Plasma modeling using accurate cross sections for Ar
R. Srivastava * , R. K. Gangwar* , L. Sharma * and A. D. Stauffer#
*Dept. of Physics, Indian Institute of Technology, Roorkee 247667 India
#
Dept. of Physics and Astronomy, York University, Toronto, Canada M3J 1P3
Abstract: In the present work we have used reliable electron excitation cross sections in a Collisional-Radiative
Model to obtain the population densities for 1s levels of argon for low temperature CCP and ICP discharges. The
model includes excitation cross sections from ground as well as excited 3p 5 4s fine structure levels to the various
higher lying manifolds. We have recently published the cross section results used in the present paper which were
calculated using our fully relativistic distorted wave approximation method. We have also calculated cross
sections for the excitation of the 3p 54s resonance levels to higher lying 3p 5 4p manifolds for use in the present
work. We have compared our results for 1s level population densit ies to the recent OES measurements and the
available other CR model calculations.
Keywords: CR Model, Relativistic effects, Distorted wave, Plasma modeling, fine structure levels
1. Introduction
Optical emission spectroscopy (OES) is one of the
most widely used diagnostic methods for low
temperature plasmas [1]. Advanced plasma
diagnostics can be performed by the combined use
of an appropriate collisional radiative model (CRM)
and OES measurements which can yield various
plasma parameters like electron temperature,
electron density and populations of excited species
[1]. In low temperature and pressure plasmas the
dominant processes are caused by electron impact.
Thus the success of optical-based plasma diagnostic
techniques depends on using accurate theoretical and
experimental electron impact cross section data [2].
There are several recently published CR models
for low temperature Ar discharges [3, 4]. These
models have used cross section data either from the
few available experimental measurements or
theoretical data calculated by empirical formula or
simple non-relativistic methods which average over
fine structure states. The experimental measurements
are very limited and unresolved for transitions which
are closely spaced in energy. Thus for reliable
plasma modeling there is a real need for accurate
theoretical cross section data for various transitions
from the ground as well as excited states.
Recently our group has reported various finestructure resolved excitation cross sections for inert
gases from the ground as well as metastable states to
the higher lying fine structure levels using fully
relativistic distorted-wave (RDW) theory [5, 6, 7].
The RDW method used for the calculations of the
cross sections utilizes the solution of the Dirac
equations for the wave functions of both the initial
and final channels for the projectile electron [5]. The
bound target states are represented as relativistic
Dirac-Fock multi-configuration wave functions. We
have obtained these wave functions for the ground
and the excited states from the relativistic atomic
structure GRASP92 code [8].
Recently Zhu et al [3] have reported experimental
results from optical emission spectroscopy (OES)
for various Ar plasma discharges along with their
theoretical calculation using a CR model. In the
present work we calculate the population distribution
of the 1s levels (in Paschen notation) for low
temperature argon capacitively coupled plasmas
(CCP) and inductively coupled plasmas (ICP) using
a CR model which includes our RDW cross section
results for the various transitions. We compare our
results with the OES and CRM results of [3].
CR model
Our collisional radiative model is similar to recently
reported models [3, 4] and incorporates 40 excited
fine-structure levels, one ground level and one ion
level as listed in table 1. Since we have considered
here only low temperature plasmas the dominant
processes are due to electron impact.
The rate coefficients kij for the electron impact
excitation from a lower energy level i to a higher
level j can be expressed as
2 
k 
  ( E ) E F ( E )dE
ij
m Eij ij
(1)
where m is the electron mass and σij is the electronimpact integrated cross section (ICS) at an electron
impact energy E. F(E) is the electron energy
probability function which is related to the electron
energy distribution function (EEDF) f(E) through the
relation f(E) = E1/2 F(E) with the property that

 E F ( E )dE  1 .
0
For the inverse process, i.e. the electron impact deexcitation from level j to level i, the rate coefficients
kji follow from the detailed balance principle [9] and
are given by
k
g 2 
 i
 ( E ) E F ( E  E )dE
ji g m Eij ij
ij
j
(2)
where g i and gj are the degeneracy of levels i and j,
respectively.
