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Random walk model and nuclear magnetic relaxation in a BCC lattice containing two nuclear species
Volume 10, number 2
P HY S I C S L E T T E R S
1 June 1964
(8), is that the expectation value of any product o p e r a t o r containing u n p a i r e d c r e a t i o n or annihilation o p e r a t o r s a+k, s or a k s as f a c t o r s v a n i s h e s . T h e s e c o n c l u s i o n s }~-~vean i m p o r t a n t b e a r i n g on the d e s c r i p t i o n of the r e c e n t l y o b s e r v e d i n t e r f e r e n c e effects between independent light b e a m s f r o m l a s e r s . It can be shown 1) that the a m p l i t u d e s of i n t e r f e r e n c e or modulation t e r m s in the e x p r e s s i o n for the total m e a n light i n t e n s i t y involve the f a c t o r s
References 1) L. Mandel, Phys.Rev. 134 (1964) A10. 2) H.Paul, W. Brunner and G.Richter, Ann. Physik 12 (1963) 325. 3) T. F. Jordan and F. Ghielmetti, to be published (1964). 4) A. Javan, E.A. Ballik and W. L. Bond, J. Opt. Soc. Am. 52 (1962) 96. 5) M.S. Lipsett and L.Mandel, Nature 199 (1963) 553. 6) G. Magyar and L.Mandel, Nature 198, 255. 7) E. C. G. Sudarshan, Phys. Rev. Letters 10 (1963) 277. 8) E. C. G. Sudarshan, Proc. Brooklyn Syrup. on Optical Masers, (John Wiley & Sons, N.Y., 1963) p. 45. 9) J.R.Klauder, Ann.Phys. 11 (1960) 123. 10) R.J. Glauber, Phys. Rev. 130 (1963) 2529. 11) R. J. Glauber, Phys: Rev. 131 (1963) 2766. 12) Y.Kano, Ann. of Phys., to be published (1964). 13) C.L.Mehta and E.Wolf, Phys.Rev., in press (1964). 14) L.Mandel, Physics Letters 7 (1963) 117. 15) M. Born and E.Wolf, Principles of Optics (Pergamon Press, London & N.Y., 2nd ed. 1964). 16) E.Wolf, Proc.of Symposium on Optical Masers (Polytechnic Inst.of Brooklyn, 1963), p.29.
RANDOM WALK MODEL AND NUCLEAR MAGNETIC RELAXATION IN A B C C L A T T I C E CONTAINING TWO NUCLEAR SPECIES D. P. TEWARI and G. S. VERMA
Department of Physics, University of Allahabad, Allahabad, India Received 5 May 1964
A random walk model is applied to dipolar relaxation via atomic diffusion in a bcc lattice containing two nuclear species. Expressions for T1 and T2 in the high field and low temperature limit as well as high temperature and low field case are derived for magnetic field along [100] and [110] directions. The results of our calculations axe as follows. For a magnetic field applied along the [100] direction and in the high field, low temperature limit T1- I = 8.99 AI ao-6 vI wi-2 + AS ao-6 (Vl+ ,,S)[0.149(wi-00S)2 + 11.124 001-2 + 12.105(wi+cOS)-2] T2-1 = 8.31 A i a o -6 yi -1 + 0.56 A S a o - 6 ( u i + US)-1 , where
AI = v 1 4 ~ 2 l ( I + l ) f l ,
A S =vl2rs2h2 S(S+I)f s .
