A theory of the electric quadrupole contribution to resonant x-ray scattering: Application to multipole ordering phases in Ce_{1-x}La_{x}B_{6}

We study the electric quadrupole (E2) contribution to resonant x-ray scattering (RXS). Under the assumption that the rotational invariance is preserved in the Hamiltonian describing the intermediate state of scattering, we derive a useful expression for the RXS amplitude. One of the advantages the derived expression possesses is the full information of the energy dependence, lacking in all the previous studies using the fast collision approximation. The expression is also helpful to classify the spectra into multipole order parameters which are brought about. The expression is suitable to investigate the RXS spectra in the localized f electron systems. We demonstrate the usefulness of the formula by calculating the RXS spectra at the Ce L_{2,3} edges in Ce_{1-x}La_{x}B_{6} on the basis of the formula. We obtain the spectra as a function of energy in agreement with the experiment of Ce_{0.7}La_{0.3}B_{6}. Analyzing the azimuthal angle dependence, we find the sixfold symmetry in the \sigma-\sigma' channel and the threefold onein the \sigma-\pi' channel not only in the antiferrooctupole (AFO) ordering phase but also in the antiferroquadrupole (AFQ) ordering phase, which behavior depends strongly on the domain distribution. The sixfold symmetry in the AFQ phase arises from the simultaneously induced hexadecapole order. Although the AFO order is plausible for phase IV in Ce_{1-x}La_{x}B_{6}, the possibility of the AFQ order may not be ruled out on the basis of azimuthal angle dependence alone.Comment: 12 pages, 6 figure

Theory of Coupled Multipole Moments Probed by X-ray Scattering in CeB6_6

A minimal model for multipole orders in CeB$_6$ shows that degeneracy of the quadrupole order parameters and strong spin-orbit coupling lead to peculiar temperature and magnetic-field dependences of the X-ray reflection intensity at superlattice Bragg points. Furthermore, the intensity depends sensitively on the surface direction. These theoretical results explain naturally recent X-ray experiments in phases II and III of CeB$_6$. It is predicted that under weak magnetic field perpendicular to the (111) surface, the reflection intensity should change non-monotonically as a function of temperature.Comment: 4 pages, 5 figure

Neutron scattering study of magnetic ordering and excitations in the ternary rare-earth diborocarbide Ce^{11}B_2C_2

Neutron scattering experiments have been performed on the ternary rare-earth diborocarbide Ce$^{11}$B$_2$C$_2$. The powder diffraction experiment confirms formation of a long-range magnetic order at $T_{\rm N} = 7.3$ K, where a sinusoidally modulated structure is realized with the modulation vector ${\bm q} = [0.167(3), 0.167(3), 0.114(3)]$. Inelastic excitation spectra in the paramagnetic phase comprise significantly broad quasielastic and inelastic peaks centered at $\hbar \omega \approx 0, 8$ and 65 meV. Crystalline-electric-field (CEF) analysis satisfactorily reproduces the observed spectra, confirming their CEF origin. The broadness of the quasielastic peak indicates strong spin fluctuations due to coupling between localized $4f$ spins and conduction electrons in the paramagnetic phase. A prominent feature is suppression of the quasielastic fluctuations, and concomitant growth of a sharp inelastic peak in a low energy region below $T_{\rm N}$. This suggests dissociation of the conduction and localized $4f$ electrons on ordering, and contrasts the presently observed incommensurate phase with spin-density-wave order frequently seen in heavy fermion compounds, such as Ce(Ru$_{1-x}$La$_x$)$_2$Si$_2$.Comment: accepted for publication in Phys. Rev.

Antiferro-quadrupole Ordering of CeB6_6 Studied by Resonant X-ray Scattering

Under zero magnetic field, a quadrupolar order parameter at q_Q=(1/2,1/2,1/2) in a typical antiferro-quadrupole (AFQ) ordering compound CeB6 has been observed for the first time by means of a resonant X-ray scattering (RXS) te\ chnique. The RXS is observed at the 2p->5d dipole transition energy of the Ce L3-edge. Using this RXS technique to observe the pure order parameter of the AFQ state, the magnetic phase diagram of Phase II is first determined.Comment: 7 pages, 4 figure

Resonant X-Ray Scattering from the Quadrupolar Ordering Phase of CeB_6

We theoretically investigate the origin of the resonant x-ray scattering (RXS) signal near the Ce $L_{III}$ absorption edge in the quadrupolar ordering phase of CeB$_6$, considering the intersite interaction between the $\Gamma_8$ states in the initial state. The anisotropic charge distribution of the $4f$ states modulates the $5d$ states through the intra-atomic Coulomb interaction and thereby generates a large RXS superlattice intensity. The temperature and magnetic field dependence indicates that the induced dipolar and octupolar orders have little influence on the RXS spectra, in good agreement with the recent experiment.Comment: 4 pages, 4 figure

