Analysis of multicomponent mass spectra applying Bayesianprobability theoryT. Schwarz-Selinger, R. Preuss, V. DoseCentre for Interdisciplinary Plasma ScienceMax-Planck-Institut f�ur Plasmaphysik, EURATOM AssociationBoltzmannstr. 2, D-85748 Garching, GermanyW. von der LindenInstitut f�ur Theoretische Physik, Technische Universit�at Graz, Austria(June 19, 2001)AbstractIn this paper we develop a method for the decomposition of mass spectraof gas mixtures together with the relevant calibration measurements. Themethod is based on Bayesian probability theory. Given a set of spectra thealgorithm returns the relative concentrations and the associated margin ofcon�dence for each component of the mixture. In addition to the concentra-tions, such a data set enables the derivation of improved values of the crackingcoe�cients of all contributing species, even for those components, for whichthe set does not contain a calibration measurement. This latter feature allowsalso to analyze mixtures which contain radicals in addition to stable mole-cules. As an example we analyze and discuss mass spectra obtained from thepyrolysis of azomethane which contain the radical CH3 apart from nitrogenand C1- and C2-hydrocarbons.Keywords: Decomposition of mass spectra, cracking coe�cients, azomethane, Bayesianprobability theory

I. INTRODUCTIONMass spectrometric analysis of gas mixtures is a common technique in plasma, vacuumand surface physics and is a powerful tool in active process control in semiconductor and thin�lm applications. Typical low resolution rf quadrupole mass spectrometers o�er, in principle,the ability to identify the constituents of a gas mixture with high sensitivity. Depending onthe experimental implementation, the total amount, the partial pressure or the ux of thespecies can be measured. The neutral molecules are ionized by electron impact in the ionsource at energies of 50 - 100 eV. This choice of impact energy is selected to obtain a highionization e�ciency but this high energy leads to the �rst complication. Consider a diatomicmolecule AB. Electron impact ionization will not only produce the parent molecular ionAB+ but also A+ and B+ ions from dissociative ionization. Also doubly charged atomic ionsA++ and B++ may occur. The quadrupole �lter will transmit these doubly charged ions atmass tuning A=2 and B=2 respectively since the �lter is sensitive to the mass-to-charge ratiorather than to the mass directly. Therefore, even a simple molecule AB produces signals inas many as �ve mass channels; this mass distribution is referred to as "cracking pattern" ofthe molecule.Cracking pattern become progressively more complicated as the number of atomic con-stituents of a polyatomic molecule increases. Quantitative detection of the constituents ofa gas mixture is therefore only simple if the cracking pattern of the species do not over-lap. This is a rare and unimportant case in most practical situations. Usually the patternoverlap and mixture signals must be disentangled. Consider a mixture of nitrogen (N2) andethane (C2H6). Such a mixture will give rise to signals in mass channels 28, 27, 30, 26, 29,15, 25, 14, 1 from the ethane component of the mixture and in channels 28, 14, 29 fromthe nitrogen component. The mass channels have been ordered by signal strength. Letus assume that the cracking patterns of nitrogen and ethane in this hypothetical experi-ment were known exactly. The pedestrian approach to the decomposition of the mixturesignal would be to choose the signal in channel 27 which originates uniquely from ethane.2

Knowing the exact cracking pattern we can then subtract the ethane contribution to allchannels from the mixture signal and are left with the nitrogen spectrum. Such a procedureassumes �rst of all, that the exact cracking patterns are known and assumes in additionthat the measurement is free of noise. The latter assumption can be relaxed if we resort toa least squares evaluation of the concentrations of the mixture components [1{4]. Since thenumber of mass channels will nearly always be larger or even much larger than the numberof mixture components, the procedure is relatively sound and often provides useful results[4]. In demanding cases, for example a noisy measurement on a mixture containing traceconstituents, the procedure described above may produce physically unreasonable solutionssuch as negative concentrations of the trace constituents.So far we have assumed that the cracking pattern are known. This is an unjusti�edassumption since the cracking pattern of a molecule is not an intrinsic property of themolecule but depends on a particular mass spectrometer for several reasons: i) the arising ionspectrum depends on the geometry of the ion source as well as for example the temperature ofthe �lament; ii) the electron impact energy will in uence the cracking pattern because of theenergy dependence of the ionization cross sections; iii) the ion extraction e�ciency dependson the ion extraction voltage in particular for fragment ions from dissociative ionizationwhich may carry a considerable Franck Condon energy; iv) the mass dependent transmissionof a quadrupole �lter also depends strongly on the resolution setting of the spectrometer;v) and �nally, even the ion detector { for single particle detection usually a channeltron{ may exhibit a mass dependent e�ciency. Published tables of cracking patterns [5,6] aretherefore only valuable to identify molecules according to their �ngerprint but are uselessfor quantitative analysis.For quantitative analysis cracking patterns of a stable mixture of constituents must bedetermined from a calibration experiment combined with an independent measurement ofthe composition of the mixture. The necessity of repeated and frequent calibrations has beenpointed out by Dobrozensky [7,8]. Such calibration measurements do not, however, providean exact cracking pattern but rather an approximation to the exact pattern deteriorated by3

