Elastic anomalies at structural phase transitions : a consistent perturbation theory. I. One component order parameter

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Elastic anomalies at structural phase transitions : a consistent perturbation theory. I. One component order

parameter

A. Levanyuk, S. Minyukov, M. Vallade

To cite this version:

A. Levanyuk, S. Minyukov, M. Vallade. Elastic anomalies at structural phase transitions : a consistent

perturbation theory. I. One component order parameter. Journal de Physique I, EDP Sciences, 1992,

2 (10), pp.1949-1963. �10.1051/jp1:1992258�. �jpa-00246675�

Classification

Physics

Abstracts

64.70K 63.75 62.65

Elastic anomalies at structural phase transitions

_{: a}

consistent

perturbation theory. I. One component order parameter

A. P.

Levanyuk ('.~),

_{S. A.}

_{Minyukov (')}

_{and M. Vallade}

₍₂₎

(')

Institute of

Crystallography,

Russian

Academy

of Sciences, 117333 Moscow, Russia (2) Universit6

Joseph

Fourier, Laboratoire de

Spectromdtrie Physique,

B-P. 87, 38402 Saint-

Martin-d'Hbres Cedex, France

(Received 28

February

1992, accepted in

final form

12 June 1992)

Rksumd. La th60rie

perturbative

des anomalies

dlastiques prbs

des transitions de

phase

structurales est rdexaminde. On montre que les

expressions

utilisdes le

plus

souvent pour

interpr6ter

les anomalies ultrasonores _{ne sont} pas correctes, en

particulier

_en dessous de la temp6rature de transition de

phase

T~. ^{Une th60rie de}

perturbation

auto-coh6rente _est dtablie _{et on}

montre que l'attdnuation du _{son est} donnde par une

expression beaucoup

moins

simple

_que celle

utilisde d'habitude. Los rdsultats

explicites

quant ^{h la}

d£pendance

de l'att£nuation ultrasonore _en fonction de la

tempdrature

et de la

frdquence,

sont donnds pour les deux _cas extrEmes suivants _:

systbmes

ordre-d£sordre et

systbmes displacifs.

Pour les transitions ordre-ddsordre, qui ne se

produisent

pas trop ^{loin du}

point tricritique,

l'attdnuation _est ddcrite par la forrnule de Landau- Khalatnikov (LK) oh l'expression du

param~tre

d'ordre n'est pas dorm£e par la thdorie du champ moyen. Pour les transitions

displacives,

la partie de type ^LK et celle due _aux fluctuations _ont la mEme

d£pendance

_en

temp£rature

(dans la r6gion ^oh le calcul de

perturbations

est valable). En

cons£quence

l'att6nuation basse

frdquence

_a le mEme indice

critique

_pour (es deux

phases,

mais AT_<r~

des

amplitudes critiques

diff£rentes. Le rapport ^de ces

amplitudes

tend _vers z£ro _au

point Ar~r~

tricritique.

En _ce point ^(es indices critiques ne sont plus dgaux entre eux.

Abstract. The

perturbation theory

of elastic anomalies _near structural

phase

transitions is revisited. It is shown that

expressions

more often used to interpret ultrasonic attenuation anomalies

are not correct,

particularly

below the

phase

transition temperature T~. A consistent

perturbation theory

is worked _out and it is shown that the sound _wave attenuation coefficient takes _a form less simple in the general case than that

usually

assumed. The

explicit

results for the temperature ^and

frequency dependence

of the sound attenuation coefficient _are

given

for two extreme cases : order-

disorder systems ^and displacive systems. ^{It is} found that for the order-disorder transitions, which are not too far from the tricritical

point

the main part of the sound attenuation

anomaly

_can be

described by the Landau-Khalatnikov (LK) formula with the order parameter exhibiting a non- mean field behaviour. For

displacive

transitions for both LK and fluctuation, contributions have the _same temperature dependence and the _same order of magnitude ^within the perturbative

region.

As _a result the low

frequency

sound attenuation coefficient has the _same

« critical index _» for the two

phases

but different _« critical

amplitudes

_», the ratio of the

amplitudes Ar~r~/Ar~r~ going

to

zero when the tricritical point is approached. At tricritical

phase

transition the

« critical index _» in the

low-temperature phase

is different from that in the

high-temperature

_one.

1950 JOURNAL DE PHYSIQUE I N° 10

1. Introduction.

Ultrasonic anomalies at structural

phase

transitions have been studied for several decades and _a great amount of

experimental

data has been obtained

(see

I

]).

To

interpret

^{these data} one often

uses the mean-field

theory

_{or a}

perturbative

method which _treats the contribution of the order parameter fluctuations _{as a} small

parameter.

^{Such theories} are

surely

invalid in the close

vicinity

of _a second order

phase

transition but

they

_may

provide

useful

approximations

outside this « critical

region

_».

Although

the basic ideas of

perturbation theory

of ultrasonic anomalies

are

quite

old

[2, 6],

_an exhaustive and consistent treatment has _never been worked out. This has sometimes led

experimentalists

to make _erroneous

interpretations.

The purpose of the present paper is _to

give

_a consistent

perturbation theory

for the _case of _a one-component ^order parameter. ^As we shall _see the

expressions

which describe the ultrasonic

properties

_are far from

being simple

in the

general

_case. That is

why

_we shall discuss in detail _{two extreme}

situations _:

displacive

and order-disorder

phase transitions, although

most

phase

transitions fall between these _two limits.

