Performance calculations of the X-ray powder diffraction beamline at NSLS-II

The X-ray Powder Diffraction (XPD) beamline at the National Synchrotron Light Source II is a multi-purpose high-energy X-ray diffraction beamline with high throughput and high resolution. The beamline uses a sagittally bent double-Laue crystal monochromator to provide X-rays over a large energy range (30–70 keV). In this paper the optical design and the calculated performance of the XPD beamline are presented. The damping wiggler source is simulated by the SRW code and a filter system is designed to optimize the photon flux as well as to reduce the heat load on the first optics. The final beamline performance under two operation modes is simulated using the SHADOW program. For the first time a multi-lamellar model is introduced and implemented in the ray tracing of the bent Laue crystal monochromator. The optimization and the optical properties of the vertical focusing mirror are also discussed. Finally, the instrumental resolution function of the XPD beamline is described in an analytical method.

1. Introduction

An X-ray Powder Diffraction (XPD) beamline is being built at the new synchrotron X-ray source (NSLS-II) at Brookhaven National Laboratory, USA. The optical scheme takes full advantage of the high flux of a 7 m-long 1.8 T damping wiggler, and uses techniques and instrumentation pioneered or under development at the NSLS [e.g. Laue optics (Zhong et al., 2001a,b) and Ge strip array detectors (Rumaiz et al., 2010)]. Using three experimental hutches, the XPD powder diffraction beamline will be a multi-instrument facility with the ability to collect diffraction data at high monochromatic X-ray energies (30–70 keV), offering rapid acquisition (sub-second) as well as high angular resolution capabilities. XPD is designed to combine high Q-space resolution diffraction measurements and high real-space resolution pair distribution function (PDF) measurements. The beam size is adjustable to match the graininess and heterogeneity scales of the samples above the micrometer scale. XPD addresses future scientific challenges in, for example, hydrogen storage, CO₂ sequestration, advanced structural ceramics, catalysis and materials processing. Such materials of high technological value often are complex, nanostructured and heterogeneous. The scientific grand challenge is to obtain robust and quantitative (micro)structural information, not only in the ground state under ambient conditions but also in situ or in operando with varying temperature, pressure, magnetic, electric or stress field, chemical environment, etc. The XPD beamline will be commissioned during 2014 and full user operation mode is due to begin in June 2015.

This paper describes the layout and calculates the expected performance of the XPD beamline. The source simulation and the beamline ray tracing are performed using the SRW (Chubar, 2001) program and the SHADOW (Welnak et al., 1994) code under XOP (Sanchez del Rio & Dejus, 2004). In particular, we developed a multi-lamellar model to simulate the performance of the sagittally bent double-Laue crystal monochromator (DLM). The instrumental resolution function (IRF) of the XPD beamline is expressed in an analytical method, which includes the contributions from each optical element of the beamline.

The purpose is first to rationalize and optimize the choice of each optical component of the XPD beamline. The simulated performance will be useful for future users to explore the capabilities of the beamline as well as to plan for early experiments. At the same time, the simulation method (e.g. the multi-lamellar modelling of the DLM and the IRF calculation) shown in this work will also benefit the design of high-energy wiggler beamlines in the future.

2. Beamline layout/description

The XPD beamline is designed to operate under two different operation modes, i.e. the high-flux mode and the high-resolution mode. The optical layouts are shown in Fig. 1. In the high-flux mode the X-ray beam generated by the damping wiggler is horizontally focused by the sagittally bent DLM and vertically focused by the vertical focusing mirror (VFM). The modest 0.1% energy bandwidth is acceptable for many high-energy experiments, e.g. in situ, time-resolved, materials science, strain and stress analysis, and PDF research. In the high-resolution mode the beam is collimated by the VFM in the vertical diffraction plane before impinging a channel-cut high-resolution monochromator (HRM). Both the induced lower vertical divergence and the higher monochromaticity (ΔE/E ≃ 10⁻⁴) improve the resolution performance for such experiments as structure solving, lattice parameter measurement and line shape analysis.

