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DOE OSTI2022

MAGNETICALLY SUSPENDED CANNED ROTOR PUMPS FOR THE INTEGRAL MOLTEN SALT REACTOR

This project developed a molten salt pump design for small modular reactors, concentrated solar, and Gen IV nuclear reactors with a magnetically levitated rotor that can operate at temperatures of up to 700 °C. This eliminates the need for rotating seals and roller element bearings which require maintenance and are prone to failure. The pump design is also more compact than existing molten salt pump designs which require long shafts to thermally isolate the electric motor. The aims of the project are to: 1) design and test the molten salt pump, 2) develop and test a high temperature position sensor needed for magnetic bearing control, and 3) develop and test high temperature coil fabrication methods for electric motor and magnetic bearing fabrication. To develop the high temperature coils, conductor materials were analyzed for oxidation, diffusion, cladding, and resistivity. Of the candidate conductor materials that met the requirements, silver was chosen. The long-term impact of radiation on silver resistivity was analyzed by developing a model of a generic molten salt fast reactor to model the conductor irradiation in worst case scenarios. The analysis showed that increased resistivity due to transmutation of silver decreases exponentially with the reactor reflector thickness. Different electrical insulation materials were tested and the final high temperature coils developed were able to withstand conductor to ground voltages up to 1800 V before insulation breakdown. Magnetic bearings require high-precision high-speed measurements of the rotor position to operate. Current commercial position sensors can only operate up to 550 °C so a novel position sensor was developed that can operate up to 800 °C. These sensors were also designed to measure the position of the metallic rotor through a thin metallic corrosion barrier that protects that coils, insulation, and magnetic cores so that the salt containment barrier does not have any penetrations. The high-temperature sensor was tested and had comparable performance to commercially available inductive position sensors. The sensor design and signal processing has led to several invention disclosures. The magnetically levitated molten salt pump design was completed and analyzed. The chloride salt test loop at Oak Ridge National Laboratory was chosen to test the pump and the pump hydraulics and power were designed for the test loop flow and pressure requirements. Classes of motor designs that do not utilize permanent magnetic were considered for the high-temperature motor. After extensive analysis of their performance, a synchronous reluctance motor was chosen for the pump. The design was refined and analyzed to optimize performance. The magnetic bearings were designed to meet the force and frequency requirements necessary to levitate the rotor and reject disturbances from rotor imbalance, motor forces, and hydraulic forces. Compliant mounts for the position sensors were designed that would not damage the ceramic sensor components and maintain sensor alignment over a wide temperature range. A thermal hydraulic analysis of the molten salt coolant flow used to cool the rotor and stator was performed to ensure that the maximum temperature in the rotor and stator will not exceed 725 °C. The corrosion barrier and casing were designed to maintain the salt fluid boundary by using compliant features to absorb large expansion mismatches due to different coefficients of thermal expansion (CTE). Finally, the salt drain and fill systems were designed along with the pump support structure, integration with the test loop, and thermal insulation and heating. Corrosion barrier fabrication experiments were performed to assess using laser cladding to apply a sub-millimeter corrosion barrier to the surface of a Fe-Co-V magnetic core material. The cladding method showed minimal mixing between materials at their interface and withstood temperature cycling without damage to the cladding. The technology commercialization is focused on the high-temperature position sensors and there has been significant interest from the nuclear industry in utilizing these sensors along with some interest from the aerospace industry. The project also led to a research collaboration with a fusion energy industrial partner to study barriers to scaling the pump design from the kW size to the MW size. This research collaboration will also study methods to create high-temperature coils that have conductor to ground breakdown voltages above 5 kV. The successful development of high-temperature electromagnetic coils and high-temperature position sensors greatly increases the temperature limits for electromagnetic devices that don’t rely on permanent magnets. This includes electric motors, magnetic bearings, linear actuators, rotary position and velocity sensing, actuated valves, and generators to name a few. This will fill a current need in molten salt reactors, concentrated solar, and fusion energy for these critical peripheral devices needed for practical reactor designs that do not currently have commercial solutions.

