Antiferroquadrupolar Phase Research Articles

Magnetic anisotropy of antiferro-quadrupole ordering in tetragonal HoB 2C 2

Elastic properties of tetragonal HoB 2C 2 showing antiferro-quadrupole (AFQ) ordering have been investigated by means of ultrasonic measurements. An anisotropic RKKY interaction would be responsible for the AFQ phase II of HoB 2C 2, which has an anisotropic field-temperature ( H– T) phase diagram. We succeeded in obtaining the H– T phase diagram for H || [0 0 1] by using a hybrid magnet up to 30 T in NIMS, and found the presence of AFQ phase II over 4 T for H || [0 0 1] in HoB 2C 2. The critical magnetic field H C[0 0 1] =26.3 T suggests a distinct stability of AFQ phase II in fields along the principal axis by contrast with relatively unstable behavior under fields H C[1 1 0] =3.9 T and H C[1 0 0] =2.0 T in basal plane.

Neutron diffraction study on the antiferroquadrupolar ordering system PrPb [formula omitted] in a magnetic field

The cubic intermetallic compound PrPb 3 is considered to exhibit an antiferro-quadrupolar (AFQ) ordering at T Q = 0.4 K . We have performed neutron diffraction experiments on PrPb 3 in order to detect a possible induced antiferromagnetic (AFM) moment in the AFQ phase. At T = 40 mK , a small magnetic reflection q = ( 1 2 δ δ ) with δ = 0.062 is observed above 3 T. The emergence of the magnetic reflection is accompanied by a large hysteresis, implying the transition to be of first order. The direction of the AFM component ( | | [ 1 1 0 ] ) suggests an underlying AFQ order of O 2 2 type.

Temperature and Field Dependence of the Order Parameter in the Antiferroquadrupolar Phase of CeB6 from μ+ Knight Shift Measurements

The Fermi contact hyperfine contribution to the Knight shift of positive muons, implanted at the interstitial 3d sites in CeB6, is found to exhibit the same temperature dependence below T(Q) in phase II as the quadrupolar order parameter determined from resonant and nonresonant x-ray scattering. Furthermore, the contact coupling parameter is shown to be anisotropic and field dependent. These unanticipated features are interpreted to arise from the RKKY induced conduction electron spin polarization, which depends on the orientation and expectation value of the ordered 4f quadrupole moments.

Magnetic Anisotropy of the Antiferroquadrupole Phase inCe0.50La0.50B6

Magnetic phase diagrams for antiferroquadrupole (AFQ) phase II and antiferromagnetic (AFM) phase III in Ce0.50La0.50B6 with a Gamma(8) ground state have been investigated by ultrasonic measurements. The hybrid magnet (Gama) in the National Institute for Materials Science was employed for high-field measurements up to 30 T and a 3He-4He dilution refrigerator was used for low-temperature experiments down to 20 mK. The phase boundary from paramagnetic phase I to AFQ phase II under [001] magnetic fields closes at H(I-II) approximately 29 T, while the boundary is still open under fields along the [110] and [111] directions even up to 30 T. This anisotropic character of phase II in fields is consistent with the theoretical calculation based on the O(xy)-type AFQ ordering. We also found that AFM phase III reduces considerably in fields turning from the [001] to [110] and [111] directions.

Quadrupolar, structural, and magnetic ordering inDyB2C2studied by symmetry analysis and neutron diffraction

