- Email: [email protected]
The line shape problem in nuclear magnetic resonance
Volume24A. n u m b e r 4
THE
LINE
PHYSICS
SHAPE
PROBLEM
IN
LETTERS
13 F e b r u a r y 1967
NUCLEAR
MAGNETIC
RESONANCE
W. A. B. E V A N S * a n d J . G . P O W L E S
The Physics Laboratory. University of Kent at Canterbury, Canterbury, England Received 31 D e c e m b e r 1966
A solution is given of the nuclear magnetic r e s o n a n c e line shape problem for static dipolar interaction. Good a g r e e m e n t is found between the theory and e x p e r i m e n t for the 19F signal in calcium fluoride.
O n e of t h e m o s t i m p o r t a n t u n s o l v e d p r o b l e m s in nuclear magnetic resonance is the calculation of t h e r e s o n a n c e l i n e s h a p e . W e p r e s e n t a s o l u t i o n of t h i s p r o b l e m f o r t h e c a s e of s t a t i c d i r e c t d i p o l a r i n t e r a c t i o n f o r a l a t t i c e of e q u i v a l e n t s p i n s o n e h a l f f o r l a r g e f i e l d s . V a n V l e c k [1] h a s s h o w n h o w to c a l c u l a t e t h e m o m e n t s of t h e a b s o r p t i o n l i n e b u t in p r a c t i c e it i s o n l y f e a s i b l e to c a l c u l a t e t h e s e c o n d a n d f o u r t h m o m e n t s , M 2 a n d M 4. We c o n s i d e r t h e a m p l i t u d e of t h e t r a n s i e n t t r a n s v e r s e m a g n e t i z a t i o n , t h e s o - c a l l e d B l o c h d e c a y [2], B(t), w h i c h i s t h e F o u r i e r t r a n s f o r m of t h e l i n e s h a p e [3]. L o w e a n d N o r b e r g [3] h a v e g i v e n a t h e o r y f o r B(t) f o r t h e s t a t i c d i p o l a r i n t e r a c t i o n of s p i n s o n e half on a simple cubic lattice and have attempted to i n t e r p r e t t h e e x p e r i m e n t a l r e s u l t s f o r t h e 1 9 F r e s o n a n c e in c a l c i u m f l u o r i d e . T h e i r t h e o r y i s a r a t h e r m o r e e l a b o r a t e e x p a n s i o n f o r B(t) t h a n t h e m o m e n t o n e . It h a s b e e n e r i t i c i s e d [ 4 , p a g . 119] b e c a u s e t h e e x p a n s i o n i s a r b i t r a r y . It i s u n s a t i s f a c t o r y b e c a u s e it d i v e r g e s f o r t i m e s not m u c h l o n g e r t h a n t h o s e f o r w h i c h it w a s c o m p u t e d [5] a n d b e c a u s e it i s not e a s y to s e e how t h e t h e o r y c a n b e i m p r o v e d . A g e n e r a l i s e d f o r m of t h i s t h e o r y [5] g i v e s r a t h e r b e t t e r c o n v e r g e n c e a n d a g r e e m e n t w i t h e x p e r i m e n t f o r C a F 2. H o w e v e r , the theory and the calculation are complex and t h e p h y s i c a l s i g n i f i c a n c e of t h e a p p r o x i m a t i o n s i s not e t e a r . We u s e t h e f o l l o w i n g f o r m u l a f o r t h e B l o c h decay for N identical spins,
B(t) = 2 1 - N ~ T r e x p ( - i ~ / ) S+~exp(i~t)S f,g f g f,g
f
* Supported by an S. R. C. R e s e a r c h Fellowship. 218
g
>,,,< C.3 LIJ Q
I (,3 0
\
,° oo,,so,
~ ,
,
,.
[1,o,o1
,',. ",~
I
,
.,,_,~-a_~
!
