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

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A SIMPLE MODEL FOR STUDYING THE GRAVITATIONALLY 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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