- Email: [email protected]
Covalency in the rare-earth trichlorides
Volume 23, number I
PHYSICS
p r o x i m a t i o n , r e s p e c t i v e l y , and A' - 4 is a c o r r e c t i o n to the e f f e c t i v e m a s s t h e o r y [6]. We a r e now in a p o s i t i o n to c o m p a r e the t h e o r y with the e x p e r i m e n t [1]. In f i g . 1 we h a v e p l o t t e d t h e o r e t i c a l and e x p e r i m e n t a l a t t e n u a t i o n c o e f f i c i e n t s in S b - , P - and A s - d o p e d Ge, r e s p e c t i v e l y . In p l o t t i n g the t h e o r e t i c a l c u r v e s we h a v e taken the f o l l o w i n g v a l u e s of p h y s i c a l p a r a m e t e r s : N o =
LETTERS
3October 1966
of c~ and p e r h a p s of N o. H o w e v e r , f u r t h e r d e t a i l e d e x p e r i m e n t s m a y c l a r i f y the t r u e o r i g i n of the d i s c r e p a n c y in fig. 1. A f t e r t h i s w o r k has b e e n c o m p l e t e d we h a v e s e e n the a b s t r a c t of the p a p e r by P . C. Kwok (to be p u b l i s h e d in P h y s . R e v . ), who has t r e a t e d the same problem as ours.
= 3 x 1015 cm -3, v s = 3.56 × 105 c m / s e c , c44 =
= 0.68 × 1012 d y n e s / c m 2, ~ u = 19 eV, a ( e f f . m a s s ) = 45 ~ , Edo = 9.2 × 10-3 eV; 4 4 = 0.57 (Sb), 2.9 (P) 4.15 (As) × 10-3 eV; 4 ' - 4 = 0.68 (Sb), 0.74 (P), 0.63 (As) x 10-3 eV. On the o t h e r hand, the e x p e r i m e n t a l v a l u e s of a h a v e b e e n o b t a i n e d by s u b t r a c t i n g the a t t e n u a t i o n in undoped Ge f r o m t h o s e of Sb-, P - and A s - d o p e d Ge. The a g r e e m e n t b e t w e e n t h e o r y and e x p e r i m e n t is f a i r l y s a t i s f a c t o r y in v i e w of e x p e r i m e n t a l u n c e r t a i n t i e s
COVALENCY
IN
THE
References 1. M. Pomerantz, Proc. IEEE 53 (1965) 1438. 2. C. Zener, Elasticity and anelasticity of metals (Univ. of Chicago Press, Chicago, 1948). 3. R.W. Keyes, IBM J. Res. Develop, 5 (1961) 266. 4. H.Hasegawa, Phys. Rev.118 (1960) 1523. 5. A. Griffin and P. Carruthers, Phys.Rev. 131 (1963) 1976. 6. J . H . R e u s z e r and P. Fisher, Phys. Rev. 135 (1964) Al125.
RARE-EARTH
TRICHLORIDES
D . J . N E W M A N and M . M . E L L I S
Department of Physics, Queen Mary College, London E. 1 Received 23 August 1966
On the basis that the zero field splitting of Gd 3 ~:LaC13 is due to spin-orbit induced covalency, it is shown that the covalency contribution to the crystal field in r a r e - e a r t h trichlorides is about 5%.
