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X-ray scattering in X-ray fluorescence spectra with X-ray monochromatic, polarised excitation – Modelling, experiment, and Monte-Carlo simulation
Nuclear Instruments and Methods in Physics Research B 269 (2011) 1493–1498
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
X-ray scattering in X-ray fluorescence spectra with X-ray monochromatic, polarised excitation – Modelling, experiment, and Monte-Carlo simulation V.-D. Hodoroaba a,⇑, M. Radtke b, U. Reinholz b, H. Riesemeier b, L. Vincze c, D. Reuter a a
BAM Federal Institute for Materials Research and Testing, Division 6.4 Surface Technologies, D-12200 Berlin, Germany BAM Federal Institute for Materials Research and Testing, Division 1.3 Structure Analysis; Polymer Analysis, D-12200 Berlin, Germany c Ghent University, Department of Analytical Chemistry, B-9000 Ghent, Belgium b
a r t i c l e
i n f o
Article history: Received 25 January 2011 Received in revised form 1 April 2011 Available online 16 April 2011 Keywords: X-ray scattering Monochromatic excitation Polarisation Modelling Monte-Carlo simulation
a b s t r a c t A systematic series of measurements has been carried out with monochromatic X-ray excitation with synchrotron radiation in order to check a physical model on X-ray scattering. The model has recently been successfully tested for the case of polychromatic, unpolarised excitation emitted by an X-ray tube. Our main purpose is the modelling of a physical background in X-ray fluorescence spectra, so that improved quantitative results can be achieved especially for strongly scattering specimens. The model includes single Rayleigh and Compton scattering in the specimen, the effect of bound electrons, the challenging Compton broadening and the polarisation degree. Representative specimens, measurement geometries and excitation energies have been selected with synchrotron monochromatic light at BAMline/BESSY II. Monte-Carlo simulations have been also carried out in order to evaluate the quality of the results achieved with the model. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction X-ray fluorescence (XRF) with excitation from an X-ray tube is a well established analytical method for determining element concentrations in solid samples for main, secondary and trace elements. Especially for light matrices such as glasses, polymers and ceramics the contribution of X-ray scattering to the XRF spectra in the form of a pronounced spectral background, scattered X-ray lines and sometimes diffractions peaks are significant. Furthermore, if the specimens are rich in elements, the background construction necessary for the background subtraction and calculation of the net peak areas of the fluorescence lines in the XRF spectra, respectively, become critical. The pure mathematical construction of the spectral background based only on peak-free regions of interest is state of the art. Altered limits of detection and high uncertainties in the determination of elements at trace level concentration are the consequence. A physical model for a quite accurate calculation of X-ray scattering spectra based on well-known X-ray tube excitation spectra and differential scattering cross sections reported in the literature has been recently successfully implemented by our group [1]. The results have also been related to metrological measurements performed with modern calibrated instrumentation, i.e. with a side-
⇑ Corresponding author. Tel.: +49 030 81043144; fax: +49 030 81041827. E-mail address: [email protected] (V.-D. Hodoroaba). 0168-583X/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2011.04.009
window X-ray tube attached to a SEM/EDS (scanning electron microscope with an energy dispersive spectrometer) system [2–5]. Initially, it was expected by the authors that the modelling of the X-ray scattering spectra in the case of polychromatic, unpolarised excitation with an X-ray tube can be better understood by comparing this case with measurements with monochromatic radiation. However, in spite of the excellent monochromaticity attained at the BAMline at the BESSY II synchrotron storage ring [6], it has turned out that the accurate description of the degree of polarisation assigned to the individual experiments is a challenging task. Thus, the degree of polarisation constitutes a new source of uncertainties which competes with those accompanying e.g. the differential scattering cross-sections or the considered (solid) angles and sampling volumes in the specimen. Nevertheless, similar systematic investigations with monochromatic, polarised radiation involving (i) measurements, (ii) modelling and (iii) Monte-Carlo simulations, and relating them to each other have not been reported upon in the literature so far. The present study addresses this purpose, so that weak points in the physical model as well as in the measurements can be identified and their uncertainty budget assessed at least in a rough form. It must be also noted that the contribution of X-ray scattering to XRF spectra with monochromatic excitation is considerably lower than that when using polychromatic excitation. The Bremsstrahlung background emitted by an X-ray tube over the whole energy range up to the excitation energy of typically 40 keV is scattered elastically as well as inelastically over the whole energy range. This
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continuous contribution is missing when the excitation is monochromatic, so that only the elastic and inelastic peak corresponding to the monochromatic excitation energy constitute the scattering ‘‘background’’ in the XRF spectra. Due to this reason the determination of the net fluorescence peak areas in the case of polychromatic excitation remains much more critical than for the monochromatic one. The apparently significant contribution of multiple X-ray scattering [7], occurring at energies lower than the excitation energies and especially for light matrices, can be simulated only with Monte-Carlo software packages. In the present study X-ray scatter spectra of representative sample matrices at various monochromatic excitations and at selected geometrical conditions have been measured and modelled. Additionally, X-ray spectra obtained by performing Monte-Carlo simulations for the same conditions have also been taken into consideration.
2. Experimental In order to check the accuracy of the X-ray scattering model implemented as described in our recent work [1], systematic measurements with monochromatic, polarised excitation have been carried out at the BAMline/BESSY II synchrotron facility, adapted for monochromatic, polarised excitation. The specimens selected have been PMMA, pure aluminium and pure copper. The specimen has been fixed on the specimen stage so that the incident X-ray beam hits the specimen surface under an incidence angle of 27°, see Fig. 1a. The incident monochromatic X-ray beam has been tuned with the double crystal monochromator (DCM) for energies at 10, 20, 30 and 40 keV, with corresponding FWHM values in the range of 1.5–6.0 eV. These values are smaller than the 10 eV energy channel width of the acquisition system. An impression of how the X-ray scattering spectra emitted towards detector at these excitations look like is given in Fig. 2, as calculated with our model in Ref.
