R-matrix Approach Research Articles

Dynamical r-matrices for integrable maps

The integrability of two symplectic maps that can be considered as discrete-time analogs of the Garnier and Neumann systems is established in the framework of the r-matrix approach, starting from their Lax representation. In contrast with the continuous case, the r-matrix for such discrete systems turns out to be of dynamical type; remarkably, the induced Poisson structure appears as a linear combination of compatible “more elementary” Poisson structures. It is also shown that the Lax matrix naturally leads to define separation variables, whose discrete and continuous dynamics are investigated.

A novel hierarchy of integrable lattices

In the framework of the reduction technique for Poisson-Nijenhuis structures, we derive a new hierarchy of integrable lattices, whose continuum limit is the AKNS hierarchy. In contrast to other differential-difference versions of the AKNS system, our hierarchy is endowed with a canonical Poisson structure and, moreover, it admits a vector generalization. We also solve the associated spectral problem and explicitly construct action-angle variables through the r-matrix approach.

Quantum group covariant noncommutative geometry

An algebraic formulation of the quantum group covariant noncommutative geometry in the framework of the R-matrix approach to the theory of quantum groups is given. Structure groups are considered taking values in the quantum groups. The notion of noncommutative connections and curvatures is introduced transforming as comodules under the ‘‘local’’ coaction of the structure group that is an exterior extension of GLq(N). These noncommutative connections and curvatures generate GLq(N)-covariant quantum algebras. For these algebras combinations of generators are found which are coinvariants under the ‘‘local’’ coaction of the exterior extension of the quantum group, and one can formally consider these coinvariants as noncommutative images of the Lagrangians for the topological Chern–Simons models, non-Abelian gauge theories, and the Einstein gravity. Also an explicit realization of such covariant quantum algebras is presented via the investigation of the coset construction GLq(N+1)/(GLq(N)⊗GLq(1)).

GENERALIZED QUON STATISTICS

Generalized quons interpolating between Bose, Fermi, para-Bose, para-Fermi and anyonic statistics are proposed. They follow from the R-matrix approach to deformed associative algebras. It is proved that generalized quons have the same main properties as quons. A new result for the number operator is presented and some physical features of generalized quons are discussed in the limit [Formula: see text].

Electron-impact excitation of complex atoms and ions

A new R-matrix approach for calculating cross sections and rate coefficients for electron-impact excitation of complex atoms and ions is described. This approach, based on an expansion of the total wavefunction in target configurations rather than in individual target states and taking advantage of the special status of the scattered electron in the collisional wavefunction, enables the angular integrals to be performed very much more efficiently than hitherto. It also enables electron correlation effects in the target and in the electron-target collision complex to be treated consistently, eliminating pseudo-resonances which have caused serious difficulties in some earlier work. A major new program package RMATRX II has been written that implements this approach and, as an example, electron-impact excitation of Fe2+ is considered where the four target configurations 3d6, 3d54s, 3d54p and 3d54d are retained in the expansion of the total wavefunction. RMATRX II is compared with the standard R-matrix program package and is found to be much more efficient showing that accurate electron scattering calculations involving complex targets, such as the astrophysically important low ionization stages of iron-peak elements, are now possible.

An R-matrix approach for electron scattering off polyatomic molecules

A variational R-matrix approach from first principles-the selected states R-matrix (SSRM)-is proposed for the study of electron scattering off polyatomic, non-linear molecules. Cartesian Gaussian-type orbitals are used exclusively to represent both valence and continuum orbitals. The two-electron integrals along with the nuclear-attraction integrals are evaluated over the entire space. In a subsequent step these integrals are effectively confined to the R-matrix sphere by subtracting their contributions from outside the sphere using a multipole expansion which approximates the projectile-target interaction in the outer region. This approach allows the use of efficient integral codes available for the calculation of molecular bound states. Correlation and polarization effects are only taken into account for selected, energetically low-lying (n+1)-particle CI roots, corresponding to the R-matrix poles. Higher-lying roots are approximated by a static exchange calculation. In order to test this method, electron scattering off N2 is considered in detail for both resonant (2 Pi g) and non-resonant (2 Sigma g+) scattering symmetries. Good accordance is found between the eigenphase sums from the SSRM calculations and results obtained with a standard R-matrix package for linear molecules as well as with reference data from the literature.

Variational calculations for the dt micro molecule including nuclear effects on sticking by means of the Bloch operator.

