System and method for x-ray absorption spectroscopy using a crystal analyzer and a plurality of detector elements

Abstract

A fluorescence mode x-ray absorption spectroscopy apparatus includes an electron bombardment source of x-rays, a crystal analyzer, the source and the crystal analyzer defining a Rowland circle having a Rowland circle radius (R), a detector, and at least one stage configured to position a sample such that at least a portion of the sample is between the crystal analyzer and the detector.

Claims

1. A fluorescence mode x-ray absorption spectroscopy apparatus comprising: a crystal analyzer; a source of x-rays comprising at least one target configured to generate x-rays upon bombardment by electrons, the source and the crystal analyzer defining a Rowland circle in a tangential plane and having a Rowland circle radius (R), the source having an effective source size less than 100 microns in a direction parallel to the tangential plane and/or having an effective source size in a range of 20 to 4000 microns in a direction perpendicular to the tangential plane; a detector; and at least one stage configured to position a sample such that at least a portion of the sample is between the crystal analyzer and the detector.

2. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 1, wherein the crystal analyzer is positioned on the Rowland circle in the tangential plane, the crystal analyzer comprising crystal planes curved along at least one direction within at least the tangential plane with a radius of curvature substantially equal to twice the Rowland circle radius (2R), the crystal planes configured to receive x-rays and to disperse the received x-rays according to Bragg's law, the detector comprising a spatially resolving detector configured to receive at least a portion of the dispersed x-rays.

3. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 2, wherein the spatially resolving detector comprises a plurality of x-ray detection elements having a tunable first x-ray energy and/or a tunable second x-ray energy, the plurality of x-ray detection elements configured to measure received dispersed x-rays having x-ray energies below the first x-ray energy while suppressing measurements of the received dispersed x-rays above the first x-ray energy and/or to measure the received dispersed x-rays having x-ray energies above the second x-ray energy while suppressing measurements of the received dispersed x-rays below the second x-ray energy, the first and second x-ray energies tunable in a range of 1.5 keV to 30 keV.

4. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 3, wherein the first and second x-ray energies differ from one another by an energy in a range of 50 eV to 5 keV.

5. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 3, wherein at least three x-ray detection elements of the plurality of x-ray detection elements are configured to measure at least some dispersed x-rays along a direction perpendicular to the tangential plane and/or at least three x-ray detection elements of the plurality of x-ray detection elements are configured to measure at least some dispersed x-rays along a direction parallel to the tangential plane.

6. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 3, wherein the at least one stage is further configured to position the crystal analyzer with respect to the source, to align the at least one direction of the crystal planes to the tangential plane, and to position the spatially resolving detector at a distance D, wherein the at least one stage is configured to rotate the crystal analyzer over a predetermined angular range while the relative positions of the source, the crystal analyzer, and the spatially resolving detector are changed to maintain Rowland circle geometry.

7. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 3, wherein each x-ray detection element of the plurality of x-ray detection elements has a linear dimension between 3 microns to 200 microns.

8. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 3, wherein the dispersed x-rays have an energy bandwidth in a range of 2 eV to 250 eV along at least one direction perpendicular to or parallel to the tangential plane.

9. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 1, wherein the detector is positioned at least partially inside the Rowland circle.

10. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 1, wherein the detector comprises an array of detector elements.

11. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 10, wherein the detector elements have an energy resolution better than 3 keV.

12. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 1, wherein the crystal analyzer comprises a cylindrically curved Johansson crystal analyzer.

13. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 1, wherein the Rowland circle radius is in a range of 50 millimeters to 1000 millimeters.

14. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 1, wherein the at least one target comprises one or more x-ray generating target materials selected from the group consisting of: Cr, Fe, Co, Ni, Cu, Mo, Rh, Pd, Ag, Ta, W, and Au.

15. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 14, wherein the at least one target comprises a diamond substrate and the one or more x-ray generating target materials are embedded on or within the diamond substrate.

16. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 1, wherein the source has an effective source size less than 50 microns in the direction parallel to the tangential plane.

17. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 1, further comprising an aperture between the crystal analyzer and the detector.

18. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 17, wherein the aperture is a slit aperture having a width in a range of 3 microns to 1000 microns.

19. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 1, further comprises a beam stop configured to prevent the x-rays from the source that are not diffracted by the crystal analyzer from being received by the detector.

20. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 1, further comprising a computer system configured to analyze signals generated by the detector, to respond to the signals by generating an x-ray spectrum of x-ray flux as a function of x-ray energy, and to convert x-ray spectra obtained with different x-ray energies to a single combined x-ray spectrum.

21. The fluorescence mode x-ray absorption spectroscopy apparatus of claim 1, further comprising a computer system configured to record x-ray absorption spectra measured by the detector and to produce a spectroscopic image of the sample under analysis.

22. A x-ray spectroscopy apparatus comprising: a crystal analyzer; a source of x-rays comprising at least one target configured to generate x-rays upon bombardment by electrons, the source and the crystal analyzer defining a Rowland circle in a tangential plane and having a Rowland circle radius (R)the source having an effective source size less than 100 microns in a direction parallel to the tangential plane and/or having an effective source size in a range of 20 to 4000 microns in a direction perpendicular to the tangential plane; a spatially resolving detector positioned at least partially inside the Rowland circle; and at least one stage configured to position a sample such that at least a portion of the sample is between the crystal analyzer and the detector.

23. The apparatus of claim 22, wherein the spatially resolving detector comprises an array of detector elements having an energy resolution better than 3 keV.

24. The apparatus of claim 22, wherein the spatially resolving detector comprises an array of detector elements, each detection element of the array of detection elements having a linear dimension between 3 microns to 200 microns.

25. The apparatus of claim 22, wherein the crystal analyzer comprises crystal atomic planes curved along at least one direction within at least the tangential plane.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 schematically illustrates two types of crystal analyzers: Johann crystal analyzers (left) and Johansson crystal analyzers (right).

(2) FIG. 2 is a plot of an example calculated energy broadening ΔE as a function of Bragg angle θ.sub.B for 8 keV x-rays with an x-ray source spot size of 400 microns and a Rowland circle diameter (2R) of 500 millimeters.

(3) FIG. 3 is a plot of the x-ray line energy and the radiative line width as a function of atomic number of elements within a sample being analyzed.

(4) FIG. 4 schematically illustrates an example apparatus in accordance with certain implementations described herein.

(5) FIG. 5 schematically illustrates simulation ray tracings of dispersed x-rays downstream from an example cylindrically curved Johansson crystal analyzer in accordance with certain implementations described herein.

(6) FIG. 6 schematically illustrates simulation ray tracings of dispersed x-rays downstream from an example spherically curved Johansson crystal analyzer in accordance with certain implementations described herein.

(7) FIG. 7 schematically illustrates simulation ray tracings of dispersed x-rays downstream from an example spherically curved Johann crystal analyzer in accordance with certain implementations described herein.

(8) FIG. 8 shows simulated x-ray spectra from a tungsten (W) target, a rhodium (Rh) target, and a molybdenum (Mo) target.

(9) FIG. 9 schematically illustrates simulation ray tracings of dispersed x-rays downstream from an example cylindrically curved Johansson crystal analyzer for various targets in accordance with certain implementations described herein.

(10) FIG. 10 schematically illustrates an example apparatus configured for XAS measurements in accordance with certain implementations described herein.

(11) FIG. 11 schematically illustrates another example apparatus configured for XAS measurements in accordance with certain implementations described herein.

(12) FIG. 12 schematically illustrates another example apparatus configured for XAS measurements in accordance with certain implementations described herein.

(13) FIG. 13A schematically illustrates the tangential plane and the sagittal plane of an example apparatus configured to have the sample between the x-ray source and the crystal analyzer in accordance with certain implementations described herein.

(14) FIG. 13B schematically illustrates the tangential plane and the sagittal plane of an example apparatus configured to have the sample between the crystal analyzer and the spatially resolving detector in accordance with certain implementations described herein.

DETAILED DESCRIPTION

(15) Overview

(16) There are several challenges for XAS systems that use a laboratory x-ray source for high quality and high throughput XAS measurement. These challenges are largely rooted in the laboratory x-ray source and existing XAS system designs. The challenges associated with the x-ray source include: low x-ray source brightness that leads to long measurement times, large x-ray source spot size that necessitates crystal analyzers operating at high Bragg angles, large source-to-crystal analyzer distance, or combination thereof, presence of narrow characteristic spectral lines over an extended x-ray energy range which are not useful for XAS measurement, and constraint on the maximum electron acceleration voltage to minimize high order harmonics that could be reflected by the crystal analyzer, reducing production of useful x-rays for XAS measurement.

