Proton Spin Lattice Relaxation in the ...

Proton Spin Lattice Relaxation in the Dimethylammonium Group K . V e n u a n d V. S. S. S a s t r y School of Physics, University of Hyderabad, Hyderabad - 500 134, India Z. Naturforsch. 48a, 713-719 (1993); received July 18, 1992 A model for the spin lattice relaxation time of the protons of dimethylammonium in the Redfield limit and common spin temperature approximation is developed. The three fold reorientations of the methyl groups, the rotation of the whole molecular group around its two fold symmetric axis and possible correlations among these motions are considered. The effect of these processes on the dipolar interactions among the protons within the same molecular group is taken into account. The resulting relaxation rate is powder averaged and used to explain the experimental data in literature on [NH 2 (CH 3 ) 2 ] 3 Sb 2 Br 9 . The analysis shows that dynamically inequivalent groups exist in this compound and that the effect of proposed correlation among the different motions on the final results is negligible.

Introduction C o m p o u n d s containing dimethylammonium (DMA) are k n o w n to undergo structural phase transitions. These phase changes are normally associated with s u b t l e c h a n g e s in t h e d y n a m i c s of t h e D M A g r o u p s [ 1 - 5 ] . P r o t o n spin lattice r e l a x a t i o n t i m e ( 7 \ ) m e a s u r e m e n t s of this m o l e c u l a r g r o u p p r o v i d e a c o n v e n i e n t a n d p o w e r f u l t o o l t o s t u d y its d y n a m i c s . H o w ever, t h e d a t a o n these s y s t e m s a r e n o t a p p r o p r i a t e l y a n a l y z e d s o far, d u e t o t h e lack of r e l e v a n t t h e o r e t i c a l c a l c u l a t i o n s in t e r m s of all d y n a m i c a l p a r a m e t e r s ass o c i a t e d w i t h this g r o u p . T h e data have been analyzed b y a s s i g n i n g t h e o b s e r v e d r e l a x a t i o n t o t h e m e t h y l g r o u p d y n a m i c s o n l y , w h e r e a s , in p r i n c i p l e , the D M A ion undergoes m o r e complicated dynamics [1]. T h i s p a p e r a i m s a t d e r i v i n g a n a n a l y t i c a l e x p r e s sion for the relaxation rates, taking into account the d y n a m i c s of t h e D M A g r o u p in a d d i t i o n t o t h e d y n a m i c s of t h e m e t h y l g r o u p . T h e o u t c o m e of t h i s calc u l a t i o n is u s e d t o a n a l y z e t h e p u b l i s h e d d a t a [2] on tris-dimethylammonium nonabromo diantimonat, y i e l d i n g f o r t h e first t i m e d y n a m i c a l i n f o r m a t i o n a b o u t t h e D M A ion a s a w h o l e .

t h e i r t h r e e fold s y m m e t r i c axes, (ii) t w o fold r o t a t i o n s of t h e D M A g r o u p a r o u n d its d i a d axis, (iii) n-fold r e o r i e n t a t i o n s of t h e D M A g r o u p a r o u n d its d i a d axis, a n d (iv) i s o t r o p i c t u m b l i n g of t h e w h o l e D M A g r o u p (see F i g u r e 1). O u t of these, (iii) a n d (iv) d o n o t p r e s e r v e t h e s y m m e t r y of t h e D M A g r o u p a n d h e n c e a r e expected to be quite hindered, taking place only at very high t e m p e r a t u r e s . In this paper, a n expression f o r t h e r e l a x a t i o n r a t e ( T ^ 1 ) of t h e g r o u p is d e r i v e d c o n s i d e r i n g o n l y t h e first t w o t y p e s of d y n a m i c s , m o d elling b o t h i n t r a - a n d i n t e r - m e t h y l p r o t o n - p r o t o n d i p o l a r i n t e r a c t i o n s a s well a s l o n e p r o t o n - m e t h y l p r o t o n i n t e r a c t i o n s . T h e g e n e r a l m e t h o d o l o g y of these c a l c u l a t i o n s is s i m i l a r t o t h e t r e a t m e n t of S j o b l o m a n d P u n k k i n e n [6] f o r t r i m e t h y l a m m o n i u m g r o u p . T h e m a g n e t i c d i p o l a r i n t e r a c t i o n ( w h i c h is t h e d o m i n a n t r e l a x a t i o n m e c h a n i s m f o r p r o t o n s in t h e D M A ion), b e t w e e n t w o p r o t o n s m a n d n is Sir/K

