research papers
Beamline simulations using monochromators with high dspacing crystals
^{a}Singapore Synchrotron Light Source, National University of Singapore, 5 Research Link, Singapore 117603, Singapore, ^{b}Department of Physics, National University of Singapore, Singapore 117576, Singapore, ^{c}Department of Engineering Physics, Fonty University of Applied Sciences, 5615DB Eindhoven, The Netherlands, ^{d}Centre for Advanced 2D Materials and Graphene Research Centre, National University of Singapore, Singapore 117546, Singapore, ^{e}NUS Graduate School for Integrative Sciences and Engineering, Singapore 117456, Singapore, and ^{f}European Synchrotron Radiation Facility, 38000 Grenoble, France
^{*}Correspondence email: slsyxj@nus.edu.sg, srio@esrf.eu, phymbhb@nus.edu.sg
Monochromators for synchrotron radiation beamlines typically use perfect crystals for the hard Xray regime and gratings for soft Xrays. There is an intermediate range, typically 1–3 keV (tender Xrays), which common perfect crystals have difficulties covering and gratings have low efficiency, although some less common crystals with high dspacing could be suitable. To evaluate the suitability of these crystals for a particular beamline, it is useful to evaluate the crystals' performance using tools such as raytracing. However, simulations for doublecrystal monochromators are only available for the most used crystals such as Si, Ge or diamond. Here, an upgrade of the SHADOW raytracing code and complementary tools in the OASYS suite are presented to simulate high dspacing crystals with arbitrary, and sometimes complex, structures such as beryl, YB_{66}, muscovite, etc. Isotropic and anisotropic temperature factors are also considered. The YB_{66} crystal with 1936 atomic sites in the is simulated, and its applicability for tender Xray monochromators is discussed in the context of new lowemittance storage rings.
Keywords: high dspacing crystal; YB_{66}; crystal monochromators; ray tracing; SHADOW; OASYS.
1. Introduction
Xray monochromators use crystals that must fulfill many requirements: they must have high perfection (no dislocations, low mosaicity); be available as large, single crystals; have high resistance to radiation damage; and have high dspacing of 3.135 Å, Si 111 is not very effective for large Bragg angles (3 keV corresponds to 41.2° and the minimum energy attained is 1.977 keV at normal incidence). Therefore, other crystals with large cell parameters must be investigated for applications using tender Xrays. Many natural crystals have been proposed. An exhaustive list of crystals with larger dspacing is given by Underwood (2001). Databases such as DABAX (Sanchez del Rio, 2011a) and Stepanov's Xray server (Stepanov, 2004) contain long lists of crystal structures that can be used in Xray monochromators. We have compiled in Table 1 a list of crystal reflections with large dspacing, including useful energy range, Darwin width (θ_{d}), relative energy resolution (ΔE/E) and peak reflectivity (R).
The ubiquitous material for Xray monochromators and analyzers is silicon; crystals of silicon are available in large sizes and with high perfection. Indeed, silicon is the most perfect large crystal in the world. Germanium, with slightly higher cell parameters, also forms a highly perfect crystal but is more expensive. Synthetic diamond is also used in Xray monochromators because of its low absorption and exceptional These are cubic facecentredcubic (f.c.c.) crystals, and the lower nonforbidden reflection is 111, which is indeed the most used reflection in synchrotron monochromators. With a

Despite such a long list, it is difficult to acquire a suitable crystal for the tender Xray regime. In particular, organic crystals cannot withstand high heat loads, and many natural crystals cannot be found with high perfection in the large size needed for monochromators. Synthetic crystals such as synthetic quartz (Cerino et al., 1980; Wong et al., 1999; Ohta et al., 1986) and sapphire (Shvyd'ko et al., 2017; Said et al., 2020) are nowadays obtained with quality and size suitable for Xray applications. However, quartz degrades very quickly under exposure to intense synchrotron radiation. Sapphire is better (Gog et al., 2018), nevertheless still sensitive enough to radiation damage. Diamond has exceptional resistance that makes it appropriate for hard Xrays. The ongoing search for good crystals must be accompanied, and in many cases driven, by computer simulations of the theoretical reflectivity profiles of the crystals, to simulate by raytracing the whole monochromator embedded in a synchrotron beamline. Computer tools for such simulations are not easy to find, despite the already long list of available crystals in tools like OASYS (Rebuffi & Sanchez del Rio, 2017), or Sergei Stepanov's Xray server (Stepanov, 2004).
