Secondary Fluorescence Enhancement in Confocal X-ray Microscopy Analysis

Secondary Fluorescence Enhancement in ConfocalX-ray Microscopy Analysis

Dimosthenis Sokaras* and Andreas-Germanos Karydas

Institute of Nuclear Physics, N.C.S.R. “Demokritos”, Athens, Greece

In the present work, the influence of the secondaryfluorescence enhancement in confocal X-ray microscopyanalysis is studied when stratified type of materials areexamined. Through a proper mathematical formalism, anexact global theoretical model is presented which ac-counts for the secondary fluorescence enhancement wheneither particle (3D-Micro particle induced X-ray emission)or photon (3D-Micro X-ray fluorescence) microbeams areused in the excitation channel. The contribution of thesecondary fluorescence effect to the confocal X-ray inten-sity profiles was calculated for some typical representativecases. In addition, the influence of several experimentalparameters was examined in terms of their influence inthe absolute intensity and shape of the secondary fluo-rescence intensity profile.

Confocal X-ray microscopy has developed considerably duringthe last 5 years due to the increasing analytical need to supportand upgrade elemental microanalysis with depth resolved capabili-ties. 3D-Micro X-ray fluorescence (3D-Micro XRF) at synchrotronfacilities1-3 but also with laboratory setups4,5 and 3D-Micro particleinduced X-ray emission (3D-Micro PIXE)6,7 at nuclear microprobefacilities8,9 have indicated the potential of confocal X-ray fluores-cence microscopy to offer depth or even 3D element specificresolved quantitative information in a few micrometer region.10-15

Confocal X-ray microscopy is based on the formation of aconfocal volume (probing microvolume) defined through theintersection of a focused excitation beam and the sensitive volumeof a polycapillary lens placed across the detection channel. Theexperimental measurements for the examined samples are ac-complished by means of sequential displacements of the probingmicrovolume across the sample depth and surface. This can beachieved by either moving the sample across the probingmicrovolume or vice versa or even both leading toward 3Delemental investigations.

The spatial resolution of confocal X-ray fluorescence micros-copy is affected from the size of the excitation beam but mainlyand more crucially by the performance of the polycapillary X-raylens over the detection channel. As far as the spatial resolution inthe excitation channel is concerned, state of the art ion beamsetups can provide beams even in the submicrometer regime(nuclear microprobes),16,17 while modern synchrotron beamlinescan nowadays form X-ray beams down to a few tenths ofnanometers size by means of proper X-ray optics devices (K-Bmirrors, refractive lenses, Fresnel zoneplates, etc.).18,19 Neverthe-less, polycapillary half-lenses are an alternative and easy-to-implement way to form X-ray microbeams over synchrotronbeamlines as well as in front of microfocus X-ray tube-basedtabletop setups. On the other hand, in the detection channel,modern polycapillary X-ray lenses (half-lens and Poly-CCC) canoffer collective focal spot sizes down to 15-30 µm at Cu-KR.20

Despite the intrinsic spatial resolution of the X-ray optics devices,confocal X-ray microscopy can offer an improved spatial resolution(an order of magnitude lower) particularly in the analysis of the

* To whom correspondence should be addressed. E-mail: [email protected].

(1) Kanngiesser, B.; Malzer, W.; Reiche, I. Nucl. Instrum. Methods Phys. Res.,Sect. B 2003, 211-212, 259.

(2) Woll, A. R.; Mass, J.; Bisulca, C.; Huang, R.; Bilderback, D. H.; Gruner, S.;Gao, N. Appl. Phys. A: Mater. Sci. Process. 2006, 83, 235–238.

(3) Janssens, K.; Proost, K.; Falkenberg, G. Spectrochim. Acta, Part B 2004,59, 1637–1645.

(4) Kanngiesser, B.; Malzer, W.; Rodriguez, A. F.; Reiche, I. Spectrochim. Acta,Part B 2005, 60, 41–47.

(5) Tsuji, K.; Nakano, K. X-Ray Spectrom. 2007, 36, 145–149.(6) Karydas, A. G.; Sokaras, D.; Zarkadas, Ch.; Grlj, N.; Pelicon, P.; Zitnik, M.;

Schutz, R.; Malzer, W.; Kanngiesser, B. J. Anal. At. Spectrom. 2007, 22,1260.

(7) Kanngiesser, B.; Karydas, A. G.; Schutz, R.; Sokaras, D.; Reiche, I.; Rohrs,S.; Pichon, L.; Salomon, J. Nucl. Instrum. Methods Phys. Res., Sect. B 2007,264, 383.

(8) Simcic, J.; Pelicon, P.; Budnar, M.; Smit, Z. Nucl. Instrum. Methods Phys.Res., Sect. B 2002, 190, 283.

(9) Pelicon, P.; Simcic, J.; Jaksic, M.; Medunic, Z.; Naab, F.; McDaniel, F. D.Nucl. Instrum. Methods Phys. Res., Sect. B 2005, 231, 53.

