8.2: Reciprocal lattice vectors

Last updated
Save as PDF

Page ID: 7829

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Every periodic structure has two lattices associated with it. The first is the real space lattice, and this describes the periodic structure. The second is the reciprocal lattice, and this determines how the periodic structure interacts with waves. This section outlines how to find the basis vectors for the reciprocal lattice from the basis vectors of the real space lattice.

Reciprocal lattice vectors, K, are defined by the following condition:

\[e^{i \mathbf{K} \cdot \mathbf{R}}=1\]

where R is a real space lattice vector. Any real lattice vector may be expressed in terms of the lattice basis vectors, a₁, a₂, a₃.

\[\mathbf{R}=c_{1} \mathbf{a}_{1}+c_{2} \mathbf{a}_{2}+c_{3} \mathbf{a}_{3}\]

in which the c_i are integers. The condition on the reciprocal lattice vectors may also be expressed as

\[\mathbf{b}_{1}=\frac{2 \pi \cdot \mathbf{a}_{2} \times \mathbf{a}_{3}}{\left|\mathbf{a}_{1} \cdot \mathbf{a}_{2} \times \mathbf{a}_{3}\right|}\]

\[\mathbf{b}_{2}=\frac{2 \pi \cdot \mathbf{a}_{3} \times \mathbf{a}_{1}}{\left|\mathbf{a}_{1} \cdot \mathbf{a}_{2} \times \mathbf{a}_{3}\right|}\]

\[\mathbf{b}_{3}=\frac{2 \pi \cdot \mathbf{a}_{1} \times \mathbf{a}_{2}}{\left|\mathbf{a}_{1} \cdot \mathbf{a}_{2} \times \mathbf{a}_{3}\right|}\]

Note that b₂ and b₃ are given by cyclic permutations of the expression for b₁ . From this expression it may be seen that the real lattice basis vectors and the reciprocal lattice basis vectors satisfy the following relation:

\[\mathbf{b}_{i} \cdot \mathbf{a}_{j}=2 \pi \delta_{i j}\]

where \(\delta_{i j}\) is the Kronecker delta, which takes the value 1 when i is equal to j, and 0 otherwise. Any reciprocal lattice vector may then be expressed as a linear sum of these reciprocal basis vectors:

\[\mathbf{K}=h \mathbf{b}_{\mathbf{1}}+k \mathbf{b}_{2}+l \mathbf{b}_{3}\]

in which h, k and l are integers. The set of all K vectors defines the reciprocal lattice.