Experimental determination of differential scattering coefficients for nickel by means of linearly polarized x-ray radiation

Performing an x-ray scattering analysis can underpin the quantitative description of a sample of interest, for example, to complement an x-ray fluorescence analysis. However, the reliability of such an analytical approach depends on good knowledge of scattering coefficients (or scattering cross-sections), which describe the probability of interaction and are characteristic of each chemical element. In this work, a metrological study of experimentally determining differential Rayleigh and Compton scattering coefficients for nickel is presented. Angular scans of the scattering intensities at different positions are enabled by a flexible experimental set-up and, therefore, allow for the robust determination of differential scattering coefficients at a wide range of forward and backward scattering angles. As a result, scattering coefficients in the range from 0.117(14)×10−3  cm2 g−1 sr−1 to 33.7(39)×10−3  cm2 g−1 sr−1 were determined in the momentum transfer range of 12.1 nm−1 to 22.4 nm−1. In addition, utilizing monochromatized and highly linearly polarized synchrotron radiation ( E0=30keV ) ensures direct comparability to theoretical databases.


Introduction
X-ray radiation scattered by a specimen may contain information about the scattering target at the level of individual electrons, whole atoms or condensed matter [1][2][3][4][5][6]. When the energy of the incident x-ray radiation is sufficient to excite x-ray fluorescence radiation, both interactions, x-ray fluorescence and x-ray scattering, may provide complementary insights because the physical processes involve different interaction details and probabilities. Consequently, investigations of x-ray scattering radiation on a quantitative level were already established during the development of x-ray radiation analysis [7][8][9][10][11]. Applications based on incorporating scattering radiation intensities into an x-ray fluorescence analysis usually focus on gaining information on light matrix elements that emit no or insufficient x-ray fluorescence signals [12][13][14][15]. The acquisition of scattered x-ray radiation in the tender to hard x-ray range with modern energy-dispersive x-ray (EDX) detectors is straightforward. These devices can record and treat x-ray fluorescence and scattering radiation simultaneously. Nevertheless, the difficulty of data evaluation and information extraction for such experiments can range from very easy to highly intricate, depending on the excitation conditions and sample complexity. Since scattering can be elastic (Rayleigh scattering) or inelastic (Compton scattering [16][17][18][19]), it can be challenging to distinguish these phenomena in EDX spectra. This is especially true when contributions from multiplescattering events are significant and increase the spectral overlap of both phenomena. In either case, accurate and reliable descriptions of differential scattering coefficients are required for any quantitative interpretation of x-ray scattering. Note also, that the scattering probabilities are highly anisotropic in general. Thus, coefficients have to be determined for different scattering angles.
Furthermore, conventional sources of monochromatic xray radiation such as isotropic x-ray fluorescence radiation or well-known radioactive sources [20][21][22] usually emit unpolarized x-ray radiation. In contrast, the x-ray radiation from modern synchrotron radiation sources is usually highly linearly polarized and therefore constitutes an ideal tool for studying the directional polarization dependency [18,[23][24][25][26][27][28]. This work presents experimentally determined whole-atom singledifferential Rayleigh and Compton scattering coefficients for pure nickel at different scattering angles and thereby varying momentum transfer (q = E0 hc sin θ 2 ) in the range of about 12 nm −1 to 22 nm −1 . The utilization of highly polarized synchrotron radiation allows the determination of polarized coefficients, which are directly comparable to theoretical databases without averaging over different polarization directions. Moreover, a flexible set-up enables the rotational variation of the angle of incidence and observation, and also both scattering angles, resulting in a robust experimental approach.

