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Isomer shift anomaly at the curie temperature of iron
Volume 20, number 2
ISOMER
SHIFT
ANOMALY
PHYSICS
LETTERS
AT
CURIE
THE
1 February 1966
TEMPERATURE
OF
IRON
S. ALEXANDER Department o f P h y s i c s , The Weizrnann Institute of Science, Rehovoth, Israel and
D. T R E V E S D e p a r t m e n t o f E l e c t r o n i c s , The Weizrnann Institute o f Science, Rehovoth, Israel Received 28 December 1965
Several explanations of the shift in the position of the centroid of the M6ssbauer spectrum in metallic iron as the specimen is heated through the Curie temperature are considered.
P r e s t o n et al [1] found t h a t in m e t a l l i c i r o n the p o s i t i o n (b) of the c e n t r o i d of the M 6 s s b a u e r s p e c t r u m c h a n g e s a b r u p t l y a s the s a m p l e is h e a t e d t h r o u g h the C u r i e t e m p e r a t u r e (Tc). The s h i f t is a p p r o x i m a t e l y - 0 . 0 2 m m s e c - 1 and o c c u r s o v e r a t e m p e r a t u r e r a n g e , 10°C. It is t h e r e f o r e c e r t a i n l y a s s o c i a t e d in s o m e w a y w i t h the m a g n e t i c phase transition. Two m e c h a n i s m s w h i c h c a n g i v e r i s e to d i s c o n t i n u i t i e s in b n e a r a p h a s e t r a n s i t i o n h a v e b e e n c o n s i d e r e d in the l i t e r a t u r e : 1. The i s o m e r s h i f t w i l l c h a n g e w h e n the a t o m i c v o l u m e c h a n g e s a b r u p t l y at the p h a s e t r a n s i t i o n [2]. 2. C h a n g e s in the D e b y e t e m p e r a t u r e can c a u s e a d i s c o n t i n u i t y in the s e c o n d o r d e r D o p p l e r s h i f t [1]. Two f u r t h e r m e c h a n i s m s w h i c h can be i m p o r t a n t f o r a m a g n e t i c p h a s e t r a n s i t i o n in a m e t a l h a v e o c c u r r e d to us. T h e s e a r e : 3. B e c a u s e of p h o n o n - m a g n o n i n t e r a c t i o n s o m e of the n u c l e a r k i n e t i c e n e r g y is c o n t a i n e d in relatively high frequency modes which are ess e n t i a l l y m a g n o n s . The c o l l a p s e of the m a g n o n s p e c t r u m at the C u r i e t e m p e r a t u r e t h e r e f o r e l e a d s to a c h a n g e in the s e c o n d o r d e r D o p p l e r shift. 4. N e a r the C u r i e t e m p e r a t u r e m a g n e t i c s p l i t t i n g of the 3d band d i s a p p e a r s . T h i s r e s u l t s in a s h i f t of the a b s o l u t e p o s i t i o n of the F e r m i l e v e l and t h e r e f o r e in a c h a n g e in the n u m b e r of 4s electrons.
