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At least 271 records · Page 15

Critical-Element-Free Permanent-Magnet Materials Based on Ce 2 Fe 14 B

Developing a critical-element-free low-cost permanent magnet is an urgent necessity in view of rapidly developing technologies and the associated huge market demand for Nd 2 Fe 14 B-based magnets. Here, inspired by the abundant and low-cost nature of Ce and these high-performance Nd 2 Fe 14 B permanent magnets, we explore whether it is in fact possible to attain a useful performance in alloys based on the sister material Ce 2 Fe 14 B, employing both experimental and theoretical efforts. Experimentally, we study Ce 2 Fe 14 B with Co, La, and Zr substitutions. The Zr substitution is explored in view of Zr’s frequent role in enhancing magnetic anisotropy in permanent magnets, while the Co and La substitutions serve to remedy the too-low Curie point of 433 K in the base alloy. While we find no Zr-related anisotropy enhancement either experimentally or theoretically, the cosubstitution of La and Co indeed improves the Curie temperature as well as the magnetization, M s , with a potential energy product as high as 38 MG Oe. These properties together suggest optimization of the alloy LaCeFe 12.7 Co 1.3 B (with only 7 wt % cobalt) as a critical-element-free permanent magnet. While the substituted elements do not enhance magnetic anisotropy, from theory, we find a substantial increase, to a first anisotropy constant, K1, as high as 4.24 MJ/m 3 , associated with Bi substitution for Ce. In this paper, our experimental and theoretical results demonstrate the great potential of La, Co, and Bi substitutions in developing low-cost and critical-element-free Ce 2 Fe 14 B-based permanent magnets.

36 MATERIALS SCIENCE↗

Solid structure of Li 2 BeF 4 (FLiBe) from room temperature to melting studied by neutron and X-ray diffraction

Molten fluoride salts such as Li 2 BeF 4 (FLiBe) are used in molten salt reactors, fluoride-salt-cooled high-temperature reactors and fusion reactors as a fuel solvent, coolant and/or tritium breeding medium. In engineered systems that use molten salt, solid-state material will be present during melting and freezing scenarios, and therefore the temperature-dependent properties of the solid and solid/liquid phase transition merit investigation. To observe the behavior of the solid state of Li 2 BeF 4 from room temperature to melting, this work used neutron and X-ray diffraction to measure the changes in the lattice parameters and volume of the crystalline unit cell and compared the results with prior low-temperature data for solid Li 2 BeF 4 . From neutron diffraction data it is also possible to identify anisotropy: centimetre-scaled crystals align preferentially with the a axes parallel to the direction of freezing front propagation, and the c axes expand 54% more than the a axes. This work provides the lattice constants as a function of temperature, quantifies the thermal expansion, and determines the equation describing the change in density for solid Li 2 BeF 4 from room temperature to 459°C to be ρ solid (kg m −3 ) = 2182 (3) − 0.115 (2) T (°C) and the volume expansion upon melting to be less than 5%. This density changes depending on molecular weight and enrichment.

36 MATERIALS SCIENCE↗

Exploring microstructures and anisotropies of serpentinites

Serpentine minerals have received a lot of attention because of their unique crystal structures, their wide occurrence in orogenic belts and their potential role in contributing seismic anisotropy in subducting slabs. Several studies have investigated crystal preferred orientation (CPO) in high temperature antigorite serpentinites from Japan, the Alps, Spain, Cuba and Tibet, documenting significant crystal alignment. However, only a limited number of lower grade serpentines have been explored to date. Mainly because of submicroscopic microstructural heterogeneities CPO cannot be measured with conventional methods such as optical microscopy and EBSD. In this study 15 serpentinites from different tectonic settings in California, the Central Alps and Northern Spain have been investigated, mainly with high energy synchrotron X-ray diffraction, to quantify bulk crystal alignment. We find that CPO is strong on sheared surfaces of fractured blocks and secondary veins but the bulk of most serpentinite samples, except high-grade recrystallized antigorite serpentinite, show only weak crystal alignment. Correspondingly calculated seismic anisotropy based on CPO is not very significant. This is supported by very heterogeneous microstructures as documented with SEM and TEM analyses.

