First-order phase transitions in spin-one Ising systems

Physica 33 689-690

Griffiths, R. B. 1967

LETTER TO THE EDITOR First-order

phase transitions

in spin-one

Ising systems*)

In a recent paper CapelI) has shown that in the molecular field approximation spin-one (S, = - l,O, + 1) Ising system described by a Hamiltonian X = --D c

(1 -

.s;i,

-

J

i

2 .s,&j (ii>

the

(1)

(the second sum is over pairs of nearest neighbours) has a first-order phase transition for D in a suitable range. That is, the energy, entropy, and spontaneous magnetization are discontinuous functions of temperature. Blumes) has used a model similar to (1) in order to explain the first-order magnetic transition in UO2. Capel’s and Blume’s results are probably qualitatively correct, even though the molecular field approximation is not always a reliable guide. In this connection it may be of interest to consider a modified form of (l),

2’

=

-D

x (1 i

Sii) -

J I; S$$.

(4

for which one may show without using any approximations the existence of an analogous first-order transition. Using at = 2Szi - 1 = f 1, let (2) be rewritten as 2’

= (+D -

+zJ) c u( -

i

4J x

oiq + constants



(3)

with z the number of nearest neighbors. The partition sum is 2 =

E 2*X(or+1) exp( -SP/kT) {or}

(4)

where the first factor appears because each spin has two states (S,i = f 1) for Q = 1. Apart from a term linear in temperature the free energy is F’ = F1*[+(kT In 2 -

D + &zJ), %J]

(5)

with FI[H, J*] the free energy for the spin &(u, = f 1) Ising model Hamiltonian with -H and -J* the coefficients of the first and second sums, respectively, on the right ‘side of (3). For temperatures below T, 0~ J*, BFIIaH is known to be a discontinues function of H at H = 0 (“spontaneous magnetization”) for spin 4 Ising models on simple twoOT three-dimensional latticess). Thus the system (2) has a first order transition at a temperature kT = (D - izJ)/ln 2, (6) and a latent heat related ‘to the spontaneous *) Supported

in part

by the U.S.A.F.O.S.R.,

-

magnetization

Grant No. 508-66,

689 -

of the corresponding

and by the N.S.F.

690

FIRST-ORDER

spin 4 Ising model,

PHASE

provided

TRANSITIONS

IN SPIN-ONE

ISING

SYSTEMS

that &zJ < D < &zJ + kT, In 2.

There is a second-order in

(7) and

diverges

a preliminary

as IT -

(no latent heat) when D is equal to the upper limit

thermodynamic

analysis

T,I- 1+1/a if the magnetization

in the corresponding A molecular

transition

spin--$ Ising

field approximation

applied

capacity isotherm

suggests

that

to (2) or, equivalently,

similar results.

$zj( 1 + In 2), and if D is equal to this quantity transition

heat

varies as HIis on the critical

the

system.

the right side of (5), yields qualitatively at the (second-order)

(7)

point,

the heat capacity

consistent

used to evaluated

The upper bound

with the classical

diverges

in (7) becomes as jT - T,I-3

result 6 = 3.

Received 5-9-66 R. B. GRIFFITHS *) Relfer Graduate School of Science Yeshiva University New York, New York, U.S.A.

REFERENCES 1) Capel, H. W., Physica 38 (1966) 966. 2) Blume, M. Phys. Rev. 141 (1966) 517. 3) Yang, C. N., Phys. Rev. 88 (1952) 808; Griffiths, R. B., Phys. Rev. 136 (1964) A437; Dobrushin, R. L., Tear. Veroyatuost. i Primenen 10 (1965) 209.

*) Permanent address: Department of Physics, Carnegie Institute of Technology, Pennsylvania 15213, U.S.A.

Pittsburgh,