Generalized susceptibilities and phonon anomalies in Pd and Pt metals

The generalized susceptibility, χ(q), in Pd and Pt for q along the [100], [110], [111], and [120] directions was determined from their APW and RAPW energy band structures, respectively, using the analytic tetrahedron linear energy scheme of Rath and Freeman. The band structures were previously found to yield Fermi surface radii, temperature dependencies of the static magnetic susceptibility, χ(T), resistivity, and a spin lattice relaxation, T₁T, in very good agreement with experiment. In the χ(q) calculations, we used 2048 tetrahedra in 1/48th irreducible BZ and the energy eigenvalues for bands 4, 5, and 6 which cross the Fermi energy as fitted to a Fourier series representation. The intraband parts of χ(q) at q = 0 for both metals are found to agree with the density of states at the Fermi energy to without 0.5%. Our results show that the dominant contribution to χ_intra arises from the dominant band 5 whose “jungle-gym” FS has strong nesting features; the main peak for Pd occurs at the same q value (= 0.65π/a) for q along the [0q0], [q, q, 0], and [q, q, q] directions. The locus of this main peak is a square in the (0, 0, 1) plane. The maximum of χ_intra for q along the [110] and [111] directions are 23% and 13%, respectively, higher than the value of χ(q) at q = 0. For q along the [010] and [120] directions, the peak is, however, lower than the value of χ_intra at q = 0. Hence, while phonon anomalies are predicted for the [110] and [111] directions, no anomaly is predicted for either the [100] or [120] direction. The predicted q value for the [110] anomaly, $\text{q}\text{= 0.65π/a}$ is close to the experimental value of ～0.7 π/a. Although there may be a hint of an anomaly at 0.56 [111] in the measurements, a more detailed investigation of this region is called for. For platinum, χ_intra for q along the [010], [110] and [111] directions has main peaks which occur at q = 0.68 π/a, 0.75 π/a, and 0.85 π/a, respectively. Here too, this main peak comes from the nesting of the jungle-gym Fermi surface which is not, however, as flat as that of palladium. Anomalies are predicted (although weaker in Pt than in Pd) along [110] and [111] but not along [100] and [120]. The [110] anomaly is close to the measured q value (～0.7–0.8 π/a). Also in agreement with experiment, we predict a weaker [110] anomaly for Pt than for Pd. In both Pd and Pt, weaker anomalies are predicted for the [111] direction than for the [110] direction.     

Antiferroquadrupolar and antiferromagnetic structures in TmGa3

Magnetization processes and magnetic phase diagrams have been studied in magnetic fields applied along the three main crystallographic directions of the cubic cell in TmGa₃. Large magnetization jumps occur at well-defined critical fields. Each of the corresponding magnetic phases has been characterized by means of neutron diffraction on monocrystalline samples. They are shown to be stabilized by antiferroquadrupolar pair interactions within the trigonal symmetry which favours multiaxial spin arrangements: in the spontaneous triple q-structure, the spins point along the four threefold axes. A comparison with the magnetic properties of dysprosium intermetallic compounds within the CsCl-type structure is given. The difficulties of studying antiferroquadrupolar interactions and orderings in rare earth intermetallics are then underlined.     

X-ray diffraction study of the CDW phase in (TaSe4)2I: Determination of the CDW modulation amplitude

An extensive X-ray diffraction study of charge density wave (CDW) phase in (TaSe₄)₂I is reported. We have observed the superstructure satellites at 2q in addition to those at q reported by Fujishita et al. The results imply a sinusoidal lattice modulation with polarization almost perpendicular to q (i.e. transverse) and the existence of CDW domains. At 15 K we have extracted an approximate value for the amplitude of the lattice modulation perpendicular to q to be μ_⊥ ∽ 0.087 Å.     

A three-peak structure for the critical relaxation function in the transverse Ising model

The dynamical properties of the transverse Ising model with an interaction of uniaxial dipolar nature are shown to exhibit a marked q-anisotropy in the critical region. For any wave vector q strictly non perpendicular to the z-axis the longitudinal relaxation shape function F_q^zz exhibits a three-peak structure and consequently a finite width of its central peak.     

Neutron inelastic scattering study of transverse spin fluctuations in CsNiF3: A soliton-only central peak

We have observed a quasi-elastic contribution to the spectrum of the transverse spin fluctuations S^perp;(Q, ω), perpendicular to an applied magnetic field in the easy plane of the one-dimensional ferromagnet CsNiF₃. According to the present theoretical understanding this contribution is due solely to soliton quasi-particles and it should not contain two-magnon scattering. The observed dependence on momentum transfer is as expected for soliton scattering with zero intensity at q_c = 0 rising through a maximum with increasing q_c.     

