Eur. Phys. J. D (2013) 67: 203DOI: 10.1140/epjd/e2013-40244-9

Regular Article

THE EUROPEANPHYSICAL JOURNAL D

Collisional-radiative model for non-Maxwellian inductivelycoupled argon plasmas using detailed fine-structure relativisticdistorted-wave cross sections

Dipti1, Reetesh Kumar Gangwar1,a, Rajesh Srivastava1,b, and Allan Daniel Stauffer2

1 Department of Physics, Indian Institute of Technology, Roorkee 247667, India2 Department of Physics and Astronomy, York University, Toronto, Ontario M3J 1P3, Canada

Received 16 April 2013 / Received in final form 20 July 2013Published online 1st October 2013 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2013

Abstract. Our recently developed collisional-radiative model which included fine-structure cross sectionscalculated with a fully relativistic distorted-wave method [R.K. Gangwar, L. Sharma, R. Srivastava, A.D.Stauffer, J. Appl. Phys. 111, 053307 (2012)] has been extended to study non-Maxwellian inductivelycoupled argon plasmas. We have added more processes to our earlier collisional-radiative model by furtherincorporating relativistic distorted-wave electron impact cross sections from the 3p54sJ = 0, 2 metastablestates, (1s3, 1s5 in Paschen’s notation) to the 3p55p (3pi) excited states. The population of various excitedlevels at different pressures in the range of 1–25 mTorr for an inductively coupled argon plasma have beencalculated and compared with the recent optical absorption spectroscopy measurements as well as emissionmodel results of Boffard et al. [Plasma Sources Sci. Technol. 19, 065001 (2010)]. We have also calculatedthe intensities of two emission lines, 420.1 nm (3p9 → 1s5) and 419.8 nm (3p5 → 1s4) and compared withmeasured intensities reported by Boffard et al. [J. Phys. D 45, 045201 (2012)]. Our results are in goodagreement with the measurements.

1 Introduction

In a low-temperature plasma, electron-impact processesplay a dominant role. Therefore, in modelling such plas-mas, the required rate coefficients for various electron-impact processes should be calculated using reliableelectron-impact cross sections obtained from either exper-imental measurements or a consistent theoretical modelas well as a suitable electron energy distribution func-tion (EEDF). The success of the model entirely dependson these two factors [1–7]. Plasma kinetic models alongwith optical emission spectroscopy measurements providea widely used means for plasma diagnostics [5,8]. Thisoptical-based, passive plasma diagnostic technique hasvarious advantages such as being non-invasive inexpen-sive and applicable in various plasma conditions whereno other technique can be employed [9,10]. Hence thedevelopment of accurate models which improve plasmadiagnostics using this technique is highly desirable asthey can provide a means to determine more accu-rate plasma parameters [11,12]. Because of the limitedamount of cross section data available, either experimen-tal or theoretical, we developed a consistent data setof electron-impact fine-structure cross sections for argon

a Present address: Departement de Physique, Universite deMontreal, Montreal, Quebec, H3C 3J7, Canada

b e-mail: rajsrfph@iitr.ernet.in

for a large number of transitions from the ground aswell as excited states using a fully relativistic distorted-wave (RDW) approach [12–17]. This data set is availablefrom the LXCAT website (http://www.lxcat.laplace.univ-tlse.fr). We also fitted our cross sections to ana-lytic formulae at higher energies so that these could beeasily adopted in any plasma model. To check the re-liability of our cross sections we compared them withpublished experimental measurements and non-relativisticand semi-relativistic theoretical calculations. We observedour RDW cross sections were generally in good agree-ment with the measurements [12–17]. Furthermore, asan application of our calculated results, we developeda collisional-radiative (CR) model [12] for low temper-ature argon plasmas where we incorporated our RDWcross sections and used a Maxwellian EEDF to calculatethe rate-coefficient of the processes involved. We appliedour model to both low temperature capacitance coupledplasma (CCP) and inductance coupled plasma (ICP) andreported the populations of the 3p54s and 3p54p levels.These were in good agreement with the optical emissionspectroscopy measurements of Zhu and Pu [18] and animprovement over earlier CR model results. However, re-cent studies showed that often the EEDF deviates fromthe Maxwellian form in the case of an ICP plasmas dueto various dynamic processes involving electrons such asnon-local electron heating in non-uniform electromagnetic

Page 2 of 9 Eur. Phys. J. D (2013) 67: 203

field or energy loss processes due to the inelastic collisionswith gas species [1–3,7,19]. Thus in the present study weexplore this fact by using a generalised form of EEDFwith our extensive RDW cross section data set for otherlow temperature argon ICP plasmas in an attempt to fur-ther establish the reliability and usefulness of our crosssection data.

