Confocal X-ray fluorescence micro-spectroscopy experiment in tilted geometry

Spectrochimica Acta Part B 97 (2014) 99–104

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Spectrochimica Acta Part B

j ourna l homepage: www.e lsev ie r .com/ locate /sab

Confocal X-ray fluorescence micro-spectroscopy experiment intilted geometry

Mateusz Czyzycki a,b,⁎, Pawel Wrobel b, Marek Lankosz b

a DESY Photon Science, Notkestr. 85, D-22607 Hamburg, Germanyb AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Al. A. Mickiewicza 30, 30-059 Krakow, Poland

⁎ Corresponding author at: DESY Photon Science, NoGermany. Tel.: +49 40 8998 6142, fax: +49 40 8998 328

E-mail address: [email protected] (M. Czyzyc

http://dx.doi.org/10.1016/j.sab.2014.05.0070584-8547/© 2014 Elsevier B.V. All rights reserved.

a b s t r a c t

Article history:Received 28 November 2013Accepted 13 May 2014Available online 22 May 2014

Keywords:Confocal X-ray fluorescence spectroscopyTilted geometryMulti-layer materials

This paper provides a generalized mathematical model to describe the intensity of primary X-ray fluorescenceradiation collected in the tilted confocal geometry mode, where the collimating optics is rotated over an anglerelative to a horizontal plane. The influence of newly introduced terms, which take into account the tiltedgeometry mode, is discussed. The model is verified with a multi-layer test sample scanned in depth. It is provedthat for low-Z matrices, the rotation of the detection channel does not induce any significant differences in areconstruction of the thickness and chemical composition of layers, so that it may safely be ignored.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

Confocal micro-beam X-ray fluorescence spectroscopy is an instru-mental technique which enables a spatial examination of the distribu-tion and characteristics of elements in samples of different origins. Forthe last decade various mathematical approaches, based strictly on thefundamental parameter model [1–11], and on stochastic Monte Carlosimulations [12] have been used to describe the intensity of primaryand secondary X-ray fluorescence radiation and multiple scatteringeffects collected in confocal geometry and to determine the amount ofelements in a sample. Any currently available model assumes that thethree axes: 1) an impinging X-ray beam axis, 2) an X-ray detector axisand 3) a depth-scanning axis, are located in one horizontal plane. Thisexperimental arrangement is referred to as the plain geometry mode.

However, due to the development of multi-segment X-ray detec-tors the assumption of plain geometry can be avoided. There can be apolycapillary half-lens transmitting X-rays generated within a sampletowards an X-ray detector and not mounted in front of its central seg-ment. This introduces the tilted experimental geometry mode and it isroutinely in operation e.g. in the micro-focusing beamline MySpot[13] at the BESSY II synchrotron radiation facility in Berlin, Germany.

Past and current mathematical approaches do not take into accountthe tilted geometry mode. This means that the rotation of a detectionchannel relative to a horizontal (ring) plane is not taken into account atall. This paper presents a concise derivation of an analytical equation

tkestr. 85, D-22607 Hamburg,2.ki).

to describe the intensity of primary X-ray fluorescence radiationrecorded in tilted confocal mode with the introduction of necessarycorrections into the fundamental parameter model. The new, general-ized procedure is experimentally verified with a home-made multi-layer test sample. The influence of the tilted geometry mode on thereconstructed chemical composition and on the thickness of individuallayers is discussed.

2. Model

The equation derived in this section describes the intensity of pri-mary X-ray fluorescence radiation probed from multi-layer samplesscanned in depth with tilted confocal geometry. The general conceptof this approach was taken from original papers of Malzer withKanngiesser [2,3] and Mantouvalou et al. [5], which cover only theplain geometry mode. To facilitate the reader to understand thisderivation, the wording is consistent with that of these papers.

