Polarization transfer in Rayleigh scattering of hard x-rays

Elastic scattering of hard x-rays by atoms is a fundamental process which is usually described as a coherent sum of scattering from the individual atomic constituents. Scattering from bound electrons is referred to as Rayleigh scattering [14] and dominates the total cross section in a broad energy range from a few keV up to the MeV range. At even higher energies, scattering from vacuum fluctuations in the atomic field (Delbrück scattering [37]) and nuclear scattering [3, 15] become important. Comprehensive reviews on the topic of elastic x-ray scattering by atoms have been given by Kane et al [23] in 1986 and Bradley et al [5] in 1999.

Recent theoretical investigations of elastic scattering focus on the effects of photon polarization [30, 31, 46, 47, 57, 58]. For all basic photon-matter processes, the features of polarization transfer from the incident to the outgoing beam are of particular importance as they enable tests of theory methods that are more stringent than measuring just total or differential interaction cross sections. Vice versa, with an established theory the incident polarization states of the photons can be reconstructed from the measured polarization of the outgoing photons. However, with respect to Rayleigh scattering, experiments involving (linearly) polarized hard x-rays were up to now restricted to scenarios where either the differential cross section for a linearly polarized incident beam [6, 7, 20, 51, 52] or the polarization of the scattered beam for an unpolarized incident beam [16, 32, 50, 53, 54, 70] was measured. Several of those experiments were compared to theoretical predictions in [44].

For linearly polarized incident x-ray beams a direct measurement of the linear polarization for the outgoing elastically scattered photons was not feasible due to technological limitations: for hard x-ray energies, polarized sources were not intense and polarimeters not efficient enough to compensate for the small elastic scattering cross section, especially at large scattering angles. These difficulties were recently overcome with the advent of third-generation synchrotron radiation facilities and the availability of highly efficient Compton polarimeters. The combination of both instruments now enables scattering experiments with independent determinations of the linear polarizations of the incident and scattered hard x-rays, therefore providing the most stringent test of the theoretical understanding of the underlying processes. Similar experiments were performed recently, where the polarization transfer from spin-polarized electrons to bremsstrahlung photons was investigated [28, 34, 60, 61]. Moreover, these measurements demonstrated that utilizing theoretical estimates of the polarization transfer characteristics enables an accurate reconstruction of the incident beam polarization. As an example, such schemes for polarization diagnostics of particle and photon beams are required for exploring atomic parity violation phenomena [59, 71].

In this report, we present the first direct measurement of the linear polarization of elastically scattered hard x-rays with a highly linearly polarized incident beam. The experiment was performed at the High Energy Materials Science Beamline P07 [48] of the third-generation synchrotron radiation facility PETRA III at DESY. Details on the setup were discussed in [4], so here we restrict ourselves to briefly describing the main aspects. A gold (high Z) scatterer and a high photon energy of 175 keV were chosen, a regime where Rayleigh scattering is the dominant elastic process and relativistic effects, such as non-vanishing contributions from higher-order multipoles, are predicted to have significant influence. The target was a foil thin enough to neglect multiple-scattering contributions (1.036 μm compared to a mean free path length of the incident photons of 400 μm). A coplanar scattering geometry was chosen, i.e. the scattered photons were observed in the polarization plane of the incident beam. The polarization of the scattered radiation was measured with a thick-crystal, double-sided segmented Si(Li) detector applied as a Compton polarimeter [42, 66], that was placed at polar scattering angles of θ = 65°, 90° and 120°. This polarimeter consists of a single 7 mm thick lithium-drifted silicon crystal with an active area of 64 mm × 64 mm which is segmented into 32 strips on the front side and—orthogonal to these—32 strips on the back side. Each of these strips acts as an individual detector with an energy resolution (FWHM) of 2.5 keV at 60 keV. The combination of energy sensitivity with the position resolution of the segmented detector enables one to apply Compton polarimetry, see [68] for a detailed description of the method. This technique for hard x-ray polarization measurements was introduced in the 1950s [38] and was widely used and improved since then, see [29] for a comprehensive overview. Examples for recent Compton polarimetry experiments in the field of atomic physics can be found in [2, 8–10, 13, 22, 49, 60–62] and, in particular with detectors of the type used in the present work, in [18, 19, 33, 34, 69]. Selected experiments on the polarimetry of hard x-rays using thick-crystal, double-sided segmented semiconductor detectors were also recently reviewed in [65].

