Equilibrium crystal shapes for lattice models with nearest-and next-nearest-neighbor interactions

Equilibrium crystal shapes for a three-dimensional ferromagnetic Ising model with both nearest-neighbor ($J_1$) and next-nearest-neighbor ($J_2=R{J}_1$) interactions are studied at nonzero temperatures. Phase diagrams and crystal shapes are first calculated via mean-field theory. Subsequently, fluctuation corrections are taken into account in a qualitative manner, incorporating known results and exploiting interconnections with other ($d=2\mathrm{}$) models, including both roughening and commensurate-incommensurate phase transitions. In the resulting picture, crystal facets appear only below appropriate roughening temperatures. Phase boundaries correspond directly to edges bounding crystal facets and may be either first order (slope discontinuity, sharp edges) or second order (no slope discontinuity, smooth edges). For $R\ge 0$, only smooth edges occur, and phase transitions are of the Gruber-Mullins—Pokrovsky-Talapov type. For $R<\mathrm{}0\mathrm{}$ additional, first-order phase transitions take place at sufficiently low temperatures.