Systematic search for spherical crystal X-ray microscopes matching 1–25 keV spectral line sources

Monochromatic x-ray microscopy using bent crystals has provided tremendous insights into the dynamics of High Energy Density Physics (HEDP) experiments, such as wire-array or liner implosions using pulsed-power devices,^1–7 hot-electron transport in short-pulse laser plasma interactions,^8–15 or capsule implosions using high-energy lasers.^16–21 

In combination with the spectral-line sources from highly charged ions (e.g., the He_α resonance or intercombination line) or from cold material (e.g., the K_α doublet), these microscopes are efficient and provide a high spatial resolution.^22–24 The crystal substrates and the Miller indices (hkl) of the Bragg plane have to be matched such that the lattice spacing d, the spectral line energy E = hc/λ with hc ≈ 12.398  keVÅ, and the reflection order m fulfill the Bragg equation

The Bragg angle ϑ should be close to 90° to reduce astigmatism.²² 

Toroidal crystals provide a high spatial resolution for a larger range of Bragg angles.^25,26 However, as the Bragg angle decreases, additional aberrations become apparent, and the point spread function becomes sensitive to whether or not the crystal surface is a mathematical toroid or a mathematical ellipsoid. Furthermore, toroidal crystals have higher manufacturing costs and their alignment is more demanding.

Other imaging concepts using spherical crystals that are also not restricted to near-normal incidence have been developed,^27–31 but these concepts require a point-like source which cannot always be generated.

The limitation to fulfill the Bragg condition at near-normal incidence seemingly limits the available choices of crystals and spectral lines. Furthermore, a spherical-crystal microscope requires that the crystal can be bent into a spherical shape with a high optical quality to obtain high-resolution images, which means that only perfect crystals such as α-quartz, mica, Ge, GaAs, or Si are suitable crystal materials, whereas, for example, mosaic crystals such as LiF or HOPG cannot be used.

Bent-crystal x-ray microscopes for the diagnostics of plasmas were first suggested in the 1970s;⁴² a literature survey reveals that since then only a few suitable matches of crystal and line combinations have been realized in applications. Table I shows some of the results. Many systems were developed for x-ray energies below 2 keV. The reason for this might be that low-Miller-index crystals such as quartz (1 0 0) or (1 0 1) were the only high-quality and high-efficiency reflectors at that time, which could be produced having a decimeter-scale radius of curvature. On the positive side, sub-2-keV backlighter systems are important for fusion research with cryogenic deuterium-tritium (DT) implosions; the low opacity of the DT shell requires a low photon energy to obtain sufficient contrast in the image.¹⁹ 

Looking at Table I and the literature suggests that spherical-crystal imaging is restricted to these few chance matches between spectral lines and crystals. Finding other combinations relies on finding the correct combination of Miller indices resulting in a suitable d-spacing and Bragg angle for a spectral line. An earlier work by Koch⁴³ describes a systematic study for matching combinations using selected quartz, LiF, Si, or Ge Miller-index combinations and a few exotic crystals such as single-crystal aluminum or tungsten, gypsum, etc. However, “no attempt was made to rank the crystals in terms of suitability for high-resolution imaging (mechanical properties, integrated reflectivity, mosaic vs. perfect crystal, etc.); the purpose of the compilation is to explore which crystal might be valuable for such further study.” Some selected results of this work were published in Table 2 in Ref. 22.

Loupias et al.³¹ identified about 40 matching pairs for x-ray energies between 1 and 16 keV for quartz and mica crystals. Similar to the work by Koch it was not further examined whether or not the found reflection orders have sufficient integrated reflectivity to be suitable for imaging.

In this work, after performing a systematic, automated search over all 9.1 × 10⁶ possible combinations for six selected crystals with arbitrary Miller-index combinations (hkl) = ([0, …, 20], [0, …, 20], [0, …, 20]), for x-ray energies between 1 and 26 keV (Z = 1–50, Ne to Sn; evaluating the He_α resonance, intercombination or K_α1,2 lines), we show that a matching, efficient crystal and spectral-line pair can be found for basically any of these sources.

In Sections II–IV, we first identify the constraints for spherical-crystal imagers as microscopes for HEDP experiments. Then a method is presented to find all possible spectral-line, crystal, and Miller-index combinations and rank them with respect to their suitability for high-resolution x-ray microscopy experiments. The results are given in a tabulated format at the end of this article.

