Double spin one-half lattice gas model

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Double spin one-half lattice gas model

J. Sivarditre

To cite this version:

J. Sivarditre. Double spin one-half lattice gas model. Journal de Physique, 1976, 37 (11), pp.1267-

1277. �10.1051/jphys:0197600370110126700�. �jpa-00208523�

DOUBLE SPIN ONE-HALF LATTICE GAS MODEL

J.

SIVARDITRE

Departement

de Recherche

Fondamentale,

Centre d’Etude Nucléaires de Grenoble 85

X,

38041 Grenoble

Cedex,

^France

(Reçu

^{le 5 avril} 1976,

accepté

^le

25 juin 1976)

Résumé. ²⁰¹⁴ Nous présentons ^un modèle cellulaire à double spin un-demi, qui reproduit ^{le compor-}

tement

thermodynamique

^d’un alliage binaire recuit dont les deux composantes ^sont magnétiques.

L’hamiltonien

ferromagnétique

^du type Ising ^est traité dans

l’approximation

^du champ moléculaire.

Divers diagrammes ^de phases ^sont obtenus, présentant ^des points critiques ^de démixion, ^des lignes critiques magnétiques, ^des points triples ^et tricritiques. ^{La mise en ordre}

magnétique

peut ^favoriser

ou retarder la séparation ^de phase induite par les forces interatomiques, ^et peut même provoquer

une

séparation

^de phases ^{dans une} solution idéale. Les résultats de

Blume-Emery-Griffiths

^sont

retrouvés dans le cas où une des composantes ^de l’alliage n’est pas

magnétique.

Le modèle décrit

également ^les mélanges ^de liquides nématiques.

Abstract. ²⁰¹⁴ A double spin one-half lattice gas model is introduced in order ^to simulate the ther-

modynamical behaviour of annealed binary alloys, ^{the two} constituents of which are magnetic.

The ferromagnetic Ising-like Hamiltonian is solved in the molecular field approximation. ^Various types ^of phase diagrams ^are found, exhibiting ^critical points ^for

phase

separation, magnetic ^cri-

tical lines, triple and tricritical points. Magnetic ordering ^{may hinder or enhance} phase separation

induced by interatomic interactions, ^{or even} induce it in an ideal mixture. The Blume-Emery-Griffiths

results are recovered in a particular ^case. The model may apply ^{to the} description ^of liquid

crystal

mixtures and the interplay ^between phase separation and nematic ordering.

Classification Physics ^Abstracts

8.514

1. Introduction. ^- The

thermodynamical

^beha-

viour of _many

cooperative physical

systems ^{can be} simulated

by

^a

spin - 1/2 Ising

model. The

following phenomena

^{can be}

described,in

^{this manner :}

magnetic

or nematic

ordering, condensation, freezing,

^and

phase separation [1-3]. They

^are characterized

by

^a

single

^order parameter. ^Other

physical

systems ^are characterized

by

^two

kinematically coupled

^order parameters ^{and the}

interplay

^{of two}

cooperative

processes, for instance a He3-He4 mixture

[4],

^{or a}

binary

^{mixture at variable} pressure

[5]. They

^{can be}

simulated

by

^a

spin -

¹

Ising

^{model. More}

compli-

cated systems, characterized

by

three order parameters,

can be simulated

by

^a

spin - 3/2 Ising

^model

[6].

In this _paper, we

investigate

^the

applications

^{of a}

double

spin - 1/2 Ising

model. This model describes

some systems characterized

by

^two

kinematically independant

^order parameters. ^The paper is

organized

as follows. In section

2,

^{we show how our lattice} gas model

applies

^to

magnetic

^{or nematic}

binary

^mixtures.

The Hamiltonian is constructed in section

3,

^{and solved}

in the molecular field

approximation

^{in section 4.}

Phase

diagrams

^are discussed in sections 5

(ideal mixtures)

^{and 6}

(non-ideal mixtures).

