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3. Measurement of Debye-Scherrer Photographs

The pattern of lines on a photograph (Fig. 2) represents possible values of the Bragg angles $\theta_{hkl}$ which satisfy Bragg's equation.

$\begin{displaymath} \lambda = 2d_{hkl} \sin \theta_{hkl}.\end{displaymath}$ (1)

Here $\lambda$ is the wavelength of the X-rays and d_hkl is the spacing of the planes (hkl). (Note that the usual symbol n is missing; the second order of reflection from the (111) planes is designated 222.) We require to derive the values of $\theta_{hkl}$ from the photograph.

**Figure 2:** Film from powder camera laid flat.
$\begin{figure} \includegraphics {fig2.ps} \end{figure}$

Three methods are illustrated in Fig. 3. For the Bradley-Jay method, the exposed part of the film is limited by knife-edges, corresponding to a Bragg angle, $\theta_{k}$ , of about 85 $^{\circ}$ (Section 2). This angle must be accurately known; it can most conveniently be found by centring the camera on the table of an optical spectrometer and focusing the cross-wire of a fixed microscope on one of the edges; the camera can then be rotated until the other knife edge coincides with the cross-wire. If the angle through which the camera is rotated is $\phi$ , then

$\begin{displaymath} \theta_{k} = 90^\circ - \textstyle{\frac{1}{4}} \phi.\end{displaymath}$ (2)

To find a Bragg angle $\theta$ for any pair of lines, we measure the distance S between the lines and the distance S_k between the knife-edge shadows; then

$\begin{displaymath} \theta = \theta_{k}S/S_{k}.\end{displaymath}$ (3)

The second method - the van-Arkel method - is rather better for high-angle lines; these are closer together than with the Bradley-Jay method and so less influenced by possible irregularities in the film.

We now have

$\begin{displaymath} \theta = 90^{\circ} - \psi S/S_{K}\end{displaymath}$ (4)

where $\psi$ is one quarter of the angle subtended by the knife edges.

**Figure 3:** Fig. 4.11, p. 87 from Lipson and Steeple. Three methods of mounting films. (a) Bradley-Jay, (b) van Arkel, (c) Straumanis.
$\begin{figure} \includegraphics {fig3.ps} \end{figure}$

The third method - the Straumanis method - has the advantage that no calibration is required; the positions on the film corresponding to $\theta$ = 0 $^{\circ}$ and $\theta$ = 90 $^{\circ}$ can be found from the measurements of pairs of lines. This advantage however is somewhat illusory. Calibration need be carried out only once for the other two methods, whereas it has to be carried out each time for the asymmetric method; moreover, the calibration then depends upon the quality of the lines on the film and so extra errors may be introduced.

All three methods take account equally well of changes in film dimensions during processing. It has to be assumed that changes - usually shrinkage - are uniform and this is probably true to a reasonable accuracy. Only the first two methods are recommended - the first for ordinary interpretive work (Section 4) and the second for high accuracy in lattice spacings (Section 5).

Next: 4. Interpretation of Powder Photographs Up: The Study of Metals and Alloys Previous: 2. The Method

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