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DOE OSTI2026

Epitaxial Electrodeposition of Fe with Controlled In-Plane Variants for a Reversible Metal Anode in an Aqueous Electrolyte

The development of reversible metal anodes is a key challenge for advancing aqueous battery technologies, particularly for scalable and safe stationary energy storage applications. Here, in this study, we demonstrate a strategy to realize epitaxial electrodeposition of iron (Fe) on single-crystal copper (Cu) substrates in aqueous electrolytes. We compare the electrodeposition behavior of Fe on polycrystalline and single-crystalline Cu substrates, revealing that the latter enables highly uniform, dense, and crystallographically aligned Fe growth. Comprehensive electron backscatter diffraction (EBSD) and X-ray diffraction (XRD) analysis confirms the formation of Fe with specific out-of-plane and in-plane orientations, including well-defined rotational variants. Our findings highlight that epitaxial electrodeposition of Fe can suppress dendritic growth and significantly enhance Coulombic efficiency during plating/stripping cycles. This approach bridges fundamental crystallography with practical electrochemical performance, providing a pathway toward high-efficiency aqueous batteries utilizing Earth-abundant materials.

aqueous battery↗

DOE OSTI2021

Microscopic piezoelectric behavior of clamped and membrane (001) PMN-30PT thin films

Bulk single-crystal relaxor-ferroelectrics, like Pb(Mg 1/3 Nb 2/3 )O- 3 -PbTiO 3 (PMN-PT), are widely known for their large piezoelectricity. This is attributed to polarization rotation, which is facilitated by the presence of various crystal symmetries for compositions near a morphotropic phase boundary. Relaxor-ferroelectric thin films, which are necessary for low-voltage applications, suffer a reduction in their piezoelectric response due to clamping by the passive substrate. To understand the microscopic behavior of this adverse phenomenon, we employ the AC electric field driven in-operando synchrotron x-ray diffraction on patterned device structures to investigate the piezoelectric domain behavior under an electric field for both a clamped (001) PMN-PT thin film on Si and a (001) PMN-PT membrane released from its substrate. In the clamped film, the substrate inhibits the field-induced rhombohedral (R) to tetragonal (T) phase transition resulting in a reversible R to Monoclinic (M) transition with a reduced longitudinal piezoelectric coefficient d 33 < 100 pm/V. Releasing the film from the substrate results in recovery of the R to T transition and results in a d 33 > 1000 pm/V. Here, using diffraction with spatial mapping, we find that lateral constraints imposed by the boundary between the active and inactive materials also inhibit the R to T transition. Phase-field calculations on both clamped and released PMN-PT thin films simulate our experimental findings. Resolving the suppression of thin film piezoelectric response is critical to their application in piezo-driven technologies.

36 MATERIALS SCIENCE↗

DOE OSTI2021

Synthesis and anisotropic magnetism in quantum spin liquid candidates A YbSe 2 ( A = K and Rb)

The quantum spin liquid (QSL) state in rare-earth triangular lattices has attracted much attention recently due to its potential application in quantum computing and communication. Here, we report the single-crystal growth synthesis, crystal structure characterizations, and magnetic properties of AYbSe 2 (A = K and Rb) compounds. The x-ray diffraction analysis shows that AYbSe 2 (A = K and Rb) crystallizes in a trigonal space group, R-3m (No. 166) with Z = 3. AYbSe 2 possesses a two-dimensional (2D) Yb–Se–Yb layered structure formed by edged-shared YbSe 6 octahedra. The magnetic properties are highly anisotropic for both title compounds, and no long-range order is found down to 0.4 K, revealing the possible QSL ground state in these compounds. The isothermal magnetization exhibits a one-third magnetization plateau when the magnetic fields are applied in the ab-plane. Heat capacity is performed along both ab-plane and c axis and features the characteristic dome for triangular magnetic lattice compounds as a function of magnetic fields. Due to the change in the interlayer and intralayer distance of Yb 3+ , the dome shifts to low fields from KYbSe 2 to RbYbSe 2 . All these results indicate that the AYbSe 2 family presents unique frustrated magnetism close to the possible QSL and noncollinear spin states.