The paper is

organized

in the

following

_way in section 2 _we make some comments on the

present theoretical situation relative to the kinetic coefficients _near

displacive

and order- disorder

phase

transitions.

Special

attention is

given

to the temperature

dependence

of the soft mode

damping

coefficient which

plays

an

important

role in ultrasonic anomalies. The

incompleteness (or inconsistency)

of the

theory

is

pointed

out. In section 3 _a

perturbation theory,

which takes into _account first order fluctuation

effects,

in _a consistent _way, is

presented.

A summary of the results and _a discussion of

possible experimental

verifications of the

theory

_are

presented

ⁱⁿ section 4.

2. Comments _on the

theory

of ultrasonic anomalies.

The ultrasonic

properties

of

isotropic

media _are described

by

a

complex frequency-dependent

elastic modulus

>(n)

=

>'(n)+

it

"(n).

^{The real} part

>'(n)

is related _to the sound

velocity

and the

imaginary

_part >

"(n

to the sound attenuation. In the _zero

frequency

limit,

>

(0)

is _a

thermodynamic

real

quantity

and its temperature

dependence

_can be

correctly

described in the frame of

thermodynamic perturbation theory.

In the ultrasonic

frequency

range, the sound

velocity

can thus be well described

by

such _a

theory,

but it is not the _case for ultrasonic attenuation which is _a

purely

kinetic effect.

For

higher frequencies (typically

those

probed

in Brillouin _or inelastic _neutron

scattering)

both the real part ^{and the}

imaginary part

^{of >}

(n

are not obtainable within the

thermodynamic theory.

Let _us first recall the condition of

applicability

of the

thermodynamic theory.

For _{a one} component

order-parameter

_~, the Landau

thermodynamic potential

is written _:

i~~~ _~~ ~~~~ ~~~~ _~~~~~j

~~

(l)

where A

=

A'(T- T~)

and B and D _are assumed to be temperature

independent.

Both the Laudau

(mean-field) theory

and the first order

perturbation theory

are valid

only

when the

following inequality

holds

[7]

_:

T T~

B~

_T~

(r(

= ~

(2)

Tc A'

D~

The coefficients in

(2)

can be estimated for two

widely

used models. For the

Ising

model with

nearest

neighbour interactions,

which is the

prototype

oforder-disorder

system,

one finds

[8]

_:

A'~l; B~T~d~; D~T~d~

where d is the lattice constant. The

inequality (2)

then reduces _{to r}

~ l. As the

expansion (I )

is not

usually

valid for

large

_{r, one} cannot define _a range of _r where the _mean field

(or perturbation theory)

^is

certainly

valid. It does not mean, of _course, that this

theory

is _never valid for order-disorder

phase

transitions _: the real

systems

are

usually

much _more

complicated

than the

Ising

model.

For _a

displacive phase

transition

(see

_e-g-

[9])

_one finds _:

A'~l; B~T~~d~; D~T~~d~

where _T~~ is _a

typical

_« atomic temperature » (T~~

~10~ -10~ _K).

_Thus

₍₂₎

_{reduces to}

:

T~

r ~

(3)

T~~

and in this _{case one can} define _a range of Tin which the Landau

theory

is

valid,

since

T~«T~~. Strictly speaking

_one has to confine oneself _to the consideration of

displacive

transitions when

using

the

perturbation theory.

The criterion defined

by (2) is,

however, _very

approximate

and _a conclusion about the

applicability

^{of the}

perturbation theory

_can

only

be

reached after _a

comparison

with _an

appropriate

set of

experimental

data.

Let _us consider _now the

perturbation theory

for kinetic coefficients. For _an order disorder

phase transition,

the order parameter

equation

of motion is of the relaxation type :

d~F~

yi

₊

j

(~ (~)

D A~ ₌ ^o

(4)

~~

eq

where

(~ )

is the

equilibrium

value of _~ and

F~

is the

homogeneous

_part of the free energy

density

in

(I).

The coefficient _y is

usually

considered _as

temperature independent. Strictly speaking equation (4)

is

only

valid _at low

frequencies.

The

general equation

contains

higher

order time derivatives _or, in other words, _yis

complex

and

frequency dependent. However,

we

are not aware of any reliable calculation of this

frequency dispersion

of _y for order-disorder

systems

(this

is unfortunate because, as we shall _see, the

high frequency dynamics

of _~ may be

important

for the low

frequency

sound

attenuation).

In the

following

_we shall consider _{y as} constant for order-disorder

systems.

For

displacive phase transitions,

the

equation

of motion of the order parameter ^{is of the}

oscillatory

type :

d~F~

m4 +

Y4

+ j

(~ (~))

-D A~ ₌ o.

(5)

d~

eq

For small

anharmonicity

the

damping

coefficient _{y can} be evaluated in the frame of

conventional lattice

dynamics.