[1] for aluminium as a scatterer and 150° scattering angle. The Xray beam was nearly linearly polarised with the polarisation vector in the horizontal (synchrotron storage ring) plane. The X-ray spectrometer used for the series of measurements reported in this paper was an EDS system NSS with a Si(Li) detector (ThermoFisher Scientific). The EDS was calibrated up to 60 keV by using synchrotron radiation from the electron storage ring BESSY II [8]. Hence, both detector efficiency and spectrometer response functions are well-known [9–11]. The EDS detector has been placed at different selected positions so that the direction of the polarisation vector is always parallel to the scattering plane, which is equivalent to the horizontal plane. The resulting scattering angles have been 30°, 60°, 120° and 150°, Fig. 1b. The geometry corresponding to a scattering angle of 90° has also been included in this series and exploited in order to check the value of the degree of polarisation, being known that there are slight deviations from the assumed ‘‘pure’’ linear polarisation. According to the theory (see Section 3 below), if the linear polarisation of the incident X-ray beam is in the scattering plane, i.e. ‘‘pure’’ parallel polarisation, then at a scattering angle of 90° the Rayleigh scattering equals zero, while the amount of Compton scattering is reduced to a basic residual. All the measurements have been performed at ambient air pressure, having a distance between the exit of the monochromatic X-ray beam from the beamline to the specimen surface of 705 mm and a distance of 120 mm from the specimen surface up to the detector crystal. In order to work with reasonable photon count rates of the Si(Li) spectrometer, the deadtime has been kept below 30%. For copper specimens and for aluminium specimens an aluminium filter of 0.5 mm thickness has been placed in front of the detector, so that the intensive fluorescence lines originating in the specimen themselves were considerably diminished. 3. Theory The elastic (Rayleigh) and inelastic (Compton) X-ray scattering with monochromatic, polarised excitation are governed by the same basic physical equations as described in detail in our recent publication for the case of polychromatic, unpolarised excitation [1]: (i) Absorption of the incident radiation in the specimen up to the depth where the scattering event takes place. (ii) Scattering event according to differential scattering cross sections (from literature [12]) for consideration of the scattering anisotropy; the influence of the bound electrons in the specimen atoms is determined by the atomic form factor and the incoherent scattering function [12]. The energy loss of
Fig. 1a. Photograph of the experimental setup at the BAMline/BESSY II, highlighting the scattering geometry. The geometry has been selected in such a way that the scattering plane (the horizontal one) contains the direction of the linearly polarised incident beam. Only one value of 27° has been selected for the incidence angle w1 and the detector has been moved in the scattering plane, yielding representative scattering angles h for forward and backward scattering, i.e. h = 30°, 60°, 120° and 150°. A geometry corresponding to h = 90° has been also taken into account for the exact determination of the degree of polarisation.
Fig. 1b. Schematic top view of the experimental setup detailing the incident angle w1 (27°) and the scattering angle h (30°, 60°, 120° and 150°).
V.-D. Hodoroaba et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 1493–1498
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polarisation, i.e. Ik = 0 and I = I\, P becomes equal to 1. The intensity of the scatter peak, SP is then a sum of the two corresponding components:
SP ¼
1þP 1P Sk þ S? ; 2 2
ð3Þ
with Sk and S\ the intensities of the scattered radiation resulting from ‘‘full’’ parallel (P = 1) and perpendicular scattering (P = 1). It can be concluded that both definitions lead to correct results, however, the degree of polarisation must be specified and the corresponding formula of P employed. The same results in scattering intensities will be derived from different degrees of polarisation, PHanson as defined in [17] and PVincze as defined in [19], respectively. The correlation between the two P0 s is:
PHanson ¼ Fig. 2. Plot of the elastically and inelastically scattered photons at various monochromatic excitation energies E0 = 10, 20, 30 and 40 keV, for aluminium and at a scattering angle h = 150°.
1 þ P Vincze : 2
ð4Þ
In this paper the definition of P according to [17] has been adopted.
Compton scattered radiation is considered in the model as well as the challenging Compton broadening [13,14]. The latter one is caused by the so-called pre-collision motion of the atomic electrons. Compilated so-called Compton profiles data have been implemented from Biggs et al. [15]. (iii) Absorption of the scattered radiation from the point of the scattering event in the specimen up to the specimen surface on the way toward EDS detector. Absorption in air or in the filter in front of the detector is also included. Mass attenuation coefficients from [16] have been used. The modelling of monochromatic radiation is in principle simpler than for the polychromatic case, because the complete Rayleigh and Compton scattering spectra result from the excitation in only one energy channel of 10 eV. As already mentioned, the energetic FWHM of the incident beam at the various excitation energies used in this study has been below 10 eV. For the case of polychromatic excitation with an X-ray tube at 40 kV, there are 4.000 ‘‘excitation’’ energy channels of 10 eV which must be considered for the calculation of the X-ray scattering spectra. However, the additional parameter which brings some potential difficulties into the description of X-ray scattering of polarised radiation is the degree of polarisation, P, respectively its exact knowledge. In the literature there are different definitions of P. In [17,18] P is defined as the fraction of the component of the polarisation vector onto the direction parallel to the scattering plane, so that the resulting intensity of the scattered radiation, SP, is given by the following sum:
SP ¼ PSk þ ð1 PÞS? ;
Fig. 3a. Dependence of the intensity of the Rayleigh and Compton scatter radiation on scattering angle when the incident X-ray beam is linearly polarised (parallel and orthogonal).
ð1Þ
with Sk and S\ the intensities of the scattered radiation corresponding to ‘‘fully’’ parallel and ‘‘fully’’ orthogonal linear polarisation. Hence, in the case of an ideal linear polarisation in the scattering plane, i.e. I = Ik and I\ = 0, the result is P = 1. If the linear polarisation vector is orthogonal to the scattering plane, i.e. if Ik = 0 and I = I\ then the resulting P = 0. In [19] P is defined according to the formula:
P¼
I k I? ; I k þ I?
ð2Þ
with Ik and I\ the parallel and orthogonal components of the excitation intensity I. One notes that according to this definition for I = Ik and I\ = 0, P = 1; however, for the case of orthogonal linear
Fig. 3b. Intensity ratio of Rayleigh to Compton scattering in dependence on the degree of polarisation, calculated for aluminium, at 30 keV energy of the incident radiation and for three representative scattering angles 60°, 90° and 150°.