Figure 1 Schematic layout of the XPD beamline. DW: damping wiggler; DLM: double-Laue crystal monochromator; VFM: vertical focusing mirror; HRM: high-resolution monochromator.

3. Performance calculations

The beamline parameters for all the XPD optical elements are listed in Table 1.

Table 1
Beamline parameters

Damping wiggler (DW100)
Number of periods	68
Period length (mm)	100
Effective K value, Keff	16.5
Electron beam size r.m.s (µm)	137 × 4.9
Electron beam divergence r.m.s (µrad)	6.6 × 1.6
Fixed mask aperture (mrad)	1.1 × 0.1
Filter assembly from source (m)	30
DLM
Distance from source, F1 (m)	32.3
Vertical offset (mm)	50
Crystal orientation	111 reflection on Si (100) crystal
Crystal asymmetric angle, χ (°)	35.26
Crystal thickness, T0 (mm)	0.7
Range of Rs (m)†	0.8–∞
Mirror
Distance from source, p (m)	40.1
Grazing angle, δ (mrad)†	1.0–2.0
Bending radius, R (km)†	12–76
Mirror length, L (mm)	1300
HRM
Distance from source (m)	53.3
Crystal orientation	Symmetric Si (111) crystal
Sample-to-source distance (m)	56.4

†The value depends on the operation energy (see Table 2 and Fig. 9 for details).

3.1. Incident beam properties

The damping wiggler source (DW100) is located in a high-β straight section of the NSLS-II storage ring. The DW100 extends the range of X-ray energies well beyond 50 keV, thus matching the scientific needs of the XPD beamline. The total power generated by the DW100 is 61 kW when the NSLS-II ring electron beam current reaches the nominal value of 500 mA [see Fig. 2(a) for the power density distribution]. The XPD beamline accepts 1.1 mrad horizontally and 0.1 mrad vertically from the DW100 radiation using a fixed aperture mask, within which the total power is 5.9 kW. Most of the thermal power of the DW100 lies inherently within the low-energy spectrum (50% occurs below the critical energy of 11.1 keV). The XPD beamline uses two fixed diamond filters (one also serves as a vacuum isolation window) and one set of adjustable silicon carbide (SiC) filters to attenuate the beam. This concept is being successfully used at the I12 beamline at Diamond Light Source (Drakopoulos, 2010). Fig. 2(b) shows the power spectrum of the incident beam after 2 mm of diamond and different thicknesses of SiC. The diamond filter, which could consist of a single 2 mm-thick window, will be made for safety reasons of two 1 mm-thick windows. The SiC is subdivided into five filters to modulate the amount of transmitted power to allow as low as 0.28 kW (with the maximum of 8 mm of SiC filters in the beam) incident on the first white-beam optics (DLM). All subsequent calculations are based on the optimized case when 2 mm of diamond and 4.8 mm of SiC are applied [see Fig. 2(b) for the power spectrum].

Figure 2 (a) Power density of the damping wiggler source (DW100) at 28 m from the source, calculated using SRW (Chubar, 2001). The gray color scale represents the power density, ranging from zero (white) to the on-axis power density of 72 W mm^−2 (black). The white box denotes the fixed mask aperture of 1.1 mrad × 0.1 mrad. (b) Power spectrum within the fixed aperture of the source (solid line), the beam after 2 mm of diamond (dashed line), and the beam after 2 mm of diamond and 4.8 mm of SiC (dotted line) or 8 mm of silicon carbide (dash-dotted line).

