A test of the spin-orbit sum rule for actinides using an ab-initio calculation

The expectation value of the angular part of the spin-orbit-coupling (SOC) operator $<l\cdot s>$ and the branching ratio for the $N_{4,5}$ X-ray-absorption spectroscopy in $\delta$-Pu, $\alpha$-U and face-centered-cubic (fcc) Th are calculated within the framework of the density-functional theory in combination with the local-spin-density approximation (LSDA), the generalized-gradient approximation (GGA) and the LDA+$U$ method. The SOC contribution is calculated in terms of the first- and the second- variational schemes. A strong variation in the magnitudes of the calculated $<l\cdot s>$ operator along the actinide series is found. The results show that the SOC sum rule for $\delta$-Pu and $\alpha$-U is reasonably valid, whereas there is a sign mismatch for the case of fcc Th. The calculated branching ratios for the case without SOC in the valence shell strongly deviate from the statistical value.