Cubic crystal symmetry

Because of the orientational freedom, plastic crystals usually crystallize in cubic structures (Table 4.2). It is significant that cubic structures are adopted even when the molecular symmetry is incompatible with the cubic crystal symmetry. For example, t-butyl chloride in the plastic crystalline state has a fee structure even though the isolated molecule has a three-fold rotation axis which is incompatible with the cubic structure. Such apparent discrepancies between the lattice symmetry and molecular symmetry provide clear indications of the rotational disorder in the plastic crystalline state. It should, however, be remarked that molecular rotation in plastic crystals is rarely free rather it appears that there is more than one minimum potential energy configuration which allows the molecules to tumble rapidly from one orientation to another, the different orientations being random in the plastic crystal. 

Because of the cubic crystal symmetry, the fluxes parallel to the directions of the gradients in the three cases (i.e., — gDn, —gD22, and —gD33) must be equal. Therefore,... 

For WZ compounds, we must consider hexagonal symmetry in the effective Hamiltonian. The Luttinger-Kohn Hamiltonian is constructed under the condition of cubic symmetry and the form reflects cubic crystal symmetry. Thus, in the analysis of WZ nitrides, we must use a k-dependent parabolic band for the conduction band state and Bir-Pikus Hamiltonians for the valence band states. The Hamiltonians for the upper six valence bands and the lowest two conduction bands are given by [28]... 

The derivative terms are known as the phonon deformation potentials, and each is a component of a full phonon deformation potential matrix For cubic crystals, symmetry considerations reduce the number of independent phonon deformation potentials to three ... 

In the case of cubic crystal symmetry the T, and F symmetry components are Raman allowed. In fig. 15 the well known vibrations of the octahedra have been included to demonstrate the corresponding symmetry. The peak near 95 cm appears only in F symmetry with the Tj and F components being zero. The symmetry analysis is consistent with the identification of the 95 cm line as due to a crystal-field excitation, but does not allow a separation of the two transitions. 

In fignre A1.3.9 the Brillouin zone for a FCC and a BCC crystal are illustrated. It is a connnon practice to label high-synnnetry point and directions by letters or symbols. For example, the k = 0 point is called the F point. For cubic crystals, there exist 48 symmetry operations and this synnnetry is maintained in the energy bands e.g., E k, k, k is mvariant under sign pennutations of (x,y, z). As such, one need only have knowledge of (k) in Tof the zone to detennine the energy band tlnoughout the zone. The part of the zone which caimot be reduced by synnnetry is called the irreducible Brillouin zone. 

For cubic crystals, which iaclude sUicon, properties described by other than a zero- or a second-rank tensor are anisotropic (17). Thus, ia principle, whether or not a particular property is anisotropic can be predicted. There are some properties, however, for which the tensor rank is not known. In addition, ia very thin crystal sections, the crystal may have two-dimensional characteristics and exhibit a different symmetry from the bulk, three-dimensional crystal (18). Table 4 is a listing of various isotropic and anisotropic sUicon properties. Table 5 gives values for the more common physical properties and for some of the thermodynamic properties. Figure 5 shows some thermal properties. 

FIG. 2. A complex of twenty Friauf polyhedra, with icosahedral symmetry (Samson, Ref. 23). This complex contains 104 atoms, if the central icosahedral position is not occupied. Most of the atoms show approximate icosahedral ligan-cy twenty atoms, at the centers of the Friauf polyhedra, have ligancy 15 or 16. The complex was first identified in Mg32(Al,Zn)<9. In the cubic crystals that form the icosatwins and decatwins these complexes are packed in such a way as to approximate an icosahedral arrangement of twelve complexes about a central one, the structure being similar to that of 0-W. 

The structure factor for the 104-atom complex with almost perfect icosahedral symmetry determines the intensities of the diffraction maxima, in correspondence with the inverse relationship between intensity in reciprocal space and the atom-pair vectors in real space that was discovered fifty years ago by Patterson.27 The icosahedral nature of the clusters in the cubic crystal explains the appearance of the Fibonacci numbers and the golden ratio. 

The growth direction of MgCl2 is independent of the substrate symmetry [93]. Growth on a cubic crystal, namely Pd(lOO), leads to hexagonal LEED pattern. However, the structure of the films is much more complex due to the... 

No ferroelectricity is possible when the dipoles in the crystal compensate each other due to the crystal symmetry. All centrosymmetric, all cubic and a few other crystal classes are... 

Hardness also depends on which face of a non-cubic crystal is being indented. The difference may be large. For a crystal with tetragonal symmetry the face that is normal to the c-axis can be expected to be different from those that are normal to the a-axes. Similarly the basal faces of hexagonal crystals are different from the prism faces. One extreme case is graphite where the resistance to indentation on the basal plane is very different than the resistance on the prism planes. 

In general, CijU is a 9 x 9 tensor with 81 terms, but symmetry reduces this considerably. Thus, for the cubic crystal system, it has only three terms (Cmi, Cm2, and C4444) and for an isotropic material only two terms remain B = bulk modulus and G = shear modulus. A further simplification is that the bulk modulus, B for the cubic system is given by (Cmi + 2Ci2i2)/3, and the two shear moduli are C44 and (Cmi - Ci2i2)/2. 

In crystals of high symmetry, there are often several sets of (hkl) planes that are identical. For example, in a cubic crystal, the (100), (010), and (001) planes are identical in every way. Similarly, in a tetragonal crystal, (110) and (110) planes are identical. Curly brackets, hkl, designate these related planes. Thus, in the cubic system, the symbol 100 represents the three sets of planes (100), (010), and... 

The relationship between directions and planes depends upon the symmetry of the crystal. In cubic crystals (and only cubic crystals), the direction [hkl] is normal to the plane (hkl). A zone is a set of planes, all of which are parallel to a single direction, called the zone axis. The zone axis [mvvv] is perpendicular to the plane (mvvv) in cubic crystals but not in crystals of other symmetry. 

At the time that Mark began x-ray work the crystal structure had not yet been determined for any organic compound. At that time the x-ray techniques could be applied with the greatest prospect of success in determining the complete structure to crystals with high symmetry, especially cubic crystals. 

If, in addition to the cubic crystal field, a component of lower symmetry is present, such as one having tetragonal or trigonal symmetry (as for the Al sites in a-AUOa), further splitting will occur as shown in Fig. 24. Crystal field splittings for other configurations in both the weak and strong field cases are summarized in a review article by Moffitt and Ballhausen... 

If the diffusion medium is isotropic in terms of diffusion, meaning that diffusion coefficient does not depend on direction in the medium, it is called diffusion in an isotropic medium. Otherwise, it is referred to as diffusion in an anisotropic medium. Isotropic diffusion medium includes gas, liquid (such as aqueous solution and silicate melts), glass, and crystalline phases with isometric symmetry (such as spinel and garnet). Anisotropic diffusion medium includes crystalline phases with lower than isometric symmetry. That is, most minerals are diffu-sionally anisotropic. An isotropic medium in terms of diffusion may not be an isotropic medium in terms of other properties. For example, cubic crystals are not isotropic in terms of elastic properties. The diffusion equations that have been presented so far (Equations 3-7 to 3-10) are all for isotropic diffusion medium. 

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