58 GEOSCIENCES↗

Axis dependent conduction polarity in the air-stable semiconductor, PdSe 2

Axis-dependent conduction polarity (ADCP) is a unique electronic phenomena in which the charge polarity of carrier conduction can differ from p-type to n-type depending on the direction of travel through the crystal. Most materials that exhibit ADCP are metals, and very few semiconducting materials exhibit this effect. Here, in this work, we establish that PdSe 2 , a ~0.5 eV band gap semiconductor that is air- and water-stable, exhibits ADCP, through the growth and characterization of the transport properties of crystals with extrinsic p- and n-type doping levels of Ir and Sb, respectively, in the 10 16 –10 18 cm –3 range. Electron doped PdSe 2 exhibits p-type conduction in the cross-plane direction and n-type conduction along the in-plane directions above an onset temperature of 100–200 K that varies with doping level. Lightly p-doped samples show p-type thermopower in all directions at low temperatures, but above ~360 K the in-plane thermopower turns negative. Density functional theory calculations indicate that the origin of ADCP arises from the complementary effective mass anisotropies in the valence and conduction bands in this material, which facilitate hole transport in the cross-plane direction, and electron transport along the in-plane directions. ADCP occurs at temperatures with sufficient thermal population of both carrier types to overcome the extrinsic doping levels to exploit the effective mass anisotropy. In total, the development of this stable semiconductor in which thermally or optically excited holes and electrons inherently migrate along different directions opens up numerous potential applications in a multitude of technologies.

36 MATERIALS SCIENCE↗

Anomalies in the electromagnetic response and peculiarities of the domain structure in the magnetic Weyl semimetal Co 3⁢ Sn 2 ⁢S 2

In this paper, we present a comprehensive study of magnetic properties of Co 3 ⁢Sn 2 ⁢S 2 (CoSnS) crystals using macroscopic characterization and observations of the domain structure (DS) in a wide range of easy- and hard-axis fields (𝐻) and temperatures (𝑇). From macroscopic measurements, we derive temperature dependences of the basic magnetic parameters of the material [saturation magnetization 𝑀 𝑠 ⁡(𝑇), uniaxial anisotropy constant 𝐾⁡(𝑇), and exchange stiffness 𝐴⁡(𝑇)] and, based on them, calculate the thermodynamic magnetization 𝑀⁡(𝐻,𝑇) curves and DS parameters for pinning-free samples. We find important qualitative and quantitative differences between the theory and experiment defined by the efficient pinning of domain walls (DWs) in our samples. It turns out that, already at temperatures close to the ferromagnetic transition 𝑇 𝑐 , the DW dynamics falls into the heavy creep regime, resulting in a strong deferral of the DS adjustment into the equilibrium state. Here, we suggest that specific magnetic defects, responsible for pinning and causing field-tuned reproducible domain nucleation patterns observed in our high-quality samples, are pointlike. Recently found magnetic polarons formed around vacancies and impurities, which carry strong diamagnetic moments of spin-orbit nature and are intrinsic for CoSnS and expected in other Weyl semimetals, are a possible source of pinning yielding peculiar 𝑀⁡(𝐻,𝑇) response of CoSnS.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Intrinsic Elastic Anisotropy of Westerly Granite Observed by Ultrasound Measurements, Microstructural Investigations, and Neutron Diffraction

Westerly granite (WG) has been generally accepted as an isotropic homogeneous rock. Here, we return to WG and observe significant elastic anisotropy using multidirectional ultrasonic sounding on spherical samples at pressures up to 400 MPa. Thermal treatment of WG leads to formation of microcracks that reduce elastic wave velocities and increase its elastic anisotropy. The 3D distribution of P-wave velocities at low pressure is close to orthorhombic symmetry. Application of hydrostatic pressure closes most of thermally induced microcracks and decreases elastic anisotropy of WG, but at high pressure the anisotropy is practically reversed compared to low pressure: maximum P-wave velocity direction at low pressures is near minimum velocity direction at high pressure and vice versa. To understand this effect, microstructures of the rock were investigated by optical and scanning electron microscopy. Preferred orientations of four major rock-forming minerals—quartz, orthoclase, plagioclase, and biotite—were measured by time-of-flight neutron diffraction, which confirms significant crystal alignment. All these data were used to numerically model anisotropic elastic properties of WG. It is shown that WG possesses weak intrinsic elastic anisotropy related mainly to the preferred orientation of feldspars formed during igneous crystallization. Observed microcracks are mostly related to the cleavage planes of feldspars and biotite, and thus also demonstrate preferred orientation. Higher preheating temperatures produce larger quantity of longer microcracks. A numerical model shows that these microcracks act against the weak intrinsic elastic anisotropy of WG, and define the elastic anisotropy at low pressures.

