Zerlegung von Massenspektren mit Hilfe der Bayes schen Datenanalyse

Transcription

1 Max-lanck-nstitut für lasmaphysik Zerlegung von Massenspektren mit Hilfe der Bayes schen Datenanalyse Thomas Schwarz-Selinger, H.D. Kang, R. reuss, V. Dose Workshop Oberflächentechnologie mit lasma- und onenstrahlprozessen Mühlleithen, March 2 nd, 2004

2 detection of neutrals: electron impact ionization first QMS ionizer: +e ---> + +2e n ions emission < 1m filament < 4 U electron energy = e U eV However: + B +e > B + +2e + B + +2e or + + B +2e B 2+ +3e :

3 detection of neutrals: electron impact ionization first QMS ionizer: CH 4 +e ---> CH e QMS signal counts / sec CH 4 E e - = 70 ev 10 3 However: mass / charge amu / e CH 4 +e ---> CH H +2e, CH H +2e CH + + 3H +2e, C + + 4H +2e H + + CH 3 +2e,...

4 What we measure: overlap of different constituents 10 6 ΓN 2 / ΓCH 4 = 1 / 1 expected constituents: signal QMS counts / sec e.g. CH 4 / N 2 C discharge CH 4 H 4 H 6 NH 3 HCN m/q amu/e CH 3 : :

5 What we want to know: 1 which species contribute to the signal expected identification? constituents: 10 6 ΓN 2 / ΓCH 4 = 1 / 1 signal QMS counts / sec e.g. CH 4 / N 2 C discharge CH 4 2 to what extent do they contribute quantification? H 4 H 6 NH 3 HCN 3 confidence interval of the result accuracy? m/q amu/e CH 3 : :

6 Decomposition of Multicomponent Mass Spectra applying Bayesian Data nalysis Outline: Treating the nverse roblem of Mass Spectrometry Basic Rules of Bayesian robability Theory Nitrogen Containing Methane lasma Summary

7 inverse problem of quadrupole mass spectrometry 1 successive substraction of constituents signal by signal only applicable if there is one channel with no overlap exact cracking patterns of constituents are needed CH 4 CH 3 O NH 3 H 4 H 6 N 2 HCN error propagation not applicable for radicals no cracking pattern, low concentration intensity a.u mass / charge amu/q

8 d = x + ε C inverse problem of quadrupole mass spectrometry 2 matrix inversion = = = = n n m n m m x x x C C C C C d d d ε ε ε :, :,, : 1 1,,1 2,2 1,1 1 Definitions: data d with m elements, representing m mass channels concentrations x with n elements, representing n species cracking matrix with n columns consisting of m elements measurement error ε with m elements counting statistics, drifts... x : relative concentrations depending on normalization of cracking matrix C

9 inverse problem of quadrupole mass spectrometry 2 matrix inversion: e.g., CH 4 : 1amu 2 amu : 15 amu 16 amu : d = C x + ε = C H : 0 0 C CH : x x H 2 CH 4 Definitions: data d with m elements, representing m mass channels concentrations x with n elements, representing n species cracking matrix C with n columns consisting of m elements measurement error ε with m elements counting statistics, drifts... x : relative concentrations depending on normalization of cracking matrix

10 inverse problem of quadrupole mass spectrometry 2 matrix inversion if: d = C x + ε : singular value decomposition d = C x : x = C -1 d exact cracking patterns of constituents are needed 10 6 ΓN 2 / ΓCH 4 = 1 / 1 negative concentrations are possible not applicable for radicals no cracking pattern, low concentration signal QMS counts / sec m/q amu/e

11 inverse problem of quadrupole mass spectrometry 2 matrix inversion if: d = C x + ε : singular value decomposition d = C x : x = C -1 d exact cracking patterns of constituents are needed 0.6 ΓN 2 / ΓCH 4 = 1 / 1 negative concentrations are possible not applicable for radicals no cracking pattern, low concentration relative signal intensity N 2 CH 4 NH 3 H 6 H 4 CH 3 HCN

