Variable Phase Approach Research Articles

Complex model optical potential for electron scattering from oxygen

We report calculations of the total (elastic plus inelastic) cross-sections for the e-O system over a wide energy range 10–5000 eV. A local complex optical potential is calculated for the system using the atomic wavefunctions at the Hartree-Fock level. The real part of potential is composed of the attractive static, correlation polarisation and exchange potential. The imaginary component of the complex potential is a function of target charge density, incident electron energy and mean excitation energy. The resulting complex potential is treated in variable phase approach to yield complex phase shifts and total cross-sections. The total and elastic cross-sections are compared with the other available results. The agreement is excellent for total cross-sections with other theoretical work at energies greater than 100 eV. We have excellent agreement for elastic cross-sections with the experimental and theoretical results at all the energies. The elastic differential cross-sections are presented at 50 eV and 1000 eV. We fit the absorption cross-section values to Bethe asymptotic formula in high energy range (≥ 5500 eV). Ionisation cross-sections above 1000 eV are also deduced from the theory.

The relativistic multi-channel variable phase method for solving asymptotic equations in electron-atom and electron-ion scattering

The multi-channel variable phase method has been generalized to the relativistic case. The problem of solving a system of coupled radial Dirac equations without exchange has been reduced to the problem of direct outward integration of a non-linear differential equation satisfied by the reaction matrix K( r). The method is suitable for solving relativistic asymptotic equations occurring in the electron-atom and electron-ion scattering theory. A matching procedure for the relativistic R-matrix method and the relativistic multi-channel variable phase approach has been described.

Perturbation theory for the one-dimensional Schrödinger scattering problem

A perturbation theory is constructed within the framework of a linear version of the variable-phase approach, with the aim of making a complete study of the problem of scattering by a superposition of the Coulomb potential and the potential V(x) which decrease faster than the centrifugal potential. As a zero approximation of the theory for regular and irregular solutions to this problem, for normalization factors, scattering phase and amplitude, use is made of the corresponding functions calculated for the potential V(x) cut off at a certain point x=b. All subsequent approximations are determined analytically by the iteration method. Perturbation theory is applied to investigate the asymptotics of the partial waves of scattering phases and amplitudes in the low-energy limit and in the limit of large angular momenta.

Perturbation theory for the one-dimensional Schrodinger scattering problem

A perturbation theory is constructed within the framework of a linear version of the variable-phase approach, with the aim of making a complete study of the problem of scattering by a superposition of the Coulomb potential and the potential V(x) which decrease faster than the centrifugal potential. As a zero approximation of the theory for regular and irregular solutions to this problem, for normalization factors, scattering phase and amplitude, use is made of the corresponding functions calculated for the potential V(x) cut off at a certain point x=b. All subsequent approximations are determined analytically by the iteration method. Perturbation theory is applied to investigate the asymptotics of the partial waves of scattering phases and amplitudes in the low-energy limit and in the limit of large angular momenta.

Semianalytical method for treatment of the N-body problem with complex total energy within the hyperharmonics approach

We develop the variable phase approach to analyze all the solutions of the N-body hyperradial equations. As a first result of this analysis, we display here how the usual cutoff of the long-range potential matrix generates unphysical solutions. As an example of such solutions we present a set of unphysical three-neutron resonances. To overcome this peculiarity of the standard hyperharmonics approach we introduce a well-defined solution of the Jost-type and construct it at large hyperradius analytically.

Total cross-sections for electron scattering from iron at 10?5000 eV using a model optical potential

We report calculations on the total (elastic plus inelastic) electron-scattering cross sections in the energy range 10–5000 eV. A model complex optical potential, composed of static, exchange, polarisation and absorption terms, is employed to describe the collision system at each electron energy. The Iron atom is described by Dirac-Hartree-Fock-Slater self-consistent charge density. The complex phase shifts are computed in a variable phase approach. The absorption cross sections are compared with the experimental results. The experimental absorption cross sections are obtained by adding the experimental ionisation cross sections and available experimental excitation cross sections for electron impact of the allowed transitions a5 D → (x,y,z)5 D 0, (w,y,z)5 P 0. We have good qualitative agreement between our results and the experimental results available below 200 eV. The Born-Bethe parameters are also calculated. Elastic differential cross-sections with and without absorption are also reported at a few selected energies.

Equivalent local potentials and phase approach to low-energy scattering parameters

It is shown that the use of the McTavish-Kermode-Melhem-type equivalent local potentials linearizes algorithms of the variable-phase approach needed to compute the low-energy scattering parameters. A case study is presented to demonstrate the merit of the linearized equations, with particular emphasis on the relationship between the interpolating scattering length function and potential functions.

Combined variable-phase supersymmetric quantum mechanics

Methods of supersymmetric quantum mechanics (SUSYQM) are used to derive relations between phase functions which occur in the context of the variable-phase approach to potential scattering. It is shown that the phase-equivalent potentials obtained by the application of SUSYQM are not off-shell equivalent. A case study is presented in support of this.

