The dual (p,q)-Alexander-Conway Hopf algebras and the associated universal ℐ-matrix

The dually conjugate Hopf algebrasFun _p,q (R) andU _p,q (R) associated with the two-parametric (p,q)-Alexander-Conway solution (R) of the Yang-Baxter equation are studied. Using the Hopf duality construction, the full Hopf structure of the quasitriangular enveloping algebraU _p,q (R) is extracted. The universal ?-matrix forsFun _p,q (R) is derived. While expressing an arbitrary group element of the quantum group characterized by the noncommuting parameters in a representation independent way, the ?-matrix generalizes the familiar exponential relation between a Lie group and its Lie algebra. The universal ?-matrix and the FRT matrix generators,L ^(±), forU _p,q (R) are derived from the ?-matrix.     

On determination of the analytical formulas for reduction matrices of tetrahedral-symmetry molecules

The formulas for the elements of ^(J)G matrices (J = 0, 1, 2 ?) determining reduction of irreducible representations D^(Jg) of the group 0(3) to the irreducible representations D^(Γ) of the T_d group have been obtained in an analytical form. It was shown how the transition was made from the G matrices of this paper to those determined by Moret-Bailly's basis [Cah. Phys., 15, 237–314 (1961)].     

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.     

Existence of a phase-transition in a one-dimensional Ising ferromagnet

Existence of a phase-transition is proved for an infinite linear chain of spins μ _j =±1, with an interaction energy $$H = - \sum J(i - j)\mu _i \mu _j ,$$ whereJ(n) is positive and monotone decreasing, and the sums ΣJ(n) and Σ (log logn) [n ³ J(n)]^?1 both converge. In particular, as conjectured byKac andThompson, a transition exists forJ(n)=n ^?α when 1 < α < 2. A possible extension of these results to Heisenberg ferromagnets is discussed.     

Crystal symmetry groups and their representations I lattice extensions of point groups in R2 and R3

We classify all point groups in R² and R³ according to their decomposition into m-fold semidirect products of cyclic groups. This product is characterized completely by means of m?1 matrices α⁽¹⁾,…, α^(m?1) with non-negative integer elements. For any arithmetic class [Pⁿ_i] in Rⁿ (n=2, 3) we define the set D(Pⁿ_i, T) of representative translations with the following properties: (a) The elements of D(Pⁿ_i, T) are in one-to-one correspondence with defining relations of Pⁿ_i; (b) D(Pⁿ_i, T) determines uniquely all equivalence classes of the factors of Pⁿ_i over T.We present: (i) a construction of all sets D(Pⁿ_i, T) in R² and R³; (ii) a construction of all algebraic classes of space groups and a derivation of all group operations which transform one algebraic class into another; (iii) a derivation of formulas expressing the multiplication table of any space group by means of the representative translations and matrices α⁽¹⁾,…, α^(m?1). The results (i), (ii) and (iii) are obtained without making use of the notion of non-primitive translations.     

Study of 170,171,173Ta isotopes at high spinṡ

High-spin data are presented for ¹⁷⁰Ta and ¹⁷¹Ta for the first time, and the known level scheme of ¹⁷³Ta is extended to higher spins. Anomalies found in the crossing frequencies for the alignment of a pair of neutrons are considered. At high spin a large and constant value of J⁽²⁾, with J⁽²⁾ > J⁽¹⁾, is found for two ¹⁷¹Ta bands and the implications of this are discussed. The πi_13/2 [660]1/2⁺, α = + 1/2 orbital is identified as it crosses the rotational band built on the [402]5/2⁺, α = + 1/2 orbital in ¹⁷¹Ta. Magnetic transition probabilities obtained from measured branching ratios show the effects of the aligning pair of neutrons and the spectator orbital. The data are interpreted using cranked shell-model calculations and a semiclassical vector-coupling scheme developed by Donau and Frauendorf.     

