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On transferred hyperfine structure in ferromagnets
Volume 25A, number 2
PHYSICS L E T T E R S
ON T R A N S F E R R E D
HYPERFINE
STRUCTURE
31 July 1967
IN F E R R O M A G N E T S
D. A. SHIRLEY Mullard Cryomagnetic Laborator3'. Clarendon Laboratom.'. Oxford. England Received 7 June 1967
It is suggested that well-known transferred hyperfine structure mechanisms are responsible for the large positive hyperfine fields observed at fluorine, iodine and xenon solutes in iron. Positive fields are predicted for alkali solutes. The hyperfine m a g n e t i c field Hhf at xenon e m bedded in an i r o n lattice has been r e p o r t e d [1] a s (+) 1.04 × 106 Oe. T h i s continues and extends the t r e n d [2,3] of Hhf in d l 0 s 2 p n solutes. A conduct i o n - e l e c t r o n p o l a r i z a t i o n model given by Daniel and F r i e d e l [4] for such solutes in f e r r o m a g n e t s a p p e a r s to p r e d i c t Hhf > 0 for xenon in i r o n if Z, the e x c e s s "valence" of xenon Cover iron) is eight. It is shown below that this model does not apply to r a r e gas and halogen solutes. An a l t e r native theory i s suggested, in which Hhf at F, I and Xe in i r o n a r i s e s from well-known " t r a n s f e r r e d hyperfine s t r u c t u r e " effects that lead to positive p o l a r i z a t i o n of the bound 552 shell. T h e s e effects should give positive c o n t r i b u t i o n s to Hhf for halogens, r a r e g a s e s and a l k a l i s in ferromagnets. In the D a n i e l - F r i e d e l model the solute is r e p r e s e n t e d by a s q u a r e - w e l l potential the depth of which is r e l a t e d to Z by the F r i e d e l sum r u l e [5]. This depth (relative to the F e r m i energy) is s p i n independent, but the bottoms of the " s p i n - u p " and "spin-down" h a l f - b a n d s a r e offset by twice the s - d exchange e n e r g y . The r e s u l t i n g difference in s - w a v e phase shifts for the two h a l f - b a n d s c r e a t e s a n o n z e r o spin density at the solute n u cleus. T h e s e r e s u l t s a r e s u m m a r i z e d [4] in the Z-dependence of the p o l a r i z a t i o n p a r a m e t e r s / ( f " ) for f r e e (bound) s e l e c t r o n s . F r e e s - w a v e e l e c t r o n s c o n t r i b u t e to Hhf in proportion t o f ' : H~Lf cc 6p' (0) c c f ' [u(0)l 2. A s i m i l a r equation holds for the bound e l e c t r o n s . The atomic functions a s s o c i a t e d with free and bound s t a t e s a r e a s s u m e d to be identical, making f ' and f " s i m p l y additive. The fact that f = f ' + f " b e c o m e s positive for l a r g e Z is taken * National Science Foundation Senior Postdoctoral Fellow. Present address: University of California, Berkeley.
