ANL-6069 Phys ics and Mathematics (TID-4500, 15th Ed.) AEG R e s e a r c h and Development Report

ARGONNE NATIONAL LABORATORY P . O. Box 299

Lemont, Illinois

THE OVERHAUSER E F F E C T IN PARAMAGNETIC SYSTEMS

by

William A. B a r k e r

Solid State Science Division

November, 1959

Operated by The Universi ty of Chicago under

Contract W-Sl-109-eng~38

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document.

TABLE OF CONTENTS

Page

INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1. Nuclear Magnetization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2. Elec t ronic Magnet izat ion. . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3. Nuclear Polar iza t ion by the Overhauser Effect . . . . . . . . . . . . 9

4. A Therinodynamic Trea tment of the Overhauser Effect . . . . . . . 13

5. Resul ts of General ized Trea tmen t s of the Overhauser Effect . . . 15

6. Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7. Metal -Ammonia Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 25

8. Liquids Containing Pa ramagne t i c Impur i t ies . . . . . . . . . . . . . . 26

9. F r e e Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

10. Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

11. Color Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

THE OVERHAUSER E F F E C T IN PARAMAGNETIC SYSTEMS

INTRODUCTION

The f i rs t half of this r epor t , in Sections 1-5 inclusive, is devoted to a d iscuss ion of the theory of the Overhauser nuclear polar izat ion effect, and pa r t i cu la r attention is paid to the expected sign of the nuclear magne t i ­zation. A negative nuclear polar izat ion in which the higher nuclear Zeeman levels a r e more populated than the lower energy levels could be used as a low-noise quantum mechanical amplif ier of radiofrequency s ignals . In the second half of the repor t , in Sections 6-11 inclusive, is considered var ious sys t ems which have or might be in-^'estigated exper imenta l ly by s imultaneous saturat ion of the e lect ron or hole spin resonance and observat ion of an appropr ia te nuclear magnet ic resonance frequency. There a r e severa l s y s ­t e m s -which should p resen t very in teres t ing r e s u l t s : Be , donors in Ge, accep to rs in Ge and Si, Ag°Ag " in KCl , F^ in L i F .

This repor t does not review all the possible paramagnet ic s y s t e m s . Fo r example, the re is no d iscuss ion of paramagnet ic resonance in c rys t a l s containing ions with an unfilled d or f shel l . These sys tems a r e of cu r r en t in te res t in connection with MASER work and the re is a definite possibi l i ty that sa turat ion of the e lec t ron spin resonance will r e su l t in an invers ion of the nuc lear population dis t r ibut ion, which could then be detected by induced emiss ion of a nuclear resonance frequency. Such a case has recent ly been repor ted and involves the negative polar izat ion of Al nuclei in ruby. This exper iment can be understood in t e r m s of the theory of dynamic polar iza t ion .

There a r e severa l impor tant double resonance techniques which have not been considered m this r e p o r t . These a r e dynamic polar iza t ion, ENDOR, and optical pumping. These a re all sufficiently closely re la ted to the Overhauser effect that a d iscuss ion of the i r re la t ive m e r i t s would be mos t ins t ruc t ive .

This r epo r t makes re fe rence to inost of the theore t ica l pape r s on the Overhauser effect and to all of the available exper imenta l p a p e r s . Some of the m o r e impor tant pape r s on other double resonance methods a r e l is ted below.

Dynamic Polar iza t ion

Ao Abragam and Wo P r o c t o r . Comptes Rendus 246, 2256 (1958). Uebersfeld, Motchane, and Erb , J . Phys . Rad. r9 , 843 (1958). J. A. Cowen, W. A. Schafer. and R. D. Spence, Phys . Rev. L e t t e r s 3,

27 (July 1959)

ENDOR

J. H. Bu rges s and R. E. Norberg , Phys . Rev. 100, 752 (1955) G. F e h e r , Phys . Rev. 1 ^ , 834 (1956)

Optical Pumping

B r o s s e l and Kas t le r , Phys . Rev. 79, 196, 225 (1950) B r o s s e l and Bi t te r , Phys . Rev. 86, 309 (1952)

Pa ramagne t i c Resonance in Trans i t ion Group Ion Impur i t ies

B. Bleaney and K. W, H. Stevens, Rep. P rog . Phys . 16_, 108 (1953) K. D. Bowers and J. Owen, Rep. P r o g . P h y s . 1_8, 304 (1955) D. M. S. Bagguley and J. Owen, Rep. P r o g . Phys . 20, 304 (1957)

SECTION 1

Nuclear Magnetization

Consider a sys tem of N ( ^ ) identical noninteract ing nuclei . Each nucleus has a total angular moinentum Ih and a magnet ic moment Mn proport ional to the angular momentum such that

? n = 7 n M , (1.1)

where 7 is the propor t ional i ty factor . In a constant ex terna l magnet ic field. Ho, each nuclear magnet has an or ientat ional potential energy:

^ m = "^n°So = - '^n*Homi , (1.2)

where -I<mT<I<, If Eg denotes the energy of a nucleus in the absence of a magnet ic field, the poss ible energy leve l s ,

E ^ j = Eo -7j^iiHomj (1.3)

of a nucleus in a magnet ic field a r e r ep re sen t ed in F ig . (la) for 7jj^^0, I = y and in F ig . (lb) for Ij^K 0, I = |-. Equation (1.3) applies to nuclei whose gyromagnet ic ra t io , y , is e i ther posi t ive or negat ive . As is indicated in F ig . 1, if 7j^>0, the ground state has the l a rges t pps_itive_ value of m, and if 7n<lO the ground state has the l a rge s t negative value of m . In t h e r m a l equi l ibr ium at t e m p e r a t u r e T the ra t io of the population of nuclei in differ­ent Zeeman levels m, m ' is given by the Boltzmann factor

N(m') " ^""P kT " ^""P kT ° ^ ' ^

m T

m r 1 2

E, rrn m .

yfiiiQ

1 "2

4 7|hHo

1_ 2

2

3 + 2

F i g . l a

7 , > 0 , 1 = 1

F i g . l b

7 n < 0 ' I

T h u s

N(3 /2 ) ^ 7n^Ho

N(1/2J " ^^'^ kT exp 7n lhHo

kT , 7 > 0 (1.4a)

and

N ( - l / 2 ) f - 7 n M o

k T

iTnlhHc

kT -,7< 0 (1.4b)

It is c l e a r tha t Eq . (1.4) a u t o m a t i c a l l y t a k e s in to accoun t n e g a t i v e a s we l l a s p o s i t i v e 7 n ' s , so t h a t the m o s t popu la t ed s t a t e i s a lways the g r o u n d s t a t e . The to t a l m a g n e t i z a t i o n MQ in the d i r e c t i o n of HQ i s s i m p l y

( ) ^ T^o = ^ n ^ I N ( m i ) m i . (1.5)

- I which , by the u s e of E q s . (1.3) and (1.4) , m a y be w r i t t e n

M i ^ ) = N H 7 t i { ^ c t n h (2I + l ) a

y c t n h ( f )} - (1.6)

6

w h e r e I

N(n) = X N(mj) ^ I

i s the to t a l n u m b e r of n u c l e i and

a :. 7 n ^ H o / k T . (1.7)

T h i s n u c l e a r m a g n e t i z a t i o n i s a l w a y s p o s i t i v e i r r e s p e c t i v e of the s ign of 7. F o r s m a l l v a l u e s of a we u s e the e x p a n s i o n

c tnhx ^ - + — X 3

to obta in the a p p r o x i m a t e v a l u e :

M(") . N(">7„t» {^1(1. l).x} = N(° W H , ^^^^^^ , j3)

The n u c l e a r p o l a r i z a t i o n , P Q , and s u s c e p t i b i l i t y , Xo ^^^ r e l a t e d to MQ by the fol lowing de f in i t ions :

pj'^) = Mj^ ) /N^^) lh7n '"^ 7nftHo(I+ l ) / 3 k T (1.9)

for 7n^Ho < < kT

xi"^^ = M ( ^ ) / H O ---^ N(^)7n^^I ( I + l ) / 3 k T . (1.10)

In t h i s s e c t i o n m a t e r i a l h a s b e e n p r e s e n t e d , in a s o m e w h a t amp l i f i ed f o r m , of a v e r y wel l known sub j ec t a s found, for e x a m p l e , in P a k e ' s r e v i e w of n u c l e a r m a g n e t i c r e s o n a n c e . ( 1 )

SECTION 2

E l e c t r o n i c M a g n e t i z a t i o n

In p r i n c i p l e e l e c t r o n p a r a m a g n e t i c r e s o n a n c e m a y t ake p l a c e in a s y s t e m in which t h e r e is an e l e c t r o n whose sp in i s u n p a i r e d . T h e r e a r e a n u m b e r of such s y s t e m s which we wil l t r e a t in de t a i l in s u b s e q u e n t s e c t i o n s of t h i s r e p o r t .

