Symmetry analysis in neutron diffraction studies of magnetic structures: 3. An example: the magnetic structure of spinels

Using the symmetry analysis technique developed in the two earlier papers of this series a study is made of the magnetic structures in spinels (space group O⁷_h). The magnetic structures with the wave vector K = 0 and those with K ≠ 0 are considered in detail. As an example, an analysis is given of the magnetic ordering in MgV₂O₄ which is characterized by the three-arm star {bdK₁₀} (in Kovalev's notation) and of that in HgCr₂S₄ where a helical structure corresponding to the star {bdK₆} has been found. For each of the three stars we have determined the composition of the magnetic representation and calculated the basis functions of the irreducible representations. For the magnetic structures determined experimentally we have specified the irreducible representations by which these structures should be described. The examples furnished illustrate the typical situations liable to occur when performing symmetry analysis of magnetic structures of crystals.     

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.     

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.     

High-pressure effect on the crystal and magnetic structure of LuMnO3: Correlation between a distortion of the triangular lattice and the symmetry of the magnetic state in hexagonal frustrated manganites

The effect of a high pressure (up to 6 GPa) on the crystal and magnetic structure of the hexagonal manganite LuMnO₃ is studied by neutron diffraction in the temperature range 10–295 K. It is found that, as the pressure increases, the ordered magnetic moment of Mn ions at T = 10 K decreases noticeably from 2.48 (0 GPa) to 1.98 μ_B (6 GPa). This decrease is due to an enhancement of the geometrical frustration effects on the triangular lattice. At the same time, the symmetry of the triangular antiferromagnetic state (the irreducible representation Γ₂) remains unchanged. A correlation is revealed between the distortion parameter of the triangular lattice formed by Mn ions and the symmetry of the antiferromagnetic state of hexagonal manganites RMnO₃. Based on this correlation, a generalized magnetic phase diagram of these compounds is constructed. The obtained phase diagram provides an explanation for the changes observed in the magnetic state of hexagonal manganites caused by high pressure and chemical substitution.     

Magnetic and ferroelectric ordering in BiMn<Subscript>2</Subscript>O<Subscript>5</Subscript> oxide

A group-theoretical analysis of the magnetic phase of BiMn₂O₅ oxide is performed using the space symmetry group of the compound. Using the projection operator method, we determine the basis functions of the irreducible representation of the space group, which are expressed in terms of the magnetic vector components. This representation can govern two phase transitions from the paramagnetic state to the antiferromagnetic phase with close temperatures and ordering of the spins of manganese ions in two crystallographic positions. It is found from renorm group analysis of these transitions that when these transitions occur as second- order transitions, the electric polarization does not appear in the system because spin fluctuations in this case elevate the symmetry of the system. Polarization appears when at least one of these transitions becomes a first-order transition as a result of spin fluctuations.     

Symmetry analysis of magnetic structures and spin-waves on the basis of a stabilizer concept

The common scheme for the group theory analysis of a crystal magnetic structure and a spin-wave spectrum is developed. This scheme is based on the representation theory of a crystal space group G (for magnetic structure analysis) and on the co-representation theory of the Shubnikov group M of a magnetically ordered crystal (for spin wave analysis).Unlike the standard calculation, the procedure proposed in this paper is based on the conception of the stabilizer-the symmetry group of single atoms, and this considerably symplified the calculation work, reducing it to the work with the elements of the stabilizer and with the representatives of the expansion of the G or M groups with respect to the group of the stabilizer. It is enough to calculate the components of the basic functions only at one atom, as for the other atoms they may be found by the action of the representatives.For the symmetrization of the spin wave Hamiltonian matrix a new mathematical procedure is used the expansion of group G over the double coset classes. It lets us easily find the symmetric relations between the different matrix elements. The method is illustrated in the example of the complicated magnetic structure of a garnet Dy₃Al₅O₁₂. The advantages of the new method are especially obvious in the case of structures with a great number of magnetic atoms in a primitive cell.     

Low-temperature neutron diffraction study of the crystal and magnetic phase transitions in DyCrO4

The crystal and magnetic structures of DyCrO₄ were studied using neutron powder diffraction. Complete diffraction data at 3.6, 17, 27, and 40 K show that a crystal structural phase transition from tetragonal I4₁/amd to orthorhombic Imma symmetry is found to take place between 27 and 40 K. This transition does not involve a significant change in the unit cell volume. Strong ferromagnetic reflections are observed at 3.6 and 17 K, and can be fit well using the magnetic model of space group Im'ma', with the moments of both Dy³⁺ and Cr⁵⁺ ions aligning along the y-axis. Detailed temperature dependent magnetic intensities of 101/011 and 211/121 peaks reveal a Curie temperature of T_c=22.35(15) K.     

Crystal and magnetic structure of KFeO2

The crystal and magnetic structures of KFeO₂ have been determined by neutron and X-ray powder-diffraction and Mössbauer-effect techniques. The crystal structure at 4.2 K and 300 K is orthorhombic and the magnetic space group is Pbca'. The Fe³⁺-ions in this structure are tetrahedrally coordinated by oxygen ions, and each Fe³⁺-ion has a magnetic moment which is antiferromagnetically coupled to the moments of four Fe³⁺-neighbours. The direction of the moments is parallel to the a-axis. A crystal phase transition has been observed near the Néel temperature?960 K.     

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.     

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.     

