High-resolution inelastic x-ray scattering at the high energy density scientific instrument at the European X-Ray Free-Electron Laser

Inelastic X-ray Scattering (IXS) is a powerful tool to measure material dynamics by comparing the energy and momentum of a photon before and after the interaction process. The accessible processes can be categorized by the value of the energy transfer and range from keV individual electron responses in Compton scattering processes^1–3 through collective electronic excitations such as plasmons and x-ray Raman scattering in the eV range^4–6 down to structural dynamics on the order of meV.^7–10 Consequently, IXS has become a well-established technique at many facilities in the past few decades.^11–13 Developments in high-resolution IXS at synchrotron radiation sources, in conjunction with neutron scattering techniques,¹⁴ have enabled the elucidation of many processes in the field of condensed matter physics. Energy transfers in the range of a few meV give access to detailed measurements of phonons in crystalline¹⁵ materials and collective dynamics in non-crystalline materials.¹⁵ The technique can also be applied to studies of phase transitions in conditions relevant to the interior of the Earth^16,17 and the effect of magnetism on lattice dynamics.¹⁸ 

With the emergence of X-Ray Free-Electron Lasers (XFELs) as a new intense radiation source with femtosecond pulse-lengths, many established techniques have successfully been transferred from synchrotron radiation sources to XFELs. The combination of XFELs and synchronized optical lasers enables pump–probe measurements of previously inaccessible processes in short-lived and extreme states of matter generated by optical lasers.^19–22 In particular, for energy transfers in the eV range, measurements of plasmon excitations in shock compressed matter on a ns timescale^23,24 and the dielectric function of isochorically heated matter²⁵ have been successfully demonstrated. Recently, high-resolution IXS has been realized at the Linac Coherent Light Source (LCLS) with tens of meV resolution, allowing measurements of phonon-modes in polycrystalline diamond.²⁶ 

Simultaneous measurement of the energy and momentum transfer in materials gives access to the dynamical structure factor S(⁠$k →$⁠, ω), where $k → = k → o u t − k → i n$ is the momentum transfer and ω is the frequency change due to the scattering process. This contains a wealth of information on the fundamental properties of the dynamic states probed, such as atomic and ionic motion,²⁷ sound velocity,²⁸ dissipative macroscopic phenomena such as viscosity, and thermal conductivity.²⁹ Moreover, this technique at high resolution can resolve small energy and momentum transfers and thus enables direct temperature measurements of temperatures below 1 eV either through Doppler broadening or by measurements of Stokes/anti-Stokes lines in detailed balance.³⁰ Additionally, by comparing these to the intensity of the elastic scattering, it is possible to determine diffusive modes in amorphous materials and plasmas.³¹ To facilitate these future measurements, a high-resolution IXS setup has been incorporated into the High Energy Density (HED) scientific instrument at the European XFEL^32,33 with a measured resolution of ∼45 meV, which should allow typical IXS features in matter at extreme conditions to be resolved while having a sufficient transmission for the required photon statistics.

The setup is conceptually based on the successful design used at the LCLS, as described by McBride et al.²⁶ It is located at the HED instrument at the hard x-ray SASE 2 undulator of the European XFEL.^34,35 The accelerator driving the light source accelerates electron macro-bunches at a base rate of 10 Hz. Each of the macro-bunches can contain between 1 and 2700 electron bunches separated by a minimum time of 220 ns. This enables x-ray pulses at a maximum repetition rate of 4.5 MHz within such a bunch train. For the experiment described here, the x-ray source was operated in a single bunch self-amplified spontaneous emission (SASE) mode at a repetition rate of 10 Hz with a photon energy centered at 7.49 keV and a bandwidth of roughly 20 eV. The average pulse energy was 1 mJ after the source, measured by a gas monitor.³⁶ 

