Wigner-Seitz

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... 

Most of the present implementations of the CPA on the ab-initio level, both for bulk and surface cases, assume a lattice occupied by atoms with equal radii of Wigner-Seitz (or muffin-tin) spheres. The effect of charge transfer which can seriously influence the alloy energetics is often neglected. Several methods were proposed to account for charge transfer effects in bulk alloys, e.g., the so-called correlated CPA , or the screened-impurity model . The application of these methods to alloy surfaces seems to be rather complicated. 

Bonding,/-character, and actinides. .. 66-74 Bonding model, Wigner-Seitz... 

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. 

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),... 

The ionic profile of the metal was modeled as a step function, since it was anticipated that it would be much narrower than the electronic profile, and the distance dx from this step to the beginning of the water monolayer, which reflects the interaction of metal ions and solvent molecules, was taken as the crystallographic radius of the metal ions, Rc. Inside the metal, and out to dl9 the relative dielectric constant was taken as unity. (It may be noted that these calculations, and subsequent ones83 which couple this model for the metal with a model for the interface, take the position of the outer layer of metal ion cores to be on the jellium edge, which is at variance with the usual interpretation in terms of Wigner-Seitz... 

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]. 

Figure 2-11 compares the observed work function, 4>, with that calculated based on the jeUium model as a function of the electron density, n.,in metals here, n, is represented in terms of the Wigner-Seitz radius which is inversely proportional to the cube root of n.. The chemical potential term (p. = —1.5 to-2.5 eV) predominates in the work function of metals of low valence electron density, while on the contrary the surface term (- e x = -0-1 -5.0 eV) predominates for... 

Kg. 2-11. Work function, 4>, observed and calculated by using the jellium model as a function of Wigner-Seitz radius, rs, for various metals rs = 3 / (4 n n, = electron... 

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.]...

We have used the procedure of Marcelja et al (22) to compute r for spherical molecules with the same charge and volume as 200 bp rodlike DNA. Each molecule is at the center of a Wigner-Seitz cell of volume (4/3)with bulk salt concentrations spanning the experimental range (2). The nonlinear Poisson-Boltzmann e( uation is solved numerically with appropriate boundary conditions at the particle surface and the cell boundary. The results are that F = 155 at about 3 mg/mL DNA (twice the experimental concentration) with no added salt, but F is always < 155 for added salt in the experimental range. For NaPSS, with dp 3800 at 1-4 X 10 mg/mL, F > 300, consistent with the observation of a structure factor maximum. 

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. 

Recently, the effective mass of the electron has been calculated [148] within a Wigner-Seitz framework [175] for Ar, Kr, and Xe. In all three liquids, m decreases with increasing density. At the triple point densities, m = in argon and m =... 

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)...

Fig. 3. Wigner-Seitz radii of d-transition metals and actinides vs atomic number Z. To the plot, elements displaying empty and full d- and f-shell have been added. In abscissae, the groups of the Periodic Chart of Elements have been indicated (see, e.g. Handbook of Chemistry and Physics). The figure shows the sudden jump in radius between Pu and Am discussed in this chapter, and, more deeply, in Chap. C...

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... 

Also shown in the Table are the relative positions, Q, in energy of the different band centres (when hybridization is neglected) and the Wigner-Seitz radii in an assumed fee structure. From Eq. (5) and Table 1 one estimates for iron, nickel, uranium, cerium, gadolinium and americium... 

Table la. Potential parameters for the d-transition series taken from Andersen and Jepsen (Ref. 9). Cl is the band centre and pi is the band mass. S is the Wigner-Seitz radius in atomic units... 

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