). complicated e x p r e s s i o n equat ion: that < n K > L . determining the value of the l o c a l number d e n s i t y denoted may be i n the s t a t e ^""'(r) a t temperature T, although i t should be noted i s only a c o a r s e , i n t u i t i v e way t o think of be s a i d I t i s possible of fermions, t o proceed for i ^ n ^ (r) to derive from the the f o l l o w i n g 104 (1.7) Ic Equation (1.7), however, i s clear on intuitive grounds, by t h i n k i n g of <n >u as being the number of fermions i n the K It local x i s reasonable to expect t h a t , deep i n s i d e the metal, the number d e n s i t y of f r e e e l e c t r o n s w i l l be independent and y. This fact emerges from equation L->°°in such a way remains that, constant. that the average I n t u i t i v e l y , therefore, i n the thermodynamic l i m i t , of L, (1.7) i n a mathematically p r e c i s e manner by t a k i n g the thermodynamic letting state particle limit: density i t i s t o be expected equation (1.7) w i l l go over to (1.8) where - u s z / L , ~r'= r^/L, the limit of n(u) n(u) i s the l i m i t of r\ ("r) i s a temperature may be determined <nit>. and performing the i n d i c a t e d summation. may then be determined section. of determining finding to equation the <n*> and the The r e s u l t for f o r the model by t a k i n g T=0; r e c a l l that the model i s f o r the metal method by According / (r ), n(n) is dependent f u n c t i o n , the temperature n(u) ? / > / } (1.8), K (r ) 1 ^(r). dependence e n t e r i n g through the , and U) n(n) in i t s ground state. This i s the one presented i n the next 105 An a l t e r n a t e method, the further simplifies potential method i s presented specification the expression for n(*tt). The this which external i s t r e a t e d as a p e r t u r b a t i o n , and the ^(r*) are found i n the p e r t u r b a t i v e A In T=0 i s made at the beginning, by using p e r t u r b a t i v e techniques. order thereafter. comparison n(-u) i s evaluated to lowest parameter. 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, considers a paper appeared that form. exceed u n i t y . 8 is The f u g a c i t y i s e i s Boltzmann's constant, the chemical Study which of this to a paper that the f u g a c i t y does It may be made. f u g a c i t y , the r e s u l t s obtained is temperature, and^a for 0<e^<l For this a r e as f o l l o w s : grand c a n o n i c a l pressure not , where T i s the absolute potential. aforementioned extension The literature i t i s very easy t o extend the r e s u l t s of the paper to the case of a Fermi gas, provided k the the problem of a n o n i n t e r a c t i n g Bose gas s u b j e c t weak e x t e r n a l p o t e n t i a l of power reveals in p i s given by range that the of the 106 (2.2) where (2.3) (2.4) (2.5) where c i s the slope 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 , and ^ i s given i n terms of the average p a r t i c l e d e n s i t y £ by (2.6) where 2 /V/TT-JO y~ * -M l z e (2.7) 107 The l o c a l p r e s s u r e and l o c a l number d e n s i t y of fermions are given by (2.8) and where IC =Z. (2.10) • " L' The ^ i n the above equations i s i n f a c t the f u g a c i t y ; =e Equation / S / * (2.6) e s t a b l i s h e s a one-to-one (2.11) correspondence between p and i~ , whence equations (2.2) and (2.6) determine a unique plot of p i n terms of ^>, f o r a given temperature. \ i s c a l l e d the thermal wavelength. in p o t e n t i a l energy between the "bottom" and the "top" , atoc = 1 . c i s the d i f f e r e n c e of the cube, at 11=0, 108 Notice that equations number d e n s i t y f o r a f r e e (2.8) a n d Fermi (2.9) a r e the p r e s s u r e gas with chemical and potential fx-cu. The appeal of the above i n v e s t i g a t i o n a t hand i s t h i s : to be true electrons for a l l^ i n a metal , results i f these n('u) in terms results could of be the shown c o u l d be o b t a i n e d f o r t h e s i m p l y by e v a l u a t i n g ( 2 . 9 ) i n the free limit (T-»0). In the remainder d e s c r i b i n g how n(n), are a of t h i s s e c t i o n , an outline is presented t h e a b o v e 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 discussion i s presented indeed v a l i d f o r a l l ^, and to i n d i c a t e that these and results the e x p r e s s i o n s are e v a l u a t e d i n t h e l i m i t fl-* *. 0 (A) R e v i e w o f Some B a s i c S t a t i s t i c a l The work t o be done h e r e c a n o n i c a l ensemble. where ft = l / k T , B The and Q(/?,N,V) be done Ideas using canonical partition T i s the temperature volume of t h e system,yU bath grand shall Mechanical of the heat i s the chemical p o t e n t i a l i s the p a r t i t i o n function the grand function bath, V of the is i s the heat f o r the c a n o n i c a l 109 ensemble: QfaWhZ The sum i n equation satisfying (1.6) and W . { 2 occupation number noninteracting, Using equation - 1 3 ) sets E ^ ^ i s as d e f i n e d p r e v i o u s l y . (2.12) and (2.13), and using equation Because the fermions are may be s i m p l i f i e d . p (2.13) i s over a l l equation Combining equations e equation (1.5), (2.13) (1.6), (2.15) It i s easy t o v e r i f y that the two summations i n equation may be r e p l a c e d , e q u i v a l e n t l y , by summing each n This gives: k (2.15) independently. 110 r S2Z ' n ie « r>» W .i may £ - 0 L assume only the values exclusion p r i n c i p l e . J 0 Equation and 1 because of the Pauli (2.15) has become (2.16) Equation (2.16) nomnteracting is the grand c a n o n i c a l p a r t i t i o n function for fermions c o n f i n e d to a volume L ; the c k r e f e r to a cube of edge l e n g t h L. The grand c a n o n i c a l 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 f ^ft-^. <N> = L z "*° )z /3z 4« i m or (2.19) I n s e r t i n g equation (2.16) i n t o equation ^>L--2 e / (^- u) 5 / + l (2.19) g i v e s ' ( 2 - 2 0 > in terms of which the average p a r t i c l e number d e n s i t y i s Q 1 pis to not a f u n t i o n of L. infinity, z IM^, L . In the thermodynamic (2.21) l i m i t , as L so does <N> , i n such a way that (p s t a y s c o n s t a n t . L The ensemble average occupation numbers are given by or goes 1 12 Use of equation (2.16) once again g i v e s * ' L " /SC^-/*) e A comparison of equations <M> L -T that thermodynamic n • ( 2 c o n s i s t s of l e t t i n g By equations equation p {^^) M (2.20) and totju , o r ^ = e ^ , iy p denotes the l i m i t of (2.23) (2.20) and (2.23) shows that < k\ limit remains c o n s t a n t . g i v e s an i m p l i c i t |' - 2 4 ) k 1 The + — — ; — Ir* 00 but such (2.21), i n terms of this :' (2.25) 113 Equations (2.25) and given temperature The limit (2.26) together g i v e p i n terms of p , a t a T. of equation (2.23), with given (2.25), g i v e s the <n > which enter i n t o equation to Case of no E x t e r n a l P o t e n t i a l ; Review of the I d e a l Fermi Gas^ When there 2) is no e x t e r n a l p o t e n t i a l , the s i n g l e e i g e n v a l u e s are "pV2m, where ^=2n^h*/L andln components over k may are equation (1.8). K (B) S p e c i a l i z a t i o n by integers. As such, is a particle vector whose i n the l i m i t L-*-°, the sums be r e p l a c e d by i n t e g r a l s over p, as i n 1 where s i s the spin of the fermion In this case, the (s=xfor expression for electrons). p, equation (2.26), becomes _(2+l)<frr 5 and the equation A f o r p, /-ICO (2.27) J equation (2.25), g i v e s fe,S*l)fn- (2.28) 114 where i s the f u g a c i t y , given by equation Equations (2.27) and (2.28) may (2.12). be w r i t t e n as and P= with A given by J \ ( ^ ) by equation I f power ' (2.3) r ^ M f ) equation (2.30) by equation (2.4) and (2.7). 