58 GEOSCIENCES↗

Tuning the Intrinsic Anisotropy with Disorder in the Ca K Fe 4 As 4 Superconductor

In this paper, we report on the anisotropy of the London penetration depth of CaKFe 4 As 4 , discussing how it relates to its electronic structure and how it modifies under introduction of disorder, both chemically induced (by Ni substitution) and irradiation induced (by 3.5-MeV protons). Indeed, CaKFe 4 As 4 is particularly suitable for the study of fundamental superconducting properties due to its stoichiometric composition, exhibiting clean-limit behavior in the pristine samples and having a fairly high critical temperature, T c ≈35 K. The London penetration depth λ L is measured with a microwave-coplanar-resonator technique that allows us to deconvolve the anisotropic contributions λ L,ab and λ L,c and obtain the anisotropy parameter γλ=λ L,c /λ L,ab . The γλ(T) found for the undoped pristine sample is in good agreement with previous literature and is here compared to ab initio density-functional-theory and Eliashberg calculations. The dependence of γλ(T) on both chemical and irradiation-induced disorder is discussed to highlight which method is more suitable to decrease the direction dependence of the electromagnetic properties while maintaining a high critical temperature. Lastly, the relevance of an intrinsic anisotropy such as γλ on application-related anisotropic parameters (critical current, pinning) is discussed in light of the recent employment of CaKFe 4 As 4 in the production of wires.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Development of MKIDs for Measurement of the Cosmic Microwave Background with the South Pole Telescope

We present details of the design, simulation, and initial test results of prototype detectors for the fourth-generation receiver of the South Pole Telescope (SPT). Optimized for the detection of key secondary anisotropies of the cosmic microwave background (CMB), SPT-3G+ will measure the temperature and polarization of the mm/sub-mm sky at 220, 285, and 345 GHz, beyond the peak of the CMB blackbody spectrum. The SPT-3G+ focal plane will be populated with microwave kinetic inductance detectors (MKIDs), allowing for significantly increased detector density with reduced cryogenic complexity. We present simulation-backed designs for single-color dual-polarization MKID pixels at each SPT-3G+ observation frequency. We further describe design choices made to promote resonator quality and uniformity, enabling us to maximize the available readout bandwidth. We also discuss aspects of the fabrication process that enable rapid production of these devices and present an initial dark characterization of a series of prototype devices.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Missing components in ΛCDM from DESI Y1 baryonic acoustic oscillation measurements: Insights from redshift remapping

We explore transformations of the Friedman-Lemaître-Robertson-Walker (FLRW) metric and cosmological parameters that align with observational data while aiming to gain insights into potential extensions of standard cosmological models. We modified the FLRW metric by introducing a scaling factor, e 2Θ(a) –the cosmological scaling function (CSF), which alters the standard relationship between cosmological redshift and the cosmic scale factor without affecting angular measurements or cosmic microwave background (CMB) anisotropies. Using data from DESI Year 1, Pantheon+ supernovae, and the Planck CMB temperature power spectrum, we constrained both the CSF and cosmological parameters through a Markov chain Monte Carlo approach. Our results indicate that the CSF model fits observational data with a lower Hubble constant (although it is compatible with the value given by Planck 2018 within 1σ) and is predominantly dark matter dominated. Additionally, the CSF model produces temperature and lensing power spectra similar to those predicted by the standard model, though with lower values in the CSF model at large scales. We also checked that when fitting a CSF model without dark energy to the data, we obtain a more negative conformal function. This suggests that the CSF model may offer hints about missing elements and opens up a new avenue for exploring physical interpretations of cosmic acceleration.

