Symmetry analysis in neutron diffraction studies of magnetic structures: 2. Changes in periodicity at magnetic phase transitions

A study is made of the translational properties of magnetic structures. The first stage of magnetic structure decoding, which is aimed at determining the magnetic lattice of the crystal, is described in detail using the information obtained from neutron diffraction patterns. Attention is given to the necessity of introducing the notion of transition channel which consists in specifying the wave vector star and the set of its arms describing a given transition. For each of the 14 Bravais lattices of the initial crystal we consider all the possible transition channels with symmetric stars and determine the corresponding magnetic lattices. The results are presented as a table which contains magnetic reflections corresponding to each channel. The notion of wave vectors of a magnetic structure is made more precise and the rule of their translation from the diffraction language into that of group theory is given.     

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.     

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.     

Symmetry analysis of surface band structures in magnetic fields

The method of invariants is used to study the structure of the surface electronic energy bands in the vicinity of symmetry points in the Brillouin zone (BZ). Each of the 17 surface groups has been examined both in the presence and in the absence of spin-orbit coupling. The symmetry points for spinless particles follow a parabolic law while conic “neutrino-like” shape turns out to be the usual shape for spin −1/2 particles. Compulsory Herring degeneracy is found to occur at the sides of the BZ normal to the glide planes. The possibility of cubic rather then linear splitting in the dispersion law in the center of the BZ for groups containing 3- and 6-fold axes is demonstrated. The electron Hamiltonian for all 17 surface groups in the presence of external magnetic fields has been derived and the splitting of the degenerate energy levels in this situation has been discussed.     

The magnetic structures of NdCu2 determined by single crystal neutron diffraction

We investigated the magnetic structure of NdCu₂ by means of neutron diffraction as a function of temperature between 1.5 K and 8 K in zero external field. The diffraction data were obtained on two single crystals with different orientations using the triple-axis-spectrometer TAS6 at the DR-3 reactor at Risø. Two magnetic phases were observed between 1.5K andT _N =6.5K. From 1.5 K to 4.1 K the magnetic reflections can be described by the commensurate wave vector =(3/5 0 0) and its higher harmonics 3 and 5. Below 2.5K the structure is completely squared-up. For 4.1 KT6.5 K the magnetic structure is incommensurate with the chemical lattice and can be described by the wave vector=(3/5 0 0) and its higher harmonies 3 and 5M. Below 2.5 K the structure is completely squared-up. For 4.1 K T 6.5 K the magnetic structure is incommensurate with the chemical lattice and can be described by the wave vector ^*=(0.62 0.044 0). In both phases the Nd-moments are oriented along the easyb-direction.     

Structural and magnetic phase transitions in TbPO4 studied by neutron diffraction

Neutron diffraction experiments on TbPO₄ single crystals have been performed in the temperature range from 1.35 to 294 K. We observe two phase transitions: the onset of antiferromagnetic ordering along the tetragonalc-axis at 2.28 K and tilting of the moments away from thec-axis below 2.15 K. The analysis of the measured reflection profiles shows that the tilting is connected with a distortion of the tetragonal zircon structure.     

Theoretical studies on magnetic atom imaging by use of photoelectron diffraction

We theoretically study possible methods how to obtain imaging of magnetic atoms by use of photoelectron diffraction (PD). We propose a novel method to apply Daimon effect where PD peaks are rotated around forward focusing peaks. In usual circular dichroism, we simply use the difference of the PD intensities for different X-ray circular polarization. In contrast to this dichroism, we rather use the difference of the PD intensities for different circular polarizations but - is used for the – circular polarization only in the data handling. This technique allows us to obtain clear atomic image only of spin polarized atoms, and to distinguish magnetic atoms with up-spins from those with down spins. Some illustrative calculations demonstrate the potential use and also the limitation of this technique.     

