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NMR resolution of the chemical shift in solids
Volume 27A, number 3
PHYSICS LETTERS
T h e c r i t i c a l c o n c e n t r a t i o n (at which T~ = 0) i s c o = 0.5. F o r t r i a n g u l a r and hexagonal l a t t i c e s the c o v a l u e s c a l c u l a t e d by the s a m e method a r e :
cT = 13 + s ~ ) / 1 8
and %~= (15 - 5 ~ ) / 1 8 .
It i s v e r y i n t e r e s t i n g to know what happens to the s p e c i f i c heat n e a r t h e c r i t i c a l point. F o r s i m p l i c i t y we s h a l l c o n s i d e r the c a s e c << 1. L e t t = t _U +5 a n d x = x ~ sx + 6 O w h e r e 6 and 60 a r e s m a l l . Then we r e c e i v e f r o m eel. (2) C
5 -6 o--~61nl61
=0
w h e r e a = ½~r(~f2+ 1) 3. The a s y m p t o t i c a l solution of this equation i s
T h e s i n g u l a r p a r t of the s p e c i f i c h e a t n e a r the c r i t i c a l point i s given by the e x p r e s s i o n : (for IT ] << exp [-ol/c] w h e r e ~" = (T - T~.)/T~ )
NMR
RESOLUTION
OF
THE
17 June 1968 C ~ {1+f2)22c K 2 I I + ~ l a
and C cc In Ira when ]r I >> e x p [ - ~ / c ] . Thus we s e e that the l o g a r i t h m i c s i n g u l a r i t y in the s p e c i fic heat i s r e p l a c e d by a cusp with v e r t i c a l t a n gent at the c r i t i c a l point. A m o r e thorough d i s c u s s i o n of this and o t h e r t o p i c s will be p u b l i s h e d e l s e w h e r e . T h e author w i s h e s to a c k n o w l e d g e d i s c u s s i o n s with D r - s . V. G. V a k s , A . I . L a r k i n and M. A. Mikulinsky.
Reference 1. G . F . Newell and E. W. Montroll, Rev. Mod. Phys. 25 (1953) 353.
CHEMICAL
SHIFT
IN SOLIDS
P. MANSFIELD and D. WARE * Department of Physics, University of Nottingham, Nottingham, UK Received 7 May 1968
A reply to criticism is given and further details of our previous calculations are presented.
O u r r e c e n t l e t t e r [1], which s u g g e s t s a method of a p p r o x i m a t e c a l c u l a t i o n f o r the r e s p o n s e of a d i p o l a r b r o a d e n e d s o l i d containing two t y p e s of c h e m i c a l l y s h i f t e d spin, h a s been c r i t i c i s e d on a n u m b e r of points by Waugh et al. [2]. We find o u r s e l v e s in d i s a g r e e m e n t with s o m e of t h e i r s t a t e m e n t s . We t r e a t the p o i n t s r a i s e d by t h e m in turn. a) The e x p e r i m e n t to which we r e f e r , i . e . , a n o n - a r t i f i c i a l l y c o n t r i v e d s y s t e m , had not been p e r f o r m e d b e f o r e and to our knowledge s t i l l r e m a i n s to be done. Waugh and H u b e r [3] p e r f o r m e d a s i m u l a t e d c h e m i c a l shift e x p e r i m e n t by moving off r e s o n a n c e , thus v e r i f y i n g e x p e r i m e n t a l l y our e a r l i e r e x p e c t a t i o n [4]. b) We r e j e c t the a s s e r t i o n that our t h e o r y i s c o n ceptually incorrect. A particular approximation * Present address: Physik-Institut der Universitat Ziirich.
