The average unit cell is the probability distribution function P(u) of atomic positions calculated against some periodic reference lattices. A Fourier transform of P(u) gives the diffraction pattern. Structural modeling within the statistical approach involves a modeling of the P(u) distribution, which is an object in physical space, and enables including all kinds of structural disorder in a derivation of the structure factor. We will show how this methodology applies for the harmonically modulated crystal (with harmonic modulation and incommensurate scattering vector), including its correspondence to a quasicrystal, and selected decagonal quasicrystals.

The method has no limit in applying it also to such complex systems, like proteins of macromolecules, where the structural disorder appears to play a crucial role, and attempts to use crystallographic methods known for periodic crystals are rather doubtful. We will shortly introduce a possible benefits from applying the statistical method to complex organic systems.

[1] Strzałka, R., Bugański, J., Wolny, J. (2016), Crystals, 6, 104.

[2] Wolny, J., Bugański, I., Strzałka, R. (2015), Philos. Mag., 96, 1344-1359.

[3] Wolny, J., Bugański, I., Strzałka, R. (2018), Cryst. Rev., 25, 22-64.Keywords: structure modeling, statistical method, complex systems