Hines, J Wesley↗

DOE OSTI2021

Adaptive variational quantum eigensolvers for highly excited states

Highly excited states of quantum many-body systems are central objects in the study of quantum dynamics and thermalization that challenge classical computational methods due to their volume-law entanglement content. In this work, we explore the potential of variational quantum algorithms to approximate such states. We propose an adaptive variational quantum eigensolver (VQE) for excited states (X) that self-generates a variational ansatz for arbitrary eigenstates of a many-body Hamiltonian H by attempting to minimize the energy variance with respect to H. We benchmark the method by applying it to an Ising spin chain with integrable and nonintegrable regimes, where we calculate various quantities of interest, including the total energy, magnetization density, and entanglement entropy. We also compare the performance of adaptive VQE-X to an adaptive variant of the folded-spectrum method. For both methods, we find a strong dependence of the algorithm's performance on the choice of operator pool used for the adaptive construction of the ansatz. In particular, an operator pool including long-range two-body gates accelerates the convergence of both algorithms in the nonintegrable regime. Here, we also study the scaling of the number of variational parameters with system size, finding that an exponentially large number of parameters may be necessary to approximate individual highly excited states. Nevertheless, we argue that these methods lay a foundation for the use of quantum algorithms to study finite-energy-density properties of many-body systems.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2024

Magnetic tunnel junction random number generators applied to dynamically tuned probability trees driven by spin orbit torque

Abstract Perpendicular magnetic tunnel junction (pMTJ)-based true-random number generators (RNGs) can consume orders of magnitude less energy per bit than CMOS pseudo-RNGs. Here, we numerically investigate with a macrospin Landau–Lifshitz-Gilbert equation solver the use of pMTJs driven by spin–orbit torque to directly sample numbers from arbitrary probability distributions with the help of a tunable probability tree. The tree operates by dynamically biasing sequences of pMTJ relaxation events, called ‘coinflips’, via an additional applied spin-transfer-torque current. Specifically, using a single, ideal pMTJ device we successfully draw integer samples on the interval [0, 255] from an exponential distribution based on p -value distribution analysis. In order to investigate device-to-device variations, the thermal stability of the pMTJs are varied based on manufactured device data. It is found that while repeatedly using a varied device inhibits ability to recover the probability distribution, the device variations average out when considering the entire set of devices as a ‘bucket’ to agnostically draw random numbers from. Further, it is noted that the device variations most significantly impact the highest level of the probability tree, with diminishing errors at lower levels. The devices are then used to draw both uniformly and exponentially distributed numbers for the Monte Carlo computation of a problem from particle transport, showing excellent data fit with the analytical solution. Finally, the devices are benchmarked against CMOS and memristor RNGs, showing faster bit generation and significantly lower energy use.

77 NANOSCIENCE AND NANOTECHNOLOGY↗

DOE OSTI2020

Slow scrambling and hidden integrability in a random rotor model

In this work, we analyze the out-of-time-order correlation functions of a solvable model of a large number $\textit{N}$ of $\textit{M}$-component quantum rotors coupled by Gaussian-distributed random, infinite-range exchange interactions. We focus on the growth of commutators of operators at a temperature $\textit{T}$ above the zero temperature quantum critical point separating the spin-glass and paramagnetic phases. In the large $\textit{N, M}$ limit, the squared commutators of the rotor fields do not display any exponential growth of commutators, in spite of the absence of any sharp quasiparticlelike excitations in the disorder-averaged theory. We show that in this limit, the problem is integrable and points out interesting connections to random-matrix theory. At leading order in 1/$\textit{M}$, there are no modifications to the critical behavior but an irrelevant term in the fixed-point action leads to a small exponential growth of the squared commutator. We also introduce and comment on a generalized model involving $\textit{p}$-pair rotor interactions.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2025

