Face-dependent Hamaker constants and surface melting or nonmelting of noncubic crystals

Surface melting is strongly influenced by long-range dispersion forces which crucially control the thickness l of the liquid layer that forms at the crystal surface, through a free-energy term of asymptotic form V(l)=H/${\mathit{l}}^2$. In isotropic crystals, the face-independent Hamaker constant H is usually calculated by a well-known formula by Lifshitz et al. that relates dispersion forces to the optical conductivity of the solid and the liquid. In this paper we generalize this result to anisotropic crystals specializing in uniaxial crystal surfaces normal or parallel to the optical c axis, and to surfaces of orthorhombic biaxial crystals normal to one principal axis. The main result is that the Hamaker constant becomes face dependent, and may, in principle, even change sign from one face to another. Using experimental data for frequency-dependent dielectric constants we have calculated Hamaker constants of the main faces for some low-melting-point anisotropic metals, namely, α-Ga, β-Sn, and Cd. It is found that H is negative and large for all faces of α-Ga, implying surface nonmelting. Values found for different faces of β-Sn are moderately negative, and those for Cd are marginally positive. The search for surface nonmelting or melting on these metals should be interesting in light of these results.