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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 April, 1970 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a D a t e ( i i ) ABSTRACT The e.p.r. spectrum for r e l a t i v e l y dilute samples of phosphorus-doped s i l i c o n (<5 x 10 donors/cm ) has been calculated in detail for an assumed random di s t r i b u t i o n of impurities. The system of donor electron spins i s treated as a collection of nearest neighbor donor pairs. An expression i s derived for the donor pair exchange energy using Kohn-Luttinger -wavefunctions and a general exchange energy expression. The resultant relationship contains an adjustable parameter a , the "effective Bohr radius", which i s determined from a comparison of the calculated spectrum and the experimental results obtained for the r a t i o , C, of the "central p a i r " and "hyperfine" l i n e i n t e n s i t i e s . The resulting expression J(R), where J represents the exchange energy and R the separation vector connecting the two pai r donors, exhibits an o s c i l l a t o r y spatial dependence due to interference from portions of the wavefunction a r i s i n g from diff e r e n t conduction band valleys. ( i i i ) TABLE OF CONTENTS PAGE ABSTRACT i i TABLE OF CONTENTS i i i L IST OF FIGURES v L IST CF TABLES v i ACKNOWLEDGEiiENTS . . v i i CHAPTER I. INTRODUCTION 1 I I . THEORY . 3 A. E l e c t r o n Spin Resonance . . . . . 3 B. The Wavefunct ions o f the Impur i ty S ta te s . . . . 4 C. The Neares t Donor Approx imat ion 5 D. The Exchange I n t e r a c t i o n . . 6 E. The Donor P a i r System . . . . . 7 I I I . CALCULATION OF THE SPECTRUM 9 A. I n t r o d u c t i o n . . . < > • 9 B. M a t r i x Element C a l c u l a t i o n s 10 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 Between T r a n s i t i o n L e v e l s 18 D. The T h e o r e t i c a l Spectrum . . 21 IV. EXPERIMENTAL METHODS AND RESULTS 22 A. Apparatus and O p e r a t i n g Cond i t i on s 22 B. Exper imenta l R e s u l t s 25 C. P o s s i b l e E r r o r 29 V . DISCUSSION OF RESULTS 30 BIBLIOGRAPHY 31 APPENDIX A. POPULATION DIFFERENCES BETWEEN TRANSITION LEVELS . . . 32 B. CALCULATION OF EXCHANGE 34 (iv) APPENDIX PAGE C. THE DISTRIBUTION OF NEAREST EXCHANGE COUPLED PAIRS . . 38 a. Calculation of the Number of Lattice Sites i n Shells of Unit Thickness Centred on the Origin 38 b. Determination of the Nearesit Neighbour 41 Distribution . D. CALCULATION OF THE THEORETICAL SPECTRUM 47 (v) LIST OF FIGURES FIGURE • ' PAGE 1. Energy Levels of a Spin System . 3 2. Energy vs. k Rel a t i o n i n [J-JOJ 0^ D i r e c t i o n 4 3. Spin System of the P a i r . . . . . . . . . . . . . . . ? 4. Spectrum of the P a i r System . . . . . . . S 5. Energy Level Diagram f o r a Phosphorus Donor P a i r . The Eigenstates are Labelled Numerically From 1 to 16 Along With the Usual Strongly Coupled P a i r States to Which Each Reduces i n the L i m i t J» A . . . « • • ^ 6o Spectral Contributions of Allowed Transitions . . . . . 12 7. Comparison of Discrete and Continuous N ( j ) ( a =16.5$. ) . 20 8. Introd u c t i o n of the Gaussian Shape Function . . . . . . 21 9. Block Diagram of Experimental Apparatus . . . . . . . 24 10. Mode Shape and Sample Cavity Resonance . . . . . . . . 22 11. Experimental ESR Derivative Trace f o r Nd»3.7 x 10 1 6/cm 3 . . 26 12. A P l o t of Experimental Points and T h e o r e t i c a l l y Calculated Values of 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 Line Represents the Calculated Ratio f o r an E f f e c t i v e Bohr Radius a*=17.3A° 27 13. A P l o t of the Normalized D i s t r i b u t i o n of P a i r J Values N(J)/No/2 as a Function of the Exchange X 6 3 Energy J f o r a 4x10 Donors /cm Sample ( S o l i d Curve) and a 6x10^ Donors /cm3 Sample (Dashed Curve) 28 14. Simple Face Centred Cubic L a t t i c e Structure 38 15. S i m p l i f i e d Energy Level Diagram of the Donor P a i r . . . 32 (vi) LIST OF TABLES TABLE PAGE I. Matrix Representation of the Hamiltonian of the Donor Pair System • 13 II* Eigenfunctions of the Donor Pair 14 II I . A Comparison of (i) the Computer Projected Donor Pair Eigenvalues M i e n the Off-diagonal Hyperfine Elements are not Neglected to ( i i ) the Functional Derivation Where These Off-diagonal Elements are Neglected IV. Donor Pair Transitions Having Non-zero Transition Probability. The Relative Transition P r o b a b i l i t i e s and the Transition Energies,A, are _ .9 1 0 Given and Ye=gBHo V. Boltzmann Population Differences Between Transition Levels 33 VI. Number of Latti c e Sites Having the Same Absolute Value for Projections on the x, y, z Axes i n a Crystal with Interlocking F.C. C. Structure . . . . . 40 ( v i i ) ACKNOWLEDGEMENTS I would l i k e to thaifcDr. John R. Marko for his suggestion of the thesis topic and subsequent encouragement through a l l phases of the problem. Thanks also go to Dr. R. Banrie, who was a c r i t i c a l sounding-board for many aspects of th i s thesis. I would l i k e to express gratitude to the National Research Council for t h e i r award of a Post Graduate Studentship for the duration of th i s work. The research for t h i s thesis was supported by the National Research Council, Grant Number 67-4624. -1-CHAPTER I INTRODUCTION The electron spin resonance (E.S.R.) spectrum of phosphorus 1 o 18 doped s i l i c o n (P-Si) in the concentration range 10 <N<10 " ' 3 impurities /cm demonstrates a "-weak centre peak" that l i e s midway between the hyperfine l i n e s of the isolated donors. Slichter* attributed this peak to coupled pairs of neighbouring impurities that act as a unit with a total 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 clusters of three or more donor atoms. The degree to which two interacting spins may be considered to act as a pair i s given by the exchange energy " J " between them. Slichter's calculation included only those pairs for which J ~ A , where A i s the hyperfine interaction. I t i s the purpose of this work to determine the functional dependence of J on " r " , the interdonor separation, for a l l values of r. This functional dependence i s constructed sa as to contain a single adjustable parameter a , the "effective Bohr radius" of the impurity electron, which i s subsequently f i t t e d to experimental resu l t s . Motivation for the determination of J arises partly from the "spin d i f f u s i o n " mechanism that can transport energy from one part 2 of a spectral l i n e to another , presumably via a f l i p - f l o p of neighbouring spins. I t has been proposed that the bulk of electronic spins relax through such a d i f f u s i o n of spin and energy to other spin centres with very short spin l a t t i c e relaxation times, "T,,f. 3 One of these fast relaxing centres i s suspected to be the previously _2 mentioned highly coupled pair, with T^04 J . A