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Spectrochimica Acta Part B
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Review
Quantification for 3D micro X-ray fluorescence
Ioanna Mantouvalou ⁎, Wolfgang Malzer, Birgit KanngießerInstitute for Optics and Atomic Physics, Technical University of Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
Article history:Received 15 June 2012Accepted 21 August 2012Available online 27 August 2012
Keywords:Quantification for 3D Micro-XRFConfocal setupDepth profilingProbing site selection3D mapping
3DmicroX-rayfluorescence analysis (3DMicro-XRF) is a non-destructivemethod for the investigation of elementalcompositions of specimenswithwhich three-dimensionally resolved information can be obtained. This is renderedpossible through the formation of a probing volume resulting from the overlap of a condensed X-ray beam and theacceptance of a polycapillary lens in front of an energy-dispersive detector. Various setup schemes have beendeveloped in the last years, which can be divided into synchrotron instrumentation andX-ray tube based spectrom-eters. Established in 2003/2004 numerous applications have been published up until now. Quantification of datathough is still a topic of considerable interest and has been reported only for limited number of publications. Thisreview aims to give an overview ofwork on quantitative 3DMicro-XRF. As themethod can be appliedwith adaptedsetups to a variety of analytical problems, quantification also has to be flexible and different schemes have beendeveloped.
For 3D Micro-XRF two X-ray optics in a confocal arrangement areutilized. The result is that the gathered information derives, as afirst approximation, exclusively from a volume, which is formed bythe overlap of the foci of both optics. The size of this probing volumeis characterized by the spot sizes of the used optics and its sensitivityfor a measured energy is a function of the transmission of the optics.By moving this probing volume through a sample three-dimensionallyresolved information can be obtained.
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Since its establishment [1,2] numerous applications in the field ofenvironmental, material and life sciences have been published upuntil now. For extraordinary specimen optimized experimental arrange-ments such as the use of a nano-sized synchrotron beam for excitation [3]can be readily found. Nevertheless, 3D Micro-XRF has been mostly usedfor Cultural Heritage (CH) related applications. Especially for this fieldthe non-destructive character of analysis and the non-restricted size forthe objects, in contrast to other elemental imaging methods, are indis-pensable features. An overview of CH applications and experimentalsetups development is given in [4].
As for the experimental part threemain types of measurement proce-dure have been developed, depth profiling, probing site selectionand three-dimensional mapping. The difference between the three
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procedures is the differing movement of the probing volume inside aspecimen. 3D mapping implies a three-dimensional movement of thesample through the probing volume. While this is the most powerfulprocedure, due to themovement of the sample measuring times can eas-ily reach very high values. For depth profiling the probing volume ismoved into the specimen perpendicular to its surface and X-ray spectraare collected as a function of depth. This is the method of choice if thedepth distribution of the elemental composition is of interest and a full3D map is too time-consuming. Probing site selection is applied, if onlya feature inside the specimen is of interest. Then the probing volume isplaced at that position and spectra with long measuring times can becollected.
The existence of the probing volume results in the following differ-ences for the quantification of 3D Micro-XRF data as compared to XRFor Micro-XRF measurements. Spectra are collected as a function of posi-tion. The quantification of spectra collected in a certain depth is onlypossible if the distance to the sample surface is known as well as thecomposition of the sample in the excitation and detection beam paths.
Only atoms in the probing volume can contribute to the measuredsignal, thus, the quantity that determines the signal intensity is thelocal density. The local density is defined by the product of the densityof the sample and the concentration of the element, yielding a measurefor the number of atoms of the species of interest.
For quantification, the characteristics of the used optics must beincluded in the calculation scheme. Especially if the characteristics ofthe optics are energy-dependent an energy-dependent setup-specificcalibration function must be found.
2. Radiation energy dependence in the confocal geometry
The X-ray optic in the detection channel in all published setups is apolycapillary lens due to the possibility for broadband radiation transport.Therefore, the following energy considerations actually hold for 3DMicro-XRF in general and explain some difficulties when interpreting3D Micro-XRF data qualitatively.
The transmission and spot size of a polycapillary lens are functionsof the radiation energy, as characterized in detail in [5]. The spot sizedecreases with increasing transported energy, which means that thesize of the probing volume also decreases with increasing fluorescenceenergy. The transmission of a polycapillary lens can be described by anasymmetric peak function, thus limiting the range of detectable ele-ments through its decrease to higher and lower energies. Additionally,the absorption of radiation through matter is energy dependent andincreases with decreasing energy.
In Fig. 1a simulated measurement example already presented in[6] is displayed. As this example was chosen to represent the simplestkind of specimen – a homogeneous bulk sample – no full 3D map will
Fig. 1. Depth profile simulation on a 200 μm thick glass samples with 50 ppm of CaO, Fe2O3,with respective attenuation depths xi; right: normalized depth profiles.
be discussed, as all depth profiles are identical. A 200 μm thick glass sam-ple is assumed with 50 ppm of CaO, Fe2O3, PbO and SrO homogeneouslydistributed in the sample matrix. The measurable fluorescence energiesof the four elements are E(Ca Kα)=3.6 keV, E(Fe Kα)=6.4 keV,E(Pb Lα)=10.55 keV and E(Sr Kα)=14.14 keV. Hence, the probingvolume is smallest for the Sr fluorescence and the absorption is mostpronounced for Ca radiation.