In a r e c e n t d i s c u s s i o n of the e x p e r i m e n t a l d a t a J o h n s t o n et al. [1] h a v e s u g g e s t e d that the obs e r v e d z e r o - f i e l d s p l i t t i n g s of the 8S½ g r o u n d l e v el of Gd 3+ in v a r i o u s h o s t c r y s t a l s m i g h t be due c o v a l e n c y . We s h o w b e l o w that the o b s e r v e d o r d e r of m a g n i t u d e of the s p l i t t i n g in LaC13 can ind e e d be e x p l a i n e d in t e r m s of c o v a l e n c y induced by the s p i n - o r b i t c o u p l i n g . T h i s is a r a t h e r s i m p l e r m e c h a n i s m than that d i s c u s s e d in [1 ] f o r it d o e s not i n v o l v e the 6p~ s t a t e s of the ion. The d e g r e e of c o v a l e n c y r e q u i r e d is shown to p r o d u c e only a s m a l l c o n t r i b u t i o n to the c r y s t a l f i e l d s p l i t ting in the r a r e - e a r t h c h l o r i d e s . L e t the v a r i o u s d e g e n e r a t e 8S~ e i g e n s t a t e s of the c o m p l e x (GdC19)6- (containing a Gd 3+ ion at its c e n t r e ) be d e n o t e d ~ M . The e x c i t e d s t a t e s , in which an e l e c t r o n t r a n s f e r s f r o m the l i g a n d s to the Gd 3+ ion, w i l l be w r i t t e n ÷ i . If p r o p e r l y o r t h o g o n a l i z e d L C A O o r b i t a l s a r e u s e d to c o n s t r u c t t h e s e f u n c t i o n s , s o that (~M I~i> = 0, the c o v a l e n c y e f f e c t s m a y be i n t r o d u c e d a s a p e r t u r b a t i v e 46
r e d u c t i o n of the g r o u n d e n e r g y
ES: to E M =EMo +~.l(&i [H I&M~ 12/AEi terms
+higher order
z
where
Ei
M =Eo -
E l , Ei
= < ÷ ~ l ~ l + i > is Eo M
In t h e s e f o r m u l a e H is the t o t a l H a m i l t o n i a n of the c o m p l e x and E oM is the " e x c l u s i o n m o d e l " e n e r g y w h i c h has a l r e a d y b e e n d i s c u s s e d in r e f . 2. We now w r i t e H = H o + //so w h e r e H o c o n t a i n s the k i n e t i c and e l e c t r o s t a t i c e n e r g y c o n t r i b u t i o n s and H s o =.~ ~(ri)! i.s i is the s p i n o r b i t c o u p l i n g . It is c l e a r that (~oMjHo I÷M> is i n d e p e n d e n t of M and h e n c e g i v e s no c o n t r i b u t i o n to the s p l i t t i n g ; a l s o ( & M I H s o J÷oM>= 0 as L = 0 in the g r o u n d level.
Volume 23, number 1
PHYSICS L E T T E R S
The s t a t e s ÷i may be divided into two types. Those in which the spin component r e t a i n s in the value M and those in which it changes to M + 1. The s u m over excited s t a t e s i n c l u d e s both types. Diagonal m a t r i x e l e m e n t s (÷M IH [ ~ / ~ have cont r i b u t i o n s f r o m Ho, but not f r o m //so. Hence the e l e c t r o s t a t i c a l l y induced covalency leaves E M i n dependent of M. However, the off-diagonal m a t r i x e l e m e n t s (~-M I//so 14~M+I ) a r e n o n - v a n i s h i n g . These depend on the value of M and hence cause a splitting of the ground level. In o r d e r to set up an effective i n t e r a c t i o n (V) in t e r m s of o p e r a t o r s which act upon the ground state spin functions alone we w r i t e
~kM = ~o~M ;
E M = (~M 1v I ~ M ) ,
where ~ M is the a p p r o p r i a t e spin function of the Gd 3+ ion and ~o contains e v e r y t h i n g e l s e . Then, to f i r s t o r d e r , •
'
r
_ + li
si+)I~
"M~:I)
.
This e x p r e s s i o n can at m o s t be q u a d r a t i c in the spin o p e r a t o r s , and m u s t at the s a m e time r e t a i n the s y m m e t r y of the complex. Thus, after e v a l u ation of the o r b i t a l i n t e g r a l s and s u m m a t i o n over i n t e r m e d i a t e spin s t a t e s , V r e d u c e s to o
V =-~b2(SS
bO2 +S S+) = 25z + c o n s t ,
where S+, S_, Sz a r e total spin o p e r a t o r s . It follows f r o m the above e x p r e s s i o n for V that b~ > 0 which a c c o r d s with e x p e r i m e n t a l r e s u l t . The e x p e r i m e n t a l v a l u e s of b° , b° and b 6 a r e O "x o U c o n s i d e r a b l y s m a l l e r than b 2 b e c a u s e these coeff i c i e n t s will only a r i s e f r o m higher o r d e r p e r t u r bation t e r m s . • , O In o r d e r to e s t i m a t e the m a g m t u d e of b 2 we w r i t e b~ = AE x Yso where 5E = LXEi is a s s u m e d to be effectively independent of which excited state is c o n s i d e r e d . A r e a s o n a b l e e s t i m a t e for AE is 1 A . U . for the r a r e - e a r t c h c h l o r i d e s . The s p i n o r b i t covalency p a r a m e t e r Yso m u s t be d i s t i n guished f r o m the e l e c t r o s t a t i c p a r a m e t e r (~¢) which has been c a l c u l a t e d in p r e v i o u s work (e. g.
