- Email: [email protected]
Dynamic polarisation in a three-spin system
Volume 2, number 1
PHYSICS
LETTERS
L a s t a b i l i t e se t r a d u i t p a r une a b s e n c e de f l u c tt~tion sur les diverses grandeurs mesurees; les 1 s o n d e s n~agnetiques d o n n e n t d e s s i g n a u x l i s s e s et :r e p r o d u c t i b l e s : fig. 1 r e p r e s e n t e la s u p e r p o s i t i o n de deux m e s u r e s e f f e c t u e e s d a n s l a couche de plasma. L e s s o n d e s de L a n g m u i r (fig. 2) i n d i q u e n t e l l e s a u s s i u n e a b s e n c e de f l u c t u a t i o n d e s c h a m p s e l e c t r i q u e s . C e l l e - c i sont d ' a l l l e u r s t r ~ s s e n s i b l e s a u x i n s t a b i l i t e s et l a m o i n d r e t u r b u l e n c e se t r a d u i t p a r de v i o l e n t e s f l u c t u a t i o n s . R a p p e l o n s que l e s s i g n a u x de s o n d e s m a g n e t i q u e s et de L a n g m u i r ont une b a n d e p a s s a n t e p e r m e t t a n t de p a s s e r c o r r e c t e m e n t p l u s i e u r s m e g a c y c l e s . Le c o n f i n e m e n t du p l a s m a p e n d a n t 60 ~s a ere confirmd par p l u s i e u r s diagnostics: la spectroscopie, l e s s o n d e s m a g n e t i q u e s , e l e c t r o s t a t i q u e s et t h e r m i q u e s , c e s d e r n i ~ r e s m o n t r e n t que la d e n s i t e globale d ' e n e r g i e p e r d u e r a d i a l e m e n t n ' e s t que le 1/20 de c e l l e p e r d u e p a r l e s e x t r e m i t e s . Ce c o n -
DYNAMIC
POLARISATION
1 August 1962
f i n e m e n t s ' e s t o m p e p r o g r e s s i v e m e n t p a r s u i t e de la d e c r o i s s a n c e d e s c h a m p s m a g n e t i q u e s et p a r i a fuite du p l a s m a p a r l e s e x t r e m i t e s . L e s p l a s m a s o b s e r v e s l o t s de c e s e x p e r i e n c e s a v a i a n t une t e m p e r a t u r e e l e c t r o n i q u e Te = 17 eV, une t e m p e r a t u r e i o n i q u e e s t i m e e a 30 eV, un p r e s sion c i n e t i q u e de 0.1 k g / c m 2 et le /3 de l ' o r d r e de 0.2. I1 faut s i g n a l e r a u s s i que le m o d e l e t h e o r i q u e a ~t~, jusqu'~t p r e s e n t , b i e n c o n f i r m ~ p a r l ' e x p e r i e n c e et que l e s l i m i t e s de s t a b i l i t e t h e o r i q u e et e x p e r i m e n t a l e sont i d e n t i q u e s .
Rdfdrences 1) Bernstein et al., Proc. Roy. Soc. (London) A 244 (1958) 17. 2) P. H. Rebut, InstabilitSs non magnStohydrodynamiques dans les plasmas ~t densit~ de courant ~lev~e, J. Nuclear Energy, ~t paraitre. 3) P.H.Rebut et J. P. Somon, J. Nuclear Energy 3 (1961) 246.