79 ASTRONOMY AND ASTROPHYSICS↗

Challenges to magnetic doping of thin films of the Dirac semimetal Cd 3 As 2

Magnetic doping of topological quantum materials provides an attractive route for studying the effects of time-reversal symmetry breaking. Thus motivated, we explore the introduction of the transition metal Mn into thin films of the Dirac semimetal Cd 3 A s2 during growth by molecular beam epitaxy. Scanning transmission electron microscopy measurements show the formation of a Mn-rich phase at the top surface of Mn-doped Cd 3 A s2 thin films grown using both uniform doping and delta doping. This suggests that Mn acts as a surfactant during epitaxial growth of Cd 3 A s2 , resulting in phase separation. Magnetometry measurements of such samples indicate a ferromagnetic phase with out-of-plane magnetic anisotropy. Electrical magneto-transport measurements of these films as a function of temperature, magnetic field, and chemical potential reveal a lower carrier density and higher electron mobility compared with pristine Cd 3 A s2 films grown under similar conditions. This suggests that the surfactant effect might also serve to remove impurities from the bulk of the film. Further, we observe robust quantum transport (Shubnikov-de Haas oscillations and an incipient integer quantum Hall effect) in very thin (7 nm) Cd 3 A s2 films despite being in direct contact with a structurally disordered surface ferromagnetic overlayer.

36 MATERIALS SCIENCE↗

Field-free switching of perpendicular magnetization in a ferrimagnetic insulator with spin reorientation transition

Writing magnetic bits through spin-orbit torque (SOT) switching is promising for fast and efficient magnetic random-access memory devices. While SOT switching of out-of-plane (OOP) magnetized states requires lateral symmetry breaking, in-plane (IP) magnetized states suffer from low storage density. Here, we demonstrate a field-free switching scheme using a 5-nanometer europium iron garnet film grown with a (110) orientation that shows a spin reorientation transition from OOP to IP above room temperature. This scheme combines the benefits of high-density storage in the OOP states at room temperature and the efficient field-free SOT switching in the IP states at elevated temperatures. While conventional switching of OOP bits faces the dilemma that high OOP anisotropy is required to improve bit stability and low OOP anisotropy is required to lower switching current density, this scheme disentangles this interdependence, allowing for low switching currents to be possible without sacrificing the bit stability, offering opportunities for future memory devices.

Science & Technology - Other Topics↗

Monolayer CrCl 3 as an Ideal Test Bed for the Universality Classes of 2D Magnetism

The monolayer halides CrX 3 (X=Cl, Br, I) attract significant attention for realizing 2D magnets with genuine long-range order (LRO), challenging the Mermin-Wagner theorem. In this work, we show that monolayer CrCl 3 has the unique benefit of exhibiting tunable magnetic anisotropy upon applying a compressive strain. This opens the possibility to use CrCl 3 for producing and studying both ferromagnetic and antiferromagnetic 2D Ising-type LRO as well as the Berezinskii-Kosterlitz-Thouless (BKT) regime of 2D magnetism with quasi-LRO. Using state-of-the-art density functional theory, we explain how realistic compressive strain could be used to tune the monolayer's magnetic properties so that it could exhibit any of these phases. Building on large-scale quantum Monte Carlo simulations, we compute the phase diagram of strained CrCl 3 , as well as the magnon spectrum with spin-wave theory. Our results highlight the eminent suitability of monolayer CrCl 3 to achieve very high BKT transition temperatures, around 50 K, due to their singular dependence on the weak easy-plane anisotropy of the material.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Progress on Producing an ODS Ferritic Alloy by High Deformation Processing of Reactive Powders

