4.5 Reciprocity of F and I lattices

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4.5 Reciprocity of F and I lattices

Let us consider a face-centered lattice. It is well known that the basic vectors a $^{\prime}$ , b $^{\prime}$ , c $^{\prime}$ , of the elementary cell are given in terms of the vectors a, b, c of the face centered cell by (Fig. 5):

$\begin{displaymath} \textbf{a}^{\prime} = \frac{\textbf{b}+\textbf{c}}{2} \quad ... ...{2} \quad \textbf{c}^{\prime} = \frac{\textbf{a}+\textbf{b}}{2}\end{displaymath}$ (4.4)

In a similar way, the basic vectors a $^{\prime\prime}$ ,b $^{\prime\prime}$ , c $^{\prime\prime}$ of the elementary cell of a body centered lattice are given in terms of the basic vectors of the multiple cell by (Fig. 6):

$\begin{displaymath} \textbf{a}^{\prime\prime} = \frac{-\textbf{a}+\textbf{b}+\te... ...{c}^{\prime\prime} = \frac{\textbf{a}+\textbf{b}-\textbf{c}}{2}\end{displaymath}$ (4.5)

Let us now look for the reciprocal lattice of the face-centered lattice. Its unit cell vectors are given by, using (2.1) and (4.4):

$\begin{displaymath} \textbf{a}^{\prime*} = \left(\frac{(\textbf{a}+\textbf{c})}{... ...(\textbf{a}^{\prime}, \textbf{b}^{\prime}, \textbf{c}^{\prime})\end{displaymath}$ (4.6)

**Figure 4:**
$\begin{figure} \includegraphics {fig5.ps} \end{figure}$

**Figure 5:**
$\begin{figure} \includegraphics {fig6.ps} \end{figure}$

Noting that the face-centered cell is of the fourth order, we find:

$\begin{displaymath} \textbf{a}^{\prime*} = \frac{\textbf{c} \wedge \textbf{b}}{(... ...{a,b,c})}+\frac{\textbf{a} \wedge \textbf{b}}{(\textbf{a,b,c})}\end{displaymath}$

We may thus express a $^{\prime}$ * in terms of the basic vectors of the reciprocal lattice of the lattice of vectors a, b, c:

$\begin{displaymath} \textbf{a}^{\prime*} = -\textbf{a}^* + \textbf{b}^* + \textbf{c}^*\end{displaymath}$

This may also be written:

$\begin{displaymath} \textbf{a}^{\prime*} = \frac{-(2\textbf{a}^*)+(2\textbf{b}^*)+(2\textbf{c}^*)}{2}\end{displaymath}$

This relation shows that the reciprocal lattice of a face-centered lattice is a body centered lattice whose multiple cell is defined by 2a*, 2b*, 2c*. If we index the reciprocal lattice defined by a*, b*, c*, that is the reciprocal lattice of the multiple lattice defined by a, b, c, we find that only the nodes such that

$\begin{displaymath} h + k = 2n \quad k + l = 2n^{\prime} \quad l + h = 2n^{\prime\prime}\end{displaymath}$

belong to the reciprocal lattice of the face-centered lattice. This shows that the only Bragg reflexions on a face-centered lattice have indices which are all of the same parity.

Next: 5. Diffraction Condition in the Reciprocal Up: 4. Crystallographic Calculations Using the Reciprocal Previous: 4.4 Zone axis of two sets

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