Fig. 3 shows the angular profiles of the DW100 emission at different energies as calculated by SRW and SHADOW. The 0.1 mrad aperture in the vertical plane matches the angular acceptance of the vertical focusing mirror. As a result, the profiles of the X-ray beam are shown in Fig. 3. Since the profiles are neither uniform nor Gaussian shapes, one has to model the emission of the full 7 m-long damping wiggler together with the fixed mask aperture. In SHADOW, the EPATH routine computes the electron trajectory in the wiggler, and then photon rays emitted at each point of the trajectory are integrated and normalized to obtain the total photon emission. The effective X-ray source size is obtained by back-tracing the output rays at the far field to the center of the wiggler. The X-ray source r.m.s. size is calculated to be 650 µm × 54 µm with a mask aperture size of 1.1 mrad × 0.1 mrad [cf. Fig. 4(a)]. The effective source size within the 0.1 mrad fixed aperture [gray region in Fig. 4(b)] linearly varies with the size of the vertical slit. The vertical slit size should match the angular acceptance of the vertical focusing mirror. For instance, at 50 keV the mirror grazing angle is 1.5 mrad to ensure total reflection. For a 1.3 m-long mirror the vertical acceptance is reduced to 0.05 mrad. As a result, the effective vertical source size is calculated to be 28 µm (r.m.s.).

Figure 3 Angular profiles of DW100 emission at 30 keV (solid line), 50 keV (dashed line) and 70 keV (dotted line) in the horizontal (a) and the vertical (b) directions, respectively. The vertical dash-dotted lines show the angular acceptance of the fixed mask, which defines a 1.1 mrad aperture horizontally and a 0.1 mrad aperture vertically.

Figure 4 (a) Effective X-ray source profile at 50 keV for a mask aperture size of 1.1 mrad × 0.1 mrad. The X-ray source r.m.s. size is 650 µm × 54 µm. (b) Effective vertical source r.m.s. size as a function of vertical aperture at 30 keV (triangles), 50 keV (circles) and 70 keV (squares).

3.2. The sagittally bent double-Laue crystal monochromator

3.2.1. Basic concept and parameters

The XPD beamline intends to use a sagittally bent double-Laue monochromator (cf. Fig. 5) for providing a focused and adjustable monochromatic beam with optimized flux at the sample. The DLM is most suited for high-energy powder diffraction beamlines (e.g. XPD) because it provides energy tunability, maximized total flux, medium energy resolution and adequate size of the beam. The focusing capability of the DLM is similar to that of the sagittal focusing by a Bragg crystal, but Laue crystals are less sensitive to the thermal load and vibration because of the smaller beam footprint and the larger angular acceptance (Suortti et al., 1990). The transmission geometry is also the best choice for high-energy wiggler beamlines in terms of reducing the crystal size and easing the alignment.

Figure 5 Schematic of the sagittally bent double-Laue crystal monochromator.

The sagittally bent double-Laue crystal monochromator (Zhong et al., 2001a,b) was pioneered at the National Synchrotron Light Source (NSLS). The focusing condition of the DLM is given by Zhong et al. (2001a),

where F₁ and F₂ are the source-to-DLM and DLM-to-sample distances (assuming the distance between the two crystals is small), respectively, θ_m is the Bragg angle of the monochromator, χ is the asymmetric angle, and R_s is the sagittal bending radius of both Laue crystals. The anticlastic (meridional) bending radius R_m is then given by R_m = R_s/Cν, where ν is the Poisson ratio of the crystal for a given orientation and C is a correction term accounting for the crystal shape and the bending mechanism (Zhong et al., 2002; Krisch et al., 1991).

Fig. 6(a) presents the sagittal radius R_s needed for the horizontal focusing as a function of χ obtained from (1). χ should not be too small (>20°), so as not to break the crystal. The Poisson ratio ν is highly dependent on the crystal orientation owing to the anisotropy of silicon (Wortman & Evans, 1965). Fig. 6(a) also shows ν as a function of χ in the plane. As a result, R_s/ν (knowing that R_m is proportional to R_s/ν) achieves the maximum at χ ≃ 40° [see the solid line in Fig. 6(b)]. Considering that the required R_m for the Rowland condition [see the dashed line in Fig. 6(b)] is given by

with the direction cosine of the incident beam γ₀ = cos(χ + θ_m) for the XPD geometry, one finds that the value χ where R_s/ν and R_m (Rowland condition) are the closest is about 35°. This optimized χ can be obtained by using the 111 reflection on the (100) crystal (χ = 35.26°). Once the asymmetric angle is chosen, the actual R_m is then a function of the parameter C, which can be experimentally determined or calculated using finite-element analysis (Shi et al., 2011). This drives the choice of the crystal dimensions (i.e. length, width and thickness) and of the bender design.

