Atomic Collision Calculations for Astrophysics Dr Cathy Ramsbottom

Modelling CLOUDY CHIANTI XSTAR Radiative Data (f- and A- values, energy levels, wavelengths) Collision data (cross sections, excitation rates, recombination rates, proton rates)

What is an Atomic Collision: Elementary Particle + Atomic System electron/photon + atom/ion/molecule Atomic System + Atomic System atom/ion/molecule + atom/ion/molecule

Types of Collisions: Consider an elementary particle A and an atomic system B A + B  A + B Elastic Scattering A + B  A + B’ Inelastic Scattering A + B  C + D + E + F + …….X Reactions

Atomic Processes Radiative Transitions * Electron-ion collisions Photoionization/recombination * Proton-ion collisions Charge exchange Atom-ion collisions Bremsstrahlung * Two-photon emission *

Atomic Processes Radiative Transitions * Electron-ion collisions Photoionization/recombination * Proton-ion collisions Charge exchange Atom-ion collisions Bremsstrahlung * Two-photon emission *

The Collision Cross Section: The results of any atomic collision are usually expressed in terms of quantities called `cross sections’. The cross section is a measure of the probability that a given type of collision will occur under given conditions. Nature of the particles The reaction being considered The mutual velocity of approach of the colliding partners The impact parameters

Resonances A resonance is in very simplistic terms a `preferred route of nature’ represented by sharp peaks or spikes on an otherwise smooth background. Autoionizing Resonances Near-threshold Shape Resonances Feshbach Resonances

Photoionization of Helium hν + He(1s 2 1 S)  e - + He + He*(2s2p 1 P o )  

In Summary: Autoionizing resonances are caused by the radiationless decay of the doubly-excited levels of atoms and ions. Because the finite lifetime of these levels is quite long, the corresponding ‘peaks’ are sharp and narrow.

Shape Resonances Electron scattering of Nitrogen e - + N(1s 2 2s 2 2p 3 4 S o )  e - + N(1s 2 2s 2 2p 3 4 S o ) N*(1s 2 2s 2 2p 4 3 P)  

In Summary: Shape resonances are caused by the temporary capture of the incident particle during the collision process. They are generally found on or just above threshold. They are mostly short-lived and hence appear in the cross section as broad peaks.

Feshbach Resonances Photoionization of Nitrogen e - + N(1s 2 2s 2 2p 3 2 D o )  e - + N + N*(1s 2 2s 2 2p 4 2 P)  

Configuration Interaction Wavefunction In LS coupling the expansion of the total wavefunction Ψ is given by

In Summary: The study of atomic collisions can conveniently be divided into two parts: Firstly, it is necessary to obtain wavefunctions which describe the target atomic bound states. Secondly, these wavefunctions must be incorporated into a description of the complete collision problem.

The R-matrix Method

The non-rel SE describing the scattering of an electron by an N- electron atom or ion with nuclear charge Z is defined by Where the (N+1)-electron Hamiltonian H (N+1) is given the atomic units by

Internal Region In the internal region the total collision wavefunction Ψ is represented as a configuration interaction expansion over a set of (N+1)-electron basis functions where

External Region In the external region the boundary radius is chosen so that electron exchange between the scattered electron and the target vanishes. The total collision wavefunction can then be expanded as The solution of the SE in each region is determined independently and the R-matrix then links the solutions on the boundary r=a

The R-matrix The R-matrix, first introduced by Wigner in (1946) in an analysis of nuclear resonance reactions, is defined by where the surface amplitudes ω ik are given by

Observables The collision strength, Ω ij, for each transition and energy can be calculated from the scattering (or S-matrix) using the following formula where g=(2L+1)(2S+1) in LS coupling and (2J+1) in intermediate or jj coupling. The collision strength is related to the cross section σ ij through the relation

The collision strength may be averaged over a Maxwellian distribution of electron velocities to produce a smooth curve of effective collision strengths for the temperature range of interest. Where T e is the electron temperature in K, k B is Boltzmann constant and E f is the final energy of the electron.

The non-rel SE describing the scattering of an electron by an N- electron atom or ion with nuclear charge Z is defined by Where the (N+1)-electron Hamiltonian H (N+1) is given the atomic units by

Relativistic Effects 1 The Breit-Pauli Hamiltonian is suitable for light to medium ions where only first order relativistic effects need to be considered. The complete relativistic Breit- Pauli Hamiltonian is defined as follows:

Relativistic Effects 2 In heavier systems, where it is essential to include all relativistic effects, the Dirac Hamiltonian must be adopted. where c is the speed of light, p i is the momentum and α and β are given by where σ are the usual Pauli spin matrices

R-matrix Methodologies: RMATRX I (LS coupling) BP RMATRX I DARC RMATRX II (LS + transformation) ICFT (LS + transformation) B-spline R-Matrix BSR

Cr II 3d 5 6 S 5/2 – 3d 4 4s 6 D 1/2 Wasson IR, Ramsbottom CA & Scott MP, 2010, a&A, 524,A35

Cr II 3d 5 6 S 5/2 – 3d 4 4s 6 D 1/2,3/2,5/2,7/2,9/2 Wasson IR, Ramsbottom CA & Scott MP, 2010, a&A, 524,A35

Mn V 3d 3 4 F 3/2 – 3d 3 2 D2 5/2 Grieve MFR, Ramsbottom CA, Hibbert A, Ferland G & Keenan FP, in preparation

Mn V 3d 3 4 F 3/2 – 3d 3 2 D2 5/2 Grieve MFR, Ramsbottom CA, Hibbert A, Ferland G & Keenan FP, in preparation

Mg VIII 2s 2 2p 2 P o 1/2 – 2p 3 2 P o 3/2

W XLV BP –present DARC 2007 DARC 2015

W XLV BP –present DARC 2007 DARC 2015

W XLV BP present RMATRX II

W XLV BP present RMATRX II

W XLV BP present RMATRX II

Chianti Database Version 7.1.3

Fe II The Rosetta Stone Element

Model LSπJπ No. ConfigsStatesChannelsStatesChannels No. Target States Max No. Channels Max size (N+1) H matrix Total No. Transitions 100 (LSπ) (Jπ) (Jπ)5076>

Model LSπJπ No. ConfigsStatesChannelsStatesChannels No. Target States Max No. Channels Max size (N+1) H matrix Total No. Transitions 100 (LSπ) (Jπ) (Jπ)5076>

Fe II 3d 6 4s 6 D 9/2 – 3d 6 4s 6 D 7/2 Ramsbottom CA, Hudson CE, Norrington PH & Scott MP, 2007, A&A, 475, 765

Fe II 3d 6 4s 6 D 9/2 – 3d 6 4s 6 D 5/2 Ramsbottom CA, Hudson CE, Norrington PH & Scott MP, 2007, A&A, 475, 765

Cloudy Models

Leighly KM, Halpern JP, Jenkins EB & Casebeer D, 2007, ApJSS, 173, 1

Fe II The Rosetta Stone Element

Latest Calculation from QUB 262 LS or 716 Jπ target states with configs 3d 6 4s, 3d 7, 3d 6 4p, 3d 5 4s 2, 3d 5 4s4p Last target threshold approximately 23eV Breit-Pauli + DARC treatments Target energies shifted to NIST where possible Highly applicable to recent quasar studies

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