Proceedings of ITC/ISHW2007

Integrated Analysis of Spectroscopic Data
Dirk Dodt1) , Andreas Dinklage1) , Rainer Fischer2) , Klaus Bartschat3) and Oleg Zatsarinny3)Max-Planck-Institut für Plasmaphysik, EURATOM Association1)
Wendelsteinstraße 1, 17489 Greifswald, Germany
2)
Boltzmanntraße 2, 85748 Garching, Germany
3)
Department of Physics and Astronomy, Drake University, Des Moines, IA 50311, USA
Spectroscopic data are analysed by fitting a collisional-radiative model to the emission spectrum of a
low-temperature plasma in the wavelength range of visible light. The inference procedure employs Bayesian
probability theory and accounts for all measurement and model uncertainties. This allows for the validation ofmodel parameters, such as atomic data obtained in recent close-coupling calculations, which are ore only partly
or not at all accessible by beam-type experiments. Initial results indicate that the spectroscopic data containsignificant information about some Einstein coefficients for spontaneous emission of less prominent spectral lines.
Keywords: Integrated Data Analysis, Optical Emission Spectroscopy, Neon, Edge Plasma, Atomic Data

1 IntroductionThe line radiation emitted by excited neutrals in a plasma
can be used to obtain information about characteristic
plasma parameters. More specifically, properties of theplasma species causing the excitation can be assessed. Theenergy dependence of the elementary processes allows, todetect deviations from thermal energy distribution functions. In the case of low-temperature plasmas, the main
excitation channel is electron impact excitation, and thus
the spectrum carries information about the electronic component of the plasma.
An established technique is to analyse the relative
magnitude of line intensities measured, for example, inneutral-beam plasma diagnostics to infer the electron density ne and the temperature T e [1]. The approach presented here uses a collisional-radiative model (CRM) toreconstruct electron energy distribution functions from awhole range of the spectrum of a cylindrical neon discharge plasma. The population of different atomic levels
is described by a set of balance equations. With these level
populations, the light emission along a line of sight can bemodelled. Hence, starting from the electron energy distribution function (EEDF), these steps lead to a full forward
model of the spectroscopic data, incorporating the intensities of 87 emission lines. The approach described in [2]is extended by a description of the full spectral data. Thebasic idea is to calculate the probability of fitting the forward model to a full spectrum. The inference is done in
the framework of Bayesian probability theory.
For the data analysis approach in Bayesian probability theory, it is necessary to describe all uncertainties involved. In addition to error statistics of the data, the uncertainties of model parameters are incorporated in the analysis. For the interpretation of spectroscopic data, Einsteinauthor’s e-mail: dirk.dodt@ipp.mpg.decoefficients and excitation cross sections are such model
parameters. The provision of a complete and validated data
set is a persistent issue in the interpretation of any spectroscopic measurement. The incorporation of the uncertainties of the extensive sets of atomic data is a challenging
issue, because it can involve the probabilistic descriptionof a great number of parameters. Since resolving this issue
is a prerequisite for any uncertainty assessment in the interpretation of such spectroscopic data, the validity of theemployed atomic dataset with respect to the spectroscopic
results is addressed below.
A particular benefit of the approach presented in thispaper results from the use of a full collisional-radiativemodel, which accounts for all correlations in the data, i.e.,
how the line intensities are affected by all transitions relevant to the observed spectrum. This enables using thespectral lines to assess the validity of the atomic data used
for the collisional-radiative modelling.
The spectroscopic data to be analysed are obtainedfrom a cylindrical neon discharge, a well-investigated system with various published results (e.g. [6, 7]) that canbe used for validation. The full set of atomic data being employed was calculated by a B-spline Breit-Pauli
R-matrix (close-coupling) model, as described in detail inrefs. [3, 4].