To reduce the incident bandwidth of the SASE beam, the x rays were sent through two stages of monochromatization (Fig. 1), which all use symmetrical Bragg reflections. The first stage comprises a two-bounce quasi channel-cut Si (111) monochromator.³⁷ This monochromator can be cryogenically cooled to 70 K to reduce the effects of thermal expansion under irradiation by the intense x rays and serves primarily as a pre-monochromator for the following monochromator. For this experiment, it was operated at room temperature. The second monochromator employs a Si (533) channel-cut crystal, which was manufactured by the x-ray optics group of the Friedrich-Schiller University Jena. Its channel width of 106 mm is adapted to the Si (111) pre-monochromator to roughly compensate the vertical offset imposed by the first monochromator at a photon energy of 7.49 keV at the corresponding Bragg angle of 87.9°. Ensuring a zero offset to the nominal beamline has the advantage that the following beamline and diagnostics, as well as laser optics in the experimental chamber, require no re-alignment when using the monochromatized beam. Under these conditions, the energy width of the incident beam would be reduced to a value of 30.8 meV or a bandwidth ΔE/E of 4 × 10⁻⁶. Offline measurements with a double crystal diffractometer showed a broadening smaller than 10% of the theoretical rocking curve width. The Si (533) reflection was chosen because it yields the necessary resolution while having no lower order reflections. This is particularly important when using the setup to study warm-dense matter states, as the plasma emission could add strongly to the noise on the IXS signal. For example, when using the Si (444) reflection at a XFEL photon energy of 7.912 keV, the scattering setup will also transmit via the Si (111) reflection at a quarter of the photon energy, 1.978 keV, which is emitted from the ablation plasma.

The Si (533) channel-cut crystal was cut from a monolithic ingot. The surfaces were sawn with a diamond blade and subsequently polished with SiC of decreasing grain size. Afterward, the crystal was etched in a mixture of hydrofluoric acid, acetic acid, and nitric acid for 20 min. To increase the flatness of the surfaces, the crystal was polished a second time with 38 μm SiC, followed by the same etching process.

Despite this, optical wavefront measurements yield a surface roughness with a root-mean-square (rms) value of ∼5 μm and a peak-to-valley (PV) value of ∼20 μm. In comparison, the Si (111) crystals of the first monochromator stage were highly polished, which is technically possible because the crystals are separated in a quasi channel-cut, and have a rms roughness of order 30 nm with a PV value on the order of 200 nm. Nevertheless, the two reflections from the Si (533) crystal preserve the spatial profile of the beam (Fig. 2),³⁸ most likely due to the high Bragg angle of ∼88°.

One of the main goals of the high-resolution IXS setup is to study extreme states of matter generated by optical drive lasers. The Helmholtz International Beamline for Extreme Fields at the European XFEL (HIBEF) user consortium has contributed two high-power lasers to the HED scientific instrument,^39,40 which runs at a maximum repetition rate of 10 Hz. Because of this, the monochromator was designed to match this repetition rate. Higher pulse rates would be a rare case for these experiments, as, even at 4.5 MHz, the extreme state will have disappeared before the next pulse arrives after 220 ns. At 10 Hz, the time between two pulses is sufficient to dissipate the heat deposited by the x-ray pulse at the location of the first reflection into the crystal bulk and support structures. Therefore, there is no loss of intensity between pulses due to the expansion of the lattice spacing at the first reflection, and the Darwin curves overlap at 10 Hz. In this early stage of instrument operation, we were not able to study the transmission at repetition rates above 10 Hz, as the cryogenic cooling of the Si (111) monochromator was not yet operational, without which the transmission of a pulse train is significantly reduced.

Both monochromators are located roughly 120 m upstream of the experimental chamber and 850 m downstream of the undulator exit. A set of beryllium compound refractive lenses (CRLs)⁴¹ is located 230 m after the undulator, which was used to collimate the x-ray beam. Repeated measurements of the beam’s spatial profile at different locations along the beam propagation have yielded a vertical divergence, which is the dispersive direction of the monochromators, of ≤1 μrad for this collimated beam. Thus, a further reduction of the divergence by asymmetric reflections, often necessary on high-resolution monochromators at synchrotron radiation sources, where the divergence can be of the order of the crystal rocking curve, is not required for this setup. Final focusing can be achieved by using one of the two CRL sets installed downstream of the monochromators. For this experiment, we used a CRL set located 9 m upstream of the interaction point. It was used to focus the beam down to a spot size of ∼25 μm at the target in the interaction chamber IC1 in the experimental hutch. Smaller spot sizes below 10 μm are possible with this lens set. Another set of CRLs, which were not used in this experiment, is located directly after the monochromators and can focus to a few tens of micrometers at the sample location.