2. Lattice model for

binary

^{mixtures. - We consi-}

der here

liquid

^{or solid}

binary mixtures,

^{both atoms A}

or B

being magnetic.

^Two

spins 1/2

^{are introduced}

at each site of the lattice ^- or in each cell of the lattice gas

description

of the fluid. The first _{one, ai =} ±

1/2, gives

^the

type

of atom, A ^or

B,

^{which is found at the} site or in the cell i. All sites or cells are

occupied,

^so

that the _pressure is

kept

^constant. The second

spin,

Ti ⁼ ±

1/2, gives

^the

orientation, positive

^or nega-

tive,

^{of the}

magnetic

moment : atoms A and B are

supposed

^{to have the same}

magnetic

^{moment. The two}

order parameters ^{are : M}

=== ( ai >

^and

Q = Ti >.

M will be called the concentration and is in fact the deviation from

equiconcentration, Q

^{is the magne-}

tization ;

the concentration xA in A is

given by :

xA ⁼

1/2

^{+ M.}

A

binary

fluid mixture of

elongated

^molecules may be described in the same way. The first

spin a gives

the type ^{of molecule A or B found in a}

given

cell of the

lattice-gas

^model

representing

^the

fluid,

^{the second one}

z the orientation of the molecule : molecules A and B

are

supposed

^{to take}

only

^two orientations.

Q

^{is here}

the nematic order

parameter

^or orientation.

Various

isotropic

^and

anisotropic

interactions will be introduced between the atoms or molecules A and

B,

^{and the}

possible phase diagrams

^{will be}

given.

We are

mainly

interested in the

interplay

^between

the two

cooperative phenomena : phase separation

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197600370110126700

TABLE I

Interactions in a

binary

^mixture

oj’magnetic

^atoms

A and B

and

magnetic (or nematic) ordering.

^{In the}

following,

we concentrate on

magnetic systems.

3. Double

spin

^{one half} hamiltonian. ^- The inter- actions between

magnetic

^{A and B atoms are defined}

in table I. The

following projection

operators ^are introduced :

The Hamiltonian of the system ^{is then :}

Introducing Zi

⁼ ai Ti and the chemical

potentials

,uA and _MB, we obtain :

A

change

^{of notation is now} introduced :

JAA, JBB

^and

JAB

^are interatomic

(isotropic)

^interac-

tions, independant

^{of the}

spin orientation; jAA, jBB

and

JAB

^are

purely magnetic

interactions. All interac- tions are taken to be

positive,

^{so that no} sub-lattice

structure has to be introduced.

We define now the interactions

K, J, L, k, j

^{and 1:}

and :

K is the mean interatomic interaction in an

equiatomic

mixture. J measures the

tendancy

^to

phase separation

in a

non-magnetic

^{mixture. L measures the}

difference

between the _pure A and B constituents

(for

^instance

the difference between their cohesive forces if A and B

are solids or their critical

temperatures

^{if A and B are}

liquids). Similarly k

^{is the mean}

magnetic

interaction in an

equiatomic mixture, j

^{measures the}

tendancy

to

phase separation

^{due to}

magnetic

interactions and I measures the difference between the

magnetic

critical

temperatures

^{of the two} constituents. We note that :

and :

Using

the notations

(5), (6)

^{and :}

the Hamiltonian JC becomes :

In the absence of

magnetic interactions,

^{Je reduces to a}

spin

^one-half

Ising

Hamiltonian :

If A and B are identical

atoms, JC

^reduces

similarly

^{to :}

4. Molecular field

equations

^{of state. - In order to}

obtain

phase diagrams,

^{we treat} the Hamiltonian

(10)

in the molecular field

approximation.

^{At the first}

stage, ^{we introduce two} molecular fields

H6

^and

Ht

and two order parameters ^M

= ai >

^and

Q = ti ),

The molecular field Hamiltonian is then :

with :

H6

^and

Ht

^are determined

using

the variation

principle

for the free energy

[7] :

From

(14)

^and

(15),

^we obtain the two

equations

^of

state. Similar

equations

have been found

by Pople

and Karasz

[8]

^{in their}

study

^of

liquid

solid transitions of molecular

crystals.

