Electron energy distribution reconstruction in low-pressure ... - CiteSeerX

Electron energy distribution reconstruction in low-pressure helium plasmas from optical measurements Plasma Phys. Control. Fusion 41 (1999) 1109-1123

R. Fischery and V. Dose

Max-Planck-Institut fur Plasmaphysik, EURATOM Association, Boltzmannstr. 2, D-85748 Garching, Germany

Abstract. Electron energy distribution functions (EEDFs) in electron-cyclotron resonance heated low-pressure helium plasmas were reconstructed form-free from 8 helium spectral line intensities. The extreme ill-posedness of the inversion problem due to the small number of noisy data is tackled with Bayesian data analysis by quanti cation of all relevant information and combining them within a probabilistic theory. The line intensities are calculated with a stationary collision-radiative model including de-excitation of the meta-stable states by wall collisions. The uncertainties of the cross sections of electron impact excitation are taken into account as they correspond in size with the errors of the measured line intensities. The most probable form-free EEDF and its con dence interval describing only the signi cant information content in the data shows a distinct non-Maxwellian form with a depressed tail in the range of inelastic collisions. PACS numbers: 52.70.-m, 52.20.-j, 02.50.-r, 07.05.Kf

y Email: [email protected]

Electron energy distribution reconstruction

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1. Introduction The electron energy distribution function (EEDF) in a plasma is one of the most important quantities for fusion and process plasmas. The knowledge of the EEDF is essential for modeling the chemical reactions in plasmas, growth processes in thin lm deposition, surface treatment with plasmas, and wall erosion and material migration in fusion experiments. Low-pressure plasmas are widely used for plasma processing such as plasma chemical vapor deposition, plasma etching or dierent surface treatments. Typical properties are a low degree of ionization, the mean kinetic energy of the electrons is much larger than that of the atoms and ions, and the EEDF is far from thermal equilibrium [1]. For a low degree of ionization (< 10?4 ) electron-atom collisions can be more eective for the shape of the EEDF than electron-electron collisions. Such weakly ionized plasmas are generally far from the equilibrium with a non-thermal EEDF deviating signi cantly from the Maxwell distribution. As various properties of the plasma depend on dierent parts of the EEDF, such as diusion coecients depend on the bulk of the EEDF and ionization or excitation rates depend on the tail of the EEDF, the determination of the whole distribution is compelling for a comprehensive characterization of the plasma. The notation of temperatures for the various parts of the EEDF often is a poor approximation for describing the whole EEDF [2]. Theoretically determined EEDFs solving the Boltzmann equation can be found in [3, 4, 5, 6, 7, 8, 9] and experimentally determined EEDFs can be found in [7, 8, 10]. Nevertheless, most low-pressure plasmas are analyzed assuming a Maxwell EEDF. The probably most common diagnostic techniques are electrical (Langmuir) probes which yield routinely local values of the electron temperature Te and electron density ne . The Langmuir probe technique is also capable of measuring the EEDF [7, 8, 10]. The drawback of a Langmuir probe is the close plasma contact resulting in various interacting eects and interpretation problems. Deterioration of the neighboring plasma, material deposition on the probe, thermal stress, and the interpretation in strong magnetic elds and the microwave eld are prominent problems with Langmuir probes. The presence of the probe in the plasma may result in increased particle losses and, in consequence, smaller electron densities and larger temperatures [1]. Spectroscopic diagnostics have no impact on the plasma. Emission spectroscopy of line intensities are mostly used for the determination of the electron temperature from line intensity ratios. Quite often the data set comprises a larger number of emission line intensities which allows the inference of more information on the EEDF and, hence, deviations from thermal equilibrium. The present paper deals with the inference of the EEDF from 8 emission line intensities from a low-pressure helium plasma generated with electron cyclotron resonance heating. Helium is used since it is one of the main constituents of fusion plasmas, it is chemically inert to avoid complications due to chemical reactions, and the atomic data are best known. For the interpretation of the measured emission line