As an illustration of the analyzed radiation in the current experiment, an energy spectrum of single hits (only one registered photon interaction inside the polarimeter per event, which is in most of the cases related to photoabsorption of the incident photon) is shown in figure 1. Its main features are the characteristic Kα and Kβ fluorescence lines from the gold target, the broad Compton peak of inelastically scattered photons and the narrow Rayleigh peak (elastic scattering) at the incident beam energy of 175 keV.

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For the polarization determination we analyze only double hits (two recorded energy depositions at different positions in the polarimeter per event) as they are candidates for Compton scattering within the detector crystal (one energy deposition being associated to the Compton-scattered photon and the other to the recoil electron). Note that according to the Klein–Nishina equation [27] Compton scattering in the direction perpendicular to the incident photon's electric field vector is preferred, while scattering in the parallel direction is less probable. Thus, the linear polarization characteristics of the x-rays impinging on the detector can be reconstructed from the azimuthal position distribution of Compton-scattered photons inside the detector. As a result of such a polarization study, one usually obtains the degree of linear polarization ${P}_{{\rm{L}}}$ and the orientation ${\varphi }_{0}$ of the electric field vector (here: with respect to the plane defined by the directions of the initial and Rayleigh-scattered photons) of the analyzed photon beam. A common way to obtain the polarization parameters is to adjust the Klein–Nishina cross section for a photon beam to the measured azimuthal Compton scattering distribution with ${P}_{{\rm{L}}}$ and ${\varphi }_{0}$ as free parameters. However, for a precise reconstruction of both values it is necessary to take into account geometrical effects resulting from the individual detector characteristics (e. g. the pixel structure) that lead to a deviation of the measured distribution from the ideal behavior of the theoretical cross section. As it was shown in [67], the observed detector response to incident x-rays can be reproduced in great detail when using a Monte Carlo simulation to generate artificial data sets that are processed by the same analysis routine as is used for the experimental data. A possible way to correct for the detector-dependent effects in the scattering distribution is presented in [68], where the measured distribution was normalized to the simulated detector response to unpolarized radiation (i.e. a uniform scattering distribution).

In the present work, we employed a slightly different approach: the measured scattering distribution was fitted with a linear combination of four Monte-Carlo-generated distributions which correspond to the detector response to 100% linearly polarized radiation, oriented in directions of 0°, 45°, 90° and 135° with respect to the reaction plane of the initial scattering in the target foil. Here the intensities I₀, I₉₀, I₄₅ and I₁₃₅ were treated as free parameters. Now it is convenient to express the (linear) polarization characteristics of the incident radiation in terms of the corresponding Stokes parameters [36, 55] ${P}_{1}={P}_{{\rm{L}}}{\rm{\cos }}(2{\varphi }_{0})$ and ${P}_{2}={P}_{{\rm{L}}}{\rm{\sin }}(2{\varphi }_{0})$. These Stokes parameters follow directly from the intensities I_χ that were obtained in the fitting procedure as ${P}_{1}=({I}_{0}-{I}_{90})/({I}_{0}+{I}_{90})$ and ${P}_{2}=({I}_{45}-{I}_{135})/({I}_{45}+{I}_{135})$. A bootstrap resampling procedure [11] of the experimental and the Monte Carlo spectra was employed to estimate the statistical uncertainties of P₁ and P₂. Note that for Rayleigh (and Compton) scattering on unpolarized target electrons and scattering in the polarization plane of the incident x-ray beam, the outgoing photon polarization is expected to be oriented perpendicular (${P}_{1}\lt 0$) or parallel (${P}_{1}\gt 0$) to the scattering plane, resulting in ${P}_{{\rm{L}}}=| {P}_{1}|$ and ${P}_{2}=0$. To distinguish the (linear) polarization characteristics of the Rayleigh-scattered radiation from the Compton-scattered radiation (both scattering events taking place in the target foil), in the following we shall denote the Stokes parameters of the former with ${P}^{{\rm{R}}}$ while those for the Compton scattering will be denoted by ${P}^{{\rm{C}}}$. Moreover, the linear polarization of the synchrotron radiation impinging on the target foil is denoted by ${P}^{{\rm{i}}}$. In order to determine the Stokes parameters ${P}_{1}^{{\rm{R}}}$ and ${P}_{2}^{{\rm{R}}}$ of the elastically scattered photons from the gold target, in the polarimetry analysis only those events are taken into account where the sum energy of the recoil electron and the Compton-scattered photon (inside the detector) lies in the energy range of the incident photon (chosen here: 173–178 keV, i.e. the Rayleigh peak in figure 1). An example of a measured scattering distribution together with the Monte Carlo fit is shown in figure 2.