a s evidence for Hhf b e c o m i n g positive. We have extended the calculation to Z > 7, c o n f i r m i n g that the t r e n d s f ' and f " continue. The s u c c e s s of the F r i e d e l model for s m a l l Z i s well-known. With i n c r e a s i n g Z, however, bound states appear and sink into the well, becoming atomic s t a t e s of the solute ion core. A s q u a r e well potential cannot s i m u l a t e an atomic s p e c t r u m , so the model m u s t u l t i m a t e l y b r e a k down. Even in atomic xenon the 55 state i s about 10 eV below the 5p state. Attempts to use the D a n i e l F r i e d e l model for F__~eXe lead to s e r i o u s i n c o n s i s t e n c i e s . F o r example, the total s - w a v e d e n sity p(0) = Pt(0) + P4(0) r e q u i r e d to give a 6p(0) c o r r e s p o n d i n g to the o b s e r v e d Hhf is 2.5 t i m e s that of a filled 552 shell. Modification of the model by a s s i g n i n g p r i m a r i l y 55 c h a r a c t e r to the a tomic component of the bound state and 6s. 75, etc. c h a r a c t e r to the f r e e s - w a v e gives Hhf < 0, b e cause t h e b o u n d state ~.s negatively polarized, and l u(0)l Ks ~ 101u(0)[6s'Z This a r g u m e n t casts doubt on the validity of the D a n i e l - F r i e d e l model when bound s t a t e s a r e p r e s e n t . In seeking an a l t e r n a t i v e explanation for the v a l u e s of Hhf o b s e r v e d at nuclei of I and Xe in i r o n , it is c l e a r that only d i r e c t p o l a r i z a t i o n , by the host, of solute s e l e c t r o n s in closed s h e l l s can c r e a t e fields of sufficient magnitude. The solute d shell is filled, o r b i t a l c o n t r i b u t i o n s from the p shell a r e s u r e l y quenched, and core p o l a r i zation by 5p e l e c t r o n s p r o d u c e s [6] an Hhf of only
3 × 105 o e / s p i n .
For n > 5, lu(0)t s is too small
to c r e a t e an Hhf of 106 Oe in iron. By c o n t r a s t , lu(°)[ sv i n c r e a s e s rapidiy with atomic n u m b e r for the sequence Sb, Te, I, Xe . . . , as does Hhf for these e l e m e n t s as solutes in iron. A 5s e l e c t r o n in n e u t r a l iodine can give a m a g n e t i c field H5s ~ 3 × 107 Oe at the n u c l e u s : thus only + 3% p o l a r i z a t i o n of the 5s shell is n e c e s s a r y to yield the o b s e r v e d [3] Hhf = 1.13 × 106 Oe on iodine in 129
Volume 25A, number 2
PHYSICS
iron. Two well-known m e c h a n i s m s can account f o r this p o l a r i z a t i o n : the " P a u l i d i s t o r t i o n effect" and covalent bonding. T h e s e e f f e c t s have been i n voked to explain s u c c e s s f u l l y (if not q u a n t i t a t i v e ly) t r a n s f e r r e d hyperfine s t r u c t u r e on ligands on d i e l e c t r i c solids such as MnF 2 and KMnF 3 [7]. Of the two m e c h a n i s m s , the d i s t o r t i o n effect is m o r e e a s i l y e s t i m a t e d and probably l a r g e r . We d i s c u s s it alone, noting that covalent bonding would enhance, r a t h e r than r e d u c e , the r e s u l t a n t Hhf. O r t h o g o n a l i z a t i o n of the host 3d and solute 5s wave functions l e a d s to spin p o l a r i z a t i o n . F o l lowing F r e e m a n and W a t s o n ' s a r g u m e n t s for the fluorine 2s e l e c t r o n s in KMnF3, we may d e s c r i b e this p r o c e s s by the transformat~or~ 13d?