Crystallographic coordinate system

Because of the restrictions imposed on the values of the rotation angles (see Table 1.4), sincp and cos(p in Cartesian basis are 0, 1 or -1 for one, two and four-fold rotations, and they are 1/2 or Vs/2 for three and six-fold rotations. However, when the same rotational transformations are considered in the appropriate crystallographic coordinate system, all matrix elements become equal to 0, -1 or 1. This simplicity (and undeniably, beauty) of the matrix representation of symmetry operations is the result of restrictions imposed by the three-dimensional periodicity of crystal lattice. The presence of rotational symmetry of any other order (e.g. five-fold rotation) will result in the non-integer values of the elements of corresponding matrices in three dimensions. 

The coordinates of oriented rigid body group are then transformed into the crystallographic coordinate system defined by the unit cell axes a,b,c and then the origin of the group is translated to the appropriate location within the unit cell. Given the crystal to Cartesian transformation matrix ... 

At the end of this subsection it should be noted, that having a set of parameters Bpq attached to the crystallographic coordinate system one gets the opportunity (1) to describe the magnetic properties of a crystal at any appropriate value and direction of applied magnetic fields (2) to obtain reasonable estimates of different effects due to electrons phonon interaction with only those model parameters which have been introduced to describe energy spectra in the static equilibrium lattice. 

The values of interionic interaction parameters, obtained from the spectral, calorimetric, and magnetic investigations of different lanthanide compounds, are given in table 11. The relative interionic positions are specified by the vectors ry in the crystallographic coordinate systems. 

To calculate the lattice sums it is advisable to use the crystallographic coordinate system, where the tensors y and diagonal, and we proceed with the... 

Let us have a crystallographic coordinate system Oxyz using the lattice translational vectors ai, aa and as as the base vectors. Arty vector r cormecting the origin O with a lattice pointX at a comer of a nnit cell can be expressed as... 

In many cases, it is necessary to specify directions and planes in a crystal lattice. It is most sensible to do so using a crystallographic coordinate system, with axes parallel to the edges of the chosen unit cell. All parallel directions and planes in a crystal are equivalent, rendering it unnecessary to state the origin of the direction vector or plane. 

For specifying directions and planes in a crystal, the origin of the crystallographic coordinate system is positioned in a lattice point, and the axes are scaled so that the length of every edge of the unit cell is one. Thus, for non-cubic lattice types, this coordinate system is non-Cartesian. 

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. 

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.

Extended-Hiickel calculations have been carried out [185] for systems such as IrCl4(NO)2-, based on a slightly distorted square pyramid of C4v symmetry (crystallographically studied 5-coordinate systems do not have a planar base but exhibit this slight distortion). Figure 2.104 shows how the... 

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

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... 

Let us suppose that we place a polarizing prism between the light source and the sample. If the sample is a single crystal in which all of the molecules have the same orientation relative to the crystallographic axes, we can so orient the crystal that the direction of the electric vector of the light will correspond to the x, y, or z direction in a coordinate system for the molecule. It is then possible that some transition may occur for only one or two of these orientations but not for all three. 

The crystallographer works back and forth between two different coordinate systems. I will review them briefly. The first system (see Fig. 2.4) is the unit cell (real space), where an atom s position is described by its coordinates x,y,z. 

A crystal structure usually is described by the unit cell dimensions, space group and coordinates of the atoms (or orientation and position of the molecules) in the asymmetric unit. This, in fact, is the order in which the information is obtained when a crystal structure is determined by X-ray or neutron diffraction experiments. However, an equivalent way to describe a structure is to place the center of a molecule at the origin of an orthogonal coordinate system and to specify its molecular surroundings. This alternative is especially powerful in crystals with one molecule per asymmetric unit because the orientations of the surrounding molecules are related to the central molecule by crystallographic symmetry. The coordination sphere or environment of the structure then is defined as those surrounding molecules which are in van der Waals contact, or nearly in contact, with the central molecule. 

The thirty-two crystal classes (crystallographic point groups) described in Section 9.1.4 can also be classified into the same seven crystal systems, depending on the most convenient coordinate system used to indicate the location and orientation of their characteristic symmetry elements, as shown in Table 9.2.1. 

The vector that defines the crystallographic direction should be situated in such a way that it passes through the origin of the lattice coordinate system. 

For a crystal having a hexagonal symmetry, a set of four numbers, [uvtw], named the Miller-Bravais coordinate system (see Figure 1.5), is used to describe the crystallographic directions, where the first three numbers, that is, u, v, t, are projections along the axes at, a2, and a3, describing the basal plane of the hexagonal structure, and w is the projection in the z-direction [2,3],... 

---