Ratio of valency electrons to atoms

The Ratio of Valence Electrons to Atoms in Metals and Intermetallic Compounds ... 

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 ... 

For the Y-alloys the ratio of valence electrons to atoms has the surprising value... 

Hume-Rothery (1926) pointed out that the appearance of a particular phase depended on the ratio of valency electrons to atoms (see Table 21). For a / -phase it is. ... 

RATIO OF VALENCY ELECTRONS TO ATOMS FOR SOME ALLOYS... 

A striking feature of this table is the variety of formulae of alloys with a particular structure. Hume-Rothery first pointed out that these formulae could be accounted for if we assume that the appearance of a particular structure is determined by the ratio of valence electrons to atoms. Thus for all the formulae in the first two columns we have an electron atom ratio of 3 2, for the third column 21 13, and for the fourth 7 4, if we assume the normal numbers of valence electrons for all the atoms except the triads in Group VIII of the Periodic Table. These fit into the scheme only if we assume that they contribute no valence electrons, as may be seen from the following examples ... 

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. 

The alloy /J NiAl is a solid with a CsCl-type structure in which one atom is located at the corners, and the second atom at the center of the unit cell. The valence-electron to atom ratio is often quoted as 1.5, using a counting scheme in which the transition metal has zero valence and the A1 is considered as trivalent. 

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... 

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. 

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 ... 

An interesting area still under debate in the field of metallurgy is the consequences of Fermi surface topology on the phase equilibria in alloy systems. Elucidation of the connection between these two, seemingly unrelated, features started with the work of William Hume-Rothery, who reported that the critical-valence electron to atom ratios. 

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. 

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. 

The answer to this question was provided by Hume-Rothery, who in 1926 made the empirical observation that the widespread occurrence of the / , y and e phases in chemically dissimilar systems and at widely differing compositions is determined, not by the chemical properties of the elements concerned, or by any arguments based on valency concepts, but solely by the relative number of valency electrons and atoms in the crystal structure. This generalization, usually known as Hume-Rothery s rule, is illustrated by the data of table 13.03, from which it will be seen that the / , y and e phases are characterized by electron atom ratios of 3 2, 21 13 and 7 4, respectively. In each case this ratio alone determines the structure, and the relative number of atoms and the particular atoms by which the electrons are contributed appear to be of... 

Inspecting the structure of copper-zinc alloys, Hume-Rothery observed that the transformation between different phases followed a change in the ratio of the number of valence electrons to the number of atoms in the Wigner-Seitz cell (Hume-Rothery, ... 

The study of QC surfaces has led to interest in the surfaces of related CMAs. QCs typically exist in a narrow composition region of the phase diagram due to the Hume-Rothery constraint of specific valence electron to atom ratio, which is related to electronic stabilization of QCs [196-198]. In the neighborhood of this composition region, phases with giant unit cells and local atomic order related to that of the QC can usually be found. The surfaces of these approximant phases offer the possibility of exploring surface structure and properties as a function of increasing complexity, which can be most simply defined in terms of atoms f>er unit cell. 

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. 

Alternatively, the effects of valency may be felt through the decrease in stacking fault energy (SFE) of fee alloys having increasing electron to atom ratio (14). 

Combination with oxygen. On the basis of the electronic theory of valency the meaning of the term has been extended to include all reactions in which there occurs an increase in the ratio of the electronegative to the electropositive atoms or groups of a substance. The controlled oxidation of natural rubber produces resinous substances called Rubbones. 

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. 

Classical Free-Electron Theory, Classical free-electron theory assumes the valence electrons to be virtually free everywhere in the metal. The periodic lattice field of the positively charged ions is evened out into a uniform potential inside the metal. The major assumptions of this model are that (1) an electron can pass from one atom to another, and (2) in the absence of an electric field, electrons move randomly in all directions and their movements obey the laws of classical mechanics and the kinetic theory of gases. In an electric field, electrons drift toward the positive direction of the field, producing an electric current in the metal. The two main successes of classical free-electron theory are that (1) it provides an explanation of the high electronic and thermal conductivities of metals in terms of the ease with which the free electrons could move, and (2) it provides an explanation of the Wiedemann-Franz law, which states that at a given temperature T, the ratio of the electrical (cr) to the thermal (k) conductivities should be the same for all metals, in near agreement with experiment ... 

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