Anisotropic spin-orbit coupling in paramagnetic resonance

Volume 2, number 3

PHYSICS

LE T T E R S

1 September 1962

References

t i o ~ s ' i n h e a v i e r n u c l e i by obtaining f o r m f a c t o r curves for magnetic scattering as well as for charge scattering.

1) R. Hofstadter, Ann. Rev. Nuclear Sct. 7 (1957) 231. 2) G.A. Peterson and W. C. Barlmr, to be published in Phys. Rev.; and Proc. of the Rutherford Jubilee Int. Conf. (Manchester, 1961), p. 631. 3) R.S.Wflley, to be published in Nuclear Phys.

ANISOTROPIC SPIN-ORBIT COUPLING IN PARAMAGNETIC RESONANCE F. K. K N E U B ~ H L Physics Department, Swiss Federal Institute of Technology, ZUrich, Switzerland

Received 11 August 1962

The s p i n - o r b i t i n t e r a c t i o n in p a r a m a g n e t i c r e s o nance i s u s u a l l y r e p r e s e n t e d by 1,2)

ULS -- x(L . s ) .

(1)

This equation is b a s e d upon the a s s u m p t i o n that the e l e c t r o n s with u n p a i r e d s p i n s move in the c e n t r a l l y s y m m e t r i c field of the nucleus. However, it will c e a s e to d e s c r i b e s p i n - o r b i t coupling in s y s t e m s with an e l e c t r i c f i e l d that d i f f e r s f r o m t h i s s y m m e t r y . The p u r p o s e of t h i s l e t t e r i s to show that f o r t h i s reasor~ ~dae s p i n - o r b i t i n t e r a c t i o n c a n n o t ' b e r e p r e s e n t e d by eq. (1) in the following i m p o r t a n t cases: a. p a r a m a g n e t i c c e n t e r s with a s y m m e t r i c g - t e n s o r s , b. p a r a m a g n e t i c c e n t e r s in a s t r o n g e x t e r n a l e l e c t r i c field. A v e r y g e n e r a l e x p r e s s i o n f o r the s p i n - o r b i t i n t e r a c t i o n is given by

En " Eo

)'

(4) ,

E~ E o )

(2)

It was d e r i v e d by using the P a u l i a p p r o x i m a t i o n 3) of a s l i g h t l y m o d i f i e d B r e i t equation. The index i l a b e l s the n e l e c t r o n s with u n p a i r e d s p i n s , and ri, Pi, si and E a r e the position v e c t o r s , the m o m e n t a , the s p i n s , and the e l e c t r i c f i e l d r e s p e c t i v e l y . The t e r m H'LS contains the i n t e r a c t i o n between spins and o r b i t s of d i f f e r e n t e l e c t r o n s . It was f i r s t d e r i v e d by H e i s e n b e r g (Condon and S h o r t l e y 4)). F o r a c e n t r a l e l e c t r i c f i e l d E ( r ) eq. (2) t r a n s f o r m s into the c o m m o n l y used e x p r e s s i o n 4) n

ttLS : ~

(~, <01ghln> Ghk = go 6hk + Cl Re " n

Ahk -- ahk + c2 Re ( ~

n

H L S - 2 n ~e c 2 /~1 "= ([E(ri) x P i ] ' s i ) + H ' L S .

a. In p o l y a t o m i c p a r a m a g n e t i c s y s t e m s , e.g., c o l o u r c e n t e r s , f r e e r a d i c a l s in s o l i d s , and s e m i c o v a l e n t m e t a l l o - o r g a n t c compounds of low s y m m e t r y the e l e c t r i c f i e l d a c t i n g on the u n p a i r e d e l e c t r o n s can d e v i a t e c o n s i d e r a b l y f r o m c e n t r a l s y m m e t r y without c o m p l e t e l y quenching the o r b i t a l m o m e n t . In t h e s e c a s e s the calculation of the g - t e n s o r and r e l a t e d t e n s o r s has to be b a s e d on eq. (2) i n s t e a d of on eq. (3). A s s u m i n g f o r the s a k e of s i m p l i c i t y one s i n g l e e l e c t r o n with u n p a i r e d spin, one can d e t e r m i n e the g - t e n s o r a s w e l l as the t e n s o r s d e s c r i b i n g the h y p e r f i n e i n t e r a c t i o n s by the p e r t u r b a t i o n p r o c e d u r e of P r y c e 1):