Here r , ~0, f and v a r e the g y r o m a g n e t i c r a t i o , r e s o n a n c e f r e q u e n c y , f r a c t i o n a l isotopic abundance and j u m p frequency. The s u b s c r i p t s I and S r e f e r to the two n u c l e a r s p e c i e s of spin I and S. a 0 is the cube edge. 168
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1 J u n e 1964
S i m i l a r l y for a m a g n e t i c field a p p l i e d along the [110] d i r e c t i o n T1-1 = 12.966 A I a o - 6 coI -2 ui + AS ao-6(vi+ VS)[2.882 ( w / - coS)-2 + 6.153 w/-2 + 14.703 (coI +coS )-2] T 2 " l = 2.928 Ai ao -6 ui-1 + 12.67 A s ao-6 (ui + US)-1 The high f i e l d and low t e m p e r a t u r e l i m i t c o r r e s p o n d to the c a s e co/~t >> 1 w h e r e co i s the a r g u m e n t of the s p e c t r a l densityJ(q)(co) for the iike n e i g h b o u r s orgt~(q)(co) for the unlike n e i g h b o u r s , co can take v a l ue.~., which m a y be m u l t i p l e s , s u m s o r d i f f e r e n c e s of coI and coS o r z e r o . ForJ~(q)(co) the j u m p p r o b a b i l i t y Jn the r e l a t i v e c o o r d i n a t e s y s t e m i s given by ~t = 2u/ and forgff(q)(co), ~ = vi+ vS. The s p e c t r a l d e n s i t y q)(co) is given by 1)
~
flq)(co) = 2~ ~ F,.(q ) _ c_1 ~ (1)Elm(q)}, CO 2 t
tt
lm
where /~rn (0) = (1 - 3 cos 2 0/)(1 - 3 cos 2 Om)/rl3 rm3
Flm (1) = sin 0 l cos 0 l sin 0rn cos 0 m cos (q~l-¢rn)/rl 3 rm 3 /~rn (2) = sin 2 0l sin 2 em cos (2~l-2q~rn)/rl 3 rm3 . H e r e rl, 0l, q~l are the p o l a r c o o r d i n a t e s of the v e c t o r r/ drawn f r o m the r e f e r e n c e spin to the point l and f i e l d d i r e c t i o n i s the p o l a r axis. The f i r s t s u m m a t i o n i s done up to t h i r d n e i g h b o u r s of the o r i g i n and the s e c o n d s u m m a t i o n i s o v e r point 1 and m which a r e n e a r e s t n e i g h b o u r s to e a c h o t h e r and e i t h e r l o r m i s a n e a r e s t neighbour to the o r i g i n . C i s the like neighbour coordination n u m b e r . The l a t t i c e s u m s which a r e n e e d e d to c a l c u l a t e the s p e c t r a l d e n s i t i e s in the l i m i t of high f r e q u e n c y a r e a s follows. F o r HI100], ao6(co2/2~t)J{q)(co) = 12.37, 0.28, 4.86 for q = 0, 1, 2 r e s p e c t i v e l y . S i m i l a r ly ao6(W2/2#)3t~(q)(w) = 0.895, 3.71, 8.07 for q = 0, 1, 2 r e s p e c t i v e l y . F o r H a l o n g [110] d i r e c t i o n , ao6(co2/2~t)~C~(w ) = 4.54, 1.18, 3.94 for q = 0, 1, 2 r e s p e c t i v e l y . Also ao6(W2/2U)s(q}(co ) = !7.29, 2.05, 9.80 for q = 0, 1, 2 r e s p e c t i v e l y . In the high t e m p e r a t u r e l i m i t which c o r r e s p o n d s to low f i e l d c a s e , T 1 and T 2 a r e i s o t r o p i c and f i e l d a n g u l a r a v e r a g e s a r e taken of_f(q) a n d S ( q ) . In this c a s e TI'I
= T2-1 = ~ A I ~¢(0)(0))+ ~ A SQ~(O)(o~ ) = 1 3 . 3 5 A I a o - 6 vi-1 + 51.2 A S ao-6( ui+ VS) -1 ,
w h e r e in the l i m i t of z e r o f r e q u e n c y 1) 2~J(0)(0) = Im ~ Flm(O) Z(rlm) ;
3
Z(rlm) = ~n P(rlm) ;
Pn(r) = ~{\nn/3 ~ exp (_ 4__~ao2) . 3 r2
