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DOE OSTI2024

Proximity-Induced Superconductivity in a 2D Kondo Lattice of an f -Electron-Based Surface Alloy

Realizing hybrids of low-dimensional Kondo lattices and superconducting substrates leads to fascinating platforms for studying the exciting physics of strongly correlated electron systems with induced superconducting pairing. Here, we report a scanning tunneling microscopy and spectroscopy study of a new type of two-dimensional (2D) La–Ce alloy grown epitaxially on a superconducting Re(0001) substrate. We observe the characteristic spectroscopic signature of a hybridization gap evidencing the coherent spin screening in the 2D Kondo lattice realized by the ultrathin La–Ce alloy film on normal conducting Re(0001). Upon lowering the temperature below the critical temperature of rhenium, a superconducting gap is induced exhibiting an energy asymmetry of the coherence peaks that arises from the interaction of residual unscreened magnetic moments with the superconducting substrate. A positive correlation between the Kondo hybridization gap and the asymmetry of the coherence peaks is found.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2021

Photoinduced Electron and Energy Transfer Pathways and Photocatalytic Mechanisms in Hybrid Plasmonic Photocatalysis

Hybrid plasmonic nanostructures are built on plasmonic metalnanostructures surrounded by catalytic metals or metal oxides. We report that recent studies have shown that hybrid plasmonic nanocatalysts can concurrently utilize thermal energy and photon stimuli and exhibit high catalytic activity, selectivity, and stability that are not attainable in conventional purely thermally activated catalytic processes. The hybrid plasmonic photocatalytic approach has recently emerged as an attractive concept for the conversion of solar energy into chemical energy, the distributed synthesis of valuable chemicals such as ammonia with little to no requirement of external heating, and the development of coke-resistant and selective catalytic processes. The field of hybrid plasmonic photocatalysis has grown tremendously in the last decade. In this review article, the advantages of visible-light-augmented hybrid plasmonic photocatalysis over conventional pure thermally activated heterogeneous catalysis are discussed. Fundamental insights are provided into photocatalytic mechanisms by which the photoexcited charge carriers (electrons and holes) are formed and transferred to adsorbates triggering chemical transformations on the surface of hybrid plasmonic nanocatalysts. Computational modeling used for predicting and understanding the photocatalytic activity and selectivity on hybrid plasmonic nanostructures is also reviewed. The review closes with a discussion of the current challenges, new opportunities, and future outlook for hybrid plasmonic photocatalysis.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Chemistry, Structure, and Function of Lone Pairs in Extended Solids

The lone pair has been a known feature of the electronic structure of molecules for over 100 years. Beginning with the pioneering work of Lewis and others that was later developed into useful guidelines for predicting molecular structure, lone pairs and their steric consequences are now taught at the very earliest stages of a chemistry education. In the crystalline solid state, lone pairs have perhaps had a less visible yet equally consequential role, with a significant impact on a range of properties and functionalities. Important properties associated with s 2 electron-derived lone pairs include their role in creating conditions favorable for ion transport, in the formation and correlation of local dipoles and the resulting polar behavior leading to ferroics and multiferroics, in increasing the refractive index of glass, in reducing the thermal conductivity of thermoelectric materials, and in breaking local symmetry permitting second-harmonic light generation.. In recent years, the role of the lone pair in developing the electronic structure of some topological quantum materials has also been recognized. While structural distortions due to lone pairs have traditionally been characterized through their crystallography, recent advances in scattering and spectroscopy have revealed the presence of local lone pair-driven distortions that do not correlate over long length scales. The role of these crystallographically “hidden” lone pairs, their detection, and their impact on properties have become a growing body of work in the literature. Hidden lone pairs are an effective argument for considering a role for lone pairs that goes beyond their being objects that occupy space in the coordination polyhedra of cations. This Account introduces the chemistry of lone pairs in extended crystalline solids, including a discussion of when they are stereochemically active, how they manifest in the structure, and how their chemistry can be tuned by the chemical environment around them. Eventually, all of these factors work in unison to help develop and tune properties of interest. Certain specific examples of structure–property relationships in materials that are driven by lone pair behavior are described here, including the potential impact of lone pairs on the optical and electronic properties of hybrid halide perovskite compounds that are relevant to their photovoltaic applications. Here, we highlight the role of lone pairs in the dielectric behavior of geometrically frustrated pyrochlores, the temperature-dependent optoelectronic behavior of halide perovskites, the polar phase transitions in lead-free ferroelectric perovskites, and the compositional insulator-to-metal transition in ruthenium pyrochlores. The theme underpinning this Account is that the lone pair can be considered to be a powerful design element for a broad range of material function.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Tunable Collective Excitations in Epitaxial Perovskite Nickelates