functional knowledge of J i s therefore essential before detailed studies of the spin d i f f u s i o n relaxation mechanism can be i n i t i a t e d . Attention has 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 of P—Si and a. which extends w e l l outside t h i s spectrum. This l i n e i s thought to a r i s e from exchange coupled p a i r s f o r which J^A . I t i s shown that a t l e a s t p art of t h i s l i n e i s due to such p a i r s . -3-CHAPTER I I THEORY A. El e c t r o n Spin Resonance The basic energy l e v e l s of the unpaired electrons of 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 spin quantum number h/12=energy d i f f e r e n c e between states N =number of electrons i n upper _ state . N =number of electrons i n lower state Figure 1. Energy l e v e l s of a spin iV system. I f t h i s system i s subjected to a microwave f i e l d at 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 absorbtion of energy (ESR) can be ( \ 4 / v. assuming no saturation) as: (1-D where C_^_9^ i s the matrix element connecting the s t a t e s , g(fl2) i s a "shape functi o n " due to the f i n i t e width of the energy l e v e l s , and n i s the steady state excess number of electrons i n the ground ss s t a t e . Assuming Boltzmann population 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 of spins, k i s Boltzmann's constant, and -4-T i s the absolute temperature i n K. Examination o f equation 1-2 r e v e a l s t h a t the 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 hence the l a r g e s t ESR s i g n a l s are obtained a t h i g h magnetic f i e l d s and low temperatures. B. The Wavefunctions o f the Impurity S t a t e s S i l i c o n has s i x conduction band minima i n "k" space t h a t l i e a l o n g each o f the d i r e c t i o n s £l,0 ,o] , [-1>0,u] , [ o , l , o ] , , _0,0,-]] as shown i n Figure 2. a= 5 . 3 A i n s i l i c o n _ 5=conduction band minimum Fig u r e 2. Energy vs k r e l a t i o n s h i p i n the [ l , 0 , o ] d i r e c t i o n . The wavefunctions of the impurity s t a t e s can be represented by the f o l l o w i n g l i n e a r combination o f the wavefunctions a t each o f these conduction band minima: (1 - 3 ) where ^ n r e p r e s e n t s the c o n t r i b u t i o n o f each minimum to the t o t a l wavefunction, and (_) i s a p e r i o d i c f u n c t i o n o f r_ corresponding t h to the n minimum. i'(_) i s an envelope f u n c t i o n s a t i s f y i n g a hydrogen-like Schroedinger 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 a t (1-5) where a and b are, r e s p e c t i v e l y , the transverse 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 radius that enables equation 1 - 5 to be rew r i t t e n as: -_ _ L _ e~** / ( i - 6 ) where a 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 considerations of symmetry and the experimentally observed ground state hyperfine s p l i t t i n g that 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 state as: iko (n) T The i n t e r f e r e n c e term e — -~ i n equation 1-7 w i l l be retained e x p l i c i t l y i n our furth e r c a l c u l a t i o n s * C. The Nearest Donor Approximation This study assumes a random d i s t r i b u t i o n of 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 prev i o u s l y been shown to be reasonably j u s t i f i e d i n the concentration range of 7 i n t e r e s t . Further, we assume that the exchange i n t e r a c t i o n J between donor electrons i s non-negligible f o r nearest neighbours only. Obviously, t h i s approximation becomes inaccurate when some impurity s i t e "A" i s the same distance from neighbouring s i t e s "B" and "C", but i t i s found t h a t , f o r our concentration range, non-n e g l i g i b l e p r o b a b i l i t i e s of such c l u s t e r s a r i s e only when A, B and -6-C are so widely separated as to act l i k e i s o l a t e d donors. This approximation i s also j u s t i f i e d by the absence of l i n e s due to such c l u s t e r s i n our spectra. 8 D. The Exchange I n t e r a c t i o n The nearest 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 of the two electrons concerned, and: - - [ [ k ^ i 1 ^ c o W o *fi*> V l i (1-9) \*(^ dv, - \X^(-£) ^  Ml. ( i - i o ) where^T^, ^ b> a r e * n e wavefunctions of electrons 1 and 2, and r i 2=l£l - r j . We can now c a l c u l a t e the exchange energy J as a function of interdonor separation x_ by s u b s t i t u t i n g the impurity wavefunctions of equation 1—7 into equation 1-9. This i s done i n Appendix I I , where we obt a i n : J The f a c t o r represents an in t e r f e r e n c e e f f e c t due to co n t r i b u t i o n s from various conduction band minima. (1-H) -7-E. 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 of large (J>">A) and small ( J « A ) exchange energy. The spin system i s given i n Figure 3. _»static magnetic f i e l d _ , _> are the i n d i v i d u a l nuclear spins _ j ? £>2 a r e the i n d i v i d u a l e l e c t r o n i c spins of the p a i r Figure 3. Spin system of the p a i r . The Hamiltonian f o r t h i s system can be w r i t t e n as: H = 3 p U [ S,_ *.§__1 + A ^ T S , + __-S_^ 4-TS;S_ (1-12) where g i s the el e c t r o n i c g f a c t o r and ^  i s the Bohr magneton. The much smaller nuclear Zeeman and nuclear spin-spin i n t e r a c t i o n s are neglected. In the l i m i t J « A , the r e s u l t i n g spectra i s obviously that of the i s o l a t e d donor, which gives two hyperfine peaks at 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 . For 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 a good quantum number. Neglecting the off-diagonal components of almost" the hyperfine 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 values of J ( c . f . Chapter I I , Section B), equation 1-12 becomes: ^ - gpHS, ^ ^ ( ^ ^ i J ^ T C ^ - ^ (1-13) w i t h eigenvalues: J> -+ A (1-14) a n d S © S • -e -2. 4s H 1 i (1-15) -8-where Ex, ES are the t r i p l e t and s i n g l e t energies (using the usual n o t a t i o n f o r the ground state of the hydrogen molecule). M i s the t o t a l e l e c t r o n i c spin p r o j e c t i o n quantum number, and mj. , m2 are the i n d i v i d u a l nuclear spin p r o j e c t i o n numbers. The i n t r o d u c t i o n of a microwave f i e l d adds a perturbation to the Hamiltonian $1 of the form: Vtr.i '- $tl< C s + - ^ *-.