Atomic Radii and Interatomic Distances in Metals

The basis of this work is a viewpoint previously suggested by the author, according to which the metallic bond can be considered as a resonant covalent bond. This means that each atom is bonded by pairs of electrons "in turn" with all its neighbors. The nature of the metallic bond, according to this conception, is determined by the bond order and is equal to the ratio of the number of valence electrons to the number of a given atom's bonded neighbors:

\[ n = \frac{\nu}{W} \]

For transition elements, the valence number \(\nu\) may be fractional. For example, in the case of iron, it is accepted that 6.78 of its electrons participate in the formation of pairing bonds. This number is obtained by subtracting 2.22 unpaired electrons (the magnetic moment of iron is 2.22 Bohr magnetons) from a total of eight. Similar considerations yield other fractional valence values appearing in the table of elements given below.

In the work being reviewed, a semi-empirical equation is proposed, relating the bond order to the atomic radius. In the reasoning leading to this equation, the known dependence of covalent atomic radius on bond order for such atoms as carbon, nitrogen, etc., is taken into account. The shortening of the bond due to the stabilizing effect is also considered.

) See also the works of J. A. Wheeler and comments by Pauli in a number of his recent papers.
*) L. Pauling, J. Am. Chem. Soc., 69, 542, 1947.

METALLIC RADII OF THE ELEMENTS

ATOMIC RADII AND INTERATOMIC DISTANCES IN METALS

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 monovalent” radius, and \(R(n)\) is the radius for a bond whose order is equal to \(n\). The formula given thus expresses the shortening of a bond when its order changes. It should be emphasized that this formula is based on facts concerning distances between atoms bound by different numbers of electron pairs. Pauling, despite this, 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) at the same number of valence electrons.

Since a considerable number of elements possess a structure with coordination number 8 (body-centered cube), Pauling first of all develops an empirical method for passing from the interatomic distances in this structure to the 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 monovalent 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 discussed in detail in the text, each separately.

Of interest are the author’s arguments concerning manganese, which, as is known, crystallizes in three modifications. But none of these modifications gives the valence 5.78. By interpolation between the values for chromium and iron we arrive at the value \(1.168\,\text{\AA}\) for the metallic monovalent radius. The closest and simplest structure of manganese is a close-packed cubic packing (with a small 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 equal to 4.16. Considering now the structure of \(\beta\)-manganese (20 atoms in the unit cell), in which there are two kinds of crystallographically distinct atoms, we find, from empirical distances for atoms of one kind, a valence of 5.88, and for atoms of the other kind, a valence of 4.00. Thus, the author concludes, in the unit 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 developed point of view. In an analogous manner the structure of \(\alpha\)-manganese and of other “anomalous” elements with respect to crystalline structure is considered.

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

The table given and Pauling’s equation may be used either for finding the orders of bonds whose sum may be equated to the expected valence (for this calculation one uses the value of the monovalent radius and the experimental interatomic distances), or for comparing experimental values of interatomic distances with the sum of 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 then considers the dependence of the new concept he has introduced—the “metallic monovalent radius”—on the atomic number. The purpose of this

FROM CURRENT LITERATURE

The analysis demonstrates that the values of the radii \( r^{(1)} \) are closely related to the values of the normal covalent radii, as well as to the tetrahedral and octahedral radii of elements. In establishing this connection, Pauling sees support for his view of metallic bonding as resonating covalent. The consideration shows that, for example, the curve of the metallic univalent radii of the elements of the two short periods continuously transitions into the curve of the normal covalent radii. Furthermore, it 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 can 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 paper—"The Nature of Interatomic Forces in Metals" (Phys. Rev., 54, 899, 1938). He hopes to continue this research further, believing that he will manage to clarify relative atomic sizes, the stressed bonds between atoms, the Hume-Rothery ratio, and similar questions.

A. I. Kitaigorodsky