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Thermal variation of a Raman line width in calcite
Volume 25A. number 7
PHYSICS
LETTERS
9 October 1967
Table 1 (Approx.) (microns) 373 337 335 311 310 284
Frequency * (GHz)
Wavelength ** (microns)
804.7509 890.7607 894.4142 964.3134 967.9658 (unmeasured)
Wavelength (calculated)*** (microns)
372.5283 336.5578 335.1831 310.8870 309.7140 284
372.55 336.50 335.24 311.83 309.70 284.01
* Estimated e r r o r 1 part in 106 {~ 1 mc). ** Speed of light assumed to be 2.997 925 x 108 m / s e c . *** The e r r o r is conservatively 3 parts in 104.
i n c h e s in d i a m e t e r , a n d e m p l o y e d a f o l d e d c o n focal m i r r o r configuration. With this l a s e r the m e a s u r e d p o w e r output e x c e e d e d 15 mW at 337p a n d h a s y i e l d e d r e c t i f i e d s i g n a l s at t h e h a r m o n i c m i x e r a s l a r g e a s 100 m V . T h e m i c r o w a v e s e t u p w h i c h w a s e m p l o y e d in t h e f r e q u e n c y m e a s u r i n g p a r t of t h i s e x p e r i m e n t w a s e s s e n t i a l l y s i m i l a r to t h o s e u s e d p r e v i o u s l y
[2,3]. W e w o u l d l i k e to a c k n o w l e d g e n u m e r o u s u s e ful d i s c u s s i o n s w i t h P r o f . D. R a m a c h a n d r a R a o a n d h i s a s s i s t a n c e in t h i s e x p e r i m e n t . W e a l s o
THERMAL
VARIATION
OF
a c k n o w l e d g e t h e t e c h n i c a l a i d of M r . L e o G e o f frion.
References 1. D . R . L i d e a n d A . G . Maki. Appl. Phys. L e t t e r s 11 (1967) 2. We are deeply indebted to Dr. Lide and Dr. Maki for providing us with this information before it was published. 2. L.O. Hocker. A. Javan. D. Ramachandra Rao. L. Frenkel and T.Sullivan. Appl. Phys. L e t t e r s 10 (1967) 5. 3. L.O. Hocker. D.Ramachandra Rao and A. Javan. Phys. L e t t e r s 24A (1967) 12.
A RAMAN
LINE
WIDTH
IN
CALCITE*:~
K. P A R K
Department of Physics. University of Oregon. Eugene. Oregon. USA Received 2 September 1967
A width m e a s u r e m e n t of the Alg Raman line in calcite (P = 1086 cm -1) over an extended t e m p e r a t u r e range (10OK to 660OK) is reported. The r e s u l t is compared with a theory based on anharrnonicity.
W e r e p o r t e d in a p r e v i o u s p a p e r [1] t h a t t h e w i d t h of t h e A l g R a m a n l i n e in c a l c i t e (F = 1086 c m -1) i s v e r y n a r r o w a n d e x h i b i t s a m a r k e d n a r r o w i n g with d e c r e a s i n g t e m p e r a t u r e . We a l s o reported several subsidiary measurements made in a n e f f o r t t o r e s o l v e t h e d i s c r e p a n c y b e t w e e n our r e s u l t and o t h e r p u b l i s h e d data [2-5]. F r o m t h e s e m e a s u r e m e n t s we concluded that t h e r e is n o s i m p l e way t o r e c o n c i l e t h e d i s c r e p a n c y . In 490
t h i s l e t t e r we r e p o r t t h e l i n e w i d t h d a t a o v e r a n e x t e n d e d t e m p e r a t u r e r a n g e **
* Work supported in p a r t by the Army R e s e a r c h Office, Durham. A portion of this work was done while the author was at Bell Telephone Labs., Murray Hill, N.J., USA. ** The experimental a r r a n g e m e n t and the details of slit width c o r r e c t i o n have been d i s c u s s e d in ref. 1.
