Coordinates crystal coordinate system

Figure 6-3. Top Structure of the T6 single crystal unit cell. The a, b, and c crystallographic axes are indicated. Molecule 1 is arbitrarily chosen, whilst the numbering of the other molecules follows the application of the factor group symmetry operations as discussed in the text. Bottom direction cosines between the molecular axes L, M, N and the orthogonal crystal coordinate system a, b, c. The a axis is orthogonal to the b monoclinic axis.

A sinusoidal plot of grf>2 vs.

crystal plane gives another set of Ks that depend on other combinations of the gy, eventually enough data are obtained to determine the six independent values of gy (g is a symmetric matrix so that gy = gy,). The g-matrix is then diagonalized to obtain the principal values and the transformation matrix, elements of which are the direction cosines of the g-matrix principal axes relative to the crystal axes. An analogous treatment of the effective hyperfine coupling constants leads to the principal values of the A2-matrix and the orientation of its principal axes in the crystal coordinate system. 

Figure 5. The relationship between molecular and crystal coordinate systems for a unit cell containing two molecules.

Three additional concepts may be introduced by means of Fig. 2A. First, all measurements of the diffracted beams are made relative to the incident x-ray beam so that the diffracted beam leaves the crystal at an angle 20, as illustrated. Thus, while the diffraction condition is determined by the angle between the incident and diffracted beams relative to a lattice plane, the measured position of the diffracted beam is determined relative to the incident beam. Second, there is a reciprocal relation between the spacing between lattice planes and the positions of the diffraction spots small lattice spacings give diffraction spots with large values of 20. This leads to two types of space. The crystal coordinate system is in real space, whereas the diffrac-... 

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. 

Since the applied field is fixed in the laboratory, the components in the crystal coordinate system will be Hz = Ho cos 0, H = Ho sin 0 cos <(>, and H, = Ho sin 0 sin , where 0 and are the usual spherical coordinate angles. Thus, for NMR in a single crystal of a metallic solid for which the Knight shift tensor has axial symmetry, the dependence of the resonance frequency v on crystal orientation with respect to the laboratory field Ho will be given by [Abragam (I960)] ... 

The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of derived quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities. 

Maximum information is obtained by making Raman measurements on oriented, transparent single crystals. The essentials of the experiment are sketched in Figure 3. The crystal is aligned with the crystallographic axes parallel to a laboratory coordinate system defined by the directions of the laser beam and the scattered beam. A useful shorthand for describing the orientational relations (the Porto notation) is illustrated in Figure 3 as z(xz) y. The first symbol is the direction of the laser beam the second symbol is the polarization direction of the laser beam the third symbol is the polarization direction of the scattered beam and the fourth symbol is the direction of the scattered beam, all with respect to the laboratory coordinate system. 

For the alkali metal doped Cgo compounds, charge transfer of one electron per M atom to the Cgo molecule occurs, resulting in M+ ions at the tetrahedral and/or octahedral symmetry interstices of the cubic Cgo host structure. For the composition MaCgg, the resulting metallic crystal has basically the fee structure (see Fig. 2). Within this structure the alkali metal ions can sit on either tetragonal symmetry (1/4,1/4,1/4) sites, which are twice as numerous as the octahedral (l/2,0,0) sites (referenced to a simple cubic coordinate system). The electron-poor alkali metal ions tend to lie adjacent to a C=C double... 

To find the equilibrium form of a crystal, the following Wullf construction [20] can be used, which will be explained here, for simplicity, in two dimensions. Set the centre of the crystal at the origin of a polar coordinate system r,6. The radius r is assumed proportional to the surface tension 7( ), where 6 defines the angle between the coordinate system of the crystal lattice and the normal direction of a point at the surface. The anisotropy here is given through the angular dependence. A cubic crystal, for example, shows in a two-dimensional cut a clover-leaf shape for 7( ). Now draw everywhere on this graph the normals to the radius vector r = The... 

S is the ionic spin and D, ID, and a are crystal field parameters describing the strength of the axial, the rhombohedral, and the cubic crystal field terms, respectively. The coordinate system of the cubic crystal field (ij, rj, g) may differ from that used to describe the axial and the rhombohedral crystal field interactions (x,y,z). 

The magnitude of k corresponds to a wave number 2n/X and therefore is measured with a unit of reciprocal length. For this reason k is said to be a vector in a reciprocal space or k space . This is a space in a mathematical sense, i.e. it is concerned with vectors in a coordinate system, the axes of which serve to plot kx, ky and kz. The directions of the axes run perpendicular to the delimiting faces of the unit cell of the crystal. 