Almost strong zero modes at finite temperature

Interacting fermionic chains exhibit extended regions of topological degeneracy of their ground states as a result of the presence of Majorana or parafermionic zero modes localized at the edges. In the opposite limit of infinite temperature, the corresponding nonintegrable spin chains, obtained via generalized Jordan-Wigner mapping, are known to host so-called almost strong zero modes, which are long-lived with respect to any bulk excitations. Here we study the fairly unexplored territory that bridges these two extreme cases of zero and infinite temperature. We blend two established techniques for states, the Lanczos series expansion and a tensor network ansatz, uplifting them to the level of operator algebra. This allows us to efficiently simulate large system sizes for arbitrarily long timescales and to extract the temperature-dependent decay rates. We observe that for the Kitaev-Hubbard model, the decay rate of the edge mode depends exponentially on the inverse temperature 𝛽, and on an effective energy scale Δ eff that is greater than the thermodynamic gap of the system Δ.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2024

Data-Informed Evaluation Framework for Integrated Energy Systems: Insights from Power, Process Heat, and Hydrogen Production Applications

The multi-criteria decision analysis (MCDA) framework provides a systematic evaluation of the diverse preferences and performance metrics associated with alternative solutions. This approach is advantageous over a single-criterion methodology, which are only valid under conditions that assume ceteris paribus or an "apples-to-apples" comparison. However, selecting suitable technologies for integrated energy systems (IES) can be likened to an "apples-to-oranges" comparison, given the heterogeneous factors at stake. These factors include economics and performance parameters, geological compatibility, and environmental impacts. Consequently, past research has often employed a mixture of qualitative and quantitative criteria tailored to the specific interests of each study. While the method proves effective in handling the intricate interplay of criteria, the resulting rankings and scores can vary from study to study. This inconsistency is introduced from the use of subjectively defined thresholds and weights. As a result, decision-makers frequently find it challenging to establish clear connections between specific criteria and the resulting scores, as the transformation of criteria into ordinal scores results in a substantial loss of information. To address this challenge, we introduce a data-informed IES evaluation framework that offers comprehensive, interpretable, and traceable evaluations backed by quantifiable rationale. First, we identified key IES evaluation criteria from a decade of literature, focusing on relevant IES applications in power, process heat, and hydrogen production. We leveraged state-of-the-art cost estimates from the Idaho National Laboratory (INL) and technical data from 78 reactor designs from the International Atomic Energy Agency (IAEA) and the Organization for Economic Co-operation and Development - Nuclear Energy Agency (OECD-NEA). Lastly, we established thresholds by analyzing the mean, variance, root mean square, and slope of values across alternatives, categorizing the preferences of decision-makers into distinct utility functions, such as linear, saturating, exponential, and stepwise. Our approach yielded two main outcomes: (1) it provided consistent assessments across different stakeholder groups and (2) it visualized uncertainties in the decision-making context via comprehensive sensitivity analysis. To demonstrate the impact of our framework, we conducted case studies on 6 reactor designs (AP1000, NuScale, BWRX-300, Xe-100, eVinci, iMSR) for the three applications. Our data-driven framework proved to be highly effective in addressing heterogenous uncertainties faced by varied decision-makers? preferences and IES applications, as well as cost and technical estimates of advanced reactors.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗

DOE OSTI2024

Data-Informed Evaluation Framework for Integrated Energy Systems: Insights from Power, Process Heat, and Hydrogen Production Applications

29 ENERGY PLANNING, POLICY, AND ECONOMY↗

DOE OSTI2020

Systematically localizable operators for quantum simulations of quantum field theories

Correlations and measures of entanglement in ground-state wave functions of relativistic quantum field theories are spatially localized over length scales set by the mass of the field. We utilize this localization to design digital quantum circuits for preparing the ground states of lattice scalar quantum field theories. Controlled rotations that are exponentially localized in their position-space extent are found to provide exponentially convergent wave function fidelity. These angles scale with the correlation between sites and the classical two-point correlation function, as opposed to the more localized mutual information or the hyperlocalized (1×1)-site negativity. We anticipate that further investigations will uncover quantum circuit designs with controlled rotations dictated by the measures of entanglement. This work is expected to impact quantum simulations of systems of importance to nuclear physics, high-energy physics, and basic energy sciences research.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2024

Simple and General Unitarity Conserving Numerical Real-Time Propagators of the Time-Dependent Schrödinger Equation Based on Magnus Expansion

