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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 955 records · Page 53
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Ethanol Electrooxidation at 1–2 nm AuPd Nanoparticles

We report a systematic study of the electrocatalytic properties and stability of a series of 1–2 nm Au, Pd, and AuPd alloy nanoparticles (NPs) for the ethanol oxidation reaction (EOR). Following EOR electrocatalysis, NP sizes and compositions were characterized using aberration-corrected scanning transmission electron microscopy (ac-STEM) and energy dispersive spectroscopy (EDS). Two main findings emerge from this study. First, alloyed AuPd NPs exhibit enhanced electrocatalytic EOR activity compared to either monometallic Au or Pd NPs. Specifically, NPs having a 3:1 ratio of Au:Pd exhibit an ~8-fold increase in peak current density compared to Pd NPs, with an onset potential shifted ~200 mV more to the negative compared to Au NPs. Second, the size and composition of AuPd alloy NPs do not (within experimental error) change following 1.0 or 2.0 h chronoamperometry experiments, while monometallic Au NPs increase in size from 2 to 5 nm under the same conditions. Notably, this report demonstrates the importance of post-catalytic ac-STEM/EDS characterization for fully evaluating NP activity and stability, especially for 1–2 nm NPs that may change in size or structure during electrocatalysis.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2017

Erratum: “A high precision gas flow cell for performing in situ neutron studies of local atomic structure in catalytic materials” [Rev. Sci. Instrum. 88, 034101 (2017)]

We report that in the original paper,1 the following chemical formula was reported erroneously: Na 78-2x Ca x Al 78 Si 144 O 384 , x ≈ 38. The correct formula for the material studied is Na 86-2x Ca x Al 86 Si 106 O 384 , x ≈ 43. The difference in formula values between this corrected composition and values in Table 1 has no effect on the refinement quality (within the reported error), the subsequent analysis, or overall conclusions of the manuscript. Specifically, the neutron diffraction data and crystallographic refinement are insensitive (within the reported error) to the corresponding change in the Si/Al ratios, and the corrected number of Ca (x ≈ 43) is found to be within the error of the value reported via crystallographic refinement of the Ca occupancy [x = 46(3)].

36 MATERIALS SCIENCE↗
DOE OSTI2024

Unusual Base Pair between Two 2-Thiouridines and Its Implication for Nonenzymatic RNA Copying

2-Thiouridine (s 2 U) is a nucleobase modification that confers enhanced efficiency and fidelity both on modern tRNA codon translation and on nonenzymatic and ribozyme-catalyzed RNA copying. We have discovered an unusual base pair between two 2-thiouridines that stabilizes an RNA duplex to a degree that is comparable to that of a native A:U base pair. High-resolution crystal structures indicate similar base-pairing geometry and stacking interactions in duplexes containing s 2 U:s 2 U compared to those with U:U pairs. Notably, the C=O···H–N hydrogen bond in the U:U pair is replaced with a C=S···H–N hydrogen bond in the s 2 U:s 2 U base pair. The thermodynamic stability of the s 2 U:s 2 U base pair suggested that this self-pairing might lead to an increased error frequency during nonenzymatic RNA copying. However, competition experiments show that s 2 U:s 2 U base-pairing induces only a low level of misincorporation during nonenzymatic RNA template copying because the correct A:s 2 U base pair outcompetes the slightly weaker s 2 U:s 2 U base pair. In addition, even if an s 2 U is incorrectly incorporated, the addition of the next base is greatly hindered. This strong stalling effect would further increase the effective fidelity of nonenzymatic RNA copying with s 2 U. Our findings suggest that s 2 U may enhance the rate and extent of nonenzymatic copying with only a minimal cost in fidelity.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2017

Viscosity of Heterogeneous Silicate Melts: A Non-Newtonian Model

The recently published viscosity data of heterogeneous silicate melts with well-documented structure and experimental conditions are critically re-analyzed and tabulated. By using these data, a non-Newtonian viscosity model incorporating solid fraction, solid shape, and shear rate is proposed on the basis of the power-law equation. This model allows calculating the viscosity of the heterogeneous silicate melts with solid fraction up to 34 vol pct. The error between the calculated and measured data is evaluated to be 32 pct, which is acceptable considering the large error in viscosity measurement of the completely liquid silicate melt.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Accurate Atomic Correlation and Total Energies for Correlation Consistent Effective Core Potentials