(17) The problems associated with crystal analyzer operating at high Bragg angles include the use of high Miller index crystal planes of the crystal analyzer with excessively narrow energy bandwidth, spectral contamination due to high order harmonics reflected by higher Miller index crystal planes of the crystal analyzer and/or low harmonics (n being a fractional integer) reflected by the lower Miller index crystal planes of the crystal analyzer, limited x-ray tuning x-ray range per crystal analyzer, and large beam size change on the sample over the x-ray energy measured. The higher and/or lower harmonics of the x-ray energy reduces the signal-to-noise ratio of XAS measurements and can lead to substantial reduction of the XAS measurement quality and throughput. The problems also impose challenges to the x-ray detector used to measure the transmitted x-rays in transmission mode XAS measurement.

(18) Additionally, several problems can hinder accurate x-ray spectrum measurements that include 1) change of the relative position between the x-ray source, the crystal analyzer, and the detector during the measurement; 2) angular stability of the crystal analyzer; and 3) change in the x-ray spectrum of the x-ray source during the measurement.

(19) Certain implementations described herein advantageously circumvent at least some of these problems to provide accurate x-ray spectrum measurements. Additionally, certain implementations described herein achieve higher data collection speed by collecting a large solid angle of x-rays emitted from the x-ray source.

(20) In the 1970s and 1980s, numerous laboratory XAS systems were developed by academic groups in recognition of the capability of x-ray absorption spectroscopy for materials analysis and convenience of laboratory based XAS systems. Rigaku developed several models of commercial laboratory XAS systems but abandoned them some time ago. Most laboratory XAS systems use an electron bombardment laboratory x-ray source, a cylindrically bent crystal designed to operate in a Rowland circle geometry, and a single element (e.g., point) x-ray detector (e.g., such as ionization chamber or proportional counter). XAS spectra are typically generated by scanning the x-ray energy point by point. Energy scans are achieved by rotating the bent crystal while moving both the crystal and a detector along the Rowland circle.

(21) Such conventional laboratory approaches suffered drawbacks that include insufficient energy resolution, poor XAS data quality, long measurement times, stringent sample preparation requirements associated with a large illuminated area, requiring use of multiple crystals for acquisition of a single spectrum, and manufacturing difficulties for standard operation (e.g., challenges associated with moving a heavy rotating anode x-ray source along the Rowland circle and/or moving the sample—which is difficult when the sample of interest is placed in in situ environments). Those drawbacks, in conjunction with increasing availability of synchrotron x-rays source based XAS facilities, led to the waning of laboratory based XAS developments.

(22) FIG. 1 schematically illustrates two types of crystal analyzers: Johann crystal analyzers (left) and Johansson crystal analyzers (right). In earlier systems, the crystals were cylindrically bent: flat in one dimension and curved in the other like a portion of a cylinder. Johann-type and Johansson-type crystals differ in the broadening error. For a Johann-type crystal, the reflection points on the crystal away from the center lie outside of the Rowland circle (see, FIG. 1), so x-rays reflected at these non-central points by the reticular planes of the crystal are focused at different points, causing focusing aberrations (e.g., Johann errors; also referred to as focusing error) arising from the geometry. These focusing aberrations lead to degradation of the energy resolution, which is dependent on the Bragg angle OB and can be expressed by the following expression:

(23) $\begin{array}{cc}\epsilon =\frac{1}{2}{E\left(\frac{l}{4R}\right)}^2{\cot }^2{\theta }_B& \left(2\right)\end{array}$
where l is the crystal size (e.g., width) along the Rowland circle, and R is the Rowland circle diameter. This relation implies that the focusing aberrations increase rapidly with lower Bragg angles and consequently, high energy resolution measurements are to be performed using high Bragg angles (usually greater than 70 degrees) where the cotangent of OB is small. In comparison, a Johansson crystal analyzer does not suffer from the Johann focusing aberrations because the crystal is ground such that its surface matches the Rowland circle and all points of the Johansson-type crystal are coincident on the Rowland circle (see, FIG. 1).

(24) An example laboratory XAS system, developed by a group led by Prof. Jerry Seidler at the University of Washington, is based on a conventional laboratory x-ray source with recently developed commercially available x-ray components, including a spherically bent (e.g., doubly curved, rather than the single curvature of cylindrically bent) Johann crystal analyzer and a silicon drift detector. Large Bragg angles (e.g., greater than or equal to 55 degrees) are used to achieve minimal energy broadening of the x-rays reflected by the crystal analyzer resulting from the Johann focusing error, the energy broadening ΔE given by:

(25) $\begin{array}{cc}\Delta E=E\cot {\theta }_B\Delta w& \left(3\right)\end{array}$
where E is the x-ray energy and Δw is the angular width of the x-ray source as seen by the crystal. The use of high Bragg angles enables high energy resolution (because source broadening from the finite source spot size and Johann errors are minimized).

(26) FIG. 2 is a plot of an example calculated energy broadening ΔE as a function of Bragg angle θ.sub.B for 8 keV x-rays with an x-ray source spot size of 400 microns and a Rowland circle diameter (2R) of 500 millimeters. The angular contribution (Δw) is the x-ray source spot size divided by the distance from the x-ray source to the crystal (which is equivalent to 2.Math.R.Math.sin θ.sub.B). As shown in FIG. 2, high Bragg angles are used for large x-ray sources, with an energy resolution of 2 eV for 8 keV x-ray being achieved at angles greater than or equal to 73 degrees.

(27) At high Bragg angles, high Miller index diffraction planes of high-quality single crystal materials (e.g., Si and Ge) are typically used, which leads to several important drawbacks. High Miller index crystal reflections have very narrow Darwin widths, which means that the energy band pass of the crystal is substantially smaller than the x-ray energy resolution desired for most XAS measurements (e.g., around 0.5 eV for XANES at about 2 keV and 1 eV to 4 eV for higher energy absorption edges of XANES). Therefore, high Miller index crystals act as an excessively narrow energy filter which leads to a large loss of useful source x-rays and thus a waste of source x-rays. Furthermore, undesirable x-rays including high order harmonics can be reflected by the crystal analyzer with higher Miller index crystal planes than lower Miller index planes, resulting lower XAS spectrum quality.

(28) Another major drawback of using high Bragg angles is the use of multiple crystals. The energy change per degree of crystal rotation at high Bragg angles is substantially smaller than at low Bragg angles, resulting in limited energy coverage per crystal analyzer. Many crystal analyzers are therefore utilized to cover an operation energy range. For example, for a Bragg angle of 30 degrees and 8 KeV x-rays, every degree of rotation of the crystal covers about 236 eV, whereas at 85 degrees every degree of rotation of the crystal covers only about 12.4 eV. Hence, for measurements of about 100 eV coverage, the use of lower Bragg angles easily satisfies the covered range but not so at higher Bragg angles. Such a system with a wide energy coverage (e.g., 2 to 20 keV) that can address a large portion of the periodic table of elements would be cumbersome and expensive by use of an impractically large number of crystals. Additionally, the beam size on a sample also changes with x-ray energy substantially faster at higher Bragg angles than at lower Bragg angles, leading to use of a homogeneous sample (e.g., good sample uniformity) or lower XAS spectrum quality with a heterogenous sample.

(29) A key challenge that existed in the 1970s and 1980s laboratory-based x-ray systems was the presence of harmonics that contaminate the signal. Such earlier systems simply ran the x-ray source at an electron accelerating voltage (e.g., kVp) that was less than the energy of the lowest higher order harmonic to avoid contamination of the spectra. Because the amount of bremsstrahlung x-rays produced by electron bombardment is proportional to the accelerating voltage, this reduction of accelerating voltage came at the expense of the x-ray source efficiency. To run the x-ray source at higher efficiency, Seidler's group use silicon drift detectors (SDDs) to circumvent the problem of harmonic contamination to the XAS spectrum, but new problems are introduced. First, the dimension of the active area of the SDDs is typically quite small—around 4 to 12 mm in diameter, which can be much smaller than the beam reflected by the crystal analyzer along the direction perpendicular to the Rowland circle plane and thus leads to long data collection time. Use of SDDs also limits their system design to using a spherically bent Johann crystal analyzer (SBCA) because the size of the detector utilizes point-to-point focusing. Due to Johann broadening error, their system is limited to operation at high Bragg angles. Another problem of SDDs is that the maximum count rate with acceptable linearity is less than 1,000,000 per second, limiting their use to systems with lower count rates.