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(13) De Samber, B.; Silversmit, G.; Evens, R.; De Schamphelaere, K.; Janssen,C.; Masschaele, B.; Van Hoorebeke, L.; Balcaen, L.; Vanhaecke, F.;Falkenberg, G.; Vincze, L. Anal. Bioanal. Chem. 2008, 390, 267–271.

(14) Zitnik, M.; Pelicon, P.; Grlj, N.; Karydas, A. G.; Sokaras, D.; Schutz, R.;Kanngiesser, B. Appl. Phys. Lett. 2008, 93, 094104, DOI: 10.1063/1.2976163.

(15) Zitnik, M. Pelicon, P. Grlj, N. Karydas, A. G. Sokaras, D. Schutz, R.;Kanngiesser, B. X-ray Spectrom. Submitted for publication.

(16) Watt, F.; van Kan, J. A.; Rajta, I.; Bettiol, A. A.; Choo, T. F.; Breese, M. B. H.;Osipowicz, T. Nucl. Instrum Methods Phys. Res., Sect. B 2003, 210, 14–20.

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(19) Kang, H. Ch.; Yan, H.; Winarski, R. P.; Holt, M. V.; Maser, J.; Liu, Ch.;Conley, R.; Vogt, St.; Macrander, A. T.; Stephenson, G. B. Appl. Phys. Lett.2008, 92, DOI: 10.1063/1.2912503.

(20) Wolff, T.; Mantouvalou, I.; Malzer, W.; Nissen, J.; Berger, D.; Zizak, I.;Sokaras, D.; Karydas, A. G.; Grlj, N.; Pelicon, P.; Schutz, R.; Zitnik, M.;Kanngiesser, B. J. Anal. At. Spectrom. 2009, 24, 669.

Anal. Chem. 2009, 81, 4946–4954

10.1021/ac900688n CCC: $40.75 2009 American Chemical Society4946 Analytical Chemistry, Vol. 81, No. 12, June 15, 2009Published on Web 05/22/2009

elemental distribution of different atomic species within thematerial.

Quantitative models have been already developed21,22 andimplemented6,7,23 in the analysis of 3D-Micro PIXE and 3D-MicroXRF data deducing local elemental densities. The formalismdescribes analytically the so-called probing microvolume and thefluorescence emission as a result of a single ionization processwithin the sample. As confocal X-ray microscopy is becoming aroutine method for material research, the second order effects inthe fluorescence emission should be included as well. Additionalinner-shell ionizations can be induced due to various second orderprocesses such as photoelectric absorption within the sample ofthe characteristic X-ray lines of an element (secondary fluores-cence), photoelectron-induced ionizations, or even ionizationsinduced by the elastically and inelastically scattered X-rays withinthe sample (XRF). Up until now, an approximate approach forthe secondary fluorescence in 3D-Micro XRF has been proposedby Smit et al.24

In this work, a detailed study of the secondary fluorescenceenhancement in confocal X-ray microscopy analysis is presented.The proposed approach is a general one and can be applied eitherto particle or photon exciting microbeams. After a propermathematical formalism for homogeneous samples is deduced,the model is extended accordingly to include the secondaryfluorescence between different layers of a multilayered sample, afavored case to be studied experimentally by confocal X-raymicroscopy techniques. The proposed model results in an analyti-cal expression that has to be calculated by means of multidimen-sional integration. Proper numerical methods are recommended,whereas the model has been independently validated by MonteCarlo simulation. Finally, the secondary fluorescence enhance-ment is calculated for certain representative cases of samples(homogeneous and multilayered structures), and the influenceof various experimental parameters is discussed.

Secondary Fluorescence in X-ray Spectrometry. In thePIXE or XRF analysis of a sample, the primary beam (chargedparticles or X-rays) may induce characteristic X-ray radiation fromthe elements that compose the sample (primary intensity). Thesecondary fluorescence effect takes place when characteristicX-rays of one element within the sample are energetically enoughto produce fluorescence emission from another element withinthe sample. Evidently, it is a phenomenon strongly dependent onthe given sample properties (composition and thickness), as wellas the characteristics of the experimental setup (type of beam,energy, geometry).

Generally, the secondary fluorescence enhancement has agreater influence in the XRF analysis compared to PIXE; this isdue to the fact that the charged particle projectiles favors theproduction of characteristic X-rays of lower energy, whereas X-raysof a particular energy exhibit higher production cross sections

for the more energetic characteristic X-rays. In this way therelative contribution of the secondary over primary fluorescenceintensity is much more enhanced in XRF analysis. Secondaryfluorescence in conventional X-ray spectrometry has been exten-sively studied in both PIXE and XRF analytical techniques eitherin the case of thick homogeneous or layered types of materials.26-33

It is interesting to note that the secondary ionizations insidethe sample may occur at regions well distant with respect to theincident beam path, i.e., where the primary ionizations areproduced. Lets examine for example the XRF analysis of a metallicmatrix (50% of Fe and 50% of Cu) that is irradiated by a spot-sizedX-ray beam of 15 keV energy. We have performed a Monte Carlocalculation25 to predict and illustrate the topology of the secondaryinduced fluorescence in the aforementioned case. The conven-tional XRF geometry of incident/takeoff angle 45°/45° wasadopted. The results are presented in Figure 1 from which canbe observed that the secondary Fe-K photons produced due tothe absorption of Cu-K radiation and subsequently emitted acrossthe X-ray beam path originate from points situated rather a fewmicrometers away.