C o n s i d e r a s y s t e i n of N ' ^ / e l e c t r o n s , e a c h with sp in sh and m a g n e t i c m o m e n t ;ig = 7gSh. In a c o n s t a n t m a g n e t i c f ie ld HQ e a c h e l e c t r o n i c m a g n e t h a s an o r i e n t a t i o n a l po t en t i a l e n e r g y

7

V. m . = - M . Hn •7pAHom, (2.1)

w h e r e nig = t y . In F i g . 2 the e l e c t r o n Z e e m a n l e v e l s a r e i n d i c a t e d . Note tha t , s i n c e 7g<0-, the nig = - ^ s t a t e i s t he g r o u n d s t a t e , c o r r e s p o n d i n g to an e l e c t r o n i c m a g n e t o r i e n t e d p a r a l l e l to HQ. L e t No(-) denote the e q u i l i b ­r i u m n u m b e r of e l e c t r o n s with sp in p a r a l l e l ( m a g n e t i c m o m e n t a n t i -p a r a l l e l ) to the m a g n e t i c f ie ld and No(+) the e q u i l i b r i u m n u m b e r of oppos i t e l y o r i e n t e d e l e c t r o n s :

Nn ;-) + No(+) = N(e) (2.2)

The exp l i c i t f o r m for the e q u i l i b r i u m m a g n e t i z a t i o n ,

M, ;e) _ 4li|7J[No(-) - No(+)] . (2.3)

depends upon w h e t h e r the e l e c t r o n s a r e t r e a t e d a s a d e g e n e r a t e F e r m i g a s , a s in a m e t a l , or c l a s s i c a l l y , u s ing B o l t z m a n n s t a t i s t i c s , a s in a non­conduc t ing p a r a m a g n e t i c .

Ef, + !7e IliHo

' m .

En -l7e|Mo

m .

belMo

2

1 2

F i g . 2

Suppose now tha t we p l a c e the s y s t e m in a m a g n e t i c f ie ld wh ich h a s a s t a t i c and a t i m e - d e p e n d e n t c o m p o n e n t at r i gh t ang le s to one a n o t h e r :

H = Hok + 2Hi cos cot i (2.4)

B l o c h ' s ( 2 ) w e l l - k n o w n p h e n o m e n o l o g i c a l f o r m u l a t i o n of t h i s p r o b l e m for n u c l e a r m a g n e t i c r e s o n a n c e is equa l ly app l i cab l e for the c a s e of e l e c t r o n p a r a m a g n e t i c r e s o n a n c e :

(e) MfK M(«)_. M^^^-M^) - ^ ^ - 7 e ( M ( e ) x H ) + ^ i + - ^ J + k = 0 . (2.5)

T h e s t e a d y - s t a t e ( s l o w p a s s a g e ) s o l u t i o n of t h e d i f f e r e n t i a l E q . (2 .5 ) f o r

t h e c o m p o n e n t s of t h e e l e c t r o n i c m a g n e t i z a t i o n M ' ® ) i s

Mt^)=iX^kTl"^ Z^O '^QJ-Z

(e) 2 H i c o s a3t(cDo-a))T2 + ZHj s i n cx>t

_1 + (a3o-a3)'Tp' +7iHi TS^^TF^,

2 H i C o s CDt - 2Hi s i n (Dt (cDo-a))T2

1 + (a3o»CD)2Ti^^^ + 7 | H f T S ^ ^ T 2 ^ ^ _

M (e) „ y ( e ) ~ A i 'Hn

1 + (U)0-(D)\1^^' / , 2„ (e )2 2^ . ,2„ (e )„ (e )

_1 + (cOo-CD) T^ ^ + 7 e H i T } ^T^ ^

(2.6)

w h e r e Xo = Mg / H Q , CDQ = | 7 g | H o , a n d T j ' a n d T2 a r e t h e e l e c t r o n s p i n

l a t t i c e a n d e l e c t r o n s p i n - s p i n r e l a x a t i o n t i m e s , r e s p e c t i v e l y .

.(e) I t i s c l e a r t h a t a s u f f i c i e n t l y l a r g e v a l u e of Hj w i l l r e d u c e M ^ t o

z e r o . T h e r e a r e s e v e r a l " s a t u r a t i o n p a r a m e t e r s " u s e d in t h e l i t e r a t u r e , w h i c h a r e d e f i n e d a s f o l l o w s :

s i = = 1 -M (e)

M^ (e)

0 < s i < l

_ 2 T T 2 m ( e ) , p ( e ) S2 = 7 e-"i-^i ^z

S3 = i + 7 M T ^ M ' ^ ] " ' = [i + s2r^

(2.7)

T h e s a t u r a t i o n p a r a m e t e r u s e d i n d i s c u s s i o n s of t h e O v e r h a u s e r n u c l e a r p o l a r i z a t i o n e f f e c t i s S j , w h i c h w e w i l l h e n c e f o r t h d e n o t e b y s .

I t i s p o s s i b l e t o o b t a i n r e l a t i v e l y s i m p l e e x p r e s s i o n s f o r Hj a n d f o r t h e p o w e r W a b s o r b e d b y t h e s u b s t a n c e i n t e r m s of s , Xd> T p ' , T ^ ^ ' a n d t h e s t a t i c f i e l d HQ- F r o m E q s . (2 .6 ) a n d ( 2 . 7 ) ,

a n d

H, -

l7el^FM^ s [ l + (C3o-cJD)'Ti^^']

1 - s

2 -1 /2

(2.8)

03 r ^ ^ A dM^^) W = - ~ H „ — 7 ^ - dt

27T I ^ d t

s.-oMr sa3xJ ^Ho , (e)

7 e l T i ^ | 7 e l T i (e)

(2.9)

9

At r e s o n a n c e , 0) = OJQ = | 7 g | H o , and t h e s e e x p r e s s i o n s r e d u c e to

H,

hel^M [l^ (2.8a)

a n d

W = sxi'^^Ho/T^^ , (2.9a)

as shown by B r o v e t t o and Cini.v-'j The s i m p l e r e s u l t e x p r e s s e d by Eq . (2.9) m a y s e e m s o m e w h a t s t r a n g e at f i r s t s ight b e c a u s e it s a y s t h a t a p a r a ­m a g n e t i c s u b s t a n c e wil l a b s o r b m o r e e n e r g y p e r uni t t i m e for v a l u e s of <X) g r e a t e r t h a n the r e s o n a n t f r e q u e n c y CDQ. H o w e v e r , t h i s is for a c o n s t a n t va lue of s a t u r a t i o n s, which is r e l a t e d to CO t h r o u g h Eq . (2.8) a s fo l lows :

Hf7|T(«)T^) / V / V ( 2 . 1 0 ) (e)T.(e) 1 + (LOO-0)2T2 + HJllrr'Tl

a t t a i n s i t s m a x i m u m va lue a t r e s o n a n c e .

SECTION 3

N u c l e a r P o l a r i z a t i o n by the O v e r h a u s e r Effect

Suppose now t h a t our N ' " - / n u c l e a r sp in s and N^®/ e l e c t r o n sp in s a r e p l a c e d s i m u l t a n e o u s l y in a m a g n e t i c f ie ld H = Hok + 2Hi cos tot i and tha t they i n t e r a c t wi th one a n o t h e r v i a the con t ac t p a r t of the hype r f i ne i n t e r a c t i o n

^ M 8 r r / 3 ) 7 e 7 n T f i ' " ^ - S"6(7) . (3.1)

r i s i l l u s t r a t e d in F i g . 3. L e t us c o n s i d e r the l i m i t i n g c a s e of c o m p l e t e s a t u r a t i o n of the e l e c t r o n sp in r e s o n a n c e , i . e . , s = 1. The e n e r g y l e v e l s of the e l e c t r o n sp in and the n u c l e a r sp in a r e g iven to a good a p p r o x i m a t i o n by

E ( m s , m i ) = - YghHoiTis - 7nhHomj + a m g m j , (3.2)

p r o v i d i n g |7g |hHo>>a. In F i g . 4 the e n e r g y l e v e l s a r e s k e t c h e d (not to s c a l e ) for I = -2> 7n.^^ ^-"-^ 7 n ^ 0 . In t h i s s k e t c h we a l s o n e g l e c t a in c o m ­p a r i s o n with | 7 n | h H o . F i g u r e s (4a) and (4b) differ in one i m p o r t a n t r e ­s p e c t . Since 7 n < 0 in (4b), the s t a t e s wi th m j = - f a r e l o w e r in e n e r g y t h a n the s t a t e s wi th m j = •^. In t h e r m a l e q u i l i b r i u m a t t e m p e r a t u r e T

10

n

o r i g i n

F i g . 3

m e m i m e m i

i

7 n RHo

( 7e - 7 n )RHo

7 n RHo

F i g . 4a

7 n > 0

1 2

1 2

1 2

1 2

1 Z

1 2

1 "2"

1 2

Ni N , I m ^ : = ^ = e x p —

l7nlliHo

' -

' J

1

7 n I i H o

( 7e - 7 n )hHo

}

. 7 n l i H o

r

1 2

1 2

1 2

2

F i g . 4b

7 n < 0

1 2

1 2

1 2

1 " 2

kT (3.3)

a n d

i7elhHo Nt Nz ijrel"-no N^ = N ; = ^ ^ P k T

(3.4)

N O W suppose tha t the t r a n s i t i o n s 1-s a t u r a t e d so that

-3 and 2-M-4 a r e s i m u l t a n e o u s l y

N1/N3 = N2/N4 = 1 , (3.5)

11

i . e . , s = 1. The of f -d iagona l e l e m e n t s of ^ , g iven in Eq . (3.1) , induce r e l a x a t i o n p r o c e s s e s in which the to ta l sp in a n g u l a r m o m e n t u m is con­s e r v e d , i . e . ,

A m s + A m i = 0 . (3.6)

T h e s e a r e t r a n s i t i o n s of the type l-^->-4 for 7 n > 0 (Eig. 4a) and of type 2-*-*-3 for 7 n < 0 (F ig . 4b) . In the s t e a d y - s t a t e t h e s e l e v e l s wi l l have a popu la t ion r a t i o g iven by the a p p r o p r i a t e B o l t z m a n n f a c t o r :

Ni ( l 7 e i + l7nl)tiHo ^ - ^ = exp ^ for 7 n > 0 k T

and

N 2 ^ ( | 7 e l - l 7 n l ) M o exp for 7 n < 0

N3 ^ kT

The r a t i o of n u c l e i wi th sp in up to t h o s e wi th sp in down is

(3.7)

N(+

and

N(-

N(+ N(-

Us ing E q s .