m a d e i s invalid, h o w e v e r , thus making the d e t a i l s of our c a l c u l a t i o n i n c o r r e c t . c) It i s t r u e , though s o m e w h a t t r i v i a l , that we w e r e f i r s t to point out the a n a l o g i e s between p u l s e d line n a r r o w i n g and o t h e r methods of line n a r r o w i n g including s p e c i m e n r o t a t i o n [4,5]. M a t h e m a t i c a l l y both the r o t a t i n g s o l i d and m u l t i p l e p u l s e e x p e r i m e n t s give n a r r o w e d l i n e s plus s i d e b a n d s , so to t h i s extent the e x p e r i m e n t s a r e s i m i l a r . We point out that a l l the m u l t i p l e p u l s e e x p e r i m e n t s a r e the n a t u r a l d e v e l o p m e n t of the much e a r l i e r two and t h r e e p u l s e s o l i d echo e x p e r i m e n t s which, t h e m s e l v e s , may be r e g a r d e d m a t h e m a t i c a l l y a s e f f e c t i v e line n a r r o w i n g . Our s t a t e m e n t was not intended a s a c l a i m on p r i o r i t y for line narrowing. A n u m b e r of points have been r a i s e d with o u r a n a l y s i s . To c l a r i f y t h i s , we r e s t a t e s o m e of our work and expand w h e r e n e c e s s a r y . We c o n s i d e r a spin s y s t e m w h e r e the effective 159
Volume27A, number 3
PHYSICS LETTERS
spin H a m i l t o n i a n in the r o t a t i n g f r a m e with no r.f. p r e s e n t is ~ = ~ ( ~ 1 + ~ . ) , ~ 1 is the dipol a r i n t e r a c t i o n and ~ c i s t h e c ~ e m i c a l shift spin m a g n e t i c i n t e r a c t i o n . The m a x i m u m n o r m a l i z e d a m p l i t u d e of the 2nth echo following 2n 90 ° p u l s e s a l t e r n a t e l y p h a s e shifted by 180 ° and of u n i f o r m spacing 2 r following at t i m e 7 an i n i t i a l p r e p a r a tory 90 ° p u l s e is
(Ix } = T r ( 0 ' n IxOn Ix)/Tr (Ix2)
where 0 = {oi9~7} {i9~ , 2~} { i 9 ~ } , ~
,
(1)
= R ~f9~R and
R is the 90 r o t a t i o n opera~or. The b r a c e symbol denotes an exponential o p e r a t o r , i.e. { i ~ z } ~--- exp (i~v). The exponential o p e r a t o r s in 0 may be s e p a r a t e d and the whole t e r m r e o r d e r e d by i n t r o d u c i n g a p p r o p r i a t e c o m m u t a t o r s , i.e.
0 = P1QPc where
P I , e = {ig~l,er} {ig~i,e2r} { i ~ l , c r } and
0=~
l
Ql"
The 0 l involve v a r i o u s t e r m s , i.e. V l = 1 + {-i9~17} {-igg~r}K1 {ig~c2r } { i ~ l r } ×
×
{-i%7}+ {-i%7}K2{-i%=7}
where K1 = [ {i~cr}, { i ~ 2 r } ]
= [{i%7} { i % ~ } ,
and K 2 =
{i%7}].
Expanding in powers
of r
Q = 1 + 2r2[A1,Ac ) + higher o r d e r t e r m s , ¢
17 June 1968
the t e r m s in the e x p r e s s i o n for U in any d i s c u s sion of t i m e r e v e r s a l s y m m e t r y . Expansion of U in powers of r and with N = 2n gives
U : 1 + ½Nr2[A1,Ac] + ½N2r2[A1,Ac] +
(3)
+ higher o r d e r t e r m s . In our l e t t e r , we took U = 1 and m e r e l y c o m mented on the c o r r e c t i o n t e r m s which appear in eq. (1) when eqs. (2) and (3) a r e substituted. In fact t h e r e a r e t e r m s in N2r 4 and N4r 4. In the l i m i t r ~ 0 with the r e a l t i m e 2Nr = t finite, the f i r s t type of c o r r e c t i o n t e r m d i s a p p e a r s , though we did not s p e c i f i c a l l y take this l i m i t , n o r did we i m p l y that the c o r r e c t i o n t e r m s could in g e n e r a l be neglected. We d i s c u s s the c a s e for s h o r t v, and this is the r e a s o n for a p p r o x i m a t i n g p~n lx P~ = Pl(t) = exp (-t/ T) 2~ I x . In an i n t e r e s t i n g p a p e r , Waugh et al. [6] show that the type of e x p e r i m e n t c o n s i d e r e d h e r e can n e v e r e n t i r e l y r e m o v e the dipolar i n t e r a c t i o n , and so completely expose the c h e m i c a l shift. They also show that in the l i m i t of v a n i s h i n g r and when ~ c << ~ 1 , the dipolar i n t e r a c t i o n can be r e d u c e d only to the point of just r e s o l v i n g the c h e m i c a l shift. It i s not c l e a r , however, without c o n s i d e r i n g the detailed f o r m of the c o r r e c t i o n t e r m s in eq. (3), whether periodic beats would be d i s c e r n i b l e . In l e s s s e v e r e c a s e s , i.e. ~ c < < ~ 1 , and where t h e r e is no obvious doublet r e s o l u t i o n in the steady state s p e c t r u m , our own p r e l i m i n a r y e x p e r i m e n t s i n d i c a t e that c h e m i c a l shifts as s m a l l as one fifth of the dipolar line width may well be r e s o l v a b l e . C o n c e r n i n g the evaluation of PcnlxPtcn; the stated n e g l e c t of s m a l l off diagonal t e r m s in Pc ' which a r e in the Z e e m a n r e p r e s e n t a t i o n , i s r e s p o n s i b l e for the d i s p a r i t y between our r e s u l t and the exact c a l c u l a t i o n of Waugh et al. [6].
