Reciprocal lattice¶

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 \(\vec{G}\cdot\vec{R} = 2\pi n\), where n is an integer and is defined as: \(k_1n_1 + k_2n_2 + k_3n_3\). 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 \(\vec{b_1} = 2\pi \frac{\vec{a_2} \times \vec{a_3}}{V_{\text{cell}}}\) and \(\vec{b_2}\) and \(\vec{b_3}\) can be obtained from cyclic permutation of 1, 2 and 3.