2 Data descriptive model of the spectroscopic measurement
The forward model maps the quantity of interest, the EEDF
~ (spectrofe (E) onto a simulation of the measured data D
meter pixels). It consists of a chain of different elements
described below. More details about the data model can befound in [5]. Schematically, we haveProceedings of ITC/ISHW2007

Z~ .
DIi j dV → L(λ) →|{z}l.o.s.
{z}measurement
X (Di − Dsim,i )2 
11~ I) = Q~ Θ,·exp −P(D|n22σ(2π)
σi iiiDifferent parameterisations of the energy distribution
are employed to determine electron collision rates for thecollisional-radiative model (CRM). Here, the data are welldescribed with a Druyvestein distribution. EEDFs derived
from hybrid modelling of neon discharges accounting fora kinetic treatment of the electrons [6, 7] are used to validate the obtained result. The CRM consists of a set ofbalance equations for the population densities ni of 31 excited states of neutral neon, taking into account populating
and depopulating elementary processes.
hi
The locally emitted power Ii j W/(m3 · sr) can bereadily obtained by multiplication with the inverse lifetimeof the excited states and the photon energy, and division by
the full solid angle (4π).
The radiation has to pass through the plasma before it
leaves the discharge device. The apparent lifetime of theexcited states is affected by the transport of photons if theplasma is optically thick, i.e., for transitions to the groundor metastable states of the atom [8]. The description of this
opacity, together with the integration along the line of sight(l.o.s.) of the spectrometer, yields the effective spectral
radiance L(λ) as a function of the wavelength λ.
The modelling of the actual measurement comprises
the translation of L(λ) into the detected signals and themapping of wavelengths to pixel numbers. This requires
details on the detector response, which were measured
with a standard light source (sensitivity calibration). The
wavelength mapping is fitted to the data within the reconstruction.
Figure 1 shows the result of the forward model together with the measurement. The parameters giving thebest fit are obtained by maximising the probability distri~ which isbution function (PDF) for the parameter vector Θ,~ D,~ I). It describes the probability
called the posterior P(Θ|of a certain parameter set to be true, given the measurement~ The posterior is set up according to Bayes’ rule:D.

The prior is describing the knowledge about the model
parameters which is not contained in the data, but originates from independent sources. For example, atomic
data that are subject to uncertainty may be described bymeans of a parameter with a prior distribution reflecting
the known confidence region.
The marginalisation theorem for probability distributions describes how to integrate out parameters ~η we are
not interested in:Z~~ ~η, I)d~η.
~~ Θ,P(D|Θ, I) =P(D|(3)fe (E)|{z}→ nikinetic theory|{z}collisional radiative model→ Ii j →|radiation transport~~ D,~ I) · P(Θ)~ I) = P(D|~ Θ,P(Θ|~P(D)(1)~ I) and the prior P(Θ).
~~ Θ,
It contains the likelihood P(D|~ is not taken into account, since it does
The evidence P(D)not affect the position maximum of the posterior and canbe deduced from the normalisation constraint. The likelihood corresponds to quantifying the probability of a certain outcome of the measurement, given the true and un~ For a Gaussian distribution
known state of the system Θ.
of the error statistic of the spectrometer pixels, it is givenby(2)In the data model there is no formal distinction between the so-called nuisance parameters ~η and the param~ The result of the projection down toeters of interest Θ.
the dimensions of the parameters of interest is a broadened
PDF compared to a posterior not taking into account additional parameters from the beginning. The parameters ofinterest in this analysis are either the EEDF or Einstein coefficients we are interested in for validation purposes. Atable with all relevant nuisance parameters can be found in[9].

3 Atomic Data for
Radiative Models

Collisional-

In the forward model a consistent set of atomic data,obtained from a B-spline Breit-Pauli R-matrix (BSRM)
model for the treatment of e−Ne collisions [3], was taken
as the starting point. The BSRM approach is based on
the close-coupling approach. In contrast to all other commonly used perturbative or non-perturbative methods togenerate such atomic data, the use of non-orthogonal,
term-dependent sets of atomic orbitals makes it possible toobtain fairly accurate descriptions of both the energy levels and the oscillator strengths with comparatively small
configuration-interaction expansions. In traditional methods with orthogonal sets of one-electron orbitals, a similar
accuracy can, in principle, be achieved by very large expansions using so-called pseudo-orbitals [3].
The major advantage of using this dataset is the fullprovision of all required data for the collisional-radiativemodel, which goes far beyond the availability in standard
databases such as [10]. Furthermore, the wavefunctions
for the basic states are derived consistently from commonstructure calculations. This means that all derived atomicdata, such as oscillator strengths and cross sections, depend on the same set of wavefunctions. From a detailed
analysis of the calculations, critical matrix elements canProceedings of ITC/ISHW2007
6107