Quantum Bundle Description of Quantum Projective Spaces

We realise Heckenberger and Kolb??s canonical calculus on quantum projective (N ? 1)-space C _q [C p ^N?1] as the restriction of a distinguished quotient of the standard bicovariant calculus for the quantum special unitary group C _q [SU _N ]. We introduce a calculus on the quantum sphere C _q [S ^2N?1] in the same way. With respect to these choices of calculi, we present C _q [C p ^N?1] as the base space of two different quantum principal bundles, one with total space C _q [SU _N ], and the other with total space C _q [S ^2N?1]. We go on to give C _q [C p ^N?1] the structure of a quantum framed manifold. More specifically, we describe the module of one-forms of Heckenberger and Kolb??s calculus as an associated vector bundle to the principal bundle with total space C _q [SU _N ]. Finally, we construct strong connections for both bundles.     

Electrodynamic features of anisotropic hard superconductors

The low-frequency electromagnetic response of superconducting plates characterized by strong anisotropy of current-carrying capacity in the plane of the sample is studied experimentally and theoretically. Measurements are made on polycrystalline textured plates of the Y-123 system with the c axis lying in the plane of the sample and on a single crystal with a single preferred direction of twinning boundaries. It is shown that the shape of the curves describing the dependence of the relative losses q on the ac field amplitude h ₀ is quite sensitive to the orientation of vector h ₀ in the sample plane. As in the case of isotropic samples, the q(h ₀) dependence is characterized by a single size peak if vector h ₀ is oriented along one of the principal symmetry directions of the anisotropic critical current density. If h ₀ deviates considerably from principal directions, two size peaks are observed on the q(h ₀) curve. A detailed analysis of the evolution of the q(h ₀) curves upon a rotation of vector h ₀ in the sample plane is carried out.     

Local-field corrections to the static dielectric function of semiconductors: A model study

We present results for the macroscopic static dielectric function at small wave vector q for semiconductors, including the local-field corrections (LFCs). We have used the Penn model for our study. Our calculations demonstrate that LFCs depend on the parameters characterizing a semiconductor. Our calculations are in agreement with the calculations based on more detailed band structures.     

Cubic helimagnets in magnetic field and at pressure

Cubic helimagnets with B20 structure display several unusual properties such as anisotropy of the spin-wave spectrum al small momenta q, rotation of the helix vector k in magnetic field and quantum phase transition at pressure. We demonstrate that first two phenomena are a result of umklapp processes mixing excitations with momenta q, q+k and q−k. At very low magnetic field perpendicular to k the helical structure remains stable due to spin-wave gap Δ. Its square is sum of two parts. The first one is a result of the magnon interaction and the second negative part stems from magneto-elastic interaction. It is suggested that competition between these parts leads to the quantum phase transition observed in MnSi and FeGe. For MnSi from rough estimations at ambient pressure was shown that both parts are comparable with the experimentally observed gap. The magneto-elastic interaction is also responsible for 2k modulation of the lattice and contributes to the magnetic anisotropy. Experimental observation by X-ray and neutron scattering of this lattice modulation allows to determine the strength of the magneto-elastic interaction responsible for above phenomena and the lattice helicity.     

Types of defect ordering in undoped and lanthanum-doped Bi2201 single crystals

Undoped and lanthanum-doped Bi2201 single crystals having a perfect average structure have been comparatively studied by x-ray diffraction. The undoped Bi2201 single crystals exhibit very narrow satellite reflections; their half-width is five to six times smaller than that of Bi2212 single crystals grown by the same technique. This narrowness indicates three-dimensional defect ordering in the former crystals. The lanthanumdoped Bi2201 single crystals with x = 0.7 and T _c = 8–10 K exhibit very broad satellite reflections consisting of two systems (modulations) misoriented with respect to each other. The modulation-vector components of these two modulations are found to be q ₁ = 0.237b* + 0.277c* and q ₂ = 0.238b* + 0.037c*. The single crystals having a perfect average structure and a homogeneous average distribution of doping lanthanum consist of 70-to 80-Å-thick layers that alternate along the c axis and have two different types of modulated superlattice. The crystals having a less perfect average structure also consist of alternating layers, but they have different lanthanum concentrations. The low value of T _c in the undoped Bi2201 single crystals (9.5 K) correlates with three-dimensional defect ordering in them, and an increase in T _c to 33 K upon lanthanum doping can be related to a thin-layer structure of these crystals and to partial substitution of lanthanum for the bismuth positions.     

A simple theory of spin-wave relaxation in ferromagnetic metals

The anisotropic exchange interaction between localized spins and conduction electrons is described by an appropriate spin hamiltonian. This is used to calculate the lifetime of magnons for arbitrary values of Λ_eq, where Λ_e is the electron mean free path and q the magnon wavevector. At Λ_eq ? 1, this lifetime depends on the angle between q and the saturation magnetization. The antisymmetric part of anisotropic exchange (Dzialoshinsky-Moriya interaction) may dominate the relaxation of spin-waves of large q. The complicated band structure of transition metals gives rise to a magnon lifetime independent of Λ_e. The contribution of isotropic exchange is also considered.     

