K-matrix Elements Research Articles

K -matrix formulation of two-particle scattering in a waveguide in the presence of one-dimensional spin-orbit coupling

The creation of artificial gauge fields in neutral ultracold atom systems has opened the possibility to study the effects of spin-orbit coupling terms in clean environments. This work considers the multi-channel scattering properties of two atoms confined by a wave guide in the presence of spin-orbit coupling terms within a K-matrix scattering framework. The tunability of resonances, induced by the interplay of the external wave guide geometry, the interactions, and the spin-orbit coupling terms, is demonstrated. Our results for the K-matrix elements as well as partial and total reflection coefficients for two identical fermions interacting through a finite-range interaction potential in the singlet channel only are compared with those obtained for a strictly one-dimensional effective low- energy Hamiltonian, which uses the effective coupling constant derived in Zhang et al. [Scientific Reports 4, 1 (2014)] and Zhang et al. [Phys. Rev. A 88, 053605 (2013)] as input. In the regime where the effective Hamiltonian is applicable, good agreement is obtained, provided the energy- dependence of the coupling constant is accounted for. Our approach naturally describes the energy regime in which the bands associated with excited transverse modes lie below a subset of the bands associated with the lowest transverse modes. The threshold behavior is discussed and scattering observables are linked to bound state properties.

Electron scattering cross sections from HCN over a broad energy range (0.1–10 000 eV): Influence of the permanent dipole moment on the scattering process

We report theoretical integral and differential cross sections for electron scattering from hydrogen cyanide derived from two ab initio scattering potential methods. For low energies (0.1-100 eV), we have used the symmetry adapted-single centre expansion method using a multichannel scattering formulation of the problem. For intermediate and high energies (10-10,000 eV), we have applied an optical potential method based on a screening corrected independent atom representation. Since HCN is a strong polar molecule, further dipole-induced excitations have been calculated in the framework of the first Born approximation and employing a transformation to a space-fixed reference frame of the calculated K-matrix elements. Results are compared with experimental data available in the literature and a complete set of recommended integral elastic, inelastic, and total scattering cross sections is provided from 0.1 to 10,000 eV.

Change in the structure of colliding deuterons

The effect of the fields of two colliding deuterons on the deuteron wave functions is investigated within the algebraic version of the resonating-group method. The problem of determining the continuum of the K-matrix elements at a given c.m. energy of the deuterons is formulated and solved on the basis of this approach.

Applications of the modified Faddeev equations: t + dμ -

We present calculations of the cross-sections for elastic as well as for muon transfer in t + dμ- collisions below the tμ-(n = 2) threshold. The appropriate mass-scaled Jacobi coordinates are used in the direct and rearrangement channels. The K-matrix elements are obtained from the solution of multichannel Faddeev equations in the configuration space formulation. A bipolar expansion and spline representation are used in the numerical solution of the modified Faddeev equations.

S-wave positron-hydrogen scattering via Faddeev equations: Elastic scattering and positronium formation.

s-wave scattering of positrons on atomic hydrogen is treated by solving the Faddeev equations. Elastic phase shifts below the Ps(n-1) excitation threshold and the K-matrix elements within the ore gap between the Ps(n=1) and H(n=2) thresholds are calculated as well as the zero-energy 2\ensuremath{\gamma} annihilation rate of the ${\mathit{e}}^{+}$${\mathit{e}}^{\mathrm{\ensuremath{-}}}$ pair.

Scattering of electrons and positrons on hydrogen using the Faddeev equations

Scattering of electrons and positrons on a hydrogen atom below the H(n=2) threshold is studied by means of directly solving the modified Faddeev equations in the total angular-momentum representation. Elastic phaseshifts of the e-+H scattering and those of the e++H scattering below the Ps(n=1) threshold are obtained. K-matrix elements for the e++H and p+Ps scattering between the Ps(n=1) and H(n=2) thresholds are calculated. The results are in a good agreement with previous variational calculations.

A new generalization of vector-coherent state theory for the SO(5)U(2) proton-neutron quasispin group

The introduction of a set of intrinsic coordinates to give an explicit construction of the intrinsic states of vector-coherent state theory has greatly simplified earlier attempts to generalize this theory to include operators lying outside the group algebra. Very explicit vector-coherent state construction of such operators can now be given in terms of vector-coupled combinations of intrinsic and collective operators. When organized into tensors which induce specific shifts in irreducible representations these lead to the reduced Wigner coefficients needed in practical calculations. The SO(5)U(2) proton-neutron quasispin algebra is used as an example to give further simplifications of earlier results. All Wigner coefficients needed to give the n,T-dependence of matrix elements in the seniority scheme can now be given through a few terms expressed solely through angular momentum recoupling coefficients and the K-matrix elements of vector-coherent state theory.

