Download Fe I electron excitation data for modelling of cool star spectra

22nd International Symposium on Plasma Chemistry
July 5-10, 2015; Antwerp, Belgium
Fe I electron excitation data for modelling of cool star spectra
J.R. Hamilton1,2, J. Tennyson1,2, P. Barklem3 and O. Zatsarinny4
1
Quantemol Ltd., University College London, London WC1E 6BT, U.K.
Department of Physics and Astronomy, University College London, London WC1E 6BT, U.K.
3
Theoterical Astrophysics, Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala,
Sweden
4
Department of Physics and Astronomy, Drake University, US-50311 Des Moines, Iowa, U.S.A.
2
Abstract: The B-Spline R-matrix (BSR) method (Zatsarinny, 2013) is used to calculate the
electronic excitation collision cross section of electrons with atomic Fe I. The BSR method
is used in conjunction with a close-coupling expansion and non-orthogonal bound and
continuum orbitals (Zatsarinny, 2006). Fe I requires a semi-relativistic treatment which is
provided using the Breit-Pauli approach. Results for excitation cross-sections will be
reported at the conference.
1. Introduction
Barklem and co-workers have performed extensive
modeling of the spectra of cool stars where the atoms
depart from local thermodynamic equilibrium (LTE) [1].
These models aim to study stellar properties such as
chemical abundances, and processes and to understand the
chemical and dynamic evolution of the galaxy. Non-LTE
modelling requires large amounts of collisional data for
the atomic species of interest.
So far much of this work has been limited to light atoms
such as Li [2], O [3], Na [4], and Mg [5, 6], as the
calculation of collision data for heavier atoms presents is
more complexities and often gives less accurate results.
Additionally heavier atoms require a relativistic treatment
which is not incorporated into many atomic collision
methods.
This work looks to produce a detailed set of collision
data of the heavy atom Fe I, using the B-spline R-matrix
(BSR) method with relativistic corrections [7].
2. Scientific Approach
After reviewing the Iron Project publications we
conclude very little is currently published on electron Fe I
collisions. Pelan et al. [8] studied the electronic excitation
to specific low-lying states of the Fe I atom using the
Belfast atomic R-matrix codes [9].
The first step in our study is to recalculate the specific
transitions considered by Pelan et al. [8] for
benchmarking. Then second step is to calculate an
electron excitation cross section for an extended set of
states: we consider the first 20 levels of Fe I. The final
step is to repeat the calculation using a much larger set of
states with the physical target augmented by pseudo-states
which represent the hundreds of higher states in Fe I.
This is necessary to get converged the results.
3. The R-Matrix Method
The R-matrix method is one of many methods for
solving the close-coupling equations. This method
divides the physical space of the problem into two
P-I-2-28
regions: an inner region containing the target atom and an
outer region containing the incident electron. The met hod
solves the Schrödinger equation in the inner region
independent of the energy of the interacting electron and
then uses this solution to solve the much simpler outer
region Schrödinger equation, which is energy dependent
the Belfast atomic R-matrix codes: FARM and STPF [10]
are used to calculate the R-matrix from the surface
amplitudes generated form the energy independent inner
region calculation. The R-matrix is then used to solve the
Schrödinger equation in the outer region for over a range
of incident electron energies.
The use of both codes will confirm numerical stability
and convergent results.
4. Multi Configuration Hartree Fock Generation of
States
The target states of the Fe I atom are calculated using in
Multi Configuration Hartree Fock (MCHF) program of
Fischer [11]. The MCHF program is used to generate the
expansion coefficients of electron configurations
including correlative effects of all states and pseudo-states
to be considered.
5. Pseudo-State Calculations
For large state calculations the BSR method employs
the use of pseudo-states to model the non-excited and
excited continuum target states. This approach involves
treating the R-matrix inner region as a potential well
where within the well the continuum states become
discrete (pseudo) states and outside of the R-matrix
boundary the continuum wave functions equal zero.
Pseudo-states are generated by treating the Fe I
continuum problem as coupled Fe II + e. The Fe II + e
scattering problem is solved and from this solution states
and pseudo states are extracted to be used as target states
for the Fe I + e calculation.
