The Wigner-Seitz Cell

Let us suppose that we are mainly interested iu electrons that inhabit the crystal. As we know, electrons glue particles of a solid. As far as electrons are concerned, it is convenient to describe the lattice by the primitive lattice translation vectors. A primitive unit cell, which can fiU up all space, is important in this case. Such a unit ceU in the real space is called the A gner-Seitz ceU. 

The choice of a unit primitive cell, that is, a ceU of the smallest possible volume for a given lattice structure is somewhat arbitrary. Instead of using a conventional 

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A guide to tire stabilities of inter-metallic compounds can be obtained from the semi-empirical model of Miedema et al. (loc. cit.), in which the heat of interaction between two elements is determined by a contribution arising from the difference in work functions, A0, of tire elements, which leads to an exothermic contribution, and tire difference in the electron concentration at tire periphery of the atoms, A w, which leads to an endothermic contribution. The latter term is referred to in metal physics as the concentration of electrons at the periphery of the Wigner-Seitz cell which contains the nucleus and elecUonic structure of each metal atom within the atomic volume in the metallic state. This term is also closely related to tire bulk modulus of each element. The work function difference is very similar to the electronegativity difference. The equation which is used in tire Miedema treatment to... 

However, a PS-fo-PI/PI blend shows direct L G transitions without appearance of the PL phase. The L microdomain is more favourable than the PL phase since the volume fraction of the PI block component and the symmetry of microdomains is increased by the addition of PI homopolymer. Hence, the PL phase may not be formed as an intermediate structure if relatively high molecular weight PI homopolymer is added. The latter is not able to effectively fill the corners of the Wigner-Seitz cells in consequence packing frustration cannot be released and the PL phase is not favoured [152]. In contrast, the addition of low molecular weight PI homopolymer to the minor component of the PL phase reduces the packing frustration imposed on the block copolymers and stabilizes it [153]. Hence, transition from the PL to the G phase indicates an epitaxial relationship between the two structures, while the direct transition between L and G yields a polydomain structure indicative of epitaxial mismatches in domain orientations [152]. 

In the theory of metals and alloys, the Wigner-Seitz cell is defined by planes perpendicular to the interatomic vectors. Analogously, the boundary between two molecules or molecular fragments can be defined by using the relative sizes RA and RB of atom A in molecule / and the adjacent atom B in molecule II. 

Figure 16.3. (a) Construction of the Wigner-Seitz cell in a 2-D hexagonal close-packed (hep) lattice, (b) Primitive unit cell of the hep lattice. 

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

It is possible, as well, to define the primitive unit cell, by surrounding the lattice points, by planes perpendicularly intersecting the translation vectors between the enclosed lattice point and its nearest neighbors [2,3], In this case, the lattice point will be included in a primitive unit cell type, which is named the Wigner-Seitz cell (see Figure 1.2). 

A concrete building procedure in three dimensions of the Wigner-Seitz cell can be achieved by representing lines from a lattice point to others in the lattice and then drawing planes that cut in half each of the represented lines, and finally taking the minimum polyhedron enclosing the lattice point surrounded by the constructed planes. 

The symmetry of the lattice will impose distinct shapes on the Brillouin zones (which by definition are the Wigner-Seitz cells of the reciprocal lattice) for each type of symmetry. Figure 8.11 shows the first Brillouin zone for a face-centered cubic structure. 

If the Wigner-Seitz cell appropriate to a metal is superimposed on the spatial charge distribution of a free atom, one finds characteristically that a quantity of charge, typically between 2/3 e and 1 e, lies outside the cell boundaries. (13) Since in the metal the cell is of course neutral, this implies that formation of the metal requires compression of the valence charge, and associated with the compression is an increase of the Coulomb interactions of the valence electrons with each other and with the ion core. A lowest order estimate of the shift associated with this effect may be based on truncation of the free atom valence orbitals at the cell radius, rws, and renormalization of the charge within the cell. For a core electron lying entirely inside the valence density the core-valence Coulomb interaction is... 

A free atom is left with a charge + 1 after photoelectron emission in a metal, however, conduction electrons act to screen the vacancy. This screening may be thought of as complete if the Wigner-Seitz cell containing the hole is essentially... 

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. 

Fig. 7. The valence Ss and 5p orbital densities of Sn 5s25pz. Shown are free atom densities (solid lines) free atom densities renormalized to the Wigner-Seitz cell of a—Sn (dashed curves) and a schematic plot (dot-dash) of the way in which the 5p conduction band orbital character deviates from simple renormalization. The Wigner-Seitz radii, rws. of a and j8 Sn are indicated...

The first Brillouin zones for the SC, BCC, and FCC lattices are shown in Figure 4.1. The inner symmetry elements for each BZ are the center, F the three-fold axis, A the four-fold axis, A and the two-fold axis, S. The symmetry points on the BZ boundary (faces) (X, M, R, etc.) depend on the type of polyhedron. The reciprocal lattice of a real-space SC lattice is itself a SC lattice. The Wigner-Seitz cell is the cube shown in Figure 4.1a. Thus, the first BZ for the SC real-space lattice is a cube with the high symmetry points shown in Table 4.3. 

The reciprocal lattice of a BCC real-space lattice is an FCC lattice. The Wigner-Seitz cell of the FCC lattice is the rhombic dodecahedron in Figure A. b. The volume enclosed by this polyhedron is the first BZ for the BCC real-space lattice. The high symmetry points are shown in Table 4.4. 

The reciprocal lattice for the FCC real-space lattice is a BCC lattice. The Wigner-Seitz cell is a truncated octahedron (Fig. 4.1c). The shapes of the BZs for the SC and... 

Figure 4.1. The Wigner-Seitz cell of reciprocal space (the first BZ) for the SC real-space lattice is itself a SC lattice (a). For the BCC real-space lattice, the first BZ Is a rhombododecahedron (6). For the FCC real-space lattice, the first BZ is a truncated octahedron (c).

BCC real-space lattices are completely determined by the condition that each inner vector, k, go over into another by all the symmetry operations. This is not the case for the tmncated octahedron. The surface of the Wigner-Seitz cell is only fixed at the truncating planes, not the octahedral planes. Nonetheless, the volume enclosed by the truncated octahedron is taken to be the first BZ for the FCC real-space lattice (Bouckaert et ak, 1936). The special high-symmetry points are shown in Table 4.5. 

A band-structure diagram is a map of the variation in the energy, or dispersion, of the extended-wave functions (called bands) for specific Ar-points within the first BZ (also called the Wigner-Seitz cell), which is the unit cell of Ar-space. 

Here it should be stressed that for condensed matter a convenient analysis of the wave functions is made in the form of cellular orbitals, these are defined inside the Wigner-Seitz cells around each atom, a band can be formed for each angular momentum beginning at the energy at which these wave functions have zero slope at the surfaces of the Wigner-Seitz cells up to the energies where the wave functions have zero value at the same points. In a sense we are speaking of a linear combination of cellular orbitals (LCCO) procedure. 

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. 

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. 

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