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Neutron line-width of longitudinal phonons in solid helium
Volume30A, number 2
PHYSICS LETTERS
A f u r t h e r r e m a r k a b l e f e a t u r e is the following. If the e l e c t r o n d e n s i t i e s and t e m p e r a t u r e s a r e a l m o s t identical, a s the above m e a s u r e m e n t s indicate, then f r o m the theory we expect both c u r v e s to coincide. In fact the o r d i n a t e s of c u r v e A a r e 1.4 t i m e s a s g r e a t a s those of c u r v e B. We have c a l i b r a t e d the signal by c o m p a r i s o n with Rayleigh s c a t t e r i n g f r o m n i t r o g e n gas and we find that for c u r v e B the a r e a u n d e r the c u r v e is in a g r e e m e n t with the t h e o r e t i c a l l y expected total s i g n a l but that c u r v e A is 1.4 t i m e s g r e a t e r than this. We a r e led to speculate that the i n i t i a l p r e s s u r e in some way c o n t r o l s t u r b u l a n c e s which c a u s e e l e c t r o n d e n s i ty fluctuations g r e a t e r than to be expected for a thermal plasma. We would like to acknowledge m a n y a n i m a t e d d i s c u s s i o n s about the i n t e r p r e t a t i o n of our r e s u l t s
22 September 1969
with Dr. W. Kegel. We also would like to e x p r e s s our gratitude to the National R e s e a r c h Council of Canada and to the U n i v e r s i t y of B r i t i s h Columbia, whose g r a n t s made it p o s s i b l e for one of u s to p a r t i c i p a t e in this work, which was p e r f o r m e d u n d e r the t e r m s of the a g r e e m e n t on a s s o c i a t i o n between the Institut fiir P l a s m a p h y s i k and E u r a tom.
References 1. H. Ringler and R. A. Nodwell, Phys. Letters 29a (1969} 151 2. H. Ringler and R. A. Nodwcll, Third Europ. Conf. on Controlled fusionand plasma physics, Utrecht (1969) 3. Mahn, Ringler, Wienecke, Witkowskiand Zankl, Z. Naturforsch. 19a (1964) 1202
*****
NEUTRON
LINE-WIDTH
OF
LONGITUDINAL
PHONONS
IN SOLID
HELIUM*
P. G. KLEMENS and F. P. LIPSCHULTZ Department of Physics and Institute of Materials Science, University of Connecticut, Storrs, Connecticut 06268, USA Received 11 August 1969
The uncertainty in frequency of phonons due to anharmonic three-phonon processes calculated for solid helium is found to be larger than the uncertainty observed experimentally. This discrepancy is attributed to dispersion. Cubic a n h a r m o n i c i t i e s c a u s e longitudinal phon o n s of frequency ,.~ to b r e a k up into two phonons of lower frequency. The r e s u l t i n g u n c e r t a i n t y in f r e q u e n c y at zero t e m p e r a t u r e has been given [1] as
AW ~ 2.5r2 W ~
(I)
where ~ is the Grtlneisen constant, coD the longitudinal Debye frequency, M the atomic mass and v the transverse sound velocity. For most solids AW/W is tOO small to be reliably measured by neutron line-width with present techniques. Solid helium, because of its low Debye temperature and its low value of M y 2 (or elastic constants) is exceptional; we would expect longer values of Aw/w. One may, of course, question whether the usual * Supported by the Air Force Office of Scientific Reaseareh under Grant No. AF-AFOSR-68-1517.