[(ri) [li" si) + H ' L S . (3) i=1 This e x p r e s s i o n is a p p l i c a b l e to f r e e a t o m s and ions and in many c a s e s to ions in c r y s t a l s . It i s equivalent to eq. (1) if R u s s e l l - S a u n d e r s coupling is p r e s e n t .

H e r e , h, k = x, y , z ; and ahk r e p r e s e n t s a s y m m e t r i c d i p o l e - d i p o l e i n t e r a c t i o n t e n s o r . These equations show that the g - t e n s o r G and the h y p e r f i n e - s t r u c t u r e t e n s o r s A can be a s y m m e t r i c . This h a s a l r e a d y been p r o v e d f o r the l a t t e r by McConnell 5) by a quite d i f f e r e n t method. A c a r e f u l examination of the s y m m e t r y p r o p e r t i e s of the s p i n - H a m i l t o n i a n r e v e a l s 6) that a s y m m e t r i c g - t e n s o r s e x i s t f o r p a r a m a g n e t l c c e n t e r s of t r i c l i n i c and m o n o c l i n i c symmetry. The p r e s e n c e of an a s y m m e t r i c g - t e n s o r m a k e s the i n t e r p r e t a t i o n of the p a r a m a g n e t i c r e s o n a n c e s p e c t r u m v e r y difficult. In the a b s e n c e of c r y s t a l field s p l i t t i n g , e . g . , f o r S =i~x the e x p e r i m e n t a l l y m e a s u r e d g - v a l u e i s given by G m e a s = 1/14 ( / / G G//)½ , where//is

G = G transposed,

the m a g n e t i c field and G the g - t e n s o r (4) 163

Volume 2, number 3

PHYSICS

entering the sptn-Hamiltonian. Because Gmeas is always symmetric, it does not contain information a b o u t t h e a s y m m e t r y of G. In a d d i t i o n , t h e r e i s no o n e - t o - o n e r e l a t i o n b e t w e e n t h e m e a s u r e d and t h e actual g-tensor. A good i l l u s t r a t i o n of t h i s e f f e c t i s t h e p a r a m a g n e t i c r e s o n a n c e of t h e O H - r a d i c a l s in KC1 and NaC1, w h i c h w a s s t u d i e d b y K ~ z i g 7). T h e s e O H 1 r a d i c a l s h a v e an e l e c t r o n s p i n of S = ~, a r e s o l v e d n u c l e a r h y p e r f i n e i n t e r a c t i o n with t h e p r o t o n , and a monoclinic symmetry C s (or m). The measured g-tensor Gmeas and affective hfs-coupling tensor A m e a s h a v e b e e n d e t e r m i n e d w i t h an a c c u r a c y of a b o u t 10-4 a n d 5 × 10-2 r e s p e c t i v e l y a s s y m m e t r i c t e n s o r s with one p r i n c i p a l a x i s in c o m m o n . The m e a s u r e d n u m e r i c a l d a t a do not f i t a n y of t h e m o s t l i k e l y q u a n t u m m e c h a n i c a l m o d e l s of t h e O H - c e n t e r , if t h e s p i n - o r b i t c o u p l i n g i s d e s c r i b e d b y e x p r e s s i o n (1). T h i s i s to b e e x p e c t e d on t h e b a s i s of our theory. b. L u d w i g a n d W o o d b u r y 8), a n d H a m 9) h a v e s t u d i e d t h e e f f e c t of a s t r o n g e x t e r n a l f i e l d on t h 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 of t r a n s i t i o n m e t a l i o n s in Si, w h i c h r e p r e s e n t p a r a m a g n e t i c c e n t e r s t h a t l a c k i n v e r s i o n s y m m e t r y . T h e y h a v e found l i n e a r c h a n g e s of t h e g - t e n s o r s , t h e h f s - t e n s o r s , a n d t h e c r y s t a l f i e l d s p l i t t i n g - t e n s o r s of o r d e r of m a g n i t u d e 10 - 6 t o 10-4 c m / k V . H i t h e r t o t h i s l i n e a r e f f e c t h a s b e e n e x p l a i n e d b y t h e a d d i t i o n of a t e r m - e(r. Eext) to t h e H a m i l t o n i a n . H o w e v e r , t h e r e i s a n o t h e r , y e t n e g l e c t e d , t e r m of t h e H a m i l t o n i a n which under the same circumstances might give l i n e a r c o n t r i b u t i o n s of a s i m i l a r o r d e r of m a g n i t u d e . T h i s i s t h e m o d i f i c a t i o n of t h e s p i n - o r b i t co~apling b y t h e e l e c t r i c f i e l d . A c c o r d i n g to eq. (2) w e h a v e for a single electron with unpaired spin