Several authors

[9-16]

have contributed to the relevant treatment and

having

summarised their results let _us

emphasize

first of

all,

that the

frequency dependence

of the

damping

coefficient _y proves to be very

important

in the

displacive

_case. For the

frequencies

which _are close _to the soft mode

eigenfrequency

_w~

=

~/(d~F/d~

_~)~~/m the

damping

coefficient _can be

roughly

estimated _as r

=

~~~°~~m ~~( ) ^~

_where

w~~~

m

~~

1952 JOURNAL DE PHYSIQUE I N° 10

10~~ _s~ ^' and _n

m ÷ 2. This estimate is valid both for low and

high

temperature

phases.

Thus the width of the soft mode neutron

scattering

line is not

expected

to be different in the two

phases.

But _{as to} the

damping

coefficient in the low

frequency region

the difference between the two

phases

becomes

significant.

For

T~T~

there exists _a

specific

contribution to

i

y

(0)

which _can be estimated _as mw~~

~

~

/.

_{Thus it}

_acquires

an order of

magnitude

of T~,

T~

w~~ ^at the

boundary

of

applicability

of the

perturbation theory already,

and within the

~at

perturbation region

it is

expected

to be

dominating.

As it is the soft mode

anharmonicity

which is the

origin

of this contribution it _can be

expressed through

the soft mode line width

(r)

and the coefficients of the Landau

thermodynamic potential [6].

In section 3 _we shall recalculate this contribution to the order parameter

damping taking

into account term nonlinear in _~

(

_~

)

in

equation (5) (in

^{the nonlinear version} of

equation (5)

the coefficient _y does _not

incorporate

the contribution of the order parameter

fluctuations).

The

aim of the above remark is to

emphasize

that in the

displacive

_{case at}

T~T~

the

« renormalization _» due to

q-fluctuations

is _more than the

« non-renormalized _» parts ^{of the}

order parameter

damping

constant unlike to that of the order-disorder _case when the

renormalization due to

~-fluctuations

is small as

long

as condition

(2)

is fulfilled.

The mean-field

theory

of sound attenuation

anomaly

_was worked _out

by

Landau and

Khalatnikov

(LK) [2].

This

anomaly

arises from _a

coupling

between the strains

u~~ and the order parameter ~. For

isotropic solids,

the

thermodynamic potential density

F takes the form _:

where K and _{p are} the bulk modulus and the

rigidity

modulus

respectively. K,

and the

coupling

constant r are assumed _to be temperature

independent.

At _zero

applied

stress the

equilibrium

value of the strain is

given by

:

K(uii)

₊

r(~ _~)

= 0

(7)

when _an ultrasonic

longitudinal

_stress with Fourier component tr~,

~

is

applied,

there is _an additional

longitudinal

strain e~,

~

= ik uk, _w, such that _:

> e~~

~

+

r(~

_~)k,

w ^"

"k,

w ~~~~

where >

=

K ₊ ~

p is the

longitudinal

elastic _constant. In the _mean field

approximation

3

equation (8a)

is linearised

18k,w +~~~~)

_{~k,w ~k,w'} ~~~~

Therefore, the ultrasonic response is

coupled

to the order parameter

only

for T _~ T~ ^{in this}

approximation. Taking

^into account the kinetic

equations (4)

or

(5)

the

complex

elastic modulus

I _(n

can be

easily

calculated.

For n _« ^" _one has _:

m

i~~(n)

= >

4

r~j~l~

= >

~~~Lj

^~~~

28(~) -iny

-i t

where

A>~~

₌

~/

_{is the difference between the static elastic moduli}

corresponding

to

clamped

and free order parameter, t ₌ "

~

is the relaxation time. For nt « _: 2 B

(~)

ij~=A>~~ntcc jT-T~j-~ _(io)

Going beyond

^this mean-field

theory approximation requires

the consideration of the _non- linear part ^{of the}

coupling

in

(8a).

For

T~T~

the fluctuation

(~~~,~ plays

_a role similar to ~~,~ ⁱⁿ

equation (8b).

Its

contribution AA

~ to the static elastic constant is well-known

(see [7]) r~ _Tc

~m ~ ^-1'2

(11)

~~F

_~

2

~D3'2A~'~

^{~ ~}

At low

frequency,

the fluctuation contribution to

I _"(n

_{takes the form}

>J(n

m

~ AAF Aw where Aw is the width of the central maximum in the

spectral density

of thermal fluctuations

~(

n

~)

^{and it}

plays

a role similar _to the

reciprocal

of the relaxation time t in

(9). (Let

us

note that this is the width of the

spectral density

of second order

light scattering).

Aw

depends

of _{course on} the

dynamics

of the order parameter. ^For order-disorder

systems

one

has _:

(Aw)~' ~Zcc [T-T~[~~ ₍₁₂₎

which leads _{to a} temperature

dependence

of the ultrasonic attenuation coefficient

[3-5]

_:

ij(n)cc jT- _{T~j-~'~ (13)}

For _a

displacive

system, the

equation

of motion is

given _by equation (5).

The soft mode is

underdamped

in the domain of

validity

of the

perturbation theory

since,

according

to

(3)

y(W T

(

T T 1/2

~

m~ ^{°~~~ j~}

^{~ ~°~}

~m ^°~~j

T~~

~ ~~~~

The

spectral density

of second order

light scattering

exhibits in this _{case a} central

peak

with _a width Aw

m r which is

approximately

temperature

independent.

Therefore _:

ij(n)

cc

jT- T~j-"~ (15)

as first shown

by

Dvorak

[6].

