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Determination of the donor pair exchange energy in phosphorus-doped silicon Cullis, Pieter Rutter 1970

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(i)  DETERMINATION OF TILE DONOR PAIR EXCHANGE ENERGY IN PHOSPHORUS-DOPED SILICON by PIETER RUTTER CULLIS  A THESIS .SUBMITTED IN PARTIAL FULFILMENT OF THE RE&UIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Physics We accept this thesis as conforming to the required standard.  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1970  In  presenting  an  advanced  the  Library  I  further  for  degree shall  agree  scholarly  by  his  of  this  written  this  thesis  in  at  University  the  make  tha  it  p u r p o s e s may  for  for  is  financial  of  Columbia,  British  by  gain  Columbia  for  the  understood  of  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  of  extensive  be g r a n t e d  It  fulfilment  available  permission.  Department  Date  freely  permission  representatives. thesis  partial  shall  requirements  reference copying  Head o f  that  not  the  copying  be a l l o w e d  agree  and  of my  I  this  that  study. thesis  Department or  for  or  publication  without  my  (ii)  ABSTRACT  The e.p.r. spectrum f o r r e l a t i v e l y d i l u t e samples o f phosphorusdoped s i l i c o n  (<5 x 10  donors/cm ) has been c a l c u l a t e d i n d e t a i l  f o r an assumed random d i s t r i b u t i o n o f i m p u r i t i e s .  The system o f  donor e l e c t r o n spins i s t r e a t e d as a c o l l e c t i o n of nearest neighbor donor p a i r s .  An expression i s derived f o r the donor p a i r exchange  energy u s i n g Kohn-Luttinger -wavefunctions and a general exchange energy e x p r e s s i o n .  The r e s u l t a n t r e l a t i o n s h i p contains an adjustable  parameter a , the " e f f e c t i v e Bohr r a d i u s " , which i s determined from a comparison of the c a l c u l a t e d spectrum and the experimental r e s u l t s obtained f o r the r a t i o , C, o f the " c e n t r a l p a i r " and "hyperfine" line intensities.  The r e s u l t i n g expression J ( R ) , where J represents  the exchange energy and R the separation v e c t o r connecting the two p a i r donors, e x h i b i t s an o s c i l l a t o r y s p a t i a l dependence due to i n t e r f e r e n c e from p o r t i o n s o f the wavefunction a r i s i n g from d i f f e r e n t conduction band v a l l e y s .  (iii)  TABLE OF CONTENTS PAGE ABSTRACT  ii  TABLE OF CONTENTS  iii  L I S T OF FIGURES  v  L I S T C F TABLES  vi  ACKNOWLEDGEiiENTS  .  .  vii  CHAPTER I. II.  III.  INTRODUCTION  1  THEORY  3  .  A.  E l e c t r o n S p i n Resonance  B.  The W a v e f u n c t i o n s o f  C.  The N e a r e s t Donor A p p r o x i m a t i o n  D.  The Exchange I n t e r a c t i o n  E.  The Donor P a i r  System  . . . . .  the I m p u r i t y  3 States  . . . .  4 5  .  .  . . . . .  7  CALCULATION OF THE SPECTRUM  9  A.  Introduction  B.  Matrix  C.  C a l c u l a t i o n o f the P o p u l a t i o n D i f f e r e n c e  . . . < > •  Element C a l c u l a t i o n s  The T h e o r e t i c a l  IV.  V.  EXPERIMENTAL  Spectrum  9 10  Between T r a n s i t i o n L e v e l s D.  6  18 . .  METHODS AND RESULTS  21 22  A.  A p p a r a t u s and O p e r a t i n g C o n d i t i o n s  22  B.  Experimental Results  25  C.  Possible  29  Error  DISCUSSION OF RESULTS  30  BIBLIOGRAPHY  31  APPENDIX A.  POPULATION DIFFERENCES  B.  CALCULATION OF EXCHANGE  BETWEEN TRANSITION LEVELS . . .  32 34  (iv) APPENDIX C.  PAGE  THE DISTRIBUTION OF NEAREST EXCHANGE COUPLED PAIRS . . a.  C a l c u l a t i o n o f the Number o f L a t t i c e S i t e s in Shells of Unit  Thickness Centred on the  Origin b.  38  38  Determination o f the Nearesit Neighbour 41  D.  Distribution CALCULATION OF THE THEORETICAL SPECTRUM  . 47  (v) LIST OF FIGURES FIGURE  • '  PAGE  1.  Energy L e v e l s o f a Spin  System  .  2.  Energy v s . k R e l a t i o n i n [J-JOJ ^ D i r e c t i o n  4  3.  S p i n System o f the P a i r  ?  4.  Spectrum o f the P a i r System  5.  Energy L e v e l Diagram f o r a Phosphorus Donor P a i r .  0  . . . . . . . . . . . . . . . . . . . . . .  3  S  The  E i g e n s t a t e s are L a b e l l e d N u m e r i c a l l y From 1 t o 16 A l o n g W i t h the Usual S t r o n g l y Coupled P a i r S t a t e s to Which Each Reduces i n the L i m i t J » A  . . . « • •  ^  6o  Spectral Contributions o f Allowed Transitions . . . . .  12  7.  Comparison o f D i s c r e t e and Continuous N ( j ) ( a =16.5$. ) .  20  8.  I n t r o d u c t i o n o f the Gaussian Shape F u n c t i o n . . . . . .  21  9.  B l o c k Diagram o f Experimental Apparatus  ....  24  . . . . .  22  . . .  10.  Mode Shape and Sample C a v i t y Resonance . . .  11.  Experimental ESR D e r i v a t i v e Trace f o r Nd»3.7 x 1 0 / c m 16  12.  . .  3  26  A P l o t o f Experimental P o i n t s and T h e o r e t i c a l l y C a l c u l a t e d Values o f the R a t i o C(°/o) o f the C e n t r a l P a i r L i n e I n t e n s i t y to the Average I n t e n s i t y o f the Hyperfine L i n e s .  The  L i n e Represents the C a l c u l a t e d R a t i o f o r an E f f e c t i v e Bohr Radius a*=17.3A° 13.  27  A P l o t o f the N o r m a l i z e d D i s t r i b u t i o n o f P a i r J Values N(J)/No/2 as a F u n c t i o n o f the Exchange X6 3 Energy J f o r a 4x10 Donors /cm Sample ( S o l i d Curve) and a 6x10^ Donors /cm  3  Sample (Dashed  Curve)  28  14.  Simple Face Centred Cubic L a t t i c e S t r u c t u r e  38  15.  S i m p l i f i e d Energy Level Diagram o f the Donor P a i r . . .  32  (vi)  LIST OF TABLES  TABLE I.  PAGE M a t r i x Representation o f the Hamiltonian of the Donor P a i r System  II* III.  •  13  E i g e n f u n c t i o n s o f the Donor P a i r  14  A Comparison o f ( i ) the Computer P r o j e c t e d Donor P a i r Eigenvalues M i e n the O f f - d i a g o n a l Hyperfine Elements are not Neglected to ( i i ) the Functional D e r i v a t i o n Where These O f f - d i a g o n a l Elements are Neglected  IV.  Donor P a i r T r a n s i t i o n s Having Non-zero T r a n s i t i o n Probability.  The R e l a t i v e T r a n s i t i o n  P r o b a b i l i t i e s and the T r a n s i t i o n Energies,A, are _  .9  10  Given and Ye=gBHo V.  Boltzmann P o p u l a t i o n D i f f e r e n c e s Between Transition Levels  VI.  33  Number of L a t t i c e S i t e s Having the Same Absolute Value f o r P r o j e c t i o n s on the x, y, z Axes i n a Crystal w i t h I n t e r l o c k i n g F.C. C. S t r u c t u r e  . . . . .  40  (vii)  ACKNOWLEDGEMENTS  I would l i k e to thaifcDr. John R. Marko f o r h i s suggestion o f the  t h e s i s t o p i c and subsequent encouragement through a l l phases  o f the problem. Thanks a l s o go to Dr. R. Banrie, who was a c r i t i c a l  sounding-  board f o r many aspects o f t h i s t h e s i s . I would l i k e to express g r a t i t u d e to the N a t i o n a l  Research  Council f o r t h e i r award o f a Post Graduate Studentship f o r the d u r a t i o n o f t h i s work. The research f o r t h i s t h e s i s was supported by the National Research C o u n c i l , Grant Number 67-4624.  -1-  CHAPTER I  INTRODUCTION  The  e l e c t r o n spin resonance (E.S.R.) spectrum of phosphorus 1o 18 doped s i l i c o n (P-Si) i n the concentration range 10 <N<10 " ' 3 i m p u r i t i e s /cm  demonstrates a "-weak centre peak" that l i e s midway  between the hyperfine  l i n e s of the i s o l a t e d donors.  Slichter*  a t t r i b u t e d t h i s peak to coupled p a i r s of neighbouring i m p u r i t i e s t h a t a c t as a u n i t with a t o t a l spin of 1. In samples with 16 3 Njj<5 x 10  /cm  a n c i l l a r y l i n e s adjacent to the centre peak have  s i m i l a r l y been explained  i n terms of c l u s t e r s of three or more  donor atoms. The  degree to which two  i n t e r a c t i n g spins may  be  considered  to a c t as a p a i r i s given by the exchange energy " J " between them. S l i c h t e r ' s c a l c u l a t i o n included where A i s the hyperfine  only those p a i r s f o r which  interaction.  J~A,  I t i s the purpose o f t h i s  work to determine the f u n c t i o n a l dependence of J on " r " , the interdonor  separation,  f o r a l l values of r .  