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intensities the collisional-radiative model was used including de-excitation of the metastable states by wall collisions. Special attention is paid for the uncertainties associated with the electron impact excitation cross sections as they correspond in size with the errors of the measured line intensities. In rejecting explicit functional forms for the EEDF we are concerned with the extreme ill-posedness of the inversion problem of inferring a form-free distribution from a small number of noisy data. To avoid spurious features in the reconstructed EEDF, which are commonly driven by the noise of the data, it is cogent to incorporate prior information about the EEDF into the data analysis. Ill-posed inversion problems with sparse data sets or incomplete information have to be tackled with probability theory. It is the intention of the paper to demonstrate that from a rather small set of emission line intensities a form-free estimation of the EEDF with con dence intervals is possible where information is recovered beyond a Maxwell distribution which is derived from Langmuir probe measurements or by analyzing intensity ratios. After a brief description of the experimental setup in section 2, the analysis of the data with a collisional-radiative model within the framework of Bayesian probability theory (BPT) is given in section 3. The results of both mock and measured data sets are summarized in the last section.

2. Experiment The low-pressure plasma is generated by electron cyclotron resonance (ECR) heating with a microwave at the frequency of 2.45 GHz in a cylindrical chamber 0.27 m in diameter and height. The magnetic eld is generated by a coaxial pair of coils, with the upper coil located at the microwave window. The chamber is evacuated by means of a turbo molecular pump to a base pressure of less than 10?3 Pa. The He pressure is adjusted with a gas ux controller in the range between 0.02 Pa and 10 Pa. The absorbed microwave power is varied in the range of 20 W to 200 W. The data presented in this paper are measured with a power of 40 W. Details of the ECR plasma reactor can be found in [11]. Optical emission spectroscopy measurements were performed using an optical multichannel analyzer and a 0.275 m spectrograph. The light emitted from the plasma passes through a quartz window and is focused by a quartz lens onto the entrance aperture of a quartz ber which guides the light to the entrance slit of the spectrograph. Two apertures between the plasma and the quartz window de ne a line-of-sight through the plasma with a width of about 5 mm. The emission lines were recorded using the 1800-lines-per-mm high resolution grating. The resolution is 0.15 nm. The intensity of an emission line was calculated by integrating the pro le over thrice the full width at half maximum to both sides of the center of the line. Spectral calibration of the system was made by recording light emission of a tungsten-band-lamp and cross-checked with the radiation of a carbon arc. No absolute calibration of the emission line intensity is done.


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The data set comprises 8 helium emission line intensities from principal quantum numbers 3 and 4 to principal quantum number 2 in the energy range 2.1-3.2 eV. There are four analyzed optical transitions in the singlet (31P, 41S, 41P, 41D) and triplet (33P, 33D, 43S, 43D) term systems of helium, each. A schematic energy level diagram of the helium atom can be found in [4]. Due to the lack of absolute intensity calibration only intensity ratios enter the analysis with eectively 7 independent data. The uncertainty of the resulting emission line intensity ratios is less than 10% in the wavelength range between 350 nm and 625 nm. Details of the optical emission spectroscopy setup can be found in [12]. The plasma parameters were measured in the center of the plasma with a Langmuir probe. The probe consist of a coaxially screened tungsten wire with a diameter of 0.2 mm and is introduced in the reactor with the cylindrical axis perpendicular to the magnetic eld lines. Assuming a Maxwell EEDF the electron part of the current is given for probe voltages V lower than the plasma potential (V < Vplasma) [13]: ! 1 j eV SPj Ie = 4 Aprobe ne e v exp ? kT ; VSP = V ? Vplasma < 0 (1) e with Aprobe the electron collection surface of the probe. Assuming that the probe is suciently negative for a sheath to form, the ion current is given by [14]: s Iion = Asheath nion exp(?0:5) mkTe : (2) ion For cylindrical geometry the area of the sheath surface is ! r sheath ; (3) Asheath = Aprobe 1 + r probe with the probe radius rprobe and the sheath edge radius rsheath. rsheath increases with the negative voltage according to the Child-Langmuir law: !3=4 j eV SPj ; (4) rsheath = 1:02 D kT e where D is the electron Debye length. The description is valid only in the absence of a magnetic eld. In our magnetized plasmas, charged particles gyrate around the magnetic eld lines. Therefore, the eective charge collecting surface of the probe wire is only the fraction perpendicular to the magnetic eld, Aprobe = 2 Lprobe rprobe; (5) where Lprobe is the length of the wire. The maximum in uence of the magnetic eld on the I (V ) curve is found in the region of the electron saturation current. Therefore, the measurements are analyzed only in the part V < (V oat + kTe e ), where the total current onto the probe consists of contributions from ions and electrons. The values of the electron temperature Te, the electron density ne, the ion density nion (which is ne for our case of low pressure plasmas) and the plasma potential Vplasma are estimated by a least squares tting algorithm.