0<^*<l , these i n t e g r a l series fj/^S) ~ J expansions. expressions may be I t i s easy to do t h i s . written The as results are: •z— :5lZ~ d' , V ^ N ' (2.31) u 1 and - l!L_r ( V • (2.32) it, N o t i c e that (2.33) 115 T h i s i s i n f a c t true f o r a l l ^ , show. Also notice the as source equations of the (2.4) and (2.7) symbol to denote the i n t e g r a l e x p r e s s i o n s (2.4) and (2.7): the s u b s c r i p t i s the power of 1/j i n the sum when 0<^<1. Equations equations (2.31) (2.29) and and (2.32) (2.30) equations (2.30) and (2.32), in are useful for expanding the l i m i t £ > A « i . may be expressed in F o r , from powers of ^>A , with the r e s u l t Zjtl 5 (2.34) VHz^l/ Use of equations (2.31) and (2.34) i n equation (2.29) then g i v e s h.C/| 4 + Observe temperature required, that ( > A « i 3 and/or order, the c l a s s i c a l > t corresponds low d e n s i t y therefore, p £ equation limit physically of (2.35) i d e a l gas law. (2.35, the Fermi reproduces, This result t o the high gas. As to l e a d i n g i s due to the 3 fact that ^A<<1 means that the average p a r t i c l e s e p a r a t i o n i s much l a r g e r than the thermal wavelength, so quantum e f f e c t s small. (2.35) to the c l a s s i c a l The corrections in equation i d e a l gas law are due to p r e c i s e l y those quantum e f f e c t s . are 1 16 Equation (2.34) may a l s o be i n s e r t e d i n t o equation (2.23), with the r e s u l t that (2.36) to l e a d i n g o r d e r . Equation (2.36) i s j u s t the Maxwell-Boltzmann distribution function. The other extreme f o r which equations be approximated is limit, £A>>1. expansion In the this case, temperature it is and/or high d e n s i t y necessary to find an f or-p3 (^) as /2 Such an expansion (2.7). low (2.29) and (2.30) may may be obtained by s t a r t i n g with equation One o b t a i n s , as k^m^rw -foe?-) 3 6 - r r Z , „ 1 Similarily, Equations (2.30) and (2.37) g i v e , to lowest order, (2.37) 117 or, u s i n g equation (2.3) f o r A , and equation /^2~U*T/ Expanding i n powers of k T/£ B F ^ P ^ f , equations (2.11) f o r ^ , - * o ° - A (2.39) (2.37) and (2.30) give (2.40) <n^> i s given by ^i>~' P>(^/^ju-) \ ' e w i t h , " given by equation <n The expansion (2.39), and ?/ (2.40). 1 ^ , (2.41) Thus, r w v { *. \+0 f o r p i s o b t a i n e d using equations • (2.29), (2.42) (2.39), (2.40): (2.43) These r e s u l t s 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 T h i s i s because wavelength J^(^) i s clear are limit which s e p a r a t i o n , and It the is ^>A>>1 corresponds 5 large compared so quantum e f f e c t s are very from equations monotonic (2.4) increasing the relationships are and for smooth, thermal particle important. (2.7) a a average functions. monotonic i n c r e a s i n g function of ^ , Moreover, to to thatj^/i. ( p and As such, p i s a given temperature. and as a r e s u l t the i d e a l Fermi gas e x h i b i t s no phase t r a n s i t i o n s . (C) Case of an E x t e r n a l L i n e a r P o t e n t i a l Consider with H i n more d e t a i l now given by equation normalized (1.4), as usual a c c o r d i n g / the Schrodinger and with equation the (1.3), eigenfunctions to |f<wf<l'r (2.44) Cube That ^ L . it) i s i n f i n i t e o u t s i d e the cube means ^ ^ ( r ) = 0 o u t s i d e the cube, and ) f u r t h e r that (r) v a n i s h on the > walls of the cube. It given is straightforward to see that the e i g e n f u n c t i o n s are by (2.45) 119 where (2.46) and the eigenvalues £ ^ are given by (2.47) The l a b e l k denotes the t r i p l e »VtT (k ,k ,k ), \, where n = 1 , 2, 3 ,... , n = 1,2,3,... y y x y z with (VTT (2.48) J^ ^(z) K must satisfy the normalization condition (2.49) as w e l l as the boundary c o n d i t i o n s 120 1<<0 The problem ( 2 . 5 0 ) may =£ specified by be s o l v e d e x a c t l y f o r J (2.50) (Q=0. equations K i (2.46), ( z ) and E . Kz ( 2 . 4 9 ) and Putting (2.51 ) and (2.52) c o n v e r t s the problem t o d Z _ Zw\c\} d ft U - (2.53) cL -1 (2.54) 121 (2.55) Equation (2.53) reveals that parameter i n the problem i s the 4 = - ^ ^ If &L though is an exact the solution in next large be e x a m i n e d is dimensionless . (2.56) 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 undertaken shall important possible. section. This theory, even approach is The e x t r e m e where 0 0 i s very below. Denoting \ K and ~ —2 •) (2.57) cL putting (2.58) 122 v=.<x 3(u- y j ; ,y (2.59) k expresses the problem as (2.60) V Z (2.61 ) O • The exact s o l u t i o n s functions and 10 ( b ) ^ to equation Aj_(v) and B^(v) d e p i c t e d (2.60) (2.62) a r e the two graphically in figures Airy 10(a) Hence, K Q^fi^W+t^g.CU) - (2.63) 123 (a) \ 1 •/» .it / / •a - / -Jo - / .% /.Ic 'A - .1 / \ / /-"? Wf /-7 1 1 1 \* V* '/ • \ 1 1 As -4 \-3 -Z 1 / /i 1 I s • '-•4 - Figure a* , z bk and 2 (2.62); }f* 2 10. The must be Airy chosen Functions -J, A (v) to satisfy 7 and By ( v ) equations (2.61) and thus: (2.64) 124 (2.65) and n' 2^ (2.66) Equations (2.64) and (2.65) combine to read (2.67) or By figure d i s c r e t e set of Equation ^lc . z two (2.67) 10(a), i t i s c l e a r values <& , z that equation where then g i v e s the r a t i o From there, equation (2.66) may labels ^/tfcz (2.68) f i x e s a those values. f o r that value of be s o l v e d f o r one c o n s t a n t s , whence the other i s a l s o known. of the In t h i s way the problem i s e x a c t l y s o l u b l e . U n f o r t u n a t e l y , the s o l u t i o n i s highly implicit. In fact, 125 to the author's knowledge, e x p r e s s i o n s a r e not a v a i l a b l e l i t e r a t u r e which g i v e the / K ' s s a t i s f y i n g equation (2.68). 2 What s h a l l be done, t h e r e f o r e , specified c*>>1. by equations study the problem be d e a l t with by u s i n g p e r t u r b a t i o n (2.53) - (2.55). The problem posed by equations been s t u d i e d by the author. which to (2.60) - (2.62) f o r the c a s e s ° < « 1 and The extreme c<<<i w i l l theory on equations is i n the Some (2.60) - (2.62) for°<>>1 has interesting results r e l a t e t o the case of a low d e n s i t y Fermi zero i n a g r a v i t a t i o n a l field. emerged gas at a b s o l u t e These r e s u l t s a r e presented in Appendix C. For notice the purposes of the present chapter, the key t h i n g to i s the emergence of the d i m e n s i o n l e s s parameter^. In a n a l y z i n g the e x p r e s s i o n s f o r Q and p, equations and (2.26), does t h i s What it L i s t o be taken imply for<=<? about c? to i n c r e a s e without Is o< to i n c r e a s e without Should c remain bound. What bound as well? constant as L i n c r e a s e s ? Or i s b e t t e r t o h o l d something e l s e f i x e d ? Van that den Berg and Lewis chose t o h o l d cL f i x e d as case c^- blows up with L l i k e L . cL f i x e d was t o ensure T h e i r reason L-»—. the system. In for holding t h a t the e f f e c t of the e x t e r n a l p o t e n t i a l would not be so extreme as to d e s t r o y the thermodynamic of (2.25) They desired this because behavior they were i n v e s t i g a t i n g the m o d i f i c a t i o n of B o s e - E i n s t e i n condensation due to the e x t e r n a l p o t e n t i a l . This as l r > o a — approach t o the Fermi gas problem — s