research papers
Temperaturegradient analyzers for nonresonant inelastic Xray scattering
^{a}Materials Dynamics Laboratory, RIKEN SPring8 Center, 111 Kouto, Sayocho, Sayogun, Hyogo 6795198, Japan, and ^{b}Precision Spectroscopy Division, Center for Synchrotron Radiation Research, Japan Synchrotron Radiation Research Institute (JASRI), 111 Kouto, Sayocho, Sayogun, Hyogo 6795148, Japan
^{*}Correspondence email: disikawa@spring8.or.jp
The detailed fabrication and performance of the temperaturegradient analyzers that were simulated by Ishikawa & Baron [(2010). J. Synchrotron Rad. 17, 12–24] are described and extended to include both quadratic and 2D gradients. The application of a temperature gradient compensates for geometric contributions to the energy resolution while allowing collection of a large solid angle, ∼50 mrad × 50 mrad, of scattered radiation. In particular, when operating relatively close to backscattering, π/2 − θ_{B} = 1.58 mrad, the application of a gradient of 1.32 K per 80 mm improves the measured total resolution from 60 to 25 meV at the full width at halfmaximum, while when operating further from backscattering, π/2 − θ_{B} = 6.56 mrad, improvement from 330 to 32 meV is observed using a combination of a gradient of 6.2 K per 80 mm and dispersion compensation with a positionsensitive detector. In both cases, the operating energy was 15.8 keV and the incident bandwidth was 22 meV. Notably, the use of a temperature gradient allows a relatively large clearance at the sample, permitting installation of more complicated sample environments.
Keywords: inelastic Xray scattering; analyzers; Xray optics; electronic excitation.
1. Introduction
Nonresonant inelastic Xray scattering (NRIXS) has become a powerful tool for studying momentumresolved atomic and electronic dynamics (Schülke, 2007; Baron, 2016, 2020). The NRIXS is simply proportional to electron so that one can investigate charge dynamics in materials straightforwardly, without the complications that can occur from intermediate states in High multipole transitions are also observable at highmomentum transfers (Larson et al., 2007; Haverkort et al., 2007), which are not accessible via In addition, the penetrating power of hard Xrays enables bulksensitive measurements and penetration into complex sample environments, e.g. to investigate samples under extreme conditions.
Spectrometers may be separated by the energy scale of the excitations that one wants to investigate, as there is usually a tradeoff between resolution and available e.g. in hydrogencontaining materials).
High resolution (∼meV) enables phonons to be probed while medium resolution (∼50 to ∼200 meV) allows access to valenceshell excitations and the multipole order of the electronic transitions. At lower resolution (∼eV) the high allows access to core levels, charge transfer and plasmon excitations, as well as band structure. Resolution in the 10 to 30 meV range is interesting to investigate the detailed structure of electronic excitations, including crystalfield transitions and more complex excitations such as orbitons. Is it also potentially useful for investigating weak/highenergy vibrational excitations (Analyzers are the most critical and difficult component of θ_{B} ≃ π/2) (Masciovecchio et al., 1996; Baron et al., 2000; Sinn, 2001; Sinn et al., 2002; Verbeni et al., 2005, 2009; Said et al., 2011). These usually operate in a Rowland circle geometry (Burkel, 1991). One problem of using a Rowland geometry close to backscattering is that the sample space is limited so the detector must be moved away from the sample towards the analyzer crystals or Bragg angles must be chosen that are far from backreflection. This results in a geometric contribution which degrades the energy resolution and has been called a `demagnification contribution' (Burkel, 1991). The impact of this contribution tends to be worse when a twodimension analyzer array is used, or when shorter sample–analyzer distances are used.
spectrometers and great effort has been invested in fabricating spherical crystal analyzers. For high (meV scale) resolution, diced spherical crystal analyzers have been utilized with higherorder Bragg backreflections (Temperature gradient (TG) analyzers have been considered in simulations (Ishikawa & Baron, 2010) as a way of reducing the demagnification contribution in a compact spectrometer. The TG analyzers reduce geometric aberration by gradually changing the d spacing over the analyzer. For highresolution spectrometers, a TG has been shown to provide modest fractional improvements in resolution, e.g. from ∼0.9 to 0.75 meV (Ishikawa et al., 2015). However, more dramatic improvements (factors of two or more) are expected in a mediumresolution setup. This performance was demonstrated by Ishikawa et al. (2017). Here, we investigate these analyzers in more detail. The present article discusses how to reduce the geometric aberration arising from an offRowland geometry while retaining large clearance (∼200 mm) between the detector and the sample. We measured the performance of analyzers installed in the 2 m mediumresolution spectrometer at BL43LXU in SPring8. The results are supported by detailed raytracing calculations.