The simulation of a depth profiling measurement with an excita-tion energy of 19 keV and a typical setup-calibration as presented in[7] is displayed in the right graph of Fig. 1. Here, the maximal fluores-cence intensity of a peak is normalized to unity in order to facilitatethe comparison of the shapes of the profiles. The exciting radiationEexc is attenuated on its way through the sample. From its surface tothe depth xexc the intensity drops to 1/e. Fluorescence is generatedin the whole path, but radiation is only transported to the detectorfrom the probing volume. This fluorescence radiation is also subjectto attenuation and leads to the fact, that there is a limited depth foreach fluorescence line from which information can still be gathered.The attenuation length in the detection path is then defined as theposition xi of the probing volume where the fluorescence intensityhas dropped to 1/e of the maximum value. More than 63% of the fluo-rescence radiation in a depth profile is detected for depth positionswhen the probing volume is closer to the surface than the attenuationlength xi of the respective fluorescence energy. In the following depthposition will be defined as the position of the middle of the probingvolume in respect to the sample surface, i.e. at depth position zerothe probing volume is halfway inside the sample. Only a limitedamount of fluorescence radiation stems from deeper positions. Theconsequence is that in our example for the Ca Kα fluorescence radiationinformation can only be gathered and interpreted to about 50 μm intothe sample. For fluorescence peaks with higher energy the informationdepth exceeds the thickness of the sample and interpretation is possiblethroughout the whole depth.
The size of the probing volume is biggest for the lowest fluorescenceenergy. That results in the fact, that the fluorescence intensities rise inthe order of their fluorescence energy, when moving the probingvolume into the sample, as seen in Fig. 1, right. In this example, the CaKα fluorescence is the first one detected in the depth scan.
The absolute fluorescence intensity is also an energy-dependentvalue as depicted in Fig. 2, because it is determined by the productof the local density of the element, the integral sensitivity of the prob-ing volume and the fluorescence production cross section. In theexample the local densities of the four elements are almost thesame. The integral sensitivity increases from Ca to Fe, and thendecreases for higher energies because its shape is mainly determinedby the transmission of the polycapillary half lens in the detectionchannel. Nevertheless, the intensity for the Sr Kα radiation is higher
PbO and SrO homogeneously distributed in the sample matrix. Left: Sketch of the setup
Fig. 2. Depth profiles of the example of Fig. 1 depicting absolute fluorescence intensities.
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than the intensity for Pb Lα due to the higher fluorescence productionprobability for K-shell radiation as opposed to L-shell radiation.
These energy effects complicate qualitative evaluation of 3D Micro-XRF measurements. They are in general the more pronounced thelower the energy of the fluorescence peak of the element of interestand the higher the absorption inside the sample.
As explained in the Introduction, there are three main applicationmodes for 3D Micro-XRF, depth profiling, probing site selection and3D mapping. In the following quantitative analysis will be discussedfor these three modes as well as for the relatively new method of3D Micro-X-ray absorption fine structure spectroscopy (3D Micro-XAFS). Thereafter a short overview over the evaluation of 3DMicro-XRF data with polychromatic excitation is given.
3. Depth profiling
When the probing volume ismoved into a sample in the direction ofits normal, spectra can be gathered as a function of depth. This measur-ing mode is called depth profiling and can be utilized in order to probethe sequence and nature of layers in stratified specimen. A prerequisitefor this measuring mode is that the specimens are in first approxima-tion laterally homogeneous, at least in the area of the radiation pathsyhom×(lateral size of the probing volume), see Fig. 3.
The reconstruction of layer thickness and composition of stratifiedspecimen from depth profiles has been elaborated for monochromaticexcitation and for primary excitation only using a polycapillary lenswith a Gaussian intensity distribution in the excitation channel [7,8].The assumption of a Gaussian distribution and monochromatic excita-tion is valid for most published synchrotron radiation setups and can,
Fig. 3. For the quantification of depth profiling measurements the sample must consistof homogeneous layers which are laterally homogeneous in an area of yhom×(lateralsize of the probing volume).
therefore, be understood as a general approach. For the reconstructionan analytical model is presented. As already mentioned above onemain difference compared to Micro-XRF quantification is, that thelocal elemental densities, i.e. the product of the local mass fraction ofan element and the overall density of the sample, determines the abso-lute intensity of the fluorescence peak. Due to the position-dependentintensity the calibration constant K of XRF is replaced by a sensitivityfunction with two characteristic parameters:
K→ η̃ i xð Þ ¼ ηi Eð Þffiffiffiffiffiffi2π
pσ x;i Eð Þ exp − x2
2σ2x;i Eð Þ
!ð1Þ
These two parameters, the size of the probing volume in the sam-ple normal direction σx,i and the integral sensitivity ηi, are functionsof excitation and fluorescence energy and can be derived through acalibration procedure described in [7]. They are defined as
with σi and Ti the spot sizes and transmission values of the X-raylenses in the excitation and detection channel, ε the detector efficien-cy, Ω the solid angle of detection and Ψ the angle between excitationchannel and sample normal in a 90° geometry (for other geometriesthe respective equations are elaborated in [9]). The width of the prob-ing volume is a measure for the depth resolution of the 3D Micro-XRFsetup. The integral sensitivity has the unit length and is present in theanalytical model as a product with the fluorescence production crosssection. This product in (g/cm3)−1 can be understood as the sensitiv-ity of the setup for a given local elemental density.
