Critical Phenomena in Films and Surfaces

The theory of critical phenomena in films (and general systems of restricted geometry) is reviewed, introducing the critical point shift exponent λ and the rounding or crossover exponent θ=1/ν, which describes the changeover from bulk behavior. The exponents α×, β×, and γ× for the surface corrections to bulk behavior are defined and discussed. The deficiencies of the “extrapolation” length concept, for representing the surface boundary conditions on the order parameter, are explained. Barber's recent scaling theory for surface properties is reformulated in terms of a gap exponent Δ1 for a surface field H1. In a range of exactly soluble models Δ1 equals 12; this is probably always a good approximation. The scaling relation β1=2−α−ν−Δ1 then predicts the critical behavior of the surface order. Comparisons of the theory are made with analytical and numerical work on Ising, Heisenberg, and spherical models and on ideal Bose fluid films. A table of exponents is presented.