Contribution of the central peak to the ultrasonic attenuation in structural phase transitions

HAL Id: jpa-00208242

https://hal.archives-ouvertes.fr/jpa-00208242

Submitted on 1 Jan 1975

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Contribution of the central peak to the ultrasonic attenuation in structural phase transitions

C. Tsallis, A. Bachellerie

To cite this version:

C. Tsallis, A. Bachellerie. Contribution of the central peak to the ultrasonic attenuation in structural phase transitions. Journal de Physique, 1975, 36 (2), pp.171-174. �10.1051/jphys:01975003602017100�.

�jpa-00208242�

CONTRIBUTION OF THE CENTRAL PEAK TO THE ULTRASONIC

ATTENUATION IN STRUCTURAL PHASE TRANSITIONS

C. TSALLIS and A. BACHELLERIE

(*)

Service de

Physique

^{du Solide et de Résonance}

Magnétique

Centre d’Etudes Nucléaires de

Saclay,

^BP

2,

⁹¹¹⁹⁰

Gif-sur-Yvette,

^France

(Reçu

^{le 30}

juillet 1974, révisé

^{le 30}

septembre 1974)

Résumé. ²⁰¹⁴ On calcule l’atténuation ultrasonore 03C3 au dessus de T0 ^en supposant : (a) ^{une inter-}

action non résonante entre les ultrasons et les phonons critiques ; (b) ^une forme lorentzienne pour le pic central; (c) ^{une forme assez}

générale

pour ^sa largeur

(d) ^une

procédure

^au

premier

^ordre pour calculer 03C3. On obtient deux régimes (03A9 ~ a ^ou 03A9 ~ a)

et les exposants ^{x et} ~ (03C3 ^{~ 03A9x} a-~) ^sont calculés dans les deux régimes ^{en fonction} de la dimension d et des exposants { ni }. ^{On montre} que

l’anisotropie

^dans

l’espace

^{k ne} joue pas ^un rôle déterminant.

On compare les résultats obtenus ^{avec ceux} expérimentaux pour

SrTiO3.

Abstract. ²⁰¹⁴ The ultrasonic attenuation 03C3 is calculated above T0 assuming : (a) ^{a non resonant}

interaction between ultrasonic and critical phonons ; (b) ^a lorentzian shape for the central peak ; (c) ^a quite general from for its width

(d) ^a first-order procedure ^{to calculate 03C3. There are two distinct} regimes, ^{03A9 ~ a, and 03A9 ~ a and}

the ^x- and ~-exponents (03C3 ^{~ 03A9x} a-~) ^are determined in both

regimes

^{as a} function of the dimen-

sionality d

^and exponents {

ni }.

Anisotropy ⁱⁿ k-space ^{is shown to} be irrelevant.

Comparison

^is

made with available experimental ^{results for} SrTiO3.

Classification

Physics ^Abstracts

7.270

1. Introduction. ^- In recent years much work has been devoted to the

study

of the central

peak

appear-

ing

ⁱⁿ the time - and

space-Fourier -

transformed correlation function

S(k, cv)

^of several structural

phase

transitions

(SrTi03, KMnF3, Tb2(MOO4)3, Gd2(MO04)3, etc.).

^{The main}

goal

^{is of course to}

obtain

knowledge

about the form and width of this central

peak, particularly

^{above the} transition tempe-

rature

To.

^{At the} present ^{date this}

problem

^remains

quite unsolved,

^at least from the

experimental

^stand-

point. Quasi-elastic

^neutron

scattering experiments [1-4] give

^{a direct} pattern ^{of the central}

peak,

^but

unfortunately adequate

resolution near

Tc

^{has not} yet been achieved. An alternative method

might

^{be to}

measure the critical contribution Q to the ultrasonic attenuation because a

change

ⁱⁿ

regime

^is

expected

when the central

peak’s

width becomes

comparable

with ultrasonic

frequency

^{Q. A certain amount of}

work

[5-13]

^has

already

^{been done within this frame}

and if we define the x- and

q’-exponents by writing

.

Typical experimental

^{values are 1}

x, il’

^2.

The main theoretical contributions have been

presented by

Tani and Tsuda

[14] (x

⁼

1, il’

⁼

3/2) ; by Pytte [15]

who critisizes Tani and Tsuda’s

results,

and

assuming

^a soft mode with constant mean life

T-1

obtains two

regimes :

F > Q

leading

^{to x = 2 and}

il = 2,

^{and r « Q}

leading

^{to x = 2 and}

r’

⁼

t) ;

by

^Rehwald

[8] (with assumptions

^{similar to those}

of Pytte he

^{obtains x =}

2 and il’

⁼

t or ’1’

⁼

5 accord-

ing

^{to two kinds of}

k-dependence

of the soft

mode)

and

by

^Schwabl

[16] (in

^the

mode-mode, coupling

framework he obtains two

regimes,

very close to and

relatively

^{far from}

To, leading

^{to x =}

2, il’ = 4

^and

q’

⁼ yd -

dVd

^{+ 2}

respectively,

^where Yd

and vd

^are

the usual

susceptibility

and correlation

length

^d-

(*) Laboratoire d’Ultrasons, Université de Paris VI, ^Tour 13, 4, place Jussieu, 75005 Paris, France.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01975003602017100

172

dimensional critical

exponents).

