Resonant inelastic X-ray scattering spectrometer with 25 meV resolution at the Cu K-edge

A high-energy-resolution spectrometer featuring a quartz crystal analyzer is presented. The manuscript includes a step-by-step recipe for manufacturing the analyzer and a detailed ray-tracing analysis.


Introduction
Resonant inelastic X-ray scattering (RIXS) has become a standard tool to probe elementary excitations in correlated electron systems. RIXS comes with unique advantages to study bosonic excitations across many Brillouin zones, including bulk sensitivity which results from the large penetration depth of high-energy photons. Since the incoming photon energy is tuned to an absorption edge of the sample, the resonantly enhanced signal reveals element-specific information (Ament et al., 2011;Schü lke, 2007;Kao et al., 1996;Hill et al., 1998;Hasan et al., 2000).
RIXS is sensitive to a wide-range of excitations in correlated systems. It has been routinely used to study chargetransfer and d-d excitations in many transition metal compounds (Grenier et al., 2005;Huotari et al., 2008), particularly in the cuprate superconductors and their parent compounds (Abbamonte et al., 1999;Kim et al., 2002). However, due to the strong elastic peak in the K-edge RIXS spectra, the low-energy excitations like magnons and phonons had been hidden under the quasi-elastic tail. With the development of high-resolution RIXS instruments, detailed studies of bi-magnon excitations in La 2 CuO 4 have become possible, albeit with longer data collection times due to still unfavorable signal to elastic background ratios (Hill et al., 2008;Ellis et al., 2010;Bisogni et al., 2012). Most recently, a glimpse of lattice excitations measured with Cu K-edge RIXS has also been reported (Yavaş et al., 2010).
High-resolution inelastic X-ray scattering (IXS) spectrometers utilize spherically bent crystal analyzers near backscattering geometry on a Rowland circle (Burkel et al., 1987). Bending the analyzer crystal introduces further broadening in ISSN 1600-5775 energy resolution due to the elastic deformation of the lattice planes. In order to mitigate this problem, the analyzer crystal is pixelated into a mosaic of unstrained small cubes (Sette et al., 1995;Baron et al., 2000;Sinn et al., 2001). With this arrangement, the energy resolution of the spectrometer is where ÁE i is the resolution of the incident beam on the sample and ÁE c is the intrinsic energy resolution of the analyzer crystal, which can be calculated using dynamical diffraction theory for perfect crystals (Batterman & Cole, 1964). ÁE g is the geometrical contribution to the overall energy resolution, and can be denoted as where E is the X-ray energy, R is the diameter of the Rowland circle, B is the Bragg angle on the analyzer crystal, and s is the effective beam size on the sample. d is the pixel size of the position sensitive detector or the twice the cube size of the analyzer crystal if the detector does not have the position sensitivity Huotari et al., 2005). The RIXS cross section depends significantly on the incident photon energy on the sample. Therefore, the design of the spectrometer needs to allow the incident energy to be tunable across an absorption edge of the sample while maintaining favorable geometrical conditions. Since the geometrical broadening ÁE g dominates all the other factors as the Bragg angle recedes from 90 [equation (2)], high-resolution RIXS measurements are constrained to the availability of back-scattering Bragg planes. Restricting analyzer fabrication to Ge and Si, as has effectively been done, has provided a very limited selection of available energies and resolutions. Noncubic crystal structures with more than one atom in the unit cell, like sapphire (Shvyd'ko & Gerdau, 1999;Yavaş et al., 2007) or quartz (Sutter et al., 2005;Hö nnicke et al., 2013), provide an increase in the number of unique back-scattering planes to choose from . The quartz (244) plane stands out as a good candidate for a high-resolution analyzer at the Cu K-edge energy with around 87 Bragg angle and close to 8 meV intrinsic energy resolution. This paper reports on simulations, development and testing of a new energy analyzer fabricated from quartz for measurements at the Cu K-edge.