X-ray imaging properties of spherical crystals for HEDP applications were already covered in Refs. 22–24, here we only describe a few basic principles of their operation. One type of x-ray microscope is a backlighter system, where the object is radiographed by the x-rays from another source, for example, a laser-generated plasma. The other commonly used configuration utilizes x-rays emitted by the object itself for the imaging and is hence called a self-emission imaging system. Both the systems obey similar imaging principles and their imaging properties are detailed in the cited references.

Fig. 1 shows a generic x-ray backlighter system with a spherically curved crystal. Image formation by this spherical x-ray mirror is described by the Coddington’s equations

where p is the object distance from the crystal, q_m (q_s) is the image distance from the crystal in the meridional (sagittal) plane, and R is the bending radius which also defines the Rowland circle by R_Rowland = R/2. The system is astigmatic, since the meridional and sagittal focal lengths are different for Bragg angles ϑ ≠ 90°. One can still achieve a good image when the crystal is operated at near-normal incidence (sinϑ ≈ 1) and when the object is placed midway between the sagittal and meridional foci, at the circle of least confusion which is at⁸ 

Here, M = q/p is the image magnification and the image distance q is calculated as

Neglecting manufacturing imperfections such as local variations of the crystal surface or bending radius, the spatial resolution σ of an imaging system with crystal aperture L is defined by astigmatism. Using the positions p and q defined above, the spatial resolution is almost equal in both the meridional and sagittal directions and scales as^22,24

The spatial resolution deteriorates quickly for decreasing Bragg angles (ϑ → 0), and it is proportional to the crystal aperture size. A self-emission imaging system utilizes the entire crystal aperture L in sagittal direction, which means the resolution can be further improved by limiting the crystal aperture at the cost of throughput.⁸ 

In a backlighter configuration, L depends on the projected size of the backlighter source onto the crystal, using the object plane as origin. If the source of diameter Δs is on the Rowland circle at a distance Rsinϑ from the crystal, L = L_BL, with

Typical backlighter source sizes using a laser to create the source are on the order of 200–400 μm,^19,24,44 which is much smaller than L. Hence, a backlighter system with the source at or near the Rowland circle has a higher spatial resolution than a self-emission imager for otherwise equal parameters.

Eqs. (6) and (7) can be used to find the smallest allowable Bragg angle for a given imaging system and desired resolution. For this work we require a spatial resolution of σ = 10 μm or better for crystal substrates with an L = 10 mm aperture. For simplicity, for backlighter geometries, we consider only configurations with the source on the Rowland circle. The source can be placed anywhere between this position and just behind the object. A source just behind the object recovers a self-emission geometry. However, placing the source on the Rowland circle leads to the smallest L_BL and therefore best spatial resolution, or vice-versa, to the smallest allowable Bragg angle.

We choose a high magnification, M ≫ 1. Therefore, the magnification plays no role in determining the limiting Bragg angle. Solving for ϑ yields

for a self-emission imaging system, and

for a backlighter configuration with a 300-μm-diameter source on the Rowland circle.

To provide the spatial resolution of 10 μm or better, the crystals need to be of optical quality when bent. Hence, we use only α-quartz, Ge, Si, mica, GaAs, and InAs as crystal materials in the search. GaAs, while essentially identical to Ge in its crystal structure and density, needs to be included because it is a two-compound crystal and has more allowed reflections. Its use as a high-quality spectrometer has been demonstrated in Ref. 45. An imaging device using an InAs crystal has not been published yet, but since single-crystal InAs is a well-known material in the semiconductor industry there is no reason to believe it could not be used as a spherical-crystal imaging device similar to Ge or GaAs.

The generated image needs a minimum number of photons per detector resolution element to provide sufficient contrast for the desired resolution.⁴⁶ Calculations for the image brightness are given in Refs. 22–26 and 47. Most of these references assumed a point source in the object plane; Refs. 23 and 24 expand these calculations for extended sources. Generally, the image fluence (photons per square-micron) at the detector I_d depends on the magnification M, the crystal reflection efficiency η, the number of photons N emitted by the source, the crystal collection solid angle Ω, and the area A_obj of the object plane²⁴ 

For any given source, crystal aperture (solid angle), magnification, and object plane, the crystal reflection efficiency η determines the overall throughput of the crystal and hence the image brightness. Under the assumption of negligible source size^22–25 

Here, R_int is the integrated reflectivity of the crystal and ΔE/E the relative linewidth of the spectral line used for imaging.

