Crystal distortion energy

As a validation tool, one may also measure the crystal distortion energy using the force fleld (11,28,30). This energy must be low, and certainly much lower than the total contribution from the packing forces. If the calculated energy of an error-free, nonionic crystal structure is high relative to the global minimum, the force fleld may be deflcient. 

In crystals with the LI2 structure (the fcc-based ordered structure), there exist three independent elastic constants-in the contracted notation, Cn, C12 and 044. A set of three independent ab initio total-energy calculations (i.e. total energy as a function of strain) is required to determine these elastic constants. We have determined the bulk modulus, Cii, and C44 from distortion energies associated with uniform hydrostatic pressure, uniaxial strain and pure shear strain, respectively. The shear moduli for the 001 plane along the [100] direction and for the 110 plane along the [110] direction, are G ooi = G44 and G no = (Cu — G12), respectively. The shear anisotropy factor, A = provides a measure of the degree of anisotropy of the electronic charge... 

The solution of these problems has been performed by calculation and analysis of crystal free energy (1). The alloy free energy has been found by the configurational method within the approximation of the pair interaction of nearest atoms without considering geometric distortion of lattice. This structure B8i has two types of tetrahedral T2,T4 interstitial sites, defined by different type site surroundings, and one type of octahedral 02 interstitial site... 

Comparison of Solid-Solubilities. The limit of solid miscibility has been related to the energy of distortion of the crystal lattice when atoms of a second component are introduced into the lattice Scott (6) and Lawson (11) have expressed the distortion energy as a function of the molal volumes of the two components. Both authors recognized that the solubility of small atoms in a lattice of large atoms is greater than the... 

The formation of banded textures in thin-film samples of solutions of hquid crystalline polymers (LCPs), subjected to shear, has been reported in the literature since 1979 [15]. Because of the symmetrical properties of the liquid crystal solutions, large domains of weU-oriented polymer chains are formed during shear flow, while defects are squeezed into small regions. The shear accounts for an additional energy stored in the solution. When the shear is stopped, the system will first relax with a characteristic time fb to a transient state. In this state the distortion energy is minimized, and the orientational order is kept, resulting in a banded stmcture. This behavior is observed only if two conditions are fulfilled [16] ... 

The elastic constants are the second derivatives of the energy with respect to the crystal distortions. Since the coefficient matrix, C, contains this information, it may also be used to compute the elastic constants. 

In the case of liquid crystal films, for which the orientation of molecules is fixed at the walls, the distortion energy of the molecular axis can dominate over the van der Waals energy, and spinodal dewetting can take place in thicker films, which makes it easier to observe (see refercncc ). 

Brooks [10] assumed the crystal is isotropic and considered the formation of a vacancy as an equivalent to creating new surface, equal to the area of one unit cell, being approximately the spherical surface of the atomic volume. He also assumed that the surface tension of the hole would shrink the vacancy size by distorting the rest of the crystal elastically. Then, the for atomic vacancy formation inside a bulk solid equals the minimum of the sum of the increased surface energy and distortion energy,... 

Generally speaking, we can divide liquid ciystalline phases into two distinctly different types the ordered and the disordered. For the ordered phase, the theoretical framework invoked for describing the physical properties of liquid crystals is closer in form to that pertaining to solids it is often called elastic continuum theory. In this case various terms and definitions typical of solid materials (e.g., elastic constant, distortion energy, torque, etc.) are commonly used. Nevertheless, the interesting fact about liquid crystals is that in such an ordered phase they still possess many properties typical of liquids. In particular, they flow like liquids and thus require hydrody-namical theories for their complete description. These are explained in further detail in the next chapter. 

The uncertainties in choice of potential function and in how to approximate the surface distortion contribution combine to make the calculated surface energies of ionic crystals rather uncertain. Some results are given in Table VII-2, but comparison between the various references cited will yield major discrepancies. Experimental verification is difficult (see Section VII-5). Qualitatively, one expects the surface energy of a solid to be distinctly higher than the surface tension of the liquid and, for example, the value of 212 ergs/cm for (100)... 

As for crystals, tire elasticity of smectic and columnar phases is analysed in tenns of displacements of tire lattice witli respect to the undistorted state, described by tire field u(r). This represents tire distortion of tire layers in a smectic phase and, tluis, u(r) is a one-dimensional vector (conventionally defined along z), whereas tire columnar phase is two dimensional, so tliat u(r) is also. The symmetry of a smectic A phase leads to an elastic free energy density of tire fonn [86]... 

Figure C2.16.6. The energy states of a metastable and bistable muonium in Si are illustrated in a configuration diagram. It plots the defect energy as a function of a coordinate which combines position and all the relaxations and distortions of the crystal. The specific example, discussed in the text, illustrates acceptor and donor levels, metastability, bistability and negative- U [50] behaviour.

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