These

equations

however do not

provide

^{a correct}

description

^{of the} system.

Indeed, equation (15b) ,

can be written :

where :

is the mean

magnetic

interaction in a

homogeneous phase

of concentration M. But the

approximation :

(Ji

Ti > = (Ji > ii ) neglects

the fact that the

magnetizations QA

^and

QB

^{of the two}

types

^{of atoms}

are not

necessarily

^{the same. This is true}

only

^if

jAA

⁼

jBB

⁼

JAB or j

^{= I =}

0,

^{and if} so,

phase

sepa- ration and

magnetic ordering

^{do not interfere.}

The

system

is in fact characterized

by

three inde-

pendant

^order parameters

M, Q

^{and R.}

According

to

(1),

the contribution of the A and B atoms to the

magnetization

^are

given by :

and

respectively. Q

⁼

QA

⁺

QB

is then the

magnetization,

and R describes the difference between the

magnetiza-

tions of the A and B atoms. The system

might

^{also be}

described

by

^a

spin - 3/2 Ising

model since each atom may be found in one of four different states :

M, Q

and R would be

replaced dipolar, quadrupolar

and

octupolar

^order

parameters (see Appendix 1).

The correct molecular field Hamiltonian is then :

The

corresponding partition

^function

Zo

^is

given by :

and the free _energy

by :

The conditions :

give

^the

following

values of the molecular fields :

whence the three

equations

^{of state :}

and the free _{energy :}

with :

Alternative

expressions

^{of the}

equations

^{of state are}

given

ⁱⁿ

appendix

^2.

Equations (23)

^{have two} types of solutions :

1) Q

^{= R =}

0 and M given by

and : ^O O

This solution describes a

paramagnetic

^mixture.

2) Q, R

^{and M} different from zero

(magnetic mixture)

^{and in}

general R # MQ.

The solution

-

Q, - R,

^{M describes a}

magnetic

domain with

magnetization

^{in the}

opposite

^direction.

Suppose

^{that H-} ± ^{oo. Then M =} ±

1/2,

R ⁼ +

Q/2 and,

^as

expected

since the mixture contains no B

(or A)

^{atoms :}

If

only

^{one solution}

(M, Q, R)

^of

equations (23a) (23b)

and

(23c)

^{is found at a}

given temperature

^{T for}

given

values

of J, k, l, j

^and

H,

^it

represents

^a

homogeneous

mixture. If n solutions

(Mi, Q1, R1), (M2, Q2, R2), ...

are found with the same free _energy,

they

represent n

homogeneous phases

^at

equilibrium.

Consequently,

^{in order to} determine the

phase diagram

^{of the}

^mixture,

^we choose fixed values of the

, interactions

J, k, l, j

and of the field

H,

^{and look for a}

first-order transition as T is varied. The two solutions at

equilibrium

determine either two

magnetic phases

with different

compositions,

^{or a}

magnetic phase

^and

a

paramagnetic phase.

CRITICAL LINE. ^- Phase

separation gives

^{rise to a}

critical

point given by : kTc

^{= 2 J. The onset of}

magnetism gives

^{rise to a} critical line. Its

equation

^can

easily

be found from

equations (23b)

^and

(23c)

^with

Q

^{and R ~ 0.}

Writing :

we obtain ’ :

whence a relation between

Q

^{and R near} the critical line :

and the

equation

of the critical line in the

(M, T) plane :

from which we find as

expected : 1 /Bi

⁼

2 jAA

^for

TRICRITICAL POINT. ^- In order to locate a

possible

tricritical

point,

^we

perform

^{a Landau}

expansion

^{of the}

free energy :

we obtain for J ⁼ 0 :

with :

Me being

the value of M at the critical

point.

