EQUATION OF STATE IN THE CRITICAL REGION

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EQUATION OF STATE IN THE CRITICAL REGION

C. Domb, D. Gaunt

To cite this version:

C. Domb, D. Gaunt. EQUATION OF STATE IN THE CRITICAL REGION. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-344-C1-345. �10.1051/jphyscol:19711116�. �jpa-00213932�

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EQUATION OF STATE IN THE CRITICAL REGION

C. DOMB and D. S. GAUNT

King's College, Strand, London, W. C . 2., England

R6sum6. - On obtient des termes d'ordre supbrieur dans l'bquation d'ktat du modkle d'Ising et Yon en fait une etude numMque. On fait une comparaison brkve avec des rksultats correspondants pour quelques modeles dont on connait la solution exacte.

Abstract. - Higher order terms in the equation of state are derived and studied numerically for the Ising model, A brief comparison is made with corresponding results for exactly soluble models.

Considerable theoretical and experimental evidence has now accumulated to support the scaling form of the equation of state of a ferromagnet in the critical region. Following Griffiths [I], this may be written

where H i s the magnetic field, M is the relative magne- tization, t is the reduced temperature (T - T,)/T,, and the critical exponents have their usual meaning 121.

Recently, the function h(x) has even been constructed numerically for the two and three dimensional Ising model 131.

In this paper the higher order terms in the equation of state are studied. An examination of the magnitudes of the correction terms as compared to the basic term should enable a rough assessment to be made of the region of validity of the scaling law form. Let us assume, therefore, that for H = 0 and T 2 T, the 2 nth field derivative of the free energy, &,, has the form

where $,(t) and cp,(t) are analytic a t t = 0. A << gap index ⁾⁾analysis analogous to that used by Domb and Hunter [4] leads to the more general equation of state

(Full details of the derivation have been given else- where [5].) As anticipated, the first term is (3) is just the standard equation of state (1). Series expansions have been used to study the leading correction terms hl(x) and kl(x) for the Ising model.

For large x, the analysis which led to (3) also indicates

where 8, and 5, depend only on +,(t) and pl(t). Thus from a refined analysis of the high temperature susceptibility serizs, we estimate

8, = - 0.38, 5, = 0.1 , ^{( 5 )}

for the plane square (SQ) lattice, and

O1

-

^-^0.14, ^c1^E^0.1^, ⁽⁶⁾

A detailed study of the critical isotherm has been made by Gaunt and Baker [6]. They suggested that the first correction term is of order M ~ + ( ~ / ~ ) , which implies hl(0) is exactly zero. We conclude that if hl(0) is non-zero, then it must be very small. More specifically, their results for the SQ lattice yield

and for the BCC lattice

On the phase boundary curve, we require a solution of (3) for which M , # 0 when H = 0. The zeroth order term corresponds to x = - x,, where

k(- x,) = 0 and

M, = x l B ( - t)' . (10) Now consider a small deviation from

- x, to - x, + EX,, where

To first order, we must solve the equation

+ M6lB kl(- x,) . ⁽¹²⁾

If we assume there are no terms in (1 1) of order (- t)Y, which is certainly true in two dimensions, we must have

k,(- xo) = 0, (13)

and

- 2

E = - ~o [hl(- xo)lhb(- xo)] (- t) - ⁽¹⁴⁾

E, and hence h,(- x,), can be calculated from the exact results in two dimensions and by series analysis in three dimensions. We find

hl(- x,) ^{~ 1 '}0.106 , 0.102, (1 5) for the SQ and BCC lattices, respectively.

Differentiating (3) with respect to M leads to for the body-centred cubic (BCC) lattice. expressions for hi(- x,) and k;(- x,), which in prin-

Along the critical isotherm x = 0, (3) predicts ciple can be estimated from the low temperature

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711116

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EQUATION OF STATE IN THE CRITICAL REGION C 1 - ³⁴⁵

susceptibility. In practice, the appropriate series are (1) Hn(x) z cn xY4" as x -+ CO, in agreement with rather difficult to analyse. (4) when one remembers that y = 2. c, like 0, is

Finally, as further support for (31, we mention that negative, while c, like l , is positive.

the corresponding equations 15, 71 for the mean-field (2) Hn(0) ⁼cn # 0 so that all the correction terms approximation (Y = 1, P ⁼3 7 6 = 3) and spherical in (7), including the first, are present.

model (y = 2, P ⁼3, 6 = 5) are both of this form,

(3) Hn(- x,) = 0. Thus, ^E in (11) and all higher with an equation for the order terms are identically zero, showing that (10) Base gas [&I. Note, however, how the sets of correction

with no correction terifis is the exact result [9] for the terms merge into one another enabling us to write spherical

00

H = M' h(x) + 1 M" '" Hn(x) . ⁽⁴⁾^{Hi(- x,)} ⁼0. This result is not expected to

n = 1 (I6) hold for the Ising model, since it reflects the anomalous

behaviour of the susceptibility (and higher order field Furthermore, the correction terms H,,(x) can be calcu- derivatives) along the phase boundary for the spherical lated explicitly for these cases. In the mean-field model [g].

approximation they are all positive, linear functions

of x for x > - x,. For the spherical model, the Acknowledgements. - We are grateful to G.-S. Joyce features pertinent to our work are as follows : and M.-F. Sykes for helpful dicusssion.

References

[I] GRIFFITHS (R. B.), Phys. Rev., 1967, 158, 176. [6] GAUNT (D. S.) and BAKER (G. ^A.,Jr), Phys. Rev. B., 121 FISHER (M. E.), Rep. Prog. Plzys., 1967, 30, 615. 1970, 1, 1184.

[3] GAUNT ^(D.s.) and DOMB (C.), J. P h ~ s . C., 1970,3, 1442. 171 D ~ M B (c.) and JOYCE (G. S.), to be published.

[4] DOMB (C.) and HUNTER (D. L.), Proc. Phys. Soc.,

1965, 86, 1147. [8] COOPER (M. J.) and GREEN (M. S.), Phys. Rev., 1968, [5] Proceedings of the Summer School on ^ccCritical 176, 302.

Phenomena I), Varenna, Italy, July 1970. [9] JOYCE (G. S.), Phys. Rev., 1966, 146, 349.