The formation of plasmons through the collective excitation of charge density has generated intense discussions, offering insights to fundamental sciences and potential applications. While the underlying physical principles have been well-established, the effects of multibody interactions and orbital hybridization on plasmonic dynamics remain understudied. Here, in this work, we present the observation of conventional metallic and correlated plasmons in epitaxial La 1-x Sr x NiO 3 (LSNO) films with varying Sr doping concentrations (x = 0, 0.125, 0.25), unveiling their intriguing evolution. Unlike samples at other doping concentrations, the x = 0.125 intermediate doping sample does not exhibit the correlated plasmons despite showing high optical conductivity. Through experimental investigation using spectroscopic ellipsometry and X-ray absorption spectroscopy, that is further supported by theoretical calculations, the O2p-Ni3d orbital hybridization for x = 0.125 is found to be significantly enhanced, alongside a considerable weakening of its effective interaction comprising long-range Coulomb and variable interaction, U*. These factors account for the absence of correlated plasmons and the high optical conductivity observed in LSNO(0.125). Our findings highlight the significant impact of orbital hybridization on the electronic structures and the formation of quasiparticles in strongly correlated systems, opening new paths for plasmonic-based engineering research.

36 MATERIALS SCIENCE↗

DOE OSTI2021

Astrophysical Explosions Revisited: Collisionless Coupling of Debris to Magnetized Plasma

The coupling between a rapidly expanding cloud of ionized debris and an ambient magnetized plasma is revisited with a hybrid (kinetic ion/fluid electron) simulation code that allows a study over a wide range of plasma parameters. Over a specified range of hypothetical conditions, simple scaling laws in terms of the total debris mass and explosion speed are derived and verified for the maximal size of the debris cloud and the fraction of debris that free-streams from the burst along the magnetic field. Furthermore, the amount of debris that escapes from the burst with minimal coupling to the background magnetic field increases with the debris gyroradius. Test cases with two different debris species—including a heavy minority species with a relatively large gyroradius—highlight how the collisionless coupling of the debris depends on the single particle trajectories as well as the overall conservation of energy and momentum.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2022

Correlative SIP-FISH-Raman-SEM-NanoSIMS links identity, morphology, biochemistry, and physiology of environmental microbes

Microscopic and spectroscopic techniques are commonly applied to study microbial cells but are typically used on separate samples, resulting in population-level datasets that are integrated across different cells with little spatial resolution. To address this shortcoming, we developed a workflow that correlates several microscopic and spectroscopic techniques to generate an in depth analysis of individual cells. By combining stable isotope probing (SIP), fluorescence in situ hybridization (FISH), scanning electron microscopy (SEM), confocal Raman microspectroscopy (Raman), and nano-scale secondary ion mass spectrometry (NanoSIMS), we illustrate how individual cells can be thoroughly interrogated to obtain information about their taxonomic identity, structure, physiology, and metabolic activity. Analysis of an artificial community demonstrated that our correlative approach was able to resolve the activity of single cells using heavy water SIP in conjunction with Raman and/or NanoSIMS and establish their taxonomy and morphology using FISH and SEM. We then applied this workflow to a sample of yet uncultured multicellular magnetotactic bacteria. In addition to establishing their identity and activity, backscatter electron microscopy (BSE), NanoSIMS, and energy-dispersive X-ray spectroscopy (EDS) were employed to characterize the magnetosomes within the cells. By integrating these techniques, we demonstrate a cohesive approach to thoroughly study environmental microbes at single cell resolution.