The arrangement of tilted geometry is sketched in Fig. 1. There arethree Cartesian coordinate systems with the same point of origin. Thefirst system is given by the vector r!A ¼ xA; yA; zAð Þ and it is related tothe incident X-ray beam,where two of its axes xA and yA span a horizon-tal ring plane. The second system given by the vector r!t

D ¼ xtD; ytD; z

tD

� �is related to the detection channel, tilted over an angle α relative to thering plane. The third system with the vector r!¼ x; y; zð Þ, where the zaxis is parallel to the zA axis, with two axes x and y being included inthe ring plane, describes the spatial coordinates within the sample.The term ϑ stands for the angle between the X-ray beam yA and theprojection of the detection channel on the ring plane. The sample nor-mal x is rotated over an angle ψ relative to the axis of X-ray beam yA.

Fig. 1. Sketch of the tilted geometry mode in a confocal experiment.

100 M. Czyzycki et al. / Spectrochimica Acta Part B 97 (2014) 99–104

If the incident X-ray beam propagates along the yA axis, the distribu-tion of the photonflux inside beamηA r!Að Þ is given by a two-dimensionalGaussian function:

ηA r!Að Þ ¼ TA

2πσ2A

exp − x2A þ z2A2σ2

A

!ð1Þ

where both terms TA and σA are energy-dependent and express thetransmission of a focussing polycapillary half-lens and the characteristicwidth of this distribution, respectively. In a similar way, since yD

t is theoptical axis of the collimating polycapillary half-lens, the distributionof the characteristic X-rays generated in the detection channel ηD r!t

D

� �is again expressed by the Gaussian:

ηD r!tD

� �¼ Ω

4πTDε exp −

xtD� �2 þ ztA

� �22σ2

D

0B@1CA ð2Þ

where:Ω is the solid angle seen by the optics, TD is the energy-dependenttransmission of the radiation emitted by a point-like source through theoptics, σD is the characteristic width of this distribution and ε is the quan-tum efficiency of the X-ray detector.

The sensitivity function of the X-ray instrumentη r!Að Þ is given by theproduct of both transmission functions for the excitation (Eq. (1)) anddetection channels (Eq. (2)):

η r!Að Þ ¼ ηA r!Að Þ � ηD r!Að Þ: ð3Þ

To do a multiplication according to Eq. (3), the spatial coordinatesrelated to thedetection channel have to be expressed by the coordinates

from the beam-related system. This conversion can be easily done bythe matrix equation given below:

ytDxtDztD

0B@1CA ¼

sinϑ cosα cosϑ cosα sinα− cosϑ sinϑ 0

− sinϑ sinα − cosϑ sinα cosα

0@ 1A �xAyAzA

0@ 1A: ð4Þ

Multiplication via Eq. (3) results in a sensitivity function given as:

η r!Að Þ ¼ TATDΩε8π2σ2

A

exp − c11x2A þ c22y

2A þ c33z

2A þ c12xAyA þ c13xAzA þ c23yAzA

� �h ið5Þ

where the coefficients at the polynomial in the exponent are given by:

c11 ¼σ2

D þ σ2A cos2ϑþ sin2ϑ sin2α� �

2σ2Aσ

2D

ð6:1Þ

c22 ¼σ2

A sin2ϑþ cos2ϑ sin2α� �

2σ2Aσ

2D

ð6:2Þ

c33 ¼ σ2D þ σ2

A cos2α

2σ2Aσ

2D

ð6:3Þ

c12 ¼ − sin2ϑ cos2α2σ2

D

ð6:4Þ

c13 ¼ − sinϑ sin2α2σ2

D

ð6:5Þ

c23 ¼ − cosϑ sin2α2σ2

D

: ð6:6Þ

Further derivation assumes an optimum case where the contribu-tion from the Compton scattering effect is minimised by the perpendic-ular alignment of the X-ray detector relative to the X-ray beam. In thiscase ϑ = 90°, and the c12 and c23 coefficients in Eq. (5) are simplyzero. An integral sensitivity term eη is introduced by the integration ofthe sensitivity function η r!Að Þ over the space. The integration results in:

eη ¼Z∞−∞

Z∞−∞

Z∞−∞

η r!Að ÞdxAdyAdzA ¼ TATDΩεffiffiffiffiffiffi8π

p σ2Dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

σ2A þ σ2

D

q ð7Þ

and it is worth highlighting that this quantity remains independent ofany rotation of the detection channel over the angle α relative to thering plane.

