Symmetry analysis in neutron diffraction studies of magnetic structures: 5. Polarization effects in the scattering of neutrons by magnetic structures

It is shown that both the cross section of elastic scattering by a magnetic structure and the polarization vector of the scattered neutrons beam are described by a set of complex axial vectors M^L. They depend on the vectors of atomic magnetic moments in a primitive cell of the crystal and the wave-vector star of the magnetic structure. Particularly, the intensity of a magnetic Bragg reflection and the polarization vector of the scattered beam of that reflection are determined only by a vector M^L connected with an arm star contribution of the magnetic structure and corresponding to the wave vector k_L of a star {k}. A single magnetic reflection allows one to determinean arm contribution of the magnetic structure if one uses polarization effects and an expansion of the magnetic structure over the basis functions of the irreducible representation of the crystal space group. In the general case minimal number of necessary reflections for the total determination of a magnetic crystal is equal to the arm number of the wave-vector star. All analyses here have been done for the single domain magnetic structure. The polarization phenomena will not be observed for many cases where there is a uniform distribution of domains.     

Symmetry analysis in neutron diffraction studies of magnetic structures: 1. A phase transition concept to describe magnetic structures in crystals

On the basis of phase transition symmetry theory there has been developed an efficient method of calculating the possible magnetic structures liable to arise from the paramagnetic phase of the crystal. Every magnetic structure is described by the superposition of the basis functions of the irreducible representation incorporated in the magnetic representation of the crystal symmetry group. Convenient formulas are given to calculate the basis functions and composition of the magnetic representation. These formulas permit calculation by referring only to the tables of space group irreducible representations. The technique is illustrated on the example of magnetic structures in crystals with the symmetry group D⁶_3d.     

Application of symmetry groups of four-dimensional space in spectroscopy of crystals

A method of classification of many-electron states that involves symmetry groups of four-dimensional space is developed. Taking into account the spatial and temporal symmetry, the selection rules for electric and magnetic dipole transitions are investigated. The reduction relations for the irreducible representations of the orthogonal group O ₄ and the group R ₄ of pure rotations of the four-dimensional space on the groups and O _h ⁴ and O ⁴ of the four-dimensional cube and of the octahedron on the three-dimensional groups O _h and D _6h of the cubic and hexagonal systems, respectively, are obtained. The four-dimensional classification of the levels of the spin-orbit interaction of rare earth ions for the intermediate reduction on the group of the four-dimensional cube is performed. With the help of the irreducible representations of the O ₄ group, the selection rules of magnetic and forbidden electric dipole transitions, as well as of intercombination transitions, are investigated.     

Magnetic structure of the Nd5Ge3 compound

Neutron diffraction measurements of the Nd₅Ge₃ intermetallic compound with a hexagonal structure (space group P6₃/mcm) have been performed at temperatures of ～10 and 293 K. The basis functions of irreducible representations of the space group D _6h ³ (P6₃/mcm), which are calculated as a result of the symmetry analysis of possible magnetic structures with the wave vector k = μb ₁, are used to facilitate the search for a real model of the magnetic structure of the compound.     

On determination of the analytical formulas for reduction matrices of tetrahedral-symmetry molecules

The formulas for the elements of ^(J)G matrices (J = 0, 1, 2 ?) determining reduction of irreducible representations D^(Jg) of the group 0(3) to the irreducible representations D^(Γ) of the T_d group have been obtained in an analytical form. It was shown how the transition was made from the G matrices of this paper to those determined by Moret-Bailly's basis [Cah. Phys., 15, 237–314 (1961)].     

Rare earth single-ion anisotropy and the magnetic structures of the RTiO3 perovskites: R = Tb,Dy, Ho,Er and Tm

Attention is drawn to common features in the magnetic structures of the isostructural RMO₃ phases where R = Tb, Dy, Ho, Er and Tm and M = Al, Ti, Cr, Fe and Co. The orientation of the rare earth magnetic moment with respect to the orthorhombic c-axis depends only on R. For R = Er and Tm the moments are parallel to the c-axis while for R = Tb, Dy and Ho they lie in the a-b plane. The in-plane moments are canted with respect to the a and b axes with both ferromagnetic and antiferromagnetic components and the canting angles are constant for a given R independent of M. Using symmetry arguments we show that the above systematics can be understood in terms of the rare earth single-ion anisotropy. Detailed calculations incorporating a crystal field of C_s symmetry determined for Er³⁺ in YAlO₃ and an isotropic molecular field of magnitude appropriate to the RTiO₃ compounds produce results in agreement with the experimental observations. Essentially the same results are obtained for a crystal field of D_4h symmetry. The B²₀O²₀ term in the crystal field Hamiltonian is identified as the factor which determines the orientation of the rare earth moment.     