V a l u e s of Z(0), Z(ao) , Z(~/2ao) and Z(~/3ao) a r e 1.293, 0.293, 0.347 and 0.142 r e s p e c t i v e l y , s i m i l a r l y f o r unlike n e i g h b o u r s . P e r f o r m i n g the s u m m a t i o n up to t h i r d n e a r e s t n e i g h b o u r s , like and unlike, and taking the c o n t r i b u t i o n of d i s t a n t points to be about 30~o, v a l u e s ofJ(0)(0) a n d S ( 0 ) ( 0 ) .a4?e obtained. F o r HI100], an 6 ½p~0)(0) = 22.14 and ao6 ½gg~0)(0) = 1.6. S i m i l a r l y for H[110], ao6 ½pJ~0)(0) = 7.81 and ao6 ½]~/'(~)(0) = 38. The f i e l d a n g u l a r a v e r a g e is taken by taking 0.8 t i m e s the value for H[110] plus 0.2 t i m e s the value for H[100]. ~ d e r i v i n g the above e x p r e s s i o n s for T1 and T2 no c o n s i d e r a t i o n has been given to the c o r r e l a t i o n e f f e c t s such a s c o r r e l a t i o n s in the d i r e c t i o n s of s u c c e s s i v e j u m p s (which r e d u c e s the diffusion coefficient f o r a bcc l a t t i c e by a f a c t o r of 0.8 2) r e l a t i v e to the r a n d o m walk value) and the tendency of j u m p s to o c c u r c l o s e t o g e t h e r , an effect, which h a s been" shown to be i m p o r t a n t for n u c l e a r r e l a x a t i o n . If t h e s e eff e c t s a r e taken into c o n s i d e r a t i o n , the n u m e r i c a l c o e f f i c i e n t s o c c u r r i n g in the e x p r e s s i o n s for T 1 and T 2 a r e r e d u c e d by a f a c t o r which v a r i e s f r o m 0.5 to 0.8. T h i s work i s s u p p o r t e d by the National B u r e a u of S t a n d a r d s , Washington, D . C . One of us (DPT) a l s o w i s h e s to e x p r e s s h i s thanks for the a w a r d of a R e s e a r c h F e l l o w s h i p .
References 1) M. E i s e n s t a d t and A . G . R e d f i e l d , Phys. Rev. 132 (1963) 635. 2) A. B. Lidiard, in: Handbueh tier Physik, edited by S. FlUgge (Springer-Verlag, Berlin, 1957). Vol. 20.
169
P HY S I C S L E T T E R S
1 June 1964
(8), is that the expectation value of any product o p e r a t o r containing u n p a i r e d c r e a t i o n or annihilation o p e r a t o r s a+k, s or a k s as f a c t o r s v a n i s h e s . T h e s e c o n c l u s i o n s }~-~vean i m p o r t a n t b e a r i n g on the d e s c r i p t i o n of the r e c e n t l y o b s e r v e d i n t e r f e r e n c e effects between independent light b e a m s f r o m l a s e r s . It can be shown 1) that the a m p l i t u d e s of i n t e r f e r e n c e or modulation t e r m s in the e x p r e s s i o n for the total m e a n light i n t e n s i t y involve the f a c t o r s
References 1) L. Mandel, Phys.Rev. 134 (1964) A10. 2) H.Paul, W. Brunner and G.Richter, Ann. Physik 12 (1963) 325. 3) T. F. Jordan and F. Ghielmetti, to be published (1964). 4) A. Javan, E.A. Ballik and W. L. Bond, J. Opt. Soc. Am. 52 (1962) 96. 5) M.S. Lipsett and L.Mandel, Nature 199 (1963) 553. 6) G. Magyar and L.Mandel, Nature 198, 255. 7) E. C. G. Sudarshan, Phys. Rev. Letters 10 (1963) 277. 8) E. C. G. Sudarshan, Proc. Brooklyn Syrup. on Optical Masers, (John Wiley & Sons, N.Y., 1963) p. 45. 9) J.R.Klauder, Ann.Phys. 11 (1960) 123. 10) R.J. Glauber, Phys. Rev. 130 (1963) 2529. 11) R. J. Glauber, Phys: Rev. 131 (1963) 2766. 12) Y.Kano, Ann. of Phys., to be published (1964). 13) C.L.Mehta and E.Wolf, Phys.Rev., in press (1964). 14) L.Mandel, Physics Letters 7 (1963) 117. 15) M. Born and E.Wolf, Principles of Optics (Pergamon Press, London & N.Y., 2nd ed. 1964). 16) E.Wolf, Proc.of Symposium on Optical Masers (Polytechnic Inst.of Brooklyn, 1963), p.29.