36 MATERIALS SCIENCE↗

DOE OSTI2022

PIONEER, a high-resolution single-crystal polarized neutron diffractometer

PIONEER is a high Q-resolution, single-crystal, polarized neutron diffractometer at the Second Target Station (STS), Oak Ridge National Laboratory. It will provide the unprecedented capability of measuring tiny crystals (0.001 mm 3 , i.e., x-ray diffraction size), ultra-thin films (10 nm thickness), and weak structural and magnetic transitions. PIONEER benefits from the increased peak brightness of STS cold-neutron sources and uses advanced Montel mirrors that are able to deliver a focused beam with a high brilliance transfer, a homogeneous profile, and a low background. Monte Carlo simulations suggest that the optimized instrument has a high theoretical peak brilliance of 2.9 × 10 12 n cm -2 sr -1 Å -1 s -1 at 2.5 Å at the sample position, within a 5 × 5 mm 2 region and a ±0.3° divergence range. The moderator-to-sample distance is 60 m, providing a nominal wavelength band of 4.3 Å with a wavelength resolution better than 0.2% in the wavelength range of 1.0–6.0 Å. PIONEER is capable of characterizing large-scale periodic structures up to 200 Å. With a sample-to-detector distance of 0.8 m, PIONEER accommodates various sample environments, including low/high temperature, high pressure, and high magnetic/electric field. Additionally, a large cylindrical detector array (4.0 sr) with a radial collimator is planned to suppress the background scattering from sample environments. Bottom detector banks provide an additional 0.4 sr coverage or can be removed if needed to accommodate special sample environments. We present virtual experimental results to demonstrate the scientific performance of PIONEER in measuring tiny samples.

47 OTHER INSTRUMENTATION↗

DOE OSTI2022

Magnetic structure and spin dynamics of the quasi-two-dimensional antiferromagnet Zn-doped copper pyrovanadate

We report magnetic properties of the antiferromagnet Zn x Cu 2-x V 2 O 7 (ZnCVO) with x ≈ 0.06 have been thoroughly investigated on powder and single-crystal samples. The crystal structure determination using powder x-ray and neutron diffraction confirms that our ZnCVO samples are isostructural with β-Cu 2 V 2 O 7 (β-CVO) with a small deviation in the lattice parameters. Macroscopic magnetic property measurements also confirm the similarity between the two compounds. The Cu 2+ spins were found to align along the crystallographic c axis, antiparallel to their nearest neighbors connected by the leading exchange interaction J 1 . Spin dynamics reveals a typical symmetric spin-wave dispersion with strong interactions in the bc plane and weak interplane coupling. The exchange interaction analysis indicates that the spin network of ZnCVO is topologically consistent with the previous DFT prediction but the values of leading exchange interactions are contradictory. Furthermore, rather than the predicted 2D honeycomb structure, the spin network in ZnCVO could be better described by the anisotropic 2D spin network composed of J 1 , J 5 , and J 6 interactions, four bonds per one spin site, coupled by weak interplane interactions.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2024

Single-crystal growth and characterization of antiferromagnetically ordered EuIn 2

Here, we report the single crystal growth and characterization of EuIn 2 , a magnetic topological semimetal candidate according to our density functional theory (DFT) calculations. We present results from electrical resistance, magnetization, Mössbauer spectroscopy, and X-ray resonant magnetic scattering (XRMS) measurements. We observe three magnetic transitions at T N1 ~ 14.2 K, T N2 ~ 12.8 K and T N3 ~ 11 K, signatures of which are consistently seen in anisotropic temperature dependent magnetic susceptibility and electrical resistance data. Mössbauer spectroscopy measurements on ground crystals suggest an incommensurate sinusoidally modulated magnetic structure below the transition at T N1 ~ 14 K, followed by the appearance of higher harmonics in the modulation on further cooling roughly below T N2 ~ 13 K, before the moment distribution squaring up below the lowest transition around T N3 ~ 11 K. XRMS measurements showed the appearance of magnetic Bragg peaks below T N1 ~ 14 K, with a propagation vector of $τ$ = ($τ_h$, $\overline{τ}_h$, 0), with $τ_h$ varying with temperature, and showing a jump at T N3 ~ 11 K. The temperature dependence of $τ_h$ between ~ 11 K and 14 K shows incommensurate values consistent with the Mössbauer data. XRMS data indicate that $τ_h$ remains incommensurate at low temperatures and locks into $τ_h$ = 0.3443(1).

