Wigner-Seitz approximation

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]. 

The radius rs is sometimes called the Wigner-Seitz radius and can be interpreted to a first approximation as the average distance between two electrons in the particular system. Regions of high density are characterized by small values of rs and vice versa. From standard electrostatics it is known that the potential of a uniformly charged sphere with radius rs is proportional to l/rs, or, equivalently, to p( r,)17 3. Hence, we arrive at the following approximate expression for Ex (Cx is a numerical constant),... 

The behaviour of the transition-metal bands as the atoms are brought together to form the solid may be evaluated within the Wigner-Seitz sphere approximation by imposing bonding, = 0, or antibonding, R, = 0,... 

Derive the contribution Ues by making the Wigner-Seitz sphere approximation, in which inter-cell electrostatic interactions are neglected and the intra-cell potential energy is approximated by that of the Wigner-Seitz sphere. 

Assumes that the driving force for reactions between metals is a function of two factors a negative one, arising from the difference in chemical potential, A y of electrons associated with each metal atom, and a positive one that is the difference in the electron density, Anws, at the boundaries of Wigner-Seitz tvpe cells surrounding each atom. Values of for the metals are approximated by the electronic work functions n ws is estimated from compressibility data. The atomic concentrations in the alloy must be included in the calculation. ... 

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 KS exchange potential coefficient aKS x(0) is essentially the image-potential value of 1/4, ranging from 0.195 to 0.274 over the metallic range of densities. Its value is precisely 0.250 for 0 = yjl, which corresponds to a Wigner-Seitz radius of rg — 4.1. The jellium model is stable for approximately this value of rs. With the assumption that the asymptotic structure of vxc (r) is the image potential, we see that the correlation contribution to this structure is an... 

Fig. 1 Asymptotic structure coefficients as(j8), c ks,xG ) crw03), a CS), and aKS gOS) as fimction of barrier height parameter )S =VW/eF, where W is the barrier height and eF the Fermi energy. Corresponding values of the Wigner-Seitz radius rs for jellium and structureless-pseudopotential models over the metallic range of densities are also given. The relationship between rs and ff is via self-consistent calculations in the local density approximation for exchange-correlation.

Fig. 6 The approximate exchance potential v pp(z) of Eq. (64) at the surface of a metal of Wigner-Seitz radius rs = 3.24. The potential in the local density approximation (LDA) is also plotted, as is the exact asymptotic structure - ks(X0J)/z of the KS exchange potential.

An approach that is very closely related to the Atomic Sphere Approximation is the Renormalized Atom Theory, introduced first by Watson, Ehrenreich, and Hodges (1970) (sec also Watson and Ehrenreich, 1970, Hodges et al., 1972, and particularly Gelatt, Ehrenreich, and Watson, 1977). The name derives from the way the potential is constructed a charge density for each atom is constructed on the basis of atomic wave functions that are truncated at the Wigner-Seitz, or atomic, sphere. The charge density from each state is then scaled up (renormalized) to make up for that density beyond the sphere which has been dropped. 

Upon hydrogenation the hydrogen atoms will bond with an A atom but they will also be in contact with B atoms. The atomic contact between A and B that was responsible for the heat of formation of the binary compound is lost. The contact surface is approximately the same for A-H and B -H thus implying that the ternary hydride AB H2m is energetically equivalent to a mechanical mixture of AH, and Bniim [37]. More specifically, this could be explained by two terms one is due to the mismatch of the electronic density of metals A and B at the boundary of their respective Wigner-Seitz cells, the other term is associated with the difference in chemical potential of the electrons in metals A and B. From these considerations, a semi-empirical relation for the heat of formation of a ternary hydride can be written as [70] ... 

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... 

From the density, it is now possible to construct the potential for the next iteration, and also to calculate total energies. During the self-consistency loop, we make a spherical approximation to the potential. This is because the computational cost for using the full potential is very high, and that the density converges rather badly in the corners of the unit cell. In order not to confuse matters with the ASA (Atomic Sphere Approximation) this is named the Spherical Cell Approximation (SC A) [65]. If we denote the volume of the Wigner-Seitz cell centered at R by QR, we have fiR = QWR — (4 /3), where w is the Wigner-Seitz radius. This means that the whole space is covered by spheres, just as in the ASA. We will soon see why this is practical, when we try to create the potential. 

The physical interpretation of the KKR-ASA equations is that the energy-band problem may be approximated by a boundary-value problem in which the surrounding lattice through the structure constants imposes a k-dependent and non-spherically symmetric boundary condition on the solutions P (E) inside the atomic Wigner-Seitz sphere. This interpretation is illustrated in Fig.2.1. 

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