EXPANDED SCALING OF MAGNETS

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Submitted on 1 Jan 1971

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EXPANDED SCALING OF MAGNETS

M. Cooper

To cite this version:

M. Cooper. EXPANDED SCALING OF MAGNETS. Journal de Physique Colloques, 1971, 32 (C1),

pp.C1-349-C1-349. �10.1051/jphyscol:19711118�. �jpa-00213934�

JOURNAL DE PHYSIQUE

Colloque C I , supplLme~zt au no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1 - 349

EXPANDED SCALING OF MAGNETS

M. J. COOPER

National Bureau of Standards Washington, D. C . 20234

R6sum6. - Les proprietes des aimants dans la

critique sont obtenues A partir d'un dkveloppement de l'energie libre. Le formalisme permet une description du phenomkne dans une large region autour du point critique.

Abstract. - The properties of magnets in the critical region are obtained from a suggested scaled expansion of the free-energy function. The formalism allows an expanded description over an enlarged region about the critical point.

The scaling approach to critical phenomena provides a remarkably good description of magnets (also fluids) near their critical point [I]. The representation is limited, however, by the fact that (1) it requires a certain intrinsic symmetry of the thermodynamic variables and (2) it is of an asymptotic form and valid only very close to the actual critical point. Because of the need for a more extensive characterization of the critical region, particularly in the analysis of data, we have proposed [2] a possible extension of thermo- dynamic scaling which overcomes these limitations.

While the scheme suggested is only one of many possible ways of modifying scaling, it does lead to certain interesting physical consequences.

Generalizing the classical thermodynamic analysis, we take the Helmholtz free-energy to be a function of the reduced temperature

(T - Tc)/Tc and a scaled variable x

t/

M 11'8, where M is the magnetiza- tion [3]

The set of functions ( f,(x) ) are to be

well-behaved

in the range min xo <

< ^a, with F(My T ) satisfying the usual criteria of thermodynamic stability.

All of the thermodynamic properties can be obtained directly from F(M, T). In particular, the equation of state H(M, T) is of the form

where

The condition H ( M , T )

- H ( - M, T) requires that all even s terms vanish identically, h,,(x)

0.

Close to the critical point, the behavior is given by the first term in this expression (s

1) and is like that of simple scaling with h,(x) having a zero at The additional terms (s > 1) represent corrections to the asymptotic form and permit an extended descrip- tion over an enlarged region about the critical point.

For example, the spontaneous magnetism (H

0, M

0, t < 0) which to lowest order varies as M - ¹

^IS also contains higher order (odd) powers in I t IS.

The anomalous temperature behavior in the zero- field specific heat (~'FI~T') is characterized by the critical exponent a'

2 - P(6 + 1) below T,. For T > T,, the specific heat varies like Fs(x) t-", where

2 - P(6 + ^s), with s being determined by the minimum value of s for which the coefficient

does not vanish in the limit x

a. This admits the possibility of a

a', such that their difference is an integer multiple of P.

The entire formalism suggested here is also appli- cable to fluid systems (replace M by the density Ap

p - p,) where it leads to an asymmetric descrip- tion of the liquid and gas phases [2].

References

[I]

VKCENTINI-M~SSONI (M.), JOSEPH (R.), GREEN (M.) and Thermodynamic Scaling in the Critical Region

LEVELT SENGERS

M. H.),

Rev., 1970, to be published.

B 1, 2312. [3]

This expansion is exact for the spherical model of

121

COOPER (M. J.), LEVELT SENGERS (3. M. H.) and ferromagnetism where the fdx) are simple poly- GREEN (M. S.),

Expanding Formulation of nomials. COOPER (M. J.) and GREEN (M. S.),

Phys. Rev., 1968,176, 302.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711118