For a depth-sensitive experiment the sensitivity function η r!Að Þ ,given by Eq. (5), has to be integrated over the surface plane of the sam-ple to get a sensitivity profile ηx(x) along depth-scanning axis x. To dothe integration the beam-related coordinate system has to be rotatedover the angle ψ around the zA axis in the horizontal (ring) plane ontothe sample-related system. This operation can be easily realized by thematrix equation given below:

xAyAzA

0@ 1A ¼sinψ − cosψ 0cosψ sinψ 00 0 1

0@ 1A �xyz

0@ 1A: ð8Þ

Fig. 2. The X-ray instrument in the MySpot end-station at BESSY II operated in confocaltilted mode. Legend: 1 — focussing polycapillary half-lens, 2 — collimating polycapillaryhalf-lens tilted over the angle of 20° relative to the ring plane, and 3— pinhole collimator.

101M. Czyzycki et al. / Spectrochimica Acta Part B 97 (2014) 99–104

The integration results in a sensitivity profile ηx(x) simply given bythe Gaussian as:

ηx xð Þ ¼Z∞−∞

Z∞−∞

η r!� �

dydz ¼ eηffiffiffiffiffiffi2π

pσx

exp − x2

2σ2x

!ð9Þ

where the following terms are given by:

σx ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiχσ2

A sin2ψþ σ2

D cos2ψ

qð10Þ

χ ¼ σ2A cos

2α þ σ2D

σ2A þ σ2

D

: ð11Þ

The term σx, given by Eq. (10), can be considered a measure of thespatial resolution of the X-ray instrument along scanning direction xin the depth-sensitive experiment. The newly introduced unitlessterm χ, given by Eq. (11), justifies the rotation of the detection channelin tilted geometry mode over angle α.

The intensity of X-ray line Φl(x) probed from depth x of the l-thlaterally infinite flat layer with finite thickness D is the convolution ofthe depth-dependentmass deposit per unit area, ρ (ζ) dζ, with the sen-sitivity function ηx(x), taking the absorption term of the Lambert–Beerlaw into account. This is expressed by the equation below:

Φl xð Þ ¼ Φ0σ F

ZD0

ηx ζ−xð Þρ ζð Þ exp −Zζ0

μ lin ξð Þdξ

264375dζ ð12Þ

whereσF is the cross section for the production of the X-ray line of inter-est, while the μlin term is an effective linear attenuation coefficient.Withthe assumption of constant mass density along the scanning direction,ρ(x)≡ρ, the integration simply results in:

Φl xð Þ ¼

Φ0eησ Fρ2

exp −μ linxð Þ exp μ linσxð Þ22

!�

� erfDþ μ linσ

2x−xffiffiffi

2p

σx

!−erf

μ linσ2x−xffiffiffi

2p

σx

!( ) : ð13Þ

If the X-ray signal is probed from a stack system of n individuallayers, where each k-th layer is characterized by two terms, μlin,k andDk, Eq. (13), following Mantouvalou et al. [5], transfers to:

Φ xð Þ ¼Xnl¼1

Φl xð Þ∏l−1

k¼1exp −μ lin;kDk

� �: ð14Þ

The term μlin with the assumption of the parallel beam approxima-tion [14] expresses the absorption of the impinging and generated radi-ation, and it also takes into account the rotation of the detection channelover angle α:

μ lin ¼Xi

ρiμ0;i

cosϑAþ μ j;i

cosα cosϑD

� �ð15Þ

where μ0,i and μj,i are the mass attenuation coefficients for the i-thsubstrate at the excitation energy and the energy of generated X-rays,respectively. The terms ϑA and ϑD are the take-in and take-off angles,respectively, measured relative to the sample normal.