The rate coefficients for the electron impact
ionization process from level i is given as
2 
k 
  ( E ) E F ( E )dE
i
m Ei i
level j to i, ( j > i ), A ji eff, are calculated from the
relation Aji eff = ji Aji.. Where Aji is the Einstein
coefficients and ji is the escape factor. By
introducing this escape factor and using an effective
radiative decay coefficients in place of the Aji , we
can partially account for the self absorption of the
emitted radiation. In the present model, assuming a
uniform distribution of emitting and absorbing
atoms, we used the Mewe approximation to calculate
escape factors [10]. In the present model the
calculation of escape factor has only been done for
transitions involving the five lowest levels. For low
temperature and pressure, the population density of
the more highly excited levels are usually low
enough so that we can neglect the radiation trapping
effect.
Now the particle balance equation which accounts
population and depopulation terms for an excited
levels j, having population density n j in steady state
is given by,
41
eff
 kij (Te )ni ne   Aij ni  ne n ne k j
i1
i j
i j
41
(3)
where σi+ is the electron impact ionization cross
section and Ei+ is the ionization energy of the ith
level.
The reverse of the ionization process, a
collision between an ionized atom and two electrons,
can cause recombination of the ion and an electron.
This process is called three-body recombination and
its rate coefficients can also be calculated by using
the detailed-balance principle and the direct
ionization cross sections [9].
For all the electron impact processes, the electron
energy distribution function (EEDF) is assumed to
be Maxwellian as used in other CR models [3, 4] so
that we can make a direct comparison with their
results.
Emission and absorption of radiation can also
contribute to the production and destruction of level
populations. There are three processes involved, viz
spontaneous emission, absorption and stimulated
emission. In this model the last two processes are
neglected and the radiative decay coefficients from
  k ji (Te )n j ne -  Aeff
n j - n j ne  k j  0 (4)
ji
i1
i j
i1, 2, 4
i j
Where n e and Te is electron density and electron
temperature of the plasma. In putting the electron
density, atom density and electron temperature at
which the experimental measurements have been
made [3], we solved these linear equations and
calculated the 1s level populations following the
method described in Mullen et al [9].
We used our recent RDW calculations [5, 6, 7] for
the various excitation cross sections from both the
ground state as well as the metastable states with
configuration 3p 5 4s to the higher lying fine-structure
levels as input to equation 1. The electron-impact
excitation cross section from the resonance 3p 5 4s
levels (1s4 and 1s2 ) to the other excited levels
(especially the 2p levels) are also as important as the
excitation from the metastables states. Our earlier
RDW calculation did not consider these excitations
so we have also calculated these in the present work.
TABLE 1. Argon energy levels considered in model.
The levels are given in paschen notation.
Number
of level
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
Level
1S0
1s5
1s4
1s3
1s2
2p10
2p9
2p8
2p7
2p6
2p5
2p4
2p3
2p2
2p1
3d12
3d11
3d10
3d9
3d8
3d7
3d6
3d5
3d4
3d3
3d2
3d1
2s5
2s4
2s3
2s2
3p10
3p9
3p8
3p7
3p6
3p5
3p4
3p3
3p2
3p1
Ion
Excitation
Energy (eV)
0
11.548
11.623
11.723
11.828
12.907
13.076
13.095
13.153
13.172
13.273
13.283
13.302
13.328
13.480
13.845
13.863
13.903
13.979
14.012
14.063
14.100
14.153
14.214
14.234
14.236
14.303
14.068
14.090
14.241
14.255
14.464
14.500
14.506
14.525
14.529
14.575
14.680
14.687
14.688
14.738
15.76
Statistical
weight
1
5
3
1
3
3
7
5
3
5
1
3
5
3
1
1
3
5
9
7
5
7
3
5
5
7
3
5
3
1
3
3
7
5
3
5
1
3
3
5
1
6
In equation 3 we have included available
experimental ionization cross sections from the
ground state [11] and theoretical data for ionization
of the 3p 5 4s metastable states [12]. The Einstien
radiative transition probabilities have been taken
from the NIST database [13] and from Wiese et al
[14].
