kept instead of fourfold symmetry without the lattice distortion. Terms of $O_4^4\equiv \frac{1}{2}(J_+^4 + J_-^4)$, which admixes the states $|M\rangle$ with $|M\pm 4\rangle$, come from the higher order expansion to make the local symmetry fourfold. The detailed study along this line, however, is beyond the scope of the present study.

In any event, the second term makes the local symmetry twofold. The lowest energy state is admixed by $|M\rangle$ with $|M|>1/2$. The quadrupole moment is ordered; $\langle\tilde O_{x^2-y^2}\rangle$ ($\equiv\frac{\sqrt{3}}{2}\langle\tilde J_x^2-\tilde J_y^2\rangle$) takes a finite value with $\langle\tilde O_{xy}\rangle$ ($\equiv\frac{\sqrt{3}}{2}\langle \tilde J_x\tilde J_y
+\tilde J_y\tilde J_x\rangle$)$=0$ in the local coordinate frame for each sublattice ($\langle\cdots\rangle$ means the average over the lowest Kramers doublet). With increasing values of $|E_f|/D_f$, the average quadrupole moment increases, as shown in Fig. \[fig.parameter\](b). It becomes largest, $\langle \tilde{O}_{x^2-y^2}\rangle=46.1$ at $D_{f}=0.1$.

Cross Section of Resonant X-Ray Scattering
==========================================

We briefly summarize here the formulae used in the calculation of the RXS spectra in the next section. The conventional RXS geometry is shown in Fig. \[fig.geom\]; photon with frequency $\omega$, momentum ${\textbf k}_i$ and polarization $\mu$ ($=\sigma$ or $\pi$) is scattered into the state with momentum ${\textbf k}_f$ and polarization $\mu'$ ($=\sigma'$ or $\pi'$). The scattering vector is 