System and method for approximating x-ray intensity for sample measured by x-ray detection system

Abstract

A solution for approximately measuring one or more X-ray intensities of a sample by an X-ray detection system that includes at least one collimator having a given angular range, a monochromator, and an X-ray detector. Measurement intensities are received from the detection system at respective one or more diffraction angles. Simulated sample intensities are calculated from an X-ray fluorescence sample model having initial sample model parameters indicative of a sample composition and / or a layer thickness of one or more sample layers. Triangular collimator corrections are applied to the simulated sample intensities, and mathematical distances between the corrected simulated sample intensities and corresponding measured intensities are determined. The sample model parameters are adjusted and the correction step is repeated until the distance change falls below the minimum distance change. Sample model parameters associated with the corrected simulated intensities are provided as approximate concentration values of the respective components contained in the measured sample and / or a layer thickness of the measured sample.

Description

Technical Field

[0001] The present invention relates generally to wavelength dispersive X-ray spectroscopy and, more particularly, to approximating the X-ray intensity of a sample measured by an X-ray detection system. Background Art

[0002] X-ray spectrometers are used to quantify the composition and / or structure of a material (sample) by relating intensity to the mass fraction of the sample's elements and / or layer thickness. This is done by modeling the interaction of the radiation with the sample and calculating the expected intensity. Starting with an arbitrary (but similar) sample composition and / or structure, the sample parameters are varied until the calculated intensity best matches the measured intensity.

[0003] The generation of X-rays in a sample can be induced by photons from an X-ray tube. These methods are called energy dispersive X-ray fluorescence (EDX) or wavelength dispersive X-ray fluorescence (WDX) spectroscopy. If the X-rays in a sample are induced by electrons from an electron emitter in a scanning electron microscope, a transmission electron microscope, or an electron probe microanalyzer (EPMA), the methods are called energy dispersive X-ray spectroscopy (EDS) and wavelength dispersive spectroscopy (WDS). The solution described here addresses wavelength dispersive detection systems and is independent of the source of the X-rays. The following description discusses methods using a WDX spectrometer.

[0004] In WDX spectroscopy, an X-ray tube generates polychromatic radiation, including characteristic lines and bremsstrahlung. This radiation is directed onto the sample to be analyzed. This radiation generates polychromatic X-ray fluorescence and scattered radiation in the sample. In wavelength-dispersive X-ray fluorescence spectrometers, the radiation θ emitted by the sample is analyzed by measuring the radiation reflected at various diffraction angles by the spectrometer's monochromator (usually a crystal) with a goniometer. The monochromator of a WDX system is typically a single crystal that achieves monochromatic radiation by diffraction. The technique is based on Bragg's law.

[0005] nλ=2dsinθ(F1)

[0006] where n is the diffraction order, λ is the wavelength, d is the distance between the lattice planes of the crystal, and θ is the angle of the incident and outgoing radiation relative to the crystal surface. The energy E is connected to the wavelength λ by

[0007]

[0008] where h is Planck's constant and c is the speed of light. This can be simplified to

[0009]

[0010] Due to diffraction order n, photons with multiples of energy E (n=1) can reach the detector.

[0011] Therefore, the detector must be able to discriminate between diffraction order photons. Most detectors are designed to define this discrimination window as a fraction of the first diffraction order energy. A lower discrimination limit of 50% combined with an upper discrimination limit of 150% means that photons with energies between half and one and a half times the first order energy recorded by the detector are summed to form the analytically relevant intensity for the spectrometer.

[0012] Historically, detectors were only able to sum all photons within a single discrimination window. Modern detectors use multi-channel analyzers (MCAs) to sort photons by energy across multiple channels. This technique makes it easier to identify additional effects that contribute to the background (intensity components that do not originate from the element being analyzed).

[0013] In summary, a WD spectrum typically displays the integral of the energy region of the monochromator's response as intensities exceeding 2θ (corresponding to first-order energies). Therefore, it also includes artifacts generated by the crystal or detector that contribute to the background. These artifacts can increase the background through fluorescence from the crystal and its holder, diffuse scattering of radiation from the sample, higher-order emission, or reduce the intensity of the analyte through pile-up and first-order emission. All of these artifacts affect the analytical performance of the detection system.

[0014] The physical processes behind the interaction of X-rays in the instrument and sample are highly complex. Multiple processes occur simultaneously within the X-ray tube, sample, and detector. The resulting spectrum contains both the actual analytical signal and a background contribution. This background contribution is a combination of scattered tube radiation and technically unavoidable contributions (artifacts) that may originate from the WDX detection system itself.

[0015] Empirical models for estimating / calculating analytically relevant intensities and backgrounds are known in the art. However, these models require calibration on the actual instrument using standard samples that closely correlate with the sample for quantification, and they lack the ability to predict the full shape of the background. Such models require the detection efficiency (the ratio between the predicted and measured intensity) to be determined individually for each crystal / collimator / detector combination.

[0016] On the other hand, fully simulated models (from first principles / ab initio) allow for a wider range of applicability of the model to a variety of sample parameters and equipment configurations. They allow prediction of analytically relevant intensities and backgrounds for virtually any sample material and shape. The most detailed calculations are based on Monte-Carlo methods, which track single photons or groups of photons through a detection system. However, these methods are very slow and complex, and are therefore not convenient for use in industrial settings.

[0017] Therefore, the WDX detection system comprising a collimator, a crystal monochromator and an actual detector is still approximated by empirical or semi-empirical methods. The complete simulation of the WDX spectrum stops just before the empirically modeled detection system. Therefore, a lot of effort is required to measure and extract the peak information (all commonly available collimators, crystals and detector variations typically result in 3*5*2=30 different settings, which must be measured and analyzed for multiple single element standards and their multiple emission lines). In addition, this must be done again for each new crystal, collimator and detector combination or any change to the equipment. In addition, the continuous background of the spectrum must be semi-empirically adjusted by the absorption of the sample and the efficiency of the previously determined WD detection system, also individually for each combination of collimator, crystal and detector. Summary of the Invention

[0018] Therefore, there is a need for systems and methods that can calculate the interactions of X-rays with the sample and the WDX detection system as a collective process, thereby achieving an optimal compromise between computational speed and accuracy. The method disclosed herein provides a fully physics-based and geometry-based model for all interactions occurring in a WDX detection system, thereby allowing for the complete simulation of WDX spectra.

[0019] Specifically, the method enables correction of the simulated spectrum by taking into account various effects of the detection system on the measured intensity, so that the corrected simulated intensity provides a reliable approximation of the measured intensity. This allows indirect measurement of the sample composition and / or layer thickness of the measured sample via model parameters related to the sample composition and / or layer thickness, as long as the simulated intensity is within a predetermined distance from the measured intensity. In a first approximation, reliable results are already achieved when, according to the independent claim, a correction is applied to the simulated intensity by taking into account the effects of the detection system's collimator on the measured intensity.

[0020] Additionally, optional embodiments disclosed herein include further corrections that take into account the following background contributions caused by the detection system (see Portnoi et al., Filtering Amplitude Spectra to Reduce Background Levels in X-ray Fluorescence Analysis, J. Anal. Chem. 60, 838 (2005)):

[0021] - radiation scattered by the sample (especially Bremsstrahlung)

[0022] - Monochromator's second-order sample scattered tube radiation

[0023] - Diffuse (non-Bragg) scattering of radiation from the sample impinging on the monochromator and

[0024] - Fluorescence from the monochromator itself and

[0025] In the method disclosed herein, the detection system for a WDX device is modeled so that the parameters to be adjusted are smaller than those in prior art methods, and the effects of the parameters are correctly linked to the corresponding components of the detection system so that changing one component does not invalidate the previously determined parameters of other components. For example, with respect to correction for the divergence of a collimator, the corresponding system component parameter is the measurable divergence of the collimator, and with respect to correction for the length of a monochromator, the length parameter is the measurable length of the crystal.

[0026] In a basic embodiment, according to claim 1, a computer-implemented method for approximating one or more X-ray intensities for a sample measured by a WD detection system is provided. The computer-implemented method can be executed by a computer system that is suitable for executing functional modules implemented by software or hardware. The corresponding software is loaded into the memory of the computer system and executed by one or more processors of the computer system. The detection system includes at least one collimator with a given angular range (divergence), a crystal monochromator and an X-ray detector. The method starts with receiving one or more measured intensities from the detection system at corresponding one or more diffraction angles. Typically, a spectrum of the sample is measured, which covers an energy range associated with a series of varying diffraction angles between the X-ray radiation and the crystal monochromator. However, the method can also be applied to single measurements at a given diffraction angle.

[0027] The system calculates one or more simulated sample intensities for the corresponding one or more diffraction angles from an X-ray fluorescence sample model having initial sample model parameters indicating the sample composition and / or layer thickness of one or more sample layers. For example, the system may include a sample simulation module that stores the sample model and corresponding model parameters for simulation purposes. Sample models that can simulate X-ray fluorescence spectra of samples containing one or more substances are known in the art. For example, in B. Beckhoff, B. Kanngieβer, N. Langhoff, R. Wedell, and H. Wolff, Handbook of Practical X-ray Fluorescence Analysis, 2006 edition (Springer, Berlin; New York, 2006), an overview of different model types is provided starting on page 309, including empirical influence coefficients, theoretical influence coefficients, classical fundamental parameters, and Monte Carlo models. The simulated sample intensities correspond to the measured intensities because they are simulated for the same diffraction angles used for the measurement. That is, given a full spectrum measured for the sample, a corresponding simulated sample spectrum can be calculated.

