ULTRASONIC RELAXATION IN MOLECULAR CRYSTALS

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ULTRASONIC RELAXATION IN MOLECULAR CRYSTALS

Bernard Perrin

To cite this version:

Bernard Perrin. ULTRASONIC RELAXATION IN MOLECULAR CRYSTALS. Journal de Physique

Colloques, 1981, 42 (C5), pp.C5-677-C5-682. �10.1051/jphyscol:19815104�. �jpa-00220972�

U L T R A S O N I C R E L A X A T I O N

IN

MOLECULAR CRYSTALS B. P e r r i n

Laboratoirc Dynamique Cm'stalline e t Ultrasons, D. R. P., Tour 22, h i v e r s i t E Pierre e t Marie Curie, 4 , place Jussieu, 75230 Paris Cedes 05, fiance

Abstract. - One of the main mechanisms of ultrasonic attenuation in molecular gases and liquids is due to a slow transfer of energy between external and internal degrees of freedom of molecules. In 1959 Liebermann propounded the extension of this relaxation to the solid state and used the approach soon derived in gaseous and liquid phases. A Boltzmann-equation approach is sug- gested here which makes it possible to consider this ultrasonic relaxation as an

anomaly

of the Akhieser loss. tloreover it is shown that anisotropy of the solid state does not act on the frequency dependence of the relaxa- tional term but mainly on its magnitude whose expression used up to now is proving incorrect

;

a new expression is given.

1.

Introduction. - It is well established that a relaxation process may occur in molecular gases and liquids due to a slow transfer of energy between the internal and external degrees of freedom of molecules perturbated by an ultrasonic wave. In

1959

Liebermann suggested that this phenomenon might also occur in the solid state;

1

moreover he Droposed the use of the phenomenological approach soon derived by Richards in the aaseous and liquid phases to relate the ultrasonic absorption coef-

2

ficient

a

and the relaxation time

T.

This relation may be expressed in the follo- win? way

:

where

:

.

^R

and s are respectively the angular frequency and the velocity of the ultrasonic wave.

. C and Cv are the specific heats at constant pressure and constant volume resnectively and P

C I

the specific heat associated with the internal degrees of free-

dom of molecules.

This pioneering work of Liebermann has entailed a certain number of ultrasonic studies on molecular crystals and at the present time an ultrasonic rela- xational behaviour has soon been identified in several com~ounds.

In this work, in contrast to Liebermann's approach, we suggest a solid state viewpoint to analyse this relaxation phenomenon. Molecular crystals may indeed

be

first of all considered such as dielectric crystals forwhichthe ultrasonic

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19815104

JOURNAL DE PHYSIQUE

a t t e n u a t i o n i s w e l l d e s c r i b e d i n t h e hydrodynamical regime o f p r o p a p a t i o n of sound by t h e c l a s s i c a l t h e o r y suggested by A k h i e s e r 3

.

So we propose t o f o l l o w t h e Boltzmann-equation a ~ ~ r o a c h o f Woodruff and E h r e n r e i c h ( w h i c h i s an imnroved v e r - 4 s i o n o f A k h i e s e r ' s t h e o r y ) t o d e s c r i b e t h i s u l t r a s o n i c r e l a x a t i o n .

2. Boltzmann-equation approach.

-

I n A k h i e s e r ' s t h e o r y t h e u l t r a s o n i c wave i s c o n s i - dered as an e x t e r n a l f i e l d which modulates i n space and t i m e t h e p o p u l a t i o n o f t h e r - mal phonons. More p r e c i s e l y t h e l a t t i c e d e f o r m a t i o n a s s o c i a t e d w i t h t h e u l t r a s o n i c wave i n d u c e a s h i f t o f t h e f r e q u e n c y oA o f phonons AX d e s c r i b a b l e by t h e Gruneisen t e n s o r s yA i . ; t h e s e Gruneisen parameters depending on t h e mode A , t h e phonon gas

1 J

i s n o t i n e q u i l i b r i u m and t h e phonon d i s t r i b u t i o n s d e p a r t f r o m Boltzmann d i s t r i b u - t i o n s (we n o t e d n X ) . N e v e r t h e l e s s t h e e v o l u t i o n o f phonons can be d e s c r i b e d by Boltzmann e q u a t i o n s which enable us t o d e t e r m i n e t h e a c t u a l phonon d i s t r i b u t i o n s . The s t a n d a r d u l t r a s o n i c experiments l e a d t o s t r a i n s w h i c h a r e t y p i c a l l y about lod7 t o and t h u s B o l tzrnann e q u a t i o n s may be l i n e a r i z e d ; under t h i s a p p r o x i - m a t i o n t h e c o l l i s i o n terms o f t h e s e e q u a t i o n s may be d e s c r i b e d by t h e phonon c o l l i - s i o n o p e r a t o r M.