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The mean free path of the atoms with respect to the plasma diameter is important for the de-excitation of meta-stable helium states by wall collisions. At the largest pressure of 2 Pa presented in this work, a gas temperature of 300 K, and a cross section of 0:5 10?20 m2 [1] the mean free path is about 40 cm, i.e. larger than the plasma radius of 13.6 cm. Hence, the ow of neutral atoms to the wall are modeled with molecular

ow conditions.

3. Theory Inferring a distribution curve f from measured data d is an inverse problem which is ill-posed if the data are deteriorated by noise or the information needed for modeling the data is incomplete. Inverse problems which cannot be uniquely solved due to uncertainties have to be tackled with probability theory. Probability theory as a logical inferring machine, usually termed Bayesian probability theory, is extensively used in various elds such as astrophysics [15], image reconstruction [16], surface science [17], and plasma physics [18]. The starting point is given by Bayes theorem P (f jd; ; I ) = P (djf;P;(dIj)I )P (f jI ) (6) which relates the searched for posterior probability density distribution (pdf) P (f jd; ; I ) to quantities that are known, namely, the likelihood pdf P (djf; ; I ) and the prior pdf P (f jI ). P (djI ) is the evidence of the data which constitutes the normalization and will not aect the conclusions within the context of a given model. The historical terms \posterior" and \prior" have a logical, rather than a temporal, meaning. They simply mean \with" and \without" the new data taken into account. Bayes theorem is easily derived from only the familiar sum and product rules of probability theory. It is fundamental to all scienti c work, as it provides a formal rule for updating knowledge in the light of new data or learning from observations. The goal is to reason as best we can with the incomplete information we have. A probability distribution in the Bayesian sense is a quanti cation of the uncertainty of our knowledge and is not a frequency distribution of a \random" quantity although frequency arguments are often important for assigning priors and frequency estimates can be derived from Bayesian probabilities. The posterior pdf encodes all the information necessary to decide how reasonable a solution f is given the data d, the uncertainty of the data and further information I abbreviating all background information relevant for the problem such as the physical model or constraints on f like positivity. For practical reasons we characterize the posterior pdf with the two quantities of interest, namely the best estimate, often given by the maximum, and its reliability, given by the width. A recommendable tutorial on BPT is given by Sivia [19].


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3.1. Likelihood The likelihood function describes the error statistics of the experiment. If the noise k = dk ? Dk (f ) is assumed to be independent and Gaussian distributed with zero mean and standard deviation k , the likelihood function is given in terms of the usual mis t statistic 2 1 P (djf; ; I ) = QNd p (7) exp ? 21 2 ; 2k k=1 !2 Nd X d k ? Dk (f ) 2 = : (8)

i For the present inversion problem the model data Dk (f ) comprise 8 emission line intensities Iij . The intensity Iij of a radiative transition from level i to level j in a plasma column of length L depends on the density of the excited state ni integrated along the line of sight, ZL 1 hc Iij = 4 Aij dx ni ; (9) k=1

ij

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where ij is the wavelength and Aij is the transition probability from level i to level j .