£ ß =

(1)

- 2

w h e r e t h e S ^ ' s a r e t h e c o m p o n e n t s of t h e s e c o n d r a n k irreducible t e n s o r o p e r a t o r s involving spin variables, a n d t h e F ^ ' s a r e t h e c o r r e s p o n d i n g

spatial

f u n c t i o n s given b y Proton Spin Lattice R e l a x a t i o n in D M A Groups Fmn T h e f o u r t y p e s of p l a u s i b l e m o t i o n s of t h e D M A i o n , r e l e v a n t in the p r e s e n t e x p e r i m e n t s , a r e [1]: (i) t h r e e f o l d r e o r i e n t a t i o n s of t h e m e t h y l g r o u p s a r o u n d Reprint requests to Prof. V. S. S. Sastry, School of Physics, University of Hyderabad, Hyderabad - 500 134, Indien.

=

r mn

COS2 6 m n ) ,

(1

F£) = F ^ * =

y

-^sine

m n

cosd

m n

e-i*-,

^ mn

mn

mn '

3

mn

mn

0932-0784 / 93 / 0500-725 $ 01.30/0. - Please order a reprint rather than making your own copy.

Unauthenticated Download Date | 11/24/15 5:32 AM

(2)

714

K. Venu and V. S. S. Sastry • Proton Spin Lattice Relaxation in the Dimethylammonium G r o u p

H e r e ( r m n , d m n , mn) is t h e v e c t o r j o i n i n g t h e p r o t o n s

I n c a l c u l a t i n g t h e v a l u e of Ti of a n y p r o t o n in t h e

m a n d n e x p r e s s e d in s p h e r i c a l c o o r d i n a t e s . T h e s e

D M A g r o u p , t h e p a i r wise i n t e r a c t i o n s b e t w e e n p r o -

quantities become time dependent due to the molecu-

t o n s c a n be r e s t r i c t e d t o a g o o d a p p r o x i m a t i o n

lar motions, a n d their fluctuations, X ^ , are

those within the D M A group.

to

L e t t h e r a t e of t h e t h r e e fold r e o r i e n t a t i o n ( C 3 r o t a -

(3)

t i o n ) of t h e m e t h y l g r o u p s be r a n d t h e r a t e of t h e t w o

T h e spin l a t t i c e r e l a x a t i o n r a t e of t h e p r o t o n m d u e

f o l d r e o r i e n t a t i o n of t h e D M A g r o u p a r o u n d its d i a d

t o its d i p o l a r i n t e r a c t i o n w i t h p r o t o n n, in t h e R e d f i e l d limit, is given by [7]

axis ( d i a d r o t a t i o n s ) b e R. N o w , t o t a k e t h e statistical i n t e r - d e p e n d e n c e of these m o t i o n s i n t o a c c o u n t , let t h e r a t e of c o u p l e d C 3 r o t a t i o n of one m e t h y l g r o u p

IcoJ),

(4)

a n d d i a d r o t a t i o n b e R', a n d t h a t of t h e c o u p l e d C 3 r o t a t i o n s of two m e t h y l g r o u p s a n d t h e d i a d r o t a t i o n b e R". R e f e r r i n g t o Fig. 1, c o n s i d e r t h e m o t i o n s of a

where

p r o t o n p a i r initially at p o s i t i o n s 1 a n d 2 ( d e n o t e d b y J

(5)

- oo

is t h e s p e c t r a l d e n s i t y f u n c t i o n , a n d t h e c o r r e s p o n d i n g a u t o c o r r e l a t i o n f u n c t i o n s a r e given b y =

+

12). A C 3 r o t a t i o n of t h e m e t h y l g r o u p can t r a n s f e r t h i s p a i r t o p o s i t i o n 23 o r 31 at a r a t e of r / 3 a n d t o p o s i t i o n 12 at a r a t e of 2 r / 3 . It c a n a l s o o c c u p y p o s i t i o n 12 o r 45 d u e t o d i a d r o t a t i o n of t h e D M A g r o u p

(6)

a t a r a t e of R/2.