dependence i s constructed parameter a , the  This  functional  sa as to contain a s i n g l e  adjustable  " e f f e c t i v e Bohr r a d i u s " of the impurity  electron,  which i s subsequently f i t t e d to experimental r e s u l t s . Motivation  f o r the determination of J a r i s e s p a r t l y from the  " s p i n d i f f u s i o n " mechanism that can t r a n s p o r t  energy from one  part  2 of a s p e c t r a l l i n e to another , presumably v i a a f l i p - f l o p neighbouring s p i n s .  I t has  of  been proposed that the bulk of e l e c t r o n i c  spins r e l a x through such a d i f f u s i o n of spin and  energy to other  s p i n centres w i t h very short spin l a t t i c e r e l a x a t i o n times, "T, . ,f  3  One  of these f a s t r e l a x i n g centres i s suspected to be the  previously  _2 mentioned h i g h l y coupled p a i r , with T^ knowledge of J i s therefore the  04  J  . A  e s s e n t i a l before d e t a i l e d studies  s p i n d i f f u s i o n r e l a x a t i o n mechanism can be A t t e n t i o n has  functional of  initiated.  also been given to a broad background l i n e  that  -2-  has been observed beneath the i s o l a t e d  donor spectrum o f P — S i and  which extends w e l l o u t s i d e t h i s spectrum. arise  a.  This l i n e i s thought to  from exchange coupled p a i r s f o r which J^A .  I t i s shown  t h a t a t l e a s t p a r t o f t h i s l i n e i s due to such p a i r s .  -3CHAPTER I I THEORY A.  E l e c t r o n Spin  Resonance  The b a s i c energy l e v e l s o f the u n p a i r e d e l e c t r o n s o f a paramagnetic sample i n a magnetic f i e l d J t l ^ R are given i n Figure 1  m =the e l e c t r o n i c s p i n quantum number h/12=energy d i f f e r e n c e between states N =number o f e l e c t r o n s i n upper _ state . N =number o f e l e c t r o n s i n lower state  F i g u r e 1.  Energy l e v e l s o f a s p i n iV system.  I f t h i s system i s subjected  to a microwave f i e l d a t frequency  V l 2 , the r e s u l t i n g e q u i l i b r i u m a b s o r b t i o n \4  ( assuming no s a t u r a t i o n )  o f energy (ESR) can be / v.  as:  (1-D where C_^_ ^ i s the m a t r i x element c o n n e c t i n g the s t a t e s , g(fl2) i s 9  a "shape f u n c t i o n " due to the f i n i t e w i d t h o f the energy l e v e l s , and n  i s the steady s t a t e excess number o f e l e c t r o n s i n the ground ss s t a t e . Assuming Boltzmann p o p u l a t i o n s t a t i s t i c s (see Appendix I )  we o b t a i n : (1-2) where N i s the t o t a l number o f s p i n s , k i s Boltzmann's constant, and  -4T i s the a b s o l u t e temperature  in  K.  Examination  o f equation  r e v e a l s t h a t t h e g r e a t e s t p o p u l a t i o n d i f f e r e n c e s and l a r g e s t ESR  s i g n a l s are o b t a i n e d a t h i g h magnetic  hence  1-2  the  f i e l d s and  low  temperatures. B.  The W a v e f u n c t i o n s S i l i c o n has  o f the Impurity  States  s i x c o n d u c t i o n band minima i n "k" £l,0,o] ,  a l o n g each o f t h e d i r e c t i o n s _0,0,-]] as shown i n F i g u r e  vs k  The w a v e f u n c t i o n s the  following linear  these  ,  ,  2.  in  silicon  _5=conduction  Energy  that l i e  [-1>0,u] , [ o , l , o ]  a=5.3A  F i g u r e 2.  space  relationship  i n the  band minimum  [l,0,o]  o f t h e i m p u r i t y s t a t e s can be  combination  o f the wavefunctions  direction.  represented a t each  by  of  c o n d u c t i o n band m i n i m a : (1-3)  where ^ n  r e p r e s e n t s t h e c o n t r i b u t i o n o f e a c h minimum t o t h e  wavefunction, th to  the n  and  minimum.  hydrogen-like  (_)  i s a periodic  i ' ( _ ) i s an e n v e l o p e  Schroedinger  total  f u n c t i o n o f r_ c o r r e s p o n d i n g function  satisfying  a  equation:  (1-4)  where m^  i s the l o n g i t u d i n a l  e f f e c t i v e mass, m^  i s the t r a n s v e r s e  -5-  e f f e c t i v e mass, e i s the e l e c t r o n i c charge and K i s the macroscopic d i e l e c t r i c constant.  Equation 1 - 4 i s non-separable, but a good  v a r i a t i o n a l s o l u t i o n to i t i s given by: to  -  '  e  v  (1-5)  a t  where a and b a r e , r e s p e c t i v e l y , the t r a n s v e r s e and l o n g i t u d i n a l " e f f e c t i v e Bohr r a d i i " . As p r e v i o u s l y mentioned, t h i s work w i l l employ a s i n g l e e f f e c t i v e Bohr r a d i u s t h a t enables equation 1 - 5 t o be r e w r i t t e n a s : -_ _ L _ where a  e~**  /  (i-6)  i s presumably some s u i t a b l e average o f a and b.  Kohn  f i n d s , from c o n s i d e r a t i o n s o f symmetry and the e x p e r i m e n t a l l y  observed ground s t a t e h y p e r f i n e s p l i t t i n g t h a t  n= '/fife f o r a l l n.  Therefore we can w r i t e equation 1-3 f o r the ground s t a t e a s :  iko  The i n t e r f e r e n c e term e —  () n  T  -~ i n equation 1-7 w i l l be r e t a i n e d  e x p l i c i t l y i n our f u r t h e r c a l c u l a t i o n s * C.  The N e a r e s t Donor  Approximation  This study assumes a random d i s t r i b u t i o n o f the phosphorus i m p u r i t i e s i n the host l a t t i c e .  This assumption has p r e v i o u s l y been  shown to be reasonably j u s t i f i e d i n the c o n c e n t r a t i o n range o f 7  interest.  F u r t h e r , we assume t h a t the exchange i n t e r a c t i o n J  between donor e l e c t r o n s i s n o n - n e g l i g i b l e f o r n e a r e s t neighbours only.  O b v i o u s l y , t h i s approximation  becomes i n a c c u r a t e when some  i m p u r i t y s i t e "A" i s the same d i s t a n c e from neighbouring  s i t e s "B"  and "C", but i t i s found t h a t , f o r our c o n c e n t r a t i o n range, nonn e g l i g i b l e p r o b a b i l i t i e s o f such c l u s t e r s a r i s e o n l y when A, B and  -6C are so w i d e l y separated as to a c t l i k e i s o l a t e d donors.  This  approximation i s a l s o j u s t i f i e d by the absence o f l i n e s due to such c l u s t e r s i n our s p e c t r a . D.  The Exchange I n t e r a c t i o n 8  The n e a r e s t neighbour exchange i n t e r a c t i o n may be w r i t t e n a s l •V .  =  e  *  (1-8)  §,•__'  where _ _ i , S>2 are the spins o f the two e l e c t r o n s concerned, and: -  -  [ [ k ^ i  ^ c o  1  W o  *fi*> (1-9)  V l i  \*(^  dv, - \X^(-£) ^ Ml.  (i-io)  where^T^, ^b> r  a r e  *  n e  wavefunctions o f e l e c t r o n s 1 and 2, and  i2=l£l - r j .  We can now c a l c u l a t e the exchange energy J as a f u n c t i o n o f i n t e r d o n o r s e p a r a t i o n x_ by s u b s t i t u t i n g the i m p u r i t y wavefunctions o f equation 1—7 i n t o equation  1-9.  This i s done i n Appendix I I ,  where we o b t a i n : J  (1-H)  The  f a c t o r represents  an i n t e r f e r e n c e e f f e c t  due to c o n t r i b u t i o n s from v a r i o u s conduction band minima.  -7E.  The Donor P a i r System I t i s i n s t r u c t i v e to f i r s t c a l c u l a t e  the donor p a i r spectrum i n  the two l i m i t i n g cases o f l a r g e (J>">A) and small ( J « A ) exchange energy.  The s p i n system i s g i v e n i n F i g u r e 3. _»static magnetic f i e l d  S© 4  S • -e  _ , _> are the i n d i v i d u a l spins  -2.  _j  s  ?  H  1  £>2 the i n d i v i d u a l s p i n s o f the p a i r a  r  e  nuclear electronic  i F i g u r e 3.  S p i n system o f the p a i r .  The H a m i l t o n i a n f o r t h i s system can be w r i t t e n a s : H  =  3  p U [ S,_ *.§__1  where g i s the e l e c t r o n i c  +  A^TS,  +  __-S_^ 4-TS;S_ (1-12)  g f a c t o r and ^ i s the Bohr magneton. The  much s m a l l e r n u c l e a r Zeeman and n u c l e a r s p i n - s p i n i n t e r a c t i o n s a r e I n the l i m i t J « A , the r e s u l t i n g s p e c t r a i s o b v i o u s l y  neglected.  t h a t o f the i s o l a t e d donor, which gives two h y p e r f i n e peaks a t JL.  E=gpHo -  .  A/2.  The more i n t e r e s t i n g l i m i t when J>>A s h a l l now be  considered i n d e t a i l . a good quantum number.  