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Debye temperature changes in microcrystals
Volume 21, number 2
DEBYE
PHYSICS L E T T E R S
TEMPERATURE
CHANGES D.
1 May 196~
IN MICROCRYST~LS
SCHROEER
Physik Depa~ment, Technische Hochschule Mi~nchen, Germany Received 2 April 1966
The variation in the Debye temperature for microcrystals, which manifests i t s e l f in changes in tb~ ~ . Waller factor of M~ssbauer spectra, is related to lattice constant changes.
There has been a r e c e n t i n t e r e s t in studying the p r o p e r t i e s of m i e r o c r y s t a l s by means of the M0ssbauer effect [1-6]. One effect observed is a change in the Debye-Waller f - f a c t o r with d e c r e a s i n g c r y s t a l size; Goldanskii [5] found it to d e c r e a s e for ll9Sn and M a r s h a l l and Wilenzick (MW)I) to i n c r e a s e for 197Au. The phenomenon in gold is explained as being due to a) modifications of the low frequency portion of the c r y s t a l phonon spectrum by the surface and the limited c r y s t a l size, and b) changes in the Debye t e m p e r a t u r e . This p i c t u r e leads to two p a r a m e t e r s which can be used to fit the experimental data. The f i r s t is a low frequency cutoff, 0M, in the phonon spectrum, depending on the extent of coupling of the m i c r o c r y s t a l s to their s u r roundings. No explanation has been given for the changes in the second p a r a m e t e r , the Debye t e m p e r a t u r e 0, which a r e r e s p o n s i b l e for the m a j o r changes in the f - f a c t o r . We would like to suggest a possible source of these significant changes in the Debye t e m p e r a t u r e in small p a r t i c l e s ; namely they can r e s u l t from changes in the value of the lattice constant with d e c r e a s i n g c r y s t a l size. I n c r e a s e s in the quadrupole splitting with d e c r e a s i n g c r y s t a l size, as well as i s o m e r shifts between bulk and small c r y s t a l l i t e s , have been found in the M0ssbauer spectrum of F e 2 0 3 by Ktindig et al. [2]. This suggests a d e c r e a s e of the lattice constant. Boswell [7] studied e v a p o r a ted l a y e r s of NaC1, Au, etc. by electron diffraction, and was able to m e a s u r e quantitatively the lattice constant a s a function of p a r t i c l e size. He found the reduction in the lattice constant to be i n v e r s e l y proportional to the p a r t i c l e d i a m e t e r , suggestive of a " s u r f a c e t e n s i o n " . Changes of the f - f a c t o r with volume have been observed at h i g h p r e s s u r e s in 161Dy by Stone, et al. [8], and in l l g s n by Panyushkin and Voronov [9];
these v a r i a t i o n s a r e quite~adequately explained by a theory developed by Hanks [10], which leads d i r e c t l y to a change in the Debye t e m p e r a t u r e with small volume changes. So one would expect v a r i a t i o n s in the lattice spacing to cause changes in the Debye t e m p e r a t u r e . We will now show how the changes in the lattice constant in gold m i c r o c r y s t a l s lead to changes in the Debye t e m p e r a t u r e of the c o r r e c t size to account for the effects observed by MW in 197Au. F r o m Boswell's data on gold we can extrapolate the lattice constant reduction, and hence the volume reduction between the gold p a r t i c l e s i z e s 200 ~ and 60 ~ in d i a m e t e r studied by MW, the value is found to be ~V/V= 1.65 ± 0.35%. Hank's equation for the change of Debye t e m p e r a t u r e with s m a l l volume changes is o' = (1 - ~ v / v )