3October 1966
Sugano and Shulman [3]). In o r d e r to get an app r o x i m a t e value we take the ratio of the ~"s to be equal to the ratio of the e l e c t r o s t a t i c and s p i n - o r bit i n t e r a c t i o n e n e r g i e s in the Gd3+ ion, i, e. ~so = [1500/(½x2"2×105)]Ve = l " 4 x 1 0 - 2 ~ e • Our c a l c u l a t i o n s for PrC13 give ?e = 0. 006, so that Vso = 0.8 × 10 .4 and b 2 = 14 x 10 -4 cm -1, which c o m p a r e s favourably with the o b s e r v e d value of 16 x 10-4 c m - 1 . The values of Ye and ~ used above give an e s t i m a t e d contribution to the c r y s t a l field splittings of ~eLXE = 8 cm -1, or about 5% in A~(r 6) and
A~(r6).-- This is in accord with our p r e v i o u s r e su~lt [2] that the "exclusion model" p r o v i d e s a good d e s c r i p t i o n of the c r y s t a l field splitting in PrC13. It a p p e a r s that e. s . r . m e a s u r e m e n t s on the Gd 3+ ion, unlike any other e x p e r i m e n t a l data, provide us with a d i r e c t m e a n s of finding the deg r e e of covalency in v a r i o u s host c r y s t a l s . F o r example, the r e s u l t s given in [1] indicate covalency c o n t r i b u t i o n s to A6(r 6) of about 80 cm -1 for Gd3+: LaBr3 and 260 cm -1 for Gd 3+ in the ethyl sulphate. This may explain the l a r g e r v a l u e s of A~(r6) o b s e r v e d for r a r e - e a r t h ions in the ethyl sulphates as c o m p a r e d with the anhydrous chlor i d e s and b r o m i d e s . T h e r e s e e m s to be no s i m p l e explanation for the negative sign of b~ for Gd 3+ in lanthanum sulphate• M. E. E. wishes to thank S. R . C . for a R e s e a r c h Studentship.
References 1. D.R.Johnston, E.Y. Wong and O.M.Stafsudd, J• Chem. Phys. 44 (1966) 2693. 2. M.M. Ellis and D.J. Newman, Phys. Letters 21 (1966) 508.. 3. S. Sugano and R. G. Shulman, Phys. Rev. 130 {1963) 517.
47
PHYSICS
p r o x i m a t i o n , r e s p e c t i v e l y , and A' - 4 is a c o r r e c t i o n to the e f f e c t i v e m a s s t h e o r y [6]. We a r e now in a p o s i t i o n to c o m p a r e the t h e o r y with the e x p e r i m e n t [1]. In f i g . 1 we h a v e p l o t t e d t h e o r e t i c a l and e x p e r i m e n t a l a t t e n u a t i o n c o e f f i c i e n t s in S b - , P - and A s - d o p e d Ge, r e s p e c t i v e l y . In p l o t t i n g the t h e o r e t i c a l c u r v e s we h a v e taken the f o l l o w i n g v a l u e s of p h y s i c a l p a r a m e t e r s : N o =
LETTERS
3October 1966
of c~ and p e r h a p s of N o. H o w e v e r , f u r t h e r d e t a i l e d e x p e r i m e n t s m a y c l a r i f y the t r u e o r i g i n of the d i s c r e p a n c y in fig. 1. A f t e r t h i s w o r k has b e e n c o m p l e t e d we h a v e s e e n the a b s t r a c t of the p a p e r by P . C. Kwok (to be p u b l i s h e d in P h y s . R e v . ), who has t r e a t e d the same problem as ours.
= 3 x 1015 cm -3, v s = 3.56 × 105 c m / s e c , c44 =
= 0.68 × 1012 d y n e s / c m 2, ~ u = 19 eV, a ( e f f . m a s s ) = 45 ~ , Edo = 9.2 × 10-3 eV; 4 4 = 0.57 (Sb), 2.9 (P) 4.15 (As) × 10-3 eV; 4 ' - 4 = 0.68 (Sb), 0.74 (P), 0.63 (As) x 10-3 eV. On the o t h e r hand, the e x p e r i m e n t a l v a l u e s of a h a v e b e e n o b t a i n e d by s u b t r a c t i n g the a t t e n u a t i o n in undoped Ge f r o m t h o s e of Sb-, P - and A s - d o p e d Ge. The a g r e e m e n t b e t w e e n t h e o r y and e x p e r i m e n t is f a i r l y s a t i s f a c t o r y in v i e w of e x p e r i m e n t a l u n c e r t a i n t i e s
COVALENCY
IN
THE
References 1. M. Pomerantz, Proc. IEEE 53 (1965) 1438. 2. C. Zener, Elasticity and anelasticity of metals (Univ. of Chicago Press, Chicago, 1948). 3. R.W. Keyes, IBM J. Res. Develop, 5 (1961) 266. 4. H.Hasegawa, Phys. Rev.118 (1960) 1523. 5. A. Griffin and P. Carruthers, Phys.Rev. 131 (1963) 1976. 6. J . H . R e u s z e r and P. Fisher, Phys. Rev. 135 (1964) Al125.
