Model based quantification of EELS spectra

J. Verbeeck a S. Van Aert a

aElectron Microscopy for Materials research (EMAT), University of Antwerp,Groenenborgerlaan 171, 2020 Antwerp, Belgium

Abstract

Recent advances in model based quantification of electron energy loss spectra (EELS)are reported. The maximum likelihood method for the estimation of physical pa-rameters describing an EELS spectrum, the validation of the model used in thisestimation procedure, and the computation of the attainable precision, that is, thetheoretical lower bound on the variance of these estimates, are discussed. Experi-mental examples on Au and GaAs samples show the power of the maximum likeli-hood method and show that the theoretical prediction of the attainable precision canbe closely approached even for spectra with overlapping edges where conventionalEELS quantification fails. To provide end-users with a low threshold alternative toconventional quantification, a user friendly program was developed which is freelyavailable under a GNU public license.

Key words: EELS, fitting, quantification, precision, maximum likelihood, modelPACS: 79.20Uv,82.80Pv,06.20Dk,43.50+y,02.50-z,07.05FB

1 Introduction

Although quantitative Electron Energy Loss Spectroscopy (EELS) has beenavailable for quite some time now, it still is no turnkey solution for the deter-mination of chemical concentrations. In comparison, Energy Dispersive X-rayAnalysis (EDX) seems to be much easier to apply and its use is much morewidespread. From a physical point of view, this seems odd because both tech-niques are so closely related. In terms of detection efficiency, EELS is evenfar superior over conventional EDX. On top of that, EDX suffers from severalartifacts like re-absorption and backscattering which complicate the quantifi-cation considerably. So, although an EEL spectrum contains more information

Email address: jo.verbeeck@ua.ac.be (J. Verbeeck).URL: http://webhost.ua.ac.be/emat/ (J. Verbeeck).

Preprint submitted to Elsevier Science 24 May 2004

compared to an EDX spectrum (for the same electron irradiation dose) [1],it appears to be difficult to extract this information quantitatively. The maindifferences between EELS and EDX are the occurrence of a strong backgroundand the complicated shape of the excitation edges in EELS compared to thevery high signal-to-background ratio in EDX together with simple expressionsfor the excitation cross sections.

In this paper we will evaluate the EELS quantification procedure in view ofthe promise that EELS holds in terms of its favourable information contentper electron irradiation dose [1].

The conventional way to treat core-loss EELS spectra consists of three steps[2]. The first step is to remove the background by extrapolating a power-lawfunction which is fitted in a region preceding the excitation edge. The secondstep is to remove the effect of multiple scattering by a deconvolution with thelow-loss spectrum. The third step is the integration of the number of counts ina certain energy region under the thus obtained excitation edge. This numberis then converted into an absolute chemical concentration making use of acalculated cross section for the same energy region. Although this procedureis most often used and also implemented in commercial programs, it has severaldisadvantages. The first step, the background removal, is the subject of quitesome debate [3–9]. The user has to choose a suitable window in order to fit thebackground. Depending on this choice the results may differ. Extrapolationunder the excitation edge can make the outcome quite sensitive to the choiceof window position. The result will depend on how far beyond the fittingregion one wants to extrapolate. Furthermore, the assumption of the powerlaw background is known to fail for wide energy regions. The second step, themultiple scattering deconvolution step, is usually based on Fourier techniquesand can also introduce severe artifacts. Again choices have to be made whichare not always apparent to the end user of commercial software. Finally, inthe third step, cross sections are needed to convert the integrated number ofcounts under the excitation edge into chemical concentrations. Cross sectionsfor excitation edges can not be calculated exactly. Therefore, approximationssuch as the widely used hydrogenic cross sections [2,10] are needed.

As an alternative to this conventional spectrum treatment, several authors[11–15] have presented more advanced fitting techniques to solve the EELSquantification problem. In this approach, the recorded number of counts inan EELS spectrum is considered as an observation from which chemical orelectronic information has to be extracted quantitatively. This can be done bymaking use of a so-called parametric statistical model of the observations. Thismodel describes both the expectations of the observations and the fluctuationsof the observations about the expectations. The physical model describing theexpectations contains the parameters of interest, which in most cases, canbe directly related to chemical concentrations in the sample. Quantitative

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EELS is done by fitting the model to the observations with respect to theparameters using a criterion of goodness of fit, such as, least squares, leastabsolute values or maximum likelihood. The outcome is a set of parametersgiving rise to the “best fit”. Thus, quantitative EELS can be regarded asa statistical parameter estimation problem. Comparing to the conventionalquantification this is a much more “natural” approach in the sense that themodel mimics all processes involving the recording of an EELS spectrum. Aslong as we can create a good model, the information, which is present in thespectrum in the form of physical parameters, should be accessible. There is noextrapolation and deconvolution involved and no fitting windows have to bechosen. All observations are taken into account on an equal footing. In viewof these advantages, it is remarkable that model based quantitative EELS hasnot gained more momentum in the EELS community.

In this paper, we present significant improvements for quantitative EELS anda user friendly program, called EELSMODEL, making advanced quantitativeEELS accessible. The program is available under the GNU public license [16]which briefly means that the user is free to use, distribute and alter the pro-gram under certain conditions. The main goal of this program is to lower thethreshold for interested experimentalists to start using model based quantita-tive EELS. We also exploit an advantage of parameter estimation theory toestimate the precision of the parameter estimates.

Finally, the validity of the model can be evaluated making use of model vali-dation techniques.

The paper is organized as follows. First the methodology for model basedquantification is explained and then two experimental examples, analyzed us-ing these techniques, prove the feasibility of this approach. Model validationhowever, points out that more work is needed to improve the currently avail-able models to obtain statistically valid descriptions of EELS spectra. Therejection of current models comes as no surprise in view of the approxima-tions used to derive them, but it becomes clear that this can lead to biasedresults. Comparing with conventional quantification we can state that modelimperfections are now the only source of bias while conventionally severalother sources of bias and additional noise existed due to improper statisticaltreatment of the experimental data.

To conclude, the possibilities of the implemented software are described.

