Deduction of lattice centering and translational symmetry elements from systemic absences

3 Deduction of lattice centering and translational symmetry elements from systemic absences 

Systematic absences (or extinctions) in the X-ray diffraction pattern of a single crystal are caused by the presence of lattice centering and translational symmetry elements, namely screw axes and glide planes. Such extinctions are extremely useful in deducing the space group of an unknown crystal. 

In Fig. 9.4.1, the planes (100), (110), and (111) are drawn in heavy shading for pairs of adjoining primitive (P), body-centered (/), and face-centered (F) unit cells. In the /-lattice, the centered points lie midway between the (100) and (111) planes, and their contribution to X-ray scattering exactly cancels out that due to points located at the corners of the unit cells. These two reflections are therefore systematically absent. However, the reflections (200), (110), and (222) are observable since each passes through all the lattice points. Similarly, in the / -lattice, the (100) and (110) reflections are systematically absent, but the (200), (220), and (111) reflections are observable. 

For subsequent discussion of the general case, we make use of the following theorem from geometry  

Consider the C-centered unit cell shown in Fig. 9.4.2. Setting OL = a + b, the line OL, which runs through the C-centered point, is divided into 5 parts by the planes (320). Thus the points located at the comers of the unit cell lie in the (320) set of planes, but the C-centered point does not. It is easily seen that, in general, the C-centered point lies in the (hkl) planes when (h + k) is even, and in between the planes when (h+k) is odd. In the latter case, X-ray scattering by the points at the corners of the unit cell is exactly out of phase with that caused by the C-centered point in other words, reflections (hkl) are systematically absent with (h+k) odd. Conversely, whenever this kind of systematic absences is observed, the presence of a C-centered lattice is indicated. 

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