X-ray Compton spectroscopy is one of the few direct probes of the electron momentum distribution of bulk materials in ambient and operando environments. We report high-resolution inelastic x-ray scattering experiments with high momentum and energy transfer performed at a storage-ring-based high-energy x-ray light source facility using an x-ray transition-edge sensor (TES) microcalorimeter detector. The performance was compared with a silicon drift detector (SDD), an energy-resolving semiconductor detector, and Compton profiles were measured for lithium and cobalt oxide powders relevant to lithium-ion battery research. Spectroscopic analysis of the measured Compton profiles demonstrates the high-sensitivity to the low-Z elements and oxidation states. The line shape analysis of the measured Compton profiles in comparison with computed Hartree–Fock profiles is usually limited by the resolution of the semiconductor detector. We have characterized an x-ray TES microcalorimeter detector for high-resolution Compton scattering experiments using a bending magnet source at the Advanced Photon Source with a double crystal monochromator, providing monochromatic photon energies near 27.5 keV. The momentum resolution below 0.16 atomic units (a.u.) was measured, yielding an improvement of more than a factor of 7 over a state-of-the-art SDD for the same scattering geometry. Furthermore, the lineshapes of narrow valence and broad core electron profiles of sealed lithium metal were clearly resolved using an x-ray TES compared to smeared and broadened lineshapes observed when using the SDD. High-resolution Compton scattering using the energy-resolving area detector shown here presents new opportunities for spatial imaging of electron momentum distributions for a wide class of materials with applications ranging from electrochemistry to condensed matter physics.

X-ray Compton scattering is a powerful technique for directly probing ground state electron momentum distributions for a wide class of materials under a variety of experimental conditions. The Compton technique offers several immediate advantages over other momentum (k)-resolved spectroscopies: high x-ray energy allows for deep penetration into encased samples, and the probe is bulk-sensitive but insensitive to crystal defects and surface effects.1 These characteristics allow for studies of many complex electronic structure material systems. Compton scattering spectroscopy has been a valuable tool for studying charge transfer2 and extracting redox orbitals3 in Li battery cathode materials as well as the orbital character of dopants4 and bulk Fermi surfaces5–8 in a broad class of condensed matter systems. The inelastic scattering experiments in the Compton limit are typically performed with a combination of a strong synchrotron x-ray source and precise energy analysis of the scattered photons either directly with energy resolving detectors or indirectly with diffraction techniques using analyzer crystals. High-resolution in momentum space along with high-energy and high-flux is necessary for the full realization of the capabilities of the Compton spectroscopic technique. In the Compton regime, within the limits of the impulse approximation, the measured energy spectrum is directly proportional to the Compton profile: J(pz) = ∬ρ(p)dpxdpy, where ρ(p) is the electron momentum density with scattering vector aligned in the direction of the pz-axis. Thus, by energy-analyzing the Compton scattered radiation using high-resolution detectors, it is possible to directly access information about the electronic ground state of the scattering system. Specifically, low-Z elements, like Li, are typically hard to detect with x-ray fluorescence (XRF) measurements since their k-edges are in the soft or tender x-ray regime. The fluorescence signal of lighter elements in bulk samples and/or in complex sample environments cannot be measured. On the other hand, the Compton signal is relatively strong from low-Z elements and only slightly lower than the incident photon energy, enabling non-destructive light element imaging even in battery cells. There have been successful efforts reported from SPring-8 for quantification of Li for in-situ and operando lithium batteries using Ge detectors; the high energy photons up to 100 keV allow deep penetration.9,10 Furthermore, the high-resolution Compton spectrum collected using a Cauchois type crystal spectrometer with resolution ∼0.14 a.u. was successfully used to study electronic states in cathode materials of Li batteries during lithiation and delithiation.11,12