We want to s h o w t h a t o n l y the l a s t m e c h a n i s m can a c c o u n t f o r the o b s e r v e d s h i f t in b. P o u n d et al. [3] h a v e d e t e r m i n e d the v o l u m e d e p e n d e n c e of b at r o o m t e m p e r a t u r e 8b/~ In V = - 1 . 4 m m s e c -1. T h i s v a l u e h a s the c o r r e c t o r d e r of m a g n i t u d e to e x p l a i n the d i s c o n t i n u i t y in b at the ot-7 t r a n s i t i o n of i r o n [2] (1200°K). It is t h e r e f o r e s a f e to a s s u m e t h a t it is a l s o r i g h t n e a r the C u r i e t e m p e r a t u r e (T c = 1042°K). With t h i s v o l u m e d e p e n d e n c e the o b s e r v e d c h a n g e in b n e a r T c w o u l d r e q u i r e a v o l u m e c h a n g e of a p p r o x i m a t e ly 1%. No s u c h a n o m a l y i s o b s e r v e d in the t h e r m a l e x p a n s i o n of i r o n [4]. T h i s s e e m s to r u l e out m e c h a n i s m 1. If the o b s e r v e d d i s c o n t i n u i t y in b is due to a c h a n g e in the s e c o n d o r d e r D o p p l e r - s h i f t it would i m p l y a 3% c h a n g e in the n u c l e a r v i b r a t i o n a l e n e r g y . P r e s t o n et al. [1] h a v e shown that s u c h a l a r g e d i s c o n t i n u i t y c a n n o t be e x p l a i n e d by any r e a s o n a b l e c h a n g e in the v i b r a t i o n s p e c t r u m b e c a u s e the C u r i e t e m p e r a t u r e (Tc) in i r o n is h i g h c o m p a r e d to the D e b y e t e m p e r a t u r e (TD). A p r o p e r d i s c u s s i o n of m e c h a n i s m 3 is e x t r e m e l y c o m p l i c a t e d and w i l l not be a t t e m p t e d h e r e . We w i l l r e s t r i c t o u r s e l v e s to an o r d e r of m a g n i t u d e e s t i m a t e of the a m o u n t of n u c l e a r v i brational energy associated with high frequency m a g n o n l i k e m o d e s . The a c t u a l d i s c o n t i n u i t y at T c w o u l d p r e s u m a b l y be c o n s i d e r a b l y s m a l l e r than t h i s and at m o s t of t h i s o r d e r of m a g n i t u d e . In the p r e s e n c e of phonon m a g n o n c o u p l i n g [5] the m a g n o n l i k e m o d e s h a v e the f o r m
* The r e s e a r c h reported in this document has been sponsored in part by the Air F o r c e Materials Laboratory Research and Technological Division AFSC through the European Office of Aerospace Research, United States Air F o r c e . 134
w h e r e C k i s the a n n i h i l a t i o n o p e r a t o r of the m a g non like m o d e of w a v e v e c t o r k and a k and c k a r e
Ck = akCk + ~kak ,
Volume 20, n u m b e r 2
PHYSICS
LETTERS
r e s p e c t i v e l y t h e a n n i h i l a t i o n o p e r a t o r s of p h o n o n and magnon modes before the coupling is introd u c e d . A s a r e s u l t a f r a c t i o n ~2 of t h e n u c l e a r vibrational energy is associated with magnon like modes, which can have relatively high frequenc i e s (h~2 k ~ k T c ) . A r o u g h e s t i m a t e of ~ f o r h i g h frequency magnons is N y/k(Tc_
TD),
w h e r e V i s a n a v e r a g e v a l u e of t h e p h o n o n m a g non coupling, i To e x p l a i n t h e o b s e r v e d d i s c o n t i n u i t y of 3% o n e would require at least Y 2 / k 2 ( T c - TD)2 ~ 0.03 . In i r o n T c ~ 1 0 0 0 ° K a n d T D N 4 0 0 ° K s o t h a t o n e w o u l d h a v e to a s s u m e V ~ 70 c m - 1 . T h i s v a l u e s e e m s t o o l a r g e b y o r d e r s of m a g n i t u d e s i n p a r t i c u l a r i n v i e w of t h e s m a l l m a g n e t o - s t r i c tion and anisotropy in iron. Let us now consider the fourth mechanism. A c c o r d i n g to W a l k e r e t a l . [7] t h e i s o m e r s h i f t p r o d u c e d b y t h e a d d i t i o n of o n e 4 s e l e c t r o n i s -1.4 mm sec -1. T h e o b s e r v e d a n o m a l y of - 0 . 