F o r t h i s s i t u a t i o n S=Sj + Sg i s e s s e n t i a l l y N e g l e c t i n g the o f f - d i a g o n a l components o f almost"  the h y p e r f i n e i n t e r a c t i o n , which i s j u s t i f i e d f o r ^ a l l v a l u e s o f J ( c . f . Chapter I I , S e c t i o n B ) , equation 1-12 becomes:  ^ - gpHS, ^ ^ ( ^ ^ i J ^ T C ^ - ^ with  (1-13)  eigenvalues: J> -+ A  (1-14)  and  (1-15)  -8where Ex, ES are the t r i p l e t and s i n g l e t energies ( u s i n g the u s u a l n o t a t i o n f o r the ground s t a t e o f the hydrogen m o l e c u l e ) .  M i s the  t o t a l e l e c t r o n i c s p i n p r o j e c t i o n quantum number, and mj. , m2 are the individual nuclear  s p i n p r o j e c t i o n numbers.  The i n t r o d u c t i o n o f a microwave f i e l d adds a p e r t u r b a t i o n to the H a m i l t o n i a n $1 o f the form:  Vtr.i '- $tl< C s - ^ +  *-.<*t)  d-16)  z  where S  i s the e l e c t r o n s p i n r a i s i n g o p e r a t o r ,  S~~ i s the e l e c t r o n  s p i n l o w e r i n g o p e r a t o r , and / 2 ^ i s the frequency o f t r a n s i t i o n . w  C l e a r l y , t h i s p e r t u r b a t i o n induces t r a n s i t i o n s such t h a t A M = - l , A(m_+m2)  =0.  The t r a n s i t i o n AS=l i s n o t allowed.  This i s due to the f a c t  t h a t the s p i n wavefunctions o f the t r i p l e t s t a t e are symmetric under i n t e r c h a n g e o f e l e c t r o n s , whereas those o f the ground s t a t e are a n t i s y m m e t r i c under the same o p e r a t i o n . t h i s interchange,  Thus "& .f ^symmetric under r  cannot connect the s i n g l e t to the t r i p l e t  of p a r i t y considerations.  because  Note t h a t when J ~ A, hoArever, S i s no 2  l o n g e r a good quantum number and we observe  such  "forbidden"  t r a n s i t i o n s from the " s i n g l e t " to the " t r i p l e t " s t a t e s . From the s e l e c t i o n r u l e s given above, the ESR spectrum w i l l have the form o f F i g u r e  4.  F i g u r e 4.  Spectrum o f the p a i r system.  Note t h a t the s p e c t r a due to the i s o l a t e d donors w i l l e f f e c t i v e l y mask the donor p a i r peaks a t E=  - A/2.  -9CHAPTER I I I CALCULATION OF THE SPECTRUM. A.  Introduction The spectrum o f P-SI i s inhomogeneously broadened due to h y p e r f i n e  i n t e r a c t i o n s w i t h Si""^ n u c l e i (4°/o n a t u r a l abundance). calculation w i l l 9  by P o r t i s .  The f o l l o w i n g  assume the e x i s t e n c e o f the " s p i n p a c k e t s "  proposed  Each such s p i n packet c o n s t i t u t e s a homogeneously  broadened l i n e c o r r e s p o n d i n g to a p a r t i c u l a r e f f e c t i v e l o c a l and has l i t t l e different local  field,  i n t e r a c t i o n w i t h those p a c k e t s c o r r e s p o n d i n g t o f i e l d s . " T h i s i s a q u e s t i o n a b l e assumption  be c o n s i d e r e d i n more d e t a i l i n f u t u r e s t u d i e s . i.  >n  9 .  and should  F o r each o f these  p a c k e t s the s u s c e p t i b i l i t y T = 7f +'» fi i s g i v e n by i fi'(u)  _ j .X **U 7_ 0  Cu -H>  (2-1)  a  0  (2-2) where  i s the s t a t i c magnetic s u s c e p t i b i l i t y , ^ i s the f r e e e l e c t r o n  gyromagnetic  r a t i o , Hi i s t h e microwave magnetic f i e l d i n t e n s i t y , T_  and T2 a r e the s p i n - l a t t i c e and s p i n - s p i n r e l a x a t i o n times r e s p e c t i v e l y , and H  Q  i s the c e n t r a l resonant f i e l d o f the s p i n  packet. E x p e r i m e n t a l l y , we observe d"fi'/dll.  Thus, r e a l i s i n g t h a t f o r our  low c o n c e n t r a t i o n samples ( < 4 x l 0 / c m ) V R^ T^T^V> 16  »  w  e  3  2  x^(^~R )^  ^hen  Q  o b t a i n from e q u a t i o n 2-1 t h a t :  and thus from e q u a t i o n 2-2,  [d w'Cv^A  -  ^xTr'X^  ( 2  "  4 )  -109  Then, g i v e n t h a t the r a t e of power absorbed by the sample i s $ A = 2 V Wo  ft"0O  we o b t a i n from equations 1-1 fdfi'fro)  a  (2-5)  and 2-4 t h a t : *  S 5  | c . ,1 f <y-«.}  (2-6)  3  where we assume a shape f u n c t i o n g(H-H^) f o r the homogeneously broadened l i n e o f each s p i n packet to be o f a Gaussian form.  A  t h e o r e t i c a l c a l c u l a t i o n of a s c a l e model o f the spectrum t h e r e f o r e i n v o l v e s d e t e r m i n i n g each o f the terms on the r i g h t hand s i d e o f e q u a t i o n 2-6 over each i n t e r v a l of H corresponding to a s p i n packet width. B.  M a t r i x Element C a l c u l a t i o n s . The c a l c u l a t i o n o f the m a t r i x elements o f the a l l o w e d t r a n s i t i o n s  n e c e s s i t a t e s a general c a l c u l a t i o n of the eigenvalues and e i g e n f u n c t i o n s o f the donor p a i r system over the whole range o f J .  F o l l o w i n g the  i n d i v i d u a l s p i n n o t a t i o n i n t r o d u c e d i n F i g u r e 3, we can w r i t e the general form o f the e i g e n f u n c t i o n s ^  =  <^ Ti«Si £,i»M = i c  as *  l^C i^ ^9>  (2-7)  H  R  where M j , Mg are the " z " components o f the e l e c t r o n i c s p i n s  and  Sg,  and m^ are the " z " components of the n u c l e a r s p i n s 1^ and __g, and theC  iiVl  J3^S are n o r m a l i z e d constants g i v i n g the c o n t r i b u t i o n to the  e i g e n f u n c t i o n from each o f the b a s i s s t a t e s . There are 16 b a s i c wavefunctions ^ i j H A =|MMm^m^> and c  i n g l y 16 C ^ ^ i ' s f o r a g i v e n H^ ° . /  r  i  v  i  The m a t r i x r e p r e s e n t a t i o n o f the  H a m i l t o n i a n o f the donor p a i r s p i n system (equation 1-12) b a s i s i s given i n Table I .  correspondi n this  This m a t r i x was-: diagonal i z e d v i a computer  and the corresponding e i g e n v a l u e s and e i g e n f u n c t i o n s determined p a r t i c u l a r v a l u e s of J .  for  The e i g e n f u n c t i o n s compiled i n Table I I have  a p a r t i c u l a r form which guarantees  t h a t they be i n v a r i a n t under the  -11siraultaneous interchange o f both e l e c t r o n and n u c l e a r s p i n s ( t h i s o p e r a t i o n commutes w i t h the Hamiltonian g i v e n i n equation 1-12.). As t h i s c a l c u l a t i o n o f t h e e i g e n v e c t o r s and eigenvalues o f f o r each v a l u e o f J was t e d i o u s and c o s t l y i n computer time, we determined t h e i r f u n c t i o n a l form by means o f a h y b r i d method.  L a b e l l i n g the  b a s i s vec t o r s f in by unprimed m,n * s, and t h e e i g e n v e c t o r s ( i n the d i a g o n a l r e p r e s e n t a t i o n ) by m', n', we w r i t e  (2-8) -I  where t h e Cmm' have a l r e a d y been c a l c u l a t e d v i a computer f o r c e r t a i n J's. Then:  Vt  -  1*0  E^.  \m'y  £  «  U  \«*> C  m  W  (2-9)  or: E . m  0%l*W> =  Cn\i^\^>  Cv™'  (2-10)  which we express more c o n c i s e l y a s : E^-  = 5^  C > ntn  u  ™  C  M  m  »  (2-n)  mn  In accord w i t h equation 2-11 the n** row o f Rnm was m u l t i p l i e d 1  by t h e column v e c t o r \tn*y and t h e equation was solved f o r the corresponding Em . 1  This c a l c u l a t i o n was f a c i l i t a t e d by n e g l e c t i n g  the o f f - d i a g o n a l elements o f t h e h y p e r f i n e i n t e r a c t i o n .  A check on  t h i s approximation was made by comparing the eigenvalues p r o j e c t e d by computer ( i n which the o f f — d i a g o n a l elements were n o t neglected) and the c a l c u l a t e d Era i n the r e g i o n J~A, w i t h g r a t i f y i n g r e s u l t s 1  (See Table I I I ) .  The f u n c t i o n a l Em' "* s a r e g i v e n i n Table I I and  the energy l e v e l s t r u c t u r e i n F i g u r e 5, where S, To, and T±i r e f e r to e l e c t r o n i c p a i r s p i n s t a t e s , and S , r e f e r  to s i m i l a r  -12n u c l e a r p a i r s p i n s t a t e s , i n the J » A  limit.  The non-zero v a l u e s o f the m a t r i x elements c o n n e c t i n g the v a r i o u s s t a t e s under the i n f l u e n c e o f the microwave r a d i a t i o n are now e a s i l y c a l c u l a t e d and are g i v e n i n Table IV, t o g e t h e r w i t h the corresponding t r a n s i t i o n energy  Examination of A r e v e a l s the  d i f f e r e n t r e g i o n s o f the spectrum to which each t r a n s i t i o n contributes.  T h i s i s i l l u s t r a t e d i n F i g u r e 6.  •  is-*  u  -=> 11 . • •  F i g u r e 6.  Spectral c o n t r i b u t i o n s of allowed t r a n s i t i o n s .  