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The results of the polarization measurement of the Rayleigh-scattered radiation are shown in figure 3. Only ${P}_{1}^{{\rm{R}}}$ is plotted because the values for ${P}_{2}^{{\rm{R}}}$ are consistent with zero at all observation angles proving that the polarimeter was aligned with the incident beam polarization plane as intended (as a consequence also ${P}_{2}^{{\rm{i}}}=0$). Due to low statistics (compared to previous experiments with the same polarimeter, for example [69]), we considered only statistical uncertainties as they dominate the overall error margin. The comparatively large uncertainty of the $\theta =90^\circ$ polarization value is due to the fact that the angle-differential Rayleigh cross section has its minimum near this observation angle (in coplanar geometry, i.e. scattering along the polarization direction of the incident beam), resulting in an even larger statistical uncertainty.

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We observe a strong depolarization of the Rayleigh-scattered radiation in the region near $\theta =90^\circ$. To explain this finding, we compared our result with fully relativistic (i.e. beyond dipole approximation) theoretical calculations. These have been performed within a Furry picture where the electron–nucleus and (partially) electron–electron interactions are included in the unperturbed Hamiltonian and the coupling to the radiation field is treated perturbatively. In such an approach, all the properties of the Rayleigh scattering can be traced back to the evaluation of the second-order amplitudes [1, 17, 45, 46, 57]. This requires a summation over the complete spectrum of the target atom, including not only bound, but also positive and negative energy continuum-states. In order to perform this non-trivial summation over many-electron states, we employed the independent particle approximation (IPA) in which the photon is scattered by a single active electron at a time, while the remaining electrons are kept 'frozen' [43, 58]. Such an approximation is known to work fairly well for high-energy photons and heavy targets [41]. In order to partially account for the interaction among the electrons in the target atom, the initial, final and intermediate single-particle states are taken as solutions of the Dirac equation with a screened potential ${V}_{{\rm{scr}}}(r)$, generated within the Dirac–Fock theory. Such a 'screened' IPA model has been successfully applied in the past for the analysis of both Rayleigh scattering [43, 58] and two-photon decay [56] of many-electron atoms and ions. In order to investigate the many-body effects beyond the IPA, we have recently applied the rigorous quantum electrodynamics approach. Based on this approach we have shown that the electron–electron interaction effects do not exceed 3%−4% for both the angular distribution and the linear polarization of the scattered photons [64]. In the angular region of the present experiment ($\theta \geqslant 65^\circ$) we do not expect significant effects which arise from the solid state of the scattering target (compared to the scenario of an isolated atom which is assumed in the theory discussed above). This can be justified by the findings in [58] which indicate that scattering from outer shells (the ones relevant for solid state effects) is only relevant for very small observation angles.

For a theoretical prediction of ${P}_{1}^{{\rm{R}}}$ the Stokes parameter ${P}_{1}^{{\rm{i}}}$ of the incident PETRA-III beam is required (${P}_{2}^{{\rm{i}}}$ is not needed for coplanar scattering). The synchrotron beam is expected to be nearly 100% linearly polarized, therefore as a first guess we use ${P}_{1}^{{\rm{i}}}=1$ as input for the theoretical calculations. This leads to ${P}_{1}^{{\rm{R}}}\approx 1$ (exactly 1 for a closed-shell atom) for all angles (dashed line in figure 3), which clearly deviates from the experimental findings. In our experiment, we were able to independently obtain ${P}_{1}^{{\rm{i}}}$ from the measured Stokes parameters ${P}_{1}^{{\rm{C}}}$ and ${P}_{2}^{{\rm{C}}}$ of the Compton-scattered photons (from the gold target), which are related to the Stokes parameters of the incident beam via the transfer matrix for Compton scattering [12] integrated over the solid angle covered by the active area of the detector. For the scattering angles θ = 65° and 120° values for ${P}_{1}^{{\rm{i}}}$ and ${P}_{2}^{{\rm{i}}}$ could be reconstructed, but not for 90° due to a singularity of the transfer matrix at this specific scattering angle. Assuming that the incident beam polarization did not vary during the experiment, we used the weighted mean values of ${P}_{1}^{{\rm{i}}}$ and ${P}_{2}^{{\rm{i}}}$ as a result. Their errors were conservatively estimated as the maximum difference between the weighted mean and the individual values at both observation angles. With this we obtained ${P}_{1}^{{\rm{i}}}=+0.9801\pm 0.0093$ and ${P}_{2}^{{\rm{i}}}=-0.011\pm 0.019$. The positive ${P}_{1}^{{\rm{i}}}$ together with the small value of ${P}_{2}^{{\rm{i}}}$, which agrees with zero, confirms again the initial assumption that the incident beam was polarized in the scattering plane (coplanar geometry). In contrary to the polarization measurement for the Rayleigh-scattered radiation, the uncertainty of the reconstructed polarization of the Compton-scattered x-rays (and consequently also the estimate for ${P}_{1}^{{\rm{i}}}$) is dominated by systematic effects, such as the unknown shape of the background under the broad Compton peak.