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For linearly polarised X-ray beams P = 1. Fig. 3a shows how sensitive are the changes in scattering angle on scattering intensities, if linearly polarised radiation is employed. Depending on the orientation of the polarisation vector, both Rayleigh and Compton scattering display considerable variations. The extreme case is the 90° scattering angle and parallel linear polarisation (P = 1), when the Rayleigh scatter intensity becomes zero and the Compton scattering nearly zero. Due to the finite expansion of the incident beam in
Fig. 4
the range of mm2, but also due to slight misadjustments when tuning each individual monochromatic excitation energy, slight deviations from P = 1 are practically unavoidable. As illustrated in Fig. 3b for a scattering angle range around 90° these cause severe consequences on both scattering intensities, elastically and inelastically. In order to calculate X-ray scattering spectra as measured by the EDS spectrometer, the calculated scattering spectra ‘‘emitted’’ by
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Fig. 5
the specimen toward the detector entrance have been multiplied with the known spectrometer efficiency and finally convoluted with the spectrometer response functions. The spectrometer employed is fully characterised and calibrated [8–11]. The Monte-Carlo simulations have been obtained using the computer code MSIM [19]. The incident monochromatic X-ray beam was the same for both the model and for Monte-Carlo simulations. The Monte-Carlo simulations yield the scatter spectra as emitted by the scattering body in the same format as the model calculations, and they are subsequently treated in the same manner as these, i.e. subjected to the detector efficiency function and response matrix convolution. Other Monte-Carlo software packages, e.g. [20], could also be taken into consideration.
4. Results and discussion All the mechanisms contributing to scattering as presented in Section 3 have been implemented in an analytical model, which has been shown to work successfully for polychromatic, unpolarised excitation. Additionally to this, for polarised excitation, the degree of polarisation has been now included into the model. For each individual excitation, i.e. 10, 20, 30 and 40 keV, individual adjustments of the incident monochromatic X-ray beam had to be carried out. Each time the degree of polarisation has been derived from ‘‘residual’’ X-ray scattering spectra at the geometry corresponding to 90° scattering angle. In order to get a reasonable overview on the results of the application of the analytical model for various conditions, a set of modelled scattering spectra at 30 keV excitation for all the three specimens considered in this study, i.e. PMMA, Al and Cu, at 60° and 150° scattering angle are presented in Fig. 4. At 30 keV energy of the incident monochromatic X-ray beam the corresponding shift for various geometries is clearly visible and the shape of the two scattering peaks, Rayleigh and Compton, can be checked rather well. The corresponding degree of polarisation derived from the 90° scattering spectrum was 0.993. The calculated X-ray scattering spectra shown in Fig. 4 are compared to measured spectra. Additionally, in order to have a complete picture, X-ray scattering spectra for identical experimental conditions have been also simulated by Monte-Carlo methods and displayed for comparison in Fig. 4. In order to have an idea how the results of the model behave at other energies of the incident X-ray beam, e.g. at 20 keV, scattering spectra for aluminium, as a scatterer of middle ‘‘strength’’, are shown in Fig. 5. The degree of polarisation for the 20 keV incident
X-ray beam, derived from the 90° scattering geometry, was 0.988. Also for a better overview the relative deviations of the Comptonto-Rayleigh intensity ratios of both the model and Monte-Carlo simulations relative to the measurement are shown in Table 1 for all the cases in Figs. 4 and 5. The strategy of calculation of the X-ray scattering spectra with the analytical model as well as by the Monte-Carlo simulations has consisted firstly of calculating the two scattering intensities Sk and S\ for Eq. (1), for both Rayleigh and Compton scattering, corresponding to ‘‘fully’’ linear parallel and orthogonal polarisation. Then, having the experimentally deduced degree of polarisation, Eq. (1) has been employed in order to calculate the resulting Rayleigh and Compton scattering intensities. From the X-ray scattering spectra in Figs. 4 and 5 one can generally state that both Rayleigh and Compton scattering – as measured – are rather well described by our analytical model. This confirms our expectations stemming from our already verified experience with polychromatic excitation. A sensibly better agreement relative to the measured scattering spectra is attained by the Monte-Carlo simulations. Note that similar to the spectra gained with the analytical model the spectra obtained by Monte-Carlo simulations have been scaled to the intensity of the Rayleigh peak in the measured X-ray scatter spectra. Hence, one main goal of this work, namely to investigate systematically the spectral distributions of X-ray scatter spectra with monochromatic excitation, could be achieved. The reliability of the model with respect to the absolute values of the X-ray scatter intensities have been already successfully demonstrated in our previous paper for polychromatic excitation [1]. The somewhat better overall agreement relative to the measurement is again not surprising due to the fact that, unlike the analytical model, the Monte-Carlo simulations
Table 1 Overview of the relative deviations of the Compton-to-Rayleigh intensity ratios of both the model and Monte-Carlo simulations relative to the measurement for all the cases in Figs. 4 and 5. ICo/IRa, Relative deviation to measurement (%) h = 60°
PMMA, 30 keV Al, 30 keV Cu, 30 keV Al, 20 keV
h = 150°
Model
MC simulation
Model
MC simulation
14 13 14 17
3 12 22 2
10 21 6 9
7 8 14 20
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include the challenging multiple X-ray scattering. This effect is present in the lower-energy part of the Compton peak and clearly visible in logarithmic representation, for PMMA, i.e. for matrices of low atomic number, and for the 60° of the scattering angle, i.e. for forward scattering (Fig. 4a). An underestimation of the intensities in the background at energies over a broad range below the energy of the Compton peak can be observed. This confirms also the findings in our recent systematic study on polychromatic, unpolarised excitation. The best agreement can be reported for copper, i.e. high Z matrices, and 150° scattering angle, i.e. backward scattering. The same general tendencies described above for 30 keV incident beam are confirmed by the 20 keV excitation. The somewhat higher spectral background in the measured scattering spectra at energies below the energy of the Compton peak could be attributed to effects additional to those included in the model and Monte-Carlosimulations. The ‘‘incapacity’’ to describe accurately the volume scattering effects that are yet registered by the X-ray detector, especially for light matrices, is one assumption. The penetration depth of photons at several tens of keV is significant in relation to the distance to the detector, so that scattered radiation at scattering angles other than the assumed one might deliver further contributions to the observed spectra. Multiple scattering in the ambient air or in the aluminium filter has neither been considered in the analytical model, and thus may contribute further to alterations of the X-ray scatter intensities. Another source of uncertainties is the spectral background produced by the photoelectrons, this being able to be especially pronounced in light matrices. Neither the analytical model nor the Monte-Carlo simulations do consider this challenging effect. 5. Conclusions The results of the present systematic study on X-ray scattering with monochromatic, polarised radiation demonstrate that the physical model implemented by our group is capable of describing the measured scattering spectra in a generally good agreement. This has been demonstrated with representative scattering geometries, scattering bodies (PMMA, pure aluminium and pure copper), and monochromatic energies of the incident X-ray beam of 10, 20, 30 and 40 keV. Significant discrepancies have been established in the energy range below the energy of the Compton peak and especially for the light matrices. It is assumed that this residual background occurs due to scattering effects in the pronounced penetration volume of low-Z specimens, so that the acceptance angle of the detector must be more carefully considered. Various scattering angles should be implemented into the model. Multiple scattering has been observed in the measured scattering spectra for the light matrices and under condition of forward scattering. In spite of the fact that Monte-Carlo simulations have multiple scattering included, the agreement to the measurements remains unsatisfactory with respect to this issue. The slight deviations of the actual degree of polarisation from the ideal linear polarisation constitute a source of uncertainties for the Rayleigh to Compton intensity ratio. It is believed by the authors that the present study is useful for other investigators working with X-ray fluorescence spectra, e.g.
