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4.2. Tetragonal and hexagonal structures

For tetragonal and hexagonal structures an extra variable - the axial ratio, c/a - is involved and therefore the problem is more complicated. For tetragonal structures, eq. (5) is replaced by

$$\sin^2 \theta_{hkl} = \frac{\lambda^2}{4a^2} \Big(h^2 + k^2 + \frac{a^2}{c^2}l^2\Big).$$ (6)

Now no precise rules can be given, but the values of sin² $\theta$ may give some hints. For example, if l = 0, the values of sin² $\theta$ are in the ratios 1, 2, 4, 5 $\dots$ corresponding to indices 100, 110, 200, 210$\dots$ If we find a set of values in these ratios we can assume that the indices are as shown, and then we have to find l from the lines that are not in this sequence. The ratio 1:2 particularly should be looked for.

For hexagonal structures, or trigonal structures referred to hexagonal axes, eq. (5) is modified to

$$\sin^2 \theta_{hkl} = \frac{\lambda^2}{4a^2} \Big(h^2 + hk + k^2 + \frac{a^2}{c^2} l^2\Big)$$ (7)

and the dominant factor is 1:3. Again, if some of the simpler lines can be accounted for in this way, trial-and-error can be applied to the others. Graphical and numerical methods have been applied to these problems; a summary is given by Lipson and Steeple (1970).

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