19 May 1954

Dr. Fred Ewing

112 East Los Flores

Altadena, Calif.

Dear Fred:

I am writing to ask if you would think over the question of revision of the system of relating bond length in alloys to bond number and valence, and calculating single-bond radii for metals. First, inasmuch as I have abandoned the use of the saturation magnetic moment as a simple method of determining the valence in iron and other metals, some way is needed to evaluate the number of metallic orbitals per atom, and consequently the valence in metals such as copper. I have obtained one value, for alloys of nickel and copper, by assuming that the composition of nickel-copper alloys at which the ferromagnetic moment just reaches zero corresponds to an electronic structure with 6 valence electrons per atom and pairs of electrons occupying the remaining orbitals, equal in number to 3 minus the number of metallic orbitals. Extrapolation of the curve of saturation magnetic moment of alloys versus average electron number of the atoms in nickel-copper alloys leads to the composition 57 percent copper for this point. There are accordingly 10.57 electrons altogether, or 4.57 more than the 6 valence electrons, that is, 2.285 electron pairs. I assume accordingly that there is 0.715 metallic orbital per atom. This leads to the valences 5.57 for copper, 4.57 for zinc, etc., instead of the earlier values 5.44, 4.44, etc.

Here the assumption is made that the number of metallic orbitals per atom remains constant. I think that we need to have some substantiation.

Perhaps one way of deciding is to examine the magnetization curve. If the valence remains constant at value 6 (part of it can be due to 1-electron bonds and can contribute to the magnetic moment), and the number of metallic orbitals remains constant, the magnetization curve should be symmetric between chromium and the nickel-copper alloy; that is, its maximum should be at electron number 8.285, and the curve should have a vertical plane of symmetry at this point. It probably would be worth while to examine the data, such as those given by Bozorth, to see to what extent this is true. I remember having decided that the curve was rounded between a point about one quarter and another about one half way from iron to cobalt. The point midway between a quarter and one half is at 0.375, rather than 0.285. However, my memory on this point is not exact.

Perhaps you can think of some other war of checking the assumption that the number or metallic orbitals remains constant.

Next we have the question of the value of the coefficient to use in the equation connecting bond length and bond number. I had decided to use the value 0.600, the change from 0.700, which holds for the bond-order equation, being made to correct for the stabilizing, and hence bond-shortening, effect of resonance energy when the bond number is less than 1. Verner Schomaker has contended, from the consideration of boron compounds, that the constant should be much less, perhaps 0.30. I have made some studies that indicate that 0.50 may be about right, but I think that we should try to find as many arguments in support of the value finally accepted as possible.

One way of finding the value of the coefficient is to make use of the bond-order equation with a correction for resonance energy. In the Proc. Roy. Soc. paper there is an argument to the effect that in the alkali metals resonance energy contributes an amount of stabilization equal to the bond energy. Accordingly the bond number must be doubled to obtain the bond order. I calculate that with ligancy 12 and bond order 1/6 use of the bond-order equation leads to a lengthening which would be given by the bond-number equation with bond number 1/12 by placing the coefficient equal to 0.50. A rough calculation that I have just made for the body-centered structure gives 0.48 as the coefficient. Perhaps the calculation for the body-centered structure should be made for each of the alkali metals, using the values for resonance energy given in the Proc. Roy. Soc. paper.

I have tried to extend this argument to magnesium and aluminum. The calculation is attended by greater uncertainties, but values obtained are about 0.5.

Another way of evaluating the coefficient is by the consideration of white tin. The value of the single-bond radius for sp^{3} tin is given by gray tin as 1.399 A. In white tin, with valence 2.57, there is resonance between the bivalent stats, forming p bonds, and the quadrivalent state, forming sp bonds, the latter contributing 28 1/2 percent. The single-bond radius for p bonds is estimated from the sequence antimony, selenium, iodine to be 1.45 of the 2.57 bonds formed by an atom, the quadrivalent state contributes 1.14, or 44.3 percent. The radius to use for the tin atom in white tin is accordingly about 1.427 A. This value, with the bond distances in white tin, leads to approximately 0.50 as the value of the coefficient.

Perhaps the same argument can be extended to indium. The alloy of indium and tin, with 80 percent tin, which has the simple hexagonal structure, and is described in the last Acta, might also be used. There is, however, some uncertainty as to the effect of d character on the radius for indium.

Another method that may be valuable is to compare the A1 and A3 structures with the A2 structures for the elements which exist in closest packing and also body-centered arrangements. I have checked up on titanium and zirconium, which lead to values about 0.5 for the coefficient. For iron a recent value of the lattice constant for the A1 structure at room temperature, obtained by friction on the surface of ordinary iron, is 3.60 A. This also gives a value of about 0.5, but the calculation is very sensitive to the exact values of the lattice constants. Lithium and sodium have been reported by Barrett to change to closest packing at very low temperatures, under the influence of mechanical strain. Unfortunately the lattice constants that be gives have an indicated error of 0.03 A, which makes them valueless for this purpose. I think, although I do not have the reference, that barium has been reported in both forms. A check of lattice constants for metals should be made, and might disclose some comparisons that might be valuable in this respect. I think that there may well be other methods of attacking the problem. Perhaps you would be interested in looking for them.

Sincerely yours,

Linus Pauling:W

P.S. Here is some additional information.

I have a reference for lithium under shear, -196º C, A1, A_{0} = 4.41 A, C. S. Barrett, Phys. Rev., 72, 245 (1947). Structure Reports 11 gives some additional information, including A3 and A2, page 152. Also Structure Reports 11, page 182, gives information about A1 for sodium.

Thallium A2 is said to have a_{0} = 3.874 A, by H. Lipson and A. R. Stokes, Nature, 148, 437 (1941). This may be compared with the value 3.423 A for interatomic distance in A1, and 3.427 A (average) for A3.

Perhaps a search should be made of recent literature to see what the
best values of the lattice constants of metals are.

L.P.