Translational symmetry, of crystal

For a centrosymmetric structure, Uhki can equal only 0 or 180° (i.e. F having signs + or respectively), for a noncentrosymmetric structure it can have any value. Owing to the translational symmetry of crystals, p xyz) can be considered as a periodic function in three-dimensional space and expressed by the Fourier transform of equation (8),... 

Point symmetry or individual molecules Is to be contrasted with the translational symmetry of crystals, to be discussed later in this chapter. 

In a real disordered matrix, there usually are several TLS s able to interact with a single molecule, as schematically shown in Fig. 1. The first case is that of defects in crystals, where a non-random distribution of flipping TLS s in space interacts with the probe. Dislocations or grain boundaries can give rise to such one- or two-dimensional distributions of TLS s (see Section 1.4.2). Because of the translational symmetry of crystals, we may assume in this case that all TLS s have the same... 

Translation Symmetry of Crystals. Point Symmetry of Bravais Lattices. Crystal Class... 

An alternative to the cluster approach are the band methods which directly take into account the translational symmetry of crystals, a feature which provides, in the case of moderately complicated crystal structures, a higher accuracy in the calculation of the electronic structure and physico-chemical characteristics of crystals. The band methods make it possible to determine a larger (compared to the cluster approach) set of the crystal electronic structure characteristics, e.g., the topology of the Fermi surface, or the energy levels in crystals as a function of wave vectors (dispersion curves). 

Brillouin-zone, of the thiee-dim isioiial crystal are lequiied. A dynamical theory of crystal lattices was established by Bom and Huang (1954) crystal vibrations were treated with Cartesian synunetry-coordinates whidi were constructed with respect to the translational-symmetry of crystal lattices. 

Crystallographic point groups are symmetry point groups which are compatible with the translational symmetry of crystal structures. The conditions imposed by translational symmetry are so restrictive that there are only 32 different crystallographic point groups (Table 5). Thus proper or improper rotation axes of orders one, two, three, four, or six are the only types of rotation axes allowed in crystal structures. Thus a true three-dimensional crystal lattice cannot have fivefold, sevenfold, or eightfold rotation axes. Ordered structures with fivefold or eightfold sytrunetry have been found in quasicrystals, discussed in Section 9. 

A. Pullet, J.-P. Matie Vibration spectra and symmetry of crystals (translation in to Russian) Mir, Moscow, 1974. 

In the above relation, quantum states of phonons are characterized by the surface-parallel wave vector kg, whereas the rest of quantum numbers are indicated by a the latter account for the polarization of a quasi-particle and its motion in the surface-normal direction, and also implicitly reflect the arrangement of atoms in the crystal unit cell. A convenient representation like this allows us to immediately take advantage of the translational symmetry of the system in the surface-parallel direction so as to define an arbitrary Cartesian projection (onto the a axis) for the... 

Landau (26) proposed that an additive electron in a dielectric can be trapped by polarization of the dielectric medium induced by the electron itself. Applying the model to electrons in the conduction band of an ionic crystal is rather complicated since the translational symmetry of the solid must be considered and the interaction of the excess electron with the lattice vibrations must be treated properly (I, 13, 14). 

The case B h 20 is one of the most important for molecular crystals. One expects to recover the vibronic molecular states satisfying the translational symmetry of the crystal (see Fig. 3u,c). In Section II.B, we shall examine their perturbation due to the presence of two-particle degenerate states. 

The working substance of the ITEP-group /9 source was a doubly tritiated crystalline amino acid, DL-valine (Mallikarjunan and Rao, 1969), which is a molecular crystal. As a result of decay, during the time 10 18 sec the valine molecule transforms into the corresponding helium-containing ion (RHe)+. Since the translational symmetry of the crystal is disturbed, no exciton states are produced, and only molecular states of the complex (RHe)+ are excited. Since in molecular crystals the intermolecular interactions are considerably weaker than the intramolecular ones, one can neglect the influence of the valine crystalline surrounding on the /S-decay process and consider only an isolated valine molecule. 

The translational symmetry of a (one-dimensional) crystal lattice demands that the potential in the crystal Hamiltonian... 

Hanoman, 1961 and Lander, Gobeli, and Morrison, 1963) this lowering of symmetry is called lecoiistniction. For the ideal unreconstructed crystal it is not difficult to construct primitive surface translations (sec Section 3-C for discussion of primitive translations), which are the smallest translations in the plane of the surface by which each atom is replaced by another atom. Let two of these translations be written and T2 Low-Energy Electron Diffraction (LHED) measurements tell what the actual translational symmetry of a surface is and frequently give a lower symmetry. For example, it may indicate primitive translations of 2T[ and T2 this is called a 2 x 1 reconstruction. Similarly, an observed translational symmetry of T, and mtj is called an n x m reconstruction. 

We should now relate what we have found in this chapter to the energy-band description introduced initially in Section 2-A the crystal used there also had the translational symmetry of the simple cube. There we also defined wave numbers for the states but restricted their domain to a Brillouin Zone similarly, in Section... 

Hartree-Fock calculations on molecules commonly exploit the symmetry of the molecular point group to simplify calculations such studies on perfectly ordered bulk crystalline solids are possible if one exploits the translational symmetry of the crystalline lattice (see Ashcroft and Mermin, 1976) as well as the local symmetry of the unit cell. From orbitals centered on various nuclei within the unit cell of the crystal Bloch orbitals are generated, as given by the formula (in one dimension) ... 

Going from the molecular picture to that of collective properties in a solid means adding translational symmetry to the point group symmetry. The theoretical description does this by introducing a phase of the distortion throughout the material, which is determined by the spatial variation of the variously distorted molecules. If, as is usual in a classical crystal, the phase of the distortion shows the translational symmetry of the solid, the so-called cooperative Jahn-Teller effect appears where the shape of one molecule and the space group determines the shape of all the others. If the distortions are not correlated, however, the phase is random and the situation is not different from that of isolated molecules. This is the dynamic Jahn-Teller effect where the distortions cannot be detected but the solid-state consequences still appear in the electronic structure [16]. 

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