5. Accurate Cell Dimensions

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Having found the unit cell, for many purposes in the study of alloys we need to know the cell dimensions as accurately as possible. It is fortunate that such accuracy is possible with remarkably few precautions. For very sharp lines, the separation of an $\alpha$ doublet ( $K\alpha_1$ and $K\alpha_2$ ) can be measured to about 1 per cent, and since the separation itself is about 0.25% of the wavelength, an accuracy of about 25 ppm should be attainable. This arises because the accuracy depends upon sin $\theta$ , which varies very slowly near $\theta$ = 90 $^{\circ}$ ; it is for this reason that lines with $\theta$ near to 90 $^{\circ}$ should be observable (Section 2).

We can derive the value of a cubic crystal from any line, by means of eq. (5). The answer would not however turn out to be constant because of certain systematic errors, but the error, as we have seen, will be less for lines with high $\theta$ . We can make use of this phenomenon by finding the values of a₀ from all the lines in a photograph and extrapolating the results to $\theta$ = 90 $^{\circ}$ . The only problem is to decide what function of $\theta$ should be used to give a straight-line extrapolation.

Several functions - $\theta$ itself, $\sin^{2}\theta$ , $\sin^2 \theta/\cos\theta$ - have been suggested, but it is now generally accepted that the Nelson-Riley function ( $\cos^2 \theta/\theta + cos^2 \theta/\sin \theta$ ) gives the best results. It is tabulated by Lipson and Steeple. A typical example is given in Fig. 5. It cannot be too strongly emphasized that at least one line should have $\theta$ > 80 $^{\circ}$ ; extrapolation over a large range is unreliable. It may even be necessary to find a $K\alpha$ radiation that gives such a line.

**Figure 5:** Fig. 6.3 (p. 167) from Lipson and Steeple. Extrapolation method for lattice parameter of aluminium at 298 $^{\circ}$ C. (Note that 1 pm (picometre) = 0.01 Å.)
$\begin{figure} \includegraphics {fig5.ps} \end{figure}$

Cubic crystals are the easiest to deal with. For lower symmetries extra problems arise. Complete rules cannot be given here, but basically the principle is to use lines with high h index for a, high k index for b, and high l index for c. Complete details of suggested procedures are given by Lipson and Steeple.