Variable-phase approach to the dirac equation—bound states

For a Dirac particle subject to a central potential, conditions for the potential are obtained such that there is a bound state with zero binding energy. This is done by means of the «variable-phase approach». The potential considered consists of a Lorentz scalar and the zeroth component of a Lorentz vector, and vanishes asymptotically.

Total (elastic and inelastic) scattering cross sections for several positron-molecule systems at 10-5000 eV: H2, H2O, NH3, CH4, N2, CO, C2H2, O2, SiH4, CO2, N2O, and CF4.

We report calculations on the total (elastic plus inelastic) positron-scattering cross sections from several diatomic and polyatomic molecules (${\mathrm{H}}_{2}$, ${\mathrm{H}}_{2}$O, ${\mathrm{NH}}_{3}$, ${\mathrm{CH}}_{4}$, ${\mathrm{N}}_{2}$, CO, ${\mathrm{C}}_{2}$${\mathrm{H}}_{2}$, ${\mathrm{O}}_{2}$, ${\mathrm{SiH}}_{4}$, ${\mathrm{CO}}_{2}$, ${\mathrm{N}}_{2}$O, and ${\mathrm{CF}}_{4}$) where experimental data are available. The impact energy (E) range is 10--5000 eV. A local spherical complex optical potential (SCOP) is calculated for each positron-molecule system from the target charge density [\ensuremath{\rho}(r)], which in turn is determined from the corresponding molecular wave function at the Hartree-Fock level. The real part of the SCOP is composed of the repulsive static and attractive positron-correlation-polarization potential of Jain [Phys. Rev. A 39, 2437 (1990)]. The imaginary component of the SCOP, the so-called absorption potential, is derived semiempirically as a function of \ensuremath{\rho}(r), E, and the mean excitation energy. The resulting complex optical potential is treated exactly in a variable-phase approach to yield complex phase shift and the total-cross-section quantities. In this intermediate- and high-energy region, the small contribution due to the nonspherical nature of the target is neglected. In addition, we fit the total-cross-section values to a simple analytic formula. For molecules possessing a permanent dipole or quadrupole moment, the present results are reliable only roughly above 50 eV.

Total (elastic plus inelastic) cross sections for electron scattering from diatomic and polyatomic molecules at 10-5000 eV: H2, Li2, HF, CH4, N2, CO, C2H2, HCN, O2, HCl, H2S, PH3, SiH4, and CO2.

By employing a previously proposed model [A. Jain, Phys. Rev. A 34, 3703 (1986); J. Phys. B 22, 905 (1988)], we report calculations on the total (elastic plus inelastic) electron-scattering cross sections in a wide energy range (10--5000 eV) from several diatomic and polyatomic molecular targets (${\mathrm{H}}_{2}$ ${\mathrm{Li}}_{2}$, HF, ${\mathrm{CH}}_{4}$, ${\mathrm{N}}_{2}$, CO, ${\mathrm{C}}_{2}$${\mathrm{H}}_{2}$, HCN, ${\mathrm{O}}_{2}$, HCl, ${\mathrm{H}}_{2}$S, ${\mathrm{PH}}_{3}$, ${\mathrm{SiH}}_{4}$, and ${\mathrm{CO}}_{2}$). A model complex optical potential (composed of static, exchange, polarization, and absorption terms) is calculated for each collision system from the corresponding molecular wave function at the Hartree-Fock level. The resulting complex optical potential, free from any adjustable parameter, is treated exactly in a variable-phase approach to yield scattering complex phase shifts and the total cross sections. In the intermediate- and high-energy region, the small contribution due to the nonspherical nature of the target is treated perturbatively in the first-order Born approximation. The present method is quite simple in nature and is able to reproduce fairly well the experimental total cross sections in the present energy region. Results are also given for individual elastic and absorption (accounting for all energetically possible inelastic processes in an approximate way) cross sections. In addition, we provide Born-Bethe parameters for all the above molecules including those of ${\mathrm{H}}_{2}$O and ${\mathrm{NH}}_{3}$. We have also examined the correlation between molecular properties and the total-cross-section parameter. For molecules possessing a permanent dipole or quadrupole moment, the present results are only roughly reliable above 100 eV.

