Lattice mixtures of fluctuating phases

Physica 144A (1987) 369-389 North-Holland, Amsterdam

LATTICE

MIXTURES

OF FLUCTUATING

PHASES

V.I. YUKALOV Joint Institute for Nuclear Research,

Received

Laboratory

of Theoretical Physics, Dubna,

9 December

USSR

1986

A method of constructing a renormalized Hamiltonian for a heterophase mixture on a lattice is formulated. Two ways of separating thermodynamic phases are discussed: a direct construction of spaces of states having desirable properties, and by introducing into the Hamiltonian the terms explicitly containing order parameters characterizing the needed phases. The methods are illustrated by heterophase models of the Ising and Heisenberg types.

1. Introduction The microscopic theory of systems with heterophase fluctuations’-4) has the most consistent foundation due to the use of the ensemble of quasi-equilibrium ensembles5) and the Gibbs method of separating surfaces6). This ensemble, for brevity, will further be called the heterophase ensemble. Numerous examples of real systems being heterophase mixtures have been given in refs. 5,7. In ref. 5 the theory of heterophase systems is formulated for the continous case. In the present article the theory is reformulated for lattice systems (section 2); only basic results are brought up, as far as the technique of proofs has been Aft er averaging elaborated earlier4’5,7). over heterophase fluctuations and finding a renormalized Hamiltonian the question arises: how to separate, in a mathematical description, the needed thermodynamic phases? In easy cases, e.g. for the two-level system, the ordered and disordered phases can be explicitly divided by constructing spaces of states (section 3) chracteristic for them. A good illustration of this situation is an Ising-type model generalized by taking into account heterophase states (section 4). In a two-dimensional space for a zero external field this model is exactly solved and it appears that the heterophase states can exist solely as metastable ones in a finite vicinity of the critical point. In more difficult cases the separation of phases may be made by adding to the Hamiltonian the sources containing order parameters of phases sought for (section 5). As is known, the order parameters enter in an explicit 037%4371/87/$03.50 0 (North-Holland Physics

Elsevier Science Publishers Publishing Division)

B.V

V.I. YUKALOV

370

way into the Hamiltonian when involving consequently, is the simplest procedure heterophase

model

mean-field (section

of the

approximation,

Heisenberg demonstrates

6). In such a model,

and exchange

- interaction

depending constants,

the mean-field for separating

type

with

arbitrary

a wide variety on a relation any states

approximation that, different phases. A spin,

even

of nontrivial

between

are possible:

crystalline stable

in the

properties - field

and met-

astable, pure and mixed. Some of the metastable states are characterized by anomalous maxima of specific heat or magnetization below the Curie point. In some stable systems the nucleation point appears being a phase-transition point between homophase and heterophase states. Thermodynamic functions can have jumps at the nucleation point. The presence of heterophase fluctuations lowers the critical temperature by four times, and can even lead to the break of a second-order phase transition to a first one.

2. Heterophase

lattice systems

Let on the lattice H =

c

2 n

I,.

Z = {x, 1i = 1,2,

H(q,

..

, x,,)

. . , L}

the Hamiltonian

(n = 1,2, . .)

I?,

be given, the operator structure of which is not important for the present. It is possible that not all lattice sites are occupied by particles, as it occurs in of particles over sites is characterlattice-gas models8). Then the distribution ized by the operator N(x;) such that the sum

gives the particle-number operator. The operators H and & pertain to the algebra ZZZof observables defined on the Hilbert space X. Consider subspaces from the space Yf?, so that 9e c Yf?((Y = 1,2, I . . ) s) which can be separated each %a is formed by vectors of states with properties ascribing a thermodynamic phase numbered by (Y. For the time being let us leave aside the question how to produce such a separation of the subspaces 9a as this will be explained in subsequent sections of the paper. The representation tim = rrcX(&) of the algebra of observables can be given on each SO. The separation of phases in the real space is done with the help of the Gibbs’) method of separating surfaces. For this the covering of the lattice {Z,} must be defined, so that

LATTICE MIXTURES OF FLUCTUATING PHASES

z=(_Lf,,

L=C

L, a

OL

Each covering manifolds,

371

is described4)

A fixed set of these

functions

by

of characteristic

will be denoted

The manifold of sets {S}, attributed topological space Yt={51Vz,,L,;CX=1,2

a set

)...)

functions

of sub-

by

to all possible

lattice

coverings,

forms the

S}.