for the evaluation of the spectral background caused by X-ray scattering. Such a systematic comparison of modelled X-ray scattering spectra, modelled and simulated scattering spectra has not been yet reported in the literature so far. Acknowledgement Financial support from ‘‘BAM Innovationsoffensive’’ research program is gratefully acknowledged. References [1] V.-D. Hodoroaba, M. Radtke, L. Vincze, V. Rackwitz, D. Reuter, X-ray scattering in X-ray fluorescence spectra with X-ray tube excitation – modelling experiment, and Monte-Carlo simulation, Nucl. Instr. Meth. Phys. Res. B 268 (2010) 3568–3575. [2] A. Bjeoumikhov, V. Arkadiev, F. Eggert, V.-D. Hodoroaba, N. Langhoff, M. Procop, J. Rabe, R. Wedell, A new microfocus X-ray source, IMOXS, for highly sensitive XRF analysis in scanning electron microscopes, X-ray Spectrom. 34 (2005) 493–497. [3] M. Procop, V.D. Hodoroaba, X-ray fluorescence as an additional analytical method for a scanning electron microscope, Microchim. Acta 161 (2008) 413– 419. [4] V.-D. Hodoroaba, M. Procop, Determination of the real transmission of an X-ray lens for micro-focus XRF at the SEM by coupling measurement with calculation of scatter spectra, X-ray Spectrom. 38 (2009) 216–221. [5] M. Procop, V.-D. Hodoroaba, A. Bjeoumikhov, R. Wedell, A. Warrikhoff, Improvements of the low-energy performance of a micro-focus X-ray source for XRF analysis with the SEM, X-ray Spectrom. 38 (2009) 308–311. [6] H. Riesemeier, K. Ecker, W. Görner, B.R. Müller, M. Radtke, M. Krumrey, Layout and first XRF applications of the BAMline at BESSY II, X-ray Spectrom. 34 (2005) 160–163. [7] M. Van Gysel, P. Lemberge, P. Van Espen, Description of Compton peaks in energy-dispersive X-ray fluorescence spectra, X-ray Spectrom. 32 (2003) 139– 147. [8] F. Scholze, B. Beckhoff, M. Kolbe, M. Krumrey, M. Müller, G. Ulm, Detector calibration and measurement of fundamental parameters for X-ray spectrometry, Microchim. Acta 155 (2006) 275–278. [9] M. Alvisi, M. Blome, M. Griepentrog, V.D. Hodoroaba, P. Karduck, M. Mostert, M. Nacucchi, M. Procop, M. Rohde, F. Scholze, P. Statham, R. Terborg, J.F. Thiot, The determination of the efficiency of energy dispersive X-ray spectrometers by a new reference material, Microsc. Microanal. 12 (2006) 406–415. [10] V. Rackwitz, A. Warrikhoff, U. Panne, V.-D. Hodoroaba, J. Anal. At. Spectrom. 8 (2009) 1034–1036. [11] F. Scholze, M. Procop, Modelling the response function of energy dispersive Xray spectrometers with silicon detectors, X-ray Spectrom. 38 (4) (2009) 312– 321. [12] J.H. Hubbell, Wm.J. Weigele, E.A. Briggs, R.T. Brown, D.T. Cromer, R.J. Howerron, Atomic form factor, incoherent scattering functions, and photon scattering cross sections, J. Phys. Chem. Ref. Data 4 (3) (1975) 471–538. [13] D. Brusa, G. Stutz, J.A. Riveros, J.M. Fernández-Varea, F. Salvat, Fast sampling algorithm for the simulation of photon Compton scattering, Nucl. Instr. Meth. Phys. Res. A 379 (1996) 167–175. [14] G.A. Carlsson, C.A. Carlsson, Calculations of scattering cross sections for increased accuracy in diagnostic radiology. I. Energy broadening of Comptonscattered photons, Med. Phys. 9 (6) (1982) 868–879. [15] F. Biggs, L.B. Mendelsohn, J.B. Mann, Hartree-Fock Compton profiles for the elements, Atom Data Nucl. Data 16 (1975) 201–309. [16] W.T. Elam, B.D. Ravel, J.R. Sieber, A new atomic database for X-ray spectroscopic calculations, Radiat. Phys. Chem. 63 (2002) 121–128. [17] A.L. Hanson, The calculation of scattering cross sections for polarized X-rays, Nucl. Instr. Meth. Phys. Res. A 243 (1986) 583–598. [18] A.L. Hanson, M. Meron, Errors associated with the measurement of scattered polarized X-rays, Nucl. Instr. Meth. Phys. Res. A 264 (1988) 488–496. [19] L. Vincze, K. Janssens, F. Adams, M.L. Rivers, K.W. Jones, A general Monte Carlo simulation of ED-XRF spectrometers. II: polarized monochromatic radiation, homogeneous samples, Spectrochim. Acta 50B (2) (1995) 127–147. [20] J.E. Fernandez, V. Scot, Unbiased Monte Carlo simulation of the Compton profile, Nucl. Instr. Meth. Phys. Res. B 263 (2007) 209–213.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
X-ray scattering in X-ray fluorescence spectra with X-ray monochromatic, polarised excitation – Modelling, experiment, and Monte-Carlo simulation V.-D. Hodoroaba a,⇑, M. Radtke b, U. Reinholz b, H. Riesemeier b, L. Vincze c, D. Reuter a a
BAM Federal Institute for Materials Research and Testing, Division 6.4 Surface Technologies, D-12200 Berlin, Germany BAM Federal Institute for Materials Research and Testing, Division 1.3 Structure Analysis; Polymer Analysis, D-12200 Berlin, Germany c Ghent University, Department of Analytical Chemistry, B-9000 Ghent, Belgium b
a r t i c l e
i n f o
Article history: Received 25 January 2011 Received in revised form 1 April 2011 Available online 16 April 2011 Keywords: X-ray scattering Monochromatic excitation Polarisation Modelling Monte-Carlo simulation
