Search NASA⌕ Search

SEARCH · Search NASA

Results for “dimensional”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 307 records · Page 17
All sourcesNASA NTRSDOE OSTI
NASA NTRS2002

Impact of left ventricular outflow tract area on systolic outflow velocity in hypertrophic cardiomyopathy: a real-time three-dimensional echocardiographic study

OBJECTIVES: The aim of this study was to use real-time three-dimensional echocardiography (3DE) to investigate the quantitative relation between minimal left ventricular (LV) outflow tract area (A(LVOT)) and maximal LV outflow tract (LVOT) velocity in patients with hypertrophic obstructive cardiomyopathy (HCM). BACKGROUND: In patients with HCM, LVOT velocity should change inversely with minimal A(LVOT) unless LVOT obstruction reduces the pumping capacity of the ventricle. METHODS: A total of 25 patients with HCM with systolic anterior motion (SAM) of the mitral valve leaflets underwent real-time 3DE. The smallest A(LVOT) during systole was measured using anatomically oriented two-dimensional "C-planes" within the pyramidal 3DE volume. Maximal velocity across LVOT was evaluated by two-dimensional Doppler echocardiography (2DE). For comparison with 3DE A(LVOT), the SAM-septal distance was determined by 2DE. RESULTS: Real-time 3DE provided unique information about the dynamic SAM-septal relation during systole, with A(LVOT) ranging from 0.6 to 5.2 cm(2) (mean: 2.2 +/- 1.4 cm(2)). Maximal velocity (v) correlated inversely with A(LVOT) (v = 496 A(LVOT)(-0.80), r = -0.95, p < 0.001), but the exponent (-0.80) was significantly different from -1.0 (95% confidence interval: -0.67 to -0.92), indicating a significant impact of small A(LVOT) on the peak LVOT flow rate. By comparison, the best correlation between velocity and 2DE SAM-septal distance was significantly (p < 0.01) poorer at -0.83, indicating the superiority of 3DE for assessing A(LVOT). CONCLUSIONS: Three-dimensional echocardiography-measured A(LVOT) provides an assessment of HCM geometry that is superior to 2DE methods. These data indicate that the peak LVOT flow rate appears to be significantly decreased by reduced A(LVOT). Real-time 3DE is a potentially valuable clinical tool for assessing patients with HCM.

NASA Discipline Cardiopulmonary↗
DOE OSTI2023

SEMI-EMPIRICAL MODELING OF IRRADIATION-INDUCED DIMENSIONAL CHANGE IN H-451 NUCLEAR GRAPHITE

Nuclear graphite has been used as a moderator material in nuclear reactor designs dating back to the first reactor to reach criticality, Chicago Pile 1, in 1942. In addition, it is anticipated to be used in the conceptual Generation four (GenIV) Molten-salt reactors (MSRs) and the High-temperature gas-cooled reactors (HTRs). The macroscopic dimensional change observed in irradiated nuclear graphite is a property change of significant importance. Largely, volumetric change provides valuable insight into the in-service lifetime of graphite components used in nuclear reactors. The dimensional change behavior varies amongst each grade of nuclear graphite due to processing techniques and the resulting microstructure. In this work, historic data for nuclear graphite H-451 is revisited. A semi-empirical methodology is proposed to describe the dimensional change behavior as a function of temperature for nuclear graphite H-451. The turnaround dose, or when there is a reversal of the dimensional change from contraction to expansion, is proposed to be a thermally activated process and thus can be described by an Arrhenius model. On the atomic scale, H-451 is sp2-bonded carbon atoms with some degree of disorder regardless of orientation. Towards that end, the activation energy is assumed to be a constant irrespective of orientation.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2020

A Three‐Dimensional Porous Organic Semiconductor Based on Fully sp 2 ‐Hybridized Graphitic Polymer

Abstract Dimensionality plays an important role in the charge transport properties of organic semiconductors. Although three‐dimensional semiconductors, such as Si, are common in inorganic materials, imparting electrical conductivity to covalent three‐dimensional organic polymers is challenging. Now, the synthesis of a three‐dimensional π‐conjugated porous organic polymer (3D p‐POP) using catalyst‐free Diels–Alder cycloaddition polymerization followed by acid‐promoted aromatization is presented. With a surface area of 801 m 2 g −1 , full conjugation throughout the carbon backbone, and an electrical conductivity of 6(2)×10 −4 S cm −1 upon treatment with I 2 vapor, the 3D p‐POP is the first member of a new class of permanently porous 3D organic semiconductors.