At first glance Eq. (11) implies that high-resolution systems close to 90^∘ are inefficient, as $limϑ→90∘(1/tanϑ)→0$⁠. However, R_int is changing with the Bragg angle as well (it becomes larger for ϑ → 90^∘). With the help of Eq. (1) and under the approximation that the integrated reflectivity R_int ≈ R_pΔϑ, where R_p is the peak reflectivity and Δϑ the width of the rocking curve, the ratio

The spectral bandwidth of the crystal ΔE_c/E_c is constant for σ-polarized x-rays reflecting off a flat crystal (Tab. IV in Ref. 48). R_p is not necessarily constant, and for π-polarized x-rays the bandwidth can change significantly with x-ray energy for any given crystal. Here, we concentrate on spherical crystals and unpolarized x-rays, hence the integrated reflectivity is a linear combination of π- and σ-polarization results. The scaling of R_int versus Bragg angle for selected spherical crystals was examined using the xcrystal_bent module of the X-ray Oriented Programs (XOP) code suite.⁴⁹ Fig. 2 shows a few examples for different crystals covering the energy range from 1 to 23 keV. With the exception of quartz (1 0 1), the ratio R_int/tanϑ slowly increases for ϑ → 90^∘; generally the values stay within about ±15% of the average for the Bragg-angle range considered here. The reflectivity of quartz (1 0 1) might be affected by the Si K-edge at 1.840 keV. Calculations predict a fast change in the reflectivity across the Si edge,⁵⁶ yet the exact contribution from the edge is a work in progress at the moment.

Since the spherical crystal bandwidth is approximately constant for the unpolarized x-rays for most of the energies and Bragg angles 75^∘ ≤ ϑ ≤ 90^∘ considered here, the only requirement for the efficiency of the imaging system is R_int > 0. However, multiple matches for a specific x-ray energy (or spectral line) that have different Bragg angles will be sorted by R_int/tanϑ to determine the most efficient matching pair.

The crystal structure and Miller-index combination determine the calculated lattice spacing. The simplest crystals in our search have cubic crystal structure. The lattice spacing d of a cubic crystal with the lattice constant a is calculated as

The resulting lattice spacing is then used with the Bragg equation (Eq. (1)) to calculate the resulting Bragg angle for each x-ray energy. However, satisfying the Bragg equation is not the only requirement for Bragg reflection, as it only defines the diffraction conditions for primitive unit cells such as cubic and tetragonal, where the atoms sit at unit-cell corners. Non-primitive unit cells with atoms at additional places can cause out-of-phase scattering to occur at certain Bragg angles. As a result, Bragg reflections do not necessarily exist for any set of planes or particular combinations of hkl. The reflective strength of a crystal plane is proportional to the structure factor |S|². If |S|² = 0, then the atomic scattering vectors cancel out and the spectral reflection is systematically extinct.

This requirement of |S|²≠0 results in selection rules for Bragg reflection. The diamond face-centered cubic crystals considered here (Ge, Si, GaAs, InAs) have |S|²≠0 when hkl are all odd, or hkl are all even and h + k + l = 4n, where n ∈ ℕ.⁵⁰ Hence, the lowest-possible hkl combination for Bragg reflection is for the (1 1 1) plane, which also determines the lowest possible x-ray energy E_min (for ϑ = 90^∘) that the crystal can reflect.

Table II lists the unit-cell parameters and minimum energies for the crystals used in our search. The maximum energy is not limited. Our limit of hkl_max = (20 20 20) and ϑ_min = 75^∘ results in maximum energies of about 30–80 keV, depending on the crystal, which is well beyond the maximum energy of the x-ray emission lines considered here.

Eq. (13) shows that any permutation of a specific set of Miller indices hkl will have identical lattice spacing—for example (2 6 8), (6 2 8), (6 8 2), and so on. Calculations with XOP reveal that for these cases also R_int is identical and hence these can be seen as identical crystal cuts.

Not only permutations of hkl are identical but any combination for which the quadratic sum h² + k² + l² is equal are identical as well, such as (3 5 13) and (1 9 11). These also have identical R_int. In the tables shown below these identical matches will be combined and marked accordingly.

A more complex crystal in our search is quartz, which has a hexagonal crystal structure. Its lattice spacing can be calculated as

Accordingly, Miller-index combinations with constant l but h and k swapped will result in identical lattice spacing. However, XOP indicates that the integrated reflectivity R_int may be different even though the lattice spacing is the same. See, for example, Ti K_α and quartz (2 0 3) and (0 2 3) in Table V.