^{The cri-}

tical line is

given by

^the condition :

b2

⁼ ac, ^{or :}

which is

equivalent

^to

(30),

and the relation between

Q

^{and R near the} critical line

by :

which is

equivalent

^to

(29)

^when

(30)

^{is satisfied.}

Q4 is

given by :

x

and y are

^defined

by (27).

The tricritical

point,

^if

it

exists,

is defined

by (30)

and the condition : _(P4 ⁼ 0.

When

(30)

^is

satisfied,

^x

and y

may be

replaced by :

with :

More

complicated

^{results are} found for J different from zero.

5. Phase

diagrams

^{of ideal mixtures}

(J = 0).

^-

5.1 PARTICULAR

CASE : j

^{= 1 = 0}

UAA = jBB

⁼

JAB).

- The mixture is ideal

(no phase separation

^{can be}

induced

by

interatomic

interactions),

^{and the A and B}

atoms have the same

magnetic properties.

^The magne- tic critical

temperature :

^{kT = 2 k is}

independant

^of

M,

and lines of constant H are

parallel

^{to the T axis}

in the

(M, T) plane.

^When

CA

^and

CB

^are the critical

points

^of pure A and

B,

the critical line

CA CB

^{is a}

straight

^line..

5.2 PARTICULAR CASE : k = l ⁼

j.

^{- In this} case, B atoms are

non-magnetic : jAB

⁼

jBB

^{= 0 or :}

JBB

⁼

JBB

^and

JAB

⁼

JAB.

^{This is}

equivalent

^{to the}

Blume-Emery-Griffiths [4]

^or

Hintermann-Rys [9]

model. The Hamiltonian

(10)

^{becomes :}

The system is characterized

by only

^{two order} para- meters : the concentration M

= 6i, >

^{and the} magne- tization of A atoms :

QA

⁼

1 j2(Q

^{+ 2}

R)

^{since :}

QB

⁼

(Q -

²

R)/2

^{= 0}

(from ⁽²²⁾

^{we see also that}

Hi

^{= 2}

H,).

Indeed the

equations

^{of state as}

given

ⁱⁿ

appendix

²

immediately

^{become :}

Q = 2 R,

^and

whence :

and :

From

(43)

^with

QA ~ 0,

^we deduce the

equation

^{of the}

magnetic

critical line :

in agreement ^with

(30).

The tricritical

point

^is

easily

located from

(36), (37)

^and

(38) :

^{B =}

0, x

= y whence : 12

M’+8M+1=0

and

M = - 1/6 (Fig. 1).

^Phase

separation

^is

magnetically

^induced.

.

FIG. 1. ^- Phase diagram ^{of the} Blume-Emery-Griffiths ^model

with J=O, k=l=j=4 U AD = jDD ⁼ 0) CA ^and CD ^{are the} magnetic ^critical points of pure A and B. The critical line CA ^C’

is a straight line, C’ is the tricritical point. ^{Lines of constant H are}

shown. P, ^and P 2 ^{are two} phases ^at equilibrium : only P2 ^is magnetic.

The

particular

^{case k}

= j

^{= - 1 describes AB} mixtures in which the A atoms are

non-magnetic : Q

^{= - 2}

R, QA

^{= 0.}

Suppose

^{now that A is}

magnetic

and B is a vacancy : the mixture is a

magnetic

gas, the _pressure of which can vary

[5].

whence :

The interactions

J AA and jAA

^{lead to} condensation and

ferromagnetism respectively.

^The pressure P is the

opposite

^{of the}

grand partition

^{function qJ.}

From

(24)

^and

(45),

^{we deduce :}

with :

whence :

in

agreement

with the results of ref.

[4].

5.3 GENERAL CASE. ^- We note that for J ⁼

0, equations (23b)

^and

(23c)

^{do not contain}

M,

^{so that}

they

^can be solved

separately. Knowing

^{a solution}

(Q, R),

^M is then calculated from

equation (22a).

