Irreducible Tensor Operators Research Articles

The spectral fine structure, zero-field splitting parameters and Jahn-Teller effect of LiNbO3∶Fe3+crystals

The 252 rank completely diagonalized Hamiltonian matrix of the 3d5 configuration with trigonal symmetry have been established by group theory, crystal field theory and irreducible tensor operator method. The spectral fine structure, zero-field splitting parameters, and Jahn-Teller effect, as well s the influence of spin doublet state and spin quartet state on the ground-state energy levels in LiNbO3∶Fe3+ crystals were studied with this matrix. The results show that the contribution of spin quartet state on the ground- state energy levels is the most important, the contribution of spin doublet state is weak but can not be neglected. The calculated results are in good agreement with the experiments. On their basis, the influence of spin-orbit interaction and spin-spin interaction on the spectra fine structure and zero-field splitting parameters were further studied, and we found that the influence of spin-orbit interaction is dominant, but the influence of spin-spin interaction can not be neglected. The research shows this substance has J-T effect in the spectral structure of spin quartet state. The reason is the combined action of spin-orbit interaction and trigonal distortion.

Preliminary analysis of CH 3D from 3250 to 3700 cm −1

The infrared spectrum of CH 3D from 3250 to 3700 cm −1 was studied for the first time to assign transitions involving the ν 2 + ν 3, ν 2 + ν 5, ν 2 + ν 6, ν 3 + 2 ν 6 and 3 ν 6 vibrational states. Line positions and intensities were measured at 0.011 cm −1 resolution using Fourier transform spectra recorded at Kitt Peak with isotopically enriched samples. Some 2852 line positions (involving over 900 upper state levels) and 874 line intensities were reproduced with RMS values of 0.0009 cm −1 and 4.6%, respectively. The strongest bands were found to be ν 2 + ν 3 at 3499.7 cm −1 and ν 2 + ν 6 at 3342.5 cm −1 with integrated strengths, respectively, of 8.17 × 10 −20 and 2.44 × 10 −20 (cm −1/molecule · cm −2) at 296 K (for 100% CH 3D). The effective Hamiltonian was expressed in terms of irreducible tensor operators and adapted to symmetric top molecules. Its present configuration in the MIRS package permitted simultaneous consideration of the four lowest polyads of CH 3D: the Ground State (G.S.), the Triad from 6.3 to 9.5 μm, the Nonad from 3.1 to 4.8 μm and now the Enneadecad (19 bands) from 2.2 to 3.1 μm. The CH 3D line parameters for this interval were calculated to create a new database for the 3 μm region.

Quantum dynamics of atomic coherence in a spin-1 condensate: Mean-field versus many-body simulation

We analyse and numerically simulate the full many-body quantum dynamics of a spin-1 condensate in the single spatial mode approximation. Initially, the condensate is in a “ferromagnetic” state with all spins aligned along the y axis and the magnetic field pointing along the z axis. In the course of evolution the spinor condensate undergoes a characteristic change of symmetry, which in a real experiment could be a signature of spin-mixing many-body interactions. The results of our simulations are conveniently visualised within the picture of irreducible tensor operators.

Geometric spin frustration for isolated plaquettes of the lattices: An extended irreducible tensor operator method

A new strategy to search for the good quantum numbers for the corner-sharing spin systems, as archetypal plaquettes of the lattices, was suggested for the first time in order to study on geometric spin frustration. The calculations on energy spectra by using the irreducible tensor operator method with the new strategy can be much reduced. As representative examples the energy spectra for the spin pentamer of the tetrahedron with a centered spin site and the spin heptamer of three corner-sharing equilateral-triangle were examined in order to confirm efficiency of the new strategy. Through our code, with automatically searching for the good quantum numbers, the projection operators S iz , S ix and S iy matrices in the ground state space for the spin heptamer were reliably constructed.

Spectrum structure and g factor of electron paramagnetic resonance of LiCoO2 crystal doped with Ni

In the weak field scheme, the complete diagonalizing Hamiltonian matrices and th e constants of crystal structure and g factor of electron paramagnetic resonanc e are derived for 3d4/3d6 ions configuration in the trigon al symmetry in terms of Racahs irreducible tensor operator method. The ground- state energy levels and the constants of crystal structure and g factor of elec tron paramagnetic resonance of the Co3+ in LiCoO2 and LiCo O2:Ni crystals are studied. The spin singlets and triplet of the grou nd-state energy levels of the Co3+ in LiCoO2 and LiCoO2:Ni crystals are taken into consideration in the calculation and applica tion of the crystal-field energy of 3d4/3d6. The changes o f the constants of local structure, which are the causes of the changes of the g round-state energy levels of the Co3+, and the values of g factor of electron paramagnetic resonance of Co3+ in LiCoO2 and LiCo O2:Ni crystals are also studied. The theoretical results are in good agreement with the experimental observations.