One crystal proposed and used in the tender Xray range is YB_{66}. The is facecentered cubic with a = 23.440 Å (Richards & Kasper, 1969). The material is refractory and has a melting point of 2100°C. It is thermally stable and can resist severe radiation and high heat load. The crystal can be fabricated with high quality (Tanaka, 2010) and has been successfully applied in doublecrystal monochromators (DCMs) in the tender energy regime (Smith et al., 1998; Rek et al., 1993; Wong et al., 1990; Ohta et al., 1986; Kitamura & Fukushima, 2004; Wong et al., 1995).
About 14 years ago, at the XAFCA beamline at Singapore Synchrotron Light Source (Yu & Moser, 2008) working in a photon energy interval of 0.85–12.8 keV, we adopted a sagittalfocusing DCM with Si (111) crystals to cover the energy range 2.15–12.8 keV. However, for energies below 3 keV in the original optical design, we looked at three crystals, InSn (111), beryl () and YB_{66}, to cover the lowerenergy range. It was difficult and expensive to acquire largesize YB_{66} crystal wafers, and beryl has a strong aluminium which is a disadvantage for the study of aluminium catalysis. A KTP (011) crystal pair was chosen in the final beamline. Raytracing calculations were performed using SHADOW (Cerrina, 1984), modified to account for crystal reflectivity interpolated from a data file, in this way inhibiting the internal crystal reflectivity calculation during ray tracing.
In recent years, larger YBlike crystals have been produced (Tanaka, 2020) at a lower cost, therefore interest in this crystal has renewed. However, no recent use of a YB_{66} crystal in a DCM beamline has been reported. One reason may be that, for highbrilliance insertiondevice synchrotron sources, soft Xray grating monochromators are now used up to 2.5 keV, close to the upper limit of tender Xrays, something that was impossible in the past (Hawthorn et al., 2011; McChesney et al., 2014; Tang et al., 2019). In terms of the resolution and reflectivity, YB_{66} is still attractive. The YB_{66} (E/ΔE) is ∼23000 for 004 and 122000 for 006 (from Table 1), with peak reflectivity of 15% and 10%, respectively. As compared with grating monochromators, the resolution can be considered as `high', and the peak reflectivity is also of the order of the efficiency obtained by good gratings. In theory, the performance obtained by a monochromator with an ideal YB_{66} crystal is at least as good as that using gratings. The use of crystals is more advantageous for large sources (wigglers or bending magnets), as gratings require a clean focusing to become effective. However, grating monochromators can easily tune the resolution by playing with the slit aperture and customizing ruling values.
To prepare for possible upgrades of synchrotron beamlines and the use of YB_{66} and other complex crystals, we have upgraded the raytracing code SHADOW3 (Sanchez del Rio, 2011b) and other tools in OASYS to include, in a seamless way, any The code algorithms and modifications are presented in this work. It is now possible with the ShadowOUI (Rebuffi & Sanchez del Río, 2016) addon of OASYS (that interfaces SHADOW) to raytrace YB_{66} crystals and any other crystal of interest for an Xray monochromator or analyser.
2. Calculation of the of any crystal structure
2.1. of a crystal
The
of a crystal iswhere h, k, l are C_{j} is the occupancy factor, T_{j} is the Debye–Waller or temperature factor, f_{j} = (f_{0} + f′ + if′′)_{j} are atomic scattering factors, and x_{j}, y_{j}, z_{j} are fractional coordinates of the atoms in the The subindex refers to the jth atom in the and the sum extends over the n atoms of the unit cell.