Spectrometer
The spectrometer is stationed at the Dynamics Beamline P01 of PETRA III at Deutsches Elektronen-Synchrotron, DESY. A liquid-nitrogen-cooled double-crystal monochromator featuring Si(311) crystals is used to monochromatize the radiation from the undulator beamline, where Si(111) is also available as an option. A four-bounce Si (553) channel-cut monochromator in ðþ À ÀþÞ geometry is used for further monochromatization. A Kirkpatrick-Baez (KB) mirror system focuses the beam to a spot size of 3 mm Â 9 mm (V Â H) at the sample position, which is confirmed by knife-edge scans in both directions.
The analyzer, the sample and the detector were placed on a Rowland circle with a 2 m diameter, which can rotate around the sample to determine the momentum transfer to the sample. The geometry was arranged such that, after being scattered from the sample in the horizontal direction, the X-ray beam was scattered from the analyzer in the vertical direction before being collected on the detector (Fig. 1).
The analyzer crystal has facets of 1.5 mm size and 0.2 mm gaps in between. The centers of the facets form a section of a sphere with 2 m radius (see Appendix A for the detailed fabrication procedures). The detector is a commercially available MYTHEN (https://www.dectris.com/mythen_over view.html) one-dimensional detector, which has 8 mm stripes with 50 mm pitch.
It should be noted that the spectrometer is a custom-made six-circle diffractometer, which allows all degrees of freedom to orient a single-crystal. The 2 arm (adjustable between 1 m and 2 m sample-to-analyzer distance) can swing in both vertical and horizontal direction (up to 130 ) for polarizationdependent measurements. The instrument is available for user proposals and the X-ray energy can be tuned between 4 keV and 11 keV for resonant measurements at other edges as well as non-resonant inelastic X-ray scattering.

Ray tracing of the ideal spectrometer
The spectrometer was modeled by means of the ray-tracing software xrt (Klementiev & Chernikov, 2014). The goal of the modeling was to obtain the figures of energy resolution and energy acceptance, where all the contributing factors, i.e. spectrometer geometry, analyzer crystal shape and intrinsic resolution, beam size on the sample and detector pixel size, are consistently taken into account.
The geometry of the photon source for the ray tracing mimicked the real experimental conditions. The beam size on the sample is taken as 3 mm Â 9 mm (V Â H) as mentioned The spectrometer geometry. The sample is at the center of rotation of a six-circle diffractometer. The design of the diffractometer allows large space around the sample for the detector. The arrows show the propagation of the X-ray beam. A helium bag, which is not shown here, was put between the sample and the analyzer and the detector in order to avoid air absorption.
above. The absorption of the primary and scattered beams in the sample was considered as negligible and therefore the intensity distribution over the source length was taken as uniform.
The response of the analyzer-detector pair to two distinct incident X-ray sources, i.e. a uniform energy distribution and a delta function, must be identified in order to determine the overall energy resolution of the spectrometer. For both cases the width of the detector image Áz is determined. The energy bandwidth of the analyzer-detector pair is where ÁE is the energy bandwidth, and Áz is the image size on the detector or the corresponding number of pixels. The size of the crystal facets of the analyzer along the dispersive direction determines the energy acceptance width ÁE uniform , which is equal to 305.2 meV (FWHM) for the given facet size of 1.5 mm (see the zoomed footprint image in Fig. 2). The energy acceptance width represents the energy range that is possible to analyze without any mechanical scanning, which is suitable for shot-by-shot measurements with emerging intense X-ray sources like free-electron lasers. The obtained ratio ÁE uniform =Áz uniform converts to the energy dispersion 5.087 meV per pixel, which was further used for the detector calibration. The image of a delta function source on the detector occupies $ 3 pixels (3 Â 5.087 ' 15.3 meV), which includes the intrinsic energy resolution of the quartz reflection (Fig. 2, right panel).
The calculated energy resolution of the analyzer-detector system, E = 15.3 meV, was verified with ray tracing by considering a source energy distribution with seven lines (delta functions) spaced E apart. As shown in Fig. 3, these seven lines are indeed spatially resolved. The definition of energy resolution adopted here gives a result similar to that given by the modified 1 Rayleigh criterion, which requires the local minima to be at most 80% of the maxima in order to be resolved (Born & Wolf, 1999).
Finally, we modeled the response of the analyzer-detector pair to the energy bandwidth of the high-resolution monochromator (HRM). The ray-tracing simulations of the Si(311) double-crystal monochromator and the four-bounce Si(553) HRM resulted in 16.4 meV, which was applied to the source seen by the analyzer. Fig. 4 shows the detector response to the 16.4 meV-wide energy bandwidth of the HRM from ray tracing. This image demonstrates the expected minimum width of the experimentally measured inelastic scattering spectra. It also justifies the selected detector pixel size (50 mm), as being sufficiently smaller than the narrowest measurable line width to not restrict total energy resolution. The overall energy resolution is calculated from the energy dispersion rate calculated earlier as 5 pixels Â 5.087 meV per