This topological space becomes a metric space4X5) by defining the corresponding measure m( 5). Under any choice of separating surfaces, which is under any covering with fixed 5, the considered heterophase system is by no means nonuniform; between phases there are always transitional layers. The needed type of nonuniformity may be given5) by functions of the inverse temperature /3,(x, 5) and of the chemical potential p,(x, 5). We are interested here in the situation when fluctuational nuclei of competing phases come into existence in an occasional way, so that there is no spatial localization of domains during the observation time. Therefore, on the average the system seems to be uniform, the observable inverse temperature p and the chemical potential p being renormalized quantities of the corresponding primary functions

(1) Eq. (1) means that the system is in equilibrium while under each fixed choice of average, conditions on equilibrium’) are not valid. The separation of phases, is in quasi-equilibrium5). 5 has the form p( 5) = eerts)lZ where

(2)

,

the quasi-equilibrium

on the average, and only on the separating surfaces the usual system, where there is a space The statistical operator at given

Hamiltonian

V.I. YUKALOV

372

-

I

l-%(X,> 5W,(x,) t-,(x,)

7

is defined on the space 9 = 0, SO, and N,(x{) is the representation operator N(x,) on the space SC,,; the statistical sum is

of the

Z = Tr em“(‘) dm( 5) B I The manifold of systems with every possible configuration forms a heterophase ensemble’). The integration over conforms to the averaging over heterophase fluctuations’).

of dividing surfaces the measure dm( 5) This integration in

the case of continuous system is done by means of functional integrals over characteristic functions of manifolds”‘5). Continuous models are the most general ones. In particular cases they come to lattice systems of completely localized particles, and to intermediate situations, when localized and delocalized particles coexist, as in mixed-valence compounds’“-“). Therefore, it would be enough to consider the averaging over heterophase fluctuations for a continuous space, and pass to a discrete case only at a final stage. However, it is not difficult to effect an analogous averaging directly for a lattice model. To do this the characteristic functions are to be presented in the form

integration and the functional i.e. s dm( e) = C,, C,, where

supposes

is to be changed the summation

that the set

is fixed, and the other

sum means

to the functional

summation’),

LATTICE

MIXTURES

OF FLUCTUATING

PHASES

373

This averaging results in the following theorem: (3) where the renormalized

Hamiltonian

and the phase probabilities potential

w, are defined by minimizing the thermodynamic

y=-ilnZ=-ilnQexp[-p(fi--pfi)] under the normalization the equation

ay -= awa

0,

(4)

condition C, wu = 1, that is, w, should be found from

$>O @we=‘).

(5)

a

Any operator 2 from the algebra of observables ~4, when averaging over heterophase fluctuations, transforms as the Hamiltonian and the number-ofparticle operator do. We explain it for an example of the n-particle operator on the space 9 = 0, Se for a 2 = Cl...’ A(+. . . , xi,). Its representation heteropha;e system with fixed dividing surfaces has the structure

‘a(5)= C

A~(x,,,...,X,~)~,(X,~)...~,(X~,).

~,...i,

Using the methods of integration over characteristic functions4’5’7) one can show that the mathematical expectation of the considered operator takes the form

(6) in which

V.I. YUKALOV

374

A”= @ A”, , A”, = wg a

c

I,..

The following

notation

A&,

.

. , xi)

1,

will further

be used:

Owing to the presence of separating surfaces the surface average value here is defined5) by the formula

Es,,= E -

c w,%(l)

energy’)

exists.

Its

,


(7)

EEq.

(I?>> E,(l)-[(&),I,,><,=,.

(5) for the phase

c

probabilities

yields

n[w;-'E;'- w:-~E~)]=p(R, -R,),

(8)

where

Re c $ C (Nm(Xi)), I

> s=supcY.

The chemical potential as a function of the density rom the equation temperature p can be extracted

cu w,Ra = 1. In the case of two phases

c [K

E',"'(l-

n

while eq. (9) becomes

wIR,+ wzRz= 1.

