Volume 20 Received 8 July 2012  Spherical analyzers and monochromators for resonant inelastic hard Xray scattering: a compilation of crystals and reflections Thomas Gog,^{a}^{*} Diego M. Casa,^{a} Ayman H. Said,^{a} Mary H. Upton,^{a} Jungho Kim,^{a} Ivan Kuzmenko,^{a} XianRong Huang^{a} and Ruben Khachatryan^{a}^{+} ^{a}Advanced Photon Source, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, USA Resonant inelastic Xray scattering (RIXS) experiments require special sets of nearbackscattering spherical diced analyzers and highresolution monochromators for every distinct absorptionedge energy and emission line. For the purpose of aiding the design and planning of efficient RIXS experiments, comprehensive lists of suitable analyzer reflections for silicon, germanium, quartz, sapphire and lithium niobate crystals were compiled for a multitude of absorption edges and emission lines. Analyzers made from lithium niobate, sapphire or quartz offer many choices of reflections with intrinsic resolutions currently unattainable from silicon or germanium. In some cases these materials offer higher intensities at comparable resolutions. While lithium niobate, sapphire or quartz analyzers are still in an early stage of development, the present compilation can serve as a computational basis for assessing expected and actual performance. With regard to highresolution monochromators, bandpass and throughput calculations for combinations of doublecrystal, highheatload and nearbackscattering highresolution channelcuts were assembled. The compilation of these analyzer and monochromator data is publicly available on a website. 
With the advent of thirdgeneration synchrotron radiation sources, resonant inelastic Xray scattering (RIXS) has become a popular technique to study collective electron phenomena in materials of great scientific and technological significance. Near an absorption edge, the technique provides resonant enhancements of inelastic scattering signals and makes measurements feasible that would not yield enough intensity in a nonresonant mode. Most noteworthy, excitation spectra of transition metal oxides with their vast collection of novel and important properties, such as highT_{c} superconductivity and multiferroic behavior, have been measured and interpreted very successfully with RIXS. For a comprehensive overview of the field, see a recent review article (Ament et al., 2011) and references therein.
Owing to its resonant character, one of the major technical challenges for RIXS measurements is the selection of analyzers and monochromators that provide the desired resolution and intensity at a specific absorptionedge or emissionline energy. Good energy resolution requires nearbackscattering analyzer reflections combined with a matched monochromator bandpass. In order to assist with an appropriate selection of these, comprehensive tables for suitable analyzer reflections in various materials are presented here. In addition, bandpass and throughput calculations for a particular monochromator concept are tabulated.
In a typical RIXS experimental setup (Gog et al., 2009; SchwoererBöhning et al., 1998), an incident monochromatic Xray beam with an energy bandpass in the meV range is prepared by a succession of highheatload and highresolution monochromators and microfocused onto the sample by a set of focusing mirrors. Scattered radiation from the sample is collected by a diced spherically shaped crystal analyzer in nearbackscattering configuration and redirected to a positionsensitive `strip' detector (Huotari et al., 2006). Sample, analyzer and detector are arranged in Rowland geometry. A schematic representation of this setup is shown in Fig. 1. The overall energy resolution, E_{tot}, of such a configuration is given by a convolution of all its resolution elements. For nearGaussian characteristics this convolution can be approximated by a square sum of the incident bandpass, E_{i}, corresponding to the selected monochromator combination, the intrinsic analyzer resolution, E_{a}, and geometric factors, E_{g},
A spherical RIXS analyzer typically consists of a flat wafer of an ideal crystal material, bonded to a glass or plastic substrate, diced into square pixels of millimeter size and bent into a spherical shape of radius R. Overall, the analyzer is thus an assembly of many flat unstrained crystallites tangent to a spherical surface. Ignoring possible deviations from a perfect spherical shape (figure errors), the intrinsic analyzer resolution, E_{a}, is determined by the incident energy, E_{i}, the angular reflection (Darwin) width, W, of the crystal reflection and the Bragg angle, _{B}. Namely,
It is apparent that the energy resolution of such an analyzer is best for reflections with a small Darwin width and nearbackscattering conditions, where the Bragg angle is close to 90° so that its cotangent approaches zero. The task is thus to identify nearbackscattering crystal reflections for every absorption edge and emission line of interest in RIXS, yielding the best resolution at reasonable reflectivities. In the past, silicon and germanium were the preferred choices for spherical analyzers since these materials yield nearly perfect crystals. However, with advances in crystal growth, other materials are becoming viable, such as lithium niobate (LiNbO_{3}), sapphire (Al_{2}O_{3}) and quartz (SiO_{2}). These materials have crystal structures of lower symmetry and thus offer many more possible reflections than silicon or germanium, with numerous choices of intrinsic resolution and throughput. Fig. 2 displays a partial map of reflections for the various crystal materials in the energy versus intrinsic resolution plane, with the size of the symbols proportional to the integrated reflectivity. It is quite apparent that lithium niobate and sapphire offer both highthroughput as well as highresolution reflections for the whole spectrum of energies. The fabrication of associated analyzers needs to be pursued to advance the technique.