A more pronounced example is shown in Figure 2 where 5%of Fe and Cu are diluted into a SiO2 matrix. In this case, thesecondary Fe-K fluorescence photons originate from a dis-tance up to 100 µm away from the incident beam path.

Taking into account that the sensitive microvolume in theconfocal X-ray microscopic techniques is extended even to smallerareas than the spatial origin of the secondary fluorescent photons,it is evident that in these cases, its relative contribution to the

(21) Sokaras, D.; Karydas, A. G.; Malzer, W.; Schutz, R.; Kanngiesser, B.; Grlj,N.; Pelicon, P.; Zitnik, M. J. Anal. At. Spectrom. 2009, 24, 611.

(22) Malzer, W.; Kanngiesser, B. Spectrochim. Acta, Part B 2005, 60, 1334–1341.

(23) Mantouvalou, I.; Malzer, W.; Schaumann, I.; Luhl, L.; Dargel, R.; Vogt, C.;Kanngiesser, B. Anal. Chem. 2008, 80, 819–826.

(24) Smit, Z.; Janssens, K.; Proost, K.; Langus, I. Nucl. Instrum. Methods Phys.Res., Sect. B 2004, 219-220, 35–40.

(25) Sokaras, D.; Kolbe, M.; Karydas, A. G.; Beckhoff, B.; Zarkadas, Ch.Unpublished work.

(26) de Boer, D. K. G. X-Ray Spectrom. 1990, 19, 145–154.(27) Mantler, M. Anal. Chim. Acta 1986, 188, 25–35.(28) Wegrzynek, D.; Holynska, B.; Pilarski, T. X-Ray Spectrom. 1993, 22, 80–

85.(29) Van Dyck, P. M.; Torok, S. B.; Van Grieken, R. E. Anal. Chem. 1986, 58,

1761–1766.(30) Karydas, A. G. X-Ray Spectrom. 2005, 34, 426–431.(31) Ryan, C. G.; Cousens, D. R.; Heinrich, C. A.; Griffin, W. L.; Sie, S. H.;

Mernagh, T. P. Nucl. Instrum. Methods Phys. Res., Sect. B 1991, 54, 292–297.

(32) Campbell, J. L.; Wang, J. X.; Maxwell, J. A.; Teesdale, W. J. Nucl. Instrum.Methods Phys. Res., Sect. B 1989, 43, 539–555.

(33) Smit, Z.; Pelicon, P.; Simcic, J.; Istenic, J. Nucl. Instrum. Methods Phys. Res.,Sect. B 2005, 239, 27–34.

Figure 1. Spatial distribution for the detected secondary inducedFe-K radiation due to the Cu-K characteristic photons. The sampleconsists of 50% Cu and 50% Fe. The Monte Carlo calculation25 isbased on the conventional XRF geometry of 45°/45° with a spot-sized exciting beam of 15 keV. The secondary Fe-K fluorescence isdistributed over an extended area around the path of the spot-sizedincident X-ray beam inside the sample.

4947Analytical Chemistry, Vol. 81, No. 12, June 15, 2009

total detected fluorescence intensity will not be constant withrespect to the position of the sample across the probingmicrovolume.

Analytical expressions that describe the secondary fluores-cence in conventional geometries have been published26-34 andwidely validated. Nevertheless, these models cannot be used overconfocal geometries due to the presence of the polycapillary lensin front of the detector which actually eliminates the majoringredient of the angular symmetry across the plane perpendicularto the sample normal vector.

Secondary Fluorescence in Confocal Geometry. For themathematical formulation of the secondary fluorescence enhance-ment, an ab initio model is developed. Initially, the model accountsfor the case of an homogeneous material. A spot-sized incidentcharged particle or X-ray beam is assumed, and next themathematical approach is generalized incorporating beam dimen-sions. As a final step, a complete formulation is presented that

describes the secondary fluorescence enhancement in the caseof multilayer-structured material.

Homogeneous Samples. Lets assume two elements A andB contained in a homogeneous sample with thickness d. Thecharacteristic photons of element A have an energy EA greaterthan the binding energy of a given inner shell of element B.The secondary fluorescence intensity (YAfB) produced by theelement B due to the characteristic radiation of element A canbe generally given as a function of the spatial coordinates ofthe incident beam (xo,zo) as well as sample’s surface (yo)position according to eq 1. (The bold symbol notation is usedfor simplicity for the long formulas indicating that the representa-tive quantity is not independent or constant but a function of oneor more integration variables which will be defined explicitly inthe flow of this article.) It should be noted here that we haveadopted a main coordination system that it is defined by x,y,z asfollows: the xz plane is parallel to the sample’s surface and its origincoincides with the focal point of the polycapillary lens (Figure 3).