^{In the} _present paper

some

simple assumptions

^are made which lead to two

regimes,

^whose

respective

^{x- and}

il-values

^{are calcu-}

lated as functions of

dimensionality d,

^{which is known}

to be a relevant parameter ^near

To; 17

is defined

by

(J oc Q’ a - "

(one

^{should note}

that q

^and

il’

^{are not}

necessarily

^the

same).

2. Model. ^- Let us describe our system

by

^a

hamiltonian

where

Hcr, Jeus

^and

Jeint

^{are the}

crystal,

^ultrasonic

and interaction hamiltonian

respectively.

anharmonic terms where 2013 k runs

throughout

^the interior of the first Brillouin zone whose

origin

^{is defined}

by

^{the wave}

vector

(corresponding

^{to k =}

0)

^{which is}

going

^to

freeze

in the ordered

phase (hence

^{k = 0 is not}

necessarily

^{at the centre of the usual Brillouin} zone ;

-

ak+ ,

ak ^are the creation and annihilation boson operators

concerning

^the

phonon

^branch

responsible

for the

transition ;

- (Ok is

supposed

^{to have a}

relatively

^smooth

departure

^{from the non}

vanishing

^value coo ^corres-

ponding

^{to k =}

0 ;

- the anharmonic terms are of course

responsible

for the appearance of the central

peak

^near

To ;

- the

crystal

is assumed to be

simple cubic,

^{but this}

assumption

^is

completely

irrelevant in this

theory

and _may be

easily

^removed.

Thus,

where

- q ^runs

throughout

the usual first Brillouin zone

and is measured from its centre.

Typically,

where c is the

crystalline

parameter ;

-

Qq

^is the ultrasonic normal

coordinate ;

- Qq

^{= vq where v} is the ultrasonic

velocity.

As interaction hamiltonian we consider the first

non

vanishing

^term.

^which,

because of conservation

laws,

^is

given by

where

U(k, q)

^is

given by

^usual

quasi-harmonic approximation (1)

^{in our case}

(1) U(k, q) ^{is a} Hamiltonian’s coefficient and should not be consi- dered as a thermally renormalised one.

This last

approximation

^{is due to the fact that}

only

modes with k - 0 are

important.

^{Therefore we}

adopt U(k, q)

⁼

WQ

⁼

^Wvq,

where W is a constant.

The ultrasonic attenuation is

given

^{to a} first order

approximation by (see

^for

example [17])

where

n(w) = (eP1J(J) - 1 ) -1

^and

x"(k, w)

is the ima-

ginary

part ^{of the}

susceptibility.

We now assume a Lorentzian

shape

^{for the corre-}

lation function

S(k, cv)

^{which is}

presently

^conside-

red

[4, 16, 18]

^{as the most realistic one. Since at}

high

temperatures

(which

^is

usually

^the

experimental case) S(k, co) ( ) oc1 (O ;("(k, co) (see ⁾⁽

^for

example

^p

^{[ ])} [19]),

^{we have}

at once

where the

norm Yk

^{which is a smooth} function of k and

T,

^{is a} real number of the order of

unity,

^{and the}

width is

given by :

where a, bi, ni

> 0

Vi,

^{and a -} 0 when T -

To.

^The

particular

^form

proposed

ⁱⁿ papers

[4]

^and

[18]

corresponds

^{to d =} 3, ni ⁼

2, Vi,

^and

b

1 :0

b2 = b3.

If

we have

and, by complex-plane integration (or by remarking

that this is a convolution

product),

^{we obtain}

where

Y

is a mean

value, 1 W 12

^an energy and u

the inverse of a

length.

(2) ^This restriction is sufficient but it might ^be uncessarily strong.

If we

define bs

⁼ sup

( { bi } ) and ns

^{the corres-}

ponding

^{value of}

the ni ’s (3),

^two

regimes

^are

phy- sically important :

(easily

attainable in ultrasonic

experiments

^{and called}

from now on the

far regime).

(difficultly

attainable in ultrasonic

experiments

^and

called from now on the close

regime).

In this case we may take the limit Q - 0 in the interior

of E

ⁱⁿ

expression (1),

^hence

k

On the other

hand,

which,

ⁱⁿ the limit a - 0 and as

long as bi

^{=1= 0}

Vi,

^becomes

We may summarize the results in the

far regime

^as follows :

where

( + 0)

^{means a}

logarithmic dependence.