Figure 6 (a) DLM sagittal radius Rs plotted as a function of the asymmetric angle χ (solid line), the Poisson ratio ν as a function of χ in the plane (dashed line), (b) Rs/ν and the meridional radius Rm that satisfies the Rowland condition as a function of χ at 50 keV, with the source-to-DLM distance F1 = 32.3 m and the DLM-to-sample distance F2 = 24.1 m.

The rocking curve width, ω₀, of the sagittally bent crystal is given by (Zhong et al., 2002, 2003)

where T₀ is the thickness of the crystal and ω_a is the Darwin width for a given material and Bragg reflection. Δθ(T₀) is the total change of Bragg condition caused by lattice distortion, and is given by

where = S_ij/S₃₃, and S_ij are the elastic compliances of the crystal (the Poisson ratio is ν = in the above equation). Since the total distortion Δθ(T₀) is proportional to the crystal thickness T₀, one has to increase the crystal thickness in order to achieve high flux to the detriment of energy resolution. However, increasing the thickness also induces more stress in the crystal up to the breaking point.

3.2.2. Shadow ray-tracing with the multi-lamellar approach

The rocking curves of sagittally bent Laue crystals can be simulated within the dynamical theory (e.g. the multi-lamellar approximation, the Penning–Polder method and the Takagi–Taupin theory) once the anisotropy of the crystal is implemented (Shi, 2011). The total angle change Δθ(T₀) is a result of two terms: (i) the change of the lattice orientation through the crystal thickness, and (ii) the angle change owing to the lattice spacing variation. For a sagittally bent Laue crystal, the former effect is one magnitude larger than the latter one. To include these effects, the DLM was ray-traced using SHADOW by dividing the bent Laue crystal into n thin lamellae with a thickness of ΔT. The Bragg plane in each lamella is tilted by an angle relative to the Bragg planes in its neighbor lamellae, that is, the asymmetric angle varies from one lamella to the next [cf. Fig. 7(a)]. The reflectivity and transmission of each lamella can be calculated from the dynamical theory, and the overall reflectivity of a crystal consisting of n lamellae is then (Shi, 2011)

where r_i is the reflectivity of the ith lamella, t_j is the transmission of the jth lamella before the ith lamella, μ is the linear absorption coefficient, and L_h = is the X-ray path of the diffracted beam through each lamella. For the XPD geometry, the direction cosine of the diffracted beam is = . In this work, the reflectivity and transmission of each lamella is simulated by means of SHADOW. The thickness of the lamella is chosen so that the tilt angle between two sequential lamellae equals the Darwin width of the perfect crystal.

Figure 7 (a) Multi-lamellar model of the sagittally bent Laue crystal. (b) The rocking curve (solid line) of the 111 reflection on a 0.7 mm-thick (100) crystal bent to Rs = 1.25 m is constructed using equation (5) with 15 lamellae of perfect crystal (rocking curves of all lamellae are shown as dashed lines). The dashed line denotes the diffraction profile of the same crystal calculated from the Penning–Polder theory. The X-ray energy is 50 keV.

Fig. 7(b) illustrates an example calculation at 50 keV for XPD. The (100) crystal, 0.7 mm thick, was sagittally bent along the direction to a radius of 1.25 m. The corresponding R_m is 28 m with ν = 0.064 and C = 0.7. The total distortion Δθ calculated from (3) is 67.6 µrad. The crystal is then divided into 15 lamellae, each of which has a thickness of 47 µm. The rocking curves of all lamellae are shown as dotted curves in Fig. 7(b). The rocking curve from each lamella is offset by 1/15 of Δθ from its neighbour lamella. Applying (5), the total rocking curve is constructed as the solid curve in Fig. 7(b). The result agrees with the diffraction profile calculated from the Penning–Polder theory (Shi, 2011) [dashed line in Fig. 7(b)]. This multi-lamellar approach using SHADOW ray tracing is applicable when the multi-lamellar approximation remains valid (Shi, 2011).

