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

The static structure factor $S\left(\overrightarrow{\mathrm{q}}\right)$ 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 $L^2$, it is sufficient to use the continuum approximation for the two finite lattice sums involved in $S\left(\overrightarrow{\mathrm{q}}\right)$ for a $\overrightarrow{\mathrm{q}}$ very close to a 2D reciprocal-lattice vector $\overrightarrow{\mathrm{K}}$. We also use the bulk phonon spectrum of an isotropic 2D elastic medium but with a lower cutoff at $k=\frac{\pi }{L}$. This is justified by a detailed study of the displacement-displacement correlation function for a square lattice with fixed boundaries. The expression for $S\left(\overrightarrow{\mathrm{q}}\right)$ 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 $|\overrightarrow{\mathrm{q}}-\overrightarrow{\mathrm{K}}|\gtrsim \frac{2\pi }{L}$. We also discuss several spherically symmetric approximations which further reduce the expression for $S\left(\overrightarrow{\mathrm{q}}\sim \overrightarrow{\mathrm{K}}\right)$ to a one-dimensional integral, such as recently suggested by Dutta and Sinha.