-1~
(1)
where 0 and 0' a r e the Debye t e m p e r a t u r e s of the small p a r t i c l e s with d i a m e t e r s 200 ~ and 60 ~ r e s p e c t i v e l y , 5V/V = (V-V')/V with the p r i m e again r e f e r r i n g to the 60 ~ d i a m e t e r p a r t i c l e s , and ~ i s the Grueneisen constant, r for gold is 3.05 [11]. We then find A~ -- ~' - 0 = 8 + 2 ° K , taking ~ = 163OK as determined by MW. This change in the Debye t e m p e r a t u r e i s in a g r e e m e n t with the 10°K variation r e q u i r e d by MW to l e a s t s q u a r e s - f i t their data. So the variation of the Debye t e m p e r a t u r e with p a r t i c l e size is understandable in t e r m s of the observed changes in the lattice constant. The source of these lattice constant changes on the other hand is not c l e a r l y understood. LennardJones and Dent [12,13] lmve put forward a theory according to which the lattice dimensions of ionic c r y s t a l s should d e c r e a s e with d e c r e a s i n g c r y s t a l size, whereas for covalent c r y s t a l s the r e v e r s e should be true. This theory, however, does not 123
Volume 21, number 2
PHYSICS LETTERS
p e r m i t anything to be said about m e t a l s , w h e r e the r a t i o of m e t a l l i c to covalent bonding is p r o bably the d e t e r m i n i n g f a c t o r in deciding the sign of the l a t t i c e p a r a m e t e r changes. To f u r t h e r c o n f i r m the i n t e r r e l a t i o n between the l a t t i c e constant and the Debye t e m p e r a t u r e in m i c r o c r y s t a l s , i s o m e r shift s tu d i e s coupled with d i r e c t high p r e s s u r e m e a s u r e m e n t s and s p e c i f i c heat i n v e s t i g a t i o n s would be p r o f i t a b l e . M o r e quantitative c a l c u l a t i o n s of the l a t t i c e p a r a m e t e r v a r i a t i o n s with c r y s t a l s i z e would a l s o be highly d e s i r a b l e . We would like to thank A . V . L e s i k a r and P. P. C r a i g for helpful c o m m e n t s .
References 1. S.W.Marshall and R.M.Wilenzick, Phys. Rev. Letters 16 (1966) 219.
AN
EMPIRICAL
1May 1966
2. W. Kffndig, H.B6ramel, G.Constabaris and R.H. Lindquist, Phys. Rev. 142 (1966) 327. 3. T.Nakamura, T.Shinjo, Y.Endoh, N.Yamamoto, M.Shinjo and Y.Nakamura, Phys. Letters 12 (1964) 178. 4. W.J.Schuele, S.Shtrikman and D.Treves, J. Appl. Phys. 36 (1965) 1010. 5. V. I. Goldanskii, private communication. 6. P.A.Flinn, S.L.Ruby and W.L.Keh], Science !43 (1964) 1434. 7. F.W.C.Boswell, Proc: Phys. Soc. (London) 64 (1951) 465. 8. J.A.Stone, M.Nicol, G. Jura and J . O . R a s m u s s e n UCRL-] 0630-Rev. 9. V.N. Panyushkin and F. [" .Voronov, : E T P Letters 2 (1965) 97. 10. R.V.Hanks, Phys. Rev. [24 (1961) 1319. 11. M.H.Rice. R.G.McQueen and J.M.Walsh, Solid State Phys. Ed. F.Seitz und D.Turnbull 6 (1958) ]. 12. J . E . Lermard-Jones and B.M.Dent, Proc. Roy. Soc. A 121 (1928) 247. 13. J . E . Lennard-Jones, Z.f. Kristallogr. 75 (1930) 215.
MOLLWO-IVEY RELATION IN ALKALI HALIDES
FOR
THE
Z 3 CENTER
J. MORT
Depart~ent of Physics, University of Wisconsin Madison, Wisconsin, U.S.A. Received 29 March 1966
A Mollwo-Ivey relation is shown to exist for the Z 3 centers in a l k a l i halides and is used to identify the Z 3
center in magnesium-doped LiF.
