Antiferroquadrupolar Phase Research Articles

Magnetic spin resonance inCeB6

Line shape of magnetic spin resonance at 60 GHz specific to the phase II (so-called antiferroquadrupole phase) of ${\text{CeB}}_{6}$ was studied. The applied procedure of data analysis has allowed obtaining $g$ factor of the oscillating magnetic moments, line width, and oscillating magnetization. It is found that the approaching to the transition temperature ${T}_{\text{I-II}}$ from phase II to the paramagnetic phase I results in strong broadening of the resonance (the line width increases three times in the range $1.8\text{ }\text{K}\ensuremath{\le}T\ensuremath{\le}3.8\text{ }\text{K}$) whereas $g$ factor $g=1.59$ remains temperature independent. Magnetic-resonance data suggests that the magnetization of ${\text{CeB}}_{6}$ in the phase II consists of several contributions, one of which is responsible for the observed magnetic resonance. This term in magnetization is missing in the paramagnetic phase and corresponds to ferromagnetically interacting localized magnetic moments. The magnitude of the oscillating part of magnetization is less than total magnetization in the range ${T}^{\ensuremath{\ast}}\ensuremath{\le}T\ensuremath{\le}{T}_{\text{I-II}}$ and coincides with the total magnetization for $T\ensuremath{\le}{T}^{\ensuremath{\ast}}$, where ${T}^{\ensuremath{\ast}}\ensuremath{\sim}2\text{ }\text{K}$. We argue that ferromagnetic correlations play a key role in the observed phenomenon in analogy with the recent experimental and theoretical results on the magnetic resonance in the dense Kondo systems. At the same time the interpretation of the magnetic-resonance data in the framework of the existing models of magnetism in ${\text{CeB}}_{6}$ faces substantial difficulties, which demands further development of the theory of static and dynamic magnetic properties of this heavy fermion metal.

Thomson scattering from the antiferroquadrupolar ordering phase in TmTe

On the basis of the recently established crystal electric field level scheme of Γ8 − Γ6 − Γ7, we investigate the Thomson scattering intensity of the superstructure reflections with h,k,l = 2n + 1 in the Γ3-type (O22, O02) antiferroquadrupolar (AFQ) ordering phase of TmTe. The intensities show strong dependence on G and the direction of the applied magnetic field H. These dependences on G and H similar to those found in the AFQ phase of CeB6, another AFQ material with Γ5-type (Oyz, Ozx, Oxy) order parameter having the Γ8 ground states, may provide us a useful tool to distinguish the component of the AFQ order of Γ3-type and Γ5-type.

Quadrupole Ordering in Clathrate Compound Ce3Pd20Si6

Low-temperature elastic properties of the clathrate compound Ce 3 Pd 20 Si 6 were investigated using ultrasonic measurements. Pronounced softening of elastic constants ( C 11 - C 12 )/2 and C 44 with decreasing temperature indicates a precursor of electric quadrupole ordering due to a Γ 8 quartet ground sate with an orbital degeneracy for 8c site of a Ce ion. A minimum of ( C 11 - C 12 )/2 at T C =170 mK indicates a transition from paramagnetic phase I to an ordered phase III. Applied magnetic fields induce an antiferro-quadrupole (AFQ) ordered phase II with highly anisotropic character depending on the field direction, which resembles the AFQ phase with O x y -type order parameter in CeB 6 with the Γ 8 ground sate. The order paramater of the phase III may be characterized by either magnetic dipole or magnetic octupole. The Γ 7 ground state at 4a site may fall into a singlet screened by conduction electrons at low temperatures without long-range magnetic ordering.

Anisotropic Phase Diagram for Quadrupole Ordering and Possible Crystal Field Level Structure in TmTe

The origin of an anisotropic phase diagram for the antiferro-quadrupolar (AFQ) order in TmTe is studied using a realistic J = 7/2 multiplet model, based on a previous analysis for the classical limit J →∞. Assuming the Γ 3 -type AFQ order parameters, a quantitative comparison of the phase diagram with the experiments is presented for a few candidates of the crystal field (CF) level schemes. It is shown that a scheme with a sequence from low- to high-energy levels, Γ 8 -Γ 6 -Γ 7 , naturally explains the observed field dependence and anisotropy of the phase diagram. The resonant x-ray scattering signals in the AFQ phase are studied to explore a possibility of identifying the order parameter and the CF scheme experimentally.