 Figure 1 Schematic layout of a typical RIXS experiment. The incident beam is first monochromated by a highheatload monochromator (Cryst 1). The beam is then passed through one or two channelcut crystals (Cryst 2 and Cryst 3) and focused onto the sample. Xray photons with energy E_{i} and bandpass E_{i} are scattered from the sample and then reflected by a diced spherical analyzer towards the detector. 
 Figure 2 Partial map of the analyzer nearbackscattering reflections. For the various relevant energies the intrinsic energy resolution is shown. The area of the markers is proportional to the integrated reflectivity. 
The geometric term, E_{g}, arises from the fact that both the spatial resolution of the detector and the beam footprint on the sample are not zero but of finite extent. These spatial extensions subtend angles, , which in turn translate into an energy spread, E,
For the detector portion, = p/2R, where p is the size of a detector element and R is the diameter of the Rowland circle. For the footprint, = s/R, where s is the size of beam on the sample projected towards the analyzer. The present calculations only lists the detector portion, since p and R are constant for a given experimental setup. In contrast, s depends on the focusing and the orientation of the sample, which may vary throughout the measurement.
The observed energy resolution of a RIXS setup may contain additional contributions arising from imperfections of the instrument or its performance. These contributions are not addressed in the current document.
Comprehensive lists of crystal reflections in silicon, germanium, lithium niobate, sapphire and quartz were compiled for a multitude of absorption edges and emission lines of interest in RIXS, together with auxiliary information and geometric factors. In the same vein, data for nearbackscattering silicon channelcut crystals as one appropriate choice for highresolution monochromators were assembled. For this monochromator concept the large angular acceptance associated with nearbackscattering reflections offers optimal throughput for the incident Xray beam with bandpass choices matched to the intrinsic resolution of selected analyzers. The compilation of these analyzer and monochromator data are made available on a publicly accessible website described in §3 and §4, respectively.
In this article, dynamical diffraction calculations for both analyzer and monochromator crystals are based on a formulation by Authier (2001), and were executed using the software package Mathematica (Wolfram, 2009). The reflectivity I_{R} of a crystal reflection is described as
In this expression, () is a generalized angular parameter, itself a function of the deviation of the angle of incidence from the Bragg angle, _{B}. For a symmetric reflection and photon polarization perpendicular to the diffraction plane (polarization), is given by
with = r_{e}^{2}/V. Here r_{e} is the classical electron radius, the wavelength and V the volume of the crystal unit cell. F_{0} and F_{H} are unitcell structure factors associated with reciprocal lattice vectors 0 and H, respectively. These in turn can be written as
where the sum extends over all atoms of the unit cell and consists of atomic scattering factors f_{j}, their anomalous dispersion corrections , a DebyeWaller factor exp(2M_{j}) and a geometric component.
For the purpose of the present tables, structure factors were determined for room temperature (RT = 293.15 K) to reflect realistic operating conditions in an experiment. Calculations of f_{j} follow the algorithm used in the software package XOP (del Rio & Dejus, 2004). A Waasmaier & Kirfellike parametrization is employed (Waasmaier & Kirfel, 1995), as evaluated by Kissel (2000) using modified relativistic form factors. Dispersion corrections were taken from the database at the Center for Xray Optics (CXRO) at Lawrence Berkeley National Laboratory (Henke et al., 1993). DebyeWaller factors in the form of (see AlsNielsen & McMorrow, 2001, or other text books on Xray physics)
are considered, where (sin_{B}/) is proportional to the momentum transfer and meansquare atomic vibrational amplitudes at room temperature, B_{RT}, were derived from Xray diffraction measurements. In particular, the vibrational amplitudes and associated Debye temperatures, _{D}, assembled in Table 1, were used. For the geometric portion of the structure factors, some crystallographic data from XOP were used.

