Analytic Form for the Static Structure Factor for a Finite Two-Dimensional Harmonic Lattice

The static structure factor $S\left(\overrightarrow{\mathrm{K}}\right)$ for a two-dimensional harmonic lattice of finite size $L$ is expressed analytically. Although one consequence of finite size is the absence of very-long-wavelength phonons, we find that the explicit introduction of a phonon cutoff has very little effect. The structure factor shows an universal behavior for all $L$, differing only by scale factors: $S\left(\overrightarrow{\mathrm{K}}\right)$ always has the infinite-size form far from a Bragg point, but is always rounded off close to the Bragg point. Implications for the interpretation of experimental results are discussed.