Fig. 4 shows the calibration results obtained for the confocal setup atthe mySpot beamline at BESSY II for two different excitation energiesusing thick multi-element glass reference samples (Breitländer GmbH).For a given excitation energy the width of the probing volume followsan exponential decay and the profile of the integral sensitivity can bedescribed by an asymmetric peak function. Uncertainties for the calibra-tion constants are in the range of 10–20% depending on the micro-homogeneity of the reference material. Both parameters can also bederived through scans of thin one-element foils. Ideally, if the thicknessof the foil is small, absorption in the sample can be neglected and theshape of the scan mirrors directly the sensitivity function, simplifyingthe extraction of the two parameters. When working with synchrotronsources, though, the use of multi-element reference samples poses avaluable time advantage as one depth scan yields calibration parametersfor each measurable fluorescence line.
A change of excitation energy influences both calibration parame-ters. In the presented example of Fig. 4 the use of an excitation energyof 16 keV is not advantageous, because the size of the probing volumeis enhanced, i.e. the depth resolution is worsened, and the sensitivityis decreased as compared to an excitation energy of 19 keV. Thisbehaviour can be explained by the fact, that the used X-ray lens in theexcitation channel has its maximum transmission for higher transportedenergies (Texc(16 keV)bTexc(19 keV) and σexc(16 keV)>σexc(19 keV)).Naturally, if the first lens is optimized for lower energies, the behaviouris changed and it will be of advantage to use lower excitation energies.
Additionally, for certain analytical questions it can be an advan-tage to use low excitation energies, f.e. if trying to investigate the dis-tribution of an element with low absorption edge energy, because thefluorescence production cross section is highest for energies close tothe respective absorption edge.
Fig. 4. The two characteristic parameters of the confocal setup are dependent on the excitation as well as the fluorescence energy. Results from scans with two excitation energies(16 and 19 keV) on four glass reference materials (Breitländer GmbH) are presented. Left: The width of the probing volume decreases with increasing fluorescence energy.Right: The integral sensitivity can be described by an asymmetric peak function.
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The expression for the intensityΦi of a specific fluorescence peak i ofa homogeneous layer as a function of depth position x can bewritten as:
Φi xð Þ ¼ Φ0 ηi σ F;iρi � expμ�lin:iσx;i
� �22
0B@
1CA� exp −μ�
lin;ix� �
�
�12
erfd2 þ μ�
lin;iσ2x;i−xffiffiffi
2p
σx;i
!−erf
d1 þ μ�lin;iσ
2x;i−xffiffiffi
2p
σ x;i
!" # ð4Þ
withΦ0 the excitation intensity of themonochromatic radiation,σF,i thefluorescence production cross section of the fluorescence peak i, ρi thelocal elemental density of the fluorescence element, μ⁎lin,i the effectivelinear mass absorption coefficient of the peak i, d1 and d2 the layerboundaries and the two calibration parameters σx,i the size of the prob-ing volume in the direction of the sample normal and the integral sen-sitivity ηi. For stratified specimen this expression is extended with theabsorption of upper layers, see [7].
Eq. (4) can be divided into four factors. The first factor is a constantterm for each fluorescence peak and determines its absolute intensity.The local elemental density is multiplied by the product of the integralsensitivity and the fluorescence production cross section (σF,i×ηi) andthe exciting intensity. The other three factors result from the existenceof a probing volume and its actual extension. In Fig. 5 these three factorsare demonstrated with the help of the simulation example alreadypresented above.
The last factor describes the convolution of the sample surfaces withthe probing volume and is a function of the size of the probing volumeand the effective linear mass absorption coefficient. Both factors aresmall for high fluorescence energies, as seen for the Sr Kα depth profileof Fig. 5. The deformation of the concentration profile is smallest for thehighest fluorescence energy, because absorption effects as well as thesize of the probing volume are small. For low fluorescence energies,the profile is distorted and shifted; see the red curves in Fig. 5 whichrepresent this erf-term.
The third factor of Eq. (4) is a conventional Lambert–Beer-termdescribing the absorption of excitation and fluorescence radiationfrom the surface of the specimen to the position of the probing volumeand is, hence, also most relevant for low fluorescence energies. Theabsorption is depicted in Fig. 5 as grey dashed curves, themultiplicationof the erf-termand the absorption term as blue lines.Whilefluorescenceradiation can be gathered for Sr Kα throughout the whole depth of thesample, Ca Kα can only be measured up until about 50 μm into theglass matrix.
The second term is a correction factor, which describes absorptionphenomena inside the probing volume. This term is independent of
the depth position and, thus, only the multiplication of the erf-term, theLambert–Beer-term and this respective term is depicted as green linesin Fig. 5. Again, this factor is highest for large probing volumes and highabsorption, i.e. for low fluorescence energies, see the difference betweenthe green and the blue curve of the Ca Kα profiles of Fig. 5.
In conclusion, the lower the fluorescence energy, the higher thedistortion of the concentration profile. For qualitative interpretationof depth profiles it is therefore advisable to use profiles of high energypeaks. Especially for the determination of layer boundaries not theposition of the maximum should be utilized, but the half value ofthe rise and fall of the depth profile with the highest fluorescenceenergy.
Quantification is possible with Eq. (4) for stratified specimen andhas been validated with the use of stratified reference material [7,10].For each fluorescence depth profile one intensity equation is derived.These equations are coupled through the absorption term and have tobe solved simultaneously. Here, as with all XRF measurements, theproblemmust be handled, that not all elements can bemeasured,main-ly due to low fluorescence energies. These elements constitute theso-called dark matrix. For 3D Micro-XRF the sensitivity of the setup islimited to both high and low energies (see Fig. 4, right). Prominentexamples for dark matrix elements are hydrogen, carbon and oxygen.Accurate knowledge about this dark matrix as well as the density andnumber of layers enable quantification results with uncertainties ofabout 20% for most elements. For light elements, i.e. high absorption,large width of the probing volume, low sensitivity, uncertainties areenhanced and only semi-quantitative results may be obtained.