[0028] The simulated sample intensity describes the X-ray fluorescence radiation that the sample would theoretically emit in response to the polychromatic X-ray radiation generated by the X-ray tube and directed at the sample. However, this is not the radiation ultimately measured by the detector, as various components of the detection system are responsible for various effects that modify the radiation emitted by the sample. Therefore, the simulated sample intensity is calibrated to account for the effects of the corresponding detection system components on the emitted radiation. Thus, the sample model parameters correspond to the properties of the sample being measured. That is, once the simulated sample intensity is calibrated and fitted to the measured intensity, the sample model parameters provide an estimate of the properties of the sample being measured.

[0029] According to the independent claim, at least a collimator correction is applied to the one or more simulated sample intensities at the corresponding one or more diffraction angles. Typically, the detection system has two collimators, one between the sample and the crystal, and another between the crystal and the detector. However, the detection system can also use only a single collimator. Advantageously, a single collimator can be installed between the sample and the crystal. The collimator correction function can be provided by a correction module of the system, which can optionally also provide further correction functions as described in the dependent claims.

[0030] For each of one or more diffraction angles, a collimator correction function determines an integral over the energy range permitted by the angular range / divergence of the at least one collimator. Note that, even for a specific diffraction angle, the detector integrates over the specific energy range from the starting energy to the ending energy at the specific diffraction angle. The relative intensity distribution within the angular range, caused by the angular tolerance of the crystal monochromator as defined by the divergence of the at least one collimator, is represented by an approximation function (advantageously, a piecewise linear function) that approximates a triangular line shape. In other words, the relative intensity is "0" at the outer limits of the angular range and has a maximum for diffraction angles associated with rays passing through the collimator parallel to the collimator walls. Connecting the maximum relative intensity with the "0" intensity at the outer limits results in a triangular line shape. An integral is then calculated over the product of the corresponding simulated sample intensity and the approximation function. The determined integral represents the corrected simulated sample intensity for the corresponding diffraction angle. The collimator correction function can then be applied to all diffraction angles for a portion or the entire simulated sample spectrum, thereby generating a corresponding corrected simulated sample spectrum.

[0031] The optimization module then determines a mathematical distance between the one or more corrected simulated sample intensities and the corresponding one or more measured intensities. Commonly known distance metrics include (weighted) Euclidean distance and Minkowski distance. Those skilled in the art may use any other distance metric suitable for measuring the similarity of the corrected simulated intensities and the measured intensities.

[0032] The system then adjusts the sample model parameters indicative of the sample composition and / or layer thickness in such a way that they reduce the distance (e.g., along the first derivative) and repeats the previous two steps of determining and adjusting until the change in distance drops below a minimum distance change (in other words, the first derivative approaches zero). The minimum distance change may be a predefined threshold or it may correspond to a predefined percentage.

[0033] Once the simulated sample intensity is sufficiently close to the measured intensity (i.e., the change in distance has fallen below a threshold), updated sample model parameters of the sample model indicating the sample composition and / or layer thickness are provided as approximate concentration values ​​of the corresponding components contained in the measured sample and / or the layer thickness of the measured sample. In other words, the sample model parameters associated with the latest corrected simulated intensity represent the results of an indirect measurement method for determining the sample composition and / or layer thickness of the sample from which the measured intensity was obtained. It should be noted that the sample may include multiple layers, in which case each layer thickness may be a parameter of the sample model.

[0034] In the following, further optional corrections for the simulated sample intensity are described according to optional embodiments of the dependent claims, which are associated with other artifacts caused by components of the detection system in the measured intensity. These further corrections can form part of the method in any combination and be performed by the system in addition to the collimator correction. In other words, the collimator correction can be combined with any further correction. Thus, with the exception of the background correction, all other further corrections provide multiplicative correction factors that are multiplied by the simulated sample intensity. In the case of the background correction, they are simply added to the results provided by the other corrections.

[0035] In one embodiment, the system further applies a polarization correction to the one or more simulated sample intensities such that for each of the one or more diffraction angles, the simulated sample intensity is calculated as the sum of the emission radiation emitted by the sample multiplied by a first polarization factor for single scattering at the monochromator and the scattered tube radiation scattered by the sample multiplied by a second polarization factor for double scattering at the sample and the monochromator.

[0036] In one embodiment, the system further applies a crystal length correction to the one or more simulated sample intensities such that for diffraction angles where the projection of the collimator height on the crystal monochromator plane is greater than the length of the crystal, the one or more simulated sample intensities are multiplied by the ratio of the length of the crystal to the length of the projection of the collimator height on the crystal monochromator plane.

[0037] In one embodiment, the detector further comprises a detector window between the crystal monochromator and the X-ray detector. In this embodiment, the method also accounts for the effect of the detector window, as the system further applies a window transmission correction to the one or more simulated sample intensities such that, for each corresponding diffraction angle, the simulated sample intensity is multiplied by a transmission factor defined by the material and thickness of the detector window. Thus, the transmission factor is defined as the ratio of the X-ray intensity between the detector window and the detector to the X-ray intensity between the crystal and the detector window.

[0038] In one embodiment, the system further applies a detector efficiency correction to the one or more simulated sample intensities such that, for each corresponding diffraction angle, the simulated sample intensity is multiplied by an absorption factor defined by the material and thickness of the detector. Thus, the absorption factor is defined as the ratio of the intensity absorbed by the active detector material to the intensity incident on the active detector material.

[0039] In one embodiment, the system further applies a crystal diffraction efficiency correction to the one or more simulated sample intensities such that for each corresponding diffraction angle, the simulated sample intensity is divided by the portion of the incident intensity absorbed by the crystal and adjusted by the crystal geometry factor.

[0040] In one embodiment, the system further applies a pile-up correction to the one or more simulated sample intensities such that for each corresponding diffraction angle, the simulated sample intensity is multiplied by an exponential function that describes the intensity loss produced by counting two photons arriving within a predefined time window as a single photon with double energy, which double energy is optionally cut off by an upper discrimination limit of the detection.

[0041] In one embodiment, the system further applies a correction to the one or more simulated sample intensities to account for intensity losses caused by escape of the photons from the active detector material and / or intensity gains caused by higher diffraction order artifacts of the monochromator, such that for each respective diffraction angle, the corresponding intensity loss and / or intensity gain is added to the corrected simulated sample intensity.

[0042] In all of the above embodiments where a correction has been applied to one or more simulated sample intensities, the corresponding corrected simulated sample intensities provide optimized quantification of sample composition based on the one or more measured intensities.

[0043] In one embodiment, a computer program product for approximating one or more X-ray intensities of a sample measured by an X-ray fluorescence detection system is provided. The computer program product includes computer-readable instructions that, when loaded into a memory of a computing device and executed by at least one processor of the computing device, cause the computing device to perform the computer-implemented method disclosed herein.

[0044] In one embodiment, a computer system is provided for approximating one or more X-ray intensities of a sample measured by an X-ray fluorescence detection system, the X-ray fluorescence detection system comprising at least one collimator having a given angular range, a crystal monochromator, and an X-ray detector. The computer system may be a computing device adapted to execute the computer program to perform the computer-implemented method disclosed herein.

[0045] Specifically, a basic embodiment of a system suitable for performing the basic embodiment of the computer-implemented method includes an interface adapted to receive one or more measured intensities from the detection system at corresponding one or more diffraction angles. Interfaces for exchanging data between an X-ray fluorescence detection system and a computer system are known in the art.

[0046] The system further has a sample simulation module adapted to calculate one or more simulated sample intensities for the corresponding one or more diffraction angles from an X-ray fluorescence sample model having initial sample model parameters regarding sample composition and / or layer thickness.

[0047] The correction module of the system is adapted to apply a collimator correction to the one or more simulated sample intensities at the corresponding one or more diffraction angles by:

[0048] - determining, for each of the one or more diffraction angles, an integral over the energy range allowed by the angular range / divergence of the at least one collimator, wherein the integral is calculated over the product of the multiplication of the corresponding simulated sample intensity and an approximation function (advantageously a piecewise linear function) that approximates the triangular shape of the relative intensity distribution in the angular range caused by the angular tolerance of the crystal monochromator defined by the divergence of the at least one collimator, the determined integral representing the corrected simulated sample intensity;

[0049] - determining a mathematical distance between the one or more corrected simulated sample intensities and the corresponding one or more measured intensities; and

[0050] - adjusting the sample model parameters indicative of the sample composition and / or layer thickness, and repeating the determining step and the adjusting step until the distance drops below a minimum distance.

[0051] The interface of the system is further adapted to provide the sample model parameters indicative of the sample composition and / or layer thickness associated with the corrected simulated intensity as approximate concentration values ​​of the respective components contained in the measured sample and / or the layer thickness of the measured sample.

