Quantifying the importance of orbital over spin correlations in $\delta \text{−}\mathrm{Pu}$ within density-functional theory

Spin and orbital electron correlations are known to be important when treating the high-temperature $\delta$ phase of plutonium within the framework of density-functional theory (DFT). One of the more successful attempts to model $\delta \text{−}\mathrm{Pu}$ with this approach [P. Söderlind, Europhys. Lett. 55, 525 (2001); P. Söderlind et al., Phys. Rev. B 66, 205109 (2002); P. Söderlind and B. Sadigh, Phys. Rev. Lett. 92, 185702 (2004)] has included condensed-matter generalizations of Hund’s three rules for atoms, i.e., spin polarization, orbital polarization, and spin-orbit coupling. Here, we perform a quantitative analysis of these interactions relative rank for the bonding and electronic structure in $\delta \text{−}\mathrm{Pu}$ within the DFT model. The result is somewhat surprising in that spin-orbit coupling and orbital polarization are far more important than spin polarization for $\delta \text{−}\mathrm{Pu}$. We show that these orbital correlations on their own, without any formation of magnetic spin moments, can account for the low atomic density of the $\delta$ phase with a reasonable equation of state. In addition, this unambiguously nonmagnetic treatment produces a one-electron spectra with resonances close to the Fermi level consistent with experimental valence band photoemission spectra.

Figure 1

Results from FPLMTO total-energy calculations of $\delta \text{−}\mathrm{Pu}$. The nonmagnetic treatments (NM) refer to calculations with no spin moment and are presented for three levels of approximation. No spin-orbit coupling (NM: No SO), with spin-orbit coupling (NM: SO), and $(\mathrm{NM}:\mathrm{SO}+\mathrm{OP})$ that also adds orbital polarization. The $(\mathrm{FM}:\mathrm{SO}+\mathrm{OP})$ theory is identical to the $(\mathrm{NM}:\mathrm{SO}+\mathrm{OP})$ with the exception of allowing for the formation of ferromagnetic spin moments. The obtained equilibrium volumes are marked with dashed vertical lines and the $593\mathrm{K}$ atomic volume of $\delta \text{−}\mathrm{Pu}$, $25{\mathrm{\AA }}^3$, is marked as $V_{\delta }$.

Figure 2

(Color online) Results from LMTO-ASA total-energy calculations of $\delta \text{−}\mathrm{Pu}$ as a function of fixed spin moment, utilizing the fixed-spin-moment method. The blue (No SO) curve corresponds to calculations without spin-orbit coupling. The red (SO) includes spin-orbit interaction and the black $(\mathrm{SO}+\mathrm{OP})$ also adds orbital polarization. The calculations are performed for a fixed atomic volume, $25{\mathrm{\AA }}^3$, corresponding to the $593\mathrm{K}$ lattice constant of $\delta \text{−}\mathrm{Pu}$ $\left(4.64\mathrm{\AA }\right)$.

Figure 3

FPLMTO total DOS for $\mathrm{SO}+\mathrm{OP}$ calculations with no (NM: $\mathrm{SO}+\mathrm{OP}$) and ferromagnetic (FM: $\mathrm{SO}+\mathrm{OP}$) spin configurations. The former is shifted up an amount of $2.4\text{states}∕\mathrm{eV}$ to enable a clearer display of the results. The location of the peaks A, B, C, and D, compares well between the two models. The Fermi level is shifted to zero energy and is marked with a vertical line.

Figure 4

(Color online) Results from LMTO-ASA total-energy calculations of $\delta \text{−}\mathrm{Pu}$ (black) and fcc curium (red). The curves are shifted to zero energy for zero spin moment. The calculations are performed for atomic volumes corresponding to the experimental lattice constant for $\delta \text{−}\mathrm{Pu}$ $\left(4.64\mathrm{\AA }\right)$ and Cm $\left(4.93\mathrm{\AA }\right)$. The electronic structures include spin-orbit coupling and orbital polarization.