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2 Methodology

The aim of quantitative EELS is to estimate parameters with a clear physicalmeaning, such as chemical concentrations, from the recorded spectra. There-fore, use can be made of statistical methods, such as the maximum likeli-hood (ML) method. The necessary steps to be made in the application of thismethod will be described one by one. To begin with, it requires the availabilityof a parametric statistical model of the observations describing the expecta-tions of these observations as well as the fluctuations of the observations aboutthe expectations, that is, the noise. In subsection 2.1, it will be shown howsuch a model can be constructed for EELS. This model can then be used fortwo purposes. First, in subsection 2.2, it will be used to determine the attain-able statistical precision or, in other words, the lower bound on the variancewith which the parameters can be estimated without bias from the observa-tions [17,18]. This is the so-called Cramer-Rao Lower Bound (CRLB). Second,in subsection 2.3, from the proposed parametric statistical model of the ob-servations, the maximum likelihood estimator will be derived. This estimatoractually achieves the CRLB asymptotically, that is, for an infinite number ofobservations. Therefore, the ML estimator is asymptotically most precise. Forthis and other reasons, the ML estimator is very important and often used inpractice. Furthermore, in subsection 2.4, a method will be presented to test thevalidity of the proposed model. It makes use of the obtained ML estimates ofthe parameters. Finally, in subsection 2.5, it will be described how confidenceintervals associated with the parameter estimates can be constructed so as tobe able to evaluate the level of confidence to be attached to these estimates.

2.1 Parametric statistical model of the observations

2.1.1 Components of the expectation model

The expectation model describes the expectations of the observations. Thismodel contains unknown parameters. In EELS applications, the model param-eters can be interpreted, for instance, in terms of chemical concentrations ifthere is a known relation between these parameters and physical parameterscharacterizing the sample. Creating an expectation model for EELS spectranecessarily requires the use of approximations and thus the model is not exact.We propose to build a model from a linear combination of components, de-scribing different aspects of the spectrum. For example, a simple GaAs EELSspectrum could be constructed from the following components: a smoothly de-caying background and two core-loss excitation edges for the Ga L2,3-edge andthe As L2,3-edge. The edges can be calculated, for example, with a hydrogenicmodel [2,19] or they could be the result of a detailed ab-initio calculation.

4

In practice, a model for core-loss spectra will consist of a few typical compo-nents like background and excitation edges. More complicated and thereforeprobably more accurate models and vice versa can be constructed by the user.Nevertheless, he or she has to realize that the choice of the model has itsimpact on the accuracy of the parameter estimates. If the model does not de-scribe the observations sufficiently well, the obtained estimates will be biased.This means that they will fluctuate about the wrong values of the parameters.For this reason, model validation is important. This topic will be discussed insubsection 2.4

2.1.1.1 Background : origin of AE−r The use of a power law back-ground of the form AE−r, with A and r being unknown parameters and Ethe energy, is very common in traditional quantitative EELS. Egerton [19,20]proposed this model based on the asymptotic behaviour of analytic functionsdescribing the plasmon peak (Drude model) and core-loss excitations (hydro-genic cross section). The power law function appeared from that time on as thede-facto standard for EELS background and was used to remove the unwantedbackground from a core-loss edge by means of extrapolating from a pre-edgefitting region. It is interesting to note that in the same paper, Egerton showsexperimental evidence that the parameter r of the power law is not constantover a wide energy region. The fact that r is changing with energy, effectivelymeans that the power law function is not a good approximation to model anexperimental background over a wide energy range. Several other authors havefocused on this effect and elaborated on the reasons [3,4,21]. In this paper, weonly want to point out that the function AE−r can be replaced by any othersmoothly monotonically decaying function as long as it allows to better modelthe real shape of the background (possibly with more than two parameters).The main merit of the function AE−r follows from the fact that it arose fromphysical approximations, and its parameters A and r might have some phys-ical meaning. In practice, however, one is not interested in the parameters ofthe background, one only likes to model it as accurately as possible.

2.1.1.2 Multiple scattering Multiple scattering occurs when several in-elastic scattering events take place in the sample, leading to an accumulatedenergy loss. Only scattering to a high probability excitation (typically theplasmon region) is likely to scatter a second time to a core-loss excitation.This can be described by convoluting the single-scattering core-loss spectrumS(E) with a low-loss spectrum L(E):

M(E) = S(E) ∗ L(E) (1)

5

An improvement to this treatment consists of taking into account the angulardistribution of the scattering events in the convolution as shown in [4]. Inpractice L(E) is recorded experimentally and has a signal to noise ratio farsuperior to the core loss spectrum S(E).

A logical step would be to substitute the model for the single scattering spec-trum S(E) in Eq.(1) [12]:

M(E) = (AE−r + p1σ1(E) + p2σ2(E)) ∗ L(E) (2)

for a simple spectrum consisting of a power-law background and two core-lossedges with cross sections σ1(E) and σ2(E). If the model is calculated in theenergy region where also the experiment was performed, a straightforwardapplication leads to edge artifacts after convolution as shown in fig. 1. Theedge artifacts are marked by numbers in fig. 1 and some remarks are givenhere:

• Artifact 1 is caused by the sudden unphysical onset of the background inthe core-loss spectrum when it is calculated over the same energy range asthe experimental core-loss spectrum. Since the model core-loss spectrum isa calculated spectrum this artifact can be solved by expanding the energyrange over which the model is calculated and thereby shifting the artifactoutside the region where the experimental spectrum is valid. This approachwas taken by Manoubi et al.[12]. We propose to solve artifact 1 by notincluding the background in the convolution. The model then becomes:

M(E) = f(E) + (p1σ1(E) + p2σ2(E) + . . .) ∗ L(E) (3)

with f(E) a model for the convoluted background. Based on the discussionon the origin of the AE−r background, there is no reason why finding amodel for a convoluted background would be more difficult than for a normalbackground as long as it is a smooth monotonically decaying function thatdescribes the experiment well.

• Artifact 2 is caused by the unphysical sudden onset of the experimental low-loss spectrum. This can be avoided by making sure the low-loss spectrumstarts smoothly at zero, by subtracting a constant.

• Artifact 3 is similar to artifact 2 but is caused by the ending edge of theexperimental low-loss spectrum. In principle, this artifact does not occuras long as the core-loss excitation edges are situated far enough beyondthe start of the experimental core-loss spectrum (more than the energydifference between the zero-loss peak and the low-loss onset). In fig. 1, thisartifact is shown only schematically since it would realistically occur outsideof the core-loss region in this case.

• Artifact 4 is similar to artifact 1, and is mainly solved by not including thebackground in the convolution as for the proposed solution of artifact 1. On

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Fig. 1. Schematic overview of the cause and effect of different artifacts when sim-ulating a multiple scattered spectrum M(E) by convoluting the low-loss spectrumL(E) with the single scattered core-loss spectrum S(E) if both are known only ina limited energy range (shaded area marks border of known area).

top of this, the excitation edges are required to have decayed considerablyat the end of the core-loss energy range, or alternatively the end of the rangecan be excluded from the fitting procedure.