Performance of the spectrometer
In order to measure the highest possible resolution, we used two layers of Scotch 1 tape as a point-like elastic scatterer. The quartz analyzer of 100 mm diameter and 2 m radius of curvature was placed on a high-precision gimbal stage approximately 2 m from the sample (see Appendix A for analyzer fabrication procedures and x2 for the details of the spectrometer). The Rowland circle was optimized experimentally for the highest energy resolution by tweaking the analyzer-sample and analyzer-detector distance.
The analyzer's energy acceptance width of approximately 300 meV (Fig. 2, middle panel) was confirmed by scanning the incoming beam energy while keeping the analyzer-detector pair fixed. This step further confirmed the calibration of the analyzer-detector pair estimated by the ray-tracing simulation as well as the dispersion rate of around 5 meV per pixel at 87 Bragg angle. The acceptance width depends on the X-ray energy around which the spectrometer is aligned. The analyzer energy can be tuned as low as 8975 eV and can go over 9 keV. However, the energy resolution broadens as the geometrical contribution starts dominating with Bragg angles away from back-scattering. The energy resolution is measured by scanning the incoming energy and recording a single detector pixel, which is cross-confirmed by passively collecting data without any mechanical scanning. The total energy resolution of the instrument was fitted with a Voigt function, which determined the full width at half-maximum (FWHM) of the quasi-elastic line as ÁE tot = 26 meV (Fig. 5). The measured value is only 3% larger than the ray-tracing results. The reason for this discrepancy could be a number of things like a slight deviation from the Rowland geometry or residual stress due to the bonding between the analyzer pixels and the spherical backing. Considering the state-of-the-art resolution available at the MERIX instrument at the Advanced Photon Source (around 100 meV at the Cu K-edge energy) , this result is a major improvement for the Cu K-edge RIXS.
In order to assess the absolute efficiency of the spectrometer we measured the scattering intensity of liquid water at room temperature, which can be calculated by the doubledifferential cross section, where q and ! are the momentum and energy transferred to the sample during the scattering process, r 0 is the classical electron radius, " " " i (" " " f ) is the incident (scattered) polarization unit vector, k i (k f ) is the magnitude of the momentum vector of incident (scattered) photons, f ðqÞ is the molecular form factor of water, and Sðq; !Þ is the dynamical structure factor. We followed the identical procedure as Sala et al. (2013) with necessary alterations due to the incident photon energy (8997 eV) and the 2 angle (25.5 ). After corrections of additional loses due to Kapton 1 windows and air absorption (29% in total), we calculated the effective reflectivity of the SiO 2 (244) plane as 51% (theoretical reflectivity is 71% based on dynamical diffraction calculations). We took the detector efficiency as 75%, which is measured and calibrated with a PIN diode. As a result, the analyzer efficiency turned out to be around 0.7%. The performance of the analyzer was further tested by measuring the phonon dispersion of a beryllium single-crystal due to its well known dynamical structure. The spectra were collected in the first and the second Brillioun zones along (00), and fitted with three peaks as was done with the previously developed sapphire analyzer (Yavaş et al., 2007)  Overall resolution function of the instrument. The data are collected by static counting on the position-sensitive detector without any energy scanning. In order to have more data points on the peak, the Bragg angle is chosen to be 88.66 (2.624 meV per pixel). The FWHM is measured to be 26 meV (with a Voigt function). A Gaussian (red solid line) and a Lorentzian (blue dashed line) with the same FWHM are shown for comparison. The y-scale is logarithmic in order to show that the quasielastic tails are much better than a Lorentzian.