RANDOM WALK MODEL AND NUCLEAR MAGNETIC RELAXATION IN A B C C L A T T I C E CONTAINING TWO NUCLEAR SPECIES D. P. TEWARI and G. S. VERMA
Department of Physics, University of Allahabad, Allahabad, India Received 5 May 1964
A random walk model is applied to dipolar relaxation via atomic diffusion in a bcc lattice containing two nuclear species. Expressions for T1 and T2 in the high field and low temperature limit as well as high temperature and low field case are derived for magnetic field along [100] and [110] directions. The results of our calculations axe as follows. For a magnetic field applied along the [100] direction and in the high field, low temperature limit T1- I = 8.99 AI ao-6 vI wi-2 + AS ao-6 (Vl+ ,,S)[0.149(wi-00S)2 + 11.124 001-2 + 12.105(wi+cOS)-2] T2-1 = 8.31 A i a o -6 yi -1 + 0.56 A S a o - 6 ( u i + US)-1 , where
AI = v 1 4 ~ 2 l ( I + l ) f l ,
A S =vl2rs2h2 S(S+I)f s .
Here r , ~0, f and v a r e the g y r o m a g n e t i c r a t i o , r e s o n a n c e f r e q u e n c y , f r a c t i o n a l isotopic abundance and j u m p frequency. The s u b s c r i p t s I and S r e f e r to the two n u c l e a r s p e c i e s of spin I and S. a 0 is the cube edge. 168
Vohune 10, number 2
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1 J u n e 1964
S i m i l a r l y for a m a g n e t i c field a p p l i e d along the [110] d i r e c t i o n T1-1 = 12.966 A I a o - 6 coI -2 ui + AS ao-6(vi+ VS)[2.882 ( w / - coS)-2 + 6.153 w/-2 + 14.703 (coI +coS )-2] T 2 " l = 2.928 Ai ao -6 ui-1 + 12.67 A s ao-6 (ui + US)-1 The high f i e l d and low t e m p e r a t u r e l i m i t c o r r e s p o n d to the c a s e co/~t >> 1 w h e r e co i s the a r g u m e n t of the s p e c t r a l densityJ(q)(co) for the iike n e i g h b o u r s orgt~(q)(co) for the unlike n e i g h b o u r s , co can take v a l ue.~., which m a y be m u l t i p l e s , s u m s o r d i f f e r e n c e s of coI and coS o r z e r o . ForJ~(q)(co) the j u m p p r o b a b i l i t y Jn the r e l a t i v e c o o r d i n a t e s y s t e m i s given by ~t = 2u/ and forgff(q)(co), ~ = vi+ vS. The s p e c t r a l d e n s i t y q)(co) is given by 1)
~
flq)(co) = 2~ ~ F,.(q ) _ c_1 ~ (1)Elm(q)}, CO 2 t
tt
lm
where /~rn (0) = (1 - 3 cos 2 0/)(1 - 3 cos 2 Om)/rl3 rm3
Flm (1) = sin 0 l cos 0 l sin 0rn cos 0 m cos (q~l-¢rn)/rl 3 rm 3 /~rn (2) = sin 2 0l sin 2 em cos (2~l-2q~rn)/rl 3 rm3 . H e r e rl, 0l, q~l are the p o l a r c o o r d i n a t e s of the v e c t o r r/ drawn f r o m the r e f e r e n c e spin to the point l and f i e l d d i r e c t i o n i s the p o l a r axis. The f i r s t s u m m a t i o n i s done up to t h i r d n e i g h b o u r s of the o r i g i n and the s e c o n d s u m m a t i o n i s o v e r point 1 and m which a r e n e a r e s t n e i g h b o u r s to e a c h o t h e r and e i t h e r l o r m i s a n e a r e s t neighbour to the o r i g i n . C i s the like neighbour coordination n u m b e r . The l a t t i c e s u m s which a r e n e e d e d to c a l c u l a