The broad background l i n e i s shown to be a t l e a s t p a r t l y due to donor p a i r s f o r which J~A.  -13TABLE I M a t r i x Representation  / \  -I"  4  of the ITarail t o n i a n o f the Donor P a i r System  ^>  o -in  -t*  -trl i  1  -T  A  ft  ir  -I"*  1 -in  -\«  A  A  7  J  -I"  o ( ?  1 -|N  H  '1" •'I  b  •v-TL  1 A A i  A 7.  "» A  T  _T  X  z -a  J 2. -T  A  3 z  *_A  2.  A i  I  _T  4  3"  A 2.  A  2.  2. _7  •*  A z  -I  X  A z.  -7  I  2.  J A i.  A  •"V  JL  A  i. •ft  X  3-  A z  "TH, *• T  A 2.  Hi  + 1  TABLE II Eigenfunctions of the Donor Pair  lNI-  N  N»i Nl"  i Nl|»/'K  i Nli Kl-  <7  i M>— NIi Nl-  Nh 1  H'Nl-  . NH 1 Ml-  1  Nllt \y K  <y  i  Kl" i fii HI-  1  i eii 1  vi-  <r  <^  <>  1  Energy YAM J + ^  2 3 4  1  T  5 6  All  7  a,  o O  a.  Ox  0a.  a. -Q,  T  % i  9  4-  _ 3 J 4  10 11 12 1 3 14 1 5 16  a*  a,  -ft,  -o.  -0,  1 '  \ All  to i  i  -15TABLE III A Comparison o f ( i ) the Computer P r o j e c t e d Donor P a i r E i g e n v a l u e s When the O f f - d i a g o n a l Hyperfine Elements are n o t N e g l e c t e d to ( i i ) the F u n c t i o n a l D e r i v a t i o n Where These O f f - d i a g o n a l Elements a r e Neglected. Jtate  J = 25 MHz  J = 400 MHz  (i)  (ii)  (i)  1  10065.25  10065.25  10159.  10159.  2  10007.59  10006.25  10101.  10100.  3  10005.59  10006.25  10099.  10100.  4  9947o9  9947.25  10041.  10041.  (ii)  5  6.59  6.25  99.7  100.  6  -18.39  -18.75  -299.6  -300.  7  54.06  54.06  108.5  108.5  8  54.06  54.06  108.5  108.5  9  5.9  6.25  100.3  100.  10  -22.46  -18.75  -300.3  -300.3  11  -66.56  -66.56  -308.5  -308.5  12  -66.56  -66.56  -308.5  -308.5  13  -10053.5  -10052.75  -9959.7  -9959.  14  -99941.1  -9993.75  -9900.  -9900.  15  -9994.1  -9993.75  -9900.  -9900.  16  -9934.75  -9934.75  -9841.  -9841.  -16TABLE I V Donor P a i r T r a n s i t i o n s Having Non-zero T r a n s i t i o n P r o b a b i l i t y .  The  R e l a t i v e T r a n s i t i o n P r o b a b i l i t i e s and the T r a n s i t i o n E n e r g i e s , A. , are Given and V =gPH . e o &  r  Relative Transition Probabil i t y  Transition  at  (l6)<t» ( 9)'  b:  (l3)<-»  Transition Energy ( A )  1  Y  e  5)  1  Y  e  c:  ( 9)e> I 4)  1  ds  ( 5)c-* [ 1)  1  e:  (l5)<-° ( 8)  f:  (14)«H>!; 7)  g:  ( 8)«- l 3)  II  hi  ( 7)<H> ;  II  i:  (I5)c— [ID  j:  (14)*-* (12)  II  k*  (12)«-*|; 2)  II  1:  ( 1 1 ) * - ( 3)  II  I  Hi  + J(J  2 +  9 A V* ~ 2 V  2  + A )"^) 2  «  +  +?r(J  Hi  2  + A )- -  ti  2)  2  J 2  it -&(J  + A )  +  J 2  it  - J ( J + A )"*) 2  II  2  it •4(J  2  + A )^ + 2  II  J 2  -17-  FIGURE 5  STATE NO.  STATE  1  —  :  —  2&3  J  IT„t,>  Ye+  A  +  A 2  IT ,t >,IT s> l  4  - —- — —  0  J  IT,,t.>  lT UIT t.i>  7&8  IT to>,!T s>  0l  0J  Oj  16  —  14&15 —  IT^OJT^) IT-^t,)  J  •  +  +  1  ' +  -re*  IS t >,lS t^>  6&I0  -J 4  IT^t.) —  13.  0j  J  A 2  4  Y e  5&9  11&I2  ENERGY fe=gpH)  J 4 1 \/j +A 2 2  2  J + A 2 J J  4  A 2  3J 4  IS.to^lS.s)  Energy L e v e l Diagram f o r a Phosphorus Donor P a i r . The E i g e n s t a t e s are L a b e l l e d N u m e r i c a l l y From 1 t o 16 A l o n g With the Usual S t r o n g l y Coupled P a i r S t a t e s to Which Each Reduces i n the L i m i t J A.  -18C.  C a l c u l a t i o n o f the P o p u l a t i o n D i f f e r e n c e Between T r a n s i t i o n L e v e l s The p o p u l a t i o n d i f f e r e n c e between the t r a n s i t i o n l e v e l s i s g i v e n  by n  S 3  as d e f i n e d by equation 1-2.  The d e t e r m i n a t i o n o f n  s s  thus  e n t a i l s a knowledge o f the r e l a t i v e number o f p a i r s t h a t c o n t r i b u t e to a g i v e n s p i n p a c k e t , and a l s o o f the nornxtlized d i s t r i b u t i o n o f s p i n s i n the corresponding upper and lower s t a t e s a t e q u i l i b r i u m . Appendix A d e a l s w i t h the l a t t e r c a l c u l a t i o n .  As the t r a n s i t i o n e n e r g i e s  /A can be w r i t t e n i n terms o f J (Table I V ) , we s h a l l c a l c u l a t e N ( j ) , (the n o r m a l i z e d d i s t r i b u t i o n o f n e a r e s t donor p a i r s h a v i n g J v a l u e s i n a c e r t a i n range) i n order to f i n d the r e l a t i v e number o f p a i r s c o n t r i b u t i n g to a g i v e n s p i n p a c k e t . The n e a r e s t neighbour d i s t r i b u t i o n f u n c t i o n f o r a random d i s t r i b u t i o n o f i m p u r i t i e s i s g i v e n by Chandresklar*^ a s : W(t)  where  dv  -  e  ~~a  (2-12)  NAT* <U  i s the c o n c e n t r a t i o n o f i m p u r i t i e s and W(r)dr i s the  p r o b a b i l i t y t h a t the n e a r e s t neighbour to a g i v e n i m p u r i t y l i e s i n the s p h e r i c a l s h e l l between r and r + d r . 4"wtyi*^/3 i  n  N o t i n g t h a t the f a c t o r  equation 2-12 expresses the number o f i m p u r i t y s i t e s  i n s i d e a r a d i u s r , we w r i t e :  where n ( r ) i s the t o t a l number o f l a t t i c e s i t e s i n s i d e " r " , and N j i s s  the c o n c e n t r a t i o n o f l a t t i c e s i t e s . *Vir  dv  •= _  Similarly:  . n£^r + du}  (2-14)  and t h e r e f o r e equation 2—12 can be r e w r i t t e n a s : .  WCv")  =  KLji ft^ + i / ) e - n N  nM  (2-15)  The p r o b a b i l i t y t h a t any p a r t i c u l a r l a t t i c e s i t e i n the s h e l l i s  -19a n e a r e s t donor i m p u r i t y s i t e i s then given by: -_.Yi(,ry  Tbo  -  _  e  N«r  (  g  _  1  6  )  Nsi  For p a r t i c u l a r values o f  and a*, the d e t e r m i n a t i o n o f N ( j )  e n t a i l s c a l c u l a t i n g J f o r each l a t t i c e s i t e i n a p a r t i c u l a r s h e l l . Then, the p r o b a b i l i t y t h a t t h i s l a t t i c e s i t e i s a n e a r e s t donor i m p u r i t y s i t e (as g i v e n by equation 2-16) i s a s s o c i a t e d w i t h t h i s value o f J .  This a n a l y s i s can then be performed over a l l p o s s i b l e  s h e l l s , and the p r o b a b i l i t i e s P ( r ) (whose J ' s f i t i n t o i n t e r v a l s ) added i n a corresponding a r r a y .  certain  Thus, f o r a p a r t i c u l a r  c o n c e n t r a t i o n and e f f e c t i v e Bohr r a d i u s , a d i s c r e t e N ( j ) i s determined. The computer a n a l y s i s necessary  to t h i s computation i s contained i n  Appendix C. I n o r d e r to make f u r t h e r c a l c u l a t i o n e a s i e r , a continuous N ( j ) was c a l c u l a t e d by t a k i n g an a p p r o p r i a t e average o f the i n t e r f e r e n c e term i n equation 1-11 ( c . f . Appendix C).  A comparison o f the d i s c r e t e 16 3 o and continuous d i s t r i b u t i o n s f o r N_=4 x 10 /cm ..and a*=10.5A i s given i n F i g u r e 7.  (The r a p i d f a l l - o f f o f the d i s c r e t e curve i n the  r e g i o n l o g J < - l i s simply due to the f a c t t h a t n o t a l l the p a i r s i n t h i s r e g i o n were c o n s i d e r e d , due to l i m i t a t i o n s o f computer time.) The problem o f determining the r e l a t i v e number o f s p i n s t h a t c o n t r i b u t e to a c e r t a i n s p i n packet i s now reduced to an i n t e g r a t i o n o f N ( j ) over the w i d t h o f the packet expressed  i n terras o f J .  i s combined w i t h the n a s a l i z e d Boltzmann p o p u l a t i o n d i f f e r e n c e s between the l e v e l s (Appendix A) to give  ^ss  o f equation 2-6.  This  -20-  10*  10" J  (MHz)  FIGURE 7 Comparison o f D i s c r e t e and Continuous N(J)(a*=16.5i)  10"  D.  The  T h e o r e t i c a l Spectrum  The  final  step i n the c a l c u l a t i o n c o n s i s t s f i r s t of i n t e g r a t i n g  the a b s o r p t i o n A over one  spin packet.  We  obtain  N-  where: (2-17)  where E ( j j ) - E ( J ) i s the spin packet width, 2  V  ft  (j,6,T) i s the  normalized  Boltzmann p o p u l a t i o n d i f f e r e n c e between the i and j  levels, M  ( j ) i s the matrix element connecting the i and j s t a t e s ,  and T i s the temperature i n °K. Gaussian  shape f u n c t i o n i s then  With reference to Figure 8,  the  introduced.  Transition Gaussian  ttS,>  ECl^  F i g u r e 8. The Gaussian  shape f u n c t i o n  I n t r o d u c t i o n of the Gaussian  t o t a l absorption  Nj, i s taken to be the area under the  shape f u n c t i o n , with h a l f width equal  width (~ 8ttW' ). i  The  shape f u n c t i o n .  