Byun, Yearin↗
DOE OSTI2025

Five-dimensional spinor helicity for all masses and spins

We develop a spinor helicity formalism for five-dimensional scattering amplitudes of any mass and spin configuration. While five-dimensional spinor helicity variables have been previously studied in the context of $\mathcal{N}$ = 2, 4 supersymmetric Yang-Mills scattering amplitudes with spin less than two [1], we propose an alternative viewpoint that stems from d-dimensional spinor helicity variables avoiding the decomposition of a massive momentum into the sum of two massless momenta. By enumerating all possible independent little group tensors, we systematically build the full space of five-dimensional three-point tree-level scattering amplitudes for any configuration of spins and masses. Furthermore, we provide a prescription for computing the high energy limit of scattering amplitudes written in our spinor helicity variables. We also expect that our formalism will be applicable to effective field theories with higher spin, in particular, the scattering of highly spinning black holes in five dimensions.

Field Theories in Higher Dimensions↗
DOE OSTI2021

Radiation damage of a two-dimensional carbon fiber composite (CFC)

Carbon fiber composites (CFC) are, among other applications, already in use as low-impedance collimating elements for intercepting TeV-level protons machines, like the CERN Large Hadron Collider. Towards a comprehensive understanding of these materials’ properties and behavior, a series of radiation damage campaigns have been undertaken in order to quantitively investigate the response of AC-150 K CFC to high proton fluences. The CFCs dimensional changes, stability, micro-structural evolution, and structural integrity below and above 5 × 10 20 p/cm 2 fluence threshold were studied. The irradiating particles were protons of kinetic energies in the range of 130–200 MeV. The effects of the irradiation temperature (~90–180°C and 280–500°C) combined with the proton fluence on the dimensional stability and the microstructural evolution of this anisotropic two-dimensional composite structure were studied using precision dilatometry and X-ray diffraction. Our results shed light on the fluence-based limitations for these composite materials. Furthermore, we compare these results with reference unirradiated graphite, and we discuss the similarities and differences in the dimensional changes and post-irradiation annealing properties.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Data-driven projection pursuit adaptation of polynomial chaos expansions for dependent high-dimensional parameters

Uncertainty quantification (UQ) and inference involving a large number of parameters are valuable tools for problems associated with heterogeneous and non-stationary behaviors. The difficulty with these problems is exacerbated when these parameters are statistically dependent requiring statistical characterization over joint measures. Probabilistic modeling methodologies stand as effective tools in the realms of UQ and inference. Among these, polynomial chaos expansions (PCE), when adapted to low-dimensional quantities of interest (QoI), provide effective yet accurate approximations for these QoI in terms of an adapted orthogonal basis. These adaptation techniques have been cast as projection pursuits in Gaussian Hilbert space in what has been referred to as a projection pursuit adaptation (PPA) by Xiaoshu Zeng and Roger Ghanem (2023). The PPA method efficiently identifies an optimal low-dimensional space for representing the QoI and simultaneously evaluates an optimal PCE within that space. The quality of this approximation clearly depends on the size of the training dataset, which is typically a function of the adapted reduced dimension. Here, the complexity of the problem is thus mediated by the complexity of the low-dimensional quantity of interest and not the complexity of the high-dimensional parameter space.

