Wigner-Seitz theory

In the context of the Wigner-Seitz theory, in 1937 Brillouin [7] gave a formal analysis of the atomic energy variations under boundary deformations using contact coordinate transformations that transform the boundary modifications into Hamiltonian transformations for a fixed region and generate the associated commutation relations. This allows application of the usual form of perturbation theory for the problem. 

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. 

Electrostatic. Virtually all colloids in solution acquire a surface charge and hence an electrical double layer. When particles interact in a concentrated region their double layers overlap resulting in a repulsive force which opposes further approach. Any theory of filtration of colloids needs to take into account the multi-particle nature of such interactions. This is best achieved by using a Wigner-Seitz cell approach combined with a numerical solution of the non-linear Poisson-Boltzmann equation, which allows calculation of a configurational force that implicitly includes the multi-body effects of a concentrated dispersion or filter cake. 

An approach that is very closely related to the Atomic Sphere Approximation is the Renormalized Atom Theory, introduced first by Watson, Ehrenreich, and Hodges (1970) (sec also Watson and Ehrenreich, 1970, Hodges et al., 1972, and particularly Gelatt, Ehrenreich, and Watson, 1977). The name derives from the way the potential is constructed a charge density for each atom is constructed on the basis of atomic wave functions that are truncated at the Wigner-Seitz, or atomic, sphere. The charge density from each state is then scaled up (renormalized) to make up for that density beyond the sphere which has been dropped. 

From a computational standpoint, the usefulness of the method relies on the simplicity of the calculations needed for the determination of the three equivalent crystals associated with each atom i. This is accomplished by building on the simple concepts of Equivalent Crystal Theory (ECT) [25,26], as will be discussed in detail below. The procedure involves the solution of one simple transcendental equation for the determination of the equilibrium Wigner-Seitz radius i WSE) of ch equivalent crystal. These equations are written in terms of a small number of parameters describing each element in its reference state, and a matrix of perturbative parameters Ay , which describe the changes in the electron density in the vicinity of atom / due to the presence of an atom j (of a different chemical species), in a neighboring site. The determination of parameters for each atom in... 

It is to be stressed that this is a crystal-like theory. The detail of the band anisotropy will arise from the assumption of a specific crystal structure. Since the wave function we are considering is S-like, a band dispersion calculated for a f.c.c. crystal should not differ too much) at least near the band minimum) from that of a fluid if the effect of disorder is completely taken into account by the use of the radial distribution function in the calculation of the dipole moment induced on the atom at the center of the Wigner-Seitz cell. 

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]. 

When applied to solids, the theory generalizes the definition of a Wigner-Seitz cell, identifying it with the smallest... 

R. J. F. Leote De Carvalho, E. Trizac, and J.-P. Hansen, Phys. Rev. E, 61, 1634 (2000). Nonlinear Poisson-Boltzmann Theory of a Wigner-Seitz Model for Swollen Clays. 

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... 

Probably one of the first physics paper that analyzed an atomic system in a bounded region was the work of Wigner and Seitz [1,2] on the theory of periodic structures, published in 1933-1934. The Schrodinger equation for an atom in a lattice was studied with the Neumann boundary conditions (named in honor of Carl Neumann, who had studied the differential equations and potential theory for such problems at the end of 19th century). 

What is the historical background Because the structure of matter is one of the basic fields of research in physics, it has been one of the core areas of theoretical physics. While it is not possible to give a history, or even to mention the many fundamental contributions, it is important to recognize the extent to which the work discussed here rests upon the ingenious creations of previous decades. In particular, the reader is referred to the paper by Wigner and Seitz in 1955, who summarize their own work and that of others in understanding the theory of cohesion in metals, and to the multi-volume work of Slater. ... 

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