Selection rules for the allowed reflections of quartz are more complex, see, e.g., Table V in Ref. 50. Reflections are forbidden when h + k + l is even and (h − k) an odd multiple of 3, and h + k + l is odd and (h − k) an even multiple of 3. The lowest possible reflection is for hkl = (1 0 0).

The most complex crystal is mica (or Muscovite), which is a monoclinic unit-cell crystal with space group C2/m. The lattice spacing is given by

Hence, each Miller-index combination results in a different lattice spacing. The empirical selection rules for allowed Bragg reflections using our XOP calculations are the following: h + k must be even and l can be arbitrary. However, if k = 0, then both h and l must be even (or zero).

Furthermore, with c = 20.097 Å, mica has the longest unit-cell parameter of the investigated crystals and therefore can be used for low-energy reflections down to 619 eV, using hkl = (0 0 2).

The constraints from above will be used in the following to find all possible spectral-line and x-ray-crystal combinations that can be used for imaging. The crystals are limited to those mentioned in Table II. X-ray sources were limited to spectral emission lines of the elements Ne (Z = 10, E ≈ 1 keV) to Sn (Z = 50, E ≈ 25 keV). For these elements the search for matching combinations only includes the brightest lines typically observed: the He-like resonance and intercombination lines for hot plasmas from pulsed-power or nanosecond-laser experiments, and the K_α doublet for fast-electron-driven targets as observed, for example, in short-pulse laser experiments. Table III in the  Appendix shows the spectral-line transitions and their corresponding energies, taken from Refs. 51 and 52.

The search relies heavily on XOP, both for the calculations of the 2d-spacing as well as for calculations of R_int by integrating the rocking curve. XOP calculates the lattice spacing using unit-cell parameters (a, b, c, α, β, γ) and the position and atomic number of the atoms in the unit cell (after symmetry operations) from its own database. It should be noted that the XOP calculations are here used as a guide to judge the relative efficiency of the found matches, and not to predict the absolute efficiency.

We use the multilamellar model^53,54 to calculate the rocking curve for bent crystals. All calculations are done for a crystal with R = 250 mm and 70-μm thickness using unpolarized x-rays. The peak crystal reflectivity in these calculations is strongly affected by the temperature (Debye-Waller) factor.⁵⁵ In most cases setting the temperature factor between 0.8 and 0.9 yields satisfying results; a lower temperature factor results in a lower integrated reflectivity. Here, we set a constant factor of 0.8 for all calculations, which may underestimate some reflection efficiencies.

Rocking curve calculations using the multilamellar model have been benchmarked for curved crystals both at the low-energy⁵⁶ and high-energy ends⁵⁵ of our range. Good agreement with the measurements in these references was found. Koch et al. have measured the integrated reflectivity of a high-Miller-index Germanium crystal with hkl = (15 7 7) using an Ag K_α source. XOP calculations agreed within a factor of two. The reference states that a bent Ge (15 7 7) crystal may have an integrated reflectivity up to above-10 μrad for Ru He_α at 19.7 keV. This is in line with our calculations, which result in R_int = 9 μrad as shown in Table IV.

In rare cases XOP results can be off by more than a factor of two. One example is quartz (4 6 8) and Zr K_α2 at 15.691 keV. Rocking curve calculations using XOP showed that this might be an efficient combination;⁴⁰ later measurements⁵⁷ showed that R_int is about five times lower than predicted. Our own measurements using a spherical quartz (4 6 8) crystal confirm these results. However, the scope of this paper is to identify the most promising spectral-line and crystal combinations for any given x-ray source within an order-of-magnitude estimate. Once a candidate has been identified, further work can be spent in characterizing the crystal.

We do not restrict the search for matches to Miller indices that are already known and tabulated in the literature (see, e.g., Table 4-1 in Ref. 51). Instead, we first use XOP to calculate the 2d-spacings for all Miller-index ranges hkl from zero to 20 each (i.e., starting at hkl = (0 0 0) and ending at hkl = (20 20 20)). Hence, higher-order reflections up to m = 20 and high-Miller-index combinations such as (15 7 7)⁴¹ and others are included in our search.

By default, XOP is run from a graphical user interface (GUI). We wrote custom routines to automatically change the input for calculations without invoking the GUI. These modifications allow us to generate a data table for each crystal material containing hkl values and their corresponding 2d-spacing if the structure factor |S|²≠0. By only saving 2d values for |S|²≠0, the calculations fulfill the selection rules for Bragg reflection. For reference, the supplemental spreadsheet file (see Sec. V) contains all the calculated 2d-spacings.

