at the $M_4$ edge.

![\[fig.alpha\_2\] Energy profiles $|\alpha_2(\omega)|^2$ at the Np $M_4$ (right panels) and $M_5$ (left panels) edges. The lines represent the calculated results for $\Gamma=0.01, 0.5$ and $2.0$ eV, respectively, from bottom to top panels. In the top right panel, The open circles and crosses are experimental data in NpO$_2$.[@Paixao02; @Caciuffo03] The peak heights of them are adjusted to that of the calculated value. ](Fig.5.eps){width="8.0cm"}

The energy profile in the dipole ordering is given by the sum of the dipole and quadrupole terms. However, $|\alpha_1(\omega)|^2$ is about two orders of magnitude larger than $|\alpha_2(\omega)|^2$. For instance, $|\alpha_1(\omega)|^2 \sim 192 \times |\alpha_2(\omega)|^2$ when $\Gamma=2.0$ eV. Thus the dipole term usually dominates the quadrupole term. Although the dipole ordering is ruled out from experiments, we show $|\alpha_1(\omega)|^2$ in Fig. \[fig.alpha\_1\] as a reference. The peak at the $M_4$ edge with $\Gamma=2$ eV is at 3847.5 eV, 0.7 eV higher than the peak position of $|\alpha_2(\omega)|^2$. Note that the spectral shape at the $M_5$ edge depends on $\Gamma$ more sensitive than that at the $M_4$ edge.

![\[fig.alpha\_1\] Energy profiles $|\alpha_1(\omega)|^2$ at the Np $M_4$ (right panels) and $M_5$ (left panels) edges. The lines represent the calculated results for $\Gamma=0.01, 0.5$ and 