Theoretical section
When x-ray radiation is scattered by a target, the energy of the incident radiation may be reduced (inelastic scattering) or remain practically unchanged (elastic scattering). Moreover, Figure 1. Schematic depiction of the geometric vectors describing a scattering process. The direction of the incident radiation (k 0 ) and the scattered radiation (k) are shown. Furthermore, the definitions of the polar scattering angle θ and the azimuthal scattering angle ϕ are illustrated. An example of a polarization direction for the incident radiation (ξ 0 ) is shown in the y direction, corresponding to linear parallel polarization. the scattered radiation may have a different direction and polarization state than the incident radiation. Not all of these changes in energy, direction, and polarization have the same probability: When describing the incident radiation by the direction of propagation k 0 and the polarization direction ξ 0 , as well as the scattered radiation by k and ξ, a specific scattering probability is given by the differential mass scattering coefficient dσ dΩ (unit cm 2 g −1 sr −1 ). Equivalently, this probability can also be described by a differential scattering crosssection (unit cm 2 sr −1 ). The scattering coefficient dσ dΩ is single differential with respect to the solid angle of detection Ω. Usually, the detector utilized to collect the scattered radiation is not sensitive to the polarization state of the radiation. Therefore, the experimentally observed scattering intensity will depend only on the relations between k 0 , k, and ξ 0 (and not ξ). These relations can be described by two scattering angles (cf figure 1). Firstly, there is the polar scattering angle θ, that is, the angle between the directions of incident x-ray radiation (k 0 ) and detected scattered radiation (k). In addition, there is the azimuthal scattering angle ϕ, that is, the angle between the scattering plane (k 0 , k) and the polarization plane (k 0 , ξ 0 ). If the incident radiation is unpolarized, the dependence on the azimuthal angle ϕ vanishes [8,9,29]. In contrast, the dependence on the azimuthal angle ϕ can be observed when utilizing linearly polarized incident radiation. Such radiation can be induced and investigated by dedicated polarimeters [30][31][32] and, moreover, is usually available at synchrotron radiation facilities [33][34][35][36].
The classical Thomson formula [37] quantitatively describes the elastic scattering of x-ray radiation by free electrons. However, intricate interaction effects in the atom must be considered if the scattering targets are bound electrons instead. In this case, the scattering process is called Rayleigh scattering. Likewise, the Klein-Nishina formula [38,39] can describe the corresponding inelastic scattering process for free electrons. In that case, atomic systems give rise to Compton scattering [16]. This work will use the following representation for the differential scattering coefficients: dσ R,C P,U dΩ . The subscripts of the coefficients describe the polarization state of the incident radiation, that is, (linearly) polarized (P) or unpolarized (U); and the superscripts describe the scattering process, that is, Rayleigh (R) or Compton (C) scattering. The following equation (1) can be used to experimentally determine the polarized-differential scattering coefficient based on the total number of (Rayleigh or Compton) scattered photons per second reaching a detector. This equation is similar to the well-known fundamental parameter equation for x-ray fluorescence radiation published by Sherman [40,41] and used for reference-free x-ray fluorescence analysis [42,43]. Assuming a completely linearly polarized and monochromatic excitation radiation of photon energy E 0 and a singular scattering event, that is, neglecting higher-order processes, the polarizeddifferential scattering coefficient is given by: Here, dσP dΩ is the polarized-differential scattering coefficient (cm 2 g −1 sr −1 ) for Rayleigh or Compton scattering; N the detectable photon flux of the scattered radiation emitted into the solid angle of the detector Ω (sr); N 0 the incident photon flux impinging on the sample; ϵ i the detection efficiency of the detector at the incidence photon energy E i = E 0 for Rayleigh scattering or at the Compton energy E i = E C (θ) for Compton scattering; ρd the mass thickness of the sample (g cm −2 ), that is, the product of the density ρ and the sample thickness d; µ = µE 0 sin α + µE i sin β the effective mass attenuation coefficient (cm 2 g −1 ) of the sample, with µ E0 and µ EC being the mass attenuation coefficients of the sample at the excitation photon energy and the Compton energy, respectively; and α and β are the angles of incidence and observation defined with respect to the sample surface plane, respectively. Equation (1) is used for measurements in reflection geometry (source and detector on the same side with regard to the sample surface), as depicted in figure 1 and is usually utilized for backward scattering (θ > 90 • ). For measurements in a transmission geometry (source and detector on different sides of the sample), which is usually employed for forward scattering (θ < 90 • ), the last factor of equation (1) is modified to account for different absorption paths in the sample: The unpolarized-differential scattering coefficients dσU(θ,E0) dΩ can be derived from the polarized-differential scattering coefficients dσP(θ,ϕ,E0) dΩ simply by averaging over polarization directions of the incident radiation, that is, averaging over the azimuthal scattering angle ϕ. Furthermore, the following experimental effects can influence the detected scattering intensities and need to be taken into account.

