Wigner-Seitz method

Each atom of the liquid is replaced by an equivalent sphere of radius rg given as 

In order to obtain the eigenvalues, it is sufficient to solve the Schrodinger equation. 

The potential V(r) is composed of a repulsive part, V p, and an attractive part, 

Vrep is of short range and it is caused by the Pauli principle. V ttr is of long range and caused by polarization. For V p the following approximation is taken  

Opoi denotes the polarizability of the atoms comprising the liquid. With the potential 

---

Wigner-Seitz method - A method of calculating electron energy levels in a solid using a model in which each electron is subject to a spherically symmetric potential. 

Correction to the Sphericized Cell Calculation. (Paper 30) Wigner also worked out the correction to the spherical cell approximation in a cubic lattice. Here his previous work on cohesive energy in crystal lattices stood him in good stead, since the Wigner-Seitz method of calculation of cohesive energy also involved sphericizing a cubic cell. 

The effective electron mass in high mobility liquids can be obtained by means of the Wigner-Seitz method (see Section 7.6.1) using high precision pseudopotentials. Usually, the effective mass is expressed in a relative way as, meff/mej. The values reported are in reasonable agreement with values extracted from experimental data (see Table 4). A general trend is obvious. The more polarizable liquids, argon, krypton and... 

Table 4 Effective Electron Masses Estimated by Means of the Wigner-Seitz Method...

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. 

Figure 1. A nonrelativistic window of the temperature—composition plane, showing electron density (n) and temperature (T). Normal conditions (on earth) for semiconductors and elemental metals and conditions on the Sun, Jupiter, and the White Dwarf are shown. Experimental methods in A, B, C, and D are Tokamak, glow-discharge, laser fusion, and degenerate strongly coupled plasma, respectively. Wigner—Seitz radii, rs, are also shown (adapted from Redmei4).

From a computational standpoint, the usefulness of the method relies on the simplicity of the calculations needed for the determination of the three equivalent crystals associated with each atom i. This is accomplished by building on the simple concepts of Equivalent Crystal Theory (ECT) [25,26], as will be discussed in detail below. The procedure involves the solution of one simple transcendental equation for the determination of the equilibrium Wigner-Seitz radius i WSE) of ch equivalent crystal. These equations are written in terms of a small number of parameters describing each element in its reference state, and a matrix of perturbative parameters Ay , which describe the changes in the electron density in the vicinity of atom / due to the presence of an atom j (of a different chemical species), in a neighboring site. The determination of parameters for each atom in... 

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. 

The investigations of Asada et al. and Christensen - were carried out with linear-muffin-tin orbitals within the atomic sphere approximation (LMTO- AS A) Within the muffin-tin model suitable s, p and d basis functions (muffin-tin orbitals, MTO) are chosen. In contrast to the APW procedure the radial wave functions chosen in the linear MTO approach are not exact solutions of the radial Schrodinger (or Dirac) equation. Furthermore, in the atomic sphere approximation (ASA) the radii of the atomic spheres are of the Wigner-Seitz type (for metals the spheres have the volume of the Wigner-Seitz cell) and therefore the atomic spheres overlap. The ASA procedure is less accurate than the APW method. However, the advantage of the ASA-LMTO method is the drastic reduction of computer time compared to the APW procedure. 

Wigner-Seitz-type cells. The singularities caused by the cusps of relativistic wave functions at the nuclear sites are eliminated by suitable transformations of the sample points, which leads to an improved numerical representation of the wave functions (Bastug et al. 1995). With this method, a total of approximately 1400 sample points is needed to achieve a relative accuracy of 10 8 in calculations for diatomic molecules. 

Most methods of band-structure calculation are based on the muffin-tin, atomic sphere approximation (ASA) or Wigner-Seitz construction for the electronic potential and... 

The unit cells described above are conventional crystallographic unit cells. However, the method of unit cell construction described is not unique. Other shapes can be found that will fill the space and reproduce the lattice. Although these are not often used in crystallography, they are encountered in other areas of science. The commonest of these is the Wigner-Seitz cell. 

---