(30) Certain implementations disclosed herein provide a laboratory XAS system (e.g., apparatus) that circumvents at least some of the problems of the previous laboratory XAS systems described above and that enable laboratory XAS systems with exceptional capabilities.

(31) In the description herein, the plane of the Rowland circle is referred to as the tangential plane, the direction along the tangential plane is referred to as the tangential direction, and the direction perpendicular to the tangential plane is referred to as the sagittal direction.

(32) The x-ray signal, given by the number of x-ray photons per unit time per unit energy bandwidth for a given sample of a certain thickness, can be approximately described by:

(33) $\begin{array}{cc}N\left(\frac{ph}{s}\right)\alpha B*T*{\Omega }_D*{\Omega }_S*R*D*S_D*S_S*\Delta E*M& \left(4\right)\end{array}$
where S.sub.D and S.sub.S are the x-ray generating spot size of the x-ray source (or the size of the detector aperture depending on the specific system design) in the tangential and sagittal directions, respectively, B is the brightness of the x-ray source (which depends on the x-ray generating spot size of the x-ray source S.sub.D and S.sub.S), T is the x-ray transmission through a sample, Ω.sub.D and Ω.sub.S are the collection angles of the crystal analyzer in the tangential plane and the sagittal direction, respectively, R is the reflectivity of the crystal analyzer (which depends upon the Miller index of crystal reflection planes and the choice of material used), D is the detection efficiency of the detector, ΔE is the energy resolution of the system, and M is the number of (energy) spectral modes measured at the same time, which is equal to the energy range simultaneously measured divided by ΔE.

(34) The energy resolution ΔE can be selected to meet a desired energy resolution for the XAS measurements. Several factors affect the energy resolution ΔE, including the geometrical broadening of the x-ray beam (which is determined by the x-ray source spot size and the distance between the x-ray source and the crystal analyzer), the Darwin width of the crystal, the penetration of the x-rays into the crystal, and the finite size of the apertures, as expressed approximately by:

(35) $\begin{array}{cc}\Delta E=E{\cot \left(\theta \right)\left[{\left(\frac{s_S+s_D}{4R\sin \left(\theta \right)}\right)}^2+{\left({\omega }_C\right)}^2+{\left(\frac{2\ln 2\cos \left(\theta \right)}{\mu 4R}\right)}^2+{\left(\frac{{\left(s_S+s_D\right)}^4}{4{\left(16R\right)}^4}{\sec }^2\left(\theta \right)cs{c}^2\left(\theta \right)\right)}^2\right]}^{1/2}& \left(5\right)\end{array}$
where ω.sub.C is the Darwin width of the crystal and μ is the linear absorption coefficient of the sample for a particular x-ray energy. For XAS measurement in the XANES region, the energy resolution ΔE can be less than the radiative line width inherent to the absorption edge due to core hole broadening. For XAS measurement in the EXAFS spectral region of energies higher than the XANES region, the energy resolution ΔE can be higher than the radiative line width and up to 10 eV.

(36) Certain implementations described herein have an energy resolution ΔE that, for a specific XAS measurement, obtains optimal trade-off between energy resolution and measurement speed. If the energy resolution is too coarse, the finer details of the XAS spectra are not obtained. However, if the energy resolution is too fine, a significant penalty is paid in the form of throughput loss by the acquisition of spectra taking too long, rendering the system impractical. Certain implementations described herein provide a judicious choice of x-ray source spot sizes, power loading, crystal choice, and the aperture openings that optimize the tradeoff between throughput and energy resolution.

(37) FIG. 3 is a plot of the x-ray line energy and the radiative line width as a function of atomic number of elements within a sample being analyzed. As shown in FIG. 3, the radiative line width is typically dependent on the energy of an x-ray absorption edge. In certain implementations described herein, the XAS system is configured to have an energy resolution that matches the desired (e.g., required) energy resolution for the appropriate XAS measurements. For example, an energy resolution between 0.2× to 1× of the radiative line width can be selected for XAS measurements in the XANES spectral region based on the specific applications. A coarser energy resolution can be selected for the EXAFS region to efficiently use source x-rays to increase throughput.

EXAMPLE IMPLEMENTATIONS

(38) FIG. 4 schematically illustrates an example apparatus 100 in accordance with certain implementations described herein. The apparatus 100 comprises an x-ray source 110 comprising a target 112 configured to generate x-rays 114 upon bombardment by electrons. The apparatus 100 further comprises a crystal analyzer 120 positioned relative to the x-ray source 110 on a Rowland circle 150 in a tangential plane and having a Rowland circle radius (R). The crystal analyzer 120 comprises crystal atomic planes 122 curved along at least one direction 124 with a radius of curvature substantially equal to twice the Rowland circle radius (2R). The crystal atomic planes 122 are configured to receive x-rays 114 from the x-ray source 110 and to disperse the received x-rays according to Bragg's law (e.g., the dispersed x-rays 126). The apparatus 100 further comprises a spatially resolving detector 130 positioned at a distance (D) from a downstream side 128 of the crystal analyzer 120, with D less than or equal to 2R. The spatially resolving detector 130 is configured to receive at least a portion of the dispersed x-rays 126, the spatially resolving detector 130 comprising a plurality of x-ray detection elements 132 having a tunable first x-ray energy and/or a tunable second x-ray energy. The plurality of x-ray detection elements 132 are configured to measure received dispersed x-rays 126 having x-ray energies below the first x-ray energy while suppressing measurements of the received dispersed x-rays 126 above the first x-ray energy and/or to measure the received dispersed x-rays 126 having x-ray energies above the second x-ray energy while suppressing measurements of the received dispersed x-rays 126 below the second x-ray energy. The first and second x-ray energies are tunable in a range of 1.5 keV to 30 keV.

(39) In certain implementations, the Rowland circle 150 is a circle tangent to a center of the surface of the crystal analyzer 120 impinged by the x-rays 114. The Rowland circle radius (R) in certain implementations is in a range of less than 100 millimeters, in a range of 100 millimeters to 200 millimeters, in a range of 200 millimeters to 300 millimeters, in a range of 300 millimeters to 500 millimeters, or in a range of 500 millimeters to 1000 millimeters.

(40) In certain implementations, the apparatus 100 further comprises at least one stage 140 configured to position the crystal analyzer 120 with respect to the x-ray source 110 on the Rowland circle 150 in the tangential plane with the curved direction 124 of the crystal atomic planes 122 aligned to the tangential plane, and to position the spatially resolving detector 130 at the distance (D) from the downstream side 128 of the crystal analyzer 120. For example, the at least one stage 140 can comprise at least one linear motion stage configured to adjust the position of the crystal analyzer 120 (e.g., along substantially perpendicular x-, y-, and z-directions) and at least one rotational motion stage configured to adjust the orientation of the crystal analyzer 120 (e.g., in substantially perpendicular pitch, yaw, roll angles about principal axes). For another example, the at least one stage 140 can comprise at least one linear motion stage configured to adjust the position of the spatially resolving detector 130 (e.g., along substantially perpendicular x-, y-, and z-directions) and at least one rotational motion stage configured to adjust the orientation of the spatially resolving detector 130 (e.g., in substantially perpendicular pitch, yaw, roll angles about principal axes).

(41) In certain implementations, the apparatus 100 further comprises a sample stage 160 configured to position a sample 162 for analysis either between the x-ray source 110 and the crystal analyzer 120 or between the crystal analyzer 120 and the spatially resolving detector 130. For example, the sample stage 160 can comprise at least one linear motion sub-stage configured to adjust the position of the sample 162 (e.g., along substantially perpendicular x-, y-, and z-directions) and at least one rotational motion sub-stage configured to adjust the orientation of the sample 162 (e.g., in substantially perpendicular pitch, yaw, roll angles about principal axes).

(42) In certain implementations, the crystal analyzer 120 is configured to be operated at low Bragg angles (e.g., in the range of 10 degrees to 60 degrees; in the range of 10 degrees to 40 degrees; in the range of 10 degrees to 30 degrees). For example, the curved crystal atomic planes 122 can comprise crystal atomic planes (e.g., atomic planes of a single crystal material selected from the group consisting of: silicon, germanium, and quartz) having low Miller indices (e.g., Si<111>; Si<220>; Ge<111>; Ge<400>) and that are bent (e.g., mechanically deformed to be curved) along the at least one direction 124 to have a radius of curvature (2R) in a range of 100 millimeters to 2000 millimeters. In certain implementations, the curved, low Miller index crystal atomic planes 122 at low Bragg angles can advantageously relax the sample uniformity constraints because at low Bragg angles, the change in the x-ray beam size on the sample 162 that contributes to the analysis is small as the crystal analyzer 120 is rotated.