[0052] Other aspects of the invention will be realized and attained by the elements and combinations particularly described in the appended claims.It should be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the invention described. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 A block diagram of an exemplary embodiment of a computer system for approximating one or more X-ray intensities for a sample measured by a wavelength dispersive X-ray detection system is included;

[0054] Figure 2 is a simplified flow chart of a computer-implemented method for approximating such X-ray intensity for the sample, according to an embodiment;

[0055] Figure 3 is an overview of applicable corrections according to the implementation plan;

[0056] Figure 4A An example of an integration width and a trigonometric correction function of an X-ray detector for considering the influence of an angular tolerance of a crystal monochromator on a simulated intensity according to an embodiment is illustrated;

[0057] Figure 4B shows an example of ray tracing results for different diffraction angles with corresponding energy distributions when a LiF200 crystal is used as a monochromator;

[0058] Figure 5A and Figure 5B Two examples demonstrating the effect of triangle correction when applied to simulated sample intensities are shown;

[0059] Figure 6 An example detection system is illustrated, which illustrates the details in the context of polarization correction;

[0060] Figure 7A 、 Figure 7B The evolution of polarization correction of LiF200 crystal in 2θ and in energy is illustrated;

[0061] Figure 8A shows an illumination projection of a monochromator crystal having a length that is less than the collimator height of a collimator for incident radiation;

[0062] Figure 8BIllustrate the effect of crystal length on strength of an exemplary LiF200 crystal;

[0063] Figure 8C Illustrate the effect of crystal length correction on triangle correction according to an embodiment;

[0064] Figure 9A is a heatmap of an example PHA spectrum obtained from a copper sample, which visualizes the additional contribution to the background;

[0065] Figure 9B The results of the integration of the PHA spectra in the heat map are shown;

[0066] Figure 10 illustrates an example source of background radiation caused by a crystal and its holder; and

[0067] Figure 11 is a diagram illustrating examples of a general-purpose computer device and a general-purpose mobile computer device that may be used with the techniques described herein. DETAILED DESCRIPTION

[0068] Figure 1 A block diagram of an exemplary embodiment of a computer system 100 for approximating one or more X-ray intensities for a sample 202 measured by an X-ray detection system 200-1 including at least one collimator 205, 206 having a given angular range / divergence, a crystal monochromator 203, and an X-ray detector 204 is included. The computer system is described in the context of a simplified flowchart of a computer-implemented method 1000 for approximating such X-ray intensities for the sample 202. Figure 1 The system 100, such as Figure 2 and Figure 3 Therefore, the following description refers to Figure 1 as well as Figure 2 and Figure 3 The system 100 is thus configured to perform the method 1000 when a corresponding computer program is loaded into a memory of the system and executed by a processing device of the system. The computer program product implements the functional modules 120 and 130 of the system 100.

[0069] exist Figure 1In an exemplary embodiment, the system 100 is communicatively coupled to a wavelength dispersive X-ray spectrometer 200. In an alternative embodiment, the computer system 100 may be an integral component of the spectrometer 200. The spectrometer 200 may be a wavelength dispersive X-ray fluorescence spectrometer. In both embodiments, the system 100 receives 1100 one or more measured intensities 212θ from the detection system 200-1 at corresponding one or more diffraction angles. The functionality of the spectrometer 200 has been largely described in the background section. Generally speaking, one skilled in the art knows how the spectrometer 200 can be used to obtain a full intensity spectrum from the sample 202 by performing a scan at a varying diffraction angle for each measurement point of the spectrum. The system can also be used to perform a single measurement for obtaining a single intensity value at a specific diffraction angle θ. However, the following reference will be made to Figure 1 Provide a brief overview of the feature.

[0070] An X-ray tube 201 generates polychromatic radiation 201-r, including characteristic lines and bremsstrahlung radiation, which is directed onto a sample 202 to be analyzed. This radiation generates polychromatic X-ray fluorescence and scattered radiation 202-r in the sample 202. The radiation 202-r emitted by the sample 202 is analyzed by measuring the radiation reflected by the spectrometer's monochromator 203 (CM) at various diffraction angles θ using a goniometer. Typically, a collimator 205 (C1) is placed between the sample 202 and the CM 203. This technique is based on Bragg's law. X-ray photons 203-r with multiples of the energy of the analytical signal reach the spectrometer's detector 204. Photons striking the detector 204 are converted into electrical pulses, where the pulse height is proportional to the photon's energy. The detector then sums only photons with energies between 50% and 150% of the expected energy at the diffraction angle to exclude photons of higher diffraction orders, thereby generating a measured intensity value 212 for each diffraction angle θ.

[0071] The system 100 has an interface (not shown) adapted to receive 1100 the one or more measured intensities 212 from the detection system 200-1. Such interfaces for the exchange of data between measurement devices (such as the spectrometer 200) and computer systems are well known in the art.

[0072] The system 100 has a sample simulation module 130 adapted to calculate 1200 one or more simulated sample intensities 112 for corresponding one or more diffraction angles θ from an X-ray fluorescence sample model 130M having initial sample model parameters 130P indicating a sample composition and / or layer thickness of one or more layers of the sample. Any sample model type disclosed in the aforementioned section of the Handbook of Practical X-ray Fluorescence Analysis may be used. A person skilled in the art may also be able to use other suitable sample model types not described in the Handbook.

[0073] The system 100 also has a correction module 120 adapted to apply 1300 at least one collimator correction function CC 121 to one or more simulated sample intensities 112 corresponding to one or more diffraction angles. In an optional embodiment, the correction module further includes optional correction functions 122 to 128 to apply corrections implied by other components of the detection system 200-1. These optional corrections are described in further detail below.

[0074] For each of the one or more diffraction angles, the collimator correction function CC 121 is determined 1320 as an integral over the energy range allowed by the angular range of the at least one collimator 205, 206. According to formula F4, for the corresponding simulated sample intensity i Sim (E) 112 and an approximation function (e.g., a piecewise linear function) T(E) that approximates a triangular line shape of the relative intensity distribution in an angular range caused by the angular tolerance of the crystal monochromator defined by the divergence of at least one collimator:

[0075]

[0076] Where E1 and E2 are the nominal positions between which the detector 204 is positioned The starting and ending energies at which the integration is performed are, and T(E) is a delta correction that takes into account the effect of the collimator on the X-ray radiation emitted by the sample. The integral determined represents the corrected simulated sample intensity i Sim (E) 112-c. Those skilled in the art may use other approximate functions such as Lorentz or Gaussian functions or (pseudo) Foucht line shapes to describe the triangle line shape.

[0077] Figure 4A The exemplary embodiment of the invention illustrates the integration width of the detector 204 and the triangular correction function 41 to take into account the angular tolerance of the crystal monochromator as defined by the divergence of one collimator C1 on the simulated intensity i Sim The energy range integrated by detector 204 is determined by the angular range / divergence allowed by collimator C1. The angular range is caused by the fact that the X-ray beam parallel to the wall of collimator C1 (e.g., the dashed arrow) ends up in the middle of the angular range between θ2 and θ1. However, light rays that pass through the collimator C1 with an oblique angle (ie, not parallel to the collimator wall, see solid arrows) are slightly deviated from the The light strikes the crystal 203 at diffraction angles between θ2 and θ1. The relative intensity over this angular range is not constant, but has a triangular profile as shown in the correction function 41. The ratio of the intensity I0 incident on the collimator C1 to the exiting intensity I(α) (where α is the divergence of the light relative to the collimator axis) is:

[0078]

[0079] in is the maximum divergence allowed by the collimator.

[0080] The detection system 200-1 has two collimators (see Figure 1 , C1 205, C2 206), where C1 is before the crystal and C2 is after the crystal, ray tracing can be used to study Figure 4B For example, a large number of rays I o A flat source before the primary collimator C1 is projected in random directions. All rays that strike the walls of the primary collimator are discarded. The remaining rays are reflected by the crystal toward the detector. Rays that strike the walls of the secondary collimator C2 are also discarded. The last remaining rays after the second collimator are then plotted against their diffraction angles on the crystal, thus providing I(α).

[0081] Figure 4B The LiF200 crystal used as a monochromator is shown. With corresponding energy distribution 401b, 402b, 403b for different Ray tracing results for the values: (cf. graph 401a), (cf. graph 402a) and (cf. graph 403a). Graph 401b shows a small deviation from the ideal triangular shape. This deviation depends on the combination of the divergence of the primary collimator C1 and the secondary collimator C2. In some cases, the relative intensity can be described as a piecewise linear function to improve the peak shape approximation (dashed and dotted graphs 401b illustrate such linear segments).

[0082] The intensity integral in equation F4 is defined in the energy E dimension. Therefore, the triangular equation of the triangular collimator correction function is transformed by the Bragg equation F3:

[0083]

[0084] The index of the limit changes from θ to E because the Bragg equation is an inverse transformation. This is shown in

[0085] ∫I(θ)dθ=-∫I(E)tanθ / EdE (F5)

[0086] The term tanθ / E is related to the Lorentz factor and can be resolved by the negative sign before the integration (see A. Iwata, K. Yuge and J. Kawai, Intensity correction of WD-XRF spectra from 2θ to energy, X-ray Spectra, 42, 16 (2013).).