3.1.1. Radiation trapping Radiation trapping in the plasma via excitation j ! i results in an eective lowering of the transition probability into the lower lying level j : A0ij := ij Aij Aij (10) The escape-factor ij depicts the fraction of the radiation which is not absorbed on the way through the plasma. For the present plasma geometry ij is approximated by the analytical expression as a function of the optical depth assuming Doppler broadening and cylindrical geometry [20]: 1:92 ? 1+1:36=5 (11) ij ( ) = ( + 0:62)( ln(1:375 + ))1=2 s

mHe ; c3 = Rnj Aij ggi 8 (12) 3 2kB TGas j ij where R = 0:136 m is the radius of the plasma column, gi and gj are the statistical weights of the levels i and j , respectively, TGas is the helium gas temperature and mHe is the helium mass. The analytical expression for a one-dimensional model approximates the results of a tree-dimensional numerical integration of the radiation transport equation quite well [21]. In the present experiments only the transitions to the ground and meta-stable states, where the lower states are considerably occupied, are signi cantly modi ed by radiation trapping.

3.1.2. Collisional-radiative model. The densities of excited levels ni are calculated with the collisional-radiative model. n in the stationary state is given by a balance of population and de-population processes with cross sections and rate coecients


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S =< v > obtained through an energy integration of the cross sections over the EEDF: e?-impact excitation (de-excitation) of level i from (to) level j with rate Sji = +nj ne < v >ji (Sij = ?nine < v >ij ) e?-impact ionization i ! 1 with rate nine < v >i1 population (de-population) of level i by radiative decay from (to) level j with rate +nj A0ji (?niA0ij ) de-excitation by wall collisions with rate ni?i , where the wall collision frequency ?i is estimated for molecular ow of the helium atoms and a de-excitation probability at the wall of 1, s v th B TGas ?i = R = 2km =R ; (13) He the mean time of ight for an atom to hit the wall is given by the plasma radius R divided by the mean thermal velocity vth. dielectronic and radiative recombination with rates DE=Rad nionne The latter can be neglected for the plasma parameter range considered (Helium neutral density 1021 m?3 and low degree of ionization 10?3 ). Furthermore, excitation transfer due to atom-atom collisions and three-body recombination processes are of minor importance in the present parameter range. The time development of the population density ni is described by the dierential equation 8 9 @ni = 6) are omitted as for the present plasma parameters already the eect of the levels n = 5 and n = 6 is negligible. The system of equations is solved iteratively starting without re-absorption of the radiation, ij = 1. The modi ed transition probabilities A0ij are calculated with the current densities ni. The solution converges after up to 10 iterations with a precision sucient to use gradient-optimization techniques, thereafter. 3.1.3. Atomic data. The energies of the excited levels and transition probabilities for spontaneous emission are taken from [22]. The transition probabilities of the eective levels n = 5 and n = 6 to the individual levels with n 4 are calculated by summation over the sub-levels of n = 5 and n = 6 multiplied with the statistical weights. The electron-impact excitation cross sections for helium are taken from various sources. The cross sections from the ground state to the S , P and D levels are from [23] and to the 41;3F levels are from [24]. The cross sections from the meta-stable state 21;3S are taken from the ADAS data base used for JET [25, 26, 27]. The cross section from 23 P to 21P is taken from [24]. Additional cross sections between excited states for spin-allowed transitions (S = 0) are from [28]. For the intercombination transitions (S = 1) we use the formula 2:3:3 from [24] where the parameter a is adjusted to t the data from [28]. The cross sections for excitation of levels n = 5 and n = 6 from the ground state and the meta-stable state are taken from [24]. The (superelastic) electron-impact cross sections for the reverse processes ij of deexcitation are related to the (inelastic) cross sections for the excitation processes ji by the Klein-Rosseland formula [29]: gi (E + Eij ) ij (E + Eij ) = gj E ji(E ) ; (17) where gi and gj are the statistical weights of the upper level i and the lower level j , respectively, and Eij = Ei ? Ej denotes their energy dierence. Ionization cross sections are taken from [23, 24]. Radiative recombination cross sections are taken from [30]. The eect of the dielectronic recombination rates are estimated using electron temperatures about Te = 10 eV [30]. Special care is taken for the uncertainties of the electron impact excitation cross sections. To nd the most important in uence of these uncertainties on the line emission intensities the cross sections of excitations and de-excitations are randomly changed with normally distributed noise : ^ij = ij (1 + ) (18) The standard deviation of the noise was altered between 0 (no noise) and 1 which is rather large. In gure 1 the variations on the data due to the variation of the cross sections is shown for the 8 measured line intensities. At low noise there is a linear correlation between the variation on the data and the uncertainty in the cross sections. This is a consequence of the fact that the cross sections enter linearly the system of dierential equations for the density of excited states.