noise. If we assume that this noise is comparable to the noise on the mixture measurementit cannot be neglected as was frequently done in previous work. On the contrary, bothsources of uncertainty will contribute about equal amounts to the uncertainty of estimatesof constituent concentrations and need a common coherent treatment. If, �nally, we discardthe assumption that the mixture we want to analyze contains only stable components, wemeet an additional and for standard mass spectrometry usually insurmountable barrier. Thecracking patterns of unstable species such as radicals are neither tabulated nor accessible bya calibration measurement. Yet the mixture signal also carries information on the concen-tration and cracking pattern of the unstable components. In favorable cases such as CH3detection in the presence of CH4, ionization threshold mass spectrometry provides a way outof the di�culty [9]. Ionization thresholds for the production of CH+3 from CH3 and CH4 are9.5 eV and 12.6 eV. An electron impact energy chosen within this window guarantees thatthe detected CH+3 ions originate from the CH3 radical only. Note the implicit assumptionthat no other hydrocarbons are in the system with threshold energy for CH+3 productionbelow or within the above energy window.After decomposing the measured spectrum to identify the species present in the recipient,in a second step the total amount of the virtual proportion of these species has to bequanti�ed. This second step is accomplished with independent calibration measurementsfor the mass spectrometer reading. However, a conclusive solution of the whole problem ofquantitative mass spectrometry is only suggestive with the �rst step performed accurately.We propose a new consistent way to solve this �rst problem in this paper.From the foregoing discussion it is clear that in standard low resolution mass spec-trometry of gas mixtures we deal in the decomposition with a set of incomplete and noisymeasurements. The proper means to deal with incomplete and noisy data is Bayesian prob-ability theory. Below, we outline the theory which will enable the analysis of mixture andcalibration measurements in a coherent way. The theory will be applied to the mass spectro-scopic analysis of azomethane ((CH3)2N2) pyrolysis which leads to C1- and C2-hydrocarbons,nitrogen and the CH3 radical. The results will be the mixture composition and the cracking4

coe�cients of the constituents based on the coherent analysis of all measurements.II. BAYESIAN DATA ANALYSISThe purpose of this section is to de�ne the probability density function nomenclaturein the rest of this paper and to recall the procedures of Bayesian probability theory. Thosewho meet for the �rst time with Bayesian probability theory are referred to the literaturefor a detailed introduction [10].Bayesian probability theory can be fully derived by quantifying the principles of logicalconsistency which immediately leads to the statement of two basic rules [11]. The �rstis the product rule. Given a probability P (D;HjI) depending on two or more variablesconditional on additional information I, the product rule allows P (D;HjI) to be brokendown into simpler probabilities or probability densities depending on only one variable. Dueto symmetry in the two variables, the expansion can be achieved in two di�erent ways.P (D;HjI) = P (HjI) � P (DjH; I) = P (DjI) � P (HjD; I) : (1)Comparison of the two alternative expansions yields Bayes theorem.P (HjD; I) = P (HjI) � P (DjH; I)P (DjI) : (2)In order to interpret (2) we associate meaning with H and D. D stands for data whichprovide information on the hypothesis H. An example for H is that a certain parameterh of a model takes on values in a certain range. Most of the time we are dealing withprobability densities, which we want to denote by small p, instead of absolute probabilitiesdenoted by P . P (H) = p(h)dh is then the probability that the hypothesis H has the valueh in an in�nitesimal range dh around h.The probability which we attach to a particular value of H before the measurement isperformed, p(hjI), is called the prior probability. This is the function which is used to encodeexpert knowledge. In the mass spectra analysis problem the cracking coe�cients from the5