a b s t r a c t A systematic series of measurements has been carried out with monochromatic X-ray excitation with synchrotron radiation in order to check a physical model on X-ray scattering. The model has recently been successfully tested for the case of polychromatic, unpolarised excitation emitted by an X-ray tube. Our main purpose is the modelling of a physical background in X-ray fluorescence spectra, so that improved quantitative results can be achieved especially for strongly scattering specimens. The model includes single Rayleigh and Compton scattering in the specimen, the effect of bound electrons, the challenging Compton broadening and the polarisation degree. Representative specimens, measurement geometries and excitation energies have been selected with synchrotron monochromatic light at BAMline/BESSY II. Monte-Carlo simulations have been also carried out in order to evaluate the quality of the results achieved with the model. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction X-ray fluorescence (XRF) with excitation from an X-ray tube is a well established analytical method for determining element concentrations in solid samples for main, secondary and trace elements. Especially for light matrices such as glasses, polymers and ceramics the contribution of X-ray scattering to the XRF spectra in the form of a pronounced spectral background, scattered X-ray lines and sometimes diffractions peaks are significant. Furthermore, if the specimens are rich in elements, the background construction necessary for the background subtraction and calculation of the net peak areas of the fluorescence lines in the XRF spectra, respectively, become critical. The pure mathematical construction of the spectral background based only on peak-free regions of interest is state of the art. Altered limits of detection and high uncertainties in the determination of elements at trace level concentration are the consequence. A physical model for a quite accurate calculation of X-ray scattering spectra based on well-known X-ray tube excitation spectra and differential scattering cross sections reported in the literature has been recently successfully implemented by our group [1]. The results have also been related to metrological measurements performed with modern calibrated instrumentation, i.e. with a side-
⇑ Corresponding author. Tel.: +49 030 81043144; fax: +49 030 81041827. E-mail address: [email protected] (V.-D. Hodoroaba). 0168-583X/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2011.04.009
window X-ray tube attached to a SEM/EDS (scanning electron microscope with an energy dispersive spectrometer) system [2–5]. Initially, it was expected by the authors that the modelling of the X-ray scattering spectra in the case of polychromatic, unpolarised excitation with an X-ray tube can be better understood by comparing this case with measurements with monochromatic radiation. However, in spite of the excellent monochromaticity attained at the BAMline at the BESSY II synchrotron storage ring [6], it has turned out that the accurate description of the degree of polarisation assigned to the individual experiments is a challenging task. Thus, the degree of polarisation constitutes a new source of uncertainties which competes with those accompanying e.g. the differential scattering cross-sections or the considered (solid) angles and sampling volumes in the specimen. Nevertheless, similar systematic investigations with monochromatic, polarised radiation involving (i) measurements, (ii) modelling and (iii) Monte-Carlo simulations, and relating them to each other have not been reported upon in the literature so far. The present study addresses this purpose, so that weak points in the physical model as well as in the measurements can be identified and their uncertainty budget assessed at least in a rough form. It must be also noted that the contribution of X-ray scattering to XRF spectra with monochromatic excitation is considerably lower than that when using polychromatic excitation. The Bremsstrahlung background emitted by an X-ray tube over the whole energy range up to the excitation energy of typically 40 keV is scattered elastically as well as inelastically over the whole energy range. This
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continuous contribution is missing when the excitation is monochromatic, so that only the elastic and inelastic peak corresponding to the monochromatic excitation energy constitute the scattering ‘‘background’’ in the XRF spectra. Due to this reason the determination of the net fluorescence peak areas in the case of polychromatic excitation remains much more critical than for the monochromatic one. The apparently significant contribution of multiple X-ray scattering [7], occurring at energies lower than the excitation energies and especially for light matrices, can be simulated only with Monte-Carlo software packages. In the present study X-ray scatter spectra of representative sample matrices at various monochromatic excitations and at selected geometrical conditions have been measured and modelled. Additionally, X-ray spectra obtained by performing Monte-Carlo simulations for the same conditions have also been taken into consideration.
2. Experimental In order to check the accuracy of the X-ray scattering model implemented as described in our recent work [1], systematic measurements with monochromatic, polarised excitation have been carried out at the BAMline/BESSY II synchrotron facility, adapted for monochromatic, polarised excitation. The specimens selected have been PMMA, pure aluminium and pure copper. The specimen has been fixed on the specimen stage so that the incident X-ray beam hits the specimen surface under an incidence angle of 27°, see Fig. 1a. The incident monochromatic X-ray beam has been tuned with the double crystal monochromator (DCM) for energies at 10, 20, 30 and 40 keV, with corresponding FWHM values in the range of 1.5–6.0 eV. These values are smaller than the 10 eV energy channel width of the acquisition system. An impression of how the X-ray scattering spectra emitted towards detector at these excitations look like is given in Fig. 2, as calculated with our model in Ref.