For T

~

T~,

^{it is}

usually

assumed that the fluctuation contribution

i~(n

_is

_nearly

_the

same

as for T

~ T~ ^{and it} can be

simply

added to the mean-field contribution

i~~(n _{(Eq. (9)).}

_As

we shall _see below this is _{not correct} in the

general

case. Let _us compare ^the

imaginary

parts ^of these two contributions. For _an order-disorder system one finds _:

it

_AA

~ i

B~

_T~ IQ

) _>LK ^" /P @'

^~~~~~

1954 JOURNAL DE PHYSIQUE I N° 10

This ratio is small when

inequality (2)

holds. For the

displacive

_{case :}

it

_TB

[A

^~'~

_

(16b)

)

^~ ₂

_{«rY (°)}

_D~~~

(we

have used t

=

" ~~~

~

and

y(0)

m y

(w~) j~~ ^/,

^{within the}

perturbation region).

2 B

(~ ) ~c

Thus the LK contribution has the _same

dependence

and the _same order of

magnitude

_as the fluctuation _one. To _our

knowledge

this conclusion has _never been taken into _account in the

interpretation

of

experimental

data _on

displacive

systems. It has been overlooked

that,

^both in

the order-disorder and in the

displacive

_cases, the LK contribution

depends

_on material

coefficients which _are temperature

independent only

in the mean-field

approximation.

To be

consistent,

the fluctuation induced correction to these coefficients has to be taken into _account since

they

_occur at the _same order of

perturbation

as the _« pure » fluctuation _terms discussed above.

Zeyher actually

considered all these corrections in _a calculation of sound attenuation in

incommensurate

phases [18].

As he focussed his attention

only

on the

phason contribution,

his results will be discussed in _a

subsequent

_paper.

3. Perturbation

theory.

For the sake of

simplicity

_we consider _an

elastically isotropic

solid and _{we assume} that the Landau

thermodynamic potential density

has the form

given

in

equation (6).

A)

T

~ T~

The

dynamical equations

_can be written _:

X~/(k>

_t°

)

_~k,

w

+ ~

~j ~kj,

_wj

~k2,

_w2

~k-kj

_-k2,

w wj w2 + ~ _~

~j

_~kj,

wj

~k-kj,

w wt ^~

~

kj, _wj kj. _wj

k2. _»2

(17a)

X

F/ _(k,

t°

)

_Sk,

w

+ ~

i _~kj,

wj ~_k kt. _{w ml} ^"

~q,

fl

~k,

q

~ fl, w

(~~~)

ki WI

where the

x,~(k, ^w)

are the

uncoupled susceptibilities

defined

by

X~~ (k, w =

iA

₊

Dk~

mw ~ i _{w y}

(w )i~ (18a)

X~~(k,

_w

=

> f

(18b)

k

(The damping

coefficient of the acoustic mode has been

neglected

in

(18b).

In first order

perturbation theory,

^the second term of

(17a)

_can be

neglected. Equations (17a, 17b)

can then be solved

(see appendix)

and _one gets :

I _(n

= > 4

r2 _{n(n (19)}

with

J7(n)= ~jX~~(k,w)((~)n _~(~) =2T~jX~~(k,w)~~~~~'~ ^~°~ ₍₂₀₎

k,w k,w

~ _~

The behaviour of

J7(n ) depends

_on the

dynamics

of the order parameter.

For _an order disorder

system (m

₌

0,

_y

independent

of

w)

the

integration

_over k and _w can be

performed.

One finds

[4, 17]

_:

~~~

~

~~2 _~l/2'j~t l~ _~~~

~~~~

In the low

frequency limit,

_one thus obtains _:

I _(n

_> ^~~

^~c

^{in y}

2

arD~'2

A~'2 ^~

fi ⁽²²⁾

For the

displacive

limit ~

» r the main contribution _to

J7(n )

_comes from the

high

~m

frequency region

_w

w~(k)

=

/

_~

^~~~

As _a result _one obtains _:

m

~~

~

2

ar~~A"~

^~

2(i~~r~)

^~~~~

We _see that in this _case the

dispersion frequency (~ r)

does not

depend

on temperature ^{and for}

high frequencies (n

_~ r

)

the attenuation coefficient has the _same

temperature dependence

_as

for the low

frequencies.

It is worth

mentioning

that in the low

frequency

limit Im

I _(n

cc r~ ^'

and _as r _cc T _{one sees} that Im

I _(n

_does

not contain the small factor T~/T~,. This

specific

feature of acoustic

damping

in _a

weakly

anharrnonic

crystal

which _was first

pointed

out

by

Sham

[19]

is

quite important

to bear in mind _: it _means that the sound attenuation anomalies _are not small in the

displacive

limit while the

thermodynamic

anomalies _are. Let _us

emphasize

that in

spite

of the fact that Im

I _{(n )}

_does

not contain the small factor Tj/T~~, ^the corrections, due _to the

higher

orders of the

hydrodynamic perturbation theory

used

here,

do contain it and the condition of

applicability

of the

perturbation theory

^{for the sound} attenuation coefficient _proves to be

practically

^the same as that for

thermodynamic quantities.

B) T~T~

The static

equilibrium

^value

(~)

and

(uii)

have _{now to} be taken into _account. To be

consistent,

they

have _to be calculated up ^to the first order of

perturbation theory (see

_e.g.

[9]).

Keeping

in mind the

possibility

of

applying

our calculations to tricritical transitions _we add the term

C~~/6

_{to the}

thermodynamic potential (6).

