The incoherent dynamical structure factor and the diffusion coefficient of the nearest neighbour (NN) jump diffusion model is studied. Diffusion of a specific particle in a static but random arrangement of background particles is considered. The virtual crystal approximation and coherent potential approximation procedures are used to find solutions. First five exact frequency moments of the spectral density are calculated and compared with those given by these theories. In simple cubic and body-centred cubic lattices four first moments are exactly reproduced by the CPA solution.
Kaski, K., Tahir-Kheli, R. A., & Elliott, R. J. (1982). CPA theory for jump diffusion of particles in periodic lattices. Journal of Physics C: Solid State Physics, 15(2), 209-220. https://doi.org/10.1088/0022-3719/15/2/006