On a reverse-painting on glass we performed depth-profiling inorder to study the corrosion processes taking place at the glass/pigmentinterface [11]. In combination with Micro-XRF measurements the com-position of the glass matrix could be reconstructed using Eq. (4) withuncertainties ranging from 8% for fluorescence lines with energieshigher than 10 keV to uncertainties of one order of magnitude higherfor low energy peaks. The diffusion of elements from the paint intothe glass was verified, which is identified as one of the causes for theloss of adhesion of the paint on the glass.
The reconstruction procedure based on Eq. (4) has until now beensuccessfully applied to glass objects [6,11], parchment fragments [12]aswell as car paint specimens [9]. For each specimen the reconstructionprocedure must be adapted due to f.e. information about the darkmatrix, the number of fluorescence elements or the density. For exam-ple, sometimes it is advisable to use the concentration values of theelements as fit parameter, if the density of the sample is well character-ized, in other cases the local elemental densities or additionally theoverall density of the layer must be varied to obtain the best possibleresults.
Fig. 5. Visualisation of Eq. (4) with the help of the simulation example of Fig. 1. The different terms of Eq. (4) distort the concentration profile. The degree of distortion isenergy-dependent.
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Woll et al. [13] present depth profiling measurements performedat the Cornell High Energy Synchrotron Source (CHESS) on a paintlayer specimen consisting of four consecutive layers with the mainelements Cd, Cr, Cu and Pb. For calibration M, L and K fluorescenceof three thin foils (Ti, Cu, Au of 0.5 μm thickness) are used in orderto extract the width of the probing volume. A 4th order polynomialfunction is utilized for interpolation in order to extract depth resolu-tion values for the fluorescence energies of the four paint layerelements. For quantification the following equation is utilized:
ID;f zð Þ ¼ ∫∞
−∞IL;f zð ÞREf
z−z′� �
dz′
with IL,f the fluorescence intensity as a function of depth for an infinites-imal small probing volume. The equation is equivalent to Eq. (4) forREf
xð Þ ¼ η̃i xð Þ. As the integral sensitivity is not calibrated or otherwisedefined, only layer thicknesses are reconstructed which are in goodagreement with visible light microscopy with uncertainties of 5–15%.
Smit et al. [2] calculate the fluorescence intensity N of an emissionline i with:
Ni ¼ j0tΔΩ4π
ΔVμ0iωi exp −Λ0−Λ ið Þεiηixi
where j0 is the excitation intensity, t the measuring time, ΔΩ the solidangle of detection, μ0iωi the fluorescence production cross section,exp(−Λ0−Λi) the absorption in the excitation and detection path,εi the detector efficiency, ηi the transmission of absorbers betweentarget and detector and xi the weight percentage of the element. ΔVrepresents the probing volume with the assumption of a sphericalshape and negligible absorption effects inside. Concentration profilesbased on a calibration procedure, which is not elaborated, are presented.
3.1. Secondary enhancement
Smit et al. [2] and Sokaras et al. [14] present theoretical models forthe investigation of the contribution of secondary enhancementmechanisms on the total fluorescence intensity. Smit et al. limit theinvestigation of the ratio of secondary to primary fluorescence forinteractions taking place inside the probing volume.
In [14] an analytical model is developed for the calculation of thesecondary enhancement for 3D Micro-XRF as well as for 3D Micro-PIXE. The primary interaction is allowed to take place throughoutthe excitation path and the secondary process in the detection path,only excitation and detections paths for 3D Micro-XRF are describedby 2D Gaussian intensity distributions with a position-independentwidth. Multiple integrations are performed using Monte-Carlo (MC)methodology. The results are compared with a full MC simulation,which is referenced to unpublished work of the authors. It would bevery interesting to learn, if in the full MC simulation the excitationand detection paths are modelled with position-dependent widthsof the Gaussian distributions, because this factor is up until now notincluded in any analytical model.
Both authors find a fluorescence enhancement of smaller than 10%for all model systems, except alloys. One has to keep in mind thatmetal samples are due to their high density and the resulting lowdepth penetrability not ideally suited for 3D Micro-XRF in general.Thus, considering the uncertainties for the calibration constants andthe difficulties of obtaining accurate values for the dark matrix andthe density of the layers, secondary enhancement inside the probingvolume may be neglected for quantification of most specimens. Thesame holds for tertiary enhancement as this is a third order effectand, thus, less pronounced than secondary enhancement.
On the other hand, an interesting point for secondary effectsmodelling would be the inclusion of all secondary effects, hence
Fig. 6. Depth profiling measurement on a car paint specimen. The first layer can only bedistinguished if evaluating the scattering depth profiles.
14 I. Mantouvalou et al. / Spectrochimica Acta Part B 77 (2012) 9–18
including scattering, which lead to fluorescence in the detection pathinitiated from outside the probing volume. While this is a big chal-lenge for an analytical model, ray-tracing codes will be the solutionhere.