(43) In certain implementations, by using such curved, low Miller index crystal atomic planes 122, the crystal analyzer 120 can have an energy resolution that is optimized according to a predetermined spectral region of an x-ray absorption spectroscopy (XAS) measurement to be made. For example, the energy resolution can be selected to be from 0.2 to 1 times a radiative line width of an element to be measured (see, e.g., FIG. 3), the radiative line width due to core hole broadening for the pre-edge spectral region before the peak absorption edge energy and the X-ray Absorption Near-Edge Structure (XANES) region, and can be selected to be from 1 to 5 times of the radiative line width for the EXAFS region. Such energy optimization of the crystal analyzer 120 can enable efficient use of source x-rays 114 and fast data acquisition speeds.

(44) In certain implementations, the curved, low Miller index crystal atomic planes 122 provide a large energy change per degree of rotation of the crystal analyzer 120, thereby enabling the crystal analyzer 120 to cover a large energy range over a given rotation angular range for XAS measurements. In certain such implementations, a small number of crystal analyzers 120 can be used for XAS measurements over a large energy range. For example, just two crystal analyzers 120, one with Ge<111> crystal atomic planes 122 and the other with Ge<200> crystal atomic planes, operating with Bragg angles in a range of 10 degrees to 50 degrees, can be sufficient for XAS measurements over an energy range of 4 keV to 20 keV.

(45) In certain implementations, the source x-ray collection angle (e.g., efficiency) in the tangential plane (e.g., dispersion plane) for the curved crystal atomic planes 122 can be larger than for flat crystal atomic planes, thereby producing a converging (e.g., focused) x-ray beam at the Rowland circle 150 in the tangential plane. For example, with an x-ray beam 126 focused on the Rowland circle 150, a single element detector can be used. In certain implementations, a slit aperture can be used on the Rowland circle 150 and at the upstream side of the detector 130 to improve energy resolution. For example, for sizes of the crystal analyzer 120 of the order of 50 millimeters to 100 millimeters can result in a collection angle of about 0.1 radian to 0.3 radian of a narrow energy bandwidth in the tangential plane, which can be over two orders of magnitude higher than with flat crystals for which the acceptance angle is determined by the Darwin width (e.g., in the range of 10 microradians to 50 microradians). In certain implementations, the crystal atomic planes 122 can also be curved (e.g., bent) in the sagittal direction, thereby increasing the collection angle of the x-rays 126 in the sagittal direction as well.

(46) In certain implementations, the apparatus 100 comprises a cylindrically curved (e.g., bent) Johansson crystal analyzer 120, which can provide a large x-ray collection angle in the tangential direction but a limited x-ray collection angle in the sagittal direction for a given x-ray energy. In certain other implementations, the apparatus 100 comprises a spherically curved (e.g., bent) Johansson crystal analyzer, a spherically curved (e.g., bent) Johann crystal analyzer, a cylindrically curved (e.g., bent) Johann crystal analyzer, or an analyzer with Wittry geometry.

(47) FIG. 5 schematically illustrates simulation ray tracings of dispersed x-rays 126 downstream from an example cylindrically curved Johansson crystal analyzer 120 in accordance with certain implementations described herein. The example cylindrically curved Johansson crystal analyzer 120 of FIG. 5 has a Rowland circle radius R equal to 300 millimeters and a crystal plane bending radius 2R equal to 600 millimeters, and the x-ray source 110 was simulated as a point source emitting x-rays 114 within an angle of ±40 milliradians in the tangential direction and ±48 milliradians in the sagittal direction, and a set of x-ray energies in a range of 8047 eV to 8054 eV, with each band in FIG. 5 corresponding to a 1 eV energy band. The horizontal axis is the distance in the tangential direction parallel to the Rowland circle 150 and the vertical axis is the distance in the sagittal direction. In the leftmost plot, each of the bands corresponds to the surface area of the crystal analyzer 120 that reflects x-rays of one energy. The central band corresponds to x-rays having energies of 8048 eV and the remaining bands correspond to x-rays with energies higher than 8048 eV. Each pairs of bands counted from the center stripe corresponds to an increase of x-ray energy by 1 eV. The other plots of FIG. 5 show example x-ray spectral distributions of the dispersed x-rays 126 at various locations relative to the Rowland circle 150 (e.g., 200 millimeters inside the Rowland circle 150, 50 millimeters inside the Rowland circle 150, 5 millimeters inside the Rowland circle 150, and on the Rowland circle 150).

(48) As seen in FIG. 5, the example cylindrically curved Johansson crystal analyzer 120 can disperse a wide energy range in the sagittal direction with high energy x-ray dispersion. The dispersed x-rays 126 can be measured by the detector 130 with at least some detection elements 132 positioned along the sagittal direction to differentiate the angularly dispersed x-rays 126 along the sagittal direction (e.g., a detector 130 with multiple detector elements 132 along the sagittal direction with sufficient spatial resolution can resolve the spectrum). In this way, the apparatus 100 can measure a spectrum over a finite x-ray energy range with sufficiently high energy resolution. Furthermore, the spatial resolution for measuring the spectrum can be significantly relaxed when the detector 130 is positioned close to the crystal analyzer 120 (e.g., within the Rowland circle 150).

(49) In certain implementations in which some or all of the detection elements 132 have at least one energy threshold (e.g., the tunable first and second x-ray energies) to define an XAS energy bandwidth of interest (e.g., 50 eV to 5 keV), the signal-to-noise ratio of the XAS spectrum can be improved by suppressing measurement of (e.g., rejecting) x-rays 126 either above the XAS energy bandwidth of interest, thereby suppressing (e.g., rejecting) one or more harmonics diffracted by the crystal analyzer 120 and/or by rejecting x-rays 126 with x-ray energies below the XAS energy bandwidth of interest by suppressing (e.g., rejecting) fluorescence x-rays.

(50) In certain implementations, the apparatus 100 comprises a spherically curved (e.g., bent) Johansson crystal analyzer 120, which can provide a large x-ray collection angle in the tangential direction and higher x-ray collection angle in the sagittal direction for a given x-ray energy resolution than can the cylindrically curved Johansson crystal analyzer 120. The spherically curved Johansson crystal analyzer 120 can disperse an x-ray energy range in both the tangential direction and the sagittal direction and the dispersed x-rays 126 can be measured with a detector 130 with a plurality of detection elements 132 configured to measure the angularly dispersed x-rays 126 along the tangential direction, achieving a spectrum over a finite x-ray energy range with high energy resolution.

(51) FIG. 6 schematically illustrates simulation ray tracings of dispersed x-rays 126 downstream from an example spherically curved Johansson crystal analyzer 120 in accordance with certain implementations described herein. The example spherically curved Johansson crystal analyzer 120 of FIG. 6 has a Rowland circle radius R equal to 300 millimeters and a crystal plane bending radius 2R equal to 600 millimeters, and the x-ray source 110 was simulated as a point source emitting x-rays 114 within an angle of ±40 milliradians in the tangential direction and ±48 milliradians in the sagittal direction, and a set of x-ray energies in a range of 8045 eV to 8049 eV, with each band in FIG. 6 corresponding to a 1 eV energy band. The horizontal axis is the distance in the tangential direction parallel to the Rowland circle 150 and the vertical axis is the distance in the sagittal direction. In the leftmost plot, each of the bands corresponds to the surface area of the crystal analyzer 120 that reflects x-rays of one energy. The central band corresponds to x-rays having energies of 8048 eV and the remaining bands correspond to x-rays with energies less than 8048 eV. Each pairs of bands counted from the center stripe corresponds to decrease of x-ray energy by 1 eV. The other plots of FIG. 6 show example x-ray spectral distributions of the dispersed x-rays 126 at various locations relative to the Rowland circle 150 (e.g., 200 millimeters inside the Rowland circle 150, 50 millimeters inside the Rowland circle 150, 5 millimeters inside the Rowland circle 150, and on the Rowland circle 150).

(52) As seen in FIG. 6, the dispersed x-rays 126 can be measured by the detector 130 with at least some detection elements 132 positioned along the sagittal direction to differentiate the angularly dispersed x-rays 126 along the sagittal direction (e.g., a detector 130 with multiple detector elements 132 along the sagittal direction with sufficient spatial resolution can resolve the spectrum). In this way, the apparatus 100 can measure a spectrum over a finite x-ray energy range with sufficiently high energy resolution. Furthermore, the spatial resolution for measuring the spectrum can be significantly relaxed when the detector 130 is positioned close to the crystal analyzer 120 (e.g., within the Rowland circle 150).