to the s p i n packet  convolution of a l l these c o n t r i b u t i n g Ga ssians  i s then c a l c u l a t e d and  the r e s u l t i n g scaled spectrum i s obtained.  Appendix D presents the computer program t h a t i s used to these  perform  operations. Subsequently, the r a t i o "C" o f the height of the centre l i n e  to the average height of the hyperfine l i n e s i s c a l c u l a t e d .  This  r a t i o i s s e n s i t i v e to the value of the " e f f e c t i v e Bohr Radius" and i s p a r t i c u l a r l y easy to determine experimentally.  a*,  -22CHAPTER I V EXPERIMENTAL METHODS AND RESULTS A.  Apparatus and Operating C o n d i t i o n s The gross f e a t u r e s o f t h e ESR spectrometer employed a r e seen i n  F i g u r e 9.  The k l y s t r o n i s a V a r i a n A s s o c i a t e s r e f l e x t y p e , d e l i v e r i n g  a p p r o x i m a t e l y 70 mW. over the frequency range 8.5 to lOGHz.The k l y s t r o n frequency was s t a b i l i z e d by a standard control  automatic  frequency  (AFC) t h a t was l o c k e d " to an e x t e r n a l c a v i t y o f h i g h g and  v a r i a b l e frequency, e n a b l i n g o b s e r v a t i o n i n the d i s p e r s i v e mode. I f we c o n s i d e r the e q u i v a l e n t c i r c u i t t o a resonant c a v i t y , %' i s p r o p o r t i o n a l to the change i n resonant frequency o f the c a v i t y as the resonant c o n d i t i o n i s swept through.  I t was found t h a t the most  s e n s i t i v e p o s i t i o n o f the e x t e r n a l c a v i t y was a p p r o x i m a t e l y one h a l f way up t h e resonant c a v i t y as shown:  Mode Shape  Sample C a v i t y  F i g u r e 10. Mode shape and sample c a v i t y , where t h e change i n r e f l e c t e d power (which i s monitored by the c r y s t a l d e t e c t o r ) p e r u n i t change i n c a v i t y frequency w i l l presumably be maximum. The b r a s s c a v i t y used was designed to operate i n the TE^^g mode and was g o l d p l a t e d by an immersion type g o l d p l a t i n g s o l u t i o n . Aluminum f o i l wrapped around i t s e x t e r i o r excluded any o p t i c a l l i g h t ,  -23and two pyrex f i l t e r s s i t u a t e d i n the waveguide j u s t above the c o u p l i n g hole s h i e l d e d the sample from i n f r a - r e d r a d i a t i o n * .  Fairly  consistent  s e m i - c r i t i c a l c o u p l i n g to the c a v i t y was a c h i e v e d as l o n g as these f i l t e r s were a t l e a s t 1 cm. above the c o u p l i n g h o l e . had dimensions  The samples used  ^ 10 x 4 x 4 mm. and were h e l d to the c e n t r e o f the  bottom o f the c a v i t y by vacuum grease. 4 Low temperatures were a c h i e v e d by pumping on l i q u i d He  t h a t was  t r a n s f e r r e d i n t o an i n n e r dewar c o n t a i n i n g the resonant c a v i t y and waveguide.  The pump used had a 150 cubic f t . / r a i n , c a p a c i t y , and  pumped on a 6" l i n e .  Temperatures  achieved were found t o be c o n s i s t e n t l y  c l o s e t o 1.05°K d u r i n g the course o f the experiments. F i e l d modulation c o i l s t h a t enabled o b s e r v a t i o n o f t h e d e r i v a t i v e of the d i s p e r s i o n were operated a t 400Hz and amplitude .5 Oersteds. The i n p u t to these c o i l s a l s o served as the r e f e r e n c e f o r the l o c k i n a m p l i f i e r , to which was f e d the output o f the c r y s t a l p r e a m p l i f i e r . The s t a t i c magnetic f i e l d was p r o v i d e d by a Newport Instruments 8" electromagnet w i t h v a r i a b l e sweep c o n t r o l .  FIGURE 9 Block Diagram o f Experimental Apparatus  for A fit.")  ro  f rt  Hoicji c T  C r y HA  uer.c *  fAoAnlo^on Coils  Amp '?'** 1  _J I  fW.o (XollftVo,  -25-  B.  Experimental  Results  The ESR spectrum was o b t a i n e d a t 1.05°K f o r f o u r samples 16 3 •with c o n c e n t r a t i o n s o f .8, 1.7, 2.3 and 3.7 x 10 donors /cm . X0 3 A t y p i c a l s p e c t r a f o r Nd=4 x 10  /cm  i s shown i n F i g u r e 1. The  experimental r a t i o n C o f the c e n t r e l i n e h e i g h t to average h y p e r f i n e l i n e h e i g h t was p l o t t e d as seen i n F i g u r e 12.  The t h e o r e t i c a l  r a t i o u s i n g an assumed e f f e c t i v e Bohr r a d i u s a* was then  calculated,  and the b e s t f i t o f t h e experimental r e s u l t s was obtained f o r a*=17.3A°. F i g u r e 12.  T h i s t h e o r e t i c a l r a t i o i s seen as the s o l i d l i n e i n U s i n g t h i s value o f a* and the t h e o r e t i c a l spectrum C XQ 3  was c a l c u l a t e d a t 4.2°K.  to be 20.2 % f o r a Nd=3.7 x 10  donors/cm  sample  This i s i n good agreement w i t h the experimental value o f  20 - 2%. U s i n g t h i s b e s t f i t v a l u e o f a*=17.3A°, N ( j ) curves were 16 3 16 3 then c a l c u l a t e d f o r Nd=4 x 10 /cm and 6 x 10 /cm , and a r e presented i n F i g u r e 13.  -26-  P1GURE 11 Experimental ESR D e r i v a t i v e Trace fcrNd=3.7 x 10  /cm  FIGURE 12 A P l o t o f Experimental P o i n t s and T h e o r e t i c a l l y C a l c u l a t e d Values o f the Ratio C( /o) of the Central P a i r Line I n t e n s i t y to the Average I n t e n s i t y of the Hyperfine L i n e s . The L i n e Represents the C a l c u l a t e d R a t i o f o r an E f f e c t i v e Bohr Radius *=17.3A.  -28-  FIGURE 13 A P l o t o f the Normalized D i s t r i b u t i o n o f P a i r J Values N(j)/No/2 as a F u n c t i o n o f the Exchange Energy J f o r a 4 x 1 0 Donors /cm Sample ( S o l i d Curve) and a 6 'x 1 0 Donors /cm Sample (Dashed Curve) 1 6  1 6  3  3  -29C.  Possible Error The measurement o f the sample c o n c e n t r a t i o n s were made through  a standard  f o u r p o i n t probe r e s i s t i v i t y technique.  had a p o s s i b l e e r r o r o f -  5jo  t  These measurements  w h i c h , together w i t h p o s s i b l e  experimental  e r r o r s i n measuring the r a t i o , a r e reflcected by the e r r o r b a r s i n Figure 12.  Values o f a* were then f i t t e d to the l i m i t s o f these  e r r o r b a r s to d e f i n e t h e accuracy o f a*. We thus obtained a*=17.3A°±.2A°. A d d i t i o n a l e r r o r s may be due to a change i n passage c o n d i t i o n s from h y p e r f i n e t o centre l i n e s due to the more h i g h l y coupled and consequently f a s t e r r e l a x i n g p a i r systems t h a t c o n t r i b u t e to the centre l i n e .  We t h e r e f o r e changed s e v e r a l parameters o f the o b s e r v a t i o n  mode, n o t a b l y the sweep r a t e and magnetic f i e l d modulation frequency, whixh was v a r i e d from 30 H z to 3000 Hz. W i t h i n experimental  error the  r e s u l t i n g v a l u e s o f the r a t i o were found to be independent o f these changes i n o b e r v a t i o n parameters. I t was a l s o found t h a t the r a t i o "C" was s i g n i f i c a n t l y f o r h i g h e r microwave powers.  increased  This can be a t t r i b u t e d t o a s p i n  3  d i f f u s i o n phenomena  t h a t would i n c r e a s e the f a s t e r r e l a x i n g centre  l i n e a b s o r p t i o n a t the expense o f t h e more s l o w l y r e l a x i n g h y p e r f i n e lines.  To a v o i d such d i f f i c u l t i e s the r a t i o C was measured a t t h e  lowest microwave powers c o n s i s t e n t w i t h useable s i g n a l to n o i s e .  -30-  CHAPTER V  DISCUSSION OF RESULTS  The  experimentally  determined e f f e c t i v e Bohr r a d i u s a*=17.3A°  i s very c l o s e to 17.2A , which i s the a r i t h m e t i c mean of the 0  l o n g i t u d i n a l and transverse e f f e c t i v e Bohr r a d i i of the a n i s o t r o p i c 12 ground s t a t e impurity wavefunction  .  This agreement would seem to  i n d i c a t e t h a t the e f f e c t i v e Bohr r a d i u s i n overlap c a l c u l a t i o n s should be set equal  to such an a r i t h m e t i c mean r a t h e r than a  13 which has been used i n previous c a l c u l a t i o n s . 14 Jerome and Winter v i a Endor techniques obtained the most 16 3 probable value of J i n a P-Sl sample where Nd=6 x 10 /cm as 4 .. 