Recently, Boffard et al. [1–3,20] have performed a de-tailed study of low temperature Ar inductively coupledplasma systems operating under a wide variety of sourceconditions (1–25 mTorr, 20–1000 W, N2 admixture). Theydeveloped a zero-dimensional extended corona emissionmodel to obtain the intensity of various lines. Instead ofusing the most basic corona model, which includes ex-citation from the ground state only, they also includedcontributions from the excited 1si levels (in Paschen’snotation) of the Ar (3p54s) configuration [1]. This in-cludes the excitations from two J = 1 resonance lev-els (1s2 and 1s4) and the J = 0, 1s3, J = 2, 1s5

metastable levels. To obtain the various rate coefficientsthey used their measured optical emission cross sectionsfor metastable excitation [21–23]. However they have notmeasured the cross sections for the excitation from theresonance levels and used instead cross sections calcu-lated with a non-relativistic first-order many-body ap-proximation (FOMBT) [1]. They also reported their mea-sured intensity ratio of two argon emission lines at 420.1and 419.8 nm for various pressures. The 419.8 nm linearises from the 3p5 → 1s4 transition and 420.1 nm is fromthe 3p9 → 1s5 transition. Then using the line ratio tech-nique they extracted the effective electron temperatureas a function of source pressure and compared the re-sults with their probe measurements [1]. In an earlier pa-per [3] for the same plasma conditions, they analyzed theoptical emission spectroscopy (OES) measurements com-bined with their emission model for the 2pj → 1si tran-sition array and reported several plasma parameters. Us-ing optical absorption spectroscopy (OAS) measurements,they also reported the population of various 1si levelsat various pressures ranging from 1–25 mTorr. A com-parison between their OES and OAS measurements forthe populations of the 1si levels was made. While theresults are in general agreement there are large discrep-ancies at low pressures (1, 2.5 mTorr) [3]. In calculatingthe rate coefficients Boffard et al. [1–3] incorporated ananalytic form of the EEDF having two adjustable param-eters and varied these parameters in the EEDF and ob-tained the most suitable values of the parameters by fittingthe modelled intensity results to the measured intensities.Recently Zhu et al. [7,19] have also investigated the pos-sibilities of determining non-Maxwellian EEDFs in low-temperature plasmas containing argon and krypton, us-ing the optical emissions measurements along with a CRmodel. They proposed a non-Maxwellian EEDF with a‘two-temperature structure’ which was obtained from theemission line-ratios of the Paschen 2pi levels of argon andkrypton atoms [7]. There are four independent parametersin their form of the EEDF. Thus it creates uncertainty inthe calculations and poses the difficulty in determining all

the four parameters accurately. On the other hand, thetwo parameters in the non-Maxwellian form suggested byBoffard et al. [1–3] are easier to determine accurately.

In the light of the above-mentioned extensive measure-ments of different plasma parameters reported by Boffardet al. [1–3,20] we have extended here our previous CRmodel [12] to study low temperature Ar ICP plasmas andcompare with their experimentally extracted results. Inorder to have a direct comparison with their results wehave utilized the same form of the EEDF as adoptedby them [1–3] but included an extensive set of our owncalculated electron-impact RDW cross sections. This willenable us to gauge the effectiveness of including our de-tailed cross sections for the required transitions betweenfine-structure levels. It is important to emphasize thatBoffard et al. [1–3] have used their experimentally mea-sured optical emission cross sections to obtain differentplasma parameters while we use our calculated RDW di-rect excitation cross sections in our CR model. Thus wehave adopted a systematic theoretical approach to extractthe plasma parameters by developing a CR model whichenables us to compare with experimental results. We firstcalculate the populations of various fine-structure levelsincorporated in our model with the chosen form of theEEDF. We obtain the effective electron temperature atvarious pressures by taking the one which yields the bestagreement of the calculated populations of the 1si lev-els with the OAS measurements. We then compare thistemperature with probe measurements as well as emis-sion model results of Boffard et al. [1–3]. We have alsocalculated the intensities of the two lines at 420.1 and419.8 nm and compared them with the measured intensi-ties of Boffard et al. [1].

2 CR model

In the present model we have incorporated 40fine-structure levels in addition to the ground state of Aras well as the ground state of Ar+. The ground state of Arhas a closed shell configuration 1s22s22p63s23p6 abbrevi-ated as 3p6 and the ion ground state has the configuration3p5. We also include the excited levels with configurations3p54s (1si,i = 2–5), 3p54p (2pi, i = 1–10), 3p53d (3di,i = 1–12), 3p55s (2si, i = 2–5) and 3p55p (3pi, i = 1–10).The various collisional and radiative processes consideredin our model are shown in Figure 1. These include elec-tron impact excitation from the ground state to all thefine–structure levels considered in our model as well asexcitation from all the 1si levels to the 2pi levels, electronimpact population transfer processes among the individ-ual 1si and 2pi levels and spontaneous radiation transferprocesses including the effects of radiation trapping. Elec-tron impact ionization from the ground state as well asfrom the 1si levels has also been included. In addition,the present model also includes the excitations from themetastable states 1s3 and 1s5 to the 3pi excited manifoldswhich were not considered in our previous CR model.