<*t) d - 1 6 ) z where S i s the electron spin r a i s i n g operator, S~~ i s the e l e c t r o n spin lowering operator, and w / 2 ^ i s the frequency of t r a n s i t i o n . C l e a r l y , t h i s perturbation induces t r a n s i t i o n s such thatAM=-l, A(m_+m2) =0. The t r a n s i t i o n AS=l i s not allowed. This i s due to the f a c t that the spin wavefunctions of the t r i p l e t state are symmetric under interchange of electrons, whereas those of the ground state are antisymmetric under the same operation. Thus "&r.f ^ symmetric under t h i s interchange, cannot connect the s i n g l e t to the t r i p l e t because of p a r i t y considerations. Note that when J ~ A, hoArever, S i s no 2 longer 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 of Figure 4. Figure 4. Spectrum of the p a i r system. Note that the spectra 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. -9-CHAPTER I I I CALCULATION OF THE SPECTRUM. A. Introduction The spectrum of P-SI i s inhomogeneously broadened due to hyperfine 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 natura l abundance). The f o l l o w i n g c a l c u l a t i o n w i l l assume the existence of the "spin packets" proposed 9 by P o r t i s . Each such spin packet c o n s t i t u t e s a homogeneously broadened l i n e corresponding 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 f i e l d , and has l i t t l e i n t e r a c t i o n w i t h those packets corresponding to d i f f e r e n t l o c a l f i e l d s . " This i s a questionable assumption and should be considered i n more d e t a i l i n future st u d i e s . For each of these i. >n 9 . packets the s u s c e p t i b i l i t y T = 7f +'» fi i s given by i fi'(u) _ j . X 0 * * U 0 7 _ Cu a -H> (2-1) (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 free e l e c t r o n gyromagnetic r a t i o , Hi i s the microwave magnetic f i e l d i n t e n s i t y , T_ and T2 are the s p i n - l a t t i c e and spin-spin 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 central resonant f i e l d of the spin packet. Experimentally, we observe d"fi'/dll. Thus, r e a l i s i n g that f o r our low concentration samples (<4xl0 1 6/cm 3) V 2R^ T^T^V> x^(^~RQ)^ ^hen » w e obtain from equation 2-1 that: and thus from equation 2-2, [d w'Cv^ A - ^ x T r ' X ^ ( 2 " 4 ) -10-9 Then, given that the rate of power absorbed by the sample i s $ A = 2 V Wo ft"0O (2-5) we obtain from equations 1-1 and 2-4 t h a t : fdfi'fro) a * S 5 | c . ,1 f 3<y-«.} (2-6) where we assume a shape funct i o n g(H-H^) f o r the homogeneously broadened l i n e of each spin packet to be of 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 scale model of the spectrum therefore involves determining each of the terms on the r i g h t hand side of equation 2-6 over each i n t e r v a l of H corresponding to a spin 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 of the matrix elements of the allowed 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 eigenfunctions of the donor p a i r system over the whole range of J . Following the i n d i v i d u a l spin notation introduced i n Figure 3, we can w r i t e the general form of the eigenfunctions as * ^ = <^ cTi«Si l ^ C H i ^ R ^ 9 > (2-7) £,i»M = i where Mj, Mg are the "z" components of the e l e c t r o n i c spins and Sg, and m^ are the "z" components of the nuclear spins 1^ and __g, and theC i i V lJ3^S are normalized constants g i v i n g the c o n t r i b u t i o n to the eigenfunction from each of the basis s t a t e s . There are 16 basic wavefunctions ^ijHA =|McMimv^ mi^ > and correspond-i n g l y 16 C ^ ^ i ' s f o r a given /H^r° . The matrix representation of the Hamiltonian of the donor p a i r spin system (equation 1-12) i n t h i s b a s i s i s given i n Table I . This matrix was-: diagonal ized v i a computer and the corresponding eigenvalues and eigenfunctions determined f o r p a r t i c u l a r values of J . The eigenfunctions compiled i n Table I I have a p a r t i c u l a r form which guarantees that they be i n v a r i a n t under the -11-siraultaneous interchange of both electron and nuclear spins ( t h i s operation commutes w i t h the Hamiltonian given i n equation 1-12.). As t h i s c a l c u l a t i o n of the eigenvectors and eigenvalues of f o r each value of J was tedious 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 of a hybrid method. L a b e l l i n g the basis vec tors f in by unprimed m,n * s, and the eigenvectors ( i n the diagonal representation) by m', n', we w r i t e (2-8) -I where the Cmm' have already been calculated 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> = C n \ i ^ \ ^ > Cv™' (2-10) which we express more co n c i s e l y as: E ^ - Cntn> = 5^ u ™ C M m » (2-n) mn In accord with equation 2-11 the n**1 row of Rnm was m u l t i p l i e d by the column vector \tn*y and the equation was solved f o r the corresponding Em1. 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 neg l e c t i n g the off-diagonal elements of the hyperfine 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 projected by computer ( i n which the off—diagonal elements were not neglected) and the ca l c u l a t e d Era1 i n the region J~A, w i t h g r a t i f y i n g r e s u l t s (See Table I I I ) . The functional Em' "* s are given i n Table I I and the energy l e v e l s tructure i n Figure 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 spin s t a t e s , and S , r e f e r to s i m i l a r -12-nuclear p a i r spin s t a t e s , i n the J » A l i m i t . The non-zero values of the matrix elements connecting the various states under the influence of 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 given i n Table IV, together with the corresponding t r a n s i t i o n energy Examination of A reveals the d i f f e r e n t regions of the spectrum to which each t r a n s i t i o n c ontributes. This i s i l l u s t r a t e d i n Figure 6. • i s - * u -=> 11 . • • Figure 6. Spectral contributions 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 at 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. -13-TABLE I Matrix Representation of the ITarail tonian of the Donor P a i r System / \ -I"4 o -in - T 1 >^ -trl i -t* -I"* ft ir 1 -in A -\« A 7 J A -I" o ( ? H '1" 1 -|N •'I •v-TL b 1 A "» A A 7. A i X _ T T z -a J 2. A 2. - T *_A 3 z A i _ T 4 I 2. A 2. A 2. 3" _7 •* A z X -I A z. I -7 2. J JL •"V A i . A i . A X • f t 3-A z "TH, *• T A 2. H i + 1 TABLE II Eigenfunctions of the Donor Pair N l -NI- N»-i N l " <7 i N l -K | -»/'-i K l -i N l -i M>— NI-i N l -< > Nh 1 H'-Nl -. NH 1 Ml-< y 1 Nl-K l -t \ y Kl" i f i -i HI-<^ i 1 i ei-i 1 vi-<r Energy 1 YAM J + ^ 2 3 4 1 5 T 6 All 7 a, Ox 0a. a. o O a . - Q , 9 % i T 4-1 0 _ 3 J 4 11 a, - f t , 1 2 a * - o . -0, 1 ' i 1 3 \ 1 4 A l l 1 5 to 1 6 i -15-TABLE III A Comparison of ( i ) the Computer Projected Donor P a i r Eigenvalues When the Off-diagonal Hyperfine Elements are not Neglected to ( i i ) the Functional D e r i v a t i o n Where These Off-diagonal Elements are Neglected. Jtate J = 25 MHz J = 400 MHz ( i ) ( i i ) ( i ) ( i 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. 