For H = 0, M = 4 QR and :

^,

In the

following

^we

always

^{take :}

2jAA

^{= 32}

(magnetic

critical

temperature

^of pure

A)

^and vary

JAB

^and

jBB (see

^{Tables II and}

III).

TABLE II

’Magnetic

interactions in

symmetrical

^mixtures

TABLE III

Magnetic

interactions in

asymmetrical

^mixtures

Symmetrical

^mixtures

(1

⁼

0).

^{- The case k =}

16, j

^{= 1 = 0 has} been considered above : the critical line

CA CB

^is

parallel

^{to the M} axis in the

(M, T) plane.

If

JAB> jAA = jBB’

the critical line

CA CB

^exhibits

a maximum

Cmax

^at

FIG. 2. ^- Phase diagram ^{of an ideal} magnetic mixture with : k ⁼ 20,1= 0, j ^{= 14} jAB > jAA ⁼ iBB)’ CA CB is the critical line.

The mixture is homogeneous ^at any temperature and lines of constant H are shown.

There is no tricritical

point

and the mixture is

always homogeneous :

ⁱⁿ

fact,

^{the onset of}

ferromagnetism

favors

homogeneity.

If

jAA

⁼

jBB

>

jAB

>

0,

^the critical line

CA Cl

exhibits a minimum

Cmin

^{at M = 0}

(Fig.

^{3a : k =}

^10, j

⁼

6).

^Moreover

phase separation

^{is found at lower}

temperature

^{in the}

magnetic phase :

the critical

point

C is the maximum of a second branch of the critical line situated in the unstable

region

^bounded

by

the binodal curve.

According

^to

(48), kTc.,. =

^{2 k}

and

kTc

⁼

2 j.

FIG. 3. ^- a) ^Phase diagram ^{of an ideal} magnetic mixture with : k ⁼ 10, ^{1 =} 0, j ^{= 6} (j,, ⁼ 7BB > jAB > 0). ^Phase separation is found in the magnetic region : P, ^and P2 ^{are two} magnetic phases

at equilibrium, ^C is the critical point. The second branch of the

critical line is represented ⁱⁿ dashed, ^{lines of constant H are shown.}

b) ^Phase diagram ^{for : k =} 8,1= 0, j ^{= 8} (jAB ⁼ 0). ^{The critical} line is made of two straight ^lines CA ^{C and} CCB. The critical line and

the binodal curve have a common point ^C.

If

jAB

⁼

0,

the critical line

CA CB

^{is made of two}

straight

lines which cross at M ⁼ 0

(Fig.

^{3b : k}

= j

⁼

8).

The minimum critical

point

^{is at the}

top

of the binodal

curve.

If JAB = k - j 0, or k j,

^and k >

0,

^a

phase diagram

^{similar to that of}

figure

3a is found. However

Q = 0

and R =A ⁰

along Cmin

^C

(we

^{have now :}

kTcmin

⁼

2 j

^and

kTc

^{= 2}

k) :

since the mixture is

equiatomic,

^{atoms A and B have}

opposite magneti-

zations

QA

^{= R and}

QB = - R,

^{so that the net}

magnetization

^{is zero.}

If I JAB is

^small

(k

>

0), phase separation

^{is found}

below

kTc : coexisting phases

have different concen-

trations and are

partially

ordered both

cristallogra- phically

^and

magnetically. If I JAB

^is

large (k 0),

atomic

ordering

^and

antiferromagnetism

^are

expected

at low

enough temperatures.

Asymmetrical

^mixtures

(10 0).

^{- We consider first}

the

case : j

^{= 0 or :}

2 jpB

⁼

jAA

⁺

iBB’ If

for instance k ⁼ 12 and 1 ⁼ 2

(Fig. 4),

^no

phase separation

^is

found in the

magnetic phase

^{even at} very low tempe-

ratures.

If the ratio

kll

^{is lower}

(Fig.

^{5 : k =}

9,

^{1 =}

3.5, j

⁼

0),

^a tricritical

point

^{C’ and}

magnetically-induced phase separation

^{are found :} the critical line is

made

of two

branches, CA

^{C’ and}

C1 CB.