Symmetry and magnetic field driven transitions in the 2D triangular lattice compound RbFe(MoO4)2

The temperature versus magnetic field phase diagram of the 2D triangular lattice and multiferroic compound RbFe(MoO(4))(2) is analysed from the point of view of symmetry. The paramagnetic space group and its irreducible representations are used in order to obtain the magnetic symmetry of the possible modulated phases and characterize the restrictions imposed by this symmetry on the corresponding magnetic structures. Superspace symmetry is considered in the case of incommensurate phases. It is shown that the experimentally observed phases correspond to different isotropy subgroups originating in the same irreducible representation of the paramagnetic symmetry group. The relevant couplings between the primary transverse spin modulation and the electric polarization, the in-plane magnetization and the secondary longitudinal magnetic modulation are discussed. The mechanisms for the destabilization of the improper ferroelectric chiral phase and the origin of the different orientation of the spins with respect to the external field in the two collinear phases are analysed from a symmetry based perspective.     

Effect of a strong magnetic field on the quadrupole and magnetic orders and crossover in the Jahn-Teller magnet DyVO<Subscript>4</Subscript>

The effect of a strong magnetic field H||[001] on magnetic properties of the Jahn-Teller compound DyVO₄ is studied. New phase transitions discovered and investigated experimentally and theoretically include the breaking of quadrupole order (enhancement of the crystal symmetry) and breaking of antiferromagnetic order as well as the effect of convergence of energy levels of the Dy³⁺ ion (crossover). The differential magnetic susceptibility of a DyVO₄ crystal is measured in fields up to 36 T in the temperature interval 1.4?15 K. The observed magnetic susceptibility anomalies and phase transitions are described using a unified theoretical approach.     

Fermion-pairing on a square lattice in extreme magnetic fields

We consider the Cooper-problem on a two-dimensional, square lattice with a uniform, perpendicular magnetic field. Only rational flux fractions are considered. An extended (real-space) Hubbard model including nearest and next nearest neighbor interactions is transformed to “k-space”, or more precisely, to the space of eigenfunctions of Harper’s equation, which constitute basis functions of the magnetic translation group for the lattice. A BCS-like truncation of the interaction term is performed. Expanding the interactions in the basis functions of the irreducible representations of the point group C_4ν of the square lattice simplify calculations. The numerical results indicate enhanced binding compared to zero magnetic field, and thus re-entrant superconducting pairing at extreme magnetic fields, well beyond the point where the usual semi-classical treatment of the magnetic field breaks down.     

Crystal structure and the magnetic state of Pr0.5Sr0.5Co0.5Fe0.5O3

The magnetic properties and crystal structure of the Pr_0.5Sr_0.5Co_0.5Fe_0.5O₃ compound are studied by neutron and x-ray diffractions using synchrotron radiation. These measurements show that this compound is a dielectric spin glass with a magnetic moment freezing temperature of about 70 K. As temperature decreases in the range 30–95 K, a structure phase transition of the first order occurs with an increase in the symmetry from orthorhombic (space group Imma) to tetragonal (space group I4/mcm). It is assumed that the transition is caused by a change in the 4f electron configuration of the Pr³⁺ ions.     

Modes of vibrations due to (U-center + additive anion impurities) in KC1—II

Defect modes due to U-centers in potassium chloride containing additive anion impurities have been computed by Green's functions techniques. Local vibrations due to U-centers in alkali halides have cubic symmetry, but when one of the twelve nearest neighbour anions is replaced by an additive impurity, the site symmetry of the system reduces from O_h to C_2v and gives rise to new vibrational modes. The phonon Green's functions have been analysed according to the irreducible representation of the point group symmetry, pertaining to the substitutional impurity. We have considered the vibrations of the U-center, additive anion impurity and their nearest neighbours i.e. 36-dimensional defect space. Using group theory, symmetry coordinates were constructed and the 36 x 36 dimensional matrix was block diagonalised into various irreducible representations. The computed local mode frequencies have been conpared with the experimental measurements. Reasonably good agreement is found between them. The computed results for resonant mode frequencies are also displayed.     

Symmetry analysis in neutron diffraction studies of magnetic structures: 4. Theoretical group analysis of exchange Hamiltonian

By analyzing the exchange Hamiltonian, we develop the technique of determining the magnetic structures liable to occur in a crystal. The technique rests on Bertaut's idea that the exchange matrix eigenfunction corresponds to some magnetic structure. A technically simple and efficient method of diagonalizing the exchange matrix is worked out using the devices of space group representation theory. A method is presented to find the magnetic structures with equal exchange energy (exchange multiplets). The occurrence of exchange multiplets results from the additional invariance of the exchange Hamiltonian under rotation of all the spins. The degeneracy within the exchange multiplet may be the reason why some magnetic structures arise not according to one irreducible representation of the space group. The theory is illustrated with reference to an example of the magnetic structure of spinels.     

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.     

Aspects of symmetry in the problem of phase transitions in improper ferroelectrics

The term ‘improper’ ferroelectrics has recently come to be used for ferroelectrics in which the spontaneous polarizationP _s is not a transition parameter. From the standpoint of symmetry therefore, the new (polar) symmetry group is no longer the highest common sub-group of the crystal symmetry and the spontaneous polarization symmetry (polar vector symmetry group ∞mm). The concept of an improper ferroelectric has therefore been revised to include transitions from polar to polar groups. It is shown that in the general cases where other parameters are permissible, the problem has to be solved using the symmetry of directions. The present analysis on the bases of irreducible representation of polar crystals establishes all the possible cases where spontaneous polarization is accompanied by spontaneous magnetisation, optical activity and more complex phenomena. The symmetry changes and the polarization configurations in Gd(MoO₄)₃, boracites, (NH₄)₂BeF₄, alkali trihydrogen selenite family have been discussed in the light of the extended concept of an ‘improper’ ferroelectric.     

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.