Schaible, George↗

DOE OSTI2022

Engineering metal–metal oxide surfaces for high-performance oxygen reduction on Ag–Mn electrocatalysts

Understanding fundamental material–property relationships in mixed-element catalyst systems is crucial to advancing the viability of renewable electrochemical energy technologies, an important part of creating a more sustainable future. Herein, we report our insight on the nature and dynamics of highly active silver–manganese oxide (Ag–MnOx) catalyst surfaces for the oxygen reduction reaction (ORR) via a combined experimental–theoretical approach. Experimentally, we synthesize well-mixed Ag–Mn co-deposited thin films that are measurably flat and smooth, despite Mn surface migration and oxidation upon air exposure and electrochemical measurements. Cyclic voltammetry in 0.1 M KOH demonstrates up to 10-fold specific activity enhancements over pure Ag at 0.8 V vs. RHE for Ag-rich films (70–95% Ag in bulk). To further understand the Ag–Mn system, separate samples were synthesized with small amounts of Mn sequentially deposited onto the surface of a pure Ag thin film (Mn@Ag), ranging from partial to full surface coverage (down to 0.3 nm Mn $cm^{-2}_{geo}$ ~ 0.2 μg Mn $cm^{-2}_{geo}$). These sequentially deposited Mn@Ag films show analogous performance to their co-deposited counterparts indicating similar enhanced active sites. With density functional theory (DFT), we calculate that this enhancement arises from the tuned d-band of these material surfaces owing to the optimal hybridization of the electronic structures in specific Ag and MnO x geometries. Together, electrochemical measurements, DFT calculations, X-ray absorption spectroscopy, and valence-band X-ray photoelectron spectroscopy suggest synergistic electronic interactions between Ag and MnOx yield enhanced oxygen adsorption, and thus ORR activity, with DFT highlighting the Ag–MnO x interface sites as the most enhanced. This work demonstrates how combined experimental–theoretical methods can help design electrocatalysts with enhanced electrocatalytic properties and understand the nature of complex mixed metal–metal oxide surfaces.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

A MeerKAT-meets-LOFAR study of Abell 1413: a moderately disturbed non-cool-core cluster hosting a ~500 kpc ‘mini’-halo

Many relaxed cool-core clusters host diffuse radio emission on scales of hundreds of kiloparsecs: mini-haloes. However, the mechanism responsible for generating them, as well as their connection with central active galactic nuclei, is elusive and many questions related to their physical properties and origins remain unanswered. This paper presents new radio observations of the galaxy cluster Abell 1413 performed with MeerKAT (L-band; 872 to 1712 MHz) and LOFAR HBA (120 to 168 MHz) as part of a statistical and homogeneous census of mini-haloes. Abell 1413 is unique among mini-halo clusters as it is a moderately disturbed non-cool-core cluster. Our study reveals an asymmetric mini-halo up to 584 kpc in size at 1283 MHz, twice as large as first reported at similar frequencies. The spectral index is flatter than previously reported, with an integrated value of α = −1.01 ± 0.06, shows significant spatial variation, and a tentative radial steepening. We studied the point-to-point X-ray/radio surface brightness correlation to investigate the thermal/non-thermal connection: our results show a strong connection between these components, with a superlinear slope of $b = 1.63^{+0.10}_{-0.10}$ at 1283 MHz and $b = 1.20^{+0.13}_{-0.11}$ at 145 MHz. We also explore the X-ray surface brightness/radio spectral index correlation, finding a slope of $b = 0.59^{+0.11}_{-0.11}$. Both investigations support the evidence of spectral steepening. Finally, in the context of understanding the particle acceleration mechanism, we present a simple theoretical model which demonstrates that hybrid scenarios – secondary electrons (re-)accelerated by turbulence – reproduce a superlinear correlation slope.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2022

