Microscopic translational symmetry

The detailed information about the symmetry properties of the complete array of atoms or molecules in a crystal can be obtained only from the combined specifications of the symmetry at a lattice point, that is, the point groups modified by the microscopic translational symmetry and the distribution in space (Bravais lattice) of those points. 

A fundamental characteristic of spatially periodic systems is the existence of a group of translational symmetry operations, by means of which the repeating pattern may be brought into self-coincidence. The translational symmetry of the array, expressing its invariance with respect to parallel displacements in different directions is represented by a lattice. This lattice consists of an array of evenly spaced points (Fig. 3-13), such that the structural elements appear the same and in the same orientation when viewed from each and every one of the lattice points. Another important property of spatially periodic arrays is the existence of two characteristic length scales, corresponding to the average microscopic distance between lattice... 

In anthracene crystal as it follows from data collected in Table 3.1 the distance between a, b plane is large in comparison with distance between nearest molecules in this plane. As the result, as we already mentioned in Section 9.1, the interactions between molecules in different planes is smaller than interaction between molecules inside the same plane. This means that anthracene crystal the same as other crystals of the its family have layered structure which we explicitly take into account in microscopical theory of surface states. We will use the same Hamiltonian (2.2) as we used in consideration of bulk states in simplest Heitler-London approximation. However, now we have to take into account that translational symmetry exists only along the surface of crystal which we assume parallel to a, b plane. In an infinite crystal the diagonalization of Hamiltonian leads to two exciton bands Eit2(k), so that the general pattern of levels is the one shown schematically in Fig. 12.4b. 

A more general microscopic model, the crystal model [370], has been formulated for dipole-dipole ET in which an angular class of ions (made up, from the translational symmetry properties of the lattice, of the set of acceptor ions having the same angular orientation with respect to the donor ion) is employed instead of a shell. The model does not include migration between donor ions, or back transfer. The phonon part is not considered, since the model is based upon the distance-dependence of the transfer rate to the angular classes of acceptor ions. The master equation for (EDV-EDV) processes is [370]... 

The non-conserved variable is a broken symmetry variable-, it is the instantaneous position of the Gibbs surface, and it is the translational symmetry in z direction that is broken by the inhomogeneity due to the liquid-vapour interface. In a more microscopic statistical mechanical approach it is related to the number density fluctuation 5p(jf,z,t) as... 

The theory of crystal symmetry and of the periodicity of microscopic structures (translational symmetry) was developed during the 18th and 19th centuries from... 

After the isotropic to nematic transition, the next step towards more ordered mesophas-es is the condensation of SmA order when the continuous translational symmetry is broken along the director. The theoretical description of the N-SmA transition begins with the identification of an order parameter. Following de Gennes and McMillan [1, 16], we notice that the layered structures of a SmA phase is characterized by a periodic modulation of all the microscopic properties along the direction z perpendicular to the layers. The electron density for instance, commonly detected by X-ray scattering can be expanded in Fourier series ... 

Taking into account these symmetry operations together with those corresponding to the translations characteristic of the different lattice types (see Fig. 3.4), it is possible to obtain 230 different combinations corresponding to the 230 space groups which describe the spatial symmetry of the structure on a microscopic... 

How many values of k are there As many as the number of translations in the crystal or, alternatively, as many as there are microscopic unit cells in the macroscopic crystal. So let us say Avogadro s number, give or take a few. There is an energy level for each value of k (actually a degenerate pair of levels for each pair of positive and negative k values. There is an easily proved theorem that E(k) = E( — k). Most representations of E(k) do not give the redundant (- ), but plot ( k ) and label it as E(k)). Also the allowed values of k are equally spaced in the space of k, which is called reciprocal or momentum space. The relationship between k = 2x7 X and momentum derives from the de Broglie relationship X = hip. Remarkably, k is not only a symmetry label and a node counter, but it is also a wave vector, and so measures momentum. 

A local thermodynamic state is determined as elementary volumes at individual points for a nonequilibrium system. These volumes are small such that the substance in them can be treated as homogeneous and contain a sufficient number of molecules for the phenomenological laws to be applicable. This local state shows microscopic reversibility that is the symmetry of all mechanical equations of motion of individual particles with respect to time. In the case of microscopic reversibility for a chemical system, when there are two alternative paths for a simple reversible reaction, and one of these paths is preferred for the backward reaction, the same path must also be preferred for the forward reaction. Onsager s derivation of the reciprocal rules is based on the assnmption of microscopic reversibUity. The reversibility of molecular behavior gives rise to a kind of symmetry in which the transport processes are coupled to each other. Although a thermodynamic system as a whole may not be in equUibrium, the local states may be in local thermodynamic equilibrium, all intensive thermodynamic variables become functions of position and time. The local equilibrium temperature is defined in terms of the average molecular translational kinetic energy within the small local volume. 

However such benefits are obtained at expenses of some additional fabrication procedures. After deposition, organic materials are centrosymmetric on a macroscopic scale and they can not be endowed of second order nonlinear properties. Poling, i.e. the orientation of the microscopic molecular dipoles, is required in order to break this symmetry. One of the major challenges concerns the effective translation of high molecular nonlinearities (/ry3), where /r is the chromophore dipole moment and p is the first molecular hyperpolarizability, into large macroscopic EO activities rss) in poled polymers with high alignment temporal stability [15,16]. 

As was pointed out in Sec. 1 of this chapter the symmetry of the phases will determine the number of independent components of the second rank electric permittivity furthermore the point group symmetry of the phase and the constituent molecules will fix the orientational order parameters that contribute to a microscopic expression for the permittivity. In order to complete the description of the low frequency or static electric permittivity of liquid crystals, it is necessary to consider the additional effects of chirality, and the translational order associated with smectic phases. 

An ideal crystal is characterized by a regular arrangement of its constituent atoms. Its microscopic structure possesses a three-dimensional (3D) translational invariance which means that a crystal can be constructed by an infinite repetition of a unit cell along three independent directions. A surface eliminates this invariance along one of the directions. It retains a symmetry with respect to two-dimensional (2D) translations along the surface itself. 

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