In this chamber, three spherical diced analyzer crystals^42–44 (Fig. 3, inset) using the same Si (533) reflection as the monochromator were mounted to a vertical curved-rail system with a radius of 1 m (Fig. 1) in a Rowland-circle geometry.^45–48 One was sitting centered over the primary x-ray beam path (blue dotted line, Fig. 1), and two others were sitting at the same height but offset to the left and right at an angle of ϕ = 9.4° to the primary beam axis. The usage of three analyzers shortens the measurement time and decreases the necessary number of samples in the case of destructive experiments, e.g., dynamic compression. Due to the off-center positions of the two analyzers, the setup measures two momentum transfer values simultaneously, which can be matched to a phonon dispersion curve as demonstrated by Descamps et al.³⁰ With the curved rail, it is possible to change the scattering angle between 4° and 32° without breaking vacuum. These angles correspond to scattering wavenumbers k = $| k → |$ from 0.26 Å⁻¹ to 2.01 Å⁻¹ at the incident photon energy of 7.49 keV. This way, it is possible to conduct measurements at different momentum transfers while keeping all other conditions constant with minimal realignment. A suitable use case is, for example, the measurement of the dispersion ω(k) of collective excitations to study the effects of local field corrections.²⁴ The analyzer crystals have a diameter of 100 mm, a bending radius of 1 m, and a dice size of 1.65 × 1.65 mm², giving a calculated spectral window of 460 meV. The foci of the analyzers in the SASE beam show the square shapes [Fig. 5(a)] expected when the Rowland-circle geometry is fulfilled. However, all three squares have a width slightly smaller than the expected 3.3 mm = 2 × 1.65 mm², from left to right 3.1 mm, 3.2 mm, and 3.2 mm. A possible reason for this is a small mismatch of the Rowland-circle, which results in a demagnification of the detector image and decreased reflectivity at the edges of the sawn dices.

Due to the large size of the analyzers, the collected signal results from an integration over a range of different k-values. In the lowest angular position, this leads to a blurring of 0.26 ± 0.19 Å⁻¹ for the middle analyzer and $0.68 − 0.05 + 0.09$ Å⁻¹ for the outside analyzers (Fig. 4). As two of the analyzers are not in line with the incident beam axis, their effective scattering angle θ_eff = arccos(cos(θ) · cos(ϕ)) is a combination of the curved rail angle θ and the fixed angle ϕ = 9.4° resulting from the mounting. Thus, the two outer analyzers have a higher minimum scattering vector than the central analyzer. Additionally, their relative k-blurring is smaller than that of the central analyzer at the cost of a lower intensity of the scattered radiation on the analyzers due to the angular dependence of the scattering cross section for horizontally polarized x rays, shown by the dotted lines (Fig. 4). Due to the spherical aperture of the crystals, there are fewer dices along the outer edges of the crystals that scatter photons at larger differences to the central k-value. Additionally, we masked the outer edge of the analyzers with aluminum foil to cover up possible damaged diced sections at the edges. Therefore, we assume that most photons contributing to the signal are reflected from the central 75% of the area (Fig. 3). Then, the blurring would be on the order of ±0.16 Å⁻¹ in the middle and $− 0.05 + 0.08$ Å⁻¹ on the left and right analyzers, as indicated by the dashed lines (Fig. 4).

Finally, the scattered radiation was recorded on an ePix100⁴⁹ detector with 768 × 704 pixel² and a pixel size of 50 μm. The detector was mounted on the innermost curved rail, 80 mm above the sample, and running at a repetition rate of 10 Hz. In this geometry, the sample, analyzers, and the detector form a Rowland circle of R = 1 m diameter. Since the center of rotation of each rail is aligned to the interaction point of x rays and the scattering target, it is possible to keep this geometry while changing the scattering angle with the curved rail system. Alternatively, the analyzers can be mounted to a rail upstream of the sample, allowing measurements at higher momentum transfers. The complete setup is kept at a typical vacuum level of 10⁻⁴ mbar to enable compatibility with high-intensity and high-power laser experiments.