Measurement of Δ1J(199Hg, 31P) in [HgPCy3(OAc)2]2 and relativistic ZORA DFT investigations of mercury–phosphorus coupling tensors

Using ³¹P solid-state NMR spectroscopy, anisotropy in the indirect ¹⁹⁹Hg-³¹P spin–spin coupling tensor (ΔJ) for powdered [HgPCy₃(OAc)₂]₂ (1) has been measured as 4700±300 Hz. Zeroth-order regular approximation (ZORA) density functional theory (DFT) calculations, including scalar and spin-orbit relativistic effects, performed on 1 and a series of other related compounds show that ΔJ(¹⁹⁹Hg, ³¹P) arises entirely from the ZORA Fermi-contact–spin-dipolar cross term. The calculations validate assumptions made in the spectral analysis of 1 and in previous determinations of ΔJ in powder samples, namely that J is axially symmetric and shares its principal axis system with the direct dipolar coupling tensor (D). Agreement between experiment and theory for various ¹⁹⁹Hg, ³¹P spin–spin coupling anisotropies is reasonable; however, experimental values of ¹J(¹⁹⁹Hg, ³¹P)_iso are significantly underestimated by the calculations. The most important improvements in the agreement were obtained as a result of including more of the crystal lattice in the model used for the calculations, e.g., a change of 43% was noted for ¹J(¹⁹⁹Hg, ³¹P)_iso in [HgPPh₃(NO₃)₂]₂ depending on whether the two or three nearest nitrate ions are included in the model. Finally, we have written a computer program to simulate the effects of non-axial symmetry in J and of non-coincidence of the J and D on powder NMR spectra. Simulations clearly show that both of these effects have a pronounced impact on the ³¹P NMR spectrum of ¹⁹⁹Hg–³¹P spin pairs, suggesting that the effects should be observable experimentally if a suitable compound can be identified.     

Contribution of the inhomogeneous waves in angular-spectrum representations

Angular-spectrum representations of wave fields separate naturally into two parts, V_h(R,t) containing only homogeneous plane waves and V_i(R,t) containing only inhomogeneous plane waves. Some properties of V_i(R,t) are presented here. We conclude that, for some problems, V_i(R,t) has several unphysical properties, and that under certain specified conditions, V_i(R,t) can not be neglected compared to V_h(R,t), even far from the sources of the field.     

EPR measurements of weak exchange interactions coupling unpaired spins in model compounds

I review electron paramagnetic resonance (EPR) measurements performed to evaluate very weak exchange interactions (defined as ?_ex(i, j) = ?J _ij S _i S _j , with 10^?3 cm^?1<|J _ij|<1 cm^?1) between unpaired spins, transmitted through long and weak chemical pathways typical of protein structures. They are performed in appropriate model compounds, mainly copper derivatives of amino acids and peptides, making use of the phenomenon of exchange narrowing and collapse of the resonances. I describe the theoretical basis and the implementations of the method to different situations, including selected experimental values of the exchange couplingsJ between metal centers, and briefly discuss correlations betweenJ and the structure of the paths. Results obtained in relatively simple EPR experiments performed at room temperature in single-crystal samples are compared with those obtained from thermodynamic magnetic measurements having higher experimental difficulties. The experimental information allows describing the role of molecular segments typical of biomolecules (H bonds, aromatic ring stacking, cation-π contacts, etc.) in the transmission of the exchange interaction. The values ofJ obtained in some model compounds are compared with those obtained in proteins to conclude that the magnitudes of the exchange interactions are useful to characterize long and weak biologically relevant chemical pathways. One observes that these exchange couplings are weakly dependent on the nature of the unpaired spins and strongly dependent on the chemical pathway. Thus, measurements of exchange couplings in model compounds may provide useful information about biological function, particularly about electron transfer in proteins.     

Diode laser spectroscopy of the ν6 band of 13CH3l and the ν″6 band of 12CH2Dl

The ν₆ band of ¹³CH₃I and the ν″₆ band of ¹²CH₂DI have been recorded under Doppler-limited resolution in the region 820–866 cm^?1 using a tunable diode laser spectrometer. For ¹³CH₃I the constants for the ν₆ band were determined by simultaneous analysis of seven ^pQ(J,K) branches and several ^pP(J,K) and ^pR(J,K) transitions. For ¹²CH₂DI, the slight asymmetry introduced by the single D atom gives rise to noticeable asymmetry effects in the spectra of some of the ^pQ(J,K) subbands. From the analysis of six such subbands, the molecular constants for the ν″₆ level were determined.     