theory on a n h a r m o n i c i n t e r a c t i o n s [2] should apply to such a n exceptional solid. One may a r g u e in favor of its applicability b e c a u s e that theory is phenomenological and t h e r e f o r e g e n e r a l , and b e cause the m e a s u r e d phonon s p e c t r a [3-6] of solid 4He do not differ g r e a t l y f r o m what is expected in conventional c a s e s . E s t i m a t i n g the a n h a r m o n icity p a r a m e t e r r r e q u i r e s a detailed a t o m i c theory [7-10] ; we shall a s s u m e ~ = 2 a s in o r d i n a r y solids. T h e r e is some i n t e r e s t in c o m p a r i n g eq. (1) with a v a i l a b l e data on n e u t r o n l i n e - w i d t h s . As will be shown below, theory c o n s i s t e n t l y o v e r e s t i m a t e s Aw/¢o. One could r e g a r d it as a f a i l u r e of conventional theory, or as an indication that solid h e l i u m has exceptionally low a n h a r m o n icity coefficients T. T h i s would not be in a c c o r d with the g e n e r a l t r e n d of t h e r m a l conductivity data [11, 12] ; the t h e r m a l conductivity of p u r e solid h e l i u m and its isotopic m i x t u r e s g e n e r a l l y support the validity of the theory of a n h a r m o n i c 127
Volume 30A, n u m b e r 2
PHYSICS
interactions. An alternative explanation, which we shall propose, is that many three-phonon proce s s e s w h i c h a r e a l l o w e d in a n i s o t r o p i c c o n t i n u u m , a n d w h i c h c o n t r i b u t e to (1), a r e f o r b i d d e n in s o l i d h e l i u m by t h e r e q u i r e m e n t s of f r e q u e n c y a n d m o m e n t u m c o n s e r v a t i o n , if t h e m e a s u r e d d i s p e r s i o n c u r v e s a r e u s e d i n s t e a d of t h e c o n t i n u u m m o d e l . T h u s a t 26 a t m p r e s s u r e , a s in t h e B r o o k h a v e n w o r k [4, 5] 0 L = 2 6 ° K , My 2 / k = 1 0 0 ° K , a n d t a k i n g ~ = 2 , we o b t a i n f r o m (1) t h a t
ACO/CO = 4 [T(CO)/40] 4
(2)
w h e n T(co) =l~co/k . In t h e c - d i r e c t i o n t h e z o n e b o u n d a r y i s c l o s e to t h e o r i g i n , t h e m a x i m u m f r e q u e n c y of t h e L A b r a n c h c o r r e s p o n d s to T(co) = = 2 6 ° K , s o t h a t ACO/CO s h o u l d b e 0 . 7 . T h e o b s e r ved width is considerably less . The same phonons w e r e m e a s u r e d a t 200 a t m p r e s s u r e by t h e A m e s g r o u p [6]. Now coD i s d e c r e a s e d b y a f a c t o r 2, My 2 by a f a c t o r 4, ACO/CO a t f i x e d co b y a f a c t o r 3 0 , b u t s i n c e co a t t h e z o n e b o u n d e r y i s i n c r e a s e d by a f a c t o r 2, ACO/W a t t h e z o n e b o u n d e r y s h o u l d b e d e c r e a s e d by a f a c t o r 2 o n l y to a b o u t 0.3. T h e o b s e r v e d w i d t h i s a b o u t 0.1. A s i m i l a r d i s c r e p a n c y e x i s t s f o r t h e IX) p h o n o n a t t h e z o n e b o u n d e r y in t h e c - d i r e c t i o n . By u s i n g an extended zone scheme, this can be treated like an LA phonon, since the energy gap at the zone b o u n d e r y i s z e r o in t h a t d i r e c t i o n [5, 6], so t h a t umklapp processes have zero matrix elements [13]. H e r e a g a i n w e w o u l d e x p e c t a t t h e h i g h e r p r e s s u r e t h a t ACO/CO = 2, i . e . t h a t t h e l i n e s h o u l d b e w a s h e d out, w h i l e t h e A m e s g r o u p f o u n d ACO/CO = 0.2 to 0.3. We suggest the following reason for the discrepancy. In the isotropic continuum the allowed proce s s e s a r e L ~*T + T o r L , - . L + T , w h e r e L a n d T denote longitudinal and transverse phonons. The l o c i of t h e p o s s i b l e p r o d u c t p h o n o n s i n q - s p a c e a r e d e s c r i b e d in r e f e r e n c e [1] - t h e i r d i m e n s i o n s e n t e r i n t o (1). I t w a s a s s u m e d t h a t v T / V L = ~,
128
LE TTERS
22 September 1969
where ~r, v L are the respective phase velocities. D i s p e r s i o n in g e n e r a l r e d u c e d v L a t t h e z o n e b o u n d e r i e s m u c h m o r e t h a n v T of h a l f t h e f r e q u e n cy. A s v L a p p r o a c h e s v T , t h e l o c i a r e g r e a t l y r e duced in extent or eliminated altogether. As a result the fractional line-widths are greatly reduced. T h i s i s n o t a p h e n o m e n o n p e c u l i a r to s o l i d h e l i u m , a s d i s p e r s i o n i s a c o m m o n p r o p e r t y of m o s t c r y s t a l s . Solid h e l i u m i s o n l y r e m a r k a b l e i n t h a t ACO/CO c a n b e o b s e r v e d a t l e a s t w i t h s u f f i c i e n t a c c u r a c y to i n d i c a t e t h a t (1) d o e s n o t h o l d a t t h e z o n e b o u n d e r y . F o r t h a t r e a s o n it w o u l d b e of i n t e r e s t to i m p r o v e l i n e - w i d t h s t u d i e s in t h i s m a t e r i a l , a n d a t t h e s a m e t i m e to c a l c u l a t e /,co/co from the conventional theory, but using the actual dispersion data.
