ATOMIC RADII AND INTERATOMIC DISTANCES IN METALS **)

This work is based on a point of view previously proposed by the author, according to which the metallic bond may be regarded as a resonating covalent bond. This means that each atom is bonded by a pair of electrons “in turn” with all its neighbors. The character of the metallic bond is determined, according to this concept, by the order of the bond \(n\), equal to the quotient obtained by dividing the number of valence electrons by the number of neighbors bonded to the given atom:

\[ n=\frac{v}{N}. \]

For transition elements the valence number \(v\) may be fractional. Thus, for iron it is assumed that 5.78 of its electrons take part in forming this bond. This number is calculated by subtracting from the total number 2.22 unpaired electrons (the magnetic moment of iron is equal to 2.22 Bohr magnetons). From similar considerations there are obtained the other fractional valence values appearing in the table of elements given below.

In the paper under review a semiempirical equation is proposed which relates the bond order to the atomic radius. In the reasoning leading to this equation, the well-known data on atoms such as carbon, nitrogen, etc., on the course of the atomic covalent radius with the bond order are taken into account. The shortening of the bond due to the stabilizing

* Cf. also the works of J. A. Wheeler and Pauli’s remarks in a number of his recent works.
*) L. Pauling, J. Am. Chem. Soc., 69*, 542, 1947.

METALLIC RADII OF THE ELEMENTS

to the resonance effect (by analogy with resonance in molecules). The equation has the form

\[ R(1)-R(n)=0.300\cdot \log n, \]

where \(R(1)\) is the “metallic univalent” radius, and \(R(n)\) is the radius for a bond whose order is \(n\). Thus the formula given yields the shortening of a bond when its order is changed. It should be emphasized that this formula is based on facts concerning distances between atoms bound by different numbers of electron pairs. Pauling, nevertheless, considers it possible to apply this equality to a metallic bond, where the number \(n\) changes owing to a change in the number of neighbors of the given atom (the coordination number), with the same number of valence electrons.

Since a considerable number of elements possess a structure with coordination number 8 (body-centered cube), Pauling first develops an empirical method for passing from interatomic distances in this structure to a radius for coordination number 12 \([R(k.\mathrm{ch}.12)]\).

In the table the author gives, for each element, its valence number \(v\), the radius for coordination number 12, and the “metallic univalent radius,” calculated from the equation given above. As for the radius for coordination number 12, this quantity is taken either directly from experiment (if the normal coordination of the element is equal or close to this number) or is recalculated by the method developed by the author. All elements whose crystal structures are unique are considered in the text in detail, each separately.

Of interest are the author’s arguments concerning manganese, which, as is known, crystallizes in three modifications. None of these modifications gives a valence of 5.78. By interpolating between the values for chromium and iron we arrive at a value of \(1.168\,\text{\AA}\) for the metallic univalent radius. The simplest structure of manganese is a close-packed cubic arrangement (with a slight tetragonal distortion). From the interatomic distances for this structure we obtain a radius for coordination number 12 equal to \(1.306\,\text{\AA}\). From the values of these two radii we find that the valence \(v\) is 4.16. Considering now the structure of \(\beta\)-manganese (20 atoms in the unit cell), in which there exist two kinds of crystallographically different atoms, we find, by empirical distances, a valence of 5.88 for atoms of one kind and a valence of 4.00 for atoms of the other kind. Thus, the author concludes, in the cell of \(\beta\)-manganese there are 8 atoms of small size and high valence and 12 atoms of large size and low valence. The geometrical details of the structure become clear if one proceeds from the point of view developed. In an analogous way the structure of \(\alpha\)-manganese and of other “abnormal” elements in the sense of crystal structure is considered.

The author shows, using the examples of cementite and the compound AuSn, how, by making use of the values of metallic univalent radii, one can predict the arrangement of atoms in these compounds.

The table and Pauling’s equation given may be used either to find the orders of bonds whose sum is expected to be equal to the expected valence (for this calculation one uses the value of the univalent radius and experimental interatomic distances), or to compare experimental values of interatomic distances with the sum of the radii for the corresponding coordination numbers. Assuming in Pauling’s formula that the valence is the same, we can without difficulty calculate the radius for any coordination number if the radius for coordination number 12 is known.

The author next considers the dependence of the new concept he introduced, the “metallic univalent radius,” on atomic number. The aim of this

consideration shows that the values of the radii \(R(1)\) are closely connected with the values of the normal covalent radii, and of the tetrahedral and octahedral radii of the elements. Pauling sees in the establishment of this connection a confirmation of his view of the possibility of considering the metallic bond as a resonating covalent bond. The discussion shows that, for example, the curve of the metallic monovalent radii of the elements of two short periods passes continuously into the curve of the normal covalent radii. It further turns out that a straight line drawn through the values of the tetrahedral radii of the elements of the first long period passes through the metallic radius of calcium. This may be regarded as confirmation of the \(sp\)-character of the bond in metallic calcium.

The author regards this work as a continuation of his 1938 work, “The Nature of Interatomic Forces in Metals” (Phys. Rev., 54, 899, 1938). The author intends to continue this investigation, believing that he will succeed in explaining relative atomic sizes, directed bonds between atoms, the Hume-Rothery relationship, etc.

A. I. Kitaigorodskii