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Numerical studies of the nuclear hyperfine interactions in manganese dichloride tetrahydrate McLeod, Beverly Ann 1983

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NUMERICAL STUDIES OF THE NUCLEAR HYPERFINE INTERACTIONS IN MANGANESE DICHLORIDE TETRAHYDRATE by BEVERLY ANN MCLEOD B . S c U n i v e r s i t y Of B r i t i s h Columbia,1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department Of P h y s i c s We a c c e p t t h i s t h e s i s as c o n f o r m i n g to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1983 © B e v e r l y Ann McLeod, 1983 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 of the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h C o lumbia, I agree t h a t the L i b r a r y s h a l l make 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 and s t u d y . I f u r t h e r agree t h a t 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 of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department or by h i s or her 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 or p u b l i c a t i o n of 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 not 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 . Department of P h y s i c s The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date: A p r i l 27,1983 A b s t r a c t The n u c l e a r magnetic resonace of o r i e n t e d n u c l e i (NMRON) i s a u s e f u l t e c h n i q u e f o r the study of magnetic s o l i d s , and i s e s p e c i a l l y i m p o r t a n t f o r i n v e s t i g a t i n g v ery d i l u t e systems. The work i n t h i s t h e s i s complements e x p e r i m e n t a l NMRON s t u d i e s of 5"Mn-MnCl^.4H 20 by the n u c l e a r o r e i n t a t i o n group a t the U n i v e r s i t y of B r i t i s h C olumbia. The manganese n u c l e u s e x p e r i e n c e s a h y p e r f i n e i n t e r a c t i o n which i s due p r i m a r i l y t o a magnetic (Zeeman) i n t e r a c t i o n , but which a l s o has a s m a l l c o n t r i b u t i o n due to the i n t e r a c t i o n between the e l e c t r i c q u a d r u p o l e moment eQ of the n u c l e u s and the e l e c t r i c f i e l d g r a d i e n t eq a r i s i n g from the s u r r o u n d i n g i o n i c c h a r g e s i n the l a t t i c e . The l a t t e r i n t e r a c t i o n can be t r e a t e d as a p e r t u r b a t i o n on the Zeeman i n t e r a c t i o n and r e s u l t s i n s h i f t s i n the n u c l e a r s p i n energy l e v e l s , so t h a t t h e r e a re 21 resonant f r e q u e n c i e s f o r s p i n I . The e l e c t r i c f i e l d g r a d i e n t i s c a l c u l a t e d by t r e a t i n g t h e s e i o n s as p o i n t c h a r g e s , and a computer program t o do t h i s c a l c u l a t o n i s d e v e l o p e d . The r e s u l t of t h i s l a t t i c e sum, eq , has t o be m u l t i p l i e d by the St e r n h e i m e r a n t i -s h i e l d i n g f a c t o r which t a k e s i n t o account the a m p l i f y i n g e f f e c t due t o the d i s t o r t i o n by the f i e l d g r a d i e n t of the e l e c t r o n s on the atom of i n t e r e s t : t h i s f a c t o r i s known f o r the manganese i o n . As a check, the case of 5 5Mn i n MnF, i s i n v e s t i g a t e d because the v a l u e of |e 2qQ| has been d e r i v e d from experiment and The r e s u l t s f o r e 2qQ/h a r e : exper iment p r e v i o u s c a l c u l a t i o n our c a l c u l a t i o n independent c a l c u l a t i o n [ 1 ] . 1 1 .7±0.03 MHz 8.8 MHz 1 8.5±0.4 MHz an For the system 5"Mn-MnCl 2.4HiO two models a r e used. In the f i r s t ( I ) , i o n i c c h a r g e s of -e and .5e are a s s i g n e d t o the 0 and H i o n s i n the water m o l e c u l e s , as su g g e s t e d by E l S a f a r [ 2 ] . In the s e c o n d ( I I ) , the wate r s a re assumed e l e c t r i c a l l y n e u t r a l . T h i s case i s more c o m p l i c a t e d because of the lower symmetry of the c r y s t a l and because the a x i s of m a g n e t i z a t i o n does n o t . c o i n c i d e w i t h a p r i n c i p a l a x i s of the e l e c t r i c f i e l d g r a d i e n t t e n s o r . The q u a n t i t y measured by experiment and c a l c u l a t e d by us i s P/12 = e 2qQ/[8hI (21 + 1 ) ] x { 3 c o s 2 6 - 1 + r\ cos2<P s i n 2 9 } . Here (6, <P ) a r e the p o l a r a n g l e s of the m a g n e t i z a t i o n a x i s i n the p r i n c i p a l e l e c t r i c f i e l d g r a d i e n t frame and f\ i s an asymmetry parameter. The r e s u l t s f o r P/12 a r e : experiment :- P/12 = +0.52±0.07 MHz our c a l c u l a t e d model I :- P/12 = +0.015±0.002 MHz our c a l c u l a t e d model I I :- P/12 = +0.21±0.02 MHz . Both models g i v e the c o r r e c t s i g n f o r P, but model I I o b v i o u s l y g i v e s much b e t t e r q u a n t i t a t i v e agreement. m o d i f i e d by u s i n g a more r e c e n t and p r e c i s e v a l u e of Q We a r e a l s o a b l e t o c a l c u l a t e the magnetic d i p o l e i n t e r a c t i o n between manganese atoms of magnetic d i p o l e moment and o b t a i n f o r a f i e l d a p p l i e d p a r a l l e l t o the easy a x i s : A n t i f e r r o m a g n e t i c regime:-y H ' B d i p / V = -0.0256 A" 3 S p i n f l o p regime ( w i t h moments c a n t e d a t a n g l e c<~ t o the easy a x i s ) : -/ ? • B d i p ^ 2 = [0.01 l 9 c o s 2 c x -0.0492coscx s i n c x -0 . 0290 sin 2<X ] A" 3 F e r r o m a g n e t i c regime:-A'*<x\?W = +0.0119 A- 3 For a f i e l d a p p l i e d p e r p e n d i c u l a r t o the easy a x i s i n the MnCl .4H O c r y s t a l B-C p l a n e we o b t a i n f o r the regime ( w i t h moments c a n t e d a t angl e & t o the d i r e c t i o n of the a p p l i e d f i e l d ) : /^'"Bcl.p^ 2 = [ 0.0137cos 2/3 -0 . 0492cos/? sin/? - 0 . 0 2 5 6 s i n 2 ^ ] A" 3 A second program has been w r i t t e n t o gen e r a t e the NMRON s p e c r t r u m s p e c i f i c a l l y f o r 5"Mn-MnC^.4H zO. However, w i t h minor m o d i f i c a t i o n s i t c o u l d be a p p l i e d g e n e r a l l y . N u m e r i c a l t e c h n i q u e s a r e n e c e s s a r y t o c a l c u l a t e the r e c o v e r y of the system by s p i n - l a t t i c e r e l a x a t i o n because 5*Mn has s p i n 1=3. The t h e o r e t i c a l spectrum has a d i s t i n c t i v e p r o f i l e f o r the m=2/m=l resonance due i n p a r t t o competing second and f o u r t h o r d e r i v m u l t i p o l e c o n t r i b u t i o n s t o the gamma-ray i n t e n s i t y and i n p a r t t o s p i n - l a t t i c e r e l a x a t i o n . T h i s p r o f i l e i s ob s e r v e d i n the e x p e r i m e n t [ 3 ] . F i t t i n g the t h e o r e t i c a l t o the e x p e r i m e n t a l s p e c t r a y i e l d s v a l u e s f o r the s p i n - l a t t i c e r e l a x a t i o n time of Ti = 3.0±0.5 x 10" seconds a t a tempe r a t u r e of T = 0.081 K and T«. = 2.3±0.3 x 10" seconds at t e m p e r a t u r e T = 0.064 K. [1] H. Yasuoka, T i n Ngwe, and V. J a c c a r i n o , Phys. Rev. 177, 667 (1969). [2] Z. M. E l S a f f a r , J . Chem. Phys. 52, 4097 (1969) . [3] A. K o t l i c k i , B. McLeod, M. S h o t t , and B. G. T u r r e l l , t o be p u b l i s h e d i n Hyp. I n t . V T a b l e of C o n t e n t s A b s t r a c t i i L i s t of T a b l e s v i i L i s t of F i g u r e s v i i i Acknowledgements i x I . INTRODUCTION 1 1.1 The N u c l e a r Quadrupole I n t e r a c t i o n 3 1.1.1 The I n t e r a c t i o n H a m i l t o n i a n 3 1.1.2 Quadrupole Energy L e v e l S p l i t t i n g 6 I I . POINT SOURCE CALCULATIONS OF NUCLEAR QUADRUPOLE AND MAGNETIC DIPOLE ENERGIES 9 2.1 I n t r o d u c t i o n 9 2.2 C a l c u l a t i o n Of V;: For A L a t t i c e Of P o i n t Charges 10 A. THE GENERAL EXPRESSION 10 B. REMOVING THE CONTRIBUTION FROM THE ORIGIN 13 C. THE FINAL EXPRESSION 14 2.3 F i n d i n g The P r i n c i p a l V ^  15 2.4 A Program To F i n d V,j 16 2.5 The St e r n h e i m e r A n t i s h i e l d i n g F a c t o r 20 2.6 S t r u c t u r e Of The L a t t i c e s 20 2.6.1 MnF A 20 2.6.2 MnCl 2 .4H 20 21 2.7 P o i n t Charge R e s u l t s And Comparison W i t h Experiment 24 2.7.1 55Mn-MnFz 24 2.7.2 54Mn-MnClj. .4H,.0 25 a. MnCl-^H^O Wi t h Water M o l e c u l e s Assumed E l e c t r i c a l l y N e u t r a l 25 b. M n C l ^ H j O With Charges Q(H) = .5e And Q(0)=-e 26 c. The E x p e r i m e n t a l V a l u e Of P For 54Mn-MnCl 2.4H 20 27 2.8 The Magnetic D i p o l e I n t e r a c t i o n Energy For M n C l j ^ H ^ O In D i f f e r e n t Magnetic Phases ..28 I I I . DESCRIPTION OF AN NMRON SPECTRUM 31 3.1 The N u c l e a r Resonance C o n d i t i o n 31 3.2 Gamma-ray E m i s s i o n 33 3.3 P o p u l a t i o n Of The S p i n L e v e l s 34 3.3.1 The E q u i l i b r i u m P o p u l a t i o n 34 3.3.2 The P o p u l a t i o n s At Resonance 35 3.3.3 The P o p u l a t i o n s D u r i n g R e l a x a t i o n 35 IV. PROGRAM TO GENERATE AN NMRON SPECTRUM APPLIED TO 54MN-MANGANESE CHLORIDE TETRAHYDRATE 37 4.1 Gamma-ray E m i s s i o n For 54-Mn 37 4.2 A Two Temperature Model 38 4.3 The Resonance C o n d i t i o n s For 54-Mn In M n C l i ^ H ^ O ..38 4.4 P o p u l a t i o n Of The N u c l e a r S p i n L e v e l s 40 4.4.1 E q u i l i b r i u m 40 4.4.2 Resonance 41 4.4.3 R e l a x a t i o n 41 4.5 Width Of The Resosnance Peaks 42 v i 4.6 The Program 42 4.7 Output Of The Program For 54Mn-MnCl .4H 0 47 V. SUMMARY 54 BIBLIOGRAPHY 57 APPENDIX A - CHECK OF EQUATION 2.19 58 APPENDIX B - PROOF THAT THE EIGENVALUES AND EIGENVECTORS OF V.j ARE PRINCIPAL FIELD GRADIENTS AND PRINCIPAL AXES 60 APPENDIX C - PROGRAM 1 61 APPENDIX D - OUTPUTS OF PROGRAM 1 FOR MANGANESE FLOURIDE AND MANGANESE CHLORIDE TETRAHYDRATE 65 APPENDIX E - PROGARM 2 69 APPENDIX F - SAMPLE OUTPUTS OF PROGRAM 2 75 v i i L i s t of T a b l e s I . P4/nmn u n i t c e l l 20 I I . p o s i t i o n parameters f o r MnCl 2.41-^0 21 v i i i L i s t of F i g u r e s 1. n u c l e u s and an e x t e r n a l charge 3 2. d e f i n i t i o n of the E u l e r a n g l e s 7 3. c o o r d i n a t e axes f o r V;j of the program 17 4. S t r u c t u r e of the l a t t i c e s d i s p l a y i n g the o r d e r i n g of Mn atomic magnetic moments below the N e e l t e m p e r a t u r e s 23 5. the a x i s of m a g n e t i z a t i o n 25 6. phases of an a n t i f e r r o m a g n e t 29 7. decay scheme f o r 54-Mn 37 8. m o d i f i c a t i o n of the Zeeman n u c l e a r s p i n energy l e v e l s by the q u a d r u p o l e i n t e r a c t i o n 39 9. onset of n u m e r i c a l i n s t a b i l i t y w i t h d e c r e a s e i n Tt ...45 10. computer g e n e r a t e d spectrum f o r 5 < tMn i n MnCl z.4H zO ...48 11. e x p e r i m e n t a l spectrum f o r 5"Mn i n MnCl 1.4H iO w i t h a t h e o r e t i c a l f i t drawn i n 49 12. the o r i g i n a l m=2/m=1 t r a n s i t i o n l i n e spectrum 51 13. the a d j u s t e d m=2/m=1 t r a n s i t i o n l i n e spectrum, w i t h a t h e o r e t i c a l f i t drawn i n 52 i x Acknowledgement I would l i k e t o thank my s u p e r v i s o r , P r o f e s s o r B r i a n T u r r e l l , f o r h i s support t h r o u g h o u t the p r o g r e s s of t h i s work. In p a r t i c u l a r I would l i k e t o e x p r e s s my a p p r e c i a t i o n f o r the e x t r a time and a s s i s t a n c e g i v e n me d u r i n g the f i n a l s t a g e s of w r i t i n g , w i t h o u t which I c o u l d not have met the d e a d l i n e . I am g r a t e f u l t o many f e l l o w s t u d e n t s f o r c o n t r i b u t i o n s as v a r i o u s as a i d w i t h FMT, t o sandwiches or good humour a t odd h o u r s . S p e c i a l t hanks t o Mark S h e g e l s k i f o r encouragement and h e l p s u p p l i e d g e n e r o u s l y a t c r i t i c a l moments. 1 I . INTRODUCTION T h i s t h e s i s d e v e l o p s and a p p l i e s two computer programs t o study the n u c l e a r h y p e r f i n e i n t e r a c t i o n s i n the system 5"Mn-MnCl^.4H zO. i ) C hapter I I d e s c r i b e s the c a l c u l a t i o n of the e l e c t r i c f i e l d g r a d i e n t and the magnetic d i p o l e f i e l d i n magnetic s o l i d s . Because of the l a r g e number of atoms i n v o l v e d , these l a t t i c e sums are bes t t a c k l e d by n u m e r i c a l t e c h n i q u e s , and a FORTRAN program has been deve l o p e d t o implement them. The e l e c t r i c f i e l d g r a d i e n t t e n s o r i s c a l c u l a t e d a t a Mn s i t e f o r MnFz and f o r the «-form of M n C l z . 4 H 2 , 0 u s i n g p o i n t charge models w i t h i o n p o s i t i o n s o b t a i n e d from neu t r o n and X-ray d i f f r a c t i o n s t u d i e s . From the s e v a l u e s the e l e c t r i c q u a d r u p o l e i n t e r a c t i o n i s found f o r the 5 5Mn n u c l e u s i n the former case and the 5"Mn n u c l e u s i n the l a t t e r , assuming a v a l u e g i v e n i n the l i t e r a t u r e f o r the Ste r n h e i m e r a n t i - s h i e l d i n g f a c t o r . The r e s u l t s a r e then compared w i t h the e x p e r i m e n t a l v a l u e s o b t a i n e d i n an NMR experiment on MnF and i n an NMRON experiment on 5"Mn-MnCl^4H Z0 r e c e n t l y performed a t the U n i v e r s i t y of B r i t i s h C olumbia. The MnF2 r e s u l t s e r v e s as a check f o r the n u m e r i c a l c a l c u l a t i o n s . The magnetic d i p o l e f i e l d s a r e c a l c u l a t e d f o r the a n t i f e r r o m a g n e t i c , the s p i n f l o p and the f e r r o m a g n e t i c phases of o r d e r e d MNCl^^H^O. 2 i i ) C hapter I I I r e v i e w s the c a l c u l a t i o n of the gamma-ray i n t e n s i t y e m i t t e d from an ensemble of r a d i o a c t i v e n u c l e i o r e i n t e d by a h y p e r f i n e i n t e r a c t i o n which i s p r e d o m i n a t e l y magnetic when the system has been f i r s t r e s o n a t e d and then a l l o w e d t o r e c o v e r by s p i n - l a t t i c e r e l a x a t i o n . f o r the case i n which the system i s o r d e r e d by a h y p e r f i n e i n t e r a c t i o n which i s p r e d o m i n a t e l y m a g n e t i c , but which has a s m a l l e l e c t r i c q u a d r u p o l e c o n t r i b u t i o n . S i n c e no a n a l y t i c s o l u t i o n e x i s t s f o r I>3/2, the time r a t e of change of the l e v e l p o p u l a t i o n s (which d e t e r m i n e the e m i t t e d i n t e n s i t y ) d u r i n g r e l a x a t i o n must be s o l v e d numer i c a l l y . In c h a p t e r IV a program i s d e v e l o p e d t o ge n e r a t e a spectrum a c c o r d i n g t o the t h e o r e t i c a l d e s c r i p t i o n of ch a p t e r I I I . The program i s w r i t t e n s p e c i f i c a l l y f o r the 5 "Mn-MnCl z. 4 H 2_0 system but w i t h s i m p l e m o d i f i c a t i o n s c o u l d be a p p l i e d g e n e r a l l y . A c h a r a c t e r i s t i c p r o f i l e i s found f o r the m=2/m=1 resonance l i n e due t o competing e f f e c t s of the second and f o u r t h o r d e r m u l t i p o l e c o n t r i b u t i o n s t o the gamma-ray i n t e n s i t y and t o s p i n - l a t t i c e r e l a x a t i o n . T h i s p r o f i l e d e f i n i t i v e l y i d e n t i f i e s the e x p e r i m e n t a l m=2/m=1 resonance l i n e . F i t t i n g of the t h e o r e t i c a l and e x p e r i m e n t a l s p e c t r a a l l o w the d e t e r m i n a t i o n of the Zeeman and quadr u p o l e c o n t r i b u t i o n s t o the h y p e r f i n e i n t e r a c t i o n and of the 3 s p i n - l a t t i c e r e l a x a t i o n t i m e . The q u a d r u p o l e i n t e r a c t i o n energy i s common t o both the problems t a b l e d and i s d i s c u s s e d i n the next s e c t i o n . 