For the initial beam polarization ${P}_{1}^{{\rm{i}}}=+0.9801$ the corresponding theoretical prediction is depicted by the solid line in figure 3. This theoretical result agrees very well with the experimental data, showing that the strong depolarization of the Rayleigh-scattered photons originates from the polarization impurity of the incident beam. This finding indicates that the polarization of the Rayleigh-scattered beam is highly sensitive to the polarization of the incident beam, especially around $\theta =90^\circ$. It should be noted that the shown prediction does not take into account the finite detector size, which for each observation angle leads to a (slight) deviation from the values calculated for an infinitely small detector. For completeness, calculations for a point-like and an extended detector with their respective uncertainties are compared with the experimental data in table 1. For the predictions of ${P}_{1}^{{\rm{R}}}$ the uncertainty of ${P}_{1}^{{\rm{i}}}$ and uncertainties of the observation angles θ and φ ($\pm 1^\circ$ for both) are taken into account. Within the resulting error bars the predictions for a point-like and an extended detector both agree with the experimental data. The large uncertainties in the predictions at $\theta =90^\circ$ are due to the previously found high sensitivity of ${P}_{1}^{{\rm{R}}}$ to ${P}_{1}^{{\rm{i}}}$, which magnifies the uncertainty resulting from ${\rm{\Delta }}{P}_{1}^{{\rm{i}}}$.

In order to explain this high sensitivity, it is convenient to decompose the incident beam into beams of photons which are polarized parallel and perpendicular to the scattering plane. These portions of the beam are scattered independently from each other with amplitudes A_∥ and A_⊥, respectively. In the form factor approximation, one has ${A}_{}={A}_{\perp }{\rm{\cos }}(\theta )$ [26], so that A_∥ vanishes at $\theta =90^\circ$, i.e. coplanar scattering is strictly forbidden at that angle (perfect polarization filter). In fully relativistic S-matrix calculations A_∥ is finite at all angles [58], but around $\theta =90^\circ$, one still has $| {A}_{}| \ll | {A}_{\perp }|$ [21] (almost perfect polarization filter). Therefore even a small admixture of perpendicularly polarized photons in the incident beam can lead to a large fraction of perpendicularly polarized photons in the scattered beam at 90° and therefore to the pronounced depolarization and polarization sensitivity that were observed. In angular regions away from $\theta =90^\circ$ the effect is weaker because $| {A}_{}|$ and $| {A}_{\perp }|$ are of the same order. In figure 4, ${P}_{1}^{{\rm{R}}}$ is shown as a function of ${P}_{1}^{{\rm{i}}}$ for the scattering angles covered in the present experiment. The sensitivity can be directly identified with the derivative and it is clearly highest for $\theta =90^\circ$ and ${P}_{1}^{{\rm{i}}}\to +1$.

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The reader might note, that the pronounced difference in the magnitude of the Rayleigh scattering amplitudes near 90° leads to a significant increase of the scattering intensity when the incident beam linear polarization deviates from 100%. Thus, it would be possible to infer a possible polarization impurity from a deviation of the measured cross section at observation angles near 90°. However, obtaining absolute cross sections is a challenging task that is often hampered by uncertainties of incident beam flux, target thickness, detector efficiency, covered solid angle, etc so that typically uncertainties are of the order of 30% or more. In contrast, the determination of the linear polarization of the Rayleigh-scattered photons boils down to the analysis of the emission pattern of Compton scattering inside the detector crystal, i.e. a relative measurement that is unaffected by the systematic uncertainties mentioned above. Nevertheless, we plan to perform cross section measurements of polarization-dependent Rayleigh scattering as a follow-up study to the present work.