3.2. Scattered radiation
One of the main challenges when reconstructing an unknownspecimen is the necessity to gain knowledge about the density ofthe sample and/or the local densities of the dark matrix. Here, onepossibility to obtain more information about a specimen is to addi-tionally use the scattered radiation. For the primary coherent andincoherent scattered radiation a similar equation to Eq. (4) can befound:
ΦSc xð Þ ¼ Φ0ηSc σ Scρ� expμ�lin;ScσSc
� �22
0B@
1CA� exp −μ�
lin;Scx� �
�
�12
erfd2 þ μ�
lin;Scσ2Sc−xffiffiffi
2p
σSc
!−erf
d1 þ μ�lin;Scσ
2Sc−xffiffiffi
2p
σ Sc
!" # ð5Þ
where the index Sc may stand for Compton or Rayleigh scattering.The scattered intensity is determined by the overall density of thelayer and the fluorescence production cross section must be replacedby the corresponding scattering cross section. Naturally, two calibra-tion constants for each scattering process must be derived.
In the case of the scattering information, secondary effects play amore pronounced role, complicating quantification. Nevertheless,qualitative interpretation of the scattered profiles may already be ofuse for specific problems. First of all, scattering occurs as soon asthe incoming radiation impinges on matter. This means that material,which consists only of dark matrix elements, can be investigated uti-lizing the scattered profiles. Fig. 6 shows a depth profile of a car paintspecimen which is composed of different layers on top of the sub-strate obtained with 19 keV excitation energy [9]. Only with thehelp of the scattered depth profiles a top coat layer (layer 1) can bediscerned.
Second, the Compton scattering cross section is as a first approxi-mation independent of the atomic number of the scattering atom,while the Rayleigh scattering cross-section increases with increasingatomic number. This means, that a comparison of the Rayleigh andCompton profile can give insight into density variations and composi-tion changes of the dark matrix. In the example of Fig. 6 the scatteringprofiles have the same shape, thus, a homogeneous top coat layermay be assumed. And third, due to the usually high energy of thescattered radiation, thicknesses can be estimated with low uncer-tainties without a full quantification. In the example of Fig. 6 the esti-mated thickness of the first layer is 36 μm, which is in perfectagreement with light microscopy values.
In [12] we use the scattered information for the investigation ofhighly heterogeneous parchment fragments from the Dead Sea Scrollcollection. A semi-quantitative approach is introduced for fragments,for which quantification is not feasible due to considerable density var-iations. In the approach the scattered intensity is utilized as an indicatorfor the overall density at a specific depth position and only point spectraare analysed, where no degradation of the parchment matrix is observ-able. Absolute concentration values fromMicro-XRFmeasurements arecompared with values obtained with the full Eq. (4) as well as withthe semi-quantitative approach and show good agreement. This isinterpreted as a sign, that Micro-XRF measurements are appropriatefor these kinds of fragments despite their high inhomogeneityinto the depth. The conducted work thus paves the way for routineprovenance studies of the fragments with portable Micro-XRFinstrumentation [15].
3.3. Ray-tracing approaches
A different approach for quantification of depth profiles is theuse of Monte-Carlo (MC) simulations. Czyzycki et al. [16] presenta MC code in which primary and secondary fluorescence andmultiphoton-scattering are taken into account. The detection channelis modelled in the way that photons emitted in the probing volumeare forced to the detector with weighing factors. The authors compareMC simulation results with experimental depth profiles as well as theresults obtained with the analytical model based on Eq. (4). They findgood agreement for both quantification schemes.
In principle it would be interesting to perform a full MC simulationwithout the constraint of forcing the photons to the detector. Addition-ally to secondary fluorescence effects especially the investigation of thescatteringdepth profileswith such aMC codewould be of great interest,as themulti-scattering effects which hamper the reconstruction of den-sity with the analytical model could be overcome. Also, when utilizingfundamental-parameter-based quantification, absorption is modelledusing Lambert–Beer-terms, inwhich thedetectable intensity is constantlyunderestimated due to the neglect of multiple scattering phenomena.With MC-methods this problem can be overcome if the trace of theX-rays is modelled correctly from the source to the detector.
4. Probing site selection
As a secondmeasuring mode probing site selection means that theprobing volume is positioned at a certain position inside a sample anda spectrum is collected. This measuring mode is interesting, whenonly a feature, preferably a dense homogeneous inclusion in a lightmatrix, at a certain depth inside a specimen is of interest. Throughthe use of 3D Micro-XRF, information, i.e. fluorescence but alsoscattered radiation, of the surrounding matter is suppressed. Alreadyby a qualitative analysis elements can be attributed to the feature ofinterest. In Fig. 7 a few selected cases are depicted. For each of thecases the quantification strategy must be adapted.
In case A the feature of interest is larger than the probing volume,and homogeneous. This case can be understood considering the depthprofiling section as a stratified sample. Themost interesting parametersare the depth of the feature inside the specimenD and the depth x of theprobing volume inside the feature in order to calculate the attenuation.Eq. (4) simplifies to:
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with μlin.i*F and μlin.i*S the effective linear mass attenuation coefficients ofthe feature and the surface layer, respectively. With a suitable calibra-tion, quantification is straight-forward and conventional codes suchas PyMCA [17] can be used.
In case B the feature of interest is sufficiently smaller than the sizeof the probing volume, so that the shape of the probing volume aswell as the absorption inside the probing volume is negligible. Ifthis feature is placed in the middle of the probing volume Eq. (4)can as a first approximation be written as:
Φi xð Þ ¼ Φ0ηiσF;iρiqffiffiffiffiffiffi
2πp
σ x;i
� exp −μ�Slin;i xþ Dð Þ
� �ð7Þ
with q the diameter of the feature. The factor q=ffiffiffiffiffiffi2π
pσ x;i describes the
size ratio of probing volume to feature size. Again a careful calibrationand the full knowledge about the composition of the surrounding ma-trix and the depth of the feature inside the specimen are necessary.