2.1.1.3 Advantages of convolution over deconvolution with the low-loss spectrum As mentioned in the introduction, the normal way of treatingmultiple scattering in traditional quantitative EELS is to deconvolute theexperimental spectrum with the low-loss spectrum and thus obtain a singlescattering spectrum that can be compared to cross section calculations which

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take into account only single scattering. This process is mostly done withFourier-ratio deconvolution:

S(E) = F−1F(J(E))

F(L(E))(4)

with F and F−1 the forward and inverse discrete Fourier transform, S(E) theestimated single scattering spectrum, J(E) the recorded spectrum, and L(E)the low-loss spectrum. The main problems of this deconvolution are describedin [2]. Basically, the solution of Eq. (4) is affected by so-called noise amplifi-cation. This means that after deconvolution, the estimated single scatteringspectrum contains a lot of high frequency components due to the fact thatthe Fourier components of L(E) get smaller for increasing frequencies so thatthe ratio amplifies the high frequency components of L(E) and J(E) whichmainly consist of noise. There are ways to solve this noise amplification butthe user has to make a choice between loss of resolution and signal-to-noiseratio (in practice this may be hidden within the commercial software). Notethat the Fourier-ratio technique given by Eq. (4) can only be applied whenthe background is already removed by extrapolation. Furthermore, the distri-bution of the statistical fluctuations of the reconstructed S(E) is unknown.The ML estimator, which has a number of favourable statistical properties,can therefore no longer be used as will be shown in subsection 2.3. For thesereasons, convolution of the single-scattering core-loss spectrum with a low-lossspectrum, as in Eq. (1), is preferred over deconvolution. This approach is alsomore natural since the model mimics all steps in the experiment instead oftrying to reverse these steps.

2.1.1.4 Notation The expectations of the EELS observations will be de-scribed as a linear combination of components. Suppose that the observationswm,m = 1, ...,M are made at bins m = 1, ...,M of the spectrometer, corre-sponding to energies E1, ..., EM . Then, the expectations of the observations,E[wm], are described by an expectation model φ(Em; θ) evaluated at energyEm and depending on the parameters θ to be estimated:

E[wm] = φ (Em; θ) = φm (θ) . (5)

In the present paper φm(θ) is proportional to M(Em):

φm (θ) ∝ M(Em), (6)

where M(Em) is a linear combination of components, which is given, for ex-ample, by Eq. (3). The proportionality in Eq. (6) depends among other thingson the incident number of electrons. The unknown parameters are represented

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by the T × 1 parameter vector θ = (θ1...θT )T . This vector includes, for exam-ple, background parameters A and r, and proportionality constants p1 and p2.Apart from the expectations, also the noise needs to be described. This willbe the subject of the following subsection.

2.1.2 Noise model

As a result of the inevitable presence of noise, EELS spectra made underthe same conditions will differ from experiment to experiment. This noise ismainly governed by Poisson statistics since the collection of an EELS spectrumis essentially a counting process of incoming, inelastic electrons. In practice,however, additional processes in the conversion of these electrons to digitalcounts v in the CCD/scintillator assembly of an EELS spectrometer will addnoise. This conversion is an extremely complicated process to describe exactly.Ishizuka et al.[22] give such a description taking into account the inter-pixelmixing of the signal due to the modulation transfer function (MTF). Theyderive an expression for the pixel value variance var(v) as a 2nd order polyno-mial in the expected number of counts E[v]. Under the condition that carefulcorrection for the pixel-gain variation is made (2nd order term) and that thedark current noise and readout noise of the CCD are negligible (constantterm), the 2nd order polynomial can be replaced by its 1st order term, whichis of the form:

var (v) = βE [v] (7)

with

E [v] = kN, (8)

where k represents the conversion efficiency and N the expected number ofinelastic electrons. The factor β in Eq. (7) can then be interpreted as the ratioof the conversion efficiency k and the Detector Quantum Efficiency (DQE)[23] since:

DQE =SNR2

out

SNR2in

=

(kN/

√βkN

)2

(N/

√N

)2 =k

β(9)

with SNRout and SNRin the signal-to-noise ratio at the output and input ofthe CCD/scintillator assembly, respectively. If we define

w =v

β(10)

9

then, it can be shown that

var(w) = E[w], (11)

which is a property of the Poisson distribution. Indeed, it can be shown thatw, as defined by Eq. (10), is Poisson distributed. This has also been testedexperimentally for a Philips CM30 UT FEG with Gatan GIF 200 CCD anda Jeol 3000F with Gatan GIF 2000 CCD (both 1k x 1k phosphor). Furthercomplications arise because of the frequency dependence of the DQE [24].However, current experiments show that the approximations made are suf-ficient. Including the full details of the CCD/scintillator conversion processwould require a much more detailed study. This is in progress.

For Poisson distributed observations, the probability that an observation wm

made at bin m is equal to ωm is given by [25]

λωmm

ωm!exp (−λm) , (12)

where the parameter λm is equal to the expectation of the observation wm,which in turn, is described by the expectation model, given by Eq. (6):

E [wm] = λm = φm (θ) . (13)

Moreover, if the observations wm = vm/β (with vm digital counts) are assumedto be statistically independent, the probability P (ω; θ) that a set of observa-tions w = (w1...wM )T is equal to ω = (ω1...ωM )T is equal to the product of allprobabilities described by Eq. (12):

P (ω; θ) =M∏

m=1

λωmm

ωm!exp (−λm) (14)

This function is called the joint probability density function of the observa-tions. It represents the parametric statistical model of the observations. Theparameters θ to be estimated enter P (ω; θ) via λm. The joint probability den-sity function of the observations will be used in subsection 2.2 to determine theattainable statistical precision and in subsection 2.3 to derive the maximumlikelihood estimator of the parameters.

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2.2 Attainable precision : the Cramer-Rao Lower Bound

The parameterized probability density function of the observations, given byEq. (14), will be used to define the Fisher information matrix and to computethe CRLB on the variance of unbiased estimators of the parameters of theexpectation model. The meaning of the CRLB is that the variance of differentunbiased estimators will never be lower than this CRLB. It is independentof any particular method of estimation. The CRLB will also be extended toinclude unbiased estimators of vectors of functions of the model parameters.The reader is referred to [17], [18], and [26] to find the details of the CRLB.The derivation of the equations related to the CRLB are outside the scope ofthis paper and will only be summarized below.

