Previous developments in the theory of metals may be divided clearly into two parts: that based principally upon the hypothesis of free electrons and dealing with conductiv-

An attempt is made to analyze the factors which determine the operation of the bubble chamber. It is concluded that the majority of bubbles in conventional chambers are nucleated by moderately energetic free electrons produced by the incident particles in Coulomb encounters. Nuclei are displaced too infrequently by Coulomb encounters to account for the observed densities of bubbles. The electrons deposit their kinetic energy in highly localized regions which then are the source of explosions which produce bubbles of greater than critical size in a time of the order of 10−10 or 10−11 sec. The bubbles grow subsequently by evaporation of the fluid. The temperature of the fluid should be sufficiently close to the critical temperature that the energy required to produce the bubble of critical size can be provided by an electron with a range comparable to or less than the diameter of the bubble of critical size. Otherwise the electron will be unable to localize its energy in a sufficiently small volume in any but highly improbable cases. It is also concluded that the viscosity of the liquid plays a very important role in determining the threshold energy for forming a bubble of critical size when it has a value near 1 centipoise or larger.

The present work represents an extension to a previous development by the same authors, on the theory of metallic sodium. In the first part of this paper a completely selfconsistent solution of Fock's equations for the sodium lattice is carried through indirectly, this being the approximation in which one-electron functions are employed. The question of the correlations between electrons with parallel spin is investigated quantitatively and the Fermi "zero-point energy" is calculated using the proper effective field. The results show that the electrons behave almost exactly as if they were entirely free, the binding energy being 9 kg cal and the lattice constant 4.86A, as compared with the observed values 26.9 kg cal and 4.23A. To complete the picture, the correlations between electrons with anti-parallel spins are investigated in the latter part, since these are not included in the Fock picture. A general discussion of this question is presented and a quantitative treatment of its e6'ect is made which yields a new binding energy of 23.2 kg cal and a lattice constant of 4.75A. The source of the remaining discrepancy is discussed. I. THE POTENTIAL INSIDE THE LATTICEIn a previous paper by the same authors a method of calculating the binding properties of metals was developed' and applied to sodium. The procedure employed was essentially one of so]ving the Fock system of equations' for the valence electrons (i.e. , the system of equations to which the Schrodinger equation reduces when one electron functions are assumed). This solution did not proceed from a forrnal investigation of Fock's differential equations, but was developed indirectly under the guiding principles of the picture afforded by the free electron theory.To begin with, the lattice was subdivided into polyhedrons of equal size and form, which we shall call s-polyhedrons, each of which surrounds one ion lying in its center, and is bounded by the planes which bisect, perpendicularly, the lines connecting the corresponding ion with its 14 neighbors (the alkali metals form body centered lattices). Since these polyhedrons closely resemble spheres, they may be replaced by spheres of equal volume for many purposes and these we shall designate as s-spheres, their radius being r, = (3vo/4s)', where v& is the atomic volume.Concerning the nature of the electronic states of the lattice, we know that there will be bands of allowed levels, no more than two electrons ' E. Wigner and F. Seitz, Phys. Rev. 43, 804 (1933}. J. C. Slater, Phys. Rev. 35, 210 (1930};V. Fock, Zeits. f. Physik 51, 126 (1930}, occupying each level because of the restrictions imposed by the Pauli principle, and that the lowest state in the lowest band will possess the symmetry of the lattice. From this it follows that its normal derivative will vanish on the boundaries of the s-polyhedrons and to obtain its wave function, it is only necessary to solve the Schrodinger equation within one polyhedron, by using a suitable effective field and this boundary condition. Approximate wave function...

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