If the feature is heterogeneous and large compared to the probingvolume, as depicted in case C in Fig. 7, the extracted information canbe the convolution of all different phases of the feature of interest.Different phases can be investigated through scanning the object,though quantification becomes more laborious, as the absorption ofthe excitation and fluorescence radiation through other parts of thefeature have to be taken into account. In principle the specimenmust be scanned and a full 3D model must be reconstructed.
For all cases, A, B and C, the knowledge of the depth of the featureis mandatory for quantification. If it is not known a priori from otheranalytical methods like light microscopy, this knowledge can bederived through a depth scanning analysis. While the assumption ofa layered system is not strictly valid, a good estimation can be derived.
Leroy et al. [18] investigated slag inclusions in medieval armourwith the combination of LA-ICP-MS and 3D Micro-XRF. In order toperform provenance studies, elemental ratios of minor constituentsfrom inclusions of sizes of a few tens of micrometers are of interest.Here, the inclusions are assumed to be homogeneous, positioned onthe surface and bigger than the probing volume (case A). For calibrationof 3D Micro-XRF, reference samples of similar composition are used andfluorescence energy-dependent sensitivity or correction coefficients aredefined in order to handle the energy-dependent transmission of thepolycapillary lens in front of the detector. Due to the use of similar refer-ence standards challenges dealing with the absorption inside the sampleare overcome. Quantification is then performed for point spectra usingthe internal standard method with the assumption of a known Mn con-centration. Comparison with Eq. (6) shows that the last exponentialterm vanishes as the inclusions are on the surface and the correction co-efficient here corresponds to the only setup-dependent parameter, theintegral sensitivity. Thus, a simple quantification is feasible in this case.Concentration values are compared to another independent method(LA-ICP-MS) yielding uncertainties of below 20%. As a result for one ofthe investigated armour pieces the Lombard area is ruled out as produc-tion site.
Fig. 7. Probing site selection; A) large homogeneous specimen; B) small homogeneous specvolume.
Sun et al. [19] present a quantification procedure for aerosol particles,which are small in relation to the probing volume and assumed to behomogeneous (case B of Fig. 7). The used X-ray excitation from a rotatinganode tube ismonochromatizedwith aNb-filter, and the probing volumeis characterized with a set of thin reference standards. Sensitivity coeffi-cients are extracted from thesemeasurements, as well. For quantificationthe quantitative X-ray analysis for thin samples of a standard softwarepackage [20] is used. In order to take into account that the sensitivitycoefficients SCexp were attained by the measurement of foils and theinvestigated objects are smaller than the probing volume a correctionfor the volume V of a particle is introduced
SCvir ¼ SC exp∫∫∫V1
2πσ1σ2exp − x2 þ z2
2σ1þ y2 þ z2
2σ2
! !dxdydz
with σ1 and σ2 the two constants, which can be obtained by knife edgescans. This correction corresponds to the factor q=
ffiffiffiffiffiffi2π
pσ x;i in Eq. (7).
The quantification was tested with standard solution drops and yieldsan accuracy of 20% and a precision better than 4%. As an applicationaerosol particles of vehicle exhaust and coal combustion are analysedand concentration values are given. Unfortunately, no information isgiven about the size, density or composition of the standard solutiondrops or how the different densities of standard foils and investigatedparticles are taken into account.
In [21] the authors reconstruct different mineral phases of Ca andAl-rich inclusions in cometary matter. Two heterogeneous particlesembedded in aerogel are analysed which are considerably largerthan the probing volume (case C) and, therefore, the confocal setupis mainly used to reduce information from the surrounding material.The quantification of point spectra is performed using a fundamentalparameter approach which is simplified by the measurement of ageological glass standard as reference. An additional absorption cor-rection is performed, based on Kα/Kβ-ratios [3]. The shape andgeometry of the particles are neglected and also the attenuation ofradiation therein. Concentration values for Ca, Ti, Cr, Mn, Fe, Co, Ni,Si and O are presented, where Si and O apparently must be obtainedthrough normalization. The uncertainties of the concentration valuesamount to about 15% due to the uncertainty of the overall density ofthe sample. The similarity of the obtained compositions of three dif-ferent volumes of the dust particles to known mineral phases is theninvestigated with principal component analysis (PCA).
4.1. 3D Micro-XAFS
A very interesting possibility when measuring in the probing siteselectionmode is to scan the excitation energy in the area of an absorp-tion edge of interest and thus perform three-dimensionally resolvedmicro X-ray absorption fine structure spectroscopy (3D Micro-XAFS).As a result non-destructive chemical speciation is feasible. Two applica-tions have already been published [22,23] exploring this novel technique,
imen; C) large heterogeneous specimen; size classification as compared to the probing
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in which the measured XAFS spectra have proven to give valuable infor-mation about the chemical state of a feature of interest. As with 3DMicro-XRF, the absorption of impinging and fluorescence radiation insidethe sample is a function of energy and depth and can deform the mea-sured XAFS-spectra considerably.
Recently Lühl et al. [24] have presented the first quantitativeapproach for the reconstruction of the original XAFS-spectrum ofstratified samples. The product of photo ionisation cross section andjump factor which represents the XAFS-signal can be extracted fromthe measured data using Eq. (4). The equation cannot be solved ana-lytically, because the XAFS-signal is not only part of the fluorescenceproduction cross section, but also part of the effective linear massattenuation coefficient. As a consequence numerical tools are used.Also, as for XAFS measurements the excitation energy is changedover an absorption edge and only one fluorescence line is analysed,the calibration must be performed accordingly.