Magnus expansion (ME) provides a general way to expand the real-time propagator of a time-dependent Hamiltonian within the exponential such that the unitarity is satisfied at any order. Here, we use this property and explicit integration of Lagrange interpolation formulas for the time-dependent Hamiltonian within each time interval and derive approximations that preserve unitarity for the differential time evolution operators of general time-dependent Hamiltonians. The resulting second-order approximation is the same as using the average of Hamiltonians for two end points of time. We identify three fourth-order approximations involving commutators of Hamiltonians at different times and also derive a sixth-order expression. A test of these approximations along with other available expressions for a two-state time-dependent Hamiltonian with sinusoidal time dependences provides information on the relative performance of these approximations and suggests that the derived expressions can serve as useful numerical tools for time evolution in time-resolved spectroscopy, quantum control, quantum sensing, real-time ab initio quantum dynamics, and open system quantum dynamics.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2026

Block encoding of the three-dimensional heterogeneous Poisson equation with application to fracture flow

Quantum linear system (QLS) algorithms offer the potential to solve large-scale linear systems exponentially faster than classical methods. However, applying QLS algorithms to real-world problems remains challenging due to issues such as state preparation, data loading, and efficient information extraction. In this work, we study the feasibility of applying QLS algorithms to solve discretized three-dimensional (3D) heterogeneous Poisson equations, with specific examples relating to groundwater flow through geologic fracture networks. We explicitly construct a block encoding for the 3D heterogeneous Poisson matrix by leveraging the sparse local structure of the discretized operator. While classical solvers benefit from preconditioning, we show that block encoding the system matrix and preconditioner separately does not improve the effective condition number that dominates the QLS run-time. This differs from classical approaches where the preconditioner and the system matrix can often be implemented independently. Nevertheless, due to the structure of the problem in three dimensions, the quantum algorithm achieves a run-time of 𝑂⁡(𝑁 2/3 polylog 𝑁 ⋅log (1/𝜖)), outperforming the best classical methods (with run times of 𝑂⁡(𝑁⁢log 𝑁 ⋅log (1/𝜖))) and offering exponential memory savings. These results highlight both the promise and limitations of QLS algorithms for practical scientific computing, and point to effective condition-number reduction as a key barrier in achieving quantum advantages.

58 GEOSCIENCES↗

DOE OSTI2021

The scalar chemical potential in cosmological collider physics

Non-analyticity in co-moving momenta within the non-Gaussian bispectrum is a distinctive sign of on-shell particle production during inflation, presenting a unique opportunity for the “direct detection” of particles with masses as large as the inflationary Hubble scale ( H ). However, the strength of such non-analyticity ordinarily drops exponentially by a Boltzmann-like factor as masses exceed H . In this paper, we study an exception provided by a dimension-5 derivative coupling of the inflaton to heavy-particle currents, applying it specifically to the case of two real scalars. The operator has a “chemical potential” form, which harnesses the large kinetic energy scale of the inflaton, $ {\overset{\cdot }{\phi}}_0^{1/2}\approx 60H $ , to act as an efficient source of scalar particle production. Derivative couplings of inflaton ensure radiative stability of the slow-roll potential, which in turn maintains (approximate) scale-invariance of the inflationary correlations. We show that a signal not suffering Boltzmann suppression can be obtained in the bispectrum with strength f NL ~ $ \mathcal{O} $ (0 . 01–10) for an extended range of scalar masses $ \lesssim {\overset{\cdot }{\phi}}_0^{1/2} $ , potentially as high as 10 15 GeV, within the sensitivity of upcoming LSS and more futuristic 21-cm experiments. The mechanism does not invoke any particular fine-tuning of parameters or breakdown of perturbation-theoretic control. The leading contribution appears at tree-level , which makes the calculation analytically tractable and removes the loop-suppression as compared to earlier chemical potential studies of non-zero spins. The steady particle production allows us to infer the effective mass of the heavy particles and the chemical potential from the variation in bispectrum oscillations as a function of co-moving momenta. Our analysis sets the stage for generalization to heavy bosons with non-zero spin.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2021

Crowded electrolytes containing redoxmers in different states of charge: Solution structure, properties, and fundamental limits on energy density

