Effective operators of the spin correlated crystalline electric field

The effective operators of the spin correlated crystal field have been formed by coupling the exchange with the crystal field. The exchange field is a scalar in spin space so the effective operators will still have the same rank as that of the crystal field but will give non-zero matrix elements between the states ^2S+1L_J of the half filled shells, contrary to the zero value obtained for a pure crystal field tensor.     

Spectra of tensor operators adapted to nonstandard basis. Qualitative features of clustering

Spectra of irreducible tensor operators for the chain of group G ? O(3) are investigated from the qualitative point of view. Conditions are found for clustering of the eigenvalues of tensor operators in the high-J limit. Approximate formulas are proposed which permit the calculation of the relative positions of the clusters. Centrifugal distortion of spherical tops, crystal-field splitting, and low-frequency bending vibrations of nonrigid molecules composed of an atom and rigid symmetrical core are considered as physical examples. Clustering of the F^(kJJ′) coefficients of Moret-Bailly are qualitatively explained as well.     

Virasoro type Lie algebras and deformations

The first and second cohomologies of Cartan Type Lie algebras with coefficients in irreducible tensor modules are calculated. The spaceH ¹(L, U) is interpreted as a space of deformations of (L, U)-modules.H ²(L, L)≠0 ifL=S ₂,S ₂ ⁺ orL=H _n ,H _n ⁺ . Lie algebra of divergenceless vector fieldsS ₂ ⁺ has only one nontrivial local deformation. The two-sided simple hamiltonian algebraH _n has 2n ²+n new local deformations in addition to Moyal cocycle. The Lie algebrasL=W _n (n>3),S _n?1(n>2),H _n (n>1),K _n+1(n>1) have 3, 1, 1, 3 nonisomorphic tensor modules with irreducible bases and nonzero 1-cohomologies; respectively, the corresponding numbers for 2-cohomologies are 9, 6, 7 and 9.     

Variational calculations of realistic models of nuclear matter

We report variational calculations of nuclear matter with a semi-realistic Reid v₁₂ model, and a realistic v₁₄ model of the two-nucleon interaction operator. The v₁₄ model fits the available nucleon-nucleon scattering data up to 425 MeV lab energy, and has relatively weak L² and $\text{(}\text{L}\text{·}\text{S}\text{)}{}^2$ interactions in addition to the standard central, tensor and ($\text{L}\text{·}\text{S}$). The L² and $\text{(}\text{L}\text{·}\text{S}\text{)}{}^2$ interactions are treated semiperturbatively; their contribution reduces the overbinding of nuclear matter. However, the equilibrium k_F = 1.7 fm^?1 and E₀ = ?17.5 MeV obtained with the v₁₄ model are both higher than their empirical values k_F = 1.33 fm^? and E₀ = ?16 MeV. We assume that the difference between the calculated and empirical E(ρ) is entirely due to three-nucleon interactions (TNI). The TNI contributions are phenomenologically added to the nuclear matter energy, and their parameters are adjusted to obtain the correct equilibrium energy, density and compressibility. The required TNI contributions appear to be of reasonable magnitude.     

Second and fourth order stark shifts in the 6snd 1 D 2 barium rydberg series

Stark splittings of⁶ snd ¹ D ₂ Rydberg states were measured betweenn=10 and 24 by quantum beat spectroscopy at very low electric field strengths, allowing the determination of tensor polarizabilities and atn=14, 19, and 24 the determination of tensor hyperpolarizabilities. The irregular behaviour of the tensor polarizabilities was explained by Coulomb approximation calculations. The fourth order contributions to the Stark Hamiltonian were written as effective operators. Representation in spherical tensor form gives tensors of rank zero, two, and four. The associated scalar and tensor hyperpolarizabilities were derived. Using these formulae, theoretical values for the hyperpolarizabilities were calculated.     

A characterization of dilatation analytic integral kernels

We characterize integral operators belonging to B(L ² (ℝ³))which are dilatation analytic in the Cartesian product of two sectors S _a ⊂ ℂ as analytic functions from S _a×S_a into B(L ²(Ω)), the space of bounded operators on square integrable functions on the unit sphere Ω, which satisfy certain norm estimates uniformly on every subsector.     