The region within which k is considered (—n/a first Brillouin zone. In the coordinate system of k space it is a polyhedron. The faces of the first Brillouin zone are oriented perpendicular to the directions from one atom to the equivalent atoms in the adjacent unit cells. The distance of a face from the origin of the k coordinate system is n/s, s being the distance between the atoms. The first Brillouin zone for a cubic-primitive crystal lattice is shown in Fig. 10.11 the symbols commonly given to certain points of the Brillouin zone are labeled. The Brillouin zone consists of a very large number of small cells, one for each electronic state. 

Crystal family Symbol Crystal system Crystallographic point groups (crystal classes) Number of space groups Conventional coordinate system Bravais lattices... 

The unit cell content. To complete the description of the crystal structure, the list of the atoms contained in the unit cell and their coordinates (fractional coordinates related to the adopted system and unit cell edges) are then reported. These are usually presented in a format such as M El in n x, y, z. In the MoSi2 structure, also reported in Table 3.2, and in Fig. 3.7, for instance, four silicon atoms... 

The ambiguity involved in assigning the absolute configuration of a chiral molecule in a chiral crystal is presented in Scheme 1. Scheme la depicts a chiral molecule of, say, configuration S, with individual atomic coordinates — x — y - z, (i = 1,. . . , n, for n atoms) in a crystal axial system a,b,c. Scheme 1 b represents the enantiomeric crystal structure containing a molecule of configuration... 

In the orthorhombic point group mm2 there is an ambiguity in the sense of the polar axis c. Conventional X-ray diffraction does not allow one to differentiate, with respect to a chosen coordinate system, between the mm2 structures of Schemes 15a and b (these two structures are, in fact, related by a rotation of 180° about the a or c axis) and therefore to fix the orientation and chirality of the enantiomers with respect to the crystal faces. Nevertheless, by determining which polar end of a given crystal (e.g., face hkl or hkl) is affected by an appropriate additive, it is possible to fix the absolute sense of the polar c axis and so the absolute structure with respect to this axis. Subsequently, the absolute configuration of a chiral resolved additive may be assigned depending on which faces of the enantiotopic pair [e.g., (hkl) and (hkl) or (hkl) and (hkl)] are affected. 

The zone axis coordinate system can be used for specifying the diffraction geometry the incident beam direction and crystal orientation. In this coordinate, an incident beam of wavevector K is specified by its tangential component on x-y plan = k x + k y, and its diffracted beam at Kt+gt, for small angle scatterings. For each point inside the CBED disk of g, the intensity is given by... 

The multipole formalism described by Stewart (1976) deviates from Eq. (3.35) in several respects. It is a deformation density formalism in which the deformation from the IAM density is described by multipole functions with Slater-type radial dependence, without the K-type expansion and contraction of the valence shell. While Eq. (3.35) is commonly applied using local atomic coordinate systems to facilitate the introduction of chemical constraints (chapter 4), Stewart s formalism has been encoded using a single crystal-coordinate system. 

Many molecules contain chemically equivalent atoms, which, though in a different crystal environment, have, to a good approximation, the same electron distribution. Such atoms may be linked, provided equivalent local coordinate systems are used in defining the multipoles. In particular, for the weakly scattering hydrogen atoms, abundant in most organic molecules, this procedure can lead to more precisely determined population parameters. 

We demonstrate the use of local coordinate systems with the molecule of tetrasulfur tetranitride, S4N4, (Fig. 4.2) as an example. It occupies a general position in its crystal s space group, with one molecule in the asymmetric unit. Thus, there are eight crystallographically independent atoms. If multipoles up to and including the hexadecapoles are included, the number of population parameters... 

In typical organic crystals, molecular pairs are easily sorted out and ab initio methods that work for gas-phase dimers can be applied to the analysis of molecular dimers in the crystal coordination sphere. The entire lattice energy can then be approximated as a sum of pairwise molecule-molecule interactions examples are crystals of benzene [40], alloxan [41], and of more complex aziridine molecules [42]. This obviously neglects cooperative and, in general, many-body effects, which seem less important in hard closed-shell systems. The positive side of this approach is that molecular coordination spheres in crystals can be dissected and bonding factors can be better analyzed, as examples in the next few sections will show. 

Figure 1.19 Definition of a coordinate system for crystal structures.

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