Here we have used the following generalized form ofthe electron energy distribution function which has two

Eur. Phys. J. D (2013) 67: 203 Page 3 of 9

Fig. 1. Energy level diagram for Arshowing various collisional and radia-tive processes included in our CRmodel. The solid lines represent exci-tations from the ground state, the dashdotted lines show excitations from the1s5 and 1s3 metastable states, the dot-ted lines excitations from the 1s2 and1s4 resonance levels while the wavylines represent radiative transitions.

adjustable parameters (x, Tx) [1–3]

fx(E) = c1T−3/2x

√Ee−c2(E/Tx)x

(1)

where

c1 = x

(23

)3/2 [Γ

(52x

)]3/2

[Γ

(32x

)]5/2

and

c2 =(

23

)x[

Γ(

52x

)Γ

(32x

)]x

(2)

and Γ (ξ) is the usual gamma function. For x = 1.0 thisform reduces to the Maxwellian distribution. Increasingthe value of x from unity represents a deviation from thesimple Maxwellian distribution and exhibits the suppres-sion of the high energy portion of the electron energy dis-tribution occurring in ICP plasmas [3]. To get an appropri-ate value of x and the corresponding electron temperatureTx we follow Boffard et al. [2,3]. We first chose a value of xstarting from 1.0 and varied the electron temperature toget the value of Tx which provides the best agreement ofthe populations for each 1si level with the OAS measure-ments of [3]. Similarly, another value of x was taken andthe corresponding Tx was evaluated and so on. Our EEDFwith x = 1.2 and the corresponding value of Tx providesthe best representation of the results obtained from boththe OAS and probe measurements. This value of x is thesame as that obtained by Boffard et al. [1–3]. We will dis-cuss this point further when we present our calculationsof the intensities of line emissions.

The modified rate coefficient kij for excitation from alower level i to an upper level j are obtained from the

following equation

kij =

√2m

∞∫Eij

σij(E)EFx(E)dE (3)

where m is mass of the electron, σij is the electron-impactexcitation cross section which is a function of electron en-ergy E, Fx(E) is the electron energy probability function(EEPF) which is related to the EEDF by the relationfx(E) = E1/2Fx(E) and Eij is the energy difference be-tween the lower level i and upper level j which is also thethreshold energy for this transition. Similarly, the rate co-efficients ki+ for the ionization of level i can be obtainedfrom equation (3) using ionization cross-sections σi+ inplace of excitation cross-section σij . For the inverse pro-cess, i.e. electron impact de-excitation and three-particlerecombination, the detailed balance principle [12,24] hasbeen used and the rate coefficient of electron-impact de-excitation process is given by

kji =gi

gj

√2m

∞∫Eij

σij(E)EFx(E − Eij)dE (4)

where gi and gj are the statistical weights of levels i andj, respectively.

We take the rate-coefficient for three particle recombi-nation using the Saha relation [24] as

k+i =gi

2g+

(h2

2πmTx

)3/2 √2m

∞∫Ei+

σi+(E)Fx(E − Ei+)dE

(5)where g+ is the statistical weight of the ion state, h isPlanck’s constant and Ei+ is ionization energy of the

Page 4 of 9 Eur. Phys. J. D (2013) 67: 203

ith level. In calculating the rate coefficients from equa-tion (3) we have used our RDW cross sections [12–17].The required electron impact ionization cross sectionshave been taken from reported measurements [25,26]. Ra-diation trapping i.e. re-absorption of emitted radiationfrom resonance and metastable levels affects the popu-lations of non-resonant levels by changing branching frac-tions [4,20]. Therefore reabsorption of radiation by popu-lations in lower levels will cause a decrease in branchingfractions for these emission lines along with an increase inother emission lines from the same upper level [20]. Thusit is important to include an accurate description of radi-ation trapping processes in the model. We have includedthis effect through changing the Einstein coefficients Aji

to the effective radiative decay coefficients Aeffji, expressed

as Aeffji = Λ(Kjiρ)Aji where Λ is the photon escape fac-

tor. There are various approximations available for Λ ofvarying complexity. In the pressure range considered inthe present work only Doppler broadening is importantIn this case assuming a uniform distribution of emittingand absorbing atoms, we can calculate the escape factorsusing the following formula given by Mewe [20]

Λ (Kjiρ) =2 − e−Kjiρ/1000

1 + Kjiρ(6)

where Kji is the reabsorption coefficient and ρ is a char-acteristic scale length of the source plasma which is25 cm [20]. Reabsorption coefficients Kji are expressedas

Kji (ni) =λ3

ji

8π3/2

gj

giniAji

√M

2RTg(7)

where M is the atomic mass, Tg is the gas temperature,R is the gas constant and ni is the number density of thelower level and λji is the wavelength of the photon for thetransition from level j to i. This formula has been usedin our earlier paper [12] as well as by Zhu and Pu [18]and Boffard et al. [20] in their work. However, as a fur-ther check on the appropriateness of this approximation,we have compared our branching ratio with the measure-ments of Boffard et al. [20]. The branching fractions Γji

have been calculated using the following equation givenby Boffard et al. [20]

