Crystallographic axes

In order to give the point group designation of this crystal, it is necessary to identify all of the symmetry elements in the crystal and to understand the relationship of these elements to the crystallographic axes. 

4/m32/m. This point group has the greatest number of synunetry elements of any of the isometric classes. The class with the most synunetry is sometimes referred to as the holohedral class (from the Greek holos meaning complete). 

In most of the other crystal systems, the axis with the highest order rotational symmetry is designated the z-axis, which is more commonly called the c-axis. The order of this axis is given first in the point group designation. In the monoclinic systan, the -axis is taken as the two-fold axis or as perpendicular to the mirror plane. 

In the oithoihombic system, the two-fold axes are taken as the crystallographic axes. If there is only one two-fold axis it is designated the c-axis. The symmetry elements are listed in order of the three axes—a, b, c. 

The Structure of Some Simple Closest-packed Compounds 

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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.

Here, we have arranged the layers on a two-dimensional structure, even though the layers are arranged in three dimensional order. Note that only two crystallographic axes are indicated. We call this the natural stacking sequence because of the nature of the hexagonal close- packed lattice. 

If a volume expansion is required, then mccisurements in three simultaneous dimensions are needed, a result experimentally difficult to achieve, to say the least. Even a slab of a single crystal does not completely solve the problem since thermal expansion in three dimensions is needed for the volume thermal expansion coefficient. The crystal has three (3) crystallographic axes and may have three (3) linear coefficients of expansion. Only if the crystal is cubic does one have the case where all three values of ol are equal. 

The crystal structures of two compounds are isotypic if their atoms are distributed in a like manner and if they have the same symmetry. One of them can be generated from the other if atoms of an element are substituted by atoms of another element without changing their positions in the crystal structure. The absolute values of the lattice dimensions and the interatomic distances may differ, and small variations are permitted for the atomic coordinates. The angles between the crystallographic axes and the relative lattice dimensions (axes ratios) must be similar. Two isotypic structures exhibit a one-to-one relation for all atomic positions and have coincident geometric conditions. If, in addition, the chemical bonding conditions are also similar, then the structures also are crystal-chemical isotypic. 

In crystalline solids, the Raman effect deals with phonons instead of molecular vibration, and it depends upon the crystal symmetry whether a phonon is Raman active or not. For each class of crystal symmetry it is possible to calculate which phonons are Raman active for a given direction of the incident and scattered light with respect to the crystallographic axes of the specimen. A table has been derived (Loudon, 1964, 1965) which presents the form of the scattering tensor for each of the 32 crystal classes, which is particularly useful in the interpretation of the Raman spectra of crystalline samples. 

Figure 18 Schematic model of a iPP and y iPP branching from a parent lamellae. The crystallographic axes are indicated. Adapted from similar schemes in Refs. [245,246], with permission of Elsevier copyright 2004.

Since the surfaces of crystals have specific symmetries (usually triangular, square, or tetragonal) and indenters have cylindrical, triangular, square, or tetragonal symmetries, the symmetries rarely match, or are rotationally misaligned. Therefore, the indentations are often anisotropic. Also, the surface symmetries of crystals vary with their orientations relative to the crystallographic axes. A result is that crystals cannot be fully characterized by single hardness numbers. 

Figure 3. HRTEM images along three main crystallographic axes. The inserts (from left to right) show their corresponding Fourier transforms and images with PI and Cmcm symmetry imposed, respectively.

A good example of a higher-than-actual symmetry is provided by hexamethyl-benzenetricarbonylchromium. In the crystal structure the threefold axes of the Cr(CO)3 groups are almost parallel both to each other and to one of the (symmetry determined) crystallographic axes (Fig. 6)65 It follows that the dipole moment... 

Plane and direction indices. Reference to the three coordinate axes (crystallographic axes) must also be made, in order to indicate (to identify) the position of a plane and the direction of a line. 

A plane is identified by its equation or by three parameters. These are related to the plane intercepts on the three chosen crystallographic axes (z, y, z) with reference to their own unit lengths (a, b and c). The three indices, generically defined by the three letters h, k and l are defined by the ratios ... 

A certain anisotropy of the refractive index along specific crystallographic axes indicates that the microstructures in the porous network are not spherical but somewhat elongated along the PS growth direction [Mi4], This birefringence is below 1% for micro PS, while it may reach values in the order of 10% for meso PS films formed on (110) oriented silicon wafers [Ko22]. 

For the higher moments, s, depends on the location of the subunit i. But in the case of the net charge, all s, are equal, provided the subunits have identical shape. When the subunits are parallelepipeds with edges of length 2dx, 25y, and 2dz parallel to the crystallographic axes, as in Fig. 6.4, the shape transform of the subunit, s0(H), is of a particularly simple form (Weiss 1966, Coppens and Hamilton 1968) ... 

Treating diffusion along each principal axis is hence relatively simple. The principal axes coincide with the crystallographic axes for crystals with at least orthorhombic symmetry (Nye, 1985). To further simplify, define... 

Fig. 12 The composite incommensurate structure of Sc-II, as viewed down the c axis. The eight-atom host framework is shown in grey, and the ID guest chains are shown in black. The insets show perspective views of (a) the body-centred guest structure of Fujihisa et al. and (b) the C-centred guest stmcture of McMahon et al. The crystallographic axes are labelled...

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