[1] for aluminium as a scatterer and 150° scattering angle. The Xray beam was nearly linearly polarised with the polarisation vector in the horizontal (synchrotron storage ring) plane. The X-ray spectrometer used for the series of measurements reported in this paper was an EDS system NSS with a Si(Li) detector (ThermoFisher Scientific). The EDS was calibrated up to 60 keV by using synchrotron radiation from the electron storage ring BESSY II [8]. Hence, both detector efficiency and spectrometer response functions are well-known [9–11]. The EDS detector has been placed at different selected positions so that the direction of the polarisation vector is always parallel to the scattering plane, which is equivalent to the horizontal plane. The resulting scattering angles have been 30°, 60°, 120° and 150°, Fig. 1b. The geometry corresponding to a scattering angle of 90° has also been included in this series and exploited in order to check the value of the degree of polarisation, being known that there are slight deviations from the assumed ‘‘pure’’ linear polarisation. According to the theory (see Section 3 below), if the linear polarisation of the incident X-ray beam is in the scattering plane, i.e. ‘‘pure’’ parallel polarisation, then at a scattering angle of 90° the Rayleigh scattering equals zero, while the amount of Compton scattering is reduced to a basic residual. All the measurements have been performed at ambient air pressure, having a distance between the exit of the monochromatic X-ray beam from the beamline to the specimen surface of 705 mm and a distance of 120 mm from the specimen surface up to the detector crystal. In order to work with reasonable photon count rates of the Si(Li) spectrometer, the deadtime has been kept below 30%. For copper specimens and for aluminium specimens an aluminium filter of 0.5 mm thickness has been placed in front of the detector, so that the intensive fluorescence lines originating in the specimen themselves were considerably diminished. 3. Theory The elastic (Rayleigh) and inelastic (Compton) X-ray scattering with monochromatic, polarised excitation are governed by the same basic physical equations as described in detail in our recent publication for the case of polychromatic, unpolarised excitation [1]: (i) Absorption of the incident radiation in the specimen up to the depth where the scattering event takes place. (ii) Scattering event according to differential scattering cross sections (from literature [12]) for consideration of the scattering anisotropy; the influence of the bound electrons in the specimen atoms is determined by the atomic form factor and the incoherent scattering function [12]. The energy loss of
Fig. 1a. Photograph of the experimental setup at the BAMline/BESSY II, highlighting the scattering geometry. The geometry has been selected in such a way that the scattering plane (the horizontal one) contains the direction of the linearly polarised incident beam. Only one value of 27° has been selected for the incidence angle w1 and the detector has been moved in the scattering plane, yielding representative scattering angles h for forward and backward scattering, i.e. h = 30°, 60°, 120° and 150°. A geometry corresponding to h = 90° has been also taken into account for the exact determination of the degree of polarisation.
Fig. 1b. Schematic top view of the experimental setup detailing the incident angle w1 (27°) and the scattering angle h (30°, 60°, 120° and 150°).
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polarisation, i.e. Ik = 0 and I = I\, P becomes equal to 1. The intensity of the scatter peak, SP is then a sum of the two corresponding components:
SP ¼
1þP 1P Sk þ S? ; 2 2
ð3Þ
with Sk and S\ the intensities of the scattered radiation resulting from ‘‘full’’ parallel (P = 1) and perpendicular scattering (P = 1). It can be concluded that both definitions lead to correct results, however, the degree of polarisation must be specified and the corresponding formula of P employed. The same results in scattering intensities will be derived from different degrees of polarisation, PHanson as defined in [17] and PVincze as defined in [19], respectively. The correlation between the two P0 s is:
PHanson ¼ Fig. 2. Plot of the elastically and inelastically scattered photons at various monochromatic excitation energies E0 = 10, 20, 30 and 40 keV, for aluminium and at a scattering angle h = 150°.
1 þ P Vincze : 2
ð4Þ
In this paper the definition of P according to [17] has been adopted.
Compton scattered radiation is considered in the model as well as the challenging Compton broadening [13,14]. The latter one is caused by the so-called pre-collision motion of the atomic electrons. Compilated so-called Compton profiles data have been implemented from Biggs et al. [15]. (iii) Absorption of the scattered radiation from the point of the scattering event in the specimen up to the specimen surface on the way toward EDS detector. Absorption in air or in the filter in front of the detector is also included. Mass attenuation coefficients from [16] have been used. The modelling of monochromatic radiation is in principle simpler than for the polychromatic case, because the complete Rayleigh and Compton scattering spectra result from the excitation in only one energy channel of 10 eV. As already mentioned, the energetic FWHM of the incident beam at the various excitation energies used in this study has been below 10 eV. For the case of polychromatic excitation with an X-ray tube at 40 kV, there are 4.000 ‘‘excitation’’ energy channels of 10 eV which must be considered for the calculation of the X-ray scattering spectra. However, the additional parameter which brings some potential difficulties into the description of X-ray scattering of polarised radiation is the degree of polarisation, P, respectively its exact knowledge. In the literature there are different definitions of P. In [17,18] P is defined as the fraction of the component of the polarisation vector onto the direction parallel to the scattering plane, so that the resulting intensity of the scattered radiation, SP, is given by the following sum:
SP ¼ PSk þ ð1 PÞS? ;
Fig. 3a. Dependence of the intensity of the Rayleigh and Compton scatter radiation on scattering angle when the incident X-ray beam is linearly polarised (parallel and orthogonal).
ð1Þ
with Sk and S\ the intensities of the scattered radiation corresponding to ‘‘fully’’ parallel and ‘‘fully’’ orthogonal linear polarisation. Hence, in the case of an ideal linear polarisation in the scattering plane, i.e. I = Ik and I\ = 0, the result is P = 1. If the linear polarisation vector is orthogonal to the scattering plane, i.e. if Ik = 0 and I = I\ then the resulting P = 0. In [19] P is defined according to the formula:
P¼
I k I? ; I k þ I?
ð2Þ
with Ik and I\ the parallel and orthogonal components of the excitation intensity I. One notes that according to this definition for I = Ik and I\ = 0, P = 1; however, for the case of orthogonal linear
Fig. 3b. Intensity ratio of Rayleigh to Compton scattering in dependence on the degree of polarisation, calculated for aluminium, at 30 keV energy of the incident radiation and for three representative scattering angles 60°, 90° and 150°.