Interference effects in the electrical conductivity tensors of doped polar semiconductors

In this paper, we develop a general procedure, based on the Kubo formula, for finding the frequency- and wavevector-dependent electrical conductivity tensor for arbitrary polarizations of the applied electric field. This procedure gives careful consideration to both the electron and the ion contributions to the current density. We find that, in addition to the standard electron and ion contributions to the conductivity tensor, there are contributions arising from the quantum interference between the electrons and the ions. These interference effects, which will affect the infrared absorption of a material, are similar to the interference effects observed by Cerdeira, Fjeldly, and Cardona (Solid State Commun.13 (1973), 325–328; Phys. Rev. B8 (1973), 4734–4745; 9 (1974), 4344–4350) in Raman scattering from p-type Si. We then evaluate these cross terms for simple models of a variety of semiconductors. We find that these interference effects are finite-q effects, with the actual dependence of the cross terms on the wavevector q being sensitive to the symmetry of the crystal. We also find that, in principle, these cross terms may be quite large near the TO-phonon frequency ω_TO. The cross terms are evaluated for both n-type and p-type semiconductors, and it is suggested that they are probably most important for p-type materials. However, we also find that the basic structure of these terms is very similar in the two cases.     

On the diffraction of neutrons by magnetic spiral structures

A beam of neutrons of wave vector K incident on a magnetic spiral structure with propagation vector q gives rise to two diffracted beams. These are peaked at $\text{K·q}\text{q}\text{～±}\text{q}\text{2}$, and their intensity varies with the direction of K with the same functional form as is derived in the dynamical theory of X-ray diffraction. In real cases the two lines are broad enough to overlap significantly, giving rise to intrinsic double diffraction effects in a multi-domain crystal. These have been observed experimentally in holmium.     

Quasi-1 structure on q-Poincaré algebras

We use braided groups to introduce a theory of 1-structures on general inhomogeneous quantum groups, which we formulate as quasi-1 Hopf algebras. This allows the construction of the tensor product of unitary representations up to a quantum cocycle isomorphism, which is a novel feature of the inhomogeneous case. Examples include q-Poincaré quantum group enveloping algebras in R-matrix from appropriate to the previous q-Euclidean and q-Minkowski space-time algebras R₂₁x₁x₂ = x₂x₁R and R₂₁u₁Ru₂ = u₂R₂₁u₁R. We obtain unitarity of the fundamental differential representations. We further show that the Euclidean and Minkowski-Poincaré quantum groups are twisting equivalent by another quantum cocycle.     

Competing magnetic structures in the DySi FeB-type phase diagram

The temperature magnetic phase diagrams, of the dimorphic DySi compound, have been studied in terms of wave vectors in the range 1.5-45 K, by neutron diffraction. The polycrystalline sample consists of 26% of CrB-type (Cmcm no. 63, all atoms at 4c site: (0, y, 1/4)) and of 74% of FeB-type (Pnma no. 62, all atoms at 4c site: (x, 1/4, z)). The CrB-ordering is described by the wave vector: q₁=(0, 0, 1/2) over the entire magnetically ordered regime with a uniaxial magnetic structure along the shortest axis c. The FeB-type magnetic phase diagram reveals three distinct regions of magnetic ordering below T_N and one first order transition at T₂=23.5 K (on heating). The ordering is described by two symmetry independent magnetic vectors q₂=(0, 1/2, 1/6) and q₃=(0, q_3y, q_3z) with a temperature variable length. At 1.5 K q_3y≈1/2 and q_3z≈1/11. The two phases coexist in the form of domains. They differ in the moment orientation of the q₃ phase that deviates by ∼22° from the b-axis in the (0, 0, 1) plane. The low temperature range (LT) 1.5 K—T₂ subdivides into two regions: (i) LT-1, between 1.5 K—T₁ where the relative amount of the two phases remains unchanged and in (ii) LT-2: T₁-T₂ where the amount of the incommensurate q₃ phase increases at the cost of the commensurate q₂ amplitude modulated structure which remains unchanged but fully disappears at the first order transition at T₂=23.5 K. The q₃ phase undergoes minor changes until 22 K and gets destabilised at T₂ where the q_3z component jumps from the LT value q_3z≈1/11 to the HT value ≈1/7 and the q_3y component increases from 0.484(1) to 0.495(1). (iii) The high temperature (HT) range T₂-T_N (T_N=40±1 K) is described by a single wave vector q₃. The disproportionation of the HT magnetic phase q₃ below T₂ into two coexisting distinct phases q₂, q₃ down to 1.5 K is an unusual phenomenon, to our knowledge observed for the first time. Various mechanisms are discussed.     