(53) In certain implementations, the apparatus 100 comprises a spherically curved (e.g., bent) Johann crystal analyzer 120, which can provide a large x-ray collection angle in the sagittal direction but limited x-ray collection angle in the tangential direction for a given x-ray energy resolution. The spherically curved Johann crystal analyzer 120 can disperse an x-ray energy range in the tangential direction and the dispersed x-rays 126 can be measured with a detector 130 with a plurality of detection elements 132 configured to measure the angularly dispersed x-rays 126 along the tangential direction, achieving a spectrum over a finite x-ray energy range with high energy resolution.

(54) FIG. 7 schematically illustrates simulation ray tracings of dispersed x-rays 126 downstream from an example spherically curved Johann crystal analyzer 120 in accordance with certain implementations described herein. The example spherically curved Johann crystal analyzer 120 of FIG. 7 has a Rowland circle radius R equal to 300 millimeters and a crystal plane bending radius 2R equal to 600 millimeters, and the x-ray source 110 was simulated as a point source emitting x-rays 114 within an angle of ±40 milliradians in the tangential direction and ±48 milliradians in the sagittal direction, and a set of x-ray energies in a range of 8028 eV to 8048 eV, with each band in FIG. 7 corresponding to a 4 eV energy band. The horizontal axis is the distance in the tangential direction parallel to the Rowland circle 150 and the vertical axis is the distance in the sagittal direction. In the leftmost plot, each of the bands corresponds to the surface area of the crystal analyzer 120 that reflects x-rays of one energy. The central band corresponds to x-rays having energies of 8048 eV and the remaining bands correspond to x-rays with energies less than 8048 eV. Each pairs of bands counted from the center stripe corresponds to decrease of x-ray energy by 4 eV. The other plots of FIG. 7 show example x-ray spectral distributions of the dispersed x-rays 126 at various locations relative to the Rowland circle 150 (e.g., 200 millimeters inside the Rowland circle 150, 50 millimeters inside the Rowland circle 150, 5 millimeters inside the Rowland circle 150, and on the Rowland circle 150).

(55) As seen in FIG. 7, the dispersed x-rays 126 can be measured by the detector 130 with at least some detection elements 132 positioned along the sagittal direction to differentiate the angularly dispersed x-rays 126 along the sagittal direction (e.g., a detector 130 with multiple detector elements 132 along the sagittal direction with sufficient spatial resolution can resolve the spectrum). In this way, the apparatus 100 can measure a spectrum over a finite x-ray energy range with sufficiently high energy resolution. Furthermore, the spatial resolution for measuring the spectrum can be significantly relaxed when the detector 130 is positioned close to the crystal analyzer 120 (e.g., within the Rowland circle 150).

(56) In certain implementations, the apparatus 100 comprises a toroidally curved (e.g., bent) Johansson crystal analyzer 120, which like the spherically curved Johansson crystal analyzer 120, can provide a large x-ray collection angle in the tangential direction but higher x-ray collection angle than the cylindrically curved Johansson crystal analyzer 120 in the sagittal direction for a given x-ray energy resolution. The toroidally curved Johansson crystal analyzer 120 can disperse the x-rays in both the tangential direction and the sagittal direction, and the dispersed x-rays 126 can be measured with a detector 130 with a plurality of detection elements 132 configured to measure the angularly dispersed x-rays 126 along the tangential direction, achieving a spectrum over a finite x-ray energy range with high energy resolution.

(57) In certain implementations, the x-ray source 110 comprises a high efficiency (e.g., high brightness) x-ray source comprising an electron source and at least one anode target 112 (e.g., having a size on the order of microns) configured to generate x-rays upon being bombarded by electrons from the electron source. The target 112 can be on or embedded in a thermally conductive substrate (e.g., comprising diamond) configured to dissipate thermal energy from the target 112 that is generated by the electron bombardment of the target 112. Examples of target 112 materials include but are not limited to, Cu, Cr, Fe, Co, Ni, Zn, Al, Rh, Mo, Pd, Ag, Ta, Au, W, SiC, MgCl, or other metals or metal-containing materials. Examples of x-ray sources 110 compatible with certain implementations described herein are disclosed by U.S. Pat. Nos. 10,658,145, 9,874,531, 9,823,203, 9,719,947, 9,594,036, 9,570,265, 9,543,109, 9,449,781, 9,448,190, and 9,390,881, each of which is incorporated in its entirety by reference herein.

(58) In certain implementations, the x-ray source 110 can comprise a plurality of targets 112 having different materials configured to provide a continuous (e.g., smooth) x-ray energy spectrum over an extended x-ray energy range for XAS measurements. Certain such implementations can overcome a limitation of single target material x-ray sources 110 in which the spectrum of the resultant x-rays 114 from a single target 112 includes characteristic lines over an extended energy range, and these characteristic lines are not suitable for XAS measurements. For example, FIG. 8 shows simulated x-ray spectra from a tungsten (W) target 112, a rhodium (Rh) target 112, and a molybdenum (Mo) target 112. The presence of the characteristic lines of the W target 112 between 7.5 keV and 13 keV makes this spectral region from the W target 112 unsuitable for XAS measurements. In certain implementations, an x-ray source 110 comprising a W target 112 and one or both of a Rh target 112 and a Mo target 112 can provide a combined continuous (e.g., smooth) energy spectrum over a range of 1.5 keV to 20 keV by using the W target 112 for the ranges of 1.5 keV to 7.5 keV and 13 keV to 20 keV and using the Rh target 112 and/or the Mo target 112 for the range of 7.5 keV to 14 keV. While W is more efficient for x-ray production, which is proportional to the atomic number Z of the target material and the electron accelerating voltage, the characteristic lines in the x-ray spectrum from the W target 112 can contaminate the spectrum between 7.5 keV and 14 keV, so the Rh and/or Mo targets 112 can be used instead for this energy range.

(59) In certain implementations, the size and shape of the target 112 are selected to optimize performance depending on the parameters and characteristics of the crystal analyzer 120. For example, for a cylindrically curved Johansson crystal analyzer 120, the x-ray source 110 can comprise a rectangular (e.g., line-shaped) target 112 having a first dimension (e.g., width; in a range of 3 microns to 100 microns) that is substantially aligned along the tangential direction and a second dimension (e.g., length; in a range of 10 microns to 4 millimeters) that is substantially aligned along the sagittal direction. The ratio of the second dimension to the first dimension for obtaining a given fractional energy resolution ΔE/E due to the size of the target 112 can be approximately equal to (ΔE/E).sup.−½.Math.cot(θ), where B is the Bragg angle. For another example, for a spherically curved Johann crystal analyzer 120 or for a spherically curved Johansson crystal analyzer 120, the ratio of the second dimension to the first dimension for obtaining an given fractional energy resolution ΔE/E due to the size of the target 112 can be approximately equal to (ΔE/E).sup.−½.Math.cot(θ).Math.sin(θ). The x-ray source spot size (e.g., the size of the electron beam spot bombarding the target 112) is the “apparent” source size when viewed at the take-off angle such that the electron footprint on the target 112 is compressed along one axis (e.g., the apparent source spot size is one-tenth the size of the electron footprint at a take-off angle of 6 degrees).

(60) FIG. 9 schematically illustrates simulation ray tracings of dispersed x-rays 126 downstream from an example cylindrically curved Johansson crystal analyzer 120 for various targets 112 in accordance with certain implementations described herein. The example cylindrically curved Johansson crystal analyzer 120 of FIG. 9 has a Rowland circle radius R equal to 300 millimeters and a crystal plane bending radius 2R equal to 600 millimeters, and the x-ray source 110 was simulated as emitting x-rays 114 within an angle of ±40 milliradians in the tangential direction and ±48 milliradians in the sagittal direction, and a set of x-ray energies in a range of 8048 eV to 8054 eV, with each band in FIG. 9 corresponding to a 1 eV energy band. For the targets 112 of FIG. 9, the first dimension substantially along the tangential direction is 20 microns and one-half of the second dimension along the sagittal direction is varied among 0 (e.g., point source), 0.2 millimeter, 1 millimeter, and 2 millimeters. The horizontal axis is the distance in the tangential direction parallel to the Rowland circle 150 and the vertical axis is the distance in the sagittal direction. In the topmost plot for the point source, each of the bands corresponds to the surface area of the crystal analyzer 120 that reflects x-rays of one energy. The central band corresponds to x-rays having energies of 8048 eV and the remaining bands correspond to x-rays with energies higher than 8048 eV. Each pairs of bands counted from the center stripe corresponds to an increase of x-ray energy by 1 eV. The other topmost plots for the other sizes of the x-ray source spot size of the target 112, the bands are less well-defined but are distinguishable even with the largest simulated x-ray source spot size, indicating that energy resolution of about 1 eV can still be achieved. The other plots in FIG. 9 show the x-ray spectral distribution at two different locations relative to the Rowland circle 150 (e.g., 5 millimeters inside the Rowland circle 150 and on the Rowland circle 150).

