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ANHARMONICITY EFFECTS IN CaF2 AT LOW TEMPERATURE
P. Tua, G. Mahan
To cite this version:
P. Tua, G. Mahan. ANHARMONICITY EFFECTS IN CaF2 AT LOW TEMPERATURE. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-122-C6-124. �10.1051/jphyscol:1981636�. �jpa-00221572�
JOURNAL DE PHYSIQUE
CoZZoque C6, suppZe'ment au n012, Tome 42, de'cembre 1981 page C6- 122
ANHARMONICITY EFFECTS I N C a F 2 A T LOW TEMPERATURE
P.F. Tua and G.D. Mahan
Physics Department, I n d i m a University, BZoomington, I N 47405, U.S.A.
A b s t r a c t . - Using t h e framework of t h e r i g i d i o n model, we have c a l c u l a t e d t h e l i f e t i m e of l o n g i t u d i n a l a c o u s t i c phonons i n CaF a t low temperature.
The decay mode 1 4 + t i s s t r o n g l y a n i s o t r o p i c and og one o r d e r of magnitude s m a l l e r t h a n t h e decay mode l+t + t . For each s i n g l e decay mode, t h e long-wavelength approximation which y i e l d s t h e v5 dependence of l / r , i s v a l i d f o r v ( 2.5 THz. A t h i g h e r f r e q u e n c i e s , t h e competing decay modes show d i f f e r e n t deviances which, c u r i o u s l y , c a n c e l each o t h e r s o t h a t t h e t o t a l l i f e - t i m e can be w e l l approximated by t h e long-wavelength l i m i t up t o 5.5 THz. We have a l s o c a l c u l a t e d t h e temperature dependence of t h e second and t h i r d o r d e r e l a s t i c c o n s t a n t s and a good agreement i s found w i t h t h e a v a i l a b l e experimental d a t a .
S e v e r a l new experimental t e c h n i q u e s have been r e c e n t l y a p p l i e d t o s t u d y t h e l i f e t i m e of high-frequency a c o u s t i c phonons a t low temperature'''. P r e l i m i n a r y r e s u l t s [ 2 1 f o r l o n g i t u d i n a l a c o u s t i c (LA) phonons i n f l u o r i t e s , a-quartz, and ruby show a l i f e t i m e l o n g e r t h a n t h e t h e o r e t i c a l p r e d i c t i o n s of Orbach and Vredevoe 131
and ~ l e m e n s ' ~ ' , which a r e based on an i s o t r o p i c model i n t h e long-wavelength approximation (LWLA) . ~ r o n [ ~ ] h a s suggested t h a t , f o r SrF2, CaFZ, and a-quartz, t h e d i s c r e p a n c y l i e s i n t h e d i f f e r e n c e s i n t h e d e n s i t y of phonon s t a t e s between t h e r e a l c r y s t a l and an i s o t r o p i c , d i s p e r s i o n L e s s Debye s o l i d . We b e l i e v e t h i s i s
F i g .
0.0 0.25 0.50
- a 2a 151
1: Example of d i s p e r s i o n c u r v e s i n C ~[81 a t 295OK. F ~
o Exp. (Ref. [9])
Fig. 2: Temperature-dependence of c44 i n CaF2.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981636
n o t t h e case. F i r s t , t h e frequency v of t h e probed LA phonons l i e s between 1 and 3.5 T H Z ' ~ ] and t h e deviances of t h e d i s p e r s i o n r e l a t i o n s from t h e LWLA a r e v e r y s m a l l i n t h a t r e g i o n , a s shown i n Fig. 1 f o r CaF2. Second, t h e form of Orbach and Vredevoe's r e l a t i o n , where l/.c i s p r o p o r t i o n a l t o v5, i s independent of t h e a n i s o t r o p y of t h e c r y s t a l i n t h e LWLA, because it i s a consequence of a s i m p l e s c a l i n g lawr5 I , t h e a n i s o t r o p y a f f e c t i n g t h e c o n s t a n t of p r o p o r t i o n a l i t y s l i g h t l y .