^Phase

separation disappears

^as T -> 0.

If j # 0,

^{and k ~ I}

~ j (Fig. 6),

^the

phase diagram

is similar to that of the

Blume-Emery-Griffiths model,

except that the critical line is

composed

^{of two bran-}

ches

CA

^{C’ and}

C1 CB. Figure

^{7 shows a similar}

situation in which the two branches are

straight

^lines.

Finally figure

^{8 shows an}

asymmetrical phase diagram

^{similar to that of}

figure

^{3a :} the binodal

curve is not

symmetric

^{due to the} asymmetry ^{of the}

magnetic

interactions

(l # 0).

FIG. 4. ^- Phase diagram ^{of an ideal} magnetic mixture with : k ⁼ 12, / = 2, j ^{= 0.} J_ > JAB> jBB > 0). The mixture is

homogeneous ^at any temperature.

FIG. 5. ^- Phase diagram ^{of an ideal} magnetic mixture with : k ⁼ 9, / = 3.5, j ^{= 0. C’ is a} tricritical point.

FIG. 6. ^- Phase diagram ^{of an ideal} magnetic ^{mixture with :}

k ⁼ 5.25, 1 = 3.75, j ^{= 3.25} ( jAA > An > Aa) ^{C’ is a} tricritical

point. The dashed line represents the unstable part of the critical line.

FIG. 7. ^- Phase diagram ^{of a} magnetic mixture with : k ⁼ 5.33, 1 ⁼ 2.66, j ⁼ 5.33. The critical lines CA ^{C’ and} CB C1 ^are straight

lines. C’ is a tricritical point.

FIG. 8. ^- Phase diagram ^{of an ideal} magnetic ^{mixture with :} k = 9, I = l,j = 5. The binodal curve is non-symmetric.

We note that the critical line is

composed

^of

straight

lines in various cases.

1)

^{If k}

= j (and

I #

0)

^or

jAB

^{= 0. From}

(30)

^we

deduce the

equations

^{of the two branches :}

The two branches intersect at the

point :

The

following

^cases have been studied : I ⁼ 0

(Fig. 3b) ;

2)

^If

kj

⁼

^l2.

^From

(30)

^{we deduce :}

A

particular

^{case is}

again : k

^{= I}

= j (Fig. 1).

6. Phase

diagrams

of non-ideal mixtures

(,l

⁼

0).

^-

6.1 PARTICULAR

CASE : j

^{= I = 0. - As}

expected

from the above

discussion,

^M is solution of

(25a)

^and

equations (23b)

^and

(23a)

^{become :}

Phase

separation

^and

magnetic ordering

^{are inde-}

pendant phenomena :

^the

magnetization

^{of an atom}

is

independant

of its chemical nature A or

B and, according

^to

(34),

^the

magnetic

^critical

temperature

^is

independant of M : kTQ

^{= 2 k.}

^Figure

⁹ illustrates the

two

possible phase diagrams

^{in the}

(M, T) plane :

C is the critical

point

^for

phase separation, kTc

^{= 2 J.}

FIG. 9. ^- Phase diagram ^{of a} magnetic ^mixture with : j ^{= 1 = 0} (jAA ⁼ jBB ⁼ As ⁼ k) : a) k > J, the critical phase ^is magnetic;

b) ^{k J.} C1 ^and C2 ^{are two critical} points ^at equilibrium.

6.2 PARTICULAR CASE : k ⁼ I ⁼

j.

^-

Figure

¹⁰

shows a

typical

^result

already given

^{in ref.}

[4] :

^we

choose : J ⁼

12 ; k

^{= I}

= j

^{= 4. As T is}

^lowered, phase separation

^is induced either

by

interatomic ^or

magnetic interactions, according

^to the initial concen-

tration. C is the critical

point

^for

phase separation,

’r a

triple point.