Commercial Ge detectors are used for this application due to their high quantum efficiency at high x-ray energy (>30 keV). However, the use of such detectors is often limited due to their low energy (momentum) resolution, typically ∼500 eV (0.55 a.u.) at 100 keV. The energy resolution of semiconductor (Ge and Si) detectors is typically limited by charge generation statistics within the detector, which is given by ΔE2.35ϵFE, where ϵ is the pair creation energy, F is the Fano factor, and E is the photon energy.13 For imaging purposes, Compton profiles with relatively low statistics are sufficient for the imaging of the characteristic elements of interest. The Compton line shape analysis can distinguish between elements fairly easily, but it still requires a sufficient energy resolution to identify the slope changes in the Compton profile. After the advent of strong synchrotron sources, high-resolution crystal analyzers with limited angular acceptance became more efficient, and the collection of Compton profiles in reasonable beamtime was made possible. Therefore, they provide a solution to the energy resolution limitation of the Ge detectors. However, their use requires rigid geometry with precision crystal-alignment, as well as the necessary components need to be placed and maintained on the Rowland circle that is defined by the analyzer crystal bent to a radius equal to the diameter of the Rowland circle. There are mainly two types of high-resolution Compton spectrometers used at the synchrotron facilities using crystal analyzers: the Cauchois type14,15 and the Johann scanning type.16 In the Cauchois type, all of the components are stationary on the Rowland circle as opposed to the scanning type, and the entire energy spectrum is recorded simultaneously as a function of position on a position sensitive detector. The scanning type spectrometers further require rotations and translations of the analyzer and the detector to scan the energy of the scattered photons. For both types, geometrical constraints further add complexity and limitations in the experiments, such as for multimodal imaging, where a detector needs to be placed in a transverse geometry to simultaneously obtain the momentum-space images via Compton scattering and the structural information via x-ray diffraction. For the linearly polarized x-ray beam, there is strong anisotropy in the azimuthal angular distribution of the Compton-scattered x-ray intensity at the scattering angle of 90°. This anisotropic nature of the scattered radiation imposes further constraints to place a Compton detector for applications where substantially higher counts are needed. At high energies near 100 keV, the bent crystal focusing and the bending of thick crystals necessary for sufficient counts also become challenging. Thus, it would be advantageous to have an energy dispersive detector that provides high-energy resolution comparable to the crystal analyzers, high signal to background ratio, and broadband energy spectrum capability without the need for intermediate dispersive crystal analyzer components on a rigid circle geometry.

For calorimetric x-ray sensor designs, the energy resolution is given by ΔE2.35kBT2C, where kB is the Boltzmann constant, T is the operating temperature and C is the heat capacity. The resolution of designs optimized for hard x rays can be made much better than the Fano limit for Si or Ge. X-ray microcalorimeters are composed of three main components: an absorber that converts the x-ray photon energy into thermal energy, a thermometer that measures the temperature rise after absorption of the photon, and a weak thermal link that controls the heat flow from the device to the thermal bath. Although early progress was reported for Compton scattering measurements using a low temperature detector based on a silicon thermistor,17 good energy resolution and scalability needed for high counting efficiency remains a challenge. One particular optimization of x-ray TES microcalorimeter arrays offers energy resolution of ∼50 eV at 100 keV and 250-pixel array count rates up to 2.5 kcps, providing an order of magnitude improvement over Ge detectors.18 Compared to crystal analyzers, the high-resolution TES arrays have the advantage of placing the detector at scattering angles of 90°, which allows multimodal measurements: transverse Compton scattering and high-energy x-ray diffraction using forward scattering. High-resolution with broadband spectrum capability using TES arrays further enables efficient studies of Fermi surface signatures via different crystal orientations while fixed detector positions. A 100-pixel array with count rates of ∼10 cps/pixel offers efficient high-resolution Compton-profile measurements. In this work, we demonstrate high-resolution Compton spectroscopy using hard x-ray TESs fabricated and tested at Argonne National Laboratory (ANL) and analyze the sensitivity of the Compton profiles for detecting low-Z elements, particularly lithium. Furthermore, we evaluate these sensors for line shape analysis, which is necessary for studying orbitals related to fundamental redox processes in lithium batteries.