Γji =Λ (Kjiρ)Aji∑l

Λ (Kjlρ)Ajl. (8)

The particle balance equation can now be expressed as

41∑i=1i�=j

kij(Tx)nine +∑i>j

Aeffij ni + nen+nek+j(Tx)

−41∑

i=1i�=j

kji(Tx)njne −∑i<j

Aeffji nj − njnekj+(Tx) = 0 (9)

where ne, n+ and Tx are the electron density (in cm−3),ion density (in cm−3) and electron temperature (in eV),

respectively. We calculate the population of various ex-cited states by solving these equations following themethod described in our previous paper [12]. Since thereabsorption factors given in equation (7) contain the un-known level populations, these can be calculated itera-tively. If we take Aeff

ji = Aji initially, equations (9) canbe solved. Using these values for the level populations,Aeff

ji can be calculated using equation (7) and inserted intoequation (9) which can be solved as before [12]. This pro-cess is repeated until converged results for the populationsare obtained.

3 Results and discussions

In Figure 2 we present the population densities of the 1si

levels at various pressures in the range of 1–25 mTorr. Wecompare our results with the OES and OAS measurementsof Boffard et al. [3]. The OAS measurements are reportedfor the populations of the 1s3, 1s4 and 1s5 levels while theOES measurements which have been obtained from thevariation in branching fractions for Ar (3p54p → 3p54s)transitions are available for the 1s4 and 1s5 levels only.Boffard et al. [3] found that the populations of the res-onance levels, viz. 1s2 and 1s4, were nearly equal andtherefore reported only the population of the 1s4 level.In order to compare their results with ours we have as-sumed the equality of their 1s2 and 1s4 populations asshown in Figure 2.

From Figure 2a, we find that our results at a pressureof 1.0 mTorr for the populations of the 1s2, 1s4 and 1s5

levels are in very good agreement with the OAS measure-ments. While the OES measurement is very close to theOAS one for the 1s5 level, it produces a much lower valuefor the population density of the 1s4 level as compared toours and the OAS measurement. We also observe that ourpopulation densities for the 1s2 and 1s4 levels are nearlyequal at this pressure as observed by Boffard et al. [3]. Infact, the population density ratio for the resonance levels1s4 and 1s2 which have very similar energies is approx-imately unity which is consistent with having the samestatistical weights as explained by Boffard et al. [20]. Fur-ther, the comparison of the population density for 1s3 levelshows that our calculation produces larger values than theOAS measurement. The reason for this is discussed laterin the paper.

Further, as we go from Figure 2a to Figure 2g i.e. asthe pressure is increased from 1.0 to 25 mTorr, we ob-serve some noticeable trends in the comparison of our re-sults with the experiments. For the 1s5 level, the goodagreement of our results with the two sets of measure-ments [3] continues within the experimental uncertainty.As the pressure increases, the OES results for the 1s4 levelbecome much closer to the OAS measurements and in verygood agreement with our calculations. We also observethat our population densities for the 1s2 level are slightlysmaller than that for the 1s4 level but both are in goodagreement with the measurements except the OES valuesat the two lowest pressures. For the 1s3 level, our popula-tion density is consistently approximately a factor of two

Eur. Phys. J. D (2013) 67: 203 Page 5 of 9

Fig. 2. Population densities of the 3p54s (1si) levels for a 600 W ICP Ar plasma at seven pressures in the range of 1–25 mTorr:(a) 1.0; (b) 2.5; (c) 5; (d) 10; (e) 15; (f) 20; (g) 25 mTorr.

higher than the OAS measured values at all pressures. Wenote that the ratio of our population densities for the 1s5

and 1s3 levels is ∼4.5 at all pressures which is close totheir statistical weight ratio of 5 and thus is quite con-sistent theoretically. Boffard et al. [20] observed the valueof this ratio as close to six from their experiments, conse-quently their population density for the 1s3 level obtainedfrom OAS measurement is lower at all pressures than ours.

We further observe from Figure 2 that for low pressuresthe population densities of the metastable states 1s5 and1s3 are larger than those of resonance 1s4 and 1s2 lev-els. However, the population densities of the resonancelevels increase with the increase in pressure and becomelarger than the population density of 1s3 metastable level.In fact, the transitions from resonance levels with J = 1to the ground state with J = 0 are allowed transitionswhereas the decay of the 1s5 and 1s3 levels with J = 2, 0 tothe ground state is dipole forbidden. With the increase in

pressure the ground state population also increases. Con-sequently, there is a considerable increase in the popula-tion densities of the resonance levels due to radiation trap-ping causing by the re-absorption of resonance photons byatoms in ground state thus enlarging their lifetimes at highpressures and high electron densities [3,20].