^For

larger

^{values of}

J,

^no tricritical

point

^{is found.}

6.3 SYMMETRICAL PHASE DIAGRAMS

(! = 0).

^-

j

^{0 : Phase}

separation

^induced

by

interatomic inter- actions is hindered

by magnetic

interactions

(Fig. 11).

If J >

k, phase separation

^{occurs as T} is reduced for M ⁼ 0. The binodal curve is narrowed as compar- ed to the

non-magnetic

^{case when the two}

phases

^at

equilibrium

^become

magnetic.

^{If J} k the binodal

curve is

entirely

^{below the}

magnetic

^{critical curve.}

For H ⁼

0,

^{M and R} become different from ^zero.

Phase

separation

^{is not found if J}

+ j

^0.

FIG. 10. ^- Phase diagram ^{of the} Blume-Emery-Griffiths ^model with J ⁼ 12, k ^{= 1} = j ^{= 4. C is the} unmixing ^critical point. ^C’

the tricritical point, ^{i the} triple point. The dotted line represents the binodal curve in the absence of magnetic interactions.

FIG. 11. - Phase diagram ^{of a} symmetrical magnetic ^mixture

with : k = 20, l = 0, j = - 4 ; a) ^{J =} 22 ; b) ^{J =} 18 The dotted lines represent the binodal curves if j ⁼ 0, the dashed line represents

the critical line if J ⁼ 0.

j

> 0 : Phase

separation

^{is enhanced}

by magnetic

interactions. Four types ^of

phase diagrams

^are

found

(Fig. 12).

. a)

^{J is}

large : phase separation

^{is found in the non-}

magnetic region

^{of the}

(M, T) plane (Fig. 12a).

b)

For lower values of

J,

^a

second

^order

phase

transition in M is followed

by

^a first-order

phase

transition

in Q

^{and R as T is} decreased for H = 0.

This leads to the

diagram

^of

figure

^{12b :}

C’

^and

C’

are two tricritical

points, A

^{is a}

quadruple point.

c)

^For still lower values of

J,

^{the two}

phase

^transi-

tions considered above coalesce and the

diagram

^of

figure

^{12c is} obtained. T is a

triple point.

d)

^{For low values of}

J,

^the

diagram

is similar to

that of

figure

^3a.

6.4 UNSYMMETRICAL PHASE DIAGRAMS

(1

⁼⁼

0).

Two different situations are shown in

figure

^{13. For}

small values of

J,

the binodal curve is

symmetrical

FIG. 12. ^- Phase diagram ^{of a} symmetrical magnetic ^mixture with:k=10,1=O,j=6:a)J=15;b)J=l2:CiandCi

are tricritical points : A ^{is a} quadruple point; c) ^{J = 9 :}

C,

^and C2

are tricritical points, ^{i is a} triple point; d ) ^{J = 5. In} (a), (b) ^and (c),

the dashed curve represents the unstable part ^{of the} magnetic critical curve.

FIG. 13. ^- Phase diagram ^{of an} asymmetrical magnetic ^mixture

with k ⁼ 12, ^{1 =} 2, j ^{= 0 :} a) ^{J =} 8 : the binodal curve is asymme- trical ; b) ^{J = 14 : below} Tc,, the binodal curve is asymmetrical.

The dotted line represents the binodal curve if I ⁼ 0.

and is

entirely

^{below the}

magnetic

^critical

^curve.

^For

large

^{values of}

J,

the critical

point

^{a for}

phase

separa- tion is non

magnetic,

^{and the}

magnetic

^{critical curve}

is made of two branches

CA C1

^and

CB C2.

7. Discussion. ^- We have shown in this paper that the

thermodynamic

behaviour of a mixture of two

magnetic

^{or nematic}

compounds

^can be simulated

by

use of a double

spin

^one-half

Ising

model with Hamil- tonian

(10).

