Multipole Tensor Analysis of the Resonant X-ray Scattering by Quadrupolar and Magnetic Order in DyB₂C₂

Abstract

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.

Resonant x-ray scattering spectra from multipole ordering: NpM4,5edges inNpO2

“…36,37,38,39,40,41,42,43 The intermediate state is created by the transition from the 2p-core to 5d states. Since the 5d states are considerably delocalized with forming energy bands, the assumption that the intermediate state preserves the rotational invariance becomes less accurate.…”

Competition of Magnetic and Quadrupolar Order Parameters in HoB4

“…In DyB 2 C 2 , the quadrupolar interaction is stronger than the magnetic one, resulting in the successive phase transitions of an antiferroquadrupolar (AFQ) order at T Q ¼ 24:7 K and an antiferromagnetic (AFM) order at T N ¼ 15:3 K. Although the magnetic interaction favors a parallel stacking of [110]-oriented magnetic moments along the c-axis, the AFQ order confines the orientation of the moments to be perpendicular to adjacent moments. 6,9,10) In HoB 2 C 2 , the Ruderman-Kittel-Kasuya-Yoshida (RKKY)-type magnetic interaction is considered to be slightly stronger than the AFQ interaction, and the system initially exhibits an AFM order at T N ¼ 5:8 K. However, because of the underlying AFQ interaction, a first-order transition into the AFQ phase occurs at T Q ¼ 5:0 K. 7,11,12) The AFM and AFQ interactions are also competitive in TbB 2 C 2 . Although TbB 2 C 2 exhibits only an AFM order at zero magnetic field, the same AFQ phase as that in DyB 2 C 2 and HoB 2 C 2 appears in a magnetic field of only 1 T. 8,13,14) What is more intriguing in TbB 2 C 2 is its magnetic structure at zero magnetic field described by q ¼ ð0; 0; 1=2Þ, which is different from the q ¼ ð0; 0; 0Þ-type structure realized commonly in the RB 2 C 2 series.…”

Resonant X-ray Scattering Experiments on the Ordering of Electronic Degrees of Freedom

“…[82][83][84][85][86][87][88][89][90] In the AFQ state, only the quadrupole moments are ordered, whereas the magnetic degree of freedom remains paramagnetic. DyB 2 C 2 is a tetragonal compound exhibiting two successive phase transitions at T Q ¼ 25 and T N ¼ 15 K. 91) Since the AFM order is observed only below T N , the intermediate phase between T N and T Q was ascribed to an AFQ-ordered phase.…”

“…18 includes detailed distributions including up to the rank-6 component, the signal observed by RXS actually reflects up to the rank-2 moment, and it is considered that the effects of the rank-4 and rank-6 moments are not reflected in the data. 88,89) Although the azimuthal dependence can be well reproduced by the rank-2 term, care should be taken with respect to the real mechanism of the resonant peak, especially for E1, which probes the 5d state. In the case of DyB 2 C 2 , displacements of the B and C atoms take place by the AFQ order below T Q .…”