A Wide-Range High-Resolution X-ray Crystal Spectrometer for Laser – Plasma Diagnostics

: Wide-range crystal spectrometers are important tools for performing X-ray spectroscopic measurements of medium-and high-Z tracer elements in research on laser-driven inertial confinement fusion (ICF) plasmas. In this paper, we propose a wide-range high-resolution crystal spectrometer based on a tandem array of crystals that have the same geometric parameters. We have developed a three-channel crystal spectrometer that covers the range of 8 – 18 keV by combining Ge<311>, Ge<331>, and Ge<531> crystals. Here, we report the design, optical simulations, and X-ray test experiments of this spectrometer. The calibration results indicate that the spectral resolution E /Δ E is greater than 2800 at 8.048 keV. By selecting appropriate Bragg angles, crystal materials, orientations, or other geometrical parameters, the wide-range crystal spectrometer developed in this paper can also be used to make measurements in other energy ranges.


Introduction
X-ray spectroscopic measurements of implosion targets containing medium-and high-Z tracer elements can be used to diagnose the shapes and intensity ratios of characteristic lines in laser-driven inertial confinement fusion (ICF) and high-energy-density physics (HEDP) experiments [1][2][3][4].Such measurements are important tools for quantitative studies of the plasma temperature, density, and other implosion information [5][6][7].An implosion target may contain multiple tracer elements separately or simultaneously, and the characteristic lines of these elements frequently range in energy from a few keV to tens of keV [8].Consequently, a wide-range and high-resolution crystal spectrometer is useful for such X-ray spectroscopic diagnostics [9,10].
The energy resolving power of a crystal spectrometer is achieved via its Bragg diffraction of X-rays; i.e., 2d sin θ = kλ, where d is the crystal interplanar spacing, λ the X-ray wavelength, k the diffraction order, and θ the Bragg angle.To obtain wide-range spectroscopic measurements, a vast range of Bragg angles is required.Seely et al. developed a multi-channel, high-energy X-ray spectrometer (HENEX) consisting of four convex reflecting crystals (mica<002>, KDP<011>, CaF2<111>, and quartz<203>) and one convex transmission crystal (quartz<100>) for the National Ignition Facility (NIF) and OMEGA [11].The respective energy ranges are 1-20 keV and 11-40 keV, and the spectral resolution E/∆E is greater than 300.Based on the HENEX instrument, Martin et al. developed a fourchannel convex crystal spectrometer (beryl<100>, quartz<100>, Si<111>, and Ge<220>) [12].They expanded the energy range to approximately 1.2-9.8keV by decreasing the radius of curvature of the crystals, and they improved the spectral resolution E/ΔE to more than 700 by increasing the object distance.
The energy range can also be broadened by modulating the crystal shape, e.g., in an elliptical crystal spectrometer [13].Compared to a conventional spherical surface, the point-to-point focusing provided by an elliptical surface not only reduces the impact of the source size on spectral resolution but also provides a broader range of Bragg angles and a wider energy range.Anderson et al. developed a three-channel, ellipsoidal, opacityzipper spectrometer (OZSPEC) that covers the energy range of 0.24-5.80keV; it achieved a spectral resolution E/∆E greater than 400, which is 2-3 times larger than that provided by a spherical surface [14].Heeter et al. developed a modified version of the OZSPEC-2 structure that further improved the spectral resolution E/ΔE to more than 500 [15].Haugh et al. developed the Supersnout-2 X-ray spectrometer for NIF, which consists of four elliptical PET crystals and which covers the energy range of 5.8-16.5 keV [16].Stoupin et al. developed a variable-sagittal-radius elliptical crystal spectrometer, which can obtain timeresolved X-ray spectroscopy in the energy range of 20-30 keV [17].In addition, a bent crystal spectrometer with an asymmetric orientation can also be used to obtain a wide energy range.Seely et al. developed an asymmetric Cauchois-type spectrometer, which used a transmission quartz<203> crystal to record Kα and Kβ spectral lines from Fe to Ag.It covered an energy range of about 6-22 keV with a spectral resolution E/ΔE of about 1000-4000 [18].
The above references commonly used the multi-channel form with either a short object distance or a convex substrate with a small curvature radius to increase the variation range of the Bragg angles while balancing the signal brightness.But when a crystal is bent to achieve a small curvature radius, the practical spectral resolution deteriorates seriously due to numerous crystal dislocations; for example, the curvature radius of the convex substrates was generally set to only 25-120 mm [11][12][14][15], which resulted in a low spectral resolution of only a few hundred.In addition, surface shapes such as elliptical or asymmetric crystals broadens the energy range, but they are more difficult to manufacture.
In many cases, a longer working distance is necessary due to space constraints, laser beam arrangements, or debris shields, and a high spectral resolution needs to be guaranteed at the same time.In this paper, we proposed a crystal spectrometer with a wide range, high spectral resolution, and long working distance.Through reasonably selecting the crystal materials or orientations, as well as the Bragg angle and surface size, multiple crystals with the same geometric parameters can be used in different energy ranges, enabling the spectrometer to perform well over a wide range of measurement energies.As proof of principle, we developed a three-channel, concave-crystal spectrometer that works in the energy range of 8-18 keV for X-ray spectroscopy diagnostics in the Shenguang-II Update laser facility.The spectrometer can be easily used for measuring other energy ranges by changing the Bragg angle, and it can be used more widely.Here, we discuss the optical design, performance simulations, and calibration experiments of this spectrometer.