Oxide dispersion-strengthened (ODS) ferritic alloys, such as 14YWT, are considered as benchmarks for fuel cladding and core structural materials in advanced nuclear energy reactor systems that require high temperature strength and creep properties as well as resistance to radiation damage. Unfortunately, the popular processing method of mechanical alloying (MA) is time-consuming, incurs high manufacturing costs, and typically produces anisotropy in the microstructure and mechanical properties of ODS ferritic alloys. These considerations hinder their acceptance in many high-performance, high-temperature materials for advanced nuclear energy reactors. This work package is pursuing an alternative solid-state processing to MA involving high deformation at high temperatures for producing an ODS ferritic alloy for fuel cladding. This report summarizes the concept for this processing method and the progress toward selecting the reactive and ferritic alloy powders, preparing 316L stainless steel cans for filling with blended powder followed by degassing, and choosing the heat treatments and forging parameters for performing the first two forging experiments. The preliminary results of optical microscopy performed on polished samples that were cut in three orthogonal directions revealed remnant porosity in both samples. The analysis of the porosity using ImageJ revealed a lesser amount of ~2.49% in the HiLoMe-1 sample that was processed with the three heat treatments and forging parameters compared to ~5.70% for the Hi-1 sample that was processed with only the first sintering heat treatment and forging parameter. These initial results will provide insights that can be leveraged to improve the densification of the powders in future forging experiments that will be continued into FY24 for determining the optimum heat treatments and forging parameters coupled with detailed microstructure characterization studies for producing the ODS ferritic alloy.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

Chemical Tuning Meets 2D Molecular Magnets

Two-dimensional (2D) magnets provoke a surge of interest in large anisotropy in reduced dimensions and are promising for next-generation information technology where dynamic magnetic tuning is essential. Until recently, the crucial metal-organic magnet Cr(pyz) 2 ∙xLiCl∙yTHF with considerable high coercivity and high-temperature magnetic order opens up a new platform to control magnetism in metal-organic materials at room temperature. Here we report an in-situ chemical tuning route to realize the controllable transformation of low-temperature magnetic order into room-temperature hard magnetism in Cr(pyz) 2 ∙xLiCl∙yTHF. The chemical tuning via electrochemical lithiation and solvation/desolvation exhibits continuously variable magnetic features from cryogenic magnetism to the room-temperature optimum performance of coercivity (H c ) of 8500 Oe and energy product of 0.6 MGOe. Such chemically flexible tunability of room-temperature magnetism is ascribed to the different degrees of lithiation and solvation that modify the stoichiometry and Cr-pyrazine coordination framework. Furthermore, the additively manufactured hybrid magnets show air stability and electromagnetic induction, providing potential applications. Our findings here suggest chemical tuning as a universal approach to control the anisotropy and magnetism of 2D hybrid magnets at room temperature, promising for data storage, magnetic refrigeration, and spintronics.

36 MATERIALS SCIENCE↗

Proximity enhanced magnetism at NiFe 2 O 4 /Graphene interface

Here, we explore the change in effective magnetic anisotropy of the ferrimagnetic (FM) insulator nickel ferrite (NFO) thin film due to the inclusion of monolayer graphene (MLG) grown on top of the NFO layer. This was done by performing radio frequency (RF) transverse susceptibility (TS) measurements on bare NFO and NFO/MLG bilayer samples for both in-plane (IP) and out-of-plane (OOP) configurations utilizing a tunnel diode oscillator technique. Our magnetometry measurements indicated an enhancement in the overall saturation magnetization of the NFO/MLG bilayer with respect to the bare NFO film. The TS measurements reveal that the inclusion of MLG reduces the effective magnetic anisotropy for both IP and OOP configurations drastically, by up to a factor of 2 over the temperature range 40 K ≤ T ≤ 280 K. Since NFO is a magnetic substrate, it is possible that NFO could induce magnetic ordering in MLG at the NFO/MLG interface via the magnetic proximity effect. Furthermore, since NFO is insulating and MLG is a semimetal, there likely exists a large conductivity difference at the interface, making charge transfer plausible. These two effects could modify the interfacial magnetism leading to a change in the effective magnetic anisotropy. These results highlight the importance of understanding the interfacial magnetism of FM/MLG heterostructures.

36 MATERIALS SCIENCE↗

Physical properties of 𝑅⁢ Co 2 ⁢Al 8 single crystals (𝑅 = La, Ce, Pr, Nd, and Sm) : An emerging structure-type for anisotropic Kondo-lattice studies