N(+

and

w-

N(+ N(-

Ni + N3 N2 + N4

N2 + N4

for 7 n > 0 (3.8)

Ni + N3

(3.5) and (3.7),

for 7 n < 0 (3.9)

2N 1 _ 2N4 exp

( l 7 e l + l7nl)tiHo

k f for 7 n > 0 (3.10)

2N2 viJ-e 2 N : = ^ " P

( b e I -l7nl)hHo kT

for 7 n < 0 (3.11)

The s t e a d y - s t a t e n u c l e a r m a g n e t i z a t i o n , 'hA^^I, is g iven by an equa t ion of e x a c t l y the s a m e f o r m as Eq . (1.6) , bu t wi th a v a s t l y d i f fe ren t va lue for a.

M(n) . N(n )7^h { ^ c tnh [ ^ a ) - \ c t n h ( | ) } .

w h e r e

- = ( i 7 e l + 7 n ) M o / k T > 0 .

(3.12)

(3.13)

12

Thus the sign of the Overhauser nuclear magnetizat ion depends ent i rely on the sign of 7n. The nuclear polar izat ion may be defined ei ther as

P = M H / N ^ ^ ) Ih |7nl

o r P = M(n)/Nin) ihY^

The la t te r definition guarantees a positive nuclear polar izat ion i r r e s p e c ­tive of the sign of M. We shall use the fo rmer definition so that P and M always have the same sign. Suppose the sys tem is now subjected to a second t ime-vary ing magnetic field of frequency CD2 close to (X)^ = | 7J^ |HO

such that the total external magnetic field is given by

H = Hok + 2Hi cos C0itT+ 2H2 cos (X)2tT . (3.14)

Let us call CDj the pumping or saturat ing frequency and 032 ^^^ detecting frequency. The power absorbed by the sys tem will now be given to a good approximation by the sum of two t e r m s :

|7elT<'> 1 + Ti°' (m„-co,)% 7„H^T;"'Tr M , ^ .rW / ^' ^ „ „^T.(n)^ n)

The f i rs t t e r m in Eq. (3.15) r ep resen t s "pumping power" absorbed and was derived previously in Eq. (2.9)- The second t e r m rep re sen t s rf power ab­sorbed or emitted depending on the sign of Ml"-). The essent ia l assumption in Eq. (3.15) is that the static nuclear magnetization, MQ [Eq. (1.6)], which appears in the usual express ion for power absorbed in nuclear magnetic resonance , be replaced by the s teady-s ta te Overhauser nuclear magnet iza­tion, M(n) [Eq. (3.12)]. Fo r Odi = |7elHo> s ~ 1, 0)2 = b n N o . and

7nH2Ti T2 << 1, Eq. (3.15) reduces to the simple form

W^xi^WT^^X^'^H^c^T^^ , (3.16)

where X '' ' is the s teady-s ta te nuclear susceptibil i ty. Fo r I --jSLnd a < < l ,

XW s N(n)7j^|7g|h2/4kT . (3.17)

With these approximations the ra t io of the power absorbed by the nuclear spin sys tem to the power absorbed by the sa tura ted electron spin sys tem is

W(n)/W(e) =N("^n l7e l^ 'H^Ti^^Ti^V4kTX^^ . (3.18)

The d i s c u s s i o n p r e s e n t e d in Sec t ion 3 of th i s r e p o r t r e p r e s e n t s a s i m p l e e x t e n s i o n of P a k e ' s d e r i v a t i o n ( l )of the O v e r h a u s e r effect to the c a s e of n e g a t i v e n u c l e a r m a g n e t i c m o i n e n t s . It c l e a r l y shows that an in ­v e r t e d n u c l e a r popu la t ion d i s t r i b u t i o n m a y be ob ta ined by s a t u r a t i n g the e l e c t r o n sp in r e s o n a n c e p r o v i d i n g t h r e e cond i t i ons a r e s a t i s f i e d :

(1) 7 e < 0 ;

(2) 7 n < 0 '

(3) Amg + A m i = 0

Condi t ion (1) is s a t i s f i e d if the " u n p a i r e d e l e c t r o n " i s a s ing le e l e c t r o n ou t s ide a c l o s e d s u b s h e l l . Condi t ion (2) is s a t i s f i e d by a c o n s i d e r a b l e n u m b e r of n u c l e i w h o s e m a g n e t i c m o m e n t s a r e n e g a t i v e . Condi t ion (3) i s s a t i s f i e d if I • S coupl ing d o m i n a t e s ove r o r d i n a r y d i p o l a r coupl ing . We wil l e x a m i n e t h e s e cond i t ions in m o r e de ta i l in s u b s e q u e n t s e c t i o n s of t h i s r e p o r t , and we wi l l find tha t o the r s e t s of cond i t ions a l s o give r i s e to n e g a t i v e n u c l e a r m a g n e t i z a t i o n . An i n v e r t e d n u c l e a r popu la t ion d i s t r i b u t i o n of t h i s type is of i n t e r e s t b e c a u s e of the p o s s i b i l i t y of u s ing such a s y s t e m a s a low n o i s e rf a m p l i f i e r .

SECTION 4

A T h e r m o d y n a m i c T r e a t m e n t of the O v e r h a u s e r Effect

It i s i n t e r e s t i n g to e x a m i n e a s econd d e r i v a t i o n of the O v e r h a u s e r effect by m e a n s of a s i m p l e t h e r m o d y n a m i c a r g u m e n t due to B r o v e t t o and Cini.^ ) C o n s i d e r aga in e l e c t r o n s and n u c l e i coupled by the c o n t a c t p a r t of the h y p e r f i n e i n t e r a c t i o n , which we sha l l w r i t e now in t e r m s of the f a m i l i a r r a i s i n g and l o w e r i n g o p e r a t o r s :

/ ^ = ( 8 7 T / 3 ) 7e7n^ i ' ( l zS2 + 21+ S_ + 2I„ S+) 6 (T) , (4.1)

w h e r e

I± = (Ix ± i I y ) / 2

and

S^ = (S^ ± iSyj, 2

The second and t h i r d t e r m s in the p a r e n t h e s i s of Eq . (4.1) induce s i m u l ­t a n e o u s n u c l e a r and e l e c t r o n i c sp in f l ips of the type

Ci + n J-«--«-e I T n j "2 "2" ~ 2 T

(4.2)

14

w h e r e e f j /2 deno te s an e l e c t r o n wi th sp in p a r a l l e l (+) and a n t i p a r a l l e l (-) to the m a g n e t i c f i e l d s ; n i 1/2 h a s a s i m i l a r m e a n i n g for n u c l e a r s p i n s . F o r s i m p l i c i t y we c h o o s e I - j - The r e s u l t s , a s in Sec t ion 3, m a y be r e a d i l y e x t e n d e d to any va lue of I. The i m p o r t a n t poin t to note h e r e i s tha t the s u b s c r i p t on e and n deno te s the sp in o r i e n t a t i o n , not the o r i e n t a ­t ion of the c o r r e s p o n d i n g m a g n e t i c m o m e n t . The t e rna I+S_ f l ips a n u c l e a r sp in f r o m down to up and an e l e c t r o n sp in f r o m up to down. The t r a n s i ­t i ons induced by t h i s o p e r a t o r a r e t h o s e which p r o c e e d f r o m left to r i g h t in E q . (4.2). The t e r m I_S-|- i n d u c e s t r a n s i t i o n s f r o m r i g h t to left in Eq . (4.2) . L e t N(+) be the n u m b e r of ni/2, and N(-) be the n u m b e r of n_i/2; l e t N(2) be the n u m b e r of ei/2 and N(-2-) be the n u m b e r of e_i/2. When the t r a n s i t i o n s r e p r e s e n t e d by E q . (4.2) c o m e to a n e q u i l i b r i u m o r s t e a d y -s t a t e the l aw of m a s s ac t ion p r o v i d e s us wi th a r e l a t i o n s h i p b e t w e e n the occupa t ion n u m b e r s of the " c h e m i c a l " c o n s t i t u e n t s :

N(4) N(+) N(

w h e r e K m a y b e ob ta ined by i n t e g r a t i n g the Van ' t Hoff equa t ion

d In K ^ _AU_ dT RT2 '

(4.4)

w h e r e R is the gas cons t an t . We m u s t be ca r e fu l about the eva lua t i on of AU. The t r a n s i t i o n s r e p r e s e n t e d by E q . (4.1) do not c o n s e r v e e n e r g y . The dominan t t r a n s i t i o n i s the one in which e n e r g y i s t aken out of the sp in s y s t e m and t r a n s f e r r e d to the l a t t i c e and not v i ce v e r s a . T h i s i s the t r a n s i t i o n induced by the o p e r a t o r I-|-S_. The change in i n t e r n a l e n e r g y of the sp in s y s t e m d u r i n g th i s p r o c e s s i s

AU = - N ( | 7 e l + 7 n ) M o f o r 7 n > 0 (4-5)

and

AU = - N ( | 7 e l - b n b ^ H o f o r 7 n < 0 ,

w h e r e N i s t a k e n a s A v o g a d r o ' s n u m b e r . T h i s r e s u l t m a y be e a s i l y o b ­t a i n e d wi th the a id of F i g . 5.

In K = In N(-i)N(+)-

LN(4)N(-) AU ( l 7 e l + 7n)hHo ^ ^ + c o n s t . = ---; + c o n s t .