In Figure 3 we present the 1s5 metastable level popu-lation fraction relative to the ground state population asa function of pressure along with the OAS and OES mea-surements of Boffard et al. [1,3]. Here again we see thevery good agreement between our results and the OASmeasurements at all pressures except the lowest one whereour results differ only by 7.1% of the measured value whichis still within the experimental uncertainty. There is somedeviation between the OES and OAS measurements atseveral pressures but these are within the experimentaluncertainties.

Page 6 of 9 Eur. Phys. J. D (2013) 67: 203

Fig. 3. Variation of the metastable fraction (n1s5/n0) as afunction of pressure for 600 W Ar ICP plasma.

Boffard et al. [20] reported the branching fractions forAr (3p54p → 3p54s) transitions obtained from photonemission measurements and compared their results withthe ones calculated using transition probabilities which donot take into account the radiation trapping effect and dis-cussed the significance of radiation trapping for the use ofthese transitions in optical plasma diagnostics. Since wehave also incorporated radiation trapping effects in ourmodel as described in Section 2, it is of interest to com-pare our calculated branching fractions with both the the-oretical and experimental results reported in [20]. We cal-culated branching fractions using equation (8) at all sevenpressures considered but as an illustration we present ourcalculated branching fractions at 1 mTorr pressure in Ta-ble 1 as well as the results of Boffard et al. [20]. FromTable 1 we observe that there is close agreement of ourcalculated branching fractions with the experimentally ob-tained values and both these results differ significantlyfrom the theoretical results obtained without radiationtrapping [20]. The branching fraction results at other pres-sures not presented here show similar behavior. This illus-trates the importance of radiation trapping as well as jus-tifying the method we use to account for it in our model.

Electron energy distribution and the mean electronenergy denoted by the electron temperature are the ba-sic parameters needed to understand the physics driv-ing the plasma. We have obtained the effective electrontemperature of the plasma at a fixed pressure by vary-ing the temperature in our model to get the best fit ofour calculated population for the 1si levels to the cor-responding population of OAS measurements of Boffardet al. [3]. Boffard et al. [1] extracted electron temperaturefrom probe measurements as well as from the measuredoptical emission of the lines at 420.1 (3p9 → 1s5) and419.8 (3p5 → 1s4) nm. The electron temperature fromthese optical measurements was extracted using a coronamodel and an extended corona model. The results ob-tained with the corona model are denoted as line pair

Table 1. Comparison of branching fractions for Ar(3p54p → 3p54s) transitions calculated from our CR modelpopulations using equation (8) with branching fractions ob-tained from photon emission measurements as well as calcu-lated using transition probabilities without taking into accountradiation trapping [20] for a 600 W ICP plasma at a pressureof 1.0 mTorr.

Transition λ (nm) Γji (no radiation Γji (our Γji [20]trapping) [20] model)

2p1 → 1s2 750.39 0.995 0.994 0.9962p1 → 1s4 667.73 0.005 0.006 0.004

2p2 → 1s2 826.45 0.434 0.541 0.5312p2 → 1s3 772.45 0.332 0.215 0.2612p2 → 1s4 727.29 0.052 0.069 0.0602p2 → 1s5 696.54 0.181 0.175 0.148

2p3 → 1s2 840.82 0.645 0.638 0.6322p3 → 1s4 738.40 0.245 0.271 0.2882p3 → 1s5 706.72 0.110 0.091 0.080

2p4 → 1s2 852.14 0.419 0.645 0.6192p4 → 1s3 794.82 0.561 0.324 0.3582p4 → 1s4 747.12 0.001 0.001 0.0002p4 → 1s5 714.70 0.019 0.030 0.022

2p5 → 1s2 858.10 0.000 0.036 0.0002p5 → 1s4 751.47 1.000 0.964 1.000

2p6 → 1s2 922.45 0.146 0.311 0.3932p6 → 1s4 800.62 0.142 0.309 0.2992p6 → 1s5 763.51 0.712 0.380 0.309

2p7 → 1s2 935.42 0.031 0.036 0.0392p7 → 1s3 866.79 0.072 0.063 0.0732p7 → 1s4 810.37 0.743 0.778 0.7532p7 → 1s5 772.38 0.154 0.123 0.135

2p8 → 1s2 978.45 0.046 0.060 0.0502p8 → 1s4 842.46 0.667 0.780 0.8162p8 → 1s5 801.48 0.288 0.160 0.133

2p9 → 1s5 811.53 1.000 1.000 1.000

2p10 → 1s2 1148.80 0.007 0.016 0.0212p10 → 1s3 1047.00 0.038 0.068 0.0872p10 → 1s4 965.78 0.213 0.452 0.4662p10 → 1s5 912.30 0.741 0.464 0.426

ratios (LPR) and with the extended corona model as LPR(advanced). They also obtained electron temperature fromthe analysis of Ar (2pi → 1si) emissions [3] in combinationwith the corona model. The LPR analysis utilizes mea-sured apparent cross sections from the ground and ex-cited 3p54s metastable levels and the metastable fraction(n1s5/n0) from OAS measurements to extract the elec-tron temperature whereas LPR (advanced) calculationsincludes additional contributions from electron excitationfrom the resonance 1s4 and 1s2 levels, radiation trappingof the 3pi → 1si radiation and electron collisions with