Eigenfunction expansions of operators admitting separation of an infinite number of variables

Let A_k be a self-adjoint operator in a Hilbert space H_k (k = 1, 2, …) and let $\text{L}$ be an operator of the form $\text{L}\text{= A}{}_{\mathrm{r}}\text{? 1 ? 1 ? … + 1 ? A}{}_2\text{? 1 ? 1 ? … + …}$ acting in the infinite tensor product $\text{?}\text{k=1}\text{∞}\text{H}{}_{\mathrm{k}}$. We construct the spectral theory of these operators. In particular, the expansion is generalized eigenvectors of this operator is constructed using the eigenvectors of the operators A_k.     

Anisotropic characteristics of the kraichnan direct cascade in two-dimensional hydrodynamic turbulence

The statistical characteristics of the Kraichnan direct cascade for two-dimensional hydrodynamic turbulence are numerically studied (with spatial resolution 8192 × 8192) in the presence of pumping and viscous-like damping. It is shown that quasi-shocks of vorticity and their Fourier partnerships in the form of jets introduce an essential influence in turbulence leading to strong angular dependencies for correlation functions. The energy distribution as a function of modulus k for each angle in the inertial interval has the Kraichnan behavior, ~k ^–4, and simultaneously a strong dependence on angles. However, angle average provides with a high accuracy the Kraichnan turbulence spectrum E _k = C _Kη^2/3k^–3, where η is the enstrophy flux and the Kraichnan constant C _K ? 1.3, in correspondence with the previous simulations. Familiar situation takes place for third-order velocity structure function S ₃ ^L which, as for the isotropic turbulence, gives the same scaling with respect to the separation length R and η, S ₃ ^L = C ₃ηR ³, but the average over the angles and time differs from its isotropic value.     

Anisotropy of the linear and quadratic magnetic birefringence in rare-earth semiconductors (γ-Ln2S3 (Ln=Dy3+, Pr3+, Gd3+, La3+)

The field and angular dependence of the magnetic linear birefringence (MLB) in noncentrosymmetric cubic (symmetry class T _d) rare-earth (RE) semiconductors γ-Ln₂S₃ (Ln=Dy³⁺, Pr³⁺, Gd³⁺, La³⁺) was studied. The field dependence of the MLB in Dy₂S₃ and Pr₂S₃ is a combination of two contributions, quadratic and linear with respect to the magnetic induction B, both possessing a strong anisotropy. The quadratic birefringence related to the Cotton-Mouton effect manifestations at a wavelength of λ=633 nm is characterized in Dy₂S₃ by the value of β=1.5 deg/(cm T²) and the anisotropy parameter a=?0.7 and in Pr₂S₃, by β=0.2 deg/(cm T²) and a=2. The non-reciprocal MLB caused by the magnetic-field-induced spatial dispersion reaches γ 0.55 and 0.71 deg/(cm T) in Dy₂S₃ and Pr₂S₃, respectively. The relationship between parameters A and g of the γ_ijkl tensor describing contributions of the B _ik_j type to the dielectric tensor ε_ij(ω, k, B) is A=2g in Dy₂S₃ (as well as in boracite crystals containing 3d ions), which is characteristic of the second-order magnetoelectric permittivity manifestations in the optical frequency range. In Pr₂S₃, the relationship A=3.3g is evidence of manifestations of the additional quadrupole mechanism. A comparison of the Cotton-Mouton and Faraday effects in Ln₂S₃ and in magnetic semiconductors Cd_1?x Mn_xTe shows a principal difference between these systems and indicates that both phenomena in Ln₂S₃ are determined by the optical transitions in RE ions rather than by the interband or exciton transitions. This is also confirmed by the comparison of the Cotton-Mouton effect manifestations in Ln₂S₃, in dielectric Dy₃Ga₅O₁₂ and Dy₃Al₅O₁₂ single crystal cubic garnets, and in Dy₂O₃. An analysis of the non-reciprocal MLB mechanisms related to manifestations of the local interconfiguration optical transitions 4f ^N→ 4f ^N?15d in RE ions showed that this phenomenon, in contrast to the Cotton-Mouton and Faraday effects, is caused by the presence of odd components of the crystal field acting upon the RE ion in Ln₂S₃. In Gd₂S₃, as well as in diamagnetic Ln₂S₃, neither the Cotton-Mouton effect nor the non-reciprocal MLB are manifested at T=294 K, which is explained by different microscopic mechanisms of the magnetooptical phenomena for ions in the S-state and diamagnetic ions, on the one hand, and RE ions with nonzero orbital moment, on the other hand.     