Working at large Bragg angles in near-backscattering geometry Θ ∼ 90° ensures a high dispersion and resolution. The exact value of the Bragg angle is strongly dependent on the photon energy near 90°, as given by Bragg’s law λ = 2d sin(Θ). Here, λ is the x-ray wavelength and 2d is twice the inter-planar distance of the reflection: 1.656 446 Å⁵⁰ in the case of Si(533). Subtle differences in the x-ray wavelength Δλ relate to significant changes in Bragg angle ΔΘ and thus positions on an area detector via the differential Bragg equation,

as near 90°, cos(Θ) is small. Furthermore, the relative spectral resolution

where Δθ is the Darwin width of the reflection and is smallest at large Bragg angles. Even though the Darwin width gets wider near 90°, this is compensated by the 1/tan Θ term approaching zero. To interpret spectral measurements, it is therefore crucial to know the absolute photon energy.

We determined the photon energy by comparing the positions of the beam after the Si (111) monochromator and the beam after both the Si (111) and Si (533) monochromators on a screen located 80 m downstream. After averaging over a few thousand shots, the Bragg angle of 87.85° ± 0.05° can be calculated geometrically from the offset and the known length of the channel-cut of 106 mm. From the Bragg angle, the photon energy was determined to be 7490.2 ± 0.2 eV.

To determine the energy dispersion of the monochromator, the same procedure was repeated for slightly different Bragg angles in a range of ±0.02°. Afterward, we scanned the same Bragg angles with a scattering target in the experimental chamber. With the differences in incident photon energy, this leads to a change of position of the radiation dispersed by the analyzer on the detector [Figs. 5(b)–5(d)]. The combination of the two measurements yields a dispersion value of 7.7 ± 0.5 meV/pixel on the detector, which is equivalent to 0.15 meV/μm. The largest contributions to the error for this measurement are the beam jitter and motor backlash of the axis, which changes the Bragg angle. On the contrary, the photon energy can be calculated from the dispersion as the dispersion is determined by photon energy E, Bragg angle Θ, and Rowland circle diameter R,

This way, the dispersion value of 0.15 meV/μm yields a photon energy of 7490.95 eV. To match the previously obtained photon energy of 7490.2 eV, the dispersion should have a value of 0.141 meV/μm. This value still lies at the edge of the 6% error of our measured value. In addition, a mismatch of the Rowland circle and inaccuracies in the analyzers’ bending radii, which we did not measure accurately, both add an uncertainty to Eq. (3).

Using various samples and changes of the scattering angle, we collected the quasielastic scattering signal from different thicknesses (50 μm, 500 μm, and 1100 μm) of polymethylmethacrylate (PMMA, Goodfellows Ltd.) and fused silica to characterize the instrument resolution for each set of conditions (Fig. 6). The effective source size is given by t · sin(θ_eff) + a · cos(θ_eff), where t is the sample thickness and a is the focus size on the sample, with θ_eff as defined earlier for the outer analyzers and equal to θ for the central analyzer. For quantification, we take the full width at half maximum (FWHM) of the quasielastic peak as the instrument function, as PMMA has only low-energy and strongly damped modes, which our setup cannot resolve. Generally, the spectra from the outer analyzers show a slightly worse resolution than the central analyzer because the projected source size is larger for the outer analyzers. This leads to a reduced resolution due to source broadening, where the analyzers collect radiation from different source points. A similar effect can be observed when thicker samples are used.

We achieved the highest resolution of 44 ± 3 meV by using a 50 μm PMMA sample [Fig. 7 (blue)] with a photon count of 3.8 photons per shot. Here, the largest contribution to the error comes from the uncertainty on the dispersion. This value is very close to the convolution of the incident bandwidth and Darwin width of the analyzers at 42.8 meV, which are the two largest contributors to the overall resolution, as can be seen from Table I. If other factors such as pixel and source size are included, the measured 44 meV is precisely the expected resolution under the assumption that these factors contribute with a Gaussian shape. We could only estimate the size and shape of the incident x-ray focus spot with 10 μm precision. Consequently, there remains some uncertainty on the source size contribution. Using the same setup, we measured inelastic scattering from phonons in single crystal diamond [Fig. 7 (red)], where an average of 0.5 photons per shot was achieved. These data were published by Descamps et al.³⁰ to demonstrate the capability of the setup to perform accurate temperature measurements via detailed balance.