With the help of dedicated reference samples with different coppercompounds in different layers the validity of this approach is presented.Reconstruction is possible for layers larger than half of the size of theprobing volume for the considered fluorescence line, i.e. in the range of5 to 20 μm.Aprerequisite for the reconstruction is a complete knowledgeabout all other parameters in Eq. (4) that means the density and compo-sition of all layers of the sample as well as the position of the probing vol-ume inside the sample, which can be derived by a prior depth profilinganalysis.
5. 3D mapping
With 3D Micro-XRF three-dimensionally resolved elemental andscattering maps can be obtained non-destructively. As opposed toconventional tomographic methods the size of the sample is irrele-vant, i.e. large objects like paintings can be investigated.
The drawback of this measuring mode is that the measuring timeseasily reach non-reasonable limits. Due to the fact, that the samplemust be moved for each data point and a spectrumwith sufficient sta-tistics must be collected, an assumption of at least 1 s per data pointcan be made. A data cube of 50×50×30 points thus results in a fullmeasuring time of more than 20 h. Ideally, small data cubes and spec-imen with high fluorescence yield should be chosen.
Generally, the spectra show a low number of counts due to the shortmeasuring times, thus, no spectral deconvolution is needed and typicallyregions of interest are used. For specimenwith very low absorption valu-able information can be obtained from this raw data with statisticalmodels like principal components analysis or k-means clustering. Theinformation gain of the gathered unprocessed 3D maps is enhanced thesmaller the probing volume is compared to the structures of the sample.
The quantification of a full 3D map of a heterogeneous sample isextremely complex. Considering Fig. 3, if the structures are at leastlarger than yhom in one lateral direction and larger than the probingvolume in the other lateral direction, the quantification for depth pro-filing can be utilized. If the extension of the probing volume can becompletely neglected (σybbyhom), only depth-dependent absorptionmust be taken into account. This absorption has to be carefully calcu-lated not only dependent on the depth, but also on the lateral posi-tion. If the lateral structures of the sample are smaller than yhom a3D model of the sample must be reconstructed in order to correctfor absorption and the size of the probing volume.
Faubel et al. [25] investigate protrusions in a painting by MaxBeckmann which threaten to destroy the artwork. Samples weretaken and a 3D Micro-XRF map of 26×101×16 voxels was gathered.No measurement time is given, but assuming 1 s per voxel, 11 h ofmeasurement are expected. The authors correct for absorption usinga constant absorption coefficient inside the whole heterogeneoussample and show a 3D rendering of the area around two blisters.Through this visualisation information is gained regarding relativeconcentration differences in the bulk compared to the blisters.
Semi-quantitative concentration values are presented. In combina-tion with other techniques (Raman, EDX, Microscopy) they identifyZn-soap formation as the cause for the blistering of the paint. Theuse of four different techniques in this work is very appropriate fora complex investigation of such a heterogeneous specimen, and theuse of the performed simple absorption correction yields valuableinformation. As the authors present scattering data in the visualisationit could be a further step to take this information into account for aposition-dependent absorption correction.
Vekemans et al. [26] scanned a data cube of 31×31×30 voxels ofan inclusion in a diamond, which must have taken more than 40 hof measuring time (5 s per voxel). The work presents the use ofchemometric data analysis methods for the quantification of the full3D map. First, with the help of high energy fluorescence lines ofselected elements with negligible absorption (Sr, Th, Zr) a firstk-means clustering was performed for a qualitative visualisation.Then, with a suitable weighing of intensities of all other fluorescencelines a second clustering with 40 means was conducted and on thisbasis three different mineral phases were identified with principalcomponent analysis (PCA). A third clustering with 10 ‘super’-clustersthen yielded the final 3D rendering of the inclusion. As a next stepthis reconstruction together with information from Raman measure-ments concerning the chemical species of the elements in the threephases was used to perform a geometrical absorption correctionwith constant absorption coefficients in the three phases and the sur-rounding diamond. Corrected data for two of the three phases ispresented. Quantification of the sum spectra of the different phasesis thus in principal rendered feasible and results are presented inVincze et al. [27].
For biological applications, certain sample preparation instrumen-tation is a prerequisite as most specimens degrade under ambientconditions. The mySpot beamline at BESSY II has been equippedwith a nitrogen cryogenic stream in order to freeze-dry the specimenand as a proof-of-principle experiment a virtual 2D cut through a rootof common duckweed could be obtained with a measurement timeof 45 min in which the specimen remained unaltered [28]. Becauseof the small density of the root, qualitative information could beobtained, though the need for an absorption correction is discussed.
In 2010, de Samber et al. [29] presented a detailed study of theecotoxical model organism Daphnia magna with 3D Micro-XRF incombination with laboratory absorption micro-tomography. Withfree standing thin films a calibration of σA is performed. Then a NISTreference standard (SRM 1577B) is used to determine minimumdetection limits and elemental yields. These values are derived onlyfor the depth position zero, i.e. where the probing volume is locatedhalfway inside the sample, and, thus, constitute the best possiblevalues for negligible absorption. On the D. magna sample sagittaland dorsoventral 2D sections are collected with measuring times of2–3 s. Additionally, in a dynamic scanning mode a full 3D map of63248 voxels of an extracted egg is collected resulting in a acquisitiontime of 20 h. Self-absorption effects are illustrated and calculatedusing the Ca Kα/Kβ-ratio with the result, that self-absorption isneglected for semi-quantitative evaluation of the Zn-distribution.Valuable biological information is extracted through the visualizationof the full sample as well as of the egg especially when comparingwith laboratory absorption micro-tomography which yields densityinformation.