Noncollinear phase matched four photon frequency mixing in water

An intense ultraviolet picosecond light pulse at ω_S = 2γ_P ? ω_L is generated in water by noncollinear phase matched nonresonant four photon frequency mixing of two input picosecond light pulses at frequencies ν~_P = 18 960 cm^-1 and ν~_L = 9480 cm^-1. An energy conversion of up to W_S/W_L = 0.07 was achieved. The nonlinear susceptibility components were determined to be χ⁽³⁾_yyyy(?ω_S; ω_P, ω_P, ? ω_L) = 7.5 × 10^-34 Cm/V³ and χ⁽³⁾_yxxy(?ω_S; ω_P, ω_P, ?ω_L) = 2.4 × 10^-34 Cm/V³.     

Investigation of LiFeAs by means of “break-junction” technique

In our tunneling investigation using Andreev superconductor-normal metal-superconductor contacts on LiFeAs single crystals we observed two reproducible independent subharmonic gap structures at dynamic conductance characteristics. From these results, we can derive the energy of the large superconducting gap ??_L = (2.5?C3.4) meV and the small gap ??_S = (0.9?C1) meV at T = 4.2 K for the T _C ^local ?? (10.5?C14) K (the contact area critical temperature which deviation causes the variation of ??_L). The BCS-ratio is found to be 2??_L/k _B T _C = 4.6?C5.6, whereas 2??_S/k _B T _C ? 3.52 results from induced superconductivity in the bands with the small gap.     

Boson peak of AsxS1−x glasses and theoretical calculations of low frequencies clusters vibrations

The results of comparison of spectral position of the LF bands of g-As₂S₃ and g-As₂₂S₇₈ with theoretical calculations of vibrational spectra of different clusters (As₂S₄, As₂S₅, As₂S₆, AsS₃-As₃) in the LF region (10-85 cm⁻¹) are given. The torsion type vibrations of the small chain-like clusters are located in the same spectral region as Boson peak of As-S glasses system. They can make several contributions to the LF spectrum. The cluster lengths by weight distribution functions f(L)*L are from 5.5 to 10.1 Å and from 4.5 to 7.5 Å for g-As₂S₃ and g-As₂₂S₇₈, respectively. The lognormal fittings of the f(L)*L functions give the most probably values about the 7.6 Å (for g-As₂S₃) and 6.2 Å (for g-As₂₂S₇₈).     

Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues

We extend some recent results of Lubinsky, Levin, Simon, and Totik from measures with compact support to spectral measures of Schrödinger operators on the half-line. In particular, we define a reproducing kernel S _L for Schrödinger operators and we use it to study the fine spacing of eigenvalues in a box of the half-line Schrödinger operator with perturbed periodic potential. We show that if solutions u(ξ, x) are bounded in x by ${e^{\epsilon x}}We extend some recent results of Lubinsky, Levin, Simon, and Totik from measures with compact support to spectral measures of Schr?dinger operators on the half-line. In particular, we define a reproducing kernel S _L for Schr?dinger operators and we use it to study the fine spacing of eigenvalues in a box of the half-line Schr?dinger operator with perturbed periodic potential. We show that if solutions u(ξ, x) are bounded in x by e^ex{e^{\epsilon x}} uniformly for ξ near the spectrum in an average sense and the spectral measure is positive and absolutely continuous in a bounded interval I in the interior of the spectrum with x₀ ? I{\xi_0\in I}, then uniformly in I,