A 3D physical layout of a TES microcalorimeter for a completed device is shown in Fig. 1(a). The sensor [Fig. 1(b)] is composed of a square (125 μm)2 proximity-coupled superconducting Mo/Cu thin-film bilayer with a “sidecar” normal-metal Au absorber with an area (750 μm)2. The bilayers were deposited using optimal sputter deposition parameters so that the thin-film internal stress in Mo is minimized from a few-GPa tensile stress to slightly compressive stress (<100 MPa). The resultant bilayer films have a good superconducting state with a sharp transition temperature of 90 mK and a normal state sheet resistance of 8 mΩ/□. The x-ray absorber and thermometer were deposited on a 0.5 μm-thick suspended silicon nitride (SiNx) membrane, which was released by a bulk silicon micromachining method. The heat bath was cooled below 50 mK using an adiabatic demagnetization refrigerator. We have tested sensors with a 1 μm thick sputtered Au absorber, a subset of pixels from a 100-pixel array, and measured energy resolution below 20 eV for x-ray energies up to 20 keV. The measured thermal conductance, G, and heat capacity, C, were ∼550 pW/K and ∼4 pJ/K, respectively, close to the target values. The measurements were consistent with those reported for a different 24-pixel array design.19 As the Compton spectrum in momentum space is not sensitive to small shifts in the absolute energy scale and the linear range of this TES design has been confirmed by a separate set of XRF measurements to cover the range measured, the energy calibration of each TES array pixel was fixed by a linear fit to the elastic peak position based on the configured beamline monochromatic source energy. The consistency of this approach was confirmed by the good agreement of peak positions when co-adding the spectra of all pixels together. In Fig. 1(c), a high-energy TES design with electroplated Au and Bi absorber fabricated at ANL is shown. The heat capacity expected for this design is 8.7 pJ/K, which increases the saturation energy (EsatC/α, where α ∝ 1/transition-width) near the Compton peak for a 100 keV incident beam.

FIG. 1.

(a) 3D view of the completed device showing color coded layer stackup for a TES pixel (not-to-scale). Optical micrographs of the fabricated sensor chips showing (b) a 100-pixel array with ∼1 μm thick sputtered Au absorber and (c) scalable high-energy TES pixels with an electroplated metal stack of ∼2 μm thick Au and ∼22 μm thick Bi absorbers incorporated in the TES fabrication process. (d) Prototype device characterization of high-energy TES design (absorber: 2 μm thick Au) using a lab x-ray tube source showing measured Mo Kα line and corresponding fit for estimating detector energy resolution.

FIG. 1.

(a) 3D view of the completed device showing color coded layer stackup for a TES pixel (not-to-scale). Optical micrographs of the fabricated sensor chips showing (b) a 100-pixel array with ∼1 μm thick sputtered Au absorber and (c) scalable high-energy TES pixels with an electroplated metal stack of ∼2 μm thick Au and ∼22 μm thick Bi absorbers incorporated in the TES fabrication process. (d) Prototype device characterization of high-energy TES design (absorber: 2 μm thick Au) using a lab x-ray tube source showing measured Mo Kα line and corresponding fit for estimating detector energy resolution.

Close modal

In Fig. 2(a), the Compton scattering geometry at the 1-BM beamline for high-resolution inelastic scattering experiments using the TES instrument is presented; a 100-pixel sensor chip was packaged and installed at the cold stage. The sample under study was set at 45° with respect to the incident beam at 27.5 keV. The size of the x-ray beam interacting with the sample was reduced to 5 × 3 mm2 (H × V) by closing incident beam slits. Scattered x-ray photons were counted and energy-resolved by two Compton detectors at two different scattering angles: the TES detector at 90° (perpendicular geometry) and SDD at 140° (backscatter geometry). The two geometries allow direct comparison of the data from TES and SDD for scattering geometries favorable for imaging and line shape analysis, respectively. Typically, the line shape and width of the Compton profile are directly related to the electron orbitals, whether the electrons are in the valence or the core shells. However, further contributions to the broadening of the profile can be associated with instrumental and geometric conditions: Δpz=pzE1ΔE12+pzE2ΔE22+pzϕΔϕ2, where E1 and E2 are the incident and scattered photon energies, respectively, and ϕ is a scattering angle. Thus, it is important to minimize different contributions adversely broadening the Compton spectrum. The source broadening (ΔE1) is the energy spread of the incident beam, which is roughly proportional to the Darwin width of the Si (111) monochromator crystal. For the 1-BM experiments using a bending magnet source, the energy spread of the incident x-ray beam was ΔE1/E1 ∼ 1.5 × 10−4 with a typical flux >1010 photons/s for energies below 30 keV. Aluminum slits and brass pinhole collimators were used to limit the broadening due to the spread in scattering angle (Δϕ) while maximizing the number of photons collected for the TES and SDD, respectively. In Table I, estimates for the total experimental momentum resolution for the scattering measurements are listed. To prevent oxidation induced broadening of the profile, air sensitive samples, particularly lithium, were carefully sealed in an Ar-filled glovebox with <0.1 ppm oxygen and <0.5 ppm water in a polyethylene pouch. Containers were chosen to ensure low incident and scattered beam attenuation and minimal background contribution. Lithium metal enables evaluation of the high-energy TES microcalorimeters due to its inherently narrow Compton profile.