Orbital Angular Momentum Contribution to the Magneto-Optical Behavior of a Binuclear Cobalt(II) Complex

We report magnetic and magnetic circular dichroism investigations of a binuclear Co(II) compound. The Hamiltonian of the system involves an isotropic exchange interaction dealing with the real spins of cobalt(II) ions, spin-orbit coupling, and a low-symmetry crystal field acting within the (4)T(1g) ground manifold of each cobalt ion. It is shown that spin-orbit coupling between this ground term and the low-lying excited ones can be taken into consideration as an effective g factor in the Zeeman part of the Hamiltonian. The value of this g factor is estimated for the averaged experimental values of Racah and cubic ligand field parameters for high-spin cobalt(II). The treatment of the Hamiltonian is performed with the use of a irreducible tensor operator technique. The results of the calculation are in good agreement with experimental observations. Both a large effective g factor for the ground state and a large temperature-independent part of the magnetic susceptibility arise because of a strong orbital contribution to the magnetic behavior of the Co(II) dimer.

Many-spin effects in inelastic neutron scattering and electron paramagnetic resonance of molecular nanomagnets

Many molecular magnetic clusters, such as single-molecule magnets, are characterized by spin ground states with defined total spin S exhibiting zero-field-splittings. In this work, the spectroscopic intensities of the transitions within the ground-state multiplet are analyzed. In particular, the effects of a mixing with higher-lying spin multiplets, which is produced by anisotropic interactions and is neglected in the standard single-spin description, are investigated systematically for the two experimental techniques of inelastic neutron scattering (INS) and electron paramagnetic resonance (EPR), with emphasis on the former technique. The spectroscopic transition intensities are calculated analytically by constructing corresponding effective spin operators perturbationally up to second order and consequently using irreducible tensor operator techniques. Three main effects of spin mixing are observed. Firstly, a pronounced dependence of the INS intensities on the momentum transfer Q, with a typical oscillatory behavior, emerges in first order, signaling the many-spin nature of the wave functions in exchange-coupled clusters. Secondly, as compared to the results of a first-order calculation, the intensities of the transitions within the spin multiplet are affected differently by spin mixing. This allows one, thirdly, to differentiate the higher-order contributions to the cluster magnetic anisotropy which come from the single-ion ligand-field terms and spin mixing, respectively. The analytical results are illustrated by means of the three examples of an antiferromagnetic heteronuclear dimer, the Mn-[3 x 3] grid molecule, and the single-molecule magnet Mn12.

Calculation of the Magnetic Properties of Low-Symmetry Polynuclear Complexes. Exchange Interactions in the Hexanuclear Chromium(III) Pivalate

A theory that allows numerical calculations and interpretation of magnetic properties of low-symmetry polynuclear complexes containing a large number of orbitally non-degenerate transition metal ions is developed by the irreducible tensor operator method. The potential of the theory was illustrated using the chromium(III) complex, Cr6O2(OH)2[O2CC(CH3)3]11 as an example.

Spherical tensor analysis of nuclear magnetic resonance signals

In a nuclear magnetic-resonance (NMR) experiment, the spin density operator may be regarded as a superposition of irreducible spherical tensor operators. Each of these spin operators evolves during the NMR experiment and may give rise to an NMR signal at a later time. The NMR signal at the end of a pulse sequence may, therefore, be regarded as a superposition of spherical components, each derived from a different spherical tensor operator. We describe an experimental method, called spherical tensor analysis (STA), which allows the complete resolution of the NMR signal into its individual spherical components. The method is demonstrated on a powder of a (13)C-labeled amino acid, exposed to a pulse sequence generating a double-quantum effective Hamiltonian. The propagation of spin order through the space of spherical tensor operators is revealed by the STA procedure, both in static and rotating solids. Possible applications of STA to the NMR of liquids, liquid crystals, and solids are discussed.

WIGNER–ECKART THEOREM FOR TENSOR OPERATORS OF HOPF ALGEBRAS

We prove Wigner–Eckart theorem for the irreducible tensor operators for arbitrary Hopf algebras, provided that tensor product of their irreducible representations is completely reducible. The proof is based on the properties of the irreducible representations of Hopf algebras, in particular on Schur lemma. Two classes of tensor operators for the Hopf algebra Ut(su(2)) are considered. The reduced matrix elements for the class of irreducible tensor operators are calculated. A construction of some elements of the center of Ut(su(2)) is given.