IN A
THREE-SPIN
SYSTEM
*
K. H. H A U S S E R and F. R E I N B O L D Max-Planek-Institut, Heidelberg Received 16 July 1962
We have o b s e r v e d the d y n a m i c p o l a r i s a t i o n due to the O v e r h a u s e r effect in a s y s t e m of t h r e e s p i n s S, I1 and I2 u s i n g double and t r i p l e r e s o n a n c e t e c h n i q u e s . We have m e a s u r e d the s i g n a l e n h a n c e m e n t of the n u c l e a r m a g n e t i c r e s o n a n c e of the C 1 3 - n u c l e i (/2) of the s o l v e n t in n a t u r a l a b u n d a n c e in a solution of the f r e e r a d i c a l 1 , 3 - b i s - d i p h e n y l e n e - p h e n y l - a l l y l in b e n z e n e and p e r d e u t e r o b e n z e n e while s a t u r a t i n g the e l e c t r o n (S) and both the e l e c t r o n s and the p r o t o n s (I1) s i m u l t a n e o u s l y . T h i s s y s t e m i s of p r a c t i c a l i n t e r e s t i n view of the a p p l i c a t i o n of the O v e r h a u s e r effect to s t u d y i n g the h i g h - r e s o l u t i o n n u c l e a r m a g n e t i c r e s o n a n c e of C 13 in n a t u r a l a b u n d a n c e in a wide v a r i e t y of m o l e c u l e s . T h e o r e t i c a l l y , it h a s the a d v a n t a g e that it i s p o s s i b l e to a good a p p r o x i m a t i o n to c a l c u l a t e in f i r s t a p p r o x i m a t i o n the i n t e r a c t i o n b e t w e e n the e l e c t r o n s and the p r o t o n s n e g l e c t i n g the c o m p a r a t i v e l y s m a l l n u m b e r s of the C 1 3 - s p i n s . In a s e c o n d a p p r o x i m a t i o n , the p o l a r i s a t i o n of the C 1 3 - s p i n s due to t h e i r i n t e r a c t i o n with the e l e c t r o n s and with the p r o t o n s can b e c a l c u l a t e d . Our e x p e r i m e n t a l a r r a n g e m e n t u s i n g the h e l i x t e c h n i q u e i n a m a g n e t i c f i e l d of about 3200 G a u s s * This work was supported by the "Deutsche Forschungsgeme ins ehaft".
has been described previously I). Since it was impossible with our set-up using a sample of about 0.5 cln 3 to measure the C 13 nuclear magnetic resonance without enhancement, we have determined the enhancement factors by comparison with the protons at the same resonance frequency. At the concentration of about 1019 spins per c m 3 used in our samples the leakage factor fl for the protons was one within the experimental accuracy. The leakage of the C 13 is negligible if their interaction with the protons is excluded. In the triple resonance experiment, the protons were fully saturated (Sp~ i) by m e a n s of a third radio-frequency field. S o m e oscilloscope traces of the C13 nuclear magnetic resonance are given in fig. 1. The observed splitting due to one proton in benzene (la) and due to one deuteron in perdeuterobenzene (lc) agrees with expectation. The enhancement factor m e a s u r e d with an a c c u r a c y of + 15% is 300 f o r b e n z e n e , 350 f o r p e r d e u t e r o b e n z e n e and 1200 for the t r i p l e r e s o n a n c e . The l a t t e r v a l u e m e a n s an i n c r e a s e d e n h a n c e m e n t of a f a c t o r 2 c o m p a r e d with b e n z e n e in addition to the t r i v i a l f a c t o r 2 c a u s e d by the c o a l e s c e n c e of two l i n e s into one. T h e o r e t i c a l l y , the 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 p r e s e n t in o u r s y s t e m l e a d s to a n u c l e a r p o l a r i s a 53
Volume 2, number 1
PHYSICS
tion I 1 ~ of the p r o t o n s in the d i r e c t i o n of the m a g n e t i c f i e l d 2)
: I1,0 (1 + ½
se fl V~i(i1) ,
LETTERS
1August 1962
S Q
(1)
