The surface scaling theory previously presented by the authors is developed further, and derived heuristically from a cluster model. Monte Carlo calculations are carried out to obtain the spatial and temperature dependence of the magnetization in Ising and Heisenberg systems with free surfaces. The exponent ${\beta }_1$ of the (surface) layer magnetization is shown to agree with the scaling value (${\beta }_1\approx \frac{2}{3}$) previously derived. In the Heisenberg system, the results at low temperature agree with a spin-wave calculation by Mills and Maradudin. Ising models with modified exchange $J_s=J(1+\Delta )\ne J$ on the surface are considered, both in mean-field theory and by means of high-temperature-series expansions. The critical value ${\Delta }_c$ for surface ordering is found from the series to be 0.6, compared to the mean-field value of 0.25. For $\Delta >{\Delta }_c$ there is a temperature region in which the surface behaves like a bulk two-dimensional Ising model near its phase transition. The critical exponents experience a crossover at $\Delta ={\Delta }_c$, which is reflected in poorly behaved series, and effective exponents differing from the true ones for $\Delta \lesssim {\Delta }_c$. In the case of weakened surface exchange ($0<\mathrm{}{J}_s<J$), the layer magnetization is shown to fit a linear temperature dependence over a large temperature range below $T_c$, thus providing a possible explanation for previous experiments. For sufficiently strong negative $J_s$, mean-field theory predicts that the surface will order antiferromagnetically while the bulk is ferromagnetic.

• Received 10 August 1973

DOI:https://doi.org/10.1103/PhysRevB.9.2194