An important parameter in the present context is the Darwin width of a reflection. It is given by
The tables for the spherical analyzer are located at http://www.aps.anl.gov/Sectors/Sector30/AnalyzerAtlas/AnalyzerAtlas.html and constitute a compilation of nearbackscattering reflections in silicon, germanium, lithium niobate (LiNbO_{3}), sapphire (Al_{2}O_{3}) and quartz (SiO_{2}) for Bragg angles in the range 70° to 90°.^{1} Absorption edges and emission lines for a selection of chemical elements of interest in RIXS, which are included in the present compilation, are indicated in Fig. 3.
 Figure 3 Absorption edges and emission lines for a selection of chemical elements included in the present data compilation. 
Partial screenshots for the toplevel menu and an example of a listing for the Cu Kedge are shown in Figs. 4 and 5. These tables are divided into two groups of crystals with Si and Ge at the top (highlighted in yellow), lithium niobate, sapphire and quartz at the bottom. Within these two groups reflections are arranged from top to bottom by strength, according to the integrated reflectivity . For lithium niobate, sapphire and quartz all equivalent reflections are listed, while for Si and Ge equivalent reflections are only listed if their indices are not simple permutations or inversions of the parent reflection. The following quantities are included in the tables:
 Figure 4 Partial screenshot of the toplevel spherical analyzer compilation. Pointing to an element highlighted in yellow will open a menu with available absorptionedge or emissionline data. 
 Figure 5 Partial screenshot of the spherical analyzer listing for an absorptionedge energy of 8.9805 keV (Cu Kedge). 
Backscattering energy, E_{B} (keV). Xray energy for which the incident beam is reflected at a Bragg angle of _{B} = 90°. It is given by
where h is Planck's constant, c is the speed of light and d_{hkl} is the diffraction plane spacing. These are the lowestenergy photons that can be reflected by a particular analyzer.
Integrated reflectivity, (µrad). As a measure of the reflection strength this quantity represents a numerical integration of the dynamical reflectivity over the entire rocking curve.
Angular reflection (Darwin) width, W (µrad). Intrinsic, dynamical (Darwin) width associated with the symmetric reflection.
Change in energy with angle, dE/d = E_{i}cot (meV µrad^{1}). From the differential Bragg law, this quantity serves as the conversion factor from angular to energy width in convenient units and is included for guidance.
Intrinsic energy resolution, E_{a} (meV). Energy resolution of the analyzer reflection owing to its intrinsic (Darwin) width, E_{a} = WE_{i}cot_{B}.
Geometric contribution, E_{g} (meV). Geometric contribution to the energy resolution, based on the analyzer radius R and the detector pitch p: E_{g} = E_{i}cot(_{B})p/2R. For the present tables the detector pitch is assumed to be p = 50 µm (Dectris `Mythen' detector) while the analyzer radius is R = 2 m.
Combined intrinsic and geometric energy resolution, E_{t} (meV). E_{t} = ( + )^{1/2}.
The tables for highresolution channelcut monochromator crystals are located at http://www.aps.anl.gov/Sectors/Sector30/AnalyzerAtlas/MonoAtlas.html and were assembled for combinations of a Si or diamond highheatload monochromator followed by one or two pairs of highresolution Si channelcut crystals, as indicated in Fig. 1.^{2} The rationale for this crystal arrangement is inspired by the fact that the angular acceptance of the highresolution portion is proportional to 1/sin(2_{B}) [equation (5)]. This term becomes large for nearbackscattering conditions and thus guarantees an optimal throughput, while many choices of reflections arise to closely match the bandpass to the analyzer resolution. The bandpass and throughput data were calculated by multiplying a Gaussian Xray source distribution with all pertinent dynamical crystal reflectivities as shown in Fig. 6. The source distribution is given by
where
are the vertical combined, electron and photon beam divergences, respectively. For a typical undulator beamline at the Advanced Photon Source (APS), = 3.3 µrad and the undulator length L_{u} is 4.8 m.