Systematic investigations of rare-earth (𝑅)-based intermetallic materials are a leading strategy to reveal the underlying mechanisms governing a range of physical phenomena, such as the formation of a Kondo lattice and competing electronic and magnetic anisotropies. Here, in this work, the magnetic, thermal, and transport properties of 𝑅⁢Co 2 ⁢Al 8 (𝑅 = La, Ce, Pr, Nd, and Sm) single crystals are presented. LaCo 2 ⁢Al 8 is characterized as a Pauli paramagnet, and transport measurements, with the current along and perpendicular to the orthorhombic 𝑐-axis (𝜌 𝑐 and 𝜌 𝑎⁢𝑏 , respectively), reveal a clear electronic anisotropy, with 𝜌 𝑎⁢𝑏 ⁢≈ (4 –7)⁢𝜌 𝑐 at 300K . We show that CeCo 2 ⁢Al 8 is a Kondo lattice for which the Kondo coherence temperature 𝑇$^*_K$, deduced from broad maximums in 𝜌 𝑐 and 𝜌 𝑎⁢𝑏 at ≈ 68 and 46 K, respectively, is also anisotropic. This finding is related to a possible underlying anisotropy of the Kondo coupling in CeCo 2⁢ Al 8 . The Pr- and Nd-based materials present strong easy-axis anisotropy (𝑐-axis) and antiferromagnetic (AFM) orders below 𝑇 = 4.84 and 8.1K , respectively. Metamagnetic transitions from this AFM to a spin-polarized paramagnetic phase state are investigated by isothermal magnetization measurements. The Sm-based compound is also an easy-axis AFM with a transition at 𝑇 = 21.6K .

Garcia, Fernando A. [Ames Laboratory (AMES), Ames,↗

Haldane topological spin-1 chains in a planar metal-organic framework

Haldane topological materials contain unique antiferromagnetic chains with symmetry-protected energy gaps. Such materials have potential applications in spintronics and future quantum computers. Haldane topological solids typically consist of spin-1 chains embedded in extended three-dimensional (3D) crystal structures. Here, we demonstrate that [Ni(μ-4,4'-bipyridine)(μ-oxalate)] n (NiBO) instead adopts a two-dimensional (2D) metal-organic framework (MOF) structure of Ni 2+ spin-1 chains weakly linked by 4,4'-bipyridine. NiBO exhibits Haldane topological properties with a gap between the singlet ground state and the triplet excited state. The latter is split by weak axial and rhombic anisotropies. Several experimental probes, including single-crystal X-ray diffraction, variable-temperature powder neutron diffraction (VT-PND), VT inelastic neutron scattering (VT-INS), DC susceptibility and specific heat measurements, high-field electron spin resonance, and unbiased quantum Monte Carlo simulations, provide a detailed, comprehensive characterization of NiBO. Vibrational (also known as phonon) properties of NiBO have been probed by INS and density-functional theory (DFT) calculations, indicating the absence of phonons near magnetic excitations in NiBO, suppressing spin-phonon coupling. The work here demonstrates that NiBO is indeed a rare 2D-MOF Haldane topological material.

36 MATERIALS SCIENCE↗

Room temperature colossal superparamagnetic order in aminoferrocene–graphene molecular magnets

Intensive studies are published for graphene-based molecular magnets due to their remarkable electric, thermal, and mechanical properties. However, to date, most of all produced molecular magnets are ligand based and subject to challenges regarding the stability of the ligand(s). The lack of long-range coupling limits high operating temperature and leads to a short-range magnetic order. Herein, we introduce an aminoferrocene-based graphene system with room temperature superparamagnetic behavior in the long-range magnetic order that exhibits colossal magnetocrystalline anisotropy of 8 × 10 5 and 3 × 10 7 J/m 3 in aminoferrocene and graphene-based aminoferrocene, respectively. These values are comparable to and even two orders of magnitude larger than pure iron metal. Aminoferrocene [C 10 H 11 FeN] + is synthesized by an electrophilic substitution reaction. It was then reacted with graphene oxide that was prepared by the modified Hammers method. The phase structure and functionalization of surface groups were characterized and confirmed by XRD, FT-IR, and Raman spectroscopy. To model the behavior of the aminoferrocene between two sheets of hydroxylated graphene, we have used density functional theory by placing the aminoferrocene molecule between two highly ordered hydroxylated sheets and allowing the structure to relax. Here, the strong bowing of the isolated graphene sheets suggests that the charge transfer and resulting magnetization could be strongly influenced by pressure effects. In contrast to strategies based on ligands surface attachment, our present work that uses interlayer intercalated aminoferrocene opens routes for future molecular magnets as well as the design of qubit arrays and quantum systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