Reciprocal lattice¶

A reciprocal lattice is regarded as a geometrical abstraction. It is essentially identical to a wave vector k-space.

Definition

The collection of all wave vectors that yield plane waves with a period of the Bravais lattice. Note: any R vector is a possible period of the Bravais lattice.
A collection of vectors $\vec{G}$ satisfying $e^{i \vec{G} \cdot \vec{R}} = 1$ or $\vec{G} \cdot \vec{R} = 2 π n$ , where n is an integer and is defined as: $k_{1} n_{1} + k_{2} n_{2} + k_{3} n_{3}$ . Here $\vec{G}$ is a reciprocal lattice vector which can be defined as $k_{1} \vec{b_{1}} + k_{2} \vec{b_{2}} + k_{3} \vec{b_{3}}$ , where $k_{1}$ , $k_{2}$ and $k_{3}$ are integers.
The reciprocal lattice vector $\vec{G}$ which generates the reciprocal lattice is constructed from the linear combination of the primitive vectors $\vec{b_{1}}$ , $\vec{b_{2}}$ and $\vec{b_{3}}$ , where $\vec{b_{1}} = 2 π \frac{\vec{a_{2}} \times \vec{a_{3}}}{V_{cell}}$ and $\vec{b_{2}}$ and $\vec{b_{3}}$ can be obtained from cyclic permutation of 1, 2 and 3.

Why do we need a reciprocal lattice?

Reciprocal lattice provides a simple geometrical basis for understanding:
1. All things of "wave nature" (like behavior of electron and lattice vibrations in crystals.
2. The geometry of x-ray and electron diffraction patterns.