Talk given by Klaus Bartschat (Drake University) at DAMOP12, Anheim, California, USA (06 June 2012)

1. BENCHMARK CALCULATIONS OF ATOMIC DATA
FOR MODELING APPLICATIONS
Klaus Bartschat, Drake University
DAMOP 2012 (GEC Session)
June 4-8, 2012
Special Thanks to:
Igor Bray, Dmitry Fursa, Arati Dasgupta,
Gustavo Garcia, Don Madison, Al Stauffer, ...
Michael Allan, Steve Buckman, Michael Brunger, Hartmut Hotop, Morty Khakoo,
and many other experimentalists who are pushing us hard ...
Andreas Dinklage, Dirk Dodt, John Giuliani, George Petrov, Leanne Pitchford, ...
Phil Burke, Charlotte Froese Fischer, Oleg Zatsarinny
National Science Foundation: PHY-0903818, PHY-1068140, TeraGrid/XSEDE

2. BENCHMARK CALCULATIONS OF ATOMIC DATA
FOR MODELING APPLICATIONS
OVERVIEW:
I. Introduction: Production and Assessment of Atomic Collision Data
II. Numerical Methods
• Special-Purpose Methods
• Distorted-Wave Methods
• The Close-Coupling Method: Recent Developments
CCC, RMPS, IERM, TDCC, BSR
III. Selected Results
• Energy Levels
• Oscillator Strengths
• Cross Sections for Electron-Impact Excitation
IV. GEC Application: What We Can Learn from Modeling a Ne Discharge
V. A Closer Look at Ionization
• Basic Idea
• Example Results
• Solution of the Excitation Mystery in e-Ne Collisions
VI. Conclusions and Outlook

3. Production and Assessment of Atomic Data
• Data for electron collisions with atoms and ions are needed for modeling processes in
• laboratory plasmas, such as discharges in lighting and lasers
• astrophysical plasmas
• planetary atmospheres
• The data are obtained through
• experiments
• valuable but expensive ($$$) benchmarks (often differential in energy, angle, spin, ...)
• often problematic when absolute (cross section) normalization is required
• calculations (Opacity Project, Iron Project, ...)
• relatively cheap
• almost any transition of interest is possible
• often restricted to particular energy ranges:
• high (→ Born-type methods)
• low (→ close-coupling-type methods)
• cross sections may peak at “intermediate energies” (→ ???)
• good (or bad ?) guesses
• Sometimes the results are (obviously) wrong or (more often) inconsistent !
Basic Question: WHO IS RIGHT ? (And WHY ???)
For complete data sets, theory is often the "only game in town"!

4. Choice of Numerical Approaches
• Which one is right for YOU?
• Perturbative (Born-type) or Non-Perturbative (close-coupling, time-
dependent, ...)?
• Semi-empirical or fully ab initio?
• How much input from experiment?
• Do you trust that input?
• Predictive power? (input ↔ output)
• The answer depends on many aspects, such as:
• How many transitions do you need? (elastic, momentum transfer, excitation,
ionization, ... how much lumping?)
• How complex is the target (H, He, Ar, W, H2 , H2 O, radical, DNA, ....)?
• Do the calculation yourself or beg/pay somebody to do it for you?
• What accuracy can you live with?
• Are you interested in numbers or “correct” numbers?
• Which numbers do really matter?

5. Who is Doing What?
The list is NOT Complete or Arranged – concentrates on “GEC folks”
• “special purpose” elastic/total scattering: Stauffer, McEachran, Garcia, ...
• inelastic (excitation and ionization): perturbative
• Madison, Stauffer, McEachran, Dasgupta, Kim (NIST), ...
• inelastic (excitation and ionization): non-perturbative
• Fursa, Bray, Stelbovics, ... (CCC)
• Burke, Badnell, Pindzola, Ballance, Griffin, ... (“Belfast” R-matrix)
• Zatsarinny, Bartschat (B-spline R-matrix)
• Colgan, Pindzola, ... (TDCC)
• Molecules: Orel, McCurdy, Rescigno, Tennyson, McKoy, Winstead,
Madison, Kim (NIST), ...
These names are all I am going to say about molecules ...

6. Classification of Numerical Approaches
• Special Purpose (elastic/total): OMP (pot. scatt.); Polarized Orbital
• Born-type methods
• PWBA, DWBA, FOMBT, PWBA2, DWBA2, ...
• fast, easy to implement, flexible target description, test physical assumptions
• two states at a time, no channel coupling, problems for low energies and optically
forbidden transitions, results depend on the choice of potentials, unitarization
• (Time-Independent) Close-coupling-type methods
• CCn, CCO, CCC, RMn, IERM, RMPS, DARC, BSR, ...
• Standard method of treating low-energy scattering; based upon the expansion
1
ΨLSπ (r1 , . . . , rN +1 ) = A
E ΦLSπ (r1 , . . . , rN , ˆ)
i r F (r)
r E,i
i
• simultaneous results for transitions between all states in the expansion;
sophisticated, publicly available codes exist; results are internally consistent
→
• expansion must be cut off (→ CCC, RMPS, IERM)
• usually, a single set of mutually orthogonal one-electron orbitals is used
→
for all states in the expansion (→ BSR with non-orthogonal orbitals)
• Time-dependent and other direct methods
• TDCC, ECS
• solve the Schr¨dinger equation directly on a grid
o
• very expensive, only possible for (quasi) one- and two-electron systems.