w h e r e /o i s t h e n u c l e a r p o l a r i s a t i o n at t e m p e r a t u r e >H0 e q u i l i b r i u m , 7S and 711 a r e t h e m a g n e t o g y r i c r a t i o s of t h e s p i n s S and I 1 r d s p e c t i v e l y , s e i s the s a t u r a 12 tion f a c t o r of the s p i n s S and f l i s a l e a k a g e f a c t o r V which t a k e s into a c c o u n t a l l r e l a x a t i o n m e c h a n i s m s of the s p i n s I1 not i n v o l v i n g the s p i n s S. The l e a k a g e f a c t o r w i l l be s e t e q u a l to one in t h e f o l l o w i n g t r e a t m e n t s i n c e the l e a k a g e w a s n e g l i g i b l e in o u r s a m • H0 ples. S u b s t i t u t i n g I2 f o r I1, eq. (1) d e s c r i b e s a l s o the p o l a r i s a t i o n of the s p i n s I2 due to t h e i r i n t e r a c t i o n 5 with t h e S. H o w e v e r , a s e c o n d t e r m h a s to be a d d e d which t a k e s into a c c o u n t the i n t e r a c t i o n of the I2 with t h e / 1 - s p i n s . T h i s i n t e r a c t i o n can b e d e s c r i b e d hy an equation s i m i l a r to (1) but l e a d i n g to a p o s i t i v e ~olarisation since the sign of ~ii is positive. ConseFig. 1. Nuclear magnetic resonance signal of C 13 in quently, the resulting polarisation I2, z is due to a natural abundance enhanced by the Overhausercompetitive effect of the interaction with the S and effect in benzene (a) and perdeutero-benzene (c). with the 11 both pushing 12 z into opposite directions. In the triple resonance experiment in benzene (b), the scale of the ordinate is reduced by a factor 2. This statement holds in general for any combmatmn of the signs of 7S, ~11 and ~I2, provided the interacs i n c e p i n c r e a s e s with i n c r e a s i n g m a g n e t i c m o m e n t tions are either all purely dipolar or all purely scalar. ~I 1 • F i n a l l y , we c o n s i d e r the n u c l e a r p o l a r i s a t i o n of In o r d e r to o b t a i n the m a g n i t u d e of the s e c o n d t e r m one h a s to c o n s i d e r that t h e I 1 a r e not s a t u r a t t h e 1 2 - s p i n s in a t r i p l e r e s o n a n c e e x p e r i m e n t w h e r e ed but h i g h l y p o l a r i s e d . T a k i n g t h i s into a c c o u n t by both t h e p o p u l a t i o n s of the S and of t h e I 1 a r e e q u a l i s e d b y a p p l y i n g two s a t u r a t i n g r a d i o - f r e q u e n c y m u l t i p l y i n g t h e s e c o n d t e r m with t h e p o l a r i s a t i o n f i e l d s . A s s u m i n g full s a t u r a t i o n of the p r o t o n s , f a c t o r f o r the I 1 f r o m eq. (1) we obtain: Sp = 1, we o b t a i n :
I6
T
: r2,0
(1
+ -2Se
Vl--2~s- ½ ~T2 -2Se ~-~ ,1
( = / 2 , 0
= 12,0 ( 1 + ~ - / ~ 2 )
.
(2)
Eq. (2) does not contain Y/I; within the approximation of (2) the magnitude of the magnetic m o m e n t of the spins/i is irrelevant for the polarisation of the spins/2- However, eq. (2) leaves out of account the relative degree of coupling between the 12-spins and the S- and/1-spins respectively. Suitable quantities which measure this coupling are the reciprocal partial,spin-lattice relaxation times o = I/Tli2~ and p = 1/TI I211. Introducing these into (2) we obtain:
(1+~
~ "s Se p "/2) ~+p~i--22- 2 o+p Se 2~ - p ~__) = I2,0 (1 + 4 ~+-p
l p "II~ Se2 ~ +~~ i 2 7s + ~2a P712/
.
(4)
Our e x p e r i m e n t a l r e s u l t s a r e in a g r e e m e n t with the t h e o r y given in eq. (3) and (4). T h e r a t i o of a / p c a l c u l a t e d f r o m o u r m e a s u r e m e n t s by m e a n s of e q s . (3) and (4) i s about one; t h i s v a l u e i s p l a u s i b l e c o n s i d e r i n g the r a t i o s of #e and pp and t h e i r d i s t a n c e s to the C 1 3 - n u c l e i . Eq. (3) can a l s o a c c o u n t f o r the e n h a n c e m e n t of H 1 and F 19 o b s e r v e d by R i c h a r d s and W h i t e 3). We thank D r . N. B l o e m b e r g e n f o r a d i s c u s s i o n of the t h e o r y .
References .
(3)
-ga
The p o l a r i s a t i o n c a l c u l a t e d f r o m eq. (3) i s , in c o n t r a s t to eq. (2), not i n d e p e n d e n t of t h e / 1 - s p i n s
54
1+
1) K.H. Hausser and F.Reinbold, Z. Naturforsch. 16a (1961) 1114. 2) A.Abragam, Phys. Rev. 98 (1955) 1729. 3) R . E . R i c h a r d s and J.W.White, Proc. Chem. Soc. 1962, p. 119.