Temperature dependence of magnetic anisotropy constant in cobalt ferrite nanoparticles

The temperature dependence of the effective magnetic anisotropy constant K(T) of CoFe₂O₄ nanoparticles is obtained based on the SQUID magnetometry measurements and Mössbauer spectroscopy. The variation of the blocking temperature T_B as a function of particle radius r is first determined by associating the particle size distribution and the anisotropy energy barrier distribution deduced from the hysteresis curve and the magnetization decay curve, respectively. Finally, the magnetic anisotropy constant at each temperature is calculated from the relation between r and T_B. The resultant effective magnetic anisotropy constant K(T) decreases markedly with increasing temperature from 1.1×10⁷ J/m³ at 5 K to 0.6×10⁵ J/m³ at 280 K. The attempt time τ₀ is also determined to be 6.1×10⁻¹² s which together with the K(T) best explains the temperature dependence of superparamagnetic fraction in Mössbauer spectra.     

Determination of the magnetic anisotropy of ferrofluids from torque magnetometry data

A theoretical analysis is presented which enables the uniaxial magnetic anisotropy constant K₁ of particles in ferrofluids, frozen in an external field, to be obtained from torque magnetometry measurements. The two-fold symmetry of the torque curve, found experimentally, is correctly predicted. An asymptotic solution is found which enables K₁ to be determined without recourse to iterative numerical methods. In this limit, the torque amplitude varies linearly with the inverse of the freezing field for large freezing fields. For all cases, extraction of K₁ requires an accurate knowledge of the particle size distribution parameters.     

The symmetric amino-wagging band of hydrazine: Assignment and analysis

For the first time a weak fundamental symmetric amino-wagging band, ν₆, was assigned in the high-resolution Fourier-transform infrared spectrum of hydrazine. The analysis of the Fermi-type resonance between the ν₆ and the third excited torsional state, 3ν₇, is presented. A global fitting was carried out taking into account 3392 lines of the ν₆ band (for K′ from 0 to 10 and for all symmetry species) and 428 lines of the 3ν₇ band (for K′ from 3 to 9 and only for the symmetry species in resonance). For all 3820 rovibrational transitions the overall standard deviation of the fit of 0.019 cm⁻¹ was obtained. The band centers of the symmetric wagging state and the third torsional were determined at 795.137 and 860.138 cm⁻¹, respectively. Individual fits were also carried out for K′ from 3 to 8 for all symmetry species with much improved standard deviations. The effective group-theoretical Hamiltonian for the coupling between inversion and torsion states was used.     

High-pressure effect on the crystal and magnetic structures of the frustrated antiferromagnet YMnO<Subscript>3</Subscript>

The high-pressure (to 5 GPa) effect on the crystal and magnetic structures of the hexagonal manganite YMnO₃ is studied by neutron diffraction in the temperature range 10–295 K. A spin-liquid state due to magnetic frustration on the triangular lattice formed by Mn ions is observed in this compound at normal pressure and T > T_N = 70 K, and an ordered triangular antiferromagnetic state with the symmetry of the irreducible representation Γ₁ arises at T < T_N. The high-pressure effect leads to a spin reorientation of Mn magnetic moments and a change in the symmetry of the antiferromagnetic structure, which can be described by a combination of the irreducible representations Γ₁ and Γ₂. In addition, it is observed that the ordered magnetic moment of Mn ions decreases from 3.27 μ_B (5 GPa) to 1.52 μ_B (5 GPa) at T = 10 K and diffuse scattering is enhanced at temperatures close to T_N. These effects can be explained within the model of the coexistence of the ordered antiferromagnetic phase and the spin-liquid state, whose volume fraction increases with pressure due to the enhancement of frustration effects.     

Dynamical symmetry of the magnetic monopole

It is demonstrated that the interaction of a charged particle with a magnetic monopole possesses a large invariance; time can be arbitrarily re-parametrized. When the interaction occurs within conventional, non-relativistic dynamics, the entire theory admits an O(2, 1) conformal group of symmetry transformations, which seems to have escaped notice. Combining this invariance group with the O(3) group of spatial rotations shows that an O(2, 1) × O(3) group of invariances is present, in analogy with the Kepler/Coulomb system. Furthermore, at fixed angular momentum, the dynamics are characterized by a single, irreducible, unitary representation of the conformal O(2, 1) symmetry group, whose Casimir eigenvalue is determined by the monopole strength. Some similar properties of the isotropic harmonic oscillator are also mentioned.     

