Diffusion coefficient

As shown in equation for Bravais lattices and for non-Bravais lattices the accelerated decay produced by diffusion in an NRS time spectrum depends on the jump frequency 1/$\tau$. The main result in an usual diffusion investigation is the jump-diffusion model and the diffusion coefficient. The diffusion model can be estimated using the angular dependence of the accelerated decay. To calculate the diffusion coefficient for uncorrelated diffusion one has to use the Einstein equation in the case of one set of equivalent jump vectors $l$ and one jump frequency 1$\tau$ [45]

$$\begin{array}{ll} D=\frac{1}{6}\frac{l^{2}}{\tau} & \textnormal {... ...D=\frac{1}{4}\frac{l^{2}}{\tau} & \textnormal {in 2D} \nonumber \\ \end{array}$$

In the case of more lattice sites an average over all sites with different symmetry is necessary [46,47]

$$D=\frac{1}{6} \sum_{i}\sum_{j}\frac{l_{ij}^{2}c_{j}}{\tau_{ij}}$$ (76)

$l_{ij}$ is the jump vector from a site with symmetry $i$ to a site with symmetry $j$, 1/$\tau_{ij}$ is the corresponding jump frequency and $c_{j}$ is the concentration of the jumping atoms at the site with symmetry $j$.

The temperature dependence in condensed matter is usually expressed using the well known Arrhenius equation [48] predicting an exponential behaviour

$$D=D_{0}e^{-\frac{E_{A}}{kT}}$$ (77)

The activation energy $E_{A}$ contains the contribution from the formation of a vacancy and the energy for the jump of an atom into the vacancy position.