h a l l be c o n s i d e r e d next. The steps h o l d i n g cL f i x e d to be outlined 126 f o l l o w those employed by Van den Berg and Lewis, ( i ) Approach due t o Van den Berg and Lewis The p h y s i c a l system c o n s i d e r e d by Van den Berg and Lewis i n their paper i s a n o n i n t e r a c t i n g Bose gas s u b j e c t t o an e x t e r n a l p o t e n t i a l of power form, c (-r)^ , w h e r e £ > 0 . potential i s linear, pOu) so,£=1.) and n (u) i n terms of (In t h i s t h e s i s , the They d e r i v e e x p r e s s i o n s f o r p, (> , i n the l i m i t L->°" with held constant. mentioned previously, the mathematical techniques employed i n t h i s paper may be e a s i l y extended t o the Fermi when the temperature and density are such t h a t ^ < 1 . complete understanding of these techniques, consult the paper itself. A brief the reader For a should sketch of the key ideas, however, i s presented here, a l o n g with the p a r t i c u l a r take gas form they f o r the case of a n o n i n t e r a c t i n g Fermi gas i n an e x t e r n a l linear potential. First, define (2.69) and (2.70) 127 where the £ ^ smallest are given by equation (2.47) and &i ) is the eigenvalue. In terms of 2f(-0 and ")7£) the occupation numbers (2.23) may be w r i t t e n as L j^(^) is * e"? + c f ; ' ( 2 CD to be determined v i a the c o n d i t i o n s (2.24) and - 7 1 ) (2.71); i.e.: P= -rr• = ~ f r | > ^ ' The f i r s t equation step i s to prove that equation (2.6) that equation g i v e s equation i n the l i m i t L—•>°". (2 (2.72) leads ' 72) to The second step is' to show ( 2 . 1 8 ) , which can be w r i t t e n as (2.2) f o r IT*-". Then, e x p r e s s i o n s (2.8) and (2.9) for p(-u) and n.(-u) have to be d e r i v e d . The s t a r t i n g p o i n t of the above p r o o f s i s to r e w r i t e p \ lL p may be expressed as and 128 or p=-(2s*IJ2 (-ir(^L)) ( 5(, , L h (2.74) where S,^) = k^ It e n>l fe > (2.75) has been assumed that J (L)<1 (which i m p l i e s ^<1) i n order to employ 0° X with x i d e n t i f i e d Similarily, AVI asJ(L)e"^. using o vv 1^ .„ _ , with the i d e n t i f i c a t i o n ^ i 3 - ^ of x as ^ ( L ) e ? , p ^ may be w r i t t e n as 1 ( ( " 0 (2.76) 129 D e f i n e G(n) v i a Combining equations (2.6), (2.32) and (2.77), one r e s u l t to be proven i s t h a t , i n the l i m i t L->-=, From equations (2.2), (2.31) and (2.77), another result to prove is (2.79) Proving equations (2.78) trivial task. (2.79) However, the equations Jb and and is by no means a 130 (2.81 ) S, Cn) = GiC*) certainly suggest that equations in the thermodynamic l i m i t . sufficient (2.79) i n themselves result The asymptotic mathematical and from Equations to show equations forms of (2.78) and (2.79) w i l l (2.80) and (2.81) are not that equations (L) and S (n) as L" ^ are r e q u i r e d . y< to prove (2.79) are somewhat lengthy, but the b a s i c (2.80) and Combining equations (2.78) and (2.74) and (2.76), r e s p e c t i v e l y . manipulations r e q u i r e d captured i n equations result equations The (2.78) idea i n v o l v e d i s (2.81). (1.7) and (2.71), 02.82) where the x- and y-dependences are omitted: they w i l l as L-*°° . disappear S i m i l a r l y , the equation f o r p (ai) i s 4) (2.83) Equation (2.82) may be c a s t i n t o the form (2.84) 131 where From equations one (2.84) and (2.85), by using a simple extension of of the lemmas s t a t e d i n the Van den Berg and Lewis paper, i t i s easy t o see that equation' (2.9) r e s u l t s , where n(lA)-|^ %) ( In a s i m i l a r f a s h i o n , equation by t a k i n g the l i m i t L->°° i n equation This (2.2), are all completes the sketch (2.6), (2.8) and (2.9). valid only f o r 0<^<1. . (2.86) (2.8) can be shown to r e s u l t (2.76). of the p r o o f s f o r equations As noted p r e v i o u s l y , the The reason for this proofs i s simply that of the p r o o f s i n the Van den Berg and Lewis paper use series extend expansions f o r the f u n c t i o n s i n q u e s t i o n : i n order t o those p r o o f s t o the case of a Fermi forms f o r J s / J J ) a n d ^ ( J ) power gas, the power s e r i e s must be v a l i d , and t h i s i s so only i f 0<^<1 . To summarize, if ^ i s i n the range [0,1], the f o l l o w i n g equations may be e a s i l y proven by extending Van the r e s u l t s of the den Berg and Lewis paper: A (2.87) 132 i (2.88) (2.89) (2.90) Since equations cL i s h e l d is fixed, (2.87) t o (2.90) r e s u l t the p h y s i c a l i n t e r p r e t a t i o n of these equations 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 t h e gas behaves a s i f i t were a f r e e p o t e n t i a l / " -cu.; The equation expected i s s o weak t h a t , Fermi gas locally, with chemical and p a r e s i m p l y t h e a v e r a g e p a r t i c l e and a v e r a g e p r e s s u r e , r e s p e c t i v e l y . this result, i n t h e l i m i t L->- i f F o r a more d i r e c t density route to s e e A p p e n d i x B. obvious requirement (2.89) i s v a l i d are particularized at this for a l l ^ . given next. stage R e a s o n s why t h i s Then, e q u a t i o n s t o t e m p e r a t u r e T=0 i s t o show K. should that be (2.87) t o (2.90) a r e 133 (ii) Analyticity o f t h e L o c a l Number D e n s i t y i n the Fugacity A p h y s i c a l reason that shall now be a s g i v e n by e q u a t i o n s niu.), presented (2.86) and which indicates (2.82), i s equal to the f u n c t i o n for a l l ^>0. Equation (2.89) i s v a l i d of J function . I f n(u) i s a l s o a n a l y t i c from t h e t h e o r y of a n a l y t i c n (u,J ) f o r 0<^<1. n(u;j[) is a t h i s must be s o . analytic i n j, i t will functions, that convincing n(u) is follow, equal to p h y s i c a l argument w h i c h i n d i c a t e s Assume t h a t n ( & ) i s n o t a n a l y t i c t h e r e must be a p h a s e t r a n s i t i o n the gas i s c o m p r i s e d the Fermi statistics between transition can o n l y r e s u l t between the transition, equation the individual if there and n(u-) must be a n a l y t i c Similar (2.88) and of n o n i n t e r a c t i n g fermions. individual particles. (2.89) i s v a l i d reasoning . Then fermions, is an give As rise whereas such, to a a phase attractive force T h u s , t h e r e c a n be no p h a s e i n 2j . This means that f o r a l l ^. can (2.90) t o i n d i c a t e f o r a l l t- . in^ i n t h e g a s f o r some v a l u e o f ^ . w h i c h t h e p a r t i c l e s obey repulsion valid an for a l l £ . There But is be applied that to a l l four equations (2.87), expressions are 134 A mathematical presented of the a n a l y t i c i t y o f n(u) i s n o t thesis. (iii) Specialization t o C a s e T=0 K Using the asymptotic forms f o r j ° 3 ( J ) a n d J ^ ( J ) (2.37) the in this proof and ( 2 . 3 8 ) , limit , respectively / v — i t i s easy when <£> >cT ^ e q u a t i o n s ( 2 . 