The Theoretical Constitution of Metallic Beryllium

Abstract

Calculations of total energy as a function of lattice constant, and of some other properties, have been made for metallic beryllium, by a self-consistent field method. Since such calculations have not been made before for any but monovalent metals, the principal object was to find out how far the various assumptions which are usually made in them remain valid for higher valencies, and to test the practicability of certain methods of making the calculations. The theoretical values of binding energy, lattice constant, and compressibility agree reasonably well with experiment. The calculated work function, however, could be made to coincide with the experimental value only by assuming a surface double layer of over 5 volts, which seems impossibly large. This suggests a very large deviation of the exchange energy from the value for completely free electrons. An investigation of the behavior of the exchange energy yields an expression for the deviation from the free electron value which is valid for low electron densities, but not for those which occur in beryllium. The distribution of the electronic states in energy is found to be of the sort needed to account for the diamagnetism of beryllium. Concerning methods of calculation, it is shown that a rather complicated procedure is necessary to obtain quantitative results when a Hartree ion core field is used (as was done in the present case), and that construction of an empirical field is preferable. The assumption ${E}_{k}={E}_{0}+\frac{\ensuremath{\alpha}{\ensuremath{\hbar}}^{2}{k}^{2}}{2m}$, for the energy of an electron with wave vector $k$, cannot be used for calculations of lattice constant or compressibility for a divalent metal; it is therefore necessary to calculate directly the energies of states near the Fermi surface. This was done by the "orthogonalized plane wave" method, which is shown by tests to be capable of fairly high accuracy, though laborious. This method suggests a simple qualitative way of understanding a number of features of the electronic energy spectrum of a metal and its manner of variation with lattice constant. Incidental results include a proof that the interaction of the $1s$ shells is entirely negligible, and a calculation of the electrostatic interaction energy of the ions as a function of the $\frac{c}{a}$ ratio.