To compute the for a particular structure we need libraries and methods to access:
using equation (1)(1) The information from the a, b, c, α, β, γ). Each atomic center in this list must contain the fractional coordinates, the nature of the center (its Z) and also the ionic charge and fractional occupation. Moreover, other information could be added to compute the temperature factor, as described below. Note that the summation goes over the n atoms in the In crystallography, the list of all atoms is created by applying the symmetry operations from the space symmetry group to a reduced number of atoms (the asymmetric unit).
itself, meaning a list of all atoms in the as well as the cell parameters ((2) The atomic scattering factors. The f_{0} depends to a good approximation only on q = (sinθ)/λ, with θ the grazing incidence angle and λ the photon wavelength. The socalled `anomalous' scattering factors f′ and f′′ depend on the nature of the atom and on the photon wavelength.
Although the f_{0} values, and the values f′ and f′′. The ab initio calculation of the scattering factors can only be performed using complex quantum mechanics calculations, and it is out of the scope of most crystallography codes. Tabulated data from some references may be used, usually linking the code to available data files or databases. This linkage makes the software structure complicated and reduces portability. This is why in the raytracing code SHADOW (and also in OASYS crystal tools) the calculation is performed in two steps: (i) a preprocessor code that accesses necessary data from databases and creates a `crystal material file' with the basic ingredients needed to build the and (ii) the calculation of the in the SHADOW kernel using only the information in the preprocessor file, without any further link to databases.
can be efficiently calculated by a single piece of code, it requires quick access to data usually stored in libraries or databases. Four sources of information are needed: unitcell information,The first version of SHADOW used internal tabulations to retrieve the f′ and f′′ values, but did not provide any f_{0} data. The old SHADOW interface in XOP (Sanchez del Rio, 2011a) took the data from DABAX, an ad hoc compiled collection of material data, available from https://ftp.esrf.eu/pub/scisoft/DabaxFiles, where several tabulations for the same kind of data (e.g. f′) coexist. The OASYS package uses xraylib (Schoonjans et al., 2011; Schoonjans, 2021) which is a compiled library that allows fast access of a large collection of Xray data. OASYS can also use DABAX where we included the new data needed for this work.
2.2. Ingredients for computing the structure factor
The F(h,k,l) [equation (1)] comes from the summation of all waves scattered by the n atoms in the in the direction defined by the hkl. Each atom j contributes to a wave whose amplitude is proportional to the f_{j}, that measures the Xray scattering power of each atom. Its main component is the factor f_{0}. The scattered power is maximum in the direction of the incident Xrays, and decreases as a function of the angle of departure. It is proportional to the number of electrons in the atom. In `electron units', f_{0} is equal to Z at the zero scattering angle (θ = 0°), and reduces to almost zero at values of q = (sinθ)/λ larger than about 2 Å^{−1} (2θ is the angle between incident and scattered Xray beam with wavelength of λ). This dependency is tabulated after some theoretical calculations and can be parametrized. Cromer & Mann (1968) proposed a sum of Gaussians parametrization with nine coefficients,
with Nc = 4. The nine coefficients (a_{1}–a_{4}, c, b_{1}–b_{4}) are obtained by fitting tabulations of f_{0} computed using theoretical models. A common tabulation is shown in International Tables of Crystallography, and includes all neutral atoms and a few ionic states. Waasmaier & Kierfel (1995) extended the number of coefficients to 11 to fit data up to q_{max} ≃ 6 Å^{−1}. In OASYS, the theoretical calculation from Kissel (2000) is fitted to obtain a list of 11 coefficients for all neutral atoms. Most typical crystals used in monochromators are covalent crystals so f_{0} data for neutral atoms is enough. However, data for ions with integer or fractional charge is sometimes needed. For example, for YB_{66} the calculation of form factors f_{0} requires the ionic states `Y^{3+}' and `B^{−.}' (the dot indicates a tiny negative charge in the B atoms. We follow the formula of Higashi et al. (1997), so the charges removed from the neutral Y atoms are equally allocated among all boron atoms. Some f_{0} coefficients for ionic states are available in International Tables for Xray Crystallography (Ibers & Hamilton, 1974) (hereafter called ITC).
