Wigner-Seitz unit-cell model

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 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 essential feature of the Wigner-Seitz model is that it represents a typical lattice element of a heterogeneous array by an equivalent unit cell. For example, if a heterogeneous core were to be composed of cylindrical fuel rods placed in a square lattice of spacing a, then a reasonable choice of the unit cell would be the circle of radius a/y/r concentric with a rod. In performing a calculation based on this model the usual practice is to specify that the net neutron current at the boundary of the... 

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]. 

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