literature constitute this knowledge for the present authors. Since expert knowledge dependson the expert, the probability p(hjI) is not unique but depends on the circumstances whichwe have chosen to be represented by I.The probability p(Djh; I) is the probability that we measure the data D assuming h isknown. It is the sampling distribution of the data given h i.e. it represents the forwardcalculation.Here we are more interested in p(Djh; I) when considered as a function of h. In this casep(Djh; I) is generally called the likelihood function.The denominator in Bayes' theorem is called the global likelihood of the data or theevidence. p(DjI) is not independent of the other probabilities but follows from the secondrule, the so-called marginalization rule of Bayesian probability theoryp(DjI) = Z p(h;DjI) dh = Z p(hjI) � p(Djh; I) dh : (3)p(DjI) provides therefore the normalization of the right hand side of Bayes' theorem calledthe posterior probability of h given D. The theorem thus constitutes a recipe of learning. Itallows to combine prior knowledge of h with results of an experimentD designed to extractnew information on h. The posterior probability p(hjD; I) is the full answer of a Bayesiananalysis. If appropriate it may be summarized in terms of its �rst and second momentshhi = Z h � p(hjD; I) dH ; h�h2i = Z (h� hhi)2 p(hjD; I) dh : (4)We now turn to the explicit de�nition of the likelihood, the posterior and the prior proba-bilities. III. THE LIKELIHOODAssuming a linear response of the mass spectrometer, the signal in each mass channeldn is the sum of all contributing species i weighted with their actual concentration xabsi andthe overall detection sensitivity �i for species i. Thus,6

dn = KXi Cni�ixabsi + �n ; (5)where Cni is the n-th component of the cracking vector ~Ci for the species i. In combustionchemistry and organic mass spectroscopy the cracking vector is usually normalized such thatits largest element is arbitrarily set to 1000. We found it convenient for the calculation inthis paper to adopt a di�erent convention. Our normalization is such that the componentsof ~C sum up to one for each species i.NXn=1Cni = 1 : (6)The freedom for such changes in normalization is due to the sensitivity factor �i which mustbe modi�ed accordingly. � is not dimensionless. It matches the di�erent dimensions of dnand xabsi where xabsi may be either partial pressure, partial number density, or partial ux.We shall call it concentration in the rest of this paper. The actual value of �i must bedetermined by calibrating the quadrupole mass spectrometer. Several di�erent pathwaysare mentioned in the literature namely the gas expansion from a calibrated volume [7],the ash �lament method [12] or the gas-injection method [13]. The common idea of allthese methods is to achieve higher accuracy in the total quantities by comparing the massspectrometer reading with the reading of a pressure gauge [7]. However, such a calibration isnot possible for unstable species such as radicals. There is no way to an absolute calibrationfor them although approximate estimates may be obtained from a comparison to similarstable molecules. Since the emphasis of this paper is on disentangling complex mass spectrawe shall not pursue the problem of absolute calibration further. It is then convenient tolump the sensitivity factor � into the absolute concentrations xabs by de�ningxi = xabsi � �i : (7)Our model equation is then in matrix notation~d = C � ~x+ ~� : (8)7

where ~� is the vector of measurement uncertainties on ~d. We now assume that the expectationvalues of h�ni vanish and that h�2ni = s2n. These two assumptions together with the principleof maximum entropy de�ne the sampling distribution of the data (the likelihood) asp(~dj~x;C; ~S; I) = 1Qn snp2� exp��12 �~d �C~x�T S�2 �~d�C~x�� ; (9)where S�2 is the diagonal matrix with elements 1=s2n. Equation (9) is the likelihood fora single measurement ~d. Our goal is to treat an arbitrary number of measurements n~djorelated to corresponding sample compositions f~xjg. We therefore needp �n~djo j f~xjg ;C;n~Sjo ; I� (10)which may be expanded using the product rule (1)p �~d1j f~xjg ;C;n~Sjo ; I� � p �n~d2:::~dJo j f~xjg ;C;n~Sjo ; I� : (11)In order to use in the notation of Eq. 8 the same cracking matrix C for all measurementsn~djo, every vector in f~xjg has been assigned the row dimension of C. Newly introducedelements in ~xj are �lled with zeros accordingly. We now employ the concept of logical inde-pendence. This implies that the data vector ~d1 depends only on the gas mixture ~x1 and onthe errors ~S1 made in the measurement of ~d1. The �rst factor of (11) is therefore independentof ~x2:::~xJ and ~S2:::~SJ. Similarly the second factor in (11) describes measurements on ~d2:::~dJand is independent of the measurement result ~d1 corresponding to ~x1 and also independentof ~S1. Deleting the non-existing conditions we now expand the second factor in (11). Byrepeated application of the product rule and the concept of logical independence we arriveat p �n~djo j f~xjg ;C;n~Sjo ; I� =Yj p �~dj j~xj;C; ~Sj; I� (12)Each factor of the product is in turn given by (9). Note that the formulation does not dis-tinguish between calibration and other measurements. Equation (12) is the �nal expressionfor the likelihood of our problem. It constitutes the basis for all subsequent conclusions andinferences in particular on ~xj and C. 8