\fracSL(x0 + a/L, x0 + b/L)SL(x0, x0)? \fracsin(pr(x0)(a - b))pr(x0)(a - b),\frac{S_L(\xi_0 + a/L, \xi_0 + b/L)}{S_L(\xi_0, \xi_0)}\rightarrow \frac{\sin(\pi\rho(\xi_0)(a - b))}{\pi\rho(\xi_0)(a - b)},      16. Nuclear hyperfine interactions in spherical top molecules      F. Michelot 《Journal of Molecular Spectroscopy》1984,106(1):77-109 The nuclear hyperfine Hamiltonian for XY4 and XY6 spherical top molecules in their ground electronic state is built up by a tensor method, and is given by a linear combination of invariant tensor products in the group L0(3) × G(G = Td or 0h); the case when an external magnetic field is applied is also considered. For the study of a given vibrational state, the appropriate restriction is easily determined; as examples were consider the ground vibronic state and a triply degenerate fundamental state.      17. Calculation of high-order rotational centrifugal-distortion matrix elements for a Hund's case (a) basis set      R.N. Zare  A.L. Schmeltekopf  D.L. Albritton  W.J. Harrop 《Journal of Molecular Spectroscopy》1973,48(1):174-180 The derivation of explicit expressions for the Hund's case (a) matrix elements of R2k is discussed, where R is the mechanical rotational angular momentum operator of the molecule. A recursion relation is developed that permits matrix elements of R2k to be expressed in terms of those of R2(k?1), thus affording a straightforward means of calculating the case (a) matrix elements of rotational centrifugal-distortion constants Dv, Hv, Lv, Mv, etc., to an arbitrarily high order. The explicit matrix elements of Lv are listed.      18. Optical and magnetooptical characterization of FeNi/Cu multilayers      Q.S. Bie  M. Lu  J. Du  H.W. Zhao  B. You  L.N. Tong  H.R. Zhai  M.L. Yan  W.Y. Lai  S.M. Zhou  L.Y. Chen 《Zeitschrift für Physik B Condensed Matter》1997,102(2):183-187 A series of [FeNi(1.7 nm)/Cu(tCu)]30/Fe(10 nm) multilayers with the range of tCu from 0.36nm to 4.32nm deposited on glass substrates was studied by measuring the effective optical constants n and k, saturation magnetooptical polar Kerr rotation θ k and ellipticity ηk and evaluating the effective complex off-diagonal elements of the dielectric tensor ε xy ′ and ε xy ″ . It was found that the four primary quantities ε xy ′ , ε xy ″ , n and k vary concurrently with respect to tCu, while the dependences of θ k and η k on tCu differ drastically. To explain this difference the contributions of both the real and imaginary parts of the off- diagonal dielectric tensor elements and the effect of complex optical constants have to be considered. No obvious correlation was found between the variation of ñ, ε~ xy and the oscillations of GMR of the multilayers.      19. Fluorescence quenching of vaporous polycyclic aromatic hydrocarbons by oxygen      G. A. Zalesskaya  F. Piuzzi  E. G. Sambor 《Optics and Spectroscopy》2007,102(4):503-509 The fluorescence quenching by oxygen of vapors of nine polycyclic aromatic hydrocarbons with strongly different oxidation potentials 0.44 eV < E ox < 1.61 eV (anthracene, 9-methylanthracene, 2-aminoanthracene, 9,10-dibromanthracene, pyrene, chrysene, phenanthrene, fluoranthene, and carbazole) is studied. From the dependences of the fluorescence decay rates and intensities on the oxygen pressure P O2, the quenching rate constants k S O2 for the excited singlet states S 1 and the fraction f S O2 of the S 1 states quenched by oxygen are estimated. At P O2 = 5 Torr, the k S O2 constants vary from 1.2 × 107 to 3.0 × 105 s?1 Torr?1, while the fraction of the quenched excited singlet states changes from 0.1 (fluoranthene) to 0.7 (chrysene) and 0.8 (pyrene). The dependences of k S O2 on the photophysical and electron-donor characteristics of the fluorescing compounds are analyzed. It is shown that, in the gas phase of anthracene and its derivatives, the magnitudes of k S O2 are limited by the rate constants of gas-kinetic collisions k gk and do not depend on the electron-donor characteristics of fluorophores, while the fraction of quenched states f S O2 changes with the oxidation potential. For compounds with k S O2 < k gk, both the rate constants k S O2 and the fraction of quenched states f S O2 depend on the E ox of sensitizers, which demonstrates an important role played by the charge-transfer interactions in quenching of the S 1 states. The dependence of the rate constants k S O2 on the free energy of electron transfer ΔG et is considered.      20. Functional properties of quantum logics      M. J. Mączyński 《International Journal of Theoretical Physics》1974,11(3):149-156 A quantum logic is defined as a setL of functions from the set of all statesS into [0, 1] satisfying the orthogonality postulate: for any sequencea 1,a 2, ... of members ofL satisfyinga i+a j≤1 fori≠j there isb∈L such thatb+a 1+a 2+...=1. Every logicL is in a natural way an orthomodular σ-orthocomplemented partially ordered set (L, ≤, ′) with members ofS inducing a full set of measures onL. It is shown that a logicL is quite full if and only if (L,≤,′) is isomorphic to an orthocomplemented set lattice of subsets ofS. Sufficient conditions are given in order that a quite full logic be representable in the set of projection quadratic formsf(u)=(Pu, u) on a complex Hilbert space, or in the set of trace functionsf(A)=Trace (AP) generated by projectionsP, where the domain off is the set of non-negative self-adjoint trace operators of trace 1 in a complex Hilbert space.

