4.2 Vector product of two direct lattice vectors

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4.2 Vector product of two direct lattice vectors

Let us consider two direct lattice vectors:

$\begin{displaymath} \textbf{n}_1 = u_1\textbf{a}+ v_1 \textbf{b} + w_1 \textbf{c... ... \textbf{n}_2 = u_2\textbf{a} + v_2 \textbf{b} + w_2 \textbf{c}\end{displaymath}$

Their vector product is equal to:

$\begin{displaymath} \textbf{n}_1 \wedge \textbf{n}_2 = \left\vert\begin{array} {... ...\ u_2 & v_2\end{array} \right\vert \textbf{a} \wedge \textbf{b}\end{displaymath}$

Using the definition (2.1) of the basic reciprocal vectors, we may write:

$\begin{displaymath} \textbf{n}_1 \wedge \textbf{n}_2 = V \left\vert\begin{array}... ... {cc} u_1 & v_1 \\ u_2 & v_2\end{array} \right\vert \textbf{c*}\end{displaymath}$ (4.1)

This shows that the vector product of two direct lattice vectors is easily expressed in terms of the basic reciprocal vectors.

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