Ferroquadrupolar ordering in PrTi${}_2$Al${}_{20}$

The origin of the nonmagnetic phase transition in PrTi${}_2$Al${}_{20}$, reported earlier in a macroscopic study, has been asserted microscopically using elastic and inelastic neutron scattering techniques. It has been shown spectroscopically that the crystalline-electric-field ground state is a nonmagnetic ${\Gamma }_3$ doublet, whereas the excited states are two triplets (${\Gamma }_4$ and ${\Gamma }_5$) and a singlet (${\Gamma }_1$). The diffraction experiment under external magnetic field shows that the nonmagnetic transition is indeed ferroquadrupolar ordering, which takes place as a consequence of cooperative removal of the ground-state-doublet degeneracy. It is therefore concluded that PrTi${}_2$Al${}_{20}$ is another rare example of Pr compounds exhibiting nonmagnetic quadrupolar order.

Figure 1

Inelastic spectra in a wide energy range up to 50 meV measured at the three temperatures $T=2.8$, 50, and 250 K, and at the momentum transfer $Q=3$ Å${}^{-1}$. We found that all the peaks above 10 meV (i.e., at 15, 20, 35, and 47 meV) are due to phonon scattering, judging from their temperature and $Q$ dependence.

Figure 2

(a)–(f) Inelastic spectra at (a) $T=50$, (b) 40, (c) 30, (d) 20, (e) 10, and (f) $4.2$ K. The momentum transfer was fixed to $Q=1.5$ Å${}^{-1}$. The solid lines stand for the total fitting results, whereas dotted lines represent each CEF peak, and dash-dotted line is the background. The upturn of the background at higher energy ($\hslash \omega >12$ meV) is due to the contamination from the main (direct) beam. See text for details. (g) Higher statistics data at $T=0.77$ K ($<T_{\mathrm{c}}$) and $T=4$ K ($>T_{\mathrm{c}}$). For these data, supplemental radiation shield was placed at low $2\theta$ to reduce the higher energy upturn of the background.

Figure 3

Temperature dependence of the width parameters ${\Gamma }_{\mathrm{L}}$ for the 6 and 9.5 meV peaks. Note that the uncertainty is considerably large for the higher-energy-peak width, prohibiting us to discuss its temperature dependence.

Figure 4

Determined CEF level scheme for PrTi${}_2$Al${}_{20}$.

Figure 5

(a) $Q$ positions where the reflection intensity was collected in the present study. (b)–(f) $\theta -2\theta$ scans at representative $Q$ positions, 220, 440, 620, 110, and 330. For the 220, 440, and 620 reflections, both the zero-field and 4 T data were collected at both $T=9\mathrm{K}>T_{\mathrm{c}}$ and $T=1.6\mathrm{K}<T_{\mathrm{c}}$, whereas for the 110 and 330 reflections only $H=4$ T data were measured.

Figure 6

(a) Temperature dependence of the 220 (closed circles) and 440 (open triangles) reflection intensity under the external magnetic field $H=4$ T. Solid lines are guides to the eyes. (b) Temperature dependence of the 220 (closed circles) and 440 (open triangles) reflection intensity under the zero external magnetic field. Absolute counting number inconsistency between the $H=4$ and 0 T results is due to the difference in the used cryostats; for the $H=4$ T experiment, we used the vertical field magnet, which has thicker radiation shields with several dark angles, whereas we used the simpler Orange cryostat for the zero-field experiment.

Figure 7

Magnetic field dependence of the 200 reflection intensity measured at the base temperature $T=1.6$ K. The nuclear scattering intensity, estimated at the zero field, is subtracted as a nuclear scattering background. Solid line stands for a result of fitting to the $H^2$ dependence.

Figure 8

Temperature dependence of $h$ scans around the 400 Bragg reflection. Open circles stand for the data taken at $T=4.1$ K ($>T_{\mathrm{c}}$), whereas closed circles for $T=0.26$ K ($<T_{\mathrm{c}}$). The solid lines are the Gaussian fitting results.