As _a result _one has _:

A ₊

h(~)~

+ C

(~)~

₊

(h

₊ 2

h

_{+10 C}

(~)~)((A~ _)~)

=

0

(24a)

luff)

=

~

l~)~

=

~

(l~~)

₊

lA~~) _(24b)

with

h=B-~~~

and

B=B-~~

(24c)

and

((A~ )~) (which

is _a function of

(~))

is _to be calculated within the Landau

theory

but with the renormalized

phase

transition

temperature

^{which is identified with the}

experimental

1956 JOURNAL DE PHYSIQUE I N° lo

one :

~

h

_{+ 2 B}

(24d)

T~ =

Tc

~, ^a

where _a is defined

by

the formula

j (A~ )~j

₌ a +

f (T

_T~*

(25)

with

f(x)

=

0 at _x

= 0. For _a

phase transition,

which is far from the tricritical

point

_{one can put}

C

= 0 and obtain _:

f (T

_T~

)

= b T

Tf (26a)

~

A'[T-Tf[ 12jj

(

_~

)

= + l + ~ b T Tm

(26b)

h

_B

In what follows _we shall consider such transitions if _not

specially

indicated.

The

dynamical equations

can then be written

X~/(k>

_{t° ~k,}

w

+ ~

~~~)

_~k,

w

+

+

~j [3~~~) _{~kj,wj ~k-ki,w-wi} ~~~~kj,wt ~k-kiw-wjl

k,wt

+ ~

~j ~kj,

wj ~k~, _w~ ^~k kt k~, w wj w~ ^~

°

(27a)

kb _ml k~, _w~

XEE~(k,

_{W Sk,}

w

+ 2 _r

(~ )

_~k,

w

+ ~

~j ~kj,

_wj>

~k-ki,

w wi

~q,

fl

~k,

q

~fl,

w

(~~~)

kj, _wj

~~

~

is the Fourier component ^of ~

(~), X~~(k, w)

is the

low-temperature uncoupled

suiceptibility

_:

X~~

(k,

_w

=

iA

₊

Dk~

₊ 3 B

j

_~

~j

+ 2 _r

puny

mw ~ i

w y

(w )j~ (28)

As in the

previous

_case, the last _term in

(27a)

can be

neglected (except

the term 3 B

(A~ _~)

_~~,

~

which can be

incorporated

into the first term of

(27a)).

The

equations (27a), (27b)

involve _a linear

coupling

between ~~,

~

and e~,

~,

and

they

can be put ^{under the}

following

matrix form _:

~~~~)

#

f)~

f)~lX

~k,w Xe~ XEE

~j ^[3

^B

(~) _~kj,

wj

~k-kj,

w wi ~ ~

~~ki

wi

~k-ki,

w

l~

x

~"~'

(29)

~

~j ~kj,

_wj> ~k _kj,

w w1 ~q, fl

~k,

q

~fl,

w

kj, _wj

where the

susceptibilities f((k,

w

correspond

to the

linearly coupled

_{system :}

j0j X~/

2 _r

(~ )

l~

~

~2rj~j

XiE~

l 4

r~(/)~

Xee X~~ ^{2 r}

(~(~ee

X~~

~ _~

~~~~~

_~~~

~~~~

Both components

~~

_n ^and

s~

_n ^involve a

part

^driven

by

the extemal _stress

tr~,

_n ^{and another}

part

^which

corresponds

to

th~

_{indirect influence of}

tr~,

n via thermal fluctuations.

The latter contribution can be calculated with

equation (29) by keeping

in the _non linear

terms of the

fight

hand side those which _are linear in the driven components.

By incorporating

these results back into

(29),

one can then calculate the driven components

(see appendix).

At the end _one gets :

~~' "

=

f~~ f~

~ ~

(31)

~q, fl X~e X

ee

"q, fl

where the renormalised

susceptibility

is

given by

:

lf~']~~

=

([28(~)~+Dq~-mn~-iny(n)] -4(~)~

x

~~2

_~4

x

9 82 ni (q,

n 12

~ ^{n~ (q,}

ⁿ

⁾

^{+ 4 ~}

n~ (q,

n A

)j

lf~

_]~~

=

lf~

_]~~

=

2 _r

(

_~

)

l 6

BJ7i (q,

ⁿ

)

₊ 4

~

J7~

(q,

n

) (32)

A

lf~

_{]~~ =} >

@

4

r~ Hi (q,

n

)

q with

°i

(q> ^lJ

)

"

~( i(~ _(k,

_n

₎ ~(

k, n _w

(33a)

H~(q,

ⁿ

)

= >

~j I(~ _(k,

w

~(_

~, n _~

~) lx~~(k,

^{w +}

xee(q

k, ⁿ w

)j (33b)

H3 (q,

n

= >

~') i(~ _(k,

w

~(_

~, n _~

~) lx ~~(k,

^w

)

+ Keg

(q k,

n _w

)j2 (33c) According

to the fluctuation

dissipation theorem,

_one has _:

~ _{~ n}

~

~)

= 2 T Im

~~

~ ~~~ ~' ~ _~°

(34)

]~ ]~~

^{is the} «

clamped

»

reciprocal susceptibility.