IV. POSTERIOR ESTIMATESIn this section we derive estimates of the cracking matrix C and of the concentrations~xk of a single measurement based on our knowledge about the whole group from the knownform of the likelihood. First, we address the problem of obtaining the concentration of asingle measurement in the light of the whole group. That is we needp �~xkjn~djo ;n~Sjo ; I� = Z d~x�k p �f~xjg jn~djo ;n~Sjo ; I� (13)The notation d~x�k means, that integration is over all f~xjg except ~xk. For reasons of conve-nience we introduce the notation n~djo � D, f~xjg � X and n~Sjo � S. Employing Bayestheorem to the integrand in (13) we obtainp (~xkjD;S; I) = 1p (DjS; I) Z d~x�k p (XjI) p (DjX;S; I) : (14)The integral in (14) is transformed further by use of the marginalization rule (3) top (~xkjD;S; I) = 1p (DjS; I) Z d~x�k p (XjI) Z dC p (D;CjX;S; I) : (15)The prior knowledge on f~xjg will usually be limited to the bounds x(i)j � 0 and x(i)j � xmax.We therefore choose a at prior in these limits. For the integrations d~x�k we shall assumethat the integrand is sharply peaked within this prior range such that the values of theintegrals do not change if we extend the integration to the range (�1;1). Since the prioron f~xjg introduces only a constant factor in (15) we drop it together with the denominatorp (DjS; I) and go over to proportionalitiesp (~xkjD;S; I) / Z d~x�k dC Yj p �~dj ;Cj~xj; ~Sj; I�= Z dC p(CjI) exp��12 �~dk �C~xk�T Sk�2 �~dk �C~xk�� �Yj 6=k Z d~xj exp��12 �~dj �C~xj�T Sj�2 �~dj �C~xj�� : (16)The inner integral can be performed analytically. Let�~dj �C~xj�T Sj�2 �~dj �C~xj� = (~xj � ~xj0)T Qj (~xj � ~xj0) +Rj : (17)9

We then obtain for a single factor of the inner integral in (16)exp��Rj2 � jdetQj j� 12 : (18)Note that Qj depends of course on C. We postpone the explicit calculation of Rj , Qj , and~xj0, and �nd �rst the expectation value h~xki in terms of ~xk0. Using (16,17,18) we obtainh~xki / Z dC p(CjI)Yj 6=k exp��Rj2 � jdetQj j� 12� Z d~xk ~xk exp��12 (~xk � ~xk0)T Qk (~xk � ~xk0)� 12Rk� : (19)The inner integral yields ~xk0 exp f�Rk=2g jdetQkj� 12 and we haveh~xki / Z dC ~xk0 p(CjI)Yj exp��Rj2 � jdetQj j� 12 : (20)The �nal integration must be done numerically. We use Markov Chain Monte Carlo(MCMC) techniques [14] with the sampling density�(C) = 1Z p(CjI)Yj exp��Rj2 � jdetQj j� 12 : (21)For MCMC techniques the normalization constant Z needs not to be known. Next we needthe second moment of ~xk. The calculation is straightforward and yieldshx2kli / Z h�Qk�1�ll + x2k0li�(C) dC (22)with the density function �(C) given by (21). Eqn. (20) and (22) are then used to calculatethe variance h�x2kli = hx2kli � hxkli2 : (23)We return now to the explicit evaluation of Rj and Qj in (17). Comparing coe�cients onboth sides of (17) yields the equationsQj = CTSj�2C~xj0 = Qj�1CTSj�2~djRj = ~dTj Sj�2~dj � ~dTj Sj�2CQj�1CTSj�2~dj : (24)10

It is convenient to simplify notation by introducing ~d�j = Sj�1~dj and Cj� = Sj�1C. Thecomplicated matrix expressions in (24) and (22) may be simpli�ed considerably employingthe singular value decomposition (SVD) [15] of Cj�.Cj� =Xi �ji ~Uji~V Tji : (25)~Uji are mutually orthogonal and have the dimension of the column vectors of C. Similarly~Vji are also mutually orthogonal and have the dimension of the row vectors of C. We makeuse of this orthogonality property to obtainQj =Xk �2jk ~Vjk ~V Tjk ; Qj�1 =Xk ~Vjk~V Tjk�2jk ; jdetQjj =Yk �2jk (26)and �Qj�1�ll =Xk V 2jkl�2jk (27)needed in (22) to calculate the variance. Vjkl denotes the l-th component of vector ~Vjk inthe SVD of Qj. Similar simpli�cations are obtained for ~xj0 and R~xj0 =Xk 1�jk ~Vjk �~UTjk ~d�j� ; (28)Rj = ~d�Tj (I �Xr ~Urj ~UTrj) ~d�j : (29)This completes the simpli�cation of the matrix expression (24). The last quantity to bespeci�ed before (22) and (20) can be evaluated is the prior on the cracking matrix C.Di�erent choices will be discussed in the section on results.Posterior inferences based on the likelihood are not restricted to the concentrations ofthe mixture constituents. By similar means we are able to arrive at improved numericalvalues of the cracking coe�cients. The distribution which we need for their estimation isp(CjD;S; I) which we obtain from an application of Bayes theoremp(CjD;S; I) = p(CjI)p(DjC;S; I)p(DjS; I) (30)in terms of the marginal likelihood 11