1.34 x 10 MHz. From Figure 13 which presents our normalized N(J) f o r geometric mean  a s i m i l a r sample, the most probable  value o f J i s seen to be  P a r t o f t h i s discrepancy may  to the complications  be due  i n t o the i n t e r p r e t a t i o n of Endor r e s u l t s due electronic  s p i n l a t t i c e r e l a x a t i o n time.  introduced  to the J dependent  But more importantly, i t  might be pointed out t h a t Endor measurements only i n v o l v e d p a r t s f o r which J?A, coupled  50 Mil .  those  thus i g n o r i n g the great bulk of more weakly  pairs.  In c o n c l u s i o n , the e x c e l l e n t agreement that i s achieved the c a l c u l a t e d and experimental confidence  r e s u l t s (Figure 13)  between  provides  i n the isotropic assumption o f the e f f e c t i v e Bohr r a d i u s , and  a l s o i n our assumption o f a random d i s t r i b u t i o n of i m p u r i t i e s .  This  u n i f i e d , s e l f c o n s i s t e n t approach should provide u s e f u l and d e t a i l e d i n f o r m a t i o n on the d i s t r i b u t i o n of exchange coupled p a i r s which has h i t h e r t o been u n a v a i l a b l e .  -31BIBLIOGRAPHY 1.  S l i c h t e r , C P . , Phys. Rev. 99, 479 (1955)  2.  Marfco, J.R., Phys. L e t t . 27A, 119  3.  Yang, G. and Honig, A., Phys. Rev. 168, 271  4.  Pake, G.E., Paramagnetic Resonance, Ed. D. P i n e s , 38 Benjarain I n c . , N.Y., 1962)  5.  Kohn, W.,  (1968) (1967) (W.A.  i n S o l i d .State P h y s i c s , e d i t e d by F. S e i t z and  D. T u r n b u l l (Academic P r e s s I n c . , N.Y.,  1957), V o l . 5.  6.  Kohn, V/. and L u t t i n g e r , J.M., Phys. Rev. 98, 915 (1955)  7. 8.  A l e x a n d e r , M.N. and Kolcomb, D.F., Rev. Mod. Phys. 40, 815 (1968) Anderson, P.W., i n S o l i d State P h y s i c s , e d i t e d by F. S e i t z and D. T u r n b u l l ( A c a d e m i c P r e s s I n c . , N.Y., 1963), V o l . 14  9.  P o r t i s , A.M.,  Phys. Rev. 91_, 1071  (1953)  10.  Chanirasekhar, S., Rev. Mod. Phys. 1_5, 1 (1943)  11.  Ronig, A., P h y s i c a 24, 1635  12.  M i l l e r , A. and Abrahams, E., Phys. Rev. 120, 745 (i960)  13.  S u g i h a r a , K. , J . Phys. Chem. S o l i d s 29, 1099  14.  Jerome, D. and W i n t e r , J.M., Phys. Rev. 134, A1001  (1958) (1968) (1964)  15., P a u l i n g , L. and W i l s o n , E.B., I n t r o d u c t i o n to Quantum Mechanics, 343 (McGraw-Hill Book Company I n c . , N.Y., 1935)  -32APPENDIX A POPULATION DIFFERENCES BETWEEN TRANSITION LEVELS Our whole c a l c u l a t i o n o f the spectrum i s based on the f a c t , t h a t a l l the l o c a l i z e d i m p u r i t y p a i r s are " d i s t i n g u i s h a b l e " . T h e r e f o r e , even though the e l e c t r o n s a r e fermions, the r e l a t i v e p o p u l a t i o n s o f the v a r i o u s p a i r l e v e l s w i l l be given by Boltzmann statistics.  W i t h r e f e r e n c e to F i g u r e 5, page 17, we d i s c o u n t the  s p l i t t i n g o f the v a r i o u s t r i p l e t l i n e s (as A  a  n  d  o b t a i n the  s i m p l i f i e d energy l e v e l diagram given i n F i g u r e 15.  4  E  3  E,i  2  _  +  H  j_  I  E ^ - K W a - I  1 F i g u r e 15.  t  j _ X /yl 31^ A " ^ 2. 1  E  1  S i m p l i f i e d energy l e v e l diagram o f the donor p a i r .  T h e r e f o r e , i f we a s s i g n a p o p u l a t i o n o f 1 to energy l e v e l 1, and a t o t a l p o p u l a t i o n of N to the whole system, the f r a c t i o n a l p o p u l a t i o n s o f the o t h e r s t a t e s are given by:  , State 1. thus: S t a t e 2.  N, =  N  e.  ^ , Ne-^  and s i m i l a r l y f o r s t a t e s 3 and 4.  H  \  .^e^-.Nft'C  (A-l-l) '  \ The r e s u l t s o f t h i s  (A-l-2) ( a - 1  _  3 )  calculation  and c o r r e s p o n d i n g p o p u l a t i o n d i f f e r e n c e s are given i n Table V.  -33TABLE V Boltzuiann P o p u l a t i o n D i f f e r e n c e s Between T r a n s i t i o n L e v e l s  Transition 15  Population  Difference  8  -  14 -» 7  7  -~>2  -  J  €vp  - *V»  2. VC\  8 -» 3  1 1 - 915  1 w.-  12- ^14  12  2  11 •  3  16  9  13  5  I  -  r 2.  W  L tell  APPENDIX B C A L C U L A T I O N OF E X C H A N G E  The term J,t.  o f e q u a t i o n 1-10 i s now t o be c a l c u l a t e d  S u b s t i t u t i n g the s h i e l d e d p o t e n t i a l - e?/\<*  xx  explicitly.  seen by t h e donor  f o r the hydrogenic p o t e n t i a l — eV<\x , and w r i t i n g  electrons  r=r_2 - £j we o b t a i n T°  *K J  As  Mr  J^gCan now be w r i t t e n :  (A-2-2)  We examine the i n n e r i n t e g r a l g i v e n by £ ^ . o f the u's  (  u (n• '}(r< i ) lt  ,  +  ,  u  ^  h  ^  O  O  )  ^fo-'ft vfefle L  As Fourier  J  U s i n g the p e r i o d i c i t y  u ^ O O (A-2-3)  h\ r>"'  WjjJ'^U^t,')  i s p e r i o d i c i n r ^ , we expand i t i n a  series: 8 „ ,„ &  (A-2-4)  Thus:  [ Then, i f U^  .'V  ^ ( t y t ).Cr.N F  <*£,/( A-2-5)  varies slowly with r ^ , ( t h i s i s  -35-  e q u i v a l e n t t o the "gentle p o t e n t i a l approximation  " b a s i c t o the  d e r i v a t i o n o f the i m p u r i t y wavefunctions) one can o b t a i n :  Ur ^-K \ 2  f  (A-2-6)  by u s i n g the common d e f i n i t i o n o f the d e l t a f u n c t i o n . As  fc^o  and  a r e both i n the f i r s t B r i l l o u i n zone, and  K ^ i s a r e c i p r o c a l l a t t i c e vecto-r, equation A-2-6 demands K^= 0 o r ~$a0  f o r non-neglibible c o n t r i b u t i o n s i n t h i s gentle p o t e n t i a l l i m i t .  Similarly,  Ho"  an<  *  must be i n the same d i r e c t i o n i n  o r d e r t h a t t h e i r d i f f e r e n c e be c l o s e to z e r o , which p r o v i d e s the c o n d i t i o n n " =n 1  iv  .  Therefore (A-2-7)  where the o r t h o n o m a l i t y  o f the  l A ^ l X ^ S has been used to o b t a i n  Thus (A-2-8)  -i  2  n" o * K O VA k  (nf  ) (v^ d , r  w h i c h , by the same type o f procedure can be w r i t t e n as  ICQ hfra%fo<M*, (A-2-9) 1^  K  N o t i n g t h a t the F^'s, F^'s are hydrogenic I s wavefunctions w i t h an e f f e c t i v e Bohr r a d i u s a*, the i n t e g r a l K i s regognized e q u i v a l e n t t o the i n t e g r a l K  i  t o be 15 evaluated by P a u l i n g and Wilson in  a treatment o f the hydrogen molecule, where a =a*.  They o b t a i n  -36-  i  K «  a"  A ' e^s)  T) =  where  (A-2-10)  ^A^e c-i^  -  1  +  c  ^/a*.  By an e x a c t l y s i m i l a r a n a l y s i s , u s i n g 42-10, 42-12 o f P a u l i n g and 1  Wilson  5  j (A-2-11)  and 2  - i In.  •r (A-2-12)  K  As  7,, -  "~ ^S.iiL-,x.  , we have as our f i n a l e x p r e s s i o n : e  \ 2CUT>)CHT> +j> )  (A-2-13)  where terms  O^;.^ ^) have been d i s c o u n t e d as they go to zero 2-  very q u i c k l y f o r ^ ' L 7  .  As [ k ^ l . 8«5T/a  , where a i s the l a t t i c e s p a c i n g , f o r a l l n,  the i n t e r f e r e n c e term i n e q u a t i o n A-2-13 can be r e w r i t t e n a s :  -37T h e r e f o r e , e q u a t i o n A-2-13 becomes: -2$  (A-2-15)  which i s our f i n a l  expression.  -38-  APPENDIX C THE DISTRIBUTION OF NEAREST EXCHANGE COUPLED PAIRS a.  C a l c u l a t i o n o f the Number o f L a t t i c e S i t e s i n S h e l l s o f U n i t Thickness Centered on the O r i g i n . S i l i c o n has a so-called'"diamond"  c r y s t a l l i n e structure  c o n s i s t i n g o f two i n t e r l o c k i n g f a c e - c e n t r e d cubic  F i g u r e 14.  Simple Face-centred  lattices.  Cubic L a t t i c e S t r u c t u r e (F.C.C.)  For the simple F.C.C. l a t t i c e noted i n the above f i g u r e , the b a s i c l a t t i c e t r a n s f o r m a t i o n s d e p i c t e d a r e given by  Q, = at* VI}  (A-3-1)  Therefore, l a t t i c e s i t e s a r e a t p o s i t i o n s R where  E  =  i• ^a i v  a  4  r  t  5  0  l  3  where ^ , n , n 2  g  are i n t e g e r s .  (A-3-2)  The second F.C.C. l a t t i c e o f s i l i c o n i s d i s p l a c e d a l o n g the body diagonal o f the cube d e p i c t e d Therefore  i n F i g u r e H by  of i t s length.  the l a t t i c e p o i n t s o f the second l a t t i c e a r e d e s c r i b e d by  -39-  R  idie r e : + 0 < -  P=B  ,  +  A  +  (A-3-3)  f e 3  R= (ji'flT) o f the l a t t i c e s i t e s from the o r i g i n i s 1  The d i s t a n c e then g i v e n bys R e f t  for  +(is +ft t 1  t  4  ^  (A-3-4)  K +«Jj"  the f i r s t l a t t i c e , and by: R'  for  ^Q\ +*?f  =  SL  ^1 + A^-t.^T ^ ( P ^ s  + -<T  ( _3_ )  * (PS^^.