Data-driven↗
DOE OSTI2022

Incorporating two-dimensional conduction modeling techniques into an energy simulation program: The EnergyPlus radiant system example

Due to the inherent complexity of multi-dimensional solutions for transient conduction in a multilayered building element, most heat balance-based algorithms assume that conduction occurs only in one dimension—through the building element between the inside and outside environments. Usually, since the length and width of the elements are much greater than the thickness of the element, this is a valid assumption. However, there are some cases such as hydronic radiant slabs where this assumption may not lead to an accurate estimate of the energy consumption of the system. In a hydronic radiant slab, water tubes are embedded in a slab, and hot or chilled water is circulated to meet any heating or cooling loads. The energy consumption of such a system will depend highly on the temperature of the water being sent to the slab (i.e., produced by the boiler or chiller). The temperature of the water that must be sent to the slab will depend on the spacing of the tubes, the multilayer construction of the building element, and the boundary conditions to which the construction is exposed. Thus, to accurately predict the energy consumption of such a system, a transient two-dimensional heat conduction model must be integrated with a heat balance-based simulation. This paper demonstrates the potential loss of accuracy associated with the one-dimensional radiant system models by providing a comparison of one- and two-dimensional solutions using EnergyPlus.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2021

Feasibility of DEIM for retrieving the initial field via dimensionality reduction

When parameter estimation is solved in a high-dimensional space, the dimensionality reduction strategy becomes the primary consideration for alleviating the tremendous computational cost. Here, the discrete empirical interpolation method (DEIM) is explored to retrieve the initial condition (IC) by combining the polynomial chaos (PC) based ensemble Kalman filter (i.e. PC-EnKF), where a non-intrusive PC expansion is considered as a surrogate model in place of the forward model in the prediction step of the ensemble Kalman filter, resulting in fewer forward model integrations but with a comparable accuracy as Monte Carlo-based approaches. The DEIM acts as a hyper-reduction tool to provide the low-dimensional input for the high-dimensional initial field, which can be reconstructed using the information on the sparse interpolation grid points that is adaptively obtained through PC-EnKF data assimilation method. Thus an innovative framework to reconstruct the IC is developed. The detailed procedure at each assimilation iteration includes: the determination of the spatial interpolation points, the estimation of the initial values on the interpolation locations using the optimal observations, and the reconstruction of IC in the full space. The current study uses the reconstruction field of initial conditions of the Navier-Stokes equations as an example to illustrate the efficacy of our method. The experimental results demonstrate the proposed algorithm achieves a satisfactory reconstruction for the initial field. The proposed method helps to extend the applicable area of DEIM in solving inverse problems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Affinity propagation clustering of full-field, high-spatial-dimensional measurements for robust output-only modal identification: A proof-of-concept study

Determination of the model order is a challenging problem in system identification, especially in output-only or operational modal identification where some modes are weakly excited. While existing methods such as the stabilization diagram method (spectral information) are effective, they do not scale to high-dimensional data, which is usually needed for high-fidelity characterization of structural dynamics and has been made available in the emerging full-field measurement techniques using optical methods such as photogrammetry and laser vibrometers. In this proof-of-concept study we present a new non-parametric, data-driven approach for robust output-only identification of high-spatial-dimensional modal parameters of basic structures by efficiently processing and interactively exploiting the full-field measurement (i.e., very dense spatial measurement points). Specifically, we first over-estimate the system model once, producing a pool of candidate modes associated with their modal frequencies and full-field, high-spatial-dimensional mode shapes. This is accomplished by a data-driven method termed affinity propagation clustering (APC), where the active clusters, which are the active modes in our formulations, emerge from the "message-passing" procedure and does not require a pre-determination of the cluster number (mode or model order). Next, rather than using the spectral information to distinguish the physical and spurious modes in the stabilization diagram method, we exploit and visualize the spatial, full-field mode shape associated with each candidate mode to do so. We conduct extensive experiments on basic structural models with comparisons to a few existing methods. The results indicate that the new method is computationally efficient for identifying high-spatial-dimensional modal parameters, and robust to identify weak modes by exploiting the full-field measurement. We also discuss its applicability and limitations for structures with complex geometry (shapes).

