Wigner-Seitz cells electron density

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 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]) ... 

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... 

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] ... 

The interaction was described by two terms with different signs. The first term took into account the ionic part of the interaction. It was proportional to the difference of the electronegativities. Only values of the electronegativities were chosen, which took into account empirical corrections. This term was principally a negative term. The second contribution came from the interaction of the electrons, which is, in principal, proportional to the electron density at the interface between the Wigner-Seitz cells. 

At the interface between the Wigner-Seitz cells of two different components the electron density must be equilibrate. This can be done by a reorganization of the orbital structure and the orbital occupation at the interface. This is, in principal, a positive contribution. This was multiplied by the molar surface area of the Wigner-Seitz cell of component A (for A in B) (T molar volume of the Wigner-Seitz cell of component A) and divided by an average of the electron density of the adjacent Wigner-Seitz cells, leading to... 

Difference of electron density on the surface of the Wigner-Seitz cells of... 

E) electron density of states Wigner-Seitz cell boundaries... 

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. ... 

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... 

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