Antiferroquadrupolar (AFQ) and antiferromagnetic (AFM) order in ${\text{DyB}}_{2}{\mathrm{C}}_{2}$ have been investigated by theoretical symmetry analysis and neutron single crystal diffraction experiment with 0 to 4 T applied magnetic field along [100]. New symmetry arguments indicate that the AFQ ordering below ${T}_{Q}=24.7\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ is accompanied by structural distortions involving boron and carbon atoms, but not dysprosium ions. In the AFQ phase a new arrangement of quadrupoles is proposed based on symmetry analysis of the long-range quadrupolar ordering, neutron diffraction study of the field-induced dipolar magnetic structure, and other available observations. This is a $90\ifmmode^\circ\else\textdegree\fi{}$ arrangement of quadrupoles of neighboring atoms in the $xy$ plane, and along $z$. For the zero-field AFM phase, the refinement of neutron single crystal diffraction data of ${\text{Dy}}^{11}{\mathrm{B}}_{2}{\mathrm{C}}_{2}$ against models proposed by Yamauchi et al. [J. Phys. Soc. Jpn. 68, 2057 (1999)] (model A) and by van Duijn, Attfield, and Suzuki [Phys. Rev. B 62, 6410 (2000)] (model B) allows to give preference to modified model A. The magnetic moments of the four Dy sublattices are equal and are confined in the $xy$ plane. The tilt of the Dy moments from the $x$ axis is $36(4)\ifmmode^\circ\else\textdegree\fi{}$ at (0 0 0) and $\ensuremath{-}112(4)\ifmmode^\circ\else\textdegree\fi{}$ at the ($\frac{1}{2}$ $\frac{1}{2}$ 0) site. The angle between the Dy moments adjacent along $z$ is $76(6)\ifmmode^\circ\else\textdegree\fi{}$ and not $90\ifmmode^\circ\else\textdegree\fi{}$ as proposed originally.

Theory of Field-Induced Phase Transition in PrOs4Sb12

We propose a theory of the field-induced ordered state in PrOs 4 Sb 12 . It is based on an antiferro-quadrupolar (AFQ) interaction in the low-lying singlet–triplet subspace. Using a pseudo-spin description, we clarify a hidden symmetry in the local Hamiltonian and its relation with possible order parameters. Stable AFQ phases are determined for the direction of magnetic fields, and the effect of T h symmetry on the anisotropic phase diagram is discussed.

Nd Ion Doping Effects on the Multipolar Interactions in CeB6

The Nd ion doping effects on the multipolar interactions in CeB 6 were studied. At high magnetic fields, the antiferro-quadrupolar (AFQ) phase II is not significantly affected by Nd doping. This may be because the Nd ion is in the paramagnetic state. However, with decreasing magnetic field, the Nd doping effect becomes apparent. This may be because Nd–Nd interaction dominates the long-range ordering at low magnetic fields. At low magnetic fields, just after T Q and T N coincide with each other when Nd ion is doped, new phases, namely, phases IIIA and V appear. Phase IIIA is the AF magnetic state where all types of quadrupolar interaction are small. The nature of phase V seems to change largely at x ∼0.5. For x 0.5, the AF magnetic state where both Ce and Nd ions contribute may be realized.

A new phase boundary in quadrupolar ordered phase of Dy 0.9Gd 0.1B 2C 2

By specific heat measurements of Dy 0.9Gd 0.1B 2C 2, we found a new phase boundary in the antiferroquadrupolar (AFQ) ordered phase with H||[1 1 0] . It is supposed that the divided AFQ phases differ in arrangements of Dy moments induced by magnetic fields, since there are four possible moment directions in relation to field direction on the complicated AFQ structure.

Quadrupole ordering and multipole interactions in Pr-based compounds

Field–temperature phase diagrams of cubic compounds PrPb3 and PrOs4Sb12with non-magnetic crystalline electric field (CEF) ground state have beenexamined by means of low-temperature magnetization measurements. PrPb3 is atypical Γ3doublet CEF ground state system which undergoes an antiferro-quadrupole(AFQ) transition at TQ = 0.4 K. In a magnetic fieldTQ(H)exhibits re-entrant behaviour depending on the field direction, which can beinterpreted in terms of interactions between induced dipole (and possibly octupole)moments in the AFQ phase. In the heavy-electron superconductor PrOs4Sb12,a field-induced ordered state has been observed above 4 T for allthe principal directions. The results are in favour of a Γ1CEF ground state model in which a level crossing with a Γ5triplet excited state induces a long-range (possibly quadrupolar) ordering in amagnetic field.