N/L and of the inverse

(9) (s = 2

eq. (8) gives

wl)'-'Er)]= p(R,- R2),

LATTICE

MIXTURES

If each size of the lattice

OF FLUCTUATING

is occupied

375

PHASES

by one particle,

that is N(xi) = 1, then

R,=l and N=L. When

the

Hamiltonian

contains

(n s 2), then for the two-phase

w,=

2.6”

*

+ Ey’ - E’,” + 2[.q’

3. Explicit construction

not

more

than

two-particle

interactions

case

p(R, - R,)

)

w,=l-w,.

(10)

+ EY)]

of spaces

The approch set forth in the previous section presupposes that in the space of states of the considered system there exist subspaces Sa characterizing concrete thermodynamic phases. In some exceptional cases such subspaces could be explicitly constructed. This has to do first of all with lattice systems whose variables take a finite number of discrete values each. We elucidate the statement by an example of a system with two-level lattice variables. In such a case the basis of the space associated with the site i consists of two functions

cp,+ = The index one-particle

0 1 0 i > ‘p,- = 0 1 i.

0

i enumerates only space is a closure

replenished The total

those sites that are occupied of the linear envelope

with a scalar product space

of the N-particle

and a norm system

generated

is formed

by particles.

The

by the scalar product.

by the direct

product

(11) Absolutely

ordered

states

are described

by the functions

(12) Absolutely

disordered

states

correspond

to functions

of the type

N cpr=

8

rand{qj+,

Pi->,

(13)

V.I. YUKALOV

376

where

in place of the ith representative

way. The functions

Let us call the pseudovacuum by means

of elementary

be constructed.

either

cp,+ or ‘p,_ is chosen

of type (13) form an equivalence that vector

excitations,

in a random

class.

of the space of states,

from which,

the total basis of the considered

In the finite-dimensional

space

space can

(11) any of its functions

can

play the role of a pseudovacuum. Elementary excitations, in the language of spin variables, are the spin overturns. It is possible to call these excitations flippons. They are generated by the ladder operators

having

the properties

s,Lp;+=o

s,+cp,_ = pj+

1

)

[s,-,s,+]+ =i, One-flippon

[s,:,s;]_=o

excitations

Many-flippon

above

excitations

&,,

. . i,) = )

s,(p,+ =

,

Sl~cp,_=o,

(i#j).

the ordered

correspond

‘p,_

states

are defined

by the vectors

to the functions

s,;...s;cp; .

(14)

The analogous procedure is to bring about for constructing flippon excitations above the disordered states characterized by the function (13): as a one-flippon vector

one has

and for the many-flippon

vector

one has

N . . ) i,) = sl; . . . s,;qJ,,

c&i,)

(15)

Any two functions of space (11) can be transformed sequence of flippon excitations. For example,

&i* Ordered

iN)=(pN, cp”(il

)...)

and

disordered

states

one into another

through

a

Q=qq.

)...,

can

be more

strictly

distinguished

if one

LATTICE MIXTURES

OF FLUCTUATING

PHASES

defines an observable quantity called the order parameter. observable quantity, in the case under consideration, is

The operator of this

the eigenvalues of which can help to separate ordered from disordered The pseudovacua (12) and (13) are eigenfunctions of this operator:

a?,#; =-N UN

d

377

states.

(iaNI<=‘, vN> ;

here the finiteness of a,,, is due to the law of high numbers. The total set of eigenfunctions of the operator eN is given by all flippon vectors for which

(

eN(p:(i,, . . . , i,) = -+ lGNcpf(il, . . . , i,) =

j!+(a,

2n N

5

1

cp!Y(i,, . . . , i,) ,

2n)cpr(i,,

. . . ,

i,) .