p(DjC;S; I) = Z p(D;xjC;S; I) dx (31)= Z p(xjI)p(Djx;C;S; I) dx : (32)The integration needed here has been done already in (15) over all f~xjg except ~xk. Thegeneralization is trivial and we obtainp(DjC;S; I) / exp8<:�12Xj Rj9=;Yj jdetQj j� 12 : (33)Remember that Rj as well as Qj depend on C. The numerator of (30) is then identicalto the previous sampling density (21) and we obtain for the �-th moment of the crackingcoe�cient Crs hC�rsi = Z C�rs�(D;S;C) dC (34)with the important special cases � = 1 for the mean value and � = 2 for the second momentfrom which we obtain the variance h�C2rsi = hC2rsi � hCrsi2.V. EXPERIMENTThe analysis of the previous sections was applied to the interpretation of mass spectraobtained from the pyrolysis of azomethane ((CH3)2N2). This molecule is known to dissociatespontaneously at temperatures above 1000 K, mainly into nitrogen (N2) and methyl (CH3).It is therefore often used as a source of methyl radicals [16,17]. In our application, a CH3beam was needed for a thin �lm growth experiment [18]. A directed beam of particles isproduced by thermal dissociation in a resistively heated tungsten capillary. Azomethaneis prepared in a small oven via thermal release from an azomethane-Cu-Cl2 complex at anoven temperature of about 320 K. The complex was synthesized before according to themethod of Remmler et al. [19]. The oven is connected to the capillary gas line. The heatingpower applied to the oven is feedback controlled by a baratron gauge, which measures theazomethane pressure at the entrance to the capillary. This pressure is stabilized at 1.5 hPato maintain a constant ow of azomethane through the capillary. The whole methyl beam12

source is mounted in an UHV system with a base pressure of 10�10 hPa. The UHV system isequipped with a di�erentially pumped mass spectrometer (Hiden HAL 201) in line-of-sightof the capillary. The mass spectrometer can be rotated by �20� around the exit ori�ce of thecapillary to measure the angular distribution of the emanating particles. A more detaileddescription of the experimental setup and the radical source can be found elsewhere [20,21].Panel [a] of �g. 1 shows a mass spectrum of azomethane with the capillary held atroom temperature. The parent molecular ion (CH3)2N+2 appears at mass 58. Mass 43corresponds to the fragmentation CH3N+2 . In the region M � 30 the most prominent peakstems from CH+3 at mass 15 from dissociative ionization of the parent molecule as well asfrom direct ionization of CH+3 cracked at the hot �lament of the ionizer. Panel [b] showsa mass spectrum of azomethane with the capillary held at a temperature of 1150 K. Thestriking di�erence compared to panel [a] is the complete absence of the parent molecularion at mass 58 and the heavy fragment at mass 43. The spectrum is restricted to massnumbers �30. The appearance of mass 16 indicates the presence of methane. The factthat mass 15 is stronger than 16 signals the presence of CH3. The group around mass 28is due to nitrogen and C2-hydrocarbons which may be formed by recombination reactionsin the capillary. Hydrocarbon impurities also contribute to the spectrum in panel [a]; theseimpurities may arise from chemical reactions at the hot �lament in the ionizer or from thechemical synthesis of the initial azomethane-CuCl2-complex itself.In summary, we analyze the spectrum in panel [b] assuming that the contributing speciesare N2, CH3, CH4, C2H2, C2H4, and C2H6. Calibration measurements were therefore per-formed on each of these components except CH3. The capillary temperature was held at1150 K during the calibration measurements to account for the change of cracking patternsas a function of gas temperature due to vibrational excitation, an e�ect previously demon-strated for methane [21]. It is known from the literature that cracking patterns can dependon gas pressure [22]. To minimize the latter in uence the calibration measurements wereperformed at signal levels close to those observed in the pyrolysis except for C2H2 and C2H4.Data for these molecules were taken at pressures such that the respective signals fell well13