While the high sensitivity makes it difficult to predict ${P}_{1}^{{\rm{R}}}$, it can be exploited when the theory is applied inversely, i.e. when ${P}_{1}^{{\rm{i}}}$ is reconstructed from the measured ${P}_{1}^{{\rm{R}}}$. In such a procedure, a high sensitivity allows to determine ${P}_{1}^{{\rm{i}}}$ very precisely, even though the uncertainty of ${P}_{1}^{{\rm{R}}}$ is relatively large. The results for the reconstruction of ${P}_{1}^{{\rm{i}}}$ in the present experiment are also shown in figure 4. Of all the angles covered in the present experiment, the sensitivity is lowest at $\theta =120^\circ$, resulting in the largest uncertainty of the reconstructed ${P}_{1}^{{\rm{i}}}$. At 65° and 90°, these uncertainties are comparable, even though the sensitivity is significantly higher at 90° where the uncertainty of the measured ${P}_{1}^{{\rm{R}}}$ is larger. The latter has two reasons: firstly, the statistics were lowest at 90°. Secondly, a significantly smaller (absolute) value of ${P}_{1}^{{\rm{R}}}$ needed to be measured (for a given number of events, the statistical uncertainty of Compton polarimetry increases when the degree of polarization decreases [40]). The reconstructed values of ${P}_{1}^{{\rm{i}}}$ at the three angles agree with each other and their weighted mean, +0.9824 ± 0.0085, which also agrees with the value derived from the linear polarization of the Compton-scattered photons. It is important to note that the precision of the ${P}_{1}^{{\rm{i}}}$ estimate based on Rayleigh scattering could be significantly increased by accumulating more statistics. In the present measurement the amount of recorded data was limited by the count rate of the polarimeter detector that could be handled by the data acquisition system (based on NIM and VME modules). We expect that a fully digital readout of the detector will enable a ten times higher count rate (about 10 kHz) which for the present experimental parameters would result in a very small uncertainty of the reconstructed synchrotron beam polarization close to 0.25%.

In summary, the polarization transfer in Rayleigh scattering of highly linearly polarized hard x-rays was studied at the synchrotron radiation source PETRA III. We observed a significant depolarization of the scattered photons at scattering angles close to $\theta =90^\circ$ which is in agreement with fully relativistic calculations when taking into account a small polarization impurity of the incident photon beam. Indeed, an imperfect linear polarization of about 98% was obtained independently from the Rayleigh measurement by analyzing the polarization of the Compton-scattered radiation from the target. Thus, the present experiment enabled a very stringent test of polarization transfer theory in Rayleigh scattering, as both the polarization features of the incident and the outgoing radiation were obtained for the first time.

This strong polarization sensitivity makes Rayleigh scattering a very promising candidate for precision polarization diagnostics for highly polarized photon beams (e.g. at synchrotron facilities) considering the following advantages: only a thin, passive target needs to be placed in the primary beam. Moreover, from tabulated S-matrix amplitudes [24, 25] we estimate that the observed polarization sensitivity is sufficiently pronounced at least over the entire range of incident photon energies where the applied model is valid, i.e. well above the K-shell ionization threshold of the target (to justify IPA) and below energies where the Delbrück amplitude is non-negligible (differential cross section data are described well without the Delbrück amplitude up to 662 keV [39]). One could also use the polarization of the Compton-scattered photons to reconstruct the incident beam polarization. In the present experiment, this method yielded a comparable accuracy as with the Rayleigh-scattered photons, even though the statistics for the Compton-scattered photons were significantly higher. We expect that Rayleigh scattering is the more suitable of the two processes for polarization diagnostics, as it features a monoenergetic peak which simplifies the background estimation in comparison to the broad Compton profile. Furthermore, in the mentioned range where Rayleigh scattering provides sufficient polarization sensitivity, Compton scattering rapidly looses sensitivity when the photon energy is increased. For lower photon energies, other polarization diagnostic schemes can be applied, for example a Compton polarimeter with a passive scatterer mounted directly in the synchrotron radiation beam was reported in [63] for the energy range of 15–40 keV. Below 20 keV, extremely polarization-sensitive channel-cut crystals can be employed [35]. We see Rayleigh scattering as a complement to the mentioned techniques as it covers higher energies ($\gt 100$ keV).

The support of Max Schwemlein and Thorsten Groß is acknowledged. Parts of this research were carried out at the light source PETRA III at DESY, a member of the Helmholtz Association (HGF). VAY acknowledges support by Russian Foundation for Basic Research (grant No. 16-02-00538).