Eur. Phys. J. D (2013) 67: 203 Page 7 of 9

Table 2. Comparison of the effective electron temperature used in our CR model with the results of Boffard et al. [1,3] for a 600 WAr ICP plasma. LPR: temperature extracted from emission model line pair ratio analysis; LPR (adv.): temperature extractedfrom emission model line pair ratio advanced analysis including additional contributions as compared to LPR; 2pi → 1si: resultsobtained from the analysis of 2pi → 1si emissions; probe: temperature extracted from Langmuir probe measurements.

Pressure Tgas ne Present Teff (eV)

(mTorr) (K) (1010 cm−3) model Present LPR LPR (adv.) 2pi →1si Probe(n1s5/n0) model [1] [1] [1,3] [1]

1.0 600 4.7 2.23 × 10−3 5.6 5.7 5.6 6.5 5.4 ± 0.542.5 670 8.3 1.45 × 10−3 4.2 4.7 4.9 5.4 4.4 ± 0.445.0 745 13 8.97 × 10−4 3.4 3.8 4.0 3.8 3.8 ± 0.3810 755 26 4.56 × 10−4 2.7 3.0 3.3 3.2 3.1 ± 0.3115 757 40 3.08 × 10−4 2.4 2.4 2.6 2.7 2.8 ± 0.2820 835 54 2.67 × 10−4 2.2 2.2 2.4 2.5 2.7 ± 0.2725 850 72 1.82 × 10−4 2.0 2.1 2.3 2.4 2.6 ± 0.26

excited 3pi atoms [1]. In Table 2, we have compared ourelectron temperature results with the four different set ofresults of Boffard et al. [1,3] for the same plasma operatingconditions. We observe that all the different sets of effec-tive electron temperature reported by Boffard et al. [1,3]as well as our theoretical calculations are in reasonableagreement among themselves and lie within the experi-mental uncertainty. In particular, the agreement of ourresults with the probe measurements is good at lower pres-sures while at higher pressures our values become lowerand are close to the LPR results. Our temperatures differfrom probe measurements by only 0.2–0.6 eV. Our tem-perature is determined from choosing an EEDF with theform given in equation (1) which produces the best overallagreement for the population of the 1si levels. While thisgives temperature values in good agreement with the mea-surements at lower pressures it appears to systematicallyunderestimate the temperature as the pressure increases.Also at higher pressures the higher levels can have signifi-cant populations and perhaps by taking into account morelevels one may get better temperature values. We intendto investigate this further by including the higher levels inour CR model. Further, we have fitted our temperature vspressure results to an analytic formula in order to enablecalculations at arbitrary pressures without correspondingexperimental measurements. We find that to a very goodapproximation our Teff results in eV can be fitted to theform Teff = aP b where P is the pressure in mTorr and thecoefficients have values a = 5.6093 and b = −0.31497.

Intensity measurements for the 419.8 nm and 420.1 nmemission lines which originate from transitions betweenthe 3p5 → 1s4 and 3p9 → 1s5 levels, respectively, were alsoreported by Boffard et al. [1] as a function of pressure. Theintensity of an emission line is directly proportional to thepopulation density of the corresponding upper level fromwhich they decay [12,27]. Consequently, we can write theintensities of the 419.8 nm and 420.1 nm emission lines,I419.8 and I420.1, as [27],

I419.8

Pg= C419.8X3p5 (10)

I420.1

Pg= C420.1X3p9 (11)

Fig. 4. Comparision of the intensities of the 419.8 nm and420.1 nm lines with the measurements of Boffard et al. [1] fora 600 W Ar ICP plasma at various pressures in the range 1 to25 mTorr.

where Pg is the gas pressure, X3p is the relative popula-tion density which is given as X3p = n3p/n0, where n0,n3p are the total atom density and the absolute popula-tion density of the 3p level respectively. C419.8 and C420.1

are calibration constants dependent on the experimentalapparatus.

From our present CR model we obtained the popula-tions of the 3p5 and 3p9 levels at different pressures. Usingequations (10) and (11) we calculated the calibration con-stants C419.8 and C420.1 using the intensities of these linesreported by Boffard et al. [1] at pressure 15 mTorr. Thisallows us to calculate the intensities of the 420.1 nm and419.8 nm lines at different pressures. The cascade con-tribution from higher levels to the population of the 3pi

levels is relatively small as compared to the 2pi levels dueto their low number densities and small transition prob-abilities [4] and thus can be neglected. Boffard et al. [1]calculated intensities of the 419.8 nm and 420.1 nm linesusing their extended corona model in which they use theOAS measurements for excitation out of the 1s5 level.

