Tripling of the scattering vector range of X-ray reflectivity on liquid surfaces using a double-crystal deflector

A straightforward method is presented for aligning a double-crystal deflector for X-ray diffraction measurements on liquid surfaces. This method allows the measurement of liquid surfaces and interfaces in a high qz range not achievable up to now.


Introduction:
Investigation of processes occurring at atomic and molecular levels at the surfaces and interfaces of liquids is of paramount importance for fundamental surface science and practical applications in physics, chemistry, and biology (Pershan, 2014;Dong et al., 2018;Zuraiqi et al., 2020;He et al., 2021;Allioux et al., 2022).However, experimental methods that allow insight into these phenomena are scarce, making synchrotron-based X-ray scattering the prime choice when sub-nanometer accuracy is needed.The high intensity of synchrotron X-ray beams, their highly compact beam size, and their very low divergence allow in situ and operando experiments with sub-second time resolution, which is impossible with standard laboratory X-ray sources.Furthermore, the recent upgrade of the European Synchrotron Radiation Facility (ESRF) allows for very demanding experiments using the extremely bright X-ray source (EBS) with unprecedented parameters (Raimondi, 2016).
One of the most widely used X-ray-based techniques for the characterization of liquid surfaces is Xray reflectivity (XRR).It relies on measurements of the intensity of the reflected X-ray beam from a surface at varying incidence angles, the so-called reflectivity curve, which is used to deduce the surface's out-of-plane electron density profile.Applications of this method are very diverse.They range from the determination of the roughness of the water surface (Braslau et al., 1985), lipid layers on the water-air interface (Helm et al., 1987), free liquid metal surfaces (Magnussen et al., 1995;Regan et al., 1995) displaying layering, polymer assemblies on water (Kago et al., 1998), to protein layers on liquid surfaces (Gidalevitz et al., 1999).Recent technical developments of advanced sample environments and methods further allowed the investigations of even more complex systems.Among these, we may cite Langmuir troughs (Yun & Bloch, 1989) and specialized reactors (Saedi et al., 2020), studies of electrochemical systems (Duval et al., 2012), layer-by-layer assembly of DNA (Erokhina et al., 2008), self-assembled layers (Bronstein et al., 2022), or 2D materials formation on liquid metal catalysts (Jankowski et al., 2021;Konovalov et al., 2022).Thus, the use of XRR, sometimes in connection with other methods like grazing-incidence small-angle scattering (GISAXS) (Geuchies et al., 2016) or X-ray absorption spectroscopy (XAS) (Konovalov et al., 2020), offers a powerful tool for the characterization of a vast family of materials on liquid surfaces.However, one general difficulty exists in performing XRR on liquid surfaces since neither the liquid sample nor the synchrotron source can be tilted.The requirement of variation of the X-ray beam grazing angle () at the sample surface to change the (vertical) scattering vector perpendicular to the surface,   = 4 −1   ( is the X-ray wavelength), brings significant experimental difficulties.
Different technical solutions were implemented to overcome this problem.The synchrotron X-ray beam can be inclined with respect to the horizontal sample plane using mirrors or single or double Bragg reflections from crystals (overview in (Pershan & Schlossman, 2012), Chapter 2).The main drawback of using a mirror is the maximum achievable qz value, usually limited to several critical angles of the total surface reflection on the mirror material.The single crystal deflector (SCD) extends this range to   = 2, where  is the Bragg angle of the chosen scattering planes of the crystal (Smilgies et al., 2005).However, the use of an SCD demands to move the sample to follow the horizontal and vertical displacement of the beam on it, concomitantly with the change of the  angle.This has the drawback to agitate the liquid surface.A more recent solution, the double crystal deflector (DCD) (Honkimäki et al., 2006), relies on a double Bragg reflection from two crystals in a geometry that does not require any sample movement with a change of the  angle, thus assuring a more stable measurement.The maximum obtainable incident grazing angle is   = 2( 2 −  1 ), where  1 and  2 are the Bragg angles of the first and second crystals, respectively, and  2 >  1 (Murphy et al., 2014).Practically, in the case of SCDs or DCDs, the maximum achievable perpendicular momentum transfer    , does not depend on the X-ray beam energy (see SI Note 1).The most typical choices of crystal sets used in realized DCDs are Ge(111)/Ge(220), Si(111)/Si(220), and InSb(111)/InSb(220).The maximum scattering vector reached for these sets is about 2.5 Å -1 (Honkimäki et al., 2006;Arnold et al., 2012;Murphy et al., 2014), which might not be sufficient for studies of some liquid metals, e.g., the surface layering peak and the first structure peak of liquid copper are present at approximately 3 Å -1 (Eder et al., 1980).
The ID10 beamline at ESRF was equipped with an SCD since 1999 (Smilgies et al., 2005).During more than 1.5 decades of operating this instrument, deep technological knowledge and experience were acquired, which led to the design and construction, in collaboration with Huber Diffraktionstechnik GmbH & Co. KG company, of a new generation instrument to study liquid surfaces and interfaces, using a DCD.The new 6+2 diffractometer, equipped with a DCD, has been operating since 2016.This diffractometer has the necessary set of rotation and translation stages to precisely align the DCD and assure its high rigidity and accuracy during operation.In this paper, we present a method of tripling the    value using a DCD by using higher-order Bragg reflections of the two crystals.In practice, we use the Ge(333)/Ge(660) reflections instead of the now standard set of Ge(111)/Ge(220) reflections.
In addition, we confirm experimentally that even with a three orders of magnitude loss of photon flux with these reflections, recording X-ray scattering at high   is still feasible thanks to the recently upgraded ESRF-EBS synchrotron beam (Raimondi, 2016).