Acceptance angle of detector
Both equations (1) and (2) are valid for situations when the differential scattering coefficient can be assumed not to change rapidly over the detector's active area with regard to both scattering angles θ and ϕ. Otherwise, integration over the observable angular range is required, as given by the acceptance angle of the detector, defined by its geometry [44]. For the detector geometry used in this work, that gives only an appreciable contribution for configurations when almost no scattering is expected (θ ≈ 90 • and ϕ ≈ 0 • ). Accordingly, a Monte Carlo simulation based on the known detector geometry was utilized for these cases to correct the measured scattering intensities. The extreme case of θ = 90 • and ϕ = 0 • results in a correction of about 12%. For most scattering angles in this work, if |θ − 90 • | ⩾ 10 • or ϕ ⩾ 10 • , this correction becomes negligible.

Partially polarized radiation
In an idealized manner, equations (1) and (2) for the polarizeddifferential scattering coefficient dσP dΩ assume completely linearly polarized incident radiation, meaning that it is entirely polarized in a single direction. If the incident radiation is only partially polarized, it can be regarded as composed of two perpendicular linear polarization states with different intensities (I ∥ and I ⊥ ). The total intensity of the incident radiation is then trivially given by I 0 = I ∥ + I ⊥ . In principle, the choice of the two directions is arbitrary and will not influence the result I 0 . For synchrotron radiation, the natural choice is to define two parallel and perpendicular directions to the storage ring plane, respectively, while also being perpendicular to the direction of propagation of the incident radiation. Quantitative contributions of both polarization directions are taken into account by the degree of (linear) polarization P [36,45,46], here defined as: The incident radiation is completely parallel linearly polarized for P = 1, and for P = 0, the radiation is unpolarized. Perpendicular linear polarization states are described by values below zero, and partial linear polarization refers to values between 0 < |P| < 1. The differential scattering coefficients for completely linear polarization can be derived even if the incident radiation is only partially polarized, provided that P is known [24,36]. In that case, the effective scattering coefficients experimentally determined in one specific direction according to equation (1), for example, the (horizontal) plane of the storage ring ( dσ ∥ dΩ ) and the direction vertical to this plane ( dσ ⊥ dΩ ) are used to derive the actual differential scattering coefficients of completely parallel and completely perpendicular linear polarization of the incident radiation according to: Here, the symbol dσP(ϕ =0 • ) dΩ shall denote the scattering coefficients for the configuration when the polarization direction is in the scattering plane (also called in-plane), and dσP(ϕ =90 • ) dΩ shall denote the coefficients for when the polarization direction is perpendicular to the scattering plane (out-of-plane).
It should be noted that equations (4) and (5) are undefined if the incident radiation is unpolarized since no polarizationdependent information can be extracted in that case. Furthermore, the utilized synchrotron radiation is mostly polarized in the horizontal plane of the storage ring. Therefore, P will be close to unity. Measurements at the polar angular position θ = 90 • are used to determine the degree of polarization P experimentally. The parallel linear Rayleigh scattering coefficient has a minimum in this polar angular region. In the formfactor approximation, this value indeed vanishes [34,47] , and equation (4) yields: While the value for real atomic systems is expected to be non-zero, it is very low. For example, for nickel, it can be calculated [48] to be about 2 × 10 −4 cm 2 g −1 sr −1 . However, considering this value in equation (6) only leads to a slight modification, with a relative difference of about 10 −4 . Accordingly, the degree of polarization was determined to be about P = 99.1% during the presented measurements.

Higher-order contributions
Contributions due to higher-order processes also need to be considered [49]. These processes include, for example, Compton-Rayleigh or Rayleigh-Rayleigh multiple-scattering events, and they are simulated utilizing XMI-MSIM, a freely available Monte Carlo code [50]. This software relies on tabulated scattering cross-sections from the xraylib database [51]. The experimentally determined photon flux N exp. is corrected by a factor h ⩾ 0 according to N = Nexp.