The mathematical expectation of the order operator GNacquires the meaning of the order parameter solely for N + 00. As is known, only in the thermodynmic limit one is able to rigorously define the notion of a thermodynamic phase and of an order parameter ascribed to it. Introduce the notation for the limiting pseudovacua (12) and (13):

(16) The limiting transition N+m, as usual, means -+const. The limits of flippon vectors (14) and (15) it is necessary to impose additional conditions excitations. In particular, as N-+m, the number n increase with N. Defining the limits

9z(i,, . . . ; 6) = ,ljT_ ptY(il, . . . , i,)

di,,

. . . , i,) = l$n_ cpt(i,, . . . , i,)

and considering tions as N+m,

that N-+m, L+m, N/L are not uniquelly defined for a number of flippon can be either finite or can

(

lim ;=a<$

,

N+m

(

1

(17)

lim X = 0 ,

N--t-

the action of the order operator

)

on the corresponding

func-

V.I. YUKALOV

378

lim GN(pz(I,, N+=

. . ,i,)=*(l-26)cp,(i

,‘...

;S)#O, (18)

lim eN(pt(i,,

It+=

.. ,i,)= 0,

we make it certain that the vectors cp,(. . .) describe ordered states, while the cpO(.. .) describe disordered ones. Composing the closures of the linear envelopes

over bases from eq. (17) we get the spaces

The limit of space

(11) can be now defined

as

The existence of two spaces X+ and FK, each of which corresponds to an ordered state and to an ordered thermodynamic phase, testifies to the possibility of a macroscopic degeneracy in the system. To take away the degeneracy, X+. The disordered phase one of these spaces must be chosen, for instance meets the space X~o. The quasi-equilibrium heterophase system is characterized by the space

Here a general scheme for the explicit construction of the space of heterophase states for a wide class of lattice systems has been built. The next section contains an illustration of how to apply this scheme to a concrete model of the Ising type.

4. Modified

Ising model

Consider a two-dimensional Ising model with the number of lattice sites equal to the average number of particles, L = N. The heterophase generalization of such a model, according to section 2, leads to the renormalized Hamiltonian

LAlTICE

MIXTURES

OF FLUCTUATING

PHASES

379

in which the summation is over nearest neighbours, and the unitary operator iu reminds that 6m is given on the space Se. The total operator (21) is defined on the total space (20). Applying to this problem the transfer-matrix method, one can check that the maxima1 eigenvalue of that matrix corresponds to an ordered state, while the minima1 eigenvalue corresponds to a disordered phase14). In such a way for the thermodynamic potential (4) one has Y = Yl + Y2 9

-

i

ln(2sinh b,) - A, ,

(22)

The functions ye(v) being defined by the equation cash ‘Y,(V)= cash 6, . coth b, - cos v , in which the positive solution is denoted as n(v) and the negative one, as X(V). The latter equation can be transformed to the following: y,(v) = (-l)“+‘{ln2-

ln(sinh b,)

57 +$

Therefore,

i 0

1n[cosh2b,-(cosv+cosv’)sinhb,]dv’

the potential

(22) yields

y=/?f(2w2+1-2w)-Q,+Q,-ln(2sinhb,)-@, where w = wr and

Q, = $

[hr[cosh2

b, - (cos Y + cos v’) sinh b,] dv dv’ .

0

According to expression

(lo), the probability

2u - B, w = 4u - B, - B, ’

of the ordered phase is

u 77’

(23)

V.I. YUKALOV

380

where

q. is the number

of nearest

neighbours,

2 Bn = ~ 2 (a;+, 4oN (I/) =I

~ ’ (-‘)““K(z~l)

sinh b - 1 sinh h” + 1 II

coth B, .

8 sinh LI,~. cash’ bCr (1 + sinh ba)J

For the spontaneous

magnetization

~=~~(,),=wu, I The expressions

one finds

for the entropy

per particle

s = Q, ~ Q, + ln(2 sinh b2) - pl[wB,

and the specific c,

= 2p’P

(24)

cr=(l-l/sinhJh,)“X

- 2u( 1 - w)(2w - l)]

(25)

heat 21/(X, - X,) + X, YJ - X,Y, y,+y,-4u

(26)

’

in which tl +, B,” tanh bcr ~ 112 sinh’ ba - 1

will be also useful when analyzing the stability of the system. The heterophase system is absolutely stable. if the specific negative, the second derivative

is positive

and the difference

heat

is non-

LATTICE MIXTURES

OF FLUCTUATING

PHASES

381

AY =Y -y(l) is negative,

y(l)-

in which

;

