Finite-size effects on the static structure factor of two-dimensional crystals

The static structure factor S(→q) of a finite two-dimensional (2D) harmonic crystal is discussed in detail. Our major interest is in the effect of finite size on the characteristic Landau-Peierls power-law Bragg peaks of an infinite crystal. For a square crystal of area L², it is sufficient to use the continuum approximation for the two finite lattice sums involved in S(→q) for a →q very close to a 2D reciprocal-lattice vector →K. We also use the bulk phonon spectrum of an isotropic 2D elastic medium but with a lower cutoff at k=πL. This is justified by a detailed study of the displacement-displacement correlation function for a square lattice with fixed boundaries. The expression for S(→q) is reduced to a simple twofold integral. Numerical calculations show how, as L increases, the power-law behavior at the Bragg positions develops out of the characteristic diffraction pattern of a finite lattice. These results indicate that if experiments are done close to the melting temperature, one can obtain information about the power-law behavior in the asymptotic region |→q-→K|2πL. We also discuss several spherically symmetric approximations which further reduce the expression for S(→q~→K) to a one-dimensional integral, such as recently suggested by Dutta and Sinha.