Performance predictions of the SPARC x-ray crystal spectrometers for ion temperature and toroidal rotation measurements Open Access

Fusion power plants (FPPs) are expected to operate in novel regimes of electron/ion temperature. To optimize the DT fusion reactivity, plasmas need to reach 10–20 keV. Estimates of the toroidal rotation velocity are in the order of 100 km/s.^1–3 It is, therefore, important to have reliable measurements of these quantities to benchmark predictions and provide feedback in the control room.

X-ray crystal spectroscopy (XRCS) will be employed on the SPARC tokamak,⁴ currently under construction, to provide such measurements. XRCS is a class of Bragg spectrometers that utilizes a crystal for the dispersive element, as the small lattice spacing is of the order suitable to diffract x-ray wavelengths. The inherent almost Delta function-like reflectivity (rocking curve) of a crystal allows for high resolving power passive spectroscopy.⁵ The exact wavelength of interest constrains the Miller plane the crystal needs to be cut at.

A carefully selected trace impurity is imaged to observe the Doppler broadening and shift of characteristic electronic transitions from high ionization states. This impurity is usually either a seeded noble gas,^6–8 such as Ar, Kr, or Xe, or intrinsically present,^5,9–11 such as Fe, Mo, or W. Typically, closed-shell charge states are utilized to maximize signal over a wide electron temperature range. Another benefit of closed-shell charge states is that the atomic structure is easier to calculate.

This paper summarizes performance predictions of this diagnostic to validate the design. Section II introduces the design of three sets of spectrometers, each with its own specialization. Section III describes and justifies the synthetic diagnostic modeling performed. Section IV details the atomic data modeling. Section V then introduces a method we are exploring to reconstruct ion temperature and toroidal rotation radial profiles. Section VI then concludes with future work.

Because of the large dynamic range of the electron temperatures FPPs are projected to operate in (1–20 keV), no single impurity charge state has sufficient abundance to provide full coverage, as has been the role of He-like Ar in modern tokamaks.^12,13 As such, it was deemed necessary to have two sets of spectrometers, one dedicated to low temperature regions of the plasma and another for high temperature. Numerous candidate lines have been considered in the literature from similar FPP XRCS design papers.^14–18 We have elected to rely on Ne-like Xe to diagnose an electron temperature range of 4–9 keV and He-like Kr for 10–20 keV. Figure 1 illustrates this decision from an argument of emissivity per radiated power (hence agnostic to electron density or impurity concentration) from the brightest line of each charge state. It should be noted here that there are known issues with the atomic data utilized, which will be described in more detail in Sec. IV.

It was determined desirable to implement a third system to act as a low-resolution survey of highly ionized (He-, H-like) common industrial metals (Cr, Fe, Co, Ni, Cu) as well as close Ne-like W lines. Throughout this paper, we will denote these three systems as XRS-HR-Kr, XRS-HR-Xe, and XRS-LR, where XRS means X-Ray Spectrometer, HR means High-Resolution, and LR means Low-Resolution.

An important design constraint for all SPARC diagnostics is the expected intense neutron fluxes. Other FPP XRCS designs have chosen to approach this problem by implementing grazing incidence mirrors.¹⁸ For SPARC, we have decided not to do the same in favor of simplifying alignment with fewer optical elements. We, therefore, could not use a spherical Johann configuration as the aperture required would be too big. Most SPARC diagnostics are located about 16 m away from the tokamak behind a thick concrete wall. The beamline diameter is 10 cm, meaning that for a spectrometer with a slit, the volume-of-sight through the plasma is very narrow. Therefore, an array of spectrometers is needed to cover the poloidal plane. A spectrometer for each system (XRS-HR-Kr, XRS-HR-Xe, and XRS-LR) is housed within a common beamline. Figure 2 illustrates how, per beamline, there is a vertical stack of three crystals for each respective system that reflect an image onto a large common camera. Due to the port space available, up to five beamlines can be installed. See the accompanying paper in these proceedings¹⁹ for more information.

Table I details the crystal chosen for each spectrometer system. Care was taken to select the highest integrated reflectivity, and hence throughput, while not costing spectral resolution by an angular width significantly larger than candidate detector pixels. This typically implies choosing Ge crystals over Quartz for the same Bragg angle. The crystals will be bent cylindrically typical of von Hámos configurations.^6,20

The notable difference of our design to the von Hámos configuration is that space constraints in the diagnostic lab forced us to compactify by forgoing the vertical focus. Moving the slit light source off the principal axis of the reflector causes the image rays to be divergent. This though allows one to move the detector closer from the center of curvature toward the focal point. This also means that the crystal radius of curvature is now a free variable. For a particular crystal-to-detector distance, decreasing the radius of curvature starting from a flat crystal conically bends and compactifies a flat image at the cost of spectral resolution at the image top/bottom from a wider wavelength integration per pixel. An example of such image for radius, R = 0.6 m, is given in Sec. V. The exact radius of curvature for each crystal is being chosen to ensure each spectrum can be measured on a common detector per beamline.