3. Experimental

The confocal experiment in tilted geometry was performed at theMySpot end-station in the BESSY II synchrotron radiation facility inBerlin, Germany. The experiment was performed in typical 45°/45°

geometry, which means that the angle between the impinging X-raybeam and the sample surface was 45°. Two polycapillary half-lenses,provided by the IfG — Institute for Scientific Instruments GmbH, Berlin,Germany, were used to create the confocal probing volume as illustratedin Fig. 2. The beam of primary radiation, monochromatized to an energyof 15 keV, was focussed with first optics to a spot size of 25 μm (fullwidth at half maximum, FWHM). The spot size was determined by asharp-edge scan [15,16] of 5 μmthick Pt foil. Themeasured intensity pro-file of the Pt-Lα line was differentiated and then fitted with a Gaussian.The second opticswas tilted over an angle of 20° relative to the horizontalring plane and it collimated the emerging radiation from the confocalvolume towards the upper segment of a seven segment X-ray detector.This device was a Sirius Si(Li) detector, provided by SGX-sensortech,Buckinghamshire, UK, equipped with seven independent counting chan-nels, each with a 30 mm2 active area, a peak/noise ratio of 18 000 and anenergy resolution of 134 eV at the Mn K-edge. Other detector segmentsequippedwith pinhole collimators were used to record the X-ray spectraprobed from thewhole irradiated volume of samples. The flux of primaryradiationwasmonitoredwith an ionisation chamber and further used forthe normalisation of the X-ray intensities measured.

The analytical parameters of the X-ray instrument were determinedby the multi-element NIST SRM 1832 and SRM 1833 standards [17].These materials were scanned in depth perpendicular to their surface,with a step size of 2 μm and a counting time of 25 s per point. Due tothe relatively low thickness of the standards, ca. 0.55 μm, they can beeasily used as a tiny probe to estimate the spatial resolution of the con-focal alignment. The intensity profiles measured for each X-ray linewere then iteratively fitted with Eq. (13) by least-squares, taking intoaccount the certified chemical composition and physical properties(mass density, thickness) of the standards. The fitting routine yieldedtwo parameters pairs, eη and σx, for each X-ray line and these quantitieswere used to calibrate the quantification procedure. The estimated spa-tial resolution of the X-ray instrument σx at the energy of Cu-Kα linewas 10.4 μm.

A system of nine layers stacked together wasmeasured to verify themodel. The test sample was home-made and based on the low-Z poly-mer matrix of poly(methyl methacrylate). Details on the preparationprocedure can be found in paper [18]. Odd-numbered layers werefilled with Cu2O oxide with nominal weight fractions of 5%. Cuprousoxide powder with a grain size less than 350 nmwas supplied by theSigma-Aldrich Co. LLC, St. Louis, MO, USA. The nanometre range ofthe Cu2O powder suppressed any grain size effect [19,20] in the X-rayfluorescence spectroscopy. The thickness of layers, adjusted by ahydraulic press, was aimed at around 15 μm. The sample was scannedin depth, perpendicular to its surface, with a step size of 2 μm and acounting time of 5 s per point. To overcome the problem of heterogene-ity, the sample was scanned in depth 15 times, encompassing 4 500

102 M. Czyzycki et al. / Spectrochimica Acta Part B 97 (2014) 99–104

probing points on the surface area of 200 μm × 400 μm. The resultingindividual depth-sensitive profiles were averaged and the final meanprofile of Cu-Kα line intensity was further analysed. The XRF spectrarecorded were processed by the AXIL tool [21,22] from the microxrf2software package.

Fig. 4. A hypothetical mono-layer containing Cu scanned in depth. The depth-sensitiveprofiles of Cu-Kα line intensity theoretically probed for different rotation angles α in therange from 0° to 60°. The FWHM illustrates thewidth of the intensity profile. The negativenumbers by xC indicate the shifts of the maxima of intensity profiles towards the X-rayoptical devices forming the confocal volume.