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2022

Charge Density Wave in Kagome Lattice Intermetallic ScV 6 Sn 6

Materials hosting kagome lattices have drawn interest for the diverse magnetic and electronic states generated by geometric frustration. In the AV 3 Sb 5 compounds (A=K, Rb, Cs), stacked vanadium kagome layers give rise to unusual charge density waves (CDW) and superconductivity. In this work, we report single-crystal growth and characterization of ScV 6 Sn 6 , a hexagonal HfFe 6 Ge 6 -type compound that shares this structural motif. We identify a first-order phase transition at 92 K. Single crystal x-ray and neutron diffraction reveal a charge density wave modulation of the atomic lattice below this temperature. This is a distinctly different structural mode than that observed in the AV 3 Sb 5 compounds, but both modes have been anticipated in kagome metals. The diverse HfFe 6 Ge 6 family offers more opportunities to tune ScV 6 Sn 6 and explore density wave order in kagome lattice materials.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Physical properties and electronic structure of single-crystal KCo 2 As 2

We present a method for producing high quality KCo 2 As 2 crystals, stable in air and suitable for a variety of measurements. X-ray diffraction, magnetic susceptibility, electrical transport and heat capacity measurements confirm the high quality and an absence of long range magnetic order down to at least 2 K. Residual resistivity values approaching 0.25 µΩ cm are representative of the high quality and low impurity scattering, and a Sommerfeld coefficient γ = 7.3 mJ/mol K 2 signifies weaker correlations than the Fe-based counterparts. Together with Hall effect measurements, angle-resolved photoemission experiments reveal a Fermi surface consisting of electron pockets at the center and corner of the Brillouin zone, in line with theoretical predictions and in contrast to the mixed carrier types of other pnictides with the ThCr 2 Si 2 structure. A large, linear magnetoresistance of 200% at 14 T, together with an observed linear and hyperbolic, rather than parabolic, band dispersions are unusual characteristics of this compound.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2022

Ferromagnetic MnBi 4 Te 7 obtained with low-concentration Sb doping: A promising platform for exploring topological quantum states

The tuning of the magnetic phase, chemical potential, and structure is crucial to observe diverse exotic topological quantum states in Mn Bi 2 Te 4 (Bi 2 Te 3 ) m (m = 0–3). Here we show a ferromagnetic (FM) phase with a chiral crystal structure in Mn (Bi 1–x Sb x ) 4 Te 7 , obtained via tuning the growth conditions and Sb concentration. Unlike previously reported Mn (Bi 1–x Sb x ) 4 Te 7 , which exhibits FM transitions only at high Sb doping levels, our samples show FM transitions (T C = 13.5 K) at 15%–27% doping levels. Furthermore, our single-crystal x-ray-diffraction structure refinements find Sb doping leads to a chiral structure with the space group of P3, contrasted with the centrosymmetric $P\bar{3}m1$ crystal structure of the parent compound MnBi 4 Te 7 . Through angle-resolved photoemission spectroscopy measurements, we also demonstrated that the nontrivial band topology is preserved in the Sb-doped FM samples. Given that the nontrivial band topology of this system remains robust for low Sb doping levels, our success in making FM Mn(Bi 1–x Sb x ) 4 Te 7 with x = 0.15, 0.175, 0.2, and 0.27 paves the way for realizing the predicted topological quantum states, such as the axion insulator and Weyl semimetals. Additionally, we also observed magnetic glassy behavior in both antiferromagnetic MnBi 4 Te 7 and FM Mn (Bi 1–x Sb x )4 Te 7 samples, which we believe originates from cluster spin-glass phases coexisting with long-range antiferromagnetic/FM orders. Further, we have also discussed how the antisite Mn ions impact the interlayer magnetic coupling and how FM interlayer coupling is stabilized in this system.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Thermal properties of the metallic delafossite PdCo⁢O 2 : A combined experimental and first-principles study