-sf  A  5  the second l a t t i c e . When t h i s f o r m u l a t i o n  i s used i n t h e 360-67 computer to c a l c u l a t e o  the number o f l a t t i c e s i t e s i n s h e l l s o f w i d t h 1 A , i t becomes o  extremely c o s t l y i n computer time t o proceed p a s t r= loo A .  It is,  however, e s s e n t i a l t o the c a l c u l a t i o n t h a t we do so. Thus the problem was r e f o r m u l a t e d  by u s i n g the symmetry p r o p e r t i e s o f the  lattice. The b a s i c q u a n t i t i e s o f i n t e r e s t a r e the absolute (for  value o f R and  t h e i n t e r f e r e n c e term i n e<^. ft-l-i5) t h e p r o j e c t i o n s o f R on the  x, y, z axes r e s p e c t i v e l y .  The i n t e r f e r e n c e term i s seen t o be an  even f u n c t i o n o f these p r o j e c t i o n s . the a b s o l u t e  Therefore we s h a l l  calculate  v a l u e o f the p r o j e c t i o n s o f v a r i o u s R on the x , y, z axes  r e s p e c t i v e l y and f i n d t h e number o f l a t t i c e s i t e s t h a t correspond t o each s e t o f p r o j e c t i o n s . conceptually  The a n a l y s i s , a l t h o u g h t e d i o u s , i s  t r i v i a l , and i n essence c o n s i s t s o f t r e a t i n g the two  i n t e r l o c k i n g F. C C . l a t t i c e s as 8 i n t e r l o c k i n g cubic l a t t i c e s . r e s u l t i n g absolute  The  v a l u e s o f the p r o j e c t i o n s and c o r r e s p o n d i n g l a t t i c e  s i t e s a r e summarized i n Table V I .  I t was then easy t o c a l c u l a t e the  number o f l a t t i c e s i t e s c o n t a i n e d i n each s h e l l of w i d t h  IA  f  and t o  subsequently o b t a i n the p r o b a b i l i t y t h a t any one s i t e i n t h a t s h e l l was an i m p u r i t y s i t e through e q u a t i o n 2 - 1 6 .  -10TABLE VI Number o f L a t t i c e S i t e s Having the Same A b s o l u t e Value f o r P r o j e c t i o n s on the x, y, z Axes i n a C r y s t a l With I n t e r l o c k i n g F.C.C. S t r u c t u r e . A b s o l u t e Value o f P r o j e c t i o n s (n^ , n^, 3 ^ - l where n^, n , n^ are integers^) n  0  X  n^a  z  7  (n -l)a  n^a  8  2  (n  2  No. o f "Equivalent"  - .5) a ,75)a  • 5)a  24  o 25) a  12  .25) a  12  .75)a  4  .25) a  4  (nj - .75)a  (n  2  -  (n  - .75)a  (n  2  - .25)a  (n^ - .75)a  (n  2  - .75)a  (n  (n  2  - .25)a  <»,-  - .5)a  0  12  x  x  - .25)a  (n^ - .5)a  (n  9  <-3<»3" ( n  3  "  n^a  n a 2  0  4  n^a  o  0  6  -41-  b«  D e t e r m i n a t i o n o f the N e a r e s t Neighbour T h i s was performed by Program I.  Distribution  The p r o b a b i l i t y t h a t a l a t t i c e  s i t e i n a p a r t i c u l a r s h e l l was a n e a r e s t doner, as c a l c u l a t e d  in  S e c t i o n A f o r a p a r t i c u l a r c o n c e n t r a t i o n Nd, was read i n as the a r r a y P. 1-10,  The exchange energy J was then c a l c u l a t e d ,  a c c o r d i n g to e q u a t i o n  f o r each c o l l e c t i o n o f " e q u i v a l e n t " s i t e s assuming an e f f e c t i v e  Bohr r a d i u s a* (=AD).  Subsequently, depending upon which range the  l o g ( j ) term corresponds t o , the p r o b a b i l i t y P m u l t i p l i e d by the number o f e q u i v a l e n t s i t e s L pT^T  array.  i s added to a c e r t a i n element of the  T h i s program i s run f o r a l l the e q u i v a l e n t s i t e s noted  i n Table VI, which e n t a i l s n i n e separate computations i n o r d e r to o b t a i n the f i n a l d i s c r e te F T £ T  , or N ( j ) , array.  -42" SLIS  THE NOJ 1 2 3. 4 5 6 7 8 9 10 11 12 13 14. 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52  100  101  2  104  6  7 8  PROGRAM I  D I M E N S I O N R R ( 2 5 0 ) , N O ( 2 5 0 ) , P R O B ( 2 5 0 ) , A L J ( 2 5 0 ) , P ( 2 5 0 ) , F T O K 2 5C REAL L G E , J J ( 2 5 0 ) AA=2.90E0 8 BB=2./9. CC=1 ./45 . ALPHA=5.3 .£=..85*3.1416 LGE=AL0G10(2.71828) READ ( 4 , 1 0 0 ) D I S T,AD FORMAT ( 2 F 1 0 . 2 ) N=DIST DO 2 I=1 ,N READ ( 5 , 1 0 1 ) RR ( I ) ,M0(I ) , P ( I ) FORMAT ( F 1 0 . 2 , I 1 0 , E 1 0 . 2 ) N O ( I ) =0' FTOT(I ) -0. . ' 'PROB(I)=0. CONTINUE M=INT(DIST/5.3) NJMAX=90 NJMIN=-15 NN=NJMAX+16 READ ( 4 , 1 0 4 ) GOD FORMAT ( F 1 0 . 2 ) I F (GOD.EO.O.) GO TO 8 DO 6 I=1,MN READ ( 3 , 1 0 2 ) J J { I ) , J J ( I + 1 ) , A L J ( I ) , A L J ( I + 1 ) , P R O B ( I ) CONTINUE DO 7 1 = 1 ,N READ ( 3 , 1 0 3 ) RR( I ) , N 0 ( I ) , F T O T ( I ) CONTINUE CONTINUE DO 1 I = 1 , M  A=I B=0 C=0 L= 6 R=ALPHA*SQRT(A**2+B**2+C**2) I F ( R . G T . D I S T + 1 ) GO TO 1 NR =R F=(COS(E*A)+COSIE*B)+C0S(E*C))**2 FTOT(NR)=FTOT(NR)+AL0G10(F)*L MO(MR)=N0(NR)+L D=R/AD FD=AA*BB*(1.+D)*(l.+D+D**2/3.)*EXP(-2.*D)/AD GD=AA*CC*(25./8.-5.75*0-3.*D**2-D**3/3i 2+(6./D)*((l,+D+D**2/3.)**2)*(.57777+ALOG(D)))*EXP(-2.*D)/AD X=F*(FD-GD) NJ = I N T ( 1 0 . * A L 0 G 1 0 ( X ) )  1  -4353 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 END OF  1  4 102  5 103  FILE  I F ( N J . L T . N J M I N ) GO TO 1 NJ =NJ+16 PROB(NJ)=PROB(NJ)+L*P(NR) CONT INUE ALJ(1)=-15 J J ( 1 ) = E X P ( A L J ( 1 ) / ( 10.*LGE ) ) ALJ(1)=-1.5 DO 4 1 = 1 , NN A L J ( T + 1) = ( 1 - 1 5 ) / 1 0 . JJ(1+1)= EXP(ALJ(I+1)/LGE) WRITE ( 6 , 1 0 2 ) J J ( I )*, J J ( 1 + 1) , A L J ( I ) ,ALJ( 1 + 1 ),PROB(I ) WRITE ( 7 , 1 0 2 ) J J ( I ) , J J ( I + 1 ) , A L J ( I ) , A L J ( 1 + 1 ) , P R O R ( I ) CONTINUE FORMAT ( 2 E 1 0 . 2 , 2 F 1 0 . 2 , 1 P E 1 0 . 2 ) DO 5 1=1 ,'N WRITE ( 6 , 1 0 3 ) R R ( I ) ,N0 ( I ) , F T O T ( I ) WRITE ( 7 , 1 0 3 ) R R ( I ) , N O ( I ),F TO T ( I ) CONTINUE FORMAT ( F 1 0 . 2 , I 1 0 , F 2 0 . 5 ) STOP END  -44A continuous approximation to t h i s N ( j ) d i s t r i b u t i o n n e c e s s i t a t e s d e t e r m i n i n g a s u i t a b l e average value f o r the i n t e r f e r e n c e  term i n  e q u a t i o n 1-10. As the d i s t r i b u t i o n o f n e a r e s t donors i s c a l c u l a t e d as a f u n c t i o n  o f l o g ( j ) r a t h e r than J , we c a l c u l a t e d  I  Co  ^V-*$)  and o b t a i n e d the a p p r o p r i a t e average value o f \S Cos. k^,""*.*") S  X  0  4663.  W r i t i n g J as a continuous f u n c t i o n  ,  to be  o f r , and assuming the  n e a r e s t donor d i s t r i b u t i o n g i v e n by equation  , the p r o b a b i l i t y N ( j )  t h a t a n e a r e s t donor p a i r has exchange energy i n the range 3", J d - i i s g i v e n by: N(J)  4 n Nji y  do =  4*  £  3  3  dlT  (A-3-6)  Therefore, upon i n t e g r a t i n g A-3-6 over a c e r t a i n range o f J (the same i n t e r v a l s as i n the d i s c r e t e  c a l c u l a t i o n are used) one should  o b t a i n a reasonable approximation to t h e d i s c r e t e d i s t r i b u t i o n . procedure di/dr  This  i s performed by Program II, where J i s denoted by X , and  by )(? .  The r e s u l t i n g p r o b a b i l i t y as c a l c u l a t e d  from equation  A-3-6 i s entered i n t o the a r r a y PNJ, which the second s e c t i o n o f the program i n t e g r a t e s  over the i n t e r v a l s o f J .  SLIS  . -45-  PROGRAM II  NOJ2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 . 27 28 29 30 31 32 33 34 35 36  C C 100  20  101 1  T H I S PROG. C A L C S . T H E F R A C T . # OF N R S T . DONOR P R S . EXCHANGE ENERGY " J " FROM. A P O I S S O N D I S T . OF N R S T . READ ( 5 , 1 0 0 ) N,AD,CON,RMIN,RMAX FORMAT .(110 , F 1 0 . 2 E 1 0 . 2 2 F 1 0 . 2 ) LOGICAL LI,L2 t  HAVING DONORS.  T  N = 300 D I M E N S I O N A L J ( 6 0 0 ) , P M J ( 6 0 0 ) .P.1(600) ...._ ... D I MENS I ON R R ( 6 0 0 ) REAL L J ( 6 0 0 ) DO 20 1 = 1 , 6 0 0 P N J ( I )=0 .0 CONTINUE AA=2 . 9 0 E 0 8 . BB=2./9. CC=l./45. F=.4663 DELR=(RMAX-RMIN)/N DO 1 I = 1 , N R = RMIN+( I - l ) * D E L R D=R/AD FD=AA*BB*(1.+D)*( l.+D+D**2/3. ) * E X P ( - 2 . * D ) / A D G D = A A * C C * ( 2 5 . / 8 . - 5 . 7 5 * D - 3 . * D * # 2 - D * * 3 / 3. + 6 . / D * ( 1. + D + D * * 2(.57722+ALOG(D) ) )*EXP.(-2.*D)/AD X= F* ( F D-GD ) XP=-2*X/AD+F*AA*EXP(-2.*D)/AD**2*(BB*((l+2*D/3)*(1+D)+ 2(l+D+D**2/3))-CC*(-5.75-6*0-D**2-6/D**2*(l+D+D**2/3)**2* 3 ( - . 4 2 27 8 + A L 0 G ( D ) ) + 1 2 / D * ( l + D + D * * 2 / 3 ) * ( l + 2 # D / 3 ) * ( . 5 7 7 + A L 0 G ( D ) ) 4)  ALJ(I)=AL0G10(X) PNJ(I}=-4.*3.1416*C0N*R**2*EXP(-4./3.*3.1416*C0N*R**3)/XP WRITE ( 7 , 1 0 1 ) R , A L J ( I ) , P N J ( I ) FORMAT ( 2 F 2 0 . 2 , E 2 0 . 5 ) CONTINUE  -46-  43 44 45 46 47 48 49 50 51 ' 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 END OF  10 11  21 ' 2 2 102 2  PJ(1)=0. • . DO 2 I = 1 , 2 0 0 L J ( I ) = (BO.