Very recently, we introduced a set of correlation consistent effective core potentials (ccECPs) constructed within full many-body approaches. By employing significantly more accurate correlated approaches, we were able to reach a new level of accuracy for the resulting effective core Hamiltonians. We also strived for simplicity of use and easy transferability into a variety of electronic structure methods in quantum chemistry and condensed matter physics. Here, as a reference for future use, we present exact or nearly exact total energy calculations for these ccECPs. The calculations cover H–Kr elements and are based on the state-of-the-art configuration interaction (CI), coupled-cluster (CC), and quantum Monte Carlo (QMC) calculations with systematically eliminated/improved errors. In particular, we carry out full CI/CCSD(T)/CCSDT(Q) calculations with cc-pVnZ with up to n = 6 basis sets and we estimate the complete basis set limits. Using combinations of these approaches, we achieved an accuracy of ≈1–10 mHa for K–Zn atoms and ≈0.1–0.3 mHa for all other elements—within about 1% or better of the ccECP total correlation energies. We also estimate the corresponding kinetic energies within the feasible limit of full CI calculations. In order to provide data for QMC calculations, we include fixed-node diffusion Monte Carlo energies for each element that give quantitative insights into the fixed-node biases for single-reference trial wave functions. The results offer a clear benchmark for future high-accuracy calculations in a broad variety of correlated wave function methods such as CI and CC as well is in stochastic approaches such as real space sampling QMC.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2019

Unravelling donor–acceptor film morphology formation for environmentally-friendly OPV ink formulations

The challenge of coating organic photovoltaics (OPV) from green solvents is to achieve the required nanostructured interpenetrating network of donor and acceptor domains based on a rational choice of solvent approach as opposed to the usual trial-and-error methods. We demonstrate here that we can achieve a bicontinuous interpenetrating network with nanoscale phase separation for the chosen donor–acceptor material system poly[2,3-bis-(3-octyloxyphenyl)quinoxaline-5,8-diyl-alt-thiophene-2,5-diyl]:phenyl-C 61 butyric acid methyl ester (TQ1:PC 61 BM) when processing from green solvent ink formulations. This structure is achieved by first calculating the Hansen solubility parameters (HSP) of the donor and acceptor materials, followed by careful choice of solvents with selective relative solubilities for the two materials based on the desired order of precipitation necessary for forming a nanostructured interdigitated network morphology. We found that the relative distances in Hansen space (R a ) between TQ1 and the primary solvent, on the one hand, and PC 61 BM and the primary solvent, on the other hand, could be correlated to the donor–acceptor morphology for the formulations based on the solvents d-limonene, anisole, and 2-methyl anisole, as well as the halogenated reference solvent o-dichlorobenzene. This nanostructured blend film morphology was characterised with scanning transmission X-ray microscopy (STXM) and transmission electron microscopy (TEM), and the film surface composition was analysed by near edge X-ray absorption fine structure (NEXAFS) spectroscopy. Hansen solubility theory, based on solution thermodynamics, has been used and we propose an HSP-based method that is a general platform for the rational design of ink formulations for solution-based organic electronics, in particular facilitating the green solvent transition of organic photovoltaics. Our results show that the bulk heterojunction morphology for a donor–acceptor system processed from customised solvent mixtures can be predicted by the HSP-based method with good reliability.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

High temperature stability of regrown and alloyed Ohmic contacts to AlGaN/GaN heterostructure up to 500 °C

This Letter reports the stability of regrown and alloyed Ohmic contacts to AlGaN/GaN-on-Si high electron mobility transistors (HEMTs) for high temperature applications up to 500 °C. Transfer length method (TLM) measurements from 25 to 500 °C in air show that the regrown contacts appear to be stable up to 500 °C during short term (approximately 1 h) testing, while alloyed contacts appear to decrease in contact resistance from 300 to 500 °C though increases in the error bounds due to increase sheet resistance make it difficult to conclude definitely. Additionally, longer term testing shows both technologies remain stable at least up to 48 h at 500 °C, after which the large increase in sheet resistance makes the measurement uncertainty too large to conclude definitively. Advanced microscopy images indicate both the regrown and alloyed contact regions remain structurally intact after prolonged high temperature exposure with no visible degradation in crystallinity or metal composition.