Metallic delafossite materials (e.g., PdCo⁢O 2 , PtCo⁢O 2 ), have attracted much recent attention due to record-high oxide conductivities, the origins of which remain unclear. Relatively little attention has been paid to their related thermal properties, however. Here, we address this via wide temperature range experimental studies of the crystal structure, thermal expansion, and specific heat of single-crystal PdCo⁢O 2 , combined with density-functional theory (DFT) calculations of the electronic and phononic densities of states, and thus thermal properties. PdCo⁢O 2 is shown to retain the $R\bar{3}m$ space group from 12 to 1000 K, exhibiting a- and c-axis thermal expansion in good quantitative agreement with DFT-based lattice dynamics calculations. The Co-O bond lengths additionally elucidate the stability of the low-spin state of the nominally Co 3+ ions, which is a notable difference between the edge-shared Co-O octahedra in PdCo⁢O 2 and the corner-shared octahedra in Co-based perovskites. Measurements of specific heat from 1.9 to 400 K provide accurate values for the Debye temperature and Sommerfeld coefficient, the phononic part being describable via a combined Debye-Einstein approach (accounting for high-frequency oxygen-related optical phonons), with excess intermediate-temperature specific heat due to a prominent low-energy peak in the phonon density of states. Most significantly, all electronic and phononic contributions to the specific heat are shown to be remarkably closely reproduced by DFT-based calculations, establishing quantitative understanding of key thermal properties of the metallic delafossite PdCo⁢O 2 .

36 MATERIALS SCIENCE↗

DOE OSTI2025

Computing virtual dark-field X-ray microscopy images of complex discrete dislocation structures from large-scale molecular dynamics simulations

Dark-field X-ray microscopy (DFXM) is a novel diffraction-based imaging technique that non-destructively maps the local deformation from crystalline defects in bulk materials. While studies have demonstrated that DFXM can spatially map 3D defect geometries, it is still challenging to interpret DFXM images of the high-dislocation-density systems relevant to macroscopic crystal plasticity. This work develops a scalable forward model to calculate virtual DFXM images for complex discrete dislocation structure(s) (DDS) obtained from atomistic simulations. Our new DDS-DFXM model integrates a non-singular formulation for calculating the local strain from the DDS and an efficient geometrical optics algorithm for computing the DFXM image from the strain field. We apply the model to complex DDS obtained from a large-scale molecular dynamics simulation of compressive loading on single-crystal silicon. Simulated DFXM images exhibit prominent contrast for dislocation features between the multiple slip systems, demonstrating the potential of DFXM to resolve features from dislocation multiplication. In conclusion, the integrated DDS-DFXM model provides a toolbox for DFXM experimental design and image interpretation in the context of bulk crystal plasticity for a range of measurements across shock plasticity and the broader materials science community.

X-ray imaging↗

DOE OSTI2023

At-wavelength characterization of X-ray wavefronts in Bragg diffraction from crystals

The advent of next-generation synchrotron radiation sources and X-ray free-electron lasers calls for high-quality Bragg-diffraction crystal optics to preserve the X-ray beam coherence and wavefront. This requirement brings new challenges in characterizing crystals in Bragg diffraction in terms of Bragg-plane height errors and wavefront phase distortions. Here, a quantitative methodology to characterize crystal optics using a state-of-the-art at-wavelength wavefront sensing technique and statistical analysis is proposed. The method was tested at the 1-BM-B optics testing beamline at the Advanced Photon Source for measuring silicon and diamond crystals in a self-referencing single-crystal mode and an absolute double-crystal mode. The phase error sensitivity of the technique is demonstrated to be at the λ/100 level required by most applications, such as the characterization of diamond crystals for cavity-based X-ray free-electron lasers.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2023

Fe 3+ /Fe T ratios of amphiboles determined by high spatial resolution single-crystal synchrotron Mössbauer spectroscopy