7. Inclusion of Target Continuum (Ionization)
• imaginary absorption potential (OMP)
• final continuum state in DWBA
• directly on the grid and projection to continuum states (TDCC, ECS)
• add square-integrable pseudo-states to the CC expansion (CCC, RMPS, ...)
Inclusion of Relativistic Effects
• Re-coupling of non-relativistic results (problematic near threshold)
• Perturbative (Breit-Pauli) approach; matrix elements calculated between non-
relativistic wavefunctions
• Dirac-based approach

8. Optical Model Potential (Blanco, Garcia) – a "Special Purpose" Approach
Numerical Methods: OMP for Atoms
• For electron-atom scattering, we solve the partial-wave equation
d2 ℓ(ℓ + 1)
− − 2Vmp (k, r) uℓ (k, r) = k 2 uℓ (k, r).
dr 2 r2
• The local model potential is taken as
Vmp (k, r) = Vstatic (r) + Vexchange (k, r) + Vpolarization (r) + iVabsorption (k, r)
with
• Vexchange (k, r) from Riley and Truhlar (J. Chem. Phys. 63 (1975) 2182);
• Vpolarization (r) from Zhang et al. (J. Phys. B 25 (1992) 1893);
• Vabsorption (k, r) from Staszewska et al. (Phys. Rev. A 28 (1983) 2740).
• Due to the imaginary absorption potential, the OMP method
• yields a complex phase shift δℓ = λℓ + iµℓ
• allows for the calculation of ICS and DCS for
• elastic scattering
• inelastic scattering (all states together) It's great if this
• the sum (total) of the two processes
is all you want!

9. Comparison with "ab initio" Close-Coupling
2 0 0
e + I (5 p 5 , J = 3 /2 )
PRA 83 (2011) 042702
1 5 0
C r o s s s e c t i o n ( a 2o )
1 0 0
B P R M -C C 2
D B S R -C C 2
5 0
D B S R -C C 2 + p o l
D A R C -C C 1 1 (W u & Y u a n )
O M P
0
0 1 2 3 4 5 6 7 8
E le c tro n e n e rg y (e V )

10. Polarized Orbital – an "Ab Initio Special Purpose" Approach

11. Extension to account for inelastic effects:
J. Phys. B 42 (2009) 075202

12. BEf-scaling; Plane-Wave Born with Experimental Optical
Oscillator Strength and Empirical Energy Shift
works well, but is limited to optically allowed transitions
Similar idea works even better for ionization of complex targets :=)

13. Semi-Relativistic DWBA
polarization and absorption potentials
may also be included

14. Ar 3p54s –> 3p54p: DWBA vs. R-matrix
unitarization problem!
(can be fixed; e.g., Dasgupta's NRL code)
Theoretical results depend on
wavefunctions and potentials

15. Relativistic DWBA; Semi-Relativistic DWBA; Experiment
Ar 3p6 –> 3p53d, PRA 81 (2010) 052707
Key Message from this (unreadable) Graph:
Sometimes BIG Differences between Theories
and HUGE Experimental Error Bars!

16. Time-Independent Close-Coupling
The (Time-Independent) Close-Coupling Expansion
• Standard method of treating low-energy scattering
• Based upon an expansion of the total wavefunction as
HΨ=EΨ
1
ΨLSπ (r1 , . . . , rN +1 ) = A
E ΦLSπ (r1 , . . . , rN , ˆ)
i r F (r)
r E,i
i
• Target states Φi diagonalize the N -electron target Hamiltonian according to
N
Φi | HT | Φi = Ei δi i
good start – remember
your QM course?
• The unknown radial wavefunctions FE,i are determined from the solution of a system of coupled integro-
differential equations given by
d2 i( i + 1)
− + k 2 FE,i (r) = 2 Vij (r) FE,j (r) + 2 Wij FE,j (r)
dr2 r2
j j
with the direct coupling potentials
N
Z 1
Vij (r) = − δij + Φi | | Φj
r |rk − r|
k=1
and the exchange terms
N
1
Wij FE,j (r) = Φi | | (A − 1) Φj FE,j
|rk − r|
k=1
Close-coupling can yield complete data sets, and the results are
internally consistent (unitary theory that conserves total flux)!