References 1. P.G. Klemens, J.Appl. Phys. 38 (1967) 4573. 2. P.G. Klemens, in Solid State Physics, Vol. 7 (Academic P r e s s , New York) p. 1. 3. M. Bitter, W. G i s s l e r and T. Springer, Phys. Status Solidi 23 (1967) K155. 4. F. P. Lipschultz, V.J. Minkiewicz, T.A. Kitchens, G. Shirane and R. Nathans, Phys. Rev. L e t t e r s 19 {1967) 1307. 5. V . J . Minkiewicz, ]?. A. Kitchens, F. P. Lipschultz, R. Nathans and G. Shirane, Phys. Rev. 174 {1968) 267. 6. r . O . Brun, S.K. Sinha, C.A. Swenson and C.R. l~ilford, in; Proc. Intern. Atomic Energy Agency Symp. on Inelastic neutron s c a t t e r i n g , Copenhagen 1968 (I. A. E. A., Vienna), to be published. 7. N.S. Gillis and N. R. W e r t h a m e r , Phys. Rev. 167 (1968) 607. 8. F.W. de Wette, L.H. Nosanow and N.R. W e r t h a m e r , Phys. Rev. 162 (1967) 824. 9. H. Horner, Z. Physik 205 (1967) 72. 10. R.A. Cowley, Reports on P r o g r e s s in Physics 31 (1968) 123. 11. B. B e r t a m , H.A. Fairbank, R.A. Guyer and C. W. White, Phys. Rev. 142 (1966)79. 12. R. B e r m a n , C. L. Bounds and S.J. Rogers, Proc. Roy. Soe. (London) A289 (1965) 66. 13. P.G. Klemens, Phys. Rev. 148 (1966)845.
PHYSICS LETTERS
A f u r t h e r r e m a r k a b l e f e a t u r e is the following. If the e l e c t r o n d e n s i t i e s and t e m p e r a t u r e s a r e a l m o s t identical, a s the above m e a s u r e m e n t s indicate, then f r o m the theory we expect both c u r v e s to coincide. In fact the o r d i n a t e s of c u r v e A a r e 1.4 t i m e s a s g r e a t a s those of c u r v e B. We have c a l i b r a t e d the signal by c o m p a r i s o n with Rayleigh s c a t t e r i n g f r o m n i t r o g e n gas and we find that for c u r v e B the a r e a u n d e r the c u r v e is in a g r e e m e n t with the t h e o r e t i c a l l y expected total s i g n a l but that c u r v e A is 1.4 t i m e s g r e a t e r than this. We a r e led to speculate that the i n i t i a l p r e s s u r e in some way c o n t r o l s t u r b u l a n c e s which c a u s e e l e c t r o n d e n s i ty fluctuations g r e a t e r than to be expected for a thermal plasma. We would like to acknowledge m a n y a n i m a t e d d i s c u s s i o n s about the i n t e r p r e t a t i o n of our r e s u l t s
22 September 1969
with Dr. W. Kegel. We also would like to e x p r e s s our gratitude to the National R e s e a r c h Council of Canada and to the U n i v e r s i t y of B r i t i s h Columbia, whose g r a n t s made it p o s s i b l e for one of u s to p a r t i c i p a t e in this work, which was p e r f o r m e d u n d e r the t e r m s of the a g r e e m e n t on a s s o c i a t i o n between the Institut fiir P l a s m a p h y s i k and E u r a tom.