RARE-EARTH
TRICHLORIDES
D . J . N E W M A N and M . M . E L L I S
Department of Physics, Queen Mary College, London E. 1 Received 23 August 1966
On the basis that the zero field splitting of Gd 3 ~:LaC13 is due to spin-orbit induced covalency, it is shown that the covalency contribution to the crystal field in r a r e - e a r t h trichlorides is about 5%.
In a r e c e n t d i s c u s s i o n of the e x p e r i m e n t a l d a t a J o h n s t o n et al. [1] h a v e s u g g e s t e d that the obs e r v e d z e r o - f i e l d s p l i t t i n g s of the 8S½ g r o u n d l e v el of Gd 3+ in v a r i o u s h o s t c r y s t a l s m i g h t be due c o v a l e n c y . We s h o w b e l o w that the o b s e r v e d o r d e r of m a g n i t u d e of the s p l i t t i n g in LaC13 can ind e e d be e x p l a i n e d in t e r m s of c o v a l e n c y induced by the s p i n - o r b i t c o u p l i n g . T h i s is a r a t h e r s i m p l e r m e c h a n i s m than that d i s c u s s e d in [1 ] f o r it d o e s not i n v o l v e the 6p~ s t a t e s of the ion. The d e g r e e of c o v a l e n c y r e q u i r e d is shown to p r o d u c e only a s m a l l c o n t r i b u t i o n to the c r y s t a l f i e l d s p l i t ting in the r a r e - e a r t h c h l o r i d e s . L e t the v a r i o u s d e g e n e r a t e 8S~ e i g e n s t a t e s of the c o m p l e x (GdC19)6- (containing a Gd 3+ ion at its c e n t r e ) be d e n o t e d ~ M . The e x c i t e d s t a t e s , in which an e l e c t r o n t r a n s f e r s f r o m the l i g a n d s to the Gd 3+ ion, w i l l be w r i t t e n ÷ i . If p r o p e r l y o r t h o g o n a l i z e d L C A O o r b i t a l s a r e u s e d to c o n s t r u c t t h e s e f u n c t i o n s , s o that (~M I~i> = 0, the c o v a l e n c y e f f e c t s m a y be i n t r o d u c e d a s a p e r t u r b a t i v e 46
r e d u c t i o n of the g r o u n d e n e r g y
ES:
+higher order
z
where
Ei
M =Eo -
E l , Ei
= < ÷ ~ l ~ l + i > is Eo M
In t h e s e f o r m u l a e H is the t o t a l H a m i l t o n i a n of the c o m p l e x and E oM is the " e x c l u s i o n m o d e l " e n e r g y w h i c h has a l r e a d y b e e n d i s c u s s e d in r e f . 2. We now w r i t e H = H o + //so w h e r e H o c o n t a i n s the k i n e t i c and e l e c t r o s t a t i c e n e r g y c o n t r i b u t i o n s and H s o =.~ ~(ri)! i.s i is the s p i n o r b i t c o u p l i n g . It is c l e a r that (~oMjHo I÷M> is i n d e p e n d e n t of M and h e n c e g i v e s no c o n t r i b u t i o n to the s p l i t t i n g ; a l s o ( & M I H s o J÷oM>= 0 as L = 0 in the g r o u n d level.