(4.6)

a t T - 00, N(-2) - N(+2) and N(+) = N ( - ) , wh ich m e a n s tha t the cons t , i s equal to 0. T h e r e f o r e ,

15

N(-i) N(+) N(+|) N(-) exp

(l7el + 7n)hHo kT

(4.7)

7 e t H o

7efeHo |7e I Mo

7e < 0

e^ 2

^ _ i 2

7nhHo

7n > 0 or 7n < 0

-n ,N(+) 2

r - n j_N(-) ' t-Ynl^o

- nj^N(+) n i_N(-) 2 2

Elec t ron Spin Zeeman Levels Fig . 5

Nuclear Spin Zeeman Levels

If the electron spin sys tem is sa tura ted by intense microwaves of frequency 03 = |7e|Ho, the t rans i t ions e_i/2 + hOi—^e+1^2 inc rease the N(+2) population, and the relaxat ion p r o c e s s ej/j + n_i/2—^^-1,2 + %, 2 will induce nuclear polarization. In the l imit for 3 = 1 , when N(+2) = N(--2), we have, from Eq. (4.7),

N(+) (l7el + 7n)hHc kT

(4.8)

in agreenaent with Eq. (3.7). This resu l t holds for ei ther sign of 7^^. F r o m Fig. 5 it is c lear that the upper nuclear Zeeman level is more populated than the lower when 7n is negative.

SECTION 5

Results of General ized Trea tments of the Overhauser Effect

There a re two papers l^ ' " ) which give a more general and r igorous t rea tment of the Overhauser effect, where resu l t s a re of importance both for the in terpreta t ion of exper iments which have been per formed and for the suggestion of exper iments which might be worthwhile doing.

In the paper by Barke r and Mencher '^/ it is shown that the the rmo­dynamic argument of Brovetto and Cini,v4j which is summar ized in Sec­tion 4, can be justified on the bas i s of the the rnao dynamics of i r r e v e r s i b l e

16

p r o c e s s e s . The Overhauser effect is found to be a steady state of minimum production of entropy. Fo rmu lae a re der ived which apply to re laxat ion p r o c e s s e s other than that due to the contact pa r t of the hyperfine interact ion. The nuclear magnetizat ion, M ( " ) , has exactly the same form as in Eqs . (1.6) and (3.12), but the a rgument a of the Bri l louin function is found to be more complex.

In a meta l where the conduction e lec t rons mus t be descr ibed using F e r m i - D i r a c s t a t i s t i c s ,

a F D - [7n + s|7e|(f^^^-f^2)^j^Ho/kT , (5.1)

where f(i) denotes the fraction of nuclei having spin r e v e r s e d which r e l ax by p r o c e s s i with i = 1,2,3. P r o c e s s 1 we have a l ready d iscussed; it is that p r o c e s s desc r ibed by the reac t ion equation

ei/2 + n m - i ^ ^ e„i/2 + nm (5.2)

in which Amj + Amg = 0. Here n^i denotes a nucleus with comiponent m of the spin angular momentum I in the direct ion of the external magnetic field HQ. P r o c e s s 2 is desc r ibed by the reac t ion equation

ei/2 + n m ^ e_i/2 + Um-i (5.3)

in which Amg + Amj = i 2 . P r o c e s s 3 is descr ibed by the reac t ion equation

nm^=^n-m-l (5-4)

in which a nuclear spin is flipped without an accompanying e lec t ron spin flip. The contact pa r t of the hyperfine in terac t ion induces nuclear r e l axa ­tion by p r o c e s s 1, in which nuc lear plus e lec t ron spin angular momentum is conserved . The dipolar pa r t of the hyperfine in teract ion induces nuclear re laxat ion by p r o c e s s e s (1), (2) and (3). There a r e of cour se other in te r ­actions which induce nuclear re laxat ions of p r o c e s s 3. We note that the nuclear spin la t t ice re laxat ion t imes a r e re la ted to the f i l l ' s as follows:

f(i) = T^VT^^^'^ (5.5)

where Tj is the total nuc lear spin la t t ice t ime and T } the nuclear spin la t t ice re laxat ion t ime assoc ia ted with p r o c e s s i. If we a s s u m e that p r o c e s s e s (1), (2) and (3) r e p r e s e n t all possible nuclear spin la t t ice r e l axa ­tion mechan i sms it is c lea r that

3 I f i) - 1 . (5.6) i=l

17

It should be noted that f(^) is included implici t ly in Eq. (5.1) through Eq. (5.6) and may considerably reduce the nuclear polar iza t ion.

In a nonmetal where the unpaired e lec t ron spins do not form a de­genera te F e r m i gas , one may use Boltzmann s ta t i s t ics for the e l ec t rons , with the r e su l t that a is given by

a^ .hn±M^'\-^^^'^l2±imi „z(f(0 .f(2))tan h^

h W 1 ( I 7elftHo) (5.7)

Note that in the l imit ing case of complete sa tura t ion of the e lec t ron spin resonance (s = l ) , '^•pT) ^^^ ^ ^^® identical and a r e given by Eq. (3.13). In the absence of a t i ine-vary ing field we set s = 0 in Eqs . (5.11) and (5.7) and recover the equi l ibr ium express ion for a , given in Eq. (1-7). For i n t e r ­mediate values of the sa tura t ion p a r a m e t e r between 0 and 1, Of-B^^FD' This means that one can obtain a higher nuclear polar iza t ion for a given s < l in a nonmetal than in a me ta l . This impor tant p rac t i ca l point was made by Brovetto and Ferroni.C^)

There is a se r ious question as to whether or not the express ion for the s t eady-s t a t e nuclear magnet izat ion as given by Eqs . (3.12), (5.1) and (5.7) is valid at low t e m p e r a t u r e s . At the p re sen t t ime this point has not yet been invest igated exper imenta l ly . F r o m the theore t ica l point of view there is an impor tant r e s t r i c t i on d i scussed both by M. Klein(8) and by Barker and Mencher.(5) These inves t iga tors find essent ia l ly that the condition

i 7 e l f t H o « k T (5.8)

mus t be sat isf ied in o rde r to obtain the quoted r e su l t s using ther inodynamic a r g u m e n t s . The r e s t r i c t i on is somiewhat l e s s seve re in the paper by Barke r and Mencher.(5) Recent calculat ions on the nuclear spin specific heat a s s o ­ciated with Overhauser polar iza t ion indicates s trongly that one mus t take these validity r equ i r emen t s se r ious ly and that the express ions for the Overhauser effect when j 7 lilHo is comparab le to kT a r e not known theo­re t ica l ly (at l e a s t using thermodynamic a rguments ) . These specific heat calculat ions have been made by Ba rke r , Desloge, and Neusel and have so far been r epor t ed only at meet ings of the Amer ican Physical Society. A m o r e thorough discuss ion of this impor tant point will be published in the Physical Review in the near future.

Detailed kinetic calculat ions using an explicit Hamiltoaiian and per turba t ion theory have been made on the Overhauser effect by Overhauser , ^"' Abragam, ( l " ) and by Barker and Mencher.v5)

We shall review here Abragam ' s r e su l t s for four ca ses of nuclear magnet ic resonance :

I. in a liquid containing paramagnet ic impur i t i e s ;

II. in a meta l ;

III. in a diamagnet ic containing paramagnet ic impur i t i e s ; and

IV. in a paramagnet ic ion.

In each case we shall consider the e lec t ron spin resonance to be completely sa tura ted . The question is to what extent, if any, does the nuclear spin sys tem acquire an enhanced posit ive or negative magnet izat ion. The basic Hamiltonian is of the form

J4'^M^+M\t) (5.9)

where the stat ic Hamiltonian

MQ = 7e&S2Ho - 7nMz^o + T- AQ • S (5.10)

de te rmines the energy levels of the sys tem and the t ime-vary ing per turbing Hamiltonian

/V' (t) = T . Al • "S + 7e*H' • S"

induces t rans i t ions between different e igens ta tes of ^ Q . Now A = AQ + Ai(t) is a symmet r i ca l t ensor with a static and a t ime-vary ing component which provides a convenient shorthand for writ ing the magnet ic in teract ion between an e lec t ron spin S and a nuclear spin I. Also H' r e p r e s e n t s the fluctuating magnet ic field produced by the lat t ice and is responsible for producing r e ­laxation of e lec t ron sp ins .

Case 1: Nuclear Resonance in a Liquid Containing Paramagne t i c Impur i t i e s

The pr incipal assumpt ions made he re a r e that (a) the e lec t ron-nuc lear spin in teract ion is the c l a s s i ca l dipole-dipole in teract ion, (b) the s tat ic par t of the e lec t ron-nuc lea r dipole-dipole in teract ion ave rages to ze ro , and (c) the nuclear re laxat ion is provided by the re la t ive motion of the paramagnet ic ion and the nucleus . The energy level scheme of an e lec t ron plus a nucleus in a liquid is given in F ig . 5. Abragam uses the notation 2A = e lec t ron energy splitting •- |7g jfiHo/kT and 26 = nuclear energy sp l i t t ing/kT. In this case 2 6 «s7 fiHo/kT and the a lgebra ic sign of 6 is the same as that of 7^ . The explicit assumpt ion is made that A and 6 a r e smal l compared to unity. The c l a s s i ca l dipole-dipole in teract ion,

19

# dipole-dipole ~7n7e^ 1 s 3(T-?)(s"-T) r^ " r^

(5.11)

contains relaxation mechanisnas of type (1), (2) and (3) [Eqs. (5.2), (5.3) and (5.4)]. This may be seen explicitly by writing Eq. (5.11) in terms of spheri­cal polar coordinates as is done in reference 10 and in the well known BPP paper.(11) It is assumed that the spectral intensities of the interactions are frequency independent or "white" for all transitions involved and that the problem is isotropic, so that one can average the squares of the matrix ele­ments over the angles. It is found(12) that f " : f ': f'"^ = 2;12;6, so that

^ . l . A = l.^—~-^ (5.11a)

in the limit of complete saturation of the electron spin resonance. To inter­pret this result we must refer again to Fig. 5. For 7n>0, Eq. (5.10) gives a negative nuclear polarization; for 7n<0, Eq. (5.10) gives a positive nuclear polarization.

