Evidence for hidden quadrupolar fluctuations behind the octupole order in ${\mathrm{Ce}}_{0.7}$La${}_{0.3}$$B_6$ from resonant x-ray diffraction in magnetic fields

The multipole ordered phase in Ce${}_{0.7}$La${}_{0.3}$B${}_6$, emerging below 1.5 K and named phase IV, has been studied by resonant x-ray diffraction in magnetic fields. By utilizing diamond x-ray phase plates to rotate the incident linear polarization and a conventional crystal analyzer system, full linear polarization analysis has been performed to identify the order parameters. The analysis shows that the ${\Gamma }_{5g}$ ($O_{yz}$, $O_{zx}$, $O_{xy}$) quadrupoles are more induced by the field than the ${\Gamma }_{3g}$ ($O_{20}$ and $O_{22}$) quadrupoles on the ${\Gamma }_{5u}$ ($T_{x+y+z}^{\beta }$) antiferro-octupole order in phase IV. The problem is that this result is contradictory to a mean-field calculation, which inevitably gives the ${\Gamma }_{3g}$ quadrupole as the main induced moment. This result indicates that the ${\Gamma }_{5g}$ quadrupole order is close in energy. We consider that a large fluctuation of the ${\Gamma }_{5g}$ quadrupole is hidden behind the primary ordering of the ${\Gamma }_{5u}$ octupole and that the multipolar fluctuation significantly affects the ordering phenomenon.

Figure 1

The $H$-$T$ phase diagram of Ce${}_{0.7}$La${}_{0.3}$B${}_6$ for $H\parallel \left[001\right]$ and the calculated electric and magnetic charge distributions of the antiferro-type ordered states [26]. The broken line is a speculation. The I-II phase boundary for CeB${}_6$ is also shown for reference.

Figure 2

Scattering geometry of the experiment and definitions of the angles.

Figure 3

(Left) X-ray energy dependence of the intensity for $\sigma$-${\sigma }^{\prime }$, $\sigma$-${\pi }^{\prime }$, and $\pi$-${\pi }^{\prime }$ channels. (Right) Temperature dependence of the $E2$ resonance intensity at 6.160 keV for the $\sigma$-${\sigma }^{\prime }$ channel. The solid line is a fit to a power law $I\propto |T-T_{\text{IV}}{|}^{2\beta }$ with $\beta =0.70\pm 0.1$.

Figure 4

${\varphi }_{\text{A}}$ dependencies of the $E2$ intensity at zero field in phase IV. The incident polarization angle $\eta$ is fixed and ${\varphi }_{\text{A}}$ is scanned. Solid lines show the calculated intensity curves assuming the ${\Gamma }_{5u}$-type AFO order of $\langle \pm T_x^{\beta }\pm T_y^{\beta }\pm T_z^{\beta }\rangle$ with four equal domain populations.

Figure 5

Magnetic-field dependencies of the $E2$ intensity for $\sigma$-${\sigma }^{\prime }$ and $\pi$-${\sigma }^{\prime }$ channels. The solid lines are the calculations described in the text. Inset in the right panel illustrates the $H$-$T$ phase diagram for $H\parallel \left[\overline{1}10\right]$ and the arrow represents the sweep line.

Figure 6

Incident polarization dependencies of the $E2$ intensity at $H=\pm 1$ T. ${\varphi }_{\text{A}}$ is fixed and the incident polarization angle $\eta$ is scanned. Solid lines are the calculations.

Figure 7

X-ray energy dependencies of the intensity for $\sigma$-${\sigma }^{\prime }$ at $H=\pm 1$ and 0 T.

Figure 8

Magnetic-field dependencies of the $E1$ and $E2$ intensities for the $\pi$-${\pi }^{\prime }$, $\pi$-${\sigma }^{\prime }$, and $\sigma$-${\sigma }^{\prime }$ channels at the $L_3$ edge. The background level for $\sigma$-${\sigma }^{\prime }$ is $\sim$0.4 counts/s.

Figure 9

X-ray energy dependencies of the intensity at 1.2 T for $\sigma$-${\sigma }^{\prime }$ and $\pi$-${\pi }^{\prime }$.

Figure 10

Magnetic-field dependence of the $E2$ and $E1$ intensities for the $\pi$-${\sigma }^{\prime }$ channel.

Figure 11

Magnetic-field dependence of the $E2$ and $E1$ intensities for the $\pi$-${\sigma }^{\prime }$ channel.

Figure 12

Incident polarization dependencies of the $E1$ intensity at $H=\pm 4$ T. Solid lines are the calculations.

Figure 13

Incident polarization dependencies of the $E2$ intensity at $H=\pm 4$ T. Solid lines are the calculations.

Figure 14

Magnetic-field dependence of the $E2$ and $E1$ intensities for the $\pi$-${\sigma }^{\prime }$ channel.

Figure 15

Incident polarization dependencies of the $E1$ intensity at $H=\pm 4$ T. Solid lines are the calculations.

Figure 16

Incident polarization dependencies of the $E2$ intensity at $H=\pm 4$ T. Solid lines are the calculations.

Figure 17

Results of the MF calculation of Eq. (1). The interaction parameters are represented by the respective transition temperatures: $K_{5g}^{\text{(q)}}$, $T_{5u}^{\left(o\right)}$, and $T_{2u}^{\left(o\right)}$. (a)–(c) $H$ dependencies of the AFM dipole, AFQ, and ${\Gamma }_{5u}$-AFO moments, respectively, for domain 1 at $T=0.1$ K. (d) and (e) $T$ dependencies of the ${\Gamma }_{5u}$-AFO and ${\Gamma }_{5g}$-FQ moments, respectively, at $H=0$ T. (f) $T$ dependencies of the domain averaged uniform magnetization. The primary ${\Gamma }_{5u}$($T^{\beta }$)-AFO is realized at zero field in all cases. Energy-level splitting of the ${\Gamma }_8$ CEF ground state and the off-diagonal matrix elements responsible for the field-induced dipole and quadrupole moments are also shown.

Figure 18

Incident polarization dependencies of the 111 fundamental Bragg reflection by nonresonant Thomson scattering. Solid lines are the calculations with Eq. (A11).