4. Results and discussion

The rotation of the detection channel over angle α in tilted geometrymodel affects two parameters. The first one is σx directly influenced bythe term χ, and the secondone is μlin. For thefixed spot size of impingingX-ray beam σA and for the fixed rotation angle α, the correction term χand, consequently, the width of probing volume σx are only affected bythe energy dependence of σD, which is an optics device-specific feature.The plots in Fig. 3 answer the question of how much the rotation of thedetection channel acts on the width of a probing volume in scanning di-rection σx. The plots show the ratio σx (α = 20°)/σx (α = 0°) betweenwidths in tilted and plain geometry modes plotted vs. the energy ofX-ray lines in a range from 3.31 keV to 10.55 keV probed by the NISTSRM 1832 and SRM 1833 standards. The dependence was illustratedfor three different sets of focussing-collimating optics available at threesynchrotron radiation beamlines in Germany: MySpot in BESSY II inBerlin, the Fluo beamline [23] at ANKA in Karlsruhe and the beamline L[24,25] at DORIS III (currently closed down) in DESY Photon Science inHamburg. The ratio is a decreasing function of X-ray energy. This is adirect result of Eqs. (10–11) and the energy dependence of σD reportedin papers [26,27]. The rotation of the detection channel over 20° resultedin a relative deviation of width σx of no more than 2.5% for the opticsdevices investigated in the given energy range of X-rays. Additionally, itis worth noticing that the sequence and the slope of plots illustrated inFig. 3 is related to the spot sizes of the focused beam σA, which wereequal to 2.6 μm, 5.3 μm and 10.6 μm for experiments conducted atbeamline L, Fluo and MySpot, respectively.

A basic consequence of Eq. (15) is that any rotation of the detectionchannel makes the effective linear attenuation coefficient μlin greater.For the angle α = 20° and an excitation energy of 15 keV, the attenua-tion of the Cu-Kα line considered in the test sample in the pure polymermatrix increased by 26% as compared to the plain geometry mode. Thecorresponding increase in the attenuation of Cu-Kα line in copper itselfwas 12%.

Another question is the extent towhichdifferent rotation anglesα inthe tilted geometry mode affect the shape of a depth-sensitive profile.For a theoretical answer, a hypothetical mono-layer mixture of Cu2O(5% wt.) and PMMA (95% wt.), with a thickness of 25 μm scanned in

Fig. 3. The ratio of widths between tilted σx (α = 20°) and plain σx (α = 0°) geometrymodes in confocal experiment vs. energy of X-ray lines.

depth was studied. For this the properties of the X-ray instrumentfrom the MySpot beamline at BESSY II were assumed. While scanningthe mono-layer in depth, the X-ray intensity profile can be describedby two parameters, FWHM and the depth position of the maximumsignal xC. The dependence of these two parameters for a Cu-Kα line in-tensity profile on angle α is illustrated in Fig. 4. As a direct consequenceof the increase in X-ray absorption with rotation angle α, the FWHMdecreases. Similarly, an increase in angle α makes the profiles shifttowards the sample surface facing the X-ray optics. This means thatthe higher the angleα, the lower the depths fromwhich theX-ray signalcan be probed. However, one can see that the order of magnitude of therelative changes in both FWHM and xC is very small.

The next problem addressed is this. If X-ray intensity is experimen-tally probed in tilted geometry, to what extent do we need to take therotation of the detection channel into account as affecting the percep-tion of sample parameters. To this end a depth-sensitive profile ofCu-Kα line intensity collected experimentally from a nine layer testsample, presented in Fig. 5, is analysed. The plot reflects the layeredstructure of the sample. Peaks corresponding to subsequent layersdoped with Cu2O oxide decreased with increasing depth of probingdue to the absorption of primary and secondary X-rays in the matrix,

Fig. 5. A depth-sensitive profile of Cu-Kα line intensity recorded from the test 9-layersystem in a confocal experiment in tilted geometry mode performed at the MySpotend-station in BESSY II.

Fig. 7. The concentration of Cu2O oxide in the test 9-layer system determined by the plain(α = 0°) and tilted (α = 20°) geometry models.

103M. Czyzycki et al. / Spectrochimica Acta Part B 97 (2014) 99–104

as expected. However, due to the low-Z polymer matrix, the Cu-Kα sig-nal could be easily detected even from the furthermost layer withoutbeing too much attenuated. Because of the limited (and insufficient)spatial resolution of the X-ray instrument, the intensity of the Cu-Kαline probed from layers unfilled with Cu2O oxide did not drop to zero,and peaks related to Cu2O-doped layers were not perfectly resolved.