High-pressure neutron diffraction studies of KH2PO4-type phase transitions

Abstract

High-pressure neutron-diffraction studies have been carried out on KH₂PO₄, KD₂PO₄, H₂C₄O₄ and PbHPO₄, using single-crystal samples. The results show that changes in the H-ordering temperature, T_C, with pressure (or deuteration) in these systems can be largely attributed to the variation of the H-site separation, δ, with pressure (or deuteration).     

X-ray diffraction microscopy of magnetic structures

We report the first proof-of-principle experiment of iterative phase retrieval from magnetic x-ray diffraction. By using the resonant x-ray excitation process and coherent x-ray scattering, we show that linearly polarized soft x rays can be used to image both the amplitude and the phase of magnetic domain structures. We recovered the magnetic structure of an amorphous terbium-cobalt thin film with a spatial resolution of about 75 nm at the Co L3 edge at 778 eV. In comparison with soft x-ray microscopy images recorded with Fresnel zone plate optics at better than 25 nm spatial resolution, we find qualitative agreement in the observed magnetic structure.     

Phase transitions in LiKSO4: Low-temperature neutron diffraction results

Results of a detailed crystallographic study of the structural phase transitions in LiKSO₄, carried out using single crystal neutron diffraction at ambient and below-ambient temperatures (300–100 K), are presented. The study indicates two clear phase transitions at 205 K (from space group P6₃ to space group P31c), at 189 K (to space group Cc) and a third transition at about 130 K in the cooling cycle. The corresponding temperatures are 250 K, 200 K and 138 K in the heating cycle. The existence of twins, mixed phases, and extremely sluggish kinetics for the various phase transitions, as well as the structural details of the various phases in this mixed salt, can explain the apparent inconsistencies in the results of several experimental investigations reported in the literature.     

Structural studies of liquid alcohols by neutron diffraction

Neutron diffraction measurements have been made on methyl alcohol at room temperature for an incident wavelength of 0·94 Å. Cross sections have been obtained for CD₃OD, CD₃OH, and mixtures of these compounds. These data are subtracted to obtain the separated structure factors for intermolecular H₀H₀, the hydroxyl components, and the non-hydroxyl components. The Fourier transformations of the structure factors show components of both intra- and intermolecular distribution functions. Width parameters obtained from model fits are too large for thermal vibrations and are interpreted as geometrical broadening due to the stretching of bonds, variations in bond angles, and rotation of the methyl group. Differences in the real space distribution function between hydrogen and deuterium are noted.     

磁性物质交换Hamiltonian中两项的竞争

对于一个N电子体系, 正确的交换Hamilton应该由两项组成，为H_ex=-2A1ii·s_j-2A₂ii·sj，而不是以往的铁磁学理论使用的H_ex=-2Aii ·s_j （其中A为A₁与A₂的代数和， A₁>0, A₂<0）, 以往的理论使用了一个不合理的交换Hamiltonian量.-2A₁ii·s_j与-2A₂ii< /sub>·s_j在数学上是同类项，但是在物理上不是 同类项，它们有不同的本征态和本征值.根据量子力学中的态叠加原理，这个电子系统的本 征态矢为X〉=1A²₁+A²₂(A₁ 1〉+A₂‖2〉),其中Dirac符号1〉表示系统所有电子 的自旋平行排列时的态（简称平行自旋态）矢量，2〉表示系统所有电子或最近邻电子的自 旋反平行排列时的态（简称反平行自旋态）矢量，H_ex的本征值（即系统的 交换能） 为E=-Nz（A₁-A₂）-2NzA₂²A₁ +A₂=-Nz（A₂-A₁）-2NzA²₁A< sub>1+A₂,其中z为最近邻电子数.当A₂=0时，X〉=1〉，E =-A₁, 系统具有Wei ss 铁磁性；当A₁ =0 时，X〉=2〉，E =-A₂，系统具有Neel 反铁磁性；当A₁ =A₂（即A=0）时，X〉=12 (1〉+2〉)，E=-A₁，系统处于自旋玻璃（spin glass）态；当A₁>A ₂时，X〉=1A²₁+A²₂［(A₁-A₂)1〉+A₂(1〉+2〉)］,平行自旋态与自旋 玻璃态共存；当A₁2时，X〉=1A²₁+A2₂［(A₂-A₁)2〉+A₁( 1〉+2〉)］，反平行自旋态与自旋玻 璃态共存.与原来理论中的Weiss铁磁态或Neel反铁磁态相比，平行自旋态与自旋玻璃态共存 或反平行自旋态与自旋玻璃态共存使系统的交换能降低.自旋玻璃态中电子自旋之间取向的 随机性或无序性是由交换Hamiltonian中-2A₁iis_j与-2A₂ii·s_j之间的竞争引起的，不是热运 动引起的. 关键词： 交换哈密顿量 铁磁态 反铁磁态 自旋玻璃态     