For the conventional meridionally bent double-Laue crystal monochromator, the surfaces of the two crystals are parallel. The geometric effect in the diffraction plane caused by the first crystal is cancelled out by the second crystal. Therefore, the virtual image after both crystals coincides with the source when the distance between the two crystals is negligible. However, for the sagittally bent DLM, the virtual image-to-crystal distance will be multiplied by a factor of when the Rowland condition is achieved for both crystals (cf. Fig. 5). Similarly, the vertical height of the exit beam is altered by the same factor. The vertical divergence of the beam is also modified by the angular width of the crystal diffraction profile. As a result, the focusing condition and the beam size in the vertical direction cannot be calculated easily with an analytical approach but with the ray tracing (e.g. SHADOW). Fig. 8 shows the beam size dimension simulated at the sample position without and with the DLM at 50 keV for the XPD geometry. In the horizontal plane, the beam is focused from a near-uniform distribution with a width of 62 mm (cf. Fig. 3) to a Gaussian-shaped profile with a FWHM of 0.6 mm. On the other hand, a detectable increase of the vertical beam size is observed following the effects mentioned above.

Figure 8 Beam spot simulated by SHADOW at the sample position (56.4 m from the source) of the XPD beamline without (a) and with (b) the DLM (located 32.3 m from the source). The X-ray beam generated by the DW100 source at 50 keV is predefined by a fixed aperture of 1.1 mrad × 0.1 mrad. See the caption of Fig. 7 for the crystal parameters.

3.3. Mirror

The function of the vertical beam optic is to re-condition the vertical monochromatic beam coming out of the DLM and to adapt the two operation modes of XPD by focusing or collimating the beam, respectively. The compound refractive lenses (CRLs) and the mirror are both suitable for the energy tunability requirement of the XPD beamline. However, a significant drawback of the CRL is the aperture-limited gain in flux for long-focal-length (low demagnification) focusing. It is also difficult to match the limited hole depth of the CRL with the large horizontal size of the beam (about 24 mm wide at the optic location with the DLM focusing the beam horizontally at the sample position).

The mirror is therefore the best choice for high-energy beamlines using wiggler sources. The XPD platinum-coated mirror provides high reflectivity for high energies from 30 keV to 70 keV with the optimized grazing angle, δ [cf. Fig. 9(a)]. Ideally, the point-to-point focusing requires an elliptical bending of the mirror in the meridional direction while the collimating mode requires a parabolic bending. However, the coma and spherical aberrations of the XPD mirror are negligible owing to the large focal length. Careful ray-tracing verification was performed to prove the adequacy of utilizing a cylindrically bent mirror. The meridional bending radius R is calculated from

with the source-to-mirror distance p = 40.1 m and the mirror-to-sample distance q = 16.3 m for the focusing mode and q = ∞ for the collimating mode. (Note that the virtual source-to-mirror distance is modified as discussed in §3.2.2.) Fig. 9(b) presents the required bending radii as a function of the photon energies calculated from (6) with the optimized grazing angle shown in Fig. 9(a).

Figure 9 (a) Maximum grazing angles to ensure a reflectivity over 90% plotted as a function of photon energies for a Pt-coated mirror with 0.2 nm r.m.s. roughness, calculated by XOP. (b) The cylindrical bending radii of the focusing (solid line) and collimating (dashed line) mirror, calculated from equation (6), plotted as a function of photon energies.