Case II. Nuclear Resonance in a Metal

The important electron spm-nuclear spin interaction in a metal is the contact part of the hyperfine interaction:

^contact = (8V3) 7e7n^^'^' S 6(T) , (5.12)

whose static part is responsible for the Knight shift and whose time-varying part produced by the relative motion of the nuclei and the conduction electrons induces relaxations in which f ~ 1 and

^ ^ 1 , 2 ^ = , , ] ^ (5.13) N_ kT ^ '

when the electron spin resonance is saturated. For 7j^^0, Eq. (5.13) gives a positive nuclear polarization; for Jj.'^^s Eq. (5.13) gives a negative nuclear polarization

Case III: Nuclear Resonance in a Diamagnetic Crystal Containing Paramagnetic Impurities

No Overhauser effect is to be expected under the following assumptions:

(1) The relaxation of the nuclear spins is produced by the random flips of the electron spins rather than through their relative motion. (1^/

(2) The dipole-dipole interaction is much smaller than the nuclear splitting 26kT. Otherwise no NMR would be observed.

20

(3) The electron-nuclear interaction is dipole-dipole rather than _^ I • S. An Overhauser effect would be observed if the interaction were I • S, but this would increase the nuclear relaxation time by a factor of 10 - - an unrealistic result.

As a consequence of these assumptions, f as 1 and therefore

N+ZN. = 1 + 26 . (5.14)

Case IV: Overhauser Effect in a Paramagnetic Ion

In this example the nucleus under consideration is in a paramagnetic ion. It is assumed that the hyperfine structure is resolved, which means that the tensor Ag is diagonal and reduces to a single constant Ag. The Hamiltonian for the electron-nuclear system is

^= 7e*HoSz + AoSzIz +|AO(S+I.+S.I+) - 7n^HoIz

+ I • Al • S + 7e^H' • S . (5,15)

Unlike the cases previously treated, the local miagnetic field due to the electron in the ion is much larger than the external magnetic field at the nucleus. Thus we neglect the fourth term in comparison with the second and third. We also consider the strong-field or Paschen-Bach case. Hence,

17 e |hHo » I Ao j » |7n I Ho . (5.16)

Now Ao has the same algebraic sign as 7^. The energy-level diagram, and eigenstates are given in Fig. 6a for 7n>0 and in Fig. 6b for 7n'^^- 1 ^'^^^ diagram

2A = ^^ , p2 + q2 = 1 k T

Y (5.17)

26 = ^ ,q « P H H ; , P « 1

The first algebraic sign in the eigenstates (a'), (b'), etc., refers to S2, the second to I^.

The eigenstates (a) and (a') are essentially nuclear spin up states while (b) and (b') are essentially nuclear spin down states. The magnetic moment of the electron produces a large magnetic field at the nucleus. Consequently the nuclear magnet orients itself with respect to this field rather than with respect to the external field. Thus for electron spin down (up) a nucleus with positive gyromagnetic ratio has nuclear spin up (down) in the lower energy state. This situation is reversed for a nucleus with a negative gyromagnetic ratio.

21

2AkT

(a ') = (+,+)

I 2 6kT

( b ' ) - p ( + , - ) + q ( - , + )

(b) = (-.-)

1 2 6kT

( a ) = p ( - , + ) -q (+ , - )

2AkT

Fig. 6a

7n > 0

Fig . 6b

7 n < 0

(b-)

2 6kT

(a-)

(a)

2 6kT

(b)

Let us assume that both electron resonances a re sa tura ted , i .e . , a = a' , b = b ' , where a, a', b , and b ' denote the populations of the c o r r e ­sponding s tates (a), (a ') , e tc . The t e r m I • Aj • S produces nuclear re laxa­tion through the relat ive motion of e lectron and nucleus. If we assume it is of the form A^ I • S, it induces t rans i t ions (b')-»—^(a). If this is the only t e r m in the t ime-dependent pa r t of Eq. (5.15) and if nuclear relaxation without accompanying electron spin flips has a very snaall probabil i ty. then it follows that f

(1) ciZ) 1, r ^ ' = 0, f ' = 0, and (3) __

N „ a + a' b + b '

a , l7elhHo = f =« 1 + 2A = 1 + —f-—-

b kT (5.18)

as in the Overhauser effect in me ta l s . The t e r m 7ghH' • S produces nu­c lear re laxat ion by randona e lec t ron spin flips. This t e r m induces s trong electronic t rans i t ions (a)-%--»(a') and (b)-.-^(b'), weak nuclear t rans i t ions (a')-*—»»-(b') and (a)-«-*-(b) because of the small admixture in the e igens ta tes , and weak t rans i t ions (b')-^-^(a). If this is the only t e r m in the t ime-dependent par t of Eq. (5.15) and if the spectruna of H' is isotropic and white. then it follows that f^ = i^^\ f^ = 0 and

N . b 1 + l7elMo

2kT (5.19)

which is just one-half the enhancement predicted in Eq. (5.18). It is c lear from Fig. 6 that Eqs . (5.18) and (5.19) predict a positive nuclear magnet i ­zation when 7j^ is positive and a negative nuclear magnetization when 7^^ is negative. Abragam(l^) t r ea t s in termediate cases where random re l a ­tive motion and random electron flips compete in producing nuclear r e l ax ­ation in a paramagnet ic ion, and he cons iders the situation where one of the resonances is sa tura ted but not the other (say a = a' but not b = b ' ) .

However, he does not consider the in te rmedia te case involved when one or both of the e lect ron spin resonances is only par t ia l ly sa tura ted (i .e. , s < l ) . His a rguments should be general ized to cover this si tuation for the following reason . Suppose an NMR exper iment is per formed, the frequency being CDj = 26kT. In F ig . 6a the power absorbed by vir tue of a>b will be emit ted by vir tue of a '>b ' and the re will be no net absorpt ion of energy if a = a' , b = b". It is reasonable to suppose that a maximuna nuclear resonance ab­sorpt ion or emiss ion effect can be produced he re when s = j .

It is c lea r from the detailed discuss ion p resen ted in Section 5 that a negative nuclear magnet izat ion may be achieved by sa tura t ing the e lec t ron spin resonance if ei ther of the following se ts of conditions a re satisfied:

(1) 7e<0 (1) 7e<0

(2) 7n<0 or (2) 7n>0

(3) Amg + Amj = 0 (3) Amg + Amj = ±2

The f i r s t set of conditions is normal ly satisfied in a meta l or a paramagnet ic ion in a solid providing, of c o u r s e , the nuclear magnet ic mojaaent is negative. The second set of conditions is normal ly sat isf ied in a liquid with p a r a m a g ­netic impur i t ies provided that the nuclear naagnetic moment is posi t ive.

In subsequent sect ions of th is r epo r t we shall examine the var ious sys t ems which have been invest igated exper imenta l ly . This should tu rn up some ideas as to what new sys tems would be profitably studied by this p a r ­t icular double resonance technique.

SECTION 6

Metals

Successful conduction e lec t ron spin resonance exper iments ( l^ ) have been made in Li , Na, K, Cs , and Be with f requencies ranging between 300 and 9000 M c / s e c . Samples have been ei ther meta l pa r t i c l e s smal l compared to the skin depth or flat p la tes thick compared to skin depth. The line widths have been found in some ins tances to be t empe ra tu r e independent (Li and Be) and in other ins tances to be t empe ra tu r e dependent (Na, K, Cs) . The widths range f rom 0.1 gauss (Na at 4°K) to 20 gauss (K at 140°K). The g values a re approximately the free e lec t ron values (g = 2.0023). Negative g shifts, Ag -(Sexp ~ 2.0023), have been observed for all c a ses except B e ' , which has a Ag = +9 ±1 X 10"^.

Successful nuclear magnet ic resonance exper iments have been made on at l eas t 19 naetals.(15) The NMR frequency in me ta l s exhibit a "Knight shift" as a r e su l t of the sma l l local naagnetic field at the nuclei due to the

23

spin pa ramagne t i sm of the conduction e l ec t rons . This has been shown to be p r i m a r i l y due to the contact par t of the hyperfine in teract ion [Eq. (5.12)] and ranges from about 0.02% (Li" ) to 2.5% (Kg" ' ) . No Knight shift has been observed in Be^. It is perhaps not so well known that the e lec t ron-nuc lea r dipole-dipole in teract ion [Eq. (5.11)] a lso makes a contribution to the Knight shift in noncubic c rys t a l s .'1 ") The effect depends on the angle between the c rys ta l l ine axes and the external magnet ic field.

Successful Overhauser effect exper iments ( l 7) have been repor t ed on Li and Na' . Enhanced nuclear resonance absorpt ion was observed at 50 k c / s e c in fields of 30 3 and 40 2 gauss , respec t ive ly The e lec t ron spin resonances were sa tura ted with the corresponding applied frequencies at 84 and 124 Mc / sec The t e m p e r a t u r e for the Li exper iment was at about 70°C and it was considerably above room tenaperature for Na. The Li NMR signal was found to i nc rease by a factor of about 110, which is cons iderably sma l l e r than the optimum theore t ica l value of j 7g | / 7n ~ 1690. The enhance­ment was reduced by other nuclear re laxat ions which contr ibuted to f(^) [see Eqs . (5 5) and (5.6)]. The Na NMR signal was found to i nc rease by a factor of about 10, which is again much sma l l e r than the theore t ica l l imi t . The r eason for this was that the e lectron resonance line width was about 12 gauss under the conditions of observat ion and an Hj of only 1 to 2 gauss was produced by the sa tura t ing osci l la tor over the sanaple volume. The sa tura t ion factor was therefore ~ 0.01 Although these exper iments were not c a r r i e d out under optimum conditions of high HQ and low T, they never the les s const i tuted a s t r iking confirmation of O v e r h a u s e r ' s theore t ica l predic t ion. No low-t e m p e r a t u r e , high-field r e su l t s have been repor ted on Li and Na as yet, nor have the re been any exper iments done on other m e t a l s .