: ,
,
,
,
1,1,0]
N IE
nO Z
. . . . . . . 1.0
o--~2o - : ;'o ' o~a~_~. 2.0
3.0
Fig. 1. The full lines a r e the theoretical [Bo(t) + Bl(t)] for the 19F r e s o n a n c e in a single c r y s t a l of calcium fluoride for the t h r e e c r y s t a l o r i e n t a t i o n s [100], [1101 and [111]o The dashed lines a r e Bo(t ). The open c i r c l e s a r e the experimental r e s u l t s of B a r n a a l and Lowe [7]. F o r s t a t i c d i p o l a r i n t e r a c t i o n we u s e V a n V l e c k ' s [1 ] t r u n c a t e d H a m i l t o n i a n ,
~:
~ B o ( S ZsZ- -~S..~) : ~5~o + ~ v i,j ij 3~
w h e r e B / j = ¼~2~(1 - 3 c o s 2 0 i i ) r : . 3 ; r~.~ i s t h e d i s zJ ~j tance between spins with gyromagnetlc ratio a n d Oij i s t h e a n g l e rij m a k e s w i t h t h e l a r g e p o l a r i z i n g f i e l d , g o a n d V c o r r e s p o n d to t h e f i r s t and second terms in ~ respectively. T h e p a r t V of ~ c o m m u t e s w i t h S + a n d s o d o e s not a f f e c t t h e t i m e d e v e l o p m e n t of S+(-t) d i r e c t l y b u t o n l y t h r o u g h t h e t e r m ~ o . We t h e r e -
Volume24A, number4
PHYSICS L E T T E R S
fore use a Dyson type e x p a n s i o n in powers of V, +
+
Sf (-t) = exp(-i~ot)Sf exp(i~ut) + t + i exp(-i~ot) f dt' [V(t'), S;] exp(i~ot) + O
J
-exp(-if~Piot)(fotdt' fOrdr" [g(t'), I V ( t " ) ,
S;]]) x
× exp(i~7(ot) + . . . w h e r e V(t) = exp(i~ot) Vexp(-ig{ot).
,
This leads to
a c o r r e s p o n d i n g s e r i e s for the Bloch decay, B(t) = Bo(t ) + Bl(t ) + B2(t ) + . . . . The r e s u l t for Bo(t) is well known [4, pag. 117]. We have evaluated B 1 (t):
So(t ) = k~j c°S(Bkjt) • Bl(t) = 3 k
Bkj s i n (Bkj O × t
k
o
g¢jJe
°J
~"
J
We have computed Bo(t) and Bl(t) for the fluorine r e s o n a n c e in c a l c i u m fluoride for the t h r e e o r i e n tations for which e x p e r i m e n t a l r e s u l t s a r e a v a i l able [3,6, 7]. The 119F n u c l e i have spin one half and a r e on a s i m p l e cubic lattice with spacing a = 2.725 A. The effect of the c a l c i u m nuclei may be neglected. The n a t u r a l unit of time is a3/72pi which is 30.32 m i c r o s e c for C a F 2. We have c o m puted Bo(t) and the f i r s t t e r m of B l ( t ) for a cube of side four lattice s p a c i n g s with the r e f e r e n c e s p i n at the c e n t r e (i. e. 124 neighbours). The s u m m a t i o n in the second t e r m in B 1 (t) was over a cube with side two lattice spacings (i. e. 26 neighbours) but the p r o d u c t was evaluated for 124 n e i g h b o u r s . Bo(t ) and [Bo(t) + Bl(t) ] a r e c o m p a r e d with the e x p e r i m e n t a l r e s u l t s of B a r n a a l and Lowe [7] in the f i g u r e . Since the zero o r d e r a p p r o x i m a tion, Bo(t), is so good, the next approximation, [Bo(t) + Bl(t)], is a l r e a d y v e r y s a t i s f a c t o r y . The d i f f e r e n c e s between theory and e x p e r i m e n t a r e
13 February 1967
