Fluctuation effects in phase segregation

PhysicaB 165&166 (1990) 815-816 North-Holland

FLUCTUATION

EFFECTS

IN PHASE

SEGREGATION

P. Kumar Department

of Physics,

University

of Florida-Gainesville,

F1.32611,

U.S.A

M. Bernier Service de Physique CEA-CEN

du Solide et de Resonance

Saclay, 91191 Gif sur Yvette,

The excess specific

heat observed

*He is attributed 3-dimensional,

Cedex,

France

above the phase separation

to the fluctuations

rich solid the fluctuations

Magnetique

that represent

temperature

the symmetry

of the 4He rich phase are 2dimensional

the characteristic

length

scale is approximately

I. INTRODUCTION

in solid mixtures

of the condensing

the interparticle

change

the non Ising

between

are significant

then derive the fluctuation

heat of mixing specific ment

heat with

measured

by EMD adds to the usual

of the solid mixture the regular

some excess

specific

solution

heat

perature

To(z),

increases

as the temperature

cess specific

where

heat

z is the molar

which

range

range

tem-

of 3He, that

To(z).

This exof

on the dimensionality

of

the *He has been known one might

order.

what

we find when lines:

analyzing

the excess

fluctuations

expect

planar

specific

order. the data heat

domains

the condensation

of spherical

range

to con-

for the 4He rich com-

Furthermore

consist

for the short

these

Gaussian

fraction

as the contribution

while the lattice

else should

fluctuations

lar concentration

is however

domains

H,

can also be simply requirements.

expected

OF THE

term.

the instability

separation,

for phase

are fully characterized HflTo(z)

[l] along

of the inhomogeneous

with

the

energy

ensures

d = 3 for z in the

proper

lattice

fluctuations in units

and Landesman

a fictitious [2]: a lattice

the 4He is characterized the occupancy +1/Z

of these regular

using

gas where

a more elaborate

a spin

models, solution

0921-4526/90/$03.50

in the mean

-l/2

energy

@ 1990 - Elsevier

gas

The Hamiltonian

field theory,

used by Edwards

reduces

to the

and Balibar

Science

the symA is indeed

fluctuations

need

the coefficient the Gaussian

must be

fluctuations

separation

only

symmetry

temperature

term

lattice

denotes

increase

(I)

defined by vectors

of the

the contribution

The above form of the

the fluctuations

is characterized

of the lattice

I(V + &)A?

away

the energy.

by a length

from

the

Each of these

scale & measured

constant.

contribution

can then be written

to the thermodynamic

free

as 151

of while

version of the lattice

Thus

fluctuations.

that

Ff = -ksT

en

n/

Where

As is the Fourier

a specific

heat 2

[4].

Publishers

d A, exp {--a

[kJ)

flf

q

by the spin component

1 is given in Ref.[3].

theory

by Bernier

the occupancy

by the spin component

of 3He is described

on a lattice;

model

spin model proposed

following

- I] A* t c&r e

The last

The fluctuation Let us consider

the

Hamiltonian

Since it must also represent

EMD [l] and the kis are the reciprocal state.

the

by

= [(T/To(z))

of EMD

FLUCTUATIONS

down

and the Gaussian

to T - To(z).

We can

and expanding

the parameter

only the first, quadratic,

condensing

can be fitted

written

Note that

to be small

exthey

where zs is the given mo-

of 3He in the mixture,

This is essentially

with the dimension

II. DESCRIPTION

as A = z-z0

for small A (51 . The fluctuation

Hamiltonian metry

atomic because

Hf by defining

Hamiltonian

where To(z) is the phase

i.e. 3-d

of 0.5 and d = 2 for z close to one.

neighborhood

variable

proportional

is hcp and thus,

anywhere

agree-

separation

T approaches

depends

In a 3He rich mixture,

for the short

fluctuation

order.

dense at the surface ponent

There

the phase

can be understood

the fluctuations, the short

theory.

above

describing

and host atoms

only at much lower temperatures.

Debye

and is in excellent

terms

the impurity

sures, on cooling, has been studied by Edwards, MC Williams specific

they are

spacing.

and Daunt (EMD) [l]. The temperature

dependent

In a 3He

while in an even mixture

We can ignore

The segregation of a 3He -4 He mixture at high pres-

of 3He and

state.