8 7 ) t o show equat ions that, t o (2.90) in become, f o r s=^z, 7 ( 2 - 9 2 ) (2.93) where 135 (2.96) (3.3) PERTURBATIVE APPROACH In t h i s s e c t i o n , ^ s h a l l be taken to be a small number, and p e r t u r b a t i o n theory w i l l be used t o n(oi) c o r r e c t t o the f i r s t be at absolute zero derive an expression for order in°^. The gas s h a l l be taken to (and i t i s t o be assumed from the outset that the number of p a r t i c i e s i n the system i s very l a r g e , i . e . , that the system i s l a r g e enough to be regarded as macroscopic). The perturbation problem is specified by the f o l l o w i n g equat i o n s : (3.1) (3.2) (3.3) (3.4) 136 [_^ f\,^K f\... E (3.5, ZE Since the system i s l a r g e , ft * (r) s h a l l be taken t o be p e r i o d i c in the x and y d i r e c t i o n s , with p e r i o d i c i t y L. j/ ^ * j 1 vanish on the planes c o n d i t i o n s i s made f o r Using z=0 and s t o z=L. T h i s c h o i c e of boundary convenience. standard p e r t u r b a t i o n theory techniques, one finds that (3.6) (3.7) where (3.9) 137 (3.10) and a l s o Fl ' (3.11) 2 " ^ jD (f) l e i ( z ) ' (3.12) where (3.13) Normalizing the e i g e n f u n c t i o n determines A^ (3.14) z By v i r t u e of equation 8 (LkJ (3.11), each energy l e v e l i s elevated 138 by the same amount, -%c, independent of k. t h i s order i n the p e r t u r b a t i o n c a l c u l a t i o n , for the gas wave number k F In consequence, the Fermi is spherical, j u s t as f o r a f r e e gas. may t h e r e f o r e be immediately expressed to surface The Fermi in terms of the average p a r t i c l e d e n s i t y ^ : k F = ( 3 i r ^ . (3.-5) (the spin s of the fermions has been taken to be £ ) . To first order i n c:, the l o c a l number d e n s i t y of fermions n(ti) i s Z (?) k: IcClcr The *2c tf% ii & , —> (3.16) summation i n equation (3.16) i s over a l l k' with magnitude less than k . F because f a c t o r 2 p r e c e d i n g the summation s i g n the fermions are spin i: Equation where The (3.16) i s e a s i l y particles. simplified to k arises 139 _ AcTT I (3.18) ~ ~~L~ Using the w e l l known formulae (3.20) 5'in X = Z it ^(l-Cos2x) (3.21) i s s t r a i g h t f o r w a r d t o check t h a t , t o dominant large number np-, equation (3.17) equations (3.13) the is first simply and ( 3 . 1 4 ) , + term (•> . on i n the the r i g h t hand s i d e of Combining equation order (3.17) this result and reduces to . (3.22) 140 Combining equations (3.19), (3.21) and the relations (3.23) - (3 24) z**A 1 0 0 1 S -5Wloc = — ic =1 < Z =, k it ^ X 6 , 0<X<ZTF Z f o l l o w s that the terms of order np " 2 in equation to order n . F (3.22) c a n c e l out. The As such, (3.25) 1 ° ^ ^ ' (3.26, i n the c u r l y parentheses H i t i s necessary to go f o l l o w i n g e x p r e s s i o n s supplement those a l r e a d y cited: 0*0 I y i Z_ ~ k=n Ic" 2 F ^ 1 — — n F a; Hp •»p (7° (3.27) 141 ^-A n F W V W shb 2 K k F c ^ 5 " ' (3.28) $ ^ x Vf (3.29) ,p O<X<2TT U s i n g these t h r e e e q u a t i o n s , e q u a t i o n (3.22) becomes, t o l e a d i n g order, (3.30) Equation (3.27) is dx/x Equations (3.28) and Equation (3.30) l o c a l number d e n s i t y . .preceding i s of would and to by showing t h a t Z L i k ' i s bound from above (3.29) a r e proven gives by dx/(x-1) . i n Appendix D. the f i r s t o r d e r e x p r e s s i o n f o r the The m a t h e m a t i c a l derivation i s valid the o r d e r of mg, seem z proven JLo i s of criterion f o r which the is<*<<1, or L«(h "/2mc)= J-* . If c 2 the o r d e r of 10 3 meters! suggest t h a t , u n l e s s c t u r n s out t o be s m a l l , e q u a t i o n (3.30) w i l l be of no use i n the This extremely present work. 142 This, in equation fact, turns out to be not so, and as s h a l l be seen, (3.30) w i l l p l a y a key r o l e i n the model metal. (3.4) LINEARITY OF THE LOCAL NUMBER DENSITY IN THE PARAMETER cV Equations Consider the (2.92) to (2.95) are v a l i d extreme & «£f . for < * » \ Expanding equations and c <f . y F (2.92) and (2.94) i n terms of the s m a l l parameter £ / c > , and r e t a i n i n g terms only up to first power in C / E p , the following expressions result: (4.1 ) (4.2) Using equation (4.2) to e l i m i n a t e £ F from equation (4.1), or, e q u i v a l e n t l y : (4.3) 143 equations ( 3 . 3 0 ) Compare and (3.18) and ( 4 . 3 ) . i t i s seen that the identical. Yet two (3.30) equation f o r n (IL) expressions was (3.15) From equations assuming <=>(« 1 , derived whereas equation ( 4 . 3 ) was d e r i v e d assuming °<>> 1 a n d c < < £ . The F condition C«tp *<<(3n2N) . translates, This using are (4.2), equation into that equation ( 4 . 3 ) i s the asymptotic means form f o r n O u ) both when c*r i s s m a l l , and when i s large, provided c*r i s not too l a r g e . The obvious i n f e r e n c e from t h i s (4.3) i s also simply, as assertion long can i n c r e a s i n g c. 7 5 that equation f o r a l l <=< i n between these extremes, o r , as ° ~«(3v -N) < 7 . The reasoning imagine increasing <=*• by holding behind this L f i x e d and As long as c i s not too l a r g e -- i . e . : as long as --, the e x t e r n a l p o t e n t i a l may be perturbation. As c increases, i f n(-u) i s l i n e a r thought n(-^) w i l l P h y s i c a l l y , one does not expect n(u) is, is i s as f o l l o w s . One ^<<(3-n 'N) valid result change of as smoothly. to f l u c t u a t e with c. i n oC when oL i s s m a l l , and a l s o when l a r g e , w i t h the same s l o p e i n both regimes, a i t i s reasonable That is to assume that n(-u) i s a l s o l i n e a r , with the same s l o p e , i n between the two extremes. In all See f i g u r e conclusion, equation s m a l l compared t o N . 3 11 t o h e l p c l a r i f y this concept. ( 4 . 3 ) i s assumed to be v a l i d f o r 1 44 h(u) I 44o( 44 N 3 2/ F i g u r e 11. The L o c a l Number Density n(u) as a F u n c t i o n of the P a r a m e t e r s (3.5) CONTRIBUTION OF INTERNAL CHARGE DENSITY TO THE AMBIENT INTERNAL ELECTRIC FIELD A nonzero i n t e r n a l charge d e n s i t y w i l l average i n t e r n a l e l e c t r i c field. contribute to the T h i s c o n t r i b u t i o n s h a l l now be calculated. The charge d e n s i t y at height u, denoted q ( u ) , terms of the number d e n s i t y of e l e c t r o n s , n ( u ) , e d e n s i t y of i o n s , n-j-(u), by the f o l l o w i n g n (u) e is given in found v i a macroscopic following equation gravitational field: turn by equation elasticity and the number equation: (4.3), while theory, i s given i n which f o r an i s o t r o p i c body s u b j e c t n ( u ) may be x produces to a the uniform 145 oXW-.h26p Ma 4 i s Poisson's mass r a t i o , Y i s Young's modulus and M<^ i s the f o r the m a t e r i a l ions when there < 5 in question. ' 2 > atomic p i s the number d e n s i t y of x i s no e x t e r n a l f i e l d , while If rijXu) i s chosen t o be (\ at u=J-, <jn i s x n (u)-p . I x then i t f o l l o w s that n(u) = p ^ p / M L U ^ > (5.3. T + I 3 where M has been r e p l a c e d by M, the i o n i c a mass. Denote by 6(u) the charge per u n i t area at height large L l i m i t . u in the Since d>(u)=Lq(u)du, 6(*)~f\y (i-u) g du. (5.4) where (5.5) Equation (5.4) follows from equations the assumption of o v e r a l l charge (4.3), (5.1), (5.3) and neutrality: (5.6) 146 where ^> i s the average number d e n s i t y of f r e e e l e c t r o n s . e Using equation charge found (5.4), the c o n t r i b u t i o n by d e n s i t y t o the average i n t e r n a l e l e c t r i c the field internal i s easily t o be Note that E ^ depends on h e i g h t . I t was c outset that E i s independent of h e i g h t . avc as t o how t o r e c o n c i l e t h i s apparent assumed a t the The q u e s t i o n conflict. R e c a l l that the o b j e c t i v e of the model i s t o develop not t o produce a p r e c i s e c a l c u l a t i o n of E is of interest the c a l c u l a t i o n s first order order ave . ideas, Specifically, i t i n t h i s chapter t o see i f the i n t e r n a l charge imbalance i n a metal c o n t r i b u t e s a f i r s t Since arises i n g, E ft(/e in this c term to E - . a v e model have been made t o only w i l l be independent of i n g i f E /j i s an order g the next chapter order quantity. height to first I t w i l l be shown i n that E ^ does, i n f a c t , v a n i s h t o c first order in g. There equation order a r e two p o s s i b l e ways t o proceed. One i s t o use ( 5 . 