SURFACE AND INTERFACE ANALYSISSurf. Interface Anal. 27, 1114–1117 (1999)

SHORT COMMUNICATION

Notes on Locating Peak Energies in XPS

Janos Vegh*Institute of Nuclear Research of the Hungarian Academy of Sciences, H-4001 Debrecen, Pf. 51, Hungary

The algorithms proposed for determination of the position in XPS are analysed. The method of bisectedchords is reformulated in a form readily available for digital-computers, another derivation of fitting aparabola to the top of peak is given and some more non-iterative methods are presented. The suggestedmethods determine different characteristics of the photopeak, such as ‘peak location’, maximum height,centre of gravity and the distribution parameter. Copyright 1999 John Wiley & Sons, Ltd.

KEYWORDS: peak location; peak position; energy calibration; bisected chord method

INTRODUCTION

One of the key points in electron spectroscopy is to deter-mine the energy of the photo-peaks. Because the energyscale calibration itself is also based on the determinationof the energies of some photopeaks, it is worth compar-ing the features of the available methods offered for thatpurpose.

METHODS USED TO LOCATE PEAKS

Let us have a measured spectrumy.E/ (a distributionof countsy as a function of energyE), i.e. y.k/ is thenumber of counts recorded in channelk of nominal energyE.k/. One can determine ‘peak position’ with severalmethods, each providing a different result, partly becausea different characteristic is interpreted as a ‘peak position’.In addition, different parts of the peaks are used in thecalculation, i.e. the peaks are truncated differently. AsHansen indicates in his work determining peak positionwith a high precision,1 the arbitrary choice of the channelsconsidered distorts the sample (which should be infinitelywide) and necessitates what in statistical language is calleda ‘bias correction’. Also, it is worth noting that the so-called ‘binning’ (the non-infinitely small channel width)can also distort the derived result (see section II. C.in Ref. 2).

The bisected chord method

The method was originally given3 in terms like

plot spectra through. . . , draw the baseline,

. . . draw horizontal chords, etc.

* Correspondence to: J. Vegh, Institute of Nuclear Research of theHungarian Academy of Sciences, H-4001 Debrecen, Pf. 51, Hungary.E-mail: J.Vegh@atomki.hu

Contract/grant sponsor: Hungarian Academy of Sciences; Con-tract/grant number: OTKA (T026514).

Because of this, it is quite difficult to compare this methodwith the present-day digital computer methods availablein the data evaluation software. Now we will reformulateit in a form readily available for use on the modern digitalcomputers.

In this method chords are drawn horizontal (formerlyparallel to the baseline). They intersect the irregular poly-gon, forming the peak envelope. Using the notation shownin Fig. 1 for thekth chord (drawn at heightyk) it happensbetween points.i, iC1/ and.j, jC1/. TheE coordinatesof this chord are

Ek,L D Ei.k/ C [Ei.k/C1� Ei.k/] yk � yi.k/yi.k/C1� yi.k/ .1/

and

Ek,R D Ej.k/C1 � [Ej.k/C1� Ej.k/] yk � yj.k/C1

yj.k/ � yj.k/C1

.2/

for the left (L) and right (R) intersections, respectively,with the two polygon sections.

Usually, the data points are equidistant (i.e.E DEi.k/C1 � Ei.k/ D Ej.k/C1 � Ej.k/), so the energy of themidpoint is

Ek D 1

2

{Ei.k/ C Ej.k/C1 CE

ð[yk � yi.k/

yi.k/C1� yi.k/ �yk � yj.k/C1

yj.k/ � yj.k/C1

]}.3/

The same calculation can be carried out on several chords,each contributing a midpoint energy. These midpoints arethen extrapolated to a value at the peak giving the peakenergy; in other words, a curve (line) is drawn through themidpoints, which is some kind of (rather arbitrary) fitting.As found by Cumpson,5 a simple average can be taken forsymmetrical peaks but the midpoints will scatter on bothsides of this line.

Without loss of generality, one can correct the calcu-lated midpoint energy of a chord with an arbitrary constantvalue, namely with the value (relative to the projectedpeak energy) of the intersection of this fitted line andthe chord. After having corrected the individualEk items,

CCC 0142–2421/99/121114–04 $17.50 Received 30 June 1999Copyright 1999 John Wiley & Sons, Ltd. Revised 23 August 1999; Accepted 23 August 1999

LOCATING PEAK ENERGIES IN XPS 1115

Figure 1. The notations used in the bisected chord method.

the energy of the peak itself is calculated as an averageof these correctedE0k values over allk chords, i.e. thealgorithm simplifies to a simple summation over properlyselected pairs of sections of the polygon. The sum can beseparated into two terms

E D

∑k

[E0i.k/ C E0j.k/C1]

2∑k

1

CE

∑k

[yk � yi.k/yi.k/C1� yi.k/ �

yk � yj.k/C1

yj.k/ � yj.k/C1

]2∑k

1.4/

The first sum is evidently the (unweighted) average energyfor all points used, whereas the second term needs moreattention. One can assume (without loss of generality) thatthe symmetricpeak is located symmetrically (i.e. suchthat its peak energyEo coincides with the nominal energyof one of the measured pointsEi). Notice that in theabsence of statistical disturbance the individual terms inthe second sum would be exactly zero. It does not holdfor asymmetric peaks, so thesecond term measures theasymmetry of the peak.This asymmetry can arise from theasymmetric placingor theasymmetric shapeof the peak,or both. Of course, the counting statistics also affect theseterms, but in a statistical sense they remain valid.