0 2 m m s e c - 1 i s t h u s e q u i v a l e n t to t h e a d d i t i o n of 0 . 0 1 4 e l e c t r o n s p e r a t o m to t h e 4 s b a n d . U s i n g t h e r e s u l t s of C o r n w e l l a n d W o h l f a r t h [8] o n t h e b a n d s t r u c t u r e of i r o n o n e f i n d s t h a t t h e f e r r o m a g n e t i c Fermi level is about 0.012 ry below the paramagn e t i c F e r m i l e v e l o n a n a b s o l u t e s c a l e . If o n e t a k e s t h e w i n g s of t h e h y b r i d 3 d - 4 s b a n d a s t h e 4 s c o n t r i b u t i o n to t h e d e n s i t y of s t a t e s a n d i n t e r p o l a t e s to t h e F e r m i l e v e l o n e g e t s t h e r e a d e n s i t y of 3 e l e c t r o n s r y - 1 s o t h a t 0 . 0 3 6 e l e c t r o n s a r e a d d e d to t h e 4 s b a n d . T h i s i s t h e d i f f e r e n c e b e t w e e n t h e low t e m p e r a t u r e a n d t h e p a r a m a g n e t i c s t a t e . If o n e a s s u m e s t h a t t h i s a d d i t i o n of e l e c t r o n s to t h e 4 s b a n d v a r i e s r o u g h l y a s t h e magnetization one would expect that about one
1 F e b r u a r y 1966
t h i r d * of t h e e f f e c t s h o u l d o c c u r i n t h e s p a n of temperature in which the anomaly is observed (i.e. w i t h i n 1 0 ° C b e l o w Tc). T h i s g i v e s i n t h e a n o m a l y r e g i o n a t r a n s f e r of 0 . 0 1 2 e l e c t r o n s to t h e 4 s b a n d a s c o m p a r e d to t h e v a l u e of 0 . 0 1 4 e s t i m a t e d f r o m t h e o b s e r v e d a n o m aly. T h e c l o s e a g r e e m e n t i s p r o b a b l y f o r t u i t i o u s i n v i e w of t h e r o u g h n a t u r e of t h e c a l c u l a t i o n a n d the simplified model used. A proper quantitative e v a l u a t i o n of t h e e x p e r i m e n t a l r e s u l t s w o u l d r e quire a much more sophisticated theoretical t r e a t m e n t . It s e e m s h o w e v e r t h a t t h i s i s i n d e e d the mechanism responsible for the anomaly. Similar measurements on other magnetic m e t a l s a n d in p a r t i c u l a r o n i r o n a l l o y s s h o u l d y i e l d i n t e r e s t i n g i n f o r m a t i o n o n t h e s h a p e of t h e electronic bands near the Fermi level and on the n a t u r e of e l e c t r o n i c s t a t e s i n m a g n e t i c t r a n s i tions. T h e a u t h o r s w i s h to t h a n k D r . A. Sz~Jke a n d D r . J. A. S u s s m a n f o r h e l p f u l d i s c u s s i o n s . i. R.S. P r e s t o n , S.S. Hanna and J. Heberle, Phys. Rev. 128 (1962) 2207. 2. D . N . P i p k o r n , C . K . E d g e , P . D e b r u b b e r , G.De P a s quali, M. G. D r i c k a m e r and H. F r a u e n f e l d e r , Phys. Rev. 135 (1964) A 1604. 3. R.V. Pound, G. B. Benedek and R. D r e v e r , Phys. Rev. L e t t e r s 7 (1961) 405. 4. A. Goldsmith, T . E . W a t e r m a n and H. J. H i r s c h h o r n , Handbook of t h e r m o p h y s i c a l p r o p e r t i e s of solid m a t e r i a l s Vol. I ( P e r g a m o n P r e s s New York, 1962) pp. 1, 371. 5. F o r details see C.Kittel, Quantum theory of solids (John Wiley and Sons Inc .New York 1963} p. 74. 6. J . K a n a m o r i , Magnetism, Vol. I (Academic P r e s s Inc.New York, 1963) p. 191. 7. L . R . Walker, G.K. W e r t h e i m and V. J a c c a r i n o , Phys. Rev. L e t t e r s 6 (1961) 98. 8. J . F . Cornwell and E. P. Wohlfarth, J. Phys. Soc. J a p a n 17, Suppl. B~l (1962) 3 2 . * See fig. 8 of ref. 1.
