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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 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Efficient Control of Atom Arrangement in Ternary Metal Chalcogenide Nanoparticles Using Precursor Oxidation State

Controlling both the concentration and the distribution of elements in a given material is often crucial to extracting and optimizing synergistic properties of the various constituents. An interesting class of such multielement materials is metal chalcogenide nanoparticles, which exhibit a wide range of composition-dependent optoelectronic properties including both bandgap-mediated processes and localized surface plasmon resonance properties, each of which is useful in applications ranging from energy conversion to sensing. Because metal chalcogenide nanoparticles can support several different metal elements in a variety of chalcogen lattices, this material class has particularly benefited from the ability to control both atom concentration and atom arrangement to tailor final particle properties. The primary method to access complex, multimetallic chalcogenide particles is via a postsynthetic cation exchange strategy. One-pot syntheses have been less explored to access these complex particles, although this route is desirable for economy and scalability. In this work, we compare the composition and morphology outcomes from cation exchange and one-pot preparation approaches using a Cu/Ag/Se system, which is already known to exhibit both binary and ternary metal chalcogenide phases. We show that at similar concentrations of the two metal cations, initial reaction conditions for the one-pot method yield multicomponent nanoparticles, whereas cation exchange yields homogeneous ternary metal chalcogenide structures. We then show that by tuning the precursor oxidation state for the one-pot method, this approach can be used to access homogeneous ternary metal chalcogenide particles that are similar in atom arrangement to the particles obtained using cation exchange. Taken together, our results demonstrate reliable synthetic methods that yield a variety of controlled compositions and composition morphologies in the Cu/Ag/Se system. Importantly, we demonstrate that this entire collection of architectures can all be accessed via a one-pot method simply by modifying metal precursor chemistry. The mechanistic insights gained and the resulting streamlined syntheses outlined indicate pathways to easily scaled, highly tailorable syntheses for rapid translation into downstream technologies.

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

Volumetric Shaping of Nanoparticle-DNA Crystals by Light-Induced Milling

DNA-programmable self-assembly enables the formation of nanoparticle crystals with controlled lattice symmetry. While this approach offers the formation of complexly ordered nanostructures for optical, mechanical, and biological applications, a mesoscale control over such nanomaterials is limited. Directing the material formation process through the assembly pathway or external fields allows for modulating crystal morphology, but achieving arbitrary morphology remains challenging. Here, we present a photothermal method for shaping 3D DNA-programmable crystals of gold nanoparticles. Through local heating of nanoparticles due to plasmonic light absorption, we induce targeted volumetric dissolution of specifically defined crystal areas with micron-scale accuracy. This technique effectively prescribes crystal shaping and creates arbitrarily shaped voids within crystals. We further investigate both computationally and experimentally the key factors governing volumetric material subtraction. The developed automated light-milling platform enables the fabrication of nanomaterials exhibiting both DNA-programmable nanoscale order and custom-designed mesoscale architecture.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Ultimate compressive strength and severe plastic deformation of equilibrated single-crystalline copper nanoparticles

Mechanical properties and deformation mechanisms of defect-free copper nanoparticles are investigated by combining experiments with atomistic simulations. The compressive strength of the particles increases with decreasing size and tends to saturate near the theoretical strength in the small-size limit. In this limit, the intrinsic size dependence of the strength is governed by the stochastic nature of dislocation nucleation near the particle surface. The particle deformation process evolves from the initial strain softening to strain hardening as the particle accumulates residual damage. The normalized strength-size relation for Cu is compared with those for Au, Ni, and Pt. The lack of universal behavior among the four FCC metals is discussed. Heavily deformed Cu nanoparticles develop polycrystalline structures and change the lattice orientation from [111] to [110]. The experiments and simulations reveal the twinning mechanism of the lattice rotation leading to the new grain formation.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Ultimate compressive strength and severe plastic deformation of equilibrated single-crystalline copper nanoparticles