Several authors

[13, 14]

have calculated this

quantity

for different _cases, but

equation (32) gives

the

fully

consistent solution.

lj/~

_{]~~ can} be identified with _a

piezoelectric

coefficient in the

particular

case of _a proper ferroelectric transition.

lj/~

_{]~~ has} the _same form _as in the

symmetrical phase but,

of _course,

J7i(q, n)

_now involves the order parameter fluctuations of the low

symmetry phase.

The renormalized elastic constant

I _(n

can be deduced from

equation (31), by taking

the limit q _- 0 in

lee(q,

n

).

It is convenient _to discuss the order-disorder and the

displacive

cases

separately.

Let _us

begin

with the former _one. As it has been mentioned in section 2 in this _case the fluctuation corrections to the order parameter

damping

constant _are small within the range of

applicability

of the

perturbation theory.

Thus _{one can}

expand

the

expression

for

I (q,

n

)

ⁱⁿ terms of

J7;(q,

ⁿ

).

One has _:

I(n)=

_>

~~l~l~

~

~~~

~

jj4B~]+my(n)j2n~(n)

2Bj~) -my(nj j2B~o-my(njj

~i~ ~i(4 ^B~ i

₊ in _Y

(n

₎₁

l/2(n

₊

~l ^l~ ~' _l/3(n )

(35)

where:

~/~A'(T-T/(/B.

JOURNAL DE PHYSIQUEi -T 2, N' IO, OCTOBER 1992 TO

1958 JOURNAL DE PHYSIQUE I N° 10

This

expression

^is very

complicated

in the

general

_case, and it _can be

easily analysed only

ⁱⁿ

some

limiting

_cases. Let _{us assume} that _p

=

0. Then

lee(k,

_w

)

= so that

>

n~(n) n~(n) nj(n)=

=

(see Eq. (33))

and

I(n)

_{takes then the}

_simpler

_form

:

I(n)=> ~~~(~~~ 4r~J7i(n)l~~~~~~~~ ~

28(~) -iny 2B~o-iny (36)

Hi (n

has been calculated in

[16].

In the low

frequency

limit

°i~~l

_»

~

(i +inyi

fl~° " 8D

(37)

with

r)

=

l~ ₍₃₈₎

2

h ~(

so that

I(n)

_>

-2r21~

⁺

^{~~~ ~} j~j

n~o arD

n

~~2 II

~

~~~

h

~

l

~

~ _~~ ~

B~ ~/

4

arD~

B

Bh (39)

We _see that the second _term in the

imaginary

_part of

I _{(n )}

may be

negative

_as well _as

positive.

This term,

however,

remains small in

comparison

to the LK correction _as

long

as condition

(2)

is fulfilled and the system ^{is far from} a tricritical

point.

Close _to the tricritical

point

it becomes

quite important

since

h

goes ^to zero. One _can note,

however,

that the contribution to

>

"(n arising

from the last term in

equation (36)

is

always finite,

_even when

h

goes ^to zero.

Therefore the most

important

contribution _comes also in this _case, from the LK _term but this includes

important

fluctuation corrections

through

the _{non mean} field behaviour of the order parameter.

(This

result has been obtained with the

assumption

_p

=

0. It is easy to see however that in the

opposite

case, p = co, one has

only

_to

change h

_{into B in}

equations (36)-(39),

_so that the above conclusions _are still valid _{one can}

conjecture

that the above result is also true in the

general case.)

For the

displacive

case, one

has,

from

equation (31)

:

I _(n

= >

~

_~~~~~

4

r2 n~ (n (40)

li l~

~

]~~]~~

and

lf~~]~~

_can then be

approximately

calculated in the two extreme cases p =

0 and _p

= co. Let _us first consider the _{case p}

= co and

neglect

in

lf~ ]~~

^{both the} term

-mn~ _(because

acoustic

frequencies

_are much smaller than

optical ones)

and the term

in _y

(n (see

Sect.

2).

One has _:

~

l 6

BJ7i (n

_)~

>(n)=>

-2r -4r ~

ni(n). (41)

B(1-18BJ7j(n))

For small

frequencies

and within the

perturbation region

_{one can}

expand equation (41)

as

function of

Hi (n ).

To _a first

approximation

_one

gets

I _(n

=

> ~ ~~

16

r2 nj (n ). (42)

In the _{case p}

= 0 one obtains instead of

(42)

:

I(n)=

_>

~~~-16r2(~ )~nj(n). ⁽⁴³⁾

B B

This result has the _same form as the _y

- 0 limit of

equation (36), (but Hi (n)

has to be calculated for the

displacive case).

Let _us

emphasize

once more that this result arises from the fact that the main contribution to the order parameter

damping

constant at small

frequencies

comes from the soft mode

anharmonicity (see

Sect.

2).

An

explicit

calculation of

Hj(n)

_can

only

be obtained in the _{two cases p} =0 _or p = co. In the latter _{case, one can} put

f(~

= X~~ ^{and use}

equation (23), substituting

A

by

28~/.

_Then, for

phase

transitions far from

tricriticality,

the _« critical indices

» for sound attenuation _are the _same for both

phases

and the

« critical

amplitude

_» ratio is

CT

>T~

/ ~B

^~~~~

It is remarkable that this ratio decreases when the tricritical

point

is

approached.

Close to the tricritical

point,

the mean-field order

parameter ~/

=

(A/C )"~

has to substituted into

equation (23).

The _« critical indices _» are then 1/2 for T _~ T~ ^and IN for T

~ T~, I.e. sound attenuation is

quite

different for the two

phases

_near a tricritical

point.

The _same conclusions also hold for _p

=

0,

the

only

difference appears in the _« critical

amplitude

ratio _» which is in this _{case :}

CT>T~ / ^B~

^~~~~

One _can

reasonably

_expect that these results can be extended _to the

general

_case,

Let _us also mention that, as for T

~

T~,

^the

high frequency

and the low

frequency

sound attenuation show the _same temperature

dependence.

4, Conclusion.

The main conclusion of this paper is that acoustic anomalies observed _near structural

phase

transitions _are less

simple

to

analyse

than

usually

assumed. The low

frequency

sound

velocity

is a

thermodynamic quantity

which exhibits _an

anomaly

similar _to that of the

specific

heat. The difficulties _met in

analysing

these

quantities

have been discussed in detail in

[20]. Conceming

the ultrasonic

attenuation,

the temperature

dependence

of the

anomaly

is

expected

to

depend

upon the material constants which characterize the

dynamics

of the system, ^{For T~}

T~,

^it

depends

on the

dynamical

behaviour of the order parameter: ^{for low}

frequencies

>"~

1960 JOURNAL DE

PHYSIQUE

I N° 10

IT

_T~ ^"~ in the

displacive

limit and > ^"

IT

T~ ^{~'~ in the} order-disorder limit. For

high

frequencies,

I.e, for

frequencies

which _{are more} than the characteristic

dispersion frequency,

the temperature

dependence

of the attenuation coefficient in the

displacive

limit is the same as at low

frequencies

and is

practically

absent for what _we called the order-disorder _case

(purely

relaxational

dynamics

of the order

parameter).

Then the

frequency dispersion

is of

simple Debye-type

^for

displacive

transitions and the characteristic

frequency

of

dispersion

does not

depend

on temperature ^{while for} order-disorder _case the

dispersion

is

non-Debye

one, and the characteristic

frequency

is

proportional

to T T~.

For T

~ T~

things

are even more

complicated

since the ultrasonic attenuation also

depends

on

the

dynamics

of the elastic

subsystem

and _on the

coupling

between the

order-parameter

and the strains. As _a

result,

_no

general

closed form

expression

_can be found for the attenuation

coefficient,

_even in the first order

perturbation theory. Relatively simple expressions

_can

only

be obtained at the

price

of additional

assumptions

_p

= 0 and _p

= co and

again

for the _two

extreme cases of the order

parameter dynamics

_{: pure} relaxation and

underdamped

oscillations.

Qualitatively,

the difference between results for the two types ^of

dynamics

_proves to be similar

to that for T

~ T~ ^{with the} reservation that there is _an additional

Debye-type

contribution.

One has _to pay attention to the fact that the conventional

representation

of the attenuation

anomaly

at T

~ T~ as a sum of the mean-field LK _term and _a

positive

fluctuation contribution is, ⁱⁿ most cases, ^not

justified. Indeed,

for _a

displacive

system ^{the LK} temperature

dependence

of the attenuation coefficient

(as

T T~ ~) is _not

expected

at any temperature.

Moreover,

for

phase

transitions which _are not far from the tricritical

point

the attenuation

anomaly

_proves to be much _more extended in the

high-temperature phase

than in the

low-temperature

_one, I.e. it has the assymmetry ^{which is}

just opposite

_to the _« naive _»

expectation.

For the order-disorder

system one can, ^to a first

approximation, si§nply ^neglect

^the fluctuation contribution in the

low-temperature phase

as

long

as

h ~16~

(B fi) _{(see Eq. (39))}

_{and derive the sound}

B

attenuation

anomaly

at T~ T~ ^from

temperature dependence

of the order parameter. ^This

anomaly, however,

_may be _{not a} mean-field _one because the order parameter may exhibit _non- classical behaviour.

It would be

quite interesting

to compare the above

predictions

with the

experimental

data.

But this

comparison

^is

already beyond

the framework of the present paper as well _as discussion of the _case when _p is finite and _non-zero and account for the elastic

anisotropy

which may be

quite important

for _a

quantitative interpretation

of

experimental

data.

To

end,

let _us not that in this paper we started with _{a set} of

hydrodynamic equations

in which the temperature was not considered _{as a}

hydrodynamic variable,

I-e- _we made _no distinction

between adiabatic and isothermal elastic moduli in the low

frequency

limit. This

difference,

which is

proportional

to the

specific heat,

is

expected

to be

particularly important

close _{to a}

tricritical

point.

The introduction of the heat transfer

phenomena

into _our

perturbation theory

would be

straightforward.

But it is

expected

to introduce

only

additional

quantitative changes

without any

qualitative

_new features.

Acknowledgments.

The authors _are

grateful

to Professor

Joseph Lajzerowicz

for fruitful discussions.

Appendix.

The non-linear

equations

which

couple

the

order-parameter

_{components ~~,~} and the

longitudinal

strain components _{e~_}

~

can be solved in the

following

_{way :}

A)

T

~ T~

Equation (17a) (without

the term

proportional

to

B)

leads to :

~~,~ =

~),~-2rX~~(k, _{w) ~j}

_e~,~

~)_~,~_~ (Al)

ki.wj

where

~),~

refers to the thermal fluctuation of the order parameter ^{without the non-linear}

coupling

to the strains.

When

inserting (Al

into

(17b)

and

keeping only

terms

quadratic

in the thermal

fluctuations,

one gets

8q, fl ^" X

ee(~>

^~

) ("q,

^{fl ~}

^{~j [~~,}

w

~~

_{k, fl}

w k.w

-4r~)~X~~(q-k, n-w) ~j

e~

~

~(_~_~~

_n-w-wj

.

(A2)

kiwj

~'

Taking

statistical average over thermal

fluctuations,

_one obtains

e~ n =

x~~(q,

n

tr~

n + 4

r~ ~j X~~(q

k, ⁿ _w

)( ~)

~

(~)

e~ n =

lee(q,

n tr~ n

k,w

(A3)

with

iie~ _(q,

n

= x

ie'(q,

n 4

r2 n(q,

n

) (A4)

and

n(q,

n

=

z

_x

~~

(q

k, n _w

)

_~

j,

~

2j ^(A5)

~, »

Taking

the limit q - 0

~n2 ⁿ²

ii/(q,~l)

m

A(~l)-j=A(~l)-4r~1I(~l)-Pj. (A6)

q~o q q

B) T~T~

Equation (29)

is used _to calculate the thermal fluctuation parts ^of ~~,

~

and e~,

~.

Keeping

only

terms linear in the driven components,

~~

~

and e~

~

one has _:

l~~'

^{~ "}

~l'

~

(k~(k,

W

))

_X

