Sequence‐adapted molecular tensors: Algebraic methods and application to crystal field theory

The first order "forbidden" transitions, $M=\ensuremath{-}\frac{1}{2}\ensuremath{\rightarrow}+\frac{1}{2}$, $\ensuremath{\Delta}m=\ifmmode\pm\else\textpm\fi{}1$, in the paramagnetic-resonance spectrum of ${\mathrm{Mn}}^{2+}$ in cubic MgO have been studied both experimentally and theoretically. A powder sample was used which permitted the resolution of the "forbidden" transitions which would ordinarily have been obscured by the full fine-structure spectrum. The analysis was based on a spin Hamiltonian that included the cubic crystal field interaction with the $^{6}S_{\frac{5}{2}}$ state of ${\mathrm{Mn}}^{2+}$, magnetic hyperfine interaction with the ${\mathrm{Mn}}^{2+}$ nucleus, and electronic and nuclear Zeeman interactions. Satisfactory agreement between the calculated and observed spectrum was achieved by carrying the hyperfine-perturbation calculation to third order. It was unnecessary to include a trigonal crystal field interaction or a quadrupole interaction in order to satisfactorily explain the spacings of the "forbidden" transitions. On the other hand, the appropriate mixing of states that is required for observation of the forbidden transitions requires the presence of trigonal or lower symmetry crystal field terms. Analysis of the relative intensity of the forbidden to the allowed transitions permitted an estimate to be made of the trigonal crystal field constant $D$ on the basis of an assumed trigonal distortion to the cubic crystal field. By inserting this value of $D$ into the angular dependent terms in the Hamiltonian, an estimate of the expected linewidths was obtained which agreed well with experiment. The contributions of the estimated trigonal distortion to shifts in the spectrum were too small to be observed for those transitions that are resolved with a powder sample.

We present magneto-optical studies of intra-center absorption band related to neutral Mn acceptor in GaN bulk crystals. The band is built of a zero-phonon line at $1.4166\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ followed by GaN phonon spectrum. Magneto-optical experiments performed in magnetic fields up to $22\phantom{\rule{0.3em}{0ex}}\mathrm{T}$ in both Faraday and Voigt configurations revealed very small influence of magnetic field on the spectrum. In particular no splitting of the zero-phonon line was observed. The only characteristic feature recorded in high magnetic fields was a step-like behavior of the spectral position of zero-phonon line measured in Faraday configuration. The step appeared at the magnetic field of about $7\phantom{\rule{0.3em}{0ex}}\mathrm{T}$, with a shift in energy of about $1.3\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$. We interpret the behavior of the zero-phonon line in magnetic field based on crystal-field model for ${\mathrm{Mn}}^{3+}({d}^{4})$ ion in trigonal crystal field, undergoing Jahn-Teller distortion and spin-orbit coupling. Applied model explains reasonably experimental results, supporting strongly localized character of Mn neutral acceptor in GaN.

In this paper, the complete algebraic structure of the finite semisimple group algebra of a normally monomial group is described. The main result is illustrated by computing the explicit Wedderburn decomposition of finite semisimple group algebras of various normally monomial groups. The automorphism groups of these group algebras are also determined.

Geometrical frustration is a central challenge in contemporary condensed\nmatter physics, a crucible favourable to the emergence of novel physics. The\npyrochlore magnets, with rare earth magnetic moments localized at the vertices\nof corner-sharing tetrahedra, play a prominent role in this field, with a rich\nvariety of exotic ground states ranging from the "spin ices" \\hoti\\ and \\dyti\\\nto the "spin liquid" and "ordered spin ice" ground states in \\tbti\\ and \\tbsn.\nInelastic neutron scattering provides valuable information for understanding\nthe nature of these ground states, shedding light on the crystal electric field\n(CEF) level scheme and on the interactions between magnetic moments. We have\nperformed such measurements with unprecedented neutron flux and energy\nresolution, in the "ordered spin ice" \\tbsn. We argue that a new interaction,\nwhich involves the spin lattice coupling through a low temperature distortion\nof the trigonal crystal field, is necessary to account for the data.\n