Similarly, different c o u p l e d m o t i o n s


K. Venu and V. S. S. Sastry • Proton Spin Lattice Relaxation in the Dimethylammonium G r o u p c a n t r a n s f e r t h e initial p a i r 12 t o 23, 31, 56, a n d 64 a t a r a t e R'/4, t o 12, 4 5 a t a r a t e R"/4, a n d t o 23, 31, 56, a n d 6 4 a t a r a t e R"/S. L e t p 0 (t) b e t h e p r o b a b i l i t y of f i n d i n g t h i s p a i r a t a p o s i t i o n ij a f t e r a t i m e t. D u e t o and s y m m e t r y , p 2 3 ( 0 = P 3 i ( 0 a n d PseiO = PeM hence only four probabilities i j = 12, 23, 45, a n d 56) a r e i n d e p e n d e n t . T h e a b o v e a r g u m e n t s c a n b e e l e g a n t l y s u m m a r i z e d in t h e f o r m of a d e t a i l e d b a l a n c e e q u a t i o n f o r t h e r a t e of c h a n g e of a p a i r p r o b a b i l i t y f u n c t i o n , s a y pl2{t), as (2

R

( R

\

4

R

R"\

(R'

R"\ + Y J ( P 5 6 ( 0 + P64W),

+

which on simplification d u e to s y m m e t r y leads to R . . . f 2 P n ( 0 = - (•j r + y

[2 r

+

3

(t

+

R'

T

3 \ -R"jp

+R'+

r

- ( j

+

(j

+R'+

+ y

+

2r

+ —+ R'+-R' R' 4

3

P23(0 P45

+

'R

( 0

TJp"(,)

PssW R'

R"\

Similar equations can be written for the remaining

3

R' 2

t J

R +

t

R

R"\

+

( R

(7)

probabilities, a n d these are s u m m a r i z e d as

R' +

t

R"

+

(t)

")Pi2(t)

f + f ) ( p » W + P,tW)

Pl2 (0 \

l 2

R"\

+

/ R Pi2(t)=

715

+

R"\

R R '

t )

2

+

R +

t

2

5

R'

R"\ t J

(2 - l 3

r +

3 R"

R 2 +

r

4

3

R"

R'

t

t

R" 3 j ,

'

R' +

R 2

T

+

j

R" +

r

\ " j

/2r ( T

R" +

+

t

R' T +

3R" 1T

+

R' t

+

R"\ t J

P12 (0 P23W

(8)

P45 ( 0

PseW

r R 3 5 - I - + — + - R' + - R" \3 2 4 8

T a k i n g s u i t a b l e l i n e a r c o m b i n a t i o n s of t h e s e p r o b a b i l i t i e s , t h e s e e q u a t i o n s c a n b e d e c o u p l e d t o give R 5R' 7R" r+ — + — + 2 4

P 1 2 - P 2 3

R

P45-P56

R' +

2 " 4

R

R'

R"

2 " 4

R"

R 5 R' 7R" r + — + —— + — 2 4 !

"

P 1 2 - P 2 3

P45-P56

(9)

and Pl2+

2

P23

P45 + 2 P 5 6

= 2- ( R + R ' + R " )

-1

+ l \ / p 1 2 + 2p23

+ 1

- V V P 4 5 + 2P 5 6

(10)

T h e s o l u t i o n s , w i t h t h e initial c o n d i t i o n p 1 2 ( 0 ) = 1, p,-(0) = 0 f o r i j = 23, 45, a n d 56, a r e w r i t t e n a s 2

1

1\

P23(0

-1

-1

1

1

P45W

2

- 2

- 1

1

I P 12

2

W \

-1

\p56(0/

1 -1

kl le~~k2t '\ e

(11)

e~k3t

l1

j

where 3 R'

3 R"

(12)

/c7 = r + R + K' + R "

(13)

and (14)