Invariant Form of Multipolar Interactions and Relation between Antiferro-Quadrupolar Order and Field-Induced Magnetic Moments

The invariant interaction form between multipoles, including the octupoles, is studied for three types of cubic lattices: s.c., b.c.c. and f.c.c. The coupling terms can be arranged in a form similar to the hopping matrices between LCAO's. By using it, order parameters in the magnetic field are classified into groups in which they couple with each other, and the symmetry of the antiferro-quadrupolar (AFQ) moment is analyzed through induced antiferromagnetic moments (AFM). In this way the table for s.c. by Shiina et al. [J. Phys. Soc. Jpn. 66 (1997) 1741] is generalized for general wave vectors of the three lattice types. Recent experimental results of TmTe having the f.c.c. structure are analyzed from this viewpoint. The nature of the ferromagnetic moment below the AFM transition in the AFQ phase is also discussed. Finally brief comments are given also on field-induced moments in PrFe 4 P 12 and NpO 2 .

Elastic constants of antiferro-quadrupole ordering system DyB 2C 2

Elastic constants of tetragonal system DyB 2C 2 with antiferro-quadrupole (AFQ) ordering have been measured by using ultrasonic method. Remarkable elastic softenings of the C 44 and ( C 11− C 12)/2 have been observed towards the AFQ transition at T Q =25 K . The present results show that a CEF ground state is a pseudo-quartet consisting of two Kramers doublets of E 1/2 and E 3/2 in C 4h site symmetry of Dy 3+ ion. The dominant softening of C 44 probably leads to the order parameter of O yz and O zx in AFQ phase. Magnetic phase diagram applied fields along the [1 0 0] direction determined by field dependence of the C 44 is presented.

Possible form of multi-polar interaction in cubic lattice

The invariant form of interaction between multi-poles, including the octupole, is studied for the simple cubic (SC), body centered and face centered cubic lattices. The coupling terms can be arranged in a way similar to that of the hopping matrix between the LCAO's. A table for SC by Shiina et al. (J. Phys. Soc. Japan 66 (1997) 1741) is generalized for the general wave number case of the three types of lattice. Recent experimental result of TmTe is thereby analyzed. The development of the ferromagnetic moment below the anti-ferromagnetic transition under the anti-ferro quadrupolar order phase is discussed in this connection.

Hyperfine-enhanced nuclear spin order of PrPb 3

The cubic compound PrPb 3 with the Cu 3Au-type structure is known to undergo an antiferro-quadrupolar (AFQ) phase transition at 0.4 K . The ground state of the trivalent Pr ion in cubic local symmetry is a nonmagnetic doublet Γ 3. We measured the magnetic susceptibility of PrPb 3 in the magnetic field of 0.94 mT down to 200 μK . This is the first investigation of the hyperfine-enhanced nuclear magnetism in the AFQ ordered PrPb 3 and we observed a peak of the magnetization at 5 mK and an abrupt decrease at 0.8 mK . These results suggest that the hyperfine-enhanced nuclear magnetic moments of Pr order antiferromagnetically below 5 mK .

Histogram Monte Carlo simulation of the geometrically frustratedXYantiferromagnet with biquadratic exchange

Histogram Monte Carlo simulation is used to investigate effects of biquadratic exchange $J_{2}$ on phase transitions of a 3D classical XY antiferromagnet with frustration induced by the antiferromagnetic exchange $J_{1}$ and the stacked triangular lattice geometry. The biquadratic exchange is considered negative (antiferroquadrupolar) within the triangular planes and positive (ferroquadrupolar) between the planes. The phase diagram obtained features a variety of interesting phenomena arising from the presence of both the biquadratic exchange and frustration. In a strong biquadratic exchange limit ($|J_{1}|/|J_{2}| \leq 0.25$), the antiferroquadrupolar phase transition which is of second order is followed by the antiferromagnetic one which can be either first or second order. The separate antiferroquadrupolar and antiferromagnetic second-order transitions are found to belong to the chiral XY and Ising universality classes, respectively. If the biquadratic exchange is reduced both transitions are found to be first order and occur simultaneously in a wide region of $|J_{1}|/|J_{2}|$. However, if $|J_{2}| \rightarrow 0$ the transition changes to the second-order one with the chiral universality class critical behavior.