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. -16-TABLE 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 P r o b a b i l i t y . The Rel 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 Given and V =gPH . e & r o T r a n s i t i o n 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 y T r a n s i t i o n Energy ( A ) at (l6)<t» ( 9)' 1 Y e 2 b: (l3)<-» I 5) 1 Y e + 2 c: ( 9)e> I 4) 1 9 A V* ~ 2 ds ( 5)c-* [ 1) 1 V « + 2 e: (l5)<-° ( 8) Hi + J ( J 2 + A 2)"^) +?r(J + A )- - J 2 f: (14)«H>! ; 7) ti it g: ( 8)«- l 3) II -&(J + A ) + J 2 hi ( 7)<H> ; 2) II it i : (I5)c— [ID Hi - J ( J 2 + A 2)"*) II j : (14)*-* (12) II it k* (12)«-*| ; 2) II • 4 ( J 2 + A 2 ) ^ + J 2 1: (11)*- ( 3) II II -17-FIGURE 5 STATE NO. STATE ENERGY fe=gpH) 1 — : — IT„t,> J Ye+A A + 2 2&3 ITl,t0>,ITJs> 4 - — - — — IT,,t.> Y e 4 A 2 5&9 lT 0 l UIT 0 J t . i> J 4 7 & 8 ITOjto>,!T0js> - J + 1 4 2 \/j2+A2 16 — IT^t.) + J + A 2 14&15 — I T ^ O J T ^ ) ' J 13. — IT-^t,) + J -re* 4 A 2 6&I0 ISJt1>,lSJt^> 3J 4 11&I2 • IS.to^lS.s) Energy Level Diagram f o r a Phosphorus Donor P a i r . The Eigenstates are Labelled Numerically From 1 to 16 Along With the Usual Strongly Coupled P a i r States to Which Each Reduces i n the L i m i t J A. -18-C. C a l c u l a t i o n of the Population Difference Between T r a n s i t i o n Levels The population 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 given by n S 3 as defined by equation 1-2. The determination of n s s thus e n t a i l s a knowledge of the r e l a t i v e number of p a i r s that contribute to a given spin packet, and also of the nornxtlized d i s t r i b u t i o n of spins i n the corresponding upper and lower states at e q u i l i b r i u m . Appendix A deals 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 energies /A can be w r i t t e n i n terms of J (Table IV), we s h a l l c a l c u l a t e N ( j ) , (the normalized d i s t r i b u t i o n of nearest donor p a i r s having J values 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 of p a i r s c o n t r i b u t i n g to a given spin packet. The nearest 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 of i m p u r i t i e s i s given by Chandresklar*^ as: W(t) dv - e ~~a NAT* <U (2-12) where i s the concentration of 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 that the nearest neighbour to a given impurity l i e s i n the sph e r i c a l s h e l l between r and r + dr. Noting that the f a c t o r 4"wtyi*^/3 i n equation 2-12 expresses the number of impurity s i t e s i n s i d e a radius r , we w r i t e : where n ( r ) i s the t o t a l number of l a t t i c e s i t e s i n s i d e " r " , and N s j i s the concentration of l a t t i c e s i t e s . S i m i l a r l y : *Vir dv •= _ . n£^r + du} (2-14) and therefore equation 2—12 can be re w r i t t e n as: . n M WCv") = KLji ft^ + i / ) e - N n (2-15) The p r o b a b i l i t y that 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 -19-a nearest donor impurity s i t e i s then given by: - _ . Y i ( , r y Tbo - _ e N«r ( g _ 1 6 ) N s i For p a r t i c u l a r values of and a*, the determination of 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 that t h i s l a t t i c e s i t e i s a nearest donor impurity s i t e (as given by equation 2-16) i s associated with t h i s value of J . This a n a l y s i s can then be performed over a l l possible 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 into c e r t a i n i n t e r v a l s ) added i n a corresponding array. Thus, for a p a r t i c u l a r concentration and e f f e c t i v e Bohr radius, 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. In order to make further 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 taking an appropriate average of the interference term i n equation 1-11 ( c . f . Appendix C). A comparison of 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 Figure 7. (The rapid f a l l - o f f of the d i s c r e t e curve i n the region logJ<-l i s simply due to the f a c t that not a l l the p a i r s i n t h i s region were considered, due to l i m i t a t i o n s of computer time.) The problem of determining the r e l a t i v e number of spins that contribute to a c e r t a i n spin packet i s now reduced to an i n t e g r a t i o n of N(j) over the width of the packet expressed i n terras of J . This i s combined with the nasalized Boltzmann population d i f f e r e n c e s between the l e v e l s (Appendix A) to give ^ss of equation 2-6. - 2 0 -10* 10" 10" J (MHz) FIGURE 7 Comparison of Discrete and Continuous N(J)(a*=16.5i) D. The Theoretical Spectrum The f i n a l step i n the calculation consists f i r s t of integrating the absorption A over one spin packet. We obtain N- where: (2-17) where E ( j j ) - E(J 2) i s the spin packet width, V f t (j,6,T) i s the normalized Boltzmann population difference between the i and j levels, M (j) i s the matrix element connecting the i and j states, and T i s the temperature in °K. With reference to Figure 8, the Gaussian shape function i s then introduced. Transition Gaussian shape function ttS,> E C l ^ Figure 8. Introduction of the Gaussian shape function. The total absorption Nj, i s taken to be the area under the Gaussian shape function, with half width equal to the spin packet width (~ 8ttW'i). The convolution of a l l these contributing Ga ssians i s then calculated and the resulting scaled spectrum i s obtained. Appendix D presents the computer program that i s used to perform these operations. Subsequently, the ratio "C" of the height of the centre lin e to the average height of the hyperfine l i n e s i s calculated. This ratio i s sensitive to the value of the "effective Bohr Radius" a*, and i s p a r t i c u l a r l y easy to determine experimentally. -22-CHAPTER IV EXPERIMENTAL METHODS AND RESULTS A. Apparatus and Operating Conditions The gross features of the ESR spectrometer employed are seen i n Figure 9. The k l y s t r o n i s a Varian Associates r e f l e x type, d e l i v e r i n g approximately 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 automatic frequency control (AFC) that was l o c k e d " to an external c a v i t y of high g and va r i a b l e frequency, enabling observation i n the di s p e r s i v e mode. I f we consider the equivalent c i r c u i t to a resonant c a v i t y , %' i s proportional to the change i n resonant frequency of the c a v i t y as the resonant condition i s swept through. I t was found that the most s e n s i t i v e p o s i t i o n of the external c a v i t y was approximately one h a l f way up the resonant c a v i t y as shown: Mode Shape Sample Cavity Figure 10. Mode shape and sample c a v i t y , where the 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 detector) per u n i t change i n c a v i t y frequency w i l l presumably be maximum. The brass c a v i t y used was designed to operate i n the TE^^g mode and was gold plated by an immersion type gold 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 , -23-and two pyrex f i l t e r s s i t u ated i n the waveguide j u s t above the coupling hole shielded the sample from i n f r a - r e d r a d i a t i o n * . F a i r l y consistent s e m i - c r i t i c a l coupling to the c a v i t y was achieved as long as these f i l t e r s were at l e a s t 1 cm. above the coupling hole. The samples used had dimensions ^ 10 x 4 x 4 mm. and were held to the centre of the bottom of the c a v i t y by vacuum grease. 