Optical Design
Figure 1 shows the optical structure of the spectrometer proposed in this paper.It consists of multiple bent crystals (C1, C2, and C3) with the same geometric parameters that are arranged in parallel.Each crystal has a large curvature radius and a long working distance, but the response spectrum and signal brightness are limited.Therefore, we use a multi-channel form to broaden the energy range and replace the convex substrate with a concave surface to improve the focusing effect.By selecting appropriate crystal parameters and crystals of different materials or with different orientations, the X-rays emitted from the target are dispersed by the multiple bent crystals in response to different narrow energy ranges (i.e., E1, E2, and E3), which are then recorded using the detector placed on the image plane, enabling the entire wide-range spectrum to be measured.Due to the multi-channel form, the X-ray spectrometer we developed requires a more complex optical design than elliptical and asymmetric crystals.In order to reduce the difficulty of assembly, we use three crystals that have the same geometrical parameters in the meridional direction.Figure 2a shows the optical path through the spectrometer in the meridional direction (corresponding to the dispersion), where a is the distance from the source to the crystal center, b the distance from the crystal center to the image plane, θ0 the Bragg diffraction angle at the crystal center, l the crystal length, and R the crystal curvature radius.The overall energy range E is determined by geometric parameters such as crystal curvature radius R, crystal length l, as well as the distances a and b.The designed energy range of the spectrometer described in this paper is 8-18 keV, which covers K-shell emission lines from highly charged ions of Cu, Zn, Ge or Zr elements on the Shenguang-II Update laser facility.In order to avoid spatial interference with the ninth beam PW laser focused by an off-axis parabola mirror in the facility chamber [19], we chose the work distance a to be 400 mm and the work distance b to be 180 mm.Although the corresponding spectral resolution is somewhat reduced compared to that obtainable with larger values of b, the signal brightness is increased.The curvature radius R of each crystal was 400 mm, larger than the previous references, to obtain better spectral resolution, as analyzed later.We chose Ge as the crystal material for all three channels because it has high diffraction efficiency in this energy range [20].Also, Ge crystals exhibit less degradation of their spectral resolution after bending than α-quartz or other crystals do [21].We selected the crystal orientations Ge<311>, Ge<331>, and Ge<531> to cover three narrow energy ranges.
The Bragg angle θ0 corresponding to the crystal center is 25°, at which angle, the central working energies E0 of the three crystals are 8.60 keV, 11.30 keV, and 15.34 keV, respectively.We chose l to be 100 mm; at this length, the Bragg angle varies from 20.78° to 29.09°, and the overall energy range E spans 8-18 keV.The specific parameters of the spectrometer are listed in Table 1.The optical path in the sagittal direction is shown in Figure 2b, where w is the crystal width.The sagittal direction perpendicular to the meridional direction has no dispersion capability but can still affect the spectrometer performance by choosing different parameters or substrate surfaces.For example, both the FSSR (Focusing Spectrometer with Spatial Resolution) with spherical mirror substrate [20] and the crystal on a toroidal substrate having different curvature radii in the meridional and sagittal directions can be optimized to focus X-rays in the sagittal direction to improve the signal brightness or realize the spatial resolution [22].This paper mainly discusses multi-channel form and crystal material selection in achieving wide-range X-ray spectroscopy; thus, we use concave spherical K9 mirrors with the curvature radius R = 400 mm in both the meridional and sagittal directions as crystal substrates.The observation angles Δα of the bent crystals in the different channels vary to some extent, and the value of Δα depends on the ratio of the spacing ΔL between the centers of the bent crystals and the working distance a.The angle Δα is relatively small if the distance a between the source and the crystal center is large.
Under the parameters set in this paper, there is a virtual image behind the crystal, and the image plane can only receive the divergent X-rays.The spatial broadening of the divergent X-rays in the sagittal direction is related to the crystal width w and the distances a and b, and its estimated value is about 16.41 mm for central working energies E0.In this paper, we designed a gap of approximately 3 mm between two adjacent crystals to allow for spectrometer adjustment.The spacing ΔL between the centers of two adjacent crystals was thus 18 mm.Correspondingly, the observation angle Δα is approximately 2.579°, and the maximum difference in observation angles across the three channels is 5.158°; thus, they form an approximately quasi-coaxial structure.