The structure of the test sample, i.e. the thickness of each layer aswell as the weight fractions of Cu2O oxide inside were determined.For the reconstruction the data points, shown in Fig. 5, were fittedwith the function given by Eq. (14). An iterative least squares procedurewas combinedwith a conjugate gradientmethod [28]. The fundamentalparameters, i.e. the cross section for the production of the Cu-Kα line aswell as the mass attenuation coefficients, were taken from the xrayliblibrary [29,30]. Two cases, α = 0° and α = 20°, were considered,which allowed us to estimate the influence of tilted geometry on thesample parameters determined. The iterative procedure resulted in achi-square number of 7.6 · 10−6 for α = 0°and 6.5 · 10−6 when α =20°. The reconstructed layer thicknesses and Cu2O weight fractions areillustrated in Figs. 6 and 7, respectively. One can see that the quantitiesgiven for these two cases were equal in the limits of their uncertainties.Individual thicknesses were predicted with a relative uncertainty of noworse than 30%, while the absolute differences between correspondingthicknesses determined for these two cases did not exceed 11 μm. Therelative uncertainties of estimated Cu2O weight fractions were in arange from 15% for the first (uppermost) layer up to 52% for the further-most layers, while the absolute discrepancies between correspondingconcentrations obtained with two approaches were as high as 2.2%.There were two contributors to the uncertainties. One was related tothe models themselves. The intensity of X-rays, according to Eq. (13),was described by non-linear functions and, hence, even relativelylarge deviations of element mass fractions might cause only minorchanges in the X-ray intensity observed. The other was related to theexperiment, where the spatial resolution of any confocal X-ray instru-ment, expressed by σx, should be significantly less than the expectedthickness of layers inside any sample investigated. Nevertheless, whenthe tilted geometry mode was neglected at the reconstruction stage,one can see it had no noticeable influence on the parameters of amulti-layer sample composed of a low-Z polymer matrix slightlydoped with detectable elements. However, due to the imperfection ofthe preparation process, the test sample used in the experiment cannotbe considered a standardmaterialwith a certified (or even known) con-centration level of Cu2O oxide and a guaranteed thickness of each layer.For this reason, nominal Cu2O weight fractions and layer thicknesses inthe test sample cannot be collated with the outputs yielded by thesetwo models.

Fig. 6. Thickness of layers in the test 9-layer system determined by the plain (α=0°) andtilted (α = 20°) geometry models.

5. Conclusions

This paper presents a generalized model for describing the intensityof primary X-ray fluorescence radiation probed from depth-scannedmulti-layer samples. This model purports to encompass and accountfor experiments with tilted geometry. The form of the final equationderived for the tilted geometry model remains fully consistent withone from original papers by Malzer with Kanngiesser, which is validonly for the plain geometry experimental mode. Only χ as a new termincorporated into the newmodel, makes, as expected, thewidth of con-focal probing volume in the scanning direction slightly smaller than itsequivalent from plain geometry. In contrast, the attenuation coefficientfor emitted X-ray lines in a sample matrix gets greater as the radiationpasses a longer way in the sample material (and in air) towards theX-ray detector.

The new model for tilted geometry was employed to determine thestructure of a multi-layer test sample. No significant differences werefound between outputs returned by plain and tilted geometry modelsfor the sample. On the other hand, the tilted geometry model was vali-dated for a test sample where layer thicknesses were comparable withthe spatial resolution of the X-ray instrument used. In the future thetilted geometry experiment should be repeated with much thickerlayers in stack samples or with an X-ray spectrometer with higherspatial resolution.

The experimental validation of the tilted geometry model shouldalso be carried out in the light of various reference standards of aknown or certified composition. Multi-layer materials of differentchemical compositions (in terms of the range of elements and their con-centration levels) and with different layer thicknesses would be condu-cive to further examination to reveal any strengths and shortcomings ofthe model presented here.

Acknowledgement

Part of this research was done at the synchrotron light source BESSYII, Helmholtz-Zentrum Berlin, Germany. The authors kindly thank Dr.Ivo Zizak for his assistance in the operation of the MySpot beamline.

The research was financially supported and realized under theauspices of the International Atomic Energy Agency, Vienna, Austriawithin the frame of research contract no. 16023.

The research has received funding from the European Community'sSeventh Framework Programme (FP7/2007-2013) under grant agree-ment no. 226716. The research was also supported by the Polish

104 M. Czyzycki et al. / Spectrochimica Acta Part B 97 (2014) 99–104

Ministry of Science and Higher Education and its grants for scientific re-search (AGH UST project no. 11.11.220.01).

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