The nuclear polar izat ion achieved by Carver and Slichter was negl i ­gible for the values of H Q / T used and, consequently, no shift in the e lec t ron resonance frequency was observed . This shift is cal led the "Overhauser shift" and is the analogue of the Knight shift. It is ve ry in te res t ing that the a lgebra ic sign and magnitude of the O^i-erhauser shift can be used to d e t e r ­mine the a lgebra ic sign and magnitude of the nuclear polar iza t ion . F u r t h e r , as the e lec t ron spin resonance is sa tura ted , it is c lear that both the i so t ropic and anis t ropic Knight shift will vanish l inear ly with s. Consequently, both the shape and posit ion of the EPR line and the NMR line a r e changed in an in te res t ing fashion in an Overhauser effect exper iment .

Beryl l ium p r e s e n t s some intr iguing p rob lems and poss ib i l i t i e s , and m e r i t s further theore t ica l and exper imenta l study. A successful Overhause r effect on bery l l ium would be mos t in te res t ing . Be^ has a nuclear spin I = •f, a negative nuclear magnet ic naoment 1-^ - -1 1774 nuclear magnetons , and a nuc lear quadrupole moment ( l^ ) of Q = 0.029 x 1 Q-^^cm^ One would expect a negative r a the r than a posi t ive Ag = +9 x 10""* as observed in EPR.'-^^^ The l ine width^l"-' is r a the r l a r g e , —13 gauss , which would make sa tura t ion of the e lec t ron spin resonance difficult. However, this width is t e m p e r a t u r e inde­pendent, which means it is probably due to impur i t i e s . This si tuation could

24

probably be improved by using small Be meta l pa r t i c l e s of high puri ty and doing the e lec t ron spin resonance at low t e m p e r a t u r e s where the line width should d e c r e a s e . In view of i ts negative nuclear magnet ic inoment, Be should exhibit a negative nuclear magnet izat ion when the e lec t ron spin r e s o ­nance is sa tura ted . This could be detected by the enhanced emiss ion of a nuclear magnet ic resonance signal and by a negative Overhauser shift. This will occur providing the I ' S in teract ion is significant for nuclear re laxat ion. The absence of a Knight shift cas t s some doubt on thiss but other evidence points to a substantial S state cha rac t e r in the meta l l ic wave function. The m e a s u r e d Knight shift in Be is 0.000% which is quite surprising.(15) The predic ted value is 0.01%. One poss ible explanation is that the Knight shift is masked by a compensating negative shift produced by the second-order nu­c lear quadrupole interaction.V-*-"). This question too could be resolved if it is possible to sa tura te the e lec t ron spin resonance in Be, since the Knight shift would d isappear but the negative nuclear quadrupole shift in the nuclear r e s ­onance would r ema in .

It would be ve ry in teres t ing to per form Overhauser effect exper iments on other me ta l s but they p re sen t severa l difficulties:

(1) In view of the high conductivity of m e t a l s , t he re is but a smal l depth of penetra t ion in a microwave field. This difficulty can be met by using pa r t i c l e s smal l compared to the skin depths Pa r t i c l e d i spers ions have been p repa red u l t rasonica l ly in paraffin wax or by precipi ta t ion in frozen m e t a l -ammonia solut ions. It is also poss ible in some cases to p r e p a r e colloidal pa r t i c l e s of a diameterf"*200 A. Thick p la tes of me ta l s have been used for EPR work. The resul t ing line shapes a r e a s y m m e t r i c a l and a re accounted for quite well by Dyson's theory.i^*^) However, Carve r and S l i c h t e r l l ' ) point out that the effects of sa tura t ion when the e lec t rons diffuse in and out of the skin depth a r e not known,

(2) Fo r a given level of saturat ion, metaJs exhibit a lower nuclear polar iza t ion than nonconducting pa ramagne t i c s . This i s the case especia l ly for small s and high H/T ra t io , as pointed out in Section 5.

(3) The mos t ser ious difficulty is the short value of the e lec t ron spin lat t ice re laxat ion t ime . In me ta l s Tj = T . and a short Tg means very broad l ines , which a r e difficult to observe and ext remely difficult to satu­r a t e . EUiott^^-'-^ has developed a theory which cons iders the probabil i ty of an e lec t ron spin flip via spin-orbi t coupling when there is a coll ision be ­tween a conduction e lec t ron and a phonon. The relaxat ion t ime is found to be

Ti = CLTj^/iAgf , (6.1)

where a ~ 1/30, Tj^ is the coll ision relaxat ion t ime which can be found from res i s t iv i ty data, andAg is the e lec t ron g shift which is a mieasure of the s t rength of the sp in-orbi t in te rac t ion . F o r Na at SOCK, Tj^ = 3 x 10" sec ,

Ag = -8 ± 2 X 10" and hence Tj '= 2 x 10" ' sec , which is in fair agreement with Ti(expt) " 9 x 10" ' s ec . The i inportant point to note is that Ag in­c r e a s e s with increas ing Z and hence Tj becomes so short for the heavy meta l s that no E P R is observable . Thus for Li"^: Ag = ^ ± 1 x 10"*; Na: Ag = (-8 ± 2) X 10"*; K: Ag = (-70 + 50) x 10"*; and for Cs : Ag = (-700 i 200) x 10~". A sat isfactory theory of the g shifts for the alkalis has been worked out by Brooks.(^^) It is possible that this difficulty can be c i rcumvented by using me ta l - ammonia solut ions. An.other possibi l i ty is to do a double r e s o ­nance on neut ra l metal a toms p resen t as impur i t ies in one of the alkali hal ides . We shall d iscuss these topics in more detail in subsequent s e c ­tions of this r epor t .

A good deal of the ma te r i a l in Section 6 on me ta l s has been obtained from an excellent survey: Microwave P r o p e r t i e s of Solids by Bagguley and Owen,(23)

SECTION 7

Meta l -Ammonia Solutions

If an alkali meta l like Na or K is dissolved in ammonia , the free atom will ionize in solution: Na-^-^-Na"^ + e. The positive meta l ion has full, closed shel ls with no unpaired e lec t rons . The e lect rons form a "free gas ." Successful e lec t ron spin resonance exper iments 1 *) have been made with such solutions. The r e m a r k a b l e feature of these exper iments is that the line widths a r e ex t remely na r row being of the order of 0.05 gauss and smal le r . A theory of m e t a l - a m m o n i a solutions has been worked out by Kaplan and Kittel.l^^) According to this theory, the e lec t rons a re t rapped in "cavi t ies" in the solution and in te rac t by the contact pa r t of the hype r -fine interact ion with the protons of the NH3 surrounding the cavity. The line widths a r e smal l due to the motional nar rowing assoc ia ted with the rotat ion and diffusion of the NH3 molecu les .

The advantage of such meta l -ammonia solutions for the observat ion of the Overhauser effect is that the EPR may be easi ly sa tura ted . One would expect from the theory a positive nuclear polar iza t ion of the pro tons , since 7e<C0, Tn' O? and I • S coupling is dominant. An Overhauser effect exper iment has been c a r r i e d out by Ca rve r and S l i c h t e r " ') on this type of solution. They used Na in NH3, ranging in concentrat ion from 0.01 to 0.9 N. The stat ic field used was 11.7 gauss . A t ime-vary ing field with H | =0,09 gaus produced saturat ion values g r e a t e r than 88% in all c a s e s . The proton NMR signal was enhanced, but only in fair ly concentrated solutions did this ap­proach the value iTe l /Tp ^ 660. Evidently nuclear spin re laxat ion by some interact ion other than 1 • S shor t c i rcui ted the expected effect at low concentra t ions .

26

There a r e th ree disadvantages to me ta l - ammonia solutions:

(1) The meta l m e r e l y provides a source of "free e lec t rons . " It is not the meta l nucleus which one would expect to be polar ized but ra ther the protons in NH3 with which the e lec t rons have an appreciable S s tate hyper -fine interact ion.

(2) If one is looking for a high polar izat ion by the Overhauser effect, r a ther than for a substantial enhancement of the NMR absorption^ it is n e c e s ­sa ry to go to low t empera tu r e . The solutions become solid at these t e m p e r a ­tu re s and the motional nar rowing of the E P R l ines would no longer be effective. Saturat ion of the wider l ines to be expected at lower t e m p e r a t u r e s would r e ­quire l a rge r values of H| and hence m o r e microwave power.

(3) Increasing the concentrat ion of the metal in NH3 is evidently n e c e s s a r y to achieve a max imum effect. However, such an inc rease broadens the e lec t ron spin line width, making sa tura t ion m o r e difficult, and introduces the possibi l i ty of e l ec t ron-pro ton dipolar coupling,l^^) which reduces the enhancement.

SECTION 8

Liquids Containing Pa ramagne t i c Impur i t ies

Three exper iments confirming the genera l fea tures of Abragam's theory(°) of the Overhauser effect in liquids containing paramagnet ic im­pur i t i es have been per formed. In each case the nucleus polar ized was the proton. The polar izat ion was negative as the theory p red ic t s for 7g<C0, 7jj^>0, and dipolar coupling.