-$lng[exp(-Pfi,)l,.,, I

thus, Ay = /3Uw(w - 1) - Q, + Q2 + Q - ln(2 sinh b2) ,

In[cosh’( PI) - ( cos v + cos II’) sinh( PZ)] dv dv’

In a low-temperature region the probability of the ordered different ways depending on the value of the crystalline-field u<; T W=l-

2(1 -2U)

’

T-

1 ~ 4JP

phase behaves in constant U; so, at

(u
exp

However, at low temperatures the entropy diverges, s-+ -x, if u < i, and the specific heat becomes negative, if u 2 4. The case of infinite entropy is to be rejected as unphysical. If C, < 0, this testifies to the instability of the system. The temperature Td, where the stability is lost, is defined as a temperature, for which C, = 0. At the temperature Td the heterophase system has to decompose. A numerical analysis of the quantities a2yldw2 and Ay shows that the thermodynamic potential of the heterophase system is always higher than that of the pure Ising model with w = 1. Consequently, at the decomposition temperature Td the heterophase system transforms into a pure ordered phase. To check if there is a temperature region between the decomposition system could temperature Td and the critical one T,, where the heterophase exist as metastable, the critical properties of the model should be investigated. The critical temperature, defined by the condition (T = 0, is equal to

V.I. YUKALOV

382

(27) This is four times as small as the critical model.

An expansion

for the probability 1

in powers of the ordered

(-7)

in the corresponding

pure

phase,

arsinh 1

W = 2 + 4Gr(u + 0.087) the order

temperature

of T = (T - T,)l T, + -0 gives the asymptotics

ln(-7)

,

(28)

parameter

(29) and the specific

c, =

heat

8 arsinh”

1

n-*(u + 0.087)

ln*(-7).

The asymptotics (29) and (30) differ from the corresponding expressions for the pure Ising model, for which u x (-T)“~ and C, aln(-7). Formula (30) ensures that the specific heat can be positive in the critical region. Hence, the temperature interval T, < T < T, can exist, inside which the heterophase system is metastable. To be more satisfied, the system of equations (23)-(26) has been numerically solved. The following values are obtained for the decomposition temperature: T, = 0.278T, at u = 0.5, Td = 0.472T, at u = 1.5; T, = 0.556T, at u = 3. The entropy (25) for these cases, unfortunately, is negative. However, we find consolation in remembering that the same situation, when the entropy is negative, occurs for the spherical model and the replica-symmetric model of the spin glass”).

5. Method of ordering sources The spaces of states explicitly constructed in section 3 are valid for an Ising model of arbitrary dimension. But it is not easy, as is known, to calculate something for the three-dimensional model. The information on the structure of state spaces does not give an effective calculating receipt. Fortunately, the detailed knowledge of this structure is not necessary at all. It is more convenient to separate thermodynamic phases not according to microscopic, but to macroscopic characteristics. Presumably, one of the simplest ways to do

LA-I-I-ICE

it is by invoking parameter from the

the

MIXTURES

order

OF FLUCTUATING

parameter.

In some

383

PHASES

approximations

the

order

appears in the process of calculations. If this is not so, then, just beginning, one may add to the Hamiltonian infinitesimal terms

containing

explicitly

phases’).

the order

In the present

ase generalization

section

parameters

with the properties

such a method

of the Heisenberg

model

is concretized

of considered for the heteroph-

with arbitrary

spin.

Let the number of lattice sites again coincide with the number of particle, N = L. The operator Sj (i = 1,2, . . . , N) of the spin S is given for each lattice site. For convenience the notation (s,),

will be further

used.

sj, = SJ, The Hamiltonian

of the model

has the form

E? is given on the space 9 = P1 @ FZ, where %i is the state space SU(3)-symmetry, and .9* is the SU(3)-symmetric space3). The

The operator with broken order

)

= (Si,)

parameter

here is (32)

In

the

ordered,

disordered,

ferromagnetic

paramagnetic

phase

phase

(ri

the order

differs

from

parameter

zero,

equals

while

in the

zero: (33)

The condition uished. Under rewritten

(33) is just the criterium by which one phase can be distingthe translational invariance over the lattice eq. (33) can be

as

The probability

of the ferromagnetic U-25,

w1 = 2(U - J, - J2) ’

phase,

Ja-~pij(si/sj.). 11

according

to eq. (lo),

is

(34)

384

V.I. YUKALOV

From the positivity and boundedness of the inequality 0 < w, < 1, it follows that heterophase U < inf{ZJ,,

2.4)

probability, fluctuations

that is from the can exist if either

,

or U > sup{2J,,

2J,} .