PHYSICS
LETTERS
L a s t a b i l i t e se t r a d u i t p a r une a b s e n c e de f l u c tt~tion sur les diverses grandeurs mesurees; les 1 s o n d e s n~agnetiques d o n n e n t d e s s i g n a u x l i s s e s et :r e p r o d u c t i b l e s : fig. 1 r e p r e s e n t e la s u p e r p o s i t i o n de deux m e s u r e s e f f e c t u e e s d a n s l a couche de plasma. L e s s o n d e s de L a n g m u i r (fig. 2) i n d i q u e n t e l l e s a u s s i u n e a b s e n c e de f l u c t u a t i o n d e s c h a m p s e l e c t r i q u e s . C e l l e - c i sont d ' a l l l e u r s t r ~ s s e n s i b l e s a u x i n s t a b i l i t e s et l a m o i n d r e t u r b u l e n c e se t r a d u i t p a r de v i o l e n t e s f l u c t u a t i o n s . R a p p e l o n s que l e s s i g n a u x de s o n d e s m a g n e t i q u e s et de L a n g m u i r ont une b a n d e p a s s a n t e p e r m e t t a n t de p a s s e r c o r r e c t e m e n t p l u s i e u r s m e g a c y c l e s . Le c o n f i n e m e n t du p l a s m a p e n d a n t 60 ~s a ere confirmd par p l u s i e u r s diagnostics: la spectroscopie, l e s s o n d e s m a g n e t i q u e s , e l e c t r o s t a t i q u e s et t h e r m i q u e s , c e s d e r n i ~ r e s m o n t r e n t que la d e n s i t e globale d ' e n e r g i e p e r d u e r a d i a l e m e n t n ' e s t que le 1/20 de c e l l e p e r d u e p a r l e s e x t r e m i t e s . Ce c o n -
DYNAMIC
POLARISATION
1 August 1962
f i n e m e n t s ' e s t o m p e p r o g r e s s i v e m e n t p a r s u i t e de la d e c r o i s s a n c e d e s c h a m p s m a g n e t i q u e s et p a r i a fuite du p l a s m a p a r l e s e x t r e m i t e s . L e s p l a s m a s o b s e r v e s l o t s de c e s e x p e r i e n c e s a v a i a n t une t e m p e r a t u r e e l e c t r o n i q u e Te = 17 eV, une t e m p e r a t u r e i o n i q u e e s t i m e e a 30 eV, un p r e s sion c i n e t i q u e de 0.1 k g / c m 2 et le /3 de l ' o r d r e de 0.2. I1 faut s i g n a l e r a u s s i que le m o d e l e t h e o r i q u e a ~t~, jusqu'~t p r e s e n t , b i e n c o n f i r m ~ p a r l ' e x p e r i e n c e et que l e s l i m i t e s de s t a b i l i t e t h e o r i q u e et e x p e r i m e n t a l e sont i d e n t i q u e s .
Rdfdrences 1) Bernstein et al., Proc. Roy. Soc. (London) A 244 (1958) 17. 2) P. H. Rebut, InstabilitSs non magnStohydrodynamiques dans les plasmas ~t densit~ de courant ~lev~e, J. Nuclear Energy, ~t paraitre. 3) P.H.Rebut et J. P. Somon, J. Nuclear Energy 3 (1961) 246.