[0087] Finally, the triangular part of equation F4 can be written as

[0088]

[0089] This piecewise linear approximation can also be extended to even more parts.This type of triangular collimator correction is new for WD detection system modeling.

[0090] Using the triangular collimator response and the derived integration limits, the line width of the characteristic X-ray lines in the simulated intensity (the simulated peak width of the corresponding element) is limited only by the collimator width and the crystal lattice plane distance, which are both known constants. In other words, the number of parameters required to describe the distribution for modeling the WD detection system is significantly reduced. In the prior art methods, typically for each peak, a parameter that must be calibrated via a corresponding standard is required. Further, the θ-dependent integration width explains why the background intensity increases from high to low θ angles, because the background is integrated in an increasing energy range from high to low angles. This is a major step towards a complete description of the WD detection system and the relationship between the background and the emission line intensity.

[0091] The angle difference Δθ = θ2 - θ1 can be converted to the corresponding energy difference ΔE = E2 - E1 by

[0092]

[0093] Now turn back to Figure 1 The correction module 120 has an optimization submodule 129 that is adapted to determine 1340 a mathematical distance between one or more corrected simulated sample intensities and the corresponding one or more measured intensities. For example, the distance can be measured using the Minkowski distance. The Minkowski distance is typically used with parameters such that the Minkowski distance corresponds to the Manhattan distance, the Euclidean distance, or the Chebyshev distance. Other suitable distance metrics may also be used by those skilled in the art.

[0094] The sample model parameters 130P of the sample model 130M with respect to the sample composition and / or layer thickness are then adjusted 1360. That is, the optimization module 129 modifies one or more of the sample parameters 130P in a manner that reduces the distance (e.g., along the first derivative). Since the effect of the sample model parameters 130P on the simulated intensity 112 is generally nonlinear, the determination steps 1320, 1340, and the parameter adjustment step 1360 are then repeated until the change in distance falls below a minimum change in distance. In other words, the first derivative approaches zero. Thus, a check step 1341 compares the determined mathematical distance change with the minimum distance change. The minimum distance change can be a given threshold or it can be a relative value.

[0095] Finally, the interface of system 100 is adapted to provide sample model parameters 130P regarding the sample composition and / or layer thickness associated with the corrected simulated intensities 112-c as approximate concentration values ​​of the corresponding components contained in the measured sample 202 and / or the layer thickness of the measured sample. In other words, once optimization of the sample model 130M has been performed such that a minimum distance change has been achieved, one or more corrected simulated intensities match the corresponding one or more measured intensities to such an extent that the model parameters 130P can be interpreted as indirect measurements of sample composition and / or layer thickness parameters of the measured (real-world) sample.

[0096] Figure 5A and Figure 5B Two examples are shown that demonstrate the effect of triangle correction when applied to simulated sample intensities. Figure 5A In the example, the measured sample contains a mixture of iron in different oxidation states (Fe(III) in Fe2O3 and Fe(II) in FeO). The measured intensity (point 50) shows a single peak of overlapping Fe(II) and Fe(III) KB lines. Therefore, in the measurement result 50, it is impossible to distinguish the contributions of Fe(II) and Fe(III) to the total measured intensity. The sample model used to provide the simulated intensity has corresponding concentration parameters for Fe(II) and Fe(III). Line 50-1 illustrates the contribution of the Fe(II) intensity to the total simulated intensity and line 50-2 shows the contribution of the Fe(III) intensity. Line 50-3 is the sum of the two contributions. Figure 5A The final result after sufficient iterations have been performed to minimize the distance between the simulated intensity 50-3 and the measured intensity 50 by adjusting the Fe(II) / Fe(III) concentration parameters accordingly is illustrated. These final Fe(II) / Fe(III) concentration parameters correspond to indirect measurements of the corresponding Fe(II) / Fe(III) concentrations in the measured sample. In this example, the Fe(III) concentration is greater than twice the Fe(II) concentration. This detailed analysis is not possible with prior art methods.

[0097] Figure 5B Shown with Figure 5A The same data as in , but with a logarithmic intensity axis. This emphasizes the potential of the method to use peak shape to separate two strongly overlapping components.

[0098] Return to Figure 1 In one embodiment, the correction module 120 may also be adapted to apply 1321 the polarization correction PC 122 to one or more simulated sample intensities such that for each of the one or more diffraction angles θ, the simulated sample intensity i Sim (E) is calculated as the sum of the emission radiation emitted by the sample multiplied by a first polarization factor of single scattering at the monochromator and the scattered tube radiation scattered by the sample multiplied by a second polarization factor of double scattering at the sample and the monochromator.

[0099] Figure 6 An example detection system 200-1a is shown having an X-ray tube 201 and a sample 202, wherein the tube generates a beam of light with an intensity i t The X-rays pass through tube window 208 and filter 209 before striking sample 202 at angle α. Figure 6 The black arrows in the figure indicate the radiation from the tube (i t ) is scattered by the sample 202 (the scattered radiation i S (E) is shown as a solid black arrow passing through collimator C1) and then scattered again at crystal 203 towards detector 204 in the same scattering plane (dashed black arrow passing through C2 and detector window 207). The grey (thick) arrows indicate the emitted radiation i from the sample. E (E) (solid grey arrow through collimator C1) is scattered only once at the crystal towards the detector (dashed grey arrow through collimator C2 and detector window 207). α is the angle of incidence, and β is the take-off angle on the sample 202, and θ is the diffraction angle on the crystal 203. As known to those skilled in the art, scattering can be represented by two vectors (as shown in the figure). The vectors always define a plane if they are not parallel or antiparallel. This plane (in which the two scattering vectors lie) is called the scattering plane. The principal component of the linear polarization of the scattered radiation is always perpendicular to this plane. In Figure 6 In the example of , all vectors are in the plane of the diagram, ie in the same plane. However, there are detection systems in which not all three scattering vectors are in one plane.

[0100] Polarization correction PC 122 addresses intensity changes due to polarization through scattering. If an unpolarized beam strikes crystal 203, the exiting beam is attenuated not only by the crystal's diffraction efficiency but also by the polarization effects of the scattering process. This effect is angle-dependent and reaches its maximum at 2θ = 90° between the incident and exiting beams. At this angle, the exiting beam is completely linearly polarized, and the polarization correction is 50%.

[0101] However, in the case of the WD detection system 200-1a, only the radiation i emitted by the sample E (E) is unpolarized. The radiation i that has been scattered at the sample S (E) has been polarized by this scattering process. Therefore, when considering polarization correction, the two contributions to the spectrum will be treated differently. In most detection systems, scattering at the sample and crystal occurs in the same plane, which simplifies PC correction. However, PC 122 can be extended to any geometry, for example by using Mueller calculus.

[0102] In the following, we address the common case of two scattering processes occurring in the same plane, where only the amplitude of the polarization vector parallel to the scattering plane is affected. For unpolarized emitted radiation, this polarization vector has the same amplitude as the perpendicular polarization vector before hitting the crystal. Therefore, the total intensity of the emission from the sample is adjusted by:

[0103] 0.5(1+cos 2 2θ)

[0104] where θ is the diffraction angle on the crystal.

[0105] In the case of already scattered radiation (black arrows), the following equation can be used for the WD detection system:

[0106]

[0107] where α is the incident angle on the sample and β is the take-off angle on the sample.

[0108] Therefore, the polarization correction equation is

[0109]

[0110] A first polarization factor for single scattering at the monochromator is multiplied by the emission radiation emitted by the sample, and

[0111]

[0112] A second polarization factor for double scattering at the sample and at the monochromator is multiplied by the scattered tube radiation scattered by the sample.

[0113] When PC 122 is also included, equation (F4) is modified to equation F4-1:

[0114]

[0115] Among them, P E (E) and P S (E) are the polarization factors of the emitted radiation and the scattered radiation, respectively, and i E (E) and i S (E) are s -1 eV -1 The emitted radiation and scattered radiation in.

[0116] The separation of the effects of polarization on the radiation components that strike the detector further improves the accuracy of the simulated intensity to which the WD detection system responds. Thus, once the minimum distance is reached, the accuracy of the model parameters is also improved. That is, when the polarization correction 122 is added to the triangulation correction 121 to calculate the corrected simulated intensity as described in Formula F4-1, the indirectly measured sample parameters (component concentration / layer thickness) become even more reliable. For example, the polarization correction also explains why the peak to background ratio (emitted radiation / scattered radiation) is different than in an energy dispersive (ED) detection system. In Figure 7A LiF200 crystal is shown in The evolution of the polarization correction in 2θ and Figure 7B The polarization correction in energy is illustrated in FIG. The dashed lines 710, 720 represent the emitted radiation i E (E) and solid lines 711 and 712 represent the scattering tube radiation i S (E)(α+β=105°).