not much l a r g e r than the d i s c r e p a n c i e s between the d i r e c t Bloch decay [3, 7] and the F o u r i e r t r a n s f o r m of the line shape [6]. The e a r l i e r e x p e r i m e n t a l r e s u l t s [3] fit our t h e o r e t i c a l c u r v e s b e t t e r for [110] than the l a t e r ones [7]. It is p e r haps s i g n i f i c a n t that the a l i g n m e n t of the c r y s t a l is most difficult for this o r i e n t a t i o n . The r e v e r s e is t r u e of [111]. It is hardly n e c e s s a r y to include B2(t ) until the e x p e r i m e n t a l r e s u l t s a r e extended to longer t i m e s a n d / o r t h e i r absolute a c c u r a c y is i m proved. The time for computation a v e r a g e d about one minute p e r t i m e point on an Atlas I (we a r e grateful to the Atlas C o m p u t e r L a b o r a t o r y , N. I, R. N. S . , Chilton, for time on t h e i r machine). Even [Bo(t ) + B l(t)] gives the c o r r e c t o r d e r of a t tenuation of the s i g n a l and the higher o r d e r t e r m s , B2(t) e t c . , m u s t give only r e l a t i v e l y s m a l l cont r i b u t i o n s to B(t). O u r method is able to explain the r e l e v a n t exp e r i m e n t a l r e s u l t s . It is not a time s e r i e s expansion and does not d i v e r g e for long t i m e s . T h e r e a r e no disposable p a r a m e t e r s . F o r m o s t p u r p o s e s it is sufficient to take only the f i r s t two t e r m s of the expansion and the r e s u l t is self c o n s i s t e n t to this o r d e r . The expansion is s t r a i g h t f o r w a r d and can be extended as needed by evaluating higher t e r m s . We believe this method to be a powerful one in solving other line shape p r o b l e m s in m a g netic r e s o n a n c e and we s h a l l s h o r t l y give f u r t h e r examples.
References 1. J.H.Van Vleck, Phys.Rev. 74 (1948) 1168. 2. F.Bloch, Phys.Rev. 70 (1946) 460. 3. I.J. Lowe and R. E. Norberg, Phys.Rev. 107 (1957) 46. 4. A.Abragam, The principles of nuclear magnetism (Clarendon Press, Oxford, 1961) N.b. in eq. (47) the argument of cos should have a factor ~. 5. S. Clough and I. R. McDonald, Proc. Phys. Soc. 86 (1966) 833. 6. C.R.Bruee, Phys.Rev.107 (1957) 43. 7. D. Barnaal and I. J. Lowe, Phys. Rev. 148 (1966) 328.
219
THE
LINE
PHYSICS
SHAPE
PROBLEM
IN
LETTERS
13 F e b r u a r y 1967
NUCLEAR
MAGNETIC
RESONANCE
W. A. B. E V A N S * a n d J . G . P O W L E S
The Physics Laboratory. University of Kent at Canterbury, Canterbury, England Received 31 D e c e m b e r 1966
A solution is given of the nuclear magnetic r e s o n a n c e line shape problem for static dipolar interaction. Good a g r e e m e n t is found between the theory and e x p e r i m e n t for the 19F signal in calcium fluoride.
O n e of t h e m o s t i m p o r t a n t u n s o l v e d p r o b l e m s in nuclear magnetic resonance is the calculation of t h e r e s o n a n c e l i n e s h a p e . W e p r e s e n t a s o l u t i o n of t h i s p r o b l e m f o r t h e c a s e of s t a t i c d i r e c t d i p o l a r i n t e r a c t i o n f o r a l a t t i c e of e q u i v a l e n t s p i n s o n e h a l f f o r l a r g e f i e l d s . V a n V l e c k [1] h a s s h o w n h o w to c a l c u l a t e t h e m o m e n t s of t h e a b s o r p t i o n l i n e b u t in p r a c t i c e it i s o n l y f e a s i b l e to c a l c u l a t e t h e s e c o n d a n d f o u r t h m o m e n t s , M 2 a n d M 4. We