states mixing with $3s$, $3p$ states of B and C atoms, which is outside our interest.

![ Absorption coefficient $A(\omega)$ (lower panel) in comparison with the $L_{\rm III}$-edge fluorescence spectra (upper panel).$^{17)}$ \[fig.absorp\]](fig.print.4.eps){width="8.0cm"}

Now we discuss the RXS spectra. The resolvent above calculated is used to calculate the scattering amplitude at the origin with the help of the relation, $$\begin{aligned}
&  \sum_{\Lambda}\frac{|\Lambda\rangle\langle\Lambda|}
       {\hbar\omega-(E_{\Lambda}-E_j)+i\Gamma} \nonumber \\
  &= \sum_{m^ds^d\lambda}\sum_{m'^ds'^d\lambda'}
|m^ds^d\lambda\rangle \nonumber \\
& \times 
 \left(\frac{1}{\hbar\omega - H_{\rm int} + i\Gamma}\right)_
{m^ds^d\lambda;m'^ds'^d\lambda'}
  \langle m'^ds'^d\lambda'| .
\label{eq.intermed}\end{aligned}$$ This expression is independent of the quadrupole ordering $4f$ states. It is inserted into eq. (\[eq.dipole\]) to calculate an RXS amplitude. The extension to general site $j$ is straightforward. In the coordinate frame fixed to crystal (not in the local coordinate frames), their forms are given by $$\begin{aligned}
\hat M(1,\omega) &=
\left( \begin{array}{ccc}
          \xi(\omega)  & \eta(\omega)  & 0 \\
          \eta(\omega) & \zeta(\omega) & 0 \\
          0            & 0             & \gamma(\omega)
       \end{array}
\right), \nonumber \\ % \label{eq.Mamp1}
%\end{equation}
%\begin{equation}
\hat M(2,\omega) &=
\left( \begin{array}{ccc}
          \zeta(\omega)  & \eta(\omega) & 0 \\
          \eta(\omega)   & \xi(\omega)  & 0 \\
          0              & 0            & \gamma(\omega)
       \end{array}
\right), \nonumber \\ %\label{eq.Mamp2}
%\end{equation}
%\begin{equation}
\hat M(3,\omega) &=
\left( \begin{array}{ccc}
          \zeta(\omega)   & -\delta(\omega) & 0 