Nonaqueous redox flow batteries use liquid electrolytes containing redox-active organic molecules (redoxmers) as their energy storage medium. To maximize energy density, the redoxmer concentration needs to be maximized while maintaining low viscosity and high ionic conductivity. During charge, a redoxmer molecule pairs with an ion in the electrolyte while another ion migrates across the membrane to maintain electric neutrality. In a crowded electrolyte, this reconstitution changes physical and chemical properties of the solution. To explore these behaviors, a phenothiazine redoxmer fully miscible with acetonitrile was used, and electrochemical charge was mimicked by chemical oxidation. The solutions were examined using small-angle X-ray scattering, nuclear magnetic resonance, and conductometry and modeled using classical molecular dynamics. Overall, our study indicates that physical and structural properties of redoxmer solutions in both states of charge make it exceedingly difficult to increase the redoxmer concentrations over 2 M at any temperature without compromising dynamic properties of such solutions. The cause for this limitation is proximity to a gel-like regime in which fluidity, diffusivity, and ionic conductivity exponentially decrease with increasing concentration. This tendency is compounded by non-Arrhenius behavior of the electrolyte: a small increase in the concentration outruns gains in fluidity and conductivity at a higher temperature. Thus the properties of crowded electrolytes generally make it impossible to operate when gel-like behavior sets in. Pushing the redoxmer concentration to 2.5-3 M might be possible for small redoxmer molecules, but it would require the use of ionic liquid electrolytes at 340-360 K. (C) 2021 Elsevier B.V. All rights reserved.

25 ENERGY STORAGE↗

DOE OSTI2024

Recycled dark matter

Here, we outline a new production mechanism for dark matter that we dub “recycling”: dark sector particles are kinematically trapped in the false vacuum during a dark phase transition; the false pockets collapse into primordial black holes (PBHs), which ultimately evaporate before Big Bang Nucleosynthesis (BBN) to reproduce the dark sector particles. The requirement that all PBHs evaporate prior to BBN necessitates high scale phase transitions and hence high scale masses for the dark sector particles in the true vacuum. Our mechanism is therefore particularly suited for the production of ultra heavy dark matter (UHDM) with masses above ~10 12 GeV. The correct relic density of UHDM is obtained because of the exponential suppression of the false pocket number density. Recycled UHDM has several novel features: the dark sector today consists of multiple decoupled species that were once in thermal equilibrium and the PBH formation stage has extended mass functions whose shape can be controlled by IR operators coupling the dark and visible sectors.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2022

Quantum Markov chain Monte Carlo with digital dissipative dynamics on quantum computers

Modeling the dynamics of a quantum system connected to the environment is critical for advancing our understanding of complex quantum processes, as most quantum processes in nature are affected by an environment. Modeling a macroscopic environment on a quantum simulator may be achieved by coupling independent ancilla qubits that facilitate energy exchange in an appropriate manner with the system and mimic an environment. This approach requires a large, and possibly exponential number of ancillary degrees of freedom which is impractical. In contrast, we develop a digital quantum algorithm that simulates interaction with an environment using a small number of ancilla qubits. By combining periodic modulation of the ancilla energies, or spectral combing, with periodic reset operations, we are able to mimic interaction with a large environment and generate thermal states of interacting many-body systems. We evaluate the algorithm by simulating preparation of thermal states of the transverse Ising model. Our algorithm can also be viewed as a quantum Markov chain Monte Carlo process that allows sampling of the Gibbs distribution of a multivariate model. To illustrate this we evaluate the accuracy of sampling Gibbs distributions of simple probabilistic graphical models using the algorithm.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2020

AdS/CFT unitarity at higher loops: high-energy string scattering

What is the space of weakly-coupled, gravitational theories which contain massive, higher-spin particles? This class of theories is highly constrained and it is conjectured their ultraviolet completion must be string theory. We provide more evidence for this conjecture by studying the Regge limit in large N, 4d CFTs with single-trace operators of unbounded spin. We show that in the Regge limit, these theories have bulk scattering amplitudes which are consistent with the string theory prediction to all orders in 1/N for large, but finite, coupling. In the language of Regge theory, we show Pomeron exchange naturally exponentiates in the 1/N expansion. To do this, we solve the bootstrap equations at tree-level and then use the Lorentzian inversion formula to find the one-loop correlator in the Regge limit. This is a unitarity method for AdS/CFT which can be repeated iteratively to make all orders statements. We also explain under what conditions the tree-level result exponentiates in the 1/N expansion at arbitrary coupling. Finally, we comment on further inelastic effects and show they give subleading contributions at large coupling. As a consistency check, we recover results from bulk Einstein gravity in the limit where all higher-spin particles decouple.