1+h . The factor h is derived from the XMI-MSIM code by comparing the intensity of a singular interaction event N (i.e. a singular photoabsorption event, Rayleigh scattering event or Compton scattering event) to that of higher-order contributions of up to the fourth order. For all employed scattering angles, this correction is typically h ≈ 2%. Upper bounds on the uncertainties of these higher-order contributions are estimated based on previous works using XMI-MSIM [15,52,53].

Experimental section
Experiments were performed at the wavelength-shifter beamline BAMline [54], located at BESSY II, an electron storage ring for synchrotron radiation. The beamline was operated with a double-crystal monochromator, providing x-ray radiation with a photon energy of E 0 = 30 keV and a beam width of about 1 mm. In general, the beamline provides highly linearly polarized and monochromatic x-ray radiation with photon energies in the energy range of about 5 keV to 60 keV. Different end-stations can be set up in an easily accessible hutch in ambient air, thus, allowing fast sample exchanges and flexible experimental set-ups. The in-house designed end-station [55] used in this work consists of a silicon-drift detector (SDD) mounted on a 2-axis manipulator, which can be rotated around the sample surface. Likewise, the sample holder is mounted on a 2-axis manipulator and can be rotated around the same pivotal point, cf figure 2. This set-up allows the orientation of the detector in space in reference to the direction of the incidence radiation while keeping the sample surface in the field of view of the detector. Thus, polar scattering angles θ in the range of about 30 • to 150 • and azimuthal scattering angles ϕ in the range of 0 • to 90 • can be investigated. For this work, polar scattering angles from 60 • to 90 • and from 90 • to 135 • were considered with a step size of 10 • and 5 • , respectively. Azimuthal scattering angles were 0 • , 45 • , and 90 • . Furthermore, the incidence and observation angle of the x-ray radiation can be selected through sample rotation. The distance from the sample surface to the detector is fixed by the design at about 5 cm. This distance is related to the trade-off between a large solid angle of detection (less sampledetector distance is better) and the effort to reduce measurement errors due to the collection of a multitude of scattering angles (more distance is better). While a larger solid angle of detection allows faster collection times, precise quantitative results require a well-defined scattering angle. The xray attenuation for the photon pathway from the sample surface to the SDD due to ambient air was taken into account by means of Lambert-Beer's law for the transmitted intensity T air = exp (−µ air (E) ρ air d air ). Experimentally determined mass attenuation coefficients µ air (E) of air were taken from the literature [56]. The air distance d air was determined with a commercial laser scanning system. In addition, for this work, a tapered collimator made out of copper was placed between the sample and the SDD to minimize contributions due to stray-light scattering.
The sample was a commercially available free-standing nickel foil of high purity with a thickness of nominally 25 µm. The SDD simultaneously collected x-ray fluorescence radiation, Rayleigh scattering radiation, and Compton scattering radiation with an average collection time of about 700 s per spectrum. The response function and spectral detection efficiency of the SDD can be determined in a wide spectral range [57][58][59]. Reliable knowledge about the detector response function can be used for the deconvolution of the spectra and even for significant spectral overlapping of Rayleigh scattering and Compton scattering contributions-assuming the individual spectral shapes of these contributions are known (i.e. the Compton profile [1]). For this work, the detection efficiency of the SDD ϵ i can be described by the thickness (d Si ) of the active volume of the detector (consisting primarily of pure silicon) and the energy-dependent absorption length (L Si ). Therefore, the detection efficiency was derived according to . For example, based on the nominal thickness d Si = 450 µm from the manufacturer, a value of ϵ E0 = 1 − exp (−0.121) = 0.114 can be obtained at the incidence photon energy E 0 = 30 keV.
