Benchmark calculations for electron collisions with ... · Benchmark calculations for electron collisions with complex atoms Oleg Zatsarinny and Klaus Bartschat Drake University,

Benchmark calculations for electron collisions with complex atoms

Oleg Zatsarinny and Klaus BartschatDrake University, Des Moines, Iowa, USA

supported by the NSF under PHY-1520970,

PHY-1212450, Teragrid/XSEDE-090031

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Benchmark calculations for electron collisions with

general B-spline R-matrix package (BSR)

OVERVIEW:

1. General features of BSR:

R−matrix method (close−coupling expansion)

B-splines as universal basis

Non-orthogonal orbitals technique

Continuum pseudo−state approach (RMPS)

2. Examples

Scattering on neutral atoms (Ar, C, N, O, F, Be, Mg)

Scattering on ions (Be-like N3+, O-like Mg4+)

4. Summary

Benchmark results:

• maximum accuracy

• possible source of uncertainties

• convergence

• uncertainty estimations

Benchmark calculations for electron collisions with

general B-spline R-matrix package (BSR)

OVERVIEW:

1. General features of BSR:

R−matrix method (close−coupling expansion)

B-splines as universal basis

Non-orthogonal orbitals technique

Continuum pseudo−state approach (RMPS)

2. Examples

Scattering on neutral atoms (Ar, C, N, O, F, Be, Mg)

Scattering on ions (Be-like N3+, O-like Mg4+)

4. Summary

The B-Spline R-Matrix (BSR) Method[O. Zatsarinny, Comp. Phys. Commun. 174, 273 (2006)]

• The method is based on the non-perturbative close-coupling expansion.

• The close-coupling equations are solved using the R-matrix method.

• Atomic-structure calculations − frozen-core approximation

Distinctive feature:

Allows for non-orthogonal orbital sets to represent both bound andcontinuum radial functions

• independent generation of target states – much more accurate target

representation (term-dependence, relaxation effects, correlation)

• no artificial orthogonality constraints for continuum orbitals –

more consistent treatment of N-electron target and (N+1)-electron

collision system −> no pseudo−resonances

• Standard method of treating low-energy scattering

• Based upon an “exact” expansion of the total scattering wavefunction over target states

• Simultaneous results for transitions between all states in the expansion

• Problems:

• close−coupling expansion must be cut off• accuracy (expansions) of target states are limited

),...,()(1

)ˆ,,...,(),...,( 11,111

Nj

LS

jiENi

LS

iN

LS rrrFr

rrrArrE

Main source of uncertainties







• Basic Idea: indirect calculations – inner ( r < a) and outer regions ( r > a).

• Complete set of inner-region solutions is found from diagonalization of total

Hamiltonian modified with Bloch operator

• Scattering parameters can then obtained from matching with solutions in external

region − allows us obtain cross sections at many energy points rather cheaply












• Computer Codes:

• RMATRX–I : Berrington et al (1995)

• PRMAT - parallelized version of RMARX-I

• Badnell's Rmax complex - http: //amdpp.phys.strath.ac.uk/,

with possibility for radiative damping

• DARC – relativistic version, http://web.am.qub.ac.uk/DARC/

• Enormous number of calculations

Principal ingredient: a single set of orthogonal one-electron orbitals

• < Pnℓ | Pn'ℓ > = 0 → difficulties to achieve accurate target representation

for different states

• < Pnℓ | Fkℓ > = 0 → large (N+1)-electron expansions needed for completeness

(may lead to appearance of pseudo-resonances)

),...,()(1

)ˆ,,...,(),...,( 11,111

Nj

LS

jiENi

LS

iN

LS rrrFr

rrrArrE

http://web.am.qub.ac.uk/DARC/












It allows us to considerably (almost completely) reduce the uncertainties

due to the accuracy of the target states









• no artificial orthogonality constraints for continuum orbitals – more

consistent treatment of N-electron target and (N+1)-electron collision

system −> (no pseudo−resonances, improved convergence)

Additional source of uncertainties

),...,()(1

)ˆ,,...,(),...,( 11,111

Nj

LS

jiENi

LS

iN

LS rrrFr

rrrArrE

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

n+k

n+1nk+1k

1

Bi(r): k = 8, n = 17, h = 0.5

• excellent numerical properties − machine accuracy with Gaussian quadratures.