Effect of interference inside surroundings on XANES. O K-shell photoabsorption in N2O

In this work an interference picture occurring as a result of inner-shell ionization of an atom in a substance is examined. We focus our attention on the interference of photoelectron scattered waves inside the surroundings of an excited atom. Such interference leads to the interdependence of amplitudes of the photoelectron waves reflected (or back scattered) by different coordination spheres and thus the surroundings act as an indivisible system. The influence of this interference is analysed here using as an example, the O K-shell XANES in the N2O molecule (the case of XANES of a terminal atom in a linear triatomic molecule). To reveal the role of the interferencial effects inside the diatomic fragment ⟨N1-N2⟩ we obtain an approximate analytical expression for XANES description. Such a description becomes possible due to (i) the localized orbitals approach, (ii) the variable phase approach and (iii) the concept of electron-optical properties of surroundings. The performed calculations show that for the correct O K-shell XANES interpretation the interference photoelectron waves inside the fragment must necessarily be taken into account. It is as well stated that the interference effects inside the surroundings explain: (1) the disturbance of the linear correlation between σ-resonance position and bond length, (2) the transformation of the EXAFS-like band to the resonance-like line in the spectrum and (3) the appearance of the extra maximum in the σ-channel of the photoabsorption cross section.

Elastic scattering of electrons by water molecules at intermediate and high energies.

Elastic differential, integral, and momentum-transfer cross sections are reported for the scattering of electrons by water molecules in the energy range of 100--1000 eV. A parameter-free model optical potential which is the sum of three spherical terms, namely the static, exchange, and polarization forces, is constructed from near--Hartree-Fock one-center-expansion water wave functions. The total optical potential is then treated exactly in a partial-wave analysis using the variable-phase approach to yield the scattering phase shifts. We employ several versions of parameter-free polarization and exchange potentials. We find that the present calculated differential cross sections reproduce all the important features (such as forward peaking, dip at middle angles, and enhanced backward scattering) observed in recent experiments. Quantitatively, the present results are in very good agreement with experiment and better than earlier calculations.

Spherical-complex-optical-potential (SCOP) model for electron–monosilane (SiH4) collisions at 30–400 eV: Total (elastic+absorption), momentum transfer, and differential cross sections

We report nonempirical quantum mechanical calculations on the total (elastic+absorption), momentum transfer, and differential cross sections for e-SiH4 collisions at intermediate and high energies (30–400 eV). A parameter-free and energy-dependent spherical-complex-optical potential (SCOP) is evaluated for the e–SiH4 system. The real part of the SCOP consists of three local terms, namely static, exchange, and polarization. The static interaction is generated very accurately from near-Hartree–Fock one-center silane wave functions, while the exchange effects are accounted for in the free-electron-gas-exchange (FEGE) model. The polarization potential is evaluated ab initio in a parameter-free approximation of Jain and Thompson. The imaginary term of the total SCOP represents loss of flux due to inelastic channels via an energy-dependent absorption potential calculated from target electron density and short-range static-exchange force in the quasifree model with Pauli blocking [Staszewska et al.; J. Phys. B 16, L281 (1983)]. Two versions of this absorption potential are employed; one with an undistorted density and the other with a polarized density determined approximately from first order target wave functions. The later version is more successful when the final results are compared with experiment. The total SCOP is treated exactly in a partial-wave analysis using the variable-phase approach to yield complex phase shifts. Our final total cross sections compare very well with the only available measurements of Sueoka and Mori. However, below 50 eV, present total cross sections overestimate the experimental data within 10%. The effect of absorption potential is to reduce the elastic cross sections significantly; this reduction is more dramatic in case of the differential cross sections (DCS); for example, the reduced DCS are exposed to more pronounced structure. Interestingly, the e–SiH4 reduced DCS are very close in shape to the corresponding e–Ar cross sections above 60 eV.

Inverse problem for the half-off-shellTmatrix

The inverse scattering problem for the off-shell $T$ matrix is formulated within the framework of a combined variable-phase---off-shell scattering theory. A simple expression is constructed for the $s$-wave half-off-shell $T$ matrix. The input information consists of binding energies and phase shifts at all energies.NUCLEAR REACTIONS Scattering theory, off-shell generalization of the variable-phase approach, inverse problem for the $T$ matrix.

Variable-phase approach to off-shell scattering by nonlocal potentials

The basic ideas of the variable-phase approach are used to develop first-order differential equations for the quasiphase parameters which describe the half-off-shell scattering amplitudes for a nonlocal potential. The on-shell equation of Calogero and Sobel’s equation for a local potential are obtained as special cases.

Interpolating amplitude equations from an embedding technique

Abstract The variable phase approach to potential scattering is regarded as an embedding technique. Algorithms are presented for studying the spatial behaviour of off-shell T matrices.

Variable-phase method in positron-hydrogen elastic scattering

The elastic scattering of a positron by a hydrogen atom is studied by using the variable-phase approach to potential scattering. Values of the scattering length and the elastic scattering phase shifts obtained in the (i) static, (ii) polarization and (iii) extended polarization approximations for the e+−H interaction are found to be in close agreement with the results of earlier workers.

Variable phase approach to many-channel relativistic scattering

Variable phase approach to many-channel relativistic scattering

Equation for the total scattering amplitude in the variable phase approach

Equation for the total scattering amplitude in the variable phase approach