References 1. H. Ringler and R. A. Nodwell, Phys. Letters 29a (1969} 151 2. H. Ringler and R. A. Nodwcll, Third Europ. Conf. on Controlled fusionand plasma physics, Utrecht (1969) 3. Mahn, Ringler, Wienecke, Witkowskiand Zankl, Z. Naturforsch. 19a (1964) 1202
*****
NEUTRON
LINE-WIDTH
OF
LONGITUDINAL
PHONONS
IN SOLID
HELIUM*
P. G. KLEMENS and F. P. LIPSCHULTZ Department of Physics and Institute of Materials Science, University of Connecticut, Storrs, Connecticut 06268, USA Received 11 August 1969
The uncertainty in frequency of phonons due to anharmonic three-phonon processes calculated for solid helium is found to be larger than the uncertainty observed experimentally. This discrepancy is attributed to dispersion. Cubic a n h a r m o n i c i t i e s c a u s e longitudinal phon o n s of frequency ,.~ to b r e a k up into two phonons of lower frequency. The r e s u l t i n g u n c e r t a i n t y in f r e q u e n c y at zero t e m p e r a t u r e has been given [1] as
AW ~ 2.5r2 W ~
(I)
where ~ is the Grtlneisen constant, coD the longitudinal Debye frequency, M the atomic mass and v the transverse sound velocity. For most solids AW/W is tOO small to be reliably measured by neutron line-width with present techniques. Solid helium, because of its low Debye temperature and its low value of M y 2 (or elastic constants) is exceptional; we would expect longer values of Aw/w. One may, of course, question whether the usual * Supported by the Air Force Office of Scientific Reaseareh under Grant No. AF-AFOSR-68-1517.
theory on a n h a r m o n i c i n t e r a c t i o n s [2] should apply to such a n exceptional solid. One may a r g u e in favor of its applicability b e c a u s e that theory is phenomenological and t h e r e f o r e g e n e r a l , and b e cause the m e a s u r e d phonon s p e c t r a [3-6] of solid 4He do not differ g r e a t l y f r o m what is expected in conventional c a s e s . E s t i m a t i n g the a n h a r m o n icity p a r a m e t e r r r e q u i r e s a detailed a t o m i c theory [7-10] ; we shall a s s u m e ~ = 2 a s in o r d i n a r y solids. T h e r e is some i n t e r e s t in c o m p a r i n g eq. (1) with a v a i l a b l e data on n e u t r o n l i n e - w i d t h s . As will be shown below, theory c o n s i s t e n t l y o v e r e s t i m a t e s Aw/¢o. One could r e g a r d it as a f a i l u r e of conventional theory, or as an indication that solid h e l i u m has exceptionally low a n h a r m o n icity coefficients T. T h i s would not be in a c c o r d with the g e n e r a l t r e n d of t h e r m a l conductivity data [11, 12] ; the t h e r m a l conductivity of p u r e solid h e l i u m and its isotopic m i x t u r e s g e n e r a l l y support the validity of the theory of a n h a r m o n i c 127
Volume 30A, n u m b e r 2
PHYSICS
interactions. An alternative explanation, which we shall propose, is that many three-phonon proce s s e s w h i c h a r e a l l o w e d in a n i s o t r o p i c c o n t i n u u m , a n d w h i c h c o n t r i b u t e to (1), a r e f o r b i d d e n in s o l i d h e l i u m by t h e r e q u i r e m e n t s of f r e q u e n c y a n d m o m e n t u m c o n s e r v a t i o n , if t h e m e a s u r e d d i s p e r s i o n c u r v e s a r e u s e d i n s t e a d of t h e c o n t i n u u m m o d e l . T h u s a t 26 a t m p r e s s u r e , a s in t h e B r o o k h a v e n w o r k [4, 5] 0 L = 2 6 ° K , My 2 / k = 1 0 0 ° K , a n d t a k i n g ~ = 2 , we o b t a i n f r o m (1) t h a t
ACO/CO = 4 [T(CO)/40] 4
(2)