Similar to the approach of Shahi et al [60], nickel Kα fluorescence radiation is used to derive the product of the incidence photon flux N 0 and the solid angle of detection Ω independently for each measurement position, that is, for each scattering angle combination. To this end, a reference-free x-ray fluorescence analysis [42,43] is employed according to the following formula: Here, N Kα is the photon flux of nickel Kα fluorescence radiation, corrected for attenuation due to air in the same manner as before; ϵ EKα is the detection efficiency of the detector at the photon energy of nickel Kα fluorescence radiation; τ is the partial mass photoelectric absorption coefficient for nickel Kα fluorescence radiation; ω the fluorescence yield for the nickel K shell; g the total transition probability for nickel Kα fluorescence radiation; and µ ′ = µE 0 sin α + µE Kα sin β is the effective mass attenuation coefficient of the sample, with µ EKα being the mass attenuation coefficient of the sample at the photon energy of nickel Kα fluorescence radiation. Once more, it is worth noting that equation (7) is used for measurements in reflection geometry, as shown in figures 1 and 2. For measurements in transmission geometry, equation (7) has to be adapted similarly to equation (2). Most of the fundamental parameters required for the x-ray fluorescence analysis were experimentally determined and are readily available [61]. Again, contributions due to higher-order processes are taken into account based on calculations with XMI-MSIM, as described above. They are in the order of about 3% for the chosen experimental conditions.
A photodiode installed downstream from the sample holder is used to measure the sample transmission in reference to an upstream ionization chamber. This approach is used to determine the sample mass thickness (ρd) independently since both processes, x-ray scattering and fluorescence, depend on this value: Here, S D and S I are the signals of the photodiode and ionization chamber, respectively, with the sample in the x-ray pathway. S 0 D , S 0 I are the signals in a reference measurement, that is, without any sample in place. It is worth noting that this measurement is not influenced by the x-ray attenuation due to air. The experimentally determined mass thickness of the available nickel specimen was ρd = 23.0(9) mg cm 2 .

Results and discussion
Polarized-differential Rayleigh scattering coefficients were determined according to equation (1) for ϕ = 45 • , to equation (4) for ϕ = 0 • , and to equation (5) for ϕ = 90 • . These results are listed in table 1 and shown in figure 3. In addition, the results of the corresponding Compton coefficients are listed in table 2 and shown in figure 4. A direct comparison to independent experimental values would be desirable, but none exist to the best of our knowledge (see also [62]). Similarly, comprehensive and accessible theoretical compilations are scarce since wide ranges of chemical elements, of incident photon energies, and of both scattering angles simultaneously (for polarized-incident radiation) pose high demands. Nevertheless,  [65] (1983). Values from the first two and the last publications [63][64][65] are derived in the often-employed form factor approximation [47]: Here, σ T P is the classical Thomson free-electron scattering coefficient for polarized-incident radiation, and F is the atomic form factor of the given element. While xraylib uses F = f 0 , with the coherent form factor f 0 as tabulated in Hubbell et al [63], EPDL2017 lists values in terms of F = f 0 + f 1 + if 2 , with anomalous scattering factors f 1 and f 2 . Values from the RTAB compilation [48] are based on more recent S-matrix calculations. They are available as unpolarized-differential Rayleigh cross-sections dσ R U dΩ and also in terms of complex scattering amplitudes for parallel and perpendicular linear polarization A p , A s . Here, these amplitudes were used to calculate polarized-differential Rayleigh coefficients based on the following relationship: The results of the Compton scattering coefficients in table 2 are compared to data from theoretical compilations by Cullen et al [66] (1997; EPDL97)-as extracted from xraylib [51] (version 4.1.3)-and by Kahane [67] (1998). Since all the cited databases are available on a finite and irregularly spaced grid, cubic spline interpolation was used to interpolate the data retrieved from them [51,68]. Although this introduces a systematic error, this is a relatively small effect because the compilations provide sufficiently fine grids.
The experimental uncertainty budget of one example of a differential scattering coefficient is given in table 3. It is calculated according to the Guide to the expression of uncertainty in measurement [69]. Care was taken to include all known influencing factors, such as experimental, instrumental, and x-ray fundamental parameters. The determination of the product of the incidence photon flux and the effective solid angle of detection through a quantitative x-ray fluorescence analysis introduces non-negligible contributions to the uncertainty. This is mainly due to the product of the fluorescence yield and the mass photoelectric absorption coefficient. In principle, the absolute uncertainty could be reduced by using a well-calibrated aperture in the detection channel, defining the solid angle of detection and avoiding the requirement of x-ray fluorescence detection. However, this would further increase the measurement durations, caused by reducing the detectable photon flux. Not surprisingly, the uncertainty of the detector efficiency contributes strongly, and an independent and accurate determination of this is essential. Generally, the detector efficiency can be determined experimentally with further reduced uncertainty by additional dedicated experiments [58].