Magnetic properties of erbium iron garnet in high magnetic fields up to 150kOe

The magnetic properties of single crystals of erbium iron garnet (ErIG) were studied in applied fields up to 150kOe between 1.4 and 300K. At low temperature, the macroscopic easy direction of the bulk magnetization is [100]; below the compensation temperature (80±2K), the magnetization presents non-linear field evolution. On the assumption of an isolated ground doublet, the anisotropy constantsK _i (i=1,2) of ErIG are given byK _i (Er)+K _i (YIG); theK _i are calculated as a function of theG andg tensor components. It is worthwhile noting that theK _i (Er) are strongly temperature dependent; so at low temperature the anisotropy of the garnet is determined by the rare earth ions, while in the 50 K regionK ₁(Er) becomes comparable toK ₁(YIG) with the opposite sign which results in a very weak anisotropy of the garnet. Above 50 K,K ₁(YIG) is predominant and the Fe³⁺ ions determine the garnet anisotropy.     

The influence of strong magnetic fields on position production in heavy-ion collisions

The strongest magnetic fields (|B|_max≈10¹⁵G) being accessible to experimental investigations are created transiently in collisions of very heavy ions. The possible interplay of these collisional magnetic fields with detected peak structures in spectra of emitted electron-position pairs is elucidated. As basis for a dynamical treatment we computed the adiabatic phase correlation diagram which exhibits no structures to cause the striking peaks. Relativistic two-centre continuum states for non-axial symmetric vector potentials are determined.     

Tensor methods for the exceptional group E6

We present a tensor formalism to describe irreducible representations of the exceptional group E₆. Irreducible tensors are characterized by covariant and contravariant indices associated with the irreducible representation 27, and a third (orthogonal-type) index associated with the 78; contractions of these indices with a set of invariant tensors are required to vanish for irreducibility. The formalism is applied to the reduction of Kronecker products of E₆ irreducible representations. As a further illustration of the method, we construct explicitly the Higgs potential for scalar fields in the E₆ representations 27, 78, 351, 351′.     

First-principles study of the effect of iron doping on the electronic and magnetic properties of TbMn2O5

We have studied the electronic and magnetic properties of TbFe_xMn_2−xO₅ (x=0, 0.125, 0.25) samples using first-principles density functional theory within the generalized gradient approximation (GGA) schemes. The crystal structure of TbMn₂O₅ is orthorhombic containing Mn⁴⁺O₆ octahedra and Mn³⁺O₅ pyramids. The structure changes to monoclinic symmetry for the Fe-doping at the Mn sites. Our spin-polarized calculations give an insulating ground state for TbMn₂O₅ and a metallic ground state for Fe-doped TbMn₂O₅. Based on the magnetic properties calculations, it is found that the magnetic moment enhances with increase in the Fe-content in TbMn₂O₅. Most interestingly, the enhanced magnetic moment is due to a substantial reduction of the magnetic moments at the Fe sites.     

Some estimates for magnetic fluctuations in a turbulent medium

A new integral relationship between the fluctuations b(r, t) of a magnetic field and its mean B ₀(r, t) is derived for the steady-state magnetic field in a turbulent medium. This formula provides the estimate 〈b?curlb〉=?B ₀?curlB ₀. Simultaneously, the coefficient of amplification of the mean magnetic field α effect) is obtained: α=(η+β)B ₀? curlB ₀/B ₀ ² . The formula for α allows for a decrease in this coefficient owing to the back action of the magnetic field on the turbulent velocity field. It is shown that the Zel’dovich’s estimate 〈 b ²〉?β/η B ₀ ² for two-dimensional turbulence holds for magnetic fields at the instant the fluctuations 〈a ²〉 of the vector potential, rather than 〈b ²〉, reach a maximum. Here, η and β are the ohmic (molecular) and turbulent diffusion coefficients, respectively. This estimate is refined with allowance made for the fact that the condition for diffusion approximation itself relates the β, b, and B ₀ quantities to each other.     