Mechanical properties and deformation mechanisms of defect-free copper nanoparticles are investigated by combining experiments with atomistic simulations. The compressive strength of the particles increases with decreasing size and tends to saturate near the theoretical strength in the small-size limit. In this limit, the intrinsic size dependence of the strength is governed by the stochastic nature of dislocation nucleation near the particle surface. The particle deformation process evolves from the initial strain softening to strain hardening as the particle accumulates residual damage. The normalized strength-size relation for Cu is compared with those for Au, Ni, and Pt. The lack of universal behavior among the four FCC metals is discussed. Heavily deformed Cu nanoparticles develop polycrystalline structures and change the lattice orientation from [111] to [110]. The experiments and simulations reveal the twinning mechanism of the lattice rotation leading to the new grain formation.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Size-Dependent Optical Band Gaps in Metal–Organic Framework Nanoparticles

Decades of research into size-dependent semiconductor optical gaps have focused on quantum confinement as the dominant mechanism. Emerging reports indicate that lattice strain─intentional or incidental─can impart optical shifts similar or greater in magnitude. Here, we report evidence of optical absorption and photoluminescence spectra of M(1,2,3-triazolate)2 (M = Mg, Cr, Mn, Fe, Co, Cu, Zn, or Cd) nanoparticles that blueshift from bulk values with decreasing particle sizes in a manner that defies explanation by conventional quantum confinement. Here, the phenomenon persists for particle sizes as large as 200 nm, whereas quantum confinement generally ceases beyond 20–30 nm diameters and follows a weaker dependence on the particle radius. Computational simulations and crystallographic analysis suggest that this behavior arises from size-dependent changes to metal–linker bonding that manifest in strain values comparable to literature reports of strain-induced optical shifts in other classes of materials. This behavior appears beyond this family of materials in other notable examples of metal–organic frameworks (MOFs), including the well-studied Cu3(trimesate)2 (CuBTC), where smaller sizes correlate with blueshifted optical gaps. Taken together, these results represent one of the few examples of size-dependent strain in crystalline materials and reinforce the emerging view that MOFs become softer materials when isolated as nanoparticles.

Electrical conductivity↗
DOE OSTI2021

Intervening Oxygen Enabled Magnetic Moment Modulation in Spinel Nanostructures

Oxygen vacancies and lattice oxygen on the surface of nanoparticles play crucial roles in the magnetism of transition metal oxides. A fundamental understanding of the interactions between the 2p orbitals of oxygen anions and the 3d orbitals of metal cations is of immense interest for applications due to specific electronic, magnetic, and chemical properties. The spinel nanostructures Co 3 O 4 ,Fe 3 O 4 , and CoFe 2 O 4 present a unique model to alter coordination and bonding due to d electrons and oxygen 2p states. We performed complementary experimental techniques and corresponding model calculations to show that the overall net magnetism differs significantly between the O 2 -rich and -deficient environments. It was found that in monometallic spinel oxides (e.g., Co 3 O 4 and Fe 3 O 4 ), the changes in the spin moment are relatively small (~0.08μ B per formula unit) compared to those in the bimetallic spinel oxide CoFe 2 O 4 (~0.40 μ B per formula unit). Finally, our study illustrates that the O 2 -rich/deficient conditions alter the exchange of O h sites and adjust the metal ion moments so as to impact the overall magnetism.

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

Unveiling Structural Heterogeneity and Imbalance of Gold Decahedral Nanoparticles using Four-dimensional Scanning Transmission Electron Microscopy