c o n s i d e r t h e a m p l i t u d e of t h e t r a n s i e n t t r a n s v e r s e m a g n e t i z a t i o n , t h e s o - c a l l e d B l o c h d e c a y [2], B(t), w h i c h i s t h e F o u r i e r t r a n s f o r m of t h e l i n e s h a p e [3]. L o w e a n d N o r b e r g [3] h a v e g i v e n a t h e o r y f o r B(t) f o r t h e s t a t i c d i p o l a r i n t e r a c t i o n of s p i n s o n e half on a simple cubic lattice and have attempted to i n t e r p r e t t h e e x p e r i m e n t a l r e s u l t s f o r t h e 1 9 F r e s o n a n c e in c a l c i u m f l u o r i d e . T h e i r t h e o r y i s a r a t h e r m o r e e l a b o r a t e e x p a n s i o n f o r B(t) t h a n t h e m o m e n t o n e . It h a s b e e n e r i t i c i s e d [ 4 , p a g . 119] b e c a u s e t h e e x p a n s i o n i s a r b i t r a r y . It i s u n s a t i s f a c t o r y b e c a u s e it d i v e r g e s f o r t i m e s not m u c h l o n g e r t h a n t h o s e f o r w h i c h it w a s c o m p u t e d [5] a n d b e c a u s e it i s not e a s y to s e e how t h e t h e o r y c a n b e i m p r o v e d . A g e n e r a l i s e d f o r m of t h i s t h e o r y [5] g i v e s r a t h e r b e t t e r c o n v e r g e n c e a n d a g r e e m e n t w i t h e x p e r i m e n t f o r C a F 2. H o w e v e r , the theory and the calculation are complex and t h e p h y s i c a l s i g n i f i c a n c e of t h e a p p r o x i m a t i o n s i s not e t e a r . We u s e t h e f o l l o w i n g f o r m u l a f o r t h e B l o c h decay for N identical spins,
B(t) = 2 1 - N ~ T r e x p ( - i ~ / ) S+~exp(i~t)S f,g f g f,g
f
* Supported by an S. R. C. R e s e a r c h Fellowship. 218
g
>,,,< C.3 LIJ Q
I (,3 0
\
,° oo,,so,
~ ,
,
,.
[1,o,o1
,',. ",~
I
,
.,,_,~-a_~
!
: ,
,
,
,
1,1,0]
N IE
nO Z
. . . . . . . 1.0
o--~2o - : ;'o ' o~a~_~. 2.0
3.0
Fig. 1. The full lines a r e the theoretical [Bo(t) + Bl(t)] for the 19F r e s o n a n c e in a single c r y s t a l of calcium fluoride for the t h r e e c r y s t a l o r i e n t a t i o n s [100], [1101 and [111]o The dashed lines a r e Bo(t ). The open c i r c l e s a r e the experimental r e s u l t s of B a r n a a l and Lowe [7]. F o r s t a t i c d i p o l a r i n t e r a c t i o n we u s e V a n V l e c k ' s [1 ] t r u n c a t e d H a m i l t o n i a n ,
~:
~ B o ( S ZsZ- -~S..~) : ~5~o + ~ v i,j ij 3~
w h e r e B / j = ¼~2~(1 - 3 c o s 2 0 i i ) r : . 3 ; r~.~ i s t h e d i s zJ ~j tance between spins with gyromagnetlc ratio a n d Oij i s t h e a n g l e rij m a k e s w i t h t h e l a r g e p o l a r i z i n g f i e l d , g o a n d V c o r r e s p o n d to t h e f i r s t and second terms in ~ respectively. T h e p a r t V of ~ c o m m u t e s w i t h S + a n d s o d o e s not a f f e c t t h e t i m e d e v e l o p m e n t of S+(-t) d i r e c t l y b u t o n l y t h r o u g h t h e t e r m ~ o . We t h e r e -
Volume24A, number4
PHYSICS L E T T E R S
fore use a Dyson type e x p a n s i o n in powers of V, +
+
Sf (-t) = exp(-i~ot)Sf exp(i~ut) + t + i exp(-i~ot) f dt' [V(t'), S;] exp(i~ot) + O
J
-exp(-if~Piot)(fotdt' fOrdr" [g(t'), I V ( t " ) ,
S;]]) x
× exp(i~7(ot) + . . . w h e r e V(t) = exp(i~ot) Vexp(-ig{ot).
