# First-principles Calculations of Atomic Diffusion in Crystalline Solids
*Written by Cheng-Wei Lee* (clee2 \[at] mines \[dot] edu)
## Atomic diffusion mechanisms
There are multiple textbook diffusion mechanisms and the vacancy-mediated diffusion is one of the most common diffusion mechanisms in ionic compounds. The associated diffusion coefficient can be written as:
$$
D=\frac{1}{6}fZ\_{f}Z\_{m}l^{2}C\_{D}\Gamma
$$
where $$\frac{1}{6}$$ is for 3D system ( $$\frac{1}{2n}$$ and $$n=3$$) and $$f$$, $$Z\_{f}$$, and $$Z\_{m}$$ are geometry factors, which depend on the crystal structure. $$l$$ is the distance the vacancy travel for each jump, $$C\_{D}$$ is the concentration of participating defects, e.g. vacancies. Lastly, $$\Gamma$$ is the successful jump frequency,
$$
\Gamma = \nu^{\*}e^{-\frac{\Delta E\_{m}}{k\mathrm{\_{B}}T}}
$$
where $$\nu^{\*}$$ is the attempted jump frequency, $$\Delta E\_{m}$$ is the migration barrier of a diffusing atom (see figure below)
A schematic for vacancy-mediated diffusion in crystalline solids. Dashed open circles indicate vacancy sites.
Therefore, the diffusion coefficient for vacancy-mediated diffusion in crystalline solids can be calculated from first principles:
1. The attempt frequency can be approximated by the local dynamic matrix of the moving atom and the force due to smaller change in moving atom can be calculated using DFT.
2. The defect concentration can be calculated using the supercell approach (see defect calculations)
3. The migration barrier can be calculated using [nudged elastic band method ](https://3d-materials-lab.gitbook.io/3dmaterialslab-tutorials/dynamics-in-crystalline-solids/broken-reference)
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