7 ) as i t i s , and t o show that A turns out t o be of g . The other i s t o use the maximum value of E ^ ( n ) , c which occurs a t U = i , and t o show t h a t A i s zero t o f i r s t in g. Both approaches l e a d t o the same r e s u l t . To be more s p e c i f i c , replace E ^ ( u ) c by E i , ( | ) . c Thus, order 147 (5.8) Using shall is equation be f o u n d used i n the next c h a p t e r . because The l a t t e r (5.8), a s e l f - c o n s i s t e n t d e p e n d s on Eave but E E ^, c e l e c t r o n s , which for Eave. "self-consistent" ^ p o i n t f o l l o w s because A i n equation c, t h e s l o p e o f t h e e x t e r n a l l i n e a r conduction The t e r m solution d e p e n d s on E a v e (5.8) depends p o t e n t i a l experienced . on by t h e i n t u r n i s g i v e n by (5.9) It to be i n o r d e r t o g i v e r i s e rise to n(u). thesis. field E 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 ) w o u l d h a v e a V e T h i s problem, The r e a s o n i n a metal metal. assumption will E a v e which (u) t o v a r y much i n t h e m e t a l Once t o be A h a v e been 'M. V a n d e n B e r g 475-494 ( 1 9 8 1 ) . i n turn gives however, i s n o t r e l e v a n t t o t h i s i s b e c a u s e one d o e s n o t e x p e c t h a s been assumed model to the approximately the interior. uniform electric As such, inside h a s been shown t o be o f o r d e r the , this verified. a n d J . T . L e w i s , Comm. M a t h . P h y s . 8 1 , 148 ^See, f o r 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 g i v e s the p a r t of the e i g e n s t a t e which can be expressed i n the p o s i t i o n r e p r e s e n t a t i o n . T h i s may be l a b e l l e d by "k". The l a b e l "k" denotes both "It" and the s p i n l a b e l of the e i g e n s t a t e . a 'M. Abramowitz and I.A. Stegun, eds. Handbook of Mathematical Functions (U.S. Government P r i n t i n g O f f i c e , Washington, 1964), p. 446. 5 At r T=0 K, JX i s equal to the Fermi Energy, £^. ° S . Gradsteyn and I.M. Ryzhik, 4th ed., ed. Alan J e f f r e y , Table of I n t e g r a l s , S e r i e s , and Products (Academic Press, New York, 1965), p.30. 7 Ibid., p.31. 149 CHAPTER 4 DETERMINATION OF E». /a The model b a s i c r e s u l t of chapter 2 was of the construction the ions and the c a l c u l a t i o n of d i ^ \ E e e S • I n of a chapter 3, a m o d e l l i n g of the f r e e e l e c t r o n s produced a r e s u l t f o r E,.^. Using these f i n d i n g s , and the equation E« E = Eavc will now be A l ? (1) .i„ L, t". •' + + determined. Write -/? Mi 4.A where /} x and ^ ' C . . x/>'r a r e found by comparing equation (2) (2) t o equation (2-10.3): A. =Substituting equation . (2) i n t o equation ( 1 ) : (4) 150 where (6) Now w r i t e ^Mh where, from equations i s (7) -A (3-5.5) and (3-5.8): (9) (10) I n s e r t i n g equation r (7) i n t o equation ( 5 ) , and s o l v i n g f o r E , A M3/i + AEact + A M - A A A : i»i w -2, /3jytfsQ/9 i s a number. of order e the /^3/^5 Q / e g of 10 For L of the order of 10 /(metres) i s of the order of 10 , . meters and Pi a t y p i c a l value f o r metals, By equations ( 4 ) and ( 6 ) , and 151 t a b l e s II and I I I , i t i s c l e a r that jh^^s%/9 1 * and so equation (11) may be r e p l a c e d by _ AM^te y ? i £ « r f ; 4 - « > L [ It I V w i l l turn out that the term i n the numerator i n v o l v i n g large compared expression t o the other for E c t v e terms. which i s c o n s i s t e n t In consequence, to f i r s t order (12) is the i n the small parameter (/BzySy^ q / g ) ' i s e Into equation (13) put r ° r c } (14) where (15) The r e s u l t a n t equation f o r Ar - ~3 or, using /4y$ q /g - 1 0 5 /6 e is , _J r _ i r w (,6) , (17) r 2 - U 152 Rewrite equation (15) as r Equations ( o J - _ £ y ^ (17) and ( 1 8 ) show that <18) i s small 6E ^ a<J compared (o) E &ve . . From tables I , II and I I I , i t i s c l e a r not be very l a r g e , and w i l l c e r t a i n l y From those f o r copper), order one number. Unless e*t E ~ not be that of will /3\y5z order 10^. (9) and (10), u s i n g t y p i c a l values f o r 6 and Y^ equations (specifically, R to is very i t f o l l o w s that /fy/M/3 i s an 5 large, A E u ^ « E a y / c , and F, To v e r i f y the c o n s i s t e n c y of t h i s r e s u l t , put equation (13) i n t o equation (7) t o o b t a i n and put equations p Equations (6) and (19) i n t o equation JJH 3 i p V r (fl) (19) and (20) combine with equation (2) t o get . <20) (1) t o say that where E 4 V 4 It has been assumed i n t h i s model that the dependence of on u i s very weak. The c o n s i s t e n c y of t h i s assumption with 153 the above r e s u l t s w i l l now be demonstrated. First, 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 g . Replacing E a . by E« <-mg/q i n equation (7), and a v v v e e (o) using equation equations (15) f o r E , g i v e s E ^=0. a v f c (14), (15), (16) Equivalently, replacing E In c words, and (7) imply that E v^ ~ 1 0 E _Vt c by a v / € other E ^'-mg/q 4v in e equation a v e . (5.5) gives A=0 (recall that c=mg+q E , ), which means that A i s of order 10 'S equation (5.7) then says that E ^ ( M . ) X 10~'^ E - e av e c Since Ev, i s very s m a l l , equation (19) i n d i c a t e s that c r ~ A text- f Combining equations E U C - Lp <j A + x a e v ( o ) < (23) ' (1), (20) and (23) along with the f a c t i s n e g l i g i b l e r e v e a l s that E in t h i s C a v a ~ E„^ that , as s t a t e d e a r l i e r chapter. (o) . Because E i , « E , , the assumption that E c the . a v e metal av e interior i s consistent with . a v e the . i s uniform . in r e s u l t s of t h i s chapter. Equation the (21) f o r E heuristic argument a v / e i s the same r e s u l t due by Dessler determines inside et. produced by to Dessler e t . a l . This r e s u l t , however, has been a r r i v e d at here was as i n a d i f f e r e n t manner than it a l . Moreover, the present approach a l s o how much of E a v € i s made up by the charge the metal, and how much by the i o n i c d i p o l e s . imbalance I t i s not s u r p r i s i n g though, that the answer obtained here f o r E ^ e same as i n the h e u r i s t i c treatment f o r the basic physics of D e s s l e r et.al., i s the i s very much the same i n each case: the l a t t i c e 154 compression electrons is are calculated treated, Moreover, differs presented density et. al. objectives a of e l e c t r o n s i n the model two i s very as c a l c u l a t e d by the h e u r i s t i c final answer for E a v e of the model, as was s t r e s s e d work. In the next compared to the has a v t locally. as c a l c u l a t e d a v e the small. in the Dessler approach. Obtaining this of ions the d i f f e r e n c e between the same as E and as i f they were f r e e In consequence, t o dominant order, E model i s the theory, t h e l o c a l number d e n s i t y l o c a l number here, elasticity basically, although from the using literature contributed gravitationally towards part was only i n the of t h i s induced e l e c t r i c of the introduction the model the problem of field. 