Generalized vector coherent state theory of sp(2N,R) vector operators and of sp(2N,R) contains/implies u(N) reduced Wigner coefficients

sp(2N,R) contains/implies u(N) reduced Wigner coefficients coupling a positive discrete series unitary irreducible representation with the 2N-dimensional non-unitary irreducible representation characterized by (1) are evaluated by making use of generalized vector coherent state techniques. General formulae are given in terms of the K-matrix elements of vector coherent state theory and of some u(N) 6j or 9j-type recoupling coefficients. Explicit analytic expressions are obtained in some important special cases. Such results should allow a determination of wsp(2N,R) approximately=w (N)(+)p(2N,R) and osp(P/2N,R) matrix representations to be completed, because both the w(N) generators and the osp(P/2N,R) odd generators transform under the sp(2N,R) irrep(1).

Wigner coefficients for the proton-neutron quasispin group: An application of vector coherent state techniques

SO(5) ⊃ U(2) reduced Wigner coefficients, needed to extract the n, T-dependence of nuclear matrix elements in the seniority scheme, are evaluated by vector coherent state techniques by casting operators other than the group generators into their Bargmann z-space realizations. Results are given, in terms of simple angular momentum recoupling coefficients and the K-matrix elements of vector coherent state theory, for SO(5) couplings involving the 4-, 5-, and 10-dimensional representations. Both a simplification of earlier results and a generalization to states of arbitrarily high seniority has been achieved.

Relative performances of the Kohn, Schwinger, and Newton variational principles in scattering theory

Comparisons of three variational principles commonly used in scattering problems, namely those due to Kohn (KVP), Schwinger (SVP), and Newton (NVP), are presented. These comparisons are conducted by computing K-matrix elements for elastic scattering from nine different interaction potentials. We represent the KVP trial functions as expansions containing two non-L2 terms that represent the asymptotic free wave, and a set of L2 functions, while the SVP and the NVP trial functions are expansions containing only the L2 terms. Three different sets of L2 functions are used to examine the effect of changing the basis on the convergence characteristics of the three methods. We find that the rates of convergence for the Kohn, Schwinger, and Newton methods are strongly dependent on the nature of the potential and the basis set used. We also find that purely repulsive potentials are, in general, easier to converge than purely attractive potentials.

Lanczos recursion, continued fractions, Padé approximants, and variational principles in quantum scattering theory

Connections are established between the Schwinger and Newton variational principles and recursive generation of the remainder in Born series expansions of individual K-matrix elements. It is shown that Lanczos development of the remainder yields results identical to either of these variational principles, depending upon the starting vectors that are used to initiate the recursion sequence. In all cases, the correction to the Born series is computed from the 1,1 element of the inversion of a small tridiagonal matrix. The relationship to Padé approximants and continued fractions is also noted. Numerical results on the convergence of elastic and inelastic K-matrix elements are presented for a model system.

Multilevel adaptive technique for quantum reactive scattering

Discretization (with localized basis functions or grid points) of the coupled integral equations for molecular collisions leads to a very large system of linear algebraic equations. New methods, which are well adapted to vector supercomputers and parallel architectures, are developed for solving this large system. The multilevel adaptive technique (MLAT) is combined with recursive and iterative techniques. First, a multichannel solution is obtained on a low level grid. The basis is then adapted to this solution and the coarse solution is projected or interpolated onto the adapted basis. The scattering amplitudes (K-matrix elements) on the high level are then developed through use of either the recursion method (for single amplitudes, or a small batch of them) or the iterative technique (for all transitions from a specified initial state). In both of these methods, the original large system of algebraic equations is projected into a much smaller subspace (an orthonormalized Krylov space) spanned by a few basis vectors. Applications to very large systems are possible because it is not necessary to store or invert a large matrix. Computational results on a model chemical reaction are presented.

A first-order non-degenerate adiabatic theory for calculating near-threshold cross sections for rovibrational excitation of molecules by electron impact

An approximate expression is derived for the inelastic K-matrix elements for rotational excitation of molecules by electron impact. This expression incorporates a Born-Oppenheimer separation of the projectile and nuclear coordinates in the continuum wavefunction but allows for different energies in the entrance and exit channels. Hence it is particularly suited to near-threshold collisions. The method is applied to pure rotational excitation of H2 by comparing approximate cross sections with those from laboratory frame close-coupling calculations.