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To evaluate the uncertainties of the data due to uncertainties in the cross sections the primary population channels of the emitting levels have to be determined. For the present plasma parameters the primary population of the levels is impact excitation from the ground state. Excitation from the meta-stable states is the next important population channel especially for the triplet terms. Cross-section uncertainties for excitations from the ground state are taken from [23]. Recently critically assessed excitation cross sections [31] showed that the cross-section uncertainties taken in the present work are conservative upper limits. 1.0 1

0.8 relative uncertainty in data

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3 P−>2 S 1 1 4 S−>2 P 1 1 4 P−>2 S 1 1 4 D−>2 P 3 3 3 P−>2 S 3 3 3 D−>2 P 3 3 4 S−>2 P 3 3 4 D−>2 P

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Figure 1. Relative uncertainty of the line emission intensities due to relative uncertainties of the cross sections. The variance k2 of the data entering the likelihood function comprise two terms which sum up quadratically. One term describes the uncertainty of the line intensities of 10% due to measurement errors and the other term re ects the eective uncertainty of the data due to the uncertainties in the cross sections determined from gure 1. 3.2. Priors The prior pdf P (f jI ) entering equation 6 has to quantify all information we have about the EEDF. The distribution has to be positive and additive in the sense that connecting two energy cells the number of electrons in the joint cell is the sum over the sub-cells. Additionally, we have the data of the Langmuir probe which are measured independent of the spectroscopic data. The resulting electron temperature Te constitutes weak prior


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information. Last but not least, the phase space volume, which is proportional to E , gives an invariant measure on the space of all distributions. The least informative prior pdf for a positive and additive distribution f is the entropic prior [32] P (f j; I ) = Z1 exp (S ) (19) ! N X f j (20) S = fj ? mj ? fj ln m j =1

Z

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j

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P (f jI ) = d P (f j; I ) P () (22) where S is the information-theory entropy relative to the default model m which represents weak prior information for f . For m we use a Maxwell distribution with a temperature Te from the independent Langmuir measurements. The extraneous parameter controls the relative weighting of the prior information and the data. If is large the solution is close to m whereas if is small the solution approaches the maximum likelihood estimate. Since we have no intrinsic interest in , it is a nuisance parameter which, in the framework of BPT, has to be marginalized according to equation 22 (see e.g. [19]). Recently, it has been demonstrated [33] that even the optimal treatment of the -marginalization leaves some residual ringing and noise tting of the form-free distribution f . The reason is the large number of degrees of freedom (DOF) of a form-free reconstruction which is greatly but not suciently restricted by the entropic prior. The entropic prior contains no correlation between neighboring cells like ad hoc regularization techniques such as minimizing derivatives which reduce eectively the DOF by correlations. Since all natural distributions have dierent length scales we have to incorporate a exibility in the form-free reconstruction which allows for adaptively choosing the smoothness in various regions of the distribution. In regions where the distribution varies slowly few DOF are sucient to describe this part of the distribution whereas in regions where the data contain cogent information for rapidly varying structures the length scale is small enforcing more DOF to describe the distribution. Common examples for distributions f where local smoothness is an important prior information are image reconstruction, density estimation and interpolation where continuous derivatives are expected. For reducing adaptively the DOF to describe only the signi cant information content of the data various methods are proposed [34, 35]. Closely related to the adaptive-kernel method in density estimation theory [36] we convolve a hidden distribution h with a smoothing kernel B but with locally varying kernel widths b [34]: ! Z x ? y f (x) = dyB b(y) h(y) (23) 2 ! !2 3 1 x ? y 1 x ? y (24) exp 4? 2 b(y) 5 B b(y) = p 2b(y)