'Note: (1 - 2 6 ) / y =1/3K where K i s the "modulus of c o m p r e s i b i l i t y " . K i s of order 10 ° J o u l e s / m e t r e f o r most metals. 1 to t h e s i s , the model i s so as to demonstrate what understanding one 3 the 155 CHAPTER 5 CONCLUSION In this thesis, a simple constructed. The gravitationally induced e l e c t r i c of the f i e l d . model has model of been used field a metal to has been calculate the i n terms of the The sources which have been c o n s i d e r e d sources explicitly are the i o n i c d i p o l e moments and the charge imbalance i n s i d e the metal; a l l other sources have been grouped t o g e t h e r i n t o E e x ^ . A b r i e f summary of t h i s model i s now presented. The ions are arranged, number of density locally, of ions decrease in a l a t t i c e pattern. l i n e a r l y with h e i g h t . g r a v i t y , these ions have a d i p o l e moments create an average electric moment p. c Because These f i e l d E j^ / 0 es The dipole i n s i d e the metal. In order the model, p and E^ip i 0 eS in g. are c a l c u l a t e d t o only Because the d e n s i t y of ions changes with h e i g h t , the ions do not a l l have the exact same d i p o l e moment. the change first in S i m i l a r l y , E ^^ )es 0 dipole moment with height i s very However, small. a l s o depends very weakly on h e i g h t . N e v e r t h e l e s s , the e x p r e s s i o n s c a l c u l a t e d f o r p and E j , - ^ ^ are correct metal. density g. to Taking of f i r s t order i n g, f o r a l l p o s i t i o n s i n s i d e the i n t o account the height dependence on the number s h e l l s would produce c o r r e c t i o n s of second order i n 156 The conduction e l e c t r o n number density, number density, like decreases l i n e a r l y with h e i g h t . the The s l o p e s of the number d e n s i t i e s d i f f e r by a t i n y amount, g i v i n g r i s e net charge produces for E density c that metal. A c shows that E ^ ( u ) « E ^ ave solution E the field E ^(u). an e l e c t r i c assumption et. inside depends a v e obtained for E ' opposite E e*t to calculation The produces. ' a g*, n ck E d a while r is a e e a E ^ 1 1 o is c f weakly on the same . In the solution height. as the final order Mg/q negligible value of the Through the gravitationally analysis, to the order of insight into how against gravity responds a v e the can f a r too simply to give a The r e a l worth of the one induced e l e c t r i c how a metal . can field comes how about. One to g r a v i t y , and how t h i s In p a r t i c u l a r , requirement give understand rise to the model on the r e v e a l s that charge n e u t r a l i t y charge imbalance the can response that the n u c l e i be h e l d up ionic dipole s i t e of the nucleus can be e x a c t l y - M g V q , the model provides moments. Moreover, the model shows very c l e a r l y how the e l e c t r i c considerably different that i t a v & i t provides. model, generates sources of E assumed; Dessler and d i r e c t e d 6 model i s not the value of E The model r e p r e s e n t s a metal i s the p h y s i c a l i n s i g h t the The i n the model. value of E<xye that can be t r u s t e d . envisage a charge d e n s i t y self-consistent only a l . h e u r i s t i c e x p r e s s i o n f o r E^ di|ooks to , thereby v e r i f y i n g the i n i t i a l a E This ionic average. i n the metal y e t be Finally, f i e l d at something the model i n t e r i o r need not be i n a metal may be shown t o be so 157 minute that E ^ makes a n e g l i g i b l e c o n t r i b u t i o n to E c The model has compressibility of gravitationally the problem demonstrated the lattice, in addition charge imbalance induced e l e c t r i c fields induced e l e c t r i c fields whether the l a t t i c e the ionic a K e moments; the i s not important. relative therefore importance suggests do Note that According to the f o r a d i e l e c t r i c ~ i s the f i e l d due effect these of lattice But i n a metal, both c o m p r e s s i b i l i t y and the i o n i c d i p o l e moments model model £ i s compressible or not! dipole into of order Mg/q , and o p p o s i t e t o g, avc the key source of E compressibility The The n predicts E in in dielectrics. reduces t o simply a l a t t i c e of ions, with Z = 1 . the model s t i l l to and i n metals. For a d i e l e c t r i c , there a r e no conduction e l e c t r o n s . model, t o the metals, the model a l s o generates some i n s i g h t the q u e s t i o n of g r a v i t y then • induced i o n i c d i p o l e s are important concepts of g r a v i t y Besides that, a v e the two are following effects lattice significant. question: have in a What semi- conductor? The model presented here i s a very simple model. needed i s t o supplant t h i s account the ionic polarization more r e f i n e d manner. be considered model with The l a t t i c e compression carefully. One some c o n c l u s i o n s about d i e l c t r i c s , a final which and the charge would proposed model possess some f l e x i b i l i t y , As one also What i s takes imbalance into in a w i l l a l s o need like so that to t o see t h i s i t could give semiconductors and metals. p o i n t , the model of t h i s t h e s i s does not d e a l with the p h y s i c s i n v o l v e d i n the support a g a i n s t g r a v i t y of the 158 ion as a u n i t . I t would be i n t e r e s t i n g to i n c l u d e i n t o a more elegant model, gravity and induced e l e c t r i c f i e l d to see what this sources feature of t h i s f e a t u r e would l e a d t o . the 159 BIBLIOGRAPHY Abramowitz, M., and Stegun, I.A., eds. Handbook of Mathematical F u n c t i o n s with Formulas, Graphs, and Mathematical T a b l e s . Washington: U.S. Government P r i n t i n g O f f i c e , 1964 D e s s l e r , A . J . ; M i c h e l , F.C.; Rorschach, H.E.; and Trammell, G.T, " 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 i n Conductors." P h y s i c a l Review 168 ( A p r i l 15 1968): 737-743. Gradstehyn, I.S., and Ryzhik, I.M. T a b l e s of I n t e g r a l s , S e r i e s , and Products, 4th ed. E d i t e d by Alan J e f f r e y . New York: Academic Press, 1965 Hanni, R.S., and Madey, J.M.J. " S h i e l d i n g by an E l e c t r o n Surface Layer." P h y s i c a l Review B 17 (February 15 1978): T976-1983. H a r r i s o n , W.A. "Force on an E l e c t r o n near a Metal i n a Gravitational Field." P h y s i c a l Review 180 ( A p r i l 25 1969): 1606-1607. H e r r i n g , C. " 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 near a Conductor, and i t s R e l a t i o n t o the S u r f a c e - S t r e s s Concept." P h y s i c a l 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. " E l e c t r o n s of the Vacuum Surface of Copper Oxide and the Screening of Patch F i e l d s . " P h y s i c a l Review B 17 (February 15 1978): 1934-1939. Leung, M.C; Papani, G.; and Rystephan i c k , R.G. "Gravity-Induced E l e c t r i c F i e l d s i n M e t a l s . " Canadian J o u r n a l of P h y s i c s 49 (1971): 2754-2767. Leung, M.C. " E l e c t r i c F i e l d s Induced by G r a v i t a t i o n a l F i e l d s in M e t a l s . " I_l Nuovo Cimento 7 (February 11 1972): 220-224. Lockhart, J.M.; Witteborn, F.C.; and Fairbank, W.M. "Evidence f o r a Temperature-Dependent Surface S h i e l d i n g E f f e c t i n Copper." P h y s i c a l Review L e t t e r s 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 P h y s i c s 46 (1968): 1-11. Peshkin, M. "Gravity-Induced E l e c t r i c F i e l d Near a Conductor." P h y s i c s L e t t e r s 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 i n M e t a l s . " P h y s i c a l 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. " G r a v i t a t i o n - I n d u c e d E l e c t r i c F i e l d Near a M e t a l . " P h y s i c a l Review 151 (November 25 1966): 1067-1071. S c h i f f , L.I. "Gravitation-Induced E l e c t r i c Near a M e t a l . " P h y s i c a l Review B 1 (June 15 1970): 4649-4654. Van Field den Berg, M., and Lewis, J.T. "On the Free Boson Gas i n a Weak E x t e r n a l P o t e n t i a l . " Communications i n Mathematical P h y s i c s 81 (1981): 475-494. Witteborn, F.C., and Fairbank,W.M. "Experimental Comparison of the G r a v i t a t i o n a l Force on F r e e l y F a l l i n g E l e c t r o n s and M e t a l l i c E l e c t r o n s . " P h y s i c a l Review L e t t e r s 19 (October 30 1967): 1049-1052. Ziman, J.M. Pr inc i p l e s of the Theory of S o l i d s , 2nd ed. Cambridge: Cambridge U n i v e r s i t y Press, 1972. 