Another p o s s i b l e e x p l a n a t i o n i s t h a t t h e three-phonon i n t e r a c t i o n m a t r i x elements might show l a r g e d e v i a t i o n s from t h e LWLA i n t h a t frequency r e g i o n , even i f t h e d i s p e r s i o n r e l a t i o n s do n o t . I n o r d e r t o i n v e s t i g a t e t h i s p o s s i b i l i t y , we have performed a numerical c a l c u l a t i o n of t h e LA phonon l i f e t i m e i n CaF f o r f r e q u e n c i e s between 2.5 and 5.5 THz. We have used t h e Landau and Rumer[g3 t h e o r y f o r t h e c a s e hv>>%T. A simple c a l c u l a t i o n r 5 J shows t h a t t h e umklapp p r o c e s s e s p l a y a minor r o l e i n t h e r e g i o n v55.5 THz, and hence t h e y have been n e g l e c t e d . S t r i c t l y speaking, t h e c a l c u l a t i o n i s a t z e r o temperature, b u t t h e LA phonon decay i s temperature-independent a t s u f f i c i e n t l y low t e m p e r a t u r e (<15"K, i n our c a s e ) , a s shown by Orbach and ~ r e d e v o e ' ~ ] . We have used t h e r i g i d i o n model where t h e i o n s have a n " e f f e c t i v e " charge and t h e short-range p a i r - p o t e n t i a l s have t h e u s u a l Born-Mayer formr7]. The parameters have been determined by f i t t i n g t h e n e u t r o n s c a t t e r i n g d a t a f B 1 and t h e temperature-dependence d a t a of t h e second-order e l a s t i c c o n s t a n t s [ g 1 . I n Fig. 2 we show t h e temperature-dependence of c14 a s a n example of t h e good agreement between t h e o r y and experimental d a t a . We have a l s o performed a long-wavelength expansion of t h e i n t e r a c t i o n m a t r i x elements and c a l c u l a t e d t h e t h i r d - o r d e r e l a s t i c c o n s t a n t s . The v a l u e s t 5 ] a r e i n s a t i s f a c t o r y agreement w i t h t h e a v a i l a b l e experimental d a t a r l o l , s i n c e t h e l a t t e r r e f e r t o t h e f i r s t - s o u n d r e g i o n where v e r t e x c o r r e c t i o n s become important['l1.
' O . O , l l DIRECTION
2.5 3.5 4.5 5.5
Frequency v (THz)
I
2.5 3.5 4.5 5!5
Frequency V (THz) Fig. 3: T r a n s i t i o n r a t e of LA phonons F i g . 4: T r a n s i t i o n r a t e of LA phonons
i n CaF2 f o r decay LA-+LA+TA. i n CaF2 f o r decay LA+TA+TA.
C6- 124 JOURNAL DE PHYSIQUE
Table I - L i f e t i m e (10-~Osec) of LA phonons w i t h h v > > s T i n CaF2.
v (THz) (O,O, 1) ( O , l , l ) ( l , l , l )
2.5 33.9 11.9 6.41
3.5 6.33 2.18 1.19
4.5 1.80 0.610 0.341
5.5 0.662 0.220 0.125
The c a l c u l a t e d t o t a l LA phonon l i f e t i m e s i n t h e t h r e e p r i n c i p a l d i r e c t i o n s i s r e p o r t e d i n Table I. A s expected, t h e l i f e t i m e depends upon t h e d i r e c t i o n , but only w i t h i n a f a c t o r of 5. The v5-dependence i s w e l l s a t i s f i e d . Nevertheless, a s shown in Figs. 3 and 4, t h e s i n g l e decay modes show d e v i a t i o n s from t h e
v5-dependence b u t t h e y c a n c e l o u t in t h e t o t a l l i f e t i m e . The s t r o n g d e v i a t i o n of t h e p r o c e s s LA+LA+TAl, where TA1 i s t h e lowest t r a n s v e r s e a c o u s t i c branch, can b e a t t r i b u t e d only t o t h e i n t e r a c t i o n m a t r i x elements, because t h e d i f f e r e n c e s between t h e two TA branches a r e minimalt8'. Besides, i n t h e (0,0,1) d i r e c t i o n t h e t r a n s i t i o n r a t e f o r LA+TA+TA i s one o r d e r of magnitude g r e a t e r t h a n t h a t f o r LA-tLA+TA, t o t h e c o n t r a r y of Orbach and Vredevoe's assumption[31. The r a t i o between t h e two decay modes becomes s m a l l e r i n t h e o t h e r two p r i n c i p a l d i r e c t i o n s .
I n conclusion, a r e a l i s t i c c a l c u l a t i o n of t h e l i f e t i m e of LA phonons in CaF2 w i t h hv>>%T and 2.5 THz <v<5.5 THz confirms t h e v5-dependence and s u g g e s t s t h a t a more c a r e f u l i n t e r p r e t a t i o n of t h e raw experimental d a t a i s needed.
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Phys. 43, 303 (1980).
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[3] R. Orbach and L.A. Vredevoe, Physics 1, 19 (1964).
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[51 P. F. Tua, Ph.D. T h e s i s , Indiana U n i v e r s i t y (unpublished).
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[7] M.P. Tosi, S o l i d S t a t e P h y s i c s % , 1 (1964).
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