The magnetic structure of the compound Ho5Sb3

We have investigated the magnetic behavior of Ho₅Sb₃ compound (Mn₅Si₃-type, hexagonal; a=0.8865(1) nm, c=0.6232(1) nm, as derived from X-ray Guinier powder pattern) by using the techniques of magnetization, electrical resistivity, heat capacity and neutron diffraction. We find that Ho₅Sb₃ exhibits a ferrimagnetic type (Ferrimagnet I) ordering below 60 K with propagation vectors K₀=[0, 0, 1] and K₁=[±K_x, 0, 0]. Below 40 K, the thermal variation of magnetic reflections and the appearance of an additional magnetic component with propagation vector K₂=[0, 1/2, 0] show the onset of an antiferromagnetic type of ordering in the magnetic structure; which evolves into yet another ferrimagnetic structure (Ferrimagnet II) as the temperature is lowered down to 2 K. The magnetic moments of the Ho atoms at the (4d) and (6g) sites with magnitudes of nearly 7.4 and 6.3 μ_B at 2 K, respectively, are inclined approximately at 70° to the c-axis.     

Nuclear magnetic resonance in noncollinear antiferromagnet Mn3Al2Ge3O12

The ⁵⁵Mn nuclear magnetic resonance spectrum of noncollinear 12-sublattice antiferromagnet Mn₃Al₂Ge₃O₁₂ has been studied in the frequency range of 200–640 MHz in the external magnetic field H ‖ [001] at T = 1.2 K. Three absorption lines have been observed in fields less than the field of the reorientation transition H _c at the polarization h ‖ H of the rf field. Two lines have been observed at H > H _c and h ⊥ H. The spectral parameters indicate that the magnetic structure of manganese garnet differs slightly from the exchange triangular 120-degree structure. The anisotropy of the spin reduction and (or) weak antiferromagnetism that are allowed by the crystal symmetry lead to the difference of ≈3% in the magnetization of sublattices in the field H < H _c. When the spin plane rotates from the orientation perpendicular to the C ₃ axis to the orientation perpendicular to the C ₄ axis, all magnetic moments of the electronic subsystem decrease by ≈2% from the average value in the zero field.     

Dynamical generation of flavour

We propose the generation of Standard Model fermion hierarchy by the extension of renormalizable SO(10) GUT with O(N_g) family gauge symmetry. In this scenario, Higgs representations of SO(10) also carry family indices and are called Yukawons. Vacuum expectation values of these Yukawon fields break GUT and family symmetry and generate MSSM Yukawa couplings dynamically. We have demonstrated this idea using \({\mathbf {10}}\oplus {\mathbf {210}} \oplus {\mathbf {126}} \oplus {\overline {\mathbf {126}}}\) Higgs irrep, ignoring the contribution of 120-plet which is, however, required for complete fitting of fermion mass-mixing data. The effective MSSM matter fermion couplings to the light Higgs pair are determined by the null eigenvectors of the MSSM-type Higgs doublet superfield mass matrix \(\mathcal {H}\). A consistency condition on the doublet ([1,2,±1]) mass matrix (\(\text {Det}(\mathcal {H})=\) 0) is required to keep one pair of Higgs doublets light in the effective MSSM. We show that the Yukawa structure generated by null eigenvectors of \(\mathcal {H}\) are of generic kind required by the MSSM. A hidden sector with a pair of (S_ab; ?_ab) fields breaks supersymmetry and facilitates \(D_{O(N_{g})}\hspace *{-1pt}=\) 0. SUSY breaking is communicated via supergravity. In this scenario, matter fermion Yukawa couplings are reduced from 15 to just 3 parameters in MSGUT with three generations.     

Ordered phases of lithium nickelite Li<Subscript>1−<Emphasis Type="Italic">x</Emphasis>−<Emphasis Type="Italic">z</Emphasis></Subscript>Ni<Subscript>1+<Emphasis Type="Italic">x</Emphasis></Subscript>O<Subscript>2</Subscript>

A symmetry analysis of ordering in lithium nickelite Li_1?x?zNi_1+xO₂ (Li_1?x?z□_yNi_1+xO₂) was performed with regard to the substitution of Li and Ni atoms and the occurrence of structural vacancies □ in the metal sublattice. For all the ordered phases, the k ₉ ⁽³⁾ ray of the Lifshitz {k₉} star is present in the order-disorder transition channel. This ray determines the consecutive alternation of atomic planes filled with only Ni atoms or only Li atoms and vacancies in the \([1\bar 11]_{B1} \) direction. It was shown that the rhombohedral ordered LiNiO₂ phase is formed in the defect-free lithium nickelite, whereas a family of three monoclinic Li₃□Ni₄O₈ (C2/m space group) and Li₂□Ni₃O₆ (C2/m and C2 space groups) superstructures arises as the concentration of structural vacancies increases. For all the superstructures, the order-disorder phase-transition channels were determined and the distribution functions of Li and Ni atoms have been calculated. The long-range order parameters describing each superstructure were found as functions of the Li_{1?x? z}Ni_1+xO₂ composition.