P-matrix and J-matrix approaches. Coulomb asymptotics in the harmonic oscillator representation of scattering theory

The relation between the R- and P-matrix approaches and the harmonic oscillator representation of the quantum scattering theory (J-matrix method) is discussed. We construct a discrete analogue of the P-matrix that is shown to be equivalent to the usual P-matrix in the quasiclassical limit. A definition of the natural channel radius is introduced. As a result, it is shown to be possible to use well-developed technique of R- and P-matrix theory for calculation of resonant states characteristics, scattering phase shifts, etc., in the approaches based on harmonic oscillator expansions, e.g., in nuclear shell-model calculations. P-matrix is used also for formulation of the method of treating Coulomb asymptotics in the scattering theory in oscillator representation.Comment: Revtex, 57 pages including 15 figures; to be published in Annals of Physic

Nuclear astrophysics studies with ultra-peripheral heavy-ion collisions

I describe in very simple terms the theoretical tools needed to investigate ultra-peripheral nuclear reactions for nuclear astrophysics purposes. For a more detailed account, see arXiv:0908.4307.Comment: 12 pages, 8 figures, Lecture notes presented at the 5th European Summer School on Experimental Nuclear Astrophysics, Sep. 20- 27, 2009, Santa Tecla, Sicily, Italy. AIP proceedings, to be publishe

Quasi-elastic neutrino charged-current scattering cross sections on oxygen

The charged-current quasi-elastic scattering of muon neutrinos on oxygen target is computed for neutrino energies between 200 MeV and 2.5 GeV using the relativistic distorted-wave impulse approximation with relativistic optical potential, which was earlier successfully applied to describe electron-nucleus data. We study both neutrino and electron processes and show that the reduced exclusive cross sections for neutrino and electron scattering are similar. The comparison with the relativistic Fermi gas model (RFGM), which is widely used in data analyses of neutrino experiments, shows that the RFGM fails completely when applied to exclusive cross section data and leads to overestimated values of inclusive and total cross sections. We also found significant nuclear-model dependence of exclusive, inclusive and total cross sections for about 1 GeV energy.Comment: 30 pages, 11 figures; final version to appear in Phys. Rev.

ECCPA: Calculation of classical and quantum cross sections for elastic collisions of charged particles with atoms

Financial support from the Spanish Ministerio de Ciencia, Innovación y Universidades / Agencia Estatal de Investigación / European Regional Development Fund, European Union (projects no. RTI2018-098117-B-C22 and PID2019-104888GB-I00 ) and from Junta de Andalucía (projects no. FQM387 and P18-RT-3237 ) is gratefully acknowledged.Program summary Program Title: eccpa CPC Library link to program files: https://doi.org/10.17632/c3tn9hyfvb.1 Licensing provisions: CC by NC 3.0 Programming language: Fortran 90/95 Nature of problem: The program computes differential cross sections (DCSs) for elastic collisions of charged particles (electrons, positrons, muons, antimuons, protons, antiprotons, and alphas) with neutral atoms. Calculations are performed within the static-field approximation with screened Coulomb potentials expressed as a sum of Yukawa terms with their parameters fitted to approximate the atomic electrostatic potentials resulting from the Thomas–Fermi model and from self-consistent Dirac–Hartree–Fock–Slater calculations. The program eccpa provides DCSs computed with four different approaches: the classical trajectory method, the Born approximation, the partial-wave expansion method with approximate phase shifts, and the eikonal approximation. The user is allowed to select the atomic number of the target atom, the potential model, the kind of projectile and its kinetic energy. Calculation results are written in a number of output files with formats suited for visualization with a plotting program. A Java graphical user interface allows running the program and visualizing the results interactively. Solution method: A relativistic extension of the classical trajectory method is formulated on the assumption that the interaction in the center-of-mass frame is central, which is a fundamental requirement of the adopted calculation schemes; the DCS in the laboratory frame is then obtained from the relativistic (Lorentz) transform of the DCS calculated in the center-of-mass frame. This scheme qualifies as semi-relativistic, because it accounts for relativistic kinematics in a rigorous way, but disregards the differences between the interactions observed from the laboratory and the center-of-mass frames. We consider the elementary quantum formulation based on the relativistic Schrödinger (or Klein–Gordon) wave equation obtained from the correspondence principle. Accurate DCSs for potential scattering can be computed by using the partial-wave expansion method, at the expense of considerable numerical work. To avoid the difficult calculation of phase shifts from the numerical solution of the radial wave equation, we adopt a simplified strategy that combines the (first) Born approximation, for both the scattering amplitude and the phase shifts, and the Wentzel–Kramers–Brillouin (WKB) approximation for the phase shifts. We also describe the semi-classical eikonal approximation, which is known to yield reliable DCSs for collisions with small scattering angles. The case of collisions of electrons and positrons is considered on similar grounds, with the scattering amplitudes obtained from the Dirac equation. The numerical work is simplified by approximating the interaction potential as a sum of Yukawa terms, which allows performing a good part of the calculations analytically. Integrals of functions given by analytical formulas are calculated by means of an adaptive algorithm that combines the 20-point Gauss-Legendre quadrature formula with a bisection scheme; this algorithm allows strict control of numerical errors and gives results with a relative accuracy better than about for well-behaved integrands. The whole calculation for a given energy of the projectile takes no longer than a few seconds on a modern personal computer, quite irrespectively of the energy and of the atomic number of the target atom. Additional comments including restrictions and unusual features: The adopted interaction potentials correspond to atoms with point nuclei. The use of a parameterization instead of numerical tables of the potential (obtained, e.g., from atomic structure calculations) has a minor effect on the calculated DCSs. This effect is limited to large scattering angles, where the actual DCS does differ from calculations with screened Coulomb potentials due to the effect of the finite size and structure of the atomic nucleus, which is disregarded here. DCSs obtained from the partial-wave expansion method and with the eikonal approximation provide a fairly accurate description of collisions with small and moderate deflection angles. They can be used, e.g., in Monte Carlo simulations of the transport of fast charged particles in matter. The information generated by the program allows assessing the accuracy of calculations with the various approaches, and permits identifying the ranges of validity of the classical trajectory method and the Born approximation.The Fortran program eccpa calculates differential and integrated cross sections for elastic collisions of charged particles with atoms by using the classical-trajectory method and several quantum methods and approximations. The collisions are described within the framework of the static-field approximation, with the interaction between the projectile and the target atom represented by the Coulomb potential of the atomic nucleus screened by the atomic electrons. To allow the use of fast and robust calculation methods, the interaction is assumed to be the same in the center-of-mass frame and in the laboratory frame. Although this assumption neglects the effect of relativity on the interaction, it allows using strict relativistic kinematics. The equation of the relative motion in the center-of-mass frame is shown to have the same form as in the non-relativistic theory, with a relativistic reduced mass and an effective potential. The wave equation for the relative motion, as obtained from the correspondence principle, is formally identical to the non-relativistic Schrödinger equation with the reduced mass and the effective potential, and it reduces to the familiar Klein-Gordon equation when the mass of the target atom is much larger than that of the projectile. Collisions of spin 1/2 projectiles are also described by solving the Dirac wave equation. Various approximate solution methods are described and applied to a generic potential represented as a sum of Yukawa terms, which allows a good part of the calculations to be performed analytically. The program eccpa is useful for assessing the validity and the relative accuracy of the various approximations, and as a pedagogical tool.Ministerio de Ciencia, Innovación y UniversidadesEuropean Commission PID2019-104888GB-I00, RTI2018-098117-B-C22European Regional Development FundJunta de Andalucía FQM387, P18-RT-3237Agencia Estatal de Investigació