Magnetic phase diagram of antiferroquadrupole ordering in HoB2C2

The magnetic phase diagram for antiferro-quadrupole (AFQ) ordering in tetragonal HoB$_2$C$_2$ has been investigated by measurements of elastic constants $C_{11}$, $C_{44}$ and $C_{66}$ in fields along the basal $x$-$y$ plane as well as the principal [001]-axis. The hybrid magnet (GAMA) in Tsukuba Magnetic Laboratory was employed for high field measurements up to 30 T. The AFQ phase is no longer observed above 26.3 T along the principal [001] axis in contrast to the relatively small critical field of 3.9 T in fields applied along the basal [110] axis. The quadrupolar intersite interaction of $O_{xy}$ and/or $O_2^2$ is consistent with the anisotropy in the magnetic phase diagram of the AFQ phase in HoB$_2$C$_2$.Comment: Phys. Rev. B. (2005) in press. approx 8 pages, 10 figure

Fluctuation Effects on the Quadrupolar Ordering in Magnetic Field

Effects of magnetic field on the quadrupolar ordering are investigated with inclusion of fluctuation of order parameters. For the simplest model with the nearest-neighbor quadrupolar interaction, the transition temperature and the specific heat are derived by the use of the recently proposed effective medium theory. It is shown that magnetic field H has two competing effects on the quadrupolar ordering; one is to encourage the ordering by suppressing the fluctuation among different components of order parameters, and the other is to block the ordering as in antiferromagnets. The former is found to be of order H^2 and the latter of order H^4. Hence the fluctuation is suppressed for weak fields, and the transition temperature increases with magnetic field. The fluctuation effect is so strong that the entropy released at the quadrupolar ordering is only about half of the full value ln 4 even without the Kondo effect.Comment: 10 pages including 3 Postscript figure

Metal-insulator transition in PrRu4_4P12_{12} and SmRu4_4P12_{12} investigated by optical spectroscopy

Electronic structures of the filled-skutterudite compounds PrRu$_4$P$_{12}$ and SmRu$_4$P$_{12}$, which undergo a metal-insulator transition (MIT) at $T_{\rm MI}$ = 60 K and 16 K, respectively, have been studied by means of optical spectroscopy. Their optical conductivity spectra develop an energy gap of $\sim$ 10 meV below $T_{\rm MI}$. The observed characteristics of the energy gap are qualitatively different from those of the Kondo semiconductors. In addition, optical phonon peaks in the spectra show anomalies upon the MIT, including broadening and shifts at $T_{\rm MI}$ and an appearance of new peaks below $T_{\rm MI}$. These results are discussed in terms of density waves or orbital ordering previously predicted for these compounds.Comment: 4pages, 4figures, submitted to Physical Review

Interpocket polarization model for magnetic structures in rare-earth hexaborides

The origin of peculiar magnetic structures in cubic rare-earth (R) hexaborides RB_6 is traced back to their characteristic band structure. The three sphere-like Fermi surfaces induce interpocket polarization of the conduction band as a part of a RKKY-type interaction. It is shown for the free-electron-like model that the interpocket polarization gives rise to a broad maximum in the intersite interaction I(q) around q=(1/4,1/4,1/2) in the Brillouin zone. This maximum is consistent with the superstructure observed in R=Ce, Gd and Dy. The wave-number dependence of I(q) is independently extracted from analysis of the spin-wave spectrum measured for NdB_6. It is found that I(q) obtained from fitting the data has a similarly to that derived by the interpocket polarization model, except that the absolute maximum now occurs at (0,0,1/2) in consistency with the A-type structure. The overall shape of I(q) gives a hint toward understanding an incommensurate structure in PrB_6 as well.Comment: 5 pages, 3 figures, submitted to J.Phys.Soc.Jp

Orbital Kondo Effect in Cex_xLa1−x_{1-x}B6_6: Scaling Analysis

Peculiarity of the Kondo effect in Ce$_x$La$_{1-x}$B$_6$ is investigated on the basis of the scaling equations up to third order. For the case where the $f^1$-$f^2$ charge fluctuation enters in addition to the $f^1$-$f^0$ one, the effective exchange interaction becomes anisotropic with respect to the orbital pseudospins which represent the two different orbitals in the $\Gamma_8$ ground state. Because of different characteristic energies for electric and magnetic tensors, scaling with the single Kondo temperature does not apply to physical quantities such as the resistivity and magnetic susceptibility. Possibility of a bizzare phase is pointed out where the RKKY interaction leads to the spin ordering without orbital ordering. This phase serves as a candidate of the phase IV which is observed to be isotropic magnetically.Comment: 10 pages, 4 eps figures, submitted to PR