The distance from the port face on SPARC to the limiting slit is 11.5 m, the distance from the slit to the crystal is 4.4 m, and the distance from the crystal to the detector is 0.4 m. Note that two slits will be installed along the field-of-view to act as a double barrier for tritium. Be is the traditional window material for its good x-ray transmission, but we are scoping using diamond instead. Diamond has a lower x-ray transmission but better toxicity, neutron, and tritium properties, as well as simpler procurement, while possibly making a thinner window. Diamonds being transparent to visible light also allows for easier alignment. The particular dimensions of the optics, such as slit size, are in progress of being optimized for photon flux and ease of analysis.

The main tool we use to simulate and analyze synthetic diagnostics is the ToFu code.²⁴ In this work, we utilize the provided tools for volume-of-sight integration, tomographic inversions, rocking curve calculations, and deterministic ray-tracing. These particular tools were developed to analyze XRCS spectra measured on the WEST tokamak, which is a spherical Johann-type imaging spectrometer.¹³ As suggested in Sec. II, an addition to the code had to be made for cylindrically curved crystals for this work.

It is important to us that our performance projections for this diagnostic are reliable, so we are in the progress of cross-validating our photon flux calculations with another popular ray-tracing code, XICSRT, which is a Monte Carlo-type calculation. XICSRT has been benchmarked against other codes for a spherical Johann-type spectrometer to great success.^18,25 Still novel though are cylindrical reflections, which are in progress of being implemented.

For both codes, rocking curves for each combination of wavelength and crystal simulated are calculated using ToFu. Used are equations from dynamical diffraction theory, here assuming a finite sized, perfect crystal at 25 °C.^21,22 Models in ToFu are included to account for thermal expansion and mosaicity but are not utilized here.

Our validation exercise consists of comparing photon flux distributions calculated for a (1) mono-energetic point source, (2) mono-energetic volumetric source, (3) multi-energy volumetric source, and (4) including a flux function velocity profile. Each of these cases will be simulated for (1) an example spherical Johann spectrometer, (2) an example von Hámos spectrometer, and (3) the as-designed spectrometer layout described in Sec. II.

At the time of writing this paper, only the case of a mono-energetic point source imaged by a spherical crystal spectrometer has been completed. The example spectrometer views the SPARC midplane with a port-to-aperture distance of 11.5 m, an aperture-to-crystal distance of 4.4 m, and a crystal-to-detector distance of 0.4 m. The aperture has a width/height of 5 mm/4 cm, the crystal has a width/height of 3.0 cm/8.5 cm and a radius of curvature of 0.4 m, and the detector has a width/height of 1.3 cm/0.7 cm. The source is isotropic with a wavelength of 1.61 Å and an emissivity of 1 ph/s/srad. Figure 3 illustrates that both codes calculate the same detector image both in terms of the number of photons detected and the distribution within ray-tracing Poisson error.

Included in the spectra modeling is Bremsstrahlung and radiative recombination with free–free and bound-free (up to principal quantum number, n = 6) Gaunt factors sourced from the ChiantiPy code.^26–28 Energy levels when calculating the radiative recombination of Kr and Xe are supplemented using the Flexible Atomic Code (FAC).²⁹ 

Cross-sections and decay rates for the line radiation emitted by Kr and Xe are sourced from FAC. The processes included in this work are spontaneous emission (overall multipoles), electron impact excitation, electron impact ionization, autoionization/dielectronic recombination, and radiative recombination. Collisional-radiative modeling is then performed using ColRadPy.³⁰ Care was taken to validate the generated data against what is available on OpenADAS^31,32 as well as in the literature.^33,34 Note that since the SPARC plasmas are expected to reach T_e > 10 keV, a significant portion of the electron population approaches relativistic energies (E∼m_ec²). FAC already solves for relativistic cross-sections, but when averaging that data into rate coefficients, one must account for (1) the relativistic energy-velocity relation and (2) a lesser extent that the maximal entropy energy distribution shifts from a Maxwell–Boltzmann to a Maxwell–Jüttner. Line radiation from other species is supplemented by what is available on OpenADAS.

Since the collisional-radiative modeling includes levels that mainly decay by two-photon emission, the continuum emission is modeled. H- and He-like rates are sourced from FAC. Energy distributions are then sourced from ChiantiPy but supplemented for Z > 30.^35,36

Spectrum modeling is managed by the Aurora code.^37,38 All lines from Kr and Xe are Voigt broadened, sourcing Einstein coefficients from FAC, with other species being Doppler broadened. Aurora also manages calculating the charge state density distribution and sourcing effective ionization and recombination data from OpenADAS. For this work, no transport was considered but dramatically peaked/hollow profiles will be considered in future work. It is important to note that we have found the effective ionization/recombination rates posted on OpenADAS to not always be accurate compared to experiments. From experience doing similar calculations for the charge state distribution of Mo in Alcator C-Mod,³⁹ it would appear that excitation–autoionization processes were neglected in the atomic data modeling, resulting in an underabundance of closed shell charge states and an overabundance of alkali-like states. In progress is recalculating these data, particularly for Ne-like Xe, similarly to what was performed in Ref. 40.