Spin dynamics and tunneling of the Néel vector in theFe10magnetic wheel

The spin dynamics of a ring-shaped molecule comprising 10 iron(III) ions $(s=5∕2)$ is studied by inelastic neutron scattering. Exchange integrals and single-ion anisotropy parameters are determined by studying several intermultiplet excitations. Difficulties associated with the dimension of the Hilbert space are overcome by exploiting both the irreducible tensor operator technique and the Lanczos algorithm. By calculating the time correlation function of the N\'eel vector we show that the low-temperature spin dynamics of ${\mathrm{Fe}}_{10}$ is only approximately described by the semiclassical picture of tunneling of the N\'eel vector.

Multipole-multimode Floquet theory in nuclear magnetic resonance

In this paper, we present a new analytical approach for describing the spin dynamics of synchronous and asynchronous time-dependent modulations in solid-state nuclear magnetic resonance experiments. The approach, based on multimode Floquet theory, employs the multipole operator basis of Sanctuary for spin description and illustrates the time evolution in the Floquet-Liouville space using the effective Hamiltonians obtained from the contact (or van Vleck) transformation procedure. Since the Hamiltonian and the density operator are expressed in terms of irreducible tensor operators, extensions to higher spin magnitudes (I>12) and multiple spins are quite straightforward and permit analytical treatments for many problems. We outline the general underlying principles involved in this approach with a brief mention of its potential application in other branches of spectroscopy.

A way of visualizing NMR experiments on quadrupolar nuclei

A graphical representation of the spin states of quadrupolar nuclei in NMR experiments is described. The spherical harmonics represent the irreducible spherical tensor operator basis set, and therefore any density matrix can be illustrated with computer graphics by converting it to a sum of spherical harmonics. Using a multiple quantum-filtered experiment as an example, we demonstrate how the phase and flip angle of RF pulses can be used to create multiple quantum coherences, to suppress the creation of particular coherences, and to select coherences by exploiting their symmetry. This visual approach complements the algebraic description of pulse sequences, providing an intuitive way of designing and analyzing pulse sequences. © 2005 Wiley Periodicals, Inc. Concepts Magn Reson Part A 25A: 40–52, 2005.

Tensor operators and Wigner–Eckart theorem for the quantum superalgebraUq[osp(1|2)

Tensor operators in graded representations of Z_{2}-graded Hopf algebras are defined and their elementary properties are derived. Wigner-Eckart theorem for irreducible tensor operators for U_{q}[osp(1\mid 2)] is proven. Examples of tensor operators in the irreducible representation space of Hopf algebra U_{q}[osp(1\mid 2)] are considered. The reduced matrix elements for the irreducible tensor operators are calculated. A construction of some elements of the center of U_{q}[osp(1\mid 2)] is given.

Inelastic neutron scattering study of the molecular grid nanomagnet Mn-[3×3]

The spin Hamiltonian of a supramolecular planar grid formed by nine Mn(II) ions in an almost square geometry has been determined from the analysis of inelastic neutron scattering spectra. The experimental data have been interpreted in the framework of a theoretical model containing all the main interactions present in the system. Difficulties related with the dimension of the Hilbert space have been overcome by exploiting both the irreducible tensor operator technique and the Lanczos algorithm. The energy spectrum consists of level multiplets with an almost definite value of the total spin S. The ground multiplet has S=5/2, and the lowest-lying excited multiplets have (in order of increasing energy) S=7/2, S=3/2, S=3/2, and S=9/2. These multiplets are separated by the isotropic exchange and split by the anisotropic interactions. The two lowest manifolds of the exchange part of the spin Hamiltonian belong to the same irreducible representation of the molecular symmetry group and substantial mixing between them is demonstrated.

Probability and spin densities of rare earth tri-positive ions

Probability and spin densities for rare-earth tri-positive ions in their ground states are calculated using irreducible tensor operators and Racah double tensors, which enable an expansion of the densities in (vector) spherical harmonics. Graphical representation of such densities reveal characteristic features lost in schematic shapes predicted by considering only the sign of the electrostatic quadruple moments. As a rule, due to the spin–orbit coupling, the spin density is not collinear. The only exceptions are found in the fully aligned ( M=± J) states of the heavy rare earths, where J= L+ S and S( r)∥ z. The implications of spin-density patterns for the nature of anisotropic exchange interactions are discussed.