Volume 25A, number7
,o
~o
~
PHYSICS L E T T E R S
~o
~o
'
~o . . . .
~oo
TEMPE I~ATURI[ ~K)
Fig. 1. Line width versus temperature. See the text for the theoretical curve (solid curve). The e r r o r s in the experimental points are typically i 10%. Fig. 1 shows the v a r i a t i o n of line width as a function of t e m p e r a t u r e . The solid c u r v e is the r e s u l t of an a t t e m p t to fit the data to the t h e o r y of the a n h a r m o n i c decay of optical phonons [6]. If an optical phonon of f r e q u e n c y w o d e c a y s into phonons of f r e q u e n c i e s w 1 and w 2 (with wave v e c t o r s k 1 and k 2 r e s p e c t i v e l y ) , then the i n v e r s e l i f e t i m e contains a t e m p e r a t u r e dependent f a c t o r [1 + N 1 + N2] , w h e r e N 1 and N 2 r e f e r to the e q u i l i b r i u m occupation n u m b e r s of the w 1 and 0)2 m o d e s [6]. In o r d e r to fit our data to this t h e o r y , we m u s t add to the above p r o c e s s (000 -~ 001 + 002) a c o n c u r r e n t d ecay p r o c e s s , o)0 -4 0) 3 + 004. The r e s u l t i n g i n v e r s e l i f e t i m e (or the line width) may be w r i t t en as follows: A ~ = 2 A [ 1 + N 1 + N 2 ] + + 2B[1 + N 3 + N4] = A[coth (Pi001/2kT) + + coth (~w2/2kT) ] + B[coth (~i003/2kT) + + coth (~iw4/akT) ]. The s o l id c u r v e in the f i g u r e
9 October 1967
is obtained with the following set of constants: w0 = 2.046 x 1013 s e c - 1 (1086 c m - 1 ) , A = 0.2185 cm -1, B = 0 . 0 0 1 5 c m -1, o)1 = 5 . 2 4 x 1 0 1 2 s e c - 1 (278 c m - 1 ) , 0) 2 = 1.52 x 1013 s e c - 1 (808 c m - 1 ) , 0)3 = 1.31 x 1010 sec -1 (1 c m - 1 ) , w4 = 2.045 x × 1013 s e c - 1 (1085 c m - 1 ) . The s e e m i n g l y good a g r e e m e n t between the data and the t h e o r y a s shown in the f i g u r e , is probably f o r t u i t o u s . One m u s t r e m e m b e r that t h e r e a r e 15 phonon b r a n c h es in c a l c i t e , and t h e r e will s u r e l y be many a d d i tional decay paths allowed by the s e l e c t i o n r u l e s . Since phonon d i s p e r s i o n c u r v e s f o r c a l c i t e a r e not a v a i l a b l e , it is i m p o s s i b l e to a t t e m p t c a l c u lations of s e l e c t i o n r u l e s or a r i g o r o u s c o m p a r i son of e x p e r i m e n t and t h eo r y . N e v e r t h e l e s s , with. in the e x p e r i m e n t a l a c c u r a c y of t h e p r e s e n t data, the a n h a r m o n i c i t y m e c h a n i s m does account f o r the t e m p e r a t u r e dependence of the width of A l g Ram an line in c a l c i t e . The author acknowledges the t e c h n i c a l a s s i s t an ce r e n d e r e d by K. Wecht.
References 1. K. Park, Phys. Letters 22 (1966) 39. 2. L.S. Ornstein and J . J . Went, Physiea 2 (1935) 503. 3. K. Venkateswarlu, Proc. Ind. Acad. Sci. A16 (1942) 45. 4. R.S. Krishnan, Proc. Ind. Acad. Sei. A22 (1945) 182. 5. P.K. Naraynanaswamy, Proc. Ind. Acad. Sci. A26 (1947) 511. 7. P.G. Klemens, Phys. Rev. 148 (1966)845.