Volume 23, number 1
PHYSICS L E T T E R S
The s t a t e s ÷i may be divided into two types. Those in which the spin component r e t a i n s in the value M and those in which it changes to M + 1. The s u m over excited s t a t e s i n c l u d e s both types. Diagonal m a t r i x e l e m e n t s (÷M IH [ ~ / ~ have cont r i b u t i o n s f r o m Ho, but not f r o m //so. Hence the e l e c t r o s t a t i c a l l y induced covalency leaves E M i n dependent of M. However, the off-diagonal m a t r i x e l e m e n t s (~-M I//so 14~M+I ) a r e n o n - v a n i s h i n g . These depend on the value of M and hence cause a splitting of the ground level. In o r d e r to set up an effective i n t e r a c t i o n (V) in t e r m s of o p e r a t o r s which act upon the ground state spin functions alone we w r i t e
~kM = ~o~M ;
E M = (~M 1v I ~ M ) ,
where ~ M is the a p p r o p r i a t e spin function of the Gd 3+ ion and ~o contains e v e r y t h i n g e l s e . Then, to f i r s t o r d e r , •
'
r
_ + li
si+)I~
"M~:I)
.
This e x p r e s s i o n can at m o s t be q u a d r a t i c in the spin o p e r a t o r s , and m u s t at the s a m e time r e t a i n the s y m m e t r y of the complex. Thus, after e v a l u ation of the o r b i t a l i n t e g r a l s and s u m m a t i o n over i n t e r m e d i a t e spin s t a t e s , V r e d u c e s to o
V =-~b2(SS
bO2 +S S+) = 25z + c o n s t ,
where S+, S_, Sz a r e total spin o p e r a t o r s . It follows f r o m the above e x p r e s s i o n for V that b~ > 0 which a c c o r d s with e x p e r i m e n t a l r e s u l t . The e x p e r i m e n t a l v a l u e s of b° , b° and b 6 a r e O "x o U c o n s i d e r a b l y s m a l l e r than b 2 b e c a u s e these coeff i c i e n t s will only a r i s e f r o m higher o r d e r p e r t u r bation t e r m s . • , O In o r d e r to e s t i m a t e the m a g m t u d e of b 2 we w r i t e b~ = AE x Yso where 5E = LXEi is a s s u m e d to be effectively independent of which excited state is c o n s i d e r e d . A r e a s o n a b l e e s t i m a t e for AE is 1 A . U . for the r a r e - e a r t c h c h l o r i d e s . The s p i n o r b i t covalency p a r a m e t e r Yso m u s t be d i s t i n guished f r o m the e l e c t r o s t a t i c p a r a m e t e r (~¢) which has been c a l c u l a t e d in p r e v i o u s work (e. g.
3October 1966
Sugano and Shulman [3]). In o r d e r to get an app r o x i m a t e value we take the ratio of the ~"s to be equal to the ratio of the e l e c t r o s t a t i c and s p i n - o r bit i n t e r a c t i o n e n e r g i e s in the Gd3+ ion, i, e. ~so = [1500/(½x2"2×105)]Ve = l " 4 x 1 0 - 2 ~ e • Our c a l c u l a t i o n s for PrC13 give ?e = 0. 006, so that Vso = 0.8 × 10 .4 and b 2 = 14 x 10 -4 cm -1, which c o m p a r e s favourably with the o b s e r v e d value of 16 x 10-4 c m - 1 . The values of Ye and ~ used above give an e s t i m a t e d contribution to the c r y s t a l field splittings of ~eLXE = 8 cm -1, or about 5% in A~(r 6) and
A~(r6).-- This is in accord with our p r e v i o u s r e su~lt [2] that the "exclusion model" p r o v i d e s a good d e s c r i p t i o n of the c r y s t a l field splitting in PrC13. It a p p e a r s that e. s . r . m e a s u r e m e n t s on the Gd 3+ ion, unlike any other e x p e r i m e n t a l data, provide us with a d i r e c t m e a n s of finding the deg r e e of covalency in v a r i o u s host c r y s t a l s . F o r example, the r e s u l t s given in [1] indicate covalency c o n t r i b u t i o n s to A6(r 6) of about 80 cm -1 for Gd3+: LaBr3 and 260 cm -1 for Gd 3+ in the ethyl sulphate. This may explain the l a r g e r v a l u e s of A~(r6) o b s e r v e d for r a r e - e a r t h ions in the ethyl sulphates as c o m p a r e d with the anhydrous chlor i d e s and b r o m i d e s . T h e r e s e e m s to be no s i m p l e explanation for the negative sign of b~ for Gd 3+ in lanthanum sulphate• M. E. E. wishes to thank S. R . C . for a R e s e a r c h Studentship.
References 1. D.R.Johnston, E.Y. Wong and O.M.Stafsudd, J• Chem. Phys. 44 (1966) 2693. 2. M.M. Ellis and D.J. Newman, Phys. Letters 21 (1966) 508.. 3. S. Sugano and R. G. Shulman, Phys. Rev. 130 {1963) 517.
47