Local magnetic moments due to loop currents in metals

We present Hartree-Fock calculations on a simple model to obtain the conditions of formation of local magnetic moments due to loop currents $L$ $o$ and spin-loop currents $L$ $s$ and compare them to the conditions of formation of local spin moments $M$ which were given long ago in a similar approximation by Anderson. A model with three degenerate orbitals sitting on an equilateral triangle, with on-site and nearest-neighbor repulsions $U$ and $V$, respectively, and intersite kinetic energy, hybridizing with conduction electrons with a parameter $Δ$ is investigated. $L$ $o$ and $L$ $s$ are promoted by large $V$/$Δ$ and their magnitude is relatively unaffected by $U$/$Δ$. Spin-magnetic moments $M$ promoted by large $U$/$Δ$ on the other hand are adversely affected by $V$/$Δ$. In this model, $L$ $o$ for $V$ multiplied by the number of neighbors is approximately the same as the $M$ promoted by $U$ in Anderson’s local model for $M$. $L$ $o$ and $L$ $s$ are degenerate if exchange interactions and Hund’s rule are neglected but $L$ $o$ is favored when they are included. Many of the qualitative results are visible in an expression for the Hartree-Fock ground state energy derived as a function of small $L$ $o$ , $L$ $s$ , and $M$. Numerical minimization of the Hartree-Fock energy is presented for larger values. Additionally, we also briefly discuss the connection and differences of the interaction-generated orbital currents and spin currents discussed here and generalized to a lattice with the topological states in metals and semiconductors.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2024

Multispecies Ion Acceleration in 3D Magnetic Reconnection with Hybrid-Kinetic Simulations

Magnetic reconnection drives multispecies particle acceleration broadly in space and astrophysics. Here we perform the first 3D hybrid simulations (fluid electrons, kinetic ions) that contain sufficient scale separation to produce nonthermal heavy-ion acceleration, with fragmented flux ropes critical for accelerating all species. We demonstrate the acceleration of all ion species (up to Fe) into power-law spectra with similar indices, by a common Fermi acceleration mechanism. The upstream ion velocities influence the first Fermi reflection for injection. The subsequent onsets of Fermi acceleration are delayed for ions with lower charge-mass ratios (Q/M), until growing flux ropes magnetize them. This leads to a species-dependent maximum energy/nucleon ∝(Q/M) α . These findings are consistent with in situ observations in reconnection regions, suggesting Fermi acceleration as the dominant multispecies ion acceleration mechanism.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2020

Direct observation of minibands in a twisted graphene/WS 2 bilayer

Stacking two-dimensional (2D) van der Waals materials with different interlayer atomic registry in a heterobilayer causes the formation of a long-range periodic superlattice that may bestow the heterostructure with properties such as new quantum fractal states or superconductivity. Recent optical measurements of transition metal dichalcogenide (TMD) heterobilayers have revealed the presence of hybridized interlayer electron-hole pair excitations at energies defined by the superlattice potential. The corresponding quasiparticle band structures, so-called minibands, have remained elusive, and no such features have been reported for heterobilayers composed of a TMD and another type of 2D material. We introduce a new x-ray capillary technology for performing microfocused angle-resolved photoemission spectroscopy with a spatial resolution of ~1 μm, and directly observe minibands at certain twist angles in mini Brillouin zones (mBZs). We discuss their origin in terms of initial and final state effects by analyzing their dispersion in distinct mBZs.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2022

Unusual electronic transport in (1 - x )Cu 2 Se-( x )CuInSe 2 hierarchical composites