1
6. The use of B-Splines
Splines are bell-shaped piece-wise polynomial
functions of order, defined by a given set of points in
some finite radial interval which can be taken to be the
R-matrix radius. B-splines form an effectively complete
basis set which avoids linear dependence problems
between the bound and continuum orbitals.
The use of B-splines as the R-matrix basis set was first
outlined by van der Hart et al. [12], who applied the
method to electron H I scattering.
7. Non-Orthodonal Orbitals
In the BSR method non-orthogonal orbitals are be used
to represent both the bound and continuum one-electron
target states. The use of non-orthogonal orbitals means
the bound and continuum states can be optimised in
separate calculations. This gives a high level of accuracy
can be achieved with compact configuration expansions
of bound states. The use of non-orthogonal continuum
orbitals can drastically reduce the pseudo-resonance
problem in continuum states [13].
The use of non-orthogonal orbitals is not widely used in
R-matrix calculations as the most time-consuming part of
atomic structure calculations is connected with the
angular integration of the orbitals. However efficient use
of orthogonal orbitals with some restricted nonorthogonality makes it possible to automate to a large
extent this part of the inner-region problem.
8. The Breit-Pauli Approach
The Breit-Pauli approach can be considered a first order
perturbative method providing a first order correction to
the non-relativistic Hamiltonian operator.
This
perturbation corrects the Hamiltonian to include
relativistic shift and relativistic splitting of the Fe I target
states including also the fine structure interactions
between the spin and orbital angular momenta of the
target states electrons. The Breit-Pauli approach has been
shown to provide accurate results for atoms with Z < 30.
3d64snl). The states Fe I (3d8 and 3d24p2) can also be
considered as perturbations of these continuum states or
from a close coupling expansion.
Final results will be presented at the conference.
10. Acknowledgments
JRH thanks STFC for a studentship and Quantemol Ltd.
for support. This collaboration was supported by the
CORINF ITN network.
11. References
[1] P.S. Barklem, et al. Astron Astrophys., 530, A94
(2011)
[2] K. Lind, M. Asplund and P.S. Barklem. Astron.
Astrophys., 503, 541-544 (2009)
[3] D. Fabbian, et al. Astron. Astrophys., 500, 12211238 (2009)
[4] K. Lind, et al. Astron. Astrophys., 528, 103 (2011)
[5] P.S. Barklem, et al. Astron. Astrophys., 541, 80
(2012)
[6] L. Mashonkina. Astron. Astrophys., 550, 28 (2013)
[7] O. Zatsarinny and K. Bartschat. J. Phys. B: At. Mol.
Opt. Phys., 46, 112001 (2013)
[8] J. Pelan and K.A. Berrington. Astron. Astrophys.,
Suppl. Ser., 122, 177 (1997)
[9] K.A. Berrington, W.B. Eissner and P.H. Norrington.
Comput. Phys. Commun., 92, 290 (1995)
[10] V.M. Burke and C.J. Noble.
Comput. Phys.
Commun., 85, 471 (1995)
[11] C.F. Fischer and O. Zatsarinny. Comput. Phys.
Commun., 180, 2041 (2009)
[12] H.W. van der Hart. J. Phys. B., 30, 452 (1997)
[13] O. Zatsarinny. Comput. Phys. Commun., 174,
273356 (2006)
9. Fe I
Fe I has ground state configuration 3d64s2 (NIST). Only
excitation of the 4s electrons from this ground state will
be considered. Because the Fe I bound state spectra is so
rich the pseudo-state calculation will not include
continuum pseudo-states. The pseudo-state method will
be used to calculate bound states only:
e + Fe I (3d64s2) →e + Fe I (3d64snl)
Therefore an electron Fe II scattering calculation of:
e + Fe II (3d7, 3d64s and 3d64p)
will be carried out to calculate the target state of Fe I
(3d74s) and correlated state (3d64s4p).
The electron Fe II scattering calculation can also
calculate the continuum states Fe I (3d64pnl, 3d64dnl and
2
P-I-2-28