Volume 38A, number 6
PHYSICS LETTERS
13 March 1972
BETSUYAKU THEORY OF FREE INDUCTION DECAY IN A M A G N E T I C A L L Y MODIFIED SOLID K. Y. YANG and T. M. WU,
Department of Physics, State University of New York at Binghan2ton, Binghamton, N. Y. 13901, USA
The Free-Induction-Decay function G(t) of a magnetically modified system with spin ½ is evaluated by using Betsuyaku's theory. The result is as expected, that the amplitude of G(t) decreases and that the nodes shift to a later time as the non-magnetic impurity increases.
In solids the nuclear magnetic resonance spectra reveal the mechanism of interactions existing between the resonating nuclear spin and its surroundings. At liquid nitrogen temperature, where the nuclear diffusion can be ignored, and thus spatial order is maintained, the broadeningof the spectral lines and the relaxation effect provide us with important information about the dynamical behavior of nuclear spin, and the properties of the materials. At temperatures of interest, the thermal energy is much greater than the Zeeman energy, and under such conditions the average magnetic moment( s ) i s given by [2]
(1)
and
H = H' - ~ S
i. H l e x p ( i w t ) ,
(2)
w h e r e H' is the t r u n c a t e d H a m i l t o n i a n for static i n t e r n u c l e a r dipole-dipole i n t e r a c t i o n given by Van Vleck [3],
H' = ~.. B i j ( S i z S j z - ½Si . S j ) ;
and B i j = ¼ N 2 ~ ( 1 - 3 c o s 2 0 i j ) r - 3 i j ;
(3)
/ / 1 exp (i¢ot), the applied r.f. field in x -y plane; k, B o l t z m a n n ' s c o n s t a n t ; T, the t e m p e r a t u r e ; x(¢o), the line s h a p e ; r~.~, the d i s t a n c e between s i t e s of the l a t t i c e ; ~, the g y r o m a g n e t i c r a t i o ; and Oii, the angle r i j makes with the applied static magnetic field along z axis. The F r e e - I n d u c t i o n - D e c a y function G(t) is p r o p o r t i o n a l to the F o u r i e r t r a n s f o r m of the n u c l e a r magnetic r e s o n a n c e line shape x(w), and is
given by G(t) = tr(Sx(t) Sx(0) / t r Sx 2 (0) .
(4)
Lowe and N o r b e r g [4] f i r s t calculated G(t) for a spin ½ s y s t e m on a s i m p l e cubic lattice for dipole i n t e r a c t i o n . Though the Lowe and N o r b e r g theory is in a g r e e m e n t with e x p e r i m e n t a l r e s u l t s in the case of CaF2, it has been s u b j e c t to c r i t i c i s m because of its a r b i t r a r y expansion s c h e m e and lack of a c l e a r cut c o n v e r g e n c e even for the r e g i o n of time of i n t e r e s t . Moreover, if one extends the Lowe and N o r b e r g theory to a m a g n e t i c a l l y modified s y s t e m , the amplitude of the f r e e - i n d u c t i o n - d e c a y beats i n c r e a s e s as the c o n c e n t r a t i o n of the n o n - m a g n e t i c i m p u r i t y i n c r e a s e s . This is not physical, however. Many other t h e o r e t i c a l works have been devoted to n u c l e a r magnetic r e s o n a n c e [5-8]. Recently, using the K u b o - T o m i t a [9] theory of magnetic r e s o n a n c e , Betsuyaku was able to expand G(t) up to second o r d e r in t e r m s of V alone:
G(t) = ~ Gn(t) : Go(t) + Gl(t) + G2(t) + . . . n
where
an(t)
= (-i) n f
Here V is defined as
t
0
dtl..,
in- 1
dtn<[[.. [Sx(t), V(tl)].. ], V(tn)]S x ) .
(5) (6)
0
455
Volume 38A, number 6
PHYSICS LETTERS
13 March 1972
1.0
~ k, . ~ * O.S 08
~
1,1-P
= 1.0
--0.4
O2
0 -0.2 I 0.2
-0.4
I
0.4
I
I
0.6
I
0.8
I
1.0
1.2
1.4
t
Fig. 1. Free Induction Decay for 19F in CaF 2 for static magnetic field atong [100] axis. The unit of time is 34-a3/72~/where a = 2.725~ is the lattice spacing.
