Theory of ultrasonic attenuation in V3Si

Volume 34A, number 3

PHYSICS L E T T E R S

a s s u m e d to be optically opaque as in [1]. Fig. 1 gives the r e s u l t s : The solid c u r v e s r e p r e s e n t the population d e n s i t i e s c a l c u l a t e d by an a p p r o x i m a t e method and published in [1]. The dotts a r e the solutions of the coupled s y s t e m of rate equations published in [2] without any a p p r o x i m a t i o n s . One sees that the a p p r o x i m a t i o n s made in [1] influence mainly the population d e n s i t i e s at low p r i n c i p a l quantum n u m b e r s . In g e n e r a l , the d i s -

22 February 1971

c r e p a n c i e s a r e s m a l l , provided T e is not too much different from T e. This conclusion is different f r o m that one given in [1] for the special c a s e s treated.

References [1] S. Suckewer, Phys. Letters 33A (1970) 110. [2] H. W. Drawin, Z. Naturforseh. 25A {1970) 145. [3] H. W. Drawin, Z. Physik 225 0969) 483.

* * * * *

THEORY

OF

ULTRASONIC

ATTENUATION

IN V 3 S i

H. SCHUSTER and W. KLOSE

lnstitut fiir Theoretische FestkSrperDhysik der Universitttt des Saarlandes D-66 Saarbriicken, Germany

Received 5 January 1971

The ultrasonic attenuation coefficient for V3Si is calculated in the linear chain model for fl -W-compounds. The result deviates from the free electron value and agrees well with the experiment.

The c u b i c - t o - t e t r a g o n a l t r a n s f o r m a t i o n of the E - t u n g s t e n - c o m p o u n d V3Si at Tm = 21°K is p r e ceded by a softening of the s h e a r mode q= (110), e = ( l i 0 ) and an i n c r e a s e of the u l t r a s o n i c a t t e n uation of this mode [1]. Though the mode softening has been explained t h e o r e t i c a l l y by s e v e r al a u t h o r s [2-4], the u l t r a s o n i c attenuation has not. In the following we use the one d i m e n s i o n a l model [2] to c a l c u l a t e the u l t r a s o n i c attenuation coefficient ~ f r o m the e l e e t r o n - r e n o r m a l i z e d phonon frequency ~k. Neglecting the s - s c r e e n e d d - d - c o u l o m b i n t e r a c t i o n [5] we use S c h r i e f f e r ' s f o r m u l a (6-22) [6] and obtain after s u m m i n g over the 3 d - b a n d s : 3 2 w>, =(~0,>, - 2C°O,~. I I j ( q , w x ) . (1)

2

2

.j~

ig~,j(q)]

Here ~ denotes the soft mode. ~ 0 , ) t and w~, a r e the u n r e n o r m a l i z e d and the r e n o r m a l i z e d phonon f r e q u e n c i e s , r e s p e c t i v e l y . g j k ( q ) = 4 (A/2w 0 X) e4q4 is the coupling factor between the s h e a r mode and the one d i m e n s m n a l d - b a n d [7]. ,

I-]j(q,¢~) 152

J

J

.

1 7 d k . af (kj/m*)qj - 7Ta2 -~o 3 ~ (~/m*)qj-~)~t -ia

is the i r r e d u c i b l e p o l a r i s a t i o n p a r t in RPA for band j and q << PF- With a the lattice constant, f the F e r m i - f u n c t i o n , m* the effective m a s s and P F the F e r m i m o m e n t u m . Eq. (1) r e l a t e s the sound velocity C s = (Re w~)/q to the attenuation coefficient = (Ira w~)/C s. By s e p a r a t i n g r e a l and i m a g i n a r y p a r t s in eq. (1) we obtain: oo

~fN(~) d~ • Cs - CO, s =½A 0f oE

(2a)

a=-½A" m* ¢°~t ~f2a2 C2" ~ (m'C2s) ,

(2b)

S

where N(~) is the one dimensional density of states. Eq. (2a) is equivalent to the result of [2-4]. Our result for a differs from Pippard's freeelectron expression [8] a ~ (¢o/C~)]~0) for q.l >> i, l = electronmean-free-path, by the t e m p e r a t u r e dependent factor - ( a f / ~ )(m'C2). T h i s factor i n c r e a s e s strongly for T--* T m and is p e c u l i a r to the one d i m e n s i o n a l model for the d - e l e c t r o n s . It is e s s e n t i a l for the explanation of the t e m p e r a t u r e v a r i a t i o n of a . To evaluate eq. (2b) we put the F e r m i - e n e r g y

PHYSICS LETTERS

Volume 34A, number 3

22 February 1971

In fig. 1 our theory is c o m p a r e d with e x p e r i m e n t a l data [1]. B e s i d e s the t e m p e r a t u r e v a r i a tion, the l i n e a r ~ - d e p e n d e n c e of ~ also a g r e e s with the e x p e r i m e n t [1]. We conclude that s i n c e the u l t r a s o n i c a t t e n u ation can be explained by the l i n e a r chain model [2] it should be p o s s i b l e to detect f e a t u r e s of a one d i m e n s i o n a l b a n d - s t r u c t u r e in the F e r m i s u r f a c e of the V 3 S i - t y p e - c o m p o u n d s , by r e peating the u l t r a s o n i c a b s o r p t i o n e x p e r i m e n t s in a magnetic field.

14

12

~t

~,o

We thank Dr. W. D i e t e r i c h for useful d i s c u s s i o n s and the Deutsche F o r s c h u n g s g e m e i n s c h a f t for financial support.

Z~ 4

2

°1o

R~JeS~'~CeS A

A

40

~0 00 ]00

200

TEMPERATURE ('K) Fig.1. Temperature dependence of attenuation of 60Mc/sec (110) shear waves with (110) polarisation in V3Si. Comparison of theoretical • and experimental O points [1]. g ( ~0) = 20°K [9] and use p a r t i c l e n u m b e r c o n s e r v a ti6n to d e t e r m i n e EF(T). F o r t e m p e r a t u r e s T>EF(O) the F e r m i - f u n c t i o n is r e p l a c e d by the B o l t z m a n n - d i s t r i b u t i o n . The product of the f a c t o r s A, m* in (2b) is d e t e r m i n e d by e x p e r i m e n t [1].

[1] L. R. Testardi and T. B. Bateman, Phys. Rev. 154 (1967) 402. [2] J. Labb~ and J. Friedel, J. Phys. Radium 27 (1966) 153, 303, 708. [3] R. W. Cohen, G.D. Cody and J. J. Halloran, Phys. Rev. Letters 19 (1967) 840. [4] E. Pytte, Phys. Rev. Letters 25 (1970) 1176. [5] W. Dieterich, Z.f. Physik, to be published. [6] J. R. Schrieffer, Theory of superconductivity (Benjamin, New York, 1964). [7] W.Klose and H. Schuster, Z.f. Physik, to be published. [8] A. B. Pippard, Phil. Mag. 46 (1955) 1104. [9] J. Labb~, S. Barisic, J. Friedel, Phys. Rev. Letters 19 (1967) 1039.

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