716

K. Venu and V. S. S. Sastry • Proton Spin Lattice Relaxation in the Dimethylammonium Group

Similar results are o b t a i n e d for the probabilities involving lone p r o t o n - methyl p r o t o n ( I p - m p ) pair, like i j = 71 (Figure 1). Time m o d u l a t i o n s of the dipolar interactions a m o n g inter-methyl pairs can similarly be taken into a c c o u n t . Considering a p r o t o n pair like 14 (Fig. 1) at t = 0, various j u m p probabilities to other pair locations connected by the a b o v e p r o p o s e d d y n a m i c s c a n be calculated. After t a k i n g symmetry considerations into a c c o u n t (i.e. Pis= Pi6 = P24 = P34 a n d P25= P26 = P 3 5 = Pie) the detailed balance equation for the independent probabilities (Pi4' Pis» a n d P25) assumes the f o r m 4r

/ -

/Pi*W\

+ r

Pi s(0

3

4r

R+R"

3 R'

R' +

R"

4

R"

2r

T

T

P25W

R"

+ R

+

R'

J

+

Solutions with the initial condition p 1 4 ( 0 ) = l Pis(0) = £25(0) = 0 are o b t a i n e d as p

1 5

| = i

-2

1

1

(e~k4t\ e-k*A

/

2r

R"

~2

I Pl4-\

R'

R"

~2

~2

R'

3 R"

\p25,

and

(16)

where

These pair correlation functions in the a b o v e e q u a t i o n are then to be evaluated m a k i n g use of j u m p p r o b a b i l ities in (11). T h e result for one of the pair functions, say «))],

c r = 3 Re [X^(0) + xr(0)

5 ( 0 ) + x r (0) ^">(0)]

(22b)

(0) (22 c)

and Ci"' = j Re [(*§•>(0) + (

(0)) X t f * (0)

+ (X 6 4 (0) + X £l (0)) X(2~lß) (0) + (X^(0)

(

+ X ^ m XiSHO)].

(22d)

Substituting for p{- s from (11) in (21) the average correlation function can be written as GXL(t)=

z Kl» e -kit


(23)

717 K. Venu and V. S. S. Sastry • Proton Spin Lattice Relaxation in the Dimethylammonium G r o u p where

f u n c t i o n (5), a n d c a l c u l a t i n g t h e o v e r a l l s p i n - l a t t i c e K = £ t 2 C 1 M) -

+ 2 C M)

2

-

- Ci"'],

2 C M)

3

(24 a)

+ C 4 M ) ]'

(24 b)

and x f

= | [C - e r -

c n

(24c)

r e l a x a t i o n r a t e of t h e D M A g r o u p u s i n g (4), o n e o b tains Tf

1

=

27 16h2

2

z [2 £ M =1 L m = l

+ 2 Y Af00 -t, " k2 +

(26) where (27 a) and L = I (4£(0)|2 + |XW(0)|2+...+ | ^ ( 0 ) | 2 ] ,

(28a)

+ £

M y n=1 "

£(0) + X ^ ( 0 ) + X(0) + (X26 (0) + X24 (0) -I- X^ (0) + X^ + (X&>(0) +X£(0)

k 2 m +(pCO 0 ) 2

5

V T '(0) +X(0) + X(0) + (X£(0)

+ X(0)],

Proton Z e e m a n Spin-Lattice R e l a x a t i o n R a t e s S u b s t i t u t i n g t h e s e c o r r e l a t i o n f u n c t i o n s ((23), (25), a n d (26)) in t h e e x p r e s s i o n f o r t h e s p e c t r a l d e n s i t y

N u m e r i c a l Calculations I n o r d e r t o o b t a i n t h e v a l u e s of t h e r e l a x a t i o n c o n s t a n t s in (30) (K[, M[, a n d L',) f o r a g i v e n p h y s i c a l situation, the b o n d lengths a n d angles for the D M A i o n given by A n d r e w a n d C a n e p a [1] a r e u s e d . T o m a k e t h e results a p p l i c a b l e t o p o l y c r y s t a l l i n e s a m p l e s ( w h i c h is typically t h e case), t h e s e r e l a x a t i o n c o n s t a n t s a r e p o w d e r a v e r a g e d . T h i s is a c c o m p l i s h e d b y first c o n s i d e r i n g o n e of t h e sides of t h e c u b e c e n t e r i n g t h e D M A g r o u p (Fig. 1) t o b e p a r a l l e l w i t h t h e m a g n e t i c field (say, z-axis), a n d f i n d i n g t h e c o o r d i n a t e s of all p r o t o n s f o r t h e given d i m e n s i o n s of t h e D M A g r o u p .


