Irreducible Tensor Operator Formalism Research Articles

Nonlinear Magnetooptical Resonance in 87Rb Vapor: The Influence of Stray Magnetic Fields and Excitation Radiation Intensity on the Basic Characteristics of the Effect in Cells with Antirelaxation Coating

The nonlinear Hanle effect is generally considered in the modern literature as a particular case of coherent population trapping (CPT) at degenerate Zeeman sublevels of the ground atomic state. The shape, amplitude, and sign of observed resonances and their dependence on the excitation parameters and observation geometry were investigated theoretically and experimentally in many studies because of the wide range of application of this effect. The sensitivity of Hanle resonances to the experimental conditions increases significantly in cells with antirelaxation coating, because the atomic ensemble retains its coherence after a large number of collisions with the cell walls. In this paper, the main attention is paid to the joint influence of the excitation radiation intensity and spurious magnetic fields on CPT resonances observed in fluorescence. The theoretical description is based on the numerical solution of an algebraic system of equations for density matrix $$\hat {\rho }$$ in the formalism of irreducible tensor operators. The parameters relating different polarization moments in each equation are visualized using the modified sparse matrix of the system. It is established by numerical simulation that the shape, width, amplitude, and sign of nonlinear magnetooptical resonances change nonmonotonically with a change in the parameters of the excitation radiation and atomic system. The presented theoretical results are in good agreement with the experimental data obtained under various conditions.

Нелинейный магнитооптический резонанс в парах -=SUP=-87-=/SUP=-Rb: влияние паразитных магнитных полей и интенсивности возбуждающего излучения на основные характеристики эффекта в ячейках с антирелаксационным покрытием

AbstractThe nonlinear Hanle effect is generally considered in the modern literature as a particular case of coherent population trapping (CPT) at degenerate Zeeman sublevels of the ground atomic state. The shape, amplitude, and sign of observed resonances and their dependence on the excitation parameters and observation geometry were investigated theoretically and experimentally in many studies because of the wide range of application of this effect. The sensitivity of Hanle resonances to the experimental conditions increases significantly in cells with antirelaxation coating, because the atomic ensemble retains its coherence after a large number of collisions with the cell walls. In this paper, the main attention is paid to the joint influence of the excitation radiation intensity and spurious magnetic fields on CPT resonances observed in fluorescence. The theoretical description is based on the numerical solution of an algebraic system of equations for density matrix $$\hat {\rho }$$ in the formalism of irreducible tensor operators. The parameters relating different polarization moments in each equation are visualized using the modified sparse matrix of the system. It is established by numerical simulation that the shape, width, amplitude, and sign of nonlinear magnetooptical resonances change nonmonotonically with a change in the parameters of the excitation radiation and atomic system. The presented theoretical results are in good agreement with the experimental data obtained under various conditions.

Parameterization of the spectra of configurations 3p4f and 3p5f of a phosphorus ion P II. Gyromagnetic ratios

In the one-configuration approximation, in the formalism of irreducible tensor operators, and in the intermediate (real) coupling scheme, numerical values of the fine-structure parameters are determined for the 3p4f and 3p5f highly excited configurations of the P II phosphorus ion with the energy-operator matrix in the LK-coupling approximation. With these values of the fine-structure parameters, the energy-operator matrix is numerically diagonalized in the LS-coupling approximation. The gyromagnetic ratios calculated in both basis sets in the absence of a field are compared with one another, as well as with their vector counterparts and the experimental g-factors available for the 3p4f configuration. The experimental and theoretical g-factors calculated with the LS basis set are in good agreement with the sole exception of the 3 F 2 level. Note that the calculation of g-factors from the Zeeman splitting in the linear region totally confirmed their agreement with the values calculated in the LS basis set (g LS ) in the absence of a field. The gyromagnetic ratios are the main objectives of this and previous papers, especially for configurations for which experimental data are absent. Apart from the g-factors, the specific features of Zeeman splitting (the crossings and anticrossings of magnetic components) in the 3p5f configurations were determined. These data are to be compared with results of future experiments. Comparison of gyromagnetic ratios calculated in the intermediate coupling scheme with their vector counterparts showed that most levels of the configurations studied are closer than in the LK-coupling scheme.