Figure 4 illustrates a spectrum calculated for He-like Kr including Li-like satellites in good agreement with measured spectra on TFTR.⁴¹ For the He-like system, FAC was run with singly excited states up to n = 10 and doubly excited states 2lnl′ up to n ≤ 6, including all possible orbital angular momenta. For the Li-like system, FAC was run with singly excited states up to n = 9 and doubly excited states 1s2lnl′ up to n ≤ 6, including all possible orbital angular momenta. Note that at this time emissivities from only excitation and recombination are calculated with ionization to be included. Only two-photon emission was included for continuum radiation in this figure. In addition, shown in the shaded magenta is a possible wavelength calibration source using second-order diffraction of the Sb K_α sequence.⁴² 

Figure 5 depicts a synthetic detector image for midplane beamline imaging He-like Kr produced from volume-of-sight integration with ToFu assuming the SPARC “Primary Reference Discharge” H-mode plasma.⁴³ The w line is the rightmost trace around horizontal bin 80. Note that bins are integrated over many pixels per candidate detector geometries. The viewed regions of the plasma get hotter from the bottom to the top of the image, illustrating the asymmetric instrumental broadening discussed in Sec. II.

Ion temperature and toroidal velocity reconstructions are being performed using the He-like Kr w line and Ne-like Xe 3D line, but will not be shared here in preference to a future publication. We will scope how well we can reconstruct profiles over an ensemble of L-/H-mode n_e/T_e/T_i/n_Kr/Xe kinetic profiles, particularly fictitious cases that are extremely hollow or peaked. Similarly, misalignments from the measured as-installed diagnostic geometry considering tilts and translations of optics with respect to each other will be explored. Care must be taken to include error bars from Poisson statistics⁴⁴ and spectral fitting.

An initial concern with reconstructing ion temperature was the strong nuclear charge scaling of the natural width, e.g., for optically allowed transitions A_ij∼Z⁴, where A_ij is the Einstein coefficient. Conveniently, the full-width half-maximum due to Doppler broadening dominates that for a Lorentzian by a factor of five above T_i = 5 keV for He-like Kr (A_w = 1.56 × 10¹⁵ s⁻¹) and above T_i = 1 keV for Ne-like Xe (A_3D = 4.62 × 10¹⁴ s⁻¹). Well below the temperature range, we can expect to have significant signals from the respective species. This allows us to use reconstruction methods such as taking spectral moments of the detector image, assuming it is a sum of Gaussians.⁴⁵ 

It is important to note that it is likely that not all five beamlines will be installed on SPARC immediately. The tentative plan for day one of operations is to have two beamlines closest to the midplane installed for a redundant measure of the line-averaged ion temperature. We may install for the second campaign another beamline furthest out toward the edge to establish a gradient. Full coverage should be available for SPARC’s third campaign. This deployment plan clearly further constrains the spatial resolution of the reconstruction, so we are scoping the quality of measured ion temperature profiles given a certain subset of the beamlines.

Electron temperature reconstructions are also being explored. This has been shown possible for Ne-like Xe using Na- and Mg-like satellites on Alcator C-Mod.⁴⁰ Historically, for He-like Ar, the ratio of the w line to the n ≥ 3 or k satellites has been used, such as on WEST.¹³ As can be seen in Fig. 4, this may be challenging since the desired lines are buried by brighter lines.

Overlapping lines from other species, such as W, degenerate with our lines of interest is a significant concern when designing our reconstruction workflow. This has been shown to complicate analyzing He-like Ar spectra on tungsten-walled machines such as WEST.⁴⁶ Indeed, if one looks at some of the photon emissivity coefficient files posted on OpenADAS, it is implied there may be a degenerate Al-like W line with the Ne-like Xe 3D line. Theoretically calculated wavelengths are notoriously wrong though, so we are exploring precise wavelength measurements of W in the Ne-like Xe spectral range of interest using the assistance of an Electron Beam Ion Trap (EBIT) facility.

The final design of the SPARC x-ray crystal spectrometers is moving along with construction of the system beginning soon. Vendors have been identified for the various components discussed in the text, with prototypes already received for review. Still, we have illustrated that atomic data needs to carefully scope spectrometer throughput and reliability to reconstruct kinetic profiles. Further optimization of the design to maximize performance and analysis is in progress.

The authors would like to acknowledge N. Pablant (PPPL), L. Delgado-Aparicio (PPPL), K. Shah (PPPL), L. Gao (PPPL), G. Brown (LLNL), F. Bombarda (ENEA), T. Odstrčil (GA), M.F. Gu (U.C. Berkeley), A. Foster (Harvard), and O. Marchuk (IEK) for conversations in benefit to this work. This work was supported by Commonwealth Fusion Systems Grant No. RPP031.

All authors are financially supported by Commonwealth Fusion Systems (CFS) either as employees or through sponsored research contracts. CFS is seeking to commercialize fusion energy and may benefit financially from the science and technologies discussed in this paper.

C. Perks: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). D. Vezinet: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). J. E. Rice: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). M. L. Reinke: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal).

The data used in this publication is available upon reasonable request to the corresponding author.