IN A
THREE-SPIN
SYSTEM
*
K. H. H A U S S E R and F. R E I N B O L D Max-Planek-Institut, Heidelberg Received 16 July 1962
We have o b s e r v e d the d y n a m i c p o l a r i s a t i o n due to the O v e r h a u s e r effect in a s y s t e m of t h r e e s p i n s S, I1 and I2 u s i n g double and t r i p l e r e s o n a n c e t e c h n i q u e s . We have m e a s u r e d the s i g n a l e n h a n c e m e n t of the n u c l e a r m a g n e t i c r e s o n a n c e of the C 1 3 - n u c l e i (/2) of the s o l v e n t in n a t u r a l a b u n d a n c e in a solution of the f r e e r a d i c a l 1 , 3 - b i s - d i p h e n y l e n e - p h e n y l - a l l y l in b e n z e n e and p e r d e u t e r o b e n z e n e while s a t u r a t i n g the e l e c t r o n (S) and both the e l e c t r o n s and the p r o t o n s (I1) s i m u l t a n e o u s l y . T h i s s y s t e m i s of p r a c t i c a l i n t e r e s t i n view of the a p p l i c a t i o n of the O v e r h a u s e r effect to s t u d y i n g the h i g h - r e s o l u t i o n n u c l e a r m a g n e t i c r e s o n a n c e of C 13 in n a t u r a l a b u n d a n c e in a wide v a r i e t y of m o l e c u l e s . T h e o r e t i c a l l y , it h a s the a d v a n t a g e that it i s p o s s i b l e to a good a p p r o x i m a t i o n to c a l c u l a t e in f i r s t a p p r o x i m a t i o n the i n t e r a c t i o n b e t w e e n the e l e c t r o n s and the p r o t o n s n e g l e c t i n g the c o m p a r a t i v e l y s m a l l n u m b e r s of the C 1 3 - s p i n s . In a s e c o n d a p p r o x i m a t i o n , the p o l a r i s a t i o n of the C 1 3 - s p i n s due to t h e i r i n t e r a c t i o n with the e l e c t r o n s and with the p r o t o n s can b e c a l c u l a t e d . Our e x p e r i m e n t a l a r r a n g e m e n t u s i n g the h e l i x t e c h n i q u e i n a m a g n e t i c f i e l d of about 3200 G a u s s * This work was supported by the "Deutsche Forschungsgeme ins ehaft".
has been described previously I). Since it was impossible with our set-up using a sample of about 0.5 cln 3 to measure the C 13 nuclear magnetic resonance without enhancement, we have determined the enhancement factors by comparison with the protons at the same resonance frequency. At the concentration of about 1019 spins per c m 3 used in our samples the leakage factor fl for the protons was one within the experimental accuracy. The leakage of the C 13 is negligible if their interaction with the protons is excluded. In the triple resonance experiment, the protons were fully saturated (Sp~ i) by m e a n s of a third radio-frequency field. S o m e oscilloscope traces of the C13 nuclear magnetic resonance are given in fig. 1. The observed splitting due to one proton in benzene (la) and due to one deuteron in perdeuterobenzene (lc) agrees with expectation. The enhancement factor m e a s u r e d with an a c c u r a c y of + 15% is 300 f o r b e n z e n e , 350 f o r p e r d e u t e r o b e n z e n e and 1200 for the t r i p l e r e s o n a n c e . The l a t t e r v a l u e m e a n s an i n c r e a s e d e n h a n c e m e n t of a f a c t o r 2 c o m p a r e d with b e n z e n e in addition to the t r i v i a l f a c t o r 2 c a u s e d by the c o a l e s c e n c e of two l i n e s into one. T h e o r e t i c a l l y , the 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 p r e s e n t in o u r s y s t e m l e a d s to a n u c l e a r p o l a r i s a 53
Volume 2, number 1
PHYSICS
tion I 1 ~ of the p r o t o n s in the d i r e c t i o n of the m a g n e t i c f i e l d 2)
: I1,0 (1 + ½
se fl V~i(i1) ,
LETTERS
1August 1962
S Q
(1)
