The highly degenerate f-electron shells in actinide and rare-earth systems provide a wealth of interesting physical phenomena, with the local degrees of freedom leading to complex phase diagrams. Whilst the order associated with magnetic dipole moments has been studied extensively for decades, more recently the importance of electric quadrupole moments has been recognised. It is now clear that the interactions between quadrupoles may be as large as the more familiar dipolar interactions, leading to novel types of ordering in f-electron systems.

UPd3 is particularly interesting because it is one of the small number of metallic materials that exhibit long range quadrupolar ordering. Four phase transitions between 7.8 K and 4.4 K have been detected by microscopic and macroscopic measurement techniques, and attributed to a succession of antiferroquadrupolar (AFQ) ordered phases. Despite intensive effort over the past 25 years, an understanding of these transitions, and the exact nature of the ordering in UPd3, has proved highly challenging. Whilst we deduced, some years ago, indirect evidence of quadrupolar order from polarised neutron diffraction measurements in a magnetic field, the first direct microscopic evidence for quadrupolar ordering was obtained in earlier X-ray resonant scattering studies on ID20, with the appearance of new superlattice peaks below T0 = 7.8 K. The measurements were carried out at the uranium MIV edge (3.726 keV), confirming that the ordering arises from the U 5f electrons.

The unit cell below T0 is orthorhombic, with the ordered quadrupole moments on the quasi-cubic sites, stacked in antiphase along the c-axis. Our results were consistent with polarised neutron data that implied the order parameter below T0 was Qx2-y2. However, at that time, we could not uniquely identify the order parameter as we were unable to perform the necessary azimuthal scans. We subsequently developed a new model that explained why there are four phase transitions in UPd3, and made predictions for the possible order parameters.

Advances in cryostat technology on ID20 have now allowed us to make azimuthal () scans in the AFQ ordered phases of UPd3. Figure 119 shows our results for the -dependence of the –’ and –’ scattering intensities of the (103) superlattice peak at T = 7.1 K. To our surprise, in both the –’ and –’ channels, the data were in clear conflict with our calculations for the -dependence of the Qx2-y2 order parameter (dashed lines in Figure 119). Moreover, calculations of the scattering intensity for the Qxy and Qyz order parameters do not agree with the data (see [1] for details). However, our results were in good qualitative agreement with calculations for the Qzx order parameter. The challenge was to reconcile this finding with the polarised neutron Qx2-y2 result. Investigation of our theoretical model for the crystal field states of UPd3 revealed that the onset of Qzx order leads, from symmetry considerations, to an accompanying small contribution of Qx2-y2 order. It is this latter contribution that was detected in our polarised neutron experiments: due to the direction of our applied magnetic field, we were insensitive to antiferromagnetic moments arising from Qzx order.

Fig. 119: The azimuthal dependence of a) the –’ and b) –’ scattering intensities of the (103) superlattice peak in UPd3 at the U MIV edge, at T = 7.1 K. The dashed line shows the calculation for the Qx2-y2 order parameter, whilst the full line shows the calculation for Qzx admixed with 15% Qx2-y2 contribution, as discussed in the text.

The solid line in Figure 119 shows that our data is fitted extremely well by a calculation for Qzx order with a small (~15%) contribution of Qx2-y2 order. The intensity from Qzx order is zero at = 900 in both channels: the finite intensity in the –’ channel around this azimuthal angle arises from the small Qx2-y2 contribution (see [2] for details).

Fig. 120: The Qzx AFQ structure in UPd3 below T0 = 7.8 K, with antiphase stacking along the z-axis in an orthorhombic unit cell. The quadrupole moments of the uranium 5f electrons on the quasi-cubic sites are represented by ellipsoids.

In conclusion, the unique properties of X-ray resonant scattering have enabled us to distinguish the order parameter associated with the highest temperature antiferroquadrupolar phase in UPd3 between 7.8 K and the next transition at 6.9 K. It is primarily Qzx with an anti-phase stacking along the c-axis, as illustrated in Figure 120, together with an accompanying Qx2-y2 contribution. We now plan to analyse the azimuthal dependence of the superlattice reflections in the lower temperature phases of UPd3 where we expect the order parameters are combinations of Qzx, Qx2-y2, Qxy and Qyz.

 

References

[1] H.C. Walker, K.A. McEwen, D.F. McMorrow, S.B. Wilkins, F. Wastin, E. Colineau, D. Fort, Phys. Rev. Lett. 97, 137203 (2006).
[2] K.A. McEwen, H.C. Walker, M.D. Le, D.F. McMorrow, E. Colineau, F. Wastin, S.B. Wilkins, J-G. Park, R.I. Bewley, D. Fort, Invited paper presented at ICM2006, to be published in J. Mag. Mag. Mater.

Authors

K.A. McEwen (a), H.C. Walker (a), D.F. McMorrow (a), S.B. Wilkins (b).
(a) University College London (UK)
(b) ESRF, now at Brookhaven National Laboratory (USA)