The performance of the mirror is also determined by the roughness and the slope error. The surface roughness, characterized by the r.m.s. value of the surface height variation, affects the mirror reflectivity by means of diffuse scattering (diffraction). For the mirror at the XPD beamline, a roughness of 0.2 nm will be sufficient to maintain 90% reflectivity at the optimized angles [cf. Fig. 9(a)] for the whole energy range. Another effect of the roughness is to generate a diffuse scattering background, which has a slightly larger size than the spot size that is obtained without considering the mirror roughness. This scattering background is detrimental to imaging, but not as important for powder diffraction. In the paper, the modeling of the roughness in the ray tracing is not applied for obtaining the beam profile. The roughness is only used to calculate the total reflectivity of the mirror and the corresponding beam flux.

On the other hand, the slope error (σ_r.m.s.), defined as the r.m.s. value of the angular variation from the ideal shape, is more critical for determining the beam size in the high-flux mode and the energy bandwidth in the high-resolution mode. Fig. 10 shows these effects from the mirror with different slope errors simulated in SHADOW, where the surface profiles are generated by the WAVINESS subroutine for a mirror that is 1.3 m long and 5 cm wide. Considering that the effective vertical source size is 28 µm (r.m.s.) at 50 keV [cf. Fig. 4(b) and text in §3.1] and the demagnification factor of the mirror is about 1:2.5, the focal size should be around 11 µm (r.m.s.) for an ideal mirror. However, the ray tracing shows a focal size of 18 µm r.m.s. (s_v0 = 38 µm FWHM) with zero slope error applied on the mirror [cf. Fig. 10(a)]. This value can only be obtained by implementing the DLM simulation mentioned in §3.2.2. Once this blurring effect of the DLM is included in determining s_v0, the real FWHM vertical beam size, s_v, can be obtained from SHADOW ray tracing [markers in Fig. 10(a)] or calculated analytically by convoluting the contribution of the slope error [solid line in Fig. 10(a)], given by

Similarly, Fig. 10(b) illustrates the energy resolution plotted as a function of σ_r.m.s. when combining the collimating mirror with the high-resolution monochromator. The FWHMs of ΔE/E extracted from SHADOW ray tracing [markers in Fig. 10(b)] agree with those calculated by [solid line in Fig. 10(b)]

where ω is the intrinsic angular acceptance of the HRM.

Figure 10 Vertical beam size (a) in the high-flux mode and the energy resolution (b) in the high-resolution mode plotted as a function of the slope error of the VFM. The markers are ray-tracing results using SHADOW and the lines are calculated analytically. See text for details.

4. Instrumental resolution function

The accurate description of the instrumental resolution function (IRF) is extremely important for characterizing the XPD beamline. In the high-flux set-up (cf. Fig. 1) the instrumental angular profile in the vertical plane is a convolution of the incident beam profile, the rocking curve of the Laue crystal monochromator, the slit (located before the VFM) function, the residual divergence after the focusing mirror, and the angular acceptance of the detector, with α_m, Δ_m, τ_s, τ_f and τ_d as their Gaussian FWHMs, respectively. The FWHM of the IRF as a function of the diffraction angle θ is deduced by means of an analytical method (Sabine, 1987; Gozzo et al., 2006), given by

with

The parameter k accounts for the mismatching of the meridional bending radius R_m of the DLM crystals and the source-to-DLM distance F₁ (e.g. k = 0 for the Rowland geometry and k = 1 for flat Laue crystals). It is clear that the slit after the DLM serves as the regulation of the incident beam divergence through the term , which reduces to when the slit is widely open (τ_s = ∞). In most of the high-throughput experiments, area detectors are preferred without utilizing the analyzer crystals in order to achieve the fast detection.

In the high-resolution set-up (cf. Fig. 1), it is assumed that the rocking curve width (Δ_m) of the DLM and the mirror acceptance (τ_s) are much larger than that of the channel-cut HRM (Δ_c) and the analyzer crystal (Δ_a). It is also assumed that Δ_m and τ_s are much larger than the residual divergence of the collimated beam (τ_f) after the mirror. The FWHM of the high-resolution IRF is then

with

where θ_c and θ_a are the Bragg angles of the HRM crystals and the analyzer crystal, respectively.