Experimental:
XRR measurements using a DCD at the ESRF beamline ID10 were performed using a monochromatic X-ray beam with an energy of 22 keV, monochromatized by Si(111) channel-cut monochromator diffracting in the vertical plane.The DCD was aligned according to the below-described procedure.
The beam intensity reaching the sample after scattering by the Ge(333) and Ge(660) reflections was 7•10 10 ph/s at a synchrotron storage ring current of 200 mA.The full width at half maximum of the beam at the sample position was measured to be 26×10 m 2 (H×V) after focusing with 29 Be parabolic lenses with a radius of 300 m, located before the DCD at 8.9 m from the sample and 36.2 m from the X-ray source.The X-ray beam reflected from the surface was measured with a CdTe MaxiPix 2D photon-counting pixel detector (pixel size: 55×55 m 2 , detector area: 28.4×28.4mm 2 , sensor: 1 mm thick CdTe) and 5 s counting time at each incident angle.
We performed XRR measurements on bare liquid copper and on a graphene layer on liquid copper in situ at T = 1400 K (above the copper melting temperature) in a specially designed reactor dedicated to chemical vapor deposition (CVD) growth of thin layers of graphene on a liquid metal catalyst (Saedi et al., 2020).Single-layer graphene was grown under the same conditions as described in (Jankowski et al., 2021).The obtained scattering data, which include non-specular components (diffuse scattering and scattering from the bulk of liquid copper), were processed following the procedure presented in (Konovalov et al., 2022), considering the spread of the beam reflected on the curved surface of the liquid metal.

Results and discussion:
The X-ray diffractometer of ID10 is a multi-function device that allows working with bulk and surfaces of solid and liquid samples using different setup geometries, see Fig.The principle of DCD operation (Honkimäki et al., 2006;Arnold et al., 2012;Murphy et al., 2014) is illustrated in Fig. 2A.The primary incident X-ray beam undergoes a double Bragg reflection by hitting two crystals at points C1 and C2, and at fixed angles θ1 and θ2, respectively, under two constraints.
The first constraint imposes that the second Bragg angle is bigger than the first one:  2 >  1 .The second constraint imposes that the incident beam and the reflected beams lie in the same plane.
When the two beams are in the vertical plane, the incident angle μ is maximum and given by   =  3 = 2( 2 −  1 ).Whatever the DCD settings, the beam illuminates the sample surface at point O.
The distances between the crystals and the sample are also fixed so that the connected intervals C1C2, C1O, and C2O form the triangle OC1C2 (see Fig. 2A).The incident angle μ is set by rotating the whole DCD setup by an angle  around its main optical axis (-axis), which is supposed to coincide with the primary beam.The angle between the beam after the second crystal and the horizontal plane of the sample is the beam grazing angle  on the liquid sample surface, given by sin =sin  sin  3 .At =0 the beam lies in the horizontal plane of the sample, thus =0 (see Fig. 2B).The increase in angle  > 0 also increases  > 0 (see Fig. 2C), to finally reach the maximum value   =  3 = 2( 2 −  1 ) at =90° (see Fig. 2D).Here we reach the crucial issue: any angular misfit between the primary incident beam and the optical axis  will lead to a progressive loss of the Bragg condition, and thus of intensity, with varying .Thus, this misfit has to be precisely measured and corrected before the XRR data collection, so that the DCD optical axis coincides with the primary beam.To overcome this issue, we calculate the angular drift analytically from the Bragg condition during the  rotation around the optical axis with a non-zero misfit and apply a quantitative correction.The described situation is presented in Fig. 3.The blue line marks the DCD optical axis , the X-ray beam propagates along the X-axis, and angles  and  are parasitic offsets of the DCD optical axis relative to the X-axis in the XY and XZ planes, respectively.The vector  ⃗ is normal to the scattering plane of the first crystal, which initially, at =0, makes an angle of  2 ⁄ +  with the X-axis, i.e., is at the Bragg condition.In general, the vector  ⃗ can be misaligned by a tilt angle  relative to the XY plane.However, we assume that =0, so that the initially diffracted beam propagates in the horizontal plane.The crystals of the DCD at ID10 are mounted on a manual stage to remove this parasitic tilt and to obtain the =0 condition when the Bragg angle rotation axis is perpendicular to the horizontal plane.The angle variation between the vector  ⃗ and the X-axis during rotation around the -axis by angle  can be easily obtained with the corresponding rotation matrix ()=  ()  ()  ()  (−)  (−) (1) Figure 3 Schematic sketch of the DCD geometry with a misfit.The black lines X, Y, and Z mark the laboratory coordinate system.The incident X-ray beam is along the X-axis.The blue line is the main DCD optical axis (-axis), the red arrow marks the vector  ⃗ ⃗ normal to the scattering plane of the first crystal.The angles  and  are parasitic angular offsets of the -axis from the X-axis (primary beam).The angle  is the rotation angle of the whole DCD setup around its main optical axis. is the angle (assumed to be zero here) between the vector  ⃗ ⃗ and the XY plane.
Here   ,  ∈ {, , } are rotation matrices around the respective coordinate axes.For an elementary rotation by an angle  around the corresponding axis, they are given by In the described geometry, the X-ray beam orientation is expressed by the vector: ), while the normal vector  ⃗ to the scattering plane lying initially in the XY plane (i.e., =0 and =0) is expressed by the vector: ).
Its coordinates are modified after rotation by the angle  around the -axis according to: ⃗ () =   () 0 ⃗⃗⃗⃗ (3) We then derive the deviation angle  from the Bragg condition during a rotation  around the -axis from the equation: The effect of the misfit between the -axis and the X-ray beam is presented in Fig. 4, which shows the plot of the Bragg deviation angle  as a function of , calculated using Equation ( 4) at  = 4.5°,  = 0.002° and  = 0.004°.