The paper, as the third in a series, presents the complete formulas for the crystal field splitting third moment of free ion states as functions of the relevant crystal field parameters in the case of the trigonal and hexagonal crystal fields. The additive structure of the expressions consists of 21 independent components for the trigonal symmetry, 12 for the hexagonal one and only 4 components for the ideal hexagonal coordination. The two scalar characteristics of the crystal field Hamiltonian HCF – the crystal field splitting second and third moments – enable us to verify the HCF parametrizations solely in an algebraic way. Such verification concerns the multipolar structure of the parametrizations as well as the signs of the crystal field parameters. Both the crystal field splitting moments can ensure us in what degree the comprehensive HCF parametrizations are adequate with respect to the individual ionic states of the electron configuration. As an example two chosen HCF parametrizations, one for the hexagonal system Nd3+:LaCl3 of the C3h symmetry, and the second for the trigonal system Tb3+:Y2O2S of the C3v symmetry are subjected to in depth analysis. The presented approach provides a critical insight into the crystal field parametrizations from a different point of view, i.e., omitting the conventional fitting procedure. After the extended verification the studied classical HCF parametrizations withstand, in principle, the test of time.

Investigation of the spin-Hamiltonian parameters for the trigonal U5+ center in CaF2 crystal

The wedge-direct sum of a sequence of semisimple algebras (over an arbitrary field) was introduced in [Xu, B. (2019). Wedge-direct sums of table algebras and applications to association schemes, I. Commun. Algebra 47(8):3309–3328]. The group algebras (over the complex field) of finite groups are a special class of semisimple algebras. In [Xu, B. (2020). Wedge-direct sums of table algebras and applications to association schemes, II. J. Pure Appl. Algebra 224(11):106402], the wedge-direct sum of a sequence of group algebras of finite p-groups is used to obtain the characterization and classification of a class of p-table algebras. In this article, we continue the research in above mentioned articles and characterize, in terms of table algebras, the wedge-direct sum of a sequence of group algebras of arbitrary finite groups. Applications to association schemes will also be discussed.

The microscopic origin of the spin Hamiltonian (SH) parameters forNi2+(3d8) ions in a trigonaltype I symmetry (C3v,D3d,D3) crystal field (CF) is studied. In addition to the spin–orbit (SO) interaction,we consider also the spin–spin (SS) and spin–other-orbit (SOO) interactions.The relative importance of the four (SO, SS, SOO, and combined SO–SS–SOO)contributions to the SH parameters is investigated using the CFA/MSH package andthe complete diagonalization method (CDM). The SO mechanism is dominantfor all CF parameter (CFP) ranges studied, except where the contributionsDSO to the zero-field splitting(ZFS) parameter D change sign. For the trigonal CFP, due to the other three mechanisms exceedsDSO. Although|DSOO| is quite small,the combined |DSO−SOO| is appreciable. The SO-based perturbation theory (PT) works generally well for theg-factors: and , while it fails for D in the vicinity of vc and for large and v>0. The high percentagediscrepancy ratio δD = 2020% forvc indicates unreliabilityof DSO (in PT). Applicationsto Ni2+ ions at trigonalsymmetry sites in LiNbO3, α-LiIO3, andAl2O3, areprovided. The theoretical SH parameters are in good agreement with the experimental data. The low symmetry(C3) effects inducedby the angle φ are tentatively studied, but appear to be quite small.

The polarized Raman scattering between the electronic levels of the $^{4}T_{1g}$ ground term of ${\mathrm{Co}}^{2+}$ in ${\mathrm{CsMgCl}}_{3}$ has been observed at 2\ifmmode^\circ\else\textdegree\fi{}K. We calculated the fine structure of $^{4}T_{1g}$ for this system as well as for ${\mathrm{Co}}^{2+}$ in ${\mathrm{CdCl}}_{2}$ (observed by Lockwood and Christie). Second-order perturbation theory has been employed for a Hamiltonian containing the trigonal crystal field $\ensuremath{\Delta}$ and spin-orbit interaction $\ensuremath{\lambda}$. The data are fit satisfactorily with $\frac{\ensuremath{\Delta}}{|\ensuremath{\lambda}|}=+4.4 \mathrm{and} \ensuremath{-}2.5$, respectively. Using the crystal-field wave functions we calculated the spectral intensities of the electronic Raman lines as a function of $\frac{\ensuremath{\Delta}}{|\ensuremath{\lambda}|}$, within the framework of Axe's theory. The theory predicts a small asymmetry between $\mathrm{xz}$ and $\mathrm{zx}$ polarizations and a very weak scattering in $\mathrm{zz}$ polarization. These predictions result mainly from the fact that the intermediate configuration $3{d}^{6}4p$ lies more than 100 000 ${\mathrm{cm}}^{\ensuremath{-}1}$ above the $3{d}^{7}$ configuration, making the closure assumption an adequate approximation. Good agreement is obtained between observed and calculated spectral intensities for both crystals.