17. Cross Section for Electron-Impact Excitation of He(1s2
Total Cross Sections for Electron-Impact Excitation )of Helium
K. Bartschat, J. Phys. B 31 (1998) L469
K. Bartschat, J. Phys. B 31 (1998) L469
n=2 n=3
6 4 2.0 1.0
Trajmar 3 experiment
5 Hall 3S CCC(75) 31 S
3 0.8
Donaldson 1.5 RMPS
4 CCC(75)
RMPS 0.6
3 2 1.0
cross section (10−18cm2)
0.4
2
1 0.5
1 0.2
23 S 21 S
0 0 0.0 0.0
4 8
cross section (10−18cm2)
1.0 1.5
33 P 31 P
3 6 0.8
1.0
0.6
2 4
0.4
0.5
1 2
0.2
23 P 21 P
0 0 0.0 0.0
20 25 30 35 40 20 25 30 35 40
0.3
projectile energy (eV) 33 D 31 D
0.3
0.2
0.2
0.1
0.1
0.0 0.0
20 25 30 35 40 20 25 30 35 40
projectile energy (eV)
In 1998,1998, Heer recommends 0.5 x (CCC+RMPS) for uncertainty of! 10%
In de deHeer recommends (CCC+RMPS)/2 for uncertainty of 10% or better
— independentof experiment)
(independent of experiment!

18. Metastable Excitation Function in Kr
Oops — maybe we need
to try a bit harder?

19. We have a great new program :):):)
General B-Spline R-Matrix (Close-Coupling) Programs (D)BSR
• Key Ideas:
• Use B-splines as universal
basis set to represent the
continuum orbitals
• Allow non-orthogonal or-
bital sets for bound and
continuum radial functions
not just the numerical basis!
O. Zatsarinny, CPC 174 (2006) 273
• Consequences:
• Much improved target description possible with small CI expansions
• Consistent description of the N -electron target and (N+1)-electron collision
problems
• No “Buttle correction” since B-spline basis is effectively complete
• Complications:
• Setting up the Hamiltonian matrix can be very complicated and lengthy record:150,000
• Generalized eigenvalue problem needs to be solved to do 50-100 times;
• Matrix size typically 50,000 and higher due to size of B-spline basis
10,000 100 - 200 kSU
• Rescue: Excellent numerical properties of B-splines; use of (SCA)LAPACK et al.
We also have to solve the problem outside the box for each energy (from 100's to 1,000,000's).

20. List of early calculations with the BSR code (rapidly growing)
List of calculations with the BSR code (rapidly growing)
hv + Li Zatsarinny O and Froese Fischer C J. Phys. B 33 313 (2000)
hv + He- Zatsarinny O, Gorczyca T W and Froese Fischer C J. Phys. B. 35 4161 (2002)
hv + C- Gibson N D et al. Phys. Rev. A 67, 030703 (2003) at least 60 more
hv + B- Zatsarinny O and Gorczyca T W Abstracts of XXII ICPEAC (2003)
hv + O- Zatsarinny O and Bartschat K Phys. Rev. A 73 022714 (2006) since 2006
hv + Ca- Zatsarinny O et al. Phys. Rev. A 74 052708 (2006)
e + He Stepanovic et al. J. Phys. B 39 1547 (2006)
Lange M et al. J. Phys. B 39 4179 (2006)
e+C Zatsarinny O, Bartschat K, Bandurina L and Gedeon V Phys. Rev. A 71 042702 (2005)
e+O Zatsarinny O and Tayal S S J. Phys. B 34 1299 (2001)
Zatsarinny O and Tayal S S J. Phys. B 35 241 (2002)
Zatsarinny O and Tayal S S As. J. S. S. 148 575 (2003)
e + Ne Zatsarinny O and Bartschat K J. Phys. B 37 2173 (2004)
Bömmels J et al. Phys. Rev. A 71, 012704 (2005)
Allan M et al. J. Phys. B 39 L139 (2006)
e + Mg Bartschat K, Zatsarinny O, Bray I, Fursa D V and Stelbovics A T J. Phys. B 37 2617 (2004)
e+S Zatsarinny O and Tayal S S J. Phys. B 34 3383 (2001)
Zatsarinny O and Tayal S S J. Phys. B 35 2493 (2002)
e + Ar Zatsarinny O and Bartschat K J. Phys. B 37 4693 (2004)
e + K (inner-shell) Borovik A A et al. Phys. Rev. A, 73 062701 (2006)
e + Zn Zatsarinny O and Bartschat K Phys. Rev. A 71 022716 (2005)
e + Fe+ Zatsarinny O and Bartschat K Phys. Rev. A 72 020702(R) (2005)
e + Kr Zatsarinny O and Bartschat K J. Phys. B 40 F43 (2007)
e + Xe Allan M, Zatsarinny O and Bartschat K Phys. Rev. A 030701(R) (2006)
Rydberg series in C Zatsarinny O and Froese Fischer C J. Phys. B 35 4669 (2002)
osc. strengths in Ar Zatsarinny O and Bartschat K J. Phys. B: At. Mol. Opt. Phys. 39 2145 (2006)
osc. strengths in S Zatsarinny O and Bartschat K J. Phys. B: At. Mol. Opt. Phys. 39 2861 (2006)
osc. strengths in Xe Dasgupta A et al. Phys. Rev. A 74 012509 (2006)

