defined as ${\textbf G}\equiv{\textbf k}_f-{\textbf k}_i$. Near the Dy $L_{\rm III}$ absorption edge, a $2p$ core electron is virtually excited to $5d$ states in the $E_1$ process. Subsequently it recombines with the core hole. Since the $2p$ states are well localized around Dy sites, it is a good approximation to describe the scattering tensor as a sum of contributions from each site of the core hole. Therefore, the cross section in the $E_1$ process is given by $$I_{\mu\to\mu'}({\textbf G},\omega) \propto
 | \sum_{\alpha\alpha'}P'^{\mu'}_{\alpha}
    M_{\alpha\alpha'}({\textbf G},\omega)P^{\mu}_{\alpha'}
 |^2,
\label{eq.cross}$$ where $$M_{\alpha\alpha'}({\textbf G},\omega) = \frac{1}{\sqrt{N}}
  \sum_j M_{\alpha\alpha'}(j,\omega) \exp(-i{\textbf G}\cdot{\textbf r}_j),
\label{eq.scatensor}$$ with $$M_{\alpha\alpha'}(j,\omega) = 
  \sum_{\Lambda}
  \frac{\langle\psi_n|x_\alpha(j)|\Lambda\rangle
  \langle \Lambda|x_{\alpha'}(j)|\psi_j\rangle}
       {\hbar\omega-(E_{\Lambda}-E_j)+i\Gamma},
\label{eq.dipole}$$

   $\alpha$   $(P^\sigma)_\alpha$    $(P'^{\sigma'})_\alpha$                  $(P'^{\pi'})_\alpha$
  ---------- ---------------------- ------------------------- ----------------------------------------------------
      1       $\cos\beta\cos\psi$      $\cos\beta\cos\psi$     $-\sin\theta\cos\beta\sin\psi+\cos\theta\sin\beta$
      2           $- \sin\psi$            $- \sin\psi$             $-\sin\theta\cos\psi+\cos\theta\sin\beta$
      3       $-\sin\beta\cos\psi$    $-\sin\beta\cos\psi$                   $\cos\theta\cos\beta$

  : Geometrical factors

\[tab.azim\]

where $N$ is the number of Dy sites. Note that the cross section is an order of $N$. The $P^\mu$ and $P'^{\mu}$ are geometrical factors for the incident and scattered photons, respectively. Their explicit forms are given in Table \[tab.azim\]. The $|\psi_j\rangle$ represents the initial state with energy $E_j$. The intermediate state $|\Lambda\rangle$ consists of an excited electron on $5d$ states and a hole on $2p$ states with 