DYNAMIC CRITICAL BEHAVIOUR IN SOUND PROPAGATION

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DYNAMIC CRITICAL BEHAVIOUR IN SOUND

PROPAGATION

R. Holt, K. Fossheim, J. Fossum

To cite this version:

R. Holt,

K. Fossheim,

J. Fossum.

DYNAMIC CRITICAL BEHAVIOUR IN SOUND

PROPAGATION.

Journal

de

Physique

Colloques,

1981,

42 (C5),

pp.C5-1097-C5-1102.

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DYNAMIC CRITICAL BEHAVIOUR IN S O U N D PROPAGATION

R.M. ~ o l t * , K. Fossheim and J . O . Fossum

Department of Physics, University of Trmdheim, The Norwegian I n s t i t u t e of Technology, 7034 NTH, Trondheim, Norway

A b s t r a c t . - R e s u l t s o f u l t r a s o n i c e x p e r i m e n t s n e a r t h e 187K

s t r u c t u r a l t r a n s i t i o n i n KP4nF3 a r e d i s c u s s e d . F o r t h e hydrodynamic regime ( w - r < < l ) t h e b e h a v i o u r b o t h i n a t t e n u a t i o n and sound v e l o c i t y measurements i s found t o b e i n a c c o r d w i t h p r e d i c t i o n s f o r t h e 3-dimensional Heisenberg model. A dynamic s c a l i n g f u n c C i o n and t h e l i f e t i m e o f o r d e r e d c l u s t e r s a r e d e t e r m i n e d from measurements above T c ' 1. 1 n t r o d u c t i o y . - C r i t i c a l f l u c t u a t i o n s a r e known t o p l a y a n i m p o r t a n t r o l e i n u l t r a s o n i c p r o p a g a t i o n n e a r s t r u c t u r a l phase t r a n s i t i o n s , a s 1

was f i r s t d e m o n s t r a t e d by B e r r e e t . a l . . T h e c r i t i c a l dynamics above

Tc i s c o n t r o l l e d by c l u s t e r r e l a x a t i o n w i t h a c h a r a c t e r i s t i c t i m e 'I

C which i n t h e low f r e q u e n c y l i m i t depends on t h e reduced t e m p e r a t u r e

t = / T-Tc l/Tc a s

Here v i s t h e c o r r e l a t i o n l e n g t h e x p o n e n t and 2-2 i s t h e dynamic exponent.

I n t h i s p a p e r , we d i s c u s s r e s u l t s o f r e c e n t measurements u t i l i z i n g t h e newly developed phonon echo t e c h n i q u e 2-5

,

by which i t h a s been p o s s i b l e t o i n v e s t i g a t e i n some d e t a i l dynamic f e a t u r e s o f sound a t t e n u a t i o n a s s o c i a t e d w i t h b o t h o f t h e above mentioned r e l a x a t i o n a l p r o c e s s e s . The e x p e r i m e n t s a r e performed i n KMnF3

,

which undergoes a weakly f i r s t o r d e r t r a n s i t i o n from a c u b i c t o a t e t r a g o n a l phase a s

t e m p e r a t u r e i s reduced below Tcw187K.

2 . Dynamic b e h a v i o u r i n u l t r a s o n i c p r o p a g a t i o n . - I n a c u b i c p e r o v s k i t e l i k e KMnF3, w i t h a q u a d r a t i c c o u p l i n q between t h e o r d e r p a r a m e t e r and t h e s t r a i n , a t t e n u a t i o n o f sound waves caused by i n t e r a c t i o n w i t h o r d e r p a r a m e t e r f l u c t u a t i o n s i s t h e o n l y a b s o r p t i o n mechanism a s s o c i a t e d w i t h

t h e phase t r a n s i t i o n which i s p r e s e n t above Tc. A s l o n q a s t h e domi- n a n t r e l a x a t i o n a l p r o s e s s i s slow compared t o t h e p e r i o d o f t h e sound wave (w-r < < I ) , r e n o r m a l i z a t i o n group c a l c u l a t i o n s 6 l e a d t o t h e f o l l o w - i n g e x p r e s s i o n f o r t h e c r i t i c a l a t t e n u a t i o n f o r a mode w i t h wavevector

and p o l a r i z a t i o n g i v e n by p:

*permanent a d d r e s s : E l e c t r o n i c s Research L a b o r a t o r y , U n i v e r s i t y of Trondheim, The Norwegian I n s t i t u t e of Technology, 7 0 3 4 NTH, Trondheim,

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JOURNAL DE PHYSIQUE

Mc i s t h e u n i t c e l l mass, kBT t h e t h e r m a l e n e r g y and v i s t h e sound v e l o c i t y . The f u n c t i o n g

( 2 ,

u )

i n c l u d e s t h e c r i t i c a l t e m p e r a t u r e dependence and i s a l i n e a r c o m b i n a t i o n o f terms o f t h e t y p e

The K ' s a r e unknown c o n s t a n t i n t e g r a l s . B . a r e t h e s t r a i n - o r d e r para-

j I ₇

m e t e r c o u p l i n g c o n s t a n t s i n t h e F e d e r - P y t t e H a m i l t o n i a n

.