-I )/10. DO 10 J = 1 , N L I = L J ( I ) .LE o A L J ( J ) L2=LJ(I).GE.ALJ(J+l) I F ( L I . A N D ,L2 ) GO TO 11 CONTINUE CONTINUE I F ( A L J ( J ) . E O . O . ) GO TO 2 i P'J ( I ) = ( L J ( I ) -AL J ( J ))=:=( P M J ( J + l ) - P N J ( J ) ) / ( A L J ( J + l ) - A L J 2)+PMJ(J) GO TO 22 CONTINUE PJU)=0. CONTINUE WRITE ( 6 , 1 0 2 ) I , L J ( I ) , P J ( I ) FORMAT ( I 1 0 , F 1 0 . 3 , E 2 0 . 5 ) CONTINUE STOP END • REAL L J ( 6 0 0 ) , J ( 6 0 0 ) , M J ( 6 0 0 ) , J O , L G E , L L J ( 2 0 0 ) NA=10 DIMENSION PMJ(200 ) COMMON P N J , L L J  200 202  '  DO 2 0 2 1 = 1 , 2 0 0 READ (5 , 2 0 0 ) K , L L J ( I ) , P N J ( I ) FORMAT ( I 1 0 , F 1 0 . 3 , E 2 0 . 5 ) CONTINUE N=200 LGE=AL0G10(2.71828) LJ(1)=80.  N  101 1 102  FILE  J(1)=EXP(LJ(1)/(10.*LGE ) ) AREA =0 . DO 1 1 = 1 , 2 0 0 LJ(I+1)=80-I J ( I + 1 ) = E X P ( L J (I + l ) / ( 1 0 . * L G E ) ) CALL SIMP ( J ( I ) , J ( I + 1 ) , N A , N J ( I ),CON,JO,ALPHA,u,T) AREA = AREA +NJ ( I ) WRITE ( 6 , 1 0 1 ) J ( I ) , J ( I + 1) , L J ( I ) , L J ( I + 1 ) , N J ( I ) FORMAT ( 2 E 1 0. 2 ,2 F 1 0 . 2 , 1PE 1 0 . 2.) CONTINUE WRITE ( 6 ,102 ) AREA FORMAT (//'AREA UNDER CURVE = « , F 1 0 . 4 ) STOP END  APPENDIX D CALCULATION OF THE THEORETICAL SPECTRUM T h i s c a l c u l a t i o n was performed by Program I I I .  The b a s i c  o p e r a t i o n s i n v o l v e d c o n s i s t e d o f b r e a k i n g up the energy i n t o i n t e r v a l s , c o n t a i n e d i n the a r r a y E, c a l c u l a t i n g the corresponding i n t e r v a l s o f exchange energy which were entered i n t o the a r r a y J , and  subsequently  i n t e g r a t i n g the e x p r e s s i o n g i v e n by equation 2-17 over these o f J to o b t a i n the a r r a y A. f o r which E= 3p^t>  intervals  S p e c i a l a t t e n t i o n was g i v e n to p o i n t s  and E = ^ p » i H  A/  2  as the exchange energy, w r i t t e n  as a f u n c t i o n o f energy, s u f f e r s d i s c o n t i n u i t i e s a t these p o i n t s .  The  elements o f t h e a r r a y A were then equated to the areas under the Gaussian  shape f u n c t i o n s g i v e n i n equation 2-17, and the h e i g h t o f  the h y p e r f i n e l i n e s and centre l i n e o b t a i n e d as t h e i r c o n v o l u t i o n .  The  value o f t h e r a t i o o f centre to average h y p e r f i n e l i n e was then obtained as RATIO.  The continuous N ( j ) courve f o r a p a r t i c u l a r c o n c e n t r a t i o n and  e f f e c t i v e Bohr r a d i u s as c a l c u a l t e d Program I I I as the a r r a y L U , PNJ.  by Program I I was read i n t o  -48PROGRAM I I I $L IS  F INAL 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 • 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 . 50  31  202 201 105  2 7 1  5  10 3  LOGICAL L1,L2,L3,LX DIMENS ION E ( 8 0 2 ) , A ( 8 0 2 ) REAL J ( 8 0 2 ) , J X ( 1 1 ) , LJ(600),J0,JH(21) DATA N , N X , N H , B / 2 0 , 5 0 , 5 0 , 1 1 8 . / CONTINUE READ ( 5 , 1 0 5 ) CON , JO,ALPHA,AD,RMAX,T IF (CON.EO.O.EO) GO TO 30 DIMENSION P N J ( 2 0 0 ) REAL L L J ( 2 0 0 ) COMMON P N J , L L J DO 202 1=1,200 READ ( 4 , 2 0 1 ) K , L L J ( I ) , P M J ( I ) CONTINUE FORMAT ( 1 1 0 , F 1 0 . 3 , E 2 0 . 5 ) FORMAT (2E10.2,4F1C4) DO 1 I=1 ,802 E ( I )=9799 . 5 + F L O A T ( I ) * . 5 IF ( E ( I ) . E Q . l . E 04 ) GO TO 2 J(I)=ABS(B**2/(4.*(E(I)-l. E 04))-E(I)+1.E04) GO TO 7 J ( I ) = E X P ( L L J ( 1 ) / A L 0 G 1 0 ( 2 . 7 1828) ) CONTINUE E(I)=E(I)+.25 CONTINUE J ( 2 83) = E X P ( L L J ( 2 0 0 ) / A L O G 1 0 ( 2 . 7 1 8 2 8 ) ) J ( 5 1 9 ) =J ( 2 8 3 ) DO 3 1 = 1 ,802 L I = E ( I ) .GE.9941 . L2=E ( I ) .GE . 1 0 0 0 0 . L3 = E ( I ).GE.10059. IF ( . N O T . L I ) M = l I F ( L I .AND..NOT.L2) M = 2 IF ( L 2 . A N 0 . . N 0 T . L 3 ) M = 3 IF ( L 3 ) M=4 LX=(I.E0.282).OR.(I.EQ.28 3).OR.(I.EQ.400).OR.(I.EO.401) 2.OR. ( I . E 0 . 5 1 8 ) . O R . ( I . E G . 5 1 9 ) IF ( L X ) GO TO 5 C A L L SIMP ( J ( I ) , J ( I + l ) , N , A ( I ) , C O N , J O , A L P H A , M , T ) GO TO 3 A(I)=0. J X (1 ) =A M IN1 ( J (I ) , J ( I +1 ) ) DO 10 K = l , 1 0 X = A L O G ( A M A X I ( J ( I ) , J ( I + 1) )/AM I M l ( J ( I ) , J ( I + 1) ) ) * F L O A T ( K ) / 1 0 J X ( K + l )=AM I M l ( J ( I ) , J ( I + l ) )*EXP ( X) C A L L SIMP (JX(K),JX(K+1),NX,AX,CON,JO,ALPHA,M,T) A(I)=A(I)+AX CONT INUE CONTINUE HYP=0. JH(1)=J(283) _  51 52 53 54 55 ij6 57 58 59. 60 61 62 63 64 65 66 67 68  69 70 71 72 '73 74 75 76 77 78 79 80 ai 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96  97 98 99 100 101  102 103 104  105 ?6  107 108 109 110  11  DO 11 K = l ,2 0 X=ALOG(J(401 )/J(283) )*FLOAT(K)/20. JH(K + l ) = J ( 2 8 3 )*EXP(X ) • CALL SIMP ( J H ( K ) , J H ( K + 1 ) , N H , A H , C O N , J O , A L P H A , 5 , T ) HYP =HYP+AH CONT INUE . HHBL =0 . DO 12 1 = 2 6 3 , 3 0 3 HHBL = H H B L + A ( I ) / ( S 0 R T ( 2 . * 3 . 1 4 1 6 ) * 8 . ) * E X P ( - • 5 * ( ( E ( 2 8 3 ) - E ( I ) ) / 8 . )**2 2)  12  13  CONTINUE HCL=0. DO 1 3 1 = 3 8 1 , 4 2 1 HCL =HCL +A ( I ) / (SORT ( 2 . * 3 . . 1 4 1 6 ) * 8. ) * E X P ( - . 5 * ( ( E ( 4 0 1 ) 2 ) CONTINUE HHBR =0 . DO  14  I = 4 9 9  -E ( I  ,539  HHBR = H H B R + A ( I ) / ( S O R T ( 2 . * 3 . 1 4 1 6 ) * 8 . ) * E X P ( - . 5 * ( ( E ( 5 1 9 ) - E ( I ) )/8 . ) 2 ) 14 CONTINUE H H L = H Y P / ( S O R T ( 2 . * 3 . 1416 ) * 8 . ) RATI0=2.*HCL/(2.*HHL+HHBL+HHBR)*100. HL=HHL+HHBL HR=HHL +HHBR WRITE ( 6 , 1 0 0 ) A D , J O , A L P H A , C O N , T , R MAX WRITE ( 6 , 1 0 2 ) N , N X-, N H , H L ,HCL,HR WRITE ( 6 , 1 0 3 ) R A T I O GO TO 31 30 CONTINUE STOP 100 FORMAT (///55X,•AD=' , F 5 . 1 , / / / 1 X , • J 0 = , E 1 0 . 2 , 9 X , 'ALPHA= ' , 2 F 1 0 . 3 , 7 X , « C O N = « , E 1 0 . 2 , 9 X , « T E M P = ' , F 1 0 . 2 , 8 X , 'RMAX=« ,F 1 0 . 1 / ) 102 FORMAT ( / 5 0 X , ' C H E C K I N G P A R A M E T E R S ' , / / I X , ' N = ' , I 8 , 3 8 X , N H = , 2 1 8 , 3 8 X , ' N H = ' , 1 8 , / / 2 X , 'HEIGHT OF L E F T H Y P E R F I N E L I N E = ' , 1 P E 1 0 . 2 , 3 / 2 X , ' H E I G H T OF C E N T R E L I N E = ' , 8 X , E 1 0 . 2 , / 2 X , 4 ' H E I G H T OF R I G H T H Y P E R F I N E L I N E = ' , E 1 0 . 2 / ) 103 FORMAT ( V / 2 X , ' R A T I O OF CENTRE L I N E TO HYP L I N E S ( B Y P E R C E N T ) 2 ) =«,F10.2,///) END S U B R O U T I N E S I M P { A , B, N, AR E A , CON, J 0 ,'AL PH A, M , T ) REAL JO REAL L L J ( 2 0 0 ) DIMENSION P N J ( 2 0 0 ) COMMON P N J , L L J AN = N H=(B-A)/AN SUM1=0.0 . SUM2=0.0 C A L L AUX ( A , Y , C O N , J O , A L PHA,M,T) YA =Y C A L L AUX {8 , Y , C O N , J O , A L P H A , M , T ) YB =Y X = A-H MN=N/2 " ' ; • DO 3 0 I = 1 ,NN X = X+2.0*H C A L L AUX (X , Y ,C ON , JO , A L P H A , M , T ) 30 SUM1=SUM1+Y . . ... X=A .. .... „• , ... •.. ... J  1  1  !  1  -50-  111 112 113 114 115 116 117 118 119 120 121 122 123 124 12 5 126 12 7 128 129 130 131 132 133 134 135 136 137 138 139 140 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 _170 171 172 173 END OF  40  300  302 301 303  2  3  4  5  1  DO 40 I = 2 ,NN X = X+2.0*H C A L L AUX ( X , Y , C O N , J O , A L P H A , M , T ) SUM2=SUM2+Y AREA = H / 3 . 0 * ( Y A + 4 . 0 * S U M 1 + 2 . 0 * S U M 2 + Y B ) AREA=ABS(AREA) RETURN END S U B R O U T I N E AUX { X , Y , C O N , J O , A L P H A , M , T) R E A L J O , MINUS DATA BOL , A H / 2 . 0 8 E 4 , 1 1 8 . / REAL L L J ( 2 0 0 ) DIMEMS I ON P N J ( 2 0 0 ) COMMON P N J , L L J ALX=AL0G10(X) I F ( A L X . L T . O . ) GO TO 3 00 N=80-INT(10.*ALX) I F ( N . L E . 1 ) GO TO 3 0 2 GO TO 301 CONTINUE N=81-INT(10,*ALX) I F ( N . G T . 2 Q 0 ) GO TO 3 0 2 GO TO 3 0 1 CONTINUE PJ=0.0 GO TO 3 0 3 CONTINUE P J = - 1 0 . * ( A L X - L L J ( N - 1 ) ) * ( P N J ( N ) - P N J ( N - 1 ) ) + P N J ( M-1) CONTINUE RT=SORT(X**2+AH**2) PLUS=.5*(1.+X/RT) MINI)S = . 5 * ( 1 . - X / R T ) I F (M-.GE .2 ) GO TO 2 Y = M I N U S * P J * A 8 S ( l . - E X P ( ( 2 . E 4 - X - R T ) / ( 2 . * B O L * T ) )) GO TO 1 CONTINUE I F ( M . G E . 3 ) GO TO 3 Y=PLUS*PJ*EXP(-(X+RT)/(2.*B0L*T))*(1.-FXP(-1.E4/(BOL*T))) GO TO 1 CONTINUE -IF (M .GE .4) GO TO 4 Y=PLUS*PJ*EXP(-(X+RT)/(2.*B0L*T))*(EXP(1.E4/(B0L*T))-1.) GO TO 1 CONTINUE I F ( M . E 0 . 5 ) GO TO 5 Y=MINUS*PJ*(1.-EXP(-(2.E4+X+RT)/(2.*B0L*T))) GO TO 1 CONTINUE Y=.5*PJ*EXP(-(X+RT)/(2.*B0L*T)) 2*(EXP(1.E4/(BOL*T))-EXP(-1.E4/(BOL*T))) CONTINUE RETURN END  FILE  

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