First, the Fisher information matrix F with respect to the elements of theT × 1 parameter vector θ = (θ1...θT )T is introduced. It is defined using matrixnotation as the T × T matrix

F = −E

[∂2 ln P (w; θ)

∂θ ∂θT

], (15)

where P (ω; θ) is the joint probability density function of the observations w =(w1...wM )T . The expression between square brackets represents the Hessianmatrix of lnP , for which the (r, s)th element is defined by ∂2 ln P (ω; θ)/∂θr∂θs.For EELS observations, where P (ω; θ) is given by Eq. (14), it can be shownby making use of Eqs. (13), (14), and (15) that the (r, s)th element of F isequal to:

Frs =M∑

m=1

1

λm

∂λm

∂θr

∂λm

∂θs. (16)

Next, it can be shown that the covariance matrix cov(θ) of any unbiasedestimator θ of θ satisfies:

cov(θ) ≥ F−1 (17)

This inequality expresses that the difference of the matrices cov(θ) and F−1

is positive semidefinite. Since the diagonal elements of cov(θ) represent thevariances of θ1, ..., θT and since the diagonal elements of a positive semidefi-nite matrix are nonnegative, these variances are larger than or equal to thecorresponding diagonal elements of F−1:

var(θr) ≥[F−1

]rr

, (18)

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where r = 1, ..., T and [F−1]rr is the (r, r)th element of the inverse of theFisher information matrix. In this sense, F−1 represents a lower bound to thevariances of all unbiased θ. The matrix F−1 is called the Cramer-Rao LowerBound on the variance of θ.

Finally, the Cramer-Rao Lower Bound can be extended to include unbiasedestimators of vectors of functions of the parameters instead of the parametersthemselves. Let γ(θ) = (γ1(θ)...γC(θ))T be such a vector and let γ be anunbiased estimator of γ(θ). Then, it can be shown that

cov (γ) ≥ ∂γ

∂θTF−1 ∂γT

∂θ(19)

where ∂γ/∂θT is the C × T Jacobian matrix defined by its (r, s)th element∂γr/∂θs [17]. The right-hand member of this inequality is the CRLB on thevariance of γ.

From the previous analysis, it follows that the CRLB is a function of the un-known parameters which seems to be a problem at first sight. Nevertheless, itremains an extremely useful tool. For nominal values of the unknown param-eters it enables one to quantify variances that might be achieved, to detectpossibly strong covariances between parameter estimates and to optimize theexperimental design [17], [27]. Moreover, as will be shown in subsection 2.5,the estimates obtained using an estimator that achieves the CRLB may besubstituted for the true parameters in the expression for the CRLB so as toget a level of confidence to be attached to these estimates [28].

2.3 Maximum likelihood estimation

In this subsection, it is discussed how the ML estimator of the parametersmay be derived from the parameterized probability density function, which isgiven by Eq. (14). This estimator is very important since it has a number offavourable statistical properties. The maximum likelihood estimator is clearlydiscussed in [18]. A summary is given here.

The ML method for estimation of the parameters consists of three steps:

(1) The available observations w = (w1...wM )T are substituted for the corre-sponding independent variables ω = (ω1...ωM )T in the probability densityfunction, for example, in Eq. (14). Since the observations are numbers,the resulting expression depends only on the elements of the parametervector θ = (θ1...θT )T .

(2) The elements of θ = (θ1...θT )T , which are the hypothetical true param-eters, are considered to be variables. To express this, they are replaced

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by t = (t1...tT )T . The logarithm of the resulting function, ln P (w; t), iscalled the log-likelihood function of the parameters t for the observationsw.

(3) The ML estimates θML of the parameters θ are defined by the values ofthe elements of t that maximize ln P (w; t), or

θML = arg maxt ln P (w; t) (20)

Furthermore, it can be shown [29] that the ML estimates γML of a vector offunctions of the parameters θ, that is, γ(θ) = (γ1(θ)...γC(θ))T , are equal toγ(θML) = (γ1(θML)...γC(θML))T .

The most important properties of the ML estimator are the following ones.First, it achieves the CRLB asymptotically, that is, for an infinite numberof observations. In this sense, the maximum likelihood estimator is asymp-totically most precise. Second, this estimator is said to be consistent, whichmeans that it converges to the true value of the parameter in a statisticallywell defined way if the number of observations increases. Third, the ML esti-mator asymptotically tends to a normal distribution with a mean value equalto the true value of the parameter and a covariance matrix equal to the CRLB.Whether these asymptotic properties also apply to a finite number of observa-tions, as for EELS observations, can be tested by means of simulations. Thiswill be done in subsection 3.1.

2.4 Model validation

From Eq. (20), it directly follows that the results obtained using ML estima-tion depend on the validity of the joint probability density function of theobservations. If the expectations of the observations or the noise is not ac-curately described by this joint probability density function, ML estimationis no longer valid. Therefore, it is important to test the validity of this func-tion before attaching confidence to the parameter estimates obtained. Variousmodel validation tests can be found in the literature. An overview of thesetests will be given in [30]. In the present paper, the so-called Likelihood Ratiotest [31] will be presented for Poisson distributed observations. In comparisonto other tests, it has the advantage that replications of the observations arenot required.

Suppose that a set of Poisson distributed observations wm,m = 1, ...,M isavailable and that we want to test the so-called null hypothesis (H0) that theexpectations of these observations can be described by the expectation modelφm (θ) against the alternative hypothesis (H1) that the expectations of theseobservations cannot be described by φm (θ). It can be shown that, when H0 is

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true, the statistic

LR = 2M∑

m=1

−wm + wmlnwm

φm

(θML

) + φm

(θML

) (21)

is approximately chi-square distributed with M − T degrees of freedom withT the number of unknown parameters as shown by den Dekker et al. [30].

φm

(θML

)in Eq. (21) represents the expectation model evaluated at the ML

estimates θML. The Likelihood Ratio test uses LR as test statistic. The hy-pothesis H0 is accepted at the significance level α (which can be freely chosen)if:

LR ≤ χ2M−T,1−α (22)

or, alternatively,

χ2M−T,α/2 ≤ LR ≤ χ2

M−T,1−α/2 (23)

with χ2M−T,q the qth quantile of a chi-square distribution with M − T degrees

of freedom (value of chi-square for which the cumulative probability is q).The meaning of the significance level α is that if H0 is true, the probabilityof rejecting H0 is α and the probability of accepting H0, making the correctdecision, is 1 − α. The Likelihood Ratio test will be applied in section 3.1.