above the noise level. All measurements are collected in the data matrix Table I. Bold faceentries are chosen for the most prominent ion of the respective molecule. Errors on theindividual entries are estimated from the noise level and from reproducibility.VI. RESULTSTo completely specify the posterior distribution, the prior distribution for the elements ofthe cracking matrix must be assigned. The information which we use to specify this prior isextracted from the tables of Cornu Massot [5] for the stable molecules. This is reproduced asthe �rst entry per mass number and molecule in Table II. The original tables of Cornu Massotcontain only point estimates of the cracking coe�cients but no error margins. The errormargins given in Table II are our own estimates and are chosen to be 10% if the crackingcoe�cient is greater or equal to 100 and is constant equal to 10 for cracking coe�cientsbelow 100. The tables do not provide data on CH3 nor do we know estimates of the CH3cracking pattern from other sources. The values which we assign to CH3 are derived frommethane assuming that the probability of producing CH4�x from CH4 is similar to CH3�xfrom CH3. This assumption will turn out to be very poor. Let us �rst consider prior valueswith con�dence intervals for the cracking coe�cient. The maximum entropy distributiongiven the �rst and second moments is a Gaussian [10]. A convenient approximation of theGaussian on the support 0 � c � 1 is the Beta(cja; b) distribution [23]Beta(cja; b) = �(a+ b)�(a)�(b)ca�1(1 � c)b�1 : (35)Its mean and variance are given byhci = aa+ b ; h�c2i = hci2 ba(a+ b+ 1) (36)from which a and b can be obtained for any of the entries in Table II.Posterior values of the cracking coe�cients and their errors can then be calculated using(34). Numerical values for the applicable mass number/molecule contribution are shown14

as the second entry in Table II. Let us focus the discussion �rst on the CH4 column. Theposterior value of the cracking coe�cient for mass 15 is well within the prior range butvastly more accurate. This is in contrast to the value for mass 14, 13, 12 where we note adiscrepancy of a factor of roughly 2, 3 and 4, respectively. One explanation for this trendis provided by the threshold energies needed to produce the respective ions. These are14.3 eV, 15.1 eV, 22.2 eV, and 25.0 eV for CH+3 , CH+2 , CH+, C+ respectively. With anelectron energy of 70 eV in the ion source we see that this corresponds to 4.9 times thethreshold energy for production of CH+3 while it means only 2.8 times the threshold energyfor C+ production. For CH+3 production the cross section is quite near to the maximumand insensitive to changes in the electron impact energy. Di�erent instruments are thereforeexpected to perform similarly. For C+ production on the other hand it is still in the risingedge [24] and considerable variations in the production rate may be expected for di�erentinstruments.The pattern exhibited by CH4 is reproduced more or less for all the other hydrocarbons.The fact that large discrepancies between prior and posterior estimate in the cracking coef-�cient arise at several places indicates that the likelihood must be very informative on theseparameters and overrules our prior belief drawn from the tables of cracking patterns givenby Cornu Massot. This applies also to the cracking elements of CH3 where we �nd, in viewof the quoted errors, a huge di�erence in the coe�cient leading to CH+2 .An alternative prior assumption is to use only the point estimates from the Cornu Massottables without assuming an additional con�dence interval. The maximum entropy distribu-tion given only the expectation value of the variable is an exponential [10] on the support0 � c � 1. p(cj�) = exp f��cg =Z : (37)Z is the normalization constant and is given byZ = (1� exp f��g) =� : (38)15

The unknown parameter � is determined such, that the expectation value hci of (38) becomesequal to the respective point estimate c0 of the cracking coe�cienthci = Z 10 dc c exp f��cg =Z ; (39)c0 = 1� � 1� (1 + �) exp(��)1 � exp(��) : (40)Fig. 2 shows p(cjc0) for three representative values of c0. This prior distribution is of coursemuch less informative than the earlier Beta distribution, since it contains only one pieceof information, the point estimate. We now use this prior distribution to fully specify theposterior. Expectation values and con�dence limits for the cracking coe�cients are shownas the third entry in Table II. Note that everywhere except for the element (CH3, m=14)the posterior estimates for the two classes of priors coincide. For the methyl CH+2 case ourprior point estimate is very far from the posterior. The more informative Beta prior pulls itslightly more to the prior value than the exponential prior. Since our error estimate on theprior for this matrix element is very poorly justi�ed the result for the exponential prior isaccepted as the "best" posterior value. Note also that the posterior error margins are quitesmall compared to those assumed in the prior. This points again to the fact that the set ofdata is very informative about all cracking coe�cients.We now turn to posterior estimates for the concentrations in the mixture measurementpanel (b) in Fig. 1. These are collected in Table III for both choices of previously discussedprior distributions on the cracking matrix elements. Two features can immediately be readfrom Table III: the choice of prior is entirely unimportant for the posterior estimate and asimpler model for the mixture namely one not including C2H2 would have been su�cient.Finally, if we assume the sensitivity factors �k appearing in (5) to be approximately equalfor all the species in Table III we obtain to the same degree of approximation that the CH3concentration relative to the sum of pyrolysis products is about 12% with a surprisinglysmall error margin. A calibration measurement for all stable gases allows the determinationof the correct values for �k. The calibration of the CH3 signal can be performed in a similarway. The surface loss probability of CH3 radicals is very small (10-3) [18] so that particle16

