Wigner-Seitz sphere radius

Fig. 2. Wigner-Seitz radii, bulk moduli, and cohesive energies of lanthanides plotted vs atomic number Z (The radius of a Wigner-Seitz sphere is related to the atomic volume by Vj, = 4/3 3t Rws)...

The second factor does depend upon the particular atomic potential and must be computed. This factor is the band mass p,i and it is proportional to the reciprocal of the radius squared at the Wigner-Seitz sphere... 

The energy of the bottom of the sodium conduction band, denoted by 2, is determined by imposing the bonding boundary condition across the Wigner-Seitz sphere of radius, Kws, namely... 

Fig. 4. Hartree-Fock free atom 4s valence electron orbital for potassium (solid line) and the 4s-like orbital, obeying the Wigner-Seitz boundary condition, appropriate to the bottom of the conduction bands in metallic potassium (dashed line). Both orbitals are normalized, for the metal, integration is limited to the Wigner-Seitz sphere of radius rws...

Liu (1961) noted that the wave functions of the 4f electrons on different rare earth (R) atoms in the solid state do not usually overlap. This is because the radius of the 4f shell is almost 0.35 A and the wave functions are therefore zero on the Wigner-Seitz sphere. There can therefore be no direct exchange, and exchange interactions between different R-magnetic moments must be mediated by the conduction electrons. Liu points out that there are two possible interaction types. In the first type the 4f magnetic moment on the R-atom polarizes the sp conduction bands of the compound via a direct s-f exchange interaction given by... 

In the renormahzation scheme one utilizes the free-atom s and d wave functions, truncates them at the radius of the Wigner-Seitz sphere and normalizes them within this sphere, thereby preserving charge neutrality. In this way the atoms are prepared approximately in the form in which they actually enter the solid metal therefore placing them together. 

In fig. 3.63 we show the radial charge distribution of 4f, 5d and 6s electrons in the Wigner-Seitz sphere of Gd (Harmon and Freeman, 1974b). The 4f charge drops off rapidly and becomes vanishingly small at the WS sphere 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),... 

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.]...

To obtain a rough estimate of the multiple scattering effects a model proposed by M. H. Cohen is useful (18). This model is based on the application of the Wigner-Seitz scheme to an electron in a helium crystal. Each helium atom is represented as a hard sphere characterized by a radius equal to the scattering length. The electron wave function will then be... 

One can impose additional spatial confinement on a Debye plasma such that the potential energy vanishes at the boundary of a given sphere of radius R. For the strongly coupled system, one can assume that no electron current passes through the boundary surface and the wavefunction must vanish at the Wigner-Seitz boundary R [154], Under such conditions, the radial one-particle wavefunction ir(r) satisfies... 

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

Results from Morruzi, Williams, and Janak (1977) for cohesive properties versus atomic number. Parts (a) and (b) show equilibrium nuclear separation in terms of the Wigner-Seitz (or atomic sphere) radius Fq. (The volume per atom is Parts (c) 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 above formulas and the following ones are often expressed using a parameter rS) equivalent to the radius of a sphere with constant charge density p and a total charge of one electron, also known as the Wigner-Seitz radius ... 

With SWP(I) = 1.0 all spheres have the same size, with radii equal to the average Wigner-Seitz radius (8.16). 

Results from Morruzi, Williams, and Janak (1977) for cohesive properties versus atomic number. Parts (a) and (b) show equilibrium nuclear separation in terras of the Wigner-Seitz (or atomic sphere) radius. (The volume per atom is 4roo/3.) Parts (c) and (dj show cohesive energy in eV per atom. Parts (e) and (f) show bulk moduli in kilobars. I he atomic number increases in steps of one from 19 to 31 in (a), (c), and (e) and from 37 to 49 in (b), (d), and (f). Measured values, at low temperature where available, are indicated by crosses. (After Morruzi, Williams, and Janak, 1977.]... 

Rb is the Wigner-Seitz radius of metal B that is, the radius of a sphere with volume Q B) and R is the cluster radius, easily obtained from Q A), P(B) and the number of atoms. 

Fig. 2 Anionic star PE with eight arms having a radius in 1 1 electrolyte solution (as indicated by the small spheres that carry a plus or a minus sign) in a spherical electroneutral Wigner-Seitz cell with radius D...

The Wigner-Seitz model of a spherical unit cell is used for calculation. This is a sphere of radius tsUo with one nucleus at the center. Each sphere has overall neutrality, since one-electron charge at the center is canceled by the positive charge inside the volume of the sphere. In this model the spheres exert no electrical forces on each other. Of course this is only an approximate model, since the unit cells are not truly spheres - spheres cannot be packed together to cover aU volume. However, the error made by the approximation is remarkably small. 

The electron gas density is often expressed in terms of the Wigner-Seitz parameter r, which is the radius, expressed in units of the Bohr radius ao, of a sphere with the averaged volume occupied by one electron ... 

The main features of the resulting band structures, densities of states and Fermi surface are close to those obtained previously by Pickett et al. (1981) The one difference in this calculation from that of Pickett et al. is the way the 5p-core electrons are treated. Min et al. (1986b) find that the 5p-electrons, which are located approximately 1 Ry below the conduction band, have a nonncgligible density outside the muffin-tin sphere, especially at smaller volumes (that is, at higher pressures). The interstitial 5p-contribution amounts to almost 0.3 electrons in the case of a-Ce (r s = 3.54a.u. is the Wigner-Seitz radius) and this has a strong... 

---