For some particular ionic states, like in the case of Y^{3+} (needed in the YB_{66} crystal), the f_{0} data are found in the ITC tabulation. However, in most cases, such as for the tiny charge of B in YB_{66}, there are no tabulated data for f_{0}. Values of some ionic states of B, such as B^{3+}, are found in a table in ITC (p. 202). We fitted B^{3+} data to obtain the nine parameters that were added to the DABAX file f0 _ InterTables.dat that contains the ITC tabulation. DABAX is incorporated into our tools for retrieving the nine parameters of the f_{0} coefficients for both neutral and ionic atoms. In cases of ions with fractional charge, like B^{−0.0455} required in the YB_{66} crystal, the nine coefficients for parametrizing f_{0} are calculated by interpolating the data from two entries in the table with different charges.
Using the form factors data f_{0}(B), f_{0}(B^{3+}), f_{0}(Y^{3+}) from this reference, and assuming `B^{−.}' carries a negative charge of δ = 0.0455, the form factors for `B^{−.}' can be calculated as a linear interpolation,
A comparison plot of f_{0} versus q for neutral atoms of Y and B is shown in Fig. 1. It shows good agreement between xraylib and DABAX for the neutral species, but the new ionic states are only available from DABAX.
The intense f_{0} can be reduced if the incident Xray radiation has a frequency close to the natural oscillation frequency of the electrons of a given atom. This effect is called (although there is nothing `anomalous') and is represented by f′ and f′′, the real and imaginary components, respectively, of the anomalous fraction of the These are functions of the photon energy. factors used in most software packages come from old calculations using quantum electrodynamics data (EPDL97, https://wwwnds.iaea.org/epdl97/), mixed experimentally evaluated data (Henke et al., 1993), or a combination of them. In the past, SHADOW used hybrid data from Henke et al. (1993) (up to 30 keV) and Cromer (1983). DABAX files contain many tabulations available in the literature. The library xraylib, used by default in OASYS, accesses a selected tabulation of data as described by Schoonjans et al. (2011). The inclusion of factors in calculations of crystal diffraction profiles using the of diffraction is essential, as it contributes to the peak intensity.
The list of atoms in the DABAX file crystals.dat. The same data are also integrated in xraylib. We added the new entry YB_{66} in DABAX, with information on the composition of the crystal unit cell.
is tabulated in the2.3. Anisotropic temperature factors
Displacement of atoms from their mean positions in the ) observed that Debye–Waller factors (also called temperature factors) for most crystals are indeed anisotropic. In addition, thermal movement from the same atoms may be different in different atomic sites (as shown in Table 2). In most cases, atoms belonging to the same group of prototypical atoms have different temperature factors. The anisotropic temperature factors of 14 symmetric sites for YB_{66} are plotted in Fig. 2; one can see that the temperature factors for boron atoms are different in 13 atomic sites, except sites B6 and B7. The anisotropic factors for all symmetric sites are necessarily included for accurate calculations; therefore the list of prototypical atoms must also reflect the site information from the temperature factors.
weakens the scattered A Debye–Waller factor takes this effect into account in the structure analysis. Lonsdale & GrenvilleWells (1956

To describe the anisotropic temperature factor when the atomic displacement follows a trivariate Gaussian probability distribution function (as usually accepted), six coefficients, β_{ij} (i, j = 1, 2, 3), are needed [equation (21) in Trueblood et al. (1996)],
In Table 2, we give an example input of anisotropic temperature data for YB_{66} (Richards & Kasper, 1969). Boron atoms occupy 13 symmetric sites, each having different anisotropic temperature factors, and yttrium atoms occupy a single site, with equivalent temperature factors. The β_{ij} coefficients are always far less than 1 for an anisotropic temperature factor. Once β_{11} is set to 1, the β_{ij} coefficients are for a different notation, with only β_{22} representing an isotropic B_{eq} factor, while the β coefficients in other columns are discarded (all set to zero). The temperaturefactor data for any crystal can be easily defined accordingly. If no input line beginning with #UANISO _ COFF is found in the crystal file, the default scale temperature factor is used.