The condition of heterophase tion w, + w2 = 1 gives

$ (U - J, -

stability,

i.e. $ylawt

r,>> w:(D,- J:, - wi(D, -

J;)

> 0 under

the normaliza-

)

(35) %))

Compose

.

the Hamiltonian (36)

containing

corresponding

the source

to the mean-field

theory,

with the order

&M&)1 = The

ferromagnetic

and paramagnetic

phases

parameter

exp[-PRWI ~exp[-PfiA41 a

differ

one from

.

another

(37)

by the

criterium

which is equivalent to eq. (33). Now, being independent of a kind of the approximation used, the problem with the Hamiltonian (36) always explicitly contains the order parameter (37) so that the phases can be distinguished by means of the criterium (38). At the end of the calculations the ordering source should be taken away. In other words, the mathematical expectation of the operator 2 is defined as a peculiar quasi-average’),

385

LA’l-l-ICE MIXTURES OF FLUCTUATING PHASES

(A”) = liiTrp(v)A”

)

exp[-pHlv)l .

p(v) =

(39)

~exp[-PW4 In particular,

for order

parameter

(32) (40)

a, = li_i U,(V).

6. Modified Heisenberg

model

The most obvious illustration for the method formulated in the previous section is the case of exactly solvable models with a long-range interaction. Consider the system having Hamiltonian (31), in which Ji, is the long-range potential satisfying the property Jij -+ 0 ,

$ c Jij + const ‘I

(N-m).

An asymptotically rigorous solution of the problem with such a potential is equivalent, as is shown, to the mean-field theory. Therefore the order parameter appears in the asymptotic Hamiltonian in a natural way. The expression (32) for the order parameters gives16), taking account of eqs. (33) and (38), la,I=(T=BS(2W%S2/T), where phase,

a,=o,

B,(e) is the Brillouin function. w = w,, follows from eq. (34),

w, = w = U/2(U - a2S2) ) Here

the following L4=-,

1

u J

notation

T-E’

The

w* = 1 - w .

probability

of the ferromagnetic

(42)

is used:

J = ;

The necessary condition requires the validity of one condition of the heterophase system is metastable, while The critical temperature temperature To in the pure

(41)

c Jlj . 11

for the system to be heterophase, that is 0 < w < 1, of the inequalities: either u < 0 or u > 2u2S2. The stability (35) shows that at u < 0 the heterophase at u > a2S2 it is stable. T, is four times as small as the standard critical system,

386

V.I. YUKALOV

T,=S(S+1)/6= Moreover, peculiarities

when

T,/4,

T,=2S(S+1)/3.

the heterophase

can occur,

system

for instance

(43)

is metastable,

the magnetization

some

thermodynamic

M = waS can have the

maximum M,=fil4 located

(u
at point

T, defined

(44) by the equation

xf=E = SB,y(S-\/-u18T,)

,

(45)

so that T, < T,. The specific heat C, can also have a maximum below the Curie point. The following interesting situation can take place. At temperatures from 0 up to T, the pure ferromagnetic phase is absolutely stable. But above T, it is more profitable for the system to be in a heterophase state. The nucleation point T,, where the nuclei of the competing phase come into existence, is given by the equation %/u/2 = SB,(d&T,)

(0 < /A< 2S2)

(46)

The nucleation process is a peculiar phase transition occurring here from a pure ferromagnetic state to a heterophase state. The order parameter, corresponding to this transition, is the probability w. In the present case this is a continuous phase transition, as far as at the nucleation temperature w( T,) = 1. In particular,

for spin

l/2

the

nucleation

temperature

found

from

eq.

(46)

is T, = G/2

artanhfi

(S = i) .