• flexibility in the choice of radial grid.

• effective completeness of B-spline basis − no Buttle correction required;

finite representation of whole Rydberg series or continuum spectra.

• avoid finite-difference algorithms − established Linear Algebra packages can be used.

i

ii rBcr )()( ScHc EEH

Why B−splines ?

• first R-matrix calculation with B-splines: e-H scattering (van der Hart 1997)

However − scattering calculations require relatively big basis: n ~ 50−100

BSR – general B-spline R-matrix package

1. First implementation: Li photoionization (2000)

2. First presentation: ICPEAC XXX, Rosario, Argentina (2005)

3. First version published: Comp.Phys.Comm. (2006)

4. Fully−relativistic version: e−Cs scattering (2008)

5. RMPS extension and e−He,Ne scattering (2011−2012)

ionization:

6. Topical review: J.Phys.B 46, 112001 (2013)

Calculations:

1. Electron scattering from neutrals (including inner−core excitations)

He, Be,C, N, O, F, S, Cl, Ne, Na, Mg, Al, Si, Ar, K, Ca, Zn,

Cu, I, Kr, Rb, Xe, Cs, Kr, Xe, Cs, Au, Hg, Pb

2. Electron−ion collisions (rate coefficients)

S II, K II, Fe II, Fe VII, Fe VIII, Fe IX

3. Photodetachment and photoionization

He−, Li−, B−, O−, Ca−, Li, K, Zn

4. Atomic structure: energies, oscillator strengths, polarizabilities

C, Ne, S, SII, F, CL, Ar, Kr, Xe

8.4 8.8 9.2 9.6 10.00.0

0.4

0.8

1.2

17 18 190.00

0.05

0.10

0.15

11.5 12.0 12.5 13.0 13.50.0

0.1

0.2

0.3

0.4

9.5 10.0 10.5 11.0 11.5 12.0 12.50.0

0.2

0.4

0.6

BPRM-41

BSR-31

Xe

Electron Energy (eV)

BPRM-31

BSR-31

Ne

C

ross S

ection

(a

2 o )

BPRM-41

BSR-31`

Ar

BPRM-31

BPRM-51

BSR-31

Kr

Cro

ss S

ection

(a

2 o )


Metastable production in Electron Collisions with noble gasesBSR calculations

BSR with pseudo−states

V(r)

r

continuum spectrum discrete spectrum

V(r)

rr = a

We use the box−based approach:

Both physical and pseudo−states are found by direct solving the close−coupling

(frozen−cores) equations for N−electron atomic wavefunctions with zero boundary

conditions.

?

),...,()(1

)ˆ,,...,(),...,( 111,

1

1111

Nj

LS

jNiE

N

NNi

LS

iN

LS rrrFr

rrrArrE

How many pseudo−states do we need in this approach to simulate

the target continuum?

Electron-impact excitation of neon at intermediate energies(PHYSICAL REVIEW A 86, 022717, 2012)

J

ji

LSij

ij

J

ji

LSij

ij

LSslrBpsaA

LSslrBpsaAJnlps

},|)()S;22({

},|)()P;22({),22(

26

,

252

,

52

Target states:

Total number of states − 457

Bound states − 87

Continuum pseudostates − 370

(l = 0 − 3)

Continuum energy coverage− 40eV

Number of channels − 2260

R−matrix radius − 30ao

Number of B−splines − 70

Hamiltonian matrix − 150000

Number of processors used − up to 1000

Resources used − 200000 PU (1 hour x 1 processor)

(for 50 partial waves)

Do we really need such big computational efforts?