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For linearly polarised X-ray beams P = 1. Fig. 3a shows how sensitive are the changes in scattering angle on scattering intensities, if linearly polarised radiation is employed. Depending on the orientation of the polarisation vector, both Rayleigh and Compton scattering display considerable variations. The extreme case is the 90° scattering angle and parallel linear polarisation (P = 1), when the Rayleigh scatter intensity becomes zero and the Compton scattering nearly zero. Due to the finite expansion of the incident beam in
Fig. 4
the range of mm2, but also due to slight misadjustments when tuning each individual monochromatic excitation energy, slight deviations from P = 1 are practically unavoidable. As illustrated in Fig. 3b for a scattering angle range around 90° these cause severe consequences on both scattering intensities, elastically and inelastically. In order to calculate X-ray scattering spectra as measured by the EDS spectrometer, the calculated scattering spectra ‘‘emitted’’ by
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Fig. 5
the specimen toward the detector entrance have been multiplied with the known spectrometer efficiency and finally convoluted with the spectrometer response functions. The spectrometer employed is fully characterised and calibrated [8–11]. The Monte-Carlo simulations have been obtained using the computer code MSIM [19]. The incident monochromatic X-ray beam was the same for both the model and for Monte-Carlo simulations. The Monte-Carlo simulations yield the scatter spectra as emitted by the scattering body in the same format as the model calculations, and they are subsequently treated in the same manner as these, i.e. subjected to the detector efficiency function and response matrix convolution. Other Monte-Carlo software packages, e.g. [20], could also be taken into consideration.
4. Results and discussion All the mechanisms contributing to scattering as presented in Section 3 have been implemented in an analytical model, which has been shown to work successfully for polychromatic, unpolarised excitation. Additionally to this, for polarised excitation, the degree of polarisation has been now included into the model. For each individual excitation, i.e. 10, 20, 30 and 40 keV, individual adjustments of the incident monochromatic X-ray beam had to be carried out. Each time the degree of polarisation has been derived from ‘‘residual’’ X-ray scattering spectra at the geometry corresponding to 90° scattering angle. In order to get a reasonable overview on the results of the application of the analytical model for various conditions, a set of modelled scattering spectra at 30 keV excitation for all the three specimens considered in this study, i.e. PMMA, Al and Cu, at 60° and 150° scattering angle are presented in Fig. 4. At 30 keV energy of the incident monochromatic X-ray beam the corresponding shift for various geometries is clearly visible and the shape of the two scattering peaks, Rayleigh and Compton, can be checked rather well. The corresponding degree of polarisation derived from the 90° scattering spectrum was 0.993. The calculated X-ray scattering spectra shown in Fig. 4 are compared to measured spectra. Additionally, in order to have a complete picture, X-ray scattering spectra for identical experimental conditions have been also simulated by Monte-Carlo methods and displayed for comparison in Fig. 4. In order to have an idea how the results of the model behave at other energies of the incident X-ray beam, e.g. at 20 keV, scattering spectra for aluminium, as a scatterer of middle ‘‘strength’’, are shown in Fig. 5. The degree of polarisation for the 20 keV incident
X-ray beam, derived from the 90° scattering geometry, was 0.988. Also for a better overview the relative deviations of the Comptonto-Rayleigh intensity ratios of both the model and Monte-Carlo simulations relative to the measurement are shown in Table 1 for all the cases in Figs. 4 and 5. The strategy of calculation of the X-ray scattering spectra with the analytical model as well as by the Monte-Carlo simulations has consisted firstly of calculating the two scattering intensities Sk and S\ for Eq. (1), for both Rayleigh and Compton scattering, corresponding to ‘‘fully’’ linear parallel and orthogonal polarisation. Then, having the experimentally deduced degree of polarisation, Eq. (1) has been employed in order to calculate the resulting Rayleigh and Compton scattering intensities. From the X-ray scattering spectra in Figs. 4 and 5 one can generally state that both Rayleigh and Compton scattering – as measured – are rather well described by our analytical model. This confirms our expectations stemming from our already verified experience with polychromatic excitation. A sensibly better agreement relative to the measured scattering spectra is attained by the Monte-Carlo simulations. Note that similar to the spectra gained with the analytical model the spectra obtained by Monte-Carlo simulations have been scaled to the intensity of the Rayleigh peak in the measured X-ray scatter spectra. Hence, one main goal of this work, namely to investigate systematically the spectral distributions of X-ray scatter spectra with monochromatic excitation, could be achieved. The reliability of the model with respect to the absolute values of the X-ray scatter intensities have been already successfully demonstrated in our previous paper for polychromatic excitation [1]. The somewhat better overall agreement relative to the measurement is again not surprising due to the fact that, unlike the analytical model, the Monte-Carlo simulations
Table 1 Overview of the relative deviations of the Compton-to-Rayleigh intensity ratios of both the model and Monte-Carlo simulations relative to the measurement for all the cases in Figs. 4 and 5. ICo/IRa, Relative deviation to measurement (%) h = 60°
PMMA, 30 keV Al, 30 keV Cu, 30 keV Al, 20 keV
h = 150°
Model
MC simulation
Model
MC simulation
14 13 14 17
3 12 22 2
10 21 6 9
7 8 14 20
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include the challenging multiple X-ray scattering. This effect is present in the lower-energy part of the Compton peak and clearly visible in logarithmic representation, for PMMA, i.e. for matrices of low atomic number, and for the 60° of the scattering angle, i.e. for forward scattering (Fig. 4a). An underestimation of the intensities in the background at energies over a broad range below the energy of the Compton peak can be observed. This confirms also the findings in our recent systematic study on polychromatic, unpolarised excitation. The best agreement can be reported for copper, i.e. high Z matrices, and 150° scattering angle, i.e. backward scattering. The same general tendencies described above for 30 keV incident beam are confirmed by the 20 keV excitation. The somewhat higher spectral background in the measured scattering spectra at energies below the energy of the Compton peak could be attributed to effects additional to those included in the model and Monte-Carlosimulations. The ‘‘incapacity’’ to describe accurately the volume scattering effects that are yet registered by the X-ray detector, especially for light matrices, is one assumption. The penetration depth of photons at several tens of keV is significant in relation to the distance to the detector, so that scattered radiation at scattering angles other than the assumed one might deliver further contributions to the observed spectra. Multiple scattering in the ambient air or in the aluminium filter has neither been considered in the analytical model, and thus may contribute further to alterations of the X-ray scatter intensities. Another source of uncertainties is the spectral background produced by the photoelectrons, this being able to be especially pronounced in light matrices. Neither the analytical model nor the Monte-Carlo simulations do consider this challenging effect. 5. Conclusions The results of the present systematic study on X-ray scattering with monochromatic, polarised radiation demonstrate that the physical model implemented by our group is capable of describing the measured scattering spectra in a generally good agreement. This has been demonstrated with representative scattering geometries, scattering bodies (PMMA, pure aluminium and pure copper), and monochromatic energies of the incident X-ray beam of 10, 20, 30 and 40 keV. Significant discrepancies have been established in the energy range below the energy of the Compton peak and especially for the light matrices. It is assumed that this residual background occurs due to scattering effects in the pronounced penetration volume of low-Z specimens, so that the acceptance angle of the detector must be more carefully considered. Various scattering angles should be implemented into the model. Multiple scattering has been observed in the measured scattering spectra for the light matrices and under condition of forward scattering. In spite of the fact that Monte-Carlo simulations have multiple scattering included, the agreement to the measurements remains unsatisfactory with respect to this issue. The slight deviations of the actual degree of polarisation from the ideal linear polarisation constitute a source of uncertainties for the Rayleigh to Compton intensity ratio. It is believed by the authors that the present study is useful for other investigators working with X-ray fluorescence spectra, e.g.