This formula accounts for the contribution of each infinitesimalregion dyj across the incident beam path, that it is a source ofEA photons, over each infinitesimal thin layer dyi across all ofthe sample’s volume. The total secondary fluorescence intensityby all regions dyj across the incident beam path on theinfinitesimal layer dyi situated at y ) yi is given by the followingexpression:

The spherical-like coordinates θ and φ are depicted in Figure4 and are introduced in order to express the coordinates of eachpoint i through the Cartesian coordinates of point j. By means ofproper transformations, the spatial coordinates of i points areexpressed as follows:

(34) Karydas, A. G.; Paradellis, Th. X-Ray Spectrom. 1993, 22, 208–215.

Figure 2. Spatial distribution for the detected secondary inducedFe-K radiation due to the Cu-K characteristic photons. The sampleconsists of 5% Cu and 5% Fe into a SiO2 matrix. The Monte Carlocalculation25 is based on the conventional XRF geometry of 45°/45°with a spot-sized exciting beam of 13 keV. The secondary Fe-Kfluorescence is distributed over a widely extended area around thepath of the spot-sized incident X-ray beam inside the sample.

Figure 3. Geometrical representation of the secondary fluorescenceprocess on homogeneous samples. The characteristic radiation ofelement A, located at the j position, enhances the characteristicradiation of element B located at the i position.

YAfB ) YAfB(xo, yo, zo) ) ∫yo-d

yodYAfB(yi;xo, yo, zo)dyi

(1)

dYAfB(yi;xo, yo, zo)dyi )

∫yo-d

yo ∫0

2π ∫f1

f2 sin θ4π

PAWASBWBKlens dθ dφdyj

cos θb

dyi

|cos θ|(2)

Figure 4. Definition of the geometrical parameters involved in eqs3a and 3b.

4948 Analytical Chemistry, Vol. 81, No. 12, June 15, 2009

where,

The incident beam coordinates xo and zo and the “intrinsic”correlation with the coordinates of point j are given as

The terms cos θb and cos θ of eq 2 results from the effectivesize of the layers dyj and dyi across the incident beam andcharacteristic photon EA direction, respectively. For the integra-tion variable θ in eq 2, the limits depend on the relative positionsyj and yi with respect to the surface as follows:

The quantity PA expresses the primary intensity of thecharacteristic radiation of element A at the interaction point j.Since we are considering both modes of excitation (chargedparticles or X-rays), this quantity has a dual analytical expres-sion depending to the mode of ionization:

Io expresses the incident beam intensity of the impact projectileenergy, σA is the particle induced ionization cross section forthe element A, which depends on the impact particle energyat the corresponding depth. (Since ion beam loses energy asit propagates into matter, the impact energy has to be calculatedat each depth yj through energy loss calculations.) For the caseof the photon excitation beam, an attenuation term for theincident radiation should be included, and µ accounts for themass attenuation coefficient (in cm2/g) while the ionizationcross section for a given shell can be calculated in terms ofthe partial photoelectric absorption cross section (τ, in cm2/g), for the particular inner-shell at the incident energy, whereasω is the fluorescence yield and R the branching ratio for thegiven characteristic line. Finally ηA is the concentration ofelement A expressed in density units g/cm3.

The intensity of the secondary fluorescence photons of elementB produced at point i per characteristic photon of element Aproduced inside the infinitesimal region dyj is given as follows:

The [(sin θ)/(4π)] term is the fraction of the solid angle thatcorresponds to the point i with respect to the region dyj. Theterm SB (eq 10) contains the fundamental parameters of elementB (ω, fluorescence yield; R, branching ratio) as well as itsconcentration η expressed in density units g/cm3. Furthermore,the term WA (eq 11) accounts for the attenuation of characteristicphotons EA from point j to i:

The fraction of the photons escaping the sample surface isgiven by the attenuation factor WB defined as

The overall density of the sample F is the summation of theelemental local density of all contained atoms.

Finally, the term Klens in eq 2 represents the response functionof the sensitive volume in front of a polycapillary half-lens (or aPoly-CCC). It has been shown that it can be expressed as a 2D-Gaussian distribution with an energy dependent sigma value(σ(EB)) multiplied by the solid angle substained by the detector((∆Ω)/(4π)) and the transmission efficiency (T(EB)) of thedetection system.20

Multilayered Samples. The characterization of layer-struc-tured samples consists of one of the most important applicationsin X-ray confocal microscopy techniques. Even when a sampleincludes inhomogeneity that is represented by a concentrationgradient, the most effective strategy to perform calculations is todivide the sample into a certain number of layers in which constantelemental concentrations (local elemental densities) can beconsidered.

