Finite-size behavior of the simple-cubic Ising lattice

Abstract

A Monte Carlo method is used to study $N\ifmmode\times\else\texttimes\fi{}N\ifmmode\times\else\texttimes\fi{}N$ simple-cubic Ising lattices with periodic boundary conditions and free edges. For both types of boundary conditions the position of the specific-heat maximum varies for large $N$ as $a{N}^{\ensuremath{-}\ensuremath{\lambda}}$, where $\ensuremath{\lambda}$ has the scaling value $\ensuremath{\lambda}={\ensuremath{\nu}}^{\ensuremath{-}1}$. Both the thermal and magnetic properties are shown to obey finite-size scaling. The free-edge data are shown to be consistent with a surface contribution described by the scaling exponents ${\ensuremath{\alpha}}_{s}=\ensuremath{\alpha}+\ensuremath{\nu}$, ${\ensuremath{\beta}}_{s}=\ensuremath{\beta}\ensuremath{-}\ensuremath{\nu}$, ${\ensuremath{\gamma}}_{s}=\ensuremath{\gamma}+\ensuremath{\nu}$. Using the free-edge data we also consider corrections to scaling in the infinite lattice and discuss rounding in real systems in terms of surface contributions from grains.