Valence electron/atom number ratio

With a further increase of the zinc concentration at CujZug and a valence electron/atom number ratio of 21 13 (1.62), a superstructure of the body centered cubic lattice is formed, the y-phase (CujZug type, Pearson symbol cI52). The superstructure consists of 3 = 27 simple cells with 52 atoms and two vacancies compared to the simple cells. The atoms and vacancies are no longer statistically distributed but form an ordered structure. 

The next phase change is observed at CuZuj and a valence electron/atom number ratio of 7 4 (1.75). At this ratio a hexagonal closed packed structure the 8-phase is formed (Mg type, Pearson symbol hP2). 

It was pointed out by Hume-Rothcry 4 in 1926 that certain interns etallic compounds with close]y related structures but apparently unrelated stoichiometric composition can be considered to have the same ratio of number of valence electron to number of. atoms,. For example, the j8 phases of the systems Cu—Zti, Cu—-Alv and Ou -Sn are analogous in structure, all being based on the -4.5 arrangement their compositions correspond closely to the formulas CuZn, CusAl, and CtttSn. Considering copper to be univalent, zinc bivalent, aluminum trivalent, and tin quadrivalent, we see that the ratio of valence electrons to atoms has the value f for each of these compounds ... 

The fact that compounds such as Mg2Si to MgjPb have such high resistances and crystallize with the antifluorite structure does not mean that they are ionic crystals. Wave-mechanical calculations show that in these crystals the number of energy states of an electron is equal to the ratio of valence electrons atoms (8/3) so that, as in other insulators, the electrons cannot become free (that is, reach the conduction band) and so conduct electricity. That the high resistance is characteristic only of the crystalline material and is not due to ionic bonds between the atoms is confirmed by the fact that the conductivity of molten MgjSn, for example, is about the same as that of molten tin. 

Hume-Rothery s rule The statement that the phase of many alloys is determined by the ratio.s of total valency electrons to the number of atoms in the empirical formula. See electron compounds. 

The valence band structure of very small metal crystallites is expected to differ from that of an infinite crystal for a number of reasons (a) with a ratio of surface to bulk atoms approaching unity (ca. 2 nm diameter), the potential seen by the nearly free valence electrons will be very different from the periodic potential of an infinite crystal (b) surface states, if they exist, would be expected to dominate the electronic density of states (DOS) (c) the electronic DOS of very small metal crystallites on a support surface will be affected by the metal-support interactions. It is essential to determine at what crystallite size (or number of atoms per crystallite) the electronic density of sates begins to depart from that of the infinite crystal, as the material state of the catalyst particle can affect changes in the surface thermodynamics which may control the catalysis and electro-catalysis of heterogeneous reactions as well as the physical properties of the catalyst particle [26]. 

Number of free valence electrons per atom Poisson s ratio... 

If magnesium, with two valence electrons to be lost, reacts with chlorine (which needs one additional electron), then magnesium will donate one valence electron to each of two chlorine atoms, forming the ionic compound MgCl2. Make sure the formula has the lowest whole number ratio of elements. 

The Hume-Rothery phases constitute an interesting and ubiquitous group of binary and complex intermetallic substances it was indeed Hume-Rothery who, already in the twenties, observed that one of the relevant parameters in rationalizing compositions and structures of a number of phases is the average number of valence electrons per atom (nJnM). An illustration of this fact may be found in Table 4.6, where a number of the Hume-Rothery structure types have been collected, together with a few more major structure types relevant to transition metal alloys. For each phase the corresponding VEC has been reported as njnai ratio, both calculated on the basis of the s and p electrons and of s, p and d electrons. 

Pettifor s structure maps additional remarks. We have seen that in a phenomenological approach to the systematics of the crystal structures (and of other phase properties) several types of coordinates, derived from physical atomic properties, have been used for the preparation of (two-, three-dimensional) stability maps. Differences, sums, ratios of properties such as electronegativities, atomic radii and valence-electron numbers have been used. These variables, however, as stressed, for instance, by Villars et al. (1989) do not always clearly differentiate between chemically different atoms. 

The most famous example of the crystal structure correlating with the average number of valence electrons per atom or band filling, N, is the Hume-Rothery alloy system of noble metals with sp-valent elements, such as Zn, Al, Si, Ge, and Sn. Assuming that Cu and Ag have a valence of 1, then the fee -phase is found to extend to a value of N around 1.38, the bcc / -phase to be stabilized around 1.48, the -phase around 1.62, and the hep e-phase around 1.75, as illustrated for the specific case of Cu-Zn alloys in Fig. 6.15. In 1936 Mott and Jones pointed out that the fee and bcc electron per atom ratios correlate with the number of electrons required for a free-electron Fermi sphere first to make contact with the fee and bcc Brillouin zone faces. The corresponding values of the Fermi vector, fcF, are given by... 

The lower halides of niobium and tantalum consist of tightly bound clusters of metal atoms, with metal-metal distances close to those found in the metal. They contain ions with average oxidation numbers between +III and +1 (Table 43). Their size depends on the valence electron concentrations (VEC) that are available on the metal atoms for M—M bonding, and on the halide-metal ratio.644 Several reviews have been devoted to the clusters of early transition metals.3,643... 

HL ME-KOTHERY RULES. When alloy systems form distinct phases, it is found that the ratio or the number of valence electrons to the numher of atoms is characteristic of Lhe phase (e.g., /), y-. s- whatever the actual elements making up the alluy. Thus, both Na, Phs and NisZiii are y-structures, with the electron-atom ratio 21 13. The rules are explained by the tendency to form a structure in which all the Brillouin rones are nearly Tull, or else entirely empty. 

To demonstrate the importance of the golden ratio it is assumed that protons and neutrons occur in the nucleus on three-dimensional spirals of opposite chirality, and balanced in the ratio Z/N = r, about a central point. The overall ratio for all nuclides, invariably bigger than r, means that a number of protons, equal to Z — Nt, will be left over when all neutrons are in place on the neutron spiral. These excess protons form a sheath around the central spiral region, analogous to the valence-electron mantle around the atomic core. The neutron spiral is sufficient to moderate the coulomb repulsion while the surface layer of protons enhances the attraction on the extranuclear electrons. 

Covalent interaction in diatomic molecules depends on the golden mean t, the interatomic distance d and the radius ratio x r /r2 of the constituent atoms, as summarized in Figure 5.6. The golden mean is a universal constant that matches the geometry and topology of space-time, the radius ratio is a known function of atomic number and dl relates to the optimal wave-mechanical distribution of valence-electron density in the diatomic system. 

However, because pure metals and alloys with the same number of valence electrons per atom tend to have the same structure, for a given structure, the density of states at the Fermi energy is a periodic function of the valence electron per atom ratio, which is discussed more in Section 4.4.2. For the Fermi energy, the corresponding N Ep) is given by ... 

Ti) solid solution and simple transition metal nitrides are classified using the radius ratio of nonmetal to metal atoms and the number of valence electrons. The relationship of the generalized number of valence electrons instead of the average number of valence electrons per atom to the thermal stability of transition metal nitride has been discussed. 

In order to examine the effect of the number of 3d electrons on the x-ray intensity ratios, we have calculated the KPfKa ratios for Cr atom as a function of the number of 3d electrons with the HS code. The calculations were made by removing the 3d electron one by one from the valence electron configuration of the Cr atom in the ground state, (3d) (4s). The results are given in Table 9 and it is found that the decrease in the number of 3d electrons increases the KpiKa ratio. This fact indicates that the KPjKa ratio by EC is smaller than that by PI if the parent Mn atom in EC and the... 

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