Antiferroquadrupolar (AFQ) ordering has been conjectured in several rare-earth compounds to explain their anomalous magnetic properties. No direct evidence for AFQ ordering, however, has been reported. Using the resonant x-ray scattering technique near the Dy L(III) absorption edge, we have succeeded in observing the AFQ order parameter in DyB2C2 and analyzing the energy and polarization dependence. The much weaker coupling between the orbital degrees of freedom and the lattice in 4f electron systems than in 3d compounds makes them an ideal platform to study orbital interactions originating from electronic mechanisms.

We have investigated the antiferroquadrupolar ͑AFQ͒ order in DyB 2 C 2 by resonant x-ray scattering. X rays with energies near the L 3 absorption edge of Dy were employed. Superlattice peaks that correspond to three kinds of propagation vectors of ͑1 0 0͒, ͑1 0 1/2͒, and ͑0 0 1/2͒ were investigated in detail with polarization analyses. The experimental results are analyzed using a formalism of resonant x-ray scattering and a model of the AFQ order. The magnetic and quadrupolar scatterings are explained by the model satisfactorily. A detailed investigation of the critical behavior of the AFQ ordering is also reported. The critical exponent ␤ is deduced to be 0.35, not far from the three-dimensional Heisenberg system. We have also succeeded in detecting the diffuse scattering above T Q .

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Experimental evidence of the d-f Coulomb interaction responsible for the resonant x-ray scattering (RXS) from antiferroquadrupolar order in DyB2C2 is presented. The energy dependences of the RXS intensity with polarization analysis are analyzed by considering the interference between the resonances of dipolar (E1) and quadrupolar (E2) transition processes. It is found that the structure factors for the E1 and E2 processes have the same sign for σ-π ′ but the opposite sign for σσ ′ channel. This result, when compared with the calculated structure factors, means that the quadrupolar moments of the 4f and 5d electrons have opposite signs to each other. Interference between nonresonant Thomson scattering from the lattice distortion and the resonant scatterings is also studied and a direct coupling between 4f and the lattice is concluded.

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Resonant x-ray scattering (RXS) experiment has been performed for the (3 0 1.5) superlattice reflection in the antiferroquadrupolar and antiferromagnetic phase of DyB2C2. Azimuthal-angle dependence of the resonance enhanced intensities for both dipolar (E1) and quadrupolar (E2) resonant processes has been measured precisely with polarization analysis. Every scattering channel exhibits distinctive azimuthal dependence, differently from the symmetric reflection at (0 0 0.5) which was studied previously. We have analyzed the results using a theory developed by Lovesey et al., which directly connects atomic tensors with the cross-section of RXS. The fitting results indicate that the azimuthal dependences can be explained well by the atomic tensors up to rank 2. Rank 3 and rank 4 tensors are reflected in the data very little. In addition, The coupling scheme among the 4f quadrupolar moment, 5d ortitals, and the lattice has been determined from the interference among the Thomson scattering from the lattice distortion and the resonant scatterings of E1 and E2 processes. It has also been established from the RXS of the (3 0 1.5) reflection that the canting of the 4f quadrupolar moments exists up to TQ. We also discuss a possible wavefunction of the ground state from the point-charge model calculation.

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