Bianchi Type-II String Cosmological Model with Magnetic Field in f (R,T) Gravity

The spatially homogeneous and totally anisotropic Bianchi type-II cosmological solutions of massive strings have been investigated in the presence of the magnetic field in the framework of f(R,T) gravity proposed by Harko et al. (Phys Rev D 84:024020, 2011). With the help of special law of variation for Hubble^’s parameter proposed by Berman (Nuovo Cimento B 74:182, 1983) cosmological model is obtained in this theory. We consider f(R,T) model and investigate the modification R+f(T) in Bianchi type-II cosmology with an appropriate choice of a function f(T)=μ T. We use the power law relation between average Hubble parameter H and average scale factor R to find the solution. The assumption of constant deceleration parameter leads to two models of universe, i.e. power law model and exponential model. Some physical and kinematical properties of the model are also discussed.     

Scattering of charged particles by atoms

A closed variant of the Born approximation for calculating differential scattering cross sections in ion-atom collisions is developed. An expression in terms of the matrix elements J _ij with respect to the single-electron states of the atom is found for the matrix element describing the target atom in the formula for the differential cross section. The matrix elements J _ij are averaged over the relative orientation of the momentum transferred in the collision and the symmetry axis of the electronic orbitals of the target atom, using the single-electron Rutaan-Hartree-Fock wave functions. The algebraic representation of the matrix elements J _ij makes it possible to perform calculations for atoms with any value of Z. The model developed is used to calculate the cross sections σ_Σ and characteristic scattering angles θ_c for the process of electron loss by H^? ions with energy E = 0.1–100 MeV in targets consisting of atoms with Z = 2–54. It is shown that σ_Σ ∝ E ^?1 and θ_c ∝ E ^?1/2 for all Z, and for fixed E the behavior of σ_Σ(Z) and θ_c(Z) is determined by the order of filling of the electronic shells of the target atoms (the ionization potential). The computational results are analyzed and compared with the experimental data and the results of other calculations.     

Analysis of the 2ν3 band of 12CD3F at 5.06 μm recorded under high resolution

The 2ν₃(A₁) band of ¹²CD₃F near 5.06 μm has been recorded with a resolution of 20–24 × 10^?3 cm^?1. The value of the parameter (α^B ? α^A) for this band was found to be very small and, therefore, the K structure of the R(J) and P(J) manifolds was unresolved for J < 15 and only partially resolved for larger J values. The band was analyzed using standard techniques and values for the following constants determined: ν₀ = 1977.178(3) cm^?1, B″ = 0.68216(9) cm^?1, D″_J = 1.10(30) × 10^?6 cm^?1, α^B = (B″ ? B′) = 3.086(7) × 10^?3 cm^?1, and β^J = (D″_J ? D′_J) = ?3.24(11) × 10^?7 cm^?1. A value of α^A = (A″ ? A′) = 2.90(5) × 10^?3 cm^?1 has been obtained through band contour simulations of the R(J) and P(J) multiplets.     

Anisotropy of the magnetoresistive properties of granular high-temperature superconductors resulting from magnetic flux compression in the intergrain medium

To elucidate the origin of the well-known anisotropy of the magnetoresistive properties of granular high-temperature superconductors (HTSs), which is related to the mutual orientation of magnetic field H and transport current j, we investigate the hysteretic dependences of magnetoresistance R(H) of the yttrium HTS sample at the perpendicular (H ⊥ j) and parallel (H || j) configurations. The hysteretic R(H) dependences are analyzed using the concept of the effective field in the intergrain boundaries through which superconducting current carriers tunnel. The effective degree of magnetic flux compression in the intergrain medium at the perpendicular configuration was found to be twice as much as at the parallel one. This approach explains well the anisotropy of the magnetoresistive properties of granular HTSs, which was previously reported by many authors, and the temperature dependences of the resistance in the resistive transition region.     

The mass-shell and gauge-fixing conditions for the free relativistic massive quantum particle

Working in the context of the Weyl group, which describes off-mass-shell relativistic particles, we impose “gauge-fixing” constraints involvingR ⁰,R ⁺, andD as matrix element conditions to be satisfied by the on-mass-shell states of a massive particle. We evaluate the matrix elements inp-space using five sets of co-ordinates: (p ²,p), (p ²,p ⁺,p _T ), (p ²,p ^?,p _T ), (p ²,π), and (p ²,π ⁺,π _T ) where \(\pi ^\mu \equiv p^\mu /(p^2 )^{\tfrac{1}{2}} \) . We find that, only in the case ofR ⁰ with (p ²,p) coordinates,R ⁺ with (p ²,p ⁺,p _T ) coordinates, andD with (p ², π) or (p ²,π ⁺,π _T ) coordinates, can the condition be satisfied by arbitrary on-mass-shell states. In all other cases, the condition can be satisfied only by states belonging to a subset of subspaces of the on-mass-shell Hilbert space, i.e it forces a violation of the superposition principle. These results constitute thep-space quantum version of Shanmugadhasan's theorem for constrained classical systems which states that there exists, at least locally in phase space, a canonical transformation to a set of variables in which the second-class constraints become canonical pairs equal to zero with the other canonical coordinates independent of the second-class constraints.     