Optical Simulations
We evaluated the spatial distribution on the image plane of the spectrometer using optical simulations created with XRT software (1.5.0 version) [23].We used a typical Xray source with a focus-spot diameter of Φ300 μm as the target.We selected twenty monochromatic energy points at equal intervals within their respective energy ranges of three bent crystals, and their spatial distributions and intensities on the image plane are shown in Figure 3.The meridional direction in Figure 3a-c corresponds to the dispersion direction.The direction perpendicular to the dispersion direction in Figure 3a-c is the sagittal direction.The spatial broadening of twenty monochromatic energy points in the sagittal direction for each crystal varies from 14.70 mm to 18.90 mm.For central working energies E0, the spatial broadening of all three crystals is 16.40 mm, which is in agreement with the estimated value of 16.41 mm in the optical design section.According to the spatial distribution of twenty monochromatic energy points, we obtained the linear dispersion rate D of three crystals, as shown in Figure 4a, which was defined as the change rate of the linear energy along the detector position.The rates are approximately constant, yielding the values of 71.7 eV/mm, 92.9 eV/mm, and 129.1 eV/mm for Ge<311>, Ge<331>, and Ge<531>, respectively.The constant dispersion rate simplifies spectrum recognition, and unknown spectral lines can be calculated quickly using certain known lines [24,25].

Experiments and Analysis
The wafer size of the three Ge crystals was designed to be 100 mm × 15 mm in this paper.Although the production technology of the common 4-inch Ge wafer is mature, it is difficult and costly to manufacture Ge wafers with special orientations.We prepared each crystal by splicing three 34 × 15 mm Ge wafers together.We calibrated the spectrometer performance using an X-ray testing platform.Since the three channels of the spectrometer work under the same geometric parameters, it is simple to install and adjust each crystal using the visible laser.As a non-imaging structure, the spectrometer does not require high alignment accuracy for the working distance.Therefore, we use a ruler to characterize the working distances a and b in calibration experiments.The X-ray source currently available in the laboratory is a Cu and Pt tube.The main characteristic lines including the L-series of Pt and the K-series of Cu were consistent with the energy ranges of Ge<311> and Ge<331>, so only Ge<311> and Ge<331> crystals were tested.
The experiment layout, as shown in Figure 5, consisted of an X-ray tube, Ge<311> and Ge<331>, a beam selection slit, and an X-ray CCD.We positioned the X-ray tube directly at the nominal object distance to achieve the largest divergence angle to cover the entire crystal aperture.We further determined the relationship between the X-ray reflection position on the crystal surface and the image position on the X-ray CCD by using a beam selection slit.Figure 7a shows the spatial broadening of each characteristic line from Figure 6.We found an FWHM of 220 μm for the Pt Lα lines and 40 μm for the Cu Kα lines.The L-series characteristic lines have larger energy-level differences than the K-series, which results in the former having broader spectral lines and larger values of FWHM.The spectral bandwidths of the lower-energy Cu Kα lines shown in Figure 7b are approximately 2.6 eV based on the linear dispersion rate of 71.7 eV/mm.Correspondingly, the spectral resolution E/ΔE is greater than 2800, which is in good agreement with the simulated result shown in Figure 4b.

Discussion
The multi-channel spectrometer can expand the energy range by adding more crystals, but it is difficult to find suitable crystals or orientations with high diffraction efficiency for a wide continuous spectral measurement.In addition, increasing the number of crystals will make the spectrometer structure more complex and create a larger difference in viewing angles.However, the multi-channel wide-range spectrometer developed in this paper is not only suitable in the energy range of 8-18 keV but also can adapt to other energy ranges by changing the Bragg angle, while still maintaining high spectral resolution.
When the Bragg angle corresponding to the crystal center θ0 is changed to 20°, the central working energies E0 of the three crystals become 10.63 keV, 13.97 keV, and 15.57keV, respectively.If other parameters remain unchanged, the overall change in Bragg angle for each crystal varies from 15.12° to 24.60°, and the energy ranges of the three crystals are 8.72-13.94keV, 11.47-18.19keV, and 18.95-24.85keV, respectively.At this time, the overall energy range E spans about 8.7-24.9keV.When the Bragg angle corresponding to the crystal center θ0 is changed to 30°, the central working energies E0 of the three crystals become 7.27 keV, 8.63 keV, and 12.97 keV, respectively.If other parameters remain unchanged, the overall change in Bragg angle for each crystal varies from 26.41° to 33.59°, and the energy ranges of the three crystals are 6.57-8.17keV, 8.63-10.74keV, and 11.72-14.56keV.At this time, the overall energy range E spans about 6.6-14.6 keV.With these two Bragg angles, we evaluated the spatial distribution of twenty monochromatic energy points at equal intervals on the image plane, as shown in Figure 8.The spectral performance of the spectrometer with the Bragg angle of 20° and 30° was listed in Table 2.When the angle is 20°, the full width at half maximum (FWHM) of twenty energy points are approximately 48.8-72.6 μm, 46.8-99.6 μm, and 48.8-91.6 μm, and the spectral bandwidth ∆E of the three crystals is approximately 3.50-6.64eV, 4.35-9.25 eV, and 6.30-11.83 eV.The overall spectral resolution E/∆E is about 2100-2500, 2000-2600, and 2100-3000 for the Ge<311>, Ge<331>, and Ge<531> crystals, respectively.When the angle is 30°, the full width at half maximum (FWHM) values of twenty energy points are approximately 44.3-94.1 μm, 44.4-91.6 μm, and 46.3-96.1 μm, and the spectral bandwidth ∆E of the three crystals is approximately 3.18-6.75eV, 4.12-8.51eV, and 5.98-12.41eV.The overall spectral resolution E/∆E is about 1200-2100, 1300-2100, and 1200-2000 for the Ge<311>, Ge<331> and Ge<531> crystals, respectively.