> ~ 13dD, 5s~>--~ 5sb>, 1 5 s t ) ~ [1-<3d[5s>Z] -~, {tSsl>-<3d 5s> 13dT>}. Only e x c e s s " s p i n - u p " 3d e l e c tron s t a t e s (denoted as ]3d[>) n e e d b e c o n s i d e r e d , as the se alone will lead to a n o n z e r o net c o n t r i b u tion to Hhf. On this m o d e l we have Hhf = = 1<3dl 5s) IzH5s. E s t i m a t e s b a s e d e i t h e r on a t o m i c s i z e or on c l a s s i c a l e l a s t i c i t y [4,8] (with s o m e extrapolation), give R ~ 6 a.u. as the s o l u t e - s o l v e n t i n t e r n u c l e a r d i s t a n c e in F__eeXe. Using H a r t r e e - F o c k f r e e - a t o m wave functions * f o r Xe and those of De C ic c o and Kitz [9] f o r i ro n , we find (3d[ 5s~ 2 ~ 10-2 for this spacing. Summing o v e r n e a r e s t n e i g h b o r s y i e l d s Hhf(Xe) ~ + 3 × 106 Oe. The s e n s i t i v i t y of(3dlSs) to R is such that Hhf changes a f a c t o r of two for 5R = 0.3 a.u. F o r s e v e r a l r e a s o n s we b e l i e v e that this e s t i m a t e of Hhf is too high, but it is c l e a r that t r a n s f e r r e d h y p e r f i n e s t r u c t u r e m e c h a n i s m s can e a s i l y account f o r the o b s e r v e d Hhf = 1.04 × 106 Oe for Xe in iron. This l a t t e r figure would be a l o w e r l i m i t if s o m e of the xenon a t o m s in the e x p e r i m e n t of N i e s e n et al. went i n to the iron l at t i ce n o n - s u b s t i t u t i o n a l l y . This model is c o n s i s t e n t with all the a v a i l a b l e
LETTERS
data on solute f i el d s. It p r e d i c t s Hhf ~ + 105 Oe f o r f l u o r i n e in i r o n (the o b s e r v e d value [10] is + 87.6 kOe). The t r a n s f e r r e d h y p er f i n e s t r u c t u r e e f f e c t s should be negligible (and a r e not o b s e r v e d ) in t r a n s i t i o n m e t a l s o l u t e s , and the m o d el g i v e s p o s i t i v e co n t r i b u t i o n s to Hhf for y t t r i u m in iron (Hhf(ObS) = + 286 kOe [11]). Finally, a t h e o r y should have p r e d i c t i v e value. The t r a n s f e r r e d h y p er f i n e s t r u c t u r e m e c h a n i s m s lead to p o s i t i v e Hhf'S in iron f o r all solute h a l o gens and r a r e g a s e s . C o m p a r i s o n of o v e r l a p i n t e g r a l s s u g g e s t s p o s i t i v e Hhf'S for solute a l k a l i s and p e r h a p s alkaline e a r t h s as well, while cond u c t i o n - e l e c t r o n p o l a r i z a t i o n alone would give Hhf < 0 for t h e s e l a t t e r two g r o u p s. R e l a t i v e o v e r l a p i n t e g r a l s s u g g e s t s a d e c r e a s e in Hhf by 1 f r o m r a r e gas to alkali. Thus a value Hhf ~ + (0.5 to 1) × 106 Oe is e x p e c t e d f o r Cs in iron. It is a p l e a s u r e to acknowledge the hospitality of the Clarendon L a b o r a t o r y during the t i m e that this work was c a r r i e d out.
References
* Mr. C. S. Fadley kindly calculated the Hartree-Fock wave functions using a program written by C. C. J. Roothaan and P. Bag'us. * * * * *
130
31 July 1967
1. L. Niessen, J. Lubbers, H. Postma, H. De Waard and S. A. Drent]e, Phys. Letters 24B {1967) 144. 2. R.B. Frankel, J . J . Huntzicker, E. Matthias, S.S. Rosenblum, D.A. Shirley and N. J. Stone, Phys. Letters 15 {1965) 163. 3. H. De Waard and S. A. Drentje, Phys. Letters 20 (1966) 38. 4. E. Daniel and J. Friedel, J. Phys. Chem. Sol. 24 (1963} 1601. 5. J. Friedel, Advanc. Phys. 3 {1954)446. 6. A.M. Clogston and V. Jaccarino, Phys. Rev. 121 (1961) 1357. 7. A.J. Freeman and R.E.Watson, in Magnetism, eds. G. T. Rado and H. Suhl (Academic Press, 1965) Vol. IIA, p. 168. 8. E.J. Blatt, Phys. Rev. 108 (1957) 285. 9. P.D. De Cicco and A. Kitz, Quarterly Progress Report No. 60, Solid-State and Molecular Theory Group, MIT, 1966, p. 9. 10. H.Korner, private communication, May, 1966. 11. M. Kontani, K. Asayama and J. Itoh, J. Phys. Soc. Japan 20 (1965) 1737.