(61) As seen in FIG. 9, the second dimension of the target 112 substantially aligned with the sagittal direction can be much greater than the first dimension of the target 112 substantially aligned with the tangential direction for the cylindrically curved Johansson crystal analyzer 120. The dispersed x-rays 126 can be measured by the detector 130 with at least some detection elements 132 positioned along the sagittal direction to differentiate the angularly dispersed x-rays 126 along the sagittal direction (e.g., a detector 130 with multiple detector elements 132 along the sagittal direction with sufficient spatial resolution can resolve the spectrum). FIG. 9 shows that the x-ray source spot size of the target 112 along the sagittal direction can be increased to increase the x-ray source power and to increase throughput while keeping the energy resolution to a desired value (e.g., 1 eV). For example, a high resolution, two-dimensional detector 130 positioned at the Rowland circle 150 can be used as the energy bands are more clearly separated. For a detector 130 with a high spatial resolution, an x-ray source 110 with an extended spot size and high power can be used.

(62) In certain implementations, the x-ray source 110 can comprise at least one grazing incidence mirror configured to substantially reflect the x-rays 114 from the target 112 in the sagittal plane. For example, the target 114 can emit a beam of x-rays 114 that has an angular range that is larger than the acceptance of the crystal analyzer 120. A pair of grazing incidence plane mirrors can be placed (e.g., one at each side of the x-ray beam) to reflect portions of the x-ray beam that would miss the crystal analyzer 120. The at least one grazing incidence mirror can be considered as at least one virtual source of x-rays positioned close to the Rowland circle 150 and that can be used to monitor the background simultaneously with the central spectrum acquisition.

(63) In certain implementations, the spatially resolving detector 130 is selected from the group consisting of: a pixel array photon counting detector, a direct conversion charge coupled device (CCD) detector (e.g., configured to operate in single photon detection mode), a complementary metal-oxide-semiconductor (CMOS) detector (e.g., configured to operate in single photon detection mode), and a plurality of silicon drift detectors (e.g., placed in close proximity to one another or integrated with one another). For example, the spatially resolving detector 130 can comprise a one-dimensional (1D) position sensitive detector (e.g., strip detector) or a two-dimensional (2D) position sensitive detector. In certain implementations, a two-dimensional (2D) spatially resolving detector 130 having sufficient spatial resolution can be used with a large x-ray source spot size to achieve sufficiently high energy resolution. While FIG. 4 schematically illustrates an example implementation in which the spatially resolving detector 130 is at least partially inside the Rowland circle 150, in certain other implementations, the spatially resolving detector 130 is at least partially on the Rowland circle 150 or at least partially outside the Rowland circle 150.

(64) In certain implementations, the detection elements 132 (e.g., pixels; complete detectors) of the spatially resolving detector 130 have spatial dimensions in at least one dimension configured to provide a predetermined energy resolution (e.g., in a range of 0.2 eV to 3 eV) and a predetermined energy range (e.g., in a range of 3 eV to 200 eV). For example, the detection elements 132 can each have a size along the tangential direction in a range of 3 microns to 200 microns and a size along the sagittal direction in a range of 3 microns to 5000 microns. In certain implementations, the detection elements 132 are spatially separated from one another such that each detection element 132 corresponds to less than 3 eV (e.g., in a range of 0.5 eV to 3 eV) of the beam spread of the dispersed x-rays 126 at the detector 130 (e.g., for XANES measurements) or can correspond to less than 10 eV (e.g., in a range of 1 eV to 10 eV) of the beam spread of the dispersed x-rays 126 at the detector 130 (e.g., for EXAFS measurements). For example, to achieve high throughput, the apparatus 100 is configured to utilize an x-ray spectrum that has a breadth covered by the detector 130 while simultaneously having a number M of spectral modes (e.g., energy bands) measured at the same time by the detector 130, where M is given by the spectral coverage at the detector 130 divided by the energy bandwidth per detection element 132. For example, for a spectral coverage of 200 eV at the detector 130 and an energy bandwidth of 2 eV per detection element 132, M is equal to 100. In certain implementations, the detection elements 132 have a linearity at a count rate exceeding 10.sup.6 photons per second (most silicon drift detectors are linear only to 0.5 million photons per second).

(65) In certain implementations, the spatially resolving detector 130 is configured to measure the distribution of the dispersed x-rays 126 from the crystal analyzer 120. For example, for a cylindrically curved Johansson crystal analyzer 120, the detection elements 132 can be configured to measure angularly dispersed x-rays 126 along the sagittal direction. For another example, for a spherically curved Johann crystal analyzer 120, the detection elements 132 can be configured to measure angularly dispersed x-rays 126 along the tangential direction. For another example, for a spherically curved Johann crystal analyzer 120, at least some of the detection elements 132 can be configured to measure angularly dispersed x-rays 126 along the tangential direction. In this example, the size of each detector element 132 can be small, with the detection element size in the tangential direction (D1) expressed by: 2.Math.R.Math.sin(θ).Math.ΔE/E≥cot(θ).Math.D1 and the size along the sagittal direction (S2) of the x-ray source spot on the target 112 expressed by: 2.Math.R.Math.sin(θ).Math.(ΔE/E).sup.1/2≥S2, where R is the Rowland circle radius and B is the Bragg angle. In certain implementations, the center of the detector 130 is positioned on the Rowland circle 150. The size along the sagittal direction (D2) of the detection elements 132 can be smaller than S2.

(66) In certain implementations, the detection elements 132 have at least one tunable energy threshold (e.g., selected by the user of the apparatus 100 or automatically by a computer-based controller) to suppress (e.g., reject) x-rays outside the energy range of interest and that degrade the XAS measurements. For example, higher order harmonics can generate a background signal contribution of 10% in the measured spectrum, which can reduce the throughput by about 3×. In certain implementations, the detection elements 132 are configured to measure x-rays 126 having x-ray energies below a tunable first x-ray energy while suppressing measurements of x-rays 126 above the tunable first x-ray energy. In this way, the detection elements 132 can suppress (e.g., reject) higher order harmonics diffracted by the crystal analyzer 120, thereby improving the quality of the measured XAS spectrum. Another benefit of suppressing measurements of higher energy x-rays is that the x-ray source 110 can be operated at higher accelerating voltages for higher x-ray flux and throughput. For another example, low energy x-rays (e.g., reflected x-rays from the crystal analyzer 120; fluorescence x-rays from the sample 162) that satisfy Bragg's law can be received by the detector 130 and can contribute to the background signal. In certain implementations, the detection elements 132 are configured to measure x-rays 126 having x-ray energies above a tunable second x-ray energy while suppressing measurements of x-rays 126 below the tunable second x-ray energy. In this way, the detection elements 132 can suppress (e.g., reject) the low energy x-rays contributing to the background signal and to provide high quality XAS spectra. In certain implementations, the detection elements 132 have both a tunable first x-ray energy and a tunable second x-ray energy.

(67) In certain implementations, the detector 130 comprises an aperture between the crystal analyzer 120 and the detection elements 132. For example, the aperture can be configured to have an adjustable size (e.g., by a user of the apparatus 100; by a computer-based controller) to controllably adjust an energy resolution of the detector 130. For another example, the aperture can be structured (e.g., have a pattern of openings) such that the shape of the x-rays 126 received by the detector 130 ensure centering of components on the Rowland circle 150. In certain implementations, in which the detection elements 132 have large dimensions and the size in the tangential direction of the x-ray source spot on the target 112 is large, a small aperture in front of the detection elements 132 can be used to achieve small detector dimensions.

(68) In certain implementations, the apparatus 100 is used to measure transmission mode XAS spectra in an improved manner as compared to conventional XAS systems. For example, using a conventional XAS system, a complete transmission mode XAS measurement can be performed by scanning the angle of the crystal analyzer over an angular range to cover the x-ray energy range desired for the XAS measurement (e.g., 50-100 eV for XANES measurements; 300-1000 eV for EXAFS measurements). In contrast, in accordance with certain implementations described herein, XAS measurements can be made by simultaneously collecting multiple narrow spectra (e.g., finite energy ranges each narrower than the desired energy range for the XAS measurement) with energy ranges that partially overlap with one another. These multiple spectra can be normalized and combined together appropriately to form the full XAS measurement. In certain implementations, collecting the multiple spectra with overlapping energy range can be used to minimize x-ray source intensity fluctuation between two spectra collected with overlapping energy range.