2.5 Confidence intervals for maximum likelihood parameter estimates

In order to evaluate the level of confidence to be attached to the ML estimatesobtained, confidence intervals associated with these estimates are required. A100(1−α)% confidence interval for an element θt of θ is an interval that coversthe true element θt with probability 1− α. Confidence intervals, also referredto as error bars, are usually presented as the estimated value of the parameterplus or minus a certain amount. In [30], many methods are presented to obtainconfidence intervals. Two of these methods, the most used and an interestingalternative, will be summarized in this paper: confidence intervals based onthe sample variance and intervals based on the CRLB. The former methodrequires the experiment to be repeated a number of times, while the latterdoesn’t. This, rather technical, section may be skipped during a first readingwithout losing the thread of this paper.

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2.5.1 Confidence intervals based on the sample variance

Ideally, confidence intervals are obtained by repeating the same experimentover and over again. Subsequently, the ML method is applied to each experi-ment resulting in a sample of ML estimates θi

ML, i = 1, ..., I (with I the totalnumber of repeated experiments). If these estimates are normally distributed,a 100(1 − α)% confidence interval for an element θr of θ can be given by:[

θiML

]r± λ1−α/2sθr (24)

with [θiML]r the rth element of the parameter vector θi

ML and λ1−α/2 the(1 − α/2)th quantile of the standard normal distribution. sθr is the standarddeviation of the sample under consideration, i.e., the square root of the samplevariance s2

θr, which is defined as:

s2θr

=1

I − 1

I∑i=1

([θi

ML

]r − θr

)2(25)

with θr the sample mean:

θr =1

I

I∑i=1

[θi

ML

]r. (26)

Alternatively, a 100(1 − α)% confidence interval for the sample mean θr isgiven by [29]:

θr ± tI−1,1−α/2sθr√

I(27)

with tI−1,1−α/2 the (1−α/2)th quantile of the t distribution with I−1 degreesof freedom. This will be used in subsection 3.1.

Finally, it should be mentioned that the sample variance s2θr

is a statistic forwhich its 100(1 − α)% confidence interval is given by [29]:(I − 1) s2

θr

χ2I−1,1−α/2

,(I − 1) s2

θr

χ2I−1,α/2

(28)

with χ2I−1,q the qth quantile of a chi-square distribution with I − 1 degrees of

freedom.

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2.5.2 Confidence intervals based on the CRLB

In the previous method, confidence intervals are obtained by repeating thesame experiment several times. In practice, however, this scenario is often notvery realistic, for example, due to specimen damage. The method that will besummarized in this subsection does not require replications and is thereforepractically more useful.

From subsection 2.3, it follows that the ML estimator θML of the parametervector θ is known to be asymptotically normally distributed with a mean equalto the true value of the parameter vector and a covariance matrix equal tothe CRLB. If these properties apply to EELS observations, an approximate100(1 − α)% confidence interval for an element θr of θ can be given by:[

θML

]r± λ1−α/2

√[F−1]rr (29)

with [θML]r the rth element of the parameter vector θML, [F−1]rr the (r, r)thelement of the CRLB (F−1), and λ1−α/2 the (1− α/2)th quantile of the stan-dard normal distribution. If the CRLB is a function of the parameters to beestimated, the confidence interval (29) is usually derived by substituting theestimated parameters for the true parameters in the expression for the CRLB[28,30].

3 Experimental examples

To test the applicability of the ML method to real-life spectra, two experimentsare described here. They answer, in a practical way, the questions whether itis possible to build valid models for EELS spectra and whether the CRLB canbe approached. The samples studied are chosen to be “blameless” samples,in the sense that the quantity which is to be measured is predictable andstable. We took care that the samples are not beam sensitive, they containno constituents which can also enter as contamination from oil or air, theyrequire no specimen preparation that can change the chemical contents of thesample and they have 2 edges which can be captured in one spectrum to allowfor accurate relative quantification.

3.1 Au M4,5 edge

A first test sample consists of a thin (≈ 10 nm) layer of Au {110} bicrystals.The Au was deposited on a {001} Ge substrate which is later removed by

16

etching as described in [32]. The Au film is then lifted and deposited on aholey carbon film. A set of 100 spectra of the Au M4,5-edge was recordedin diffraction mode EELS from a micron sized region of the specimen. Allspectra were taken under exactly the same conditions; they only differ by thenoise. They were recorded on a Jeol 3000F with a GIF2000 spectrometer.The collection angle was approximately 1.4 mrad and the convergence anglewas negligible compared to this. The extraction voltage of the FEG was setto 1.7kV to avoid the inelastic spectrum created by the gun itself affectingthe spectral region of the Au M-edge [33]. Core loss spectra were recorded in10s while low loss spectra were acquired with 0.016s exposure both with 1eVdispersion. The expectation model was constructed from a background powerlaw with 2nd order polynomial exponent and two Hartree Slater cross sections,calculated with Digital Micrograph for the M4 and M5 edge in Au, convolutedwith the experimental low loss spectrum to simulate multiple scattering. Thestatistical noise is modeled as described in subsection 2.1.2 with β = 3.6obtained experimentally. The Au M-edge is formed by exciting the 3d5

2(Au

M5) and the 3d32(Au M4) atomic levels to unoccupied levels. In single particle

theory the ratio in cross section between both edges is given by the ratioof their ground state occupation number and should therefore be M5/M4 =6/4 = 1.5. Many particle corrections can, however, lead to a slight deviationfrom this ideal value (deviations can be quite large in L-edges for instance andare known as anomalous white line ratios [34]).

In addition to the experimentally obtained spectra, simulated spectra werecreated to verify whether the ML estimator is an unbiased estimator andwhether it attains the CRLB for an ‘ideal’ situation. The ideal situation iscreated by simulating 100 different, Poisson distributed, spectral observationsfor a given expectation model. The true parameters of this model were assumedto be equal to the estimated parameters obtained from an experimental EELSspectrum to mimic as closely as possible the experimental situation. Somemembers of the set of 100 spectra are shown in fig.3 for both simulated andexperimental data. The main difference between these simulated data and realexperimental data is that the model is known exactly for simulations whilefor experiments we have to propose a model and test whether it describes theexperiment well. The ML method was then applied to the real and simulateddata resulting into 100 estimates for the M5/M4 ratio for each data set. Thedistribution of the M5/M4 ratio estimates is shown as a histogram in fig. 2. Aninteresting thing which can clearly be seen from this figure is the fact that (aswith all measurements) one should never trust a single EELS measurementbecause of the finite chance to find parameters far from the true parameters.