We investigate the effects of nuclear forces on the eigenvalues and sticking fractions of the ground and excited J=0 states of the dt\ensuremath{\mu} molecule using a variational R-matrix approach. The nuclear boundary condition is imposed on the entire variational wave function, rather than on the individual elements of the basis, by means of the Bloch operator. This formulation results in rigorously symmetric variational matrix elements and improved flexibility of the basis at small d-t separations. Our results for the fusion rates and initial sticking fractions are ${\ensuremath{\lambda}}_{\mathit{f}}$=1.25\ifmmode\times\else\texttimes\fi{}${10}^{12}$ ${\mathrm{sec}}^{\mathrm{\ensuremath{-}}1}$ and ${\mathrm{\ensuremath{\omega}}}_{\mathit{s}}^{0}$=0.912% for the v=0 state and ${\ensuremath{\lambda}}_{\mathit{f}}$=1.05\ifmmode\times\else\texttimes\fi{}${10}^{12}$ ${\mathrm{sec}}^{\mathrm{\ensuremath{-}}1}$ and ${\mathrm{\ensuremath{\omega}}}_{\mathit{s}}^{0}$=0.912% for the v=1 state.

Fine structure of cluster decays.

Within the one level R-matrix approach, expressions are derived for the hindrance factors of cluster radioactive decays in which y the shell model with effective residual interactions [e.g.,thelar in the Michigan State University version for nearly spherical nuclei, or the enlarged superfluid model (ESM) recently proposed for deformed nuclei]. The exterior wave functions are calculated from a cluster-nucleus double-folding model potential obtained with the M3Y interaction. As examples of the cluster decay fine structure we analyzed the particular cases of \ensuremath{\alpha} decay of $^{241}\mathrm{Am}$ and $^{243}\mathrm{Cm}$, $^{14}\mathrm{C}$ decay of $^{223}\mathrm{Ra}$, and $^{34}\mathrm{Si}$ decay of $^{243}\mathrm{Cm}$. Good agreement with the experimental data is obtained.

R-matrix analysis of the 6d2 and 7p2 autoionizing resonances of barium

A method is presented for adapting the streamlined eigenchannel R-matrix approach to the analysis of 6d2 and 7p2 autoionizing resonances of Ba. The doubly excited states are assumed to be confined within the reaction volume. R-matrix calculations are carried in LS coupling for the 1S, 3P, 1D, 3F and 1G symmetries and in jj coupling for J=0e to J=4e. For each symmetry, an R-matrix restricted to open channels is constructed using a variational basis allowing the identification of the doubly excited states. From the R-matrix the authors deduce a scattering S-matrix which is expressed as the sum of resonant and non-resonant parts, i.e. in a form suited to resonance analysis. The positions and autoionization widths of 6d2 and 7p2 levels are predicted from the energy dependence of the density of states matrix, equivalent to the time-delay matrix. The approach is connected with other theories, in particular with Fano's configuration interaction method which treats bound levels embedded in continua.

An R-matrix approach to the quantization of the Euclidean group E(2)

The R-matrices for two different deformations of the Euclidean group E(2), calculated in a two-dimensional representation, are used to determine the deformed Hopf algebra of the representative functions. The duality of the latter with the initial quantum algebras is explicitly proved and the relationship between the two quantum groups is discussed and clarified.

Positron scattering by atomic hydrogen including positronium formation

Positron scattering by atomic hydrogen including positronium formation has been formulated using the R-matrix method and a general computer code written. Partial wave elastic and ground state positronium formation cross sections have been calculated for L<or=6 using a six-state approximation which includes the ground state and the 2s and 2p pseudostates of both hydrogen and positronium. The elastic scattering results obtained are in good agreement with those derived from a highly accurate calculation based upon the intermediate energy R-matrix approach. As in a previous coupled-channel static calculation, resonance effects are observed at intermediate energies in the S-wave positronium formation cross section. However, in the present results, the dominant resonance arises in the P-wave cross sections at an energy of 2.73 Ryd and with a width of 0.19 Ryd. The mechanism for the formation of this resonance and its possible experimental verification are briefly discussed. Finally, the summed elastic and positronium formation cross sections are presented and the latter compared with experiment for energies ranging from near threshold to 4.0 Ryd.

Quantum group gauge theories and covariant quantum algebras

An algebraic formulation of the quantum group gauge models in the framework of the R-matrix approach to the theory of quantum groups is given. We consider gauge groups taking values in the quantum groups and noncommutative gauge fields transformed as comodules under the coaction of the gauge quantum group G q . Using this approach we construct the quantum deformations of the topological Chern-Simons models, non-abelian gauge theories and Einstein gravity. The noncommutative fields in these models generate G q -covariant quantum algebras.

Alpha decay of 16O excited states.

The one-level R-matrix approach is used to calculate alpha decay widths for several low-lying ${0}^{+}$, ${2}^{+}$, and ${4}^{+}$ excited states of the $^{16}\mathrm{O}$ nucleus. The interior wave functions are calculated within the ZBM (${\mathit{p}}_{1/2}$,${\mathit{d}}_{5/2}$,${\mathit{s}}_{1/2}$) shell-model space with various effective interactions. The exterior wave functions are calculated from an \ensuremath{\alpha}${+}^{12}$C folding model potential obtained with the M3Y interaction. A new method for calculating the channel radius using the matching of the interior and exterior wave functions in the \ensuremath{\alpha}-decay channel is presented.