Effect of Channel Coupling to Discrete and Continuum spectrum in Ar

10 1000

4

8

12

10 1000

10

20

30

10 20 30 40 500

2

4

6

8

10

10 20 30 40 500.0

0.4

0.8

1.2

1.6

2.0

BSR-31

BSR-500


50205020 300

4s[3/2]1

Cro

ss S

ection

(10-1

8 c

m-2

)

Kato et al (2011)

Khakoo et al (2004)

Filipovic et al (2000)

Chutjian & Cartwright (1981)

4s'[1/2]1


Cro

ss S

ectio

n (

10

-18 c

m-2

)

4s[3/2]2

300

4s'[1/2]0

How to estimate uncertainties here?

10 1000

4

8

12

10 1000

10

20

30

10 20 30 40 500

2

4

6

8

10

10 20 30 40 500.0

0.4

0.8

1.2

1.6

2.0


50205020 300

4s[3/2]1

Cro

ss S

ectio

n (

10

-18

cm

-2)

4s'[1/2]1


BSR-5

BSR-31

BSR-78

BSR-160

BSR-500

BSR-600

Cro

ss S

ectio

n (

10

-18 c

m-2

)

4s[3/2]2

300

4s'[1/2]0

Total Cross Section for Electron Impact Excitation of ArgonConvergence study

Final uncertainty within 5−10 percent, just based on the convergence study.

• Ne(2p6) , Ar(3p6) - extremely big influence of target continuum.

• What about other atoms with open p-shells:

B(2p), C(2p2), N(2p3), O(2p4), F(2p5) - ?

20 40 60 80 1000

1

2

3

4

5

BSR-690 - Gedeon et al. (2014)

BSR-39 - the same target states

RM-11 - Boliyan & Bhatia (1994)

Cro

ss S

ectio

n (

10

-16 c

m2) F (2p5 2Po - 3s' 2D)


20 40 60 80 1000

2

4

6

8

Vaughan & Doering (1987)

Kanik et al. (2001)

BSR-26 - Tayal & Zatsarinny (2002)

RMPS-52 - Tayal (2005)


O (2p4 3P - 2p33s' 3Do)


Cro

ss S

ectio

n

(10

-18cm

2)

20 40 60 80 1000.0

0.2

0.4

0.6

0.8

expt. - Doering & Goembel (1992)


BSR-39 - polarized pseudostates

BSR-690 - Wang et al. (20015)

BSR-61 - the same target states


Cro

ss S

ectio

n (

10

-16 c

m2)

N (2p3 4So - 2p23s 4P)

15 30 45 600

1

2C

ross S

ectio

n (

10

-16 c

m2)

RM-28 Dunseath et al. (1997)

BSR-28 Zatsarinny et al. (2005)

BSR-696 Wang et al. (2013)

BSR-25 (the same target states)

C (2p2 3P - 2p3s 3Po)


• Inclusion of the target continuum – mandatory condition for the neutral atoms, at least.

• But first check target − then CC convergence !

• Uncertainties – the upper limit – may be estimated as difference between CC models with the

same target w.f.

Motivation:

1. Big interest in plasma applications (JET project, ITER).

2. No experimental data.

3. What accuracy can be achieved for this ‘simple’ system?

4. Complete set of scattering data (first 21 levels).

5. Joint work of CCC and BSR teams.

0 20 40 60 80 1000

4

8

12

RMPS-280

CCC-409

BSR-660

Cro

ss S

ection

(1016 c

m2)

(2s2)1S - (2s2p)1Po

0 20 40 60 80 1000

2

4

6

Cro

ss S

ection

(1016 c

m2)

Electron Energy (eV) Electron Energy (eV)

(2s2p)3Po - (2p2)3P

0.0 0.5 1.0 1.5 2.0 2.50

100

200

300

400 BSR-01

BSR-02

BSR-03

BSR-29

BSR-660

CCC-409

model potential

Ela

stic C

ross S

ection

(1

0-1

6 c

m2)


0 20 40 60 80 1000

1

2

3

CCC

BSR

RMPS (Ballance et al. 2003)

TDCC (Colgan et al. 2003)

Be+(2p)Io

niz

atio

n C

ross S

ectio

n (

10

-16 c

m2)


e - Be(2s2)

• A complete and consistent set of scattering data for Be, with a high level of

confidence in a small uncertainty (5-10 %).