New measurements and global analysis of chloromethane in the region from 0 to 1800 cm −1

New high resolution Fourier transform spectra of pure 12CH 3 35Cl and 12CH 3 37Cl isotopomers of chloromethane have been recorded in Wuppertal covering the region from 600 to 3800 cm −1. New rotational transitions within the v 2=1, v 5=1, and v 3=2 states have been measured at Lille. A first global analysis of the lower four band systems of the molecule (700–1800 cm −1) is reported. The model was based on an effective Hamiltonian and dipole moment expressed in terms of irreducible tensor operators. A common set of 125 effective hamiltonian parameters (sixth order) has been adjusted to fit simultaneously some 11 000 IR data for each of the isotopomers including 153 mm wave data for 12 CH 3 35 Cl. The assignments involve 12 sets of transitions (6 cold bands, 3 hot bands, and 3 pure rotational systems for 12CH 3 35Cl). The standard deviation was on average 0.00014 cm −1 and 175 kHz for the IR and MMW data, respectively. The v 3= v 6=1 state was analysed for the first time principally from observed hot band transitions.

Problem of the magnetic anisotropy in orbitally degenerate exchange and mixed-valence clusters

This contribution summarizes the results obtained in the problem of orbital degeneracy of the metal ions in exchange coupled and mixed-valence (MV) clusters. The theory of the double exchange is generalized and the orbitally degenerate systems are considered. The orbitally dependent double exchange parameter is deduced for the singlet–triplet and triplet–triplet transition metal pairs in three high-symmetric topologies. A new effective Hamiltonian of the magnetic exchange between the ions with unquenched orbital angular momenta is discussed. The technique of the irreducible tensor operators is applied to the problem of the kinetic exchange in these kind of metal clusters. Strong magnetic anisotropy is shown to appear in the exchange and MV clusters consisting of orbitally degenerate ions. The influence of the vibronic pseudo Jahn–Teller interaction on the degree of the magnetic anisotropy is discussed.

Selective cross-polarization in solution state nuclear magnetic resonance of scalar coupled spin 12 and quadrupolar nuclei

Semiselective heteronuclear cross-polarization for achieving coherence transfer between a spin I=12 scalar coupled to a spin S⩾12 in isotropic solution is considered. The expansion of the density operator as products of irreducible tensor operators provides a compact formalism for describing cross-polarization involving scalar coupled quadrupolar nuclei. An analytical description of cross-polarization is presented in the limit of strong radio-frequency (rf) fields, with respect to the scalar-coupling constant. Numerical simulations show that reducing the rf field amplitudes does not have a detrimental effect on the efficiency of the transfer provided they are comparable to or greater than the scalar-coupling constant. The use of weak rf fields largely circumvents the reduced efficacy due to Hartmann–Hahn mismatch. Applications of the method for selective observation of scalar-coupled quadrupolar nuclei are considered and experimental results are presented for a mixture of beryllium fluoride complexes.

Orbitally dependent magnetic coupling between cobalt(II) ions: The problem of the magnetic anisotropy

A comprehensive theoretical study of the magnetic exchange between Co2+ ions is reported. Using the microscopic background we deduce the general Hamiltonian for a corner shared bioctahedral system involving kinetic exchange, spin–orbit coupling, and low-symmetry local crystal field. This Hamiltonian acting within the orbitally degenerate ground manifold (4T1g)A⊗(4T1g)B of the cobalt pair is expressed in terms of orbital and spin operators. The treatment of the Hamiltonian is performed with the use of the irreducible tensor operator technique. We elucidate the major electronic factors controlling the magnetic anisotropy in the Co(II) pairs. The degree of the exchange anisotropy is shown to depend on the strength of the cubic crystal field and on the relative efficiency of two kinds of electron transfer pathways (e–e and t2–t2) contributing to the kinetic exchange. An unusual role of spin–orbit interaction is revealed. This interaction tends to reduce the anisotropy caused by the orbitally dependent exchange. Special attention is paid on the topical case when spin–orbit coupling exceeds the exchange interaction. In this case the effective Hamiltonian in its general form is projected onto the subspace of low-lying Kramers doublets and in such a way the pseudo-spin-12 Hamiltonian is derived. Unlike the commonly accepted phenomenological approaches based merely on the symmetry arguments the proposed procedure is grounded on the microscopic consideration and hence allows us to establish the interrelation between the idem parameters of the system and the effective parameters of the pseudo-spin-12 Hamiltonian. Finally, we discuss the applicability of the conventional Lines’ model.