Multi-twinned structures have been observed in technologically important crystal systems, for example diamond cubic and face-centered cubic (FCC) lattices, that include materials such as diamond, silicon, a wide range of noble metals, and their nanoscale counterparts. Beyond atomic building blocks, the special arrangements also occur in the self-assembly of nanoparticles (NP) and μm-sized colloidal particles and occupy parts of their phase diagrams. Spanning a wide range of length scales, the universality of the structures arises when the systems attempt to achieve multitwinned structures by overcoming geometric misfits during minimizing surface energies with entirely {111} or close-packing facets. While it is fundamental to understand how strain is sustained upon twinned structures and symmetry breaking, the knowledge will be paramount in practical aspects such as guiding and controlling the thin film growth, anisotropic NP growth, and self-assembly of NPs. Au decahedral (Dh) NP, as the most prevalent multi-twinned model system, fits five tetrahedral motifs into a circle by sharing an axis resulting in a geometric misfit angle of 7.35°, or a disclination with power of -7.35°. Postulating how Au FCC lattice adopts the misfit, theoretical models have been developed to address the underlying lattice symmetry and inhomogeneous strain distribution separately. Yet, experimental reports regarding the former have been limited due to the relatively large X-ray beam sizes that do not fit the sizes of NPs. On the other hand, though the latter has been widely adapted in the thermodynamics of small (<10 nm) multi-twinned nanoparticles, previous literature has shown that, at edge length of 17 nm, the theory’s is invalidated by shear strain that is observed in a defect-free Au Dh NP by high-resolution transmission electron microscopy (HR-TEM) imaging. Though the advancement of aberrationcorrected scanning transmission electron microscopy (AC-STEM) imaging and ab initio calculation techniques brings new opportunities, along with challenges in complicated image analysis and limitation in particle size (usually below 10 nm), the gap between nanoscale and mesoscale has never been extended to gain insight from atomic system with straightforward interaction potentials.

4D-STEM↗
DOE OSTI2025

Ultrafast Photoflash Synthesis of High-Entropy Oxide Nanoparticles

High-entropy metal oxides (HEOs) have recently received growing attention for broad energy conversion and storage applications due to their tunable properties. HEOs typically involve the combination of multiple metal cations in a single oxide lattice, thus bringing distinctive structures, controllable elemental composition, and tunable functional properties. Many synthesis methods for HEOs have been reported, such as solid-state reactions and carbon thermal shock methods. These methods frequently are energy-intensive or require relatively expensive heating equipment. In this work, we report an ultrafast photoflash synthesis method for HEO nanoparticles on diverse substrates. The energy input is provided by a commercial Xe photoflash unit, which triggers exothermic reactions to convert metal salt precursors to HEO nanoparticles within tens of milliseconds. The formation of HEO nanoparticles is attributed to the ultrafast heating (∼106 K/s) and cooling (∼105 K/s) rates of the photoflash and overall high temperature (>1000 K) during the ultrafast synthesis process. When the synthesized CoNiFeCrMn oxide (HEO) is tested as an oxygen evolution reaction electrocatalyst, it shows similar activity to similar materials prepared by other methods. We believe this photoflash synthesis provides a simple method for many others to synthesize diverse HEOs and explore their properties and potential applications.

exothermic reaction synthesis↗
DOE OSTI2025

Alternate InP synthesis with aminophosphines: solution–liquid–solid nanowire growth

Indium phosphide nanowires are important components in high-speed electronics and optoelectronics, including photodetectors and photovoltaics. However, most syntheses either use high-temperature and costly vapor-phase methodology or highly toxic and pyrophoric tris(trimethylsilyl)phosphine. To expand on the success of the aminophosphine-based InP colloidal quantum dot synthesis, we developed a synthesis for thin (~11 nm) zinc blende InP nanowires at 180 °C using indium tris(trifluoroacetate) and tris(diethylamino)phosphine. A flat nanoribbon morphology was identified by transmission electron and atomic force microscopy analysis, with the stoichiometric (110) lattice plane exposed. Nanowire growth proceeded through a solution–liquid–solid mechanism from in situ-formed indium metal nanoparticles. Molecular byproducts of tris(oleylamino)phosphine oxide and N-oleyltrifluoroacetamide observed by 31 P and 19 F NMR spectroscopy inform a proposed mechanism of indium reduction by the aminophosphine. Morphological control over the nanowire product was achieved by varying the phosphorus injection to control the aspect ratio, the In : P ratio to toggle between nanowires and multipods, and the pre-hot injection evacuation step to favor a quantum dot product. Furthermore, replacing the indium precursor with indium tris(trifluoromethanesulfonate) was found to make bulk zinc blende InP nanowires with an average diameter of >250 nm and tens of microns in length.