21. [One of] our apparatus ...
recently moved to Lonestar

22. Structure Calculations with the BSR Code
Phys. Scr. T134 (2009) 014009

23. Energy Levels in Heavy Noble Gases
This is VERY GOOD for a
subsequent collision calculation!

24. Oscillator Strengths in Neon

25. Summary of structure work
Conclusions
• The non-orthogonal orbital technique allows us account for term-dependence and
relaxation effects practically to full extent. At the same time, this reduce the size of
the configuration expansions, because we use specific non-orthogonal sets of
correlation orbitals for different kinds of correlation effects.
• B-spline multi-channel models allow us to treat entire Rydberg series and can be
used for accurate calculations of oscillator strengths for states with intermediate
and high n-values. For such states, it is difficult to apply standard CI or MCHF
methods.
• The accuracy obtained for the low-lying states is close to that reached in large-scale
MCHF calculations.
• Good agreement with experiment was obtained for the transitions from the ground
states and also for transitions between excited states.
• Calculations performed in this work: s-, p-, d-, and f-levels up to n = 12.
Ne – 299 states – 11300 transitions
Ar – 359 states – 19000 transitions
Kr – 212 states – 6450 transitions
Xe – 125 states – 2550 transitions
• All calculations are fully ab initio.
• The computer code BSR used in the present calculations and the results for Ar
were recently published:
BSR: O. Zatsarinny, Comp. Phys. Commun. 174 (2006) 273
Ar: O. Zatsarinny and K. Bartschat, J. Phys. B 39 (2006) 2145

26. Move on to Collisions ...
Introduction
Metastable yield in e-Ne collisions
• Using our semi-relativistic B-spline R-matrix (BSR) method [Zatsarinny and
Bartschat, J. Phys. B 37, 2173 (2004)], we achieved unprecendented agreement
with experiment for angle-integrated cross sections in e−Ne collisions.

27. Resonances in the excitation of the Ne (2p53p) states
Allan, Franz, Hotop, Zatsarinny, Bartschat (2009), J. Phys. B 42, 044009
f2
1
3
expt. S0
f1
BSR31 n g1 g2 q = 135°
p 1 p
2
n2
Cross Section (pm2/sr)
3p2
1 }
d1 it's looking
0 e d2 4p2 good :):):)
3
}
20 S1
g1 g2
3p 4s 4s 4p
4s2 q = 180°
p
}
p
}
1
S0 n1 n2 f1 f2
10
4p
3d
4s
0 3p
18.5 19.0 19.5 20.0 20.5
Electron Energy (eV)
Expanded view of the resonant features in selected cross sections for the excitation
of the 3p states. Experiment is shown by the more ragged red line, theory by the
smooth blue line. The present experimental energies, labels (using the notation
of Buckman et al. (1983), and configurations of the resonances are given above the
spectra. Threshold energies are indicated below the lower spectrum.

28. Metastable Excitation Function in Kr
Do you remember this one?

29. Metastable Excitation Function in Kr
JPB 43 (2010) 074031
What a difference :):):)
–> let's go to more detail!

30. excitation at fixed angle
1 5 0 e -K r θ= 3 0 o 5 s ’[ 1 / 2 ] 1
1 0 0
5 0
M. Allan, O. Zatsarinny, and K.B.,
0 J. Phys. B 44 (2011) 065201
(selected as a JPB "highlight")
5 s ’[ 1 / 2 ] 0
1 5
1 0
C ro s s S e c tio n (p m 2 /s r)
5
0
Allan expt.
2 0 0 5 s [3 /2 ]1
DBSR calculation
1 0 0
P h illip s (1 9 8 2 )
A lla n (2 0 1 0 )
D B S R 6 9
0
5 s [3 /2 ]2
1 0 0
5 0
0
1 0 1 1 1 2 1 3 1 4 1 5
E le c tro n E n e rg y (e V )

31. excitation at fixed energy
15 eV 5s[3/2]2 5s'[1/2]0
0 .0 4
0 .0 0 6
2/s r)
o
0 .0 0 4
D C S (a
0 .0 2
0 .0 0 2
0 .0 0 0 .0 0 0
0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0
5s[3/2]1 now confirmed 5s'[1/2]1
T ra jm a r e t a l. (1 9 8 1 ) B S R 4 9
G u o e t a l. (2 0 0 0 ) × 0 .3 7
/s r)
-1 1 0 -1 D B S R 6 9
1 0 A lla n (2 0 1 0 )
2
o
D C S (a
-2
1 0
-2
1 0
0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0
Scattering Angle (deg) Scattering Angle (deg)

32. A Classic GEC Application !!!
Forward Model
Let's use these data in a real plasma model! (Dirk Dodt's PhD Thesis)
EEDF emission coefficient
Einstein's coefficients line of sight
collisional - integration
radiative model (radiation transport)
150 cross sections 350 Einstein
population densities coefficients
radiance
line profile
intensity calibration (aparatus width)
spectrometer pixels spectral radiance
Dirk Dodt for the 5th Workshop on Data Validation 5

33. Collisional Radiative Model
● Balance equation for each
excited state density
● Elementary processes:
– Electron impact (de)excitation
– Radiative transitions
– Atom collisions
– Wall losses
● Linear equation for
Dirk Dodt for the 5th Workshop on Data Validation 6

34. Results: Treatment of Various Processes in the Model

35. Although we are are [almost] perfect, let's allow for some uncertainty ...
Uncertainty Estimate of Oscillator Strengths
∆ Aki / Aki 0.24
0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0 4
10 105 106 107 108
Aki [s-1]