for the evaluation of the spectral background caused by X-ray scattering. Such a systematic comparison of modelled X-ray scattering spectra, modelled and simulated scattering spectra has not been yet reported in the literature so far. Acknowledgement Financial support from ‘‘BAM Innovationsoffensive’’ research program is gratefully acknowledged. References [1] V.-D. Hodoroaba, M. Radtke, L. Vincze, V. Rackwitz, D. Reuter, X-ray scattering in X-ray fluorescence spectra with X-ray tube excitation – modelling experiment, and Monte-Carlo simulation, Nucl. Instr. Meth. Phys. Res. B 268 (2010) 3568–3575. [2] A. Bjeoumikhov, V. Arkadiev, F. Eggert, V.-D. Hodoroaba, N. Langhoff, M. Procop, J. Rabe, R. Wedell, A new microfocus X-ray source, IMOXS, for highly sensitive XRF analysis in scanning electron microscopes, X-ray Spectrom. 34 (2005) 493–497. [3] M. Procop, V.D. Hodoroaba, X-ray fluorescence as an additional analytical method for a scanning electron microscope, Microchim. Acta 161 (2008) 413– 419. [4] V.-D. Hodoroaba, M. Procop, Determination of the real transmission of an X-ray lens for micro-focus XRF at the SEM by coupling measurement with calculation of scatter spectra, X-ray Spectrom. 38 (2009) 216–221. [5] M. Procop, V.-D. Hodoroaba, A. Bjeoumikhov, R. Wedell, A. Warrikhoff, Improvements of the low-energy performance of a micro-focus X-ray source for XRF analysis with the SEM, X-ray Spectrom. 38 (2009) 308–311. [6] H. Riesemeier, K. Ecker, W. Görner, B.R. Müller, M. Radtke, M. Krumrey, Layout and first XRF applications of the BAMline at BESSY II, X-ray Spectrom. 34 (2005) 160–163. [7] M. Van Gysel, P. Lemberge, P. Van Espen, Description of Compton peaks in energy-dispersive X-ray fluorescence spectra, X-ray Spectrom. 32 (2003) 139– 147. [8] F. Scholze, B. Beckhoff, M. Kolbe, M. Krumrey, M. Müller, G. Ulm, Detector calibration and measurement of fundamental parameters for X-ray spectrometry, Microchim. Acta 155 (2006) 275–278. [9] M. Alvisi, M. Blome, M. Griepentrog, V.D. Hodoroaba, P. Karduck, M. Mostert, M. Nacucchi, M. Procop, M. Rohde, F. Scholze, P. Statham, R. Terborg, J.F. Thiot, The determination of the efficiency of energy dispersive X-ray spectrometers by a new reference material, Microsc. Microanal. 12 (2006) 406–415. [10] V. Rackwitz, A. Warrikhoff, U. Panne, V.-D. Hodoroaba, J. Anal. At. Spectrom. 8 (2009) 1034–1036. [11] F. Scholze, M. Procop, Modelling the response function of energy dispersive Xray spectrometers with silicon detectors, X-ray Spectrom. 38 (4) (2009) 312– 321. [12] J.H. Hubbell, Wm.J. Weigele, E.A. Briggs, R.T. Brown, D.T. Cromer, R.J. Howerron, Atomic form factor, incoherent scattering functions, and photon scattering cross sections, J. Phys. Chem. Ref. Data 4 (3) (1975) 471–538. [13] D. Brusa, G. Stutz, J.A. Riveros, J.M. Fernández-Varea, F. Salvat, Fast sampling algorithm for the simulation of photon Compton scattering, Nucl. Instr. Meth. Phys. Res. A 379 (1996) 167–175. [14] G.A. Carlsson, C.A. Carlsson, Calculations of scattering cross sections for increased accuracy in diagnostic radiology. I. Energy broadening of Comptonscattered photons, Med. Phys. 9 (6) (1982) 868–879. [15] F. Biggs, L.B. Mendelsohn, J.B. Mann, Hartree-Fock Compton profiles for the elements, Atom Data Nucl. Data 16 (1975) 201–309. [16] W.T. Elam, B.D. Ravel, J.R. Sieber, A new atomic database for X-ray spectroscopic calculations, Radiat. Phys. Chem. 63 (2002) 121–128. [17] A.L. Hanson, The calculation of scattering cross sections for polarized X-rays, Nucl. Instr. Meth. Phys. Res. A 243 (1986) 583–598. [18] A.L. Hanson, M. Meron, Errors associated with the measurement of scattered polarized X-rays, Nucl. Instr. Meth. Phys. Res. A 264 (1988) 488–496. [19] L. Vincze, K. Janssens, F. Adams, M.L. Rivers, K.W. Jones, A general Monte Carlo simulation of ED-XRF spectrometers. II: polarized monochromatic radiation, homogeneous samples, Spectrochim. Acta 50B (2) (1995) 127–147. [20] J.E. Fernandez, V. Scot, Unbiased Monte Carlo simulation of the Compton profile, Nucl. Instr. Meth. Phys. Res. B 263 (2007) 209–213.