Cell, Wigner-Seitz

A guide to tire stabilities of inter-metallic compounds can be obtained from the semi-empirical model of Miedema et al. (loc. cit.), in which the heat of interaction between two elements is determined by a contribution arising from the difference in work functions, A0, of tire elements, which leads to an exothermic contribution, and tire difference in the electron concentration at tire periphery of the atoms, A w, which leads to an endothermic contribution. The latter term is referred to in metal physics as the concentration of electrons at the periphery of the Wigner-Seitz cell which contains the nucleus and elecUonic structure of each metal atom within the atomic volume in the metallic state. This term is also closely related to tire bulk modulus of each element. The work function difference is very similar to the electronegativity difference. The equation which is used in tire Miedema treatment to... 

Van der Woude and Miedema [335] have proposed a model for the interpretation of the isomer shift of Ru, lr, Pt, and Au in transition metal alloys. The proposed isomer shift is that derived from a change in boundary conditions for the atomic (Wigner-Seitz) cell and is correlated with the cell boundary electron density and with the electronegativity of the alloying partner element. It was also suggested that the electron density mismatch at the cell boundaries shared by dissimilar atoms is primarily compensated by s —> electron conversion, in agreement with results of self-consistent band structure calculations. 

However, a PS-fo-PI/PI blend shows direct L G transitions without appearance of the PL phase. The L microdomain is more favourable than the PL phase since the volume fraction of the PI block component and the symmetry of microdomains is increased by the addition of PI homopolymer. Hence, the PL phase may not be formed as an intermediate structure if relatively high molecular weight PI homopolymer is added. The latter is not able to effectively fill the corners of the Wigner-Seitz cells in consequence packing frustration cannot be released and the PL phase is not favoured [152]. In contrast, the addition of low molecular weight PI homopolymer to the minor component of the PL phase reduces the packing frustration imposed on the block copolymers and stabilizes it [153]. Hence, transition from the PL to the G phase indicates an epitaxial relationship between the two structures, while the direct transition between L and G yields a polydomain structure indicative of epitaxial mismatches in domain orientations [152]. 

We have used the procedure of Marcelja et al (22) to compute r for spherical molecules with the same charge and volume as 200 bp rodlike DNA. Each molecule is at the center of a Wigner-Seitz cell of volume (4/3)with bulk salt concentrations spanning the experimental range (2). The nonlinear Poisson-Boltzmann e( uation is solved numerically with appropriate boundary conditions at the particle surface and the cell boundary. The results are that F = 155 at about 3 mg/mL DNA (twice the experimental concentration) with no added salt, but F is always < 155 for added salt in the experimental range. For NaPSS, with dp 3800 at 1-4 X 10 mg/mL, F > 300, consistent with the observation of a structure factor maximum. 

We previously introduced the concept of a primitive cell as being the supercell that contains the minimum number of atoms necessary to fully define a periodic material with infinite extent. A more general way of thinking about the primitive cell is that it is a cell that is minimal in terms of volume but still contains all the information we need. This concept can be made more precise by considering the so-called Wigner-Seitz cell. We will not go into... 

In the theory of metals and alloys, the Wigner-Seitz cell is defined by planes perpendicular to the interatomic vectors. Analogously, the boundary between two molecules or molecular fragments can be defined by using the relative sizes RA and RB of atom A in molecule / and the adjacent atom B in molecule II. 

Wigner-Seitz cell by that of a sphere. Thus,... 

Figure 16.3. (a) Construction of the Wigner-Seitz cell in a 2-D hexagonal close-packed (hep) lattice, (b) Primitive unit cell of the hep lattice. 

The Voronoi deformation density approach, is based on the partitioning of space into the Voronoi cells of each atom A, that is, the region of space that is closer to that atom than to any other atom (cf. Wigner-Seitz cells in crystals see Chapter 1 of Ref. 202). The VDD charge of an atom A is then calculated as the difference between the (numerical) integral of the electron density p of the real molecule and the superposition of atomic densities SpB of the promolecule in its Voronoi cell (Eq. [42]) ... 

The surface matching theorem makes it possible to generalize the idea of muffin-tin orbitals to a nonspherical Wigner-Seitz cell r. Each local basis orbital is represented as (p =a x + V on the cell surface a, where y and p are the auxiliary functions defined by the surface matching theorem. An atomic-cell orbital (ACO) is defined as the function — y, regular inside r. By construction, the smooth continuation of this ACO outside r is the function p. The specific functional forms are... 

In the last two equations Lws Is the radius of the sphere touching the Wigner-Seitz cell of the simulation cell. 

It is possible, as well, to define the primitive unit cell, by surrounding the lattice points, by planes perpendicularly intersecting the translation vectors between the enclosed lattice point and its nearest neighbors [2,3], In this case, the lattice point will be included in a primitive unit cell type, which is named the Wigner-Seitz cell (see Figure 1.2). 

A concrete building procedure in three dimensions of the Wigner-Seitz cell can be achieved by representing lines from a lattice point to others in the lattice and then drawing planes that cut in half each of the represented lines, and finally taking the minimum polyhedron enclosing the lattice point surrounded by the constructed planes. 

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