Now i t can be shown i n a v e r y g e n e r a l way t h a t t h e u l t r a s o n i c absorp- t i o n c o e f f i c i e n t may be expressed i n terms o f t h e e i g e n v e c t o r s Xi and e i g e n v a l u e s -l/ri (i = 1,p) o f t h e c o l l i s i o n o p e r a t o r F? i n t h e f o l l o w i n g way :

h e r e :

.

t h e e i g e n v e c t o r s X i a r e such t h a t t h e e i g e n v e c t o r s

xi

d e f i n e d by :

a r e n o r m a l i z e d v e c t o r s .

.

^p i s t h e d e n s i t y and y X i s d e f i n e d as f o l l o w s :

where and

k

a r e r e s p e c t i v e l y t h e d i r e c t i o n s o f p o l a r i z a t i o n and o f p r o p a g a t i o n o f t h e u l t r a s o n i c wave.

Eq. ( 2 ) has been o b t a i n e d by n e g l e c t i n g t h e group v e l o c i t y o f phonons which would l e a d t o t h e c l a s s i c a l t h e r m o e l a s t i c l o s s due t o t h e t h e r m a l c o n d u c t i o n between t h e compressed and r a r e f i e d r e g i o n s .

F u r t h e r i n f o r m a t i o n s about ^T~ and Xi a r e necessary t o g e t an a v a i l a b l e e x p r e s s i o n o f a.

r

The i n d e x " A " means t h a t t h e phonon X belongs t o a mode o f wave v e c t o r and p o l a r i z a t i o n a .

-

Hard d i e l e c t r i c c r y s t a l s would be those i n which the s t r e n g h t s of t h e v a r i - ous bonds have t h e same o r d e r of magnitude.

-

Molecular c r y s t a l s would be those i n which s t r o n g bonds forming d i s c e r n i b l e u n i t s which may be termed "molecules" ( i n a broad s e n s e ) c o e x i s t which weak bonds which l i n k t h e s e molecules.

Now, we may assume t h a t i n hard d i e l e c t r i c c r y s t a l s t h e homogeneity of t h e s t r e n q h t s of t h e various bonds l e a d s , on t h e anharmonic l e v e l , t o t h e homoge- n e i t y of t h e i n t e r a c t i o n s between t h e nodes o r between t h e phonons. This assumption makes i t p o s s i b l e t o use t h e c o l l i s i o n time approximation which i s g e n e r a l l y w r i t t e n

i n t h e following way :

In f a c t t h i s formulation i s s l i g h t y i n c o r r e c t because i t can be shown 7 t h a t PI must possess a zero eigenvalue. The corresponding eigenvector i s given by :

where

Cx

i s the s p e c i f i c h e a t a s s o c i a t e d with t h e mode A. This property of M comes from a conservation law of t h e deformation energy ( t o t h e f i r s t order in s t r a i n ) which M obeys.

Thus i t i s more c o n s i s t e n t t o formulate t h e c o l l i s i o n time approximation by saying t h a t t h e c o l l i s i o n o p e r a t o r has t h e zero eigenvalue eigenvector X1 and a second eigenvalue (-1/G) whose eigensubspace has a degeneracy equal t o ( N - 1 ) ; X i s t h e number of modes and f? i s t h e mean l i f e t i m e of thermal phonons. With t h e s e as- sumptions, Eq. 2 gives :

which i s t h e wellknown expression of t h e Akhieser l o s s . W have put e :

