Laue point

If the construction is now performed taking into account the mean inner potential of the crystal, then the center Q of the sphere is determined such that the magnitude of the wavevectors to 0(000) and to G khl) is K. It can be seen from Eq. (4.15) that K>k, so the point Q is somewhere to the left of the Laue point L in Figure 4.1. The Ewald sphere construction with Q as center and radius K is shown in Figure 4.2. The diagram also includes the point L, but the distance between Q and L has been greatly... 

An experimental teclmique that is usefiil for structure studies of biological macromolecules and other crystals with large unit cells uses neither the broad, white , spectrum characteristic of Lane methods nor a sharp, monocliromatic spectrum, but rather a spectral band with AX/X 20%. Because of its relation to the Lane method, this teclmique is called quasi-Laue. It was believed for many years diat the Lane method was not usefiil for structure studies because reflections of different orders would be superposed on the same point of a film or an image plate. It was realized recently, however, that, if there is a definite minimum wavelengdi in the spectral band, more than 80% of all reflections would contain only a single order. Quasi-Laue methods are now used with both neutrons and x-rays, particularly x-rays from synclirotron sources, which give an intense, white spectrum. 

Figure 6 CBED patterns of aluminum oxynitride spinel along the [001] direction. Symmetries in the patterns contributed to the determination of the point group and space group (a) whole pattern showing 1st Laue zone ring and (b) 0th order Laue zone. Both patterns show a fourfold rotation axis and two mirror planes parallel to the axis. (Courtesy of V. P. Dravid)...

Diffraction is usefiil whenever there is a distinct phase relationship between scattering units. The greater the order, the better defined are the diffraction features. For example, the reciprocal lattice of a 3D crystal is a set of points, because three Laue conditions have to be exactly satisfied. The diffraction pattern is a set of sharp spots. If disorder is introduced into the structure, the spots broaden and weaken. Two-dimensional structures give diffraction rods, because only two Laue conditions have to be satisfied. The diffraction pattern is again a set of sharp spots, because the Ewald sphere cuts these rods at precise places. Disorder in the plane broadens the rods and, hence, the diffraction spots in x and y. The existence of streaks, broad spots, and additional diffuse intensity in the pattern is a common... 

Oscillation and Laue photographs were prepared with tetrahedra 1—2 mm. on edge of zunyite from the Zuni Mine. A Laue photograph taken with the incident beam of X-rays normal to (111), reproduced in Fig. 1, shows three planes of symmetry and a three-fold axis, requiring the point-group symmetry of the crystal to be Td, 0, or 0h, of which... 

The Laue data (Table I) contain first-order reflections only from planes with all indices odd. This fact, together with the absence of reflections with mixed indices on oscillation photographs, shows the lattice to be face-centered. Of the two face-centered space groups isomorphous with point group Td, Td and Td, the latter requires that no odd-order reflections occur from planes (khl) with h — lc. The numerous observed... 

Like paraldehyde, metaldehyde can be preserved, and, when freshly prepared, is odourless. It also has no aldehydic properties. On keeping, however, a distinct odour of acetaldehyde becomes evident —a sign that here also an equilibrium is slowly being established. Metaldehyde can be completely depolymerised by heating. Molecular weight determinations (in phenol) show that metaldehyde is tetra-molecular (Hantzsch) the examination of the space lattice of crystals by the method of Laue and Bragg points to the same conclusion (Mark). 

By applying a variant of the extremely powerful convolution theorem stated above, computing the overlap integral of one scalar field (e.g., an electron density), translated by t relative to another scalar field for all possible translations t, simplifies to computing the product of the two Fourier-transformed scalar fields. Furthermore, if periodic boundary conditions can be imposed (artificially), the computation simplifies further to the evaluation of these products at only a discrete set of integral points (Laue vectors) in Fourier space. 

The Laue symmetry of the diffraction pattern is now reduced to the symmetry of the non-centrosymmetric point group to which the crystal belongs (see Table 2 a listing of symmetry-equivalent reflections in the non-centrosymmetric point groups is available26). Under these circumstances, it is possible to determine different manifestations of non-centrosymmetry in a crystal, such as ... 

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

The Friedel pairs are a subset of the so-called Bijvoet pairs, i.e., pairs of any two reflections which are equivalent by Laue symmetry but not by the point group symmetry of the crystal (see Table 2 a full listing of these reflections is available4 1). All Bijvoet pairs may be used for a comparison, but since the measured intensity differences between general Bijvoet pairs are more likely to be affected by systematic error, the conclusion based on their comparison may be less reliable than in the case of Friedel pairs. 

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). 

System Symmetry Symmetry Point Groups (Laue) Symmetry... 

The term diffraction symmetry, which is often called Laue symmetry because it can be most conspicuous in a type of X-ray photograph called a Laue photograph, is applied to those point groups that are recognizably different in the diffraction pattern. 

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... 

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