Some effects of spin-orbit interaction on rotational levels and rotational line intensities in vibrationally unexcited 2A, 2E, and 2F electronic states of open-shell XY4 molecules

Rotational energy levels in vibronic ground states of ²A, ²E, and ²F electronic states of open-shell XY₄ molecules, as well as rotational line intensities for allowed transitions between such states, are discussed, including the effects of spin-orbit interaction and tetrahedral splittings. Jahn-Teller effects are assumed to be small, and are only taken into account implicitly, through their contributions to various parameters in the effective Hamiltonian. Qualitative information is obtained by considering several limiting-case coupling schemes among the electron spin angular momentum S, the electron orbital angular momentum L, and the pure rotational angular momentum R. These limiting cases are similar in spirit to Hund's coupling cases in diatomic molecules, but differ sufficiently from the latter to make detailed correspondences unhelpful. Quantitative information on rotational energy levels and line intensities is obtained numerically by diagonalizing a Hamiltonian matrix set up in a basis set characterized by uncoupled moleculefixed projections of S, L, and the total angular momentum J, and symmetrized so that all basis set functions belong to a definite species in the subgroup D_2d of the true point group T_d. Hamiltonian matrix elements are determined by ladder operator techniques. Three sample calculated spectra, corresponding to $\text{p(}{}^2\text{F}{}_2\text{)-s(}{}^2\text{A}{}_1\text{)}$, $\text{d(}{}^2\text{E)-p(}{}^2\text{F}{}_2\text{)}$, and $\text{d(}{}^2\text{F}{}_2\text{)-p(}{}^2\text{F}{}_2\text{)}$ are presented. As one might expect, when the spin-orbit constant A is set equal to zero, then both qualitative and quantitative aspects of the rotational-electronic problem in open-shell XY₄ molecules can be mapped easily onto discussions of the rotation-vibration problem from the CH₄ literature.     

Contact terms in E.M. decays of mesons

The ratios R_J(P)Γ(J)→P⁺P^?), P = π or K, have been evaluated ad due to contact interactions genrated by gauge invariance, for mesons with natural J^P. R_J increases rapidly with J. This provides unambigous tests for the model and indicates that contact terms are prolific sources of photons and leptons in hadron collisions.     

The perpendicular band ν14 of benzene near 10 μm

The perpendicular band of the E_1u vibration ν₁₄ of C₆H₆, with band origin at 1038.2670 cm^?1, was measured on a Bomem high-resolution Fourier transform spectrometer with a 246-cm optical path difference. The spectrum was deconvolved to an effective linewidth of ～0.002 cm^?1. A total of 135 subbands have been assigned in the ν₁₄ band. The band was found to be remarkably free of perturbations by other states. A Hamiltonian matrix of order 2, including merely the rotational l-type resonances between the ν₁₄⁺(J, K) and ν₁₄^?(J, K ? 2) levels, was used for the treatment of the upper states of this band. A set of accurate spectroscopic constants was obtained, which reproduces a total of 3258 measured lines with an over-all standard deviation of 0.0006 cm^?1.     

Random-field induced interface widths in Ising systems

Ad-dimensional Ising system with interactionJ at zero temperature in random fields ±h is considered, where due to boundary conditions an interface between states of positive and negative magnetization is maintained over a cross-sectionL ^d?1. The root mean square interface widthw(L,h) is found to behave asw(L, h)∝(h/J)² L(d=2) andw(L, h)≦L exp [?const(J/h)²] (d=3). Ford=2 the interface energy is negative forL→∞, and hence domains form spontaneously. Nonzero temperatures are discussed qualitatively, and consequences for the interpretation of experiments on random-field systems are outlined. Ford=2 and nonzero temperatures our treatment agrees with recent results of Grinstein and Ma and Villain.