The peak position of the F a b s o r p t i o n band in alkali h a l i d e s having the NaC1 s t r u c t u r e was f i r s t r e l a t e d e m p i r i c a l l y to the n e a r e s t ionic neighbour d i s t a n c e d by Mollwo [1]. Subsequently a m o r e accurate empirical relationship was der i v e d by Ivey [2] who found that f o r the F a b s o r p tion band ~tma x = 703 d 1"84, w h e r e ~t and d a r e in ~ . S i m i l a r r e l a t i o n s h i p s w e r e found f o r the M and R a b s o r p t i o n bands [2]. In a l l the a l k a l i h a l i d e s so f a r studied one i m p o r t a n t e x p e r i m e n tal o b s e r v a t i o n e m e r g e s c o n c e r n i n g the peak of the Z 3 a b s o r p t i o n band v i z . that it i s a l w a y s found to l i e at slightly h i g h e r e n e r g i e s than the F band [3]. Since the F band a b s o r p t i o n peak shifts c o n s i d e r a b l y f r o m one alkali halide to a n o t h e r such an o b s e r v a t i o n s u g g e s t s that a M o l l w o - I v e y type r e l a t i o n might be found f o r the Z 3 c e n t e r . 124
Th e p u r p o s e of this l e t t e r is to point out that such an e m p i r i c a l r e l a t i o n does indeed e x i s t . T h i s fact in turn i s shown to p r o v i d e s t r o n g e v i d e n c e in support of the r e c e n t suggestion that the 5.5 eV a b s o r p t i o n band p r e s e n t in X - i r r a d i a t e d m a g n e s i u m - d o p e d L i F i s a Z - t y p e c e n t e r [4]. F u r t h e r m o r e it a l l o w s the m o r e s p e c i f i c a s s i g n m e n t of the 5.5 eV band a s a Z 3 c e n t e r . It is p e r h a p s w o r t h pointing out that this i n f o r m a t i o n i s p a r t i c u l a r l y valuable b e c a u s e of the difficulty in d e t e r m i n g unambiguously w h et h er c e n t e r s in L i F a r e due to t r a p p e d e l e c t r o n s or t r a p p e d h o l e s . T h i s am b i g u i t y i s a l w a y s p r e s e n t when working with L i F b e c a u s e of the f a i l u r e so f a r to a d d i t i v e ly c o l o r this salt [5]. As a r e s u l t all work on c o l o r c e n t e r s in L i F m u s t be c a r r i e d out with i r r a d i a t e d s a m p l e s so that both e l e c t r o n and hole c e n t e r s a r e s i m i l t a n e o u s l y p r o d u ced .
DEBYE
PHYSICS L E T T E R S
TEMPERATURE
CHANGES D.
1 May 196~
IN MICROCRYST~LS
SCHROEER
Physik Depa~ment, Technische Hochschule Mi~nchen, Germany Received 2 April 1966
The variation in the Debye temperature for microcrystals, which manifests i t s e l f in changes in tb~ ~ . Waller factor of M~ssbauer spectra, is related to lattice constant changes.
There has been a r e c e n t i n t e r e s t in studying the p r o p e r t i e s of m i e r o c r y s t a l s by means of the M0ssbauer effect [1-6]. One effect observed is a change in the Debye-Waller f - f a c t o r with d e c r e a s i n g c r y s t a l size; Goldanskii [5] found it to d e c r e a s e for ll9Sn and M a r s h a l l and Wilenzick (MW)I) to i n c r e a s e for 197Au. The phenomenon in gold is explained as being due to a) modifications of the low frequency portion of the c r y s t a l phonon spectrum by the surface and the limited c r y s t a l size, and b) changes in the Debye t e m p e r a t u r e . This p i c t u r e leads to two p a r a m e t e r s which can be used to fit the experimental data. The f i r s t is a low frequency cutoff, 0M, in the phonon spectrum, depending on the extent of coupling of the m i c r o c r y s t a l s to their s u r roundings. No explanation has been given for the changes in the second p a r a m e t e r , the Debye t e m p e r a t u r e 0, which a r e r e s p o n s i b l e for the m a j o r changes in the f - f a c t o r . We would like to suggest a possible source of these significant changes in the Debye t e m p e r a t u r e in small p a r t i c l e s ; namely they can r e s u l t from changes