Let us assume a general case of a layer-structured material.The characteristic radiation of element A contained into layer minduces secondary fluorescence on element B contained into layerk; in accordance to the formulation developed on the previoussection, the secondary fluorescence intensity will be given by thefollowing expression (see Figure 5):

The above expression has been based on eq 1, while properintegration limits have been taken into account in order for thelayered structure to be formulated. The total contribution of thewhole layer m over an infinitesimal region dyi into layer k can bedescribed as follows.

xi ) xj + ro cos φ sin θ (3a)

zi ) zj + ro sin sin θ (3b)

ro ) |yi - yj

cos θ | (4)

xj ) xo + yj tan θb (5a)

zj ) zo (5b)

f1 ) f1(yi, yj) ) 0 if yj < yiπ2

if yj > yi(6)

f2 ) f2(yi, yj) ) π2

if yj < yi

π if yj > yi

(7)

PA ) PA(yj, xo, yo, zo)

) Io(xo, zo)σA(E(|yo - yj

cos θb|F))ωARAηA, if ion beam

Io(xo, zo)exp[-µ(Eo)|yo - yj

cos θb|F]τA

(Eo)rA(Eo)ωARAηA,if X-ray beam

(8)

SF ) sin θ4π

WASB (9)

SB ) τB(EA)rB(EA)ωBRBηB (10)

WA ) WA(yi, yj, yo, θ) ) exp[-µ(EA)roF] (11)

WB ) WB(yi, yo) ) exp[-µ(EB)|yo - yi|cos θl

F] (12)

F ) ∑m)1

N

ηm (13)

Klens ) Klens(xi, yi, zi)

) ∆Ω4π

T(EB)exp[-(cos θlxi - sin θlyi)2

2σ(EB)2 ]exp[- zi2

2σ(EB)2](14)

YAmfBk) YAmfBk

(xo, yo, zo) ) ∫yo-Dk

yo-Dk-1dYAmfBk

(yi;xo, yo, zo)dyi

(15)

4949Analytical Chemistry, Vol. 81, No. 12, June 15, 2009

The attenuation factor WAm|k of the fluorescence photons of

element A between points j and i depends on the relativeposition of layers m and k with respect to the surface:

The analytical expressions for the attenuation WAm|k include the

attenuation of photons EA into three regions; the layers m andk and within the intermediate layers, if any. The correspondingrelationships are following:

In the above relationships, Dκ expresses the distance of thelayer κ from the sample’s surface:

The term WBk in eq 15 accounts for the attenuation of the X-ray

photons of B from the production point i to the detector includingthe intermediate layers of the sample, if any, as follows:

It should be noted that the term PAm in eq 15 is expressed in

accordance with the corresponding term in the previous section.However the attenuation of X-rays or the energy loss of thecharged particles beam has to be taken into account for all thelayers from the surface up to point j. All the remaining terms ineq 16 have been already defined previously.

In conclusion, in order to calculate the total secondaryfluorescence intensity of element B due to all possible interferingelements A (actually including all the group of characteristic X-rayseligible to produce secondary fluorescence) that might be con-tained in one or more layers of a N-multilayered sample, a propersummation over the number of interfering elements per layer Aλ

and over all possible layers has to be implemented as follows:

In the particular case of the tube-excited 3D-Micro XRF, theexcitation radiation it is not monochromatic. Therefore anadditional summation over the whole excitation spectral dis-tribution has to be carried out as well.

Incident Beam Size Contribution. The mathematical for-mulation developed up until now takes into account a spot-sizedincident beam. This assumption holds as long as the beam sizeis much less than the sensitive volume of the polycapillary lensplaced across the detection channel.21 For example, in the caseof 3D-Micro PIXE, the proton microbeam is formed by means ofa nuclear microprobe able to provide a microbeam with a sizedown to 1 × 1 µm2 in vacuum; nowadays this size is much lessthan the spatial response function of a modern polycapillarylens (fwhm ∼20 µm at Cu-KR). The same approximation canalso hold in the case of 3D-Micro XRF when the excitationbeam is formed at special synchrotron beamlines wheresubmicrometer beam sizes can be really attained. It should benoted, however, that the size of an ion microbeam of a fewMeV of energy, due to the so-called lateral straggling effect,becomes larger as the ion beam propagates inside the analyzedmaterial. The magnitude of this effect is dependent on boththe type of ion and the material composition, whereas it isenhanced close to the range of the ion. Analytical calculationshave shown that the lateral straggling contribution in thebroadening of an ion microbeam becomes important close toits range, but still the induced beam size would be muchsmaller than the available X-ray lens resolution. For example,for a 3 MeV proton beam, even at the end of its range (i.e.,where inner-shell ionizations are practically diminished), the

Figure 5. Graphical representation for the secondary fluorescenceprocess in a multilayer sample.

dYAmfBk(yi, xo, yo, zo)dyi )

∫yo-Dm

yo-Dm-1 ∫0

2π ∫f1

f2 sin θ4π

PAmWA

m|kSBWBk Klens dθ dφ

dyj

cos θb

dyi

|cos θ|

(16)

