\\
          -\delta(\omega) & \xi(\omega)     & 0 \\
          0               & 0               & \gamma(\omega)
       \end{array}
\right), \nonumber \\ %\label{eq.Mamp3}
%\end{equation}
%\begin{equation}
\hat M(4,\omega) &=
\left( \begin{array}{ccc}
          \xi(\omega)     & -\delta(\omega) & 0 \\
          -\delta(\omega) & \zeta(\omega)   & 0 \\
          0               & 0               & \gamma(\omega)
       \end{array}
\right). \label{eq.Mamp4}\end{aligned}$$

Owing to the factor $\exp(-i{\textbf G}\cdot{\textbf r}_j)$ in eq. (\[eq.scatensor\]), the total amplitude is given by a combination of $\hat M(1,\omega)+\hat M(2,\omega)-\hat M(3,\omega)-\hat M(4,\omega)$. Thus, we have the final form, $$\frac{\hat M({\textbf G},\omega)}{\sqrt{N}} =
\left( \begin{array}{ccc}
          0              & \eta(\omega)   & 0 \\
          \eta(\omega)   & 0              & 0 \\
          0              & 0              & 0
       \end{array}
\right).
\label{eq.final1}$$ The geometrical factors are given by setting $\beta=0$ in Table \[tab.azim\], which are combined to eq. (\[eq.final1\]) to calculate the scattering intensity. We have the RXS intensity as a function of azimuthal angle $\psi$ as $$\begin{aligned}
 I_{\sigma\to\sigma'}({\textbf G},\omega) &\propto |\eta(\omega)|^2\sin^2 2\psi,
\nonumber \\
 I_{\sigma\to\pi'}({\textbf G},\omega) &\propto
   |\eta(\omega)|^2\sin^2\theta\cos^2 2\psi,
\label{eq.azim1}\end{aligned}$$ with $\theta$ the Bragg angle. Here $\psi$ is defined such that $\psi=0$ corresponds to the scattering plane containing the $b$ axis.

Figure \[fig.spec\] shows the calculated RXS spectra as a function of photon energy in comparison with the experiment ($\psi=-45^\circ$).[@Hirota] The crystal field parameter is set to be $E_d=-0.1$ eV in 