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The particular shape of the kernel is not very important as long as it is smooth. For reasons of convenience, we assume a Gaussian kernel. The rules of BPT are used to determine the posterior probability density by marginalization of the hyper-parameters: hidden image h, blurring widths b and . In regions where the distribution f is at the method will automatically choose large kernel widths b and in regions where the data contain signi cant information enforcing sharp structures the kernel widths b are small. The basic requirement of keep solutions simple whenever the data do not compel a re ned solution is supported intrinsically in BPT via marginalization over the parameters. For details see our recent papers [34, 37] and references therein. p The reconstructed distribution f (E ) is the phase-space adjusted EEDF f (E ) E . Thus in case of weak data constraints the best estimation of f will be close to a homogeneous distribution in phase space: f const. So far only the data d, the noise and the quantity of interest f enters the Bayes theorem (6). The additional parameters of the collisional-radiative model, namely ne , p, TGas, and R, are extraneous (nuisance) parameters. Their values and uncertainties have to be quanti ed with priors and subsequently marginalized. The priors for p and R are rather sharply peaked at the measured values and can be assumed to be constant. TGas is assumed to be 300 K which is a reasonable assumption for the present plasma parameters with a low degree of ionization. The prior for the electron density ne is chosen to be Gaussian centered at the value determined by the Langmuir probe measurements with uncertainty of 25 ? 85%. To avoid an exorbitant increase in computing time we omit the marginalization over ne and simultaneously optimize the joint posterior pdf P (f; nejd; ; I ) which is valid if the pdf P (ne jd; I ) marginalized over f is sharply peaked relative to P (f jd; ; ne ; I ).

4. Results and discussion 4.1. Mock data set The number of parameters of the form-free distribution exceeds by far the number of data. The question arises which information gain can be achieved from 8 helium emission line intensities beyond the information encoded in one temperature parameter of a Maxwell distribution. A mock data set is generated from an EEDF which is typical for various plasma parameters (see next chapter). In gure 2 the dashed line depicts a Maxwell distribution with an electron temperature Te = 10 eV which is depleted in the energy region above 20 eV according to the sum of the excitation and ionization cross sections from the ground state. The dierence between the Maxwell distribution (dotted p line), which comprises the default model m, and the best estimate of the EEDF f (E ) E (thick solid line) re ects the information gain due to 8 helium emission line intensities which is not supported by the one parameter Te. The thin solid line depicts the con dence interval determined from the variance

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Electron energy distribution reconstruction 0

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Figure 2. Form-free reconstruction of the EEDF (thick solid line) with con dence

interval (thin solid line) compared with the true distribution (dashed line). The dotted line shows the default model assuming the weak information of an electron temperature T = 10 eV. The inset shows on a double-logarithmic scale the uncertainty of the reconstruction which spans a range of 6 orders of magnitude. e

of the posterior pdf. The small con dence interval in the inelastic region above 20 eV clearly shows the signi cant information supported by the data beyond the Maxwell distribution. The inset shows on a double-logarithmic scale the large range of the con dence interval over 6 orders of magnitude. For the elastic region below about 20 eV the information supported by the data is sparse due to the very small electron impact population and de-population contributions between the excited states. Please note that the con dence interval above about 2 eV is smaller than below 2 eV. This is due to the excitation of meta-stable states above the threshold of about 2 eV. Their populations are more than 3 orders of magnitude larger than the populations of the other excited states with optically allowed decay channels. The decrease of the EEDF at small energies is not due to signi cant data constraints but due to the phase-space adjusted reconstruction of the EEDF. 4.2. Real data set A typical EEDF inverted from 8 emission line intensities is shown in gure 3. The plasma pressure was 0.5 Pa which corresponds to a helium density of 1:2 1020 m?3 assuming a

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neutral gas temperature of 300 K. The estimate for the electron density ne = 4 1016 m?3 determined from the maximum of the posterior distribution coincides with the value determined from the Langmuir probe measurement. The degree of ionization is 3 10?5 . The relative singlet and triplet meta-stable densities are 6 10?6 and 3 10?5 , respectively. The relative densities of the states 21 P and 23P are about 2 10?8 and of the higher lying states below 10?9. 0