1.1 The N u c l e a r Quadrupole I n t e r a c t i o n 1.1.1 The I n t e r a c t i o n H a m i l t o n i a n C o n s i d e r a n u c l e u s w i t h a l o c a l i z e d charge d e n s i t y p(x*) c e n t e r e d about the o r i g i n i n an e l e c t r i c p o t e n t i a l due t o e x t e r n a l charges as p i c t u r e d i n f i g u r e 1. F i g u r e 1 - n u c l e u s and an e x t e r n a l charge The t o t a l e l e c t r i c p o t e n t i a l a t a p o i n t x i n the n u c l e u s due t o e l e c t r i c c h arges q ( x k ) e x t e r n a l t o the n u c l e u s w i l l be L e t the e x t e r n a l c h a r g e s be due t o a l l the i o n s i n a (1.1) k 4 l a t t i c e e x c l u d i n g the e l e c t r o n s of the atom t o which the n u c l e u s b e l o n g s . N o t i n g t h a t i n t e r a t o m i c d i s t a n c e s a r e much g r e a t e r than the d i a m e t e r of a n u c l e u s , e q u a t i o n (1.1) can be expanded i n a T a y l o r s e r i e s about the o r i g i n . 7=3 (1.2) T h i s p e r m i t s a m u l t i p o l e e x p a n s i o n f o r the c h a r a c t e r i s t i c energy of the system. £ ~J Vonpm dV - V(si p ( i r ) d'x N u c l e a r (1.3) L e t drw 5. cg be denoted by V ; j . S i n c e the p o t e n t i a l due t o e x t r a - n u c l e a r c h a r g e s s a t i s f i e s L a p l a c e ' s e q u a t i o n a t the o r i g i n Z VJ V&- - o ( 1.4) the t h i r d term i n the e x p a n s i o n ( 1 . 5 ) , c a l l e d the qua d r u p o l e energy, E Q , can be w r i t t e n E , - fc E V:> Q l i (1.5) where (1.6) i s c a l l e d the quadrupole moment t e n s o r 5 In quantum mechanics E Q and Qli c o r r e s p o n d t o the e x p e c t a t i o n v a l u e s of quantum m e c h a n i c a l o p e r a t o r s . These e x p e c t a t i o n v a l u e s depend upon the quantum s t a t e of the nu c l e u s which i s c h a r a c t e r i z e d by i t s s p i n , T, and the p r o j e c t i o n of the s p i n a l o n g some a x i s , I 2 . For a g i v e n s p i n , t h e e x p e c t a t i o n v a l u e of the i j - t h component of the qu a d r u p o l e t e n s o r i s {iKlQ'MlIz) = <Hzl JfsVxi " S^lSfD/otf) d'x III*') (1.7) U s i n g the W i g n e r - E c k a r t theorem, t h i s can be w r i t t e n [10] as <ii2iQ fMii;> - * f e r = y <nj(3ri i * i ii)-s , " i r | t i i > (1.8) where Q • < l I | e L ( - rk* ) S(*x\) | T I> k (1.9) Thus the H a m i l t o n i a n f o r the qua d r u p o l e i n t e r a c t i o n i s i ^ i u t . j - , . d . i o ) By c h o o s i n g x 1 = X , x 2=Y, x 3=Z t o be the p r i n c i p a l axes d e f i n e d by V.. = 0 f o r i * j (1.11) and u s i n g L a p l a c e ' s e q u a t i o n ( 1 . 4 ) , the e l e c t r o s t a t i c q u a d r u p o l e H a m i l t o n i a n f o r a s p i n I n u c l e u s reduces t o H.'SftiVi) L ^ " ^ - n d / - ^ ) ] (,.12) where 6 eQ = V 2 z and r\ = (V x x - V y y ) / \ x . ( 1 . 1 3 ) By c o n v e n t i o n the p r i n c i p a l axes a r e l a b e l l e d X, Y and Z i n a c c o r d a n c e w i t h l v x x I - ' vvy I - l v z x I < 1- 1 4> The H a m i l t o n i a n i s o f t e n e x p r e s s e d i n the c o n v e n i e n t form H q - i \ [ 3 i ; - I 1 - * i . a ) ] (1.15) where I and I_ are the s t a n d a r d a n g u l a r momentum r a i s i n g and l o w e r i n g o p e r a t o r s I + = I x + i I y , I _ = I y - i I y (1.16) and ^ 21(21 + 0 21 (21- l) ( 1 > 1 7 ) 1.1.2 Quadrupole Energy L e v e l S p l i t t i n g To de t e r m i n e the energy of the magnetic s u b s t a t e s , m, i t i s d e s i r a b l e t o w r i t e the H a m i l t o n i a n , H , i n terms of o p e r a t o r s f o r which t h e s e s t a t e s a r e e i g e n v e c t o r s . For an a x i s of m a g n e t i z a t i o n , z, which l i e s a t p o l a r a n g l e 0 and a z i m u t h a l a n g l e w i t h r e s p e c t t o the quadr u p o l e i n t e r a c t i o n p r i n c i p a l axes X, Y, and Z, the m a g n e t i z a t i o n axes can be o b t a i n e d ' from the p r i n c i p a l axes by a p p l i c a t i o n of the i n v e r s e r o t a t i o n [R(06<P)]~ 1, where R(«pV) = R z(«) R u ( ^ ) R z ( y ) (1.18) i s a r o t a t i o n t h r o u g h the E u l e r a n g l e s (a,^,^) of f i g u r e 2. 7 F i g u r e 2 - d e f i n i t i o n of the E u l e r a n g l e s A p p l y i n g the i d e n t i t i e s [7] (1.19) and (1.20) to e x p r e s s i o n (1.1 ) f o r the q u a d r u p o l a r H a m i l t o n i a n g i v e s H 0 Bi?Q{i(3c«xe-/)(3ill-ii) + IA A~xb - iB^e^e < 1 , 2 1 ) + intend [we ( 3 1 / - r ) + A -B^ec^e j j where A - 1/ - I- 2 B • * d** ( 1. 2 2 ) and I T , I = I +il> , I = I -il„ , a r e now a n g u l a r momentum o p e r a t o r s w i t h r e s p e c t t o an o r t h o g o n a l C a r t e s i a n 8 c o o r d i n a t e system i n which the z - a x i s l i e s a l o n g the a x i s of m a g n e t i z a t i o n . The q u a d r u p o l e i n t e r a c t i o n energy of a s p i n T n u c l e u s i n the s t a t e I = mh i s now r e a d i l y found: E Q(ml • ( I m | r l Q l l m > ( 1 ' 2 3 ) = 4 ? Q [ 3 w j e - I - rj c« 2<P * [ 3 ^ - 1(1 + 0 ] . D e f i n i n g P by ? * [1 ( 3 c o s 1 6 - 1 ) + | c o s s in* e ] ( 1 . 2 4 ) the q u a d r u p o l e energy d i f f e r e n c e between a d j a c e n t s p i n l e v e l s i s o b t a i n e d : E Q ( m l - E Q (ir»-1) = P Cm- ± ) , ( 1.25) 9 I I . POINT SOURCE CALCULATIONS OF NUCLEAR QUADRUPOLE AND MAGNETIC DIPOLE ENERGIES 2.1 I n t r o d u c t i o n To f i n d the s p l i t t i n g of the n u c l e a r s p i n energy l e v e l s of e q u a t i o n ( 1 . 2 5 ) , the second o r d e r s p a t i a l d e r i v a t i v e s of the e l e c t r o s t a t i c p o t e n t i a l , V(. , must be known. A s i m p l e and o b v i o u s approach i s t o c o n s i d e r the c r y s t a l l i n e s o l i d t o be a l a t t i c e of c l a s s i c a l p o i n t c h a r g e s . The p o s i t i o n s of the p o i n t c h a r g e s i n the c a l c u l a t i o n s are taken t o be the t i m e - a v e r a g e d i o n p o s i t i o n s d e t e r m i n e d by X-ray and n e u t r o n d i f f r a c t i o n e x p e r i m e n t s . At the time t h i s work was u n d e r t a k e n , s p l i t t i n g of the magnetic en-ergy l e v e l s due t o the q u a d r u p o l a r i n t e r a c t i o n had been measured i n MnF2 , but i n M nCl 2.4H z0 o n l y one resonance l i n e had been observed u s i n g the n u c l e a r o r i e n t a t i o n t e c h n i q u e t o be d i s c u s s e d i n c h a p t e r I I I . S i n c e the p o i n t charge c a l c u l a t i o n f o r the parameter P Q , which d e t e r m i n e s the magnitude of the q u a d r u p o l a r i n t e r a c t i o n , was seen t o agree i n o r d e r of magnitude w i t h experiment i n MnF2 , i t was hoped t h a t t h i s approach would a l s o g i v e a good e s t i m a t e of P f o r MnCl 2.4H a0. I t t u r n s out t h a t the q u a n t i t i e s V - r e q u i r e d t o o b t a i n P can a l s o be used t o e s t i m a t e the magnetic d i p o l e i n t e r a c t i o n due t o the s u b l a t t i c e s of magnetic i o n s i n the d i f f e r e n t phases of a c r y s t a l . The d i p o l a r f i e l d a c t i n g 10 a t the n u c l e a r s i t e i s a c o n t r i b u t i o n t o the t o t a l h y p e r f i n e f i e l d and needs t o be known i n a n a l y z i n g the e f f e c t of a p p l i e d f i e l d s on the n u c l e a r magnetic resonance f requency. We w i l l i n t r o d u c e a p r a c t i c a l method f o r c a l c u l a t i n g due t o a l a t t i c e of p o i n t c h a r g e s , and remind the r e a d e r of how the p r i n c i p a l V- may be found from the v a l u e s of V- thus o b t a i n e d . A FORTRAN program t o implement the above , which i s l i s t e d i n Appendix C, i s p r e s e n t e d and a p p l i e d t o the two l a t t i c e s MnF^ and M n C l j ^ H ^ O . The r e s u l t s of the p o i n t charge c a l c u l a t i o n s a r e then compared t o e x p e r i m e n t . F i n a l l y , the program i s used t o f i n d the d i p o l e i n t e r a c t i o n e n e r g i e s f o r the magnetic phases of M n C l a.4H 20. 2.2 C a l c u l a t i o n Of Vr For A L a t t i c e Of P o i n t Charges A. The G e n e r a l E x p r e s s i o n To e v a l u a t e the q u a d r u p o l e i n t e r a c t i o n e n e r g i e s due t o the H a m i l t o n i a n ( 1 . 1 5 ) , the v a l u e s of the second o r d e r s p a t i a l d e r i v a t i v e s of the e l e c t r o s t a t i c p o t e n t i a l , V-- , a r e needed. For a l a t t i c e of p o i n t c h a r g e s , i s g i v e n by A l a t t i c e can be g e n e t r a t e d from the p r i m i t i v e c e l l by t r a n s l a t i o n of the p r i m i t i v e or u n i t c e l l by the l a t t i c e v e c t o r s 1. L e t u be the p o s i t i o n c o o r d i n a t e s of (2.1 ) 11 the p o i n t charges q(if) i n the l a t t i c e u n i t c e l l . Then V/j may be w r i t t e n w . y (2.2) T h i s sum converges s l o w l y so t h a t d i r e c t summation i s at b e s t m a r g i n a l l y p r a c t i c a b l e due t o r o u n d o f f e r r o r . An a l t e r n a t e e x p r e s s i o n f o r e q u a t i o n (2.2) w i t h much b e t t e r convergence p r o p e r t i e s can be o b t a i n e d by use of an i n t e g r a l i d e n t i t y and F o u r i e r a n a l y s i s . The i d e n t i t y i s " nTtT ] iyi ~ t  e dp (2.3) D i f f e r e n t i a t i o n of t h i s y i e l d s (2.4) N o t i c e t h a t the r i g h t hand s i d e of e q u a t i o n (2.4) d e c r e a s e s r a p i d l y w i t h i n c r e a s i n g v a l u e s of |5f| f o r l a r g e v a l u e s of p . Now lo o k a t the expansion of V,-- i n F o u r i e r space. S u b s t i t u t i o n of (2.4) i n t o (2.2) g i v e s w 0 L ^ J (2.5) By the p e r i o d i c i t y of the l a t t i c e the i n t e g r a n d can be expanded i n a F o u r i e r s e r i e s 2 y I I - l ^ j r t l V _ y c U . u * * * (2.6) where 1 2 € = 1 (2.7) - i k • u . _ F G " V l L J [ 3 x ' ^ e J e du (2.8) Here V L i s the volume of the l a t t i c e , V L = 8 L ' L 2 L 3 . Making the change of v a r i a b l e s u -> 1+u and u s i n g e q u a t i o n (2.7) t h i s becomes 1 - LtJ dx i75 (2.9) U s i n g the f o l l o w i n g p r o p e r t y of F o u r i e r t r a n s f o r m s - t . ( r n l U) ; k ) = (-,•*)" K I ( £(x) J k) (2.10) g i v e s 7^ e (2.11) <« -IT? S i n c e t h i s i n t e g r a l c o nverges r a p i d l y , r e p l a c i n g the l i m i t s of i n t e g r a t i o n by ±oo produces r- ~ . • Ml H - |Je*l (2.12) where N i s the number of u n i t c e l l s i n the l a t t i c e of volume V L . S u b s t i t u t i o n of e q u a t i o n (2.12) back i n t o e q u a t i o n (2.6) y i e l d s \frr e j e P 3 k p» (2.13) where v = V,./NL i s the u n i t c e l l volume. N o t i c i n g t h a t the r i g h t hand s i d e of (2.13) f a l l s o f f r a p i d l y w i t h 13 i n c r e a s i n g k for s m a l l p , one s p l i t s the i n t e g r a l of e q u a t i o n (2 .4) i n t o two p a r t s by the use of an i n t e r m e d i a t e l i m i t of i n t e g r a t i o n , G , and e v a l u a t e s the i n t e g r a l i n F o u r i e r space for s m a l l v a l u e s of p and in ' r e a l ' space f o r l a r g e v a l u e s of p . Together e q u a t i o n s ( 2 . 2 ) , ( 2 . 4 ) , and (2 .13) g i v e c~ ^ k ; If; e J J dp (2.14) V 9v '3v J ImT J e dp J E v a l u a t i o n of the i n t e g r a l s y i e l d s v , - * £ ^ u ) ( ^ V - e e ^ /X()?r) t(?r) ; y + P l V r n ( 2 * 1 5 ) -4- D er~fc ( & I7xl) ] J where N (2.16) B. Removing The C o n t r i b u t i o n From The O r i g i n The d i v e r g e n t c o n t r i b u t i o n due to T = 1+0 = 0 i s s t i l l c o n t a i n e d i n e q u a t i o n ( 2 . 1 6 ) . T h i s may be remedied by s u b t r a c t i n g | x P from the r i g h t hand s i d e of e q u a t i o n (2 .2) where the i n t e g r a l now has lower l i m i t of i n t e g r a t i o n G . W r i t i n g b=rp and expanding the i n t e g r a n d i n a T a y l o r s e r i e s about b=0, one ge t s 1 4 Doing the i n t e g r a l s f o r each term i n the e x p a n s i o n g i v e s Now a p p l y p a r t i a l d e r i v a t i v e s w i t h r e s p e c t t o x L and x J t o o b t a i n a ^ T ^ l ^ n r j dp r ; v 3 ( 2 . 1 7 ) In the l i m i t as r goes t o z e r o t h i s becomes r —» o Ca ( 2 . 1 8 ) C. The F i n a l E x p r e s s i o n Thus the f i n a l e x p r e s s i o n f o r the second o r d e r p a r t i a l d e r i v a t i v e s of the e l e c t r o s t a t i c p o t e n t i a l a t the o r i g i n due t o a l a t t i c e of p o i n t charges e x c l u d i n g any charge a t t h e o r i g i n i t s e l f , i s V; , - I I V ^ j ^ i f f * e e 4 = 0 L {-Tv>3 t \ * /• r_ , • i \ ~ l l ( 2 . 1 9 ) If 6* C I W d,'J where e r f c i s the complementary e r r o r f u n c t i o n g i v e n by 2 P°° - x 1 e r f C ( \j ) = j j p ^ j e dx • T h i s e x p r e s s i o n , but w i t h the k=0 term m i s s i n g , was 15 f i r s t d e r i v e d in 1924 [ 5 ] . However, c h e c k i n g e q u a t i o n (2.19) for a s imple c u b i c l a t t i c e c o n f i r m s tha t the "k*=0 term must be p r e s e n t . T h i s a l g e b r a i c v e r i f i c a t i o n i s p r e s e n t e d in Appendix A . 