This article is organized as follows. Section 2 reviews the basic concept of the TG analyzer. Section 3 describes the analyzer fabrication methods. Section 4 presents raytracing calculations and discusses TG contributions to the geometric aberrations. Section 5 explains details of TG control. The experimental results are discussed in Section 6 and our conclusions are given in Section 7.
2. Analyzer
We used diced analyzers operating within 7 mrad of backscattering. As the NRIXS
tends to be small compared with either phonon or resonant inelastic Xray scattering (RIXS) cross sections, we accepted a large solid angle, ∼50 mrad × 50 mrad, to maximize count rates.2.1. Analytic formulae for temperaturecompensation correction
We briefly recall some basic concepts of the TG analyzer from the work of Ishikawa & Baron (2010). The total energy resolution of an spectrometer ΔE_{tot} may be estimated as
where ΔE_{inc}, ΔE_{int} and ΔE_{geom} are the incidentenergy resolution (bandwidth from the monochromator), the intrinsic analyzer reflection width and a geometric contribution, respectively. Here, we focus on ΔE_{geom}.
One contribution to the fractional energy resolution ɛ ≡ (ΔE/E)_{geom} of pixelated spherical analyzers operating on a Rowland geometry with a singleelement detector (SED) is related to the analyzer crystallite size [see Fig. 1(a)], c, by
where E is the Xray energy, R is the radius of curvature and δ_{0} (≡ π/2 − θ_{B}) is the deviation angle from exact backscattering at the analyzer center. This can be improved by using a pixelated detector (Huotari et al., 2006) with pixel size p, with
Assuming we desire ɛ ≃ 1 × 10^{−7} using a compact spectrometer (R < ∼2 m) and p = 0.1 mm, then δ_{0} must be <1 mrad [d < 4 mm, Fig. 1(a)]. This severely constrains sample environments. As a result, the pure Rowland geometry does not practically allow operation close to backscattering geometry in many cases. Therefore, the detector must be moved away from the sample, usually towards the analyzer crystal, so it is placed in front of the sample environments when viewed from the analyzer. This violates the Rowland circle condition and the resolution becomes worse owing to a demagnification contribution (Burkel, 1991; Ishikawa & Baron, 2010). This [see Fig. 1(a)] is estimated as
where Δδ is the magnitude of distribution over the analyzer, Ω is the solid angle of the analyzer in the analyzer and M (≡ L_{2}/L_{1}) is the magnification of the focusing geometry. Here, L_{1} and L_{2} are the sample–analyzer and analyzer–detector distance, respectively. L_{1} and L_{2} satisfy 1/L_{1} + 1/L_{2} = 1/(2R cos δ_{0}) ≃ 1/2R.
A TG may be used to reduce the geometric contribution of equation (4). The TG creates a dspacing variation of crystallites over the analyzer that compensates for the angle change of the beam reflected into the detector. The analytic formula for the required temperature correction, ΔT (= T − T_{0}), as a function of analyzer vertical direction (y_{a}) is given as (Ishikawa & Baron, 2010)
where T_{0} is the temperature at the center of the analyzer and α(T_{0}) ≅ 2.63 × 10^{−6} K^{−1} is the thermalexpansion coefficient of Si at T_{0} = 300 K calculated from references Okada & Tokumaru (1984), Watanabe et al. (2004) and Mohr et al. (2016). Note that the TG has a term linear in the vertical position, y_{a}, so must be inverted depending on the reflecting direction (upward or downward) of the analyzer as shown in Fig. 10.