Positronium formation in s-wave positron-hydrogen scattering

Detailed investigations are made of the convergence of the K-matrix elements and the cross sections for elastic scattering and positronium formation in the s-wave scattering of positrons by atomic hydrogen in the energy range 6.8-10.2 eV. The Kohn and inverse Kohn variational methods are used with trial functions containing many linear variational parameters and associated short-range correlation terms, and results are obtained for a set of systematically improved trial functions. The diagonal elements of the K matrix and the eigenphase shifts obtained with these trial functions can be fitted very accurately to a convergence formula which permits extrapolation to the essentially exact results corresponding to an infinite number of short-range correlation functions. No such convergence formula applies to the off-diagonal matrix element K12, nor to the cross sections, but the very slight changes in the results as the trial functions undergo their final improvements strongly suggest that the most accurate values obtained here for K12 and the cross sections for positronium formation and elastic scattering are in error by no more than 5%, 10% and 5% respectively.

Padé-approximant corrections to general variational expressions of scattering theory: Application to5σphotoionization of carbon monoxide

We discuss a method for systematically correcting results obtained using variational expressions in scattering theory. The approach taken is to compute a sequence of Pade approximants of the form [N/N] for the error in an initial variational estimate obtained using a basis-set expansion. The relationship between the Pade-approximant approach and the iterative Schwinger method for correcting variational estimates is also examined. We discuss a large class of general variational expressions to which the Pade-approximant approach can be applied. The variational expressions considered include those for the wave function, for photoionization transition matrix elements, as well as for scattering matrix (K-matrix) elements. We have applied this approach to the 5σ photoionization of CO using the frozen-core Hartree-Fock and fixed-nuclei approximations. We find that the Pade-approximant method converges rapidly and reliably. Both total photoionization cross sections and photoelectron angular distributions from threshold to 40 eV are presented and compared to previous experimental and theoretical results. We find major quantitative discrepancies between the present results for the total cross section and previous theoretical results.

Vibrational excitation of symmetric and bending modes of H2O by slow electron impact

Electron impact vibrational excitation cross sections (differential, integral and momentum transfer) for the symmetric (100) and the bending (010) modes of the H2O molecule are calculated in the fixed-nuclei and the impulse approximations from threshold to 10 eV. The normal coordinate dependent K-matrix elements are augmented with the laboratory frame first Born matrix elements for higher angular momenta in order to obtain converged cross sections via a closure formula. A sharp threshold peak and a broad resonance structure is seen around 6-8 eV in the integral cross section in agreement with the recent observations: while the A1 scattering state plays a not unimportant role, the major contribution to this broad feature is due to the B2 state. All results compare well with the crossed-beam experimental results of Seng and Linder (1974).

Multichannel threshold structures in scattering theory

The theory of multichannel threshold structures is expressed in a form that separates background scattering from the effects of resonance or virtual state singularities. The leading terms of K-matrix elements at an excitation threshold are shown to determine the characteristic parameters of a virtual state or resonance below the threshold and the cusp structures and excitation peaks at the threshold.

The pion mass difference in threshold pion reactions on light nuclei

We investigate the effect of the pion mass difference in scattering and photoproduction of pions on light nuclei near threshold. The effect is found to be most easily described in a nuclear K-matrix formalism where the nuclear K-matrix elements are shown to be independent of the mass difference.

Variationally corrected discrete-basis-set calculation for electron-molecule scattering in the static-exchange approximation

We present a method for including a variational correction in the discrete-basis-set method for electron-molecule scattering introduced by Rescigno, McCurdy, and McKoy. Our prescription for variationally corrected partial-wave K-matrix elements is based on Kohn's formula for the variationally stable scattering amplitude in three dimensions. The method is applied to e/sup -/-H/sub 2/ scattering in the static-exchange approximation, and the effect of choosing different trial basis sets is shown. Our results are in good agreement with the results of other accurate calculations.

Multipole Amplitude Analysis of Elastic Photon-Proton Scattering below 500 MeV and the One-Particle-Exchange Model

On the assumption that the partial wave K-matrix elements can be equated with the lowest order amplitudes (case A) or the modified one (case B), r·P elastic scattering below 500 MeV is investigated. Coupling constants are determined from other related processes phenomenologically. Contributions from competing channel are evaluated by using the rN-'>rrN multipole solution and the rr·N phase shifts. There are eight possible solutions with respect to the signs of meson-exchange contributions for case A, while additional two para­ meters are necessary for the modification in case B. These are determined from the energy dependence of dajd!J(90). A solution for case A gives satisfactory fit to experiments below 300 MeV but not above. For case B, good agreements with existing data are obtained below 500 MeV. Some predictions based upon the solution are presented.