FIG. 2.

Characterization of x-ray TES microcalorimeters for lithium detection and high-resolution line shape analysis. (a) Schematic of an experimental setup at the 1-BM-C beamline showing the TES instrument fixed with respect to the beam and Hitachi Vortex SDD with a pin-hole collimator. The TES is positioned in the correct geometry for three-dimensional Compton imaging. The TES array is mounted at the cold stage typically held at 50 mK. The scattered photons, in a narrow solid angle defined by a slit, pass through several x-ray windows before getting detected by the sensor. Each pixel measures the Compton signal. (b) Raw counts of scattered photon energy resolved using TES for Li metal and Li2O. (c) Normalized-to-wings (|pz| = 2 to 6 a.u.). Compton spectrum of Li metal and Li2O measured with TES showing high-sensitivity of detecting Li concentration. Inset: narrow elastic line monitors the instrument resolution. (d) Normalized-to-peak Compton spectrum of Li metal measured with TES showing a clear non-Gaussian line shape corresponding to the superposition of valence and core electron profiles, while SDD poor detector resolution caused broadening and smearing of the line shape.

FIG. 2.

Characterization of x-ray TES microcalorimeters for lithium detection and high-resolution line shape analysis. (a) Schematic of an experimental setup at the 1-BM-C beamline showing the TES instrument fixed with respect to the beam and Hitachi Vortex SDD with a pin-hole collimator. The TES is positioned in the correct geometry for three-dimensional Compton imaging. The TES array is mounted at the cold stage typically held at 50 mK. The scattered photons, in a narrow solid angle defined by a slit, pass through several x-ray windows before getting detected by the sensor. Each pixel measures the Compton signal. (b) Raw counts of scattered photon energy resolved using TES for Li metal and Li2O. (c) Normalized-to-wings (|pz| = 2 to 6 a.u.). Compton spectrum of Li metal and Li2O measured with TES showing high-sensitivity of detecting Li concentration. Inset: narrow elastic line monitors the instrument resolution. (d) Normalized-to-peak Compton spectrum of Li metal measured with TES showing a clear non-Gaussian line shape corresponding to the superposition of valence and core electron profiles, while SDD poor detector resolution caused broadening and smearing of the line shape.

Close modal
TABLE I.

Estimated experimental momentum resolution of detectors used: TES (ϕ = 90°) and SDD (ϕ = 140°).

Detectordpz/ (a.u./rad)dpz/dE2 (a.u./keV)dpz/dE1 (a.u./keV)Δpz (a.u.)
TES 5.1 3.81 3.43 0.18 
SDD 2.41 3.03 2.53 0.92 
Detectordpz/ (a.u./rad)dpz/dE2 (a.u./keV)dpz/dE1 (a.u./keV)Δpz (a.u.)
TES 5.1 3.81 3.43 0.18 
SDD 2.41 3.03 2.53 0.92 

Compton profiles were measured for lithium metal, lithium oxide, lithium cobalt(III) oxide, cobalt(II, III) oxide, and cobalt(II) oxide powders (Li, Li2O, LiCoO2, Co3O4, CoO), relevant to lithium-ion battery research. Figure 2(b) shows the raw energy spectra from Li metal and Li2O measured with the TES. The measured energy spectra were then converted to the electron momentum scale using the relation:1pzmc=E2E1+(E2E1/mc2)(1cosϕ)E12+E222E1E2cosϕ, where m is the electron mass and c is the speed of light. The area under the Compton profile subjects to the normalization rule: +J(pz)dpz=Z. Figures 2(c) and 3(a) show measured Compton profiles of the battery materials using TES and SDD, respectively. As outlined by the S-parameter approach,9 all of the measured profiles were then normalized to the wings (|pz| = 2 to 6 a.u.) to identify the sharpness of the distributions. The amplitudes of the resulting Compton profiles clearly show high detection sensitivity, particularly for low-Z elements like lithium where the electron momentum distribution is narrowly concentrated around pz = 0 (primarily, |pz| = 0 to 2 a.u. for Li). Compton profiles with broad distributions, with a small change in amplitudes, were also observed for the two stable oxidation states of cobalt(III) and cobalt(II, III) oxide.