Theoretical studies of EPR parameters and defect structures for Ni2+ ions in corundum

A 57 Fe Mössbauer spectroscopic study of natural eosphorite, (Mn,Fe 2+ )AlPO 4 (OH) 2 H 2 O, is reported. The Mössbauer spectra were collected at temperatures between 295 and 4.2 K. At temperatures exceeding 30 K, the Mössbauer spectra consist of a somewhat broadened quadrupole doublet with a splitting of 1.73 mm/s at 295 K. From the spectrum recorded at 250 K in an external magnetic field of 60 kOe, it is derived that the sign of the principal component of the electric field gradient (EFG) is negative and that the EFG’s asymmetry parameter is large (~0.5). From these findings, it is concluded that the octahedral coordination of the ferrous cations exhibits in first order a trigonal compression with a further lowering from axial symmetry due to an additional distortion. The spectrum collected at 4.2 K shows the existence of magnetic ordering. It was analyzed in terms of the full hyperfine interaction Hamiltonian, yielding a magnetic hyperfine field value of 146 kOe and EFG characteristics that are fully in line with those obtained from the external-field spectrum. In the temperature range from 10 to 30 K, the spectra indicate the occurrence of relaxation effects. They could be satisfactorily described in terms of the Blume-Tjon (BT) model for electronic relaxation, assuming a superposition of a slow- and a fast-relaxation component. The observed temperature dependence of the isomer shift is adequately described by the Debye model for the lattice vibrations. The characteristic Mössbauer temperature was found as (360 ± 20) K and the zero-Kelvin intrinsic isomer shift as (1.480 ± 0.005) mm/s. The variation of the quadrupole splitting with temperature is explained by the thermal population of the Fe 2+ electronic states within the T 2g orbital triplet, which is split by the trigonal crystal field in a singlet ground state and an upper doublet state, the latter being further split into two orbital singlets by an additional distortion. The energies of these latter excited states with respect to the ground state are calculated to be 280 ± 20 and 970 ± 50 cm -1 , respectively, based on a point-charge calculation of the 5 D level scheme. This calculation confirms the sign of the V zz component of the electric field gradient being negative as it is determined from the external magnetic field measurement and from the magnetic spectrum acquired at 4.2 K. Iron phosphates are widely spread minerals in the Earth’s crust and are expected to occur in soils and rocks on Mars as well. Mössbauer spectroscopy as complementary source of information to the results of other analytical techniques, could therefore be useful to identify and characterize the environmental Fe-bearing phosphates.

Complete matrices have been calculated for the configuration d2, when both a trigonal crystal field and spin–orbit coupling are present. In particular a value has been found for the trigonal field parameter, which gives the previously reported experimental value for the splitting of the ground state of vanadium corundum, Al2O3(V3+).

We report on the characterization of ternary Zn1–xCoxO alloy crystalline films grown by electrodeposition onto FTO‐coated substrates. The Zn1–xCoxO films have hexagonal wurtzite structure as shown by X‐ray diffraction measurements. The Co incorporation into the ZnO lattice is confirmed by the presence of absorption peaks assigned to Co in trigonal crystal field. X‐ray photoemission spectroscopy indicates that as‐grown films have a considerable concentration of not‐fully oxidized metallic Co in the surface that correlates with the O concentration in the surface. Finally, Raman measurements of as‐grown films indicate that they are polycrystalline with grains of nanometric size showing short‐range order but no long‐range order. The Raman spectra show no trace of first‐order phonons and resemble the one‐and two‐phonon density of states of ZnO (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Ground state properties of CsCoCl 3

I construct an algebraic model for a typical fiber on a 1 + 1 dimensional spacetime. The vector space comprising the fiber is composed of elements x ⊗ x formed from the direct product of two copies of an element x in the D 2 = C 2 ⊗ C 2 finite group algebra over the real numbers. The fiber contains subspaces whose elements can be associated with the tangent and momentum vectors of trajectories in the manifold. The fiber also contains a subspace whose elements are associated with the local flow of action of each trajectory. The condition of minimum action translates into a constraint on the original vector x in the direct product structure.KeywordsTangent spacemanifoldfiber differential geometry1 + 1 spacetime