Can a periodic boundary model reproduce the longer-range density fluctuations in a real amorphous material?

Abstract Whereas the conventional definition of the static structure factor, S(Q), means that, for any sample or structural model, its value at zero Q, S(0), is identically equal to zero, the structures of ideally-disordered materials, such as single-phase liquids and amorphous solids, incorporate long-range density fluctuations that are characterised by a non-zero limiting value (S0) of S(Q ≠ 0) as Q → 0. An analysis of these density fluctuations in terms of their Fourier components leads to the definition of an ideally-disordered material as one that exhibits a continuous, isotropic distribution of Fourier wavelengths, A(Λ), that decays asymptotically to zero at Λ = ∞. On the other hand, a similar analysis for a periodic boundary model reveals that the form of the intermediate-range order at higher inter-atomic distances, r, and that of the long-range density fluctuations are fundamentally different from those of a real amorphous material. The severely limited number of (especially the longer) allowed Fourier wavelengths, Λ, coupled with their strictly defined orientations within the unit cell of a periodic boundary model, means that such a model is inherently crystalline, and that no amount of orientational (polycrystalline) averaging can overcome this problem. The various methods of deriving S(Q) for both periodic-boundary and cluster models are discussed, and it is shown that, since a periodic boundary model is not ideally-disordered, a polycrystalline average does not yield a consistent value for S0, but one that is dependent on its exact method of calculation. It is therefore concluded that, to investigate the longer-range density fluctuations in amorphous materials, it is essential to employ a cluster model, rather than one generated with a periodic boundary.