Bennett and Tor rey l^" ) invest igated a solution of Na and naphthalene in 1, 2 dime thoxy ethane. The naphthalene in the so lu t i on i s ionized by the addition of an e lec t ron fornaing an ionic f ree radical.l^"^) The e lec t ron spin resonance of this f ree rad ica l was detected at 50 M c / s e c in a field of 17.8 gauss . The line width was 2.6 gauss . The proton resonance of the solvent e ther was detected at 76.8 k c / s e c . By increas ing the 5 0 - M c / s e c voltage from 0 to 400 volts a c r o s s a coil within which was located the sample and the nuc lear detection coil, values of s f rom 0 to 20% were ob­tained. (28) Let A r e p r e s e n t the amplitude of the NMR signal and A© the value of A when s = 0. Then according to the theoryl") we can wri te

A 1 I've! ^^ = 1 + E = 1 - i - ^ s , (8.1)

where the enhancement in this case is given by E = --^ (''Ye'/'^n) ^' Clear ly E is not the same as the nuclear polar izat ion, P , but it is re la ted to it.

27

(Somietimes in the l i t e ra tu re enhancement and polarizat ion a re used in ter ­changeably. It is possible to obtain an appreciable enhancement at any t empera tu re , but appreciable polar izat ion can only be obtained at t e m p e r ­a tures in the liquid He range.) It is c lear that, for s = 0, the detection of the NMR is by absorption of power. As s i n c r e a s e s , the nuclear po lar iza­tion inc reases in the negative sense , corresponding to a l a rge r number of nuclei in the upper energy s ta tes . Now NMR detection is by st imulated emiss ion suggesting, as the authors point out, that the Overhauser effect could be used for a low-noise molecular ainplifier. This is i l lus t ra ted in Fig. 7. Bennett and To r r ey observed a substantial ly complete effect for each degree of saturat ion, which is in good agreement with the a s sump­tion that the proton spins re lax via dipolar coupling with the free e lect ron.

j_Ao

s = 0 S - pff S^ 0.2 E = 0 E - - 1 E . . - 0 . 5 ( 6 5 8 ) ( 0 . 2 ) - - 6 6

Absorption Absorption Emiss ion

Fig . 7

Abragam, Combrisson and Solomoni^V) investigated a solution of disulfonate of potass ium in water . The stat ic magnetic field used was 3000 gauss . The e lec t ron resonance presented a s t ruc ture of three l ines of about 1-gauss width. One of these l ines was par t ia l ly sa tura ted . The proton resonance signal was found to dec rea se , go through zero and be­come negative with a negative enhancement of about 10. In this exper i ­ment, unlike that of Bennett and Tor rey , the proton spin relaxat ion is par t ly dipolar and par t ly without accompanying e lec t ron spin flips.

J. KrebsW^) has made a p re l iminary study at the Naval Resea rch Labora tory of a 7% molar concentration of the DPPH free radical in ben­zene. The static magnetic field was about 3400 gauss with a cor respond­ing ESR frequency of 10,000 M c / s e c , and a proton NMR frequency of 15.3 M c / s e c . The ESR line width was of the order of 12 gauss and only about 3% saturat ion had been achieved. The proton resonance was found to decrease and invert with a negative enhancement of about 5. The proton observed was the proton in benzene, not that in the free radica l .

28

These exper iments on liquids c l ea r ly confirm the predict ions of the theory regarding an " inverted Overhauser effect." However, only ex t remely smal l nuclear polar izat ions can be obtained at room t e m p e r a t u r e . Liquids a r e c lea r ly not appropr ia te sy s t ems for l ow- t empera tu re work.

SECTION 9

F r e e Radicals

All free rad ica ls have unpaired e lec t rons assoc ia ted with them. Elec t ron paramagnet ic resonance has been used extensively to detect and observe these s t r u c t u r e s . The line widths a re usually quite small and hence signals of strong intensity a r e obtained f rom smal l amiounts. This pe rmi t s the detection of very smal l concentra t ions . A considerable amount of work has been done on the organic salt a , a-diphenyl ^ t r ini t rophenyl hydrazyl : (C^Hg), N - NC6H2(N02)-- This DPPH radica l has an e lec t ron g value of 2.0036 and a line width of 2.9 gauss in a solid.

Ingrami^-"-) gives an excellent summary of paramagnet ic resonance in free r ad ica l s ,

A successful Overhauser effect in the DPPH free radical was r e ­ported in an ea r ly exper iment by Be l j e r s , Van der Kint and Van •Wieringen.(32) They used a s tat ic magnet ic field of 3300 gauss , a microwave frequency of 9000 M c / s e c and an rf frequency of 14 M c / s e c . As the microwave power in thei r 70-watt k lys t ron inc reased above 3 wat t s , they observed an inc rease in the proton resonance absorption. It is in teres t ing to cornpare this resu l t with the experiraent of Krebsi^") on the same free rad ica l dissolved in ben­zene. These two exper iments indicate that the proton spin lat t ice relaxat ion in a solid is via the contact p a r t of the hyperfine in teract ion, and in a liquid is via the dipolar par t of the hyperfine interact ion. Ingram points out that the free e lec t ron in DPPH is mainly at tached to the ni t rogen r a the r than to the hydrogen atom of the molecule and that a g r ea t e r change could have been observed by the Dutch group if they had looked at the ni trogen ra ther than the proton resonance . The abundant isotope of ni t rogen N has 1 = 1 and a nuclear moment of +0.405 nuclear magneton. One would expect a positive nuc lear polar izat ion in the case of N if this exper iment were c a r r i e d out.

SECTION 10

Semiconductors

Successful e lec t ron spin resonance exper iments on donors in Si and Ge have been made. The work on Si is reviewed by Bagguley and Owen.^^j) The work on Ge was published quite recent ly and makes an in teres t ing

29

compar ison between Si and Ge.(33) So far all a t tempts to observe spin resonance from accep tors in Si and Ge have not been successful .

The donor impur i t ies in both Si and Ge have been P , As and Sb. The g shifts a r e all negative and a r e considerably l a r g e r in Ge than in Si. The line widths in Si a r e about 2-3 gauss and in Ge about 10-11 gauss at liquid He t e m p e r a t u r e s The evidence indicates that the e lec t rons move in la rge S-like orb i t s .

The Overhauser effect has been observed in phosphorus-doped s i l i ­con by Abragam et al.(34) The weak conductivity of the sample simplified considerably the problein of the penetrat ion of the sa tura t ing field, p a r t i c ­u lar ly at high f requencies . As a resu l t of the smal l number of conduction e lec t rons , the nuclear spin- la t t ice re laxat ion t imes a re much longer than in m e t a l s . This pe rmi t t ed the Saclay group to separa te completely in space and t ime the two operat ions: (1) polar izat ion of the nuclei by appli­cation of an intense e lect ronic frequency, and (2) observat ion of this po la r i ­zation at the nuclear resonamce frequency. Si ' is the only isotope of si l icon which has a nonzero spin and magnet ic moment : 1 = 2 . Mn ^ -0.555 nuclear magneton , | 7 e l / 7 n - -3300. Si^^ has a na tura l abundance of 4.68%.

The exper iment of Abragam et al . was per formed in a s ta t ic field of HQ = 3300 gauss , a microwave frequency of 9200 M c / s e c , and at a t e m ­pe ra tu re of 77°K. The e lec t ron spin resonance line width was 4 gauss . A saturat ion s ^ 3.8% was es t imated . The sample contained 5 x 1 0 atoms of P per cm where the relaxat ion t ime of the Si spins was five minutes . The observed enhancement was negative and equal in absolute value to 100, about 20% sma l l e r than the optinaum expected: E = s iTel /Yn ~ - ( 3 . 8 ) x l 0 " ^ X 3300 • -120.

It should be poss ible , although somewhat more difficult, to obtain a s imi la r inverted Overhauser effect in Ge. The only isotope of g e r m a -nium with a nonzero spin is Ge where 1 =-g and /i = -0.877 nuc lear mag­neton. The b roader line widths in the e lec t ron spin resonance l ines in Ge make saturat ion much m o r e difficult. The Ge nucleus has a quadrupole moment Q = -0.2 x 10" cra^ and the nuclear quadrupole in teract ion would probably provide a means of nuclear relaxat ion which would compete with the I • S in teract ion. The enhancement can be wr i t ten as

E = i l ^ f s - : ; ^ ^ ^ -18,900 f s = ^ ^ . (10.1) 7n V Tn t o t a l / \ T^ t o t a l / ^

Although the factor in pa ren theses in Eq. (10,1) will notably be sma l l e r in Ge than in Si for the r easons given, this is compensated in pa r t by the fact that | 7 e l / 7 n is 5.7 t imes g rea t e r and that Ge has a natural abundance of 7.67%, which is somewhat g r ea t e r than that of Si^'.

30

A Knight shift of 0.018% has been observedi^^) in Si^'. This increase in the nuclear magnetic resonance frequency depends, of course , on the con­centrat ion of conduction e lec t rons . A measu remen t of the quenching of this shift should provide an independent value of the saturat ion pa rame te r in an Overhauser effect experiment .

It is r a the r intriguing to speculate about the spin resonance of accep­to r s in Si and Ge. The "unpaired electron" in this case is a "hole." Dr. B. Smaller of this Labora tory has suggested that the magnetic moraent of a "hole" is posit ive. Usually spin resonance exper iments a re performed using a t ime-vary ing field which is l inear ly polar ized and at r ight angles to the static field: H = HQ k + 2Hi cos wt i. The sample absorbs energy whether the magnetic moment is positive or negative. It is well known, of course , that the a lgebraic sign of the magnetic moment can be determined by using a c i rcu la r ly polar ized magnetic field. An Overhauser effect experiment on holes in Ge and Si would be quite in teres t ing . If Sma l l e r ' s suggestion is co r r ec t and if the holes have S-like orbits as do the e lec t rons in donors , then one should observe a positive nuclear polarizat ion in Ge and Si be ­cause now there is I • S coupling between a positive 7^ and a negative 7j^. Possibly the "hole" spin resonance has not been observed in Si and Ge be ­cause of a s trong spin-orbi t interact ion leading to a la rge positive Ag and a broad line width.