2.3 F i n d i n g The P r i n c i p a l V i i Once the v a l u e s of V - are known wi th r e s p e c t to one set of C a r t e s i a n axes , i t i s s t r a i g h t forward to f i n d the p r i n c i p a l second d e r i v a t i v e s of the e l e c t r i c p o t e n t i a l and the p r i n c i p a l axes which d e f i n e the c o o r d i n a t e s w i th r e s p e c t to which the p r i n c i p a l d e r i v a t i v e s are t a k e n . C o n s i d e r i n g the V • • to be the components of a symmetric m a t r i x , V , one s o l v e s the s e c u l a r e q u a t i o n for i t s e i g e n v a l u e s , ^ k , def ( V - X I ) = O k - i . i , S ) (2 .20) (here I i s the 3x3 u n i t m a t r i x ) then s u b s t i t u t e s these e i g e n v a l u e s back i n t o the o r i g i n a l e i g e n v a l u e e q u a t i o n ( v - x i ) £ k - 0 (2.21) and s o l v e s for the e i g e n v e c t o r s E k c o r r e s p o n d i n g to each 7ik' •. I f the e i g e n v a l u e s are not d i s t i n c t , the procedu re for f i n d i n g t h r e e l i n e a r l y independent e i g e n v e c t o r s i s s l i g h t y d i f f e r e n t , but can be found in any s t a n d a r d t ex t on l i n e a r a l g e b r a . I t can be shown that the t h r e e e i g e n v a l u e s \ are the p r i n c i p a l d e r i v a t i v e s of the p o t e n t i a l , V x x , V y y , and V Z 2 , and tha t the c o r r e s p o n d i n g e i g e n v e c t o r s , E ( , Ez, and E 3 , a r e p a r a l l e l to the p r i n c i p a l axes X, Y , a n d Z . A proof i s 16 p r e s e n t e d i n Appendix B. 2.4 A Program To F i n d V:j W r i t i n g the l a t t i c e v e c t o r s , I , and r e c i p r o c a l l a t t i c e v e c t o r s , F, i n terms of t h e i r b a s i s v e c t o r s , J " f K = n,A + n j •+ n3 C (2.22) IT - k n « n .7 , - n i t - n , K 3 (2.23) where Tf. - w ( 3 - ct) ? i = ^ ( C ' y ) (2.24) j<5 = UL ( J , ? ) vcEa the summations i n e q u a t i o n (2.19) over 1 and "k* can be r e -e x p r e s s e d as summations over the i n t e g e r s n, , n 2 , n 3 . S i n c e V- i s r e a l , e' can be r e p l a c e d by c o s ( k . u ) . Making t h e s e s u b s t i t u t i o n s i n e q u a t i o n (2.19) g i v e s the f i n a l e x p r e s s i o n used i n the FORTRAN program of Appendix C t o e v a l u a t e V ;j : w _ dj: v n,,«i.,rx* L 114 " 1 (2.25) j~?u+o -f- D (iff •+ u ) The program was w r i t t e n t o accommodate l a t t i c e s where at most one of the a n g l e s between the l a t t i c e b a s i s 17 v e c t o r s A, *B, ~C, ( s p e c i f i c a l l y , the a n g l e between A and C) i s not n i n e t y degrees. However, i t can be e a s i l y m o d i f i e d t o p e r m i t g e n e r a l l a t t i c e s , by r e a d i n g i n t h r e e a p p r o p r i a t e a n g l e s between the l a t t i c e b a s i s v e c t o r s and r e p l a c i n g the c o o r d i n a t e and v e c t o r e x p r e s s i o n s by t h e i r most g e n e r a l forms. The r e c t a n g u l a r C a r t e s i a n axes w i t h r e s p e c t t o which the d e r i v a t i v e s v.- are o b t a i n e d a r e t h o s e such t h a t x l i e s a l o n g the A * - a x i s , y l i e s a l o n g the B - a x i s and z l i e s a l o n g the C - a x i s . The A * - a x i s i s d i r e c t e d a l o n g the v e c t o r BxC*. z t F i g u r e 3 - c o o r d i n a t e axes f o r V ;j of the program An o u t l i n e of the program f o l l o w s . i ) F i r s t N i s read, where N-1 i s the upper l i m i t of summation i n e q u a t i o n ( 2 . 2 5 ) . Note t h a t an i n c r e a s e i n N from M-1 t o M r e q u i r e s t h a t a f u r t h u r (12(m-1)+2) x C(N=1) com p u t a t u i o n s be performed, where C(1) i s the number of c o m p u t a t i o n s r e q u i r e d f o r N=1. In a l l l a t t i c e s so f a r 18 t e s t e d , N=4 has g i v e n convergence t o the f o u r t h f i g u r e . i i ) Next the l a t t i c e b a s i s parameters and u n i t c e l l p o s i t i o n s i n l a t t i c e c o o r d i n a t e s a r e r e a d . A t o t a l of 35 i o n s i n the u n i t c e l l i s p e r m i t t e d . The c o o r d i n a t e s a r e r e - e x p r e s s e d i n terms of the r e c t a n g u l a r C a r t e s i a n c o o r d i n a t e system d e f i n e d above and t r a n s l a t e d so t h a t the f i r s t p o s i t i o n c o o r d i n a t e read becomes the o r i g i n of the u n i t c e l l . i i i ) The s i x indpendent p a r t i a l d e r i v a t i v e s of the e l e c t r i c p o t e n t i a l are then c a l c u l a t e d a t t h i s o r i g i n due t o the l a t t i c e g e n e r a t e d by t h i s u n i t c e l l of p o i n t c h a r g e s t h r o u g h e q u a t i o n ( 2 . 2 5 ) , t o an a c c u r a c y d e t e r m i n e d by N. i v ) F i n a l l y the p r i n c i p a l d e r i v a t i v e s V ;j a r e found by the method of s e c t i o n 2.3. To m i n i m i z e r o u n d o f f e r r o r , the e r r o r due t o the f i n i t e number of b i n a r y d i g i t s w i t h which a number i s r e p r e s e n t e d by the computer, double p r e c i s i o n i s used, and the summation over u i n e q u a t i o n (2.25) i s done i n s i d e the summation over n. The amount of r o u n d o f f e r r o r p r e s e n t i n c r e a s e s w i t h N, and may be d e t e c t e d by c h e c k i n g the a c c u r a c y t o which L a p l a c e ' s e q u a t i o n , e q u a t i o n ( 1 . 4 ) , i s s a t i s f i e d . Note t h a t G i n e q u a t i o n (2.25) i s j u s t the i n t e r m e d i a t e l i m i t of i n t e g r a t i o n which s p l i t s the i n t e g r a l i d e n t i t y of e q u a t i o n (2.4) i n t o p a r t s t o be e v a l u a t e d i n F o u r i e r space and i n r e a l space. Thus G 19 s h o u l d be chosen so t h a t the two sums, one i n r e c i p r o c a l or F o u r i e r space and one i n l a t t i c e space, converge a t the same r a t e . The v a l u e of G has been s e t t o g i v e e q u a l convergence f o r l a t t i c e s of c u b i c symmetry. The program was t e s t e d by f o u r methods . F i r s t V.. f o r MnF2 was done by d i r e c t summation and found t o match the program r e s u l t . S e c o n d l y , V,-: was checked t o be z e r o f o r l a t t i c e s of c u b i c symmetry. T h i r d l y , v a r i a t i o n of G was found t o g i v e no change of r e s u l t s . F i n a l l y , L a p l a c e ' s e q u a t i o n was found t o be obeyed t o w i t h i n the e x p e c t e d r o u n d - o f f e r r o r . I f the e l e c t r i c f i e l d g r a d i e n t s a t a n o n - l a t t i c e p o i n t a r e d e s i r e d , they may be found by r e a d i n g i n t h i s l o c a t i o n as the f i r s t c o o r d i n a t e i n the u n i t c e l l , and a s s o c i a t i n g w i t h i t a charge of z e r o . The z e r o charge w i l l ensure t h a t the p o i n t makes no c o n t r i b u t i o n to V;- , w h i l e i t s p o s i t i o n w i l l be taken as the o r i g i n a t which the d e r i v a t i v e s V- a r e t o be d e t e r m i n e d . For c o n v e n i e n c e , output f o r m a t s have been chosen t o be c o m p a t i b l e w i t h l e n g t h s g i v e n i n angstroms and c h a r g e s g i v e n i n m u l t i p l e s of the p r o t o n c h a r g e . A s i d e from f o r m a t , however, the c h o i c e of u n i t s i s a r b i t r a r y , and by a change of format code may be l e f t t o the d i s c r e t i o n of the u s e r . 20 2.5 The Ster n h e i m e r A n t i s h i e l d i n q F a c t o r The v a l u e of V;- a t a n u c l e u s due t o a l a t t i c e of p o i n t charge " i o n s " n e g l e c t s the e l e c t r o n s of t h e atom i n which the n u c l e u s r e s i d e s . I n t e r - i o n i c and i n t r a - a t o m i c e l e c t r o n i n t e r a c t i o n s can d i s t o r t the c l o s e d s h e l l e l e c t r o n s about the n u c l e u s from s p h e r i c a l symmetry, and s i n c e they a re so c l o s e , t h e s e c l o s e d s h e l l e l e c t r o n s make an i m p o r t a n t c o n t r i b u t i o n t o V{- . The a c t u a l f i e l d g r a d i e n t a t the n u c l e u s i s g i v e n by [12] ( v7 . I . , = < 1 _ * ~ ) v 7 7 (2.26) 22 Actual 11 where ^ i s c a l l e d the St e r n h e i m e r a n t i - s h i e l d i n g f a c t o r . I t can be c a l c u l a t e d t h rough p e r t u r b a t i o n t h e o r y , and t u r n s out i n g e n e r a l t o be a n e g a t i v e number. The v a l u e of 16 „ used f o r the Mn 2* i o n i n the c a l c u l a t i o n s which f o l l o w - i s taken from r e f e r e n c e [ 1 5 ] . 2.6 S t r u c t u r e Of The L a t t i c e s 2.6.1 MnFj_ MnF z i s an o r t h o r h o m b i c c r y s t a l d e s c r i b e d by the space group P4/nmn . The l a t t i c e b a s i s v e c t o r s have l e n g t h s A = B = 4.865 A, C = 3.284 A, and the u n i t c e l l p arameter, U , of Ta b l e I has the v a l u e U = .31 [ 1 3 ] . X/A Y/B Z/C Mn ( 0 0 0 ) Mn ( .5 .5 .5 ) F ±( U U 0 ) F ±(U+.5 .5-U .5 ) Table I - P4/nmn u n i t c e l l 21 2.6.2 MnCl .4H 0 The form of manganous c h l o r i d e t e t r a h y d r a t e i s a m o n o c l i n i c c r y s t a l w i t h space group P2,/n. The u n i t c e l l has b a s i s v e c t o r l e n g t h s A = 11.186 A, B 9.523 A, and C = 6.186 A, and the a n g l e between A and B i s = 99.74°. There are f o u r u n i t s of MnCl 2.4H 20 per u n i t c e l l , l o c a t e d i n terms of l a t t i c e c o o r d i n a t e s a t (X/A, Y/B, Z/C) = ( x , y , z ; .5+x,.5-y,.5+z) (See F i g u r e 4 ) . The x, y, z p o s i t o n s f o r the Mn, C I , and 0 atoms are t a k e n from an X-ray d i f f r a c t i o n study [ 1 6 ] , w h i l e the hydrogen p o s i t i o n s come from n e u t r o n d i f f r a c t i o n a n a l y s i s [ 2 ] . ATOM x/A y/B z/C Mn 0. 2329 0. 1714 0. 9865 C M 1 ) 0. 061 0 0. 3076 0. 0938 C l ( 2 ) 0. 3817 0. 3662 0. 0355 0(1 ) 0. 3010 0. 1 1 27 0. 3334 0(2) 0. 1 568 0. 2280 0. 6446 0(3) 0. 1323 0. 9736 0. 9590 0(4) 0. 3695 0. 0381 0. 8764 H( 1 ) 1 0. 3831 0. 1418 0. 3882 H( 1 )2 0. 3005 0. 01 54 0. 3657 H(2) 1 0. 0728 0. 2000 0. 5993 H(2)2 0. 1989 0. 1 967 0. 5288 H ( 3 ) 1 0. 1 105 0. 9277 0. 8199 H(3)2 0. 0996 0. 9292 0. 0681 H(4) 1 0. 4356 0. 0716 0. 8061 H(4)2 0. 361 6 0. 9389 0. 8540 T a b l e I I - p o s i t i o n parameters f o r MnCl .4H 0 The experiment t o i n v e s t i g a t e the q u a d r u p o l e i n t e r a c t i o n was c a r r i e d out a t c r y s t a l t e m p e r a t u r e s of l e s s than 1 K, whereas the d a t a g i v e n i n T a b l e I I a r e t i m e - a v e r a g e d room temperature p o s i t i o n s . However, an 22 o r d e r - o f - m a g n i t u d e a n a l y s i s u s i n g the a p p r o p r i a t e e x p a n s i o n c o e f f i c i e n t s i n d i c a t e s t h a t the s h i f t i n p o s i t i o n s due t o t h i s t e m p e r a t u r e d i f f e r e n c e w i l l cause changes i n the c a l c u l a t e d P v a l u e s by amounts s m a l l e r than the e x p e r i m e n t a l e r r o r s . 23 F i g u r e 4 - S t r u c t u r e of the l a t t i c e s d i s p l a y i n g the o r d e r i n g of Mn atomic magnetic moments below the N e e l t e m p e r a t u r e s 24 2.7 P o i n t Charge R e s u l t s And Comparison W i t h Experiment 2.7.1 SSMn-MnF^ The output of the program d e s c r i b e d i n s e c t i o n 2.4 f o r MnF2 i s l i s t e d i n Appendix D, and g i v e s v a l u e s of the p r i n c i p a l e l e c t r i c f i e l d d e r i v a t i v e s of V x x /e = -0. 179 A' 3 V y y /e = -0.499 A' 3 (2.27) V 7 I /e = +0.678 A" 3 U s i n g these v a l u e s i n e q u a t i o n (2.21) y i e l d s p r i n c i p a l a x i s d i r e c t i o n s (see f i g u r e 4) . Z (2.28). Here, and i n the r e s t of s e c t i o n 2.7, A, B, C, w i l l denote u n i t v e c t o r s a l o n g the b a s i s v e c t o r s of the l a t t i c e under c o n s i d e r a t i o n . 5 5Mn has s p i n I = 5/2 and a qua d r u p o l e moment Q = 0.4±0 . 02x10 2 * cm 2 [ 3 ] . T a k i n g a St e r n h e i m e r ant i s h i e l d i n g f a c t o r ( 1 - Xo. ) = 9 [15] t o account f o r enhancement of VJZ due t o the e l e c t r o n s of the Mn i o n i n which the n u c l e u s r e s i d e s , g i v e s e 2qQ/h = 8.5±0.5 MHz (2.29) For 5 5Mn i n MnF Yasuoka et a l . [15] measure |e 2qQ/h| = 11.7±0.03 MHz (2.30) and c a l c u l a t e e 2qQ/h = 8.8 MHz (where the c a l c u l a t i o n has been m o d i f i e d by u s i n g the more r e c e n t v a l u e of Q g i v e n a b o v e ) , i n good agreement w i t h our 25 e s t i mate. 2.7.2 54Mn-MnCl,,4H a0 The a x i s of m a g n e t i z a t i o n f o r MnCl^^H^O l i e s a t p o l a r a n g l e s 6 = 7.6° and <t> = 3° i n the A*-B-C frame [ 1 ] . c F i g u r e 5 - the a x i s of m a g n e t i z a t i o n In terms of u n i t v e c t o r s a l o n g the A*, B and C axes of the c r y s t a l , the u n i t v e c t o r l y i n g a l o n g the a x i s of m a g n e t i z a t i o n can be w r i t t e n m = 0.13 A* + 0.0069 B + 0.99 C (2.31) a. MnCl a.4H,0 Wi t h Water M o l e c u l e s Assumed  E l e c t r i c a l l y N e u t r a l I f the water m o l e c u l e s i n MnCl^^H^O a r e d i s r e g a r d e d i n the p o i n t charge sum, then the p r i n c i p a l d e r i v a t i v e s of the e l e c t r i c p o t e n t i a l o b t a i n e d from the program output l i s t i n g i n Appendix D are V x y /e = -0.0162 A" 3 V y y /e = -0.112 A" 3 (2.32) V z z /e = +0.128 A" 3 and from e q u a t i o n (2.21) a r e found t o c o r r e s p o n d t o 26 p r i n c i p a l a x i s d i r e c t i o n s X = -0 .152 A + 0.882 B + 0.445 C Y = -0 .938 A * - 0.271 B + 0.217 C (2.33) Z = 0.312 A * - 0. 385 B + 0.869 C . U s i n g the s p i n , 1= 3, and the e l e c t r i c quadrupo le moment, Q = 0 . 4 ± 0 . 0 4 x 1 0 2 U cm 2 f or 5 "Mn [ 9 ] , one f i n d s PQ = 0 . 5 3 ± 0 . 1 2 MHz, and fl = 0.746 (2.34) S u b s t i t u t i o n of ( 2 . 3 1 ) , (2 .33) and (2.34) i n t o e q u a t i o n (1 .24) y i e l d s P = 2 . 3 ± 0 . 2 MHz . (2.35) where the u n c e r t a i n t y i s due to Q, and u n c e r t a i n t y due to Yo» n a s been n e g l e c t e d . b . M n C l , . 4 H , 0 With Charges Q(H)=.5e And Q(Q)=-e I t i s not c l e a r , however, that the water molecu le s may be i g n o r e d e l e c t r o s t a t i c a l l y . M i n i m i z a t o n of the p o i n t charge e l e c t r o s t a t i c energy in the u n i t c e l l wi th r e s p e c t t o the hydrogen atom p o s i t i o n s has been found [2] to g ive good agreement w i th the H atom p o s i t i o n s as de termined by neutron d i f f r a c t i o n when charges of -e and .5e were a s s i g n e d to the oxygen atoms and hydrogen atoms r e s p e c t i v e l y . T a k i n g the above charges for the hydrogen and oxygen atoms in M n C l a . 4 H z O g i v e s p r i n c i p a l p o t e n t i a l d e r i v a t i v e s (see Appendix D) V ™ /e = +0.003 A " 3 V y y / e = +0.225 A " 3 (2.36) V z z / e = -0 .258 k-3 and p r i n c i p a l axes 27 X = 0.056 A* + 0.681 B - 0.730 C Y =' -0.242 A* + 0.718 B + 0.652 C (2.37) Z = 0 . 9 6 9 A* + 0.140 B + 0.205 C . The d e f i n i t o n s , e q u a t i o n s (1.13) and (1.17) and the v a l u e s (2.35) y i e l d Pq = -0.1110.01 MHz, and i\ = 0 . 9 8 (2.38) From ( 2 . 3 1 ) , ( 2 . 3 7 ) , ( 2 . 3 8 ) , and (1.24) we f i n d P = +0.18±0.02 MHz . (2.39) c. The E x p e r i m e n t a l V a l u e Of P For 54Mn-MnCl .4H 0 The e x p e r i m e n t a l v a l u e of P i s 6.2±0.8 MHz [ 6 ] . T h i s i s about 35 times the p o i n t charge r e s u l t o b t a i n e d when the hydrogen and oxygen atoms are a s s i g n e d c h a r g e s of .5e and -e r e s p e c t i v e l y . The d i s c r e p a n c y between the s e v a l u e s may i n d i c a t e t h a t the p r o p o r t i o n of i o n i c t o n o n - i o n i c bonding of the water m o l e c u l e s i m p l i c i t i n t h i s model i s t o o g r e a t . Indeed the p o i n t charge r e s u l t f o r the case where the water m o l e c u l e s are n e g l e c t e d e n t i r e l y i s r e a s o n a b l y c l o s e t o the e x p e r i m e n t a l v a l u e . Note t h a t both p o i n t charge models gave the c o r r e c t s i g n f o r P. An u n c e r t a i n amount of e r r o r has been i n t r o d u c e d i n c o n v e r t i n g the r e s u l t s f o r the p r i n c i p a l p o t e n t i a l d e r i v a t i v e s , V-- , t o the e x p e r i m e n t a l l y d e t e r m i n e d parameter, P, because the a c c u r a c y of the t h e o r e t i c a l a n t i s h i e l d i n g f a c t o r i s unknown. 