[0117] Temporarily return to Figure 1 In one embodiment, the correction module 120 may also be adapted to apply 1322 a crystal length correction CL-C 123 to one or more simulated sample intensities 112. Figures 8A to 8C The details of CL-C123 are discussed in the context of . Figure 8A A crystal 203 with a length l1 is shown. The collimator has a fixed height h1, and the transmitted intensity illuminating the plane of the crystal 203 is the projection of this height onto the crystal plane. The length of the projection depends on the diffraction angle θ. If the illumination length is greater than the finite crystal length, intensity is lost because only the beam that strikes the crystal surface (grey arrows) is scattered toward the detector. Figure 8BThis effect on intensity is illustrated for a LiF200 crystal with a collimator height of 2 cm and a crystal length of 7 cm. At energies exceeding the dashed energy threshold 810, the scattered intensity 811 drops off rapidly. In other words, at higher diffraction angles θ (below the energy threshold 810), the illumination projection h1' is smaller than the crystal length l1 and all the intensity leaving the collimator is reflected by the crystal:

[0118]

[0119] Using the Bragg equation F3 provides:

[0120]

[0121] Where h1 is the collimator height, h1 ′ as the illumination projection, and d as the lattice plane distance of the crystal. The crystal length correction CL-C 123 is then:

[0122]

[0123] Where l1 is the length of the crystal.

[0124] When CL-C 123 is also included, Equation F4-1 is modified to Equation F4-2:

[0125]

[0126] In other words, for a diffraction angle θ where the projection h1′ of the collimator height h1 on the crystal monochromator plane is greater than the length l1 of the crystal, then one or more simulated sample intensities are multiplied by the ratio of the length l1 of the crystal to the length of the projection h1′ of the collimator height on the crystal monochromator plane to obtain the corrected simulated intensity 112-c.

[0127] It should be noted that in one embodiment, crystal length correction 123 may be applied only with triangulation correction 120. In general, triangulation correction may be combined with any of optional corrections 121-128, or with any subset of the optional corrections.

[0128] Figure 8C The effect of finite crystal length on the triangle correction 120 is illustrated. When finite crystal length is taken into account during ray tracing of linear shapes, it results in a small asymmetry for low diffraction angles θ, which can be corrected by moving the nodes of the piecewise linear function accordingly. Graphs 801 to 804 show the effect of finite crystal length on the triangle correction 120. When finite crystal length is taken into account during ray tracing of linear shapes, it results in a small asymmetry for low diffraction angles θ, which can be corrected by moving the nodes of the piecewise linear function accordingly. Ray tracing results of 801, 803 and LiF200 crystal The corresponding energy distributions 802, 804 are shown in Figures 803, 804. The upper graphs 801, 802 are calculated for an infinite crystal. The lower graphs are calculated for a finite crystal. The small asymmetry introduced by the finite crystal length is visible in the graphs 803, 804 in the areas indicated by the corresponding arrows 803-1, 804-1.

[0129] As from Figure 1 , the detection system 200-1 may further include a detector window 207 between the crystal monochromator 203 and the X-ray detector 204. In this case, the correction module 120 may further include a window transmission correction function WT-C 124, which may be applied 1323 to the one or more simulated sample intensities 112 such that, for each corresponding diffraction angle θ, the simulated sample intensity is multiplied by a transmission factor defined by the material and thickness of the detector window 207.

[0130] The detector window transmission is defined as the intensity behind the window and before the window and can be expressed in terms of the absorption properties of the material:

[0131]

[0132] where the energy-dependent mass absorption coefficient μ of the window material W (E), density ρ W and thickness t W .

[0133] When WT-C 124 is also included, Equation F4-2 is modified to Equation F4-3:

[0134]

[0135] A silicon-like structure supports the grid to prevent cracking of foil or multilayer windows and can also be modeled by combining multiple window transmission terms (additive and multiplicative) separately.The optional window in front of the primary collimator 205 can be treated identically.

[0136] The correction module 120 may also be adapted to apply 1324 a detector efficiency correction function DE-C 125 to one or more simulated sample intensities such that for each respective diffraction angle θ, the simulated sample intensity is multiplied by an absorption factor defined by the material and thickness of the detector.

[0137] The detector efficiency is defined as the intensity of the absorption in the active detector material The intensity incident on the active detector material and is the complementary portion of the transmission of the active detector material:

[0138]

[0139] where the energy-dependent mass absorption coefficient μ of the active detector material D (E), density ρ D and thickness t D .

[0140] Different angles of incidence β on the detector can be accounted for by dividing the thickness t by sinβ. However, an angle of incidence of 90° is commonly used in WD detection systems. The basic calibration model assumes that the detector 204 is infinite in its dimension perpendicular to the incidence of the light beam. This is an approximation that is sufficient for current equipment. However, the model can also be adjusted to account for the actual size of the detector.

[0141] When DE-C 125 is also included, Equation F4-3 is modified to Equation F4-4:

[0142]

[0143] Correction module 120 may also be adapted to apply 1325 a crystal diffraction efficiency correction function CE-C 126 to one or more simulated sample intensities. In this embodiment, for each diffraction angle θ, the simulated sample intensity is divided by the portion of the incident intensity absorbed by the crystal and adjusted by the crystal geometry factor.

[0144] Starting from the "Master Equation of Crystallography" (5.30) and substituting (5.28) from J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed. (Wiley, Hoboken, 2011), the integrated intensity I SC Can be calculated as:

[0145]

[0146] where Φ0 is the incident flux, as a term resulting from integration over θ (related to the Lorentz factor), and As the differential elastic scattering cross section without polarization. In the aforementioned reference by Als-Nielsen et al., polarization is part of the differential scattering cross section, but in the method disclosed herein, polarization is treated separately for scattered tube radiation and emission. The differential scattering cross section without polarization is scattering angle dependent and energy independent.

[0147] G. Remond, P. Coutures, C. Gilles and D. Massiot, Analytical description of X-ray peaks: LX-ray spectroscopy of the lanthanides by electron probe microanalyzers, scanning microscopy applications, 3, (1989) Equation (5.31) describes the absorption effect of the crystal for the case of infinite thickness. This can be adjusted to handle crystals with finite thickness. However, for typical crystals used in WD detection systems, the crystal can be considered to be infinitely thick in the energy range they are used, resulting in

[0148]

[0149] On page 176 of the above-mentioned Redmond et al. reference, the dependence of the structure factor (which is part of the differential scattering cross section) on the thermal roughening of the reciprocal lattice is described by the following term:

[0150] e- M

[0151] in

[0152]

[0153] This can be transformed by using the Bragg equation F1 into:

[0154]

[0155] The Debye-Waller factor B T is constant and all other parameters are constant. The scattering cross section can be replaced by a single universal scaling factor g that is adjusted only once for each crystal.

[0156] Transforming the above equations into the energy dimension using the Bragg equation again leads to:

[0157]

[0158] In other words, the absorption part is considered when modeling the WD detection system, which is another step towards a comprehensive description of the efficiency of the WD detection system. represents a transformation of the integral as shown in the previously mentioned Iwata et al. reference and is related to the Lorentz factor.

[0159] When CE-C 126 is also included, Equation F4-4 is modified to Equation F4-5:

[0160]

[0161] The correction module 120 may also be adapted to apply 1326 a pile-up correction PU-C 127 to one or more simulated sample intensities such that for each corresponding diffraction angle, the simulated sample intensity is multiplied by an exponential function that describes the intensity loss resulting from counting two photons arriving within a predefined time window as a single photon with double energy.

[0162] Detector 204 cannot distinguish whether the signal is generated by two photons striking the detector at approximately the same time or by a single photon with double energy. Therefore, a pile-up intensity peak PU1 is expected at double the energy of the first diffraction order of the monochromator crystal (similarly, the second order corresponds to double energy). Due to the detector electronics optimized for high count rates, this effect is blurred between the first order O1 and the second order O2. The effect depends nonlinearly on the intensity. The higher the intensity, the better the effect. For low intensities, it is insignificant.

[0163] However, the upper discrimination limit of the PHA discriminator window, PHA 150, may pass right through the pileup as part of the analytical signal (with the correction that each counted photon in the pileup corresponds to two analytically relevant photons). Again, intensity is lost, and at high count rates the detector efficiency is reduced by a factor of

[0164]

[0165] in is the intensity of equation (3) without pile-up correction, and p is a parameter that must be adjusted by calibration to account for the effect of the detector's upper discrimination limit. The exponential expression is a known equation for pile-up (see G. Blaj et al., Optimal Pulse Processing, Pile-up Decomposition, and Applications for Silicon Drift Detectors, in LCLS, IEEE Transactions on Nuclear Science 64, 2854 (2017)).

[0166] When PU-C 127 is also included, Equation F4-5 is modified to Equation F4-6:

[0167]

[0168] Taking the pile-up into account in the efficiency calculation of the WD detection system allows the line shape in the measurement with high count rate to be modeled. When measuring the peak in the WD 2θ scan, the highest measurement point is weakened relative to other points of the same peak. This causes the peak in the 2θ scan to have a flatter tip of the peak. Therefore, the line shape used by the prior art (e.g., pseudo-Fokkert) is significantly different from the measurement under high count rate conditions, and it is necessary to reduce the count rate by reducing the power and therefore reducing the performance of the device. The use of the PU-C127 correction function overcomes the problem and allows measurements to be performed at high count rates without suffering the shortcomings of the prior art.