The Fe 3+ /Fe T ratios (Fe 3+ /[Fe 2+ +Fe 3+ ]) in minerals can be used to understand their crystallization and post-crystallization conditions. However, as natural minerals are often zoned and contain inclusions, bulk techniques, e.g., wet chemistry, may not provide accurate Fe 3+ /Fe T values for a single phase of interest. We determined Fe 3+ /Fe T ratios of amphiboles in different crystallographic orientations by single-crystal synchrotron Mössbauer spectroscopy (SMS) in energy and time domain modes from four volcanic localities (Long Valley Caldera, Mount St. Helens, Lassen Volcanic Center, U.S.A., and Mt. Pinatubo, Philippines). The high spatial resolution (as low as 12 × 12 μm spot size) and standard-free nature of SMS allow the detection of intra-grain compositional heterogeneities in Fe 3+ /Fe T with relatively low uncertainties. Here, we combine SMS with major element compositions, water contents, and hydrogen isotope compositions to document the Fe 3+ /Fe T ratios as a function of mineral composition and post-crystallization dehydrogenation. Spectra were fitted with up to five distinct sites: ferrous iron on M(1), M(2), M(3), and ferric iron on M(2) and M(3), consistent with X-ray diffraction studies on single crystals of amphibole. The Fe 3+ /Fe T ratios range from 0.14 ± 0.03 (Long Valley Caldera), 0.51 to 0.63 ± 0.02 (representing intra-grain heterogeneities, Mount St. Helens) to 0.86 ± 0.03 (Lassen Volcanic Center). The latter grain experienced post-crystallization dehydrogenation, shown by its low water content (0.6 ± 0.05 wt%) and its elevated hydrogen isotope composition (δD = +25 ± 3‰ relative to SMOW). The Fe 3+ /Fe T ratios of 0.62 ± 0.01 and 0.20 ± 0.01 of two Mt. Pinatubo grains correlate with high-Al 2 O 3 cores and low-Al 2 O 3 rims and smaller phenocrysts in the sample, respectively. This study shows that SMS is capable of distinguishing two different domains with dissimilar Fe 3+ /Fe T values formed under different crystallization conditions, demonstrating that SMS in combination with major element, water, and hydrogen isotope compositions allows the interpretation of amphibole Fe 3+ /Fe T ratios in the context of crystallization and post-crystallization processes.

47 OTHER INSTRUMENTATION↗

DOE OSTI2026

Synthesis of Lanthanide-Based Intermetallic Silicides

Lanthanide-containing materials are of interest because of their useful magnetic and electronic behavior. To study this intrinsic behavior, researchers need high-quality single crystals to avoid interference from grain boundaries or defects commonly found in polycrystalline samples. One method for growing single crystals is metal-flux synthesis, an approach that promotes crystal growth at lower reaction temperatures thus providing access to complex intermetallic phases. My project focuses on synthesizing novel lanthanide-containing intermetallic silicides related to previously reported f-element compounds with the Gd1??Fe4Si10?? structure type. I investigated reactions targeting lanthanum, praseodymium, neodymium, samarium, europium, and gadolinium-containing products, and used scanning electron microscopy with energy-dispersive X-ray spectroscopy (SEM-EDS) to identify promising lanthanide-containing phases. I prepared reactions by loading lanthanide oxides, Co, Si, Al, and Ga into alumina crucibles in a 0.2:1:1:10:10 Ln2O3/Co/Si/Al/Ga mmol ratio, where Ln = La, Nd, Pr, Sm, Eu, or Gd. The reactions were sealed in quartz tubes, heated in a furnace, and centrifuged to separate crystals from excess flux. Afterwards, I mechanically isolated single crystals and prepared them for characterization. Pr-, Nd-, and Sm-containing reactions were confirmed through SEM-EDS to have formed the targeted quaternary phase. Single crystal X-ray diffraction (SC-XRD) data showed the Nd-containing crystals exhibited a hexagonal unit cell associated with the Gd1??Fe4Si10?? structure type. La- and Eu-containing reactions instead formed competing Co(Al/Ga/Si)3 phase crystals. Single-crystal XRD, powder XRD, and magnetic susceptibility and heat-capacity measurements will be collected for the remaining lanthanide-containing crystals. This data will help determine the complete crystal structure of the lanthanide analogues, their phase purity and whether they exhibit localized magnetic behavior, magnetic ordering, or other electronic signatures that can be compared with the previously reported actinide analogues. The disordered structure of this intermetallic family is associated with unusual magnetic and electronic behavior meaning lanthanide analogues may show potential as thermoelectric materials. Developing new thermoelectric candidate materials is relevant to energy efficiency and waste-heat recovery, which supports DOE’s broader goal of advancing science and technology solutions for America’s energy needs.