2.2. Extension to 2D case
In addition to the quadratic term given in equation (5), there is another quadratic term related to the horizontal position on the analyzer, x_{a}, given by
Hence, to second order, the ideal temperature correction as a function of analyzer position (x_{a}, y_{a}) can be written as
where A = −δ_{0}η/[2α(T_{0})], B = C = η^{2}/[8α(T_{0})], η = (1 − M)/L_{1}M and T_{0} is the temperature at the analyzer center (x_{a} = y_{a} = 0). The geometric contribution to the energy resolution is given by ΔE/E = Δd_{h}/d_{h} = α(T)ΔT, where d_{h} is the d spacing of the diffraction plane. So one has
where A′ = −δ_{0}η/2 and B′ = C′ = η^{2}/8. An example of the impact of these terms is shown in Fig. 1(b) for offRowland geometry with L_{1} = 2005 mm, M = 0.9 and d = 6 mm. To correct the aberration, the temperature must be shifted as shown in Fig. 1(c). We consider four cases in detail: (a) uniform temperature (A, B, C ≠ 0), (b) 1D linear TG (A = 0), (c) 1D quadratic TG (A = B = 0) and (d) 2D quadratic TG (A = B = C = 0).
Even with the proper TG applied, the resolution will still be limited by the pixel size if one uses an SED. An equation analogous to equation (1) above gives the best fractional energy resolution with an SED as
By contrast, the fractional resolution with a positionsensitive detector (PSD) using dispersion compensation (DC) (Huotari et al., 2006) is estimated to be (Ishikawa & Baron, 2010)
where 2c′ [≡ c(1 + M)] is the focalspot size in the offRowland geometry.
Practically, while a fully 2D quadratic TG is the best temperature correction, it is difficult to achieve. Therefore, here we focus primarily on 1D corrections, both linear and quadratic.
2.3. Spectrometer parameters
The mediumresolution ΔE_{tot} ≃ 25 meV with a short (∼2 m) arm radius. The incident bandwidth from the mediumresolution monochromator is ΔE_{inc} = 22 meV. To improve the tails of the resolution, the Si(888) backreflection, ΔE_{int} = 4.4 meV, was used for the analyzers. To obtain large space at the sample area, the analyzers were operated off the Rowland geometry with magnification M ≃ 0.9. The offRowland geometry is also advantageous for background reduction as the beam size remains small so the detector area can also be kept small, reducing the rate of background events from natural sources (cosmic rays). The radius of curvature of the spherical analyzer was selected to be R = 1.9 m.
spectrometer at BL43LXU was designed to give a total resolution ofTo improve measurement efficiency, a 2D array of analyzers was used. This in turn forces larger deviations from exact backscattering and necessitates a large temperature correction. When the backscattering angle is δ_{0} = 1–10 mrad, the parameter A in equation (7) is estimated to be ∼−0.01 to −0.1 K mm^{−1}. Therefore, ΔT ≃ 1–10 K is needed over an ∼100 mm analyzer size. The required heat flow, Q, can be estimated by Q = λSΔT/h. Here, λ is the S refers to the ΔT is the temperature offset and h refers to the distance to apply the TG. Then, a substrate material must be selected that has appropriate i.e. the conductivity of Si is too high and would require too high a power to generate the needed gradient. We selected Invar alloy, λ = 13 W m^{−1} K^{−1}, for the substrate. When applying ΔT ≃ 1–10 K over the analyzer using the Invar substrate (cross section S = 100 mm × 15 mm, height h = 95 mm), a 50 Ω resistance heater and a lowvoltage power supply (5–23 V), one can control 0.5–10 W for creating the TG. If we had selected Si (148 W m^{−1} K^{−1}) as the substrate material, a factor of greater than ten larger heating and cooling capacity would be needed for creating the steadystate TG. On the other hand, choosing a glass (∼1 W m^{−1} K^{−1}) would require a factor of ten smaller power, which might have a slow response and be difficult to stabilize.
3. Analyzer fabrication
Analyzer fabrication requires solving a number of technical problems while staying within the constraints needed for operation. In the present case, our constraints were (i) using a pixelated spherical analyzer on (ii) an Invar substrate with (iii) a small (∼2 m) radius of curvature. In order to achieve good resolution, the Si also must be etched. To meet these conditions, we adopted a multistep process with the following main steps (the details are given below). First, a flat Si wafer was attached to a glass wafer by anodic bonding. Then the Si wafer was diced and etched to eliminate residual strain. Finally, the bonded wafers were glued to a spherical substrate. It is important that these successive processing steps do not adversely affect the analyzer focusing properties, thus each step must not introduce large slope errors, <∼10 µrad (r.m.s.) is highly desirable. The details of the fabrication process are as follows (see Fig. 2 for a schematic of each step and Fig. 3 for photographs at different stages in the fabrication process).