[0169] There are intensity contributions that arrive at the detector in a different way than described in the above description. Such contributions are called background contributions. Figure 9A 、 Figure 9B Visualize the results of a 2θ scan of a copper (Cu) sample, where each point of the spectrum is recorded by a detector with a multi-channel analyzer (energy dispersion) in such a way that the expected photon energy (first order of the monochromator - see Bragg equation F3) is always at the same channel number. This refers to 100% of the expected energy. The Cu sample was measured as a 2θ scan at 50 kV using a Rh tube, a 500 μm Al filter, a LiF200 crystal monochromator, and an Ar gas detector. Figure 9A A heat map 95 of the obtained PHA spectra is shown, visualizing the additional contribution to the background. Figure 9B The 2θ spectrum 90 shown is the result of integrating the PHA between 50% and 150% of the expected energy (O1) (PHA50 and PHA150, respectively). The intensity in the heat map 95 is logarithmic and is illustrated by different gray levels 95-1. Markers 1 to 8 (shown as black dots with corresponding numbers) are explained in detail below. Therefore, marks 1, 4, and 6 in the spectrum 90 correspond to the corresponding marks in the heat map 95.

[0170] At marker 1 in the thermal map 95, the Cu KA line gives a strong signal (100% PHA) at the position of the 1st order O1. To the left of marker 1 is another high intensity of Cu KB. In the following, Cu KA is used as an example of the process that occurs in the detector for any photon (emission or sample scattered radiation) that hits the detector.

[0171] Two additional intensity maxima can be observed above and below the first-order peak of Cu KA at marker 1 in the thermal map 95. One maximum is close to the lower identification limit PHA50 (50% PHA) indicated by marker 2, and the other maximum extends from the first-order O1 through the upper identification limit PHA150 toward the second-order O2 at marker 3. The latter is the pile-up PU1 already explained in the context of the PU-C correction function 127.

[0172] The additional intensity at marker 2 is due to the escape of the fluorescent photon from the active detector material (Ar in this example) after the incident photon has interacted with the active detector material. This additional peak at marker 2 is therefore part of the analytically relevant information and can be integrated. However, as can be seen by the dashed first-order Ar escape line O1E, the relative distance between the first-order portion O1 of the analytical intensity and the escape intensity increases with increasing 2θ. Since the integration range is fixed (from PHA50 to PHA150), the escape peak O1E in PHA will move past the lower identification limit PHA50 with increasing 2θ, thereby reducing the analytically relevant intensity that is integrated.

[0173] As mentioned above, the monochromator crystal allows multiple diffraction orders and therefore multiples of the first order energy (O1 in 95) to strike the X-ray detector. Marker 5 in 95 is the 2nd order peak of Cu KA, with the same additional intensity maximum as discussed before. Here, the Ar escape peak falls partially into the discriminator window (between PHA50 and PHA150). However, now it is not part of the analytically relevant intensity because the 1st order energy is here half the Cu KA energy. This happens again for all photons that strike the X-ray detector. This is indicated by the dotted 2nd order Ar escape line O2E. With increasing 2θ, the escape peak O2E in PHA will move into the integration range from the higher discrimination limit PHA150, thus adding to the background caused by detector artifacts.

[0174] The correction module 120 may also be adapted to apply 1327 a background correction function 128 to one or more simulated sample intensities to further account for intensity losses caused by photons escaping from the active detector material and / or intensity gains caused by higher diffraction order artifacts of the monochromator. Thus, for each respective diffraction angle, the corresponding intensity loss and / or intensity gain is added to the corrected simulated sample intensity 112-c.

[0175] This reduction in efficiency depends on the discrimination limit and the detector material and occurs only when the photon energy is above the edge energy of the escape fluorescence line of the detector material. In the following, this escape effect (i.e., loss b Escape (E)) is calculated exemplarily for a gas counter detector having argon as active material.

[0176]

[0177] Among them E Ar K As the K-edge energy of argon and e Ar KA is the probability of Ar Kα emission escaping the detector. This can be energy-dependent, but is not in the case of gas counter detectors. i(E) is the expected intensity impinging on the detector. This is followed by the integration of the Gaussian line profile from 0% PHA to l, which is the lower identification limit (here 50% PHA). E Ar KA is the energy of the Ar KA emission, and σ PHA (E) is the standard deviation of the PHA peak. This expression is negative because if the argon evolution peak is cut off by the lower identification limit, there is a loss of intensity.

[0178] Using equation F3, 2θ = 40° corresponds to 8 keV (Cu KA energy), and 2θ = 100° corresponds to 4 keV (both 1st order). However, for diffraction order n = 2, the energy is again 8 keV (Cu KA energy). Accordingly, an intensity profile (similar to that at marker 1) is also visible at marker 5 (the pileup PU2 also extends further upward). In addition, here, an escape peak is visible at marker 6 and reaches the integration range.

[0179] However, in this case the intensity is not analytically relevant at 2θ = 100° (4 keV) but is still integrated. This contribution is similar to that of b above. escape , but with a positive sign. That is, it adds to the entire background.

[0180]

[0181] with b escape Compared to the expression for , double the energy is used for the origin of the intensity and the integration is performed from the lower identification limit l (PHA50) to the upper identification limit u (PHA150). This correction can also be relevant for higher orders, where the "2" in the above equation is replaced by the corresponding diffraction order.

[0182] The above-described escape effect with clearly visible peaks at markers 1 and 5 also applies to spectral regions without clear peaks, such as at marker 4, where the main contribution at the 1st order energy O1 is due to scattered tube bremsstrahlung from the sample.

[0183] All background contributions b(E) (including b Escape (E),b 2nd order (E), etc.) can be simply added to the corrected simulated intensity in any of equations F4*. When background correction function B-C 128 is also included, equation F4-6 is modified to equation F4-7, which then includes all corrections provided by correction module 120 in a comprehensive embodiment (modeling all potential effects of detection system 200-1 on the simulated intensity):

[0184]

[0185] Figure 10 Another background source that can be corrected is illustrated by Figure 9A The contribution indicated by reference 7 in FIG is due to diffuse (non-Bragg) scattering DS of the sample emission from the crystal and the sample scattered radiation originating from the sample. Reference 8 indicates the fluorescence radiation FL originating from the crystal holder and the crystal itself (in this example, only the crystal holder fluorescence is visible). The intensity of the iron fluorescence FL increases with increasing 2θ because the slope of the incident and outgoing radiation on the crystal becomes steeper with 2θ and thus reduces the absorption of the crystal.

[0186] In this embodiment, the model of the WD detection system also calculates the contributions of fluorescence and diffuse scattering to the background of the WD detector response. Crystal 203 is thus mounted on holder 203-1. Thick arrows 11, 11-1 indicate radiation from the sample and then scattered by crystal 203 and holder 203-1 toward detector 204. Dashed arrow 11-2 indicates emitted radiation from crystal 203 and holder 203-1.

[0187] These two effects can be calculated based on a simplified prior art EDX modeling solution, where the crystal 203 and its holder 203-1 are described as a multilayer sample and the excitation radiation is the radiation from the sample. The line shape can be a line with a standard deviation σ PHA The simple Gaussian line shape of . The incident and exit beam angles vary with 2θ. The background contribution to the WD response can be calculated by integrating the energies between the lower PHA discrimination limit (l*E) and the upper PHA discrimination limit (u*E).

[0188] Using the solution detailed above, the background contribution described by the reference cited above by Portnoi et al. can be calculated. In addition, scattering effects and fluorescence from the crystal holder are also included. Other detectors may have other ways to integrate the intensity. The above background contribution calculation can be adjusted accordingly. Other effects, such as efficiency holes due to the Renninger effect, or photoelectrons disappearing from the crystal, can be modeled in a similar manner.

[0189] To solve the integrals of F4-7 (and similarly for the other integrals F*) to obtain the corresponding corrected simulated intensities, one can either integrate numerically or find a solution for the analytical integral. The former has the advantage that it allows integration of approximate functions that cannot be integrated analytically. The latter has the advantage of being significantly faster and more widely applicable. To this end, all contributions that are nearly linear within the integral range can be approximated by a constant value of the central energy.

[0190]

[0191] The integration of the continuous part of the scattered radiation is simple because both the continuous part and the description of the triangle are piecewise polynomials that can be multiplied and piecewise integrated. The integral of the multiplication of the piecewise linear triangular function with the pseudo-Vokt line shape of the emission and sample scattering characteristics can be solved by partial integration of each segment of the piecewise linear function of the triangle.

[0192]

[0193] The integral of a piecewise linear function multiplied by any function f(E) can be solved by the following formula:

[0194]

[0195] In the case of the triangle correction function T(E), there are two segments in the piecewise linear function: the ascending part and the descending part of the triangle, which are respectively "1" at all and "0" at E1 and E2.

[0196]

[0197] get

[0198]

[0199] in

[0200]

[0201] The integral is reduced to

[0202]

[0203] To solve this reduced integral, the Cauchy (Lorentz line type)

[0204]

[0205] and normal (Gaussian) distribution

[0206]

[0207] The cumulative probability function of is integrated.

