THE NUMBER OF VALENCE ELECTRONS AND THE STRUCTURE OF METALS AND INTERMETALLIC COMPOUNDS¹

As is known, metals can form compounds with one another whose formula, at first glance, has nothing in common with chemical valence, for example: NaCd₂, KHg₁₃, Cu₅Zn₈, Cu₃Sn₈, Cu₉Al₄, Fe₃Zn₂₁. It is also interesting that, say, the last four compounds, though possessing apparently quite different formulas, have very much in common in their properties and structure and belong to one class of so-called γ-phases.

Considering binary alloys—for definiteness we shall speak of the zinc–copper system—one can find the following. The first phase, richest in copper, is a face-centered lattice; with an increased zinc content a body-centered cubic β-phase arises, then follows the γ-phase (a complex cubic lattice), the ε-phase, which is a close-packed hexagonal lattice with an axial ratio less than $(8/3)^{1/2}$. Next comes the η-phase, which is likewise a close-packed hexagonal lattice, but with an axial ratio greater than the above value. This phase passes into pure zinc. Analogous phases have also been found in other alloys, and there is always a similarity in properties and structure between the phases.

The first explanation of these regularities was proposed by Hume-Rothery in 1926. This investigator observed that in all beta phases the ratio of the number of valence electrons to the number of atoms contained in the elementary cell of the crystal is the same for different intermetallic compounds and is equal to $3/2$.

For example:

\[ \mathrm{CuSn} : (2+1)/2=\frac{3}{2};\quad \mathrm{Cu_3Al} : (3+3)/4=\frac{3}{2};\quad \mathrm{Cu_5Sn} : (5+4)/6=\frac{3}{2}. \]

This rule was generalized in subsequent years. It turned out that for other phases this ratio is likewise constant, namely, it is equal to \(21/13\) for the gamma phase, and \(7/4\) for the \(\varepsilon\)-phase (the \(\alpha\)- and \(\eta\)-phases are solid solutions and are not subject to consideration). A very large number of compounds obey this simple rule. Exceptions are encountered quite often among beta phases. A whole series of alloys has a Hume-Rothery ratio equal to 1; 2; 2.5; 0.5; 4. This, however, is not surprising, since beta phases possess a very simple structure of the body-centered cell, while other phases possess very complex cells with a large number of structural units in the cell. It is not difficult to imagine that in elementary structures the dimensions of the atomic radii are much more direct factors of crystallization than the Hume-Rothery ratio. In complexly constructed phases this ratio is the determining one.

The connection of the empirical Hume-Rothery rule with the theory of metallic crystals was shown by Jones. The reciprocal-lattice space of a crystal is at the same time the space of the moments of the electrons. In this space one can construct, for any index of a reciprocal-lattice point, the so-called Brillouin polyhedra. Taking into account that two electrons occur per volume \(h^3\) of this space, one can calculate the number of electrons falling within the polyhedron. On the other hand, it follows from theoretical considerations that the number of electrons in the cell must be determined by those polyhedra which are constructed for reciprocal-lattice points having the largest structural factors for electron beams. Knowing the results of X-ray structural studies of alloys, and assuming that the maximum structural factors for X-rays and electron beams coincide in general, it was possible theoretically to calculate the Hume-Rothery ratio. The agreement proved to be rather good. However, the author of the paper under review considers it not entirely satisfactory, since in the majority of cases the calculated theoretical number was, as a rule, smaller than that given by the Hume-Rothery rule.

It should be remembered that the Hume-Rothery rule is built upon a definite system of valences of metals. In general this is a perfectly natural system, if one does not count the circumstance that it was necessary to assign zero valence to the transition elements—manganese, iron, cobalt, and nickel.

The author of the paper under review, as early as 1938,\(^2\) proposed an entirely different system of valences of metallic atoms, based on a certain theoretical interpretation of the magnetic moments of ferromagnetic metals. It was accepted that 9 orbitals (among them five \(3d\)-orbitals) are hybridized. In a metallic bond the \(4s\) and \(3d\) electrons take part in this way. According to Pauling, in elements 19–23 the bond is effected by one, two ... five electrons. In elements 24–28 the bond is effected by 5.78 electrons, in element 29—by 5.44, and further by 4.44, 3.44, and 2.44 electrons. The justification of these figures is given in the cited work.

The counting of valence electrons according to this scheme leads, of course, to different ratios; however, as the author indicates, the number of valence electrons per one elementary cell is in better agreement with the numbers obtained from Brillouin polyhedra if the numbers of valence electrons are counted according to his scheme.

It turns out that in most structures one can find several strong interferences whose indices may serve for constructing Brillouin polyhedra. If one of these polyhedra gives a number of electrons in good agreement with the previous valence scheme, then another polyhedron agrees well with Pauling’s numbers.

The author believes that this double agreement cannot be considered accidental. For what is involved is the agreement of these two valence schemes with such a,

…would be a remote thing, like the intensity of X-ray reflections from certain crystallographic planes. However, the reasons why both valence schemes give results coinciding with experiment, and the interrelation between the two valence schemes, the author considers unclear and awaits the resolution of this question from the further development of the theory of the electronic structure of the metal.

This double coincidence is illustrated by the example of several gamma alloys, beta alloys, and also alpha- and beta-manganese.

In order to be able to form an idea of the character of this coincidence, we shall give an example of one of the most important structures: gamma brass.

As is evident from the table, the first series of strong interferences occurs at $\Sigma h^2$ equal to 18. In the Brillouin polyhedron bounded by the planes 330 and 411 there is an electron density corresponding to 90 electrons per elementary cell. The next series of strong interferences occurs at $\Sigma h^2$ equal to 36. The authors constructed a polyhedron on the planes 600 and 442. The calculated density gives the number 255.60 electrons per elementary cell. The cell contains 4 molecules of $\mathrm{Cu}_5\mathrm{Zn}_8$. According to Pauling’s scheme, copper has valence 5.44 and zinc 4.44. This gives 250.88 electrons per elementary cell. The authors emphasize that in the new scheme the polyhedron is filled to 98.5%, and in the old one to 93.5%. It is thought that this distinction is illusory. But in any case, the coincidence with the two theoretical valence schemes is evident.

The authors, more decisively than has been done hitherto, emphasize that the structure of beta alloys is explained neither by the old nor by the new valence schemes. They consider that in these very simple structures the determining factor is geometry. For example, for beta brass the number of valence electrons is equal to 3 according to the old scheme and to 9.88 according to the new one (per molecule). The Brillouin polyhedron for a body-centered cell contains 4, 8, 12 ... electrons. For structures of this type the polyhedron will contain 2, 4, 6, 8, 10 ... electrons. The first structure would occur if the copper and zinc atoms were distributed over the points \(000\) and \(\frac{1}{2}\ \frac{1}{2}\ \frac{1}{2}\) at random; the second, if they were distributed in an ordered manner. In any case, Hume-Rothery’s figure 3 does not fit.

The paper under review is, in a certain sense, a continuation of the work already set forth in our journal[^3], where Pauling applied the valence scheme he proposed to the calculation of interatomic distances.

A. K.

CITED LITERATURE

1. L. Pauling and F. Ewing, Rev. Mod. Phys. 20, 112 (1948).
2. L. Pauling, Phys. Rev. 54, 899 (1938).
3. A. I. Kitaigorodskii, UFN, 33, issue 3, 443 (1947).