t
w h e r e A 1 = ~ 1 + ~ 1 a n d A c = ~ c + ~ c " By f u r t h e r i n t r o d u c t i o n of c o m m u t a t o r s , we may set the o p e r a t o r
References where
U=Qn+
n-1
pm.-n(p c)m C ' + CmQ) x m=l
1
~
1 "~
m
× Pc(PlOP c
Cm
=
[Qmpm, p1]
and
C m' = p[ r oc ,V] •
Since eq. (2) is an identity, one m u s t i n c l u d e all
160
1. D. Ware and P. Mansfield, Phys. Letters 25A (1967) 651. 2. J. S, Waugh, L.M. Huber and E. D. Ostroff, Phys. Letters 26A (1968) 211. 3. J.S. Waugh and L. M. Huber, J. Chem. Phys. 47 (1967) lS62. 4. P.Mansfield and D.Ware, Phys. Letters 22 (1966) 133. 5. E.D. Ostroff and J. S. Waugh, Phys. Rev. Letters 16 (1966) 1097. 6. J. S. Waugh, C.H. Wang, L.M. Huber and R. L. Vold, J~, Chem. Phys., to be published.
PHYSICS LETTERS
T h e c r i t i c a l c o n c e n t r a t i o n (at which T~ = 0) i s c o = 0.5. F o r t r i a n g u l a r and hexagonal l a t t i c e s the c o v a l u e s c a l c u l a t e d by the s a m e method a r e :
cT = 13 + s ~ ) / 1 8
and %~= (15 - 5 ~ ) / 1 8 .
It i s v e r y i n t e r e s t i n g to know what happens to the s p e c i f i c heat n e a r t h e c r i t i c a l point. F o r s i m p l i c i t y we s h a l l c o n s i d e r the c a s e c << 1. L e t t = t _U +5 a n d x = x ~ sx + 6 O w h e r e 6 and 60 a r e s m a l l . Then we r e c e i v e f r o m eel. (2) C
5 -6 o--~61nl61
=0
w h e r e a = ½~r(~f2+ 1) 3. The a s y m p t o t i c a l solution of this equation i s
T h e s i n g u l a r p a r t of the s p e c i f i c h e a t n e a r the c r i t i c a l point i s given by the e x p r e s s i o n : (for IT ] << exp [-ol/c] w h e r e ~" = (T - T~.)/T~ )
NMR
RESOLUTION
OF
THE
17 June 1968 C ~ {1+f2)22c K 2 I I + ~ l a
and C cc In Ira when ]r I >> e x p [ - ~ / c ] . Thus we s e e that the l o g a r i t h m i c s i n g u l a r i t y in the s p e c i fic heat i s r e p l a c e d by a cusp with v e r t i c a l t a n gent at the c r i t i c a l point. A m o r e thorough d i s c u s s i o n of this and o t h e r t o p i c s will be p u b l i s h e d e l s e w h e r e . T h e author w i s h e s to a c k n o w l e d g e d i s c u s s i o n s with D r - s . V. G. V a k s , A . I . L a r k i n and M. A. Mikulinsky.
Reference 1. G . F . Newell and E. W. Montroll, Rev. Mod. Phys. 25 (1953) 353.