718

K. Venu and V. S. S. Sastry • Proton Spin Lattice Relaxation in the Dimethylammonium G r o u p

Table 1. The relaxation constants ( x 10 8 sec 2 ) in (30) for the dimethylammonium molecular group. L'(.(M)

i 1 2 3 4 5 1 2 3 4 5

1.

2.

30.59 25.43 11.88

3.00 1.68 4.50

122.34 101.74 47.5

12.36 7.08 17.64

1.23 1.62

5.07 6.17

T h e n , f o r t h i s o r i e n t a t i o n K[, M[, a n d L\ c a n be calc u l a t e d u s i n g (24) a n d (27). N o w d i f f e r e n t o r i e n t a t i o n s of t h e D M A g r o u p a r e o b t a i n e d b y c h a n g i n g t h e p o l a r

/ -1 1000/T ( K )

c o o r d i n a t e s (6 a n d ) of t h e side t h a t is p a r a l l e l t o t h e m a g n e t i c field, in s m a l l s t e p s ( 0 - 7 t f o r 6, a n d 0 - 2 n f o r ). F o r e a c h o r i e n t a t i o n t h e c o o r d i n a t e s of all p r o tons are calculated using appropriate Wigner rotation matrices, a n d corresponding relaxation constants are

Fig. 2. Tj data of ( D M A ) 3 S b 2 B r 9 (points) at 90 M H z [2] fitted to (30) (solid line) with R' and R" — 0 and with the relaxation constants given in Table 1 (under the assumed inequivalence as explained in the text).

c a l c u l a t e d . F i n a l l y , t h e r e l a x a t i o n coefficients t h u s o b t a i n e d f o r all t h e s e c h o s e n o r i e n t a t i o n s a r e a v e r a g e d . T a b l e 1 gives t h e v a l u e s of t h e s e p a r a m e t e r s , o b t a i n e d a f t e r a v e r a g i n g o v e r 2 5 6 i s o t r o p i c a l l y d i s t r i b u t e d ori-

respectively. T h e c o n s i s t e n c y of this m o d e l is c h e c k e d

e n t a t i o n s of t h e D M A m o l e c u l e . It h a s b e e n o b s e r v e d

by e n s u r i n g t h a t t h e v a l u e of t h e Ti m i n i m u m c o r r e -

t h a t a v e r a g i n g o v e r o n l y 64 o r i e n t a t i o n s l e a d s t o a

sponding to the C H 3 dynamics, calculated by taking

negligible e r r o r (less t h a n 1 % ) in t h e r e l a x a t i o n coeffi-

t h e a p p r o p r i a t e l i m i t i n g c a s e (R = R' = R" = 0) of t h e

cients.

a b o v e m o d e l , a g r e e s v e r y well w i t h t h e e x p e c t e d re-

T h i s m o d e l is t e s t e d b y a n a l y z i n g t h e r e l a x a t i o n d a t a of I d z i a k a n d J a k u b a s [2], c o l l e c t e d o n a p o l y -

sults [6]. T h e

7\

data

of I d z i a k

and Jakubas

at

90 M H z , h o w e v e r , s h o w a b r o a d m i n i m u m of a b o u t

c r y s t a l l i n e s a m p l e of ( D M A ) 3 S b 2 B r 9 . In this p r o c e s s

100 msec, a n d t h i s v a l u e d o e s n o t m a t c h w i t h e i t h e r of

it is a s s u m e d t h a t t h e s e m o l e c u l a r m o t i o n s a r e t h e r -

t h e c o m p u t e d m i n i m a , s u g g e s t i n g t h e p r e s e n c e of d y -

mally activated a n d hence have the Arrhenius type of t e m p e r a t u r e • e x p { — Ea/kBTj,