28 2.8 The Magnetic D i p o l e I n t e r a c t i o n Energy For MnCl,.4H^0 In D i f f e r e n t Magnetic Phases A l t h o u g h the prime o b j e c t i v e of t h i s work was the c a l c u l a t i o n of the e l e c t r i c q u a d r u p o l e i n t e r a c t i o n , the magnetic d i p o l e i n t e r a c t i o n can a l s o be e s t i m a t e d u s i n g the program d e s c r i b e d i n s e c t i o n 2.4. The magnetic d i p o l e i n t e r a c t i o n energy due t o two magnetic s u b l a t t i c e s w i t h magnetic moments /a, and /T2 a t the s i t e of an atom b e l o n g i n g t o the f i r s t s u b l a t t i c e i s g i v e n by ° ' P ' x S u b l a t t i c e 1 s u b l * t l i « Z * ' (2 40) k J W r i t i n g the d i p o l a r magnetic f i e l d , B d i p a t the o r i g i n due t o a magnetic d i p l e moment /< a d i s t a n c e r from the o r i g i n i n terms of the d i p o l e v e c t o r p o t e n t i a l ^^.p., ^ d , P ~ V * X d i p. where and a p p l y i n g the v e c t o r i d e n t i t y V * ( F * 7 $ ) - ( G - V ) F - £ ( V - F ) - ( f - v ) e - F ( v-<5) a l o n g w i t h L a p l a c e ' s e q u a t i o n , ( e q u a t i o n 1.4) g i v e s (2.41 ) (2.42) (2.43 d , p" 7 (2.44) S u b s t i t u t i n g t h i s e x p r e s s i o n f o r B d i p back i n t o e q u a t i o n ( 2 . 4 0 ) , one o b t a i n s the magnetic d i p o l e i n t e r a c t i o n energy 29 at an atom of moment i n terms of the q u a n t i t i e s V;. E a . - - t [ ) 1 ( /*.)' V.j t - C A ^ ^ ^ ^ - J > P ^ ' (2.45) where V ; i i s d e f i n e d , f o r the pupose of t h i s s e c t i o n o n l y , w i t h o u t the charge f a c t o r q ( r ) , 3* w V 7 1 1 — J — a , (2.46) and the s u b s c r i p t 1 or 2 denotes the magnetic s u b l a t t i c e over which the sum i s c a r r i e d o u t . When an i n c r e a s i n g magnetic f i e l d i s a p p l i e d p a r a l l e l t o the easy a x i s of an o r d e r e d magnet w i t h a n t i f e r r o m a g n e t i c exchange i n t e r a c t i o n , the system passes t h r o u g h t h r e e phases: the a n t i f e r r o m a g n e t i c , the s p i n - f l o p and the f e r r o m a g n e t i c regimes (see f i g u r e 6 ) . EASY AXIS B. E A S Y A X I S B. EASY AXIS B. A A ' ANTIF FERROMAGNETIC SPIN FLOP FERROMAGNETIC T R A N S V E R S E FIELD F i g u r e 6 - phases of an a n t i f e r r o m a g n e t For a b i a x i a l system i n the s p i n f l o p regime, the m a g n e t i z a t i o n v e c t o r s and l i e i n the p l a n e d e f i n e d by the e a s i e s t and n e x t - e a s i e s t a x i s (the y z - p l a n e ) . For a f i e l d a p p l i e d p e r p e n d i c u l a r t o the easy a x i s i n the y z - p l a n e the magnetic moments a r e c a n t e d by angl e 30 t o the f i e l d up t o a c r i t i c a l f i e l d v a l u e when the system a g a i n i s f e r r o m a g n e t i c a l l y o r d e r e d (see f i g u r e 6 ) . From e q u a t i o n (2.45) and v a l u e s of V- c a l c u l a t e d a t a Mn atom f o r the two magnetic s u b l a t t i c e s of MnCl 4.4H 20 shown i n f i g u r e 6, the e x p r e s s i o n s f o r E i n the s p i n f l o p and f e r r o m a g n e t i c regimes f o r a f i e l d a p p l i e d p a r a l l e l t o the easy a x i s can be summarised by E = f\  2 ( .01 1 9 cos 2« -.04 92cosoc s i n a -. 0290s i n 2 « )A" 3 , (2.48) where <x =0 c o r r e s p o n d s t o the f e r r o m a g n e t i c phase. For the case of a p e r p e n d i c u l a r a p p l i e d f i e l d , the magnetic d i p o l e i n t e r a c t i o n energy f o r the phase shown i n f i g u r e t u r n s out t o be E =/4M n 2 (.01 3 7 c o s 2 0 -.0492cos^3 sin(3 -.0256sin 2(S ) % ' 3 . (2.49) Note t h a t /S =0 c o r r e s p o n d s t o the a n t i f e r r o m a g n e t i c phase ( f i e l d a p p l i e d p a r a l l e l t o the easy a x i s ) . 31 I I I . DESCRIPTION OF AN NMRON SPECTRUM 3.1 The N u c l e a r Resonance C o n d i t i o n The h a m i l t o n i a n of a s p i n I n u c l e u s i n an o r d e r e d magnet w i t h the e f f e c t i v e magnetic f i e l d a t the n u c l e u s p a r a l l e l t o the z - a x i s i s g i v e n by H = H •+ H + H n h K n Q " S N (3.1) Here H^ i s the Zeeman magnetic i n t e r a c t i o n where A i s the magnetic h y p e r f i n e i n t e r a c t i o n t e n s o r and <S> i s the t h e r m a l average of the e l e c t r o n s p i n moment. For an i s o t r o p i c i n t e r a c t i o n H can be w r i t t e n (3.2) where g N i s the n u c l e a r gyromagnetic r a t i o , /iu i s the n u c l e a r magneton, and B*N i s the magnetic h y p e r f i n e f i e l d . H g i s the q u a d r u p o l e i n t e r a c t i o n h a m i l t o n i a n of e q u a t i o n ( 1 . 1 5 ) , and H $ N i s the Suhl-Nakamura h a m i l t o n i a n [ 8 ] where (3.4) a r e c o u p l i n g c o n s t a n t s . Here Y„ i s the number of o p p o s i t e s u b l a t t i c e n e a r e s t n e i g h b o r s w i t h which t h e r e i s an exchange i n t e r a c t i o n J , A i s the t r a n s v e r s e component of the h y p e r f i n e c o n s t a n t , and f ( r ) i s the range of the 32 i n d i r e c t s p i n - s p i n i n t e r a c t i o n which f o r l a r g e r ( u s i n g the n o t a t i o n of r e f e r e n c e 8) goes a s y m p t o t i c a l l y as Ur) ~ r e , oc ~ 1 ( 3 i 5 ) N o t i c e t h a t f ( r ) f a l l s o f f e x p o n e n t i a l l y f o r l a r g e v a l u e s of r . L e t us c o n f i n e our a t t e n t i o n t o the case of v e r y d i l u t e r a d i o a c t i v e n u c l e i i n a s o l i d . S i n c e the S u h l -Nakamura i n t e r a c t i o n i s e f f e c t i v e o n l y between l i k e n u c l e a r s p i n s , the f i r s t term i n H S N i s n e g l i g i b l e when the d i s t a n c e between s p i n s i s l a r g e , as i t i s i n t h i s c a s e . The second term i s p r o p o r t i o n a l t o the number of s p i n s of a g i v e n t y p e , and i s a l s o s u p p r e s s e d . Thus H S N can be n e g l e c t e d t o f i r s t o r d e r i n the s p i n -s t a t e e n e r g i e s of the r a d i a c t i v e n u c l e i , which from e q u a t i o n s ( 1 . 2 3 ) , (1.24) and (3.2) are found t o be E ^ ~ < I ™ | H M + H p | I m> (3.6) - a m + i [ ? m z - I ( I + I )) g i v i n g the energy d i f f e r e n c e between a d j a c e n t s p i n s t a t e s E „ - E _ - a - i P [ 2 - - i ] (3.7) ( a - V/2) - P m From the e x p r e s s i o n f o r photon energy, E = hv, where h i s P l a n k ' s c o n s t a n t and p i s the f r e q u e n c y of the e l e c t r o m a g n e t i c r a d i a t i o n , (3.7) y i e l d s the 21 f i r s t 33 o r d e r resonance c o n d i t i o n s h v„ = ( a - p / 2 ) -+ ? m (3.8) 3.2 Gamma-ray E m i s s i o n The n o r m a l i s e d gamma r a d i a t i o n i n t e n s i t y e m i t t e d at an a n g l e 6 t o the q u a n t i z a t i o n a x i s of a system of n u c l e i o r i e n t e d by the h y p e r f i n e i n t e r a c t i o n of e q u a t i o n (3.2) i s [11] W(Q) ' E A ( m, 9) P l m ) (3.9) where P(m) i s the n o r m a l i s e d p o p u l a t i o n of the s p i n s t a t e m and A(m,9) i s d e f i n e d by A (m , e ) = E U 2 J F ^ P Z J ! (cos e) f ^ n r v f i i T T 1 0 ) * I -m m o / Here F and U O J a r e a n g u l a r momentum c o e f f i c i e n t s c o r r e s p o n d i n g r e s p e c t i v e l y t o the obs e r v e d t r a n s i t i o n and t o any decays which may precede i t , P 2 < i s a Legendre p o l y n o m i a l of or d e r 21, and the l a s t f a c t o r i n the e x p r e s s i o n i s a 3j symbol [ 7 ] . The summation runs over i n t e g e r s from z e r o t o the l e s s e r of 2L or 2 I M , where I M l N i s the l o w e s t s p i n v a l u e i n the cascade which precedes the obse r v e d gamma t r a n s i t o n of m u l t i p o l a r i t y L. The t e c h n i q u e of n u c l e a r magnetic resonance of o r i e n t e d n u c l e i (NMRON) r e l i e s on m o n i t o r i n g the change of t h i s a n i s o t r o p i c r a d i a t i o n when the p o p u l a t i o n 34 d i s t r i b u t i o n of the s p i n l e v e l s i s a l t e r e d , whereas i n c o n v e n t i o n a l NMR e x p e r i m e n t s , i t i s the emf in d u c e d by the a l t e r n a t i n g f l u x due t o p r e c e s s i n g n u c l e a r s p i n s which i s d e t e c t e d . An advantage of the NMRON t e c h n i q u e , i s t h a t a s p i n system of a p p r o x i m a t e l y 1 0 1 1 p a r t i c l e s can produce enough "ft - r a d i a t i o n so t h a t changes i n i n t e n s i t y a re d e t e c t a b l e , as opposed t o the a p p r o x i m a t e l y 1 0 2 0 n u c l e i r e q u i r e d t o induce an o b s e r v a b l e v o l t a g e change i n s t a n d a r d NMR e x p e r i m e n t s . 3.3 P o p u l a t i o n Of The S p i n L e v e l s The p o u l a t i o n s of the s p i n s t a t e s of a system of o r i e n t e d n u c l e i a re governed by t h r e e e q u a t i o n s of e v o l u t i o n depending upon whether the s p i n system i s i n t h e r m a l e q u i l i b r i u m , i s b e i n g e x c i t e d by incoming r a d i a t i o n a t a resonance f r e q u e n c y , or i s r e l a x i n g toward e q u i l i b r i u m a f t e r e x c i t a t i o n . 3.3.1 The E q u i l i b r i u m P o p u l a t i o n The p o p u l a t i o n s of the s p i n s t a t e s , m, f o r a n u c l e a r s p i n system i n th e r m a l e q u i l i b r i u m w i t h a t h e r m a l r e s e v o i r a t t e m p e r a t u r e T, obey the Boltzmann d i s t r i b u t i o n _ E en) /|<t ? b < ^ = £ -e<~>/kT { 3 * 1 1 ) where E(m) i s the energy of s p i n s u b s t a t e m and k i s the Boltzmann c o n s t a n t . 35 3.3.2 The P o p u l a t i o n s At Resonance E l e c t r o m a g n e t i c r a d i a t i o n s a t i s f y i n g the resonance c o n d i t i o n of e q u a t i o n (3.8) i n d u c e s t r a n s i t i o n s between the m-1 and m n u c l e a r s p i n l e v e l s . I f the RF r a d i a t i o n has enough i n t e n s i t y , the p o p u l a t i o n s of t h e s e two l e v e l s w i l l be e q u a l i z e d ( s a t u r a t i o n c o n d i t i o n ) , each t a k i n g on the v a l u e of the average of t h e i r p r e v i o u s p o p u l a t i o n s . I f the time i n t e r v a l d u r i n g which the resonance f r e q u e n c y i s a p p l i e d i s s h o r t r e l a t i v e t o fhe r e l a x a t i o n t i m e , the p o p u l a t i o n s of the o t h e r l e v e l s a r e c o m p a r a t i v e l y una f f e c t e d . 3.3.3 The P o p u l a t i o n s D u r i n g R e l a x a t i o n Once RF e x c i t a t i o n s between the n u c l e a r s p i n l e v e l s c e a s e , i n t e r a c t i o n s w i t h the l a t t i c e cause the s p i n l e v e l p o p u l a t i o n s t o r e l a x toward t h e i r e q u i l i b r i u m d i s t r i b u t i o n s . The r a t e of change of the l e v e l p o p u l a t i o n s w i t h time d u r i n g r e l a x a t i o n i s g i v e n by 1 (3.12) d P c * 0 di (3.13) where W7 and VH a r e d e f i n e d i n terms of the s p i n - l a t t i c e 36 r e l a x t i o n time T, (3.14) <Irrm|lt I I | * Here, the square magnitudes of the m a t r i x elements are In e q u a t i o n s (3.13) and (3.14) i t has been assumed t h a t the s p i n l e v e l e n e r g i e s i n c r e a s e w i t h i n c r e a s i n g m. Note t h a t the time d e r i v a t i v e of the p o p u l a t i o n of one l e v e l i s a f u n c t i o n of the p o p u l a t i o n of i t s own and a t l e a s t one a d j a c e n t l e v e l . For m = - I , -1+1, ... ,1-1, I (3.13) i s a n o n d i a g o n a l system of 21+1 f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n s . For g e n e r a l T and/or T, i n c a s e s where 1^2, t h i s system can o n l y be s o l v e d n u m e r i c a l l y , s i n c e s o l u t i o n of the c o r r e s p o n d i n g s e c u l a r e q u a t i o n r e q u i r e s f i n d i n g the r o o t s of a 21+1 o r d e r p o l y n o m i a l . , > r = i < I m+.i i + i i ^ > r = I ( I + I ) - (M-+ I ) (3.15) 37 IV. PROGRAM TO GENERATE AN NMRON SPECTRUM APPLIED TO  54MN-MANGANESE CHLORIDE TETRAHYDRATE 4.1 Gamma-ray E m i s s i o n For 54-Mn The 5 uMn n u c l e u s has s p i n 1=3. I t decays by e l e c t r o n c a p t u r e and then by an E2 gamma-ray t o the ground s t a t e of the da u g h t e r , 5 * C r [ 4 ] . 54 Mn 3 + E 2 5 4 C r _ 2 + 0 F i g u r e 7 - decay scheme f o r 54-Mn For t h i s decay, the v a l u e s of the a n g u l a r momentum c o e f f i c i e n t s U J t, and Flt of e q u a t i o n (3.10) a r e found t o be U z=0.828, U 4=0.418, F a =-0.598, and F 4 =-1.069. S u b s t i t u t i n g these v a l u e s i n t o e q u a t i o n s (3.9) and (3.30) y i e l d s the n o r m a l i z e d gamma r a d i a t i o n i n t e n s i t y e m i t t e d a t an an g l e 9 t o the q u a n t i z a t i o n a x i s of an o r i e n t e d system of 54-Mn n u c l e i , W(6). For 6=0 t h i s i s W(0) = 1 + .333[P(1)+P(-1)] + .667[P(2)+P(-2)] - [P(3)+P(-3) ] (4.1) 38 4.2 A Two Temperature Model I f a f r a c t i o n of the r a d i o a c t i v e n u c l e i i n a s o l i d a r e c o o l e d t o a temperature T low enough t o produce a p p r e c i a b l e o r i e n t a t i o n of the n u c l e a r s p i n s , w h i l e the r e s t remain warm, then the gamma-ray i n t e n s i t y measured a l o n g the s p i n q u a n t i z a t i o n a x i s w i l l be y - m { ( i - -px) + £ t yco)} f N t ( 4 - 2 ) = WA { I + ( WCo) - I J f T ] where WA i s the count measured when the sample i s warm, and f T i s the f r a c t i o n of r a d i o a c t i v e n u c l e i which a re at t e m p e r a t u r e T. 4.3 The Resonance C o n d i t i o n s For 54-Mn In MnCl,.4H,0 The Zeeman i n t e r a c t i o n parameter, a, of e q u a t i o n (3.2) f o r a Mn n u c l e u s i n MnCl 1.4H i0 i s n e g a t i v e and much g r e a t e r i n magnitude than the qu a d r u p o l e i n t e r a c t o n c o n s t a n t , P, c a l c u l a t e d i n c h a p t e r I I . S i n c e the r e s u l t s of c h a p t e r I I i n d i c a t e t h a t P i s p o s i t i v e , e q u a t i o n (3.6) p r e d i c t s the s p i n l e v e l energy diagram of f i g u r e 8, i n which the quad r u p o l e i n t e r a c t i o n energy has been e x a g g e r a t e d f o r c l a r i t y , and the s p i n s t a t e s m a r e be l a b e l l e d i n o r d e r of a s c e n d i n g e n e r g i e s by a new i n t e g e r l a b e l n d e f i n e d by n = 3 - m (4.3) ( M o t i v a t i o n f o r t h i s l a b e l l i n g comes from the f a c t t h a t a r r a y s i n the programming language FORTRAN must be indexed 39 by p o s i t i v e i n t e g e r s . ) R e w r i t n g e q u a t i o n (3.7) i n terms of n g i v e s (4.4) A E N E R G Y \ | m « -3 n • b \ l \ I M \ , m-2 n-5 m=-l n«4 • 0 n-3 m -- 1 n-2 m-2 n = l \ r \ i r3 n-0 •M F +F F i g u r e 8 - m o d i f i c a t i o n of the Zeeman n u c l e a r s p i n energy l e v e l s by the quadrupole i n t e r a c t i o n The minimum energy d i f f e r e n c e between s p i n l e v e l s o c c u r s between the n=0 and n=1, or m=3 and m=2 s t a t e s . A E , al - ! ? (4.5) 40 From e q u a t i o n s (4.4) and (4.5) the t r a n s i t i o n e n e r g i e s between a d j a c e n t s p i n l e v e l s can be w r i t t e n i n terms of t h i s minimum energy d i f f e r e n c e as 2 A E „ = A E ^ •+ P (n - I ) n» I, 2, ••• , ( 4 > 6 ) D i v i d i n g by P l a n c k ' s c o n s t a n t the f i r s t o r d e r resonance f r e q u e n c i e s a r e o b t a i n e d (4.7) n « i, i, where 4.4 P o p u l a t i o n Of The N u c l e a r S p i n L e v e l s 4.4.1 E q u i l i b r i u m S i n c e |P| << |a|, the energy of the s p i n l e v e l s i n e q u a t i o n (3.11) may be a p p r o x i m a t e d by E(m)=-ma f o r m = -3 ,-2,...,2,3 g i v i n g - n a / k T ?3 ( n l m ^ w ^ (4.8) n The e x p e r i m e n t a l v a l u e of a/h i s -508.2 MHz [6] which can be w r i t t e n i n t h e r m a l u n i t s as a/k=-0.0244 K e l v i n s . 