~~~~ ~~~

6B(~)~~-q,w-fl

_~q,fl

+~~[~~-q,w-fl

_{~q,fl +}

~)-q,w-fl q,fllj

X

~~~)

2r~k-q,w-fl

~

_~q,fl

where

~i,

w

and

ei,

w

are the thermal fluctuations in the absence of non-linear

coupling.

These fluctuations components are put back into

equation (29)

to calculate the driven components :

~~~~~ _(i~(q, l))l~ ^~~~~~~

^~~'~

^~~

^~'~~~

^{~~~~k,w 8q_k ~_~]}

q>fl

~

~

_~~>

~

~q-k,

fl

w

+ ~T~ ~

~~~~

~'~

1962 JOURNAL DE

PHYSIQUE

I N° 10

Taking

the statistical average over thermal

fluctuations,

the second _term of the

right-hand

side of

(A8)

is written.

[368~(~)~Pj(q, _n)

₊

_{24Br(~) P~(q,} n)+4r~P~(q, _{n)] ~~}

n

[12

Br

(

_~

) Pi (q,

n ₊ 4

r~

_P

~

(q,

ⁿ

)]

_e~,

n

Ii

2 Br

j

_~

) Pi (q,

n ₊ 4

r2

_P

~

(q,

n _~

~, ⁿ

4

r2 Pi (q,

n e~, n ~r~, n

(A9)

with _:

~l(~, ~)~ ~j i~~(k, t°)((~~-q,w-fl(~)

k,w

~2(~> ~)

_"

~j li~~(k> t°)(8~-q,w-fl ~~-k,fl-w)

₊

i~~(k> t°)((~~-q,fl-w (~)l

k,w

~3(~> ~)

~

~j li~~(k> t°)((8~-q,w-fl(~) +~i~~(k> t°)(8~-q,fl-w ~~-k,w-fl)

₊

k,w

+I(e(k,w)([~)-q,w-n(~)l. _(Aio)

Using

the

fluctuation-dissipation

theorem to evaluate the thermal averages, and

using equation (30),

one gets :

Pi

(q,

n

)

=

Hi (q,

n

(A

I

la)

P~(q,

n

=

~ _~

j~l _n~(q,

n

) (Al16)

P3

(q,

n

m

~ ~~

~/

^~

_{H3 (q,}

n

)

₊ ^~~

~

(Al

I

c)

A A

with the definitions of the

J7,(q,

n

given

in

equation (33).

(A

term

~j

Keg

(q

k, _w n

) ~)

~

~)

^{has been}

neglected

in

(Al lc).

This _term is

~ _~

>

exactly

zero when _p

=

0 and it

gives

a

negligible

contribution when

Eq. (2) holds.)

When

equations (A8), (A9)

and

(Al I)

_are combined _one finds

equation (32).

References

II REHWALD W., Adv. Phys. 22 (1973) 721 (Ferroelectrics 12 (1976) 105).

[2] LANDAU L. D. and LIFSHITz E. M.,

Physical

Kinetics (Nauka, Moscow, 1979)

English

translation

Pergarnon

Press, Oxford.

[3] FIXMAN M., J. Chem.

Phys.

36 (1962) 310.

[4] LEVANYUK A. P., ZhETF 49 (1965) 1304 (Sov.

Phys.

JETP 22 (1966) 901).

[5] KAWASAKI K. and TANAKA M., Proc.

Phys.

Sac. London 90 (1967) 791.

[6] DVORAK V., Czech. J.

Phys.

B

21(1971)

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Physics

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English

translation Pergamon Press, Oxford.

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Theory

of

Magnetism

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Solid State 12

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Light scattering

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phase

transitions,

H. Z. Cummins and A. P.

Levanyuk

Eds. (North-Holland, Amsterdam-New York-Oxford,

1983)

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[18] ZEYHER R., Ferroelectrics (1986) 66 217.

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Phys.

Rev. 156 (1967) 494.

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Phys..

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