WAm|k ) WAV

m|k(yi, yj, yo, θ) if k < mWA(yi, yj, yo, θ) if k ) mWAv

m|k(yi, yj, yo, θ) if k > m(17)

WAVm|k(yi, yj, yo, θ) ) exp[-µm(EA)|yj - (yo - Dm-1)

cos θ |Fm -

∑λ)k+1, |k-m|g2

m-1 (µλ(EA)dλ

|cos θ|Fλ)- - µk(EA)|yi - (yo - Dk)

cos θ |Fk](18)

WAvm|k(yi, yj, yo, θ) ) exp[-µm(EA)|yj - (yo - Dm)

cos θ |Fm -

∑λ)m+1, |k-m|g2

k-1 (µλ(EA)dλ

|cos θ|Fλ) -

µk(EA)|yi - (yo - Dk-1)cos θ |Fk] (19)

Dκ ) ∑λ)1

κ

dλ (20)

WBk ) WB

k (yi, yo) )

exp[-µk(EA)|yi - (yo - Dk-1)|

cos θlFk - ∑

λ)1, kg2

k-1 (µλ(EB)dλ

cos θlFλ)](21)

YBk) ∑

λ)1

N

∑Aλ

YAλfBk(22)

4950 Analytical Chemistry, Vol. 81, No. 12, June 15, 2009

beam broadening is about 4 µm inside a metallic or a glasslikematrix.35

In the case that the exciting X-ray microbeam is formed bymeans of a polycapillary lens or when an external (in air) nuclearmicroprobe setup is used, the optimum achievable incident beamsize is rather comparable with the sensitive volume across thedetection channel. Therefore the size of the incident beam cannotbe neglected anymore in the calculations and its contributionshould be taken into account. The finite incident beam size canbe adopted in our formalism by means of a double integrationover the cross section of the beam in the xz plane of the maincoordination system as follows:

Herewith a symmetrically shaped beam has been assumed withits center situated at the coordinate (xc,0,zc), while the horizontaland vertical dimensions are expressed as 2R and 2, respec-tively, for a projected cross section on the xz plane. The fluxvariation of the beam with respect to its center can beexpressed through the I(xo,zo) term (eq 8) that has beendescribed in the case of an excitation X-ray beam (formed bymeans of a polycapillary lens) as a Gaussian distribution andcorrespondigly as a rectangular-shaped distribution for a particlemicroprobe. For example, the flux density for a Gaussian X-raybeam in terms of the main system coordinates (xyz) at the y ) 0position is given by the following formula.

The parameter I represents the total flux (number of photons perunit of time), while θb is the incident angle of the beam withrespect to the sample normal.

RESULTS AND DISCUSSIONThe secondary fluorescence contribution in confocal X-ray

microscopy has been described in previous paragraphs forhomogeneous and multilayer-structured samples by means of aformulation that requires multiple integrations. Under certainexperimental conditions and the type of the analyzed sample,simplifications of the proposed formalism may result in a signifi-cant reduction of the complexity. However, for the treatment ofthe most general case, means of calculations that would result tofast and accurate results need to be implemented.

In numerical analysis, several methods have been developedto deal with a multiple integration. Various quadrature techniquesbased on equally or not spaced calculation points can be veryefficient tools, when low dimension integrals need to be calculated.However, when the integral dimensions are increased, MonteCarlo based techniques are more appropriate. In particular, whenthe dimension of the integration is more than four, Monte Carlointegration seems to be the most efficient computational method.The traditional Monte Carlo integration algorithms evaluate theintegrand quantity uniformly over the defined integration region.Nevertheless sophisticated transformations have been developed

(adaptive algorithms) like VEGAS37 and MISER.36 These algo-rithms select evaluation points for the integrand function withinthe multidimensional space, following either the importancesampling (VEGAS) or the stratified sampling (MISER) method.In this way, the accuracy of the results is optimized with respectto the number of sampling points.

Another important advantage of the Monte Carlo basedintegrations compared to the other numerical techniques is theinherent flexibility to treat functions with discontinuities oranomalous regions. For example in eq 2, the integration variableθ is restricted to the [0,(π/2)] or [(π/2),π] interval, respectively;therefore at the limit θ ) (π/2), both terms tan θ and ro go toinfinity. The implementation of quadrature based numericalmethods (Gauss-Legendre or Gauss-Kronrod) to evaluate theintegral of eq 15 has led to a nonconvergent solution, whereasthe Monte Carlo integration was robust. In the next paragraphs,all calculations are based on Monte Carlo integration adaptingthe VEGAS algorithm.

