Hopping Directions@

General Discussion@

\(\mathbb{KSOME}\) is an on-lattice, object KMC code. Hence the coordinates of defect locations and hop directions are specified in integer form. The integer coordinate system is expressed as some multiple of materials lattice constant. Integer coordinates are expressed in lattice unit, which is a dimensionless unit. The use of integer instead of real (or floating-point) coordinates was a design choice for the sake of simplicity and, it is possible to modify KSOME to use floating-point coordinates and directions¹. On this page, the use of integer coordinate system for fcc, bcc and hcp lattices is briefly described. Though the hop directions described below follow the crystal structure, it is important to note that there is no underlying lattice structure in the OKMC simulations (similar to the rate-equation theory)². Nevertheless, it is essential to capture any anisotropy in a defect's diffusion or in the host crystal structure to perform radiation damage simulations correctly (or rigorously)

3D Isotropic Diffusion@

If an object's⁴ diffusion is 3D and isotropic, then it can be made to diffuse along the six \(\langle 100 \rangle\) directions regardless of how the object diffuses in the the host material. However, the rates of new hops, especially the prefactor, should be adjusted accordingly such that the object's diffusivity (in Ų/s) remains unchanged.

Example@

A mono-vacancy in a bcc lattice is expected to hop along the eight \(\langle 111 \rangle\) directions i.e. between nearest-neighbor lattice sites. Vacancy hopping rate at a given temperature T between the nearest-neighbor sites is given as \(\Gamma = D_0\exp\big(-\frac{E_m}{k_BT}\big)\), where \(E_m\) and \(D_0\) are the activation energy barrier (in eV) and the prefactor (in s^-1). Then the vacancy's diffusivity (\(D\) in Ų/s) is given as

Now, if diffusion of the mono-vacancy is simulated using hops along the six \(\langle 100\rangle\) directions, then \(z^{\prime} = 6\) and \(l^{\prime} = a_0/2\). , then the diffusivity \((D^{\prime})\) for this case would be

Irrespective of how the object is moves in the OKMC simulation, its diffusivity should be the same in eq. (\(\ref{eq2}\)), i.e. \(D^{\prime} \overset{!}{=} D\). Hence,

As noted earlier, as long as the diffusion occurs via nearest-neighbor hops, the diffusivity of an object both in bcc and fcc lattices can be calculated using eq. (\(\ref{eq2}\)). As such, one can write a general expression for the new prefactor as

If the diffusion of the object is simulated via hops along \(\langle 100 \rangle\) hops with a hop-length that is some multiple (or a fraction) of the lattice-constant, then using eq. (\(\ref{feq}\)) one can write the diffusivity along the \(\langle 100\rangle\) directions as

Cubic Lattices@

For cubic lattices, most often, the obvious choice for the lattice unit is half-of-the-lattice constant (i.e. two lattice units = \(a_0\)) as shown in the figures below³.

HCP (Hexagonal Close Packed)@

Unlike fcc and bcc lattices, hcp lattice is not a Bravais lattice. Hence, the hcp lattice should be converted to a Bravais lattice. In the case of atomistic KMC simulations, the conversion from non-Bravais to Bravais lattice should be done rigorously. However, for OKMC simulations, since there is no underlying lattice, it is not necessary to transform strictly. The primary aspect of the hcp lattice that should be captured is the asymmetry along the c-axis. There are different ways one can incorporate the asymmetry. Below I am describing the easiest approach.

Approach 1@

In this approach, one considers the hcp simulation cell as a stack of 2D triangular lattice (triangular lattice as shown figure 1), with lattice units equal to \(a_0\) along the basal plane and \(c/2\) along the c-axis.

Figure. 1 Mapping of a triangular lattice onto a square lattice (2D)

Within a given 2D-plane, if \((x_{int},y_{int})\) are the integer coordinates of an object on the square lattice, then the corresponding coordinates \((x_{real},y_{real})\) in the triangular lattice are given as \(x_{\text {real }}=a_{0}\left(x_{\text {int }}+\frac{y_{\text {int }}}{2}\right)\) and \(y_{\text {real }}=a_{0}\left(\frac{\sqrt{3}}{2} y_{\text {int }}\right)\). Note that in this case, the lattice unit along the basal plane is equal to \(a_0\).

Figure 2 Mapping of a hcp onto a cubic grid (3D)

With this approach, there will be eight directions for an object to diffuse, six in the basal plane and two to move out the basal plane.

• The six directions along the basal plane are 0 (1,0,0) 1 (1, -1, 0) 2(0,-1,0) 3(-1,0,0) 4(-1, 1, 0) 5 (0,1,0),
• The two directions along the c-axis are 6(0,0,1) 7(0,0,-1).

Note that along the basal plane, integer coordinates should be converted to real coordinates as described above. While along the c-axis, real coordinates can be obtained by multiplying the integer coordinates with \(c/2\) (a post-processing step).

Hopping Rates@

As mentioned earlier, since the object is made to diffuse along eight directions instead of the standard twelve directions, the hopping rates have to be readjusted accordingly to keep the diffusivities the same as in an actual hcp lattice. For a hcp lattice the diffusivities along the c-axis (\(D_{\parallel}\)) and along the basal plane (\(D_{\perp}\)) are given as