Overall, good agreement can be observed between the experimentally determined Rayleigh coefficients and the values from Kissel [48] and Cullen [64] if the given uncertainty interval is taken into account. For increasing backward scattering angles θ > 90 • , deviations from the values from Schaupp et al [65] and Hubbell et al [63] increase for ϕ = 0 • . In this case, the influence of the anomalous scattering factors increases and uncertainty in the axis of abscissae dominates the experimental uncertainties. This deviation is still noticeable for azimuthal scattering angles ϕ > 0 • , but it is not as distinct as for ϕ = 0 • because of increased experimental uncertainties. The regime of very low scattering intensities in the vicinity of θ = 90 • and ϕ = 0 • shows large deviations in comparison to all four theoretical databases. This regime is susceptible to minor experimental errors, for example, regarding the determination of the degree of polarization and the acceptance angle of the detector. Moreover,the Compton coefficient results generally show good agreement with both databases Table 2. Experimental results for the whole-atom polarized-differential Compton scattering coefficient of nickel for different scattering angles. The momentum transfer q = E0 hc sin θ 2 is given for convenience (E 0 = 30 keV). Additionally, values from the following theoretical compilations are given for the sake of comparison: Cullen et al [66]; (1997; EPDL97, as extracted from xraylib [51]) and Kahane [67] (1998). (Cullen et al [66] and Kahane [67]). Similar to the results of the Rayleigh coefficients, larger deviations are unavoidable for the regime of low scattering intensities. Apart from this, the agreement between theoretical and experimental values is in the order of a few per cent and is within the determined experimental uncertainty.
In principle, Bragg diffraction can lead to a modification of the detected intensity because it is not unlikely that at least polycrystalline structures are present in the nickel sample. Nevertheless, no sudden or oscillatory intensity changes were observable when changing the angle of incidence, as would be expected for diffraction effects. Furthermore, these effects would be expected for momentum transfer values smaller than those employed in this work [70]. Additionally, it was concluded that unwanted scattering signals from the ambient air did not have any noticeable influence: A Monte Carlo simulation based on the XMI-MSIM software and a measurement without any sample in place indicated no significant contributions. Experimental results for the whole-atom polarized-differential Compton scattering coefficient of nickel for different scattering angles. Additionally, values from the following theoretical compilations are given for the sake of comparison: Cullen et al [66] (1997; EPDL97, as extracted from xraylib [51]) and Kahane [67] (1998).

Conclusion
This work presents experimentally determined polarizeddifferential Rayleigh and Compton scattering coefficients for nickel at E 0 = 30 keV and at multiple polar and azimuthal scattering angles. Consequently, scattering coefficients in the momentum transfer range of 12.1 nm −1 to 22.4 nm −1 are accessible, and the determined coefficients range from 0.200(30) × 10 −3 cm 2 g −1 sr −1 to 33.7(39) × 10 −3 cm 2 g −1 sr −1 for Rayleigh scattering and 0.117(14) × 10 −3 cm 2 g −1 sr −1 to 17.3(12) × 10 −3 cm 2 g −1 sr −1 for Compton scattering. Deviations from theoretical compilations are generally within the range of experimental uncertainty. Nevertheless, a possible reduction of experimental uncertainty is desirable. Here, the detection channel presents itself as the main uncertainty contribution. The effective solid angle of detection, the detection efficiency and, for small polar scattering angles, the spectral deconvolution are the primary sources of experimental uncertainties for this work. While a calibrated aperture in front of the detector can reduce contributions from the solid angle of detection, this would simultaneously further reduce the available x-ray scattering photon flux and increase collection times. Similarly, improvements in the determination of the detection efficiency are generally feasible but will also encompass an increase in the required experimental effort. As shown, several experimental aspects pertaining to alterations in the scattering intensities need to be taken into account, such as the acceptance angle of the detector and contributions of higher-order scattering effects. Nevertheless, this work highlights the capabilities of a quantitative and reference-free x-ray fluorescence analysis by incorporating complementary information based on x-ray scattering.

Data availability statement
The data generated for this work is openly available in the Open Access Repository of the PTB at www.doi.org/10.7795/ 720.20230131.