Our proposed formalism and method of calculation for thesecondary fluorescence enhancement in confocal X-ray micros-copy analysis was compared to a full Monte Carlo simulation (ray-tracing)25 in the case of a 3D-Micro XRF analysis for a homoge-neous alloy with 50% Fe and 50% Cu. It should be noted that in all3D-Micro XRF calculations, an incident spot-sized 18 keV beamwas assumed. The fwhm of the polycapillary lens on the detectionchannel was selected to vary between 20 and 50 µm for 3-10keV, in accordance with relevant experimental data.20 Thegeometry that has been adopted in all our calculations andsimulations is -45°/45° for the incident/takeoff angles, respec-tively. The results are presented in terms of the profiles of primaryand secondary fluorescence intensities (with reference to the leftaxis of the graph) but also as the relative secondary to primaryintensity contribution (with reference to the right axis of thegraph). The primary fluorescence intensity profiles were calcu-lated for both 3D-Micro XRF and 3D-Micro PIXE by means of aformalism that has been previously developed by Malzer et al.22

and Sokaras et al.,21 respectively. In Figure 6, an excellentagreement is observed between the developed formalism and thefull Monte Carlo simulation.

Next, the influence of the secondary fluorescence effect inX-ray confocal microscopy analysis has been evaluated for somerepresentative cases for both ion and X-ray beam excitation modes.As a first example, the secondary fluorescence enhancement forthe case of a bulk homogeneous multielemental glasslike material(see Table 1 for contained compounds and respective concentra-tions) has been considered for both excitation modes. Morespecifically, a spot-sized 3 MeV proton beam was assumed forthe case of 3D-Micro PIXE, whereas the polycapillary X-rays lensfwhm was selected to be similar to the case of the 3D-Micro XRF.A typical geometry of 0°/45° for the incident/takeoff angles wasused for the 3D-Micro PIXE setup. Generally, standard referenceglass materials are very useful in X-ray confocal microscopyexperiments due to the simultaneous analysis of several elements.Thus a fast calibration over a wide energy range with respect tothe spatial resolution and the overall transmission efficiency ofthe experimental setup can be attained.

(35) Ziegler, J. F.; Biersack, J. P.; Littmark, U. The Stopping and Ranges of Ionsin Matter, Vol. 1; Pergamon Press: New York, 1985.

(36) Press, W. H.; Farrar, G. R. Comput. Phys. 1990, 4, 190–195.(37) Lepage, G. P. J. Comput. Phys. 1978, 27, 192–203.

YAfB ) ∫xc-R

xc+R ∫zc-

zc+YAfB(xo, yo, zo)dzo dxo (23)

Io(xo, zo) ) I2πσ2cos θb

exp[-(cos θbxo)2 + zo2

2σ2 ] (24)

4951Analytical Chemistry, Vol. 81, No. 12, June 15, 2009

In Figures 7 and 8, the primary and secondary fluorescenceintensity profiles together with the secondary to primary fluores-cence relative contribution are presented for both the character-istic Ca and Fe KR intensities in the case of 3D-Micro XRF and3D-Micro PIXE analyses of the glasslike sample. The extendedX-ray beam was described in the corresponding calculations(represented by a dotted line in Figures 7 and 9) by eq 24 with σ) 6.4 µm. In Figure 8, in particular, the secondary fluorescenceintensity profile and its relative contribution to the primaryintensity profile are also presented with dotted lines for apolycapillary half-lens with the half of the spatial resolution (25-10µm fwhm for 3-10 keV, respectively) used in all the othercalculations.

The secondary fluorescence contribution was calculated for amultilayered structure in the case of a thin Fe (5 µm) film on aCu substrate. It is a typical example where confocal X-raymicroscopy analysis can be advantageous. Figures 9 and 10present the primary and secondary intensity profiles of Fe-KRX-rays as well as the secondary to primary fluorescence contribu-tion to Fe-KR, for both 3D-Micro XRF (spot and broad beamsize) and 3D-Micro PIXE modes.

From all the results presented in Figures 7-10, some generalremarks can be stated. The ratio of the secondary to primaryfluorescence is described by a strong asymmetric profile, whichhas a minimum close to the centered region of the primaryintensity distribution and maximum at both tails. This behaviorexpresses the fundamental nature of the secondary fluorescenceprocess in the confocal X-ray microscopy, i.e., the detected

secondary fluorescence photons originate from an extended areaaround the incident beam path (depending on the matrix composi-tion) which can be very asymmetric for a particular sampleposition within the probing microvolume. Thus, for samplepositions that correspond to the tails of the primary intensityprofiles, it is more likely to detect relatively more secondaryfluorescence photons since they can originate from a region withinthe sample closer to the optical axis of the lens.

It is interesting to note the influence of the confocal setup fwhmto the secondary fluorescence contribution; as the lens or/andincident beam fwhm become smaller (i.e., the size of the probingmicrovolume is reduced), the secondary to primary fluorescencecontribution shows more pronounced asymmetric variation be-tween the sample positions around the setup focus region andthe one being further away (Figures 7-9, dotted lines). Morespecifically, as the setup fwhm decreases, the magnitude of thesecondary fluorescence contribution becomes smaller for thesample positions around the focus region, whereas it is signifi-cantly increased at the position that corresponds to the tails ofthe intensity profile.

