Wigner Lattice Model

Wigner Lattice Model.—Marcel j a et al. have considered a Wigner lattice model in which each particle moves about in a sphere containing an equal and opposite charge centred on the lattice site. The surface potential of each particle in the dispersion is determined by solving the Poisson-Boltzmann 

There is no statistical theory of the solid-liquid transition, even for hard spheres. So, to estimate volume fractions of coexisting phases, Marcelja et al. resorted to an intuitive approach based on Lindemann s empirical rule, which states that a solid will melt when the root mean-square displacement of particles about their equilibrium positions exceeds some characteristic fraction /l of the lattice spacing. For potentials of functional form r (n 4) it is found that /l 0.10. For particles of charge zq occupying volume 4ira /3 on a f.c.c. lattice with spacing b, the Lindemann ratio is 

Monte Carlo studies show that the Wigner lattice melts when the parameter r is about 155 10 (corresponding to /l 0.08). Thus, whatever the form of the effective repulsive interaction, from very soft (unscreened Coulomb) to hard (say r ), the empirical Lindemann rule provides melting curves that are, at worst, qualitatively correct. 

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Ham [508] considered that the growth of a random array of precipitating particles could be approximated to a simple cubic lattice of spherical sinks of radius R to which more material diffused from the supersaturated solution. A model of the type is very similar to those models discussed by Reek and Prager [507] and Lebenhaft and Kapral [492], The analysis which Ham introduced highlights a similarity between the competitive effect and the Wigner—Seitz model of metals. 

The essential feature of the Wigner-Seitz model is that it represents a typical lattice element of a heterogeneous array by an equivalent unit cell. For example, if a heterogeneous core were to be composed of cylindrical fuel rods placed in a square lattice of spacing a, then a reasonable choice of the unit cell would be the circle of radius a/y/r concentric with a rod. In performing a calculation based on this model the usual practice is to specify that the net neutron current at the boundary of the... 

Most of the present implementations of the CPA on the ab-initio level, both for bulk and surface cases, assume a lattice occupied by atoms with equal radii of Wigner-Seitz (or muffin-tin) spheres. The effect of charge transfer which can seriously influence the alloy energetics is often neglected. Several methods were proposed to account for charge transfer effects in bulk alloys, e.g., the so-called correlated CPA , or the screened-impurity model . The application of these methods to alloy surfaces seems to be rather complicated. 

Consistent [50] with this model is the observation [51] that incorporation of Cd " as an impurity in the AgjCOj lattice increases the reaction rate, because this introduction of divalent ions must be accompanied by formation of cation vacancies. Wydeven et al. [52-54] decomposed pure AgjCOj and compared behaviour with that of the salt doped with or Gd. They also studied the effect of water vapour. The kinetic observations fitted the Polanyi-Wigner equation and it was concluded that decomposition proceeds by an interface mechanism. 

Like in the ball and stick models of cm, in the rep model also the positional coordinates of spheres in the assembly are carefully determined. Space filling polyhedra can be built around each sphere in the assembly, in the manner of building Wigner-Seitz cell in a reciprocal lattice that is, by erecting perpendicular planes at the mid points of the lines connecting the... 

Furthermore, within the (R)APW method the so called muffin-tin approach is used for calculating V(f). According to this model the volume of the unit cell to is separated into the volume tOy of non overlapping and approximately touching atomic spheres (muffin-tin spheres, cf. Fig. 4) centred at the lattice sites y and the volume to between the spheres. In Table 5 the radii ry and the volumes o), which are used for the RAPW calculations of Zintl phases are given. Because of the arrangement of the atoms in the crystal shown in Sect. B, the volumes to a and Wigner-Seitz volumes are listed too. 

Figure 5.6 Scheme of the near-free electron model of simple metals. The white circles represent the Wigner-Seitz cells, in which the point-positive charges are located. The lattice of the cells Is immersed in a free-electron gas. 

A simplified theory was proposed by Brandt, Berko and Walker [104] in which the positron of Ps wave function in the field of the electron was replaced by the wave function of the Ps atom. The Ps wave function was then calculated for different lattice structures in the Wigner-Seitz approximation. This approximation is generally referred to as the free volume model, since the free volume is used as one of the parameters in the calculation. This model relates o-Ps lifetime to the average free volume hole size of the medium, and results construed that the o-Ps lifetime would measure the lattice-Ps interaction. Later, Tabata et al. [105] and Ogata and Tao [106] each adopted similar - but different - approaches by considering a unit cell and Ps located at the center instead of the center of the molecule, as used by Brandt et al. [104]. 

It turns out that for the s and p block metals, a simple model, namely, thejdlium model provides useful insight. In this model, the discrete nature of the ionic lattice is replaced with a smeared out uniform positive background exactly equal to that of the valence electron gas. In jellium, each element is completely specified by just the electron density n = N/V, where N is the number of electrons in the crystal and V is its volume. Often, the electron density is given in terms of the so-called Wigner-Seitz radius, tj, where tg = which corresponds to the spherical... 

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