Magnetization anisotropy in the AFM and SDW phases of CeB6

The angular dependences of the magnetization and Hall resistance have been investigated by the method of the sample rotation in the magnetic field in the high-quality single-crystal samples in the paramagnetic and magnetically ordered phases of CeB6 in the magnetic field up to 60 kOe. It has been shown that, as CeB6 undergoes the transition from the antiferromagnetic modulated phase to the so-called antiferroquadrupolar phase, the easy-magnetization axis in the [110] plane changes from 〈100〉 to 〈110〉. The magnetic field dependences of the anisotropic component of the magnetization differ radically in these magnetically ordered phases. The analysis provides evidence in favor of the formation of a state with the spin density wave (SDW phase) in the temperature range TN ≈ 2.3 K < T < TQ ≈ 3.3 K in CeB6.

Neutron diffraction study of multipole order in light rare-earth hexaborides

Multipole interactions are known to play a central role in the unconventional properties of light rare-earth hexaborides and especially of CeB6. Substituting Pr at the Ce sites has the effect of enhancing exchange interactions and changing the symmetry of the local 4f charge distribution, while suppressing the octupole moment. The (T, H) magnetic phase diagrams of the CexPr1 − xB6 compounds display a large variety of ordered phases involving magnetic and/or charge degrees of freedom. Here we focus on the compound Ce0.7Pr0.3B6, which is located slightly beyond the Pr concentration where the antiferroquadrupolar phase of pure CeB6 is suppressed in zero field. The different magnetic structures have been characterized by neutron diffraction and their origin is discussed in connection with recent non-resonant X-ray results by Tanaka et al.

X-ray resonant scattering determination of the antiferroquadrupolar ordering inUPd3 at low temperatures

We have used x-ray resonant scattering techniques to make a comprehensive study of the quadrupolarorder in UPd3. New measurements reveal that the order parameter describing the in-phase stacking along thec-axis of the quadrupole moments of the uranium ions on the quasi-cubic sites is primarilyQxy in the two lowest temperature phases. These results are discussedwithin the context of a three-level crystal field model for the uranium5f2 electrons, the allowed quadrupolar operators, and the constraints deducedfrom the analysis of heat capacity measurements. We have now deduced, andsummarize, the order parameters for each of the four antiferroquadrupolar phases ofUPd3.

Magnetic order and multipole interactions inCexPr1−xB6solid solutions

Magnetic ordering phenomena in ${\text{Ce}}_{x}{\text{Pr}}_{1\ensuremath{-}x}{\text{B}}_{6}$ solid solutions have been studied using both powder and single-crystal neutron diffraction. A variety of magnetic structures are observed depending on temperature and Ce concentration. Over a broad composition range $(x\ensuremath{\le}0.7)$, Pr-Pr interactions play a dominant role, giving rise to incommensurate structures with wave vectors of the form ${\mathbf{k}}_{\text{IC}1}=(1/4\ensuremath{-}\ensuremath{\delta},1/4,1/2)$ or ${\mathbf{k}}_{\text{IC}2}=(1/4\ensuremath{-}\ensuremath{\delta},1/4\ensuremath{-}\ensuremath{\delta},1/2)$. The crossover to a ${\text{CeB}}_{6}$-like regime takes place near $x=0.7--0.8$. For the latter composition, the antiferroquadrupolar phase transition observed in transport measurements precedes the onset, at lower temperature, of a commensurate magnetic order similar to that existing in ${\text{CeB}}_{6}$. However, unlike in the pure compound, an incommensurate magnetic order is formed at even higher temperature and persists in the antiferroquadrupolar phase down to the lock-in transition. These results are shown to reflect the interplay between various type of dipole exchange and higher multipole interactions in this series of compounds.