losses in the entrance of the mass spectrometer and the ionizer can be neglected. Since thecross sections for direct ionization of methane and methyl are similar [25], a ux of CH3radicals can be inferred from the absolute ux of CH4 molecules measured in the calibrationexperiment, assuming the mass transmission in the quadrupole analyzer is the same for CH+3and CH+4 . With this assumption the relative concentration of CH3 in the particle beam is11.7% instead of 12.1% for the uncorrected value.As mentioned already in the introduction in the special case of the CH3 radical thereexists another means of selective detection, namely ionization threshold mass spectrometry.This approach has also been performed and has been published elsewhere [21]. For thepresent context it is su�cient to quote that by this technique we arrive at a particle uxthat corresponds to a methyl fraction in the beam of (13 � 4)% a value in accord with thepresent result but very much less precise.VII. SUMMARY AND OUTLOOKIn this paper we have shown how to combine mixture and calibration measurements froma quadrupole mass spectrometer in a consistent way in order to derive mixture componentconcentrations and improved estimates of the cracking coe�cients. The proposed procedureis also well suited for the analysis of mixtures containing radicals for which no calibrationmeasurements can be made. The case considered here was the simplest conceivable: namely,a mixture of stable gases containing the single radical CH3. The single case was chosenbecause it allows for a cross check with an experiment employing ionization threshold massspectrometry. Future application of the presented procedure will include the analysis ofneutral uxes from low temperature plasmas employed in thin �lm deposition and etching.Another area which we shall look at more closely is the proper analysis of vector mass spectra[26] in particular hydrogen helium gas mixtures as arise in nuclear fusion machines.17

REFERENCES[1] BLACKBURN, J., Anal. Chem. 8 (1965) 1000.[2] DOBROZEMSKY, R. and F�ARBER, W., Vakuumtechnik 20 (1971) 231.[3] D.L. RAMONDI, H.F. WINTERS, P. G. and CLARKE, D., IBM J. Res. Dev. 15 (1971)307.[4] DOBROZEMSKY, R., J. Vac. Sci. Technol. 9 (1972) 220.[5] CORNU, A. and MASSOT, R., Compilation of Mass Spectral Data, Heyden London,1979.[6] STENHAGEN, E., ABRAHAMSSON, S., and Mc La�erty, F., Registry of Mass SpectralData, John Wiley & Sons, New York, 1974.[7] DOBROZEMSKY, R., Vacuum 41 (1990) 2109.[8] DOBROZEMSKY, R., Rev. Brazil. Apl. Vac. 5 (1985) 145.[9] ELTENTON, G., J. Chem. Phys. 15 (1947) 455.[10] SIVIA, D., Data Analysis: A Bayesian Tutorial, Oxford University Press, 1996.[11] COX, R., Am. J. Phys. 14 (1946) 1.[12] WINKLER, A., J. Vac. Sci. Technol. A 2 (1984) 1393.[13] KENDALL, B., J. Vac. Sci. Technol. A 5 (1987) 143.[14] GILKS, W., RICHARDSON, S., and SPIEGELHALTER, D., editors, Markov ChainMonte Carlo in Practice, Chapman and Hall, London, 1996.[15] PRESS, W., TEUKOLSKY, S., VETTERLING, W., and FLANNERY, B., NumericalRecipes in Fortan, Cambridge University Press, Cambridge, 1992.[16] HEYDTMANN, H. and BOGLU, D., Chem. Phys. 168 (1992) 293.18