420

106-2-4-6

105
560

580

600

620

640

residual/σLλ [W ]
m2 sr mMeasured Dataλ [nm]
64
20-2-4-6

residual/σ

6
3d9 - 3p6
43d8 - 3p6
23d11 - 3p60residual/σLλ [ 2 W ]m sr mMeasured Data

107106105680700720740760

λ [nm]

Lλ [ 2 W ]m sr mMeasured Data107

3d8 - 3p9
3d5 - 3p9
3d8 - 3p8 3d4 - 3p5

3d3 - 3p8

106-2-4-6

105800820840860880

λ [nm]

Fig. 1 Result of the forward model. The red line depicts the modelled spectrum, black crosses show the measurement and its uncertainties,
and the dashed line is the difference between the two in units of standard deviations. The labels in the lower part of the spectrum
mark lines whose Einstein coefficients are extracted from the spectrum.

Reconstructed EEDF

f(E) × Ne[m-3 eV-1]

be identified and, if appropriate, a larger uncertainty may
be assigned to these transitions.

15Druyvenstein

10

Maxwellian1014

4 Validation of Einstein Coefficients
Some Einstein coefficients entering the model are notavailable in the literature. An independent, experimentally
based validation of the result of the structure calculations
mentioned above is desirable and also needed before thedata are included in databases like [10].
For the analysis of the EEDF, the Einstein coefficientsare treated as nuisance parameters. Prior PDFs are assigned to them accounting for the uncertainty stated in [10]or specified according to a discussion of the BSRM results.
The chosen uncertainty contributes to the uncertainty of thereconstructed EEDF and is accounted for by the probabilistic approach providing a non-Gaussian error propagation.
Figure 2 shows the result of the reconstructed EEDF
from the spectroscopic data displayed in fig. 1, where theinferred EEDF is capable to describe all spectral features.
The reconstructed EEDF agrees within the error margin
with an EEDF derived from hybrid modelling of neon discharges [6, 7] up to energies of about 20 eV. The disagreement for higher energies was found to depend on the chosen parameterisation of the EEDF. This issue requires further studies, but the rate coefficients of the CRM that depend on the energy integral are barely affected by thesedeviations. Given this consistent result of the data reconstruction, i.e., within the relevant error margins, the emission spectrum is expected to contain also information onReference131010121011
101091005101520253035404550Energy [eV]Fig. 2 Result for the reconstructed EEDF. On the ordinate axis
the EEDF, multiplied with the reconstructed electron density, is plotted. The Maxwellian distribution shown forcomparison is not able to reproduce the reference distribution from hybrid modelling [6].

the atomic data.
For the validation of the Einstein coefficients they
are assigned a scale invariant prior probability distribution(Jeffrey’s prior [11]). This means the results of the BSRM
calculations are not included for these parameters and thewidth of the posterior distribution purely reflects the significance of the spectroscopic data for the considered coefficients.
The result of the analysis for the full set of available
Einstein coefficients is shown in fig. 3 Displayed are theprobabilities of the respective Einstein coefficients derived

Proceedings of ITC/ISHW2007

probability [a.u.]

Amodel / ABSRM

Einstein Coefficients Extracted from Spectroscopic Data

10210
110-1

10-3

4s4_3p104s2_3p103d12_3p10
3d4_3p104s4_3p93d9_3p9
3d8_3p9
3d6_3p9
3d5_3p9
3d3_3p9
3d2_3p94s4_3p84s3_3p84s1_3p8
3d11_3p8
3d8_3p8
3d7_3p8
3d6_3p8
3d5_3p8
3d4_3p8
3d3_3p8
3d1_3p84s4_3p74s2_3p7
3d12_3p7
3d8_3p7
3d6_3p7
3d4_3p74s4_3p6
4s3_3p64s1_3p6
3d11_3p6
3d9_3p6
3d8_3p6
3d7_3p63d6_3p6
3d5_3p6
3d4_3p6
3d3_3p6
3d1_3p64s4_3p54s2_3p5
3d12_3p5
3d8_3p53d6_3p5
3d4_3p54s4_3p44s3_3p44s1_3p4
3d11_3p4
3d9_3p4
3d8_3p4
3d7_3p4
3d6_3p4
3d5_3p4
3d4_3p4
3d1_3p44s3_3p3
4s1_3p3
3d11_3p3
3d7_3p3
3d1_3p3
4s4_3p24s2_3p2
3d12_3p2
3d8_3p2
3d4_3p2
4s3_3p14s1_3p1
3d11_3p1
3d7_3p13d1_3p1