(T) d e p e n d e n c e ,

i.e. r = r~1 = Zq 1

w h e r e t c is t h e c o r r e l a t i o n time. I n

n a m i c a l l y i n e q u i v a l e n t m o l e c u l a r g r o u p s . In fact, t h e X-ray studies on this a n d other related

compounds

s h o w t h e p r e s e n c e of i n e q u i v a l e n t D M A i o n s in 2 : 1

order to calculate 7\ m i n i m a with the present model

r a t i o [8]. T h e e a r l i e r e x p l a n a t i o n of t h e s e d a t a is in-

(at a g i v e n L a r m o r f r e q u e n c y ) a r i s i n g d u e t o D M A

d e e d b a s e d o n s u c h a n i n e q u i v a l e n c e [2], b u t o n l y t h e

a n d C H 3 d y n a m i c s s e p a r a t e l y , s u b s t a n t i a l l y different

m e t h y l g r o u p d y n a m i c s , p r e s u m a b l y in t h e a b s e n c e of

dynamical p a r a m e t e r s for the methyl g r o u p reorienta-

a relevant model, has been taken into consideration.

t i o n a n d t h e d i a d axis r e o r i e n t a t i o n h a v e b e e n as-

H o w e v e r , d u e t o t h e l i m i t a t i o n of t h e m o d e l u s e d in

s u m e d t o m i n i m i z e t h e c o n t r i b u t i o n of o n e d y n a m i c s

t h e earlier a n a l y s i s (i.e., t h e a b s e n c e of t h e d y n a m i c s of

a t t h e T j m i n i m u m of t h e o t h e r . As a result, well

t h e D M A ion), t h e r e l a x a t i o n coefficients c o r r e s p o n d -

separated

m i n i m a o n t h e t e m p e r a t u r e axis h a v e

i n g t o b o t h t y p e s of m e t h y l g r o u p s w e r e f o u n d t o b e

b e e n o b t a i n e d (cross c o u p l i n g s a r e n e g l e c t e d for sim-

considerably different, a n d b o t h are smaller t h a n the-

plicity, i.e. R' a n d R" = 0). T h e 7 \ m i n i m a a r e c a l c u -

o r e t i c a l l y e x p e c t e d (8.05 x 10 9 sec 2 f o r a p r o t o n - p r o -

l a t e d f r o m (30) w i t h t h e r e l a x a t i o n c o n s t a n t s given in

t o n d i s t a n c e of 1.78 Ä) [9]. T h e i n e q u i v a l e n c e in t h e

T a b l e 1 a n d a r e f o u n d t o b e 63.9 m s e c f o r C H 3 d y n a m -

d y n a m i c s of t h e s e g r o u p s is n o t e x p e c t e d t o result in

ics a n d 2 4 3 m s e c f o r D M A d y n a m i c s (at 90 M H z ) ,

d i f f e r e n t r e l a x a t i o n c o e f f i c i e n t s b e c a u s e t h e o r i g i n of


719 K. Venu and V. S. S. Sastry • Proton Spin Lattice Relaxation in the Dimethylammonium G r o u p t h e s e coefficients lies m o r e w i t h t h e d e t a i l s of t h e p h y s ical d i m e n s i o n s of t h e m o l e c u l a r g r o u p s t h a n w i t h t h e surrounding dynamic environment. N o w , in o r d e r t o a c c o u n t f o r t h e o b s e r v e d b r o a d m i n i m u m a t s o m e i n t e r m e d i a t e v a l u e of 100 msec, it is a s s u m e d h e r e t h a t i n e q u i v a l e n t D M A g r o u p s a s well a s i n e q u i v a l e n t C H 3 g r o u p s exist in 2 : 1 r a t i o in t h i s c o m p o u n d . T h e o b s e r v e d b r o a d m i n i m u m t h e n is a t t r i b u t e d t o t h e d y n a m i c s of 1/3 of t h e D M A g r o u p s a n d 2 / 3 of t h e m e t h y l g r o u p s (this is t h e c o m b i n a t i o n w h i c h r e s u l t s in t h e o b s e r v e d v a l u e of t h e m i n i m u m ) . A m i n i m u m d u e t o t h e r e m a i n i n g 2 / 3 D M A g r o u p s is expected to occur at temperatures above the range c o v e r e d in t h e e x p e r i m e n t ( t h e d a t a a t h i g h t e m p e r a tures s h o w this tendency), a n d the r e m a i n i n g 1/3 m e t h y l g r o u p s p e r h a p s lead to a m i n i m u m at very low t e m p e r a t u r e s n o t c o v e r e d in t h i s e x p e r i m e n t . W i t h t h i s a s s i g n m e n t of i n e q u i v a l e n c e t h e d a t a a r e a n a l y s e d u s i n g a n o n l i n e a r least s q u a r e s m e t h o d w i t h t h e relaxa t i o n p a r a m e t e r s given in T a b l e 1, a n d t h e a g r e e m e n t b e t w e e n t h e e x p e r i m e n t a l a n d t h e o r e t i c a l v a l u e s is r e a s o n a b l y g o o d , a s s h o w n in F i g u r e 2. T h e a c t i v a t i o n energy and preexponential factor, obtained are 2.2 + 0.3 k c a l / m o l e a n d 1.1 x 1 0 " 1 2 sec, respectively, f o r t h e 2 / 3 D M A g r o u p s , w h i l e t h o s e of t h e 1 / 3 C H 3 g r o u p s a r e 1.8 ± 0 . 3 k c a l / m o l e a n d 1.5 x 1 0 " 1 2 sec, re-