CHEMICAL
SHIFT
IN SOLIDS
P. MANSFIELD and D. WARE * Department of Physics, University of Nottingham, Nottingham, UK Received 7 May 1968
A reply to criticism is given and further details of our previous calculations are presented.
O u r r e c e n t l e t t e r [1], which s u g g e s t s a method of a p p r o x i m a t e c a l c u l a t i o n f o r the r e s p o n s e of a d i p o l a r b r o a d e n e d s o l i d containing two t y p e s of c h e m i c a l l y s h i f t e d spin, h a s been c r i t i c i s e d on a n u m b e r of points by Waugh et al. [2]. We find o u r s e l v e s in d i s a g r e e m e n t with s o m e of t h e i r s t a t e m e n t s . We t r e a t the p o i n t s r a i s e d by t h e m in turn. a) The e x p e r i m e n t to which we r e f e r , i . e . , a n o n - a r t i f i c i a l l y c o n t r i v e d s y s t e m , had not been p e r f o r m e d b e f o r e and to our knowledge s t i l l r e m a i n s to be done. Waugh and H u b e r [3] p e r f o r m e d a s i m u l a t e d c h e m i c a l shift e x p e r i m e n t by moving off r e s o n a n c e , thus v e r i f y i n g e x p e r i m e n t a l l y our e a r l i e r e x p e c t a t i o n [4]. b) We r e j e c t the a s s e r t i o n that our t h e o r y i s c o n ceptually incorrect. A particular approximation * Present address: Physik-Institut der Universitat Ziirich.
m a d e i s invalid, h o w e v e r , thus making the d e t a i l s of our c a l c u l a t i o n i n c o r r e c t . c) It i s t r u e , though s o m e w h a t t r i v i a l , that we w e r e f i r s t to point out the a n a l o g i e s between p u l s e d line n a r r o w i n g and o t h e r methods of line n a r r o w i n g including s p e c i m e n r o t a t i o n [4,5]. M a t h e m a t i c a l l y both the r o t a t i n g s o l i d and m u l t i p l e p u l s e e x p e r i m e n t s give n a r r o w e d l i n e s plus s i d e b a n d s , so to t h i s extent the e x p e r i m e n t s a r e s i m i l a r . We point out that a l l the m u l t i p l e p u l s e e x p e r i m e n t s a r e the n a t u r a l d e v e l o p m e n t of the much e a r l i e r two and t h r e e p u l s e s o l i d echo e x p e r i m e n t s which, t h e m s e l v e s , may be r e g a r d e d m a t h e m a t i c a l l y a s e f f e c t i v e line n a r r o w i n g . Our s t a t e m e n t was not intended a s a c l a i m on p r i o r i t y for line narrowing. A n u m b e r of points have been r a i s e d with o u r a n a l y s i s . To c l a r i f y t h i s , we r e s t a t e s o m e of our work and expand w h e r e n e c e s s a r y . We c o n s i d e r a spin s y s t e m w h e r e the effective 159
Volume27A, number 3
PHYSICS LETTERS
spin H a m i l t o n i a n in the r o t a t i n g f r a m e with no r.f. p r e s e n t is ~ = ~ ( ~ 1 + ~ . ) , ~ 1 is the dipol a r i n t e r a c t i o n and ~ c i s t h e c ~ e m i c a l shift spin m a g n e t i c i n t e r a c t i o n . The m a x i m u m n o r m a l i z e d a m p l i t u d e of the 2nth echo following 2n 90 ° p u l s e s a l t e r n a t e l y p h a s e shifted by 180 ° and of u n i f o r m spacing 2 r following at t i m e 7 an i n i t i a l p r e p a r a tory 90 ° p u l s e is
(Ix } = T r ( 0 ' n IxOn Ix)/Tr (Ix2)
where 0 = {oi9~7} {i9~ , 2~} { i 9 ~ } , ~
,
(1)
= R ~f9~R and
R is the 90 r o t a t i o n opera~or. The b r a c e symbol denotes an exponential o p e r a t o r , i.e. { i ~ z } ~--- exp (i~v). The exponential o p e r a t o r s in 0 may be s e p a r a t e d and the whole t e r m r e o r d e r e d by i n t r o d u c i n g a p p r o p r i a t e c o m m u t a t o r s , i.e.