77 NANOSCIENCE AND NANOTECHNOLOGY↗
DOE OSTI2023

Influence of aluminum incorporation and aqueous conditions on metal ion release of high-Ni transition metal oxide nanomaterials

Developing a materials perspective of how to control the degradation and negative impact of complex metal oxides requires an integrated understanding of how these nanomaterials transform in the environment and interact with biological systems. Doping with aluminum is known to stabilize oxide materials, but has not been assessed cohesively from synthesis to environmental fate and biological impact. In the present study, the influence of aluminum doping on metal ion release from transition metal oxides was investigated by comparing aqueous transformations of lithium nickel cobalt aluminum oxide (LiNi 0.82 Co 0.15 Al 0.03 O 2 ; NCA) and lithium nickel cobalt oxide (LiNi 0.80 Co 0.20 O 2 ; NC) nanoparticles and by calculating the energetics of metal release using a density functional theory (DFT) and thermodynamics method. Two model environmental organisms were used to assess biological impact, and metal ion release was compared for NCA and NC nanoparticles incubated in their respective growth media: moderately hard reconstituted water (MHRW) for the freshwater invertebrate Daphnia magna (D. magna) and minimal growth medium for the Gram-negative bacterium Shewanella oneidensis MR-1 (S. oneidensis). Further, the amount of metal ions released was reduced for NCA compared to NC in MHRW, which correlated to changes in the modeled energetics of release upon Al substitution in the lattice. In minimal medium, metal ion release was approximately an order of magnitude higher compared to MHRW and was similar to the stoichiometry of the bulk nanoparticles for both NCA and NC. Interpretation of the release profiles and modeling indicated that the increase in total metal ion release and the reduced influence of Al doping arises from lactate complexation of metal ions in solution. The relative biological impacts of NC and NCA exposure for both S. oneidensis and D. magna were consistent with the metal release trends observed for minimal medium and MHRW, respectively. Together, these results demonstrate how a combined experimental and computational approach provides valuable insight into the aqueous transformations and biological impacts of complex metal oxide nanoparticles.

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

Interfacial Reactivity-Triggered Oscillatory Lattice Strains of Nanoalloys

Understanding the structure evolution of nanoalloys under reaction conditions is vital to the design of active and durable catalysts. Herein, we report an operando measurement of the dynamic lattice strains of dual-noble-metal alloyed with an earth-abundant metal as a model electrocatalyst in a working proton-exchange membrane fuel cell using synchrotron high-energy X-ray diffraction coupled with pair distribution function analysis. Additionally, the results reveal an interfacial reaction-triggered oscillatory lattice strain in the alloy nanoparticles upon surface dealloying. Analysis of the lattice strains with an apparent oscillatory irregularity in terms of frequency and amplitude using time-frequency domain transformation and theoretical calculation reveals its origin from a metal atom vacancy diffusion pathway to facilitate realloying upon dealloying. This process, coupled with surface metal partial oxidation, constitutes a key factor for the nanoalloy's durability under the electrocatalytic oxygen reduction reaction condition, which serves as a new guiding principle for engineering durable or self-healable electrocatalysts for sustainable fuel cell energy conversion.

Wu, Zhi-Peng [State University of New York (SUNY),↗
DOE OSTI2026

Electro-chemo-mechanically Driven Ni Exsolution from (Pr,Ce,Ni)O 2−δ : Controlled Nucleation Density and Enhanced Electrode Kinetics