6

p a r e t h e c r i t i c a l e x p o n e n t s a s d e r i v e d by Murata :

Here a d e n o t e s t h e s p e c i f i c h e a t exponent. Q 2 and Q 3 a r e a n i s o t r o p y c r o s s o v e r e x p o n e n t s . I n E q u a t i o n ( 3 ) t h e dynamic s c a l i n g f u n c t i o n 8

G . ( W T ) h a s been i n c l u d e d . L i t t l e i s known t h e o r e t i c a l l y a b o u t t h e I

form o f t h i s f u n c t i o n , a p a r t from i t s v a l u e G .=I f o r a l l j i n t h e

3

t r u l y hydrodynamic ( W T < < I ) regime. I t i s e x p e c t e d t o behave a s

G . ( u s ) N ( W T ) - K j f o r U T > > I

I ( 5 )

The l i m i t i n g form o f G . a s d e t e r m i n e d by t h e v a l u e o f K i s somewhat

I j

u n c e r t a i n . I t h a s been s u g g e s t e d l 5 t h a t K .=o . / v z . For a 3 - d i m e n s i e 3 3

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L . a r e unknown c o n s t a n t i n t e g r a l s ( L .+K . )

.

The c r i t i c a l e x p o n e n t s y j

3 3 3

t a k e t h e v a l u e s

For a 3-dimensional Heisenberg system y =-0.06, ₁ y ₂=y =0.46. ₃ W e have i n c l u d e d i n E q u a t i o n ( 8 ) a s c a l i n g f u n c t i o n H . ( w ~ )

.

H .=I f o r w ~ < < l ,

3 7

w h i l e i t s form f o r w ~ ? l i s unknown. The c o n n e c t i o n between G and H i s g i v e n by t h e Kramers-Kronig r e l a t i o n s between t h e r e a l a n d i m a g i n a r y p a r t s o f t h e l i n e a r r e s p o n s e f u n c t i o n , which i n t h i s c a s e i s t h e e l a s t i c c o n s t a n t .

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JOURNAL DE PHYSIQUE Longitudinal 5;11[110]

KMng

(SarwkIiTl 0 95 MHz 155MHz A 3 4 5 M H z A 470 MHz 305 MHz O.l

_dl

I I I 1 I , I 0.2 a 3 0.5 1 2 3 5 10 F i g . 1 : Dynamic s c a l i n g f o r u l t r a s o n i c a t t e n u a t i o n a s measured4 i n KMnF3. Data p o i n t s r e f e r t o t h e l o n g i t u d i n a l [ I 101-mode, w h i l e t h e f u l l y drawn c u r v e i s t h e a v e r a g e o v e r

all

samples, modes and f r e - q u e n c i e s .

~ w ~ t - ' i s t h e a t t e n u a t i o n e x t r a p o l a t e d from measurements i n t h e hydrodynamic regime, w h i l e

sobs

i s t h e measured a b s o r p t i o n . W i t h i n l i m i t s o f e x p e r i m e n t a l u n c e r t a i n t y , t h e r e s u l t s f o r a l l modes and samples f a l l a l o n g t h e same c u r v e . The a b s o l u t e v a l u e o f t h e t i m e c o n s t a n t T = T =T t - v Z

C CO h a s been d e r i v e d by s e t t i n g UT,=I a t t h e t e m p e r a t u r e where t h e c r i t i c a l a t t e n u a t i o n a t a g i v e n f r e q u e n c y s t a r t s t o r o l l

-1.87

o v e r from t h e t dependence. T h i s g i v e s o f c o u r s e a r a t h e r rough e s t i m a t e a s f a r a s t h e n u m e r i c a l c o n s t a n t T~~ i s concerned. The t e m p e r a t u r e dependence t-" h a s however been s a t i s f a c t o r i l y demon- s t r a t e d i n t h i s manner. W e f i n d rc0s9 . I

o - ' ~ s ,

which means t h a t rc=1 . 4 . 1 0 - ~ s a t 1K above t h e t r a n s i t i o n p o i n t . The observed r e l a x a - t i o n ?recess i s much t o o slow t o be a s s o c i a t e d w i t h t h e s o f t phonon. We i n t e r p r e t i t a s t h e l i f e t i m e o f o r d e r e d c l u s t e r s e x i s t i n g a s a p r e c u r s o r o r d e r above T c . According t o t h e model o f S c h n e i d e r and

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could be performed throughout t h e whole c r i t i c a l r e g i o n . The tempera- t u r e o f maximum a t t e n u a t i o n i s now frequency dependent. A t 155 MHz, two maxima a r e i n f a c t observed. One, which i s c e n t e r e d a t Tc, i s t h e f l u c t u a t i o n a l damping d i s c u s s e d above. The o t h e r i s i n t e r p r e t e d a s a maximum given by w ~ = l i n t h e Landau-Khalatnikov e x p r e s s i o n

(Equation ( 6 ) )

.