FIG. 3.

SDD-measured and computed Compton profiles of the battery materials. (a) The normalized Compton profiles for Li, Li2O, LiCoO2, Co3O4, and CoO showing high-sensitivity to the low-Z elements for which the electron momentum distribution is narrow and focused around pz = 0. (b) Comparison of the measured (solid) Compton profiles for the data shown in (a) and computed (dashed) Compton profiles using HF theory.20 Oxidation states of Li and Co were used for the computed profiles with the charge transfer to O (2p) orbitals.

FIG. 3.

SDD-measured and computed Compton profiles of the battery materials. (a) The normalized Compton profiles for Li, Li2O, LiCoO2, Co3O4, and CoO showing high-sensitivity to the low-Z elements for which the electron momentum distribution is narrow and focused around pz = 0. (b) Comparison of the measured (solid) Compton profiles for the data shown in (a) and computed (dashed) Compton profiles using HF theory.20 Oxidation states of Li and Co were used for the computed profiles with the charge transfer to O (2p) orbitals.

Close modal

In Fig. 3(b), the comparison of SDD-measured and theoretical Compton profiles shows good agreement with Hartree–Fock (HF) theory.20 We obtained HF profiles for each compound using Biggs’s table, e.g., JLi_2O: 2JLi[1s2] + JO[1s2] + JO[2s2] + JO[2p6], where the total Compton profile for Li2O was obtained by the weighted sum of orbital Compton profiles as listed in Biggs’s table. It should be noted that the electron in Li(2s) orbital was counted as an electron of the O(2p) orbital. However, simultaneous measurements with the TES array show that the widths, amplitudes, and lineshapes of the Compton profiles collected with the SDD are limited by the detector resolution, hindering the ability to collect profiles that accurately display the electronic structures intrinsic to the materials and also reducing the element-resolving Compton contrast for imaging.

As shown in Fig. 2(c), the Compton profile line shape from lithium metal was clearly resolved by the TES detector, showing a clear slope change between narrow valence and broad core electron profiles. An approximation using the free-electron gas model for Li metal follows: J2s(pz)=3(pF2pz2)/4pF3 for pzpF and 0 for pz>pF, where pF is the Fermi momentum. The measured line shape shows the expected inverted parabola with two inflection points on both sides of the Compton profile. This indicates a cross-over where the dominant contribution to the Compton profile shifts from valence electrons to core electrons. A valence electron profile with an inverted-parabola shape was clearly observed using the TES microcalorimeter due to its high energy resolution. High resolution also enabled separating the elastic peak from the Compton peak, which would otherwise be difficult when using SDD for a perpendicular geometry with a relatively low incident beam energy of 27.5 keV.

For the elastic peak, the measured FWHM momentum (energy) resolution with SDD was 0.91 a.u. (∼300 eV) for backscattering geometry, which corresponds to 1.15 a.u. for the perpendicular geometry. For the SDD, in addition to charge generation statistics (ϵ ∼ 3.63 eV, F ∼ 0.135, ΔE ∼ 273 eV),21 other components including charge carrier collection and electrical noise affect the overall resolution via quadrature sum. For the TESs, the measured resolution was below 0.16 a.u., which is comparable to using a crystal analyzer for the same scattering geometry and a factor of 7 better than when using SDD. An additional factor of 2 improvements is within reach of the sensor and readout capability along with a further reduction in Δϕ.