Nuclear hyperfine interactions in spherical top molecules

The nuclear hyperfine Hamiltonian for XY₄ and XY₆ spherical top molecules in their ground electronic state is built up by a tensor method, and is given by a linear combination of invariant tensor products in the group ^L0(3) × G(G = T_d or 0_h); the case when an external magnetic field is applied is also considered. For the study of a given vibrational state, the appropriate restriction is easily determined; as examples were consider the ground vibronic state and a triply degenerate fundamental state.     

Calculation of high-order rotational centrifugal-distortion matrix elements for a Hund's case (a) basis set

The derivation of explicit expressions for the Hund's case (a) matrix elements of R^2k is discussed, where R is the mechanical rotational angular momentum operator of the molecule. A recursion relation is developed that permits matrix elements of R^2k to be expressed in terms of those of R^2(k?1), thus affording a straightforward means of calculating the case (a) matrix elements of rotational centrifugal-distortion constants D_v, H_v, L_v, M_v, etc., to an arbitrarily high order. The explicit matrix elements of L_v are listed.     

Optical and magnetooptical characterization of FeNi/Cu multilayers

A series of [FeNi(1.7 nm)/Cu(t_Cu)]₃₀/Fe(10 nm) multilayers with the range of t_Cu from 0.36nm to 4.32nm deposited on glass substrates was studied by measuring the effective optical constants n and k, saturation magnetooptical polar Kerr rotation θ _k and ellipticity η_k and evaluating the effective complex off-diagonal elements of the dielectric tensor ε _xy ^′ and ε _xy ^″ . It was found that the four primary quantities ε _xy ^′ , ε _xy ^″ , n and k vary concurrently with respect to t_Cu, while the dependences of θ _k and η _k on t_Cu differ drastically. To explain this difference the contributions of both the real and imaginary parts of the off- diagonal dielectric tensor elements and the effect of complex optical constants have to be considered. No obvious correlation was found between the variation of ñ, ε~ _xy and the oscillations of GMR of the multilayers.     

Fluorescence quenching of vaporous polycyclic aromatic hydrocarbons by oxygen

The fluorescence quenching by oxygen of vapors of nine polycyclic aromatic hydrocarbons with strongly different oxidation potentials 0.44 eV < E _ox < 1.61 eV (anthracene, 9-methylanthracene, 2-aminoanthracene, 9,10-dibromanthracene, pyrene, chrysene, phenanthrene, fluoranthene, and carbazole) is studied. From the dependences of the fluorescence decay rates and intensities on the oxygen pressure P _O2, the quenching rate constants k _S ^O2 for the excited singlet states S ₁ and the fraction f _S ^O2 of the S ₁ states quenched by oxygen are estimated. At P _O2 = 5 Torr, the k _S ^O2 constants vary from 1.2 × 10⁷ to 3.0 × 10⁵ s^?1 Torr^?1, while the fraction of the quenched excited singlet states changes from 0.1 (fluoranthene) to 0.7 (chrysene) and 0.8 (pyrene). The dependences of k _S ^O2 on the photophysical and electron-donor characteristics of the fluorescing compounds are analyzed. It is shown that, in the gas phase of anthracene and its derivatives, the magnitudes of k _S ^O2 are limited by the rate constants of gas-kinetic collisions k _gk and do not depend on the electron-donor characteristics of fluorophores, while the fraction of quenched states f _S ^O2 changes with the oxidation potential. For compounds with k _S ^O2 < k gk, both the rate constants k _S ^O2 and the fraction of quenched states f _S ^O2 depend on the E _ox of sensitizers, which demonstrates an important role played by the charge-transfer interactions in quenching of the S ₁ states. The dependence of the rate constants k _S ^O2 on the free energy of electron transfer ΔG _et is considered.     

Functional properties of quantum logics

A quantum logic is defined as a setL of functions from the set of all statesS into [0, 1] satisfying the orthogonality postulate: for any sequencea ₁,a ₂, ... of members ofL satisfyinga _i+a _j≤1 fori≠j there isb∈L such thatb+a ₁+a ₂+...=1. Every logicL is in a natural way an orthomodular σ-orthocomplemented partially ordered set (L, ≤, ′) with members ofS inducing a full set of measures onL. It is shown that a logicL is quite full if and only if (L,≤,′) is isomorphic to an orthocomplemented set lattice of subsets ofS. Sufficient conditions are given in order that a quite full logic be representable in the set of projection quadratic formsf(u)=(Pu, u) on a complex Hilbert space, or in the set of trace functionsf(A)=Trace (AP) generated by projectionsP, where the domain off is the set of non-negative self-adjoint trace operators of trace 1 in a complex Hilbert space.