Theory of the electronic structure of the alloys of the actinides

An earlier theory of the electronic structure of the actinides is improved, generalized to alloys of actinides, and applied to plutonium and its alloys with other trivalent metals. The theory combines the s and d electrons as free electrons, treating them to second order in an empty-core pseudopotential. The f levels produce bands in this theory, which are actually broadened by electron correlations, but their role in the bonding is decreased. The effects both of the coupling between neighboring f shells and of the correlation are included in a second moment in terms of which the total energy is estimated using a Friedel model. This allows all contributions to be rewritten in terms of near-neighbor interactions and on-site terms. It is found that f-shell contribution alone favors a much smaller spacing than the free-electron contribution alone, and the competition makes the resulting structure very sensitive to the parameters of the theory. Adjustment of the parameters seems essential. An f-shell radius is adjusted, within the range of previous estimates, so that the minimum in the f-shell contributions for plutonium comes at a spacing of 2.5 \AA{}, and the total is scaled back such that the minimum is of the same depth. The same scaling factors are applied to other actinides. Then the pseudopotential core radius is adjusted for each actinide and trivalent simple metal in the face-centered-cubic structure to yield the observed atomic volume (that for the delta structure in the case of plutonium). The resulting electronic structure is tested successfully by using it to predict the bulk modulus of all of these metals. Using the same parameters for plutonium, the energy is calculated in the \ensuremath{\alpha} structure, found to have higher energy than the fcc structure at the atomic volume of \ensuremath{\delta} plutonium, but lower energy at a reduced volume, near that of the observed low-temperature \ensuremath{\alpha} structure. The same formulas and parameters are applicable to random alloys of any of these metals, evaluating each term in an alloy of x atomic fraction of B in A with the appropriate weighting of x, $x(1\ensuremath{-}x),$ or $(1\ensuremath{-}{x)}^{2}.$ This does not predict correct variations with concentration x of the lattice spacings unless we introduce the interaction between simple-metal core d states and actinide valence d states, absent in the pure materials. A moderate scaling of previously predicted d-state radii brings the dilute-alloy spacings into accord with experiment for plutonium alloys with gallium, indium, and thallium. Aluminum, without core d states, was initially in accord and is not affected. The ordered alloys, ${\mathrm{Pu}}_{3}\mathrm{Ga},$ ${\mathrm{Pu}}_{3}\mathrm{In},$ and ${\mathrm{Pu}}_{3}\mathrm{Al},$ are found to have larger spacings than the random alloys, in accord with experiment, though ${\mathrm{Pu}}_{3}\mathrm{Al}$ is not found to be the stable structure. For alloys of two actinides, or with rare earths, the effects of d-state couplings are expected to be negligible, but additional terms in the f-state energy arise from the contribution of f levels of different energy to the global f bands. With this effect included, cerium, americium, and curium are found to increase the spacing of \ensuremath{\delta} plutonium, in accord with experiment, but the lighter actinides, as well as the type-B simple metals, Al, Ga, In, and Tl decrease the spacing, also in accord with experiment. Alloys with the type A metals Sc, Y, and La are found to increase the spacing. Initial studies of the displacement of plutonium neighbors to substituted gallium atoms showed \ensuremath{\alpha}-like reconstructions, as in pure plutonium.