w h e n T(co) =l~co/k . In t h e c - d i r e c t i o n t h e z o n e b o u n d a r y i s c l o s e to t h e o r i g i n , t h e m a x i m u m f r e q u e n c y of t h e L A b r a n c h c o r r e s p o n d s to T(co) = = 2 6 ° K , s o t h a t ACO/CO s h o u l d b e 0 . 7 . T h e o b s e r ved width is considerably less . The same phonons w e r e m e a s u r e d a t 200 a t m p r e s s u r e by t h e A m e s g r o u p [6]. Now coD i s d e c r e a s e d b y a f a c t o r 2, My 2 by a f a c t o r 4, ACO/CO a t f i x e d co b y a f a c t o r 3 0 , b u t s i n c e co a t t h e z o n e b o u n d e r y i s i n c r e a s e d by a f a c t o r 2, ACO/W a t t h e z o n e b o u n d e r y s h o u l d b e d e c r e a s e d by a f a c t o r 2 o n l y to a b o u t 0.3. T h e o b s e r v e d w i d t h i s a b o u t 0.1. A s i m i l a r d i s c r e p a n c y e x i s t s f o r t h e IX) p h o n o n a t t h e z o n e b o u n d e r y in t h e c - d i r e c t i o n . By u s i n g an extended zone scheme, this can be treated like an LA phonon, since the energy gap at the zone b o u n d e r y i s z e r o in t h a t d i r e c t i o n [5, 6], so t h a t umklapp processes have zero matrix elements [13]. H e r e a g a i n w e w o u l d e x p e c t a t t h e h i g h e r p r e s s u r e t h a t ACO/CO = 2, i . e . t h a t t h e l i n e s h o u l d b e w a s h e d out, w h i l e t h e A m e s g r o u p f o u n d ACO/CO = 0.2 to 0.3. We suggest the following reason for the discrepancy. In the isotropic continuum the allowed proce s s e s a r e L ~*T + T o r L , - . L + T , w h e r e L a n d T denote longitudinal and transverse phonons. The l o c i of t h e p o s s i b l e p r o d u c t p h o n o n s i n q - s p a c e a r e d e s c r i b e d in r e f e r e n c e [1] - t h e i r d i m e n s i o n s e n t e r i n t o (1). I t w a s a s s u m e d t h a t v T / V L = ~,
128
LE TTERS
22 September 1969
where ~r, v L are the respective phase velocities. D i s p e r s i o n in g e n e r a l r e d u c e d v L a t t h e z o n e b o u n d e r i e s m u c h m o r e t h a n v T of h a l f t h e f r e q u e n cy. A s v L a p p r o a c h e s v T , t h e l o c i a r e g r e a t l y r e duced in extent or eliminated altogether. As a result the fractional line-widths are greatly reduced. T h i s i s n o t a p h e n o m e n o n p e c u l i a r to s o l i d h e l i u m , a s d i s p e r s i o n i s a c o m m o n p r o p e r t y of m o s t c r y s t a l s . Solid h e l i u m i s o n l y r e m a r k a b l e i n t h a t ACO/CO c a n b e o b s e r v e d a t l e a s t w i t h s u f f i c i e n t a c c u r a c y to i n d i c a t e t h a t (1) d o e s n o t h o l d a t t h e z o n e b o u n d e r y . F o r t h a t r e a s o n it w o u l d b e of i n t e r e s t to i m p r o v e l i n e - w i d t h s t u d i e s in t h i s m a t e r i a l , a n d a t t h e s a m e t i m e to c a l c u l a t e /,co/co from the conventional theory, but using the actual dispersion data.
References 1. P.G. Klemens, J.Appl. Phys. 38 (1967) 4573. 2. P.G. Klemens, in Solid State Physics, Vol. 7 (Academic P r e s s , New York) p. 1. 3. M. Bitter, W. G i s s l e r and T. Springer, Phys. Status Solidi 23 (1967) K155. 4. F. P. Lipschultz, V.J. Minkiewicz, T.A. Kitchens, G. Shirane and R. Nathans, Phys. Rev. L e t t e r s 19 {1967) 1307. 5. V . J . Minkiewicz, ]?. A. Kitchens, F. P. Lipschultz, R. Nathans and G. Shirane, Phys. Rev. 174 {1968) 267. 6. r . O . Brun, S.K. Sinha, C.A. Swenson and C.R. l~ilford, in; Proc. Intern. Atomic Energy Agency Symp. on Inelastic neutron s c a t t e r i n g , Copenhagen 1968 (I. A. E. A., Vienna), to be published. 7. N.S. Gillis and N. R. W e r t h a m e r , Phys. Rev. 167 (1968) 607. 8. F.W. de Wette, L.H. Nosanow and N.R. W e r t h a m e r , Phys. Rev. 162 (1967) 824. 9. H. Horner, Z. Physik 205 (1967) 72. 10. R.A. Cowley, Reports on P r o g r e s s in Physics 31 (1968) 123. 11. B. B e r t a m , H.A. Fairbank, R.A. Guyer and C. W. White, Phys. Rev. 142 (1966)79. 12. R. B e r m a n , C. L. Bounds and S.J. Rogers, Proc. Roy. Soe. (London) A289 (1965) 66. 13. P.G. Klemens, Phys. Rev. 148 (1966)845.