42 ENGINEERING↗
DOE OSTI2025

Bayesian learning with Gaussian processes for low-dimensional representations of time-dependent nonlinear systems

This work presents a data-driven method for learning low-dimensional time-dependent physics-based surrogate models whose predictions are endowed with uncertainty estimates. We use the operator inference approach to model reduction that poses the problem of learning low-dimensional model terms as a regression of state space data and corresponding time derivatives by minimizing the residual of reduced system equations. Standard operator inference models perform well with accurate training data that are dense in time, but producing stable and accurate models when the state data are noisy and/or sparse in time remains a challenge. Another challenge is the lack of uncertainty estimation for the predictions from the operator inference models. Our approach addresses these challenges by incorporating Gaussian process surrogates into the operator inference framework to (1) probabilistically describe uncertainties in the state predictions and (2) procure analytical time derivative estimates with quantified uncertainties. The formulation leads to a generalized least-squares regression and, ultimately, reduced-order models that are described probabilistically with a closed-form expression for the posterior distribution of the operators. The resulting probabilistic surrogate model propagates uncertainties from the observed state data to reduced-order predictions. Furthermore, we demonstrate the method is effective for constructing low-dimensional models of two nonlinear partial differential equations representing a compressible flow and a nonlinear diffusion–reaction process, as well as for estimating the parameters of a low-dimensional system of nonlinear ordinary differential equations representing compartmental models in epidemiology.

Data-driven model reduction↗
DOE OSTI2021

Quasi-One-Dimensional Structure and Possible Helical Antiferromagnetism of RbMn 6 Bi 5

Quasi-one-dimensional materials exhibit not only unique crystal structure but also abundant physical properties such as charge density wave, Luttinger liquid, and superconductivity. In this study, we report the discovery, structure, and physical properties of a new manganese-based quasi-one-dimensional material RbMn 6 Bi 5 , which crystallizes in a monoclinic space group C 2/ m (No. 12) with lattice parameters a = 23.286(5) Å, b = 4.6215(9) Å, c = 13.631(3) Å, and β = 125.00(3)°. The structure features [Mn 6 Bi 5 ] –1 double-walled column extending along the [010] direction, together with Bi–Bi homoatomic bonds linking the columns and the countercation Rb + . The temperature-dependent resistivity clearly indicates a significant resistivity anisotropy for RbMn 6 Bi 5 , whereas the magnetic susceptibility and specific heat measurements show that RbMn 6 Bi 5 is antiferromagnetic below 82 K. The density functional theory calculations indicate that RbMn 6 Bi 5 is a quasi-one-dimensional metal with possible helical antiferromagnetic configuration. The discovery of RbMn 6 Bi 5 confirms the viability of discovering new quasi-one-dimensional materials in manganese-based compounds.

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

Dependence of Exciton Spin Dynamics on Quantum Confinement Dimensionality in CsPbBr 3 Nanocrystals

Semiconductor nanomaterials offer a promising platform to produce optically addressable spins for use in quantum technologies. Here, in this study, we examine CsPbBr 3 nanospheres, cubes, and rods spanning the zero-dimensional (0D) to three-dimensional (3D) transition to investigate the influence of dimensionality and shape on exciton spin decoherence. Using circularly polarized transient absorption spectroscopy, we find that the spin relaxation rate is independent of the surface to volume ratio and instead follows a dependence on the length of the shortest dimension. Additionally, differences in surface quality and termination appear to have no effect on the spin relaxation rate for measured materials, and instead the spin relaxation rate is most clearly correlated with the exciton binding energy. Finally, decreasing the dimensionality of the nanomaterials decreases the influence of multiexciton interactions on the spin relaxation rate.

CsPbBr3 nanocrystals↗
DOE OSTI2021

Single-Particle Spectroscopy as a Versatile Tool to Explore Lower-Dimensional Structures of Inorganic Perovskites

