Laue classes symmetry

C-centering). Its Laue class was determined to be mmm since the SAED patterns show mmm symmetry. Thus, the IM-5 structure is C-centered orthorhombic. 

Radiation and particles, i.e. x-rays, neutrons and electrons, interact with a crystal in a way that the resulting diffraction pattern is always centrosymmetric, regardless of whether an inversion center is present in the crystal or not. This leads to another classification of crystallographic point groups, called Laue classes. The Laue class defines the symmetry of the diffraction pattern produced by a single crystal, and can be easily inferred from a point group by adding the center of inversion (see Table 1.10). 

If both the crystal and sample symmetry are triclinic, there are (2/ + 1) coefficients for a given value of /. For higher symmetries the number of coefficients is reduced, some coefficients being zero and some being correlated. Before finding the selection rules of the coefficients for all Laue classes, we must... 

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 12.11 Selection rules imposed by the crystal symmetry for the Laue classes 3 and 3m. For 3m there are two distinct situations for m even, at the left-hand side of the vertical bar, and for m odd, at the right-hand side of the bar.

Table 12.12 Selection rules imposed by the crystal symmetry for the Laue classes 6/m and 6/mmm.

The intensities of Friedel pairs will be equal. This will cause the diffraction pattern from a crystal to appear centrosymmetric even for crystals that lack a centre of symmetry. Diffraction is thus a centrosymmetric physical property, which means that the point symmetry of any diffraction pattern will belong to one of the 11 Laue classes, (see Section 4.7). 

The symmetry of the diffraction pattern is known as the Laue class of which there are 11 types (i.e. excluding the nature of the lattice centering). 

The symmetry of the diffraction pattern can be derived from the symmetry of the crystal. The inversion centre (hkt hkl) is introduced as an additional symmetry element because, in the absence of anomalous scattering, Fhke 2= FhHi 2 this is known as Friedel s law. In the presence of anomalous scattering this law is broken (see equations (2.17) and (2.18)). The Laue class is that set of possible diffraction pattern symmetries. There are 11 possible Laue classes, see table 2.3. 

The symmetry of the diflfractogram from a single crystal in principle allows us to classify the crystal according to one of the 11 Laue classes (Section 2.5.7 and 3.7.3). For a CTystal belonging to the Laue class 4/m, the reflections hkl, khl, hkl khl, hkl, khl, hkl, khl are equivalent and hence have the same intensity. The Laue class 4/m is composed of the crystal classes 4,4 and 4/m. The symmetry of a Laue photograph taken with the fourfold axis oriented parallel to the incident X-ray beam is tetragonal, plane group 4. 

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 ... 

Several lattice planes are characterized by the same value ofs = h - -k - -P (Section 3.5.3). The indices hkl given in Table 5.3 represent one of them with all positive indices. If the Laue class is m3m, these planes are symmetry equivalent. Clearly, the diffractograms do not allow the five cubic crystal classes or the two cubic Laue classes to be distinguished. The lattice constants may be calculated by using equation (3.47), (X 2 = sin ... 

The information obtainable from the Laue symmetry is meagre it consists simply in the distinction, between crystal classes, and then only in the more symmetrical systems—cubic, tetragonal, hexagonal, and trigonal (see Table VI). But it is useful in cases in which morphological features do not give clear evidence on this point. 

The statistical methods are valuable because they detect symmetry elements which are not revealed by a consideration of absent reflections, or by Laue symmetry5. In principle, it is possible to distinguish between all the crystal classes (point-groups) by statistical methods in fact, as Rogers (1950) has shown, it is possible by X-ray diffraction methods alone (using absent reflections as well as statistical methods) to distinguish between nearly all the space-groups (see p. 269). 

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. 

Table III. 1. Crystal class, Laue symmetry, and number of possible different orientations of the electric field gradient tensor for the nuclei at the general point position in the crystal lattice...

Crystal class Laue symmetry Number of different orientations of the EFG tensor for the general point position... 

The diffraction pattern of a crystal has its own symmetry (known as Laue symmetry), related to the symmetry of the structure, thus in the absence of systematic errors (particularly absorption), reflections with different, but related, indices should have equal intensities. According to Friedel s law, the diffraction pattern of any crystal has a center of inversion, whether the crystal itself is centrosymmetric or not, that is, reflections with indices hkl and hkl ( Friedel equivalents ) are equal. Therefore Laue symmetry is equal to the point-group symmetry of a crystal plus the inversion center (if it is not already present). There are 11 Laue symmetry classes. For example, if a crystal is monoclinic p 90°), then I hkl) = I hkl) = I hid) = I hkl) I hkl). For an orthorhombic crystal, reflections hkl, hkl, hkl, hid and their Friedel equivalents are equal. If by chance a monoclinic crystal has fi 90°, it can be mistaken for an orthorhombic, but Laue symmetry will show the error. 

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