36. Uncertainty Estimate of Cross Sections
hard to believe, but we
will see that this was
(FAR) too optimistic !

37. J. Phys. D 41 (2008) 205207
Modeled Spectrum With “Reasonable Uncertainty in Atomic Data”
W ]
residual/σ
8
sr m
7 6
10 4
2
m2
6 0
10
Lλ [
-2
5 -4
10 -6
-8
560 580 600 620 640 λ [nm]
There are measured points behind the red line!
W ]
residual/σ
8
sr m
6
107 4
2
m2
6 0
10
Lλ [
-2
5 -4
10 -6
-8
680 700 720 740 760 λ [nm]
W ]
residual/σ
8
m sr m
7 6
10 4
2
2
6 0
10
Lλ [
-2
5 -4
10 -6
-8
800 820 840 860 880 λ [nm]

38. This is the real check!
Results: Are the Atomic Data Consistent?
3s3->3d8 3s4->3d1
3s3->3d9 3s4->3d2
3s3->3d10 3s4->3d3
3s3->3d11 3s4->3d4
3s3->3d12 3s4->3d5
3s3->4s1 3s4->3d6
3s3->4s2
3s3->4s3
3s3->4s4
3s4->3d7
3s4->3d8
3s4->3d9
looks pretty good;
3s1->3p1 3s4->3d10
3s1->3p2
3s1->3p3
3s1->3p4
3s4->3d11
3s4->3d12
3s4->4s1
correction factors
3s1->3p5
3s1->3p6
3s1->3p7
3s4->4s2
3s4->4s3
3s4->4s4
mostly consistent
2p1->3d1
3s1->3p8
3s1->3p9
3s1->3p10
2p1->3d2
2p1->3d3
2p1->3d4
with 1.0
3s2->3p1 2p1->3d5
3s2->3p2 2p1->3d6
3s2->3p3 2p1->3d7
3s2->3p4 2p1->3d8
3s2->3p5 2p1->3d9
3s2->3p6 2p1->3d10
3s2->3p7 2p1->3d11
3s2->3p8 2p1->3d12
3s2->3p9 2p1->4s1
3s2->3p10 2p1->4s2
3s3->3p1 2p1->4s3
3s3->3p2 2p1->4s4
3s3->3p3 3s1->3d1
3s3->3p4 3s1->3d2
3s3->3p5 3s1->3d3
3s3->3p6 3s1->3d4
3s3->3p7 3s1->3d5
3s3->3p8 3s1->3d6
3s3->3p9 3s1->3d7
3s3->3p10 3s1->3d8
3s2->3s1 3s1->3d9
3s3->3s1 3s1->3d10
3s3->3s2 3s1->3d11
3s4->3s1 3s1->3d12
3s4->3s2 3s1->4s1
3s4->3s3 3s1->4s2
3s4->3p1 3s1->4s3
3s4->3p2 3s1->4s4
3s4->3p3 3s2->3d1
3s4->3p4 3s2->3d2
3s4->3p5 3s2->3d3
3s4->3p6 3s2->3d4
3s4->3p7 3s2->3d5
3s2->3d6
3s4->3p8
3s4->3p9 3s2->3d7
3s2->3d8
3s4->3p10
3s2->3d9
2p1->3p1
3s2->3d10
2p1->3p2
3s2->3d11
2p1->3p3 3s2->3d12
2p1->3p4 3s2->4s1
2p1->3p5 3s2->4s2
2p1->3p6 3s2->4s3
2p1->3p7 3s2->4s4
2p1->3p8 3s3->3d1
2p1->3p9 3s3->3d2
2p1->3p10 3s3->3d3
2p1->3s1 3s3->3d4
2p1->3s2 3s3->3d5
2p1->3s3 3s3->3d6
2p1->3s4 3s3->3d7
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5

39. New Journal of Physics 12 (2010) 073018
Let’s Zoom In ...
3s4->3p2 3s4->3d1
3s4->3p3 3s4->3d2
3s4->3d3
3s4->3p4
3s4->3d4
3s4->3p5 3s4->3d5
3s4->3p6 3s4->3d6
3s4->3d7
3s4->3p7
3s4->3d8
3s4->3p8
3s4->3d9
3s4->3p9 3s4->3d10
3s4->3p10 3s4->3d11
3s4->3d12
2p1->3p1
2p1->3d1
2p1->3p2 2p1->3d2
2p1->3p3 2p1->3d3
2p1->3p4 2p1->3d4
2p1->3d5
2p1->3p5
2p1->3d6
2p1->3p6 2p1->3d7
2p1->3p7 2p1->3d8
2p1->3d9
2p1->3p8
2p1->3d10
2p1->3p9
2p1->3d11
2p1->3p10 2p1->3d12
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
OOPS ?!?!?!

40. Results: Significant Correction Factors for a Few Transitions
Explanation: The results for these transitions are very
sensitive to the inclusion of continuum coupling!
(already suggested in 2004 by Ballance and Griffin)
So let's work on that ...

