Incoherent neutron scattering functions for diffusion inside two concentric spheres

We consider the diffusion of a particle inside two concentric spheres as a model for diffusing motions inside and between two semipermeable subregions or sublayers of a confining region. Analytic expressions for the elastic incoherent structure factor (EISF) and the incoherent scattering correlation function, C(Q,t), are derived for this model. It is found that, as a result of interferences between regions, the EISF is a weighted coherent summation over amplitudes corresponding to subregions and the relaxation rate of C(Q,t) has a turnover behavior with a minimum at Q(m). In addition to the single exponential behavior for Q/=Q(m). The rate of decay at short times is related to diffusion over length scales under consideration while at long times it is given by the relaxation to equilibrium in fluctuations of population between subregions.