JOURNAL DE PHYSIQUE

4. Molecular c r y s t a l s .

-

^{I t} i s c l e a r t h a t t h e c o l l i s i o n time approximation f a i l s i n molecular c r y s t a l s f o r which t h e homogeneity o f the s t r e n g t h o f t h e v a r i o u s phonon i n t e r a c t i o n s no l o n g e r holds. Nevertheless i t i s p o s s i b l e t o o b t a i n i n t e r e s t i n g i n f o r m a t i o n s about t h e behaviour o f u l t r a s o n i c a t t e n u a t i o n i n molecular c r y s t a l s under l i g h t and r e a l i s t i c approximations : we may d e f i n e a " c o l l i s i o n t i m e appro- ximation" r e s t r i c t e d t o the modes which a r e e f f i c i e n t l y coupled w i t h long wave- l e n g t h a c o u s t i c modes. These modes X would be termed " e x t e r n a l " modes and d e f i n e a subspace E w h i l e t h e o t h e r modes would be termed " i n t e r n a l " modes and d e f i n e a sub- space I .

Thus we may assume t h a t t h e c o l l i s i o n operator has a zero eigenvalue

( - 1 / ~ ~ = 0 ) , a main eigenvalue ( - 1 / ~ ~ = - 1 / 0 ) and t h a t t h e o t h e r eigenvalues ( - 1 / ~ ~ ; i > 2 ) correspond t o slow r e l a x a t i o n times and a r e much lower than(-1/8). Two consequences may be deduced from t h e assumptions above :

i ) Phonons belonging t o t h e e x t e r n a l modes a r e thermalized i n f a s t times

(e)

a f t e r a l a t t i c e deformation. Thus we may w r i t e : i >2

XEE

where p . a r e n o r m a l i z i n g f a c t o r s which now c o n t a i n a l l t h e hindered i n f o r m a t i o n s 1

about t h e c o l l i s i o n s between phonons. (We r e c a l l t h a t we normalize t h e v e c t o r s Ti).

i i ) GrUneisen parameters o f the i n t e r n a l modes may be neglected.

Thus Eq. ( 2 ) may be w r i t t e n i n t h e f o l l o w i n g way :

where nA i s the l o s s due t o t h e f a s t r e l a x a t i o n t i m e 0 and aR i s a r e l a x a t i o n a l term given by :

Though normal i i i n g f a c t o r s :U a r e n o t e x p l i c i t , we know t h a t they o n l y depend on i n t r i n s i c p r o p e r t i e s of t h e c o l l i s i o n operator and thus we may deduce an i n t e r e s t i n g i n f o r m a t i o n about t h e r e l a x a t i o n w i t h the expression above : whatever would be t h e r e l a x a t i o n spectrum, t h e frequency dependence o f aR must s t a y inva- r i a n t under any change o f the d i r e c t i o n o f propagation o f t h e u l t r a s o n i c wave. Thus t h e a n i s o t r o p y o f the s o l i d s t a t e can o n l y a f f e c t t h e magnitude o f t h e r e l a x a t i o n .

Now i t would he i n t e r e s t i n g t o compare t h e expression o f t h e magnitude given by Eq. (11) w i t h t h a t proposed by Liebermann (Eq. ( 1 ) ) w h i c h has been used i n mostly previous papers on t h i s f i e l d .

5. Comparison w i t h Liebermann's approach.

-

Liebermann's r e l a t i o n i s i m p l i c i t l y b u i l t on t h e assumption o f a s i n g l e r e l a x a t i o n time which leads i n our approach

ponds to the r a t e of t r a n s f e r of energy between external modes (carrying the energy

U E )

and internal modes (carrying the energy UI)

;

thus

:

-

a (": ^-

^{U I )}

;

-;($ ^U1)

a t 5 c o l l . 5

where

:

C

= C

- C

E v I (13)

Now i t has been discussed in Ref.

( 5 )

how the following expression of

Xg

can be derived from Eq. (12)

:

with

: cX=

( c ~ c ~ ) ~ / ~ ,

XEE

and

: =

- (cEc1)1/2 , XEI

E I

^{( 1 5}

which leads t o

:

2 2 - C I

p3 = 'ACE

-

To get an expression of

aR

i n terms of thermoelastic q u a n t i t i e s available in the experimental f i e l d we can express the generalized Griineisen t e n s o r < y . . > in

1 J

terms of the s p e c i f i c heat a t constant volume C v , the e l a s t i c constants

C i j k l

and the thermal expansion tensor Bkl using the following relation valid in the quasi- harmonic approximation

8 :