10-2

Fig. 3 The marginal distributions of various Einstein coefficients. The value of the coefficient is divided by the result of the BSRM
calculations, with a value of 1.0 corresponding to full agreement (vertical left axis). The probability for a certain value of thecoefficient to be true, given the measured data and model assumptions, is color-coded on the right, with violet/blue indicating a
low and orange/red indicating a high probablity. The respective transition is given on the horizontal axis. The intervals depictedwith the black lines show the root-mean-square variance of the distributions.

from the spectroscopic measurement and normalised to theBSRM results. There are a few lines that can be determinedrather well, i.e., a narrow marginal posterior distribution isobtained. For example, a result consistent with the BSRM
predictions is found for the 3d8 → 3p6 transition, whereas
the value for 3d8 → 3p8 extracted from the spectroscopic
data deviates significantly from the BSRM result. Furtherstudies are needed to clarify whether this can be attributed
to shortcomings in the spectroscopic analysis or problems
with the BSRM calculations.

6 Acknowledgements

Figure 3 also reflects that the spectrum is not informative for many transitions, as indicated by a broad probability distribution for the respective transitions. Note thatsome transitions with a wavelength outside the measured
spectrum are also included in the fit. They influence thespectrum through their effect on the collisional radiativemodel. However, as one might expect, these transitions
have a particularly broad posterior distribution.

[1] B. Schweer, G. Mank, A. Pospieszczyk, B. Brosda, andB. Pohlmeyer, J. Nucl. Mat. 196, 174 (1992).
[2] R. Fischer and V. Dose, Plasma Phys. Control. Fusion 411109 (1999)[3] O. Zatsarinny and K. Bartschat, J. Phys. B: At. Mol. Opt.
Phys. 37, 2173 (2004)[4] O. Zatsarinny, Comp. Phys. Commun. 174, 273 (2006).
[5] D. Dodt, A. Dinklage, R. Fischer, and D. Loffhagen, AIP
CP 872 264 (2006)[6] D. Uhrlandt and S. Franke, J. Phys. D: Appl. Phys. 35, 680(2002)[7] S. Franke, diploma thesis Bilanzgleichungen zur Modellierung der Dynamik einer Neon-Glimmentladung, Greifswald 1996[8] J.E. Lawler and J.J. Curry, J. Phys. D: Appl. Phys. 31, 3235
(1998)[9] D. Dodt, A. Dinklage, and R. Fischer, to be published inAIP CP (2007)[10] Yu Ralchenko, F.-C. Jou, D.E. Kelleher, A.E. Kramida,A. Musgrove, J. Reader, W.L. Wiese, and K. Olsen (2007).
NIST Atomic Spectra Database (version 3.1.3), [Online].
Available: http://physics.nist.gov/asd3 [2007, October 11].
National Institute of Standards and Technology, Gaithersburg, MD, USA.
[11] D.S. Sivia: Data Analysis: A Bayesian Tutorial, 2nd ed.
(Oxford University Press, USA, 2006)

5 Conclusions
A full forward model for an emission spectroscopic measurement of a low-temperature plasma was used in a
Bayesian data analysis. The electron energy distribution of
the plasma was inferred and its error band resulting fromuncertainties in the underlying atomic data was described.
The extensive data set of excitation cross section obtainedby recent close-coupling calculations allowed for the validation of Einstein coefficients for spontaneous emission,
which have not been measured to date.

This work was funded by Deutsche Forschungsgemeinschaft through Sonderforschungsbereich Trans Regio 24(Fundamentals of Complex Plasmas). KB and OZ were
supported by the United States National Science Foundation under grants PHY-0244470 and PHY-0555226.

7 References