2 Note t h a t i f the s i g n of P i s n e g a t i v e , the minimum energy d i f f e r e n c e o c c u r s between the two h i g h e s t energy l e v e l s . I f the l e v e l s are l a b e l l e d by i n t e g e r s , n, which i n c r e a s e w i t h l e v e l e nergy, then i n eq. 4.6, P(n-1) — > 3P(2I-n) 41 4 .4 .2 Resonance Assuming that t ime i n t e r v a l d u r i n g which resonance frequency r a d i a t i o n i s a p p l i e d , t , i s s m a l l r e l a t i v e to the s p i n - l a t t i c e r e l a x a t i o n time T, , the p o p u l a t i o n s of the n u c l e a r s p i n l e v e l s d u r i n g the t ime when e q u a t i o n (4 .7) i s s a t i s f i e d for n=nK can be w r i t t e n e x p l i c i t l y as •for n - n ^ n = n R - i 2 (4 .9) 4 .4 .3 R e l a x a t ion One n u m e r i c a l a p p r o x i m a t i o n to the system of l i n k e d d i f f e r e n t i a l e q u a t i o n s (3.13) i s g i v e n by - Wf in ,1^ +1) . r n (4.10) - W K n . n - i J l P t . r t W - F g i m ] where the e x p r e s s i o n s for W|(n+1,n) and Wf(n,n+1) are o b t a i n e d by s u b s t i t u t i n g d e f i n i t i o n (4 .1) i n t o (3.13) and (3.14) and n o t i n g that l e v e l energy i n c r e a s e s w i th n , not w i t h m as p r e v i o u s l y assumed. Doing t h i s for the matr ix element terms y i e l d s k i - < i i - i i n > r = \ < i « u . i i n * , > r (4.11) 42 4.5 Width Of The Resosnance Peaks I n t e r a c t i o n s of the n u c l e a r s p i n s w i t h the l o c a l f i e l d s about them i n a s o l i d produce a f i n i t e w i d t h t o the resonance gamma-emission l i n e s . The Suhl-Nakamura i n t e r a c t i o n and an inhomogeneous spread i n the Zeeman and qu a d r u p o l e i n e r a c t i o n s w i l l make near g a u s s i a n c o n t r i b u t i o n s t o the l i n e w i d t h . I f i t i s assumed t h a t the net e f f e c t may be app r o x i m a t e d by a g a u s s i a n p r o f i l e then e q u a t i o n (4.2) i s r e p l a c e d by (4.12) 4.6 The Program The FORTRAN program l i s t e d i n appendix E g e n e r a t e s a spectrum of n u c l e a r magnetic resonance l i n e s c o r r e s p o n d i n g t o t r a n s i t i o n s between s p i n l e v e l s s p l i t by the h y p e r f i n e i n t e r a c t i o n a c c o r d i n g t o the g o v e r n i n g e q u a t i o n s of the p r e v i o u s s e c t i o n s of t h i s c h a p t e r . The e x p r e s s i o n f o r W(0), e q u a t i o n ( 4 . 1 ) , used i n the program i s s p e c i f i c t o 5"Mn and the upper l i m i t s on the DO-loops have been set f o r the 5"Mn s p i n of 3, but thes e can e a s i l y be changed t o accommodate any m a g n e t i c a l l y o r i e n t e d n u c l e a r s p i n system which i s governed by the h a m i l t o n i a n e q u a t i o n (3.1) i n which H s t r o n g l y dominates. The o t h e r spectrum parameters a r e read i n so t h a t they may be v a r i e d t o f i t a p a r t i c u l a r e x p e r i m e n t a l spectrum. Some sample o u t p u t s f o r d i f f e r e n t i n p u t parameters a re l i s t e d i n Appendix F. 43 The s p e c t r a a r e g e n e r a t e d a c c o r d i n g t o a l i n e a r change of f r e q u e n c y w i t h t i m e ; both an i n c r e a s e and d e c r e a s e w i t h time a r e p e r m i t t e d . Frequency m o d u l a t i o n i s not i n c l u d e d . Output may be i n one of two forms; the g e n e r a t e d spectrum may be p r e s e n t e d as a t a b l e of number of c o u n t s v e r s u s f r e q u e n c y , accompanied by a t a b u l a t i o n of the p o p u l a t i o n of the s p i n l e v e l s and the n o r m a l i z e d gamma e m i s s i o n i n t e n s i t y w i t h t i m e , or i t may be output as a p l o t of number of 'counts' v e r s u s f r e q u e n c y . As d i f f e r e n t p l o t t i n g f a c i l i t i e s and r o u t i n e s are a v a i l a b l e a t d i f f e r e n t computer c e n t e r s , p l o t s u b r o u t i n e s have not been i n c l u d e d i n the l i s t i n g of Appendix E. The FORTRAN s u b r o u t i n e s used t o produce the s p e c t r a p l o t s of f i g u r e s 9 t o 13 were w r i t t e n by Mary Ann P o t t s . The v e r t i c a l b a r s i n t h e s e s p e c t r a a r e s t a n d a r d d e v i a t i o n b a r s . The s e t of d i f f e r e n c e e q u a t i o n s (4.10) which i s used t o n u m e r i c a l l y " s o l v e " the system of d i f f e r e n t i a l e q u a t i o n s which govern the l e v e l p o p u l a t i o n s d u r i n g s p i n -l a t t i c e r e l a x a t i o n becomes n u m e r i c a l l y u n s t a b l e f o r a range of T, below a c e r t a i n v a l u e which depends upon the i n p u t v a l u e of the temperature and the f r e q u e n c y s t e p s i z e . F i g u r e 9 d i s p l a y s the onset of t h i s i n s t a b i l i t y . I t i s m a n i f e s t e d as the u n p h y s i c a l o s c i l l a t i o n i n the number of c o u n t s which i s j u s t b e g i n n i n g t o appear a t T, = 11000 seconds and i s q u i t e n o t i c a b l e a t T, = 10000 seconds f o r the s p e c t r a of f i g u r e 9. The s p e c t r a of t h i s f i g u r e have 44 the same i n p u t parameters except f o r the v a l u e s of T, . For even lower v a l u e s of T, the a m p l i t u d e of the o s c i l l a t i o n becomes so g r e a t t h a t n e g a t i v e v a l u e s of "the number of c o u n t s " a r e produced. 45 240000 239200 238400 237600 to r— 5 g 236800 u_ o 236000 235200 234400 233600 232600 |-232000 MNCL2—T1=11000-SECONDS a x . i FREQUENCY (MHZ) 2 4 0 0 0 0 239200 238400 237600 \ 236900 5 • 236000 c 235200 234400 233600 232800 232000 MNCL2--Tl=1000u-SECQNDS S12.I 5 M . 1 FREQUENCY IMHZ) F i g u r e 9 - onset of n u m e r i c a l i n s t a b i l i t y w i th decrease i n T, 46 Users of the program s h o u l d a l s o note t h a t the d i f f e r e n c e e q u a t i o n s (4.10) become worse a p p r o x i a m t i o n s t o the o r i g i n a l d i f f e r e n t i a l system as A t i n c r e a s e s . However too s m a l l a v a l u e of A t s h o u l d not be chosen due t o round-o f f e r r o r . A change t o a h i g h e r p r e c i s i o n r e p r e s e n t a i o n of the numbers at machine l e v e l ( e . g . from REAL*4 t o REAL*16) w i l l p e r m i t s m a l l e r v a l u e s of A t . Another way t o improve a c c u r a c y i s t o r e p l a c e the 2 - p o i n t d i f f e r e n c e f o r m u l a of e q u a t i o n (4.10) w i t h a t h r e e or h i g h e r p o i n t d i f f e r e n c e f o r m u l a t o approximate the d e r i v a t i v e s of e q u a t i o n ( 3 . 1 ) . T h i s may a l l e v i a t e the above mentioned s t a b i l i t y problem t o some degree. 47 4.7 Output Of The Program For 54Mn-MnCl, .4H,0 The program was used i n two ways. F i r s t , i n s e r t i n g e s t i m a t e d v a l u e s of the par a m e t e r s , i n i t i a l resonance p r o f i l e s were g e n e r a t e d . These were then used t o i d e n t i f y the r e s o n a n c e s , so t h a t more d e t a i l e d runs c o u l d be f i t t e d t o the e x p e r i m e n t a l s p e c t r a , a l l o w i n g o t h e r s p e c t r a parameters ( e . g . T, ) t o be o b t a i n e d . The ex p e r i m e n t s r e f e r r e d t o i n t h i s s e c t i o n are those of r e f e r e n c e F i g u r e 10 shows* the n u m e r i c a l l y produced gamma e m i s s i o n spectrum a l o n g the magnetic q u a n t i z a t i o n a x i s f o r 5 "MnCl a . 4 ^ 0 . From e q u a t i o n (3.7) and the energy l e v e l d iagram of f i g u r e 8, the resonance l i n e s of the spectrum may be i d e n t i f i e d . The s m a l l e s t energu d i f f e r e c e and hence l o w e s t f r e q u e n c y resonance c o r r e s p o n d s t o the m=3/m=2 t r a n s i t i o n , the next l o w e s t f r e q u e n c y b e l o n g s t o the m=2/m=1 t r a n s i t i o n and so on up t o the h i g h e s t f r e q u e n c y m=-2/m=-3 t r a n s i t i o n l i n e . Only t h e f i r s t t h r e e resonances show up i n the e x p e r i m e n t a l s p e c t r a , see f i g u r e 11, due t o the s m a l l d i f f e r e n c e s between the e q u i l i b r i u m p o p u l a t i o n s of the h i g h e r energy s p i n l e v e l s and the r e s o l v i n g c a p a b i l i t y of the a p p a r a t u s . 48 F i g u r e 10 - computer generated spectrum for 5 a M n i n M n C l 1 . 4 H 2 . 0 I, '1 ! / I — ' 1 1 1 1 1 1— 1 1 1 I I I I I I 460 .9 S00 .1 5 0 1 . 3 5 0 2 . 5 503 .7 5 0 4 . B 50B .0 5 0 7 . 2 50B .4 5 0 9 . B 511 .8 512.1 513 .2 5 1 4 . 4 515 8 518.6 FREQUENCY (MHZ) 49 F i g u r e 11 - e x p e r i m e n t a l spectrum for 5 f t M n i n M n C l 2 . 4 H i O w i t h a t h e o r e t i c a l f i t drawn i n FREQUENCY (MHZ) 50 Note t h a t i n the m=2/m=1 t r a n s i t i o n t h e r e i s a s h a r p drop i n W(0) f o l l o w e d by a r i s e above and then r e c o v e r y t o the e q u i l i b r i u m v a l u e . Such a p o s t - r e s o n a n c e bump had not been p r e v i o u s l y a n t i c i p a t e d . I t s e r v e s as a s i g n a t u r e f o r t h i s l i n e and a l l o w e d the p o s i t i v e i d e n t i f i c a t i o n of the resonance i n the e x p e r i m e n t a l spectrum. F i g u r e 11 shows an e a r l y run i n which a s u s p e c t e d m=2/m=1 resonance was obs e r v e d . R e f i n e d measurements w i t h a s m a l l f r e q u e n c y m o d u l a t i o n then r e v e a l e d the p r o f i l e seen i n f i g u r e 12. The c o n t i n u o u s l i n e drawn t h r o u g h the e x p e r i m e n t a l data b a r s i n f i g u r e 11 i s the b e s t - f i t g e n e r a t e d spectrum. I t c o r r e s p o n d s t o a s p i n - l a t t i c e r e l a x a t i o n time of T, = 3.0±0.5 x 10* seconds and a temperature T = 0.081 K. Si n c e the program d e s c r i b e d i n the l a s t s e c t i o n does not a l l o w f o r fr e q u e n c y m o d u l a t i o n , d i s c r e p a n c i e s between the ge n e r a t e d and e x p e r i m e n t a l s p e c t r a can be acc o u n t e d f o r by the 0.3 MHz m o d u a l t i o n and 0.14 MHz fr e q u e n c y s t e p s i z e used i n the e x p e r i m e n t . I f fre q u e n c y i s modulated over the time i n t e r v a l d u r i n g which gamma-ray coun t s a r e t o t a l l e d , then because W(0) of e q u a t i o n (4.1) i s a f u n c t i o n of f r e q u e n c y , the accumulated count v a l u e w i l l be some weigh t e d average over the fre q u e n c y range of the m o d u l a t i o n . In p a r t i c u l a r t h i s c o u l d e x p l a i n the h e i g h t d i f f e r e n c e of the m=3/m=2 resonance l i n e s of the two s p e c t r a . 51 Figure 12 - the o r i g i n a l m=2/m=1 t r a n s i t i o n l i n e spectrum 243260 242478 241696 240914 5240132 ; 239350 |-238568 237786 237004 236222 235440 NNCL2 V H=0 100QS * 5 4 JL _i_ 502.1 502.7 303.9 503. B FREQUENCY (MHZ) 504.8 52 F i g u r e 13 - the a d j u s t e d m=2/m=1 t r a n s i t i o n l i n e spectrum, w i t h a t h e o r e t i c a l f i t drawn i n 240000 239200 238400 237600 CO g 236800 ; 236000 \-235200 234400 233600 232800 232000 502.1 _ 1 _ _ 1 _ 303.1 304.1 FREQUENCY (MHZ) SOS.l 53 More d i f f i c u l t y was encountered i n t r y i n g t o a n a l y z e the spectrum of f i g u r e 12. In the b e s t o b t a i n a b l e f i t , t h e h e i g h t of the peak above the e q u i l i b r i u m l e v e l f o r the t h e o r e t i c a l spectrum was l e s s than h a l f t h a t of the e x p e r i m e n t a l v a l u e . However, i n s p e c t i o n of f i g u r e 12, shows t h a t t h e r e i s an upward s l o p e i n the number of c o u n t s b e f o r e the resonance d i p , which i s c o n s i s t e n t w i t h a warming of the sample d u r i n g the r u n . When the i n c r e a s e i n c o u n t s due t o the s u s p e c t e d temperature i n c r e a s e i s s u b t r a c t e d o f f , the spectrum of f i g u r e 13 i s o b t a i n e d . As can be seen, the f i t w i t h the n u m e r i c a l spectrum, which i s the smooth c u r v e superimposed on the d a t a b a r s of the a d j u s t e d spectrum, i s e x c e l l e n t . I t c o r r e s p o n d s t o T, = 2.3±0.3 x 10" seconds and a temperatute T = 0.064 K e l v i n s . 54 V. SUMMARY Two computer programs a r e de v e l o p e d as a i d s i n the stud y of h y p e r f i n e i n t e r a c t i o n s i n magnetic s o l i d s . The f i r s t program f i n d s the p r i n c i p a l second o r d e r d e r i v a t i v e s of the e l e c t r i c p o t e n t i a l a t any p o s i t i o n i n a l a t t i c e of p o i n t c h a r g e s . I t i s used t o p r o v i d e an e s t i m a t e of the magnitude and s i g n of the n u c l e a r - l a t t i c e q u a d r u p o l e i n t e r a c t i o n c o n s t a n t s P and Pq . T h i s program can a l s o be used t o de t e r m i n e the magnetic d i p o l e i n t e r a c t i o n e n e r g i e s f o r the phases of an o r d e r e d a n t i f e r r o m a g n e t . The second program g e n e r a t e s a spectrum of n u c l e a r magnetic resonance l i n e s s h i f t e d by quadr u p o l e i n t e r a c t i o n w i t h subsequent s p i n - l a t t i c e r e l a x a t i o n , and t h i s program i s a p p l i e d t o the case of 5 "Mn-MnCl^ . 