3. Results & Discussion
Fig 1 and 2 display the cross section results for the
excitation from the resonance 3p5 4s levels to levels
of the 3p5 4p manifold. Since these results have not
been previously published we show these in detail
for use in future plasma modeling calculations
Figure 1. The ICS results calculated using RDW theory for the
excitation of the 3p 54s resonance level 1s 2 (J=1) to the 3p 54p
manifold.
Figure 2. The ICS results calculated using RDW theory for the
excitation of the 3p 54s resonance level 1s 4 (J=1) to the 3p 54p
manifold.
In figures 3 and 4 we have shown the comparison of
our results with the recent OES measurement and
CR model results reported by Zhu et al [3]. Our
calculations are overall in better agreement with the
experimental OES measurements, especially those
for the CCP plasma, than the calculations reported in
[3].
We are very thankful to Prof. J A M van der Mullen,
Dr. Wouter Graef, Dr. X M Zhu and Prof. Y K Pu
for their useful suggestions and discussions. R.K.G.
is thankful to CSIR, New Delhi for financial
assistance. R.S. would like to acknowledge research
grants in support of this work from Council of
Scientific and Industrial Research (CSIR), New
Delhi and IAEA Vienna. ADS acknowledges a grant
from NSERC Canada.
References
Figure 3. Ratios of population densities of 3p 54s levels to the
respective degeneracy for a CCP argon plasma.
Figure 4. Ratios of population densities of 3p 54s levels to the
respective degeneracy for a ICP argon plasma.
4. Conclusion
Results of a CRM using reliable cross section data
for argon are presented for low temperature CCP and
ICP plasmas. Our modeling results for the
population distribution of the 1s levels are in good
agreement with the OES measurements of Zhu et al
[3]. From the present work we have demonstrated
the necessity of using reliable cross sections for the
fine-structure transitions in order to obtain accurate
plasma population levels.
5. Acknowledgements
[1] Xi-M ing Zhu and Y K Pu, J. Phys. D: Appl. Phys. 43
403001(2010).
[2] J B Bo ffard, C C Lin and C A DeJoseph, J. Phys. D:
Appl. Phys. 37 R143 (2004).
[3] Xi-M ing Zhu and Y K Pu, J. Phys. D: Appl. Phys. 43
015204 (2010).
[4] A Palmero, E D Hattu m, H Rudolph and F H P M
Habraken, J. Phys. D: Appl. Phys. 101 053306 (2007).
[5]R K Gangwar, L Sharma, R Srivastava, A D Stauffer
Phys. Rev. A 81 052707 ( 2010).
[6] M A Khakoo et al, J. Phys. B: At. Mol. Opt. Phys., 37,
247 (2004).
[7] R Srivastava, A D Stauffer and L Sharma, Phys. Rev.
A, 74, 012715 (2006).
[8] F A Parpia, C F Fischer and I P Grant, Comput. Phys.
Commun., 94, 249 (1996).
[9] A Hartgers, J van Dijk, J Jonkers, J A M van der
Mullen, Comput. Phys. Commun. 135 199 (2001).
[10] H C Straub, P Renault, B G Lindsay, K A Smith and
R F Stebbings, Phys. Rev. A, 52, 1115 (1995).
[11] M Asgar Ali, P M Stone, International Journal of
Mass Spectrometry, 271, 51 (2008).
[12] J B Boffard, R O Jung, C C Lin and A E Wendt
Plasma Sources Sci. Technol. 18 035017 (2009).
[13]
Nist
2010
Ato mic
Spectra
database
(http://www.nist.gov/pml/data/asd.cfm)
[14] W L Wiese, J W Brault, K Danzmann, V Helbig and
M Kock, Phys. Rev A, 39 2461 (1989).