Probing the nodal structures of heavy electron superconductors by means of specific heat measurements in magnetic fields

Measurements of the angle-resolved specific heat C ( H , θ ) have been performed on PrOs 4 Sb 12 and URu 2 Si 2 in order to examine their superconducting gap structures. In the filled skutterudite PrOs 4 Sb 12 , a clear fourfold angular dependence is observed with H rotated in the (1 0 0) planes, whose field variation suggests gap minima along [1 0 0] directions. Interestingly, a reversed angular oscillation of C ( H , θ ) has been observed in the normal state above H c 2 which continues to grow as the high-field antiferroquadrupole phase is approached. In URu 2 Si 2 , C ( H , θ ) with H rotated around the tetragonal c-axis suggests that the nodal structure has rotational symmetry. Comparison between the C ( H ) data for a- and c-axes indicates the existence of point nodes along the [0 0 1] direction.

Crystalline electric field contribution to the magnetoresistance ofPr1−xLaxOs4Sb12

Resistivity measurements were performed on ${\mathrm{Pr}}_{1\ensuremath{-}x}{\mathrm{La}}_{x}{\mathrm{Os}}_{4}{\mathrm{Sb}}_{12}$ single crystals at temperatures down to $20\phantom{\rule{0.3em}{0ex}}\mathrm{mK}$ and in fields up to $18\phantom{\rule{0.3em}{0ex}}\mathrm{T}$. The results for dilute-Pr samples ($x=0.3$ and 0.67) are consistent with model calculations performed assuming a singlet crystalline-electric-field (CEF) ground state. The residual resistivity of these crystals features a smeared step centered around $9\phantom{\rule{0.3em}{0ex}}\mathrm{T}$, the predicted crossing field for the lowest CEF levels. The CEF contribution to the magnetoresistance has a weaker-than-calculated dependence on the field direction, suggesting that interactions omitted from the CEF model lead to avoided crossing in the effective levels of the ${\mathrm{Pr}}^{3+}$ ion. The dome-shaped magnetoresistance observed for $x=0$ and 0.05 cannot be reproduced by the CEF model, and likely results from fluctuations in the field-induced antiferroquadrupolar phase.

Magnetic phase diagrams of [formula omitted] in high magnetic fields

We have performed ultrasonic measurements under high magnetic fields up to 30 T by using the hybrid magnet at the National Institute for Materials Science to investigate the magnetic phase diagram for antiferroquadrupole (AFQ) phase II in Ce x La 1 - x B 6 . With increasing Ce concentration x from x = 0.50 , the AFQ phase transition temperatures T Q indicate an almost linear increase in various fields. The large magnetic anisotropy of AFQ phase II, in which T Q H ∥ [ 0 0 1 ] is much smaller than T Q H ∥ [ 1 1 0 ] and T Q H ∥ [ 1 1 1 ] in high magnetic fields, is revealed in x = 0.75 , 0.60 as well as in x = 0.50 . These experimental results support the theoretical calculation based on the Γ 5 -type AFQ ordering and the magnetic field induced octupole T xyz .

Electric quadrupole contribution to resonant x-ray scattering: Application to multipole ordering phases inCe1−xLaxB6

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.

Quadrupolar Phases of theS=1Bilinear-Biquadratic Heisenberg Model on the Triangular Lattice

Using mean-field theory, exact diagonalizations, and SU(3) flavor theory, we have precisely mapped out the phase diagram of the S = 1 bilinear-biquadratic Heisenberg model on the triangular lattice in a magnetic field, with emphasis on the quadrupolar phases and their excitations. In particular, we show that ferroquadrupolar order can coexist with short-range helical magnetic order, and that the antiferroquadrupolar phase is characterized by a remarkable 2/3 magnetization plateau, in which one site per triangle retains quadrupolar order while the other two are polarized along the field. Implications for actual S=1 magnets are discussed.

Pressure effect on the long-range order in [formula omitted

The pressure effect of CeB 6 was investigated. The pressure dependence of the Néel temperature, T N and the critical field from the antiferro-magnetic phase III to antiferro-quadrupolar phase II, H c III – II of CeB 6 exhibits the unusual pressure dependence that the suppression rate of H c III – II is much larger than that of T N . In order to explain this unusual result, we have performed the mean field calculation for the 4-sublattice model assuming that the pressure dependence of T N , the antiferro-octupolar and quadrupolar temperatures, T oct and T Q as follows; d T N / d P < 0 , d T oct / d P > d T Q / d P > 0 and could explain the unusual pressure dependence of T N and H c III – II .

