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On the Binding Forces in the Alkali and Alkaline Earth Metals According to the Free Electron Theory
1.See, e.g., Slater, Phys. Rev. 35, 509 (1930);
1.Taylor, Eyring, and Sherman, J. Chem. Phys. 1, 68 (1933);
1.Wigner and Seitz, paper presented at the Washington meeting of the American Physical Society, April 28, 1933,
1.Phys. Rev. 43, 1048 (1933).
1.(Just after this paper was submitted their article, Taylor, Eyring, and Sherman, Phys. Rev. 43, 804 (1933) was received.)
2.Frenkel, Zeits. f. Physik 49, 31 (1928) especially p. 42;
2.Wave Mechanics, Elementary Theory, Oxford University Press, 1932, p. 221 ff.
3.After I had worked out the consequences of these assumptions, Professor J. C. Slater told me that he had previously attempted a somewhat similar correction. He did not publish any results, because of the difficulty of explaining the compressibilities and the heats of sublimation of the various metals at the same time. I believe, however, that I have shown later in the paper that this difficulty is only an apparent one. I am very much indebted to Professors Slater and Kemble and others of the group in Cambridge for helpful discussions of this work. More recently I have had the opportunity of discussing the paper with Dr. Irving Langmuir. From him I learned that he, also, had previously made some calculations of a similar nature (though differing somewhat in details) which remain as yet unpublished. The question as to the exact physical cause of the existence of the intrinsic ionic volume is one about which it seems best not to make any definite statement at present. It has been suggested that it is due to the fact that the only way that a valence electron can get into the region near the nucleus of one of the positive ions is by being a penetrating electron, but such an electron (to speak in terms of the old quantum theory) can, on account of its high velocity while there, spend but a relatively short time in the neighborhood of the nucleus, and this region therefore acts to some extent like a region into which it cannot go. In addition to this, the electron may tend to get “promoted” out of such a penetrating orbit, and thus not go into the neighborhood of the positive nuclei at all, for such promotion, though requiring energy when the atoms are isolated, may actually result in a lowering of energy in the fairly closely packed metallic structure. It may be well to mention at this point another repulsive potential which comes in, namely, the Heitler‐London type repulsion of ions. On account of the large distances between the ions this contributes negligibly to the energy of the system, as is shown by a calculation very kindly sent to me by Dr. J. E. Mayer.
4.Simon and Vohsen, Zeits. f. Physik. Chemie 133, 165 (1928).
5.Wyckoff, The Structure of Crystals, 2nd edition, Chemical Catalog Co., 1931, p. 204.
6.Sherman, Chem. Rev. 11, 95 (1932). Table on p. 107. Our A corresponds to Sherman’s
7.Frenkel, Zeits. f. Physik 50, 234 (1928), has shown that for a system in which all the forces are electrical the kinetic energy will be minus twice the potential energy. In this case, this amounts practically to It will be observed that this is not the case in Table I. This is presumably because the introduction of an intrinsic ionic volume implies forces which for our purposes are not electrical forces.
8.(a) Sherman, reference 6, p. 138;
8.(b) ibid., p. 136.
9.The following values of obtained in the calculation, may be of interest: Li, 0.315; Na, 0.419; K, 0.465; Rb, 0.487; Cs, 0.509.
10.Bridgman, Proc. Am. Acad. 58, 202 (1923);
10.Bridgman, 60, 399, 409 (1925).
11.Landolt‐Börnstein Tabellen. The “mean values” of the densities, listed at the side of the tables, were those used.
12.Professor Slater has pointed out to me that this discrepancy is in the direction to be expected. For if we gradually separate the atoms of a metal and let the electrons go where they will, they will not be uniformly distributed through the space allotted to them, but each one will tend to attach itself to an atom. We thus do not finally arrive at a state of ions and free electrons, but we have instead simply separated atoms. The energy involved in this process is just the energy of sublimation, which is a much smaller quantity than E. If this process is already starting in the actual metal we should expect the minimum of the potential curve to be flattened out, and hence will be smaller than calculated. There is nothing in the correction made below which is inconsistent with this view of the behavior of the metal. The Heitler‐London type repulsion mentioned3 will, if taken into account, tend to increase the discrepancy in the compressibilities, but this is presumably not as important as the effect just considered.
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