12 inverse problem of quadrupole mass spectrometry 2 matrix inversion if: d = C x + ε : singular value decomposition d = C x : x = C -1 d exact cracking patterns of constituents are needed negative concentrations are possible not applicable for radicals no cracking pattern, low concentration intensity a.u d C x mass / charge amu/e

13 inverse problem of quadrupole mass spectrometry 3 least square evaluation χ 2 -fits: d = C x + ε : forward calculation: assumes exact measurements needs exact cracking patterns of constituents best fit best result minimize: d x C fitting noise? intensity a.u d C x not applicable for radicals no cracking pattern, low concentration mass / charge amu/e

14 Bayes basics parameter estimation powerful tool to solve inverse problems, incorporating consistently further information provides the most probable result under the current state of knowledge: < x >= x p x dx p x dx expectation values

15 Bayes basics parameter estimation powerful tool to solve inverse problems, incorporating consistently further information provides the most probable result under the current state of knowledge: < x k >= x H k p xk d, σ d, dx H p x d, σ, dx k d k k expectation values provides the confidence interval of the result: 2 2 = < xi < xi > > = < xi > < xi 2 σ x > i 2 standard deviation H Strategy: evaluate p x d, σ, with the help of simple rules k d

16 Bayes basics roduct Rule rot,ferrari = rot,ferrari Ferrari = Ferrari,rot rot

17 Bayes basics,,, B B B B = =,,, B B B = Bayes theorem roduct Rule

18 Bayes basics tells us how to update our prior knowledge H about the physical hypothesis H in the light of data D which we collected from our experiment,, D H D H D H = Hypothesis, B Data

19 Bayes basics Sum Rule, = + marginalization = db B, getting rid of nuisance parameters

20 Bayes basics likelihood: foreward calculation D H, 1 D M exp 2πσ 2σ = 2 2 prior: e.g. if you know only a point estimate µ from maximum Entropy 1 x p x µ, = exp µ µ

21 nitrogen containing methane plasma final goal: decomposing mass spectra of reactive plasmas e.g. CH 4 / N 2 C discharge expected constituents: 10 6 ΓN 2 = 20 sccm signal QMS ΓCH 4 = 20 sccm mass amu CH 4 H 4 H 6 NH 3 HCN CH 3 : no : calibration available

22 nitrogen containing methane plasma input: final goal: decomposing mass spectra of reactive plasmas e.g. CH 4 / N 2 C discharge 1 intensities of 16 mass channels for 7 mixtures 10 6 ΓN 2 = 20 sccm H 2 cracking pattern for 10 expected ΓCH 4 = 20 sccmspecies tables 2 CH signal QMS expected constituents: 3 calibration measurements for all species except: NH 3 HCN CH 3 H 4 H 6 NH 3 HCN mass amu CH 3 : no : calibration available

23 nitrogen containing methane plasma N 2 CH 4 Bayesian analysis rel. concentration weighting factors: 0.4 CH N N 2 fraction %

24 nitrogen containing methane plasma rel. concentration HCN NH 3 CH 3 Bayesian analysis no calibration available! weighting factors: HCN NH CH N 2 fraction %

25 nitrogen containing methane plasma 0.04 H 4 H 6 Bayesian analysis rel. concentration weighting factors: 1 H H N 2 fraction %

26 model comparison: Radicals in a Methane lasma output: model comparison with Occams Razor CH 3, H 5, H ln {ED,S,} d - C x number of radicals

27 conclusion Bayesian data analysis allows to decompose multicomponent mass spectra incorporating further information e.g. calibration measurements parameter estimation delivers the expectation value as well as the confidence interval for the relative signal intensities as well as the cracking patterns species with unknown cracking pattern can be included model comparison allows to determine the species present in the mixture the analysis can help to optimise the experiment

28 conclusion the method has to fail, when - we under- / overestimate our errors - we make wrong assumptions like inappropriate calibration measurements excluding species the method cannot do wonders its simply common sense reduced to calculation

29 general problem of quadrupole mass spectrometry "a mass spectrometrist is someone, who figures out what something is, by smashing it with a hammer and looking at the pieces"