41. A very recent development [PRL 107 (2011) 023203]
Fully Nonperturbative Treatment of Ionization
and Ionization with Excitation of Noble Gases
Overview:
I. Motivation: Is Theory Really, Really, Really BAD?
II. Theoretical Formulations
• Perturbative Methods
• Plane-Wave and Distorted-Wave Treatments of the Projectile
• Second-Order Effects
• Ejected-Electron−Residual-Ion Interaction: The Hybrid Approach
You'll seeNon-Perturbative Methods
• Fully in a minute how important this is for the
• Time-Dependent Close-Coupling (TDCC), well! Complex Scaling (ECS)
EXCITATION problem as Exterior
• Convergent Close-Coupling (CCC)
• Convergent B-Spline R-matrix with Pseudostates (BSRMPS)
III. Example Results
• Ionization of He(1s2 ) → He+ (1s), He+ (n = 2)
n
• Ionization of Ne(2s2 2p6 ) → Ne+ (2s2 2p5 ), (2s2p6 )
• Ionization of Ar(3s2 3p6 ) → Ar+ (3s2 3p5 ), (3s3p6 )
IV. Conclusions and Outlook

42. Ionization in the Close-Coupling Formulation
The “Straightforward” Close-Coupling Formalism
• Recall: We are interested in the ionization process
e0 (k0 , µ0 ) + A(L0 , M0 ; S0 , MS0 ) → e1 (k1 , µ1 ) + e2 (k2 , µ2 ) + A+ (Lf , Mf ; Sf , MSf )
• We need the ionization amplitude
f (L0 , M0 , S0 ; k0 → Lf , Mf , Sf ; k1 , k2 )
• We employ the B-spline R-matrix method of Zatsarinny (CPC 174 (2006) 273)
with a large number of pseudo-states:
• These pseudo-states simulate the effect of the continuum.
• The scattering amplitudes for excitation of these pseudo-states are used to
form the ionization amplitude: This is the essential idea – we'll
still have to work on the details.
Ψk2 |Φ(L S ) f (L , M , S ; k → L , M , S ; k ).
−
f (L , M , S ; k → L , M , S ; k , k ) =
0 0 0 0 f f f 1 2 f p p 0 0 0 0 p p p 1p
p
• Both the true continuum state |Ψf k2 − (with the appropriate multi-channel
asymptotic boundary condition) and the pseudo-states |Φ(Lp Sp ) are consistently
Total (ionization) cross sections are much easier to get – just
calculated with the same close-coupling expansion.
add up the results for all the (positive-energy) pseudo-states.
• In contrast to single-channel problems, where the T -matrix elements can be
interpolated, direct projection is essential to extract the information in multi-
channel problems. This is the essential idea – just do it!
• For total ionization, we still add up all the excitation cross sections for the
pseudo-states.

43. Total Cross Section in e-H Ionization
Total Cross Section and Spin Asymmetry in e−H Ionization
K. Bartschat (from I. Bray, J. Phys. 1996) (1996) L577
and Bartschat and Bray B 29
4
5
9
T R Q PI & H % 1 G 0 # # # H 2 G
S
4
69 79
4
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F §
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4
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This is the proof-of-principle case,
£
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¥
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(2 10 ) ) (# '
so let's build on that.
79
4 4
A
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% $ #
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39
4 4
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9
8
7
3
4
4
6
4
5
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!

44. Phys. Rev. Lett. 107 (2011) 023203
Total and Single-Differential Cross Section
0.5
e - He E=100 eV
correlation drops
cm )
2
e - He
3
the theoretical ICS
-16
0.4
Müller-Fiedler et al (1986)
Ionization Cross Section (10
cm /eV)
BSR227 - interpolation
BSR227 - projection
2
0.3
2
-18
SDCS (10
definitely looks o.k.
0.2
Montague et al. (1984)
Rejoub et al. (2002)
1
Sorokin et al. (2004)
0.1 BSR-525 That's a lot of states!
2
BSR with 1s correlation
2
CCC with 1s correlation
0
0.0
30 100 300
0 10 20 30 40 50 60 70
Electron Energy (eV) Secondary Energy (eV)
• Including correlation in the ground state reduces the theoretical result.
• Interpolation yields smoother result, but direct projection is acceptable.
• DIRECT PROJECTION is NECESSARY for MULTI-CHANNEL cases!
So far, so good ... Let's go for more detail!

45. BSRMPS works great: PRL 107 (2011) 023203
Triple-Differential Cross Section Ratio
experiment: Bellm, Lower, Weigold; measured directly
150 150
150 o o o
= 24 = 36 = 48
1 1 1
100 100 100
50 expt.
50 E =44 eV
50
1
BSR
E =44 eV
2
n=1/n=2 Cross Section Ratio
DWB2-RMPS
0 0 0
20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120
150
150 o
o o
= 28 = 40 = 52
1
100
1 1
100 100
50 50
50
0DWB+RM hybrid method 0 not appropriate 0
20 40 symmetric kinematics40 all! 80 100 120
for 60 80 100 120 150
20 at 60 20 40 60 80 100 120
150
o o
100
o
= 32 = 44 = 56
1 1 1
100 100
50
50 50
0 0 0
20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120
(deg)
2

46. What about more complex targets?
Ren et al., Phys. Rev. A 83 (2011) 052714
... the magnitude [of the TDCS] appears to be the most significant problem.
... the physics behind the electron-impact ionization of many-
electron systems needs to be treated more accurately by theory.