After generating all the 2d-spacing tables with XOP, a Python script is used to perform the actual search for matching combinations. Fig. 3 shows a flow diagram of the script. After initialization with the desired parameters, the list of spectral lines and the 2d-spacing data tables is read into memory. The script iterates through all Miller-index and spectral-line combinations for each crystal, looks up the corresponding 2d spacing, calculates the Bragg angle using Eq. (1), and checks if the Bragg angle is within the limits discussed in Sec. II A. If a matching combination is found, the Python script communicates the parameters to custom-written routines that prepare the input for the module cryst_ml from XOP. This module uses the multilamellar model to calculate the rocking curve of a bent crystal for the respective x-ray energy. The resulting rocking curve is then numerically integrated using the custom routine to obtain R_int. This value is then communicated back to the Python script, which, if R_int > 2.0 × 10⁻⁵, saves the result in a spreadsheet for post-processing.

The total number of tested combinations is 9 112 824. Out of these more than 9.1 × 10⁶ combinations, the script identified 37 265 possible matches with 75^∘ ≤ ϑ ≤ 90^∘. After testing for the integrated reflectivity, 10 652 matches were confirmed for He-like spectral lines and 10 628 for K_α lines. Interestingly, an almost equal number of matches for both hot (He_α) and cold (K_α) emission lines was found. The script further found 1058 combinations that could be used as self-emission imagers with 87^∘ ≤ ϑ ≤ 89^∘.

In Sections IV A and IV B, we present the results in a tabulated form, separated by combinations for self-emission imaging and backlighting systems. The data are further down-selected for each element by sorting for R_int/tanϑ in descending order. If more than three possible matching combinations are found, only the top three matches are shown for clarity. For cases where the top three matches have low Bragg angles close to 75^∘, more than three matches are shown to avoid that the reader gets the impression that many matches only work for Bragg angles near 75^∘, which is not the case. For some elements we show the top matches up to crystals that are used or have been tested in applications (e.g., quartz (2 2 3) for Mn or quartz (5 0 2) for Ni). In backlighter cases where both the w and y lines are promising but with very similar parameters, we only show the match with the higher Bragg angle for clarity and to show the variation in crystal materials that can be used.

A spreadsheet containing all identified combinations can be found in the supplementary material to this article.

Tables IV and V show matching combinations with ϑ ≥ 87^∘, suitable for self-emission imaging. Except for some elements that emit at a low energy, such as Ne or Al, the script identified matching combinations for almost every element. The lowest possible helium-like match is using a mica (0 2 4) crystal with the Si intercombination line at 1.854 keV, at a Bragg angle of ϑ = 88.58^∘. The lowest possible K_α imager is using Ca K_α2 at 3.688 keV and a mica (2 4 1) crystal, at a Bragg angle of ϑ = 88.01^∘. Mica and quartz were found to be the most efficient crystals up to about 13–15 keV. The higher-Z materials Ge, GaAs, and InAs were found to be the most efficient crystals above 15 keV.

Some matches were found for a single crystal material but with multiple combinations of hkl, having different Bragg angles and/or reflectivity R_int. For example, a source emitting the Ge K_α1 line at 9.886 keV can be imaged using a Si crystal with either the (1 1 1) reflection in fifth order or (1 5 7) in first order and permutations thereof. The XOP calculation suggests that R_int is equal for both combinations. In this example, Si (1 1 1) reflects x-ray energies in integer multiples of 1.977 keV, whereas Si (1 5 7) reflects x-ray energies in integer multiples of 9.886 keV. Hence, to avoid both lower- and higher-order reflections contaminating the image, the (1 5 7) reflection would be preferred.

Tables VI and VII show matching combinations with ϑ ≥ 75^∘, suitable for imaging with the source on the Rowland circle. Again the script identified matching combinations for almost every element. In many cases the script identified first-order reflections using high Miller-index combinations. As described above, a first-order reflection is preferred over an otherwise identical higher-order reflection due to potential contamination of the image by lower-order reflections.

Furthermore, for K_α x-ray sources, we only show the K_α1 results since the K_α2 line is spectrally very close and the resulting Bragg angle is very close to the K_α1 result. In most cases, the slight difference in the Bragg angle is compensated by R_int such that R_int/tanϑ is nearly identical. On top of that the K_α1 line has a 50% higher amplitude, rendering a K_α1 imaging system more effective than a K_α2 system.

