# Primitive Translations in Diamond Lattice

The Diamond Lattice (e.g. here) is an fcc-lattice. For an fcc-lattice, one set of primitive translation vectors could be

$$\left\{ \frac{a}{2}\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}, \frac{a}{2}\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \frac{a}{2}\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} \right\}.$$

However, this set of vectors must be slightly modified for the Diamond Lattice, and I'm not quite sure how. Any hints would be appreciated!

One question I am asking myself is: If I translated the crystal by the vector $$\frac{a}{4}\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix},$$ would it be invariant under this translation (I cannot imagine it right now)?

• Diamond cubic is an fcc lattice with a two atom basis. Jul 4, 2020 at 16:05

$$\begin{eqnarray} \mathbf{a}_1=\frac{a}{2}(1,1,0), \\ \mathbf{a}_2=\frac{a}{2}(1,0,1), \\ \mathbf{a}_3=\frac{a}{2}(0,1,1), \end{eqnarray}$$ then you can place the atoms at the following positions to obtain the diamond structure:
$$\begin{eqnarray} (0,0,0), \\ \frac{a}{4}(1,1,1). \\ \end{eqnarray}$$
The vector $$\mathbf{v}=\frac{a}{4}(1,1,1)$$ is the vector connecting the two atoms in the basis, but is not a translation vector of the lattice. For the latter, you need a linear combination of the lattice vectors.