Laue groups

One complication occurs with polar space groups, where all the possibilities must be tried in the TF. If there is an ambiguity in the extinctions, for instance in the Laue group Pmmm, again all possibilities must be searched. This is usually done T>y hand, going through all different possibilities one by one. 

Given the nascent nature of these software tools, SGX developed its own method to evaluate or score the diffraction quality. The SGX system is based on two established software programs, d TREK (Pflugrath, 1999) and Mosflm (Leslie, 1992). These programs index diffraction images to determine the appropriate Laue group. In addition, they provide an analysis of the properties of the... 

A consequence of Neumann s symmetry principle is that direct tensor Onsager coefficients (such as in the diffusivity tensor) must be symmetric. This is equivalent to the addition of a center of symmetry (an inversion center) to a material s point group. Thus, the direct tensor properties of crystalline materials must have one of the point symmetries of the 11 Laue groups. Neumann s principle can impose additional relationships between the diffusivity tensor coefficients Dij in Eq. 4.57. For a hexagonal crystal, the diffusivity tensor in the principal coordinate system has the form... 

As a consequence of Friedel s law, the diffraction pattern exhibits the symmetry of a centrosymmetric crystal class. For example, a crystal in class 2, on account of the 1 symmetry imposed on its diffraction pattern, will appear to be in class 2/m. The same result also holds for crystals in class m. Therefore, it is not possible to distinguish the classes 2, m, and 2/m from their diffraction patterns. The same effect occurs in other crystal systems, so that the 32 crystal classes are classified into only 11 distinct Laue groups according to the symmetry of the diffraction pattern, as shown in Table 9.4.1. 

Knowledge of the diffraction symmetry of a crystal is useful for its classification. If the Laue group is observed to be 4/mmm, the crystal system is tetragonal, the crystal class must be chosen from 422,4mm, 42m, and 4/mmm, and the space group is one of those associated with these four crystallographic point groups. 

Crystal system Crystal class Laue group... 

The number of different orientations of the EFG tensor due to the symmetry operations of the Laue group is lower if the nuclei occupy special crystallographic sites (special point positions). Information about such special sites of nuclei can be found from NQR single-crystal Zeeman spectroscopy. This is... 

For adequate simplicity, we return to consider the case when there is only a single Zeeman splitting term in the spin-Hamiltonian. For a given matrix g, much depends on the relations of its principal values. If all three such values are identical, then of course only one spectral line is observed, at all orientations of n = B/B, where B > 0. Except in that isotropic case, it becomes important what crystal symmetry is at hand there are 11 distinct cases (Laue groups).12... 

The evidence for the existence of screw axis symmetry is manifested in certain subclasses of reflections that are systematically absent. These systematic absences, we will see, fall along axial lines (/tOO, OkO, 00/) in reciprocal space and clearly signal not only whether an axis in real space is a screw axis or a pure rotation axis, but what kind of a screw axis it is, for instance, 4i or 42, 6i or 63. Thus the inherent symmetry of the diffraction pattern (which we call the Laue group), plus the systematic absences, allow us to unambiguously identify (except for a few odd cases) the space group of any crystal. 

What is the symmetry of the reciprocal lattice That is, what are the symmetry operators that relate sets of identical intensities The symmetry group that we observe for a crystal in reciprocal space, namely the diffraction pattern symmetry, is called the Laue symmetry, or Laue group. 

The structure type description contains a character for Laue group and numbers counting the atoms of the components in the primitive cell. The characters are... 

The eombination of the information on the Laue group with the analysis of the systematieally absent reflections allows the determination of the so-called Extinction symbol fES). In the International Tables for Crystallography the list of extinetion symbols is given per erystal system. There are 14 ES for the... 

To give a practical example, Figure 7.3 shows a small 26 interval of the experimental pattern of a P2]/n crystal structure. The three vertical bars, generated in the Laue group P2lm, correspond to the positions of the reflections (20-1), (210) and (201). The reflection (20-1) corresponds to a systematically absent reflection, but its intensity is ambiguous because it is overlapped with (210). 

Selection Rules for all Laue Classes. The selection rules for the harmonic coefficients are derived from the invariance of the pole distribution to the operations of the crystal and sample Laue groups. The invariance conditions are applied to every function t/ (h, y) from Equation (36), as the terms of different / in this equation are independent. If we compare Equations (38) and (39) with (37) we observe that they have an identical structure. On the other side the sample and the crystal coordinate systems were similarly defined. As a consequence the selection rules for the coefficients of", flf", and respectively y) ", resulting from the sample symmetry must be identical with the selection rules for the coefficients A P and resulting from the crystal symmetry, if the sample and the crystal Laue groups are the same. The exception is the case of cylindrical sample symmetry that has no correspondence with the crystal symmetry. In this case, only the coefficients af and y l are different from zero, if they are not forbidden by the crystal symmetry. 

Table 12.3 Selection rules for the non-cubic Laue groups. The symbols A and B designate Af (y) and Bf (y) for the crystal symmetry or of", jS and yT ", for the sample symmetry.

Obviously, Equation (97) is much more convenient than Equation (96) as there are only 36 integrals to calculate in place of 1296. Behnken and Hauk adopted a derivation path starting from the strain components in the sample reference system. In this report the condition of invariance of the peak shift to the operations of the point group is violated for some Laue classes. Later, Popa reported invariant expressions for all Laue groups that were derived starting from Equation (97). Here we follow this derivation. 

As expected, the peak shift is independent of direction in sample. The peak shift must be invariant to the operations of the Laue group that imposes constraints on the strains e,. For crystal symmetries higher than triclinic the peak shift is given in Table 12.8. 

X3 = 5,54 = 55 = 56 = 0, one obtains (sh) = (3Ai + 82/2)3. There is no dependence of (sh) on the Miller indices for the Voigt and the Kroner model. For the Reuss model the dependence is similar to those from Equation (115) and Table 12.8 but with only one refinable parameter for all Laue groups, the macrostress 5, and very probably the refined value of 5 will be wrong. 

Table 12.8 Peak shift (sh) caused by hydrostatic stress in isotropic samples for all Laue groups higher than triclinic. Parameters s, are refinable in the Rietveld codes.

Tables 12.9-12.12 give the selection rules imposed by the crystal symmetry for the non-cubic Laue groups. 

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