161 APPENDIX A ALTERNATE WAY Define See OF DERIVING AN EQUATION FOR ^ E ^ c o o r d i n a t e s from the center of the s h e l l , x„,y , z . e f i g u r e A.1. Let^(rt) a be the ground s t a t e e i g e n f u n c t i o n f o r Shell F i g u r e A . I . Coordinates from the Center of the S h e l l . Versus C o o r d i n a t e s from the Nucleus. the p e r t u r b e d boundary problem, where "rt denotes the center is s t i l l of the s h e l l . given by equation The (ground (1.30). position s t a t e ) energy from eigenvalue I t i s a l s o given by (A. 1 ) where the i n t e g r a l that i s over the r e g i o n i s , over a l l & and a l l f a interior to the (the p o l a r and azimuthal shell, angles, 162 as measured from the c e n t e r of the s h e l l ) and over 0<r <R, and o where H(r ) i s the Hamiltonian expressed 0 i n c o o r d i n a t e s from the c e n t e r of the s h e l l : <LVY\ > (A.2) W7> 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 r e s p e c t to ex and determining the order one term: 1 Notice that depend on o<. introduced. ^4^*hlj>. (A. 3) °^ — the l i m i t s of i n t e g r a t i o n i n equation (A.3) do not This H(rt), }(ri) is precisely why and were D i f f e r e n t i a t i n g the i n t e g r a l once (A.4) Write I n s i d e the s h e l l , 163 (A.5) 41 It follows that As a r e s u l t , the r i g h t hand s i d e of equation (A.4) becomes (A.6) R Equation (A.6) f o l l o w s ^R. because cV Jo 3^1 (I which i m p l i e s that r d« 0*. r 164 Differentiating a second time: (A.7) Expressions equation for and h^/lot- are £V/h«? required. From (A.5), = 1<Z o * or i l l - 1,7 Z Q where r and 5 a r e measured from C£S0 d r A > 8 ) the nucleus. Notice at t h i s point that, since f f d 3 ( 3 r =^ / d r 3 0 — where 3 c and d r refer to the same p h y s i c a l volume element — 165 equation (A.8) may be plugged The order oi- term on the l e f t (2.40): A U/<) x SU Soc -l dz, 3 2 1 V v.' JY-rM-? AY 1 vu »-^ V and (A.6) here i s thereby r e c o v e r i n g equation Calculating i n t o equation so, u s i n g equation (A.6), z z i ' i' to g i v e 166 (A.9) CM where, a g a i n , 6 and r are measured from the I n s e r t equations this ( A . 8 ) and (A.9) nucleus. into (A.7), and equate to (A.10) It i s r e q u i r e d to f i n d the order one of equation Notice (A.10). that § d r r e p l a c e d by f*fd** to The d r. term on the r i g h t hand s i d e 3 e in the i f the i n t e g r a l l a t t e r two i s a l s o changed i n t e g r a l with the d e l t a Sfx, f o r which the order one from be C f u n c t i o n then becomes W>8c?) term i s i n t e g r a l s may d r 3 0 167 As such, the second term i n equation (A.10) makes an order one c o n t r i b u t i o n of (A. 1 1 ) R e p l a c i n g 4}% d r by ^^6 r 3 c makes the f i n a l i n t e g r a l i n equation (A.10) 9 for which the order one p o r t i o n i s (I T h i s vanishes on account of the B i n t e g r a t i o n . The one i n t e g r a l i n equation contribution. two ot, first parts. To (A.10) a l s o makes an see t h i s , break up the i n t e g r a t i o n One p a r t c o n s i s t s of a sphere with a r a d i u s but greater than<=<. order The i n t e g r a t i o n within into of order t h i s sphere will o give a contribution discarded. of order higher than ^ , and For the r e s t of the i n t e g r a t i o n , expand ^ as may be 168 o o Only -\o<^i) \o(5) r $> and need to be known 0 order one term being sought. t/ / out in (A.12) \ powers of in To o b t a i n order $ a and, of course, to obtain the and $, , expand pi and i n terms of v r off . & 0 0 Doing t h i s g i v e s e (A.13) and ((VZ)e-P- cos9 /z 0 To lowest order i n oc: cos9 Plugging equations integral i n equation (A.14) (A.13), a (A.15) (A.14) and (A.15) i n t o the f i r s t (A.10) g i v e s the order one term sought: 169 d ^ ^(l)>-2 l -s_-z„ 4 2 ) C 2 -P^ "J>^^ ^<^^ P & ' , 6 > ^j^.(p)(2f,) -r- Combining equation c (A.11) and (A.16) g i v e s term which i s r e q u i r e d t o determine ^ . or *P - §E 7 M his? M j u s t as b e f o r e . U 2 the order Using equation one (A.10), 170 APPENDIX B A QUICK DERIVATION OF THE LOCAL NUMBER DENSITY IN A FERMI GAS SUBJECT TO AN EXTERNAL LINEAR,POTENTIAL This appendix d e a l s with a c e n t r a l problem of Chapter 3 of t h i s t h e s i s , namely t h a t of d e t e r m i n i n g the l o c a l number d e n s i t y n(oi) of fermions which are s u b j e c t The approach here assumes, to from the p r o p e r t i e s of the Fermi gas are the same Fermi gas. p o t e n t i a l y- there i s no g r a d i e n t .yu where external potential. o u t s e t , that the l o c a l as those of a free As such, n(u) i s given by where a l l these symbols chemical an a r e the same as i n Chapter 3. The l o c a l i s determined by the requirement i n the l o c a l p o t e n t i a l energy, CU) A C i t y * - r; i i s the chemical p o t e n t i a l of the gas. T h i s i s the same as equation (3-2.89). Thus: that i . e . , that 171 APPENDIX C LOW DENSITY FERMI GAS IN AN EXTERNAL LINEAR POTENTIAL AT ABSOLUTE ZERO T h i s appendix d e a l s with Fermi gas which Expressions and is some p r o p e r t i e s of a subject s h a l l be d e r i v e d to an e x t e r n a l l i n e a r p a r t i c l e density. absolute zero. The equation the asymptotic potential. the gas in terms of i t s The treatment s h a l l be r e s t r i c t e d t o s t a r t i n g point i s part (2.65) density f o r the Fermi energy, the p r e s s u r e , the i n t e r n a l energy d e n s i t y of average low i n the case ^ c of 1 and i n the l i m i t < 2 section 3.2. forms f o r the A i r y f u n c t i o n s A i ( v ) and Consider Using B^(v) as (1) ( C D (C.2) equation (2.65) i m p l i e s that (C.3) 172 From equations (2.64) and (C.3) i t f o l l o w s that (C.4) The asymptotic forms of A* (v) and B; (v) as^-^are^ r^nH-v) "* A3 1 « ~i4^ B tV, Equations where v„ (C.4), (C.5) z denotes figure negative. is the (C.6) demand that', to lowest n z zero of A ^ ( v ) , and order, has been E^/Z. r e p l a c e d by As and (C6) 10(a) shows, From equation all of the zeros (C.5), the asymptotic of Ai(v) form of the are zeros 173 ~ The integer labels of (C.8) the zeros have been chosen to i n c r e a s e with ascending a b s o l u t e magnitude of the z e r o . is an e x c e l l e n t approximation this equation appendix, (C.8) no significant i n equation C L l z (C.7). ( 2 ^ c "*"zA where the d e f i n i t i o n of gives limit error For the purposes i s i n c u r r e d by using This gives (c.9) \ 3 ( ^ - 0 ^ ^ 8 ) has been used. The regarded as h o l d i n g 'c f i x e d and (C.9) (C.8) to the p r e c i s e value of the zeros of A i ( v ) , even f o r those c l o s e to the o r i g i n . of Equation L ' l e t t i n g L- «'. those e i g e n v a l u e s 5> limit may As such, equation which are l e s s than c i n the L A mathematical (C.9) p o i n t should now be clarified. Equation i s v a l i d only f o r • This is because the employed i n equation functions d* 1 be (1 - valid. Let In consequence, i f E n not be l a r g e , and n z s i d e of equation forms (C.I) and equation denote the value of n (C.9) i s equal to c, z z - c (C.2) may (2c.22) only i f the arguments of are l a r g e . ) may asymptotic ( the , 0 ) be Airy i s too c l o s e t o c, (C.9) will not be f o r which the r i g h t hand 174 (C. 1 1 ) ' Equation that Z 4 U i 3-TT f o r E -2- provided l e s s than n-^ so as t o render equation (C.9) w i l l be a good n^ is sufficiently (C.10) t r u e . ' 1 approximation To determine how much l e s s , From equations (C.9) and (C.11), n write i n the l a r g e L l i m i t , ^n must satisfy ^n»^ cx' ^ L / (c -' 3> ITT in order that equation by how much n be z (C.10) h o l d . must be l e s s than n z Equation (C.13) i n d i c a t e s i n order that equation (C.9) valid. In terms of l equation =!k» n -12 3 (C.14) (C.9) may be w r i t t e n as \3 2/ Using equation (C.15) i n equation (C.15) (2.47): I 3 ] . (C.16) 175 Equation (C.16) may be energy s u r f a c e s i n k-space. used to determine the constant Let - ^kx'V. <C For the constant energy s u r f a c e £„ , k z + n ,7) i s given i n terms of X' by 0-3 (C.18) where C)Z 1 The l i m i t i n g equation " 2. HA C" 1 u form as L-^°°of the f u n c t i o n k ( X ) x (C.18) i s given g r a p h i c a l l y s p e c i f i e d by i n f i g u r e C.1. Figure C.2 T h i s i n f o r m a t i o n about the constant energy s u r f a c e s may be shows the constant energy s u r f a c e £ . 0 used to determine energy d e n s i t y energy . the average number d e n s i t y p, , and the pressure The preceding the F in terms of r e s u l t s apply p r o v i d e d that than c by enough to make equation £ p the internal the Fermi i s less (C.13) t r u e . i s determined i n terms of N and L, i n the l i m i t L equation , by 176 F i g u r e C. 1 . k as a Function of K i n the L-»°°Limit f o r the Energy Surface £ 2 Q F i g u r e C.2. A glance at The Energy Surface figure C.2 £> i n L - ^ 0 ° L i m i t indicates that equivalent to z i n k-Space equation (C.20) is 177 where X^-a^-ia^f , 3 . (c 22) (C.23) E v a l u a t i n g equation (C.21) g i v e s (C.24) where ^=N/L . Equation (C.24) may be s o l v e d f o r £ F (C.25) In a similar f a s h i o n , the equation f o r 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 (C.27) J or U= y e F p . (C.28) p i s c a l c u l a t e d from U(N,V), V=L , v i a S »_^U(H ) 0 - ( C 2 9 ) V The result is (C.30) For comparison, the c o r r e s p o n d i n g r e s u l t s f o r ix and the i d e a l Fermi gas , - (c 31) j>=ff p.- ( c F only was claimed f o r a low equation (C.25) that that the density (C.25). for (no e x t e r n a l p o t e n t i a l ) are u= It p Fermi In order r e s u l t s of t h i s appendix are gas. This t h a t £p- <c, claim emerges - 3 2 ) valid from i t f o l l o w s from equation 179 (C.33) where 0 = z ( L ~ £ y * ( c To get an idea of the magnitude of p„ , c o n s i d e r and L=10 meters. T h i s g i v e s ^>=756 cm . the case t y p i c a l number d e n s i t i e s i n metals, of the order densities of 10 M. Abramowitz and I.A. Stegun, eds. Handbook of Mathematical Functions (U.S. Government P r i n t i n g O f f T c e , Washinton, 1964), pp. 448-449. Ibid., c=mg -3 cm ! f o r which the r e s u l t s of t h i s appendix apply are very small indeed. 2 3 4 ) Compare t h i s to the It- The . pp.448-449. 180 APPENDIX D "PROOFS" OF EQUATIONS ( 3 ~ 3 . 2 8 ) AND (3-3.24) Consider the function C«TK/ Putting 0 ^ a . (D.1 ) u=kx, i t i s e a s y t o s e e t h a t £o, rip") ^ * £ °° Co$^ ^ „ ( D - 2 ) or From e q u a t i o n (D.3), i t follows that the asymptotic f o r m o f S' a s n ->°° i s F 5CX This ;n ) F c l a i m may be v e r i f i e d 5CX;0P>^-^ and showing ^ S f h C ^ } * s f x - > 0 (D.4) by w r i t i n g ^^Y/V-fV; n ) ) ^ C p (D.5) that J a ; Equation -1 (D.4) v e r i f i e s - o(J- ) «s n -> p equation F ( 3 - 3 . 2 8 ) f o r s m a l l x. ( D < 6 ) 181 Consider S(x;n ) F f o r x=n. in this case, (D.7) Since, 4L_ < ? _ J o (V2,)' t ( M Z O N i N 3 it l f o l l o w s that -L < ! J\ < r l (r\ < (d,8) ds F ^ ' . 1 (D.9) Similarly, <x 2(M) From equations ^ (D.9) and <: 1 ^ O ( ^ F + ZT + I ) 1v (D.10) I ZtV^l) ' (D.10), (D.1 1 ) T h i s treatment etcetera. as n ~>°° . F f o r x = n- can be repeated f o r x=£, In each case, be found theS i s of order Since $ i s t h i s order f o r a l l these d i s t i n c t and a l s o f o r a continuous by i twill x=f, intuition to x-V^ n T p points, range of x when x i s s m a l l , one i s l e d expect i t t o be true f o r a l l x i n the range 0<X<2TT. As such, equation If equation (3-3.28) has been (3-3.28) i s true, then "proven". a shows that the f u n c t i o n oo lc^ i s of order n ~' as n - ^ . F p F k Accordingly, write similar procedure 182 F(x;M= jj-WCx;fl) • F Using equation It is satisfied (3-3.24), F i t follows straightforward to - (D ,3> that verify that equation (D.14) is by f„ t * \ ) H £- Thus, which i s equation C O (3-3.29). z. (D. 15)
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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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Title | A simple model for studying the gravitationally induced electric field inside a metal |
Creator |
Shegelski, Mark Raymond Alphonse |
Date Issued | 1982 |
Description | If a metal object is placed in a gravitational field, .the nuclei and electrons in the metal will sink. This will produce a new charge distribution inside the metal. A modified charge distribution implies a modified electric field in the metal interior. This thesis investigates some possible physical processes which give rise to the gravitationally induced electric field inside a metal. To this end, a simple model of a metal is constructed. Comprising the model are ions, arranged on a differentially compressed lattice, and a gas of conduction electrons. An ion is represented by a nucleus and an electron which are confined together inside a hard, massless, spherical shell. The nucleus is treated as a point particle while the electron is represented by a wave function. The conduction electron constituent is modelled as a gas of non-interacting fermions which is subject to an external linear potential, The design of the model facilitates the investigation of two possible sources of the electric field: gravitationally induced ionic dipole moments, and the charge imbalance in the metal. To first order in g, only the first source matters, contributing approximately –Mg/q[sub=e] to the electric field, where M is the ionic mass, g is the acceleration due to gravity, and q[sub=e] is the electronic charge. The net gravitationally induced electric field is also found to be approximately -Mg/q[sub=e], |
Subject |
Electric fields Metals -- Magnetic properties |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-03-31 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0095583 |
URI | http://hdl.handle.net/2429/23234 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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