PHYSICS L E T T E R S
ON T R A N S F E R R E D
HYPERFINE
STRUCTURE
31 July 1967
IN F E R R O M A G N E T S
D. A. SHIRLEY Mullard Cryomagnetic Laborator3'. Clarendon Laboratom.'. Oxford. England Received 7 June 1967
It is suggested that well-known transferred hyperfine structure mechanisms are responsible for the large positive hyperfine fields observed at fluorine, iodine and xenon solutes in iron. Positive fields are predicted for alkali solutes. The hyperfine m a g n e t i c field Hhf at xenon e m bedded in an i r o n lattice has been r e p o r t e d [1] a s (+) 1.04 × 106 Oe. T h i s continues and extends the t r e n d [2,3] of Hhf in d l 0 s 2 p n solutes. A conduct i o n - e l e c t r o n p o l a r i z a t i o n model given by Daniel and F r i e d e l [4] for such solutes in f e r r o m a g n e t s a p p e a r s to p r e d i c t Hhf > 0 for xenon in i r o n if Z, the e x c e s s "valence" of xenon Cover iron) is eight. It is shown below that this model does not apply to r a r e gas and halogen solutes. An a l t e r native theory i s suggested, in which Hhf at F, I and Xe in i r o n a r i s e s from well-known " t r a n s f e r r e d hyperfine s t r u c t u r e " effects that lead to positive p o l a r i z a t i o n of the bound 552 shell. T h e s e effects should give positive c o n t r i b u t i o n s to Hhf for halogens, r a r e g a s e s and a l k a l i s in ferromagnets. In the D a n i e l - F r i e d e l model the solute is r e p r e s e n t e d by a s q u a r e - w e l l potential the depth of which is r e l a t e d to Z by the F r i e d e l sum r u l e [5]. This depth (relative to the F e r m i energy) is s p i n independent, but the bottoms of the " s p i n - u p " and "spin-down" h a l f - b a n d s a r e offset by twice the s - d exchange e n e r g y . The r e s u l t i n g difference in s - w a v e phase shifts for the two h a l f - b a n d s c r e a t e s a n o n z e r o spin density at the solute n u cleus. T h e s e r e s u l t s a r e s u m m a r i z e d [4] in the Z-dependence of the p o l a r i z a t i o n p a r a m e t e r s / ( f " ) for f r e e (bound) s e l e c t r o n s . F r e e s - w a v e e l e c t r o n s c o n t r i b u t e to Hhf in proportion t o f ' : H~Lf cc 6p' (0) c c f ' [u(0)l 2. A s i m i l a r equation holds for the bound e l e c t r o n s . The atomic functions a s s o c i a t e d with free and bound s t a t e s a r e a s s u m e d to be identical, making f ' and f " s i m p l y additive. The fact that f = f ' + f " b e c o m e s positive for l a r g e Z is taken * National Science Foundation Senior Postdoctoral Fellow. Present address: University of California, Berkeley.
