Analytic form for the static structure factor for a finite two-dimensional harmonic lattice

The static structure factor S(K) 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(K) 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.