• A good example of team work (CCC + BSR) to get accurate results.

• New similar calculations for e-Mg are now in progress

Electron-impact excitation of N3+ (2s2)

(2snl, 2pnl, n ≤ 8)

RM-238 (ICFT - Intermediate Coupling Frame Transformation)

Fernandez-Menchero et al. 2014, Astron. Astroph. 566 A104)

DARC-238 (Dirac Atomic R-matrix Code)

Aggarwal et al. 2016 Mon. Not. R. Astr. Soc. 461 3997)

BSR-238 with improved structure description

(submitted to J.Phys.B)

n=2 transitions

n=2 -> higher levels

All transitions

Electron-impact excitation of Mg4+ (2p4)

Big difference between existing datasets:

• RM-37 (Hudson et al., 2009, A \& A 494, 729)

• BSR-86 (Tayal & Sossah, 2015, A&A, 574, 87) { 2s22p4, 2s2p5, 2s22p33l }

• DARC-86 (Aggarwal & Keenan, 2016, Can. J. Phys., online)

statement: DARC calculations are much more accurate;

BSR results are completely wrong (possibly because

BSR code is not able to avoid pseudo resonances)



• RM-37 (Hudson et al., 2009, A \& A 494, 729)

• BSR-86(TS) (Tayal & Sossah, 2015, A&A, 574, 87) { 2s22p4, 2s2p5, 2s22p33l }

• DARC-86 (Aggarwal & Keenan, 2016, Can. J. Phys., online)

statement: DARC calculations are much more accurate;

BSR results are completely wrong (possibly because

BSR code is not able to avoid pseudo resonances )

Really ???

New calculations:

• BSR-86 with improved structure description

• BSR-316 with additional 2s22p34l and 2s2p53l states

10-5 10-4 10-3 10-2 10-1 10010-5

10-4

10-3

10-2

10-1

100

GRASP (DARC)

BSR

OT

HE

RS

MCHF

f-values

0 10 20 30 40 50 60 70 80 900.0

0.5

1.0

1.5

2.0

GRASP

BSR

Ra

tio

to

MC

HF

Level Index

lifetimes

f-values and lifetimes of Mg4+ (2p4)

MCHF - Tachiev & Froese Fischer (2004)

GRASP - Aggarwal & Keenan (2016), DARC-86

Effective collision strengths of Mg4+ (2p4)

BSR-86 vs. BSR-86(TS), DARC-86, RM-37

BSR(TS)

DARC

RM

BSR


7.2 7.4 7.6 7.8 8.00

1x10-4

2x10-4

3x10-4

4x10-4

7.2 7.4 7.6 7.8 8.00

1x10-4

2x10-4

3x10-4

4x10-4

Colli

sio

n S

trength

(10 - 30) (2p6)1S0 - (2s22p33p)3D3

BSR-86

BSR-86 (TS)

Colli

sio

n S

trength

Electron Energy (Ry)

DARC

Effective collision strengths of Mg4+ (2p4)

CC – convergence: BSR-86 vs. BSR-316

The 2s22p4 → 2s22p33l are mainly converged.

Difference mainly for weak two-electron transitions 2s2p5, 2p6 → 2s22p33l

or for transitions between close-lying levels



• RM-37 (Hudson et al., 2009, A \& A 494, 729)

• BSR-86 (Tayal & Sossah, 2015, A&A, 574, 87) { 2s22p4, 2s2p5, 2s22p33l }

• DARC-86 (Aggarwal & Keenan, 2016, Can. J. Phys., 11, 111)

statement: DARC calculations are much more accurate and the best;

BSR results are completely wrong

BSR code is not able to avoid pseudo resonances

New calculations

• BSR-86 with improved structure description

• BSR-316 with additional 2s22p34l and 2s2p53l states

Our conclusions:

• Good agreement our oscillator strengths with the extensive MCHF calculations indicate

that our target description is more accurate than used before.

