Wigner-Seitz radius

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),... 

Figure 2-11 compares the observed work function, 4>, with that calculated based on the jeUium model as a function of the electron density, n.,in metals here, n, is represented in terms of the Wigner-Seitz radius which is inversely proportional to the cube root of n.. The chemical potential term (p. = —1.5 to-2.5 eV) predominates in the work function of metals of low valence electron density, while on the contrary the surface term (- e x = -0-1 -5.0 eV) predominates for... 

Kg. 2-11. Work function, 4>, observed and calculated by using the jellium model as a function of Wigner-Seitz radius, rs, for various metals rs = 3 / (4 n n, = electron... 

Table 6-3. The effective image plane position of a metal in vacuum estimated as a function of electron density in metal x, distance at the effective image plane fiom the jellium metal edge rws = Wigner-Seitz radius (a sphere containing one electron) which is related to electron density n, in metal (1 / n, = 4 n / 3 ) au = atomic unit (0.529 A). [From Schmickler, 1993.]...

Table la. Potential parameters for the d-transition series taken from Andersen and Jepsen (Ref. 9). Cl is the band centre and pi is the band mass. S is the Wigner-Seitz radius in atomic units... 

Fig. 5.14 The binding energy U as a function of the Wigner-Seitz radius fiws for sodium. The bottom of the conduction band, 1 is given by the lower curve to which is added the average kinetic energy per electron (the shaded region). (After Wigner and Seitz (1933).)...

In general, the equilibrium Wigner-Seitz radius R%s can be found from eqn (5.59) by requiring that U is stationary with respect to Rws. It is found to depend explicitly on the core radius Re through the equation... 

Fig. 6.7 Interatomic pair potentials for (a) Na, (b) Mg, and (c) Al as a function of the interatomic separation in units of the Wigner-Seitz radius / ws. The positions of the fee first and bcc first and second nearest neighbours are marked. Since fee and ideal hep structures have identical first and second nearest neighbours, their relative structural stability is determined by the more distant neighbours marked in the figures. (After McMahan and Moriarty (1983).)...

The argument kFR is independent of electron density or atomic volume since it can be written from eqn (2.40) in terms of the Wigner-Seitz radius Kws as... 

Fig. 7.8 The energy bands as a function of Wigner-Seitz radius / ws for (a) Y, (b) Tc, and (c) Ag. The observed equilibrium Wigner-Seitz radii are marked eq. d, Ev and Eb mark the centre of gravity, and top and bottom of the d band respectively. (After Pettifor (1977).)...

Fig. 7.9 The d band occupancy Nd as a function of the Wigner-Seitz radius, / ws. The circles mark the equilibrium values.

Fig. 7.12 The theoretical ( ) and experimental (x) values of the equilibrium band width, Wigner-Seitz radius, cohesive energy, and bulk modulus of the 4d transition metals. (From Pettifbr (1987).)...

This linear dependence is not unexpected from the linear variation in the free atomic d level that is observed across the 4d series in Fig. 2.17. The ratio, ajb, in eqn (7.44) is obtained by fitting the observed Wigner-Seitz radius of molybdenum, giving ajb = 18.0. It follows that a = 216 eV and b = 12 eV for the 4d series. 

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.

The Sn 5 s and 5p radial functions, from a nonrelativistic calculation for the free 5sz5pz atom, are plotted in Fig. 7. Roughly 8% of the 5s charge extends outside the Wigner-Seitz radius, rws, for / —Sn the 5s orbital, with much of its density in a region in which Zen is about equal to the valence, is actually somewhat in the interior of the atom. It is not unlike the d orbitals of transition metals, which, as earlier noted, maintain much of their atomic quality in a metal. Thus it is quite plausible that the valence s character in Sn is much like the free atom 5 s, except for a renormalization within the Wigner-Seitz cell. The much more extended 5p component, on the other hand, is not subject to simple renormalization the p character near the bottom of the band takes on a form more like the dot-dash curve of Fig. 7. It nevertheless appears useful to account for charge terms of a pseudo P component and a renormalized s. 

Derivations of exchange and correlation energy formulas are in general based on the r value, (the Wigner-Seitz radius) which is related to the electron density by,... 

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... 

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