Reciprocal Lattice

Also known as reciprocal-space or $k$-space

Lattice

A lattice ${\mathbb{R}}^n$ is a infinite set of point generated by linear combinations of basis vectors with integer coefficients. If you have linearly independent vectors $n\_1\overrightarrow{a\_1}+...+n\_d\overrightarrow{a\_d}|n\_i\in {\mathbb{R}}^n$ then the lattice is

$$L=(n\_a\overrightarrow{a\_1}+...+n\_d\overrightarrow{a\_d}|n\_i\in \mathbb{Z})$$

Dual Lattice

The dual lattice is the set of vectors whose dot product with every lattice vector is an integer. If $L=a\mathbb{Z}$ then the dual is $L^{\text{*}}=\frac{1}{a}\mathbb{Z}$

If $L$ is generated by $\overrightarrow{a\_1},...$ then the dual lattice $L^{\text{*}}$ is generated by vectors $\overrightarrow{b\_1},...$ such that

$$\overrightarrow{a\_i}\cdot \overrightarrow{b\_j}=\delta \_ij$$

Where $\delta$ is called Kronecker delta and defined as basically the identity for components.

$$\delta \_ij=\{\begin{array}{ll}1& \text{if }i=j\\ 0& \text{if }i\ne j\end{array}$$