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Figure 3. Form-free reconstruction of the EEDF (thick solid line) compared with the Maxwell distribution (T =6.4 eV) (dashed line) obtained from Langmuir measurement. The thin solid lines depict the con dence interval of the reconstruction. The plateau region at 30 eV and the additional peak at 200 eV are barely signi cant. e

The dashed line depicts the weak prior knowledge m of a Maxwell distribution with electron temperature Te = 6:4 eV determined with the Langmuir probe. The thick solid line shows the form-free estimation of the EEDF. The thin solid lines indicate the one standard deviation of the form-free estimation calculated from the posterior pdf. The data are well described by the Maxwell distribution in the elastic energy region below 20 eV. Indicated by the rather large con dence interval the data contain only sparse information about the EEDF below the rst excitation energy since electron impact population and de-population of the radiating levels from other excited states with smaller energy thresholds is of minor importance. Above the rst excitation energy the form-free estimation of the EEDF deviates signi cantly from the Maxwell distribution. There is a steep decrease at the energy


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where excitation from the ground state into excited states and ionization occurs. The con dence interval becomes narrow above this energy threshold due to the dominant population channel by inelastic electron impact on helium atoms in the ground state. The steep decrease of the EEDF is interpreted as distinct loss of electrons in the plasma due to inelastic excitation and ionization collisions with atoms in the ground state. To support this interpretation the mock data set of the previous chapter was generated simulating the depletion. The form-free estimation of the EEDF from this mock data set signi cantly recovers the inelastic loss channel (see gure 2). The importance of inelastic collisions is reasonable since for the present plasma parameters the ratio of elastic electron-electron and electron-ion collision times relative to inelastic collision times assuming a Maxwell distribution is of the order of one. For a ratio much smaller than one thermalization by elastic collisions occurs while for a ratio much larger than one inelastic collisions dominate and result in deviations from the Maxwell distribution. The ultimate importance of elastic electron-electron collisions can only be determined via a self-consistent solution of the Boltzmann equation for the present plasma parameters. A distinct depletion of electrons relative to the Maxwell distribution is found solving the Boltzmann equation within a collisional-radiative model for microwave discharges in helium at low degrees of ionization [4]. The dierent behavior below and above the energy threshold may be modeled with a Druyvesteyn-type distribution [38]. For nonequilibrium situations the generalized Druyvesteyn p distribution characterized by the electron temperature Te and an exponent , f (E ) / E exp(?(E=kTe) ), is used [39]. The exponent allows for a steep decrease in an energy region where energy losses due to excitation and ionization provide a loss channel for the EEDF. In order to describe multiple energy loss channels a superposition of two or more such functions may be required. A similar self-consistent solution of the Boltzmann equation for argon plasmas can be found in [3]. In the case of negligible electron-electron collisions a depletion of the EEDF of many orders of magnitude occurs. For the present plasma parameters the main contributions of the power transfer of the electrons to the atoms are the inelastic collisions of excitation and ionization. Elastic collisions are negligible for the present plasma parameters [3]. The inset of gure 3 shows on a double-logarithmic scale two more structures in the EEDF which are barely signi cant. The plateau at 25-35 eV resembles the plateau at 20-40 eV resulting from a Boltzmann equation solution [5, 9] due to \repeated" superelastic collisional processes where electrons gain energy by electron impact induced de-excitation. This eect was predicted for relative meta-stable concentrations > 10?6 . The peak at an energy of about 200 eV counter-eects the depletion of the tail of the EEDF. A natural interpretation arises if we assume an inecient thermalization which yields an accumulation of high energy electrons due to the ECR heating mechanism. Small inelastic collision times of the electrons allow a high energy gain via acceleration in the microwave eld. The interpretation of this fast electron structure is hardly possible