[0208] The above is a solution for a triangle, and the solution for any other piecewise linear function is very similar. Using other approximate functions (such as Lorentz or Gaussian functions) for triangular linear shapes can also produce similar results.

[0209] Figure 11 is a diagram illustrating an example of a general purpose computer device 900 and a general purpose mobile computer device 950 that may be used with the techniques described herein. In some embodiments, the computing device 900 may be related to the system 100 (see Figure 1). Such a computer device 900 may be implemented as an integrated component of the X-ray detection system 200 (e.g., a wavelength dispersive X-ray fluorescence spectrometer). The computing device 950 is intended to represent various forms of mobile devices, such as personal digital assistants, cellular phones, smart phones, and other similar computing devices. In the context of the present disclosure, the computing device 950 may provide I / O components for a user to interact with the computing device 950 (e.g., for receiving sample model parameters 130P and / or corrected simulated spectra 112-c). The components shown here, their connections and relationships, and their functions are meant to be exemplary only and are not meant to limit the specific implementations of the inventions described and / or claimed in this document.

[0210] The computing device 900 includes a processor 902, a memory 904, a storage device 906, a high-speed interface 908 connected to the memory 904 and a high-speed expansion port 910, and a low-speed interface 912 connected to a low-speed bus 914 and the storage device 906. Each of the components 902, 904, 906, 908, 910, and 912 is interconnected using various buses and can be mounted on a common motherboard or other suitable means. The processor 902 can process instructions for execution within the computing device 900, including instructions stored in the memory 904 or on the storage device 906 to display graphical information of a GUI on an external input / output device (such as a display 916 coupled to the high-speed interface 908). In other specific implementations, multiple processors and / or multiple buses can be used as appropriate, along with multiple memories and multiple types of memory. In addition, multiple computing devices 900 can be connected, with each device providing a portion of the necessary operations (e.g., as a server cluster, a blade server group, or a multi-processor system).

[0211] The memory 904 stores information within the computing device 900. In one embodiment, the memory 904 is one or more volatile memory units. In another embodiment, the memory 904 is one or more non-volatile memory units. The memory 904 may also be another form of computer-readable medium, such as a magnetic disk or optical disk.

[0212] The storage device 906 can provide mass storage for the computing device 900. In one embodiment, the storage device 906 can be or include a computer-readable medium, such as a floppy disk device, a hard disk device, an optical disk device, or a magnetic tape device, a flash memory or other similar solid-state memory device, or a device array, including a device in a storage area network or other configuration. A computer program product can be tangibly embodied in an information carrier. The computer program product can also include instructions for performing one or more methods (such as those described above) when executed. The information carrier is a computer or machine-readable medium, such as the memory 904, the storage device 906, or a memory on the processor 902.

[0213] The high-speed controller 908 manages bandwidth-intensive operations of the computing device 900, while the low-speed controller 912 manages less bandwidth-intensive operations. This allocation of functions is exemplary only. In one embodiment, the high-speed controller 908 is coupled to the memory 904, the display 916 (e.g., via a graphics processor or accelerator), and to the high-speed expansion port 910 that can accept various expansion cards (not shown). In this embodiment, the low-speed controller 912 is coupled to the storage device 906 and the low-speed expansion port 914. The low-speed expansion port, which can include various communication ports (e.g., USB, Bluetooth, Ethernet, wireless Ethernet), can be coupled to one or more input / output devices, such as a keyboard, pointing device, scanner, or networking device (such as a switch or router), for example, via a network adapter.

[0214] The computing device 900 may be implemented in several different forms, as shown. For example, it may be implemented as a standard server 920, or multiple implementations in a group of such servers. It may also be implemented as part of a rack server system 924. In addition, it may be implemented in a personal computer such as a laptop computer 922. Alternatively, components from the computing device 900 may be combined with other components in a mobile device (not shown) such as device 950. Each of these devices may contain one or more of the computing devices 900 and 950, and the entire system may be composed of multiple computing devices 900 and 950 communicating with each other.

[0215] The computing device 950 includes a processor 952, a memory 964, an input / output device such as a display 954, a communication interface 966, and a transceiver 968, among other components. The device 950 may also be provided with a storage device (such as a microdrive or other device) to provide additional storage. Each of the components 950, 952, 964, 954, 966, and 968 is interconnected using various buses, and several of the components may be mounted on a common motherboard or in other suitable ways.

[0216] The processor 952 can execute instructions within the computing device 950, including instructions stored in the memory 964. The processor can be implemented as a chipset including independent or multiple analog and digital processors. For example, the processor can provide coordination of other components of the device 950, such as a user interface, applications executed by the device 950, and control of wireless communications performed by the device 950.

[0217] The processor 952 can communicate with the user through a control interface 958 and a display interface 956 coupled to the display 954. For example, the display 954 can be a TFT LCD (thin film transistor liquid crystal display) or an OLED (organic light emitting diode) display, or other suitable display technology. The display interface 956 may include appropriate circuits for driving the display 954 to present graphics and other information to the user. The control interface 958 can receive commands from the user and convert them for submission to the processor 952. In addition, an external interface 962 in communication with the processor 952 can be provided to enable near-area communication of the device 950 with other devices. The external interface 962 can, for example, provide wired communication in some specific implementations, or provide wireless communication in other specific implementations, and multiple interfaces can also be used.

[0218] Memory 964 stores information within the computing device 950. Memory 964 can be implemented as one or more computer-readable media, one or more volatile memory units, or one or more non-volatile memory units. An expansion memory 984 can also be provided and connected to the device 950 via an expansion interface 982, which can include, for example, a SIMM (single inline memory module) card interface. This expansion memory 984 can provide additional storage space for the device 950 or can also store applications or other information for the device 950. Specifically, the expansion memory 984 can include instructions for executing or supplementing the above-mentioned processes and can also include security information. Therefore, for example, the expansion memory 984 can serve as a security module for the device 950 and can be programmed by instructions that allow the safe use of the device 950. In addition, security applications can be provided via a SIMM card along with additional information, such as placing identification information on the SIMM card in a way that cannot be hacked.

[0219] For example, the memory may include flash memory and / or NVRAM memory, as discussed below. In one embodiment, a computer program product is tangibly embodied in an information carrier. The computer program product includes instructions that, when executed, perform one or more methods, such as those described above. The information carrier is a computer or machine readable medium, such as the memory 964, the expansion memory 984, or memory on the processor 952, which may be received, for example, via the transceiver 968 or the external interface 962.

[0220] Device 950 can communicate wirelessly via communication interface 966, which may include digital signal processing circuitry, where necessary. Communication interface 966 can provide communication in various modes or protocols, such as GSM voice calls, SMS, EMS, or MMS messaging, CDMA, TDMA, PDC, WCDMA, CDMA2000, or GPRS. Such communication can be performed, for example, via radio frequency transceiver 968. In addition, short-range communication can be performed, such as using Bluetooth, WiFi, or other such transceivers (not shown). In addition, GPS (Global Positioning System) receiver module 980 can provide additional navigation and location-related wireless data to device 950, which can be used as appropriate by applications running on device 950.

[0221] Device 950 may also communicate audibly using an audio codec 960 that receives spoken information from a user and converts it into usable digital information. Audio codec 960 may also generate audible sound for the user, such as through a speaker (e.g., in a handset of device 950). This sound may include sound from a voice phone call, may include recorded sound (e.g., voice messages, music files, etc.), and may also include sound generated by applications operating on device 950.

[0222] The computing device 950 can be implemented in a variety of different forms, as shown in the figure. For example, it can be implemented as a cellular phone 980. It can also be implemented as a part of a smart phone 982, a personal digital assistant, or other similar mobile devices.

[0223] Various implementations of the systems and techniques described herein can be realized in digital electronic circuitry, integrated circuitry, specially designed ASICs (application specific integrated circuits), computer hardware, firmware, software, and / or combinations thereof. These various implementations can include implementations in one or more computer programs executable and / or interpretable on a programmable system comprising at least one programmable processor, which can be special purpose or general purpose, coupled to receive data and instructions from a storage system, at least one input device, and at least one output device, and to transmit data and instructions to the storage system, at least one input device, and at least one output device.

[0224] These computer programs (also referred to as programs, software, software applications, or code) include machine instructions for a programmable processor and may be implemented in high-level procedural and / or object-oriented programming languages ​​and / or assembly / machine languages. As used herein, the terms "machine-readable medium," "computer-readable medium," and "machine-readable medium" refer to any computer program product, apparatus, and / or device (e.g., a disk, optical disk, memory, programmable logic device (PLD)) for providing machine instructions and / or data to a programmable processor, including a machine-readable medium that receives machine instructions as a machine-readable signal. The term "machine-readable signal" refers to any signal used to provide machine instructions and / or data to a programmable processor.

[0225] To provide interaction with a user, the systems and techniques described herein can be implemented on a computer having a display device (e.g., a CRT (cathode ray tube) or LCD (liquid crystal display) monitor) for displaying information to the user, and a keyboard and pointing device (e.g., a mouse or trackball) that the user can use to provide input to the computer. Other types of devices can also be used to provide interaction with the user; for example, the feedback provided to the user can be any form of sensory feedback (e.g., visual feedback, auditory feedback, or tactile feedback); and input from the user can be received in any form (including sound, voice, or tactile input).