4 Low temperatures were achieved by pumping on l i q u i d He that was tra n s f e r r e d into an inner dewar containing the resonant c a v i t y and waveguide. The pump used had a 150 cubic f t . / r a i n , capacity, and pumped on a 6" l i n e . Temperatures achieved were found to be c o n s i s t e n t l y close to 1.05°K during the course of the experiments. F i e l d modulation c o i l s that enabled observation of the d e r i v a t i v e of the di s p e r s i o n were operated at 400Hz and amplitude .5 Oersteds. The input to these c o i l s also served as the reference for the l o c k i n a m p l i f i e r , to which was fed the output of 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 provided by a Newport Instruments 8" electromagnet with v a r i a b l e sweep c o n t r o l . FIGURE 9 Block Diagram of Experimental Apparatus for A fit.") Hoicji c T Cry HA f A o A n l o ^ o n C o i l s Amp 1 ' ? ' ** _J I rt uer.c * fW.o (XollftVo, ro f - 2 5 -B. Experimental Results The ESR spectrum was obtained at 1.05°K f o r four samples 16 3 •with concentrations of .8, 1.7, 2.3 and 3.7 x 10 donors /cm . X 0 3 A t y p i c a l spectra for Nd=4 x 10 /cm i s shown i n Figure 1. The experimental r a t i o n C of the centre l i n e height to average hyperfine l i n e height was p l o t t e d as seen i n Figure 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 radius a* was then c a l c u l a t e d , and the best f i t of the experimental r e s u l t s was obtained f o r a*=17.3A°. This 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 Figure 12. Using t h i s value of a* and the t h e o r e t i c a l spectrum C X Q 3 was c a l c u l a t e d to be 20.2 % f o r a Nd=3.7 x 10 donors/cm sample at 4.2°K. This i s i n good agreement with the experimental value of 20 - 2%. Using t h i s best f i t value of 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 are presented i n Figure 13. - 2 6 -P1GURE 11 Experimental ESR Derivative Trace fcrNd=3.7 x 10 /cm FIGURE 12 A P l o t of Experimental Points and T h e o r e t i c a l l y Calculated Values of the Ratio C( /o) of the Central P a i r Line I n t e n s i t y to the Average In t e n s i t y of the Hyperfine Lines. The Line Represents the Calculated Ratio f o r an E f f e c t i v e Bohr Radius *=17.3A. - 2 8 -FIGURE 13 A P l o t o f the Normalized D i s t r i b u t i o n of P a i r J Values N(j)/No/2 as a Function of the Exchange Energy J f o r a 4 x 1 0 1 6 Donors /cm 3 Sample ( S o l i d Curve) and a 6 'x 1 0 1 6 Donors /cm 3 Sample (Dashed Curve) -29-C. P o s s i b l e Error The measurement of the sample concentrations were made through a standard four point probe r e s i s t i v i t y technique. These measurements had a p o s s i b l e error of - 5jot which, together with p o s s i b l e experimental errors i n measuring the r a t i o , are reflcected by the er r o r bars i n Figure 12. Values of a* were then f i t t e d to the l i m i t s of these error bars to define the accuracy of a*. We thus obtained a*=17.3A°±.2A°. Ad d i t i o n a l errors may be due to a change i n passage conditions from hyperfine to centre l i n e s due to the more highly coupled and consequently f a s t e r r e l a x i n g p a i r systems that contribute to the centre l i n e . We therefore changed several parameters of the observation mode, notably the sweep rate and magnetic f i e l d modulation frequency, whixh was var i e d from 30 Hz to 3000 Hz. Within experimental error the r e s u l t i n g values of the r a t i o were found to be independent of these changes i n obervation parameters. I t was also found that the r a t i o "C" was s i g n i f i c a n t l y increased f o r higher microwave powers. This can be a t t r i b u t e d to a spin 3 d i f f u s i o n phenomena that would increase the f a s t e r r e l a x i n g centre l i n e absorption at the expense of the more slowly r e l a x i n g hyperfine l i n e s . To avoid such d i f f i c u l t i e s the r a t i o C was measured at the lowest microwave powers consistent w i t h useable signal to noise. -30-CHAPTER V DISCUSSION OF RESULTS The experimentally determined effective Bohr radius a*=17.3A° i s very close to 17.2A0, which i s the arithmetic mean of the longitudinal and transverse effective Bohr r a d i i of the anisotropic 12 ground state impurity wavefunction . This agreement would seem to indicate that the effective Bohr radius i n overlap calculations should be set equal to such an arithmetic mean rather than a 13 geometric mean which has been used in previous calculations . 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) for a similar sample, the most probable value of J i s seen to be 50 Mil . Part of this discrepancy may be due to the complications introduced into the interpretation of Endor results due to the J dependent electronic spin l a t t i c e relaxation time. But more importantly, i t might be pointed out that Endor measurements only involved those parts for which J?A, thus ignoring the great bulk of more weakly coupled p a i r s . In conclusion, the excellent agreement that i s achieved between the calculated and experimental results (Figure 13) provides confidence i n the isotropic assumption of the effective Bohr radius, and also in our assumption of a random d i s t r i b u t i o n of impurities. This u n i f i e d , s e l f consistent approach should provide useful and detailed information on the d i s t r i b u t i o n of exchange coupled pairs which has hitherto been unavailable. -31-BIBLIOGRAPHY 1. S l i c h t e r , CP., Phys. Rev. 99, 479 (1955) 2. Marfco, J.R., Phys. L e t t . 27A, 119 (1968) 3. Yang, G. and Honig, A., Phys. Rev. 168, 271 (1967) 4. Pake, G.E., Paramagnetic Resonance, Ed. D. Pines, 38 (W.A. Benjarain Inc., N.Y., 1962) 5. Kohn, W., i n S o l i d .State Physics, edited by F. Se i t z and D. Turnbull (Academic Press Inc., 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. Alexander, M.N. and Kolcomb, D.F., Rev. Mod. Phys. 40, 815 (1968) 8. Anderson, P.W., i n S o l i d State Physics, edited by F. S e i t z and D. Turnbull(Academic Press Inc., 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., Physica 24, 1635 (1958) 12. M i l l e r , A. and Abrahams, E., Phys. Rev. 120, 745 (i960) 13. Sugihara, K. , J . Phys. Chem. S o l i d s 29, 1099 (1968) 14. Jerome, D. and Winter, J.M., Phys. Rev. 134, A1001 (1964) 15., P a u l i n g , L. and Wilson, E.B., Introduction to Quantum Mechanics, 343 (McGraw-Hill Book Company Inc., N.Y., 1935) -32-APPENDIX A POPULATION DIFFERENCES BETWEEN TRANSITION LEVELS Our whole c a l c u l a t i o n of the spectrum i s based on the fact, that a l l the l o c a l i z e d impurity p a i r s are " d i s t i n g u i s h a b l e " . Therefore, even though the electrons are fermions, the r e l a t i v e populations of the various p a i r l e v e l s w i l l be given by Boltzmann s t a t i s t i c s . With reference to Figure 5, page 17, we discount the s p l i t t i n g of the various t r i p l e t l i n e s (as A a n d obtain the s i m p l i f i e d energy l e v e l diagram given i n Figure 15. 