In the limit a -> 0 and if u --> oo we may take k ⁼ 0 and a ⁼ 0

everywhere

ⁱⁿ

expression (1)

except in the factor

1/Yk’

^hence

With the definition

we have . and

if q -

^{0 then :}

where nI- ⁼ inf

( {ni

^such

as bi e i =1= 0 } ) (4)

^and

bI, Oh

^{is the} associated coefficient.

Hence

(independent

^of

Q)

^if

Furthermore

performing

^a calculation similar to that of part

a)

^{we obtain}

(this

^{result must be}

rejected

^as inconsistent with the

assumption

^{u -+}

oo)

^’

174

We may summarize the results in the close

regime

as follows :

and and and

and

and

Conclusion. ^- We may remark that for all fixed values of d

and {ni}

^{we have}

il

(close regime) r (far regime) .

This fact seems to be confirmed

by experiments [7, 12].

We also see that

k-anisotropy (at

^{least two différent}

bg s) plays

^a minor role as

long

^{as all}

bg

^{s are}

sufficiently large

^so

that,

^for

pratical

purposes, the

dimensionality

of the

problem

^{is not}

changed.

Finally,

^{let us make a}

quantitative comparison

^with

the

experimental

^results.

Following

^Ref.

[4]

^and

[18]

we way propose for

SrTi03

where a ⁼

A(T - To)

in the range

Numerical values deduced from estimates in ref.

[4]

are

hence the smallest value of a in the temperature domain considered is a ~ 1.6 x 109 s-1 and

On the other hand ultrasonic

experiments reported

in ref.

[5]

^and

[8]

^{were done in the same} temperature domain with 0.3 ^x

10g s-1 Q

^{2 x 108}

s-1,

^so

Q « a and we are indeed in the

far regime.

^Our

theory gives

^{in this case}

(where d

^{= 3 and} ni ⁼ 2

Vi)

ri

= YI’

⁼

.;i 21 X

⁼

^2,

^{which are in}

good

agreement with the

experimental

^results

[5, 8]

^for

longitudinal

waves :

We have the

pleasure

^to

acknowledge

^fruitful

comments from B.

Jouvet,

^R.

Pick,

^R.

Bidaux,

^{J. M.}

Courdille and J.

Dumas,

and critical

reading

^{of the}

manuscript by

^A. Levelut and J. Joffrin.

References [1] RISTE, T., SAMUELSEN, ^E. J., OTNES, K., FEDER, J., Solid State

Commun. 9 (1971) ^1455.

[2] AXE, ^J. D., DORNER, B., SHIRANE, G., Phys. ^{Rev. Lett. 26} (1971) 519.

[3] DORNER, B., AXE, ^J. D., SHIRANE, G., Phys. ^{Rev. B6} (1972)

1950.

[4] SHAPIRO, S. M., AXE, ^J. D., SHIRANE, G., RISTE, T., Phys.

Rev. B6 (1972) ^{4332. ,}

[5] BERRE, B., FOSSHEIM, K., MÜLLER, ^K. A., Phys. Rev. Lett.

23 (1969) ^589.

[6] FURUKAWA, M., FUJIMORI, Y., HIRAKAWA, K., ^J. Phys. ^Soc.

Japan ²⁹ (1970) ^1528.

[7] COURDILLE, ^J. M., DUMAS, J., Doctoral Thesis and Solid State Commun. 9 (1971) ^609.

[8] REHWALD, W., Phys. Kondens. Materie 14 (1971) ^{21. Ed.} by Springer-Verlag ^1971.

[9] CHIZHIKOV, S. I., SOROKIN, ^N. G., OSTROVSKII, ^B. I., ^MELE-

SHINA, V. A., Zhetf ^{Pis. Red. 14} (1971) ^490.

[10] DOMB, ^E. R., MIHALISIN, T., ^SKALYO Jr, J., Phys. ^{Rev. B8} (1973) 5837.

[11] COURDILLE, ^J. M., DUMAS, J., ^{to be} published.

[12] FOSSHEIM, K., MARTINSEN, D., NAESO, A. (See ^ref. [10].) FOSSHEIM, K., MARTINSEN, D., LINZ, A., Proceedings ^{of the}

NATO A.S.I. Meeting ^1973.

[13] GOLDING, B., Phys. ^{Rev. Lett. 25} (1970) ^1439.

[14] TANI, K., TSUDA, N., ^J. Phys. ^Soc. Japan ²⁶ (1969) ^113.

[15] PYTTE, E., Phys. ^{Rev. B 1} (1970) ^924.

[16] SCHWABL, F., Phys. ^{Rev. B 7} (1973) ^2038.

[17] BACHELLERIE, A., JOFFRIN, J., LEVELUT, A., ^J. Physique ³² (1971) 993.

[18] FEDER, J., Proceedings of International School of Physics

Enrico Fermi ^{on «} Local properties ^{at Phase} transitions »

9-21, July, ^1972.

[19] KwoK, ^P. C., Sol. Stat. Phys. 20, ^213.