4H i0. I t was w r i t t e n because the r e d i s t r i b u t i o n of the p o p u l a t i o n s of the n u c l e a r s p i n l e v e l s (which d e t e r m i n e s the gamma r a y e m i s s i o n ) cannot be s o l v e d a n a l y t i c a l l y . The n u m e r i c a l c o m p u t a t i o n a l l o w s the p r o f i l e s of the resonance l i n e s and subsequent r e l a x a t i o n s t o be deduced. Comparison w i t h experiment a l l o w s d e t e r m i n a t i o n of parameters such as the s p i n - l a t t i c e r e l a x a t i o n t i m e . For M n C l 2 . 4 H 2 0 the p o i n t charge c a l c u l a t i o n s gave V 2 2_/e = 0. 1 277 A" 3 and an asymmetry parameter of f\ = 0.746 when the water m o l e c u l e s were c o n s i d e r e d as e l e c t r i c a l l y n e u t r a l u n i t s , and VL^/e = 0. 02551 A" 3 and f\ =0.98 when the oxygen and hydrogen atoms were a s s i g n e d c h a r g e s of -e and +e/2 r e s p e c t i v e l y . 55 Comparison of the n u m e r i c a l r e s u l t w i t h the e x p e r i m e n t a l l y o b t a i n e d q u a n t i t y , P, i s dependent upon a t h e o r e t i c a l a n t i s h i e l d i n g f a c t o r whose a c c u r a c y i s d i f f i c u l t t o d e t e r m i n e . I n c o r p o r a t i n g t h i s f a c t o r i n t o V , the above v a l u e s c o r r e s p o n d t o P = 2.3±0.2 MHz and P = 0.18±0.02 MHz r e s p e c t i v e l y . Experiment gave P = 6.2±0.8 MHz. Two e x p r e s s i o n s summarize the Mn magnetic d i p o l e i n t e r a c t i o n e n e r g i e s c a l c u l a t e d f o r the d i f f e r e n t phases of MnCl i.4H zO: , = At* \ 0. 0 111 eos1^. - 0. 0«-<?2cos ot s.« oc -0.0290 sm1*"] A i s the d i p o l e energy i n the case of a f i e l d a p p l i e d a l o n g the easy a x i s f o r the s p i n f l o p phase where the magnetic moments a r e c a n t e d a t a n g l e s of ±« t o the easy a x i s . Note t h a t o(=0 c o r r e s p o n d s t o the f e r r o m a g n e t i c phase. In the case of a f i e l d a p p l i e d t r a n s v e r s e t o the easy a x i s i n the c r y s t a l B-C p l a n e , the e x p r e s s i o n o b t a i n e d f o r the d i p o l e i n t e r a c t i o n energy f o r the phase i n which the Mn d i p o l e moments a r e o r i e n t e d i n d i r e c t i o n s of ±/5 w i t h r e s p e c t t o the a p p l i e d f i e l d d i r e c t i o n i s -^d.pote. = / V n [ 0.0 13? COS*j3 -0.04 92 COS /S - 0.0 25<p sm 1^] A j6 =0 g i v e s the magnetic d i p o l e i n t e r a c t i o n energy f o r the a n t i f e r r o m a g n e t i c phase ( f i e l d a p p l i e d a l o n g t he easy a x i s ) . The n u m e r i c a l l y g e n e r a t e d NMRON spectrum f o r 5 4Mn i n 56 MnCl z.4H J L0 produced an unexpected p r o f i l e f o r the m=2/m=1 resonance l i n e : a f t e r an i n i t i a l d e c r e a s e , the gamma-ray i n t e n s i t y then r o s e above the i n i t i a l e q u i l i b r i u m v a l u e and s u b s e q u e n t l y r e l a x e d back t o i t . T h i s b e h a v i o r can be take n as a s i g n a t u r e t o the resonance and indeed was observed i n the e x p e r i m e n t a l spectrum when the m o d u l a t i o n f r e q u e n c y was reduced t o a s m a l l v a l u e . The t h e o r e t i c a l and e x p e r i m e n t a l s p e c t r a can be f i t t e d r e a s o n a b l y w e l l , a l l o w i n g h y p e r f i n e i n t e r a c t i o n p a rameters t o be e v a l u a t e d . Of the s i x resonance l i n e s p r e d i c t e d due t o the e x i s t e n c e of a qu a d r u p o l e i n t e r a c t i o n , o n l y t h r e e were w i t h i n the r e s o l v i n g c a p a b i l i t y of the experiment owing t o the s m a l l e q u i l i b r i u m p o p u l a t i o n d i f f e r e n c e s of the l e v e l s , and o n l y the f i r s t two had d e f i n i t i v e p r o f i l e s . A l l o w i n g f o r t h e e f f e c t s of the e x p e r i m e n t a l m o d u l a t i o n f r e q u e n c y , a r e a s o n a b l e f i t was o b t a i n e d f o r the l o w e s t energy and s t r o n g e s t resonance, the m=3/m=2 t r a n s i t i o n l i n e . From t h i s v a l u e s of T, = 3.0±0.5 x 10" seconds and T = 0.081 K e l v i n s were o b t a i n e d . I t was not p o s s i b l e t o get a f i t w i t h i n t he e x p e r i m e n t a l u n c e r t a i n t i e s f o r the raw m=2/m=1 t r a n s i t i o n s pectrum. Adjustment of the spectrum t o tak e i n t o account a s u s p e c t e d warming of the sample over the d u r a t i o n of the ru n , p e r m i t t e d a good f i t f o r T, = 2.3±0.3 x 10* seconds and an i n i t i a l t e m p e r a t u r e of 0.064 K e l v i n s . Note t h a t t h i s v a l u e f o r T does not i n c l u d e any u n c e r t a i n t y f o r adjustment of the spectrum. 57 BIBLIOGRAPHY 1. A l t m a n , R. F.; Spooner, S.; Landau, D. P.; and R i v e s , J . E.; Phys. Rev. B, 11, 606 (1958). 2. E l S a f f a r , Z. M.; J . Chem Phys. 52, 4097 (1970). 3. H a n d r i c h , E.; S t e u d e l , A.; and W a l t h e r , H.; Phys. L e t t . 29A, 486 (1969). 4. K a r l s s o n , E.; Wappling, R.; eds. H y p e r f i n e I n t e r a c t i o n s : S t u d i e s i n N u c l e a r R e a c t i o n s and Decays, ( A l m q u i s t and W i k s e l l I n t e r n a t i o n a l ; S w e d e n ) . 5. K o r n f e l d , H.; Z e i t s . P h y s i k . 22 (1940). 6. K o t l i c k i , A.; McLeod, B.; S h o t t , M. ; and T u r r e l l , B. G.; t o be p u b l i s h e d , Hyp. I n t . 7. M e s s i a h , A.; Quantum Mechanics V o l . 1 1 , ( N o r t h -H o l l a n d ) , Appendix C. 8. Nakamura, T.; P r o g r . T h e o r e t . P h y s . ( K y o t o ) 20, 542 (1958). 9. N i s s e n , L.; and Hiuskamp, W. J . ; P h y s i c a 50, 259 ( 1970). 10. S l i c h t e r , C h a r l e s P.; ' P r i n c i p l e s of Mag n e t i c Resonance', pp 164-170, (Harper & Row; 1963). 11. S t e f f e n , R. M.; and A l d e r K.; i n 'The E l e c t r o m a g n e t i c I n t e r a c t i o n i n N u c l e a r S p e c t r o s c o p y ' , ed. W. D. H a m i l t o n ( N o r t h H o l l a n d ; 1975) Ch. 12. 12. S t e r n h e i m e r , R. M.; Phys. Rev. 84, 244 (1951); Phys. Rev. 86, 316 (1952); Phys. Rev. 95, 736 (1954) . 13. S t o u t , J . W.; G r i f f e l , M.; Phys. Rev. 76, 144 (1949); J . Chem. Phys. 18, 1455 (1950). 14. S u h l , H.; Phys. Rev. 109, 606 (1958). 15. Yasuoka, H.; Ngwe, T i n ; J a c c a r i n o , V., Phys. Rev. 177, 667 (1969). 16. Z a l k i n , A.; F o r r e s t e r , J . D.; Templeton, D.; I n o r g . Chem. 3, 529 (1964). 58 APPENDIX A - CHECK OF EQUATION 2 . 1 9 F i r s t we show d i r e c t l y t h a t V v a n i s h e s a t the o r i g i n f o r a s i m p l e c u b i c l a t t i c e i n a c u b i c volume c e n t e r e d on the o r i g i n . V i s d e f i n e d by v 2 2 - Zs & ( fcj) E v a l u a t i n g the d e r i v a t i v e , t h i s i s = h* (A. 1 ) The l a t t i c e v e c t o r s f o r a s i m p l e c u b i c l a t t i c e of s i d e 'a' are r = a( n , i + n 2 j + n 3k ) ( A . 2 ) where n, , n 2 , n 5 a r e i n t e g e r s . S u b s t i t u t i n g t h i s i n t o ( A . 1 ) g i v e s , , x " 3n»* - ( to,2" + ru* + in»a ) = - N I f we d i v i d e the denominator i n t o terms i n n , n and n i t i s c l e a r t h a t t h i s sum i s z e r o . "N _ v ~ N "I I f e q u a t i o n ( 2 . 1 9 ) f o r Vrj i s c o r r e c t , then the r i g h t hand s i d e of t h i s e q u a t i o n must a l s o v a n i s h f o r a s i m p l e c u b i c l a t t i c e . For such a l a t t i c e the r e c i p r o c a l l a t t i c e v e c t o r s a r e g = 2 / a ( n , i + n x 5 + n * k ) (A.3) and the volume of the u n i t c e l l i s v £ £ L t= a 3 (A.4) U s i n g the f a c t t h a t the e x p r e s s i o n i s independent of the v a l u e of G, as l o n g as G i s p o s i t i v e , we choose parameter G so t h a t the e x p o n e n t i a l terms i n the two sums of e q u a t i o n (2.19) are e q u a l . T h i s r e q u i r e s G =v/Tf/a (A. 5) The g=0 term i s found by s e t t i n g n 1=n 2=n 3=n i n e q u a t i o n (A.3) and t a k i n g the l i m i t as n goes t o z e r o . S u b s t i t u t i n g ( A . 2 ) , ( A . 3 ) , ( A . 4 ) , and (A.5) i n t o 59 (2.19) and u s i n g the f a c t t h a t f o r a s i m p l e c u b i c l a t t i c e 0 o as was shown above f o r the s p e c i a l case m=5, g i v e s -+ V ; =r 2. II ) = 0 Thus the r i g h t hand s i d e of e q u a t i o n (2.19) does indeed v a n i s h f o r a s i m p l e c u b i c l a t t i c e . T h i s would not be t r u e i f the g*=0 c o n t r i b u t i o n of 1/3 were not p r e s e n t . 60 APPENDIX B - PROOF THAT THE EIGENVALUES AND EIGENVECTORS OF V ARE PRINCIPAL FIELD GRADIENTS AND PRINCIPAL AXES L e t the u n i t e i g e n v e c t o r s E\ of the m a t r i x which has components V;j be the b a s i s v e c t o r s of the x c o o r d i n a t e system, and l e t e K be u n i t b a s i s v e c t o r s of the x c o o r d i n a t e system, such t h a t >x"" 2* J ( B . I ) then - U . - ^ U x - E , - ) V . i ( B - 2 ) By w r i t n g the e i g e n v e c t o r s i n terms of t h e i r components i n the x - c o o r d i n a t e system K - ( E w Y t - (£*Y&* - ( £ „ 1 * 6 , (B.3) the r i g h t hand s i d e of (B.3) becomes v , . - fcYW y k l ( B. 4 ) A but s i n c e Ej i s an e i g e n v e c t o r t h i s i s V„ - (£<)"{ii)' ^ (B.5> By the commutative p r o p e r t y of p a r t i a l d i f f e r e n t i a t i o n and the r e a l n e s s of the e l e c t r i c f i e l d , t he m a t r i x w i t h components V i s symmetric. I t i s t r i v i a l t o show t h a t the e i g e n v e c t o r s of a symmetric m a t r i x f o r d i s t i n c t e i g e n v a l u e s a r e o r t h o g o n a l (and i f the e i g e n v a l u e s a r e not d i s t i n t , can be chosen o r t h o g o n a l ) . Hence i t i s shown V 'j ' $ <j ^ j (B.6) The d e r i v a t i v e s of the e l e c t r i c p o t e n t i a l V w i t h r e s p e c t t o the c o o r d i n a t e system which has the ( o r t h o g o n a l ) e i g e n v e c t o r s of the m a t r i x w i t h components VCj as i t s b a s i s , a r e the p r i n c i p a l d e r i v a t i v e s . 61 APPENDIX C - PROGRAM 1 C PROGRAM TO SUM BY EWALDS METHOD, THE SECOND ORDER PARTIAL DERIVATIVES * C OF THE ELECTRIC POTENTIAL AT THE FIRST INPUT COORDINATE POSITION DUE * C TO POINT CHARGES AT THE LATTICE POINTS, FOR LATTICES IN WHICH AT MOST * C ONE OF THE ANGLES BETWEEN LATTICE BASIS VECTORS IS NOT NINETY DEGREES * IMPLICIT REAL*8(A-H,0-Z) REAL U ( 6 , 3 5 , 3 ) , C H A R G E ( 1 0 ) , G , G S 0 , X ( 3 ) , R L ( 3 ) , S 1 Z E R 0 ( 6 ) , S 1 ( 6 ) . S 2 ( 6 ) REAL 0 1 ( 6 ) . 0 2 ( 6 ) , V I J ( 6 , 1 0 ) , S U M 1(6 ) ,SUM2(6) ,ORIG IN(3 ) INTEGER NEACH(10) PI=3. 141592653 ROOTPI=DSQRT(PI) C THE SUMS FOR VIO WILL BE COMPUTED UP TO 1 NX 1=1 NY 1 = 1N21=N-1 READ(5 ,30) N C INPUT LATTICE BASIS VECTOR LENGTHS IN ANGSTROMS C AND THE ANGLE IN RADIANS BETWEEN VECTORS A AND C READ(5 ,20) A , B , C , B E T A C G IS THE INTERMEDIATE LIMIT OF INTEGRATION WHICH SPLITS THE INTEGRAL C INTO A FOURIER OR RECIPROCAL SPACE PART (SUM 1) AND A REAL SPACE PART (SUM2) C THIS CHOICE OF G GIVES GOOD CONVERGENCE RATES FOR BOTH PARTS C FOR LATTICES WITH NEAR CUBIC SYMMETRY. G= ROOTPI/C C PREPARE CONSTANTS TO BE USED IN THE SUMMATION LOOPS GSO=G»G SBETA=DSIN(BETA) CBET A = DCOS(BET A) AX = A * SB ETA AZ=A*CBETA WRITE ( 6 , 7 7 ) A . B .C . SBETA C0NST1=P I/ (AX*B*C )* (-4 . ) C INPUT AND OUTPUT CHARGE AND POSITIONS OF THE IONS IN THE C PRIMITVE CELL WHICH ARE USED TO GENERATE THE LATTICE READ(5,20) NTYPES READ(5 ,20) (CHARGE(K) ,K=1.NTYPES) READ(5 ,20) (NEACH(K) , K =1,NTYPES) WRITE ( 6 ,26 ) WRITE(6,28) DO 11 1=1.NTYPES NIONS = NEACH(I ) WRITE(6,27) CHARGE(I) DO 11 U=1.NI0NS READ(5 ,22) XCOORD.Y.Z WRITE(6,29) XCOORD.Y.Z C RE-EXPRESS LATTICE COORDINATES IN ORTHOGONAL CARTESIAN COORDINATES U ( I , J ,3 )=C*Z+AZ*XC00RD U ( I , J . 1 )=AX*XC00RD 11 U(I ,<J,2)=B*Y C THE DERIVATIVES V I J OF THE ELECTRIC POTENTIAL ARE DETERMINED AT THE ORIGIN. C TRANSLATE THE PRIMITIVE CELL SO THAT THE FIRST ION READ IN, C THE ION LABELLED 0=1, OF CHARGE LABELLED 1=1. L IES AT THE ORIGIN. DO 133 K=1 .3 133 OR IG IN (K )=U(1 ,1 ,K ) DO 12 1=1,NTYPES NIONS=NEACH(I) DO 12 0=1,NIONS DO 12 K=1,3 12 U ( I ,U ,K )=U( I ,U ,K )-OR IG IN (K ) C RECIPROCAL LATTICE BASIS FOR A MONOCLINIC LATTICE R1=2.*PI/AX R2=2.*PI/B 62 R3Z = 2.*P I/C R3X=-2.*P I*CBETA/(C*SBETA) RX=R1+R3X RYS0=R2*R2 RZS0=R3Z*R3Z RLSQ=RX*RX+RYSQ+RZSQ C FIRST TERM IN THE RECIPROCAL LATTICE SUM S1ZERO(1 )=RX*RX/RLSQ S1ZERO(2)=RYSQ/RLSQ S1ZER0(3)=RZSQ/RLSQ S1ZER0(4)=RX*R2/RLSQ S1ZERO(5)=RX*R3Z/RLSO S1ZERO(6)=R2*R3Z/RLSQ C WRITE TABLE HEADINGS WRITE(6,31 ) WRITE(6,32) WRITE(6,33) WRITE(6,23) C INT LABELS THE TERM IN THE SUMMATION WHICH IS STORED IN VI J C 0 1 , 0 2 , CORRESPOND TO THE CURRENT TOTALS OF Q ( I ) SUM1( I ) , 0 ( I ) SUM2( I ) INT=0 DO 615 1=1,6 02 ( I )=0 . 615 01 ( I )=0 . DO 116 1=1,3 116 Q2( I )=4.*GSQ*G/(3 .*ROOTPI )*CHARGE(1) C LOOK AT UNIT CELLS UP TO N-1 CELLS DISTANT DO 5 NX= 1 ,N N1=NX-1 55 CONTINUE DO 4 NY =1,N N2=NY-1 54 CONTINUE DO 3 NZ=1,N N3=NZ-1 53 CONTINUE C FIND CONTRIBUTIONS FROM EACH TYPE OF ION DO 2 1=1,NTYPES DO 91 K=1 ,6 SUM1(K)=0. 91 SUM2(K)=0. C SUM CONTRIBUTIONS FROM ALL IONS OF ONE TYPE NIONS=NEACH(I) DO 1 J=1,NI0NS C ION CARTESIAN COORDINATES FROM THE PRIMITIVE CELL ORIGIN U1.U2.U3 U1=U( I ,U , 1 ) U2=U( I , d ,2 ) U3=U( I , d ,3 ) C DO RECIPROCAL LATTICE SUM, SUM 1 RL(1)=N1*R1+N3*R3X RL(2)=N2*R2 RL(3)=N3*R3Z RLSO=RL(1 )*RL (1 )+RL (2 )*RL (2 )+RL (3 )*RL (3 ) IF (RLSO.EO.O) GO TO 14 REXP=RLSQ/(4.*GSO) IF (REXP .GT .174) GO TO 15 S=DCOS(RL(1)*U1+RL(2)*U2+RL(3)*U3)/(RLSQ*DEXP(REXP) ) DO 17 IS=1,3 17 S1 ( I S )=RL ( I S ) *RL ( I S ) *S S1 (4 )=RL (1 ) *RL (2 ) *S 63 S1(5) = RL(1 )*RL(3)*S S1 (6 )=RL (2 ) *RL (3 ) *S GO TO 16 14 DO 140 IS=1,6 140 S1( IS )=S1ZER0( IS ) 16 DO 160 IS=1,6 160 SUM1(IS)=SUM1(IS)+S1( IS) 15 CONTINUE C DO REAL SPACE LATTICE SUM, SUM2 C ION COORDINATES WRT ORIGIN ABOUT WHICH MOMENT IS MEASURED ARE DENOTED X( I ) X(1 )=N1*AX+U1 X(2)=N2*B+U2 X(3)=N1*AZ+N3*C+U3 RR=X(1 )*X (1 )+X(2 ) *X (2 )+X(3 ) *X (3 ) IF ( RR . EO .O . ) GO TO 10 R=DSQRT(RR) IF ( (GSO*RR ) .GT .174 . ) GO TO 6 GREXP=1./DEXP(GSQ*RR)*2./R00TPI*G GO TO 66 6 GREXP=0. 66 EF = DERFC(G*R ) DO 400 IS=1,3 DSO=X( IS )*X( IS ) 0UADRU=(3.*DSQ-RR)/(RR*RR) 400 S2(IS)=2.*DSQ/RR*GSQ*GREXP+QUADRU*(GREXP+EF/R) RRR=RR*R TERM=(GREXP*(2.*GSQ*G+3./RRR)+3./RRR*EF)/R S2(4)=X( 1 ) * X ( 2 ) » T E R M S2(5)=X( 1)*X(3)*TERM S2 (6 )=X(2 )*X (3 )*TERM DO 120 IS=1,6 120 SUM2(IS)=SUM2(IS)+S2( IS) 10 CONTINUE 1 CONTINUE 0=CHARGE(I) DO 123 K=1 ,6 01(K ) = 0* SUM1(K) + Q1(K) 123 02(K)=0*SUM2(K)+02(K) 2 CONTINUE IF ( ( N 1 . N E . N 2 ) . 0 R . ( N 2 . N E . N 3 ) ) GO TO 44 INT=INT+1 DO 126 K=1 ,6 126 V I J (K , INT)=C0NST1*Q1(K)+Q2(K) 44 CONTINUE IF ( N 3 . L E . 0 ) GO TO 3 N3=-1*N3 GO TO 53 3 CONTINUE IF ( N 2 . L E . 0 ) GO TO 4 N2=-1*N2 GO TO 54 4 CONTINUE IF ( N 1 . L E . 