37 - INORGANIC, ORGANIC, PHYSICAL AND ANALYTICAL C↗

DOE OSTI2022

SrNi(VO 4 )(OH): The High-Temperature Hydrothermal Synthesis and Magnetic Properties of an Adelite-Descloizite-Type Structure

Single crystals of a new transition metal adelite-descloizite-type structure were synthesized using a high temperature (580 °C) high-pressure hydrothermal technique. Single crystal X-ray diffraction and energy dispersive X-ray analysis (EDX) were used to investigate the structure and elemental composition, respectively. SrNi(VO 4 )(OH) crystallizes in an acentric orthorhombic crystal system in the space group P2 1 2 1 2 1 (no. 19); Z = 4, a = 5.9952(4) Å, b = 7.5844(4) Å, c = 9.2240(5) Å. The structure is comprised of a Ni–O–V framework where Sr 2+ ions reside inside the channels. Single-crystal magnetic measurements display a significant anisotropy in both temperature- and field-dependent data. The temperature dependent magnetic measurement shows antiferromagnetic behavior at T N ~8 K. Overall, the magnetic properties indicate the presence of competing antiferromagnetic and ferromagnetic interactions of SrNi(VO 4 )(OH).

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1977

Measurements of the frequency spectrum of transition radiation

We report a measurement of the frequency spectrum of X-ray transition radiation. X rays were generated by electrons of 5 and 9 GeV in radiators of multiple polypropylene foils, and detected in the range 4 to 30 keV with a calibrated single-crystal Bragg spectrometer. The experimental results closely reproduce the features of the theoretically predicted spectrum. In particular, the pronounced interference pattern of multifoil radiators and the expected hardening of the radiation with increasing foil thickness are clearly observed. The overall intensity of the radiation is somewhat lower than predicted by calculations.

Cherry, M. L.↗

NASA NTRS1982

XPS, AES and friction studies of single-crystal silicon carbide

The surface chemistry and friction behavior of a single crystal silicon carbide surface parallel to the 0001 plane in sliding contact with iron at various temperatures to 1500 C in a vacuum of 3 x 10 nPa are investigated using X-ray photoelectron and Auger electron spectroscopy. Results show that graphite and carbide-type carbon are seen primarily on the silicon carbide surface in addition to silicon at temperatures to 800 C by both types of spectroscopy. The coefficients of friction for iron sliding against a silicon carbide surface parallel to the 0001 plane surface are found to be high at temperatures up to 800 C, with the silicon and carbide-type carbon at maximum intensity in the X-ray photoelectron spectroscopy at 800 C. The concentration of the graphite increases rapidly on the surface as the temperature is increased above 800 C, while the concentrations of the carbide-type carbon and silicon decrease rapidly and this presence of graphite is accompanied by a significant decrease in friction. Preheating the surfaces to 1500 C also gives dramatically lower coefficients of friction when reheating in the sliding temperature range of from room temperature to 1200 C, with this reduction in friction due to the graphite layer on the silicon carbide surface.

Miyoshi, K.↗

NASA NTRS1991

Stability of Y-Ba-Cu-oxide phases in salt flux

The results of an investigation into the possibility of growing YBa2Cu3O7 (or Y123) single crystals from a NaCl-KCl flux (salt flux) are presented. They are compared with results for Y123 crystals grown from a Ba-Cu-oxide flux system. The chosen salt flux composition was 1:1 (NaCl:KCl). The preparation method of the solutes is presented, with particular attention given to the compounds surrounding the Y123 phase, such as Y2BaCuO6 (Y211). Optical microscopy was introduced to study the morphology of the sample; X-ray diffraction was used for phase identification; and a chemical analysis was effected. The Y123, Y211, and BaCuO2 phases were shown to be dissociated in the NaCl-KCl flux, and could not be recrystallized. A small amount of the flux could be added to create large-grain Y123 crystals, but single-crystal growth was not possible in a top-seeded NaCl-KCl solution. This is attributed to significant differences in solubility of the crystal elements, and phase separation caused by CuO crystal formation.

Tao, Y. K.↗