a s evidence for Hhf b e c o m i n g positive. We have extended the calculation to Z > 7, c o n f i r m i n g that the t r e n d s f ' and f " continue. The s u c c e s s of the F r i e d e l model for s m a l l Z i s well-known. With i n c r e a s i n g Z, however, bound states appear and sink into the well, becoming atomic s t a t e s of the solute ion core. A s q u a r e well potential cannot s i m u l a t e an atomic s p e c t r u m , so the model m u s t u l t i m a t e l y b r e a k down. Even in atomic xenon the 55 state i s about 10 eV below the 5p state. Attempts to use the D a n i e l F r i e d e l model for F__~eXe lead to s e r i o u s i n c o n s i s t e n c i e s . F o r example, the total s - w a v e d e n sity p(0) = Pt(0) + P4(0) r e q u i r e d to give a 6p(0) c o r r e s p o n d i n g to the o b s e r v e d Hhf is 2.5 t i m e s that of a filled 552 shell. Modification of the model by a s s i g n i n g p r i m a r i l y 55 c h a r a c t e r to the a tomic component of the bound state and 6s. 75, etc. c h a r a c t e r to the f r e e s - w a v e gives Hhf < 0, b e cause t h e b o u n d state ~.s negatively polarized, and l u(0)l Ks ~ 101u(0)[6s'Z This a r g u m e n t casts doubt on the validity of the D a n i e l - F r i e d e l model when bound s t a t e s a r e p r e s e n t . In seeking an a l t e r n a t i v e explanation for the v a l u e s of Hhf o b s e r v e d at nuclei of I and Xe in i r o n , it is c l e a r that only d i r e c t p o l a r i z a t i o n , by the host, of solute s e l e c t r o n s in closed s h e l l s can c r e a t e fields of sufficient magnitude. The solute d shell is filled, o r b i t a l c o n t r i b u t i o n s from the p shell a r e s u r e l y quenched, and core p o l a r i zation by 5p e l e c t r o n s p r o d u c e s [6] an Hhf of only
3 × 105 o e / s p i n .
For n > 5, lu(0)t s is too small
to c r e a t e an Hhf of 106 Oe in iron. By c o n t r a s t , lu(°)[ sv i n c r e a s e s rapidiy with atomic n u m b e r for the sequence Sb, Te, I, Xe . . . , as does Hhf for these e l e m e n t s as solutes in iron. A 5s e l e c t r o n in n e u t r a l iodine can give a m a g n e t i c field H5s ~ 3 × 107 Oe at the n u c l e u s : thus only + 3% p o l a r i z a t i o n of the 5s shell is n e c e s s a r y to yield the o b s e r v e d [3] Hhf = 1.13 × 106 Oe on iodine in 129
Volume 25A, number 2
PHYSICS
iron. Two well-known m e c h a n i s m s can account f o r this p o l a r i z a t i o n : the " P a u l i d i s t o r t i o n effect" and covalent bonding. T h e s e e f f e c t s have been i n voked to explain s u c c e s s f u l l y (if not q u a n t i t a t i v e ly) t r a n s f e r r e d hyperfine s t r u c t u r e on ligands on d i e l e c t r i c solids such as MnF 2 and KMnF 3 [7]. Of the two m e c h a n i s m s , the d i s t o r t i o n effect is m o r e e a s i l y e s t i m a t e d and probably l a r g e r . We d i s c u s s it alone, noting that covalent bonding would enhance, r a t h e r than r e d u c e , the r e s u l t a n t Hhf. O r t h o g o n a l i z a t i o n of the host 3d and solute 5s wave functions l e a d s to spin p o l a r i z a t i o n . F o l lowing F r e e m a n and W a t s o n ' s a r g u m e n t s for the fluorine 2s e l e c t r o n s in KMnF3, we may d e s c r i b e this p r o c e s s by the transformat~or~ 13d?