In Fig. 2(d), smearing and Gaussian broadening of the line shape are visually demonstrated for the SDD, but a clearly resolved line shape is evident when using TESs. Due to detector resolution convolution, the FWHM of the measured lithium metal Compton profile with the SDD was ∼1.8 a.u. compared to 1.2 a.u., which is close to the intrinsic width, measured with TES, which is comparable to the directional profiles collected using crystal analyzers.8 In Fig. 4(a), the total Compton profile measured with TES is presented for Li, which is in close agreement with the resolution-broadened theoretical HF profile with correlation corrections.22 The discrepancy could be attributed to the multiple scattering events and residual oxidation in Li. In Fig. 4(b), a comparison is presented for normalized Compton profiles for Li2O measured with TES, computed HF neutral atom, and HF total orbitals with charge sharing between lithium (2s) and oxygen (2p) orbitals based on their oxidation state. Excellent agreement was found between the measured profile using TES and the HF theory incorporating charge sharing.

FIG. 4.

(a) Measured and theoretical22 total Compton profiles for Li metal. The theoretical profile was normalized and convoluted with the TES experimental resolution. pF indicates the Fermi momentum for Li metal. (b) Comparison of the Compton profile measured with TES and the computed weighted sum of atomic profiles of lithium and oxygen using HF theory20 for Li2O with and without charge sharing in outer orbitals.

FIG. 4.

(a) Measured and theoretical22 total Compton profiles for Li metal. The theoretical profile was normalized and convoluted with the TES experimental resolution. pF indicates the Fermi momentum for Li metal. (b) Comparison of the Compton profile measured with TES and the computed weighted sum of atomic profiles of lithium and oxygen using HF theory20 for Li2O with and without charge sharing in outer orbitals.

Close modal

In further developments of the sensor design with high Esat, initial characterization shows measured energy resolution below 30 eV for the Mo Kα (17.5 keV) line [Fig. 1(d)]; the expected benchmark performance has recently been reported for high-energy designs with C up to 7 pJ/K obtained with evaporated Au.23 Thus, a high-energy TES instrument at a high-energy (>100 keV) beamline will be able to serve as a microscopic probe to measure redox orbitals-induced changes in operando batteries and potentially as an imaging tool after scaling up array size of the TES sensor and cold readout circuits. Hence, it would open the door to new experiments not only for Compton imaging of light elements but also for high-resolution x-ray spectroscopies for the entire energy range of 20–100 keV, including monitoring physical phenomena, such as metal to insulator transition via spherically averaged Compton signal,24,25 and Fermi surface structures in bulk materials.7,12 After the APS upgrade, experiments will further benefit from highly focused beams with flux and brilliance necessary to increase counts in a narrow solid angle needed for high-resolution measurements.

In summary, we have demonstrated the use of the x-ray TES as a Compton detector for high-resolution inelastic scattering experiments. The x-ray sensors were evaluated for energies up to 27.5 keV for a perpendicular scattering geometry required for three-dimensional imaging and a backscattering geometry favorable for high-resolution line shape analysis. The x-ray TES improves the momentum resolution by more than a factor of 7, yielding an improvement in measuring Compton profiles even at low energies and challenging scattering geometries. The high-resolution Compton spectroscopy using TESs showed high-sensitivity to the low-Z elements in battery materials, and the measured lineshapes were in good agreement with the computed HF profiles. We further demonstrated smearing of the line shape for the measured Compton profile of lithium metal when using SDD, but clear narrow lineshapes of valence and core electron profiles were observed when using TES. The momentum resolution is anticipated to further improve for high-energy sensors at high-energy scattering beamlines. The Compton statistics are also expected to improve as our TES pixels and readout packaging grow in array size, enabling highly efficient scattering experiments.

This work was supported by the Laboratory Directed Research and Development Program (2021-0059) at Argonne National Laboratory. This research used resources of the Advanced Photon Source and Center for Nanoscale Materials, U.S. Department of Energy, Office of Science User Facilities operated for the DOE Office of Science by the Argonne National Laboratory under Contract No. DE-AC02-06CH11357. This work made use of the Pritzker Nanofabrication Facility of the Institute for Molecular Engineering at the University of Chicago, which receives support from Soft and Hybrid Nanotechnology Experimental Resource (Grant No. NSF ECCS-2025633), a node of the National Science Foundation’s National Nanotechnology Coordinated Infrastructure. The authors would like to thank M. Wojcik for assistance with the beamline setup, R. Divan for advice on Bi electroplating, T. Cecil and P. Duda for discussions on the DRIE method, and members of the Quantum Sensors Group, NIST (Boulder, CO USA) for discussions on TES fabrication and cold readout circuits.