Figure 4
Histogram of the counts on the detector from a ray-tracing simulation. The discrete nature is due to the pixel size of the position-sensitive detector (8 mm Â 50 mm). (Fig. 6). The detailed balance between the phonon creation and annihilation due to Boltzmann statistics was observed carefully. The resulting dispersion curve is in good agreement with previous measurements within experimental error (Stedman et al., 1976;Alatas et al., 2008) (Fig. 7).
The current report confirms that the spectrometer's resolving power is good enough to resolve excitations at energies as low as 16 meV (Fig. 6). The instrument owes this recordbreaking resolution to its quartz analyzer, which offers a suitable back-scattering Bragg plane for the Cu K-edge energy. It is expected that the many back-scattering planes available due to quartz's lower-symmetry structure, combined with the well established processing methods described here, will make quartz a viable material for high-resolution X-ray analyzers.

APPENDIX A Fabrication steps
High-quality monolithic quartz wafers were grown, oriented and sliced by Tokyo Denpa Co. Ltd, and delivered by their German distributer Coftech GmbH. The wafer was diced with a high-precision diamond saw (DISCO Kiru-Kezuru-Migaku Technologies) with a depth of 500 mm leaving a back wall of 1.5 mm. The un-diced surface of the crystal was coated with hydrofluoric acid resistant coating (HFRC) from Allresist GmbH to protect this side of the crystal before etching. The diced quartz was etched with 48% hydrofluoric acid (HF) for 1 h to remove the residual damage due to dicing. HFRC was cleaned off the un-diced surface after etching and the grooves were filled with HFRC. Piranha etch (sulfuric acid-hydrogen peroxide) was applied to make the surface of the 300 mm-thick silicon wafer hydrophilic before gluing to the diced surface of the quartz wafer using an epoxy (EPO-TEK 1 301-2). The epoxy was cured at room temperature for three days. The quartz crystal was diced completely with a depth of 1.5 mm from the other side and HFRC was applied around the wafer, before 2 h of final HF etching. HFRC was utilized to protect EPO-TEK 1 301-2 which is not resistant to HF. The whole flat assembly was sandwiched between plano-concave and planoconvex glass substrates (Borofloat 1 -33) of 2 m radius. EPO-TEK 1 301-2 was applied between the silicon wafer and the concave glass substrate. In order to avoid both air bubbles and dust in the glue layer, gluing was performed in the clean room and extreme care was taken to distribute the glue uniformly. A thin layer of rubber was placed under the convex glass substrate to distribute the pressure uniformly and 75 mm-thick Kapton 1 foil was placed so that the rubber could be removed easily at the end. For bending process, approximately 1.8 tons were applied during the hardening of the epoxy. The procedure is summarized in Fig. 8.

Figure 7
Dispersion of phonons in a Be single-crystal along [00] in the first and second Brillouin zones. Red circles with error bars are the results of this current study and black squares are taken from previous inelastic neutron scattering measurements (Stedman et al., 1976;Alatas et al., 2008). The black line denotes the energy resolution (26 meV) of the current work.

Figure 6
Two representative spectra. The measurements were performed by statically collecting without any mechanical scans. denotes the momentum transfer values in units of 2/a, where a is the lattice parameter along the (0 0 l) direction. The peak in the middle (zero energy loss) is the elastic peak, and the peaks on the right and left are the phonon creation and annihilation peaks, respectively. Excitations at energies as low as 16 meV can be resolved (inset). Each spectrum was recorded in 300 s.