Wigner-Seitz sphere

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... 

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. 

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...

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. 

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... 

The normalization chosen corresponds to the correct normalization [Eq. (2)] of the ionization cross section of the atom. In the case of a slightly localized valence state, i.e., jS -> 0, the normalizing constant is simply determined by the atomic volume = (47r/3)Po, i.e., by the volume of the Wigner-Seitz sphere. In the... 

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. 

As follows from Fig. 2.20, chemical bond formation in UC is accompanied by an appreciable electron density transfer to the outer region of the atom. Most probably, the electron density near the Wigner-Seitz spheres is still overestimated in the LMTO caleulations (see Section 3.5). Hybridisation effects in the UC band structure ean be seen from the dispersion curves (Fig. 2.21) along the F - direction of the Brillouin zone. For the lattice constant a = 5.01 A, the hybridised C2p-U5/Aj and As bands are clearly seen, which cross the Aj-S/ state band. The bands... 

Table 4.4 Atomic state population n, pressures p, and total charges in Wigner-Seitz spheres Q.

---