Quadrupolar ordering in DyPd 3S 4

We performed the measurements of the specific heat and the magnetization along the principal axes of single crystals of DyPd 3S 4 and found that these crystals showed a quadrupolar ordering at T Q =2.7 K, which is a little lower than that of polycrystalline samples. The magnetization curves were very anisotropic and exhibited one or two anomalies in the principal directions. We have confirmed by comparing with the peaks of specific heat that these anomalies correspond to the phase boundaries between the paramagnetic and antiferroquadrupolar phase. A preliminary H– T phase diagram has been constructed along the three principal axes by plotting these magnetization anomalies together with those of the specific-heat data.

Anomalous Field and Temperature Dependence of the dHvA Oscillations in PrPb3

We have performed wave form analysis of the dHvA oscillations in PrPb 3 to investigate the interaction between conduction electrons and quadrupoles. It is found that each dHvA oscillation consists of two component oscillations which most likely originate from the up and down spin bands, and the effective masses of the two component oscillations differ from each other by about a factor of two or more, although the frequencies are close to each other. Analysis of the γ oscillation indicates that the effective mass is enhanced near the phase boundary between para- and antiferro-quadrupolar phases and decreases with increasing field at higher magnetic fields.

X-ray resonant scattering study of the quadrupolar order in UPd3.

Quadrupolar ordering in a 5f electron system has been observed directly for the first time, using x-ray scattering techniques. In UPd (3) at low temperatures satellite peaks appear at (1,0,l) (orthorhombic notation) with l odd and even. Both sets of peaks show a resonant enhancement of the scattering at the M(IV) edge of U. At resonance, the dominant scattering of the l odd peaks occurs in the unrotated polarization channel, whereas for l even a significant rotated component is found. These results are discussed in terms of possible structures of the antiferroquadrupolar phases.

Antiferro-Quadrupolar Ordering and Multipole Interactions in PrPb3

Antiferro-quadrupolar (AFQ) ordering in the cubic Γ 3 ground state system PrPb 3 has been studied by means of magnetization and specific heat measurements under magnetic fields H . Field variation of the AFQ transition temperature T Q (0.4 K at H =0) has been determined for the three crystallographic directions < 100>, <110> and <111>. The obtained T Q ( H ) is strongly anisotropic, with T Q < 100> ( H ) the largest (∼0.7 K at 6 T). The AFQ phase diagram and the magnetization behavior observed for H ∥<100> are well explained by a model that incorporates both antiferromagnetic and antiferroquadrupolar interactions. The interactions between field-induced staggered magnetic moments stabilize the AFQ order. This simple model, however, seriously fails to explain the observed anisotropy: T Q <100> ( H )> T Q <110> ( H )> T Q <111> ( H ). Reconsidering the field-induced multipole moments in the AFQ phase, we propose that inclusion of a weak ferro-octupolar interaction between Γ 5 type octupole moments is important in reproducing the anisotropic phase diagram of PrPb 3 .

Spin Dynamics of 4fElectrons in CeB6Studied by Muon Spin Relaxation

The fluctuation rate of Ce-4 f moments in the paramagnetic phase of CeB 6 near zero field was determined by longitudinal muon spin relaxation. The rate scales perfectly with resistivity in the entire paramagnetic phase, indicating that the fluctuation is predominantly determined by the Kondo effect. The absence of magnetic order in the presumed antiferro-quadrupolar (AFQ) phase was also confirmed, while no evidence for coupling between the AFQ order parameter and electric field gradient at the B site was found near zero field by the muon nuclear level-crossing resonance technique.

Ultrasonic study of Kondo compounds Ce xLa 1− xB 6 (0.60⩽ x⩽0.75)

We have performed the ultrasonic measurements to examine systematic change of the magnetic-phase diagrams of the Kondo compound Ce x La 1− x B 6 with the Γ 8 ground state as a function of cerium concentration x. A long-range ordered phase IV is observed in addition to an antiferroquadrupolar phase II and an antiferromagnetic phase III in the compounds of x=0.75, 0.70 and 0.65. The huge softenings of C 44 in phase IV indicate considerable enhancement of the quadrupolar fluctuation with Γ 5 symmetry. With decreasing cerium concentration x, the ground state of Ce x La 1− x B 6 at low temperatures under low-magnetic fields changes from the ordered phase IV to a non-magnetic state due to the Kondo effect in vicinity of x∼0.60.