612 Critical phenomena
universality class by studying its physically simplest members.
We start by introducing a set of critical exponents, known as theprimary expo-
nents, and the properties they characterize. We will illustrate these exponents first
using the gas–liquid critical point we have been discussing; however,we will see very
shortly how these definitions carry over to a different type of physical system in the
same universality class. The first exponent pertains to the behavior of the constant-
volume (or constant-pressure) heat capacity at the critical pressure and density as the
critical temperatureTcis approached from above. Recall that the heat capacity at
constant volume isCV= (∂E/∂T)V=kβ^2 (∂^2 lnQ(N,V,T)/∂β^2 )V. AsT→Tcfrom
above,CVis observed to diverge with the power-law form
CV∼|T−Tc|−α. (16.2.1)The critical exponentα, therefore, characterizes the divergence inCV asT→Tc.
The second exponentγpertains to the divergence of the isothermal compressibility
κT, defined asκT=−(1/V)(∂V/∂P)T(see eqn. (4.7.40)). At the critical pressure and
volume, asT→Tcfrom above, the isothermal compressibility is observed to diverge
following the power law
κT∼|T−Tc|−γ. (16.2.2)
The third exponent characterizes the shape of the critical isotherm near the inflec-
tion point where the density and pressure approach their critical valuesρcandPc. In
particular, it is seen that for values ofPandρnear their critical values atT=Tc,
P−Pc∼(ρ−ρc)δsign(ρ−ρc), (16.2.3)where sign(ρ−ρc) = (ρ−ρc)/|ρ−ρc|is just the sign of (ρ−ρc).
Finally, the fourth exponent refers to the dependence of the differenceρL−ρG
on temperature asTcis approached from below. HereρLandρGrefer to the liquid
and gas density values when the discontinuous change occurs. Is itobserved that this
difference obeys the power law
ρL−ρG∼|Tc−T|β. (16.2.4)α,β,γ,δare the primary critical exponents.
16.3 Magnetic systems and the Ising model
The gas–liquid critical point we have been discussing is not the simplestsystem in
its universality class due to the complexity of the interactions and the complicated
ensemble distribution they generate. The problem would be simplified considerably
if we could restrict the particles to specific points in space, specifically the points
of a regular lattice, and use variables that take on discrete values rather than the
continuous Cartesian positionsr 1 ,...,rNthat characterize liquids and gases. The only
requirements that we place on the simplified system is that it possessa critical point
and that it belong to the same universality class.
Fortunately, it is possible to fulfill these requirements by considering the phe-
nomenon of magnetization (the formation of magnetic ordering in a ferromagnetic