[0226] The systems and techniques described herein can be implemented in a computing device that includes a back-end component (e.g., as a data server), or includes a middleware component (e.g., an application server), or includes a front-end component (e.g., a client computer having a graphical user interface or a web browser through which a user can interact with a specific implementation of the systems and techniques described herein), or any combination of such back-end, middleware, or front-end components. The components of the system can be interconnected by any form or medium of digital data communication (e.g., a communication network). Examples of communication networks include a local area network ("LAN"), a wide area network ("WAN"), and the Internet.

[0227] Computing devices may include clients and servers. A client and server are generally remote from each other and typically interact through a communication network. The relationship of client and server arises by virtue of computer programs running on the respective computers and having a client-server relationship to each other.

Claims

1. A computer-implemented method (1000) for approximating one or more X-ray intensities for a sample (202) measured by an X-ray detection system (200-1), the X-ray detection system comprising at least one collimator (205, 206) having a given angular range, a crystal monochromator (203), and an X-ray detector (204), the method comprising: receiving (1100) one or more measured intensities (212) from the detection system (200-1) at corresponding one or more diffraction angles (θ); calculating (1200) one or more simulated sample intensities (112) for the corresponding one or more diffraction angles from an X-ray fluorescence sample model (130M) having initial sample model parameters (130P), the initial sample model parameters being indicative of the sample composition and / or layer thickness of one or more sample layers; Applying (1300) a collimator correction (121) to the one or more simulated sample intensities (112) at the corresponding one or more diffraction angles by: determining (1320) for each of the one or more diffraction angles an integral over the energy range allowed by the angular range of the at least one collimator (205, 206), wherein the integral is calculated over the product of the corresponding simulated sample intensity (112) and an approximation function that approximates a triangular line shape of the relative intensity distribution in the angular range caused by the angular tolerance of the crystal monochromator defined by the divergence of the at least one collimator, the determined integral representing a corrected simulated sample intensity (112-c); determining (1340) a mathematical distance between the one or more corrected simulated sample intensities and the corresponding one or more measured intensities; adjusting (1360) the sample model parameters (130P) with respect to the sample composition and / or the layer thickness, and repeating the determining steps (1320, 1340) and the adjusting steps (1360) until the change in the distance drops below a minimum distance change; as well as The sample model parameters (130P) regarding the sample composition and / or the layer thickness associated with the corrected simulated intensity (112-c) are provided (1400) as approximate concentration values ​​of the corresponding components contained in the measured sample (202) and / or the layer thickness of the measured sample.

2. The method of claim 1 , wherein applying (1300) collimator correction further comprises: A polarization correction (122) is applied (1321) to the one or more simulated sample intensities (112) such that for each of the one or more diffraction angles, the simulated sample intensity is calculated as the sum of emitted radiation emitted by the sample multiplied by a first polarization factor for single scattering at the monochromator and scattered tube radiation scattered by the sample multiplied by a second polarization factor for double scattering at the sample and the monochromator.

3. The method of claim 1 or 2, wherein applying (1300) a collimator correction further comprises: A crystal length correction (123) is applied (1322) to the one or more simulated sample intensities (112) such that for diffraction angles where the projection (h1') of the collimator height (h1) on the crystal monochromator plane is greater than the length (l1) of the crystal, the one or more simulated sample intensities are multiplied by a ratio of the length (l1) of the crystal to the length of the projection (h1') of the collimator height on the crystal monochromator plane.

4. The method of any one of the preceding claims, further taking into account the effect of a detector window (207) between the crystal monochromator (203) and the X-ray detector (204), wherein applying (1300) a collimator correction further comprises: A window transmission correction (124) is applied (1323) to the one or more simulated sample intensities (112) such that, for each respective diffraction angle, the simulated sample intensity is multiplied by a transmission factor defined by the material and thickness of the detector window.

5. The method of any one of the preceding claims, wherein applying (1300) a collimator correction further comprises: A detector efficiency correction (125) is applied (1324) to the one or more simulated sample intensities such that, for each corresponding diffraction angle, the simulated sample intensity is multiplied by an absorption factor defined by the material and thickness of the detector.

6. The method of any one of the preceding claims, wherein applying (1300) a collimator correction further comprises: A crystal diffraction efficiency correction (126) is applied (1325) to the one or more simulated sample intensities such that, for each respective diffraction angle, the simulated sample intensity is divided by the portion of the incident intensity absorbed by the crystal and adjusted by a crystal geometry factor.

7. The method of any one of the preceding claims, wherein applying (1300) a collimator correction further comprises: A pile-up correction (127) is applied (1326) to the one or more simulated sample intensities such that, for each corresponding diffraction angle, the simulated sample intensity is multiplied by an exponential function that describes the intensity loss resulting from counting two photons arriving within a predefined time window as a single photon with double energy.

8. The method of any one of the preceding claims, wherein applying (1300) a collimator correction further comprises: A background correction (128) is applied (1327) to the one or more simulated sample intensities to account for intensity losses caused by escape of the photons from the active detector material and / or intensity gains caused by higher diffraction order artifacts of the monochromator, such that for each respective diffraction angle, the corresponding intensity losses and / or intensity gains are added to the corrected simulated sample intensities (112-c).

9. The method according to all preceding claims, wherein the corrected simulated sample intensity provides an optimized quantification of the composition of the sample and / or layer thickness of one or more sample layers based on the one or more measured intensities.

10. A computer program product for approximating one or more X-ray intensities of a sample (202) measured by an X-ray detection system (200-1), the computer program product comprising computer-readable instructions that, when loaded into a memory of a computing device and executed by at least one processor of the computing device, cause the computing device to perform a computer-implemented method according to any one of the preceding claims.

11. A computer system (100) for approximating one or more X-ray intensities for a sample (202) measured by an X-ray detection system (200-1), the X-ray detection system comprising at least one collimator (205, 206) having a given angular range, a crystal monochromator (203), and an X-ray detector (204), the system comprising: an interface adapted to receive one or more measured intensities (212) from the detection system (200-1) at corresponding one or more diffraction angles (θ); a sample simulation module (130) adapted to calculate one or more simulated sample intensities (112) for the corresponding one or more diffraction angles from an X-ray fluorescence sample model (130M) having initial sample model parameters (130P), the initial sample model parameters being indicative of the sample composition and / or layer thickness of one or more sample layers; A correction module (120) adapted to apply a collimator correction (121) to the one or more simulated sample intensities (112) at the corresponding one or more diffraction angles by: determining, for each of the one or more diffraction angles, an integral over the energy range allowed by the angular range of the at least one collimator (205, 206), wherein the integral is calculated over the product of the corresponding simulated sample intensity (112) and the multiplication of an approximation function that approximates a triangular line shape of the relative intensity distribution in the angular range caused by the angular tolerance of the crystal monochromator defined by the divergence of the at least one collimator, the determined integral representing a corrected simulated sample intensity (112-c); determining a mathematical distance between the one or more corrected simulated sample intensities and the corresponding one or more measured intensities; adjusting the sample model parameters (130P) regarding the sample composition and / or the layer thickness, and repeating the determining and adjusting steps until the change in the distance falls below a minimum distance change; as well as The interface is further adapted to provide the sample model parameters (130P) regarding the sample composition and / or the layer thickness associated with the corrected simulated intensity (112-c) as approximate concentration values ​​of the corresponding components contained in the measured sample (202) and / or the layer thickness of the measured sample.

12. The system of claim 11 , wherein the correction module ( 120 ) is further adapted to apply a polarization correction ( 122 ) to the one or more simulated sample intensities ( 112 ) such that, for each of the one or more diffraction angles, the simulated sample intensity is calculated as the sum of the emission radiation emitted by the sample multiplied by a first polarization factor for single scattering at the monochromator and the scattered tube radiation scattered by the sample multiplied by a second polarization factor for double scattering at the sample and the monochromator.

13. The system of claim 11 or 12, wherein the correction module (120) is further adapted to apply a crystal length correction (123) to the one or more simulated sample intensities (112) such that for diffraction angles where a projection (h1') of the collimator height (h1) on the crystal monochromator plane is greater than a length (l1) of the crystal, the one or more simulated sample intensities are multiplied by a ratio of the length (l1) of the crystal to the length of the projection (h1') of the collimator height on the crystal monochromator plane.

14. The system according to any one of claims 11 to 13, further taking into account the influence of a detector window (207) between the crystal monochromator (203) and the X-ray detector (204), wherein the correction module (120) is further adapted to apply a window transmission correction (124) to the one or more simulated sample intensities (112), such that for each corresponding diffraction angle, the simulated sample intensity is multiplied by a transmission factor defined by the material and thickness of the detector window.

15. The system according to any one of claims 11 to 14, wherein the correction module (120) is further adapted to apply a detector efficiency correction (125) to the one or more simulated sample intensities such that for each respective diffraction angle, the simulated sample intensity is multiplied by an absorption factor defined by the material and thickness of the detector.

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