• We essentially confirm the previous BSR calculations by Tayal & Sossah (2015)

(except #10 and #86 levels – computational bugs)

• Big disagreement with DARC calculations is mainly due to the target description

(and possible pseudo resonances).

• RM-37 calculations of Hudson et al. (2009) for some transitions are clearly wrong

due to evident appearance of pseudo resonances.

• Comparison of BSR-86 and BSR-316 indicate convergence for the lower levels.

For complex targets, the BSR method with non-orthogonal orbitals has advantages:

• highly accurate target description

• more consistent description of (N+1)-electron scattering system

• considerable improvement for low-energy region and near-threshold resonance phenomena

• improvement for intermediate energies with RMPS approach (MPI version)

• BSR code is able to provide benchmark results for uncertainty estimation

► considerable reduction (elimination) of target description uncertainty

► absence of pseudo resonances

► systematic check of convergence (including target continuum)

Summary

requires large-scale computations using supercomputers (i.e., extensive and expensive)










Summary

requires large-scale computations using supercomputers (i.e., extensive and expensive)

My main point: The importance of accurate target descriptions is

highly underestimated in most scattering calculations.

This leads to lower accuracy and larger uncertainties.

THANK YOU For Your Attention!

10-5 10-4 10-3 10-2 10-1 100 101 10210-5

10-4

10-3

10-2

10-1

100

101

102

DW (Bhatia et al. 2006)

RM-37 (Hudson et al. 2009)

BSR-316

(

OT

HE

RS

)

(BSR-86)

E = 20 Ry


Background collision strengths: BSR-86 vs. DW, RM, BSR-316

0 1 2 3 4 5 6 7 80

1

2

3

4

5

Colli

sio

n s

trength


(1 - 2) (2s22p4)3P2 - (2s22p4)3P1

4 5 6 7 8

2

3

4

5

Colli

sio

n S

treng

th (

)


(4 - 9) (2s22p4)1D2 - (2s2p5)1P1


• We used 71000 energy points to resolve resonance structure

Description of target states

(GRASP_2K – DBSR_CI, DBSR_HF)

target1.c, target1.bsw


……………………….

knot.dat dbsr_par target_jj

DBSR_PREP

DBSR_CONF

DBSR_BREIT

DBSR_MAT

DBSR_HD

DBSR_MULT

DBSR_DMAT

h.nnn, bound.nnn

jnt_bnk.nnn

cfg.001

cfg.002

…….

cfg.nnn

target_jj..bsw

dbsr_mat.nnn

H.DAT

FARM,STGF, etc. d.nnn BSR_PHOT zf_res

rsol.nnn

mult_bnk.nnn

Energies

Oscillator strengths Cross sections

MPI

DBSR

Description of target states

(HF, MCHF, CI_BNK. BSR_HF)



……………………….

knot.dat bsr_par target

BSR_PREP

BSR_CONF

BSR_BREIT

BSR_MAT

BSR_HD

BSR_MULT

BSR_DMAT

h.nnn, bound.nnn

jnt_bnk.nnn

cfg.001

cfg.002

…….

cfg.nnn

target.bsw

dbsr_mat.nnn

H.DAT

FARM, STGF, etc. d.nnn BSR_PHOT zf_res

rsol.nnn

mult_bnk.nnn

Energies

Oscillator strengths Electron-impact

cross sections

MPI

Photoionization

BSR

[O. Zatsarinny, Comp. Phys. Commun. 174, 273 (2006)]

Many existing databases for neutrals should be re−evaluated

Comparison of different codes and models is crucial.

• Repeating calculations are encouraged.

• Non−critical using the existing codes − expert proof is still important.

Each atom − unique!