Calculation of the Vibrational-Rotational Energy Structure of Molecules with Tetrahedral Symmetry of the Type XY 4

On the basis of the formalism of irreducible tensor operators in symmetrized form, a Hamiltonian has been constructed, an algorithm has been developed, and a software package has been created that allows the calculation of high-resolution spectra for various polyads of interacting vibrational states and also makes it possible to interpret the spectra and solve the inverse spectroscopic problem for molecules with tetrahedral symmetry.

On the Determination of the Spectroscopic Parameters of Axially Symmetric XYZ 3 (C 3v) Molecules

Based on the formalism of irreducible tensor operators, the operator matrix of the effective rotational operator is determined in the general form for XYZ 3-type molecules of axial symmetry. Based on the symmetric properties of the Hamiltonian and the vibrational wave functions of molecules of this type, a number of previously unknown relationships are derived for the spectroscopic parameters.

Magneto structural correlations of a ‘stepped-wise’ discrete tetranuclear unit of copper(II), [Cu 4(μ 2-OH) 2((μ 3-OH) 2(2,2′-bipy) 4Cl 2]Cl 2·6H 2O

A tetranuclear ‘stepped wise’ copper(II) complex, [Cu 4(μ 2-OH) 2(μ 3-OH) 2(2,2′-bipy) 4Cl 2]Cl 2·6H 2O has been synthesized and the crystal structure of the complex redetermined. The magnetic behavior has been investigated in the temperature range 300–4.2 K based on its novel binding modes of the OH −, i.e. OH − anions bridge the copper(II) centers in a bidentate (μ-2) as well as tridentate (μ-3) fashion and this shows ferromagnetic interactions. The fit of the experimental data was made by the irreducible tensor operator formalism (ITO) using the clumag program based on the Hamiltonian; H=− J 1( S 1 S 2+ S 3 S 4)− J 2( S 1 S 4)− J 3( S 1 S 3+ S 2 S 4 ) and the best fit parameters, assuming the g factors are identical and J 2= J 3, leads to J 1=22.71 cm −1; J 2= J 3=0.34 cm −1 and g=2.10. The magnetic results are interpreted with structural parameters.

Synthesis, Characterization, and Magnetic Properties of Self-assembled Compounds Based on Discrete Homotrinuclear Complexes of Cu(II)

