reconstruction of electron energy distribution …ddodt/validation_08_dodt_as_held.pdfdirk dodt for...

Dirk Dodt1, Andreas Dinklage1, Rainer Fischer1, Klaus Bartschat2,Oleg

Zatsarinny2

[email protected]

Reconstruction of Electron Energy Distribution Functions from Optical Emission Spectroscopy

1Max-Planck-Institut für Plasmaphysik (IPP)

2Drake University, Des Moines

Dirk Dodt for the 5th Workshop on Data Validation 2

Outline

● Motivation

● Model system: neon discharge

● Integrated Data Analysis of spectroscopic data:

– Physical Model

– Statistical Model● Error statistics of measurement● Systematics of measurement

– Uncertainty of spectrometer response (apparatus function)– Uncertainties of atomic data (Plasma Model)

● Results for reconstructed EEDF

● Summary


Motivation

● Low-temperature plasmas: Lighting, plasma processing

– EEDF is key quantity for control and optimization

● Edge of fusion plasmas (neutral beam diagnostics)

[1] B. Schweer, G.Mank and A.Pospieszczyk, B Brosda,B. Pohlmeier: J. of Nucl. Mat. 196-198, p.174 (1992)

[2] R. Fischer and V. Dose: Plasma Phys. Contr. Fusion 41, 1109 (1999)

– Divertor physics, impurities, ...

● Spectroscopy is non-invasive alternative to probe measurements


Neon Discharge

[2] D. Uhrlandt and St. Franke , J. Phys. D: Appl. Phys. 35 (2002) 680–688

– Well investigated system:● Validation with modeling

and measurements– Wide range of spectrum is

accessible– Well reproducible, easy to

handle

Discharge parameters: p = 0.89 mbarR = 1.5 cm,I = 10 mA


Forward Model

intensity calibration

emission coefficient

radiance

spectral radiancespectrometer pixels

line of sightintegration

line profile (aparatus width)

Einstein's coefficients(radiation transport)

EEDF

population densities

collisional - radiative model


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


Consistent Set of Cross-Sections and Einstein's Coefficients

● Excitation cross sections and Einstein's coefficients for 32 states of neon:

● ~150 cross sections

[3] O. Zatsarinny,K. Bartschat, J. of Phys. B 37 (10): 2173 (2004)

[4] Wetzel et al, Phys. Rev. A 35, 559 (1987)

● ~350 Einstein's coefficient

● Scale of cross section is more uncertain than shape


optical depth of transitions to metastable levels

● Transitions to metastable levels:

– absorber density not constant

– introduce effective densities

● Description using a single effective density per metastable level is possible!

● Local Emissivity:


Statistical Model: Likelihood

● Error Statistics of each pixel:

– fluctuation of dark current, photon statistics negligible● Gaussian approximation: Independent, normally

distributed errors:

Model

Measurement

Difference

● Scale of residuals much to big!


Uncertainty of Transfer Function

– Combined Uncertainty:

● Shifted and scaled isolated lines from different spectra

[6] V. Dose, R. Fischer, and W. von der Linden, Maximum Entropy and Bayesian Methods, edited by G. Erickson (Kluwer Academic, Dordrecht, 1998).

Model

Measurement

Difference

● Scale of residuals reasonable.


Uncertainty of Plasma Model

● Consistent fit of line intensities to spectrum

● Bayes Rule:

● Introduce nuisance parameters and marginalize out

– Consistent description of spectrum

– Propagation of uncertainties onto reconstruction result

Likelihood

probability independent from measurement (Prior)

Evidence: Normalization constant

I showed to you now the uncertainties of the modelling of the spectrometric measurements, which was leading to a consistent description of a measured spectrum

Now I want to discuss another example of a systematic effect taken into consideration, which is the assessment of uncertainties of atomic data

they are taken into account by using bayes rule to incoporate prior information


Uncertainties of Atomic Data

● Discrepancy between length and velocity form of dipole operator

Einstein's coefficients:

Uncertainty of cross-sections: Scale factor


Uncertainties of Model Parameters

Error propagation for ~ 300 model parameters


Result of Forward Model

Model Measurement Difference


Error band of EEDF

● Posterior

● Expectation Value for Parameter :

● Expectation Value for EEDF:

Likelihood

Priors


Form Free Reconstruction

[2] D. Uhrlandt and St. Franke , J. Phys. D: Appl. Phys. 35 (2002) 680–688

[2]● Spline allows arbitrary

deviations from Maxwellian distribution

● Log scale for spline

● Error band reflects information content of spectrum


Variation of EEDF

● Vary piecewise constant EEDF

● Effect on population densities:

● Excitation from ground state

● Stepwise excitation via metastable levels

● Cascades from higher levels


Variation of EEDF


Effect of Parameterizations

Maxwellian distribution

[2]

[2]

Druyvestein: two temperature distribution

● Parametrization with little flexibility reveals small error band: Inconsistent with reference!


Summary

● Spectroscopic Approach to EEDF interesting for physics of Plasma Edge as well industrial applications

● Model of spectroscopy on neon discharge, enhancements allow form-free reconstruction

● Integrated Data Analysis allows description of uncertainties of reconstruction

● Effect of different parameterizations was shown

● Results validated with results from hybrid modeling


Radiation Transport

● Emitted photons may be reabsorbed by atom in final state of transition

● Increase of apparent lifetime of transition:

[5] J E Lawler and J J Curry: J. Phys. D: Appl. Phys. 31 (1998) 3235–3242.

● Transitions to metastables:

– Radial variation of absorber density

– Introduce effective densities

– Two effective densities for 12 transitions


Line of Sight Integration

● Local emissivity:

● Line of sight integration

● Convolution with apparatus function

● Incorporation of intensity calibration


Outlook

● Apply EEDF reconstruction to spatial inhomogeneities in neon discharge

● Validate atomic data with improved spectral measurement

● Use approach to reconstruct oxygen ion impact dissociative cross-section from spectroscopic data


Dissociative Excitation Cross-section for Oxygen

Energy [eV]

cross

-sect

ion [

m2]

[8] Matyash, K.; Schneider, R.; Dittmann, K.; Meichsner, J.; Bronold, F. X.; Tskhakaya, D.: J. Phys. D: Appl. Phys., 40 (2007) 6601-6607


Error band of EEDF


IDA RECIPE

● clear statement of the problem and context information (D, I)

● data model

● quantification of prior knowledge

● inference: apply bayes theorem

● focusing: marginalisation

●

● errors are the key: quantification for automation


IDA: Problem Statement

● Spectroscopic Data: Overlapping Lines, low statistical error, calibration


Collisional-Radiative-Model

description of coupling of all population densities ( line intensities )


Convolution with Transfer Function


Statistical Approach

● Likelihood:

● Error statistics: probability to obtain measured spectrum for a set of model parameters

● But: need pdf for not

Gaussian distribution for every pixel


Bayes Rule

● Apply sum and product rule for conditional probabilities:

● Integrate out parameters not interested in:

● Additional parameters broaden

● Propagation of uncertainty of model parameters

Likelihood

probability independent from measurement

Normalization constant


Markov Chain Monte Carlo

● Single component Metropolis Hastings algorithm:

● Sample from distribution

● Estimate moments of marginal distributions


optical depth of transitions to metastable levels

● Transitions to metastable levels:

– absorber density not constant

– introduce effective densities

● Description using a single effective density per metastable level is possible!

● Local Emissivity:

reconstruction of electron energy distribution …ddodt/validation_08_dodt_as_held.pdfdirk dodt for...

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