Before discussing the results of the ML method, the LR test of subsection 2.4has been applied to check the validity of the proposed expectation model forboth the simulated and experimental data set and to gain insight in the mean-ing of the significance level. Table 1 shows the results. Since the model for the

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1.2 1.3 1.4 1.5 1.6M

5/M

4 ratio [/]

0

10

20

30

40

freq

uenc

y [/

]

simulated results (mean=1.443; stddev=0.0443)experimental results (mean=1.453; stddev=0.069)normal distribution (mean=1.439; stddev=0.047)

Fig. 2. Histogram of ML estimated Au M5/M4 ratios from 100 experimental spec-tra (grey) and 100 simulated spectra (hatched), together with a normal distributionwith mean and standard deviation equal to the true M5/M4 ratio used in the sim-ulated spectra and square root of the CRLB computed from the known model usedin these spectra, respectively. The experimental spectra produce a slightly broaderdistribution than this normal distribution

simulated data is known, the probability of rejecting this model should, bydefinition, be equal to the significance level α and the probability of accept-ing the model should be equal to 1 − α. From the comparison of the chosensignificance levels with the percentage of accepted models when the LR test isapplied to simulated spectra, it follows that the LR test works well. However,from the comparison of the chosen significance levels with the percentage ofaccepted models when the LR test is applied to experimental spectra, it followsthat the model is rejected more often than the predicted value. This indicatesthat the model is not a perfect description of the experiment and that modelsshould be improved in the future. Of course, the model is necessarily a simpli-fication of the physics behind the experiment. Furthermore, outliers exist inthe data. These unavoidable outliers can be caused by electronic glitches andcosmic rays and are not described by Poisson statistics as is assumed in thepresented LR test. Nevertheless, even though the model seems to be rejectedon statistical grounds, the present analysis will be continued to show how themethods of section 2 work and what the impact of imperfect models may be.

Next, the mean and standard deviation of the estimates obtained from the sim-ulated data set were computed and compared to the true M5/M4 ratio and thelower bound on the standard deviation, that is, the square root of the CRLB,

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2100 2200 2300 2400 2500 2600energy loss [eV]

4000

6000

8000

10000

12000

14000

elec

tron

cou

nts

[/]

model

simulated 1

simulated 2

simulated 3

experimental 1

experimental 2

experimental 3

Au M5 Au M

4

Fig. 3. Some Au M4,5 EELS spectra selected from both simulated and experimentaldata sets together with the model used to generate the simulated data. The spectraare set apart by adding a multiple of 1000 electrons with respect to the model toimprove visibility in this plot. Note how noisy these spectra are while still allowingapproximately 5% relative precision in determining the Au M5/M4 ratio. Note alsothe qualitative difference between the simulated noise and the experimental noise,the latter having clearly less high frequency components due to frequency dependenteffects in the CCD/scintillator assembly.

Table 1Results of the LR test applied to a set of 100 simulated and experimental spectra.

significance level α accepted models from accepted models from

simulated spectra experimental spectra

0.05 93% 77%

0.25 75% 49%

0.50 51% 28%

0.75 22% 11%

0.95 6% 3%

respectively. The CRLB was computed by substituting the true values of theparameters (which are known for simulations) into the expression given bythe right-hand member of Eq. (19). The results are presented in table 2. Also95% confidence intervals are presented using Eq. (27) and the square roots ofEq. (28). From the comparison of these results, it follows that the mean of theML estimator agrees with the true M5/M4 ratio and that its standard devia-tion agrees with the lower bound on the standard deviation. Hence, the twoasymptotic properties of the ML estimator, asymptotically most precise andunbiased, seem to be fulfilled. Furthermore, the normal distribution shown infig. 2 with mean equal to the true M5/M4 ratio and standard deviation equal to

19

the square root of the CRLB makes plausible that the estimates are normallydistributed. This is another asymptotic property of the ML estimator.

Table 2Comparison of M5/M4 ratio and lower bound on the standard deviation (stddev)with estimated mean and standard deviation of 100 ML estimates obtained from asimulated data set, respectively. Furthermore, 95% confidence intervals are provided.

M5/M4 ratio estimated mean 95% confidence interval

1.439 1.443 (1.434, 1.452)

lower bound on stddev estimated stddev 95% confidence interval

0.047 0.044 (0.039, 0.051)

Furthermore, the standard deviation of the estimates obtained from the exper-imental data set and its 95% confidence interval have been computed using thesquare roots of Eqs. (25) and (28), respectively. Additionally, the lower boundon the standard deviation, that is, the square root of the CRLB, has beencomputed following the procedure of subsection 2.5.2. For each experimentalspectrum, the CRLB has been estimated by substituting the ML estimatedparameters into the expression given by the right-hand member of Eq. (19).From the thus obtained set of lower bounds, the mean and 95% confidence in-terval have been computed using the equivalent expressions for the mean andits confidence interval of estimated parameters given by Eqs. (26) and(27),respectively. The results are shown in table 3. From this table, it follows thatthe experimental standard deviation is slightly higher (statistically significant)than the computed lower bound on the standard deviation. The reason for thiscan be due to model imperfections as follows from the results of the LR test. Ifthe parametric statistical model proposed does not describe the observationscorrectly, the expressions obtained for the CRLB are no longer valid and theestimated values for the CRLB will deviate. It should be noticed that thedeviation of the lower bound from the estimated standard deviation is notdue to the estimator used. By means of simulations it has been shown thatthe ML estimator attains the CRLB. In the present experiment, however, theestimated lower bound on the standard deviation still gives the right order ofmagnitude for the error bar to be expected from an experiment even if makinga perfect model with a limited amount of meaningful parameters is not yetpossible. Moreover, one should realize that statistically rejected models mayhave an impact on the accuracy as well. If the model does not describe theobservations correctly, there is a possibility that the results are biased. In thepresent experiment, the bias, which might be introduced, seems to be smallsince it follows from the histogram of fig. 2 that the ML estimates are close to1.5 as expected from a theoretical point of view.

As a conclusion to this experiment, it can be stated that the methods of section

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Table 3Comparison of estimated lower bound on the standard deviation (stddev) and esti-mated standard deviation of 100 ML estimates of the M5/M4 ratio obtained froman experimental data set. Furthermore, 95% confidence intervals are provided.

estimated stddev 95% confidence interval

of M5/M4 ratio

0.069 (0.0606, 0.0802)

estimated lower bound on 95% confidence interval

stddev of M5/M4 ratio

0.047 (0.0466, 0.0474)

2 work well as follows from simulations for which the model is known exactly.For correct models, the ML estimator is unbiased and attains the CRLB.Therefore, the CRLB is a good way to estimate the precision. Furthermore,from the simulations, it follows that the LR test is an effective method formodel validation. Although correct models are not yet available in the EELScommunity, as follows from the LR test applied to experiments as well, theexperimental results are promising. In our carefully conducted experiment,the estimated lower bound on the standard deviation does not yet agree withthe experimental standard deviation but it is of the right order of magnitude.Furthermore, the possibly introduced bias due to model imperfections, seemsto be small. In the future, the existing problems are expected to be diminishedup to the level where the model becomes statistically acceptable, given thatEELS models are continuously improving. In the assessment of the qualityof the models, the methods presented in this paper, including the LR testfor model validation, will be indispensable. It is interesting to note that theprecision is of the order of 5% in the present experiment with overlapping edgesand a very low signal-to-noise ratio in the spectra. This would be a challengingif not an impossible spectrum for conventional quantification techniques.