0 ) GO TO 5 N1=-1*N1 GO TO 55 5 CONTINUE C OUTPUT THE 6 INDEPENDENT PARTIAL DERIVATIVES OF THE ELECTROSTATIC POTENTIAL LAST=2*N-1 WRITE(6,60) ( ( V I J ( L , K ) , L = 1 , 6 ) . K = 1 . L A S T ) CALL P R I N C ( V I U ( 1 , L A S T ) . V I J ( 2 , L A S T ) , V I d ( 3 , L A S T ) , V I d ( 4 . L A S T ) , V I J ( 5 , L A S T ) . V I J ( 6 , L A S T ) ) 64 60 FORMAT(6F13.6) 20 FORMAT(6G7.4) 22 F0RMAT(3E13.4) 23 FORMAT(/7X, 'VXX ' , 10X, 'VYY ' , 10X, ' V Z Z ' , 10X, 'VXY ' ,10X, ' VXZ ' , 10X, ' VYZ ' , / , 7 8 ( ' * ' )) 24 FORMAT(13,3E17.4 ) 25 FORMAT(G5.2 ) 26 FORMAT ( // . ' ION POSITIONS IN THE UNIT CELL IN LATTICE COORDINATES') 27 FORMAT(/, ' IONS OF CHARGE ' ,F6 . 2 ) 28 FORMAT( ' X/A Y/B Z/C ' , / , 5 3 ( ' * ' ) ) 29 FORMAT(3F10.4) 30 FORMAT(15) 31 FORMAT(// '2ND ORDER PARTIAL DERIVATIVES OF THE ELECTROSTATIC POTENTIAL AT THE POSITION' ) 32 FORMAT('OF THE FIRST ION, IN UNITS OF ANGSTROMS, WHERE X LIES ALONG THE A*-AX I S , ' ) 33 FORMAT('Y LIES ALONG THE B-AXIS, AND Z LIES ALONG THE C-AXIS OF THE LATT ICE ' ) 77 FORMAT(// , ' LATT ICE PARAMETERS: A, B, C ( A N G S T R O M S ) ' , 3 F 9 . 3 / , 2 0 X , ' S I N ( B E T A ) ' , F 9 . 4 ) END C SUBROUTINE PR INC (VXX ,VYY ,VZZ ,VXY ,VXZ ,VYZ ) c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C FINDS AND OUTPUTS THE PRINCIPAL ELECTRIC FIELD GRADIENTS VII FOR GIVEN VI J * c * * * * * * » * * » » * * » * , , * * , » * * * » » » » * » * , * * * * * » * * * * * » » * * » * * . * * * * » * * * * * * * , » , , „ * » » * * * * » * COMPLEX ROOT,S1 , S 2 , V ( 3 ) , IM,RE A2=VXY*VXY B2=VXZ*VXZ C2=VYZ*VYZ X1=VXX*VYY+VXX*VZZ+VYY*VZZ-A2-B2-C2 XO=A2*VZZ+B2*VYY+C2*VXX-2*VXY*VXZ*VYZ-VXX*VYY*VZZ 0=X1/3. P=X0/(-2 . ) ROOT=CSQRT(CMPLX(Q*Q*Q+P*P.O.)) S1=CEXP( (CLOG(P+ROOT))/3. ) S2 = CEXP( (CLOG(P-ROOT))/3 . ) V( 1 )=S1+S2 IM=(0. , 1 . )*SQRT(3. ) /2 . * (S1-S2) RE = - . 5*V(1) V(2)=RE+IM V(3)=RE-IM DO 15 1=1,3 15 IF ( A IMAG (V ( I ) ) .NE .O . ) WRITE(6,16) 16 FORMAT( ' A PRINCIPAL VII IS COMPLEX: INPUT VI J ARE NOT PHYSICALLY VAL ID ' ) V1=REAL(V( 1 ) ) V2 = REAL(V (2 ) ) V3 = REAL(V(3 ) ) WRITE(6,2) V 1 . V 2 . V 3 2 FORMAT(// 'THE PRINCIPAL DERIVATIVES A R E : ' , / , ' V 1 1 =' ,E 1 2 . 4/ , ' V22 = ' ,E 1 2 . 4/, 'V33 = ' , E 1 2 . 4 ) RETURN END APPENDIX D - OUTPUTS OF PROGRAM 1 FOR MANGANESE FLOURIDE AND MANGANESE CHLORIDE TETRAHYDRATE 1 MANGANESE FLOURIDE 2 3 LATTICE PARAMETERS: A, B, C (ANGSTROMS) 4 .870 4 .870 3 .310 4 S IN(BETA) 1.0000 5 6 7 ION POSITIONS IN THE UNIT CELL IN LATTICE COORDINATES 8 X/A Y/B Z/C g * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 10 11 IONS OF CHARGE 2 .00 12 0 . 0 0 . 0 0 . 0 13 0 . 5 0 0 0 0 .5000 0 . 5 0 0 0 14 15 IONS OF CHARGE - 1.00 16 0 . 3 1 0 0 0 . 3 1 0 0 0 . 0 17 -0 .3100 -0 .3100 0 . 0 18 O .8100 0 .1900 O .5000 19 - 0 . 8 1 0 0 -0 .1900 -0 .5000 20 21 22 2ND ORDER PARTIAL DERIVATIVES OF THE ELECTROSTATIC POTENTIAL AT THE POSITION 23 OF THE FIRST ION, IN UNITS OF ANGSTROMS, WHERE X LIES ALONG THE A*-AXIS, 24 Y L IES ALONG THE B-AXIS, AND Z LIES ALONG THE C-AXIS OF THE LATTICE 25 26 VXX VY Y VZZ VXY VXZ VYZ 27 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 28 0 .092355 0 .095193 0 .325637 -0 .275065 0 .004786 0.006808 29 0 .066398 0 .050975 0 .158975 -0.175824 -0 .000489 -0.000492 30 0 .031702 0 .024977 -0 .049819 -0 .043569 -0 .000020 -0.000023 31 0.028311 0 .024937 -0.049842 -0 .043093 0.000001 -0.000005 32 0 .024924 0 .024924 -0 .049859 -0.042854 0.000001 -0.000005 33 34 35 THE PRINCIPAL DERIVATIVES ARE: 36 V11 = 0.6778E-01 37 V22 = -0.4985E-01 38 V33 = -0.1793E-01 66 1 • 2 MnC12*4H20 n e g l e c t i n g the water m o l e c u l e s 3 LATTICE PARAMETERS: A, B, C (ANGSTROMS) 11.860 9 .513 6 . 186 1 4 5 SIN(BETA) 0 .9856 6 7 ION POSITIONS IN THE UNIT CELL IN LATTICE COORDINATES 8 X/A Y/B Z/C 9 10 1 1 IONS OF CHARGE 2 0 . 0 0 12 0 .2329 0, .1714 0 . 9865 13 0 .7329 0. 3286 0 . 4865 14 0.7671 0. 8286 0 . 0135 15 0.2671 0. 6714 0. 5135 16 17 IONS OF CHARGE 0 . 0 18 0 . 0 6 1 0 0 . 3076 0. 0938 19 0 . 5 6 1 0 0. 1924 0. 5938 20 0 . 9 3 9 0 0. 6924 0. 9062 21 0 . 4 3 9 0 0. 8076 0 . 4062 22 0 .3817 0 . 3662 0. 0355 23 0 .8817 0. 1338 0. 5355 24 0 .6183 0 . 6338 0 . 9645 25 0 .1183 0 . 8662 0 . 4645 26 27 28 2ND ORDER PARTIAL DERIVATIVES OF THE ELECTROSTATIC POTENTIAL AT THE POSITION 29 OF THE FIRST ION, IN UNITS OF ANGSTROMS, WHERE X LIES ALONG THE A*-AXIS, 30 3 1 Y LIES ALONG THE B-AXIS, AND Z LIES ALONG THE C-AXIS OF THE LATTICE 32 VXX VYY VZZ VXY VXZ VYZ 33 * * * * * * * * * * * * * * * * * * * * 34 -0 .027778 0 . 105880 -0.386016 -0.329176 -0 .5596 18 -0. 596682 35 -0 .029913 0. 292075 -0 .211707 -0 .335763 -0 .528808 -0 . 493323 36 0 .048122 0 .278755 -0.237241 -0.312874 -0 .518889 -0 . 500053 37 0 .005525 0 .274889 -0 .237235 -0.313642 -0 .519308 -0 . 500403 38 -0 .036709 0 .274417 -0 .237536 -0 .314513 -0 .519076 -0 . 500361 39 -0 .036745 0 .274413 -0 .237549 -0.314495 -0 .519050 -0 . 500358 40 4 1 -0.036781 0 .274408 -0.237562 -0 .314475 -0 .519024 -0 . 500355 42 43 THE PRINCIPAL DERIVATIVES , ARE : 44 V11 = 0.5804E+00 45 V22 = -0.9479E+00 46 V33 = 0.3675E+OQ 67 1 MANGANESE CHLORIDE TETRAHYDRATE 2 3 LATTICE PARAMETERS: A, B, C (ANGSTROMS) 11.860 9.513 6.186 4 SIN(BETA) 0 .9856 5 6 7 I ON POSITIONS IN THE UNIT CELL IN LATTICE COORDINATES 8 X/A Y/B Z/C g ***************************************************** 10 1 1 IONS OF CHARGE 2 .00 12 O.2329 0 .17 14 0 .9865 13 0 .7329 0 .3286 0 .4865 14 0.7671 0 .8286 0 .0135 15 0.2671 0.6714 0 .5135 16 17 IONS OF CHARGE - 1 .00 18 0 . 0 6 1 0 0 .3076 0 .0938 19 0 . 5 6 1 0 0 .1924 0 .5938 20 0 . 9 3 9 0 0 .6924 0 .9062 2 1 0 . 4 3 9 0 0 .8076 0 .4062 22 0 .3817 0 .3662 0 .0355 23 0 .8817 0 .1338 0 .5355 24 0 .6183 0 .6338 0 .9645 25 0 .1183 0 .8662 0 .4645 26 27 IONS OF CHARGE - 1 . 0 0 28 0 .3010 0 .1127 0 .3334 29 0 . 8 0 1 0 0 .3873 0 .8334 30 0 . 6 9 9 0 0 .8873 0 .6666 31 O .1990 0 .6127 0 .1666 32 0 .1568 0 .2280 0 .6446 33 . 0 .6568 0 .2720 0 .1446 34 0 .8432 0 .7720 0 .3554 35 0 .3432 0 .7280 0 .8554 36 0 .1323 0 .9736 0 .9590 37 0 .6323 0 .5264 0 .4590 38 0 .8677 0 .0264 0 .0410 39 0 .3677 0 .4736 0 .5410 40 0 .3695 0.0381 0 .8764 41 0 .8695 0 .4619 0 .3764 42 0 .6305 0 .9619 0 .1236 43 0 .1305 0.5381 0 .6236 44 45 IONS OF CHARGE 0 . 5 0 46 0.3831 0 .1418 0 .3882 47 0.8831 0 .3582 0 .8882 48 0 .6169 0 .8582 0 .6118 49 O.1169 0 .6418 0 .1118 50 0 .3005 0 .0154 0 .3657 51 0 .8005 0 .4846 0 .8657 52 0 .6995 0 .9846 0.G343 53 0 .1995 0 .5154 0 .1343 54 0 .0728 0 .2000 0 .5993 55 0 .5728 0 .3000 0 .0993 56 0 .9272 0 .8000 0 .4007 57 0 .4272 0 .7000 0 .9007 58 0 .1989 0 .1967 0 .5288 59 0 .6989 0 .3033 0 .0288 60 0.8011 0 .8033 0 .4712 L 6 8 61 0.3011 0 .6967 0 .9712 62 0 .1105 0 .9277 0 .8199 63 0 .6105 0 .5723 0 .3199 64 0 .8895 0 .0723 -0.1801 65 0 .3895 0 .4277 0.6801 66 0 .0996 0 .9292 0.0681 67 0 .5996 0 .5708 0.5681 68 0 .9004 0 .0708 0 .9319 69 0 .4004 0 .4292 0 .4319 70 0 .4356 0 .0716 0.8061 71 0 .9356 0 .4284 0.3061 72 0 .5644 0 .9284 0 .1939 73 0 .0644 0 .5716 0 .6939 74 0 .3616 0 .9389 0 .8540 75 0 .8616 0.5611 0 .3540 76 0 .6384 0.0611 0 .1460 77 0 .1384 0 .4389 0 .6460 78 79 80 2ND ORDER PARTIAL DERIVATIVES OF THE ELECTROSTATIC POTENTIAL AT THE POSITION 81 OF THE FIRST ION. IN UNITS OF ANGSTROMS, WHERE X LIES ALONG THE A*-AXIS. 82 Y LIES ALONG THE B-AXIS. AND Z LIES ALONG THE C-AXIS OF THE LATTICE 83 84 VXX VY Y VZZ VXY VXZ VYZ 85 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 86 0 .025639 0 .063927 -0.013792 0 .151370 0 .007280 -0.020507 87 -0 .019269 0.015972 0.013264 -0.001291 -0.000324 0.010725 88 -0 .016840 0 .013157 0.009904 -0 .007849 -0.009113 0.011133 89 -0 .019769 0 .012807 0.009922 -0 .007880 -0.009166 0.011092 90 -0 .022666 0 .012773 0.009911 -0.007921 -0.009157 0.011094 91 -0 .022672 0 .012772 0 .009908 -0.007921 -0.009153 0.011094 92 -0 .022677 O.012772 0 .009905 -0.007922 -0 .009149 0.011094 93 94 95 THE PRINCIPAL DERIVATIVES ARE: 96 V11 = 0.2551E-01 97 V22 = -0.2576E-01 98 V33 = 0 .2438E-03 6 9 APPENDIX E - PROGARM 2 C PROGRAM TO GENERATE THE NMRON SPECTRUM FOR 54-MN IN MNCL2*(4H20) * C WITH QUADRUPOLE SPLIT RESONANCES AND SPIN-LATTICE RELAXATION * c * * * » . * . » , * * » * * « . * » » * * * * * * » * * * , * » » , * * , » . * * * . , * * • * * * * , * * * * » . , * « , * • * * * * * C c***.***********************************************»**********»********,***** C MAIN PROGRAM * C READS SPECTRUM PARAMETERS, FINDS AND OUTPUTS NUMBER OF COUNTS VS FREQUENCY * c » * * * * * . * * * * * * * * * * * * . * * * * * * * * * * * * * * * * * * * * * * * * * « » * * * * * * * * * * » * * * * * * * * * * . » * * » * * * * COMMON P ( 8 ) , P 1 ( 8 ) , W D ( 8 , 8 ) , P T 1 ( 8 ) , W U ( 7 , 8 ) , INT , LEX , *LSEP ,KEND,WA,A ,B ,L INE ,COLD CDMMON/BC/SIGMA COMMON/CC/STEP C C VARIABLE TYPES LOGICAL*1 LABEL(25) R E A L M X(4096) , Y (4096) INTEGER A . B . C C C INPUT AND ECHO SPECTRUM PARAMETERS £****«»***************************** WRITE(6,101) 101 FORMAT( 'ENTER(REAL) LEVEL SEPARATION (KELVINS) & "COLD" LATTICE TEMPERATURE (KELV INS ) ' ) READ(5 , 1001 ) EK,T WRITE(G,600) EK,T WRITE(G,201) 201 FORMAT( 'ENTER(REAL) RELAXATION TIME & WARM COUNT') R E AD(5 . 1001) T1,WA WRITE(6,600) T1.WA 1001 FORMAT(4G13.5) WRITE(6,102) 102 FORMAT( 'ENTER(INTEGER) DWELL TIME, TIME & NUMBER OF FIRST EXCITATION' ) READ(5 ,1002) INT ,LEX ,L INE WRITE(G,601) INT ,LEX ,L INE WRITE(6, 104) 104 FORMAT( 'ENTER(REAL) FREQUENCY STEPS IZE, FREQUENCY OF FIRST EXCITATION IN THE SPECTRUM') READ(5 ,1004) STEP,EXFREQ WRITE(6,600) STEP,EXFREQ 1004 FORMAT(2G13.5) WRITE(6,202) 202 FORMAT( 'ENTER( INTEGER) EXCITATION SEPARATION IN TIME, END OF RUN TIME (SECONDS) ' ) READ(5 ,1002) LSEP.KEND WRITE(6,601) LSEP.KEND 1002 F0RMAT(4I8) WRITE(6, 103) 103 FORMAT('ENTER 1 IF FORWARD RUN, 0 IF BACKWARD RUN ' ) READ(5 ,1003) A WRITE(G,203) 203 FORMAT('ENTER 1 FOR WRITEOUT 0 OTHERWISE') READ(5 ,1003) B WRITE(6,303) 303 FORMAT('ENTER 1 IF PLOT IS DESIRED OTHERWISE 0 ' ) READ(5 ,1003) C WRITE(G,106) 106 FORMAT('ENTER 1 IF THE EXCIT. FREQ. DISCONTINUITY IS TO BE SMOOTHED, 0 OTHERWISE') READ(5 ,1003) NOJUMP WRITE(6,602) A,B,C.NOOUMP 1003 F0RMAT(3I3) WRITE(6,105) 3WI11-X=31 I N I ' X ' t = 3 w i n L o a (\+lNI/(X31 - O N 3 X ))*d31S-03adX3=iSN00 6 O i 0 3 ( r '03'V) d l U * * * * * * * * * * * * * * * * * * 0 ( o = v ) Nna aavwxova o ( 9 0 9 ' 9 ) 3 i i a r t (909'9)31ia« u o i o o ( o o a a j d i * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 0 U = a ) a s i s a n o s a s i i n d i n o ivoiaswnN v J I S O N I O V S H s i a v i s n a w o t+iNI / 0 N 3>l = lSVT i +aN3X = >( # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 0 N o u o a a i a Nna iviN3wia3dX3 O S I V O I O N I 3Hi aod ( x ) A 0 N 3 n 0 3 d d S A (A)SiNnoo/*' l a s o 3 o = ( 8 ' i ) n « • o = ( i ' 8 ) a « z n o a » ( w ' u w ) a « = ( t + w ' w ) n « s A ' t =w G oa ( ( i - w ) * ( t - i - w ) - 1 * ( + i ) )*O M = ( W' t +w)a« P L 'i=w f o a e = i avaHG s i s n s i o n N S H I do W H I N S W O W avinoNv i v i o i a ( z n o a + i . ) / n / i = o « ( i/X3 )dX3/l- = z i~ioa t+w <- w a o d ( t + w ' w ) n « o W <- l+W aOd (W'l+W)0M :S13A3~I l+W S W N 3 3 M 1 3 3 N O I i l S N V d i dO 3iVd 0 x v i 3 a S N i i n o a a n s N I 3sn aod aavdsad 3 3 w n s / ( w ) d= ( w ) d e z.' i=w e o a ( w ) d + wns=wns z L ' t =w s o a •o=wns (1/»3*W*'t-)dX3=(W)d t i ' t = w i o a w A 3 a3~mavn S13A31 Nids 3Hi aod (w)d N O i i n a i a i s i a N v w z i i o a 3 x v i s a O N V SiNnoo s s N i i n o a a n s N I ssn aod aavaaad 3 3 0 = ( 8 ) l i d 0 = ( 8 ) I d 0=(8)d * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * D NldS Hi8, 3Hi dO NOIiVlRdOd ONIHSINVA V SOnaOdiNI 3 xvT3a 3Niinoaans N I SNOiss3aax3 A d i i a w i s 01 o 3 VW9IS*' I -= V W 9 I S ( 0 ' 0 3 ' V ) d l ***************************+*******************************3 s s n v o 3 N i i n o a a n s a o d v w o i s do N O I S a s a a A a a Nna aavwxova d i o 3 a n o D ( 0 0 9 ' 9 ) 3 i i a M ano3 (voov ' s ) a v a a (-(3Aoav i 3aniva3dW3i i v ) a i o o 3av HOIH/A SNias do N 0 i i 3 v a d (nvsa)a3iN3-)ivwaod sot ( 8 0 1 ' 9 ) 3 i i a « VWOIS (009'9)3ild« v w s i s ( t o o t ' s ) a v 3 a (/ (S0N033S) N O i i n a i a i S i a N v i s s n v o 3Hi do N O i i v i A a a swa ( i v 3 a ) a 3 i N 3 , U v w a o d g o t OL 71 CALL COUNTS(LE.YY) INDEX=(LTIME-1)/INT+1 X(INDEX)=CONST+INDEX*STEP Y(INDEX)=YY 7 CONTINUE GO TO 19 C FORWARD RUN (A=1) £•* * * *• * * * * * * * , » , » * * * 9 CONST=EXFREO-STEP*(1+LEX/INT) DO 8 LT1ME=1,K. INT LE = LTIME- 1 NL=LE/INT+1 CALL COUNTS(LE,YY) X(NL)=CONST+STEP*NL Y(NL)=YY 8 CONTINUE C C SMOOTH THE EXCITATION DISCONTINUITIES IN ^COUNTS IF REQUESTED ( ; , * » « » * * * » * » » » * » « * * * * * * * * , » » » * * « * » » « » » » * , * * • • » , » * . * , * » * * » » » * * « » 19 IF (NOJUMP.EO.O) GO TO 500 DO 50 L= 1 ,LAST YL=Y(L) 50 Y(L ) = SMOOTH(L,YL) C C PLOT NUMBER OF COUNTS VS FREQUENCY IF REQUESTED (C=1) C MOVEC, FREAD, PICT ARE SUBROUTINES WRITTEN BY MARY ANN POTTS (;;*****»*******»*».»»***»*•***»*.*,»*,*••*******»**«,»»*****»*, 500 IF ( C . EO .O ) GO TO 89 CALL M0VEC (25 , ' ' , L A B E L ( 1 ) ) CALL F R E A D ( 6 , ' S : ' . L A B E L , 2 5 ) DX = X(2 )-X( 1 ) CALL P I C T ( L A S T , X , Y , D X , L A B E L ) C C OUTPUT NUMBER TABLE OF COUNTS VS FREQUENCY IF REQUESTED (B=1) ; ; » , » * * * * « » * * * * • * * * * « * » * * * , » * , » * « « , , * , , * » , , * » * • * » » » * • , , * • * . * » , * * * IF ( B . EQ .O ) GO TO 99 89 WRITE(6 ,70) WRITE(6,80) (X( I ) ,Y(I ) ,1 = 1,LAST) 99 RETURN 10 F O R M A T ( 2 ( F 7 . 5 . 2 X ) , F 7 . 1 , 2 X , 4 ( 1 5 , 2 X ) , F 9 . 1,3(2X, I 1)) 20 FORMAT( ' INPUT DATA ' ,/2E 1 3 . 5 ./2E 1 3 . 5 , /218 , /218 , /31 3 , /2E13 .5 ) 30 FORMAT(' PT1 = ' , 7 E 1 3 . 5 , / ) 60 F0RMAT(25A1,// ) 70 FORMAT ( / /2X , ' FREQUENCY ^COUNTS' , /2X , ' * * * * * * * * * * * * * * * * * * * * * * * » * * ' ) 80 F 0 R M A T ( E 1 3 . 5 , 2 X , E 1 3 . 5 ) 600 F0RMAT(2E13.5) 601 F0RMAT(3I8) 602 F0RMAT(4I3) 605 F0RMAT(//2X , ' T IME ' , 4X , 'NORMAL IZED INTENS ITY ' , 6X , ' POP PROB OF THE SPIN LEVELS ' ) 606 FORMAT(30X,6X, ' P 1 ' , 1 1X, ' P 2 ' . 1 1X, ' P 3 ' , 11X, ' P 4 ' , 11X, ' P 5 ' , 11X, ' P 6 ' , 11X, ' P 7 ' ) END C SUBROUTINE COUNTS(L .Y) c » » » * „ * « * * * * * * » » » * » * * * » * » » * * » « « » » * » * » , » * * . , * , * » * * » » « , * « » * * , • * • * * • • * . * * * C CALLED FROM MAIN C FINDS THE EXPECTED NUMBER OF COUNTS FOR THE FREQUENCY LABELLED BY L * COMMON P(8) , P1 (8 ) , W D ( 8 , 8 ) , P T 1 ( 8 ) , W U ( 7 , 8 ) . I N T , L E X , *LSEP ,KEND,WA.A ,B ,L INE ,COLD C0MM0N/AC/YMAX(6),YMIN(6) 72 * * * * * * * * * * * * COMMON/BC/SIGMA INTEGER A,B IF ( A . E O . 