Translation Invariance Properties in a Crystal

What is the main difference between studying the electronic structure of a molecule in the gas phase and in condensed phase In the gas phase, because of the low density and large kinetic energies, molecules interact only during collisions, which may promote them to an excited state. However, either before or after such a brief collision, a molecule is essentially not influenced by the other molecules. Thus, as far as we are not interested in molecular dynamics and thermodynamic properties, the electronic structure in the ground or in any excited state can be studied for only one isolated molecule. 

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The solid under study with a periodic program is infinite and translation invariant it is a perfect crystal. Despite that no real crystal is a perfect crystal, this model is suitable in most cases, and indeed, experimental evidence of crystal periodicity exists in x-ray, neutron, and electron diffraction patterns, which are hardly affected by the presence of the surface, unless the experiment is done in special conditions. Translation invariance has a series of interesting properties with important consequences on simplification of the problem and the implementation of efficient algorithms. 

A real crystal is a finite macroscopic object made of a finite, although extremely large, number of atoms. However, the ratio of the number of atoms at the surface to the total number of atoms in the crystal, N, is very small, and proportional to. When N is large and the surface is neutral, the perturbation caused by the presence of the boundary is limited to only a few surface layers and, therefore, has no influence on the bulk properties. For this reason, a macroscopic crystal mostly exhibits properties and features of the bulk material, and unless attention is deliberately focused onto the crystal boundary, surface effects can be thoroughly neglected. If this is the case, the crystallographic model of an infinite and translation-invariant crystal fits in the aim of studying bulk properties. 

The total energy is important and useful to us for answering this question. As discussed, the total energy of an infinite crystal, like in our model, is infinite. Therefore, the total energy per cell is definitely a preferable choice, for it is a finite well-defined property, because of translation invariance. For the sake of clarity, we remind the reader that, although the total energy per cell is defined within the direct lattice context (Eq. [39]), its calculation depends on knowing the density matrix, which in our scheme is obtained from Eq. [38]. 

The crystallization of a liquid is a change of phase in which symmetry is broken a spatially periodic state arises from one that was invariant under translations. A full description of such a phase change would involve three components (1) an understanding of the equilibrium aspects of the transition (at what temperature and pressure it takes place, what changes in thermodynamic properties such as volume and entropy accompany it), (2) a micro-... 

The simplest material symmetry beyond isotropy is cubic symmetry, a property of crystals that possess three fourfold axes of rotational symmetry, the cube axes, and four threefold axes of rotational symmetry, the cube diagonals. Alternatively, cubic symmetry may be described as invariance of material structure under a translation of a certain distance in any of three mutually orthogonal directions these directions are usually identified as the cube axes. Consider a cubic material for which the [100], [010] and [001] cube axes are parallel to the axes of an underlying rectangular a i,0 2,0 3—coordinate system. For this case, it is evident that... 

As we have seen, most liquid crystals have too high a symmetry to be macroscop-ically polar if they obey the n - -n invariance (which all civilized liquid crystals do, that is, all liquid crystal phases that are currently studied and well understood). The highest symmetry allowed is C2 (monoclinic), which may be achieved in materials which are liquid-like at most in two dimensions. Even then external surfaces are required. Generally speaking, a polar liquid crystal tends to use its liquid translational degrees of freedom so as to macroscopical-ly cancel its external field, i.e., achieve some kind of antiferroelectric order. For more liquid-like liquids, piezo-, pyro-, ferro-, and antiferroelectricity are a fortiori ruled out as bulk properties. These phenomena... 

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