Fig. 11 shows the instrumental resolution as a function of Q in the reciprocal space [Q = (4πsinθ/λ), λ is the X-ray wavelength] at different energies in the high-flux mode and the high-resolution mode calculated from (9) and (10), respectively. The values of the profile FWHMs of the optics for the XPD beamline are listed in Table 2. The incident beam divergence α_m is determined by the fixed mask aperture. Δ_m is calculated analytically from (3) and (4). τ_s is taken as the vertical angular acceptance of the mirror after the DLM [τ_s = Lsinδ/p, with the mirror length L = 1.3 m and the grazing angle δ from Fig. 9(a)] and τ_f is given by Lsinδ/q. In the high-resolution mode, τ_f is assumed to be 15 µrad (Gozzo et al., 2006). Δ_a and Δ_c are calculated using the dynamical theory for the silicon 111 reflection in symmetric Bragg geometry. The energy resolution at 30 keV is comparable with existing high-resolution powder diffraction beamlines (Gozzo et al., 2006; Masson et al., 2001; Wang et al., 2008), which are all optimized for energies below 40 keV.

Table 2
Values of the profile FWHM of the optics and parameters needed to generate Fig. 11 using equations (9) and (10) in §4

Energy, E (keV)	70	60	50	40	30
Bragg angles, θm, θc, θa (°)	1.62	1.89	2.27	2.83	3.78
DLM sagittal radius, Rs (m)	0.886	1.04	1.25	1.57	2.09
DLM meridional radius, Rm (m)	25.0	27.1	30.1	34.6	42.1
VFM grazing angle, δ (mrad)	1.14	1.32	1.52	1.75	2.00
αm (µrad)†	100	100	100	100	100
Δm (µrad)†	99.6	82.7	66.4	50.9	36.5
τs (µrad)†	37.0	42.8	49.3	56.7	64.8
τf (µrad), focusing†	90.9	105	121	140	160
τd (µrad)†	50	50	50	50	50
τf (µrad), collimating†	15.0	15.0	15.0	15.0	15.0
Δa, Δc (µrad)†	4.00	4.66	5.60	7.01	9.37

†α_m, Δ_m, τ_s, τ_f, τ_d, Δ_a and Δ_c are the FWHMs of the incident beam profile, the rocking curve of the Laue crystal monochromator, the slit (located before the VFM) function, the residual divergence after the VFM, the angular acceptance of the detector, the rocking curve of the HRM and the rocking curve of the analyzer crystal, respectively.

Figure 11 Instrumental resolution plotted as a function of Q at different energies (30 keV, 40 keV, 50 keV, 60 keV and 70 keV) in the high-flux mode (ΔHF) and the high-resolution mode (ΔHR), calculated from equations (9) and (10), respectively. The input parameters for the calculations are listed in Table 2.

5. Discussion

Table 3 lists the SHADOW ray-tracing results for the XPD beamline in both the high-flux mode and the high-resolution mode. The calculation parameters are from Table 1, Table 2 and Fig. 9. The r.m.s. slope error of the Pt-coated mirror is assumed to be 0.5 µrad and the roughness is 0.2 nm. The total fluxes at the sample were calculated from

where is the total incident flux [photons s⁻¹ (0.1% bandwidth)⁻¹] after the fixed aperture mask and the filter system (2 mm of diamond and 4.8 mm of SiC), N_i is the number of initial rays, N_f is the number of final rays at the sample location recorded by SHADOW, ΔE_i is the input photon energy bandwidth over which the ray tracing is performed, and E is the central photon energy. The output energy bandwidth, ΔE, is obtained by fitting the intensity-weighted energy histogram from the SHADOW output.