3.2 GaAs cleaved wedge

A somewhat more practically applicable experiment was conducted on a cleavedsample of GaAs. A 90◦ wedge was produced by cleaving a GaAs substrate alonga {110} plane. A STEM probe φ ≈ 1nm was scanned approximately alongthe < 001 > zone axis, from the hole progressively into thicker regions of thesample. The Ga/As ratio is estimated using the ML method. The proposedexpectation model consists of 2 hydrogenic L-edges and a power law with 2ndorder polynomial exponent for the background. Convolution with the low lossspectrum is included to model multiple scattering. The spectra are recordedin diffraction mode EELS in a Jeol 3000F and a GIF2000 spectrometer with

21

a collection angle of 5 mrad. The sample is tilted approximately 3◦ from the〈100〉 zone axis to avoid problems with channeling conditions. The quantifica-tion of these spectra is traditionally a very difficult task because both edgesare very close (GaL2,3 1115 eV and AsL2,3 1323 eV) and lie at high energiesresulting in low cross sections and bad signal-to-noise ratios in the spectra.The thickness of the sample at each point is calculated from the low loss spec-trum with Digital Micrograph giving a relative thickness t/λ with λ the meanfree path for inelastic scattering (for GaAs roughly λ = 100 nm[2]). Note thatthis thickness is the total sample thickness, and may therefore include thethickness of a thin amorphous carbon layer at the surface.

First, the LR test has been applied to each experimental spectrum to check thevalidity of the proposed model. From this test it follows that the percentageof spectra for which the model seems an accurate description is 71% for asignificance level of 5%. This points to an imperfection of the expectation ornoise model.

The obtained ML estimates for the Ga/As ratio for 2 scans in different regionsof the sample are shown in fig. 4. Furthermore, 95% confidence intervals basedon the CRLB are provided using the procedure of subsection 2.5.2. It canbe observed that most confidence intervals coincide and contain the expectedvalue of 1 for the Ga/As ratio, even for t/λ ≈ 2.3, which usually is consideredtoo thick for quantification. In fig. 4, the mean of the estimated Ga/As ratiosis indicated by a line. Note that although the sample mean is slightly differentfor both sets (0.99 and 1.012), their 95% confidence intervals, which can becomputed using Eq. (27), coincide. Furthermore, for the noisy spectra in thepresent experiment, a precision, in terms of the lower bound on the standarddeviation, of better than 5% is obtainable. Nevertheless, since the model isnot perfect, one has to remain alert to interpret the results.

4 EELSMODEL software

To promote model based quantitative EELS, software has been developed us-ing the C++ language and Trolltech Qt3.0 [35] for the user interface. Thegoal is to give users a low threshold alternative to the conventional quan-tification algorithms that are implemented in commercial software packages.The software is freely available under the GNU public license [16] from theEELSMODEL web site [36]. A screen shot of the program is given in fig. 5.

The program is written to be platform independent and offers the user a choicebetween several estimators:

• ML estimator for independent and Poisson distributed observations

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Fig. 4. (A) Plot of the ML estimated Ga/As ratio as a function of thickness (ex-pressed with respect to the mean free path (mfp) for plasmon excitations) for acleaved sample of GaAs. Two different regions are scanned. The setup of the exper-iment is shown as an inset. (B) One of the experimental spectra together with theexpectation model evaluated at the ML estimates, showing the poor signal-to-noiseratio.

• Weighted least squares estimator (= ML estimator for independent andnormally distributed observations)

• Uniformly weighted least squares estimator (= ML estimator for indepen-dent and identically normally distributed observations)

The user can build an expectation model from the following components (canbe expanded by the user to include any component which is expressible inC++):

• Background models:· Power law background AE−r

· Power law with polynomial exponent AEr1+r2E+r3E2+...

· Exponential background Aexp(−αE)• Core-loss excitation models:· Hydrogenic cross sections SigmaK, Sigmal [2,10]· Relativistic K-edge cross sections SigmaKrel [37]· Edges stored in a MSA file: can be any excitation edge calculated with Ab

23

Fig. 5. A screen shot of the EELSMODEL program, showing the results of fittinga model for a Au M4,5 edge to an experimental spectrum using the ML method.

initio packages or Hartree Fock methods. It can even be an experimentaledge shape from a reference sample.

• Fine structure functions:· Lorentzian profiles· Gaussian profiles

New components and improved models for excitation cross sections can easilybe added to test new ideas.

The program offers the possibility to lock certain parameters which will notbe estimated and to couple parameters to other parameters. Furthermore,the precision of the parameter can be estimated using the CRLB. Ratios,differences and sums of two parameters can be monitored and the associatedCRLB is estimated. Also, the LR test is implemented to find out how well themodel describes the experiment. More detailed information can be found atthe EELSMODEL web site [36].

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5 Conclusion

This paper shows that model based quantitative EELS is a powerful approachto obtain estimates of physical parameters describing an EELS spectrum. Byusing the proper statistical model of the observations describing both the ex-pectations of the observations and the noise statistics, the ML method can beapplied to estimate both the parameters and the attainable precision, that is,the CRLB on the variance of these estimates. Expectation models were buildfrom a superposition of different available components describing the differentcontributions to the spectrum. The background component was omitted fromthe multiple scattering convolution to solve several numerical artifacts on thebasis that one can model equally well a multiple scattered background as theusual single scattered background.

It was shown by means of simulations, for which the model is known exactly,that the ML estimator is unbiased and that it attains the CRLB. For statisti-cally validated models, the ML estimator is unbiased and attains the CRLB.Therefore, the CRLB is a good way to estimate the precision. Furthermore,from the simulations, it follows that the LR test is an effective method formodel validation.