[1] E. R. Andrew and P. C. Canepa, J. Magn. Reson. 7, 429 (1972). [2] S. Idziak and R. Jakubas, Ferroelectrics 80, 75 (1988). [3] R. Sobiestianks, J. Grigas, V. Samulionis, and E. F. Andreyev, Phase Transitions 29, 167 (1991). [4] B. Jagadeesh, P. K. Rajan, K. Venu, and V. S. S. Sastry (to be published). [5] H. Yamamoto, A. Ishikawa, T. Asaji, and D. Nakamura, Z. Naturforsch. 45 a, 464 (1989).

spectively. O b v i o u s l y t h e a s s i g n m e n t of m i n i m a c a n b e d o n e m o r e definitively if t h e m i n i m a a r e b e t t e r resolved. E x p e r i m e n t s a t l o w e r L a r m o r f r e q u e n c i e s normally provide better resolution, a n d perhaps such e x p e r i m e n t s a r e called f o r t o c o n f i r m t h e p r o p o s e d m o d e l for this c o m p o u n d . S u c h a m u l t i f r e q u e n c y inv e s t i g a t i o n of 7 \ d a t a in o t h e r c o m p o u n d s i n v o l v i n g D M A g r o u p s is in p r o g r e s s .

Conclusions An expression for the average spin lattice relaxation r a t e of p r o t o n s in t h e D M A g r o u p , u n d e r g o i n g 3 f o l d r e o r i e n t a t i o n of t h e C H 3 g r o u p s a n d 2 fold flips of t h e w h o l e i o n a r o u n d t h e d i a d axis is d e r i v e d w i t h i n t h e R e d f i e l d limit, a s s u m i n g c o m m o n s p i n t e m p e r a t u r e . T h e results h a v e b e e n s u i t a b l y a v e r a g e d t o m a k e t h e m o d e l readily applicable to polycrystalline samples. Existing experimental d a t a o n ( D M A ) 3 S b 2 B r 9 have been reanalysed using this m o d e l a n d bringing o u t a q u a l i t a t i v e l y d i f f e r e n t p i c t u r e of t h e d y n a m i c a l p r o cesses of t h e D M A i o n in t h i s s y s t e m . T h e r e a r e seve r a l i n t e r e s t i n g solids w h e r e t h i s m o d e l c a n b e a p p l i e d t o s t u d y D M A d y n a m i c s t h r o u g h a n a n a l y s i s of p r o t o n 7\ d a t a .

[6] R. Sjoblom and M. Punkkinen, J. Magn. Reson. 20, 491 (1975). [7] A. Abragam, Principles of Nuclear Magnetism, Oxford Press, 1961 [8] R. Jakubas and L. Sobczyk, Phase Transitions 20, 163 (1990). [9] S. Albert, H. S. Gutowsky, and J. A. Ripmeester, J. Chem. Phys. 56, 3672 (1972).