(1) Material preparation. An Si wafer (95 mm × 100 mm × t × 3.0 mm) was cut from highpurity singlecrystal Si ingot (float zone, resistivity ≃ 4 kΩ cm). Then the surface of the wafer was polished strain free within a flatness of <2–3 fringes over the plane. In parallel, a TEMPAX Float (TPX) (Ø150 mm × t × 0.7 mm) wafer was polished with a flatness of <2–3 fringes (<1–1.5 λ). These wafers were ultrasonically cleaned with acetone and ethanol followed by a pure H_{2}O (<1 µS cm^{−1}) rinse. The wafers were dried using blown N_{2} gas. UV/O_{3} cleaning was then applied to eliminate inorganic residues followed by a rinse using pure H_{2}O and drying again using N_{2} gas.
(2) Anodic bonding [see also Fig. 4(a)]. The assembly was performed using a desktop open clean bench (KOKEN LTD, KOACH) under ISO class 1 to eliminate dust. The anode electrode was a t = 20 µm SUS304 foil and the cathode electrode was a t = 0.2 mm graphite sheet. The t = 0.7 mm TPX wafer was attached to another TPX (t = 1 mm) cover to protect the 0.7 mm wafer from surface damage by Na and K aggregations coming from inside the Tempax wafer. A small amount of silicone was used as releasing agent for the TPX wafers. Anodic bonding of Si/TPX wafer was performed at T = 330°C, I_{max} = 3.0 mA and V = 0–2.5 kV.
(3) Dicing. The Si/TPX wafer was diced on a 1.0 mm pitch with an 80 µm groove width. The groove depth was carefully controlled to be through the Si but only 0.03 mm into the TPX wafer.
(4) Etching. The diced Si was etched to remove the residual strain. A _{3} (61%) : CH_{3}COOH (99.7%) = 3 : 5 : 3 was used as the etchant. The etching temperature and time were ∼20°C and 90 s, respectively. The back of the TPX wafer was protected by attaching another piece of glass (see Fig. 2) that was then sealed on the edge by polytetrafluoroethylene (PTFE) tape during the etching. This was sufficient to prevent damage to the bonded glass wafer. The final crystallite size was typically 0.87 mm × 0.87 mm × ∼3 mm after etching. The cover glass was removed and then the Si/TPX wafer was cleaned as in (2).
of HF (46%) : HNO(5) Invar substrate. The specification of the substrate is given in Table 1. The radius of curvature and slope error were R = 1900 mm and <10 µrad (r.m.s.), respectively. The spherical surface was cross grooved on a 15–20 mm pitch, 0.175 mm width and >0.6 mm depth [Figs. 3(a) and 3(b)] to provide an escape for the residual glue between the substrate and the Si/TPX wafer. Making a uniform epoxy glue bond was difficult and required several iterations to determine a successful protocol. For the first analyzers, no grooves were put in the substrate and the surface reflection profile had many voids from nonuniform gluing. Then we introduced grooves onto the substrate to eliminate residual glue. However, owing to the difficulty in machining Invar, it was difficult to make the grooves without disturbing the surface. We finally used spherical polishing after grooving to obtain a uniform response over the analyzer surface. Epoxy glue (EPOTEK 3012) was used for the second (spherical) bonding. A uniform amount of glue was potted on each grid on the substrate. This process was also performed using the clean bench mentioned in (2).

(6) Gluing. The setup for the gluing is shown in Fig. 4(b). A Kapton sheet (t = 60 µm) and a Viton sheet (t = 0.5 mm) were used for the interface materials. A standard convex jig (R = 1900 mm) and dead weight were inserted before applying external forces. A bench frame hydraulic press (ENERPAC CPF5P142) was used to apply forces to the unit with a maximum of 0.7–1.0 kN. The force was applied for 5–7 days at room temperature until the glue cured. One should emphasize that this procedure required some fine tuning as, when we started, we used very low viscosity epoxy glue (EPOTEK 3012FL) with a large pressure (>1.5 kN) and essentially forced most of the glue into the grooves so the Si/TPX wafers sometimes `popped up' from the spherical substrate after about 1 year.
(7) Trimming. The residual TPX was cut away from the rectangular substrate. The TPX was cut out 1 mm from the edge of the substrate, thus the final active area is 98 mm × 93 mm. Fig. 3(d) shows a completed analyzer.