HLS

e

0~ext) - 2 n ~ c 2 ( [ E e X t × P] .S) .

(6)

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

LETTERS

1 September 1962

l a t e d t e n s o r s , which can b e d e t e r m i n e d b y h m p ~ ; o c e d u r e of P r y c e 1). The c o n t r i b u t i o n to t h e g - t e n s o r can be w r i t t e n a s

AGhk = rn2c2 ~ Eext,u ~v euvk x Re (~n

Eo

),

h,k,u,v= x,y,z; Curk=+ l f o r even and p e r m u t a t i o n s ~ yv zk ); a n d Euvk = 0 o t h e r w i s e . where

('r) odd

A n a c c u r a t e d e t e r m i n a t i o n of t h e c o e f f i c i e n t s in t h i s equation i s v e r y d i f f i c u l t . A c r u d e but r e a s o n a b l e e s t i m a t i o n s h o w s t h a t t h i s c o n t r i b u t i o n i s of t h e o r d e r of 10-7 to 10-5 c m / k V . T h i s t e r m i s t h e r e f o r e not m u c h s m a l l e r than t h e one c o n s i d e r e d b y H a m 9). T h e i m p o r t a n t d i f f e r e n c e b e t w e e n t h e s e two, however, is that the spin-orbit modification a l o n e l e a d s to a s y m m e t r i c g - t e n s o r s . The a u t h o r i s v e r y m u c h i n d e b t e d to t h e S w i s s National Science Foundation for financial support and to P r o f . D r . W. Ir~nzig and D r . W. B a l t e n s p e r g e r for helpful discussions.

References 1) M . H . L . P r y c e , Proc. Phys. Soe. A 63 (1950) 25. 2) A.Abragam and M . H . L . Pryce, Proc. Roy. Soc. A 205 (1951) 235. 3) H.A. Bethe and E. E. Salpeter, Encyclopedia of Physics, edited by S. Fltigge, vol. XXX'V (Springer Verlag, B e r lin, 1957). 4) E.U. Condon and G. H. Shortley, The theory of atomic s p e c t r a (Cambridge University P r e s s , 1935). 5) H.M.McConnell, l>roc. Natl. Acad. Sci. U.S. 64 (1958) 766. 6) F.K.Kneubilhl, to be published. 7) W. K~nzig, private communication. 8) G.W. Ludwig and H.H.Woodbury, Phys. Rev. Letters 7 (1961) 240. 9) F . S . H a m , Phys. Rev. Letters 7 (1961) 242.

ERRATUM

B. H . A r m i t & g e and R . E . M e a d s , The 5.18 M e V s t a t e in B10, P h y s i c s L e t t e r s 1 ( 1 9 6 2 ) p . 284. The g r o u p l a b e l l e d B10(5.04) in t h e f i g u r e s h o u l d b e labelled Bl1(5.04).

164

[(OlLhln)--(nlPv[O)