Generalized Drinfeld-Sokolov reductions and KdV type hierarchies

Generalized Drinfeld-Sokolov (DS) hierarchies are constructed through local reductions of Hamiltonian flows generated by monodromy invariants on the dual of a loop algebra. Following earlier work of De Groot et al., reductions based upon graded regular elements of arbitrary Heisenberg subalgebras are considered. We show that, in the case of the nontwisted loop algebra ℓ(gl n ), graded regular elements exist only in those Heisenberg subalgebras which correspond either to the partitions ofn into the sum of equal numbersn=pr or to equal numbers plus onen=pr+1. We prove that the reduction belonging to the grade 1 regular elements in the casen=pr yields thep×p matrix version of the Gelfand-Dickeyr-KdV hierarchy, generalizing the scalar casep=1 considered by DS. The methods of DS are utilized throughout the analysis, but formulating the reduction entirely within the Hamiltonian framework provided by the classical r-matrix approach leads to some simplifications even forp=1.

Electron impact excitation of Si XIV: collision strengths and rate coefficients

Collision strengths for all transitions up to and including the n = 5 levels of Si XIV have been computed in the LS coupling scheme using R-matrix approach. Care has been taken to ensure the convergence of collision strengths and resonances have been resolved in a fine mesh of electron energy in the threshold energy region. These resonances are shown for a few selected transitions, particularly for those among the lower excited levels, but collision strengths are tabulated for all transitions in the energy range 189-215 Ry. Additionally, these are integrated over a Maxwellian distribution of electron velocities to obtain the effective collision strengths from which the excitation and deexcitation rate coefficients can be easily obtained. The results for effective collision strengths are tabulated in the electron temperature range of log Te = 4.0-6.40 K and are compared with the earlier available data.

Atomic data for opacity calculations. XV. Fe I-IV

Photoexcitation and photoionization processes are examined for ground and excited terms of Fe I, Fe II, Fe III and Fe IV. An LS coupling R-matrix approach is used in a seventeen target term expansion for Fe I, including 3d7, 3d6(5D)4S, 3d6(5D)4p and 3d5(6S)4s2 configurations; and sixteen term expansions for Fe II, Fe III and Fe IV, including 3d6, 3d5 and 3d4 configurations respectively. Data for even and odd LS terms having S<or=3; L<or=7 and effective principal quantum numbers up to 10 are calculated. Over six thousand term energies and nearly half a million dipole oscillator strengths are calculated. The photoionization cross section is tabulated for all bound terms over a fine mesh of photon energies to elucidate the low-energy resonance structure.

R-matrix calculation of atomic hydrogen photoionization in a strong magnetic field.

We describe atomic hydrogen diamagnetism within the framework of multichannel quantum-defect theory using an R-matrix approach. The calculated photoionization spectrum in the range of magnetic field B=${10}^{3}$--${10}^{4}$ T shows that the quasi-Landau resonances are broad interlopers that perturb high Rydberg states converging to the Landau thresholds, forming complex resonances. A partial-cross-section analysis indicates that electron population of different Landau channels depends on the azimuthal quantum number and parity of the final states. For odd-total-parity final states with m=1, the photoionized electron is predicted to escape predominantly in the higher Landau channels, while for final states with m=0, it escapes in the lower channels. This property is reflected in the shape of autoionizing resonances, which are more like peaks for m=1, but are more like dips (window resonances) for m=0.

Electron impact excitation of C VI: collision strengths and rate coefficients

Using the R-matrix approach electron impact excitation of C VI has been studied. Collision strengths for all transitions between the lowest fifteen excited states up to the n=5 level have been computed in the energy range below 50 Ryd. All partial waves with angular momentum L<or=12 have been included and the contributions of L>12 have been obtained from the recently developed no-exchange program. The resonances have been delineated and collision strengths are compared with the earlier available data. Additionally, they have been integrated over a Maxwellian distribution of electron velocities to obtain the effective collision strengths which are tabulated at a series of temperatures below log T3=6.2 K.

Multiparameter R-matrices and their quantum groups

Multiparameter solutions of the Yang-Baxter equation associated with the groups of types A, B, C, D are given as generalizations of the well-known one-parameter solutions. The R-matrix approach to quantum groups is related to the algebraic one of Rheshitikhin (1990).

A model describing inelastic processes in low-energy electron collisions with methyl chloride

Dissociative attachment and vibrational excitation processes in low-energy electron collisions with methyl chloride are considered. A theory based on a combination of the resonance R-matrix approach and the non-local-complex-potential approach is applied for the calculation of the cross sections. For the input data of the theory the author uses Falcetta and Jordan's calculation of the adiabatic potential curve for the CH3Cl- anion. The calculated dissociative attachment cross sections averaged over the thermal vibrational distribution are about three orders of magnitude less than those obtained in a beam experiment of Chu et al. (1990). Possible reasons for this discrepancy are analysed. Vibrational excitation cross sections exhibit a sharp threshold peak and a broad resonance with the position corresponding to the vertical attachment energy. The author also investigates the sensitivity of the results to the input parameters.