Magnetic Field Induced $4f$ Octupole in ${\mathrm{CeB}}_6$ Probed by Resonant X-Ray Diffraction

A mysterious antiferroquadrupolar ordered phase of ${\mathrm{CeB}}_6$ is considered as originating from the $T_{xyz}$-type magnetic octupole moment in magnetic fields. By resonant x-ray diffraction, we have verified that the $T_{xyz}$-type octupole is indeed induced in the $4f$ orbital of Ce with a propagation vector $(\frac{1}{2},\frac{1}{2},\frac{1}{2})$, thereby supporting the theory. We observed an asymmetric field dependence of the intensity for an electric quadrupole ($E2$) resonance when the field was reversed and extracted a field dependence of the octupole by utilizing the interference with an electric dipole ($E1$) resonance. The result is in good agreement with that of the NMR-line splitting, which reflects the transferred hyperfine field at the boron nucleus from the anisotropic spin distribution of Ce with an $O_{xy}$-type quadrupole.

Figure 1

Top: Fluorescence spectrum of ${\mathrm{CeB}}_6$. Bottom: X-ray energy spectra of the $(\frac{3}{2},\frac{3}{2},\frac{1}{2})$ superlattice reflection in magnetic fields with reversed directions. The triangle shows the background due to the fluorescence.

Figure 2

Top: Magnetic-field dependence of the integrated intensity of the $(\frac{3}{2},\frac{3}{2},\frac{1}{2})$ resonant Bragg diffraction measured at $E=5.724$ (left) and 5.718 keV (right), corresponding to the $E1$ and $E2$ processes, respectively. Bottom: Symmetric (circles) and asymmetric (squares) components of the intensity for the respective processes.

Figure 3

X-ray energy dependence of the symmetric and asymmetric components deduced from the data for $\pm 2\mathrm{T}$. The solid lines are the fits to the data with Lorentzian components. The single-dotted, double-dotted, and dotted lines represent the 1st, 2nd, and 3rd term in Eq. (4), respectively.

Figure 4

Magnetic-field dependences of the AFQ and AFO moments deduced from the symmetric $E1$ and asymmetric $E2$ intensities in Fig. 2. The solid line is a guide for the eye. The crosses represent the THF as deduced from NMR.