The ability to control the relative density of electronic point defects as well as their energy distribution in semiconductors could afford a systematic modulation of their electronic, optical, and optoelectronic properties. Using a model binary hybrid system Cu 2 Se–CuInSe 2 , we have investigated the correlation between phase composition, microstructure, and electronic transport behavior in the synthesized composites. We found that both Cu 2 Se and CuInSe 2 phases coexist at multiple length scales, ranging from sub-ten nanometer to several micrometers, leading to the formation of a hybrid hierarchical microstructure. Astonishingly, the electronic phase diagram of the (1 - x)Cu 2 Se–(x)CuInSe 2 (15% ≤ x ≤ 100%) hierarchical composites remarkably deviates from the trend normally expected for composites between a heavily doped semiconductor (Cu 2 Se) and a poorly conducting phase (CuInSe 2 ). A sudden 3-fold increase in the electrical conductivity and carrier concentration along with a marginal increase in the carrier mobility is observed for composites at the vicinity of equimolar composition (48% ≤ x ≤ 52%). The carrier concentration increases from ~1.5 × 10 20 cm -3 for the composites with x ≤ 45% to 5.0 × 10 20 cm -3 for x = 50%, and remains constant at 4.5 × 10 20 cm -3 with x value in the range of 52% < x ≤ 90%, then quickly drops to 8 × 10 18 cm -3 for pristine CuInSe 2 phase (x = 100%). The atypical electronic behavior was rationalized in the light of the formation of an inter-band (IB) within the band gap, which arises from the hybridization of native electronic point defects from both Cu 2 Se and CuInSe 2 phases in the resulting hierarchical composites. The result points to a new strategy to modulate the electronic structure of semiconductor composites to maximize interaction and coupling between two fundamentally contrasting properties enabling access to electronic hybrid systems with potential applications as interactive and stimuli-responsive multifunctional materials.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Hybridization and deconfinement in colloidal quantum dot molecules

The structural, electronic, and optical properties of CdSe/CdS core–shell colloidal quantum dot molecules, a new class of coupled quantum dot dimers, are explored using atomistic approaches. Unlike the case of dimers grown by molecular beam epitaxy, simulated strain profile maps of free-standing colloidal dimers show negligible additional strain resulting from the attachment. Further, the electronic properties of the relaxed dimers are described within a semiempirical pseudopotential model combined with the Bethe–Salpeter equation within the static screening approximation to account for electron–hole correlations. The interplay of strain, hybridization (tunneling splitting), quantum confinement, and electron–hole binding energies on the optical properties is analyzed and discussed. The effects of the dimensions of the neck connecting the two quantum dot building blocks, as well as the shell thickness, are studied.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2022

Quantum electron liquid and its possible phase transition

Purely quantum electron systems exhibit intriguing correlated electronic phases by virtue of quantum fluctuations in addition to electron–electron interactions. To realize such quantum electron systems, a key ingredient is dense electrons decoupled from other degrees of freedom. Here, we report the discovery of a pure quantum electron liquid that spreads up to ~3 Å in a vacuum on the surface of an electride crystal. Its extremely high electron density and weak hybridization with buried atomic orbitals show the quantum and pure nature of the electrons, which exhibit a polarized liquid phase, as demonstrated by our spin-dependent measurement. Furthermore, upon enhancing the electron correlation strength, the dynamics of the quantum electrons change to that of a non-Fermi liquid along with an anomalous band deformation, suggestive of a transition to a hexatic liquid crystal phase. Here, our findings develop the frontier of quantum electron systems and serve as a platform for exploring correlated electronic phases in a pure fashion.

36 MATERIALS SCIENCE↗

DOE OSTI2023

Splitting photons: Singlet fission in nanocrystal-molecule hybrid structures

The goal of this research is to enable all the energy contained in sunlight to be harvested by making full use of the energy contained in the blue and green wavelengths of light. Current systems are unable to extract all of the energy available from photons in this wavelength range due to rapid relaxation processes that dissipate a fraction of the energy as heat. In this work, inexpensive, earth-abundant components capable of supporting multi-excitonic processes involving more than one tightly bound excited state are investigated as a way to exceed the Shockley-Queisser limit. This research will examine hybrid organic-inorganic nanostructures capable of singlet fission, a process by which one high-energy spin-singlet state is converted into two lower-energy spin-triplet states, and subsequent triplet exciton transfer. Specifically, the organic molecules diphenylhexatriene and tetracene that maximally absorb blue light and are known to exhibit singlet fission will be bound to lead chalcogenide nanocrystals. A variety of steady-state and time-resolved spectroscopic techniques will be used to study the transfer of energy from spin-triplet excitons that are created in the organic molecules to the nanoparticle acceptors. The hybrid nanostructures here will be fully characterized both in solution or thin film via electronic absorption and photoluminescence spectroscopy, nuclear magnetic resonance spectroscopy, high-resolution mass spectrometry, transmission electron microscopy, photoelectron spectroscopy, time correlated single photon counting and transient absorption experiments. The effect of molecular and nanocrystal structure on the electronic coupling between the hybrid components will be examined to establish fundamental relationships between structure and triplet energy transfer efficiency. The findings will be directly applicable to a potential tetracene-silicon platform that may ultimately enhance the power conversion efficiency of silicon solar cells.