(47)

This temperature can take the values from T,, = 0 at u = 2S2 up to T, = T,) at u = 0. The phase probability w has a cusp at T = T,; as a result, the specific as experiments heat has a jump at the nucleation point. In real matters, witness, the nucleation temperature strongly varies depending on the kind of matter, the value of T, can be placed either close to the transition temperatures, or quite far from it. Thus, in the case of the spine1 Ni, ,,Zn,,,Fe,O, it has been shown”) that ferromagnetic and paramagnetic phases coexist beginning from T,, = 375 K, while T, = 516 K, that is (T, - T,)/ T, = 0.273. This has been done by means of the Mossbauer effect. The first-order

phase

transition

takes

place

in the heterophase

system

if

LATTICE MIXTURES OF FLUCTUATING PHASES

o
u,=

S2(S+ 1)’ s2+(s+1)2.

(20> 3

387

(48)

The temperature T, of the first-order transition lies in the interval (T,, T,,). The metastable state of the overheated ferromagnetic exists to the right of T, close to the spinodal-decomposition temperature T,, defined by the condition 8W -+m dT

T,).

(T+

For the spin l/2 this gives T, = 2

(4~ + 3~7;)(1-

a;)

(s = 1) ,

where o5 and w, are solutions of eqs. (41) and (42) at T = T,. When u > u,, the phase transition becomes of second order. The point at which the phase transition changes its order is a tricritical point. Here it is given by the equalities T,=T,,

u=u,.

If U, from eq. (48) is considered as a function of spin S, then U, = u,(S) defines the tricritical line. For large S this is the parabola u,-lOS73

(SSl).

At the tricritical specific heat

t-4”

point the critical indices are abruptly

So, the

U#Uf,

3

C&la (-T)y2 {

changed.

u = u,

)

)

and the magnetization

(-w2 > u # u,

Ma

i

(-T)1’4

)

2

u = u, .

Let us introduce a new critical index characterizing the phase probability, w=

$ + A,(-+

.

the critical behaviour of

V.I. YUKALOV

388

This index E = lim In w/ln(-r) 7+-O

at the tricritical

1

E=

1 , 7,

1

point

ufu,

is also abruptly

changed,

3

u = u,

The very existence of the tricritical point is due to the presence in Hamiltonian (31) of terms having opposite signs, of the positive crystalline field (first term) and the negative exchange interaction (second term). There is a number of other examples when the occurence in the Hamiltonian of terms that are opposite in sign to each other also leads to the appearance of a tricritical point. Such a situation takes place in the Blume-Cape1 modelix~“‘), into which side by side with a negative exchange interaction a positive term describing the one-axis anisotropy enters too. In conclusion let us find the averge surface energy. Substituting into formula (7) the expression E =

NJ[u(w* -

E,(l)

= NJ(;

w +

w2S%*],

;) -

- rr;S’)

,

E,(l)

= NJ ;

)

where a0 = B,(2S2qJT) is the order

parameter

Es,,,= -NJ

of the pure

ferromagnetic

phase,

we obtain

u(u - 2a:,SL)

(50)

4(u - aZS2)

As is clear, the existence of the surface energy value of the crystalline-field constant u.

(50) is also caused

by a nonzero

Acknowledgements I am grateful for discussions of some hmeteli, V.B. Kislinsky, V.K. Mitryushkin, Haar and A.S. Shumovsky.

aspects of this work to A.M. AkS.A. Pikin, N.M. Plakida, D. ter

LATTICE

MIXTURES

OF FLUCTUATING

389

PHASES

References 1) 2) 3) 4) 5) 6) 7) 8) 9)

V.I. Yukalov, Teor. Mat. Fiz. 26 (1976) 403. V.I. Yukalov, Physica 108A (1981) 402. V.I. Yukalov, Physica 1lOA (1982) 247. V.I. Yukalov, Phys. Rev. B32 (1985) 436. V.I. Yukalov, JINR P17-86-262, Dubna, 1986. J.W. Gibbs, Collected Works, vol. 1 (Longmans, New York, 1928). V.I. Yukalov, JINR P17-85-370, Dubna, 1985. H. Kawamura, Progr. Theor. Phys. 68 (1982) 764. D. ter Haar and H. Wergeland, Elements of Thermodynamics (Addison-Wesley, MA, 1967). 10) A.F. Barabanov, K.A. Kikoin abnd L.A. Maksimov, Teor. Mat. Fiz. 20 (1974) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)

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