The result above points to one more requirement: wecannot select the chords arbitrarily. When drawing chordsat some equidistant heights,some measured points mightbe used in calculating the midpoints for more than onechord (or just the opposite: it might not be used atall), which means that some measured points are usedmore than once in calculatingE, and thepeak energybecomes biased. According to the mathematical statistics,one cannot use any of the four points used in evaluatingE.k/, E.kC1/ or E.k�1/. However, in practice one cannot beso strict. For example, in spectra published by Powell4

only nine points fall in the [100%, 84%] height interval,so only (maximum) two such chords could be derived.When drawing statistical conclusions, we have to bearthis fact in mind.

Notice that the way the chords are drawn affectsthe final result. The linear interpolation used above forapproximating the upper part of the distribution seemsreasonable and the need for some spline or other fitted

envelope does not arise. It is easy to see that if the abso-lute value of the slopes of the linear sections [i, iC1] and[j, j C 1], respectively, are different (and of course theyare), the resulting midpoint energyEk will depend on theactual value ofyk, so the sampling frequency and rangeshould be given in the algorithm. An alternative could beto start at the maximum height plus the two neighbour-ing points and then extend the region on either side untilthe new point allows a new chord. In this caseyk couldbe the middle height of the narrower height interval: forexample, in Fig. 1yk D .yi C yj//2. The final resultingpeak energy could be derived as the energy coordinateof the point where the line, fitted to the chord midpoints,intersects the peak envelope.

Considering the uncertainty ofE, it can be seen easilythat the closer the points are to the peak maximum, thecloser the counts in the adjacent channels are to eachother, i.e. their contribution to the uncertainty is greater.This fact suggests that care be used when determining thepeak energy that the midpoints are extrapolated to. Onthe other hand, although the uncertainty ofE improveswith the number of the terms averaged, the differenceof the counts in adjacent channels will decrease sharplyand so also will the influence of systematic errors (suchas neighbouring peaks, asymmetric shape, etc), thusanincrease in the number of the chords will result in muchmore uncertainty. This analysis supports the suggestionfrom practical experience that this algoritm should useonly points in the [100%, 84%] height interval.

In summary, the method is equivalent to theunweightedaverageof the energiesEk, where the method of selectingthe points in summing greatly improves the precision,focusing the summing to the points near to the peakmaximum, so that practically the maximum of the peakheight is evaluated as the peak location.

Non-iterative fitting methods

These methods rely heavily on the measured points beingequidistant, so in this sectionEi stands for the sequencenumberi. Note that in calculating the value of the trans-formed function, at some point the neighbours of the pointare used; thus, in calculating the uncertainty, special careshould be taken so as not to use multiple data points.

Parabola fitting. A method for fitting a parabola (of theform yi D aE2

iCbEiCc) to the top of the peak is describedby Cumpsonet al.5 Another simple derivation of the sameresult can be given using a transformation and a non-iterative linear fitting. Let us consider the top few pointsof the peak and transform the intensities as

Q.Ei/ D yiC1 � yi�1 .5/

This leads toQ.Ei/ D 4aEi C 2b .6/

i.e. the transformed function is linear in energy. To thistransformed function a straight line can be fitted (usingthe weightsWi D [�.yiC1/2C �.yi�1/2]�1), as described instandard textbooks.6 The fitting results in the peak position

Eo D � b2a

.7/

Copyright 1999 John Wiley & Sons, Ltd. Surf. Interface Anal. 27, 1114–1117 (1999)

1116 J. VEGH

in full accordance with Eqn (14) of Ref. 5. From thisderivation it is also clear immediately that the constantterm c of the parabola (corresponding to a constant back-ground) does not have any role: however,a sloping back-ground will shift the peak position.

Polynomial fitting. A more generalized fitting procedurecan be derived easily. Having equidistant points, let us fitto the point with maximum height plus 2m neighbouringpoints annth order polinom of the form

fj Dn∑kD0

bnkjk .8/

in such a way that

M DCm∑jD�m

.fj � yj/2 DCm∑jD�m

(n∑kD0

bnkjk � yj

)2

.9/

is minimum. The corresponding condition is

∂M

∂bnrD 2

n∑kD0

bnk

Cm∑jD�m

jkCr � 2n∑kD0

Cm∑jD�m

yjjr D 0 .10/

for r D 0, . . . , n. After changing the order of the summa-tion and introducing the notationsSrCk D

∑CmkD�m i

kCr andFr D

∑CmrD�m yij

r , one arrives at the system of equationsof the form

n∑kD0

bnkSrCk D Fr

BecauseSrCk � 0 for odd values ofr C k, then whenfitting a second-order polynomial to the top five points ofthe peak, i.e.n D 2 andm D 2, the system simplifies to

b20S0 C b22S2 D F0

b20S2 C b22S4 D F2

b21S2 D F1

and the coefficients of the parabola are readily available.