6. Polychromatic excitation
In comparison to monochromatic excitation, polychromatic exci-tation (X-ray tubes, white light at synchrotron facilities) furthercomplicates evaluation of 3D Micro-XRF data. Due to the broadbandexcitation spectrum the transmission characteristics of the first lenshave to be taken into account. This means, that Eq. (4) must be inte-grated over the energy interval [Eabs,Eend] with Eabs the absorption
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edge energy of the fluorescence line of interest and Eend the highestenergy of the excitation spectrum. The characteristic parameters ofthe confocal setup are functions of this excitation energy as well asthe excitation intensity, the fluorescence production cross sectionand the effective linear mass absorption coefficient.
Up until now little work has been published which validatesquantification procedures for polychromatic excitation. Most pub-lished work constitutes proof of principle-experiments [30,31] forthe achieved depth resolution.
In a previous work [6] we have demonstrated the possibility to differ-entiate between bulk and layered material with the help of a laboratorysetup without the need for a full quantification. The measurement on areverse-painting on glass called ‘Lüneburger Meditationstafel’ showed aclear layered structure because the intensities of the fluorescence peaksof normalized depth profiles did not increase according to their energy.Using additional synchrotron-based measurements as validation wecould prove that on this object the black contour colour was not firedinto the glass.
Mazel et al. [32] performed confocal measurements with a laborato-ry setup on pharmaceutical tablets. For the evaluation of the distribu-tion of Zn stearate in a laboratory-made tablet a simplified absorptioncorrection is presented. Assuming a simple Lambert–Beer type absorp-tion of the Rayleigh scattering intensity in an energy region close to thefluorescence line of interest (Compton scattering is neglected), a con-stant effective linear attenuation coefficient along the depth is extractedand then used for the correction of a 2D map of the Zn distribution.Additionally, a commercially available tablet is investigated and thethickness of a Ti coating is estimated using a procedure based onEq. (4). For this a homogeneous Ti coating layer is assumed. With athin Ti foil they extract the width of the probing volume and with thealready explained Rayleigh scattering procedure they obtain the ab-sorption correction and, thus, are able to reconstruct the thickness ofthe coating. Thickness values are compared with SEM photographsand are in excellent agreement. As only the thickness and not the com-position of the layer is of interest in this work, the assumptions are justi-fiable and this work proves again, that even without a full quantificationvaluable information is attainable with a laboratory setup.
7. Conclusion
In this reviewwepresenteddifferent quantification approaches for 3DMicro-XRF along its principal usage modes: depth profiling, 3D mappingand probing site selection. For these modes different quantificationapproaches are necessary. Even for the same mode different proceduresmight be indispensable which are adapted to the specimen and questionto be investigated. Comparing the status of the quantification proceduresfor the three modes the ones for depth profiling are the most advanced.Here one may speak of an already generalized quantification procedurefor stratified samples and the depth profiling mode. This includes alsothe 3DMicro-XAFS quantificationwhere original XAFS-spectra of layeredspecimen can be reconstructed.
As for 3D mapping the quantification procedures published so far areespecially adapted to the specific investigation. A full quantitative 3Dreconstruction has not been published up until now, due to the complex-ity of the task. A more generalized semi-quantitative procedure for X-raytube excitation measurements might be possible by including Rayleighscattering as proposed by Mazel et al. [32].
Experimental and quantification procedures for probing site selec-tion measurements are the most difficult ones to generalize. As dem-onstrated in the respective chapter the size of the probing volumeplays an even more pronounced role which leads to adapted quanti-fication procedures for the specific specimen and question.
Improvements in terms of lower uncertainties might be obtainedby including secondary, and in some special cases tertiary, enhance-ment and especially scattering. This work has still to be done andmight be carried out by Monte Carlo simulations.
Valuable information can already be gained by qualitative or semi-quantitative analysis, when carefully including the energy effectsdiscussed in Section 2. Layer thicknesses can be estimated using highenergy fluorescence or scattering peaks and elemental concentrationratios can be obtained when using similar reference standards and/orif absorption is negligible. Nevertheless, there exist questions whichcan only be answered with a full quantification. Especially the morepronounced the absorption effects are, the more careful the measureddata has to be interpreted.
In summary quantification procedures for 3D Micro-XRF havealready reached a quite matured state for a large category of specimen,especially layered ones. This holds for monochromatic excitation,whereas for polychromatic excitation a general solution has still to bepublished. Quantification for polychromatic 3D Micro-XRF will be thelast big step in this field as it will contribute to a larger usage of themethod with respect to laboratory equipment. Still a pure qualitativeuse of the method will contribute to valuable results.
Acknowledgments
The car paint sample of Fig. 6 was provided by Ina Schaumann. Theauthors would like to thank Ivo Zizak for his support at the mySpotbeamline at BESSY II andCarla Vogt for thework on the adapted referencematerials. Lars Lühl and Timo Wolff contributed substantially to theunderstanding of the method. Parts of this work were conducted inthe framework of the PhD Thesis of Ioanna Mantouvalou funded by theGerman Research Foundation, grant KA 925/7-1.
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