Carefully conducted experiments were performed using Au and GaAs sam-ples. From the LR test, it follows that the proposed models do not describethe observations exactly. The reason being that perfect models are not yetavailable in the EELS community. Nevertheless, the experimental results arepromising. Although, the estimated lower bound on the standard deviationdoes not yet agree with the experimental standard deviation, it is of the rightorder of magnitude. In these experiments, the precision is much better thancommonly expected. Furthermore, the possibly introduced bias due to modelimperfections, seems to be small. In the future, the existing problems resultingfrom model imperfections are expected to be reduced up to the level wherestatistically valid models can be produced given the continuous improvementin EELS models. Better inelastic atomic cross section models, preferably pa-rameterized for fast calculation, are required. These cross sections don’t needto produce the fine structure of the excitation edges but the general shape andhow the edge decays with increasing energy are important. Techniques whichgo beyond atomic cross sections are not generally applicable since in principlethe sample is unknown at the time a quantitative analysis is performed. In theassessment of the quality of the models, the methods presented in this paper,including the LR test for model validation, will be indispensable.

A major advantage of quantitative EELS as described in this paper is that itworks on non-processed data (unlike for instance first-difference techniques).The model really predicts what comes out of the spectrometer, including every

25

single step. Even the last step of converting electrons to electronic countsmeasured with the spectrometer CCD can be included in a direct way butfurther work is needed to include the spatial correlation of the statistical noisedue to this process. Compared to the conventional quantification technique,the presented methods work for overlapping edges and noisy spectra as well.

The ideas presented in this paper are combined in a free user friendly programto stimulate the use of model based quantification.

6 Acknowledgements

J. Verbeeck would like to thank A. Rosenauer for discussions and help with thefirst version of the EELSMODEL program, S. Hens for making the Au sampleand IUAP V-I for financial support. S. Van Aert gratefully acknowledges thefinancial support of the Fund for Scientific Research - Flanders (FWO). Bothauthors would like to acknowledge G. Van Tendeloo and D. Van Dyck forstimulating discussions.

References

[1] R. Henderson, Quarterly Reviews of Biophysics 28 (1995) 171-193.

[2] R. F. Egerton, Electron Energy Loss spectroscopy in the electron microscope,Plenum Press, New York, 2nd edition (1996).

[3] R. F. Egerton, M. Malac, Ultramicroscopy 92 (2002) 47-56.

[4] D. S. Su, E. Zeitler, Phys. Rev. B vol 47-22 (1993) 14734-14740.

[5] T. Pun, J. R. Ellis, M. Eden, Journal of Microscopy 135 (1984) 295-316.

[6] T. Pun, J. R. Ellis, M. Eden, Journal of Microscopy 137 (1985) 93-100.

[7] R. F. Egerton, Ultramicroscopy 9-4 (1982) 387-390.

[8] M. Unser, J. R. Ellis, T. Pun, M. Eden, Journal of Microscopy, 145 (1987)245-256.

[9] D. R. Liu, L. M. Brown, Journal of Microscopy 147 (1987) 37-49.

[10] H. Bethe, Annalen Der Physik 5 (1930) 325-400.

[11] R. Door, D. Gangler, Ultramicroscopy 58-2 (1995) 197-210.

[12] T. Manoubi, M. Tence, M. G. Walls, C. Colliex, Microscopy MicroanalysisMicrostructures 1-1 (1995) 23-39.

26

[13] H. Shuman, A. P. Somlyo, Ultramicroscopy 21-1 (1987) 23-32.

[14] R. D. Leapman, C. R. Swyt, Ultramicroscopy 26 (1988) 393-404.

[15] Y. Y. Wang, R. Ho, Z. Shao, A. P. Somlyo, Ultramicroscopy 41 (1992) 11-31.

[16] http://www.gnu.org/licenses/gpl.html

[17] A. van den Bos, Parameter estimation. 1982. In Handbook of MeasurementScience, ed. P.H. Sydenham. Vol. 1 Chicester: Wiley pp. 331-377.

[18] A. van den Bos and A.J. den Dekker. 2001. Resolution reconsidered -Conventional approaches and an alternative. In Advances in Imaging andElectron Physics, ed. P.W. Hawkes. Vol. 117 San Diego: Academic Press pp.241-360.

[19] R. F. Egerton, Ultramicroscopy 4 (1979) 169-179.

[20] R. F. Egerton, Philosphical Magazine 31-1 (1975) 199-215.

[21] D. M. Maher, D. C. Joy, R. F. Egerton, P. Mochel, Journal of Applied Physics50-8 (1979) 5105-5109.

[22] K. Ishizuka, Ultramicroscopy 52 (1993) 7-20.

[23] K. -H. Hermann, D. Krahl, Journal of Microscopy 137-1 (1982) 17-28.

[24] R. R. Meyer, A. I. Kirkland, R. E. Dunin-Borkowski, J. L. Hutchison,Ultramicroscopy 85 (2000) 9-13.

[25] A. Papoulis. 1965. Probability, Random Variables, and Stochastic Processes.New York: McGraw-Hill.

[26] B.R. Frieden. 1998. Physics from Fisher Information - A Unification.Cambridge: Cambridge University Press.

[27] S. Van Aert, A.J. den Dekker, A. van den Bos, and D. Van Dyck.2004. Statistical Experimental Design for Quantitative Atomic ResolutionTransmission Electron Microscopy. In Advances in Imaging and ElectronPhysics, vol. 130, ed. P.W. Hawkes. San Diego: Academic Press.,1-164

[28] A.J. den Dekker and S. Van Aert. 2002. Quantitative high resolution electronmicroscopy and Fisher information. In Proceedings of the 15th InternationalCongress on Electron Microscopy, Interdisciplinary and Technical ForumAbstracts 2002 in Durban, South Africa, ed. R. Cross. Vol. 3 Onderstepoort:Microscopy Society of Southern Africa pp. 185-186.

[29] A.M. Mood, F.A. Graybill and D.C. Boes. 1974. Introduction to the Theory ofStatistics. 3rd ed. Singapore: McGraw-Hill.

[30] A.J. den Dekker, S. Van Aert, A. van den Bos, and D. Van Dyck. 2004.Maximum likelihood estimation of structure parameters from high resolutionelectron microscopy images. Part I: a theoretical framework. In preparation.

27

[31] G.A.F. Seber and C.J. Wild. Nonlinear Regression. John Wiley and Sons, NewYork, 1989.

[32] T. Radetic, U. Dahmen, Mat. Res. Soc. Symp. Vol. 695 (2002) 47-52.

[33] D. W. McComb, G. C. Weatherly, Ultramicroscopy 68 (1997) 61-67.

[34] B. T. Thole, G. van der Laan, Physical Review B 38-5 (1988). 3158-3171

[35] http://www.trolltech.com/products/qt/index.html

[36] http://webhost.ua.ac.be/eelsmod/eelsmodel.htm

[37] R. Knippelmeyer, P. Wahlbring, H. Kohl, Untramicroscopy 68 (1997) 25-41.

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