Conclusions
In this paper, we have proposed a wide-range crystal spectrometer based on a tandem array structure of crystals that have the same geometric parameters.Based on the measurement of K-shell emission lines from highly charged ions of Cu, Zn, Ge or Zr elements, we have used a combination of Ge<311>, Ge<331>, and Ge<531> crystals to develop a three-channel spectrometer that spans the wide energy range of 8-18 keV.X-ray testing experiments show that the spectral resolution E/ΔE of this spectrometer is greater than 2800 at the Cu Kα lines, which indicates that its performance is close to the theoretical prediction.The simulation results show that such a multi-channel crystal spectrometer also can be easily used to measure other X-ray energy ranges by changing the Bragg angle.In addition, selecting other appropriate crystal materials, orientations, and geometrical parameters can also expand applications of such spectrometers in laser-driven ICF and HEDP experiments.

Figure 1 .
Figure 1.Schematic illustration of the multi-channel, wide-range crystal spectrometer.

Figure 2 .
Figure 2. Optical paths of the wide-range crystal spectrometer: (a) in the meridional direction; (b) in the sagittal direction.

Figure 3 .
Figure 3. Spatial distributions and intensities of twenty monochromatic energy points at equal intervals on the image planes of three Ge crystals: the upper part is the spatial distribution; the lower part is the intensity: (a) Ge<311>; (b) Ge<331>; (c) Ge<531>.

Figure 4 .
Figure 4. Spectral performance of the wide-range crystal spectrometer: (a) The linear dispersion rates D; (b) The spectral bandwidth ∆E and spectral resolution E/ΔE.

Figure 5 .Figure 6 .
Figure 5. Experimental arrangement of offline X-ray tuning platform.We placed the X-ray CCD detector (PIXIS-XO-2048B) at the image plane to record the experimental images shown in Figure 6a,b, with an exposure time of 20 s.The operating voltage and current of the Pt and Cu X-ray tubes were set at 30 kV and 15 mA.Except for distorted areas at the edges of the wafers, the multiple characteristic lines of elemental Pt and Cu are straight and sharp.The main lines shown in Figure 6 are the lines Lα (Lα1, 9442.3 eV; Lα2, 9361.8 eV) and Lβ (Lβ1, 11070.7 eV; Lβ2, 11,250.5 eV) of elemental Pt and the lines Kα (Kα1, 8047.8 eV; Kα2, 8027.8 eV) and Kβ (Kβ1, 8905.3 eV) of elemental Cu.Using XRT software to simulate the theoretical value and relative intensity of each characteristic line [27], we obtained the results shown in Figure 6c, d, respectively.These figures show the distributions of the aforementioned spectral lines in the image plane after reflection from the Ge<311> and Ge<331> crystals.The relative positions of the characteristic lines are consistent in the simulations and experiments; i.e., the experimental results are as expected.

2 Figure 7 .
Figure 7. (a) Spatial distribution of Pt and Cu characteristic lines on the image plane of the Ge<311> crystal and (b) magnified image of the Cu Kα1 and Kα2 lines.

Figure 8 .
Figure 8. Spatial distributions and intensities of twenty monochromatic energy points at equal intervals on the image planes of three Ge crystal; the upper picture is 20°, and the lower picture is 30°: (a,d) Ge<311>; (b,e) Ge<331>; (c,f) Ge<531>.

Table 1 .
Optical parameters of the three-channel, wide-range crystal spectrometer.

Table 2 .
Spectral performance of the spectrometer with the Bragg angle of 20° and 30°.