4. Ray tracing
We carried out ray tracing in order to estimate analyzer performance and to confirm our understanding of our results. We used the three geometric conditions (d values) given in Table 2. The rays were assumed to uniformly irradiate the analyzer surface. We took the analyzer slope error to be 20 µrad × 20 µrad (r.m.s.) with a source size at the sample position of 5 µm × 5 µm (r.m.s.). We considered different gradient models including uniform temperature (no gradient), 1D linear, 1D quadratic and 2D quadratic (where the 1D gradients are always in the analyzer scattering plane), as given in Table 3. The temperature profile of the analyzer surface was determined by calculating several pixel temperatures and fitting to linear or quadratic equations. The parameters directly calculated from equation (7) are also listed in brackets and are generally close to the values estimated from the ray tracing. Each crystallite was assumed to have uniform temperature. To compare the experimental results, as presented in Section 6, a 2 mm × 2 mm SED was assumed in the case d = 6 mm, while a pixel size of 0.172 mm was used in the cases d = 10 and 25 mm to match the experimental conditions (PILATUS detector). The analyzer shape and size were taken as given in Table 1. The calculations were carried out assuming the incident bandwidth was a delta function which is then scanned (i.e. there is no convolution with the incident monochromator bandwidth).

‡The values in the parentheses are given by equation (10). 
4.1. d = 6 mm (singleelement detector)
Fig. 5 shows how the TG improves the geometric term of energy resolution in the case d = 6 mm. The lefthand panels show the magnitude of aberration for rays onto the analyzer surface as a function of vertical position (y_{a}). To clarify the origin of this, four horizontal positions (x_{a} = 0, 20, 30 and 45 mm) are plotted. The righthand panels show geometric energy resolution from all rays. Using uniform temperature, the energy resolution was obtained at ΔE_{geom} = 43.8 meV (FWHM, full width at halfmaximum), as shown in Fig. 5(a). It is clear that the asymmetric line shape largely arises from the vertical analyzer extent and the horizontal geometric aberrations have only a small impact. Applying a linear TG correction, the energy resolution improved to ΔE_{geom} = 18.9 meV (FWHM) [Fig. 5(b)]. In addition, when applying a 1D quadratic TG, the energy resolution was slightly improved from the linear state as ΔE_{geom} = 15.5 meV (FWHM) and line shapes became more symmetric [Fig. 5(c)]. A 2D quadratic TG resulted in ΔE_{geom} = 11.1 meV (FWHM). The 2D quadratic TG yields the best resolution in the four cases, as shown in Fig. 5(d), and the geometric energy resolution for an SED is determined by the crystallite size.
4.2. d = 10 mm (dispersion compensation)
The calculation for d = 10 mm was performed in the same way as for d = 6 mm. The lefthand panels in Fig. 6 show the magnitude of aberration for selected x_{a} positions. The righthand panels in Fig. 6 show the geometric resolution with an effective SED, i.e. integral over the spot defined by 11 × 11 pixels (∼1.9 mm × 1.9 mm). The energy resolution improved from ΔE_{geom} = 101 meV (FWHM) at uniform temperature to 17 meV (FWHM) at the 2D quadratic TG. The `effective SED' resolution can be slightly improved by DC with a PSD. The lefthand panels in Fig. 7 show the geometric aberrations as a function of detector vertical position (y_{d}). Here, four selected horizontal positions (x_{a} = 0, 20, 30 and 45 mm) were plotted. The energy–position correlation at detector vertical position (y_{d}) is not completely linear in the offRowland geometry. However, the relation can be approximated as linear in the 2D TG. The righthand panels in Fig. 7 represent geometric resolution in each detector pixel. The thick lines show DCcorrected results in each TG condition. The resolutions are summarized in Table 3. The DC correction does not work without a TG [Fig. 7(a)], but, by using both, the geometric resolution was much improved with ΔE_{geom} = 22, 15 and 9 meV (FWHM) for 1D linear, 1D quadratic and 2D quadratic TGs, respectively [Figs. 7(b)–7(d)].