SECTION 11

Color Centers

Spin resonance has been observed in F , V3 and U centers and in Ag" Ag° in KCl, but not in Vj cen te r s . A diagram of some of these centers in an alkali halide is given in Fig. 8. A D denotes a positive or negative ion vacancy; e denotes an electron and h denotes a hole. An F center is a negative ion vacancy and an unpaired electron. A V3 center consis ts of two adjacent positive ion vacancies and a hole. A Vj center is a positive ion vacancy and a hole. A U center is a negative ion vacancy occupied by an H~ ion.

F center

U c e n t e r / + - + - + | H ^ +

Fig. 8

31

The g shift is negative for e lec t rons and positive for holes . The line widths depend on concentrat ion and on t empera tu re . Many of the line widths repor ted a r e quite broad being A/50 gauss .

There a re two sys tems which have been investigated at the Argonne National Labora tory l^") and which appear to hold some p romise for an Overhauser effect exper iment .

1. Ag°Ag"'" in KCl, This is p r epa red by mixing AgCl with KCl in the mel t and then growing single c ry s t a l s which contain about 10''® Ag i o n s / cm in the host . The c rys ta l is i r r ad ia t ed with X rays or with 7 rays f rom Co to produce Ag°Ag"'", The s i lver ion, of cou r se , has no unpaired e lec t ron. The Ag" has an unpaired e lec t ron as in the meta l a tom, which makes possible observat ion of the spin resonance . Both stable isotopes of s i lver , Ag and Ag , have spin values of-j and negative nuclear magnetic moments : jj.jj = -0.113 nuclear magneton, jJ,ji = -0.130 nuclear magneton. The observed e lec t ron spin line widths a r e evidently na r row enough {'^•1 gauss) so that appreciable sa tura t ion of the ESR could be achieved. If the s i lver nuclear spin is coupled to the e lec t ron by the 1 • S interact ion, which is the most plausible assumption, the saturat ion of the ESR should produce a negative nuc lear magnet izat ion which could be detected by the induced emiss ion of an NMR signal. The expected enhancement would be

E ' " = + TJW^ ^ - 1 6 , 2 0 0 s 'n

J7e 7n

E^°' = + 77S- s ^ -14,100s

(11.1)

where s is the sa tura t ion p a r a m e t e r . The essent ia l difficulty involves the number of s i lver nuclei p resen t and the unfavorable ra t io of Ag° to Ag . The Ag° concentrat ion can probably be inc reased to ^^5 x lO- 'ycm' by i r rad ia t ing the samiple for a longer t ime. Another possibi l i ty is to put in Ag atoms by additive coloring. Although this might produce a higher concentrat ion of Ag a toms , once these atoms become aggregated in colloidal c lu s t e r s the spin- la t t ice re laxat ion t ime might become ex t remely shor t making sa tura t ion of the resonance more difficult.

2. Fz in L i F . This sys t em is of the same type as that studied by Kanzig(37) and by Smal le r , Delbecq and Yuster .(3^ '38) ^he unpaired holes a r e produced by X or 7 i r rad ia t ion . The g shifts a r e positive as expected for a hole. One would expect the magnet ic moment of the hole to be posi t ive, as in acceptors in Ge, according to Sma l l e r ' s suggestion. This could be tes ted di rect ly by using a c i r cu l a r ly polar ized, r a the r than a l inear ly polar ized, t ime-vary ing magnet ic field. If the hole resonance is sa tu ra ted one should obtain a negative nuclear polar izat ion of the fluo­r ine nuclei . F ' has a spin of y* a- magnet ic moment jl^ = +2.13 nuclear

32

m a g n e t o n , and a n a t u r a l abundance of 100%. The t h e o r e t i c a l e n h a n c e m e n t a s s u m i n g I • S coupl ing of the F " n u c l e i to the h o l e s is

I7 I E = — s = - 7 8 0 s . (11.2)

7 n

The n u c l e u s h a s no q u a d r u p o l e m o m e n t , hav ing a sp in 1 = 2' and c o n s e ­quen t ly n u c l e a r r e l a x a t i o n v ia the q u a d r u p o l e i n t e r a c t i o n would not c o m p e t e wi th the I • S i n t e r a c t i o n . T h e r e m i g h t h o w e v e r be o t h e r c o n t r i b u t i o n s to f'- ' (See Sec t ion 5). S i m i l a r e x p e r i m e n t s cou ld be done wi th CI2" in KCl o r NaCl and Br2" in K B r . V a l u e s of I, / i ^ in n u c l e a r m a g n e t o n s , IQ/I-O,

and Q in b a r n s a r e g iven for the c h l o r i n e and b r o m i n e i s o t o p e s in T a b l e 1. One u n f a v o r a b l e f e a t u r e of t h i s p r o p o s e d e x p e r i i n e n t is the fact tha t the u n p a i r e d ho le is p r o b a b l y s h a r e d p r i m a r i l y by the s i x n e a r e s t . n e g a t i v e ions and t h a t the n u c l e a r r e s o n a n c e of the ha l i de ions which do not " s e e " the hole m i g h t m a s k the p o l a r i z a t i o n of t h o s e which do. Th i s m a t t e r shou ld be looked in to f u r t h e r , wi th p a r t i c u l a r r e f e r e n c e to the p a p e r s of Kanzigl-^''') and of Inui , H a r a s a w a r a and Obata.v- '")

T A B L E 1

Ci35 3 2

B r

Br

37 3 2

79 3

Cl^

2 81 3_

2

Mn

0.82

0.68

2.11

2.27

I'eAn

6700

8100

2600

2400

-0

-0

+0

+0

Q

.079

.062

.26

.21

N a t u r a l Abundance

75.4%

24.6%

50.52%

49 .48%

C o n c l u s i o n

The r e s u l t s of t h i s r e p o r t can be s u m m a r i z e d as fo l lows . P a r a ­m a g n e t i c r e s o n a n c e c a n be o b s e r v e d in any s y s t e m which h a s an u n p a i r e d e l e c t r o n o r ho le . A p p r e c i a b l e s a t u r a t i o n of t h i s r e s o n a n c e can be a c h i e v e d p r o v i d i n g the l ine wid th is not too b r o a d . Nuc l e i with n o n z e r o m a g n e t i c m o m e n t s wh ich r e l a x at l e a s t in p a r t v ia t h e i r coupl ing with the e l e c t r o n s wil l b e c o m e p o l a r i z e d . In g e n e r a l , one can expec t the p o l a r i z a t i o n to be l a r g e r the l o w e r the t e m p e r a t u r e and the h i g h e r the s t a t i c m a g n e t i c f ie ld. The s i g n s of the p o l a r i z a t i o n depend on w h e t h e r the coupl ing i s p r i m a r i l y I • S or d ipo la r and on the s igns of 7 e and 7n - ^^ so l i d s and in m e t a l -a m m o n i a so lu t ions the e l e c t r o n s have a r e a s o n a b l y h igh p r o b a b i l i t y of be ing in the i m m e d i a t e v i c in i t y of the nuc l eus w h e r e the c o n t a c t i n t e r a c t i o n is i m p o r t a n t . In l i qu ids the e l e c t r o n s have a m u c h l e s s h igh ly l o c a l i z e d

33

position and one can expect dipolar coupling to be the important nuclear relaxation mechanism. In Table 2 the various possible combinations a re l is ted. The examples encirc led have not yet been observed. The other examples have been observed.

T A B L E 2

+

+

+

+

7j^ Coupling

I • S

T- s" Dipolar

Dipolar

T-s" T-s" Dipolar

Dipolar

N u c l e a r P o l a r i z a t i o n

E x a m p l e

Li^, Na^^ H '

(S^, (Ge-

( g ) , ( g ) , ^^, (BT^, (BI

Acknowledgments

It is a p leasure to acknowledge the hospitali ty of the Solid State Science Division of the Argonne National Labora tory where this work was done while I was a summer r e s e a r c h associa te . I ain also grateful for profitable discussions w i t h D r s . B. Smal ler , J. Robinson, K. Singwi, T. Gilbert , C. Delbecq, R. Knox and W. Hayes.

NOTE ADDED IN PROOF

There a r e two important points in this repor t which further d i scus ­sion has clarified and cor rec ted .

F i r s t , the gyromagnetic rat io of a "hole" in a solid should have the same algebraic sign as the gyromagnetic ratio of an e lect ron. It is c lear , of course , that the gyromagnetic rat io of a filled shell plus one is negative, since the only contribution to the magnetic moment comes from the extra unpaired e lect ron. One is easi ly led to believe that the gyromagnetic rat io of a filled shell minus one is positive - a belief re inforced by analogy with the posi t ron. However, a single e lectron in an S shell poses obvious diffi­cul t ies . Kittel , in a pr ivate communication, has pointed out that the a lge­braic sign of the gyromagnetic ra t io is independent of the condition of the shel l .

7 2 j ^ 2: Si

Second, the Ag" Ag^ and F2 in L iF sys tems a r e not as promising for Overhauser effect exper iments as appeared at f i r s t sight. The reason for this is that the unpaired e lectron or hole gives r i se to a r a the r la rge local magnetic field at the nucleus: | AQ !>>|7nl'fi Ho The nuclear magnetic resonance frequency is thus tUj = I6 kT = AQ/% ra the r than ITn IHQ and the expected enhancement would b e ' ^ s | i e I HO/AQ ra the r than s | 7e i/v as given in Eqs . I 1.1 and 1 1 2 .

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36

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