Table 3
Beamline performance simulated by SHADOW ray tracing

E (keV)	70	60	50	40	30
Incident flux, [10^12 photons s^−1 (0.1% bandwidth)^−1]†	16	33	62	93	78
High-flux mode
ΔE/E (10^−3)	2.3	1.6	0.99	0.59	0.40
Beam size, sh × sv (mm)	0.78 × 0.064	0.64 × 0.055	0.61 × 0.057	0.81 × 0.060	0.71 × 0.080
Flux, (10^12 photons s^−1)	4.4	8.9	17	19	12
Intensity, I (10^14 photons s^−1 mm^−2)	0.89	2.5	4.9	3.9	2.2
Intensity gain‡	25	35	50	30	35
High-resolution mode
ΔE/E (10^−3)	0.14	0.13	0.12	0.12	0.10
Beam size, sh × sv (m)	0.64 × 1.5	0.72 × 1.7	0.80 × 2.0	0.65 × 2.3	0.65 × 2.6
Flux, (10^12 photons s^−1)	0.20	0.69	1.8	3.7	2.6
Intensity, I (10^12 photons s^−1 mm^−2)	0.21	0.56	1.2	2.5	1.5

†Flux after the fixed aperture mask and the filter system (2 mm of diamond and 4.8 mm of SiC).
‡Intensity per bandwidth [photons s⁻¹ mm⁻² (0.1% bandwidth)⁻¹] ratio between the high-flux mode and the high-resolution mode.

The simulation results indicate an optimized photon flux at the sample around 50 keV. The beam size and flux numbers, however, cannot be described in a simple pattern owing to the complexity introduced by the DLM. Ideally, the anticlastic bending radii (R_m) of both crystals are the same for the same crystal shape and size. However, they also depend on the bending mechanism, the bender manufacturing error and, most importantly, the heat load. Since the first crystal operates under the white beam and extreme cooling (liquid nitrogen), the resulting R_m1 will vary from that of the second crystal (R_m2).

Fig. 12 shows the relative flux per energy bandwidth at 50 keV with different R_m1 and R_m2 combinations keeping R_s1 and R_s2 constant (1.25 m), from which the optimized bending conditions for both crystals can be obtained. Note that each point on the surface is extracted from the ray tracing using the multi-lamellar model described in §3.2.2. As a result, it is important to have a simple adjustment of the meridional bending radius while the sagittal radius is pre-determined from the focusing condition. Previous studies (Shi et al., 2011) show that the R_s/R_m ratio varies with the crystal shape (i.e. the aspect ratio). The XPD beamline intends to use a four-bar bender design where the spacing between the inner bars is adjustable. The actual bending of the crystal changes under the effect of the thermal load. Tuning the inner bar spacing can minimize this change and therefore ensures the optimization of the output flux (e.g. at the peak position as shown in Fig. 12).

Figure 12 Normalized flux per energy bandwidth at 50 keV plotted as a function of the radius ratio Rs1/Rm1 and Rs2/Rm2 of the two DLM crystals, where the Rs values are fixed (1.25 m) to satisfy the focusing condition and only the Rm values vary. The fluxes and energy bandwidths are extracted from the SHADOW ray tracing using the multi-lamellar model of the DLM crystals.

6. Conclusion

The performance of the X-ray Powder Diffraction beamline at NSLS-II is simulated using the SHADOW ray tracing. The major design challenge relies on the simulation of the DLM, of which the unique features have a huge impact on the design of the downstream optics (i.e. the VFM) and dominate the final performance of the beamline. In this paper we proposed a multi-lamellar model to reveal the optical characteristics of the DLM in the ray-tracing process. The simulations show that the expected beamline performance will serve the proposed scientific objectives. Based on the present analysis, the XPD beamline is designed to deliver high fluxes at the sample in variable millimeter-size focuses over the tunable energy range of 30 to 70 keV. The energy resolution will be 10⁻³ in the high-flux mode and up to 10⁻⁴ in the high-resolution mode. Future upgrades will include the secondary focus of the beam down to a size of 10 µm and the implementation of a dedicated PDF endstation.