[17] PENG, X., VISWANATHAN, R., SMUDDE, G., and STAIR, P., Rev. Sci. Instr. 63(1992) 3930.[18] VON KEUDELL, A., SCHWARZ-SELINGER, T., and JACOB, W., Appl. Phys. Lett.76 (2000) 676.[19] REMMLER, M., ONDRUSCHKA, B., and ZIMMERMANN,G., J. f. prakt. Chem. 327(1985) 868.[20] SCHWARZ-SELINGER, T., von Keudell, A., and JACOB, W., J. Vac. Sci. Technol. A18 (2000) 995.[21] SCHWARZ-SELINGER, T., DOSE, V., JACOB, W., and von Keudell, A., J. Vac. Sci.Technol. A 19 (2001) 110.[22] BRETH, A. and DOBROZEMSKY, R., Deviation of cracking patterns and their in u-ence on rga accuracy, in SASP 82, Symposium on atomic and surface physics, editedby W. LINDINGER, F. HOWORKA, T. M. and EGGER, F., page 75, 1982.[23] GELMAN, A., CARLIN, J., STERN, H., and RUBIN, D., Bayesian Data Analysis,Chapman & Hall, London, 1995.[24] DOSE, V., PECHER, P., and PREUSS, R., J. Phys. Chem. Ref. Data 29(2000) 1157.[25] TARNOVSKY, V., LEVIN, A., DEUTSCH, H., and BECKER, K., J. Phys. B 29 (1996)139.[26] DOBROZEMSKY, R. and SCHWARZINGER, G., J. Vac. Sci. Technol. A 10 (1992)2661.19

TABLESMass N2 CH4 C2H2 C2H4 C2H6 (CH3)2N212 0 0.142�0.012 0.193�0.015 0.072�0.005 0 0.069�0.00513 0 0.435�0.02 1.27�0.05 0.135�0.005 0.042�0.005 0.257�0.0114 1.17�0.1 1.64�0.05 0 0.342�0.005 0.28�0.01 2.33�0.115 0 19.96�0.2 0 0 0.57�0.01 7.86�0.0516 0 24.87�0.2 0 0 0 4.74�0.0524 0 0 1.26�0.05 0.239�0.01 0 0.027�0.00525 0 0 5.95�0.1 0.829�0.02 0.26�0.01 0.145�0.0126 0 0 32.86�0.2 8.23�0.1 2.12�0.05 1.29�0.0527 0 0 0 7.98�0.1 3.62�0.07 1.86�0.0528 38.46�0.2 0 0 14.65�0.1 12.1�0.1 21.73�0.129 0.4�0.1 0 0 0 3.87�0.06 1.85�0.0430 0 0 0 0 2.5�0.05 0.907�0.01TABLE I. Mass signals from �ve calibration gases and from the mixture arising fromazomethane pyrolysis.

20

Mass N2 CH4 C2H2 C2H4 C2H6 CH321�10 15�10 15�10 65�1012 5�1 6�1 5�1 14�16�1 6�1 5�1 10�165�10 51�10 26�10 11�10 136�1313 18�1 39�1 9�1 3�1 47�318�1 39�1 9�1 3�1 38�352�10 136�13 50�10 26�10 823�8214 31�2 69�2 23�1 23�1 396�1830�3 66�2 23�1 23�1 358�20823�82 37�10 1000�10015 813�6 47�1 1000�18802�6 47�1 1000�201000�10016 1000�61000�6 56�10 29�1024 39�1 17�138�1 16�1202�20 104�10 34�1025 181�3 58�1 21�1181�3 57�1 21�11000�100 584�58 216�2126 1000�3 562�6 175�41000�3 564�5 174�4609�60 320�3227 545�6 305�4544�5 303�41000�100 1000�60 1000�10028 1000�3 1000�6 1000�61000�3 1000�5 1000�67�10 222�2229 16�2 323�416�2 327�4268�2630 199�4199�3TABLE II. Prior information on the cracking elements with estimated error margin (�rst entry),posterior expectation of the cracking coe�cients using a Beta prior (second entry) or an exponentialprior (third entry) respectively. 21

N2 CH4 C2H2 C2H4 C2H6 CH3Beta 0.384�0.002 0.204�0.002 0.001�0.002 0.041�0.006 0.217�0.005 0.121�0.003exp 0.384�0.002 0.201�0.002 0.001�0.002 0.042�0.006 0.217�0.004 0.121�0.004TABLE III. Posterior fraction of the mixture constituents using either an exponential or a Betaprior on the cracking elements.

22

FIGURES

10 20 30 40 50 60

mass (amu)

T capillary

= 1150 K

T capillary

= 300 K

sign

al Q

MS

(a.

u.)

(b)

(a)

FIG. 1. Mass spectrum of the (CH3)2N2- molecule (panel a) and the mixture arising frompyrolysis (panel b).23

0 0.2 0.4 0.6 0.8 1c

0

1

2

3

4

5

6

7

p(c

|c0,

I)

c0=0.26

c0=0.85

c0=0.5FIG. 2. The exponential prior for an interval [0,1] for three di�erent mean values hci.

24