Physics↗
DOE OSTI2023

Spatio-Temporal Deep Graph Network for Event Detection, Localization, and Classification in Cyber-Physical Electric Distribution System

This work proposes a deep graph learning framework to identify, locate, and classify power, cyber, and cyber power events at the distribution system level. The proposed algorithm jointly exploits spatial, temporal, and node-level cyber and physical data features. The developed graph neural network, together with a deep autoencoder, utilizes physical measurements from distribution level phasor measurement units and cyber data from communication network logs. The spatial structure of the synchrophasor measurements and network is incorporated through a weighted adjacency matrix. The temporal structure is incorporated by defining a spatial operation in the gated recurrent unit. This spatio-temporal learning element resides inside a power event detection, localization, and classification module that provides the degree of confidence for an event label. To accurately pinpoint the location of an event to the nearest bus equipped with a measurement unit, a combination of squared error and proximity score is utilized. Also included is a cyber event detection module that employs heteroskedasticity to analyze the significance of various cyber features during different types of attacks. Finally, a dual-bit cyber-power decision table determines the nature of the event. The proposed method is validated on two distribution systems modeled in OPAL-RT/Hypersim with limited phasor measurement units for different possible physical and cyber events. Further analyses include comparison with other state-of-the-art methods and validation in the presence of measurement noise. As a result, our method outperforms existing approaches and achieves an average detection accuracy of 97.97%, F1-score of 96.88%, precision of 96.53%, and recall of 98.57%.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2023

Mechanism of structural colors in binary mixtures of nanoparticle-based supraballs

Inspired by structural colors in avian species, various synthetic strategies have been developed to produce noniridescent, saturated colors using nanoparticle assemblies. Nanoparticle mixtures varying in particle chemistry and size have additional emergent properties that affect the color produced. For complex multicomponent systems, understanding the assembled structure and a robust optical modeling tool can empower scientists to identify structure-color relationships and fabricate designer materials with tailored color. Here, we demonstrate how we can reconstruct the assembled structure from small-angle scattering measurements using the computational reverse-engineering analysis for scattering experiments method and use the reconstructed structure in finite-difference time-domain calculations to predict color. We successfully, quantitatively predict experimentally observed color in mixtures containing strongly absorbing nanoparticles and demonstrate the influence of a single layer of segregated nanoparticles on color produced. The versatile computational approach that we present is useful for engineering synthetic materials with desired colors without laborious trial-and-error experiments.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Geophysical and Environmental Monitoring Data, and Subsurface Flow Modelling Results for Chicken Bone Meadow, Mt. Snodgrass, Crested Butte, CO