47. Electron Impact Ionization of Ar(3p) with E0 = 71 eV
BSR at least gets the shape.
Other methods are way off!
Magnitude still
problematic!
There may be an
experimental issue!
(– PRA 85 (2012) 032708)

48. Back to a problem at the heart of the GEC community: e-Ne collisions
Electron Impact Ionization of Neon (from ground state)
10
- - +
e + Ne - 2e + Ne
cm )
8
2
-17
Cross section (10
6
Absolute experiments are tricky!
4
Rejoub et al. (2002)
Krishnakumar Strivastava (1988)
2 Pindzola et al. (2000)TDCC; target description?
BSR-679 That's again a lot of states!
0
0 20 40 60 80 100 120 140 160 180 200
Electron energy (eV)

49. Electron Impact Ionization of Neon (from metastable states)
80
- 5 3 - + 5
e + Ne*(2p 3s P) - 2e + Ne (2p )
70
cm )
2
60
-17
Johnston et al. (1996)
50
Cross section (10
RMPS-243, Ballance et al. (2004)
BSR-679
40
30
20
10
0
0 20 40 60 80 100 120 140 160
Electron energy (eV)

50. Total Cross Section for Electron Impact Excitation of Neon
RMPS calculations by Ballance and Griffin (2004)
3.0 5 3 12 5 1
(2p 3s) P (2p 3s) P
2.0 8
The Excitation Mystery!
1.0 4
0.0 0
16 24 32 40 16 24 32 40
cm )
2
5 3 1.6 5 3
0.6 (2p 3p) S (2p 3p) D
-18
Cross section (10
1.2
0.4
There is a BIG drop in the
0.8
cross sections due to the
0.2 coupling to pseudo-states!
0.4
0.0 0.0
16 24 32 40 16 24 32 40
0.16 2.0
5 3 5 1
(2p 3d) P (2p 3d) P
1.6
0.12
1.2
Here we (under)estimated
0.08 an error of only 60% !?!?!?
RM-61
0.8
RMPS-243
0.04 BSR-679
0.4
0.00 0.0
16 24 32 40 16 24 32 40
Electron energy (eV)
Electron energy (eV)

51. Total Cross Section for Electron Impact Excitation of Neon
Effect of Channel Coupling to Discrete and Continuum Spectrum
2.0 0.4
cm )
2
3s[3/2] 3s'[1/2]
2 0
1.5 0.3
-18
Khakoo et al.
BSR-457
Cross Section (10
Register et al.
BSR-31
Phillips et al.
1.0 BSR-5 0.2
0.5 0.1
0.0 0.0
20 40 60 80 100 20 40 60 80 100
2.0 15
cm )
2
3s[3/2] 3s'[1/2]
1 1
12
1.5
-18
Cross Section (10
9
1.0
6
0.5 Kato
3
Suzuki et al.
0.0 0
20 50 100 300 20 50 100 300
Electron Energy (eV) BIG drop also in fine-structure-resolved
Electron Energy (eV)
results, especially at intermediate energies.

52. Total Cross Section for Electron Impact Excitation of Neon
Effect of Channel Coupling to Discrete and Continuum Spectrum
cm )
2
80 3p[1/2]
50 3p'[1/2]
1 1
40
-20
60
BSR-31
Cross Section (10
BSR-457
RMPS-235 30
40 Chilton et al.
20
20 10
0 0
Again, a BIG drop!
120
cm )
2
3p[5/2]
2
300 3p'[1/2]
0
-20
Cross Section (10
80 200
40 100
0
20 40 60 80 100 20 40 60 80 100
Electron Energy (eV) Electron Energy (eV)

53. Total Cross Section for Electron Impact Excitation of Neon
Effect of Channel Coupling to Discrete and Continuum Spectrum
cm )
2
3d[1/2] 3d[3/2]
3 0
12 2
-20
Cross Sction (10
BSR-31
2 BSR-46 8
BSR-457
RMPS-235
1 4
0 0
20 40 60 80 100 20 40 60 80 100
The 2p53d states are really affected - factors of 5-10!!!
cm )
2
30 3d[1/2]
80 3d[3/2]
1
1
-20
60
Cross Sction (10
20
40
10
20
0 0
20 40 100 200 20 40 100 200
Electron Energy (eV) Electron Energy (eV)

54. Total Cross Section for Electron Impact Excitation of Metastable Neon
Effect of Cascades
3
cm )
2 3p[3/2]
3p'[3/2] 2
10
2
-16
Cross Section (10
2 BSR-457
+ cascade
Boffard et al.
5
1
0 0
0 50 100 150 200 250 300 0 50 100 150 200 250 300
25
cm )
2
6 3p[5/2] 3p[5/2]
3
2
20
-16
Cross Section (10
4 15
10
2
5
0 0
0 50 100 150 200 250 300 0 50 100 150 200 250 300
Electron Energy (eV) Electron Energy (eV)