4 E t H + j_ 3 E , - I 2 i _ E ^ - K W a - I 1 E j _ X /yl 31^ A1" 1 ^ 2. Figure 15. S i m p l i f i e d energy l e v e l diagram of the donor p a i r . Therefore, i f we assign a population of 1 to energy l e v e l 1, and a t o t a l population of N to the whole system, the f r a c t i o n a l populations of the other states are given by: , State 1. N, = N e. H \ ( A - l - l ) thus: ^ . ^ e ^ - . N f t ' C ' (A-l-2) State 2. , N e - ^ \ ( a - 1 _ 3 ) and s i m i l a r l y f o r states 3 and 4. The r e s u l t s of t h i s c a l c u l a t i o n and corresponding population d i f f e r e n c e s are given i n Table V. -33-TABLE V Boltzuiann Population Differences Between T r a n s i t i o n Levels T r a n s i t i o n 15 8 14 -» 7 7 -~>2 8 -» 3 11- 915 12- ^ 14 12 11 • 16 13 2 3 9 5 Population Difference - J 2. VC\ - € v p - *V» I -1 w.-2. W r L tell APPENDIX B CALCULATION OF EXCHANGE The term J,t. of equation 1-10 i s now to be calc u l a t e d e x p l i c i t l y . S u b s t i t u t i n g the shielded p o t e n t i a l - e?/\<*xx seen by the donor electrons for the hydrogenic p o t e n t i a l — eV<\x , and w r i t i n g r=r_2 - £j we obtain T° *-K J As Mr J^gCan now be w r i t t e n : (A-2-2) We examine the inner i n t e g r a l given by £ ^ . Using the p e r i o d i c i t y of the u's ( ult(n•,'}(r<+i,) - u ^ h ^ O O ) L J h\ r>"' As W j j J ' ^ U ^ t , ' ) Four i e r s e r i e s : ^fo-'ft vfefle u ^ O O (A-2-3) i s p e r i o d i c i n r ^ , we expand i t i n a 8 „ ,„ & (A-2-4) Thus: [ Then, i f U^ ^( t y t ) F.Cr.N v a r i e s slowly with r ^ , ( t h i s i s .'V <*£,/( A-2-5) -35-equivalent to the "gentle p o t e n t i a l approximation " basic to the de r i v a t i o n of the impurity wavefunctions) one can obtain: Ur2^-Kf\ (A-2-6) by using the common d e f i n i t i o n of the d e l t a f u n c t i o n . As fc^o and are 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 or ~$a0 f o r non-neglibible c o n t r i b u t i o n s i n t h i s gentle p o t e n t i a l l i m i t . S i m i l a r l y , Ho" a n <* must be i n the same d i r e c t i o n i n order that t h e i r d i f f e r e n c e be close to zero, which provides the condi t i o n n 1" =niv . Therefore where the o r t h o n o m a l i t y of the l A ^ l X ^ S has been used to obtain (A-2-7) Thus -i 2 n" (A-2-8) o* kKO VA (nf) (v^ d r, which, by the same type of procedure can be w r i t t e n as 1^  K ICQ hfra%fo<M*, (A-2-9) Noting that the F^'s, F^'s are hydrogenic I s wavefunctions with an e f f e c t i v e Bohr radius a*, the i n t e g r a l K i s regognized to be i 15 equivalent to the i n t e g r a l K evaluated by Pau l i n g and Wilson i n a treatment of the hydrogen molecule, where a =a*. They obtain -36-K « i a" (A-2-10) + A ' 1 e ^ s ) - ^ A ^ e c c - i ^ where T) = ^/a*. By an exact l y s i m i l a r a n a l y s i s , using 42-10, 42-12 of Pauling and 1 5 Wilson j (A-2-11) and 2 - i In. • r As 7,, - "~ ^ S.iiL-,x. , we have as our f i n a l expression: K (A-2-12) e \ 2CUT>)CHT> +j> ) (A-2-13) where terms O^;.^2-^) have been discounted as they go to zero very q u i c k l y f o r ^  7' L . As [ k ^ l -. 8«5T/a , where a i s the l a t t i c e spacing, f o r a l l n, the interf e r e n c e term i n equation A-2-13 can be r e w r i t t e n as: -37-Therefore, equation A-2-13 becomes: -2$ (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 of the Number of L a t t i c e S i t e s i n S h e l l s of 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 of two i n t e r l o c k i n g face-centred cubic l a t t i c e s . Figure 14. Simple Face-centred Cubic L a t t i c e Structure (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 basic l a t t i c e transformations depicted are given by Q, = at* VI} (A-3-1) Therefore, l a t t i c e s i t e s are at p o s i t i o n s R where E = i • v ^ a a i 4 r t 5 0 l 3 where ^ , n 2 , n g are integer s . (A-3-2) The second F.C.C. l a t t i c e of s i l i c o n i s displaced along the body diagonal of the cube depicted i n Figure H by of i t s length. Therefore the l a t t i c e points of the second l a t t i c e are described by - 3 9 -R idie r e : P = B + 0 < - , + A + f e 3 (A-3 -3 ) The distance R= (ji'flT)1 of the l a t t i c e s i t e s from the o r i g i n i s then given bys R e f t ^Q\t+*?f + ( i s 1 + f t 4 t ^ K + « J j " (A-3-4 ) f o r the f i r s t l a t t i c e , and by: R' = SL ^1 + A^-t.^T ^ ( P ^ s + -<T * (PS^^.-sf ( A _3_ 5 ) f o r the second l a t t i c e . When t h i s formulation i s used i n the 360-67 computer to c a l c u l a t e o the number of l a t t i c e s i t e s i n s h e l l s of width 1 A , i t becomes o extremely c o s t l y i n computer time to proceed past r= loo A . I t i s , however, e s s e n t i a l to the c a l c u l a t i o n that we do so. Thus the problem was reformulated by using the symmetry pr o p e r t i e s of the l a t t i c e . The basic q u a n t i t i e s of i n t e r e s t are the absolute value of R and (f o r the interfer e n c e term i n e<^ . ft-l-i5) the pr o j e c t i o n s of R on the x, y, z axes r e s p e c t i v e l y . The interfere n c e term i s seen to be an even f u n c t i o n of these p r o j e c t i o n s . Therefore we s h a l l c a l c u l a t e the absolute value of the p r o j e c t i o n s of various R on the x, y, z axes r e s p e c t i v e l y and f i n d the number of l a t t i c e s i t e s that correspond to each set of p r o j e c t i o n s . The a n a l y s i s , although tedious, i s conceptually t r i v i a l , and i n essence c o n s i s t s of t r e a t i n g the two i n t e r l o c k i n g F. CC. 