R

B.V. (North-Holland)

transform

of A(r).

I

(2)

Eq.(2) leads to

[5] _ 1 Ad

X(1 -2)

-2pgEgd

T

1

d”-2

-[To(x) 1

(3)

816

P. Kumar, M. Bernier

Here Ad is the surface sions,

area of a unit sphere

.$ is a characteristic

herence

length

ity of the dominant

as a function

fluctuations.

this result is better

approximation

In critical

near one, the condensing tuations

has a planar

symmetry plane,

dimension

the log-log

distance

temperature

is closer

The length

tures,

based

support

Balibar

z, there

z range.

Panczyk

phase

diagram

analyzed

length

to be published

in .I. Low Temp.

Phys.

diagram

proposed

pressure

I In

: The excess specific heat as a function

of 7’-To(r)

for

the data of EMD (Ref.[l])

at a

z = l/2.

The crosses denote

pressure

of 3.58 MPa and the circles represent

of 2.7 MPa.

The dashed

line represents

the pressure

an exponent

a =

0.5.

by

solid mix-

specific

heat

in

the dimension-

be 3 for z < 0.7 and 2 for analyzed

[6] have reported however

here are

in the small results

the data

way for the excess

discussion

Fig.1

scale for the density

of the excess

region,

in the same

d = 2.

of the interparticle

is a need for measurements et al.

in Fig.2.

distance.

the phase

should

in that

more detailed

and M. Bernier,

(1990)

close

from the experimental

2. In as much as most of the data

for large

Adams,

heat

is clearly

by EMD [I]. We find that

ality of the fluctuations larger

E.D.

and the Debye

are shown

[4] for the high

on an analysis

the experiments

specific

for z = 0.997 and 0.999.

the interparticle

results and

and

z

to be 1.2 while for smaller

all cases it is the characteristic

These

Gonano

to be

to one indicating

scale in units

it is 1.3 and 0.67 respectively

Edwards

J.R.

(1968)

d = 3. The data from Ref.[l] for


namely

594

2.7 and 3.6 MPa as

The exponent

for z = 0.5 is found

variation

Scribner,

Auc-

scales are supposed

pressures,

The transition

time the exponent as well.

79

21,

is then d = 2. For z

2 = 0.997 and 0.999 at P = 3.6 MPa

data

P. Kumar

R.A.

Lett.

by a length

plot of the excess

heat are from Ref.[l].

The coefficient

diagram

and the dominant

in Eq.(3)

to 0.5 giving the dimension This

[7]

Benjamin,

takes place in the more isotropic

for z = 0.5 at two different given in Ref.[l].

[5]

In the

Rev.

Phenomena”,

d = 3.

the effective

specific

Panczyk,

Phys.

of Critical

goes into an hcp lat-

characterized

bee phase for which all the length Fig.1 shows

of the phase

fluctuation

near 0.5 the condensation similar;

phenomena o = 2-du.

M.F.

Theory

excess specific heat becomes

are in the basal

(11. The effective

[6]

“Modern (1974)

used here v = 0.5 and therefore

of d. In the region

a clear measure

specific

of d, the dimensional-

known as the exponent

the power law of the diverging

tice which

the fluctuation

S. K. Ma, New York

of & and Y is the co-

In Eq.(3)

heat has been expressed

Gaussian

average

exponent.

[5]

in d dimen-

for the

cannot

be

heat.

A

specific

can be found in Ref.[7].

REFERENCES

[ll

D.O.

Edwards,

L&t.

9,

I4 M.

AS.

McWilliams

and

J.G.

Daunt,

Phys.

Rev.

10-X

195 (1962)

Bernier

and A. Landesman,

Jour.

de Physique

32,

(1971)

Fig.2

[31J. Lajzerowicz and references

[41D.O.

Edwards

and

J. Sivardiere,

Phys.

Rev.

All,

2079

therein. and S. Balibar,

(1975)

Rev.

39,

4083

10-l

AT(K)

: The excess specific heat at small z. The pressure

3.58 MPa.

The circles represent

z = 0.997 and the crosses Phys.

10.’

C5a-213

(1989)

line in this figure represents

the excess specific

are for I = 0.999. an exponent

The dashed

o = 1.

is

heat for