491
PHYSICS
LETTERS
9 October 1967
Table 1 (Approx.) (microns) 373 337 335 311 310 284
Frequency * (GHz)
Wavelength ** (microns)
804.7509 890.7607 894.4142 964.3134 967.9658 (unmeasured)
Wavelength (calculated)*** (microns)
372.5283 336.5578 335.1831 310.8870 309.7140 284
372.55 336.50 335.24 311.83 309.70 284.01
* Estimated e r r o r 1 part in 106 {~ 1 mc). ** Speed of light assumed to be 2.997 925 x 108 m / s e c . *** The e r r o r is conservatively 3 parts in 104.
i n c h e s in d i a m e t e r , a n d e m p l o y e d a f o l d e d c o n focal m i r r o r configuration. With this l a s e r the m e a s u r e d p o w e r output e x c e e d e d 15 mW at 337p a n d h a s y i e l d e d r e c t i f i e d s i g n a l s at t h e h a r m o n i c m i x e r a s l a r g e a s 100 m V . T h e m i c r o w a v e s e t u p w h i c h w a s e m p l o y e d in t h e f r e q u e n c y m e a s u r i n g p a r t of t h i s e x p e r i m e n t w a s e s s e n t i a l l y s i m i l a r to t h o s e u s e d p r e v i o u s l y
[2,3]. W e w o u l d l i k e to a c k n o w l e d g e n u m e r o u s u s e ful d i s c u s s i o n s w i t h P r o f . D. R a m a c h a n d r a R a o a n d h i s a s s i s t a n c e in t h i s e x p e r i m e n t . W e a l s o
THERMAL
VARIATION
OF
a c k n o w l e d g e t h e t e c h n i c a l a i d of M r . L e o G e o f frion.
References 1. D . R . L i d e a n d A . G . Maki. Appl. Phys. L e t t e r s 11 (1967) 2. We are deeply indebted to Dr. Lide and Dr. Maki for providing us with this information before it was published. 2. L.O. Hocker. A. Javan. D. Ramachandra Rao. L. Frenkel and T.Sullivan. Appl. Phys. L e t t e r s 10 (1967) 5. 3. L.O. Hocker. D.Ramachandra Rao and A. Javan. Phys. L e t t e r s 24A (1967) 12.
A RAMAN
LINE
WIDTH
IN
CALCITE*:~
K. P A R K
Department of Physics. University of Oregon. Eugene. Oregon. USA Received 2 September 1967
A width m e a s u r e m e n t of the Alg Raman line in calcite (P = 1086 cm -1) over an extended t e m p e r a t u r e range (10OK to 660OK) is reported. The r e s u l t is compared with a theory based on anharrnonicity.
W e r e p o r t e d in a p r e v i o u s p a p e r [1] t h a t t h e w i d t h of t h e A l g R a m a n l i n e in c a l c i t e (F = 1086 c m -1) i s v e r y n a r r o w a n d e x h i b i t s a m a r k e d n a r r o w i n g with d e c r e a s i n g t e m p e r a t u r e . We a l s o reported several subsidiary measurements made in a n e f f o r t t o r e s o l v e t h e d i s c r e p a n c y b e t w e e n our r e s u l t and o t h e r p u b l i s h e d data [2-5]. F r o m t h e s e m e a s u r e m e n t s we concluded that t h e r e is n o s i m p l e way t o r e c o n c i l e t h e d i s c r e p a n c y . In 490
t h i s l e t t e r we r e p o r t t h e l i n e w i d t h d a t a o v e r a n e x t e n d e d t e m p e r a t u r e r a n g e **
* Work supported in p a r t by the Army R e s e a r c h Office, Durham. A portion of this work was done while the author was at Bell Telephone Labs., Murray Hill, N.J., USA. ** The experimental a r r a n g e m e n t and the details of slit width c o r r e c t i o n have been d i s c u s s e d in ref. 1.