in the value of the lattice constant with d e c r e a s i n g c r y s t a l size. I n c r e a s e s in the quadrupole splitting with d e c r e a s i n g c r y s t a l size, as well as i s o m e r shifts between bulk and small c r y s t a l l i t e s , have been found in the M0ssbauer spectrum of F e 2 0 3 by Ktindig et al. [2]. This suggests a d e c r e a s e of the lattice constant. Boswell [7] studied e v a p o r a ted l a y e r s of NaC1, Au, etc. by electron diffraction, and was able to m e a s u r e quantitatively the lattice constant a s a function of p a r t i c l e size. He found the reduction in the lattice constant to be i n v e r s e l y proportional to the p a r t i c l e d i a m e t e r , suggestive of a " s u r f a c e t e n s i o n " . Changes of the f - f a c t o r with volume have been observed at h i g h p r e s s u r e s in 161Dy by Stone, et al. [8], and in l l g s n by Panyushkin and Voronov [9];
these v a r i a t i o n s a r e quite~adequately explained by a theory developed by Hanks [10], which leads d i r e c t l y to a change in the Debye t e m p e r a t u r e with small volume changes. So one would expect v a r i a t i o n s in the lattice spacing to cause changes in the Debye t e m p e r a t u r e . We will now show how the changes in the lattice constant in gold m i c r o c r y s t a l s lead to changes in the Debye t e m p e r a t u r e of the c o r r e c t size to account for the effects observed by MW in 197Au. F r o m Boswell's data on gold we can extrapolate the lattice constant reduction, and hence the volume reduction between the gold p a r t i c l e s i z e s 200 ~ and 60 ~ in d i a m e t e r studied by MW, the value is found to be ~V/V= 1.65 ± 0.35%. Hank's equation for the change of Debye t e m p e r a t u r e with s m a l l volume changes is o' = (1 - ~ v / v )
-1~
(1)
where 0 and 0' a r e the Debye t e m p e r a t u r e s of the small p a r t i c l e s with d i a m e t e r s 200 ~ and 60 ~ r e s p e c t i v e l y , 5V/V = (V-V')/V with the p r i m e again r e f e r r i n g to the 60 ~ d i a m e t e r p a r t i c l e s , and ~ i s the Grueneisen constant, r for gold is 3.05 [11]. We then find A~ -- ~' - 0 = 8 + 2 ° K , taking ~ = 163OK as determined by MW. This change in the Debye t e m p e r a t u r e i s in a g r e e m e n t with the 10°K variation r e q u i r e d by MW to l e a s t s q u a r e s - f i t their data. So the variation of the Debye t e m p e r a t u r e with p a r t i c l e size is understandable in t e r m s of the observed changes in the lattice constant. The source of these lattice constant changes on the other hand is not c l e a r l y understood. LennardJones and Dent [12,13] lmve put forward a theory according to which the lattice dimensions of ionic c r y s t a l s should d e c r e a s e with d e c r e a s i n g c r y s t a l size, whereas for covalent c r y s t a l s the r e v e r s e should be true. This theory, however, does not 123
Volume 21, number 2
PHYSICS LETTERS
p e r m i t anything to be said about m e t a l s , w h e r e the r a t i o of m e t a l l i c to covalent bonding is p r o bably the d e t e r m i n i n g f a c t o r in deciding the sign of the l a t t i c e p a r a m e t e r changes. To f u r t h e r c o n f i r m the i n t e r r e l a t i o n between the l a t t i c e constant and the Debye t e m p e r a t u r e in m i c r o c r y s t a l s , i s o m e r shift s tu d i e s coupled with d i r e c t high p r e s s u r e m e a s u r e m e n t s and s p e c i f i c heat i n v e s t i g a t i o n s would be p r o f i t a b l e . M o r e quantitative c a l c u l a t i o n s of the l a t t i c e p a r a m e t e r v a r i a t i o n s with c r y s t a l s i z e would a l s o be highly d e s i r a b l e . We would like to thank A . V . L e s i k a r and P. P. C r a i g for helpful c o m m e n t s .
References 1. S.W.Marshall and R.M.Wilenzick, Phys. Rev. Letters 16 (1966) 219.