Conclusions
We have analytically described the misalignment correction of a double crystal deflecting system used to tilt the incident synchrotron X-ray beam with respect to the sample surface for grazing incidence scattering experiments on liquid surfaces.The proposed method is fast and straightforward, considering the complexity of the system and the demand for very high accuracy.In addition, we have developed a procedure that significantly extends the maximum range of momentum transfer perpendicular to the surface   , from ~2.5 Å −1 to ~7 Å −1 .The new procedure is demonstrated for a bare and graphene-covered liquid copper surface.The recorded signal intensity is enhanced by the recent upgrade of ESRF to an EBS, allowing for more demanding measurements.The proposed method and the ESRF technical upgrade allow for new experiments with liquid metal surfaces and other systems.The measurements of out-of-plane crystallinity and order, i.e., Bragg peaks, Laue fringes, and strain effects, of materials like thin layers, nanoparticles, and quantum dots, supported on liquid surfaces, are now possible in the extended range of momentum transfer perpendicular to the surface.
1A and 1B.The X-ray detectors are mounted on the  and δ circles (see Fig.1C), allowing their movements around the diffractometer center in horizontal and vertical planes.The available beamline detectors are MaxiPix 2x2 CdTe, Dectris Eiger 4M CdTe, Pilatus 300k Si, Mythen 1K, and Mythen2 2K.The detector holder's construction allows the simultaneous use of these detectors in different configurations during an experiment.The diffractometer comprises two sample stages in horizontal or vertical geometry configuration, see Fig.1B.The horizontal stage is typically used for the investigation of liquid sample surfaces and comprises three circles θ, χ, and φ, and a z-, x-, and y-sample translation stage, marked in Fig.1C.Similarly, the vertical stage is mounted on the θ circle and comprises three circles ω, χ', φ', and a z-, x-, and y-sample translation stage.The diffractometer can be used in two modes.In the first mode, the beam is fixed on the instrument's optical axis, while in the second, the DCD is used to tilt the incoming X-ray beam around the sample plane, see Fig.1C.The first mode is routinely used to measure solid samples and when the use of a bulky and heavy sample environment is needed, whereas the DCD is used for investigations of liquid surfaces and interfaces.

Figure 1 :
Figure 1: A) Photo of the diffractometer with the mounted Langmuir trough on the antivibration table.Two detectors mounted on the diffractometer arm allow XRR and GIWAXS/GID experiments.B) The 3D drawing of the diffractometer with labeled horizontal and vertical stages.C) Schematic representation of the configuration of the diffractometer circles.

Figure 2 A
Figure 2 A) The geometrical sketch of the side view (vertical plane) on the DCD crystals assembly and sample at =90°.B) The 3D drawing of the DCD configuration corresponding to μ=0° (i.e., =0°), C) the intermediate situation when μ>0° (0° <  < 90°), and D) at maximum μmax (=90°), situation corresponding to Fig. 1A.The arrow in B) shows the direction of the rotation of the crystals around the optical axis .

Figure 6
Figure6A) Plot of the scattering intensity as a function of   recorded from bare liquid copper (blue curve) and graphenecovered liquid copper (orange curve) at 1400 K. B) Specular rod, obtained after diffuse background subtraction, of bare liquid copper (red curve) and graphene-covered liquid copper (black curve) at 1400 K.