.~***
135
ISOMER
SHIFT
ANOMALY
PHYSICS
LETTERS
AT
CURIE
THE
1 February 1966
TEMPERATURE
OF
IRON
S. ALEXANDER Department o f P h y s i c s , The Weizrnann Institute of Science, Rehovoth, Israel and
D. T R E V E S D e p a r t m e n t o f E l e c t r o n i c s , The Weizrnann Institute o f Science, Rehovoth, Israel Received 28 December 1965
Several explanations of the shift in the position of the centroid of the M6ssbauer spectrum in metallic iron as the specimen is heated through the Curie temperature are considered.
P r e s t o n et al [1] found t h a t in m e t a l l i c i r o n the p o s i t i o n (b) of the c e n t r o i d of the M 6 s s b a u e r s p e c t r u m c h a n g e s a b r u p t l y a s the s a m p l e is h e a t e d t h r o u g h the C u r i e t e m p e r a t u r e (Tc). The s h i f t is a p p r o x i m a t e l y - 0 . 0 2 m m s e c - 1 and o c c u r s o v e r a t e m p e r a t u r e r a n g e , 10°C. It is t h e r e f o r e c e r t a i n l y a s s o c i a t e d in s o m e w a y w i t h the m a g n e t i c phase transition. Two m e c h a n i s m s w h i c h c a n g i v e r i s e to d i s c o n t i n u i t i e s in b n e a r a p h a s e t r a n s i t i o n h a v e b e e n c o n s i d e r e d in the l i t e r a t u r e : 1. The i s o m e r s h i f t w i l l c h a n g e w h e n the a t o m i c v o l u m e c h a n g e s a b r u p t l y at the p h a s e t r a n s i t i o n [2]. 2. C h a n g e s in the D e b y e t e m p e r a t u r e can c a u s e a d i s c o n t i n u i t y in the s e c o n d o r d e r D o p p l e r s h i f t [1]. Two f u r t h e r m e c h a n i s m s w h i c h can be i m p o r t a n t f o r a m a g n e t i c p h a s e t r a n s i t i o n in a m e t a l h a v e o c c u r r e d to us. T h e s e a r e : 3. B e c a u s e of p h o n o n - m a g n o n i n t e r a c t i o n s o m e of the n u c l e a r k i n e t i c e n e r g y is c o n t a i n e d in relatively high frequency modes which are ess e n t i a l l y m a g n o n s . The c o l l a p s e of the m a g n o n s p e c t r u m at the C u r i e t e m p e r a t u r e t h e r e f o r e l e a d s to a c h a n g e in the s e c o n d o r d e r D o p p l e r shift. 4. N e a r the C u r i e t e m p e r a t u r e m a g n e t i c s p l i t t i n g of the 3d band d i s a p p e a r s . T h i s r e s u l t s in a s h i f t of the a b s o l u t e p o s i t i o n of the F e r m i l e v e l and t h e r e f o r e in a c h a n g e in the n u m b e r of 4s electrons.
