Laue classes, “powder

Crystal system Laue class "Powder" Laue class Point groups... 

As seen in Table 1.10, there is one powder Laue class per crystal system, except for the trigonal and hexagonal crystal systems, which share the same powder Laue class, 6/mmm. In other words, not every Laue class can be established from a simple visual analysis of powder diffraction data. This occurs because certain diffraction peaks with potentially different intensities (the property which enables us to differentiate between I ue classes 4/m and 4/mmm 3, 3m, 6/m and 6/mmm m3 and m3m) completely overlap since they are observed at identical Bragg angles. Hence, only Laue classes that differ from one another in the shape of the unit cell (see Table 1.11, below), are ab initio discernible from powder diffraction data without complete structure determination. 

The minimum and maximum values of Miller indices in three dimensions are fully determined by the symmetrically independent fraction of the reciprocal lattice as shown schematically in Figure 5.6 for the six distinguishable powder Laue classes. The same conditions are also listed in Table 5.7. ... 

Both Table 5.7 and Figure 5.6 account for the differences among powder Laue classes, which are distinguishable at this stage, and are suitable for indexing of powder diffraction patterns. For example, in Laue classes 6/m and 4/m ( powder Laue classes 6/mmm and 4/mmm, respectively), the intensities of hkl and khl reflections are different, although the... 

Figure 5.6. Schematic representations of the fractions of the volume of the sphere (r = 1/X) in the reciprocal space in which the list of hkl triplets should be generated in six powder Laue classes to ensure that all symmetrically independent points in the reciprocal lattice have been included in the calculation of Bragg angles using a proper form of Eq. 5.2. The monoclinic crystal system is shown in the alternative setting, i.e. with the unique two-fold axis parallel to c instead of the standard setting, where the two-fold axis is parallel to b. ...

Table 5.7. Symmetrically independent combinations of indices in six "powder" Laue classes.

Powder Laue class Range of indices and limiting fraction of... 

In the case of the powder method, all the symmetry equivalent reflections superimpose because they all have the same spacing (Section 3.5.3). Thus, this method does not allow us to observe the Laue symmetry, only the metric of the unit cell. For the same reason, the rotation method is also poorly adapted for the determination of the symmetry. There are other diffraction methods, which are not described in this work, by which all the reflections may be individually observed without superposition from which it is also possible to determine the Laue class of a crystal and consequently the crystal system. Is it possible to obtain other information about the symmetry and, in particular, the Bravais class and the space group ... 

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