,
This leads to
a c o r r e s p o n d i n g s e r i e s for the Bloch decay, B(t) = Bo(t ) + Bl(t ) + B2(t ) + . . . . The r e s u l t for Bo(t) is well known [4, pag. 117]. We have evaluated B 1 (t):
So(t ) = k~j c°S(Bkjt) • Bl(t) = 3 k
Bkj s i n (Bkj O × t
k
o
g¢jJe
°J
~"
J
We have computed Bo(t) and Bl(t) for the fluorine r e s o n a n c e in c a l c i u m fluoride for the t h r e e o r i e n tations for which e x p e r i m e n t a l r e s u l t s a r e a v a i l able [3,6, 7]. The 119F n u c l e i have spin one half and a r e on a s i m p l e cubic lattice with spacing a = 2.725 A. The effect of the c a l c i u m nuclei may be neglected. The n a t u r a l unit of time is a3/72pi which is 30.32 m i c r o s e c for C a F 2. We have c o m puted Bo(t) and the f i r s t t e r m of B l ( t ) for a cube of side four lattice s p a c i n g s with the r e f e r e n c e s p i n at the c e n t r e (i. e. 124 neighbours). The s u m m a t i o n in the second t e r m in B 1 (t) was over a cube with side two lattice spacings (i. e. 26 neighbours) but the p r o d u c t was evaluated for 124 n e i g h b o u r s . Bo(t ) and [Bo(t) + Bl(t) ] a r e c o m p a r e d with the e x p e r i m e n t a l r e s u l t s of B a r n a a l and Lowe [7] in the f i g u r e . Since the zero o r d e r a p p r o x i m a tion, Bo(t), is so good, the next approximation, [Bo(t) + Bl(t)], is a l r e a d y v e r y s a t i s f a c t o r y . The d i f f e r e n c e s between theory and e x p e r i m e n t a r e
13 February 1967
not much l a r g e r than the d i s c r e p a n c i e s between the d i r e c t Bloch decay [3, 7] and the F o u r i e r t r a n s f o r m of the line shape [6]. The e a r l i e r e x p e r i m e n t a l r e s u l t s [3] fit our t h e o r e t i c a l c u r v e s b e t t e r for [110] than the l a t e r ones [7]. It is p e r haps s i g n i f i c a n t that the a l i g n m e n t of the c r y s t a l is most difficult for this o r i e n t a t i o n . The r e v e r s e is t r u e of [111]. It is hardly n e c e s s a r y to include B2(t ) until the e x p e r i m e n t a l r e s u l t s a r e extended to longer t i m e s a n d / o r t h e i r absolute a c c u r a c y is i m proved. The time for computation a v e r a g e d about one minute p e r t i m e point on an Atlas I (we a r e grateful to the Atlas C o m p u t e r L a b o r a t o r y , N. I, R. N. S . , Chilton, for time on t h e i r machine). Even [Bo(t ) + B l(t)] gives the c o r r e c t o r d e r of a t tenuation of the s i g n a l and the higher o r d e r t e r m s , B2(t) e t c . , m u s t give only r e l a t i v e l y s m a l l cont r i b u t i o n s to B(t). O u r method is able to explain the r e l e v a n t exp e r i m e n t a l r e s u l t s . It is not a time s e r i e s expansion and does not d i v e r g e for long t i m e s . T h e r e a r e no disposable p a r a m e t e r s . F o r m o s t p u r p o s e s it is sufficient to take only the f i r s t two t e r m s of the expansion and the r e s u l t is self c o n s i s t e n t to this o r d e r . The expansion is s t r a i g h t f o r w a r d and can be extended as needed by evaluating higher t e r m s . We believe this method to be a powerful one in solving other line shape p r o b l e m s in m a g netic r e s o n a n c e and we s h a l l s h o r t l y give f u r t h e r examples.
References 1. J.H.Van Vleck, Phys.Rev. 74 (1948) 1168. 2. F.Bloch, Phys.Rev. 70 (1946) 460. 3. I.J. Lowe and R. E. Norberg, Phys.Rev. 107 (1957) 46. 4. A.Abragam, The principles of nuclear magnetism (Clarendon Press, Oxford, 1961) N.b. in eq. (47) the argument of cos should have a factor ~. 5. S. Clough and I. R. McDonald, Proc. Phys. Soc. 86 (1966) 833. 6. C.R.Bruee, Phys.Rev.107 (1957) 43. 7. D. Barnaal and I. J. Lowe, Phys. Rev. 148 (1966) 328.
219