The authors have no conflicts to disclose.

U. Patel: Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (lead). T. Guruswamy: Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). A. J. Krzysko: Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). H. Charalambous: Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). L. Gades: Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). K. Wiaderek: Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). O. Quaranta: Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Y. Ren: Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). A. Yakovenko: Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). U. Ruett: Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). A. Miceli: Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
M.
Cooper
,
P.
Mijnarends
,
N.
Shiotani
,
N.
Sakai
, and
A.
Bansil
,
X-Ray Compton Scattering
(
Oxford University Press
,
Oxford
,
2004
), Vol. 366, pp.
134
139
.
2.
S.
Chabaud
,
C.
Bellin
,
F.
Mauri
,
G.
Loupias
,
S.
Rabii
,
L.
Croguennec
,
C.
Pouillerie
,
C.
Delmas
, and
T.
Buslaps
,
J. Phys. Chem. Solids
65
,
241
(
2004
).
3.
K.
Suzuki
,
B.
Barbiellini
,
Y.
Orikasa
,
N.
Go
,
H.
Sakurai
,
S.
Kaprzyk
,
M.
Itou
,
K.
Yamamoto
,
Y.
Uchimoto
,
Y. J.
Wang
,
H.
Hafiz
,
A.
Bansil
, and
Y.
Sakurai
,
Phys. Rev. Lett.
114
,
087401
(
2015
).
4.
Y.
Sakurai
,
M.
Itou
,
B.
Barbiellini
,
P. E.
Mijnarends
,
R. S.
Markiewicz
,
S.
Kaprzyk
,
J.-M.
Gillet
,
S.
Wakimoto
,
M.
Fujita
,
S.
Basak
,
Y. J.
Wang
,
W.
Al-Sawai
,
H.
Lin
,
A.
Bansil
, and
K.
Yamada
,
Science
332
,
698
(
2011
).
5.
S. B.
Dugdale
,
R. J.
Watts
,
J.
Laverock
,
Zs.
Major
,
M. A.
Alam
,
M.
Samsel-Czekala
,
G.
Kontrym-Sznajd
,
Y.
Sakurai
,
M.
Itou
, and
D.
Fort
,
Phys. Rev. Lett.
96
,
046406
(
2006
).
6.
N.
Hiraoka
,
Y.
Yang
,
T.
Hagiya
,
A.
Niozu
,
K.
Matsuda
,
S.
Huotari
,
M.
Holzmann
, and
D. M.
Ceperley
,
Phys. Rev. B
101
,
165124
(
2020
).
7.
W.
Al-Sawai
,
B.
Barbiellini
,
Y.
Sakurai
,
M.
Itou
,
P. E.
Mijnarends
,
R. S.
Markiewicz
,
S.
Kaprzyk
,
S.
Wakimoto
,
M.
Fujita
,
S.
Basak
,
H.
Lin
,
Y. J.
Wang
,
S. W. H.
Eijt
,
H.
Schut
,
K.
Yamada
, and
A.
Bansil
,
Phys. Rev. B
85
,
115109
(
2012
).
8.
Y.
Sakurai
,
Y.
Tanaka
,
A.
Bansil
,
S.
Kaprzyk
,
A. T.
Stewart
,
Y.
Nagashima
,
T.
Hyodo
,
S.
Nanao
,
H.
Kawata
, and
N.
Shiotani
,
Phys. Rev. Lett.
74
,
2252
(
1995
).
9.
K.
Suzuki
,
B.
Barbiellini
,
Y.
Orikasa
,
S.
Kaprzyk
,
M.
Itou
,
K.
Yamamoto
,
Y. J.
Wang
,
H.
Hafiz
,
Y.
Uchimoto
,
A.
Bansil
,
Y.
Sakurai
, and
H.