 Figure 6 The product of a typical Gaussian source distribution and crystal reflectivities (left panel) leads to a reflectivity profile (right panel), which is the basis for the bandpass and throughput calculations. 
The intensity profile resulting from the product is numerically integrated over angle and energy. A partial screenshot of a listing at the Cu Kedge is shown in Fig. 7. The following quantities are included in the tables:
 Figure 7 Partial screenshot of a monochromator listing for an absorptionedge energy of 8.9805 keV (Cu Kedge). 
Overall energy resolution, E (meV), for the entire four or sixbounce combination of crystals (FWHM).
Integrated throughput, (µrad meV).
Figureofmerit (FOM) (µrad), ratio of throughput per energy resolution.
While parts of the calculations presented here involve parameters specific to a particular beamline or facility, the resulting tables are general enough to be a useful guide for RIXS beamlines everywhere. In the spherical analyzer section only the last two columns contain such parameters. Here the term E_{g} scales linearly with p and 1/2R as prescribed above and can easily be modified for a different geometry. In the same vein, the term E_{t} can be expanded to include additional terms in a squaresum fashion. The monochromator tables involve source parameters specific to the APS. Nevertheless, they still provide useful guidance generally in as much as source characteristics of thirdgeneration synchrotron sources are rather similar.
For the purpose of illustrating the utility of the analyzer and monochromator tables presented here, a specific example of an optical configuration for RIXS measurements at the Ir L_{3} absorption edge (E = 11.215 keV) is discussed. In recent years there has been a rapidly increasing interest in 5d transition metal oxides (TMOs), in particular the iridium compounds. These materials feature a rich spectrum of exotic electronic excitations, which are expected to lead to many novel phenomena of scientific and technological significance (Kim et al., 2012; Liu et al., 2012). In order to observe these excitations with RIXS, an energy resolution of 30 to 40 meV or better is required, substantially exceeding the typical conditions for this technique prior to this effort.
As explained in §1, the total energy resolution of the experimental setup shown in Fig. 1 is chiefly given by the square root of a sum of squares, consisting of monochromator bandwidth, intrinsic analyzer resolution and geometric contributions owing to the detector pitch and source size at the sample. According to the present tables, a monochromator as shown in Fig. 1 and using a combination of Si(111) followed by either one or two Si(448) channelcut crystals has a bandwidth of E_{i} = 15.8 meV and 8.956 meV, respectively. Furthermore, a Si(448) reflection at 11.215 keV has an intrinsic resolution of E = 14.57 meV, and a geometric detector contribution of E_{g} = 10.47 meV, given a Rowland circle with a diameter of 2 m and a detector pitch of 50 µm. The associated source contribution from a beam size of 50 µm is E_{s} = 20.95 meV. Altogether, the predicted energy resolution for such an experimental RIXS setup is then E_{tot} = 31.8 meV and 29.0 meV for the single and double monochromator combinations, respectively.
The two configurations described above have been implemented on RIXS instruments at the undulator beamlines 9ID and 30ID at the APS at Argonne National Laboratory. Measurements of elastic lines from a `standard' scatterer, Scotch Brand #810 Magic Mending Tape, which is composed of a cellulose acetate carrier and an acrylic polymer adhesive, are shown in Fig. 8. The dots represent measured data and the line is a Voigtfunction fit. The full width at halfmaximum was determined to be 34.4 meV and 33.8 meV, respectively, for the single and double monochromator, well within 10% of what was predicted based on the tables.
 Figure 8 Measurements of elastic lines from a `standard' scatterer, Scotch Brand #810 Magic Mending Tape. Measured FWHM and calculated values based on present tables are well within 10% of each other. 
Identifying nearbackscattering reflections for spherical analyzers combined with matching monochromator characteristics, yielding a required energyresolution with optimal throughput, has been a daunting challenge for RIXS experiments. The present compilation of analyzer reflections and channelcut monochromator combinations was designed to aid the selection of crystals suitable for RIXS measurements at a particular absorption edge or for a particular emission line. In addition, the inclusion of lithium niobate, sapphire and quartz for spherical analyzers provides a theoretical basis for the characteristics that can be expected from these unusual crystal materials and can help to assess their actual performance in an experiment.
Use of the Advanced Photon Source at Argonne National Laboratory is supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DEAC0206CH11357.
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