Mean-field theory of magnetic transitions in semi-infinite Ising models

The semi-infinite Ising model, for S=(1/2) and with an arbitrary number of surface magnetic couplings $J_{\mathrm{mn}}$ different from the bulk J, is solved in the mean-field approximation. Exact expressions for the critical couplings $J_{\mathrm{mn}}$,C leading to a surface Curie temperature $T_{\mathrm{CS}}$ higher than the bulk $T_C$ are obtained. The value for $T_{\mathrm{CS}}$ in the semi-infinite crystal is obtained by means of a continued-fraction method. The model is applied to explain recent experimental results on the (0001) surface of gadolinium.