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 . The r e s u l t i n g absolute values o f the pr o j e c t i o n s and corresponding l a t t i c e s i t e s are summarized i n Table VI. I t was then easy to c a l c u l a t e the number of l a t t i c e s i t e s contained i n each s h e l l of width IA f and to subsequently obtain the p r o b a b i l i t y that any one s i t e i n that s h e l l was an impurity s i t e through equation 2 -16 . -10-TABLE VI Number of L a t t i c e S i t e s Having the Same Absolute Value f o r Pr o j e c t i o n s on the x, y, z Axes i n a Crystal With I n t e r l o c k i n g F.C.C. Structure. Absolute Value of Pr o j e c t i o n s No. of (n^ , n^, n 3 ^ - l where n^, n 0 , n^ are integers^) "Equivalent" X 7 z n^a ( n 2 - l ) a 8 n^a ( n 2 - .5) a • 5)a 24 (nj - .75)a ( n 2 - ,75)a <-3 - o 25) a 12 ( n x - .75)a ( n 2 - .25)a < » 3 " .25) a 12 (n^ - .75)a ( n 2 - .75)a ( n 3 " .75)a 4 ( n x - .25)a ( n 2 - .25)a <»,- .25) a 4 (n^ - .5)a ( n 9 - .5)a 0 12 n^a n 2 a 0 4 n^a o 0 6 -41-b« Determination of the Nearest Neighbour D i s t r i b u t i o n This was performed by Program I. The p r o b a b i l i t y that 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 nearest doner, as c a l c u l a t e d i n Section A f o r a p a r t i c u l a r concentration Nd, was read i n as the array P. The exchange energy J was then c a l c u l a t e d , according to equation 1-10, f o r each c o l l e c t i o n of "equivalent" s i t e s assuming an e f f e c t i v e Bohr radius 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 of equivalent s i t e s L i s added to a c e r t a i n element of the pT^T array. This program i s run f o r a l l the equivalent s i t e s noted i n Table VI, which e n t a i l s nine separate computations i n order to obtain the f i n a l d i s c r e te F T £ T , or N ( j ) , array. - 4 2 -" PROGRAM I S L I S THE NOJ 1 DIMENSION 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 2 REAL L G E , J J ( 2 5 0 ) 3. AA=2.90E0 8 4 BB=2./9. 5 CC=1 ./45 . 6 ALPHA=5.3 7 .£=..85*3.1416 8 LGE=AL0G10(2.71828) 9 READ ( 4 , 1 0 0 ) DIS T,AD 10 100 FORMAT ( 2 F 1 0 . 2 ) 11 N=DIST 12 DO 2 I=1 ,N 13 READ ( 5 , 1 0 1 ) RR ( I ) ,M0(I ) , P ( I ) 14. 101 FORMAT ( F 1 0 . 2 , I 10,E10.2) 15 NO(I ) =0' 16 FTOT(I ) - 0 . . ' 17 'PROB(I)=0. 18 2 CONTINUE 19 M=INT(DIST/5.3) 20 NJMAX=90 21 NJMIN=-15 22 NN=NJMAX+16 23 READ ( 4 , 1 0 4 ) GOD 24 104 FORMAT ( F 1 0 . 2 ) 25 IF (GOD.EO.O.) GO TO 8 2 6 DO 6 I=1,MN 27 READ ( 3 , 1 0 2 ) J J { I ) , J J ( I + 1 ) , A L J ( I ) , A L J ( I + 1),PR O B ( I ) 28 6 CONTINUE 29 DO 7 1 = 1 ,N 30 READ ( 3 , 1 0 3 ) RR( I ) , N 0 ( I ) , F T O T ( I ) 31 7 CONTINUE 32 8 CONTINUE 33 DO 1 I = 1 , M 34 35 36 A=I 37 B=0 38 C = 0 39 L = 6 40 R=ALPHA*SQRT(A**2+B**2+C**2) 41 IF (R.GT.DIST+1) GO TO 1 42 NR =R 43 F = ( C O S ( E * A ) + C O S I E * B ) + C 0 S ( E * C ) ) * * 2 44 F T O T ( N R ) = F T O T ( N R ) + A L 0 G 1 0 ( F ) * L 45 MO(MR)=N0(NR)+L 46 D=R/AD 47 F D = A A * B B * ( 1 . + D ) * ( l . + D + D * * 2 / 3 . ) * E X P ( - 2 . * D ) / A D 48 GD=AA*CC*(25./8.-5.75*0-3.*D**2-D**3/3i 49 2 + ( 6 . / D ) * ( ( l , + D + D * * 2 / 3 . ) * * 2 ) * ( . 5 7 7 7 7 + A L O G ( D ) ) ) * E X P ( - 2 . * D ) / A D 50 X=F*(FD-GD) 51 52 NJ = I N T ( 1 0 . * A L 0 G 1 0 ( X ) ) 1 -43-53 IF (NJ .LT.NJMIN) GO TO 1 54 NJ =NJ+16 55 P R O B ( N J ) = P R O B ( N J ) + L * P ( N R ) 56 1 CONT INUE 57 A L J ( 1 ) = - 1 5 58 J J ( 1 ) = E X P ( A L J ( 1 ) / ( 10.*LGE ) ) 59 A L J ( 1 ) = - 1 . 5 60 DO 4 1 = 1 , NN 61 A L J ( T + 1) = ( 1 - 1 5 ) / 10. 62 J J ( 1 + 1 ) = E X P ( A L J ( I + 1 ) / L G E ) 63 WRITE ( 6 , 1 0 2 ) J J ( I )*, J J ( 1 + 1) , A L J ( I ) , A L J ( 1 + 1 ),PROB(I ) 64 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 ) 65 4 CONTINUE -66 102 FORMAT ( 2 E 1 0 . 2 , 2 F 1 0 . 2 , 1 P E 1 0 . 2 ) 67 DO 5 1=1 ,'N 68 WRITE ( 6 , 1 0 3 ) R R ( I ) ,N0 ( I ) ,FTOT( I ) 69 WRITE ( 7,103) RR(I ) ,NO(I ),F TO T ( I ) 70 5 CONTINUE 71 103 FORMAT ( F 1 0 . 2 , I 10,F20.5) 72 STOP 73 END END OF F I L E -44-A continuous approximation to t h i s N(j) d i s t r i b u t i o n n e c essitates determining a s u i t a b l e average value f o r the interference term i n equation 1-10. As the d i s t r i b u t i o n of nearest donors i s calc u l a t e d as a functi o n o f l o g ( j ) rather than J , we calc u l a t e d I Co^V-*$) , and obtained the appropriate average value of \SSX Cos. k^,""*.*") to be 04663. Writing J as a continuous function of r, and assuming the nearest donor d i s t r i b u t i o n given by equation , the p r o b a b i l i t y N(j) that a nearest donor p a i r has exchange energy i n the range 3", J d - i i s given by: 4n Nji y 3 N(J) do = 4* £ 3 d l T (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 of 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 obtain a reasonable approximation to the d i s c r e t e d i s t r i b u t i o n . This procedure i s performed by Program II, where J i s denoted by X , and di/dr by )(? . The r e s u l t i n g p r o b a b i l i t y as cal c u l a t e d from equation A-3-6 i s entered into the array PNJ, which the second section of the program integrates over the i n t e r v a l s of J . . -45-PROGRAM II SLIS NOJ2 1 C THIS PROG. CALCS.THE FRACT. # OF NRST. DONOR PRS. HAVING 2 C EXCHANGE ENERGY " J " FROM. A POISSON DIST. OF NRST. DONORS. 3 READ ( 5 , 1 0 0 ) N,AD,CON,RMIN,RMAX 4 100 FORMAT .(110 ,F10.2 tE 1 0 . 2 T 2 F 10.2) 5 LOGICAL L I , L 2 6 7 8 N = 300 9 DIMENSION A L J (600) , P M J ( 6 0 0 ) .P.1(600) ...._ .... 10 D I MENS I ON R R ( 6 0 0 ) 11 REAL L J ( 6 0 0 ) 12 DO 20 1=1,600 13 P N J ( I )=0 .0 14 20 CONTINUE 15 AA=2 .90E08 16 . BB=2./9. 17 C C = l . / 4 5 . 18 F=.4663 19 DELR=(RMAX-RMIN)/N 20 DO 1 I=1,N 21 R = RMIN+( I - l ) * D E L R 22 D=R/AD 23 FD=AA*BB*(1.+D)*( l.+D+D**2/3. ) * E X P ( - 2 . * D ) / A D 2 4 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 / 3 . ) * 25 2(.57722+ALOG(D) ) )*EXP.(-2.*D)/AD 2 6 . X= F* ( F D-GD ) 27 X P = - 2 * X / A D + F * A A * E X P ( - 2 . * D ) / A D * * 2 * ( B B * ( ( l + 2 * D / 3 ) * ( 1 + D ) + 28 2 ( l + D + D * * 2 / 3 ) ) - C C * ( - 5 . 7 5 - 6 * 0 - D * * 2 - 6 / D * * 2 * ( l + D + D * * 2 / 3 ) * * 2 * 29 3 ( - . 4 2 27 8 + A L 0 G ( D ) ) + 1 2 / D * ( l + D + D * * 2 / 3 ) * ( l + 2 # D / 3 ) * ( . 5 77+AL0G(D)) 30 4) 31 32 A L J ( I ) = A L 0 G 1 0 ( X ) 33 P N J ( I } = - 4 . * 3 . 1 4 1 6 * C 0 N * R * * 2 * E X P ( - 4 . / 3 . * 3 . 1 4 1 6 * C 0 N * R * * 3 ) / X P 34 WRITE (7 , 1 0 1 ) R , A L J ( I ) , PNJ( I ) 35 101 FORMAT ( 2 F 2 0 . 2 , E 2 0 . 5 ) 36 1 CONTINUE -46-43 P J ( 1 ) = 0 . • . 44 DO 2 I=1,200 45 L J ( I ) = (BO.-I )/10. 46 DO 10 J = 1 , N 47 L I = L J ( I ) .LE o A L J ( J ) 48 L 2 = L J ( I ) . G E . A L J ( J + l ) 49 IF (LI.AND ,L2 ) GO TO 11 50 10 CONTINUE 51 11 CONTINUE ' 52 IF ( A L J ( J ) . E O . O . ) GO TO 2 i 5 3 P'J ( I ) = ( L J ( I ) -AL J ( J ))=:=( PMJ ( J + l ) -PNJ( J ) ) / ( A L J ( J + l ) - A L J { J ) 54 2 ) + P M J ( J ) 55 GO TO 22 56 21 CONTINUE 57 P J U ) = 0 . 5 8 ' 2 2 CONTINUE 59 WRITE ( 6 , 1 0 2 ) I , L J ( I ) , P J ( I ) 60 102 FORMAT (I 1 0 , F 1 0 . 3 , E 2 0 . 5 ) 61 2 CONTINUE 62 STOP 63 END 64 • 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 ) 65 NA=10 66 67 DIMENSION P M J ( 2 0 0 ) ' 68 COMMON P N J , L L J 70 71 DO 202 1=1,200 72 READ (5 ,200) K , L L J ( I ) , P N J ( I ) 73 200 FORMAT (I 1 0 , F 1 0 . 3 , E 2 0 . 5 ) 74 202 CONTINUE 75 76 N=200 77 L G E = A L 0 G 1 0 ( 2 . 7 1 8 2 8 ) 78 L J ( 1 ) = 8 0 . 79 80 J ( 1 ) = E X P ( L J ( 1 ) / ( 1 0 . * L G E ) ) 81 N AREA =0 . 