(69) For another example, the apparatus 100 can be used to provide EXAFS spectra with higher resolution than conventional EXAFS systems. In accordance with certain implementations described herein, an XAS spectrum can be collected in the pre-edge region and in the XANES spectral region with energy resolution equal to or better than the radiative line width (e.g., by selecting the material of the crystal analyzer 120 and the Miller indices of the crystal atomic planes 112 for higher energy resolution) and in the full EXAFS spectral region (e.g., in the spectral region away from the absorption edge) with coarser energy resolution (e.g., 3-10 eV) but higher x-ray flux. The spectral region of the full EXAFS by the spectrum can be replaced by the pre-edge and XANES regions, with appropriate intensity normalization and stitching to generate a spectrum.

(70) FIG. 10 schematically illustrates an example apparatus 100 configured for XAS measurements in accordance with certain implementations described herein. In certain such implementations, the apparatus 100 comprises an x-ray source 110 comprising a target 112 having a small x-ray source spot size in the tangential direction, a cylindrically curved Johansson crystal analyzer 120, and a spatially resolving detector 130 with a plurality of detection elements 132 that are spatially resolving in at least one of the tangential direction and the sagittal direction. The target 112 of the x-ray source 110 can have a linear shape (e.g., a size along the tangential direction in a range of 3 microns to 50 microns and a size along the sagittal direction in a range of 20 microns to 4000 microns), and the x-ray source 110 can comprise multiple target materials. The cylindrically curved Johansson crystal analyzer 120 can comprise a single crystal of Si, Ge, or quartz, can be used at low Bragg angles (e.g., in a range of 15 degrees to 40 degrees), can have a Rowland circle radius R in a range of 50 millimeters to 1000 millimeters, and can have a crystal size less than or equal to 100 millimeters in both the tangential direction and the sagittal direction. The spatially resolving detector 130 can be one-dimensional (1D) or two-dimensional (2D) and comprising at least some detection elements 132 along the sagittal direction. The detection elements 132 can have at least one energy threshold to suppress (e.g., reject) x-rays having energies above or below the XAS measurement x-ray energy range.

(71) FIG. 10 also schematically illustrates simulation ray tracings of dispersed x-rays 126 downstream from a cylindrically curved Johansson crystal analyzer 120 for a spatially resolving detector 130 at two different positions: (a) on the Rowland circle 150 for a point source target 112 and for an elliptical source target 112, and (b) 50 millimeters inside the Rowland circle 150. The cylindrically curved Johansson crystal analyzer 120 is that of FIG. 5 (having a Rowland circle radius R equal to 300 millimeters and a crystal plane bending radius 2R equal to 600 millimeters), and a set of x-ray energies in a range of 8047 eV to 8054 eV, with each band in FIG. 10 corresponding to a 1 eV energy band and the central band corresponding to x-rays having energies of 8048 eV. The horizontal axis in the three plots of FIG. 10 is the distance in the tangential direction parallel to the Rowland circle 150, the vertical axis is the distance in the sagittal direction, and the dimensions are in millimeters. As seen in FIG. 10, the x-ray energy spread is along the sagittal direction, which is the long dimension of the spatially resolving detector 130. Sub-eV resolution is achieved by using detection elements 132 of sufficiently small size. For an elliptical source, the spatially resolving detector 130 positioned at the Rowland circle 150 can be two-dimensional (2D), and the spatially resolving detector 130 positioned within the Rowland circle 150 can be one-dimensional (1D).

(72) In certain implementations, the apparatus 100 of FIG. 10 comprising the cylindrically curved Johansson crystal analyzer 120 is configured to achieve a given energy resolving power of (ΔE/E).sup.−1 by meeting the following conditions: 2.Math.R.Math.sin(θ).Math.ΔE/E≥cot(θ).Math.S1 and 2.Math.R.Math.sin(θ).Math.(ΔE/E).sup.1/2≥S2, where R is the Rowland circle radius, θ is the Bragg angle, and S1 and S2 are the x-ray source spot sizes of the target 112 along the tangential direction and the sagittal direction, respectively. The width of the cylindrically curved Johansson crystal analyzer 120 along the sagittal direction can be equal to N.Math.S2, and the cylindrically curved Johansson crystal analyzer 120 can be configured to receive and disperse x-rays 126 along the sagittal direction over an x-ray energy range approximately equal to N times of the energy resolution ΔE (e.g., N in a range of 2 to 100). The number of detection elements 132 along the sagittal direction can be equal to or greater than N. The spatially resolving detector 130 can be positioned downstream from the crystal analyzer 120 (e.g., at a distance D from a downstream side of the crystal analyzer 120, with the distance D in a range less than 2R (e.g., twice the Rowland circle radius R). For the spatially resolving detector 130 positioned at the Rowland circle 150, the size of the detection elements 132 in the tangential plane can be comparable (e.g., for better signal-to-noise ratio) or can be larger than S1 (with potentially lower signal-to-noise ratios due to stray x-rays being measured). For detection elements 132 larger than S1, the spatially resolving detector 130 can comprise a slit aperture at an upstream side of the spatially resolving detector 130 (e.g., between the crystal analyzer 120 and the detection elements 132) to suppress (e.g., reject; reduce) the number of stray x-rays being measured.

(73) FIG. 11 schematically illustrates another example apparatus 100 configured for XAS measurements in accordance with certain implementations described herein. In certain such implementations, the apparatus 100 comprises an x-ray source 110 comprising a target 112 having a large x-ray spot size in the tangential direction, a cylindrically curved Johansson crystal analyzer 120, and a spatially resolving detector 130 with a plurality of detection elements 132 (e.g., 2 to 2000) that are spatially resolving in at least the sagittal direction. The target 112 of the x-ray source 110 can have a linear shape (e.g., a size along the tangential direction in a range of 3 microns to 50 microns and a size along the sagittal direction in a range of 20 microns to 4000 microns), and the x-ray source 110 can comprise multiple target materials. The cylindrically curved Johansson crystal analyzer 120 can comprise a single crystal of Si, Ge, or quartz, can be used at low Bragg angles (e.g., in a range of 15 degrees to 40 degrees), can have a Rowland circle radius R in a range of 50 millimeters to 1000 millimeters, and can have a crystal size less than or equal to 100 millimeters in both the tangential direction and the sagittal direction. The spatially resolving detector 130 can be one-dimensional (1D) or two-dimensional (2D) and comprising at least some detection elements 132 along the sagittal direction. The detection elements 132 can have at least one energy threshold to suppress (e.g., reject) x-rays having energies above or below the XAS measurement x-ray energy range.

(74) In certain implementations, the apparatus 100 of FIG. 11 comprising the cylindrically curved Johansson crystal analyzer 120 is configured to achieve a given energy resolving power of (ΔE/E).sup.−1 by meeting the following conditions: 2.Math.R.Math.sin(θ)ΔE/E≥cot(θ).Math.D1 and 2.Math.R.Math.sin(θ).Math.(ΔE/E).sup.1/2≥S2, where R is the Rowland circle radius, B is the Bragg angle, D1 is the size of the detection elements 132 along the tangential direction, and S2 is the size along the sagittal direction of the x-ray source spot on the target 112. The width of the cylindrically curved Johansson crystal analyzer 120 along the sagittal direction can be equal to N.Math.S2, and the cylindrically curved Johansson crystal analyzer 120 can be configured to receive and disperse x-rays 126 along the sagittal direction over an x-ray energy range approximately equal to N times of the energy resolution ΔE (e.g., Nin a range of 2 to 100). The number of detection elements 132 along the sagittal direction can be equal to or greater than N. The spatially resolving detector 130 can be positioned downstream from the crystal analyzer 120 (e.g., at a distance D from a downstream side of the crystal analyzer 120, with the distance D in a range less than 2R (e.g., twice the Rowland circle radius R). For the spatially resolving detector 130 with its center positioned at the Rowland circle 150, the size D2 of the detection elements 132 along the sagittal direction can be smaller than S2. For detection elements 132 with D1 larger than 2.Math.R.Math.sin(θ).Math.ΔE/E/cot(θ).Math., the spatially resolving detector 130 can comprise a slit aperture at an upstream side of the spatially resolving detector 130 (e.g., between the crystal analyzer 120 and the detection elements 132) and having a long dimension of the slit aperture aligned along the sagittal direction to suppress (e.g., reject; reduce) the number of stray x-rays being measured.