t e the s p e c t r a l d e n s i t i e s in the l i m i t of high f r e q u e n c y a r e a s follows. F o r HI100], ao6(co2/2~t)J{q)(co) = 12.37, 0.28, 4.86 for q = 0, 1, 2 r e s p e c t i v e l y . S i m i l a r ly ao6(W2/2#)3t~(q)(w) = 0.895, 3.71, 8.07 for q = 0, 1, 2 r e s p e c t i v e l y . F o r H a l o n g [110] d i r e c t i o n , ao6(co2/2~t)~C~(w ) = 4.54, 1.18, 3.94 for q = 0, 1, 2 r e s p e c t i v e l y . Also ao6(W2/2U)s(q}(co ) = !7.29, 2.05, 9.80 for q = 0, 1, 2 r e s p e c t i v e l y . In the high t e m p e r a t u r e l i m i t which c o r r e s p o n d s to low f i e l d c a s e , T 1 and T 2 a r e i s o t r o p i c and f i e l d a n g u l a r a v e r a g e s a r e taken of_f(q) a n d S ( q ) . In this c a s e TI'I
= T2-1 = ~ A I ~¢(0)(0))+ ~ A SQ~(O)(o~ ) = 1 3 . 3 5 A I a o - 6 vi-1 + 51.2 A S ao-6( ui+ VS) -1 ,
w h e r e in the l i m i t of z e r o f r e q u e n c y 1) 2~J(0)(0) = Im ~ Flm(O) Z(rlm) ;
3
Z(rlm) = ~n P(rlm) ;
Pn(r) = ~{\nn/3 ~ exp (_ 4__~ao2) . 3 r2
V a l u e s of Z(0), Z(ao) , Z(~/2ao) and Z(~/3ao) a r e 1.293, 0.293, 0.347 and 0.142 r e s p e c t i v e l y , s i m i l a r l y f o r unlike n e i g h b o u r s . P e r f o r m i n g the s u m m a t i o n up to t h i r d n e a r e s t n e i g h b o u r s , like and unlike, and taking the c o n t r i b u t i o n of d i s t a n t points to be about 30~o, v a l u e s ofJ(0)(0) a n d S ( 0 ) ( 0 ) .a4?e obtained. F o r HI100], an 6 ½p~0)(0) = 22.14 and ao6 ½gg~0)(0) = 1.6. S i m i l a r l y for H[110], ao6 ½pJ~0)(0) = 7.81 and ao6 ½]~/'(~)(0) = 38. The f i e l d a n g u l a r a v e r a g e is taken by taking 0.8 t i m e s the value for H[110] plus 0.2 t i m e s the value for H[100]. ~ d e r i v i n g the above e x p r e s s i o n s for T1 and T2 no c o n s i d e r a t i o n has been given to the c o r r e l a t i o n e f f e c t s such a s c o r r e l a t i o n s in the d i r e c t i o n s of s u c c e s s i v e j u m p s (which r e d u c e s the diffusion coefficient f o r a bcc l a t t i c e by a f a c t o r of 0.8 2) r e l a t i v e to the r a n d o m walk value) and the tendency of j u m p s to o c c u r c l o s e t o g e t h e r , an effect, which h a s been" shown to be i m p o r t a n t for n u c l e a r r e l a x a t i o n . If t h e s e eff e c t s a r e taken into c o n s i d e r a t i o n , the n u m e r i c a l c o e f f i c i e n t s o c c u r r i n g in the e x p r e s s i o n s for T 1 and T 2 a r e r e d u c e d by a f a c t o r which v a r i e s f r o m 0.5 to 0.8. T h i s work i s s u p p o r t e d by the National B u r e a u of S t a n d a r d s , Washington, D . C . One of us (DPT) a l s o w i s h e s to e x p r e s s h i s thanks for the a w a r d of a R e s e a r c h F e l l o w s h i p .
References 1) M. E i s e n s t a d t and A . G . R e d f i e l d , Phys. Rev. 132 (1963) 635. 2) A. B. Lidiard, in: Handbueh tier Physik, edited by S. FlUgge (Springer-Verlag, Berlin, 1957). Vol. 20.
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