Crystallographic calculations

Vectors & metric tensors

Vectors

Direct space vectors or directions are denoted by square brackets, e.g., “ $[abc]$ ”; reciprocal space vectors or planes, are denoted by round brackets, e.g., “ $(hkl)$ ”.

Metric tensors

Metric tensor is the dot product of structure vectors, it is useful for calculating vector dot products. By construction, it is a symmetric 3 x 3 tensor.

The direct space metric tensor is given by

$g_{ij} = a^T_{ik} \, a_{kj} = \begin{bmatrix} \mathbf{a}\cdot\mathbf{a} & \mathbf{a}\cdot\mathbf{b} & \mathbf{a}\cdot\mathbf{c} \\ \mathbf{b}\cdot\mathbf{a} & \mathbf{b}\cdot\mathbf{b} & \mathbf{b}\cdot\mathbf{c} \\ \mathbf{c}\cdot\mathbf{a} & \mathbf{c}\cdot\mathbf{b} & \mathbf{c}\cdot\mathbf{c} \\ \end{bmatrix}$

Similarly, we can have reciprocal space metric tensors for reciprocal lattices.

Vector operations

Dot product

For two direct space vectors $\mathbf{r}$ , dot-producting itself is

$\mathbf{r} \cdot \mathbf{r} = r_i \, g_{ij} \, r_j$

and, for two reciprocal vectors $\mathbf{g}^*$ , dot-producting itself is

$\mathbf{g}^* \cdot \mathbf{g}^* = g_i \, g^*_{ij} \, g_j$

where $g$ and $g^*$ are the direct-space and reciprocal space metric tensors, respectively.

Using dot product, we can calculate:

The length of the vector: $||\mathbf{r}|| = \sqrt{\mathbf{r} \cdot \mathbf{r}}$
The angle of the vector
Distance between the planes

Cross product

$\mathbf{g}^* \times \mathbf{h}^* = \mathbf{t}$

The cross product of two direct space vectors is a reciprocal space vector; and the dot product of two reciprocal space vectors is a direct space vector.