Free energy lattice

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]... 

Bruce A D, Wilding N B and Ackland G J 1997 Free energies of crystalline solids a lattice-switch Monte-Carlo method Phys. Rev. Lett. 79 3002-5... 

Watson G W, P Tschaufeser, A Wall, R A Jackson and S C Parker 1997. Lattice Energy and Free Energy Minimisation Techniques. Computer Modelling in Inorganic Crystallography. San Diego, Academic Press, pp. 55-81. 

In pure and stoichiometric compounds, intrinsic defects are formed for energetic reasons. Intrinsic ionic conduction, or creation of thermal vacancies by Frenkel, ie, vacancy plus interstitial lattice defects, or by Schottky, cation and anion vacancies, mechanisms can be expressed in terms of an equilibrium constant and, therefore, as a free energy for the formation of defects, If the ion is to jump into a normally occupied lattice site, a term for... 

The solution for the discretized model of the continuous functional is obtained with a certain accuracy which depends on the value of the lattice spacing h and the number of points N. The accuracy of our results is checked by calculating the free energy and the surface area of (r) = 0 for a few different sizes of the lattice. The calculation of the free energy is done with sufficient accuracy for N = 129, which results in over 2 million points per unit cell. The calculation of the surface area of (r) = 0 is sufficiently accurate even for a smaller lattice size. 

The reason for the formation of a lattice can be the isotropic repulsive force between the atoms in some simple models for the crystalhzation of metals, where the densely packed structure has the lowest free energy. Alternatively, directed bonds often arise in organic materials or semiconductors, allowing for more complicated lattice structures. Ultimately, quantum-mechanical effects are responsible for the arrangements of atoms in the regular arrays of a crystal. 

Crystals have spatially preferred directions relative to their internal lattice structure with consequences for orientation-dependent physico-chemical properties i.e., they are anisotropic. This anisotropy is the reason for the typical formation of flat facetted faces. For the configuration of the facets the so-called Wullf theorem [20] was formulated as in a crystal in equihbrium the distances of the facets from the centre of the crystal are proportional to their surface free energies. ... 

In the PPF, the first factor Pi describes the statistical average of non-correlated spin fiip events over entire lattice points, and the second factor P2 is the conventional thermal activation factor. Hence, the product of P and P2 corresponds to the Boltzmann factor in the free energy and gives the probability that on<= of the paths specified by a set of path variables occurs. The third factor P3 characterizes the PPM. One may see the similarity with the configurational entropy term of the CVM (see eq.(5)), which gives the multiplicity, i.e. the number of equivalent states. In a similar sense, P can be viewed as the number of equivalent paths, i.e. the degrees of freedom of the microscopic evolution from one state to another. As was pointed out in the Introduction section, mathematical representation of P3 depends on the mechanism of elementary kinetics. It is noted that eqs.(8)-(10) are valid only for a spin kinetics. 

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