82 DO 1 1=1,200 83 L J ( I + 1 ) = 8 0 - I 84 J ( I + 1 ) = E X P ( L J (I + l ) / ( 1 0 . * L G E ) ) 85 CALL SIMP ( J ( I ) , J ( I + 1 ) , N A , N J ( I ),CON,JO,ALPHA,u,T) 86 AREA = AREA +NJ (I ) 87 WRITE ( 6 , 1 0 1 ) J ( I ) , J ( I + 1) , L J ( I ) , L J ( I + 1 ) , N J ( I ) 88 101 FORMAT ( 2 E 1 0. 2 ,2 F 10. 2 , 1PE 10. 2.) 89 1 CONTINUE 90 WRITE (6 ,102 ) AREA 91 102 FORMAT (//'AREA UNDER CURVE =«,F10.4) 92 STOP 93 END END OF F I L E APPENDIX D CALCULATION OF THE THEORETICAL SPECTRUM This c a l c u l a t i o n was performed by Program I I I . The basic operations involved consisted of breaking up the energy into i n t e r v a l s , contained i n the array E, c a l c u l a t i n g the corresponding i n t e r v a l s of exchange energy which were entered into the array J , and subsequently i n t e g r a t i n g the expression given by equation 2-17 over these i n t e r v a l s of J to obtain the array A. Special a t t e n t i o n was given to poi n t s for which E= 3p^t> and E=^p H»i A/ 2 as the exchange energy, w r i t t e n as a function of energy, s u f f e r s d i s c o n t i n u i t i e s at these p o i n t s . The elements of the array A were then equated to the areas under the Gaussian shape functions given i n equation 2-17, and the height of the hyperfine l i n e s and centre l i n e obtained as t h e i r convolution. The value of the r a t i o of centre to average hyperfine 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 concentration and e f f e c t i v e Bohr radius as c a l c u a l t e d by Program I I was read into Program I I I as the array L U , PNJ. -48-$L IS PROGRAM III F INAL 1 LOGICAL L1,L2,L3,LX 2 DIMENS ION E ( 8 0 2 ) , A ( 8 0 2 ) 3 REAL J ( 8 0 2 ) , J X ( 1 1 ) , L J ( 6 0 0 ) , J 0 , J H ( 2 1 ) 4 DATA N,NX,NH,B/20,50,50,118./ 5 31 CONTINUE 6 READ (5,105) CON , JO,ALPHA,AD,RMAX,T 7 IF (CON.EO.O.EO) GO TO 30 8 DIMENSION PNJ(200) 9 REAL L L J ( 2 0 0 ) 10 COMMON P N J , L L J 11 DO 202 1=1,200 12 READ (4,201) K , L L J ( I ) , P M J ( I ) 13 202 CONTINUE 14 201 FORMAT ( 110, F10.3,E20.5) 15 105 FORMAT ( 2 E 1 0 . 2 , 4 F 1 C 4 ) 16 DO 1 I=1 ,802 17 E ( I )=9799 .5+FLOAT(I )*.5 18 IF ( E ( I ) . E Q . l . E 04 ) GO TO 2 19 J ( I ) = A B S ( B * * 2 / ( 4 . * ( E ( I ) - l . E 0 4 ) ) - E ( I ) + 1 . E 0 4 ) 20 GO TO 7 21 2 J ( I ) = E X P ( L L J ( 1 ) / A L 0 G 1 0 ( 2 . 7 1828) ) 22 7 CONTINUE 23 E ( I ) = E ( I ) + . 2 5 24 1 CONTINUE 25 J ( 2 83) = E X P ( L L J ( 2 00)/ALOG10(2.71828) ) 26 J (519 ) =J (283 ) 27 DO 3 1 = 1 ,802 28 L I = E ( I ) .GE.9941 . 29 L2=E ( I ) .GE .10000. 30 L3 = E ( I ).GE.10059. 31 IF (.NOT.LI) M = l 32 IF ( L I .AND..NOT.L2) M = 2 33 IF (L2.AN0. .N0T.L3) M = 3 34 • IF ( L 3 ) M=4 35 L X = ( I . E 0 . 2 8 2 ) . O R . ( I . E Q . 2 8 3 ) . O R . ( I . E Q . 4 0 0 ) . O R . ( I . E O . 4 0 1 ) 36 2.OR. ( I .E0.518 ) .OR.(I.EG. 519) 37 IF (LX) GO TO 5 38 CALL SIMP ( J ( I ) , J ( I + l ) , N , A ( I ) , C O N , J O , A L P H A , M , T ) 39 GO TO 3 40 5 A ( I ) = 0 . 41 J X (1 ) =A M IN1 ( J (I ) , J ( I +1 ) ) 42 DO 10 K=l,10 43 X=ALOG(AMAXI(J(I ) , J ( I + 1) )/AM I M l ( J ( I ) , J ( I + 1) ) )*FLOAT(K) / 1 0 44 JX(K + l )=AM I M l ( J ( I ) , J ( I + l ) )*EXP ( X) 45 CALL SIMP (JX(K),JX(K+1),NX,AX,CON,JO,ALPHA,M,T) 46 A ( I ) = A ( I ) + A X 47 10 CONT INUE 48 3 CONTINUE 49 . HYP=0. 50 J H ( 1 ) = J ( 2 8 3 ) _ 51 DO 11 K = l ,2 0 52 X = A L O G ( J ( 4 0 1 ) / J ( 2 8 3 ) ) * F L O A T ( K ) / 2 0 . 53 J H ( K + l ) = J ( 2 8 3 )*EXP(X ) • 5 4 CALL SIMP (JH(K),JH(K+1),NH,AH,CON,JO,ALPHA,5,T) 55 HYP =HYP+AH ij6 11 CONT INUE . 57 HHBL =0 . 58 DO 12 1=263,303 5 9 . 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 60 2 ) 61 12 CONTINUE 62 HCL=0. 63 DO 13 1=381,421 64 HCL =HCL +A ( I ) / (SORT ( 2 . * 3 . . 1 4 1 6 ) * 8. ) * E X P ( - . 5 * ( ( E( 4 0 1 ) -E ( I ) ) / 8 . ) **2 65 2 ) 66 13 CONTINUE 67 HHBR =0 . 68 DO 14 I = 4 9 9 ,539 69 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 70 2 ) 71 14 CONTINUE 72 HHL=HYP/(SORT(2.*3. 1416 ) * 8 . ) '73 RATI0=2.*HCL/(2.*HHL+HHBL+HHBR)*100. 74 HL=HHL+HHBL 75 HR=HHL +HHBR 76 WRITE ( 6 , 1 0 0 ) AD,JO,ALPHA,CON,T,R MAX 77 WRITE ( 6 , 1 0 2 ) N , N X-, N H , H L ,HCL,HR 7 8 WRITE ( 6 , 1 0 3 ) RATIO 79 GO TO 31 80 30 CONTINUE ai STOP J 82 1 0 0 FORMAT (///55X,•AD=' , F 5 . 1 , / / / 1 X , • J 0 = 1 , E 1 0 . 2 , 9 X , 'ALPHA= ' , 83 2 F 1 0 . 3 , 7 X , «CON=«,E10.2,9X,«TEMP=',F10.2,8X, 'RMAX=« ,F 10.1/) 8 4 102 FORMAT ( /50 X ,'CHECKING PARAMETERS',//IX,' N = ' , I 8 , 3 8 X , 1NH = 1 , 85 2 18,38X,'NH=' ,18,//2X, 'HEIGHT OF LEFT HYPERFINE L I N E =',1PE10.2, 8 6 3/2X,'HEIGHT OF CENTRE LINE = ' ,8X,E10.2,/2X, 87 4'HEIGHT OF RIGHT HYPERFINE LINE =',E10.2/) 88 1 0 3 FORMAT (V/2X,'RATIO OF CENTRE LINE TO HYP L I N E S (BY PERCENT) 89 2 ) =«,F10.2,///) 90 END ! 91 SUBROUTINE SIMP { A , B, N, AR E A , CON, J 0 ,'AL PH A, M , T ) 92 REAL JO 9 3 REAL L L J ( 2 0 0 ) 9 4 DIMENSION P N J ( 2 0 0 ) 95 COMMON P N J , L L J 9 6 AN = N 97 H=(B-A)/AN 9 8 SUM1=0.0 . 99 SUM2=0.0 100 CALL AUX (A,Y,CON,JO,AL PHA,M,T) 101 YA =Y 102 CALL AUX {8 , Y ,CON,JO,ALPHA,M,T) 103 YB =Y 104 X = A-H 105 MN=N/2 " ' ; • ?6 DO 3 0 I = 1 ,NN 107 X = X + 2 . 0*H 1 0 8 CALL AUX (X , Y ,C ON , JO , ALPH A , M , T ) 1 0 9 30 SUM1=SUM1+Y 1 1 0 . . ... X=A . . .... „• , ... •.. ... - 5 0 -111 DO 40 I = 2 ,NN 112 X = X+2.0*H 113 CALL AUX (X,Y,CON,JO,ALPHA,M,T) 114 40 SUM2=SUM2+Y 115 AREA = H/3.0*(YA+4.0*SUM1+2.0*SUM2+YB) 116 AREA=ABS(AREA) 117 RETURN 118 END 119 SUBROUTINE AUX {X,Y,CON,JO,ALPHA,M, T) 120 REAL JO , MINUS 121 DATA BOL ,AH/2.08E4,118./ 122 123 REAL L L J ( 2 0 0 ) 124 DIMEMS I ON P N J ( 2 0 0 ) 12 5 COMMON P N J , L L J 126 ALX=AL0G10(X) 12 7 IF (ALX.LT.O.) GO TO 3 00 128 N = 8 0 - I N T ( 1 0 . * A L X ) 129 IF (N.LE.1) GO TO 302 130 GO TO 301 131 300 CONTINUE 132 N = 8 1 - I N T ( 1 0 , * A L X ) 133 IF (N.GT.2Q0) GO TO 302 134 GO TO 301 135 302 CONTINUE 136 PJ=0.0 137 GO TO 303 138 301 CONTINUE 139 PJ = - 1 0 . * ( A L X - L L J (N-1) ) * ( P N J ( N ) - P N J ( N - 1 ) ) + P N J ( M-1) 140 303 CONTINUE 150 RT=SORT(X**2+AH**2) 151 PLUS=.5*(1.+X/RT) 152 MINI)S = .5*(1.-X/RT) 153 IF (M-.GE .2 ) GO TO 2 154 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 ) )) 155 GO TO 1 156 2 CONTINUE 157 IF (M.GE.3) GO TO 3 158 Y = P L U S * P J * E X P ( - ( X + R T ) / ( 2 . * B 0 L * T ) ) * ( 1 . - F X P ( - 1 . E 4 / ( B O L * T ) ) ) 159 GO TO 1 160 3 CONTINUE 161 -IF (M .GE .4) GO TO 4 162 Y = P L U S * P J * E X P ( - ( X + R T ) / ( 2 . * B 0 L * T ) ) * ( E X P ( 1 . E 4 / ( B 0 L * T ) ) - 1 . ) 163 GO TO 1 164 4 CONTINUE 165 IF (M.E0.5) GO TO 5 166 Y = M I N U S * P J * ( 1 . - E X P ( - ( 2 . E 4 + X + R T ) / ( 2 . * B 0 L * T ) ) ) 167 GO TO 1 168 5 CONTINUE 169 Y = . 5 * P J * E X P ( - ( X + R T ) / ( 2 . * B 0 L * T ) ) _170 2 * ( E X P ( 1 . E 4 / ( B O L * T ) ) - E X P ( - 1 . E 4 / ( B O L * T ) ) ) 171 1 CONTINUE 172 RETURN 173 E N D END OF F I L E 

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