Collisions of nucleons with atoms: calculated cross sections and Monte Carlo simulation

After a summary description of the theory of elastic collisions of nucleons with atoms, we present the calculation of a generic database of differential and integrated cross sections for the simulation of multiple elastic collisions of protons and neutrons with kinetic energies larger than 100 keV. The relativistic plane-wave Born approximation, with binding and Coulomb-deflection corrections, has been used to calculate a database of proton-impact ionization of K-shell and L-, M-, and N-subshells of neutral atoms These databases cover the whole energy range of interest for all the elements in the periodic system, from hydrogen to einsteinium (Z = 1-99); they are provided as part of the penh distribution package. The Monte Carlo code system penh for the simulation of coupled electron-photon-proton transport is extended to account for the effect of the transport of neutrons (released in proton-induced nuclear reactions) in calculations of dose distributions from proton beams. A simplified description of neutron transport, in which neutron-induced nuclear reactions are described as a fractionally absorbing process, is shown to give simulated depth-dose distributions in good agreement with those generated by the Geant4 code. The proton-impact ionization database, combined with the description of atomic relaxation data and electron transport in penelope, allows the simulation of proton-induced x-ray emission spectra from targets with complex geometries

Collisions of Nucleons with Atoms: Calculated Cross Sections and Monte Carlo Simulation

After a summary description of the theory of elastic collisions of nucleons with atoms, we present the calculation of a generic database of differential and integrated cross sections for the simulation of multiple elastic collisions of protons and neutrons with kinetic energies larger than 100 keV. The relativistic plane-wave Born approximation, with binding and Coulomb-deflection corrections, has been used to calculate a database of proton-impact ionization of K-shell and L-, M-, and N-subshells of neutral atoms These databases cover the whole energy range of interest for all the elements in the periodic system, from hydrogen to einsteinium (Z = 1–99); they are provided as part of the penh distribution package. The Monte Carlo code system penh for the simulation of coupled electron-photon-proton transport is extended to account for the effect of the transport of neutrons (released in proton-induced nuclear reactions) in calculations of dose distributions from proton beams. A simplified description of neutron transport, in which neutron-induced nuclear reactions are described as a fractionally absorbing process, is shown to give simulated depth-dose distributions in good agreement with those generated by the Geant4 code. The proton-impact ionization database, combined with the description of atomic relaxation data and electron transport in penelope, allows the simulation of proton-induced x-ray emission spectra from targets with complex geometries.Ministerio de Ciencia, Innovación y Universidades RTI2018-098117-B-C21, RTI2018-098117-BC2