Wigner-Seitz atomic cells

The second term represents the analog of the repulsive term in the heat of formation in Miedema s model (Miedema et al., 1980). The sign of the coefficient Q follows from the contention that the mismatch in electron density at the Wigner-Seitz atomic cell boundaries (/i,, ) in transition metal alloys can be removed by means of s-d intra-atomic electron conversion. The s electrons reside predominantly in the outside regions of the atomic cell. Conversion of s-type electrons into d-type electrons will therefore result in a decrease of n. It follows then that P and Q are of opposite sign. 

The second contribution originates from the discontinuity in electron density at the interface between dissimilar atoms when R and M atoms are combined. This difference in electron density at the Wigner-Seitz atomic cell boundaries = —makes a positive contribution to the heat of compound formation... 

Assumes that the driving force for reactions between metals is a function of two factors a negative one, arising from the difference in chemical potential, A y of electrons associated with each metal atom, and a positive one that is the difference in the electron density, Anws, at the boundaries of Wigner-Seitz tvpe cells surrounding each atom. Values of for the metals are approximated by the electronic work functions n ws is estimated from compressibility data. The atomic concentrations in the alloy must be included in the calculation. ... 

For the approximate DM to have the proper point symmetry, the LUC should be taken to be the Wigner-Seitz (WS) cell. In this case, however, the symmetry can be broken if on the boundary of the WS cell, there are atoms of the crystal. Indeed, if an atom lies on the WS cell boundary, then there is one or several equivalent atoms that also lie on the boundary of the cell and their position vectors differ from that of the former atom by a super lattice vector A. When constructing the approximate density matrix p we assigned only one of several equivalent atoms to the WS ceU. In other words, in the set there are no two vectors that differ from each other by a... 

The cycUc cluster can be chosen in the form of the Wigner-Seitz unit cell (shown by the dotted Une in Fig. 6.1 with the center at boron atom 1). A connection of the interaction range to this ceUs atoms was discussed in [301]. 

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... 

Van der Woude and Miedema [335] have proposed a model for the interpretation of the isomer shift of Ru, lr, Pt, and Au in transition metal alloys. The proposed isomer shift is that derived from a change in boundary conditions for the atomic (Wigner-Seitz) cell and is correlated with the cell boundary electron density and with the electronegativity of the alloying partner element. It was also suggested that the electron density mismatch at the cell boundaries shared by dissimilar atoms is primarily compensated by s —> electron conversion, in agreement with results of self-consistent band structure calculations. 

We previously introduced the concept of a primitive cell as being the supercell that contains the minimum number of atoms necessary to fully define a periodic material with infinite extent. A more general way of thinking about the primitive cell is that it is a cell that is minimal in terms of volume but still contains all the information we need. This concept can be made more precise by considering the so-called Wigner-Seitz cell. We will not go into... 

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. 

The Voronoi deformation density approach, is based on the partitioning of space into the Voronoi cells of each atom A, that is, the region of space that is closer to that atom than to any other atom (cf. Wigner-Seitz cells in crystals see Chapter 1 of Ref. 202). The VDD charge of an atom A is then calculated as the difference between the (numerical) integral of the electron density p of the real molecule and the superposition of atomic densities SpB of the promolecule in its Voronoi cell (Eq. [42]) ... 

The surface matching theorem makes it possible to generalize the idea of muffin-tin orbitals to a nonspherical Wigner-Seitz cell r. Each local basis orbital is represented as (p =a x + V on the cell surface a, where y and p are the auxiliary functions defined by the surface matching theorem. An atomic-cell orbital (ACO) is defined as the function — y, regular inside r. By construction, the smooth continuation of this ACO outside r is the function p. The specific functional forms are... 

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...

Muffin-Tin Orbital theory is in the spirit of the very early treatment of alkali metals by Wigner and Seitz (1934), who focused on a single atomic cell (those points nearer the atom being studied than any other atom) in which the potential is nearly spherically symmetric. They then replaced the cell by a sphere of equal volume, the sphere of radius /q that we introduced in the discussion of simple metal.s. This is illustrated in Fig. 20-12 for a face-centered cubic lattice. Wigner... 

Upon hydrogenation the hydrogen atoms will bond with an A atom but they will also be in contact with B atoms. The atomic contact between A and B that was responsible for the heat of formation of the binary compound is lost. The contact surface is approximately the same for A-H and B -H thus implying that the ternary hydride AB H2m is energetically equivalent to a mechanical mixture of AH, and Bniim [37]. More specifically, this could be explained by two terms one is due to the mismatch of the electronic density of metals A and B at the boundary of their respective Wigner-Seitz cells, the other term is associated with the difference in chemical potential of the electrons in metals A and B. From these considerations, a semi-empirical relation for the heat of formation of a ternary hydride can be written as [70] ... 

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