AN
EMPIRICAL
1May 1966
2. W. Kffndig, H.B6ramel, G.Constabaris and R.H. Lindquist, Phys. Rev. 142 (1966) 327. 3. T.Nakamura, T.Shinjo, Y.Endoh, N.Yamamoto, M.Shinjo and Y.Nakamura, Phys. Letters 12 (1964) 178. 4. W.J.Schuele, S.Shtrikman and D.Treves, J. Appl. Phys. 36 (1965) 1010. 5. V. I. Goldanskii, private communication. 6. P.A.Flinn, S.L.Ruby and W.L.Keh], Science !43 (1964) 1434. 7. F.W.C.Boswell, Proc: Phys. Soc. (London) 64 (1951) 465. 8. J.A.Stone, M.Nicol, G. Jura and J . O . R a s m u s s e n UCRL-] 0630-Rev. 9. V.N. Panyushkin and F. [" .Voronov, : E T P Letters 2 (1965) 97. 10. R.V.Hanks, Phys. Rev. [24 (1961) 1319. 11. M.H.Rice. R.G.McQueen and J.M.Walsh, Solid State Phys. Ed. F.Seitz und D.Turnbull 6 (1958) ]. 12. J . E . Lermard-Jones and B.M.Dent, Proc. Roy. Soc. A 121 (1928) 247. 13. J . E . Lennard-Jones, Z.f. Kristallogr. 75 (1930) 215.
MOLLWO-IVEY RELATION IN ALKALI HALIDES
FOR
THE
Z 3 CENTER
J. MORT
Depart~ent of Physics, University of Wisconsin Madison, Wisconsin, U.S.A. Received 29 March 1966
A Mollwo-Ivey relation is shown to exist for the Z 3 centers in a l k a l i halides and is used to identify the Z 3
center in magnesium-doped LiF.
The peak position of the F a b s o r p t i o n band in alkali h a l i d e s having the NaC1 s t r u c t u r e was f i r s t r e l a t e d e m p i r i c a l l y to the n e a r e s t ionic neighbour d i s t a n c e d by Mollwo [1]. Subsequently a m o r e accurate empirical relationship was der i v e d by Ivey [2] who found that f o r the F a b s o r p tion band ~tma x = 703 d 1"84, w h e r e ~t and d a r e in ~ . S i m i l a r r e l a t i o n s h i p s w e r e found f o r the M and R a b s o r p t i o n bands [2]. In a l l the a l k a l i h a l i d e s so f a r studied one i m p o r t a n t e x p e r i m e n tal o b s e r v a t i o n e m e r g e s c o n c e r n i n g the peak of the Z 3 a b s o r p t i o n band v i z . that it i s a l w a y s found to l i e at slightly h i g h e r e n e r g i e s than the F band [3]. Since the F band a b s o r p t i o n peak shifts c o n s i d e r a b l y f r o m one alkali halide to a n o t h e r such an o b s e r v a t i o n s u g g e s t s that a M o l l w o - I v e y type r e l a t i o n might be found f o r the Z 3 c e n t e r . 124
Th e p u r p o s e of this l e t t e r is to point out that such an e m p i r i c a l r e l a t i o n does indeed e x i s t . T h i s fact in turn i s shown to p r o v i d e s t r o n g e v i d e n c e in support of the r e c e n t suggestion that the 5.5 eV a b s o r p t i o n band p r e s e n t in X - i r r a d i a t e d m a g n e s i u m - d o p e d L i F i s a Z - t y p e c e n t e r [4]. F u r t h e r m o r e it a l l o w s the m o r e s p e c i f i c a s s i g n m e n t of the 5.5 eV band a s a Z 3 c e n t e r . It is p e r h a p s w o r t h pointing out that this i n f o r m a t i o n i s p a r t i c u l a r l y valuable b e c a u s e of the difficulty in d e t e r m i n g unambiguously w h et h er c e n t e r s in L i F a r e due to t r a p p e d e l e c t r o n s or t r a p p e d h o l e s . T h i s am b i g u i t y i s a l w a y s p r e s e n t when working with L i F b e c a u s e of the f a i l u r e so f a r to a d d i t i v e ly c o l o r this salt [5]. As a r e s u l t all work on c o l o r c e n t e r s in L i F m u s t be c a r r i e d out with i r r a d i a t e d s a m p l e s so that both e l e c t r o n and hole c e n t e r s a r e s i m i l t a n e o u s l y p r o d u ced .