(75) In certain implementations, the apparatus 100 comprises an x-ray source 110 comprising a target 112 having a small x-ray spot size in the tangential direction, a spherically curved Johansson crystal analyzer 120, and a spatially resolving detector 130 with a plurality of detection elements 132 (e.g., 2 to 2000) that are spatially resolving in at least the sagittal direction. The target 112 of the x-ray source 110 can have a size along the tangential direction in a range of 3 microns to 50 microns and a size along the sagittal direction in a range of 20 microns to 1000 microns, and the x-ray source 110 can comprise multiple target materials. The spherically curved Johansson crystal analyzer 120 can comprise a single crystal of Si, Ge, or quartz, can have a Rowland circle radius R in a range of 50 millimeters to 1000 millimeters, and can have a crystal size less than or equal to 100 millimeters in both the tangential direction and the sagittal direction. The spatially resolving detector 130 can be one-dimensional (1D) or two-dimensional (2D) and comprising at least some detection elements 132 along the sagittal direction. The detection elements 132 can have at least one energy threshold to suppress (e.g., reject) x-rays having energies above or below the XAS measurement x-ray energy range.

(76) In certain implementations, the apparatus 100 comprising the spherically curved Johansson crystal analyzer 120 is configured to achieve a given energy resolving power of (ΔE/E).sup.−1 by meeting the following conditions: 2.Math.R.Math.sin(θ).Math.ΔE/E≤cot(θ).Math.S1 and 2.Math.R.Math.sin.sup.2(θ).Math.(ΔE/E).sup.1/2≥S2, where R is the Rowland circle radius, θ is the Bragg angle, and S1 and S2 are the sizes of the x-ray source spot on the target 112 along the tangential direction and the sagittal direction, respectively. The spatially resolving detector 130 can be positioned downstream from the crystal analyzer 120 (e.g., at a distance D from a downstream side of the crystal analyzer 120, with the distance D in a range less than 2R (e.g., twice the Rowland circle radius R). The detection elements 132 can be configured to measure dispersed x-rays 126 received by the spatially resolving detector 130 along the tangential direction. In certain implementations, the spatially resolving detector 130 is a one-dimensional (1D) position sensitive detector positioned close to the crystal analyzer 120 (e.g., within the Rowland circle 150) with the detection elements 132 aligned along the tangential direction, while in certain other implementations, the spatially resolving detector 130 is a two-dimensional (2D) position sensitive detector positioned close to the Rowland circle 150 (e.g., on the Rowland circle 150).

(77) FIG. 12 schematically illustrates another example apparatus 100 configured for XAS measurements in accordance with certain implementations described herein. In certain such implementations, the apparatus 100 comprises an x-ray source 110, a spherically curved Johann crystal analyzer 120, and a spatially resolving detector 130 with a plurality of detection elements 132. FIG. 12 also schematically illustrates simulation ray tracings of dispersed x-rays 126 downstream from the spherically curved Johann crystal analyzer 120 for a spatially resolving detector 130 at two different positions: (a) on the Rowland circle 150 and (b) 200 millimeters inside the Rowland circle 150. The spherically curved Johann crystal analyzer 120 has a Rowland circle radius R equal to 300 millimeters and a crystal plane bending radius 2R equal to 600 millimeters. The set of x-ray energies of the bands in FIG. 12 are in a range of 8028 eV to 8052 eV, with each band in FIG. 12 corresponding to a 1 eV energy band and the central band corresponding to x-rays having energies of 8048 eV. For the spatially resolving detector 130 positioned on the Rowland circle 150, the dispersion of the range of x-ray energies is only over about 400 microns and spectral blurring can occur that degrades the energy resolution of the measured XAS spectra. For the spatially resolving detector 130 positioned 200 millimeters inside the Rowland circle, the corresponding x-ray energy dispersion stretches over a few millimeters and can be resolved using a pixelated detector. Energy dispersion is achieved in certain implementations by placing the spatially resolving detector 130 sufficiently far from the focal point on the Rowland circle 150 (e.g., either inside the Rowland circle 150 or outside the Rowland circle 150).

(78) In certain implementations, the x-ray energy illuminating a sample 162 can vary depending on the location of the sample 162. FIG. 13A schematically illustrates the tangential plane and the sagittal plane of an example apparatus 100 configured to have the sample 162 between the x-ray source 110 and the crystal analyzer 120 in accordance with certain implementations described herein. FIG. 13B schematically illustrates the tangential plane and the sagittal plane of an example apparatus 100 configured to have the sample 162 between the crystal analyzer 120 and the spatially resolving detector 130 in accordance with certain implementations described herein. For the example apparatus 100 of FIGS. 13A and 13B, the detection elements 132 of the spatially resolving detector 130 can measure x-ray absorption corresponding to a respective region of the sample 162. In certain implementations, by scanning the x-ray energy over a sufficiently large range, a spectral image of the sample 162 can be obtained (e.g., x-ray imaging spectroscopy).

(79) Although commonly used terms are used to describe the systems and methods of certain implementations for ease of understanding, these terms are used herein to have their broadest reasonable interpretations. Although various aspects of the disclosure are described with regard to illustrative examples and implementations, the disclosed examples and implementations should not be construed as limiting. Conditional language, such as “can,” “could,” “might,” or “may,” unless specifically stated otherwise, or otherwise understood within the context as used, is generally intended to convey that certain implementations include, while other implementations do not include, certain features, elements, and/or steps. Thus, such conditional language is not generally intended to imply that features, elements, and/or steps are in any way required for one or more implementations. In particular, the terms “comprises” and “comprising” should be interpreted as referring to elements, components, or steps in a non-exclusive manner, indicating that the referenced elements, components, or steps may be present, or utilized, or combined with other elements, components, or steps that are not expressly referenced.

(80) Conjunctive language such as the phrase “at least one of X, Y, and Z,” unless specifically stated otherwise, is to be understood within the context used in general to convey that an item, term, etc. may be either X, Y, or Z. Thus, such conjunctive language is not generally intended to imply that certain implementations require the presence of at least one of X, at least one of Y, and at least one of Z.

(81) Language of degree, as used herein, such as the terms “approximately,” “about,” “generally,” and “substantially,” represent a value, amount, or characteristic close to the stated value, amount, or characteristic that still performs a desired function or achieves a desired result. For example, the terms “approximately,” “about,” “generally,” and “substantially” may refer to an amount that is within ±10% of, within ±5% of, within ±2% of, within ±1% of, or within ±0.1% of the stated amount. As another example, the terms “generally parallel” and “substantially parallel” refer to a value, amount, or characteristic that departs from exactly parallel by ±10 degrees, by ±5 degrees, by ±2 degrees, by ±1 degree, or by ±0.1 degree, and the terms “generally perpendicular” and “substantially perpendicular” refer to a value, amount, or characteristic that departs from exactly perpendicular by ±10 degrees, by ±5 degrees, by ±2 degrees, by ±1 degree, or by ±0.1 degree. The ranges disclosed herein also encompass any and all overlap, sub-ranges, and combinations thereof. Language such as “up to,” “at least,” “greater than,” “less than,” “between,” and the like includes the number recited. As used herein, the meaning of “a,” “an,” and “said” includes plural reference unless the context clearly dictates otherwise. While the structures and/or methods are discussed herein in terms of elements labeled by ordinal adjectives (e.g., first, second, etc.), the ordinal adjectives are used merely as labels to distinguish one element from another, and the ordinal adjectives are not used to denote an order of these elements or of their use.

(82) Various configurations have been described above. It is to be appreciated that the implementations disclosed herein are not mutually exclusive and may be combined with one another in various arrangements. Although this invention has been described with reference to these specific configurations, the descriptions are intended to be illustrative of the invention and are not intended to be limiting. Various modifications and applications may occur to those skilled in the art without departing from the true spirit and scope of the invention. Thus, for example, in any method or process disclosed herein, the acts or operations making up the method/process may be performed in any suitable sequence and are not necessarily limited to any particular disclosed sequence. Features or elements from various implementations and examples discussed above may be combined with one another to produce alternative configurations compatible with implementations disclosed herein. Various aspects and advantages of the implementations have been described where appropriate. It is to be understood that not necessarily all such aspects or advantages may be achieved in accordance with any particular implementation. Thus, for example, it should be recognized that the various implementations may be carried out in a manner that achieves or optimizes one advantage or group of advantages as taught herein without necessarily achieving other aspects or advantages as may be taught or suggested herein.