Unit cells translational periodicity

From Figure 3.2, a crystal emerges as a virtually infinite array of identical unit cells that repeat in three-dimensional space in a completely periodic manner. Like a simple sine wave in one dimension, it repeats itself identically after a period of a, b, or c along each of the three axes. A crystal is in fact a three-dimensional periodic function in space, a three-dimensional wave. The period of the wave in each direction is one unit cell translation, and the value of the function at any point xj, yj, Zj within the cell, or period, is the density of electrons at that point, which we designate p(xj, yj, Zj). 

Fig. 2. Depiction of conformal mapping of graphene lattice to [4,3] nanotube. B denotes [4,3] lattice vector that transforms to circumference of nanotube, and H transforms into the helical operator yielding the minimum unit cell size under helical symmetry. The numerals indicate the ordering of the helical steps necessary to obtain one-dimensional translation periodicity.

We refer to the planes which close the box parallel to the boundary as the terminating planes. Between the terminating planes, the box includes one or more translational unit cells of the system parallel to the boundary. We therefore have to assume that the system is periodic parallel to the boundary, but this assumption is anyway necessary for practical schemes of calculation. The terminating planes should be far enough away from the interface to be locally in the undistorted bulk material. 

The SCF method for molecules has been extended into the Crystal Orbital (CO) method for systems with ID- or 3D- translational periodicityiMi). The CO method is in fact the band theory method of solid state theory applied in the spirit of molecular orbital methods. It is used to obtain the band structure as a means to explain the conductivity in these materials, and we have done so in our study of polyacetylene. There are however some difficulties associated with the use of the CO method to describe impurities or defects in polymers. The periodicity assumed in the CO formalism implies that impurities have the same periodicity. Thus the unit cell on which the translational periodicity is applied must be chosen carefully in such a way that the repeating impurities do not interact. In general this requirement implies that the unit cell be very large, a feature which results in extremely demanding computations and thus hinders the use of the CO method for the study of impurities. 

The structure of a vapor-quenched alloy may be either crystalline, in which the periodicity of the unit cell is repeated within the crystallites, or amorphous, in which there is no translational periodicity even over a distance of several lattice spacings. Mader (64) has given the following criteria for the formation of an amorphous structure the equilibrium diagram must show limited terminal solubilities of the two components, and a size difference of greater than 10% should exist between the component atoms. A ball model simulation experiment has been used to illustrate the effects of size difference and rate of deposition on the structure of quench-cooled alloy films (68). Concentrated alloys of Cu-Ag (35-65%... 

In passing from the first to the second problem, a feature of importance should be borne in mind. The periods of the orientational structure (3.1.9) can exceed those of the basic Bravais sublattice, Ai, A2. If this is the case, the unit cell A, A2 should be enlarged so that conditions (3.1.10) can be met and translations onto the new vectors R can reproduce the orientations of adsorbed molecules. Then the excitation Hamiltonian (3.1.3) can be represented in the Fourier form with respect to the wave-vector K as... 

Molecules, in general, have some nontrivial symmetry which simplifies mathematical analysis of the vibrational spectrum. Even when this is not the case, the number of atoms is often sufficiently small that brute force numerical solution using a digital computer provides the information wanted. Of course, crystals have translational symmetry between unit cells, and other elements of symmetry within a unit cell. For such a periodic structure the Hamiltonian matrix has a recurrent pattern, so the problem of calculating its eigenvectors and eigenvalues can be reduced to one associated with a much smaller matrix (i.e. much smaller than 3N X 3N where N is the number of atoms in the crystal). 

According to Eq. (1.16), the elastic coherent X-ray scattering amplitude is the Fourier transform of the electron density in the crystal. The crystal is a three-dimensional periodic function described by the convolution of the unit cell density and the periodic translation lattice. For an infinitely extended lattice,... 

Surfaces possess periodicity in two dimensions. Hence, two base vectors are sufficient for describing the periodic structure of a crystal surface. This does not imply that a surface must be flat, as even the unit cells of simple surfaces such as fee (110) and bcc (111) have three-dimensional structure on the atomic scale. However, to construct a surface from this unit cell by translation, one only needs two vectors. 