1 ) GO TO 11 LP=LEX+5*LSEP C FOR BACKWARD RUN IF FREQUENCY > HIGHEST FREQUENCY OF EXCITATION C POPULATION OF THE SPIN LEVELS IS GOVERNED BY THE BOLTZMAN DISTRIBUTION * £ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * IF ( L . G T . L P ) GO TO 12 GO TO 14 Q************************************************************************* C FOR FORWARD RUN, IF FREQUENCY < LOWEST FREQUENCY OF EXCITATION C POPULATION OF THE SPIN LEVELS IS GOVERNED BY THE BOLTZMAN DISTRIBUTION * ^************************************************************************* 11 IF ( L . L T . L E X ) GO TO 12 GO TO 14 12 IF ( ( L . N E . O ) . A N D . ( L , , NE.KEND)) GO TO•33 DO 13 M=1 ,7 13 P1(M)=P(M) CALL EMISSN(Y) GO TO 33 C C IF L CORRESPONDS TO THE TRANSITION ENERGY BETWEEN 2 ADJACENT C I . E. CORRESPONDS TO ONE OF THE 6 RESONANCE FREQUENCIES C AVERAGE AND EQUALIZE THE POPULATIONS OF THESE TWO LEVELS Q******************************************************************** 14 N=1 15 LEX2 = LEX+(N-1 )*LSEP IF ( ( L . G T . L E X 2 - 1 ) . A N D . ( L . LT . LEX2+1 ) ) GO TO 16 GO TO 18 16 DO 38 M= 1 ,7 38 PT1(M)=P1(M) LEVEL=LINE+N AVRAGE = (P1(LEVEL-1 ) + P1(LEVEL ) )/2 PT1(LEVEL)=AVRAGE PT1(LEVEL-1)=AVRAGE DO 39 M=1,7 39 P1(M)=PT1(M) Q********************************************************** C STORE ^COUNTS IMMEDIATELY BEFORE AND AFTER RESONANCE(N) * C IN YMIN(N), YMAX(N) FOR USE IN SUBROUTINE SMOOTH * Q********************************************************** YMIN(N)=Y CALL EMISSN(Y) YMAX(N) = Y GO TO 33 18 N=N+1 IF ( N . L T . 8 - L I N E ) GO TO 15 C C IF NOT A RESONANCE FREQUENCY THEN LEVEL POPULATIONS RELAX TOWARD EQUILIBRIUM Q***************************************************************************** CALL RELAX CALL EMISSN(Y) C C PRINT A NUMERICAL TABLE OF POPULATION OF LEVELS VS TIME IF REQUESTED Q********************************************************************* 33 IF ( B . EQ .1 ) CALL WRIT(L.Y) RETURN END C 73 SUBROUTINE RELAX C * * * * . * * * * » . * . . . « * * * * * * * * . * * » , * * . * * * , * * » * . * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C CALLED FROM SUBROUTINE COUNTS * C FINDS NORMALISED POPULATIONS P1(M) , OF THE SPIN LEVELS DURING SPIN-LATTICE RELAXATION * C * * * * * * * * * * * * * * * * * * * * * * * * * * « * * * * * * * * * * * * * ^ COMMON P (8 ) , P1 (8 ) .WD(8,8) , P T 1 ( 8 ) , W U ( 7 , 8 ) , I N T , L E X , *LSEP,KEND,WA,A,B PT1(1 )=P1(1 )+ INT* (WD (2,1 ) * ( P1 (2 ) -P (2 ) ) -WU(1 , 2 ) * ( P1(1 ) - P(1))) DO 41 M=2,7 4 1 PT1(M)=P1(M)+INT*(WD(M+1,M)*(P1(M+1)-P(M+1))-WU(M,M+1)*(P1(M) *-P(M))+WU(M-1,M)*(P1(M-1 )-P(M-1) )-WD(M,M-1)*(P1(M)-P(M))) DO 42 M= 1 ,7 42 P1(M)=PT1(M) RETURN END C SUBROUTINE EMISSN(Y) c * * * * * * * » * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * , * * * * * * * * * * * * * * * * » * * * * * * * . * * * » * C CALLED FROM SUBROUTINE COUNTS * C FINDS THE NUMBER OF COUNTS EXPECTED FOR THE POPULATION OF LEVELS P1(M) * c * * * * * * » * * * * * »********************** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * COMMON P ( 8 ) , P 1 ( 8 ) , W D ( 8 , 8 ) , P T 1 ( 8 ) , W U ( 7 , 8 ) , I N T , L E X , * LSEP ,KEND,WA,A ,B ,L INE ,COLD c**************************************************************** C GAMMA+1 = W(O), IS THE NORMALISED GAMMA-RAY INTENSITY EMITTED * C ALONG THE AXIS OF MAGNETISATION FOR COLD=1, * C WHERE COLD IS THE FRACTION OF MN-54 NUCLEI AT TEMPERATURE T * Q***********************************,**************************** GAMMA= .333* ( P1 (3 ) + P1(5) ) + . 667 *(P 1 (2 )+ P 1 (6 ) ) - (P1 (1 )+P1(7 ) ) c * * * * * * » « « » , * « , » * » . * . * » * * » » * , « * * * * * » * * * * * , * * * * * * * * * * * * * * * * * * * * * * * * * * C THE NORMALISED GAMMA-RAY INTENSITY EMITTED FOR COLD ARBITRARY IS * C Y/WA = (1-COLD) + COLD*W(0) * c*«*****„*.**»*»»***,*****,*»*».*.»«*»*»»***»********,»******»,*».*, Y = WA*( 1+COLD*GAMMA ) RETURN END C SUBROUTINE WRIT(LTI ME,Y) Q******************************************************************************** C CALLED FROM SUBROUTINE COUNTS C PRINTS OUT A TABLE OF POPULATION OF LEVELS P1(M) AND RELATIVE ^COUNTS VS TIME * Q******************************************************************************** COMMON P ( 8 ) , P 1 ( 8 ) , W D ( 8 , 8 ) , P T 1 ( 8 ) , W U ( 7 , 8 ) , I N T , L E X , * LS E P,KEND,WA,A,B INTEGER A , B REL=Y/WA WRITE(6,40) LTIME.REL WRITE(6,30) (P1(M) ,M=1 ,7 ) 40 F0RMAT ( I 6 ,3X , E13 .5 ) 30 F0RMAT(30X ,7E13 .5 ) RETURN END C FUNCTION GAUSS(N.NEX) Q*********************************************************** C CALLED FROM SUBROUTINE SMOOTH * C GENERATES GAUSSIAN DISTRIBUTION WITH RMS DEVIATION SIGMA * Q* ********************************************************* * COMMON P ( 8 ) , P 1 ( 8 ) , W D ( 8 , 8 ) , P T 1 ( 8 ) , W U ( 7 , 8 ) , I N T , L E X , *LSEP,KEND,WA,A,B 74 COMMON/BC/SIGMA COMMON/CC/STEP XPON=(NEX-N)/INT*STEP/SIGMA XPON=XPON*XPON/2. IF (XPON.LT .174 ) GO TO 87 GAUSS=0. GO TO 88 87 GAUSS=1 ./EXP(XPON) 88 RETURN END C FUNCTION SMOOTH(LS,Y) C CALLED FROM MAIN * C SMOOTHS THE EXCITATION FREOUENCY DISCONTINUITY WITH A GAUSSIAN * COMMON P ( 8 ) , P 1 ( 8 ) , W D ( 8 , 8 ) , P T 1 ( 8 ) , W U ( 7 , 8 ) , I N T , L E X , *LSEP ,KEND,WA,A,B C0MM0N/AC/YMAX(6),YMIN(6) IF (A . EO .O ) GO TO 114 L=(LS-1)* INT IF ( L .GE .LEX+5*LSEP ) GO TO 118 N= 1 116 LEXN=LEX+(N-1)*LSEP IF ( L . L T . L E X N ) GO TO 117 N = N+1 GO TO 116 114 L=KEND-INT*(LS-1) IF (L . LE . LEX) GO TO 118 N = 6 115 LEXN=LEX+(N-1)*LSEP IF ( L .GT . LEXN ) GO TO 117 N = N-1 GO TO 115 117 SMOOTH=GAUSS(L,LEXN)*(YMAX(N)-YMIN(N))+Y GO TO 111 118 SMOOTH=Y 111 RETURN END 75 APPENDIX F - SAMPLE OUTPUTS OF PROGRAM 2 1 ENTER(REAL) LEVEL SEPARATION (KELVINS) & "COLD" LATTICE TEMPERATURE (KELVINS) 2 0.24000E-01 0.70000E-01 3 ENTER(REAL) RELAXATION TIME & WARM COUNT 4 0.40000E+03 0.2S422E+06 5 ENTER(INTEGER) DWELL TIME, TIME 8. NUMBER OF FIRST EXCITATION 6 20 300 1 7 ENTER(REAL) FREQUENCY STEPS IZE , FREQUENCY OF FIRST EXCITATION IN THE SPECTRUM 8 0.15000E+00 0.50040E+03 9 ENTER(INTEGER) EXCITATION SEPARATION IN TIME, END OF RUN TIME (SECONDS) 10 400 2400 11 ENTER 1 IF FORWARD RUN, 0 IF BACKWARD RUN 12 ENTER 1 FOR WRITEOUT 0 OTHERWISE 13 ENTER 1 IF PLOT IS DESIRED OTHERWISE 0 14 ENTER 1 IF THE EXCIT. FREQ. DISCONTINUITY IS TO BE SMOOTHED, 0 OTHERWISE 1 5 1 O 1 O 16 ENTER(REAL) RMS DEVIATION OF THE GAUSSIAN DISTRIBUTION (SECONDS) 17 0 .20000E+00 18 ENTER(REAL) FRACTION OF SPINS WHICH ARE COLD (AT TEMPERATURE T ABOVE) 19 0.1O0O0E+01 261348 258859 256370 253881 S251392 ; 248903 -246414 -243925 -241436 -238947 236458 I-FORWRRD—NO-SMOOTHING ''I f^fimH rl,[BniHnin jiffliiHnrHnn'innBiBBinHii1-_ l _ _1_ _ l _ _1_ 3.1 4 3 9 . 8 5 0 0 . 5 S O I . 7 502 .9 504 .1 5 0 5 . 3 3 0 6 . 5 507 .7 3 0 6 . B 51D.1 511 .5 512 .5 513 .7 514 .9 518.1 FREQUENCY (MHZ) 76 1 ENTER(REAL) LEVEL SEPARATION (KELVINS) & "COLD" LATTICE TEMPERATURE (KELVINS) 2 0.24000E-01 0.70000E-01 3 ENTER(REAL) RELAXATION TIME & WARM COUNT 4 0 .40000E+03 0.26422E+06 5 ENTER(INTEGER) DWELL TIME, TIME & NUMBER OF FIRST EXCITATION 6 20 200 1 7 ENTER(REAL) FREQUENCY STEPS IZE , FREQUENCY OF FIRST EXCITATION IN THE SPECTRUM 8 0.15000E+00 0.5O040E+O3 9 ENTER(INTEGER) EXCITATION SEPARATION IN TIME, END OF RUN TIME (SECONDS) 10 400 2400 11 ENTER 1 IF FORWARD RUN, 0 IF BACKWARD RUN 12 ENTER 1 FOR WRITEOUT 0 OTHERWISE 13 ENTER 1 IF PLOT IS DESIRED OTHERWISE 0 14 ENTER 1 IF THE EXCIT. FREO. DISCONTINUITY IS TO BE SMOOTHED, 0 OTHERWISE 15 1 0 1 1 16 ENTER(REAL) RMS DEVIATION OF THE GAUSSIAN DISTRIBUTION (SECONDS) 17 0 .20000E+00 18 ENTER(REAL) FRACTION OF SPINS WHICH ARE COLD (AT TEMPERATURE T ABOVE) 19 0.10000E+01 E n d o f f i l e 261348 -258859 -256370 253881 g251392 |-•248903 246414 |-243925 241436 238947 236458 GRUSSIRN-SMQQTHING I .9 500.1 581.3 S02.5 503.7 504.B 506.0 507.2 50B.4 509.B 511.fl 312.1 513 2 314.4 SI J • M » FREQUENCY (MHZ) 77 1 ENTER(REAL) LEVEL SEPARATION (KELVINS) & "COLD" LATTICE TEMPERATURE (KELVINS) 2 0.24000E-01 0.70000E-01 3 ENTER(REAL) RELAXATION TIME & WARM COUNT 4 0.40000E+03 0.26422E+06 5 ENTER(INTEGER) DWELL TIME, TIME & NUMBER OF FIRST EXCITATION 6 20 300 1 7 ENTER(REAL) FREQUENCY STEPS IZE , FREQUENCY OF FIRST EXCITATION IN THE SPECTRUM 8 0 .15000E+00 0.50040E+03 9 ENTER(INTEGER) EXCITATION SEPARATION IN TIME, END OF RUN TIME (SECONDS) 10 400 2400 11 ENTER 1 IF FORWARD RUN, 0 IF BACKWARD RUN 12 ENTER 1 FOR WRITEOUT 0 OTHERWISE 13 ENTER 1 IF PLOT IS DESIRED OTHERWISE 0 14 ENTER 1 IF THE EXCIT . FREQ. DISCONTINUITY IS TO BE SMOOTHED, 0 OTHERWISE 15 0 0 1 0 1G ENTER(REAL) RMS DEVIATION OF THE GAUSSIAN DISTRIBUTION (SECONDS) 17 0 .20000E+00 18 ENTER(REAL) FRACTION OF SPINS WHICH ARE COLD (AT TEMPERATURE T ABOVE) 19 0.10000E+01 261500 258977 256454 253931 to • 251408 ; 248865 246362 243839 241316 238793 236270 BRCKWRRD-RUN inn u^iiiHEiiiHH^ EmiiiniiEiiinE iiiHiimiiimiii jiinnMniimii' \ JL JL J_ JL JL JL J_ J L JL J_ 464 .8 4B5 .B 4D7.0 4 9 0 . 2 409 4 4 3 0 . B 431 .8 433.B 494 .2 4 3 5 . 4 498 .0 FREQUENCY (MHZ) 437 .8 499 .0 500 .2 5 0 1 . 4 SDZ.8 78 1 ENTER(REAL) LEVEL SEPARATION (KELVINS) & "COLD" LATTICE TEMPERATURE (KELVINS) 2 0 .24000E-01 0.7OOOOE-01 3 ENTER(REAL) RELAXATION TIME & WARM COUNT 4 0.40000E+03 0.26422E+06 5 ENTER(INTEGER) DWELL TIME. TIME & NUMBER OF FIRST EXCITATION 6 20 300 1 7 ENTER(REAL) FREQUENCY STEPS IZE , FREQUENCY OF FIRST EXCITATION IN THE SPECTRUM 8 0 .15000E+00 0.50040E+03 9 ENTER(INTEGER) EXCITATION SEPARATION IN TIME, END OF RUN TIME (SECONDS) 10 400 2400 11 ENTER 1 IF FORWARD RUN, 0 IF BACKWARD RUN 12 ENTER 1 FOR WRITEOUT O OTHERWISE 13 ENTER 1 IF PLOT IS DESIRED OTHERWISE 0 14 ENTER 1 IF THE EXCIT. FREQ. DISCONTINUITY IS TO BE SMOOTHED, 0 OTHERWISE 15 0 0 0 1 16 ENTER(REAL) RMS DEVIATION OF THE GAUSSIAN DISTRIBUTION (SECONDS) 17 0 .20000E+00 18 ENTER(REAL) FRACTION OF SPINS WHICH ARE COLD (AT TEMPERATURE T ABOVE) 19 0.10000E+01 20 21 22 FREQUENCY /fCOUNTS 23 * * * * * * * * * * * * * * * * * * * * * * * * * * 24 0.48465E+03 0.24043E+06 25 0.48480E+03 0.24039E+06 26 0.48495E+03 0.24014E+06 27 0.48510E+03 0.23924E+06 28 O.48525E+03 0.23766E+06 . 29 0.48540E+03 0.23676E+06 30 0.48555E+03 0.23807E+06 . 31 0.48570E+03 0.23883E+06 32 0.48585E+03 0.23927E+06 . 33 0 .48600E+03 0.23954E+06 34 0.48615E+03 0.23972E+06 . 35 0.48630E+03 0.23985E+06 36 0.48645E+03 0.23996E+06 37 0.48660E+03 0.24004E+06 38 0.48675E+03 0.24012E+06 39 0 .48690E+03 O.24018E+06 40 0.48705E+03 0.24024E+06 41 0.48720E+03 O.24028E+06 42 0.48735E+03 0.24032E+06 43 0.48750E+03 0.24035E+06 44 0.48765E+03 0.24038E+06 45 0 .48780E+03 0.24042E+06 46 0.48795E+03 0.24050E+06 47 0.48810E+03 0.24077E+06 48 0.48825E+03 0.24123E+06 49 0.48840E+03 0.24148E+06 50 0.48855E+03 0.24070E+06 51 0 .48870E+03 0.24027E+06 52 0.48885E+03 0.24009E+06 53 0.48900E+03 0.24006E+06 54 0.48915E+03 0.24009E+06 55 0.48930E+03 0.24015E+06 56 0.48945E+03 0.24022E+06 57 0.48960E+03 0.24028E+06 58 0.48975E+03 0.24034E+06 59 0.48990E+03 0.24038E+06 60 0.49005E+03 0.24042E+06 ENTER(REAL) LEVEL SEPARATION (KELVINS) 6 "COLD" LATTICE TEMPERATURE (KELVINS) 0.24000E-01 0.64000E-01 ENTER(REAL) RELAXATION TIME 8. WARM COUNT 0.23000E+05 0.26422E+06 ENTER(INTEGER) DWELL TIME. TIME 8 NUMBER OF FIRST EXCITATION 1000 11000 2 ENTER(REAL) FREQUENCY STEPSI2E, FREOUENCY OF FIRST EXCITATION IN THE SPECTRUM 012500E+00 0.50350E+03 ENTER(INTEGER) EXCITATION SEPARATION IN TIME. END OF RUN TIME (SECONDS) 39000 31000 ENTER 1 IF FORWARD RUN, 0 IF BACKWARD RUN ENTER 1 FOR WRITEOUT O OTHERWISE ENTER 1 IF PLOT IS DESIRED OTHERWISE O ENTER 1 IF THE EXCIT. FREQ. DISCONTINUITY IS TO BE SMOOTHED, 0 OTHERWISE 1 1 0 0 ENTER(REAL) RMS DEVIATION OF THE GAUSSIAN DISTRIBUTION (SECONDS) 0.20000E-01 ENTER(REAL) FRACTION OF SPINS WHICH ARE COLD (AT TEMPERATURE T ABOVE) O.10000E+01 -J TIME NORMALIZED INTENSITY POP PROB OF THE SPIN LEVELS 0 0 .89445E+00 P1 P2 P3 P4 P5 P6 P7 1000 0 .89445E+00 0 . 33713E+00 0 . 23171E+00 0 .15925E+00 0 .10945E+00 0 .75225E -01 O .51701E -01 0 .35534E -01 2000 0 .89445E+00 0 .33713E+00 0 .23171E+00 0 .15925E+00 0 .10945E+00 0 .75225E -01 0 .51701E -01 0 .35534E -01 3000 0 .89445E+00 0 .33713E+00 0 . 23171E+00 0. .15925E+00 0. .10945E+00 0 .75225E -01 0 .51701E -01 0 . 35534E -01 4CO0 O .89445E+00 0. .33713E+00 0 .23171E+00 0, .15925E+00 0. 10945E+00 0 .75225E -01 0 .51701E -01 0. . 35534E -01 5000 0 89445E+00 0. .33713E+00 0, 23171E+00 0. 15925E+00 0. 10945E+00 0 75225E -01 0 51701E -01 0. .35534E -01 6000 0. 89445E+00 0. 33713E+00 0. .23171E+00 0. 15925E+00 0. 10945E+00 0 .75225E -01 0. 51701E -01 0. 35534E -01 7000 0. 89445E+00 0. 33713E+00 0. 23171E+00 0. 15925E+00 0. 10945E+00 0. 75225E -01 0. 51701E -01 0. 35534E -01 8000 0. 89445E+00 0. 33713E+00 0. 23171E+00 0. 15925E+00 0. 10945E+00 O. 75225E -01 0. 51701E -01 0. 35534E -01 9000 0. 89445E+00 0. 33713E+00 0. 23171E+00 0. 15925E+00 0. 10945E+00 0. 75225E -01 0. 51701E -01 0. 35534E -01 lOOOO 0. 89445E+00 0. 33713E+00 0. 23171E+00 0. 15925E+00 0. 10945E+00 0. 75225E -01 0. 5170IE -01 0. 35534E -01 110O0 0. 88235E+00 0. 33713E+00 0. 23171E+00 0. 15925E+0O 0. 10945E+00 0. 75225E -01 0. 51701E -01 0. 35534E -01 12000 0. 89438E+00 0. 33713E+00 0. 19548E+00 0. 19548E+00 0. 10945E+00 0. 75225E -01 0. 51701E -01 0. 35534E -01 13000 0. 89964E+00 0. 33153E+00 0. 21683E+00 0. 17203E+00 0. 11715E+00 0. 75225E--01 0. 51701E--01 0. 35534E--01 14000 0. 90178E+00 0. 32983E+00 0. 2244GE+00 0. 16577E+00 0. 11585E+00 0. 76861E--01 0. 51701E -01 o. 35534E--Ol 15000 0.90242E+0O 16000 0.90234E+00 170O0 0.90193E+00 18000 0.90137E+00 19000 0.90078E+00 20000 0.90020E+00 21000 0.89966E+00 22000 0.89916E+00 23000 0.89871E+00 24000 0.89830E+00 25000 0.89794E+00 26000 0.89762E+00 27000 0.89733E+00 28000 0.89708E+00 29000 0.89685E+00 30000 0.89664E+00 31000 0.89646E+00 FREQUENCY 0COUNTS ************************** 0.50213E+03 0.50225E+03 0.50238E+03 0.50250E+03 0.50263E+03 0.50275E+03 0.50288E+03 O.50300E+03 0.50313E+03 O.50325E+03 0.50338E+03 0.50350E+03 0.50363E+03 0.50375E+03 0.23633E+06 O. 23633E+06 0.23633E+06 0.23633E+06 0.23633E+06 0.23633E+06 0.23633E+06 0.23633E+06 0. 23633E+06 0. 23633E+06 O. 23633E+06 O.23314E+06 0.23632E+06 0.23770E+06 O.32948E+00 0.32969E+00 0.33011E+00 0.33061E+00 0.33112E+00 0.33161E+00 0.33206E+00 0.33248E+00 O.33285E+00 0.33320E+00 0.33351E+00 0.33379E+00 0.33404E+00 0.33427E+00 0.33448E+00 O.334G7E+00 0.33484E+00 0.33500E+00 O.22777E+00 0.22933E+00 O.2301lE+OO 0.23051E+00 0.23073E+00 0.23085E+00 0.23091E+00 0.23096E+00 0.23099E+00 0.23101E+00 0.23103E+00 0.23105E+00 0.23107E+00 0.23109E+00 0.23111E+00 0.23113E+00 0. 23115E+00 0.23117E+00 0. 16340E+00 0. 16228E+00 O. 16165E+00 O. 16123E+00 0.16093E+00 0.16069E+00 0.16049E+00 0.16032E+00 0.16018E+00 O.16006E+00 0.15995E+00 0.15986E+00 0.15978E+00 0.15971E+00 0.15965E+00 0.15960E+00 0.15956E+00 0.15952E+00 0.11440E+00 0.11338E+00 O.11267E+00 0.11216E+00 0.11178E+00 0.11149E+00 0.11125E+00 0.11105E+00 0.11088E+00 0.11074E+00 0.1 1061E+00 0.11050E+00 O.11040E+00 0.11032E+00 0.11024E+00 0.11017E+00 0.11011E+00 0.11006E+00 O.77425E-01 0.77482E-01 0.77367E-01 0.77208E-01 0.77047E-01 0.76898E-01 0.76762E-01 0.76640E-01 O.76531E-01 0.76432E-01 0.76342E-01 0.76261E-01 0.76188E-01 0.76121E-01 0.76060E-01 0.76004E-01 0.75953E-01 0.75906E-01 0.51991E-01 0.52275E-01 0.52471E-01 0. 52583E-01 0.52637E-01 0.52654E-01 0.52648E-01 O.52630E-01 0.52603E-01 0.52573E-01 0.52540E-01 0.52507E-01 0.52474E-01 0.52442E-01 0.52410E-01 0.52378E-01 0.52348E-01 0.52319E-01 O.35534E-01 0.35564E-01 0.35621E-01 0.35689E-01 0.35759E-01 0.35823E-01 O.35880E-01 O.35927E-01 O.35965E-01 O.35994E-01 0.36015E-01 O.36030E-01 O.36039E-01 O.36043E-01 O.36043E-01 0.36040E-01 0.36033E-01 0.36025E-01 

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