This dataset includes geoelectrical monitoring data acquired between October 2021 and November 2022, soil moisture and temperature data, groundwater data obtained from borehole SNIB covering the period from June 2021 to September 2022, and hydrological modelling results. The data were acquired to investigate how variations in bedrock type and topography, and vegetation cover control subsurface flow dynamics. To provide insights into the subsurface flow dynamics and their controls, a monitoring transect was installed at the Chicken Bone Meadow, Mt. Snodgrass, Crested Butte, CO, measuring the spatio-temporal variations of soil moisture, soil and snow temperature, subsurface electrical resistivity variations, and groundwater dynamics. Field data are organized in a folder structure, with Electrical Resistivity Tomography (ERT) data being provided as one file per measurement, and data of the soil moisture and temperature sensors being provided as text files covering the entire monitoring period. The ‘Locations.csv’ file contains the location of all sensors, given in NAD83 – UTM Zone 13N. ERT monitoring data has been processed to filter data based on reciprocal errors (data with errors > 30% were removed), a linear error model was fitted to each survey, and to ensure a constant set of measurements for time-lapse inversion, filtered data were interpolated and assigned a 100% measurement error. Soil moisture and temperature data were acquired at 15 min intervals, and averaged to provide 1h data. Weather data and borehole data (groundwater depth, conductivity and temperature) were acquired at 30 min intervals, and are provided as daily measurements; all measurements are averaged, except of precipitation values, which are given as daily accumulation. The hydrological model was set up along the ERT monitoring transect, and net infiltration was used as surface boundary condition and derived from the weather data. Four different results are provided, (1) results for a parameterization using hydraulic permeability and porosity as derived from the ERT data through petrophysical relationships, and (2) three simplified model results, using 1 to 3 geological layers above the bedrock. Modelling was performed using PFLOTRAN, and for each model the PFLOTRAN input files are provided. The result files include weekly hydrological modelling results (e.g., saturation, velocities, pressures), as well as the model parameterization. The dataset additionally includes a file-level metadata (flmd.csv) file that lists each file contained in the dataset with associated metadata; and a data dictionary (dd.csv) file that contains column/row headers used throughout the files along with a definition, units, and data type.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2020

Machine learning applied to classifying neutron resonances

The performance of nuclear reactors and other nuclear systems depends on a precise understanding of the neutron interaction cross sections for materials used in these systems. These cross sections exhibit resonance structure whose shape is determined in part by the angular momentum quantum numbers of the resonances. The correct assignment of the quantum numbers of neutron resonances is therefore of paramount importance. In this project, we apply a machine learning technique, namely decision trees, to automate the quantum number assignments. The tree is trained from simulated data generated to mimic the errors found in real data. We explore the use of several physics-motivated features for training our trees, including the nearest neighbor spacing distribution, cumulative level distribution, and channel width distributions. Initial results using random matrix theory motivated fits which demonstrated that we can determine resonance spin groups somewhat reliably. If we use these fits as features in our trees, we can train them to spot outliers corresponding to misassigned resonances. We found that with the large number of features used in this project that the decision tree tended to over t training data resulting in poor performance with respect to the test data. By reducing the number of features, we can achieve nearly perfect assignment of quantum numbers with our training data.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2026

Fundamentals of Solar PV Bolted Joint Loosening and Prevention

The photovoltaic (PV) industry has long reported anecdotal accounts of systems exhibiting intermittent or chronic fastener loosening, including joints that fail to maintain preload despite multiple re-tightening attempts. These occurrences are frequently, and often incorrectly, attributed to installer errors, vibration, or loading beyond design expectations (e.g., extreme weather events). Loose fasteners have serious implications and can significantly impact solar PV systems’ performance, reliability, and safety. However, effective bolted joint design and proper assembly practices can mitigate or eliminate loosening. Until the US Department of Energy’s Solar Energy Technologies Office (SETO) funded research on fasteners and solar PV structures, there was a notable gap in understanding the causes of loosening in solar PV systems.

14 SOLAR ENERGY↗
DOE OSTI2018

3D Measurement Simulation and Relative Pointing Error Verification of the Telescope Mount Assembly Subsystem for the Large Synoptic Survey Telescope

An engineering validation of a large optical telescope consists of executing major performing tests at the subsystem level to verify the overall engineering performance of the observatory. Thus, the relative pointing error verification of the telescope mount assembly subsystem is of special interest to guarantee the absolute pointing performance of the large synoptic survey telescope. This paper presents a new verification method for the relative pointing error assessment of the telescope mount assembly, based on laser tracker technology and several fiducial points fixed to the floor. Monte-Carlo-based simulation results show that the presented methodology is fit for purpose, even if floor movement occurs due to temperature variation during the measurement acquisition process. A further research about laser tracker technology integration into the telescope structure may suggest that such laser tracker technology could be permanently installed in the telescope in order to provide an active alignment system that aims to detect and correct possible misalignment between mirrors or to provide the required mirror positioning verification accuracy after maintenance activities. The obtained results show that two on-board laser tracker systems combined with eight measurement targets could result in measurement uncertainties that are better than 1 arcsec, which would provide a reliable built-in metrology tool for large telescopes.