In the calculations, two temperature factors are implemented – the anisotropic [equation (4)] and the isotropic temperature factors. In the last case, the displacement amplitudes are equivalent for all directions, and the temperature factor depends on the isotropic B_{eq} factor (Trueblood et al., 1996; Higashi et al., 1997),
where
When calculating the anisotropic temperature factor for YB_{66} 400, according to the formula, only β_{11} contributes to the temperature factor, and there are two identical pairs of β_{11}; therefore we only have 12 different anisotropic temperature factors. For isotropic factor B_{eq}, there are two identical factors for sites B2 and B7, so there are 13 different temperature factors (see Fig. 2).
For some crystal structures, for instance some of the YB_{66} variants in Higashi et al. (1997), only B_{eq} factors are given. We handle such cases using the #UANISO _ COFF coefficients, with the convention of setting β_{11} to a number equal to or greater than one, and setting β_{22} to B_{eq}.
2.4. The new crystal preprocessor for OASYS and SHADOW
The preprocessor file in SHADOW and CRYSTAL contains the elements of the [equation (1)]. The idea is to group atoms that are identical in all parameters except in (x, y, z) in a single `prototypical' atom. With this, the number of terms in the sum is reduced and the calculation is faster.
The new code bragg_calc2 is used for generating the crystal material file used in both SHADOW and OASYS (CRYSTAL and FH widgets). This code can use two libraries for material constants (xraylib and DABAX), at the users' choice. When calculating YB_{66}, DABAX is the only option, as it was upgraded with its crystal unitcell structure (in the file crystals.dat) and f_{0} ionic states (in the file f0 _ InterTables.dat).
The format of the new crystal material file evolved from the original one to minimize changes of source code; the old and new formats are shown in Figs. 3(a) and 3(b), respectively. In the old format preprocessor file, a maximum of two kinds of atoms and two symmetric sites can be input, and a maximum of eight atoms in a Bravais The first item, named i_ LATTICE in the first line, identifies the file format. Integer values of 0 to 5 denote this file in the old format containing data for the types of zinc blende, rock salt, simple f.c.c. and CsCl structure, and two hexagonal (closepacked structure and graphite structure), respectively. RN and D_ SPACE are the product of the inverse of the volume of a and classical electron radius, and the distance between crystalline planes, respectively.
In the new file format, there are three main differences for crystal input. First, there is no limit on the number of prototypical or siteatoms used; each one will have corresponding values of G and G _ BAR (complex conjugate of G) geometrical factors [corresponding to the exponential in equation (1)], and coefficients for f_{0}. Second, the atoms in a crystal can carry integer or fractional charges, ATOM _ A and ATOM _ B in Fig. 3(a). The integer atomic numbers of the constituent atoms are replaced by a float number of scattering electrons [atomic number plus ionic charge Z1, Z2, Zn in Fig. 3(b)]. Third, temperature factors are included for different symmetric sites.
In the new file format, each prototypical atom has a different identifier. This contains the becomes a sum to M prototypical atoms,
site occupation, fractional charge and temperature factor. Two atoms in the belonging to the same prototype only differ in the coordinates. Therefore, equation (1)with G a geometrical factor that contains the sum of atoms belonging to each prototypical group,
The new file format can safely replace the old file format without introducing errors for any crystal allowed in the old SHADOW file, because the raytracing calculated results are identical for the existing crystals represented by the two formats as demonstrated in Section 4.
2.5. New crystals in OASYS widgets
In the OASYS suite, the widgets that perform crystal calculations are FH and CRYSTAL in the XOPPY addon, and BRAGG preprocessor in the ShadowOUI addon. FH and CRYSTAL widgets are used for calculating parameters, for instance, the Darwin width, etc. and the Xray reflectivity for crystals. Previously these applications accepted crystals composed of neutral atoms with single scale temperature factors for all atomic sites. A new software component bragg _ calc2 has been introduced to calculate arbitrary crystals including charged atoms, and isotropic or anisotropic temperature factors in different sites. Fig. 4 shows the reflectivity of mica 111 at 8 keV calculated with the new bragg _ calc2 and compared with calculations using Stepanov's Xray server. Both curves are in good agreement.