> ~ 13dD, 5s~>--~ 5sb>, 1 5 s t ) ~ [1-<3d[5s>Z] -~, {tSsl>-<3d 5s> 13dT>}. Only e x c e s s " s p i n - u p " 3d e l e c tron s t a t e s (denoted as ]3d[>) n e e d b e c o n s i d e r e d , as the se alone will lead to a n o n z e r o net c o n t r i b u tion to Hhf. On this m o d e l we have Hhf = = 1<3dl 5s) IzH5s. E s t i m a t e s b a s e d e i t h e r on a t o m i c s i z e or on c l a s s i c a l e l a s t i c i t y [4,8] (with s o m e extrapolation), give R ~ 6 a.u. as the s o l u t e - s o l v e n t i n t e r n u c l e a r d i s t a n c e in F__eeXe. Using H a r t r e e - F o c k f r e e - a t o m wave functions * f o r Xe and those of De C ic c o and Kitz [9] f o r i ro n , we find (3d[ 5s~ 2 ~ 10-2 for this spacing. Summing o v e r n e a r e s t n e i g h b o r s y i e l d s Hhf(Xe) ~ + 3 × 106 Oe. The s e n s i t i v i t y of(3dlSs) to R is such that Hhf changes a f a c t o r of two for 5R = 0.3 a.u. F o r s e v e r a l r e a s o n s we b e l i e v e that this e s t i m a t e of Hhf is too high, but it is c l e a r that t r a n s f e r r e d h y p e r f i n e s t r u c t u r e m e c h a n i s m s can e a s i l y account f o r the o b s e r v e d Hhf = 1.04 × 106 Oe for Xe in iron. This l a t t e r figure would be a l o w e r l i m i t if s o m e of the xenon a t o m s in the e x p e r i m e n t of N i e s e n et al. went i n to the iron l at t i ce n o n - s u b s t i t u t i o n a l l y . This model is c o n s i s t e n t with all the a v a i l a b l e
LETTERS
data on solute f i el d s. It p r e d i c t s Hhf ~ + 105 Oe f o r f l u o r i n e in i r o n (the o b s e r v e d value [10] is + 87.6 kOe). The t r a n s f e r r e d h y p er f i n e s t r u c t u r e e f f e c t s should be negligible (and a r e not o b s e r v e d ) in t r a n s i t i o n m e t a l s o l u t e s , and the m o d el g i v e s p o s i t i v e co n t r i b u t i o n s to Hhf for y t t r i u m in iron (Hhf(ObS) = + 286 kOe [11]). Finally, a t h e o r y should have p r e d i c t i v e value. The t r a n s f e r r e d h y p er f i n e s t r u c t u r e m e c h a n i s m s lead to p o s i t i v e Hhf'S in iron f o r all solute h a l o gens and r a r e g a s e s . C o m p a r i s o n of o v e r l a p i n t e g r a l s s u g g e s t s p o s i t i v e Hhf'S for solute a l k a l i s and p e r h a p s alkaline e a r t h s as well, while cond u c t i o n - e l e c t r o n p o l a r i z a t i o n alone would give Hhf < 0 for t h e s e l a t t e r two g r o u p s. R e l a t i v e o v e r l a p i n t e g r a l s s u g g e s t s a d e c r e a s e in Hhf by 1 f r o m r a r e gas to alkali. Thus a value Hhf ~ + (0.5 to 1) × 106 Oe is e x p e c t e d f o r Cs in iron. It is a p l e a s u r e to acknowledge the hospitality of the Clarendon L a b o r a t o r y during the t i m e that this work was c a r r i e d out.
References
* Mr. C. S. Fadley kindly calculated the Hartree-Fock wave functions using a program written by C. C. J. Roothaan and P. Bag'us. * * * * *
130
31 July 1967
1. L. Niessen, J. Lubbers, H. Postma, H. De Waard and S. A. Drent]e, Phys. Letters 24B {1967) 144. 2. R.B. Frankel, J . J . Huntzicker, E. Matthias, S.S. Rosenblum, D.A. Shirley and N. J. Stone, Phys. Letters 15 {1965) 163. 3. H. De Waard and S. A. Drentje, Phys. Letters 20 (1966) 38. 4. E. Daniel and J. Friedel, J. Phys. Chem. Sol. 24 (1963} 1601. 5. J. Friedel, Advanc. Phys. 3 {1954)446. 6. A.M. Clogston and V. Jaccarino, Phys. Rev. 121 (1961) 1357. 7. A.J. Freeman and R.E.Watson, in Magnetism, eds. G. T. Rado and H. Suhl (Academic Press, 1965) Vol. IIA, p. 168. 8. E.J. Blatt, Phys. Rev. 108 (1957) 285. 9. P.D. De Cicco and A. Kitz, Quarterly Progress Report No. 60, Solid-State and Molecular Theory Group, MIT, 1966, p. 9. 10. H.Korner, private communication, May, 1966. 11. M. Kontani, K. Asayama and J. Itoh, J. Phys. Soc. Japan 20 (1965) 1737.