We want to s h o w t h a t o n l y the l a s t m e c h a n i s m can a c c o u n t f o r the o b s e r v e d s h i f t in b. P o u n d et al. [3] h a v e d e t e r m i n e d the v o l u m e d e p e n d e n c e of b at r o o m t e m p e r a t u r e 8b/~ In V = - 1 . 4 m m s e c -1. T h i s v a l u e h a s the c o r r e c t o r d e r of m a g n i t u d e to e x p l a i n the d i s c o n t i n u i t y in b at the ot-7 t r a n s i t i o n of i r o n [2] (1200°K). It is t h e r e f o r e s a f e to a s s u m e t h a t it is a l s o r i g h t n e a r the C u r i e t e m p e r a t u r e (T c = 1042°K). With t h i s v o l u m e d e p e n d e n c e the o b s e r v e d c h a n g e in b n e a r T c w o u l d r e q u i r e a v o l u m e c h a n g e of a p p r o x i m a t e ly 1%. No s u c h a n o m a l y i s o b s e r v e d in the t h e r m a l e x p a n s i o n of i r o n [4]. T h i s s e e m s to r u l e out m e c h a n i s m 1. If the o b s e r v e d d i s c o n t i n u i t y in b is due to a c h a n g e in the s e c o n d o r d e r D o p p l e r - s h i f t it would i m p l y a 3% c h a n g e in the n u c l e a r v i b r a t i o n a l e n e r g y . P r e s t o n et al. [1] h a v e shown that s u c h a l a r g e d i s c o n t i n u i t y c a n n o t be e x p l a i n e d by any r e a s o n a b l e c h a n g e in the v i b r a t i o n s p e c t r u m b e c a u s e the C u r i e t e m p e r a t u r e (Tc) in i r o n is h i g h c o m p a r e d to the D e b y e t e m p e r a t u r e (TD). A p r o p e r d i s c u s s i o n of m e c h a n i s m 3 is e x t r e m e l y c o m p l i c a t e d and w i l l not be a t t e m p t e d h e r e . We w i l l r e s t r i c t o u r s e l v e s to an o r d e r of m a g n i t u d e e s t i m a t e of the a m o u n t of n u c l e a r v i brational energy associated with high frequency m a g n o n l i k e m o d e s . The a c t u a l d i s c o n t i n u i t y at T c w o u l d p r e s u m a b l y be c o n s i d e r a b l y s m a l l e r than t h i s and at m o s t of t h i s o r d e r of m a g n i t u d e . In the p r e s e n c e of phonon m a g n o n c o u p l i n g [5] the m a g n o n l i k e m o d e s h a v e the f o r m
* The r e s e a r c h reported in this document has been sponsored in part by the Air F o r c e Materials Laboratory Research and Technological Division AFSC through the European Office of Aerospace Research, United States Air F o r c e . 134
w h e r e C k i s the a n n i h i l a t i o n o p e r a t o r of the m a g non like m o d e of w a v e v e c t o r k and a k and c k a r e
Ck = akCk + ~kak ,
Volume 20, n u m b e r 2
PHYSICS
LETTERS
r e s p e c t i v e l y t h e a n n i h i l a t i o n o p e r a t o r s of p h o n o n and magnon modes before the coupling is introd u c e d . A s a r e s u l t a f r a c t i o n ~2 of t h e n u c l e a r vibrational energy is associated with magnon like modes, which can have relatively high frequenc i e s (h~2 k ~ k T c ) . A r o u g h e s t i m a t e of ~ f o r h i g h frequency magnons is N y/k(Tc_
TD),
w h e r e V i s a n a v e r a g e v a l u e of t h e p h o n o n m a g non coupling, i To e x p l a i n t h e o b s e r v e d d i s c o n t i n u i t y of 3% o n e would require at least Y 2 / k 2 ( T c - TD)2 ~ 0.03 . In i r o n T c ~ 1 0 0 0 ° K a n d T D N 4 0 0 ° K s o t h a t o n e w o u l d h a v e to a s s u m e V ~ 70 c m - 1 . T h i s v a l u e s e e m s t o o l a r g e b y o r d e r s of m a g n i t u d e s i n p a r t i c u l a r i n v i e w of t h e s m a l l m a g n e t o - s t r i c tion and anisotropy in iron. Let us now consider the fourth mechanism. A c c o r d i n g to W a l k e r e t a l . [7] t h e i s o m e r s h i f t p r o d u c e d b y t h e a d d i t i o n of o n e 4 s e l e c t r o n i s -1.4 mm sec -1. T h e o b s e r v e d a n o m a l y of - 0 . 