Volume 25A, number7
,o
~o
~
PHYSICS L E T T E R S
~o
~o
'
~o . . . .
~oo
TEMPE I~ATURI[ ~K)
Fig. 1. Line width versus temperature. See the text for the theoretical curve (solid curve). The e r r o r s in the experimental points are typically i 10%. Fig. 1 shows the v a r i a t i o n of line width as a function of t e m p e r a t u r e . The solid c u r v e is the r e s u l t of an a t t e m p t to fit the data to the t h e o r y of the a n h a r m o n i c decay of optical phonons [6]. If an optical phonon of f r e q u e n c y w o d e c a y s into phonons of f r e q u e n c i e s w 1 and w 2 (with wave v e c t o r s k 1 and k 2 r e s p e c t i v e l y ) , then the i n v e r s e l i f e t i m e contains a t e m p e r a t u r e dependent f a c t o r [1 + N 1 + N2] , w h e r e N 1 and N 2 r e f e r to the e q u i l i b r i u m occupation n u m b e r s of the w 1 and 0)2 m o d e s [6]. In o r d e r to fit our data to this t h e o r y , we m u s t add to the above p r o c e s s (000 -~ 001 + 002) a c o n c u r r e n t d ecay p r o c e s s , o)0 -4 0) 3 + 004. The r e s u l t i n g i n v e r s e l i f e t i m e (or the line width) may be w r i t t en as follows: A ~ = 2 A [ 1 + N 1 + N 2 ] + + 2B[1 + N 3 + N4] = A[coth (Pi001/2kT) + + coth (~w2/2kT) ] + B[coth (~i003/2kT) + + coth (~iw4/akT) ]. The s o l id c u r v e in the f i g u r e
9 October 1967
is obtained with the following set of constants: w0 = 2.046 x 1013 s e c - 1 (1086 c m - 1 ) , A = 0.2185 cm -1, B = 0 . 0 0 1 5 c m -1, o)1 = 5 . 2 4 x 1 0 1 2 s e c - 1 (278 c m - 1 ) , 0) 2 = 1.52 x 1013 s e c - 1 (808 c m - 1 ) , 0)3 = 1.31 x 1010 sec -1 (1 c m - 1 ) , w4 = 2.045 x × 1013 s e c - 1 (1085 c m - 1 ) . The s e e m i n g l y good a g r e e m e n t between the data and the t h e o r y a s shown in the f i g u r e , is probably f o r t u i t o u s . One m u s t r e m e m b e r that t h e r e a r e 15 phonon b r a n c h es in c a l c i t e , and t h e r e will s u r e l y be many a d d i tional decay paths allowed by the s e l e c t i o n r u l e s . Since phonon d i s p e r s i o n c u r v e s f o r c a l c i t e a r e not a v a i l a b l e , it is i m p o s s i b l e to a t t e m p t c a l c u lations of s e l e c t i o n r u l e s or a r i g o r o u s c o m p a r i son of e x p e r i m e n t and t h eo r y . N e v e r t h e l e s s , with. in the e x p e r i m e n t a l a c c u r a c y of t h e p r e s e n t data, the a n h a r m o n i c i t y m e c h a n i s m does account f o r the t e m p e r a t u r e dependence of the width of A l g Ram an line in c a l c i t e . The author acknowledges the t e c h n i c a l a s s i s t an ce r e n d e r e d by K. Wecht.
References 1. K. Park, Phys. Letters 22 (1966) 39. 2. L.S. Ornstein and J . J . Went, Physiea 2 (1935) 503. 3. K. Venkateswarlu, Proc. Ind. Acad. Sci. A16 (1942) 45. 4. R.S. Krishnan, Proc. Ind. Acad. Sei. A22 (1945) 182. 5. P.K. Naraynanaswamy, Proc. Ind. Acad. Sci. A26 (1947) 511. 7. P.G. Klemens, Phys. Rev. 148 (1966)845.
491