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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 up previous
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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