The SHADOW preprocessor widget BRAGG is used for creating the crystal input file for SHADOW. This widget has been updated to use bragg _ calc2 code for new crystals such as YB_{66}.
3. Benchmarking and diffraction profiles for YB_{66} and other crystals
3.1. Validation of the model for the YB_{66} crystal
To validate the generated model (lattice constants, atomic coordinators list and temperature factor, etc.) of YB_{66} crystal insertion in the file crystals.dat for the OASYS suite, the structure factors F(0,0,0) and F(H,K,L) are calculated with the updated FH widget under XOPPY Optics in the OASYS suite (see Fig. 5). Some calculated structure factors for the index planes with practical synchrotron radiation applications are tabulated in Table 3 for comparison with data from Richards & Kasper (1969), who tabulated F_{CALC} (calculated) and F_{OBS} (measured) with Cu Kα radiation (∼8040 eV). A photon energy of 8040 eV was used in our calculations. The calculated F(0,0,0) of 8846 is in agreement with the calculated value of 8841 by Richard & Kaspar (1969). However, to make the overall comparison consistent, the F(H,K,L) values by Richard & Kaspar (1969) must be multiplied by four, because their calculation included the factor x (1/4) for YB_{66} (Tanaka, 2020).

3.2. Diffraction profiles
The calculated structure factors for YB_{66} are compared in Table 3 with those of Richards & Kasper (1969). The updated CRYSTAL widget can calculate the reflectivity for an arbitrary crystal. The reflectivity plot of YB_{66} 004 (see Fig. 6) takes into account the different anisotropic temperature factors at 14 symmetric sites and ionic atoms at 8.04 keV. No data for YB_{66} reflectivities are available in the literature for comparison. The YB_{66} 400 reflection is used in the following section in SHADOW raytracing simulation calculations, and the results are compared with the measurement data from the references.
4. Demonstration of raytracing simulations for YB_{66} crystal
The SHADOW code is a well known raytracing engine for beamline design in the synchrotron radiation community. Almost all current beamlines benefitted from the help of SHADOW. We use SHADOW3 code interfaced in the ShadowOUI addon of OASYS to demonstrate the raytracing results of a DCM with neutral atomic crystal and ionic atomic crystals represented by new SHADOW preprocessor files, in comparison with the data in the references. Both preprocessor files are created with the updated BRAGG widget.
To check the consistency of the modifications included in SHADOW, we performed ray tracing of a DCM with InSb (111) crystal pair with the old and new preprocessor file formats at a photon energy around 1.8 keV. The results [Figs. 7(a) and 7(b)] show an excellent agreement in reflectance profile and full width at halfmaximum (FWHM).
A second test aimed to compare simulations for beryl (Al_{2}Si_{6}Be_{3}O_{18}) with bibliographic data. Fig. 7(c) shows the raytracing result of a singlecrystal reflection for beryl . This includes four different elements and was not possible to simulate with the old SHADOW. The system comprises a point source with angular divergence of 0.3 mrad (FWHM) positioned 2 m upstream of a beryl crystal. Ray tracing gives a resolution of 0.72 eV (r.m.s.), which would correspond to a FWHM of 1.7 eV if the profile was supposed to follow a Gaussian. This is in good agreement with the measurement result of 1.6 eV FWHM [see Fig. 5(b) of Wong et al. (1999)]. Moreover, the peak reflectivity of 15.6% [see Fig. 7(c)] quantitatively agrees with the measurement [12.8% in Wong et al. (1999), Fig. 5(a)]. Here, the angular divergence has been intentionally set to 130 µrad RMS because this information is not found in that reference.