8•i j siz
H' =
- I ~
icj
s i " s j (2) =
v
÷
(7)
i~j
and with no r e s t r i c t i o n in time. We will extend B e t s u y a k u ' s theory to the case of a m a g n e t i c a l l y modified s y s t e m . F o r the sake of s i m p l i c i t y , we c o n s i d e r an i s o t r o p i c cubic lattice as having a f r a c t i o n p of lattice s i t e s occupied by spin ½ n u c l e i , and the r e m a i n i n g s i t e s occupied by spin zero nuclei. As a r e s u l t , G(t) now b e c o m e s G(P,t). Modifying B e t s u y a k u ' s theory, G(p,t) is now given by
c(p, t) = Co(P,t) + Cl(P,t) + G2(P, t) where
Go(P't) = rV[cj [(1-p) + pc°s(Bj rt)] = r~j
u(p, t)
(8) t
Cl(P,t) = ~ k~Cj p Bjk sin(Bjkt) { t 17 v(p,t) r¢j,k (1)
(2)
(3)
(4)
-
f 0
(9) dt 1
I-[
re j, k
[(1-P)+pcos(Bjrt+Bkrtl-Bjrtl)]}
(5) (10)
G2(P, t) = G2(P,t) + G2(P,t) + G2(P,t) + G2(P,t) + G2(P,t)
(t) G2(P,t) = -~- ~ pBjk2C°s(Bj kt) {-~t2 r Ne j , k u ( p , t ) kCj t _ f d t l t 1 17 [ ( 1 - P ) + p c o s ( B j r t . Bjrtl+Bkrtl)] } 0 r~j,k
with
(2) ~ l c~j , k p2 SJ k sin(Sjkt){½t2 S j lsin(sjlt) v2(P't) = ~1 kCj
t - f dtl(Bjlt+Bkltl-Sjl 0 456
+
r~j,k,l [1 u(p,t) +
t l ) s i n ( B j l t + S k l t l - B j l t l ) re~ j, k,l [ ( 1 . p ) + P c o s ( S j r t + B l r r t l - B j r t l ) ] }
Volume 38A, number 6
PHYSICS LETTERS
13 March 1972
k#j lcj,k p2Bkl 2 6 d t I (t-tl)cos[(Bjk-Bjl)(t-tl)] + sin [ (Bjk - Bjl )( t - t 1)] Bjle Bjl r~j ~,I [(1 -p) +p cos(Bit t+ Bkrt 1 B lrtl )]
G2(P,t)
-
G 4tp t)
--
"1
'
~ ~ kCj l c j , k
-
l, :% lB ,,,,z /
0
t1 dt1 f 0
dt 2
sin(Bjkt+Bkl tl_Bjktl ) ×
×sin(Bjlt+Bjkt2-Bjlt2) G25~'t) =
[-[ [(1-p)+pcos(Bjrt+Blrtl-Bjrtl +Bkrt2-Blrt2) ] r #j , k,l t ~ kcj ~ l*j ~,k p2BjlBkl fo d t l /10 dt2sin[(Bjk-Bjl)(t- tl)] x
× sin(Bjkt2-Bkl %) r-[ [(1-p)+pcos(Bjrt+Bkrtl-Blrtl+Blrt2-Bjrt2) ] . r , j , h,l We plot G(p,t) as a function of t for P = 1.0, 0.8, 0.6 r e s p e c t i v e l y . While valuating the p r o d u c t s and s u m m a t i o n s , we include up to the third n e a r e s t n e i g h b o r s . As is seen f r o m fig. 1, G(p,t) does behave as one expects, i . e . , the amplitude i n c r e a s e s and the nodes shift to l a r g e r time as p i n c r e a s e s .
Re/e~'ences [1] H. Betsuyaku, Phys. Rev. Letters 24 (1970) 934. [2] A. Abragam, The principles of nuclear magnetism (Oxford Clarendon Press, 1961). [3] J.H. VanVleck, Phys. Rev. 74 (1948) 1168. [4] I.J. Lowe and R. E. Norberg, Phys. Rev. 107 (1957) 46. [5] S. Clough and L R. McDonald, Proc. Phys. Soe. (London) 86 (1965) 833, [6] W. A. B. Evans and J. G. Powles, Phys. Letters 24A (1967) 218. [7] S. Gade and I. J. Lowe, Phys. Rev. 148 (1966) 382. [8] M. Lee, D. Tse, W.I. Goldburg and I. d. Lowe, Phys. Rev. 158 (1967) 246. [9] R. Kubo and K. Tomita, J. Phys. Soe.dapan 9 (1954) 888.
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