Fitting Gaussian and Lorentzian.Note that fitting a parabolato the top of the peak essentially corresponds to expandingthe lineshape as a Taylor series by.E�Eo/, neglecting thethird- and higher order terms. Obviously, only the very topof the peak is similar to a parabola, so a real lineshape canbe more effectively approximated with either a Gaussianor a Lorentzian. Although the peaks in the XPS are neitherpurely Lorentzian nor Gaussian, they are much closer toeither of the two than to a parabola. Because of this, muchmore of the peak (unlike the top few per cent of the peak atthe parabola fitting) can be used in the calculation, althoughthe fitting region is usually limited to the points whosecontents are greater than half-maximum.

It is a well-known fact that—similar to the parabola fit-ting—the frequently used Gaussian and Lorentzian peakshapes can also be linearized, i.e. peak parameters suchas position can be calculated non-iteratively. The trick isagain the same as at fitting the parabola: using the equidis-tant nature of the recorded spectra, transform the functioninto a linear function using the immediate neighbours of

the individual data points and calculate the parametersof the peak from the parameters of this linear function.Because these methods can also be used to set up theother initial parameters (FWHM and height) for (at leastrelatively) separated peaks, the further steps necessary toderive them are also described there.

For a Gaussian of the form

y.Ei/ D yo exp

[�(Ei � Eop

2�

)2]

.11/

the transformation is7

Q.Ei/ D ln..yi�1//.yiC1// D 2Ei � Eo

�2.12/

For a Lorentzian of the form

y.Ei/ D yo

1C(Ei � Eo

0.5

)2 .13/

the transformation is8

Q.Ei/ D 1

yiC1

� 1

yi�1

D 16

yo2.Ei � Eo/ .14/

Three-point Gaussian method.Another quick and simplemethod was proposed recently by Liet al.9 Three mea-sured points are used in the calculations, where the peakenergy is calculated as

E D 1

2

ln(yE2j�E2

o

i Ð yE2i �E2

jo Ð yE2

o�E2i

j

)ln(yEj�Eoi Ð yEi�Ejo Ð yEo�Ei

j

) .15/

whereEi, yi, Ej and yj are as shown on Fig. 1, andEo

andyo denote the energy and height, respectively, of thepoint with the highest count.

The moments method

It is worth drawing attention to the momentum methodtoo. In this method2 the moments of the distributiony.E/are used. Theith moment is defined as the expectationvalue of Ei. The peak position will be the first centralmoment (see Eqn (9) in Ref. 2: the simpleweightedaverage of the spectrum data points is taken)

E D

∑k

Ekyk∑k

yk.16/

for some points near to the peak energy. To calculate thestatistical uncertainty ofE we also need to calculate thesecond central moment (see Eqn (11) in Ref. 2) and obtain

�2.E/ D

∑k

E2kyk∑

k

yk� E2

∑k

yk.17/

Surf. Interface Anal. 27, 1114–1117 (1999) Copyright 1999 John Wiley & Sons, Ltd.

LOCATING PEAK ENERGIES IN XPS 1117

Note that the calculated centre of gravityE is indepen-dent of the actual shape of the symmetric distribution. Themethod is a well-established mathematical technique, sim-ple enough to use even on hand-held calculators and canbe included readily in evaluation software. However, asHansen points out in his work for determining the peakposition with high precision,1 when using a truncatedline to estimate the correct line position a bias correction

should be applied. Under the conditions used in the energycalibration measurements, this correction is of the orderof 2–5 meV.

Acknowledgements

The author kindly acknowledges Dr C. J. Powell for providing themeasurement data used in his work.4 The support of Hungarian grantOTKA (T026514) is kindly acknowledged.

REFERENCES

1. Hansen PG. Nucl. Instrum. Methods 1978; 154: 321.2. Valentine DJ, Rana EA. IEEE Trans. Nucl. Sci. 1996; 43: 2501.3. Surf. Interface Anal 1987; 10: 55; ASTM Standard E 902-88;

Surf. Interface Anal 1991; 17: 889.4. Powell CJ. Surf. Interface Anal. 1997; 25: 777.5. Cumpson PJ, Seah MP, Spencer SJ. Surf. Interface Anal.

1996; 24: 687.

6. Bevington PR. Data Reduction and Error Analysis for thePhysical Sciences (McGraw-Hill, New York, 1969) Ch. 6.

7. Mukoyama T. Nucl. Instrum. Methods 1975; 125: 289.8. Mukoyama T, Vegh J. Nucl. Instrum. Methods 1980; 173:

345.9. Li J, Valentine JD, Rana AE. Nucl. Instrum. Methods 1999;

A422: 438.

Copyright 1999 John Wiley & Sons, Ltd. Surf. Interface Anal. 27, 1114–1117 (1999)