w h e r e /o i s t h e n u c l e a r p o l a r i s a t i o n at t e m p e r a t u r e >H0 e q u i l i b r i u m , 7S and 711 a r e t h e m a g n e t o g y r i c r a t i o s of t h e s p i n s S and I 1 r d s p e c t i v e l y , s e i s the s a t u r a 12 tion f a c t o r of the s p i n s S and f l i s a l e a k a g e f a c t o r V which t a k e s into a c c o u n t a l l r e l a x a t i o n m e c h a n i s m s of the s p i n s I1 not i n v o l v i n g the s p i n s S. The l e a k a g e f a c t o r w i l l be s e t e q u a l to one in t h e f o l l o w i n g t r e a t m e n t s i n c e the l e a k a g e w a s n e g l i g i b l e in o u r s a m • H0 ples. S u b s t i t u t i n g I2 f o r I1, eq. (1) d e s c r i b e s a l s o the p o l a r i s a t i o n of the s p i n s I2 due to t h e i r i n t e r a c t i o n 5 with t h e S. H o w e v e r , a s e c o n d t e r m h a s to be a d d e d which t a k e s into a c c o u n t the i n t e r a c t i o n of the I2 with t h e / 1 - s p i n s . T h i s i n t e r a c t i o n can b e d e s c r i b e d hy an equation s i m i l a r to (1) but l e a d i n g to a p o s i t i v e ~olarisation since the sign of ~ii is positive. ConseFig. 1. Nuclear magnetic resonance signal of C 13 in quently, the resulting polarisation I2, z is due to a natural abundance enhanced by the Overhausercompetitive effect of the interaction with the S and effect in benzene (a) and perdeutero-benzene (c). with the 11 both pushing 12 z into opposite directions. In the triple resonance experiment in benzene (b), the scale of the ordinate is reduced by a factor 2. This statement holds in general for any combmatmn of the signs of 7S, ~11 and ~I2, provided the interacs i n c e p i n c r e a s e s with i n c r e a s i n g m a g n e t i c m o m e n t tions are either all purely dipolar or all purely scalar. ~I 1 • F i n a l l y , we c o n s i d e r the n u c l e a r p o l a r i s a t i o n of In o r d e r to o b t a i n the m a g n i t u d e of the s e c o n d t e r m one h a s to c o n s i d e r that t h e I 1 a r e not s a t u r a t t h e 1 2 - s p i n s in a t r i p l e r e s o n a n c e e x p e r i m e n t w h e r e ed but h i g h l y p o l a r i s e d . T a k i n g t h i s into a c c o u n t by both t h e p o p u l a t i o n s of the S and of t h e I 1 a r e e q u a l i s e d b y a p p l y i n g two s a t u r a t i n g r a d i o - f r e q u e n c y m u l t i p l y i n g t h e s e c o n d t e r m with t h e p o l a r i s a t i o n f i e l d s . A s s u m i n g full s a t u r a t i o n of the p r o t o n s , f a c t o r f o r the I 1 f r o m eq. (1) we obtain: Sp = 1, we o b t a i n :
I6
T
: r2,0
(1
+ -2Se
Vl--2~s- ½ ~T2 -2Se ~-~ ,1
( = / 2 , 0
= 12,0 ( 1 + ~ - / ~ 2 )
.
(2)
Eq. (2) does not contain Y/I; within the approximation of (2) the magnitude of the magnetic m o m e n t of the spins/i is irrelevant for the polarisation of the spins/2- However, eq. (2) leaves out of account the relative degree of coupling between the 12-spins and the S- and/1-spins respectively. Suitable quantities which measure this coupling are the reciprocal partial,spin-lattice relaxation times o = I/Tli2~ and p = 1/TI I211. Introducing these into (2) we obtain:
(1+~
~ "s Se p "/2) ~+p~i--22- 2 o+p Se 2~ - p ~__) = I2,0 (1 + 4 ~+-p
l p "II~ Se2 ~ +~~ i 2 7s + ~2a P712/
.
(4)
Our e x p e r i m e n t a l r e s u l t s a r e in a g r e e m e n t with the t h e o r y given in eq. (3) and (4). T h e r a t i o of a / p c a l c u l a t e d f r o m o u r m e a s u r e m e n t s by m e a n s of e q s . (3) and (4) i s about one; t h i s v a l u e i s p l a u s i b l e c o n s i d e r i n g the r a t i o s of #e and pp and t h e i r d i s t a n c e s to the C 1 3 - n u c l e i . Eq. (3) can a l s o a c c o u n t f o r the e n h a n c e m e n t of H 1 and F 19 o b s e r v e d by R i c h a r d s and W h i t e 3). We thank D r . N. B l o e m b e r g e n f o r a d i s c u s s i o n of the t h e o r y .
References .
(3)
-ga
The p o l a r i s a t i o n c a l c u l a t e d f r o m eq. (3) i s , in c o n t r a s t to eq. (2), not i n d e p e n d e n t of t h e / 1 - s p i n s
54
1+
1) K.H. Hausser and F.Reinbold, Z. Naturforsch. 16a (1961) 1114. 2) A.Abragam, Phys. Rev. 98 (1955) 1729. 3) R . E . R i c h a r d s and J.W.White, Proc. Chem. Soc. 1962, p. 119.