4.3. d = 25 mm (dispersion compensation)
TGdependent geometrical aberrations in the case d = 25 mm are shown in Fig. 8. Compared with d = 10 mm, the geometric energy resolutions have broader features in all the TG conditions. The energy resolution with an effective SED improved from ΔE_{geom} = 288 meV (FWHM) at uniform temperature to 45 meV (FWHM) at the 1D linear TG. However, no further improvements were obtained in the 1D and 2D quadratic TGs. The resolution with an SED can be improved by DC with a PSD. The energy–position correlation is not completely linear in the offRowland geometry. However, the relation can be approximated as linear in the 2D quadratic TG similar to d = 10 mm. The resolutions are summarized in Table 3. The DC correction does not work without a TG [Fig. 9(a)], but, by using both, the geometric contribution to the resolution was much improved with ΔE_{geom} = 22, 16 and 9 meV (FWHM) for 1D linear, 1D quadratic and 2D quadratic TGs, respectively [Figs. 9(b)–9(d)]. The best estimated resolution 2D quadratic TGs in d = 10 and 25 mm are limited by the 0.172 mm pixel size of the detector.
5. Establishing and controlling the temperature gradient
We applied a 1D quadratic TG for the et al. (2015) with some improvements. The analyzer is placed between two separated brackets. The top and bottom of the substrate are polished flat to better than 0.02 mm with the surfaces parallel to 0.2 mrad. To increase thermal contact conductance, GaInSn eutectic was used. Two heaters were used: the main heater for the base temperature control and the other as an offset heater for controlling the TG. The arrows in Fig. 10 indicate the thermal flow of the system. Eight temperature sensors (twowire readout, ϕ ≃ 2 mmdiameter glassencapsulated thermistors, OMEGA 55016) were attached on the side of the analyzer substrate. Feedback for the temperature control used a multichannel switching digital multimeter (Keithley 3706 and 3724) and a DC power supply (Wiener MPOD). Five temperature sensors (T_{1} → T_{5}, hot → cold, with a 20 mm pitch) are mounted on one side of the substrate and three temperature sensors (T_{6} → T_{8}, hot → cold, 40 mm pitch) were mounted on the other side [Figs. 11(a) and 11(b)]. Here, T_{3} and T_{7} were placed in the middle of the analyzer in the main TG direction. The feedback parameters for the base temperature and the offset temperature were T_{0} = (T_{3} + T_{7})/2 and ΔT_{g} = 0.5[T_{1} + T_{6} − (T_{5} + T_{8})] K per 80 mm, respectively. The magnitude of the gradient used here is much larger than the ∼0.01°C per 100 mm used with highresolution analyzers (Ishikawa et al., 2015). In addition, the nonlinear TG must be taken into account. The nonlinear TG is not simple but it is possible by changing the thermal flow of the system. One way is to add a triangular prism object onto the backside of the analyzer substrate, as seen in Fig. 12(a) where a finiteelement analysis using ANSYS is shown. Practically, owing to the complexity of the thermal contact conductance depending on surface roughness, contact pressure and interface material, the material of the nonlinear TG object was determined by experiment. The geometry for the 2D TG is presented in Fig. 12(b) where the additional heater and jig may be seen.
analyzers. The basic concept is the thermal circuit of IshikawaIn the case d = 6 mm, SK3 (a carbonsteel defined Japan Industrial Standard), with a of 35 W m^{−1} K^{−1}, was used to control thermal flow for creating a 1D quadratic temperature curve. Note that in the y_{a} direction there is more than a factor of two difference in T: ΔT(y_{a} = −40 mm) = 1.11°C, while ΔT(y_{a} = +40 mm) = −0.46°C. The ideal analyzer temperature is given by equation (7) as A = −1.648 × 10^{−2} K mm^{−1}, B = 1.433 × 10^{−4} K mm^{−2} and C = 0 K mm^{−2}. The elastic temperature and magnitude of TG were set at T_{0} = 27°C and ΔT_{g} = 1.32 K per 80 mm, respectively. The feedback parameters for the base heater and the offset heater were T_{0} and ΔT_{g}, respectively. The typical applied power for the offset heater was ∼0.8 W, while the chiller temperature was set at 25°C. Fig. 11(c) (left) shows the temperature of each sensor as a function of analyzer position. The magnitude of the desired TG is 1.32 K per 80 mm. The measured temperatures using the 1D quadratic TG agree well with the ideal curve. In the cases d = 10 and 25 mm, the magnitudes of ithe ideal ΔT_{g} are 2.47 K per 80 mm and 6.2 K per 80 mm, respectively. The materials of the attachment jig for creating the 1D quadratic TG are given in Table 4. Despite a large temperature offset, substrate temperature can be controlled well for more than a month, as shown in Fig. 11(c) (right).