1/N Expansion↗

DOE OSTI2024

Simulating nonlinear optical processes on a superconducting quantum device

Simulating plasma physics on quantum computers is difficult because most problems of interest are nonlinear, but quantum computers are not naturally suitable for nonlinear operations. In weakly nonlinear regimes, plasma problems can be modelled as wave–wave interactions. In this paper, we develop a quantization approach to convert nonlinear wave–wave interaction problems to Hamiltonian simulation problems. We demonstrate our approach using two qubits on a superconducting device. Unlike a photonic device, a superconducting device does not naturally have the desired interactions in its native Hamiltonian. Nevertheless, Hamiltonian simulations can still be performed by decomposing required unitary operations into native gates. To improve experimental results, we employ a range of error-mitigation techniques. Apart from readout error mitigation, we use randomized compilation to transform undiagnosed coherent errors into well-behaved stochastic Pauli channels. Moreover, to compensate for stochastic noise, we rescale exponentially decaying probability amplitudes using rates measured from cycle benchmarking. We carefully consider how different choices of product-formula algorithms affect the overall error and show how a trade-off can be made to best utilize limited quantum resources. This study provides an example of how plasma problems may be solved on near-term quantum computing platforms.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2025

Randomized Adiabatic Quantum Linear Solver Algorithm with Optimal Complexity Scaling and Detailed Running Costs

Solving linear systems of equations is a fundamental problem with a wide variety of applications across many fields of science, and there is increasing effort to develop quantum linear solver algorithms. Subaşı et al. [Phys. Rev. Lett. 122, 060504 (2019)] proposed a randomized algorithm inspired by adiabatic quantum computing, based on a sequence of random Hamiltonian simulation steps, with suboptimal scaling in the condition number 𝜅 of the linear system and the target error 𝜖. Here we go beyond these results in several ways. Firstly, using filtering [Lin and Tong, Quantum 4, 361 (2020)] and Poissonization techniques [Cunningham and Roland, ArXiv:2406.03972 (2024)], the algorithm complexity is improved to the optimal scaling 𝑂⁡(𝜅⁢log (1/𝜖))—an exponential improvement in 𝜖, and a shaving of a log 𝜅 scaling factor in 𝜅. Secondly, the algorithm is further modified to achieve constant factor improvements, which are vital as we progress towards hardware implementations on fault-tolerant devices. We introduce a cheaper randomized walk operator method replacing Hamiltonian simulation—which also removes the need for potentially challenging classical precomputations; randomized routines are sampled over optimized random variables; circuit constructions are improved. We obtain a closed formula rigorously upper bounding the expected number of times one needs to apply a block-encoding of the linear system matrix to output a quantum state encoding the solution to the linear system. The upper bound is 837⁢𝜅 at 𝜖 = 10 −10 for Hermitian matrices.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Theorem on extensive spectral degeneracy for systems with rigid higher symmetries in general dimensions

We establish, in the spirit of the Lieb-Schultz-Mattis theorem, lower bounds on the spectral degeneracy of quantum systems with higher (gaugelike) symmetries with rather generic physical boundary conditions in an arbitrary number of spatial dimensions. Contrary to applying twists or equivalent adiabatic operations, we exploit the effects of modified boundary conditions. We find that when a general choice of boundary geometry is immaterial in determining spectral degeneracies while approaching the thermodynamic limit, systems that exhibit rigid noncommuting gaugelike symmetries, such as the orbital compass model, must have an exponential (in the size of the boundary) degeneracy of each of their spectral levels.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