0 = P1QPc where
P I , e = {ig~l,er} {ig~i,e2r} { i ~ l , c r } and
0=~
l
Ql"
The 0 l involve v a r i o u s t e r m s , i.e. V l = 1 + {-i9~17} {-igg~r}K1 {ig~c2r } { i ~ l r } ×
×
{-i%7}+ {-i%7}K2{-i%=7}
where K1 = [ {i~cr}, { i ~ 2 r } ]
= [{i%7} { i % ~ } ,
and K 2 =
{i%7}].
Expanding in powers
of r
Q = 1 + 2r2[A1,Ac ) + higher o r d e r t e r m s , ¢
17 June 1968
the t e r m s in the e x p r e s s i o n for U in any d i s c u s sion of t i m e r e v e r s a l s y m m e t r y . Expansion of U in powers of r and with N = 2n gives
U : 1 + ½Nr2[A1,Ac] + ½N2r2[A1,Ac] +
(3)
+ higher o r d e r t e r m s . In our l e t t e r , we took U = 1 and m e r e l y c o m mented on the c o r r e c t i o n t e r m s which appear in eq. (1) when eqs. (2) and (3) a r e substituted. In fact t h e r e a r e t e r m s in N2r 4 and N4r 4. In the l i m i t r ~ 0 with the r e a l t i m e 2Nr = t finite, the f i r s t type of c o r r e c t i o n t e r m d i s a p p e a r s , though we did not s p e c i f i c a l l y take this l i m i t , n o r did we i m p l y that the c o r r e c t i o n t e r m s could in g e n e r a l be neglected. We d i s c u s s the c a s e for s h o r t v, and this is the r e a s o n for a p p r o x i m a t i n g p~n lx P~ = Pl(t) = exp (-t/ T) 2~ I x . In an i n t e r e s t i n g p a p e r , Waugh et al. [6] show that the type of e x p e r i m e n t c o n s i d e r e d h e r e can n e v e r e n t i r e l y r e m o v e the dipolar i n t e r a c t i o n , and so completely expose the c h e m i c a l shift. They also show that in the l i m i t of v a n i s h i n g r and when ~ c << ~ 1 , the dipolar i n t e r a c t i o n can be r e d u c e d only to the point of just r e s o l v i n g the c h e m i c a l shift. It i s not c l e a r , however, without c o n s i d e r i n g the detailed f o r m of the c o r r e c t i o n t e r m s in eq. (3), whether periodic beats would be d i s c e r n i b l e . In l e s s s e v e r e c a s e s , i.e. ~ c < < ~ 1 , and where t h e r e is no obvious doublet r e s o l u t i o n in the steady state s p e c t r u m , our own p r e l i m i n a r y e x p e r i m e n t s i n d i c a t e that c h e m i c a l shifts as s m a l l as one fifth of the dipolar line width may well be r e s o l v a b l e . C o n c e r n i n g the evaluation of PcnlxPtcn; the stated n e g l e c t of s m a l l off diagonal t e r m s in Pc ' which a r e in the Z e e m a n r e p r e s e n t a t i o n , i s r e s p o n s i b l e for the d i s p a r i t y between our r e s u l t and the exact c a l c u l a t i o n of Waugh et al. [6].
t
w h e r e A 1 = ~ 1 + ~ 1 a n d A c = ~ c + ~ c " By f u r t h e r i n t r o d u c t i o n of c o m m u t a t o r s , we may set the o p e r a t o r
References where
U=Qn+
n-1
pm.-n(p c)m C ' + CmQ) x m=l
1
~
1 "~
m
× Pc(PlOP c
Cm
=
[Qmpm, p1]
and
C m' = p[ r oc ,V] •
Since eq. (2) is an identity, one m u s t i n c l u d e all
160
1. D. Ware and P. Mansfield, Phys. Letters 25A (1967) 651. 2. J. S, Waugh, L.M. Huber and E. D. Ostroff, Phys. Letters 26A (1968) 211. 3. J.S. Waugh and L. M. Huber, J. Chem. Phys. 47 (1967) lS62. 4. P.Mansfield and D.Ware, Phys. Letters 22 (1966) 133. 5. E.D. Ostroff and J. S. Waugh, Phys. Rev. Letters 16 (1966) 1097. 6. J. S. Waugh, C.H. Wang, L.M. Huber and R. L. Vold, J~, Chem. Phys., to be published.