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Table 1. Wavelengths of the measured spectral lines, escape factors , and

population ratios with and without radiation trapping for the upper and lower levels for the plasma of gure 3 population ratio transition [nm] upper lower 21P ! 11S 0.0013 600 31P ! 11S 0.0058 36 41P ! 11S 0.016 19 21P ! 21S 0.88 600 1.8 31P ! 21S 501.7 0.98 36 1.8 1 1 4 P ! 2 S 396.6 0.99 19 1.8 41S ! 21P 504.9 1 1.1 600 41D ! 21P 492.3 1 1.4 600 23P ! 23S 0.72 1.4 1 3 3 3 P ! 2 S 389.0 0.97 1.05 1 33D ! 23P 587.7 1 1.1 1.4 43S ! 23P 471.5 1 1 1.4 43D ! 23P 447.3 1 1.1 1.4

without a detailed study of the heating mechanism with ECR. Both structures, the plateau as well as the high energy peak are barely signi cant. However, the errors for the line intensity data as well as for the cross sections are conservative upper limits. For these structures the determination of con dence intervals as supplied by Bayesian analysis is cogent for reliable data interpretation. Please note the large range of uncertainties reaching from 0.1 at low energies to 10?6 at high energies. The widths of the EEDF posterior pdf for dierent energy regions re ect the information content in the data. The data are informative above the energy of the excitation thresholds from the ground state whereas electron induced population and de-population of excited states with lower thresholds are of minor importance. Re-absorption of resonance lines results in signi cant population enhancements of excited states with a radiating transitions to the ground state and the meta-stable states. Typical escape factors and population ratios with and without radiation trapping for the upper and lower levels for the plasma of gure 3 are given in table 1. Escape factors for the other transitions are very close to one due to the small population densities. A series of EEDFs for helium pressures 0.1-2.0 Pa is shown in gure 4. The absorbed heating power is constant. The mean electron energy decreases with increasing pressure. This eect results in a decrease of the electron temperature derived only from line emission intensity ratios assuming a Maxwell distribution. In the stationary state the anti-correlation of pressure and mean electron energy is due to the balance of the net production of charged particles in the bulk plasma speci ed by the ionization rate coecient and the recombination losses at the reactor walls speci ed by the electron con nement times [1]. Additionally, the loss of electrons with energies above 20 eV is correlated with

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0

10

1

2

10

10 E [eV]

Figure 4. Form-free reconstruction of the EEDFs for various helium pressures. The mean electron energy decreases with increasing pressure. The depletion of electrons with energies above 20 eV is correlated with the pressure.

the helium pressure. Such a correlation of the pressure with the exhaustion of electrons above the excitation thresholds was previously observed for a capacitively coupled rf discharge of helium [10]. The inelastic collision time due to electron impact excitation and ionization of helium is inversely proportional to the ground state density e = 1=(n0 S ), where S is the sum over the rate coecients. Hence, an increase in the pressure results in a decrease of e, that is a larger depletion. On the other hand, thermalization of the electrons depends on the electron density and the electron energy distribution via the elastic Coulomb collisions. Since the electron density variation is comparable to the pressure variation the dominant eect can only be determined via a detailed calculation. Please note the additional peaks above an energy of 100 eV. These are barely signi cant considering the uncertainty (not shown) although there is some systematic enhancement of the line intensities of the triple term transitions. Neglecting the crosssection uncertainties, which are rather large for the triplet term system, these structures become far more signi cant.


17

5. Conclusions The electron energy distribution function in low-pressure helium plasmas heated with electron-cyclotron resonance were reconstructed form-free from 8 helium spectral line intensities. The ill-posed inversion problem of inferring the form-free EEDF is tackled within the Bayesian probability theory using the entropic prior in conjunction with the adaptive kernel concept to obtain the most uninformative solution compatible with the prior information and the data. Probability theory based on only the sum and product rules provides a consistent framework for logical inference on uncertain grounds. The condensed results are given by a best estimate for the EEDF and its con dence range. It is the latter, which is of special importance in the reconstruction, because it allows to discriminate between signi cant and spurious structures. The emission line intensities are calculated with a collision-radiative model including de-excitation of the meta-stable states by wall collisions. As the relative uncertainties of the data are comparable with the relative uncertainties of the electron impact excitation cross sections a combined error is compelling in data analysis. The results for the EEDF show a non-Maxwellian distribution with distinct depletion in the inelastic energy region above the thresholds for excitation and ionization of helium in the ground state.

Acknowledgments The authors would like to thank W. Jacob for supplying the data and valuable discussions.

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