By f i t t i n g t h i s c o n d i t i o n w i t h T =T t-" f o r f o u r P PO

d i f f e r e n t f r e q u e n c i e s , T - 4 - 1 0 - ' ~ s i s found. This i s b e l i e v e d t o be PO

t h e r e l a x a t i o n time f o r t h e o r d e r parameter, s i n c e domain damping i s a

1 2

much slower p r o c e s s

.

Improved temperature r e s o l u t i o n and measurements a t more frequen- c i e s a r e necessary t o o b t a i n a q u a n t i t a t i v e l y more c o r r e c t r e l a x a t i o n time.

5. Sound v e ' l o c i t y measuremeh't's.- Sound v e l o c i t y measurements a t 1 5 m z a r e shown i n F i g u r e 2 .

1000

I I I I I

I

I I

10 -

15 MHz

A I 1

I

I I

I

I I

0.1 0,2

0,s

1

2

5

10

20

50 AT

[Kl

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JOURNAL DE PHYSIQUE

F o r t h e t r a n s v e r s a l and l o n g i t u d i n a l [loo]-mode t h e c r i t i c a l e x p o n e n t was found t o b e ~ = 0 , 4 6 * 0 , 0 5 . For t h e l o n g i t u d i n a l [ I 1 0 1 and [ I l l ] modes i s s l i g h t l y s m a l l e r t h a n 0,46 which i n d i c a t e s i n f l u e n c e from t h e h l t e r m . A l l t h e s e o b s e r v a t i o n s a g r e e w i t h P l u r a t a ' s t h e o r y . R a t i o s between h l , h 2 and h a r e n o t p o s s i b l e t o e x t r a c t from t h e s e

3

e x p e r i m e n t s a l o n e , d u e t o t h e u n c e r t a i n t i e s i n t h e backqround v e l o c i t y . F u r t h e r v e l o c i t y e x p e r i m e n t s and comparison w i t h a t t e n u a t i o n measurements t h r o u g h t h e Kramers-Kronig r e l a t i o n s a r e n e c e s s a r y .

Nonhydrodynamic e f f e c t s are n o t s e e n . T h i s i s n o t s u r p r i s i n g , s i n c e t h e e x p e c t e d d e v i a t i o n from t h e b e h a v i o u r p r e d i c t e d by P u r a t a t a k e s p l a c e w i t h i n T-Tc-0,3K a t 15MHz. So f a r , e x p e r i m e n t s c o u l d n o t b e p e r f o r m e d c l o s e r t h a n O r 5 K from Tc. Measurements a t h i g h e r f r e q u e n - cies and c l o s e r t o Tc a r e i n p r o g r e s s . The a p p a r e n t i n c r e a s e i n t h e t e m p e r a t u r e e x p o n e n t f a r from Tc (T-T$I OK) i s n o t y e t u n d e r s t o o d .

R e f e r e n c e s :

1 . B . B e r r e , K . Fossheim, a n d K.A. M i i l l e r , P h y s . R e v . L e t t e r s

23,

589 (1969)

2. R.M. H o l t and K . Fossheim, F e r r o e l e c t r i c s 25, 515(1980)

_-

3. K . Fossheim and R.M. H o l t , P h y s i c a l A c o u s t i c s Vol. XVI ( t o be

p u b l i s h e d )

4. K . Fossheim a n d R.N. H o l t r P h y s . R e v . L e t t e r s

45,

730 (1980) 5 . R.M. H o l t and K . Fossheim, Phys.Rev.B ( t o b e p u b l i s h e d ) 6 . K.K. Murata, 'Phys.Rev.B

2,

4015 (1976)

7 . A. Aharony, P r o c . 1 3 t h IUPAP C o n f e r e n c e on S t a t i s t i c a l P h y s i c s , Ann. I s r a e l Phys. Soc.

2,

1 3 (1978)

8 . P.C. Hohenberg a n d B . I . H a l p e r i n , Rev.Mod.Phys.

49,

435 (1977) 9. L.D. Landau a n d I.M. K h a l a t n i k o v , Dokl. Akad. Nauk SSSR

96,

469

(1959)