16.3 CRITICAL EXPONENTS

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The critical exponents describe the nature of the singularities in various measurable quantities at the critical point. Six are commonly recognized, denoted by (at this point) a rather dull list of Greek letters: a, /3, y, 8, T/, P. Denote the critical temperature by Tc ' and introduce the quantity

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(16.20)

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We suppose that, in the limit t -+ 0, any thermodynamic quantity can be decomposed into a "regular" part, which remains finite (but not necessary continuous), plus a "singular" part that may be divergent, or have divergent derivatives. The singular part is assumed to be proportional to some power of t, generally fractional. The first four critical exponents are defined as follows: Heat capacity: Order parameter: Susceptibility: Equation of state (t = 0):

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c- Itl- a

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(16.21) (16.22) (16.23) (16.24)

Itl.8 X - Itl- Y

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M - H

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Here - means" has a singular part proportional to." Since the first three relations all refer to a phase transition, it is understood that H = O. The last one, on the other hand, specifically refers to the case H =1= O.

CRITICAL PHENOMENA

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We should keep in mind that these behaviors refer only to the singular part. For example, a = 0 means that the heat capacity has no singular part; but it may still have a finite discontinuity at 1 = O. The definitions above implicitly assume that the singularities are of the same type, whether we approach the critical point from above or from below. This has been borne out both theoretically and experimentally, except for M, which is identically zero above the critical point by definition. Thus obviously (16.22) makes sense only for 1 < O. We shall not bother to make this qualification every time. The last two in the Greek alphabet soup of exponents concern the correlation function, which we shall assume to have the Ornstein-Zernike form

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/-.0

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(16.25)

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Then, v and

are defined as follows: Correlation length: Power-law decay at

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(16.26)

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d - 2

(16.27)

The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of Ir..agnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear c:.nmbinations of them have more fundamental significance, as we shall see later. Only two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Fisher: Rushbrooke: Widom: Josephson:

= v(2 -

(16.28) (16.29) (16.30) (16.31)

a + 2/3 + y = 2

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vd = 2 - a

where, in the last relation, d is the dimensionality of space. Table 16.2 summarizes the experimental values of the critical exponents as well as the results from some theoretical models. We can see that the scaling laws seem to be universal, but the individual exponents show definite deviations from truly universal behavior. The theory based on the renormalization group, as we shall discuss in 18, suggests that systems fall into" universality classes," and that the critical indices are the same only within a universality class. To give actual examples of universality, we show in Fig. 16.2 a plot of the reduced temperature T/~. vs. reduced density nine for eight substances in the gas-liquid coexistence region. The critical data are quite varied, as Table 16.3 shows; but the reduced data points fall on a universal curve. The right branch refers to the liquid phase, and the left branch to the gas phase, and both come together at the critical point. The solid curve is Guggenheim's fit,* which

'E. A. Guggenheim,

Chern. Phys. 13,253 (1945).

SPECIAL TOPICS IN STATISTICAL MECHANICS

Table 16.2 Critical Exponents ,b

Exponent

EXPT

ISING2

ISING3

HEIS3

a P y

a + 2P + y

(P/) - y)/P (2 - 1I)1'/y

(2 - a)/I'd

2 1 1 1

0-0,14 0.32-0.39 LJ-1.4 4-5 0,6-0,7 0,05 2,00 0,0l 0.93 0.08 1,02 0.05

0 1/2 1 3 1/2 0 2 1 1 4/d

0 1/8 7/4 15 1 1/4 2 1 1 1

0,12 0.31 US 5 0,64 0,05 2 1 1 1

-0,14 0.3 1,4 0,7 0,04 2 1 1

"TH, theoretical values (from scaling laws); EXPT, experimental values (from a variety of systems); MFT, mean field theory; ISINGd, Ising model in d dimension; HEIS3, classical Heisenberg model, d = 3, bFor more details and documentation see A. Z. Patashinskii and V. L. Pokrovskii, Fluctuation Theory of Phase Transitions (Pergamon, Oxford, 1979), Table 3, pp. 42-43.

1.00 0.95 0.90 0.85