An estimation of the error that it is finally introduced indeducing elemental local concentrations, if the secondary fluo-

Figure 6. Comparison between the proposed analytical formalismand a full Monte Carlo simulation25 for the secondary fluorescenceover 3D-Micro XRF analysis. The sample consists of 50% Fe and50% Cu while a 18 keV incident beam was assumed.

Table 1. Chemical Composition for a Glasslike Matrix

compound concentration (%)

Na2O 10Al2O3 10SiO2 50K2O 3CaO 10TiO2 2MnO 3Fe2O3 6PbO 6

Figure 7. Calculated profiles of primary22 and secondary (eq 1)characteristic X-rays intensities (Ca-KR (a) and Fe-KR (b)). Withrespect to the right axis of the graph, the calculated relative secondaryto primary intensity profile is presented for Ca-KR (a) and Fe-KR(b). With dotted lines, the respective calculations for an extended size(σ ) 6.4 µm) X-ray incident beam are shown (3D-Micro XRF, 18 keVincident beam on a glasslike matrix).

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rescence correction is neglected, cannot be answered in astraightforward way. It can be estimated accurately only by means

of a minimization quantification algorithm that solves the inverseproblem. However, a rough estimation is that the secondaryfluorescence correction should be at least at the level of itsmagnitude around the maximum of the intensity profile.

It is evident that in the case of the 3D-Micro XRF analysis, thesecondary fluorescence contribution is much more pronounced whencompared to the 3D-Micro PIXE and can vary significantly inmagnitude according to the type of the sample and the experimentalsetup parameters. For many samples, the secondary fluorescencecontribution will be negligible, and thus there is no need to beincluded into the “heavy” calculations required for both 3D-MicroPIXE and 3D-Micro XRF quantification procedures, based up untilnow only on the primary fluorescence intensity.21,22 For particularlayer-structured materials, the incorporation of the secondary fluo-rescence enhancement results in a significant improvement in thequantitative confocal X-ray microscopy analysis.

In the case of 3D-Micro XRF analysis, the scattered incidentradiation might also produce a secondary fluorescence enhance-ment, particularly when light homogeneous matrixes or multilayersamples with thick light substrates are analyzed. Further inves-tigation of this contribution is necessary.

CONCLUSIONSWe have developed a global model that exactly describes the

secondary fluorescence enhancement effect in X-ray confocalmicroscopy when stratified types of material are analyzed. Withthe use of an ab initio mathematical formulation for all the involvedexperimental parameters (excitation/detection channel, samplestructure), a proper multidimensional integral expression has beendeduced. Further on, a numerical procedure has been recom-mended to ensure fast and accurate results but also a feasibleimplementation of the model into quantification minimizationalgorithms. It has been shown that the secondary fluorescenceeffect in confocal X-ray microscopy is strongly dependent on thesample, the geometry of the experimental setup, as well as theexperimental conditions. In certain cases, the enhancement canbe important with significant spatial variation (at different positionsof the sample within the probing microvolume); however, in othercases, its influence can be negligible. For certain type of samples,

Figure 8. Calculated profiles of primary21 and secondary (eq 1)characteristic X-rays intensities (Ca-KR (a) and Fe-KR (b)). Withrespect to the right axis of the graph, the calculated relative secondaryto primary intensity profile is presented for Ca-KR (a) and Fe-KR(b) (3D-Micro PIXE, 3 MeV incident beam on a glasslike matrix). Withdotted lines, the respective calculations for a polycapillary lens withthe half spatial resolution (25-10 µm fwhm for 3-10 keV, respec-tively) are shown.

Figure 9. Calculated profiles of primary22 and secondary (eq 15)characteristic X-rays intensities (Fe-KR). With respect to the rightaxis of the graph, the calculated relative secondary to primary intensityprofile is presented for Fe-KR. With a dotted line, the respectivecalculations for an extended size (σ ) 6.4 µm) X-ray incident beamis shown (3D-Micro XRF, 18 keV incident beam on 5 µm of Fe overthe Cu substrate).

Figure 10. Calculated profiles of primary21 and secondary (eq 15)characteristic X-rays intensities (Fe-KR). With respect to the rightaxis of the graph, the calculated relative secondary to primary intensityprofile is presented for Fe-KR (3D-Micro PIXE, 3 MeV incident beamon 5 µm of Fe over the Cu substrate).

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the implementation of the proposed formalism within the quan-tification analysis procedure is expected to provide a majorimprovement over the analytical accuracy of confocal X-raymicroscopy.

ACKNOWLEDGMENTThis work is supported by the FP7/REGPOT LIBRA Project

(Grant 230123) and by the Project ATT_29, PEP Attikis, cofunded

by the Greek General Secreteriat of Research, Ministry ofDevelopment and EU.

Received for review April 1, 2009. Accepted May 5, 2009.

AC900688N

4954 Analytical Chemistry, Vol. 81, No. 12, June 15, 2009