The remarkable defect tolerant nature of inorganic cesium halide perovskites leading to near unity photoluminescence (PL) quantum yield (QY) and narrow emission linewidth across the entire visible spectrum, have provided a tantalizing platform for a development of a plethora of light-emitting applications. Recently, lower dimensional (2D, 1D, and 0D) perovskites have attracted further attention due to the enhanced thermal, photo and chemical stability as compared to their three-dimensional (3D) analogues. Combination of the external size quantization and internal octahedral organization provides a unique opportunity to study and harness “multidimensional” electronic properties engineered both on atomic scale and the nanoscale. However, crucial research to understand the elementary charge carrier dynamics in lower dimensional perovskites lags far behind the enormous effort to incorporate them into optoelectronic devices. In this Perspective, we provide a review of recent developments that focus on studies of the dynamics of excitonic complexes in Cs-based perovskite nanocrystals using single-particle time-resolved PL spectroscopy and photon correlation measurements. Single photon statistical studies not only offer the unprecedented level of detail to directly assess various recombination pathways but also provide insights into specifics of carrier’s localization. Further, we discuss the underlying physicochemical processes that govern PL emission and draw attention to a number of attributes within this class of the materials, especially lower-dimensional perovskites that may indicate the common origin of the PL emission, as well as provide a route map for the vast unexplored territories where single particle spectroscopy can be a powerful tool to unravel crucial information.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Scaling Laws of Exciton Recombination Kinetics in Low Dimensional Halide Perovskite Nanostructures

Carrier recombination is a crucial process governing the optical properties of a semiconductor. Although various theoretical approaches have been utilized to describe carrier behaviors, a quantitative understanding of the impact of defects and interfaces in low dimensional semiconductor systems is still elusive. Here, we develop a model system consisting of chemically tunable, highly luminescent halide perovskite nanocrystals to illustrate the role of carrier diffusion and material dimensionality on the carrier recombination kinetics and luminescence efficiency. Our advanced synthetic methods provide a well-controlled colloidal system consisting of nanocrystals with different aspect ratios, halide compositions, and surface conditions. Using this system, we reveal the scaling laws of photoluminescence quantum yield and radiative lifetime with respect to the aspect ratio of nanocrystals. The scaling laws derived herein are not only a phenomenological observation but proved a powerful tool disentangling the carrier dynamics of microscopic systems in a quantitative and interpretable manner. The investigation of our model system and theoretical formulation bring to light the dimensionality, as a hidden constraint on carrier dynamics, and identify the diffusion length as an important parameter that distinguishes nanoscale and macroscale carrier behaviors. Furthermore, the conceptual distinction in carrier dynamics in different dimensionality regimes informs new design rules for optical devices where complex microstructures are involved.

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

Dimensional Evolution Guides Property Control in the A n Cu 4– n TiS 4 Semiconductor Series

Through progressive reduction of the three-dimensional (3D) covalent network of Cu 4 TiS 4 , we isolate seven new members of the A n Cu 4–n TiS 4 family (A = alkali metal; n = 0–4), spanning 3D, 2D, 1D, and 0D structural fragments. The dimensional reduction is rational, as it preserves the edge-sharing connectivity between [CuS 4 ] 7– and [TiS 4 ] 4– tetrahedra across the series. This structural evolution is driven by the stepwise substitution of Cu with alkali metals, guiding the formation of fragments with reduced dimensionality. The effects of “n” and “A” on the crystal structures, stabilities, electronic structures, and optoelectronic properties are profound, demonstrating that the manipulation of alkali metal size and A n Cu 4–n TiS 4 stoichiometry enables predictable variations in structure and properties. For example, the n = 0 and n = 4 end members of the A n Cu 4–n TiS 4 family set the range of achievable band gaps with 2.00 eV for Cu 4 TiS 4 , 2.60 eV for Na 4 TiS 4 , and intermediate values for the n = 1–3 members. Notably, CsCu 3 TiS 4 exhibits exceptional air stability and congruent melting, with density functional theory (DFT) calculating moderate hole and electron effective masses in specific crystallographic directions (mh = 1.24m 0 , me = 0.87m 0 ). Additionally, A 3 CuTiS 4 (A = Na, K, Rb) displays direct band gap behavior and long photoluminescence lifetimes of 2.3–8.6 μs, and K 3 CuTiS 4 has a PLQY of 5.19%. These findings underscore the potential of the A n Cu 4–n TiS 4 family for applications in optoelectronics and demonstrate widely applicable design concepts that unveil rational stoichiometries within a given composition space to generate a series of crystal structures related through an evolving covalent dimensionality that corresponds to a predictable electronic structure and property progression.