The lowest-energy backlighter match for an x-ray source with helium-like line emission is using a mica (1 1 2) crystal with the Al intercombination line at 1.588 keV, at a Bragg angle of ϑ = 79.06^∘. The lowest-energy K_α imager is using Mg K_α1 at 1254 keV and a mica (0 0 4) crystal, at a Bragg angle of ϑ = 81.19^∘.

Interestingly, the 6.151-keV backlighter system using a quartz (2 2 3) crystal, which is a standard diagnostic at Sandia’s Z Pulsed Power Facility,³⁸ is only the fourth-most efficient choice for this element. However, the match using a quartz (2 1 4) crystal, while having an almost equal integrated reflectivity as quartz (2 2 3), has a less favorable Bragg angle that results in inferior spatial resolution. Remembering that InAs up to now has not been used for x-ray imaging, the combination using quartz (2 2 3) is actually the most-efficient working combination.

We have used a computer script, coupled to some modules of XOP, to find all possible spectral-line and x-ray-crystal combinations that can be used for self-emission imaging and backlighting using x-ray sources emitting either K_α1,2 spectral lines or the resonance or intercombination lines from He-like ions. The used crystals in this search were restricted to Ge, GaAs, InAs, Si, quartz, and mica because the optical quality of these crystals is near-perfect and images with a high spatial resolution have been demonstrated. It may be possible to use other crystals such as KAP and RAP for high-resolution imaging, however this still needs to be demonstrated and is left for future work.

Out of the more than 9.1 × 10⁶ possible combinations, the computer script identified about 21 000 matching combinations that fulfill all the constraints on the Bragg angle, which defines the spatial resolution, on reflectivity, which defines the image brightness, and on crystal structure, which defines the lattice spacing and allowed reflections.

Tables IV–VII show the results, separated by combinations for self-emission imaging and backlighting systems. The data are sorted for R_int/tan ϑ in descending order for each element. If more than three possible matching combinations are found, only the top three matches are shown for clarity. For cases where the top three matches have low Bragg angles close to 75^∘, more than three matches are shown to avoid that the reader gets the impression that many matches only work for Bragg angles near 75^∘, which is not the case. A spreadsheet containing all identified combinations can be found in the supplementary material to this article.

A general trend of more possible matches for higher x-ray energies could be observed. For example, only two helium-like backlighter matching combinations were found at 1.6 keV (mica (1 1 2) and Al w or y) versus 588 possible matches for He-like Sn.

Higher x-ray energies require higher Miller-index reflections to reduce the lattice spacing d as described in Sec. II C, which can be satisfied by many first-order, high-Miller-index combinations for each crystal, eliminating the need to use higher-order reflections of low-Miller-index cuts that can be difficult to interpret. Recently, the integrated reflectivity of a high-Miller-index Germanium crystal was found to be in a good agreement with the calculations, demonstrating that arbitrary choices of Miller indices in Ge crystals can be used.⁴¹ 

Spectral lines and crystals that are often used in experiments, such as quartz (2 2 3) and Mn He_α, quartz (1 0 1) and Si He_α for backlighting, and quartz (2 0 3) and Ti K_α1 or quartz (4 2 2) and Cu K_α1, were confirmed to be the most efficient (or practical) combinations for these elements.

These results may help to identify new applications for backlighting and self-emission imaging. Using the data presented here, it should be possible to find a suitable imaging combination using an x-ray energy or x-ray source that is specifically selected for a particular purpose, instead of relying on the limited number of existing crystal imaging systems from the literature. For example, a backlighter experiment may be performed using an x-ray energy that results in optimum transmission at the object, leading to the best-possible contrast in the image or at a Bragg angle that supports the required spatial resolution.

See supplementary material for a spreadsheet containing all identified matching spectral-line and crystal combinations, as well as for the calculated 2d-spacings for the crystals in Table II.

We thank Tommy Ao, Ross E. Falcon, Matthias Geissel, Eric C. Harding, John L. Porter, Patrick K. Rambo, Jens Schwarz, Daniel B. Sinars, and Christopher S. Speas at Sandia National Laboratories, and Jeffrey A. Koch at National Security Technologies LLC for the support and helpful discussions. Sandia National Laboratories is a multi-mission laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under Contract No. DE-AC04-94AL85000.