Page 8 of 9 Eur. Phys. J. D (2013) 67: 203

In Figure 4, we present the comparison of our calcu-lated intensities of the 419.8 nm and 420.1 nm lines withthose reported by Boffard et al. [1]. We observe that ourresults at all pressures for both the lines are in reason-ably good agreement with the measurements except atthe lowest two pressures where our values for the 419.8 nmand 420.1 nm lines, respectively, are approximately 31%and 48% lower than the measurements. To investigatethis difference between our calculations and the experi-mental measurements, we compare our RDW electron im-pact cross sections [13,15] from the ground state (1S0) andmetastable state 1s5 to the 3p5 and 3p9 levels with the ex-perimental cross sections of Boffard et al. [21–23] at higherincident electron energies (which correspond to lower pres-sures). As reported in our previous papers [13,15], thereis very good agreement between our RDW cross sectionsand experimental values for the dipole allowed 1s5 (J = 2)to 3p9(J = 3) transition. However, for 1s5 (J = 2) to3p5 (J = 0) forbidden transition our results are lowerthan the experimental results and decrease at a fasterrate with increasing incident electron energy. A similardiscrepancy is observed for the forbidden transition fromthe ground state (J = 0) to the 3p9(J = 3) level as ex-plained in [13,15].

In Figure 5a we present the intensity line pair ratiofor the 420.1 and 419.8 nm lines as a function of pres-sure and compare with the measurements of line ratios ofBoffard et al. [1]. We show our intensity line pair ratioresults calculated with three different values of x viz. 1.0,1.2 and 1.4 in the EEDF to ascertain if x = 1.2 is stillthe best value for the pressure range considered in thepresent study. We observe that our calculations with anEEDF with x = 1.2 clearly produces the best agreementwith the measurements of Boffard et al. [1] at higher pres-sures while at lower pressures, the EEDF with all threedifferent values of x give very similar results. Thus we canconclude that the form of the EEDF is not the reason forthe discrepancy at lower pressures and is more likely tobe the high energy form of our forbidden cross sections.

Further, in Figure 5b we have plotted a comparison ofour line ratios calculated using a non-Maxwellian EEDFwith x = 1.2 with the measurements [1] as a function ofelectron temperature. From Figure 5 we see the expecteddirect relationship of the line ratio with pressure but aninverse relationship with electron temperature similar tothat observed by Boffard et al. [1]. However, in Figure 5bwe find our results are shifted to lower temperatures com-pared to the measurements. We have discussed above apossible reason for this difference at higher temperatures(which corresponds to lower pressures) while the differ-ence at lower temperatures (which corresponds to higherpressures) is due to the fact that our calculated temper-ature are lower than the probe measurements at higherpressures as explained in our discussion of Table 2.

4 Conclusions

In the present paper we have extended our earlier CRmodel [12] for an inductively coupled argon plasma byusing a non-Maxwellian EEDF and compared our results

Fig. 5. Comparison of our calculated intensity ratioI420.1/I419.8 with measurements [1] for a 600 W Ar ICP plasma(a) as a function of pressure (b) as a function of effective elec-tron temperature. Error bars represent uncertainties in theprobe measurements.

with the detailed experimental measurements reported byBoffard et al. [1–3,20] in the pressure range of 1–25 mTorr.The agreement of our calculated 1s4 and 1s5 populationdensities with the optical absorption and emission mea-surements is good at all the pressures and within theexperimental uncertaintes while our calculated popula-tion of the 1s3 level is somewhat larger than the ex-perimental values [3] but consistent with the statisticalweight ratio of 1:5 with respect to the population of the1s5 level. Our calculated branching fractions for the Ar(3p54p → 3p54s) transitions show good agreement withthe measured values of Boffard et al. [20] and thus confirmthat we have incorporated radiation trapping effect prop-erly in our model. The electron temperatures obtainedfrom our fitting of the EEDF show good agreement withthe probe measuements [1] except at higher pressures. Wefeel that the results at higher pressures may be improvedby including levels above 3pi with their related processes.

Our calculated intensities for the 419.8 and 420.1 nmlines show a similar variation of the ratio of these lines asa function of both pressure and electron temperature asreported by Boffard et al. [1] and are in good agreement

Eur. Phys. J. D (2013) 67: 203 Page 9 of 9

with the measured results except at the lowest two val-ues of the pressure. We find that at these pressures theresults are insensitive to the form of the EEDF thus thedisagreement with the measurements are more likely dueto the differences in the cross sections used in our modelas compared to the measured values.