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

Dimensional Reduction Guides Electronic Structure Evolution in the A n Cu 4–n SnS 4 Semiconductor Series

The search for new functional materials with tunable properties remains a central challenge in chemistry, particularly for applications in energy and electronics. In this work, we present a framework for predictive crystal design in alkali metal chalcogenides that enables controlled dimensional reduction of a parent covalent motif, yielding a broad range of electronic structures, which systematically evolve from one parent to the other. We present 11 new members of the A n Cu 4–n SnS 4 family (A = alkali metal; n = 0–4), which reduce the three-dimensional (3D) covalent network of Cu 4 SnS 4 into various 3D, 2D, 1D, and 0D [Cu 4–n SnS 4 ] n− motifs through the substitution of Cu with alkali metals of various radii. The end members of the family set the range in achievable band gaps at 0.99 eV for fully covalent Cu 4 SnS 4 (n = 0) and 3.38 eV for K 4 SnS 4 (n = 4) with 0D [SnS 4 ] n− tetrahedra. As the dimensionality of [Cu 4–n SnS 4 ] n− systematically reduces within A n Cu 4–n SnS 4 (n = 1–3), a stepwise increase in band gap energy occurs through a gradual decrease in the energy of the valence band maximum and an increase in the conduction band minimum, with an increase in the effective masses of charge carriers. Furthermore, irrespective of the alkali metal, the thermal stability decreases with decreasing [Cu 4–n SnS 4 ] n− dimensionality within the quaternary members. Most importantly, we demonstrate that predictable crystal structure and property evolution for a given composition space is possible by deriving a general formula based on substituting the covalent metals of a parent structure with alkali metals.

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

Graphene-driven correlated electronic states in one dimensional defects within WS2

Tomonaga-Luttinger liquid (TLL) behavior in one-dimensional systems has been predicted and shown to occur at semiconductor-to-metal transitions within two-dimensional materials. Reports of one-dimensional defects hosting a Fermi liquid or a TLL have suggested a dependence on the underlying substrate, however, unveiling the physical details of electronic contributions from the substrate require cross-correlative investigation. Here, we study TLL formation within defectively engineered WS2 atop graphene, where band structure and the atomic environment is visualized with nano angle-resolved photoelectron spectroscopy, scanning tunneling microscopy and spectroscopy, and non-contact atomic force microscopy. Correlations between the local density of states and electronic band dispersion elucidated the electron transfer from graphene into a TLL hosted by one-dimensional metal (1DM) defects. It appears that the vertical heterostructure with graphene and the induced charge transfer from graphene into the 1DM is critical for the formation of a TLL.

Rossi, Antonio↗
DOE OSTI2024

Two-dimensional-lattice-confined single-molecule-like aggregates

Intermolecular distance largely determines the optoelectronic properties of organic matter. Conventional organic luminescent molecules are commonly used either as aggregates or as single molecules that are diluted in a foreigner matrix. They have garnered great research interest in recent decades for a variety of applications, including light-emitting diodes, lasers and quantum technologies, among others. However, there is still a knowledge gap on how these molecules behave between the aggregation and dilution states. Here we report an unprecedented phase of molecular aggregate that forms in a two-dimensional hybrid perovskite superlattice with a near-equilibrium distance, which we refer to as a single-molecule-like aggregate (SMA). By implementing two-dimensional superlattices, the organic emitters are held in proximity, but, surprisingly, remain electronically isolated, thereby resulting in a near-unity photoluminescence quantum yield, akin to that of single molecules. Moreover, the emitters within the perovskite superlattices demonstrate strong alignment and dense packing resembling aggregates, allowing for the observation of robust directional emission, substantially enhanced radiative recombination and efficient lasing. Molecular dynamics simulations together with single-crystal structure analysis emphasize the critical role of the internal rotational and vibrational degrees of freedom of the molecules in the two-dimensional lattice for creating the exclusive SMA phase. Furthermore, this two-dimensional superlattice unifies the paradoxical properties of single molecules and aggregates, thus offering exciting possibilities for advanced spectroscopic and photonic applications.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