Three new supramolecular entities of Cu(II) were synthesized and characterized: [(Cu(H(2)O)(tmen))(2)(mu-Cu(H(2)O)(opba))](2)[(ClO(4))(2)](2).2H(2)O (1), [(Cu(H(2)O)(tmen))(2) (mu-Cu(H(2)O) (Me(2)pba))](2)[(ClO(4))(2)](2) (2), and [(Cu(H(2)O)(tmen))(Cu(tmen))(mu-Cu(OHpba))](n)() ((ClO(4))(2))(n)().nH(2)O (3), where opba = o-phenylenbis(oxamato), Me(2)pba = 2,2-dimethyl-1,3-propylenbis(oxamato), OHpba = 2-hydroxy-1,3-propylenbis(oxamato), and tmen = N,N,N'N'-tetramethylethylenediamine. The crystal structures of 1, 2, and 3 were solved. Complex 1 crystallizes in the monoclinic system, space group C2/c with a = 20.572(4) A, b = 17.279(6) A, c = 22.023(19) A, beta = 103.13(4) degrees, and Z = 8. Complex 2 crystallizes in the monoclinic system, space group P2(1)/c, with a = 16.7555(7) A, b = 13.5173(5) A, c = 17.1240(7) A, beta = 104.9840(10) degrees, and Z = 4. Complex 3 crystallizes in the orthorhombic system, space group Pca2(1) with a = 21.2859(4) A, b = 12.8286(10) A, c = 12.6456(2) A, and Z = 4. The three complexes are very similar in structure: a trinuclear Cu(II) complex with the two terminal Cu(II) ions blocked by N,N,N',N'-tetramethylethylenediamine, but with a different environment in the Cu(II) central ion. In the case of complex 1, two of these trinuclear entities are packed with a short distance between the central Cu(II) ions of two separate entities forming a hexanuclear-type compound. In the case of 2, two of these trinuclear entities are linked by a hydrogen bond between a water molecule of one terminal Cu(II) and one oxygen atom of the oxamato ligand of the neighboring entity, also forming a hexanuclear complex. In the case of complex 3, the intermolecular linkages give a one-dimensional system where the OH groups of the OHpba entities are linked to the terminal Cu(II) of the neighboring entities. The magnetic properties of the three complexes were studied by susceptibility measurements vs temperature. For complex 1, an intramolecular J value of -312.1 cm(-)(1) and a contact dipolar interaction of -0.44K were found. For complex 2 and 3 the fit was made by the irreducible tensor operator formalism (ITO). The values obtained were as follows: J(1) = -333.9 cm(-)(1) and J(2) = 0.67 cm(-)(1) for 2 and J(1) = -335.9 cm(-)(1) and J(2) = 3.5 cm(-)(1) for 3.

Angular Distribution of Photoelectrons upon Ionization of Oriented Diatomic Molecules

Angular distribution of photoelectrons emitted during ionization of oriented diatomic molecules was studied. The expression for angular distribution was derived using the formalism of irreducible tensor operators. Two specific cases of angular distribution were considered, the isotropic orientation of molecules and the fixed orientation of molecules along the electric vector of light E.

Thermometry by degenerate four-wave mixing: J dependence of laser-induced grating reflectivity

Irreducible tensor operator formalism is applied to the theory of resonant degenerate four-wave mixing spectroscopy, and third-order perturbation theory is used to obtain an expression for the molecular polarisation. This is applied to predict the J dependence of the signal strength expected for Hund's case (a) molecules. Comparison with the corresponding one-photon line-strength factor is made, and the effect on temperature determinations by degenerate four-wave mixing in combustion studies is discussed. For Hund's case (a) molecules in the weak-field limit our results show that degenerate four-wave mixing spectroscopy will lead to a significant underestimate of the flame temperature.

Resonant effects on the light emitted by a diatomic molecular gas excited by electronic bombardment: theory and possibilities

Density matrix and irreducible tensor operator formalism have been used to describe optical pumping or similar experiments by electronic excitation in an atomic gas. The authors show that this theory may be extended to diatomic molecules. Experimental possibilities are considered.

Polarization of the fluorescent light of Ne atoms interacting with a laser beam measurement of several relaxation times

The fluorescent light from two atomic levels is studied assuming that the atoms are subjected to a gas discharge, to a static magnetic field, and, simultaneously, to a laser beam having the transition frequency between the atomic levels. Calculations are developed in the formalism of irreducible tensor operators. One takes into account the existence of isotropic relaxations and cascade effects due to spontaneous emission between the upper and the lower levels. Following Lamb, the laser beam is described classically by its electric field, and its effects are calculated by perturbation theory. It is predicted that the σ component of fluorescent light exhibits the Hanle effect, which allows one to measure the relaxation rates <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Gamma(0)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Gamma(2)</tex> of the population and of the alignment of the emitting level, respectively. Experimental agreement with the theoretical analysis has been found for the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3s_{2}, 2s_{2}</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2P_{4}</tex> levels of Ne. Among the experimental results obtained by this method are the natural widths and the collision cross sections for population (quenching collision) and for alignment. Several transition probabilities are also derived. Furthermore, the theory of coherent imprisonment of the fluorescent line has been found valid even if laser pumping light is used.