47 OTHER INSTRUMENTATION↗
DOE OSTI2026

Bond Dissociation Energy, Ionization Energy, and Electronic Structure of Thorium Dimer

Diatomic thorium, Th 2 , has been investigated using a laser ablation, supersonic expansion source to produce the molecule and resonant two-photon ionization spectroscopy to measure its bond dissociation energy (BDE) and ionization energy (IE). The molecule has a high density of states in the vicinity of its bond dissociation energy, leading to rapid predissociation as soon as this energy is exceeded. The BDE is identified from this predissociation threshold as D 0 (Th 2 ) = 2.857(7) eV, where the assigned error limit is provided in parentheses in units of the last quoted digit. Similarly, the one-photon ionization threshold has been measured, providing the ionization energy IE(Th 2 ) = 5.042(4) eV. Together with a thermochemical cycle and the atomic ionization energy, these values provide the BDE of the cation, giving D 0 (Th 2 + ) = 4.122(8) eV. Computations show that Th 2 has three nearly degenerate low-lying electronic states (1 3 Σ u + , 1 1 Σ g + , and 1 3 Δ g ) with bonding dominated by 7s and 6d orbitals, indicating predominantly transition-metal-like behavior. The 1 3 Σ u + state exhibits a triple bond, whereas the 1 1 Σ g + and 1 3 Δ g states possess quadruple-bond character and correspondingly shorter bonds. Although 1 3 Σ u + is predicted to be the lowest state without spin–orbit coupling, the large spin–orbit stabilization of the 1 3 Δ g state makes its Ω = 1 g component the ground state. Furthermore, the calculated dissociation energy (2.840 eV) and ionization energy of Th 2 (5.098 eV) are both in excellent agreement with experiment.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Monitoring of corrosion effects in pipes with multi-mode acoustic signals

Current commercial long-range guided wave inspection and monitoring technologies for pipes usually require establishment of certain specific acoustic modes for wave propagation and use a large number of sensors in a certain geometric arrangement (for example, uniformly-spaced arrays of sensors on a collar, placed around a pipe). This paper, in contrast, presents a long-range corrosion detection method in pipes based on multi-mode guided acoustic waves, generated and received using simple, single-element sensors. In this technique, available guided wave modes in any interrogated zone of the structure are automatically down-selected by the medium, and changes in energy distribution over the selected modes are analyzed through signal processing to monitor material loss or addition. Here, to investigate the feasibility of the proposed method, we have performed both experimental and numerical simulation studies on 50.8-mm (2-in)-diameter schedule-40 carbon steel pipes. In these studies, a linear chirp signal is injected in compression mode, perpendicular to the surface of the pipe, generating multi-mode, multi-directional guided waves that travel within the pipe wall. Corrosion effects in the pipes are simulated numerically and experimentally. To reduce errors in the measurement technique due to changes in the local or global temperature of the measured media, a universal temperature compensation algorithm is introduced. The experimental results show good agreement with simulation results, indicating that the proposed multi-mode wave generation and detection in a pipe could be used for long-range detection and monitoring of corrosion effects, such as pipe wall losses or solid material deposits.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Evaluation of average leaf inclination angle quantified by indirect optical instruments in crop fields