High-pressure X-ray diffraction study of PrFe 4P 12

The filled skutterudite PrFe 4P 12 undergoes an antiferro-quadrupolar (AFQ) ordering below T Q = 6.5 K . In a recent resistivity measurement under high pressures, it was reported that T Q is suppressed with applied pressure and an insulating phase appears above 2.4 GPa. To obtain microscopic information about the AFQ phase and the insulating state under pressure, we performed X-ray diffraction experiments. We have succeeded in observing superlattice reflections at the wave vector q = ( 1 , 0 , 0 ) with the intensity being ∼ 10 - 4 of the fundamental Bragg's ones. At 1.4 GPa, T Q is suppressed to 5.5 K, while it is found out that the calculated Fe-ion displacement is almost the same as that at ambient pressure. In the insulating state, on the other hand no superlattice reflections at q = ( 1 , 0 , 0 ) are observed within the experimental error.

Magnetic resonance in cerium hexaboride caused by quadrupolar ordering

Experimental evidence of the magnetic resonance in the antiferro-quadrupole phase of CeB 6 is reported. We have shown that below orbital ordering temperature a new magnetic contribution emerges and gives rise to the observed resonant phenomenon.

Dilatometric Measurements and Multipole Ordering in DyB2C2 and HoB2C2

We have performed dilatometric measurements by the capacitance method on tetragonal rare earth compounds DyB 2 C 2 and HoB 2 C 2 that show antiferro-quadrupole (AFQ) ordering. The tetragonal structure is invariant across the paramagnetic phase to the AFQ phase at T Q =24.7 K in DyB 2 C 2 . In the case of intermediate phase IV of HoB 2 C 2 , between T C1 =5.9 K and T C2 =5.0 K, the lattice length along the [101] direction reveals an abrupt increase, while the lengths along the [100] and [110] directions shrink slightly. This lattice distortion is described in terms of triclinic strain e y z = e z x ≠0 indicating the spontaneous ferro-quadrupole moment = ≠0. A plausible model, based on an octupole ordering = = ≠0, in phase IV of HoB 2 C 2 , is argued.

Magnetic phase diagram of antiferroquadrupole ordering inHoB2C2

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$_2$.

Antiferro-quadrupole ordering of Ce [formula omitted]La [formula omitted]B [formula omitted] under high magnetic fields

We have investigated the magnetic phase diagrams for antiferro-quadrupole (AFQ) phase II in Ce x La 1 - x B 6 by ultrasonic measurements under high magnetic fields up to 30 T by using the hybrid-magnet (Gama) at National Institute for Materials Science in Tsukuba. Ce 0.50 La 0.50 B 6 reveals that the AFQ phase boundary in applied magnetic fields along the [0 0 1] direction closes at about 29 T and the boundaries in fields along the [1 1 0] and [1 1 1] directions are still open even at 30 T. The AFQ phase transition temperature in Ce 0.75 La 0.25 B 6 shows maximum at about 25 T in H ∥ [ 0 0 1 ] . These experimental results support the idea that the multipole RKKY-type inter-site interaction of the quadrupoles O yz , O zx , O xy and the field-induced octupole T xyz play an important role in the highly anisotropic magnetic phase diagram of phase II of Ce x La 1 - x B 6 in high magnetic fields.

Phase diagrams and ground-state magnetic properties of Pr-based filled skutterudites

Phase diagrams and the ground-state magnetic properties of PrOs 4 Sb 12 , PrRu 4 P 12 and PrFe 4 P 12 are studied by means of low-temperature magnetization and specific heat measurements. Field-induced ordering observed in PrOs 4 Sb 12 for all directions above 4.5 T is well explained by a level crossing in the Γ 1 – Γ 4 ( T h ) pseudo-quartet states with inclusion of an antiferroquadrupolar (AFQ) interaction. In PrRu 4 P 12 , a Curie-like increase of the susceptibility observed down to below 300 mK indicates that a half of Pr 3 + ions have the magnetic Γ 4 ( T h ) triplet ground state, in accordance with a dramatic change in the crystal field splitting below the metal–insulator transition at 63 K recently observed by an inelastic neutron scattering experiment. In PrFe 4 P 12 , a new ordered phase is observed for H ( ∥ [ 1 1 1 ] ) > 7.5 T above the AFQ phase below 0.6 K, which is probably responsible for unusual heavy electron behavior observed in this field direction.