Sakurai
,
J. Appl. Phys.
119
,
025103
(
2016
).
10.
K.
Suzuki
,
A.-P.
Honkanen
,
N.
Tsuji
,
K.
Jalkanen
,
J.
Koskinen
,
H.
Morimoto
,
D.
Hiramoto
,
A.
Terasaka
,
H.
Hafiz
,
Y.
Sakurai
,
M.
Kanninen
,
S.
Huotari
,
A.
Bansil
,
H.
Sakurai
, and
B.
Barbiellini
,
Condens. Matter
4
,
66
(
2019
).
11.
H.
Hafiz
,
K.
Suzuki
,
B.
Barbiellini
,
N.
Tsuji
,
N.
Yabuuchi
,
K.
Yamamoto
,
Y.
Orikasa
,
Y.
Uchimoto
,
Y.
Sakurai
,
H.
Sakurai
,
A.
Bansil
, and
V.
Viswanathan
,
Nature
594
,
213
(
2021
).
12.
H.
Hafiz
,
K.
Suzuki
,
B.
Barbiellini
,
Y.
Orikasa
,
V.
Callewaert
,
S.
Kaprzyk
,
M.
Itou
,
K.
Yamamoto
,
R.
Yamada
,
Y.
Uchimoto
,
Y.
Sakurai
,
H.
Sakurai
, and
A.
Bansil
,
Sci. Adv.
3
,
e1700971
(
2017
).
13.
G. F.
Knoll
,
Radiation Detection and Measurement
(
John Wiley
,
2010
), Vol. 427, p.
476
.
14.
G.
Loupias
,
J.
Petiau
,
A.
Issolah
, and
M.
Schneider
,
Phys. Status Solidi B
102
,
79
(
1980
).
15.
N.
Hiraoka
,
M.
Itou
,
T.
Ohata
,
M.
Mizumaki
,
Y.
Sakurai
, and
N.
Sakai
,
J. Synchrotron Radiat.
8
,
26
(
2001
).
16.
P.
Suortti
,
T.
Buslaps
,
P.
Fajardo
,
V.
Honkimäki
,
M.
Kretzschmer
,
U.
Lienert
,
J. E.
McCarthy
,
M.
Renier
,
A.
Shukla
,
T.
Tschentscher
, and
T.
Meinander
,
J. Synchrotron Radiat.
6
,
69
(
1999
).
17.
C. K.
Stahle
,
D.
Osheroff
,
R. L.
Kelley
,
S.
Harvey Moseley
, and
A. E.
Szymkowiak
,
Nucl. Instrum. Methods Phys. Res., Sect. A
319
,
393
(
1992
).
18.
J. N.
Ullom
and
D. A.
Bennett
,
Supercond. Sci. Technol.
28
,
084003
(
2015
).
19.
T.
Guruswamy
,
L.
Gades
,
A.
Miceli
,
U.
Patel
, and
O.
Quaranta
,
IEEE Trans. Appl. Supercond.
31
,
2101605
(
2021
).
20.
F.
Biggs
,
L. B.
Mendelsohn
, and
J. B.
Mann
,
At. Data Nucl. Data Tables
16
,
201
(
1975
).
21.
F.
Perotti
and
C.
Fiorini
,
Nucl. Instrum. Methods Phys. Res., Sect. A
423
,
356
(
1999
).
22.
R.
Dovesi
,
E.
Ferrero
,
C.
Pisani
, and
C.
Roetti
,
Z. Phys. B
51
,
195
(
1983
).
23.
D.
Yan
,
J. C.
Weber
,
K. M.
Morgan
,
A. L.
Wessels
,
D. A.
Bennett
,
C. G.
Pappas
,
J. A.
Mates
,
J. D.
Gard
,
D. T.
Becker
,
J. W.
Fowler
,
D. S.
Swetz
,
D. R.
Schmidt
,
J. N.
Ullom
,
T.
Okumura
,
T.
Isobe
,
T.
Azuma
,
S.
Yamada
,
S.
Okada
,
T.
Hashimoto
,
N.
Paul
,
G.
Bian
, and
P.
Indelicato
,
IEEE Trans. Appl. Supercond.
31
,
2100505
(
2021
).
24.
K. O.
Ruotsalainen
,
J.
Inkinen
,
T.
Pylkkänen
,
T.
Buslaps
,
M.
Hakala
,
K.
Hämäläinen
, and
S.
Huotari
,
Eur. Phys. J. B
91
,
225
(
2018
).
25.
I.
Kylänpää
,
Y.
Luo
,
O.
Heinonen
,
R.
Paul
,
C.
Kent
, and
J. T.
Krogel
,
Phys. Rev. B
99
,
075154
(
2019
).