Ultra-short electromagnetic pulses, such as the ones from a XFEL, require a sufficiently broad frequency spectrum. Thus, strong monochromatization may lead to an increase in the pulse duration. We know that the pulses after the monochromators have a bandwidth of 30.5 meV. If we assume a Gaussian profile, a Fourier-transform-limited pulse of 1 fs duration has a bandwidth of 1.83 eV. Therefore, a non-chirped FEL pulse with a Gaussian distribution monochromatized with a bandwidth of 30.5 meV FWHM would correspond to a duration of about 60 fs. Any chirp present in the monochromatized pulse would increase its duration for a fixed bandwidth, but such an effect is estimated to be small. In comparison, we can take the length of the electron bunches, which was measured to be slightly shorter than 50 fs,³³ as a maximum duration for any SASE pulse. We thus conclude that, during our experiment, the FEL pulse duration was determined by the bandwidth of the monochromator and that the pulses are somewhat lengthened due to monochromatization. However, after monochromatization, the pulses are closer to transform-limited and possess an increased temporal coherence. The slightly longer pulses do not pose a limitation for any planned experiment on structural properties.

We have successfully demonstrated a new capability of the HED instrument to resolve inelastic x-ray scattering with a spectral resolution of 44 meV. This resolution was achieved through a two-stage monochromatization of the SASE beam at 7.49 keV: first by a Si (111) quasi channel-cut, followed by a Si (533) channel-cut monochromator. The scattered radiation was collected by three Si (533) diced analyzer crystals and focused on a position resolving detector. This setup can be used to measure collective features such as acoustic phonon modes, for example, in diamond and determine its temperature by detailed balance, as demonstrated by Descamps et al.³⁰ The target chamber also offers the possibility to mount an analyzer in backscattering geometry, allowing the measurement of meV-broadening due to scattering from individual electrons in the non-collective regime. In the near future, self-seeding⁵¹ will be available at the SASE2 undulator. As the following calculation demonstrates, this will lead to a higher number of photons after monochromatization. The Si (111) monochromator will be able to transmit almost the complete incident spectrum with several 100 μJ pulse energy and ∼1 eV seeded bandwidth. This will result in a pulse energy of order 10 μJ or 8.3 × 10⁹ photons per pulse in 35 meV bandwidth after the Si (533) monochromator in comparison to the average of 1 μJ from the SASE pulses. This capability together with the short pulse-lengths of XFELs will enable the production of snapshots from materials under extreme conditions and will be uniquely suited to constraining equation-of-state (EOS) measurements on highly transient plasmas generated with the HIBEF high-intensity and high-energy lasers at the HED instrument at the European XFEL.

Raw data were generated at the European XFEL. Derived data supporting the findings of this study (10.22003/XFEL.EU-DATA-002191-00 and 10.22003/XFEL.EU-DATA-900099-00) are available from the corresponding author upon reasonable request.

We acknowledge the European XFEL in Schenefeld, Germany, for provision of x-ray free-electron laser beamtime at the Scientific Instrument HED (High Energy Density Science) and would like to thank the staff for their assistance. The authors are indebted to the HIBEF user consortium for the provision of instrumentation and staff that enabled this experiment. K.A., T.T., and R.R. acknowledge support from the DFG (Grant No. FOR 2440). Z.K. and C.P. acknowledge support from the DFG Project KO-5262/1. A.D. acknowledges funding from the National Science Foundation under Grant No. 163270. We further acknowledge that this work was supported by the Department of Energy, Laboratory Directed Research and Development program at the SLAC National Accelerator Laboratory under Contract No. DE-AC02-76SF00515 and as part of the Panofsky Fellowship awarded to E.E.M. Furthermore, J.S.W. acknowledges support from the UK EPSRC under Grant No. EP/S025065/1. O.S.H. and O.K. acknowledge support from the Oxford Centre for High Energy Density Science (OxCHEDS). This material is partially based on the work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Science under Award No. DE-SC0019268.