Laue class

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). 

Max von Laue (1879-1960). German physicist who was the first to observe and explain the phenomenon of x-ray diffraction in 1912 Laue was awarded Nobel Prize in Physics in 1914 for his discovery of the diffraction of x-rays by crystals . For more information about Laue see http //wvw.nobel.se/physics/laureates/1914/ 

Sir William Henry Bragg (1862-1942). British physicist and mathematician who together with his son William Lawrence Bragg (1890-1971) founded x-ray diffraction science in 1913-1914. Both were awarded Nobel Prize in Physics in 1915 for their services in the 

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. 

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. 

Unique axis is b, which leads to C2/c alternative cell choices are described in other pages. Three projections of the unit cell down b, a, and c (showing the symmetry elements), and a projection down b (showing the general equivalent positions). 

Q is generated from Q by an operation of the second kind (inversion, reflection, and glides) +, — coordinate above or below the plane of projection along the unique b axis. 

Numbering (in parentheses) and location of the operations (identity, twofold rotation, inversion, c glide, translation, and n glide). 

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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. 

Table 2. Crystal Systems, Laue Classes, Non-Centrosymmetric Crystal Classes (Point Groups) and the Occurrence of Enantiomorphism and Optical Activity 31...

Crystal System Laue Class" Non-Centrosymmetric Crystal Classesa,b Enantiomorphism Optical Activity"... 

Laue class Symbol from systematic absences Possible space groups ... 

Crystal System Crystal Class Laue Class Range of Space Groups ... 

The 11 Laue Classes, indicated by +, summarize the preceding classes by introducing an inversion center. Inversion center - no, + yea... 

Table 2.13. Reflection conditions for the orthorhombic crystal system (Laue class mmm).

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... 

Let us now refer to the cubic system. For all the space groups belonging to the Laue class m3 the 24 equivalent reflections are ... 

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... 

Table 12.2 gives the crystal system (x ) and the corresponding direction cosines for all Laue classes. 

Table 12.2 Crystal axes (x ) and the direction cosines (a,) as function of the lattice parameters and Miller indices for all Laue classes d is the interplanar distance, Ijm (c) means monoclinic unique axis c, and 3(i ) denotes trigonal in rhombohedral setting.

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. 

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. 

Selection Rules for all Laue Classes. To find the selection rules for all Laue classes the invariance conditions to rotations are applied to the peak shift weighted by texture (ah(y))Fh(y). As the terms of different I in Equation (118) are independent, the invariance conditions must be applied to every //. 

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.

Table 12.15 Invariant polynomials g for all Laue classes. Additional terms that should be added to mmm to obtain 2jm, etc. are enclosed in square brackets.

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