In situ exsolution of metal nanoparticles is a promising strategy to prepare electrocatalysts with enhanced activity and resistance to agglomeration for efficient chemical transformations and energy conversion. Achieving a high nucleation density of nanoparticles under mild conditions and understanding how to tailor the process is important for performance of these electrodes in electrochemical cells. In this work, we demonstrate facile exsolution of Ni nanoparticles using fluorite-structured (Pr,Ce)O 2−δ as the support oxide, driven by electrochemical potential and aided by the metastability of Ni in the solid solution (elastic driving force). We prepare single-phase oriented thin films of (Pr,Ce,Ni)O 2−δ (NPCO) on (Zr,Y)O 2−δ (YSZ) substrates by pulsed laser deposition. With the aid of a high-throughput electrochemical cell that provides a lateral gradient in Nernst voltage, we apply in situ near-ambient pressure synchrotron X-ray photoelectron spectroscopy and ex situ atomic force microscopy to investigate the impact of electrochemical potential on Ni nucleation density. We find that metallic Ni can be successfully exsolved at 550 °C upon cathodic biasing in 20 mTorr O 2 , and its nucleation density increases with increasing electrochemical driving force/decreasing oxygen chemical potential. We further evaluate the electrochemical performance under highly reducing (fuel electrode) conditions by electrochemical impedance spectroscopy. With the exsolved Ni nanoparticles, the surface exchange coefficient of the NPCO is found to be ∼4× higher than for PCO without exsolution. This work confirms mixed conducting fluorites as beneficial host lattices for facile transition-metal exsolution and suggests the possibility for constructing an all ceria-based electrochemical cell with PCO serving as both the cathode and the anode.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Discovery and Characterization of a Metastable Cubic Interstitial Nickel–Carbon System with an Expanded Lattice

Metastable, i.e., kinetically favored but thermodynamically not stable, interstitial solid solutions of carbon in iron are well-understood. Carbon can occupy the interstitial atoms of the host metal, altering its properties. Alloying of the host metal results in the stabilization of the FeC x phases, widening its application. Pure nickel finds niche applications, mainly focusing on catalysis, while nickel alloys are widely applied, e.g., in gas turbines, reactors, and seawater piping. Nickel carbide (Ni 3 C) is the well-known stable Ni–C system displaying a rhombohedral (R3̅c) crystal structure. Some reports describe an elusive cubic Ni–C system, observed during certain catalytic reactions occurring on nickel and formed by the occupation of the interstitials of the metal with carbon: to date, the stabilization and characterization of this phase have not been accomplished. Hereby, we report on the synthesis of a cubic metastable NiC x phase using chemical vapor deposition of methane on supported nickel nanoparticles. The structure was predicted by DFT/ReaxFF, synthesized and monitored with in situ time-resolved synchrotron XRD, and experimentally confirmed by Rietveld refinement and (S)TEM-EELS under ambient conditions. The results show an Fm3̅m phase with a lattice parameter of a = 3.749 ± 0.037 Å at room temperature, with the highest ever reported atomic percentage of carbon occupying the octahedral interstices of 23.1%, resulting in a NiC 0.3 phase. The degree of occupation of the interstitial voids by carbon can be controlled, enabling the tuning of the host metal’s d-spacing and composition, highlighting the applicability of this synthesis route for catalytic nanoparticle preparation.

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

Metallization of colloidal crystals

Colloidal crystals formed by size-asymmetric binary particles coassemble into a wide variety of colloidal compounds with lattices akin to ionic crystals. Recently, a transition from a compound phase with a sublattice of small particles to a metal-like phase in which the small particles are delocalized has been predicted computationally and observed experimentally. In this colloidal metallic phase, the small particles roam the crystal maintaining the integrity of the lattice of large particles, as electrons do in metals. A similar transition also occurs in superionic crystals, termed sublattice melting. Here, we use energetic principles and a generalized molecular dynamics model of a binary system of functionalized nanoparticles to analyze the transition to sublattice delocalization in different coassembled crystal phases as a function of temperature (T), number of grafted chains on the small particles, and number ratio between the small and large particles n s : n l . Here we find that n s : n l is the primary determinant of crystal type due to energetic interactions and interstitial site filling, while the number of grafted chains per small particle determines the stability of these crystals. We observe first-order sublattice delocalization transitions as T increases, in which the host lattice transforms from low- to high-symmetry crystal structures, including A20 → bct → bcc, A d → bct → bcc, and bcc → bcc/fcc → fcc transitions and lattices. Analogous sublattice transitions driven primarily by lattice vibrations have been seen in some atomic materials exhibiting an insulator-metal transition also referred to as metallization. We also find minima in the lattice vibrations and diffusion coefficient of small particles as a function of n s : n l , indicating enhanced stability of certain crystal structures for n s : n l values that form compounds.