The last simulations concern YB_{66}, and are also benchmarked against published data. The DCM of YB_{66} (004) crystals has important glitches around 1385.6 eV and 1438 eV (Smith et al., 1998; Wong et al., 1999). There is an increase of intensity due to the superposition of the 006 reflection at the yttrium L_{3} and L_{2} absorption edges of 2080 eV and 2156 eV, respectively. Tanaka et al. (1997) numerically reproduced the glitches with their custommade program. The at 1385.6 eV is about 80% higher for the (004) reflection due to the added contribution of intensity at 2080 eV from the 006 reflection. We have simulated the anomalous enhanced at 1385.6 eV by raytracing calculations for the JUMBO beamline at SSRL (Cerino et al., 1980) where the abnormal reflectivity was measured (Wong et al., 1999). measurement at JUMBO was made using a gold mesh detector (Cerino et al., 1980); therefore the of gold (Krumrey et al., 1988) must be taken into account in the calculation. For creating the YB_{66} preprocessor files around 2080 eV it is important to use a dense energy grid. The factor (f′, f′′) must be sampled correctly, which is guaranteed using Henke data (Henke et al., 1993) from the CXRO website (https://henke.lbl.gov/optical_constants/sf/y.nff), available in DABAX.
For the JUMBO beamline, there was a Ptcoated mirror at an incidence angle of 89° followed by a DCM of YB_{66} (004), and a gold screen detector downstream of the DCM. The calculation parameters are listed in Table 4. The is calculated using parameters of the SPEAR storage ring at SSRL (Baltay et al., 1991), for a beam energy of 3 GeV and bending magnetic field of 0.84 T, 0.6 mrad vertical acceptance. The at 1385.6 eV is normalized to unity in Table 4. The plots of raytracing results by the DCM of YB_{66} (004) and YB_{66} (006) at 1385.6 eV and 2080 eV, respectively, are shown in Fig. 8. The source bandwidth for source generation in the ray tracing around 1385.6 eV and 2080 eV are chosen to be 0.097% (1385–1386.35 eV) and 0.086% (2079.2–2081 eV), respectively, in order to match the DCM resolution. The obtained raytracing intensities are 9220 and 20479 for a source with five million rays.

The I) created by photons incident on the gold screen detector is proportional to the product of (F), bandwidth (BW) used in ray tracing, mirror reflectivity (R), (QE) and intensity (WI). WI is found in Fig. 8 and the actual bandwidth (BW) is normalized with the bandwidth (0.1%) used in calculation of the relative (F) in Table 4,
intensity (The yield ratio at the intensities at 1385.6 eV and 2080 eV can be calculated with this equation using data in Table 4 and Fig. 7, thus resulting in I(006)/I(004) = (1.01 × 0.086% × 0.59 × 0.027 × 20479) / (1.0 × 0.097% × 0.77 × 0.045 × 9220) = 0.91. This value of 91% increase in at 1385.6 eV quantitatively reproduces the anomalous glitch (80%) at 1385.6 eV for the DCM of YB_{66} (004) reported by Wong et al. (1999).
5. Summary and conclusions
The use of crystals with high dspacing in synchrotron radiation monochromators is still a challenge due to the poor quality of most of the suitable crystals. However, improving technology in growing crystals makes it possible to use some of them for tender Xray monochromators. We extend the current methods of simulating Xray crystal reflectivity, and perform raytracing simulations with crystals constituting charged atoms arranged in any crystalline structure and including isotropic or anisotropic temperature factors. This will open more opportunities for numerical calculations for simulating present or future Xray monochromators. We presented examples of Xray reflectivity for mica, InSb and beryl. Furthermore, we analyzed in detail the YB_{66} crystal with a complex structure of 1936 atomic centers. Raytracing simulations were performed for a JUMBO monochromator retrieving values consistent with the experimental results. In summary, the tools available in OASYS for crystal reflectivity and raytracing crystal monochromators are upgraded to account for any crystal, once the is known. The new open source software tools developed here are available for supporting accurate calculations in the design and optimization of new Xray monochromators.
The OASYS workspaces and scripts implementing the simulations presented in this paper are available at https://github.com/91902078/yb66.
Acknowledgements
The authors would like to acknowledge the Singapore Synchrotron Light Source for providing the facility necessary for conducting the research. The laboratory is a National Research Infrastructure under the National Research Foundation, Singapore. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of National Research Foundation, Singapore. The authors are grateful to Arthur Leung and Herbert O. Moser for interesting discussions.
Funding information
The following funding is acknowledged: National Research Foundation Singapore; European Union's Horizon 2020 Research and Innovation programme [award No. 823852 (PaNOSC)].
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