Thus our present CR model with our detailed the-oretical relativistic electron-impact cross sections alongwith a non-Maxwellian EEDF as given in equation (1)with x = 1.2 describes reasonably well the various exper-imental results for an ICP plasma in the pressure range1–25 mTorr. It also confirms the improvement over ourprevious CR model where we used a Maxwellian EEDFi.e. taking x = 1. However, in view of some of the disagree-ments of our results with the experiment, there are furtherimprovements which could be made to our present model.We can include more processes, e.g. electron excitationto and from the 3pi and higher levels using our relativis-tic distorted wave method to calculate the required crosssections. At present there are no theoretical data availablefor these transitions. Radiative decay from levels above 3pi

should also be included and possibly the effects of atom-atom collisions as well. We will also investigate methodsof more accurately determining the electron temperaturefrom our present form of the EEDF or use a more generalform such as the two temperature model [7,19] mentionedearlier as part of our future work on plasma modeling.

The authors wish to thank Dr. J.B. Boffard for sending us theresults of their OAS measurements in numerical form. AuthorR.S. would like to acknowledge research grants in support ofthis work from IAEA Vienna. R.K.G. and Dipti are thankfulto the Council of Scientific and Industrial Research (CSIR),New Delhi for providing financial assistance. A.D.S. wishes toacknowledge a grant from NSERC Canada.

References

1. J.B. Boffard, R.O. Jung, C.C. Lin, L.E. Aneskavich, A.E.Wendt, J. Phys. D 45, 045201 (2012)

2. J.B. Boffard, R.O. Jung, C.C. Lin, L.E. Aneskavich, A.E.Wendt, Plasma Sources Sci. Technol. 20, 055006 (2011)

3. J.B. Boffard, R.O. Jung, C.C. Lin, A.E. Wendt, PlasmaSources Sci. Technol. 19, 065001 (2010)

4. J.B. Boffard, C.C. Lin, C.A. De Joseph Jr., J. Phys. D 37,R143 (2004)

5. V.M. Donelly, J. Phys. D 37, R217 (2004)6. E. Gargioni, B. Grosswendt, Rev. Mod. Phys. 80, 451

(2008)7. X.M. Zhu, Y.K. Pu, Plasma Sci. Technol. 13, 267 (2011)8. X.M. Zhu, Y.K. Pu, J. Phys. D 43, 403001 (2010)9. U. Fantz, Plasma Sources Sci. Technol. 15, S137 (2006)

10. S. Samukawa, M. Hori, S. Rauf, K. Tachibana, P.Bruggeman, G. Kroesen, J.C. Whitehead, A.B. Murphy,A.F. Gutso, S. Starikovskaia, U. Kortshagen, J.P. Boeuf,T.J. Sommerer, M.J. Kushner, U. Czarnetzki, N. Mason,J. Phys. D 45, 253001 (2012)

11. R.A. Dressler, Y.H. Chiu, O. Zatsarinny, K. Bartschat, R.Srivastava, L. Sharma, J. Phys. D 42, 185203 (2009)

12. R.K. Gangwar, L. Sharma, R. Srivastava, A.D. Stauffer,J. Appl. Phys. 111, 053307 (2012)

13. R.K. Gangwar, L. Sharma, R. Srivastava, A.D. Stauffer,Phys. Rev. A 81, 052707 (2010)

14. R. Srivastava, A.D. Stauffer, L. Sharma, Phys. Rev. A 74,012715 (2006)

15. L. Sharma, R. Srivastava, A.D. Stauffer, Phys. Rev. A 76,024701 (2007)

16. L. Sharma, R. Srivastava, A.D. Stauffer, Eur. Phys. J.D 62, 399 (2011)

17. S. Kaur, R. Srivastava, R.P. McEachran, A.D. Stauffer, J.Phys. B 31, 4833 (1998)

18. X.M. Zhu, Y.K. Pu, J. Phys. D 43, 015204 (2010)19. X.M. Zhu, Y.K. Pu, Y. Celik, S. Siepa, E. Schungel,

D. Luggenholscher, U. Czarnetzki, Plasma Sources Sci.Technol. 21, 024003 (2012)

20. J.B. Boffard, R.O. Jung, C.C. Lin, A.E. Wendt, PlasmaSources Sci. Technol. 18, 035017 (2009)

21. T. Weber, J.B. Boffard, C.C. Lin, Phys. Rev. A 68, 032719(2003)

22. R.O. Jung, J.B. Boffard, L.W. Anderson, C.C. Lin, Phys.Rev. A 75, 052707 (2007)

23. J.B. Boffard, B. Chiaro, T. Weber, C.C. Lin, At. DataNucl. Data Tables 93, 831 (2007)

24. A. Hartgers, J. van Dijk, J. Jonkers, J.A.M. van derMullen, Comput. Phys. Commun. 135, 199 (2001)

25. H.C. Straub, P. Renault, B.G. Lindsay, K.A. Smith, R.F.Stebbings, Phys. Rev. A 177, 597 (1995)

26. M.A. Ali, P.M. Stone, Int. J. Mass Spectrom. 271, 51(2008)

27. A. Palmero, E.D. Hattum, H. Rudolph, F.H.P.M.Habraken, J. Phys. D 101, 053306 (2007)