36 MATERIALS SCIENCE↗
DOE OSTI2026

Controlling Exsolution Dynamics in High‐Entropy Oxides for Highly Active and Selective Acetylene Semi‐Hydrogenation

Exsolution-derived catalysts feature robust metal–support interactions that enhance catalytic performance; yet achieving precise control over exsolution dynamics in multicomponent oxides remains challenging. In this study, we demonstrate that exsolution behavior in high-entropy oxides (HEOs) can be rationally tuned through coupled lattice- and valence-engineering to create a highly active and selective catalyst for acetylene semi-hydrogenation. Incorporation of Li + into a rock salt-structured HEO (LiNiMgCuZnCoO x and LiHEO) induces local lattice distortion, generates oxygen vacancies, and partially oxidizes Co sites from Co 2+ to Co 3+ , collectively modulating local charge redistribution. This strategy enables facilitated Cu nanoparticle exsolution and alters the exsolution sequence from Cu 0 > Ni 0 > Co 0 in pristine HEO to Cu 0 > Co 0 > Ni 0 in the LiHEO. The resulting catalyst via controlled exsolution exhibits superior activity and ethylene selectivity, outperforming state-of-the-art transition metal systems. This work establishes entropy-enabled lattice and valence engineering as a facile route to programmable exsolution for enhanced catalysis.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Investigating Cu-Site Doped Cu–Sb–S Nanoparticles Using Photoelectron and Electron Paramagnetic Resonance Spectroscopy

Tetrahedrite (Cu 12 Sb 4 S 13 ) and famatinite (Cu 3 SbS 4 ) are good candidates for green energy applications because they possess promising thermoelectric and photovoltaic properties as well as contain earth-abundant and nontoxic constituents. Herein, X-ray photoelectron spectroscopy (XPS), ultraviolet photoelectron spectroscopy (UPS), and electron paramagnetic resonance spectroscopy (EPR) methods examined inherent electronic properties and interatomic magnetic interactions of Cu-site doped tetrahedrite and famatinite nanomaterials. An energy-efficient modified polyol method was utilized for the synthesis of tetrahedrite and famatinite nanoparticles doped on the Cu-site with Zn, Fe, Ni, Mn, and Co. This is the first parallel study of tetrahedrite and famatinite nanomaterials with XPS, UPS, and EPR methods alongside a systematic analysis of dopant-dependent effects on the electronic structure and magnetic interactions for each material. XPS showed that the Cu and Sb species in tetrahedrite and famatinite possess different oxidation states, while UPS characterization reveals larger dopant-dependent shifts in the work function for tetrahedrite nanoparticles (4.21 to 4.79 eV) than for famatinite nanoparticles (4.57 to 4.77 eV). Finally, all famatinite nanoparticles display an EPR signal, indicating trace amounts of paramagnetic Cu(II) present below the detection limit of XPS. For tetrahedrite, EPR signatures were observed only for the Zn-doped and Mn-doped nanoparticles, suggesting signal broadening from Cu–Cu spin exchange or spin–lattice relaxation. This study demonstrates the complementary nature of XPS and EPR techniques for studying the oxidation states of metals in solid-state nanomaterials. Comparing the electronic and magnetic properties of tetrahedrite and famatinite while studying the impact of dopant incorporation will guide future endeavors in designing sustainable, high-performance materials for renewable energy applications.

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

In-plane plasmon coupling in topological insulator Bi 2 Se 3 thin films

The surface states of the 3D topological insulator (TI), Bi 2 Se 3 , are known to host two-dimensional Dirac plasmon polaritons (DPPs) in the terahertz spectral range. In TI thin films, the DPPs excited on the top and bottom surfaces couple, leading to an acoustic mode and an optical plasmon mode. Vertical coupling in these materials is, therefore, reasonably well-understood, but in-plane coupling among localized TI DPPs has yet to be investigated. In this paper, we demonstrate in-plane DPP coupling in TI stripe arrays and show that they exhibit dipole–dipole type coupling. Here, the coupling becomes negligible when the lattice constant is greater than approximately 2.8 times the stripe width, which is comparable to results for in-plane coupling of localized plasmons excited on metallic nanoparticles or graphene plasmon polaritons. This understanding could be leveraged for the creation of TI-based metasurfaces.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