55. Total Cross Section for e-Ne Collisions
5
e
-
+ Ne BSR-679 can do this as a SINGLE theory
for energies between 0.01 eV and 200 eV!
cm )
2
4
-16
Wagenaar de Heer (1980)
Total Cross Section (10
Gulley et al. (1994)
Szmitkowski et al. (1996)
3
Baek Grosswendt (2003)
BSR-679, total
elastic
+ excitation
2
1
0
0.01 0.1 1 10 100 1000
Electron Energy (eV)

56. Differential Cross Section for Electron Impact Excitation of Neon at 30 eV
Experiment: Khakoo group
0.6
cm /sr)
30 eV 3s[3/2] 3s'[1/2]
3.0 2 0
2
0.4
-19
2.0
DCS (10
0.2
1.0
0.0 0.0
0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140
1 2
10
cm /sr)
10 3s'[1/2]
3s[3/2] 1
1
2
Khakoo et al. x 0.55
Here is the effect on angle-
BSR-457
differential cross sections.
-19
1
0 10
10 BSR-31
DCS (10
renormalize
experimental DCS?
0
-1 10
10
0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140
Scattering Angle (deg) Scattering Angle (deg)

57. Differential Cross Section for Electron Impact Excitation of Neon at 40 eV
Experiment: Khakoo group
cm /sr) 3.0 0.6
40 eV 3s[3/2] 3s'[1/2]
2 0
2
2.0 0.4
-19
DCS (10
1.0 0.2
0.0 0.0
0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140
cm /sr)
3s[3/2] 3s'[1/2]
1 1
1 2
10 10
2
Khakoo et al. x 0.7
-19
BSR-457
BSR-31
DCS (10
0 1
10 10
0
-1 10
10
0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140
Scattering Angle (deg) Scattering Angle (deg)

58. DCS Ratios for Electron Impact Excitation of Neon at 30 and 40 eV
Experiment: Khakoo group
10 10
r 30 eV 40 eV
8 8
6 6
4 4
BSR-31
2 2 3 3
BSR-457 P / P
2 0
Khakoo et al.
0 0
0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140
0.8 0.5
3 1
r' P / P
0.4 1 1
0.6
0.3
0.4
0.2
0.2
0.1
0.0 0.0
0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140
2.0
This is our problem –
3
P
2
/
3
P
1
oscillator strength for
r'' 1.5
1.5 3P is a bit too large in
1
1.0
1.0
this model!
0.5
0.5
0.0 0.0
0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140
Scattering Angle (deg) Scattering Angle (deg)

59. DCS Ratios for Electron Impact Excitation of Neon at 50 and 100 eV
Experiment: Khakoo group
10 10
r 50 eV 100 eV
8 8
6 6 (5 x 0) / (1 x 0) is not
4 4 easy to measure!
2 2
3 3
P / P
2 0
0 0
0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140
3 1
0.3 0.6 P / P
1 1
r'
0.2 0.4
0.1 BSR-457 0.2
Khakoo et al.
0.0 0.0
0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140
3 3
P / P
1.5 2 1
1.5
r''
1.0
1.0
0.5
0.5
0.0 0.0
0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140
Scattering Angle (deg) Scattering Angle (deg)

60. Conclusions and Outlook
• Much progress has been made in calculating the structure of and electron-induced
collision processes in noble gases — and other complex targets (C, O, F, ..., Cu,
Zn, Cs, Au, Hg, Pb).
• Time-independent close-coupling (R-matrix) is the way to go for complete datasets.
• Other sophisticated methods (CCC, TDCC, ECS, ...) are still restricted to
simple targets. These methods are have essentially solved the (quasi-)one- and
(quasi-)two-electron problems for excitation and ionization.
• Special-Purpose and Perturbative (Born) methods are still being used, especially for
complex targets!
• For complex, open-shell targets, the structure problem should not be ignored !
• For such complex targets, the BSR method with non-orthonal orbitals has achieved
a breakthrough in the description of near-threshold phenomena. The major
advantages are:
• highly accurate target description
• consistent description of the N -electron target and (N+1)-electron collision problems
• ability to give “complete” datasets for atomic structure and electron collisions
• The most recent extension to an RMPS version allows for the treatment of ionization
processes.

61. Conclusions and Outlook (continued)
• After setting up a detailed model of a Ne discharge with a large number of observed
lines, we validated most atomic BSR input data (oscillator strengths and collision
cross sections) – and we got hints about areas to improve!
• The validation method provides a valuable alternative for data assessment and
detailed plasma diagnostics — especially when no absolute cross sections from beam
experiments are available.
• This is a nice example of a very fruitful and mutually beneficial collaboration between
data producers and data users.
• We can provide atomic data for (almost) any system. Practical limitations are:
• expertise of the operator/developer
• continued code development (parallelization, new architectures)
• computational resources (Many Thanks to Teragrid/XSEDE!)
• We are currently preparing fully relativistic R-matrix with pseudo-states
calculations on massively parallel platforms (→ Xe, W, ...).
→
• We don’t have a company (yet). If you really want/need specific data, you can
• ask us really nicely and hope for the best (preferred by most colleagues)
• give us $$$ and increase your priority on the list (preferred by us)
THANK YOU !