Returning to the surface, let s assume a specific surface plane cleaved out, frozen in geometry, from the bulk. That piece of solid is periodic in two dimensions, semi-infinite, and aperiodic in the direction perpendicular to the surface. Half of infinity is much more painful to deal with than all of infinity because translational symmetry is lost in that third dimension. And that symmetry is essential in simplifying the problem—one doesn t want to be diagonalizing matrices of the degree of Avogadro s number with translational symmetry and the apparatus of the theory of group representations, one can reduce the problem to the size of the number of orbitals in the unit cell. 

Discrete translations on a lattice. A periodic lattice structure allows all possible translations to be understood as ending in a confined space known as the unit cell, exemplified in one dimension by the clock dial. In order to generate a three-dimensional lattice, parallel displacements of the unit cell in three dimensions must generate a space-filling object, commonly known as a crystal. To ensure that an arbitrary displacement starts and ends in the same unit cell it is necessary to identify opposite points in the surface of the cell. A general translation through the surface then re-enters the unit cell from the opposite side. 

Since a crystal stractrrre corrstitutes a regular repetition of a rrrrit of structure, the unit cell, we may say that a crystal stracture is periodic in three dimerrsions. The periodicity of a crystal stractrrre may be represented by a point lattice in three dimensions. This is an array of points that is invariant to all the translations that leave the crystal stractrrre invariant and to no others. We shall find the lattice useful in deriving the conditiorrs for x-ray diffraction. 

In this equation, N is equal to the number of unit cells in the crystal. Note how the function in Eq. 5.27 is the same as that of Eq. 5.19 for cyclic tt molecules, if a new index is defined ask = liij/Na. Bloch sums are simply symmetry-adapted linear combinations of atomic orbitals. However, whereas the exponential term in Eq. 5.19 is the character of the yth irreducible representation of the cychc group to which the molecule belongs, in Eq. 5.27 the exponential term is related to the character of the Mi irreducible representation of the cychc group of infinite order (Albright, 1985). This, in turn, may be replaced with the infinite linear translation group because of the periodic boundary conditions. It turns out that SALCs for any system with translational symmetry are con-stmcted in this same manner. Thus, as with cychc tt systems, there should never be a need to use the projection operators referred to earher to generate a Bloch sum. 

In the layered misfit structures each layer set A and B can be described in terms of three basic translations, i.e. by its own component lattice. [The existence of the third vector is contingent upon a strict sequence in the layer stacking. If this is absent, the two three-dimensional subcells/lattices will, in the following discussion, be replaced by two two-dimensional subcells, i.e. by submeshes (nets) built only on the intralayer vectors.] In normal layered structures the unit cells of A and B are commensurate, i.e. their unit vectors are commensurable and the periodicity of the entire structure may be described in terms of a single unit cell. In contrast, we deal with those less-frequent cases in which this is not so at least one of the basic periodicities of A and of B are incommensurate. Then the component unit cell of set A has at least one intralayer unit cell parameter which is not commensurable with the corresponding parameter of set B. Such structures have two incommensurate, interpenetrating, component lattices and can be defined as composite) layered structures with two incommensurate component unit cells. Intermediate cases, in which the nodes of the two component lattices coincide at relatively large... 

Sohds are just infinite molecules. Thus, at least in principle, there is no reason why their electronic structures could not be studied using the same techniques and concepts currently used for molecules. However, there is an obvious technical problem because the size of the matrices to be handled is, in fact, infinite In the case of periodic solids this problem can be easily circumvented by fully exploiting their translational syimnetry. Once this is done, the size of the matrices is reduced to that of the unit cell and the problem disappears. A specially simple and useful technique developed by solid-state physicists is the so-called tight-binding method see Tight-binding Approximation), in which the wavefimctions describing the motion of an electron in the solid (i.e. the Crystal... 

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