Ultrasonic attenuation determination of Hc1 and Hc2 for ErRh4B4 at 1.5 K

The attenuation coefficient of longitudinal sound waves propagating in ErRh₄B₄ has been measured as a function of applied magnetic field where the propagation direction (q) of the sound waves was oriented either parallel or perpendicular to H_applied. For both orientations there is evidence of a type II-1 superconducting transition at H_c1 for T ≈ 1.5 K. In addition, when q ⊥ H an increase in attenuation is evident at H_c2, which does not appear when q 6 H, consistent with theories developed by Tachiki et al. utilizing supercurrent screening of the internal magnetic fields.     

Local field corrections in trigonal Se

The inverse dielectric tensor ?^-1₀₀(q) in the static limit isevaluated in crystalline Se along two symmetry directions, by using a proper model Hamiltonian describing the covalent bond and modified sp³ orbitals as basis states. By comparing the results in diagonal approximation with these obtained including the off diagonal elements (ODE) of the dielectric matrix (DM) it is found that the local field corrections are very large and essential to reproduce the experimental dielectric tensor.     

Magnetic phase diagrams of the CrB- and FeB-type HoSi compounds

The temperature magnetic phase diagrams of the dimorphic HoSi compound were studied by neutron diffraction. The sample comprises 35.5% CrB- (Cmcm) and 64.5% FeB-type (Pnma) of structure. Both phases order antiferromagnetically below T_N=25 K and undergo first-order magnetic transitions at T_ic=16.5 K. Their T-phase diagrams comprise a low temperature (LT) 2.7 K−T_ic and a high temperature (HT) range T_ic−T_N with distinct wave vectors.The LT magnetic ordering of the CrB-type HoSi with the wave vector q₁=(1/2, 0, 1/2) corresponds to a uniaxial magnetic structure, with the Ho moments along the shortest axis c. At 2.7 K the ordered moment value is 8.6(2) μ_B/Ho atom. The HT ordering, described by the wave vector q₂=(q_2x, 0, q_2z) with a T-variable length, corresponds to an amplitude modulated structure.The magnetic ordering of the FeB-type HoSi requires two symmetry independent vectors q₃=(0, q_3y, q_3z) for the LT- and q₄=(q_4x, q_4y, 0) for the HT range. Both vectors correspond to sine wave modulated structures with the Ho magnetic moments confined along the shortest axis b. The q₃ vector has an almost invariable length vs. T close to ≈(0, 9/17, 1/11). At 2.7 K the amplitude of the wave is 10.9(1) μ_B/Ho atom. At T_icq₃ jumps to the wave vector q₄=(q_4x, q_4y, 0) with a T-variable length. At 17 K q₄=(0.092(1), 0.538(3), 0). Around T_ic there is a narrow coexistence range of the q₃ and q₄ competing phases. Various models are discussed and compared with the isomorphic RSi (R=rare earth) compounds counterparts of HoSi, a comparison that has led us to briefly review the magnetic structures available in the literature for this interesting class of compounds.     

Tsallis' quantum q-fields

We generalize several well known quantum equations to a Tsallis' q-scenario, and provide a quantum version of some classical fields associated with them in the recent literature. We refer to the q-Schro¨dinger, q-KleinGordon, q-Dirac, and q-Proca equations advanced in, respectively, Phys. Rev. Lett. 106, 140601(2011), EPL 118,61004(2017) and references therein. We also introduce here equations corresponding to q-Yang-Mills fields, both in the Abelian and non-Abelian instances. We show how to define the q-quantum field theories corresponding to the above equations, introduce the pertinent actions, and obtain equations of motion via the minimum action principle.These q-fields are meaningful at very high energies(Te V scale) for q = 1.15, high energies(Ge V scale) for q = 1.001,and low energies(Me V scale) for q =1.000001 [Nucl. Phys. A 955(2016) 16 and references therein].(See the ALICE experiment at the LHC). Surprisingly enough, these q-fields are simultaneously q-exponential functions of the usual linear fields' logarithms.     

Band structure of platinum from angle resolved photoemission experiments

Angle-resolved and angle-dispersed ultraviolet photoelectron spectroscopy (ARUPS) has been used to determine the experimental band structure of platinum at various points along the ΓX, ΓKX and ΓL directions in the Brillouin zone. Various methods have been employed to obtain the E(k) points, among them absolute methods which determine the energy and the momentum independently without any assumption about the final state dispersion. The experimental points are compared with a self-consistent relativistic band structure calculation for energies below and above the Fermi energy. Good agreement between theory and experiment is found.