X-ray spectra provide a wealth of information on high temperature plasmas; for example electron temperature and density can be inferred from line intensity ratios. By using a Johann spectrometer viewing the plasma off-axis, it is possible to construct profiles of plasma parameters such as density, temperature, and velocity inside the plasma with good spatial and time resolution^1,2. This manuscript presents the operation of the High Resolution X-Ray Crystal Imaging Spectrometer with Spatial Resolution (HIREXSR), a high wavelength resolution spatially imaging x-ray spectrometer used to view hydrogen- and helium-like ions of medium atomic number elements in a tokamak plasma.

HIREXSR is deployed on Alcator C-Mod, a tokamak fusion device with a major and minor radius of 0.67 m and 0.22 m respectively. It typically operates with deuterium plasmas lasting ~2 sec with average densities between 0.2-8.0 x 10²⁰ m^-3 and central electron temperatures between 1-9 keV³. Under these conditions, medium to high Z impurity elements become highly ionized and radiate in the x-ray range, which HIREXSR measures. Benchmarking atomic code modeling of x-ray spectra obtained from well diagnosed laboratory plasmas is important to justify use of such spectra to determine plasma parameters when other independent diagnostics are not available⁴.

Every spectrometer is built for its desired use. Accordingly, a general description about the machine and its related concepts is necessary to fully comprehend these powerful tools⁵. Bragg reflection occurs when a photon reflects off adjacent layers of a crystal and travels a distance that is a multiple of its wavelength. Figure 1 depicts this phenomenon. This condition is expressed by the equation nλ = 2d sin θ_b, where n is the order of reflection, λ is the wavelength of the photon, d is the separation between adjacent layers of the crystal and θ_b is the Bragg angle. A one to one correspondence between λ and θ_b indicates that all photons at a specific point of the detector plane travel with the same wavelength. In practice, however, absorption and precision limitations manifest as a deviation from the Bragg angle. This results in only a small range of angles that produce significant constructive interference, represented by a rocking curve⁶. Figure 2 is an example curve for a calcite crystal.

HIREXSR is a Johann spectrometer with a spherically bent crystal⁷. Before describing this kind of device, a discussion of a simpler, circular spectrometer is appropriate. This set up consists of a bent crystal that reflects incoming photons at their respective Bragg angles towards an array of single x-ray photon counting pixel detectors. The crystal and the detector lay tangent to the Rowland circle, as displayed in Figure 3. The diameter of the Rowland circle is equal to the radius of curvature of the crystal. All rays from a given point on the circumference to any point on the crystal have the same incident angle with respect to the crystal itself.

In the case of HIREXSR, a spherically bent crystal permits spatial resolution in the meridional plane, illustrated in Figure 4. The meridional focus f_m is defined as: f_m = R_c sin θ_b, where R_c is the radius of curvature of the crystal. The sagittal focus f_s is defined as: f_s = −f_m/cos 2θ_b. The spatial resolution of the spectrometer Δx is given by: , where L_cp is the distance between the crystal and the plasma, and d is the height of the crystal. Because the 2-dimensional spacing of the crystal layers is discrete, this must be taken into consideration when choosing a material. Since the detector surfaces are planar, they can only be tangent to the Rowland circle at one point, which consequently gives rise to error since the detected rays are not landing precisely on their corresponding points on the Rowland circle. Physically, this misalignment manifests as a "smearing" of photons of specific energy on the detector. This Johann error is defined as , where l is the width of the crystal. If the detector pixel width δx_p is much larger than the Johann error, then the spectral resolution is independent of it. If they are of comparable size, then the total error can be approximated by . The resolving power of the crystal spectrometer is given by: , where . Instead of placing the detector tangent to a point on the Rowland circle however, in HIREXSR the detector is angled slightly to sacrifice accuracy for spectral range, as shown in Figure 5. This error analysis has been experimentally verified and conforms to expectation⁸.

There are two crucial parameters to consider when designing a Johann spectrometer. First, the imaging range determines what the spectrometer will be observing. For studying plasmas, it is highly desirable to view its entire cross section in order to distinguish between line shifts caused by poloidal and toroidal rotation. HIREXSR is mounted such that it can view the whole plasma, and is tilted slightly off-axis by ∼8° (illustrated in Figure 6) to allow for accurate toroidal measurements. Second, time resolution regulates the minimum time between events that the spectrometer can record. For Alcator C-Mod, desirable values are below 20 msec, shorter than the energy and particle confinement times. The x-ray counting pixel detectors that HIREXSR uses can support a time resolution of 6 to 20 msec or larger⁹. Table 1 summarizes all of the module specifications.

For perturbative plasma studies, the laser blow-off system on Alcator C-Mod is used to deliver multiple ablations with precise timing¹⁰. The laser is a Nd:YAG (neodymium-doped yttrium aluminum garnet) operating at up to 10 Hz. The laser is incident on a remote-controlled optical train as shown in Figure 7 that focuses and steers the beam to the desired location on the slide. Spot sizes of the laser need to be controlled so the injection doesn't disrupt the plasma. A long focal length (1146 mm) converging lens is translated along the optical axis via a remote controlled linear stage to allow ablated spot sizes to vary from ∼0.5 to 7 mm. Fast beam steering is achieved via a 2D piezoelectric mirror. This piezoelectric system is mounted to an RS232 driven mirror mount capable. In addition to the Nd:YAG laser, a 633 nm diode laser is used to indicate the location of the main (infrared) beam. The beams are made to be collinear through the first mirror.