0 2 m m s e c - 1 i s t h u s e q u i v a l e n t to t h e a d d i t i o n of 0 . 0 1 4 e l e c t r o n s p e r a t o m to t h e 4 s b a n d . U s i n g t h e r e s u l t s of C o r n w e l l a n d W o h l f a r t h [8] o n t h e b a n d s t r u c t u r e of i r o n o n e f i n d s t h a t t h e f e r r o m a g n e t i c Fermi level is about 0.012 ry below the paramagn e t i c F e r m i l e v e l o n a n a b s o l u t e s c a l e . If o n e t a k e s t h e w i n g s of t h e h y b r i d 3 d - 4 s b a n d a s t h e 4 s c o n t r i b u t i o n to t h e d e n s i t y of s t a t e s a n d i n t e r p o l a t e s to t h e F e r m i l e v e l o n e g e t s t h e r e a d e n s i t y of 3 e l e c t r o n s r y - 1 s o t h a t 0 . 0 3 6 e l e c t r o n s a r e a d d e d to t h e 4 s b a n d . T h i s i s t h e d i f f e r e n c e b e t w e e n t h e low t e m p e r a t u r e a n d t h e p a r a m a g n e t i c s t a t e . If o n e a s s u m e s t h a t t h i s a d d i t i o n of e l e c t r o n s to t h e 4 s b a n d v a r i e s r o u g h l y a s t h e magnetization one would expect that about one
1 F e b r u a r y 1966
t h i r d * of t h e e f f e c t s h o u l d o c c u r i n t h e s p a n of temperature in which the anomaly is observed (i.e. w i t h i n 1 0 ° C b e l o w Tc). T h i s g i v e s i n t h e a n o m a l y r e g i o n a t r a n s f e r of 0 . 0 1 2 e l e c t r o n s to t h e 4 s b a n d a s c o m p a r e d to t h e v a l u e of 0 . 0 1 4 e s t i m a t e d f r o m t h e o b s e r v e d a n o m aly. T h e c l o s e a g r e e m e n t i s p r o b a b l y f o r t u i t i o u s i n v i e w of t h e r o u g h n a t u r e of t h e c a l c u l a t i o n a n d the simplified model used. A proper quantitative e v a l u a t i o n of t h e e x p e r i m e n t a l r e s u l t s w o u l d r e quire a much more sophisticated theoretical t r e a t m e n t . It s e e m s h o w e v e r t h a t t h i s i s i n d e e d the mechanism responsible for the anomaly. Similar measurements on other magnetic m e t a l s a n d in p a r t i c u l a r o n i r o n a l l o y s s h o u l d y i e l d i n t e r e s t i n g i n f o r m a t i o n o n t h e s h a p e of t h e electronic bands near the Fermi level and on the n a t u r e of e l e c t r o n i c s t a t e s i n m a g n e t i c t r a n s i tions. T h e a u t h o r s w i s h to t h a n k D r . A. Sz~Jke a n d D r . J. A. S u s s m a n f o r h e l p f u l d i s c u s s i o n s . i. R.S. P r e s t o n , S.S. Hanna and J. Heberle, Phys. Rev. 128 (1962) 2207. 2. D . N . P i p k o r n , C . K . E d g e , P . D e b r u b b e r , G.De P a s quali, M. G. D r i c k a m e r and H. F r a u e n f e l d e r , Phys. Rev. 135 (1964) A 1604. 3. R.V. Pound, G. B. Benedek and R. D r e v e r , Phys. Rev. L e t t e r s 7 (1961) 405. 4. A. Goldsmith, T . E . W a t e r m a n and H. J. H i r s c h h o r n , Handbook of t h e r m o p h y s i c a l p r o p e r t i e s of solid m a t e r i a l s Vol. I ( P e r g a m o n P r e s s New York, 1962) pp. 1, 371. 5. F o r details see C.Kittel, Quantum theory of solids (John Wiley and Sons Inc .New York 1963} p. 74. 6. J . K a n a m o r i , Magnetism, Vol. I (Academic P r e s s Inc.New York, 1963) p. 191. 7. L . R . Walker, G.K. W e r t h e i m and V. J a c c a r i n o , Phys. Rev. L e t t e r s 6 (1961) 98. 8. J . F . Cornwell and E. P. Wohlfarth, J. Phys. Soc. J a p a n 17, Suppl. B~l (1962) 3 2 . * See fig. 8 of ref. 1.
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