6. Results and discussion
Tests were performed at the mediumresolution spectrometer of BL43LXU (Baron, 2010; Ishikawa et al., 2017) at the RIKEN SPring8 Center in Japan. The bandwidth of the Xrays from the undulator was reduced first by a liquidnitrogencooled highheatload mirror, then a highheatload Si(111) monochromator followed by a nestedchannelcut mediumresolution monochromator consisting of Si(440) and Si(660) crystals. The focalspot size at the sample was 25 µm (V) × 30 µm (H) (FWHM) after focusing by an elliptically bent cylindrical mirror. The total energy resolution of the spectrometer was measured using a 2 mmthick polymethylmethacrylate sample with the analyzer placed at the structurefactor maximum. The full analyzer surface was illuminated, corresponding to momentum resolution of ΔQ ≃ 3.6 nm^{−1} (full width) at 15.816 keV. The incident energy was scanned by changing both channelcut crystal angles, while the analyzer temperatures and angle were kept constant. The analyzers were mounted inside a vacuum chamber on the 2θ arm. The energy scale was calibrated using a diamond phonon at 164.7 meV as described by Fukui et al. (2008). Three different analyzer geometries in the cases d = 6, 10 and 25 mm were tested.
The TG is effective when used with DC and is particularly advantageous for a multianalyzer array that requires large deviations from the exact Rowland circle condition. In the present spectrometer, Xrays from second and third rows were detected by a PSD using DECTRIS PILATUS 100KSP8 (0.172 mm pixel^{−1}), as shown in Fig. 10. The detector was specially fabricated to allow operation in vacuum with nearly no border on one side allowing a relatively tight clearance between the beam and the detector. The analyzer focus was near the edge of the active element in the vertical direction (y_{a}) to keep the as close as possible to backscattering (smaller δ_{0}); while the horizontal axis (x_{a}) was the same as the sample–analyzer axis. Note that the TG direction (arrows in Fig. 10) inverts depending on downward or upward reflection.
6.1. Results at d = 6 mm (singleelement detector)
The first (bottom) row of analyzers (see Table 2 and Fig. 10) focused Xrays onto three CdZnTe SEDs of 2 mm × 2 mm. The experimental energy resolution was measured to be 60 meV (FWHM, 49 mrad × 46 mrad) with uniform temperature. The line shape was asymmetric similar to the simulation results in Fig. 5(a). To improve total energy resolution, a 1D quadratic TG was applied as explained in Section 5. The total energy resolution improved from 60 to 25 meV (FWHM) after the TG application (Fig. 13). Not only the line width but also the line shape improved. Investigation of d–d excitations in NiO (Ishikawa et al., 2017) revealed both lattice and magnetic effects using this 25 meV resolution. For Bragg angles close to backscattering, δ_{0} < ∼2 mrad, an SED provides good resolution using only an appropriate TG. However, when further from backscattering, δ_{0} > ∼2 mrad, only a TG was not sufficient, so DC was applied. This is particularly important for multianalyzer geometries as explained in Section 6.2.
6.2. Results at d = 25 mm (dispersion compensation)
The second analyzer row focused the beam onto the area detector with d = 25 mm (Table 2). Of the three analyzer rows, the secondrow analyzer has the largest deviation from backscattering so requires the largest TG, ∼7 K per 95 mm. The total energy resolution with uniform temperature results in ΔE_{tot} = 330 meV for an effective SED (11 × 11 pixels) and DC (11 × 1 pixels), as shown in Fig. 14(a). The observed focalspot image at elastic energy is shown in Fig. 15(a) and is comparable with the raytracing results shown in Fig. 15(c). Thus, it is clear that DC does not improve energy resolution without a TG. We applied a 1D quadratic TG for this geometry. The required TG is ΔT_{g} = 6.2 K per 80 mm, as listed in Table 4. A SUS304 jig^{1} was used for the 1D quadratic TG (Table 4). To align elastic energy with firstrow analyzers, the central analyzer temperature T_{0} was set to 36.3°C, as is needed to put the elastic peak at the same energy as the firstrow analyzers. With the additional jig, the feedback parameters T_{0} = 36.3°C and ΔT_{g} = 6.2 K per 80 mm create close to the desired 1D quadratic TG given in equation (7), where A and B are listed in Table 4. The required power to apply the TG was ∼5.7 W for the offset heater, when cooling water temperature was kept at 32.5°C. Fig. 11(c) shows temperature stability – each temperature was stable for more than a month. Applying the 1D quadratic TG, the focalspot image on the detector became narrower [Fig. 15(b)] owing to the correction of chromatic aberration. The line shape of the resolution also drastically narrowed [Fig. 14(a)]. It is worth noting that the integrated intensity is conserved before and after the TG. The energy resolution with the TG was found to be ΔE_{tot} = 57.0 meV (FWHM) with an effective SED [Fig. 14(b)]. The energy resolution of each pixel improved to 30–38 meV (FWHM). As a consequence, the energy resolution with DC^{2} yields ΔE_{tot} = 32.4 meV (FWHM) [Fig. 14(b)]. The discrepancy from calculation may be largely from imperfect temperature correction away from the center of the analyzer. We obtained ΔE_{tot} = 23.7 meV (FWHM) resolution when the analyzer acceptance was reduced by a factor of three in each direction.