Average leaf inclination angle ($\overline{θ}$ L ) is an important canopy structure variable that influences light regime, photosynthesis, and evapotranspiration of plants. $\overline{θ}$ L can be measured through direct methods (e.g., protractor), which are labor-intensive and time-consuming, or through indirect optical instruments, which are more efficient than the direct methods. However, uncertainties of different indirect optical instruments for quantifying $\overline{θ}$ L remain largely unquantified. In this study, we evaluated and compared the performances of three major indirect optical instruments: (1) LAI-2200, (2) 30°-tilted camera, and (3) digital hemispherical photography (DHP), in different crop fields over a growing season, benchmarked with direct measurements. LAI-2200 and 30°-tilted camera showed higher agreement with direct $\overline{θ}$ measurements (R 2 = 0.54, RMSE = 7.37°; R 2 = 0.58, RMSE = 8.08°) than DHP (R 2 = 0.14, RMSE = 13.96°). Different performances of indirect optical instruments could be attributed to the accuracy of gap fraction measurement and the performance of the $\overline{θ}$ L quantification algorithms. When using the LAI-2200 algorithm, larger gap fraction gradients over view zenith angles led to larger $\overline{θ}$ L values, and smaller gap fraction gradients led to smaller $\overline{θ}$ L values. Such error propagation was larger in sparse canopy than in dense canopy. The Wilson G function of the LAI-2200 algorithm performed better in estimating $\overline{θ}$ L than the G function based on the ellipsoidal LAD function used by the CAN_EYE algorithm. We also proposed a modification of the LAI-2200 algorithm, which further improved the performance of LAI-2200 and 30°-tilted cameras in estimating $\overline{θ}$ L . We envision that the low-cost 30°-tilted cameras provide a promising sensor solution to continuously monitor canopy structure for various ecosystems.

30°-tilted camera↗
DOE OSTI2022

Data-Centric Development of Lignin Structure–Solubility Relationships in Deep Eutectic Solvents Using Molecular Simulations

Lignin is a natural source of aromatic chemicals with significant potential as an abundant, renewable feedstock for value-added products. Deep eutectic solvents (DES)–solvents composed of a hydrogen bond donor (HBD) and acceptor (HBA) in varying ratios–have emerged as a highly tunable class of solvents for lignin solubilization. However, the variety of possible DES compositions and limited molecular-scale understanding of lignin solubility makes solvent selection a challenge without laborious trial-and-error experimentation. To address these challenges, we use classical molecular dynamics (MD) simulations to study the interactions of lignin model compounds with various DES–water systems. Quantitative parameters (descriptors) were calculated by postprocessing the MD results and used to train a regression model that predicts experimentally determined solubilities of lignin model compounds. This approach revealed that the most important descriptors of solubility are the system temperature, solute hydrophilicity, and metrics quantifying hydrogen bonding. Maximizing the interactions between solute–HBD (hydrophobic group), water–HBD (hydrophilic group), and water–HBA molecules led to the highest model compound solubility. Our results support a hydrotropic mechanism in which extensive DES–water hydrogen bonding and favorable HBD interactions with the solute promote high solubility. We applied the regression model derived using model compounds to predict the solubility of representative lignin oligomers. The model predicted lignin oligomers’ solubilities in good agreement with experiments, indicating that the simulations of model compounds can be extended to predict the solubility of larger lignin compounds across a range of solvent compositions and temperatures. Furthermore, these findings provide new molecular-scale insight into lignin solubilization mechanisms and a new method for computationally screening potential solvent systems for lignin valorization.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

On the discretization error of the discrete generalized quantum master equation

The transfer tensor method (TTM) [Cerrillo and Cao, Phys. Rev. Lett. 112 , 110401 (2014)] can be considered a discrete-time formulation of the Nakajima–Zwanzig quantum master equation (NZ-QME) for modeling non-Markovian quantum dynamics. A recent paper [Makri, J. Chem. Theory Comput. 21 , 5037 (2025)] raised concerns regarding the consistency of the TTM discretization, particularly a spurious term at the initial time t = 0. Here, this work presents a detailed analysis of the discretization structure of the TTM, clarifying the origin of the initial-time correction and establishing a consistent relationship between the TTM discrete-time memory kernel K N and the continuous-time NZ-QME kernel $\mathscr{K}$( N Δ t ). This relationship is validated numerically using the spin-boson model, demonstrating convergence of reconstructed memory kernels and accurate dynamical evolution as Δ t → 0. While the TTM provides a consistent discretization, we note that alternative schemes are also viable, such as the midpoint derivative/midpoint integral scheme proposed in Makri’s work. The relative performance of various schemes for either computing accurate $\mathscr{K}$( N Δ t ) from exact dynamics or obtaining accurate dynamics from exact $\mathscr{K}$( N Δ t ) warrants further investigation.

Density-matrix↗