1

<Y

> = < v i j > e i K j

=

-

v 'ijkl Bkl e i K j (17)

Then with the help of Eqs. ( l l ) , (16) and (17) we obtain

^:

2

2

1 (Cijkl B k l eiKj) T

C I R T

aR =

- (18)

2 s Ps2 vKz7

I t i s impossible t o introduce in a natural way the quantity t h e equation above except f o r the i s o t r o p i c and cubic symmetries

we may write

:

-

The main difference

w i t h

used

by

Liebermann i s the existence

in E q . (19) of the s t r u c t u r e f a c t o r which r e s u l t s from the actual

C5-682 JOURNAL DE PHYSIQUE

n a t u r e o f t h e s o l i d s t a t e . An a n a l y s i s o f e x p e r i m e n t a l r e s u l t s o b t a i n e d on c u b i c c r y s t a l s has been made w i t h b o t h r e l a t i o n s ( 1 ) and ( 1 9 ) and t h e r e s u l t s p l e a d i n 5 f a v o u r o f Eq. ( 1 9 ) .

Now i t i s p o s s i b l e t o u s e t h e f o l l o w i n g thermodynamic r e l a t i o n :

T - T

C

^{C T . . T}

"i j k l

-

i j k l - mnrs 1j.n k l r s 'mn 'rs ( 2 0 ) which l e a d s t o t h i s e x p r e s s i o n o f aR :

cIss-sT QT

" R = S

q~ a

( t h e indexes "S" and "T" mean r e s p e c t i v e l y a d i a b a t i c and i s o t h e r m a l ) . T h i s a t t e n u a - t i o n i s r e l a t e d t o a d i s p e r s i o n on t h e u l t r a s o n i c v e l o c i t y g i v e n b y :

s-s S T

CT S -S

0

= -

-

s (22)

C~

x

We t h i n k t h a t Eqs. (21) and ( 2 2 ) w h i c h a r e v a l i d f o r any s y m e t r y c o u l d be o b t a i n e d s t r a i g h t b y a p u r e l y phenomenological approach b u i l t on t h e assumptions d i s c u s s e d i n t h i s work.

6. Conclusion.

-

We have shown i n t h i s work t h a t u l t r a s o n i c r e l a x a t i o n i n m o l e c u l a r c r y s t a l s may be d e s c r i b e d i n t h e same way as A k h i e s e r l o s s i n d i e l e c t r i c c r y s t a l s b y a Boltzmann-equation approach ; i n f a c t t h e s e two mechanisms o f sound a b s o r p t i o n seem t o have q u i t e t h e same s t a t u s i n t h e hydrodynamical regime o f p r o p a g a t i o n o f sound. The two main p r o d u c t s o f t h i s approach a r e :

-

F i r s t l y , f o r any r e l a x a t i o n spectrum i t has been shown t h a t t h e a n i s o t r o p y o f t h e s o l i d s t a t e m a i n l y a c t s on t h e magnitude o f t h e r e l a x a t i o n b u t n o t on i t s frequency dependence.

-

Secondly, f o r t h e s p e c i a l case o f a s i n g l e r e l a x a t i o n t i m e i t has been p r o - ved t h a t t h e r e l a t i o n used p r e v i o u s l y i n t h i s f i e l d i s i n c o r r e c t and a new r e l a t i o n which t a k e s i n t o a c c o u n t t h e a c t u a l n a t u r e o f t h e s o l i d s t a t e has been g i v e n .

REFERENCES.

/1/ L.N. LIEBERMANN, Phys. Rev.

113,

1052 (1959) /2/ W.T. RICHARDS, Revs. Modern Phys.

11,

36 (1939) /3/ A. AKHIESER, J . Phys. (USSR)

1,

277 (1939)

/ 4 / T.O. WOODRUFF and H. EHRENREICH, Phys. Rev.

-

123, 1553 (1961) / 5 / B. PERRIN, t o be p u b l i s h e d

/6/ A.E. VICTOR, H.E. ALTMANN J r . a n d R.T. BEYER, J. Phys. (PARIS), Co1 l o q u e C 6, tome

23,

166 (1972)

/7/ H.J. MARIS, Phys. Rev.

188,

1303 (1969)

/8/ W. LUDWIG, S p r i n g e r T r a c t s i n Modern Physics,

43,

93 (1967)