6.3. Results at d = 10 mm (dispersion compensation)
The third analyzer row was also designed to focus onto the area detector as shown in Fig. 10. The experimental total energy resolution at uniform temperature was obtained at ΔE_{tot} = 100 meV (FWHM) with an effective SED or with DC. The results agree well with the calculated ΔE_{geom} = 101 meV (FWHM). Similar to the d = 25 mm case of Section 6.2, DC does not improve energy resolution without a TG. To apply a nonlinear TG, pieces of stainless steel (SUS440C, λ = 24 W m^{−1} K^{−1}) were attached to the backside of the Invar substrates [see Fig. 11(a)]. To align elastic energy with firstrow analyzers, T_{0} was set to ∼29.0°C. The feedback parameters T_{0} = 29.0°C and ΔT_{g} = 2.47 K per 80 mm enabled the 1D quadratic TG that is listed in Table 4. The practical power for the TG was typically ∼1.6 W for the offset heater, when the temperatures of the base and the cooling water were 29°C and 25°C, respectively. By applying the 1D quadratic TG, the energy resolution improved to ΔE_{tot} = 31.7 meV (FWHM) using an effective SED (11 × 11 pixels), as shown in Fig. 16. Using DC, the energy resolution improved to 25–27 meV (FWHM) depending on the detector pixel positions. Consequently, the final corrected resolution^{3} ΔE_{tot} = 25.4 meV (FWHM) was obtained.
7. Conclusions
We have shown that careful application of temperature gradients on diced spherical analyzers in a Si(888) backscattering geometry allowed us to improve the energy resolution between two and ten times without loss of signal intensity. The temperature gradient allows relaxing of the Rowland geometry (magnification of 0.9) and allows a large space for the sample, making more complicated sample environments possible. We obtained a factor of 2.4 improvement in resolution, with a best case of 25 meV resolution (FWHM, 49 mrad × 46 mrad acceptance) with a singleelement detector. A combination of DC correction and a temperature gradient improved the energy resolution by a factor of ten, with a best case of 32 meV FWHM. In both cases, the measured resolution includes the incidentbeam contribution of ΔE_{inc} = 22 meV. Note that further improvement of the overall energy resolution would be possible if the incident bandwidth and/or detector pixel size were reduced. We expect a similar temperaturegradient approach may improve in hard Xray highresolution RIXS (ΔE < ∼30 meV) with spherical analyzers.
Footnotes
^{1}Practically, the jig has a slightly larger quadratic coefficient than predicted, so a thin layer of Kapton is added between the jig and the analyzer to reduce thermal contact conductance.
^{2}Strictly, the energy–position correlation is not completely linear in the offRowland geometry (Ishikawa & Baron, 2010). However, a linear energy–position correlation is almost valid in these geometries. Positionsensitive data were shifted (y_{d} direction) by 5.6 meV pixel^{−1} (d = 25 mm) and summed.
^{3}Positionsensitive data were shifted (y_{d} direction) by 2.6 meV pixel^{−1} (d = 10 mm) and summed.
Acknowledgements
The authors acknowledge K. Miura, Y. Senba and H. Ohashi for evaluating the surface profile of wafers and substrates. DI thanks S. Takahashi and Y. Senba for help with the finiteelement tools.
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