energy $E_{\Lambda}$. The $\Gamma$ is the life-time broadening width of the core hole. The dipole operators $x_\alpha(j)$’s are defined as $x_1(j)=x$, $x_2(j)=y$, and $x_3(j)=z$ in the coordinate frame fixed to the crystal axes with the origin located at the center of site $j$. The scattering amplitude $M_{\alpha\alpha'}({\textbf G},\omega)$ contains the square of the dipole matrix element $A_{dp}=\langle 5d|r|2p\rangle =
   \int_0^{\infty} R_{5d}(r)rR_{2p}(r)r^2{\rm d}r $ with $R_{5d}(r)$ and $R_{2p}(r)$ being the radial wavefunctions for the $5d$ and $2p$ states. For Dy$^{3+}$ atom, it is estimated as $2.97\times 10^{-11}$ cm in the $4f^9$ configuration within the HF approximation.[@Cowan]

![ Scattering geometry. Incident photon with wave vector ${\textbf k}_i$ and polarization $\sigma$ or $\pi$ is scattered into the state with wave vector ${\textbf k}_f$ and polarization $\sigma'$ or $\pi'$ at Bragg angle $\theta$. The sample crystal is rotated by azimuthal angle $\psi$ around the scattering vector ${\textbf G}={\textbf k}_f-{\textbf k}_i$. \[fig.geom\]](fig.print.3.eps){width="8.0cm"}

Calculation of RXS spectra
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In the $E_1$ transition, an electron is excited from $2p$ states to $5d$ states at a Dy site. A $2p$ core hole is left behind, and its state is split into the states of $j_p=3/2$ and $j_p=1/2$ ($j_p$ is the total angular momentum) due to the strong spin-orbit 