(the automatic calculations for whole iso-electronic or iso−nuclear series can be deceptive)

Comparison of published results or different databases is very time consuming:

(It is desirable to encourage development of new software for extracting and

comparison of data in different databases and sharing the results)

For more confident results - comprehensive comparison with included ALL transitions,

not just the selected ones.

Uncertainty’s questions

My main point: The importance of accurate target descriptions is

very much underestimated in most scattering calculations. This

leads to lower accuracy and larger uncertainties.

Electron-impact excitation of Mg (3s2)

(in progress)

• Numerous measurements and calculations but for selected transitions or energies

(DCS)

• Modeling of non-equilibrium plasma requires complete set of data

(Mauas et al. 1988 - collection from different sources with unclear uncertainties)

• BSR-37 (Zatsarinny et al. 2009) - most complete set of data for all transitions

between first 37 levels (n ≤ 8) except triplet-triplet transitions

• Merle et al. (2015) - rate coefficients based on BSR-37

• Osorio et al. (2015) – new RMPS calculations for transitions between first 10 levels

(noticeable differences with existing results)

• 2016, new project – parallel CCC and BSR-712 (RMPS) calculations to get

complete set with uncertainty estimations

0 20 40 60 80 1000

4

8

12

16

20

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

0 20 40 60 80 1000.0

0.5

1.0

1.5

2.0

0 20 40 60 80 1000.0

0.5

1.0

1.5

2.0

BSR-712

BSR-37 - Zatsarinny et al. (2009)

CCC - Bray (2016)

RMPS - Osorio et al. (2015)

3s2 1S - 3s3p 1P 3s2 1S - 3s4s 1S


3s2 1S - 3s4p 1P

Cro

ss S

ectio

n (

10

16cm

2)


3s2 1S - 3s3d 1D


0 20 40 60 80 1000

1

2

3

4

5

6

7

Boivin & Strivastava (1998)

Freund et al. (1990)

BSR

CCC (Bray et al. 2015)

RMPS (Ludlow et al. 2009)

TDCC (Ludlow et al. 2009)

Cro

ss S

ectio

n (

10

-16 c

m2)


e + Mg - Mg+ + 2e

Electron-impact ionization of Mg (3s2)


• The agreement between the CCC and BSR712 calculations differ by on average

only 6%, with a scatter of 27% for ALL transitions between first 25 levels.

• Another example of team work to get accurate results from independent

calculations

T = 5000 K










Summary

Goal: can we finally get accurate cross sections for complex atoms?

my main point: the important of accurate target description is very

underestimated in the most scattering calculations and leads to

lower accuracy and big uncertainties



• Fernandez-Menchero et al. 2014, Astron. Astroph. 566 A104


• Aggarwal et al. 2016 Mon. Not. R. Astr. Soc. 461 3997)

DARC -238 (Dirac Atomic R-matrix Code)

statement: DARC calculations are much more accurate and

the ICFT results might even be wrong

• BSR-238 with improved structure description (2snl, 2pnl, n ≤ 8)

N3+ - comparison of oscillator strengths






• Fernandez-Menchero et al. 2014, Astron. Astroph. 566 A104


• Aggarwal et al. 2016 Mon. Not. R. Astr. Soc. 461 3997)

DARC -87 (Dirac Atomic R-matrix Code)

statement: DARC calculations are much more accurate and

the ICFT results might even be wrong

Our conclusions:• The differences in the final results for the collision strengths are mainly due to

differences in the structure description, specifically the inclusion of correlation

effects, rather than the treatment of relativistic effects.

• Hence there is no indication that one approach is superior to another,

until the convergence of both the target configuration and the close-coupling

expansions have been fully established.

• Due to the superior target structure generated, we believe the BSR results are the

currently best for electron collisions with N+3 .

• The differences between the BSR and the DARC / ICFT results may in fact serve as an

uncertainty estimate for the available excitation rates.

Goal: can we finally get accurate cross sections for complex atoms?

Documents

Benchmark calculations for electron collisions with ... · Benchmark calculations for electron collisions with complex atoms Oleg Zatsarinny and Klaus Bartschat Drake University,