Antiferromagnetic domains in nickel oxide by magnetic neutron laue diffraction

Two kinds of domains had been observed in antiferromagnetic nickel oxide: T-domains, associated with the four 〈111〉 propagation vectors and S-domains, associated with the three equivalent antiferromagnetic directions within a T-domain. Taking advantage of the direct interaction of neutrons with the arrangement and direction of magnetic moments, the T and S antiferromagnetic domain distributions are investigated in several single crystal samples by magnetic neutron Laue diffraction (MNLD). The T-domain behaviour when a stress is applied is qualitatively investigated. In white beam a direct topographic observation and identification of T-domains are presented. By simultaneously measuring the relative intensities of magnetic Bragg reflections from a sample with preferential domain population it is possible to confirm the direction of sublattice magnetization which had been obtained by an optical method. The S-domain distribution is studied as a function of the magnetic field on the one hand, and as a function of time on the other. This last phenomenon is very similar to the well-known magnetic after-effect which occurs in most ferromagnets. It is concluded that MNLD could be a very valuable technique to investigate many topics of magnetism.     

The determination of the magnetic structure of DyCrO4 by neutron diffraction

At room temperature DyCrO₄ has the tetragonal zircon structure. Susceptibility measurements show that DyCrO₄ is ferromagnetic below 23 K. Neutron diffraction from a powder sample yield the magnetic structure. Using a computer program, which was developed for the determination of non-collinear and spiral structures, several models allowed by symmetry were compared with the experiment. It turns out that the Dy-moments and the Cr-moments as well order parallel in the a-b-plane. For the Dy-moment a value of 12μ_B and for the ratio of the Cr- to the Dy-moment a value of $\text{1}\text{10}$ was found.     

Understanding the magnetic ground state of rare-earth intermetallic compound Ce4Sb3: Magnetization and neutron diffraction studies

Magnetization and neutron diffraction studies have been performed on Ce₄Sb₃ compound (cubic Th₃P₄-type, space group I4¯3d, no. 220). Magnetization of Ce₄Sb₃ reveals a ferromagnetic transition at ∼5 K, the temperature below which the zero-field-cooled and field-cooled magnetization bifurcate in low applied fields. However, a saturation magnetization (M_S) value of only ∼0.93μ_B/Ce³⁺ is observed at 1.8 K, suggesting possible presence of crystal field effects and a paramagnetic/antiferromagnetic Ce³⁺ moment. Magnetocaloric effect in this compound has been computed using the magnetization vs. field data obtained in the vicinity of the magnetic transition, and a maximum magnetic entropy change, −ΔS_M, of ∼8.9 J/kg/K is obtained at 5 K for a field change of 5 T. Inverse magnetocaloric effect occurs at ∼2 K in 5 T indicating the presence of antiferromagnetic component. This has been further confirmed by the neutron diffraction study that evidences commensurate antiferromagnetic ordering at 2 K in zero magnetic field. A magnetic moment of ∼1.24μ_B/Ce³⁺ is obtained at 2 K and the magnetic moments are directed along Z-axis.