30 DIRECT ENERGY CONVERSION↗

DOE OSTI2021

Electronic structure modulation of MoS2 by substitutional Se incorporation and interfacial MoO3 hybridization: Implications of Fermi engineering for electrocatalytic hydrogen evolution and oxygen evolution

The design of earth-abundant electrocatalysts that can facilitate water splitting at low overpotentials, provide high current densities, and enable prolonged operational lifetimes is central to the production of sustainable fuels. The distinctive atomistic and electronic structure characteristics of the edges of MoS2 imbue high reactivity toward the hydrogen evolution reaction. MoS2 is nevertheless characterized by significantly high overpotentials as compared to platinum. Here, we demonstrate that modulation of the electronic structure of MoS2 through interfacial hybridization with MoO3 and alloying of selenium on the anion sublattice allows for systematic lowering of the conduction band edge and raising of the valence band edge, respectively. The former promotes enhanced electrocatalytic activity toward hydrogen evolution, whereas the latter promotes enhanced activity toward the oxygen evolution reaction. Such alloyed heterostructures prepared by sol-gel reactions and hydrothermal selenization expose a high density of edge sites. The alloyed heterostructures exhibit low overpotential, high current density, high turnover frequency, and prolonged operational lifetime. The mechanistic origins of catalytic activity have been established based on electronic structure calculations and x-ray absorption and emission spectroscopy probes of electronic structure, which suggest that interfacial hybridization at the MoO3 interface yields low-lying conduction band states that facilitate hydrogen adsorption. In contrast, shallow Se 4p-derived states give rise to a raised effective valence band maximum, which facilitates adsorption of oxygen intermediates and engenders a low overpotential for the oxygen evolution reaction. The findings illustrate the use of electronic structure modulation through interfacial hybridization and alloying to systematically improve electrocatalytic activity.

Parija, Abhishek↗

DOE OSTI2023

Efficient charge to spin conversion in iridium oxide thin films

Many 5d transition metal oxides have a unique electronic structure, where the density of states near the Fermi level is dominated by only 5d electrons with strong spin–orbit coupling. IrO 2 , a Dirac nodal line semi-metal, is the simplest of these oxides. The presence of 5d electrons and gap opening of Dirac nodal lines via strong spin–orbit coupling allows for the hybridization of the 5d electrons of the oxide with the itinerant d electrons of a ferromagnet, while simultaneously increasing the intrinsic spin Hall effect. We report large charge-to-spin conversion in thin films of this material using spin-torque ferromagnetic resonance experiments. Finally, by independently performing line shape analysis and linewidth modulation experiments, we conclusively determine the spin Hall angle of optimized IrO 2 films to be ∼8 times larger than that of Pt.

Ferromagnetic materials↗

DOE OSTI2020

A Novel Magnetic Material by Design: Observation of Yb 3+ with Spin-1/2 in Yb x Pt 5 P

The localized f-electrons enrich the magnetic properties in rare-earth-based intermetallics. Among those, compounds with heavier 4d and 5d transition metals are even more fascinating because anomalous electronic properties may be induced by the hybridization of 4f and itinerant conduction electrons primarily from the d orbitals. Here, we describe the observation of trivalent Yb 3+ with S = 1/2 at low temperatures in Yb x Pt 5 P, the first of a new family of materials. Yb x Pt 5 P (0.23 ≤ x ≤ 0.96) phases were synthesized and structurally characterized. They exhibit a large homogeneity width with the Yb ratio exclusively occupying the 1a site in the anti-CeCoIn 5 structure. Moreover, a sudden resistivity drop could be found in Yb x Pt 5 P below ~0.6 K, which requires further investigation. First-principles electronic structure calculations substantiate the antiferromagnetic ground state and indicate that two-dimensional nesting around the Fermi level may give rise to exotic physical properties, such as superconductivity. Yb x Pt 5 P appears to be a unique case among materials.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