^{Such a} system is characterized

by

^three

order parameters

M, Q

^{and R but}

only

^{the concen-}

tration M and the

magnetization

^or orientation

Q

are

generally

observable. Phase

diagrams

^similar

to that of He3-He4 mixtures have been obtained :

phase separation

^{can be}

triggered by

interatomic or

intermolecular

interactions,

^or

by

^{the onset of} coope- rative

magnetic

^{or nematic}

ordering.

An external

magnetic

^field

applied

^{to a} paramagne- tic mixture can also

trigger phase separation.

As discussed

by

^Wortis

[10]

^{our model of a solid}

magnetic binary

mixture is valid if the atoms are

mobile and their kinetic _energy is

neglected.

^{Indeed in}

the case of a dilute

Ising ferromagnet,

^two types ^of

impurities

^must be considered : random

impurities

which are frozen in and mobile

impurities.

^{In the}

first

situation,

the critical

temperature

^{decreases as the}

impurity

concentration is increased until the critical

percolation

concentration : above this

concentration,

the system ^is

always paramagnetic [ 11 ].

^{In the second}

situation,

^a tricritical

point

^is found above a certain

impurity

concentration

[10] : phase separation

^occurs

at low

temperature,

^as in He3-He4 mixtures. The

mobility

^{of atoms} is in fact

implied by

^the

pseudo- spin description

^{of a} solid mixture. Our model can

also describe

liquid ferromagnetic alloys :

^the

Ising

like character of the

magnetic

interaction is not well

adapted

^{to such an}

isotropic

^{system, but we are}

interested in the

topology

^{of the}

phase diagram

^more

than in detailed a

description

^{of the}

magnetic

pro-

perties.

As mentioned

above,

^our

lattice-gas

model also

applies

^{to the}

description

of mixtures of nematic

compounds.

^The

possibility

^of

finding

tricritical

points

in dilute

liquid crystals

^has

already

^been

suggested [12].

^{In our}

simple ^model,

^the

isotropic

^to

nematic

phase

transition is a second order one. In real

systems

^{it is}

slightly

first order in the sense that the

discontinuity

in the order

parameter

is small and there

are

significant

fluctuation

phenomena

^{on both sides}

of the transition. We

hope

however that the model will

give

^{a correct}

qualitative description

^{of the}

phase diagram.

First order transitions would be found if the

elongated

^{molecule were allowed to} take three

mutually perpendicular

orientations : the

spin-one-

half

(r)

^{would be}

replaced by

^a

spin-one (Potts model).

The

double-spin 1/2

^{model has been}

applied

^{to the}

description

^of

magnetic liquids [13], singlet-triplet magnetic

systems

[14],

^the

degenerate

^{Hubbard mo-}

del

[15],

Jahn-Teller systems

[16], magnetic

systems

with

spatial degrees

of freedom

[17],

and the Ashkin- Teller model

[18].

^{It can be}

applied

^{also to the}

descrip-

tion of solidification of

liquid-mixtures,

^{or the nema-}

tic-smectic A transition in

liquid crystal

^mixtures

[19].

Acknowledgments.

^{- I} thank Prof. J.

Lajzerowicz

for useful comments.

Appendix

^{1 :}

Spin

^{S =}

3/2 description

^{of a}

magnetic

mixture. ^- A double

spin

one-half manifold is

equi-

valent to a

spin 3/2

^{manifold. We shall use} the follow-

ing relationships :

so that the

dipolar operator

^is

given by :

and the

quadrupolar

operator

by :

and the

octupolar

operator

by :

the

eigenvalues

^{of c are}

1/2, - 3/2,

⁺

3/2, - 1/2.

Reciprocally

^{we have :}

We note that :

of the form

given by

Tahir-Kheli and

Taggart [20].

The

eigenvalues

^{of r + 2 6i and i - 2 as are}

(1, - ^{1, 0,} 0)

^and

(0, 0, - 1, 1).

^The Hamiltonian

(10), using

the above formula can be written in the form

given by

Tahir-Kheli and

Taggart.

Appendix

^{2 : The}

equations

^{of state}

(23)

^can also be written :

whence :

References

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