Search NASA⌕ Search

SEARCH · Search NASA

Results for “Spectral theory”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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

At least 325 records · Page 18
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Flattening of the EFT-hedron: supersymmetric positivity bounds and the search for string theory

We examine universal positivity constraints on 2 → 2 scattering in 4d planar N = 4 supersymmetric Yang-Mills theory with higher-derivative corrections. We present numerical evidence that the convex region of allowed Wilson coefficients (the “EFT-hedron”) flattens completely along about one-third of its dimensions when an increasing number of constraints on the spectral density from crossing-symmetry are included. Our analysis relies on the formulation of the positivity constraints as a linear optimization problem, which we implement using two numerical solvers, SDPB and CPLEX. Motivated by the flattening, we propose a novel partially resummed low-energy expansion of the 2 → 2 amplitude. As part of the analysis, we provide additional evidence in favor of the conjecture [1] that the Veneziano amplitude is the only amplitude compatible with both S-matrix bootstrap constraints and string monodromy.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

Random Tensor Networks with Non-trivial Links

Abstract Random tensor networks are a powerful toy model for understanding the entanglement structure of holographic quantum gravity. However, unlike holographic quantum gravity, their entanglement spectra are flat. It has therefore been argued that a better model consists of random tensor networks with link states that are not maximally entangled, i.e., have non-trivial spectra. In this work, we initiate a systematic study of the entanglement properties of these networks. We employ tools from free probability, random matrix theory, and one-shot quantum information theory to study random tensor networks with bounded and unbounded variation in link spectra, and in cases where a subsystem has one or multiple minimal cuts. If the link states have bounded spectral variation, the limiting entanglement spectrum of a subsystem with two minimal cuts can be expressed as a free product of the entanglement spectra of each cut, along with a Marchenko–Pastur distribution. For a class of states with unbounded spectral variation, analogous to semiclassical states in quantum gravity, we relate the limiting entanglement spectrum of a subsystem with two minimal cuts to the distribution of the minimal entanglement across the two cuts. In doing so, we draw connections to previous work on split transfer protocols, entanglement negativity in random tensor networks, and Euclidean path integrals in quantum gravity.

Physics↗
DOE OSTI2022

Evidence for nitrogen binding to surface defects for topological insulator Bi 2 Se 3

Using scanning tunneling spectroscopy and theoretical simulations we have studied the effects of nitrogen gas exposure on the electronic density of states of Bi 2 Se 3 , a well-studied topological insulator. In carefully controlled measurements, Bi 2 Se 3 crystals were initially cleaved in a helium gas environment and then exposed to a 22 SCFH flow of ultra-high purity N 2 gas. We observe a resulting change in the spectral curves, with the exposure effect saturating after approximately 50 min, ultimately bringing the Dirac point about 50 meV closer to the Fermi level. Furthermore, these results are compared to density functional theory calculations, which support a picture of N 2 molecules physisorbing near Se vacancies. Furthermore, ab initio molecular dynamics simulations aided by a Blue Moon ensemble method reveal the dissociative adsorption of N 2 molecules which then bind strongly to Se vacancies at the surface. In this scenario, the binding of the N atom to a Se vacancy site removes the surface defect state created by the vacancy and changes the position of the Fermi energy with respect to the Dirac point.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Sub-diffractional cavity modes of terahertz hyperbolic phonon polaritons in tin oxide

Abstract Hyperbolic phonon polaritons have recently attracted considerable attention in nanophotonics mostly due to their intrinsic strong electromagnetic field confinement, ultraslow polariton group velocities, and long lifetimes. Here we introduce tin oxide (SnO 2 ) nanobelts as a photonic platform for the transport of surface and volume phonon polaritons in the mid- to far-infrared frequency range. This report brings a comprehensive description of the polaritonic properties of SnO 2 as a nanometer-sized dielectric and also as an engineered material in the form of a waveguide. By combining accelerator-based IR-THz sources (synchrotron and free-electron laser) with s-SNOM, we employed nanoscale far-infrared hyper-spectral-imaging to uncover a Fabry–Perot cavity mechanism in SnO 2 nanobelts via direct detection of phonon-polariton standing waves. Our experimental findings are accurately supported by notable convergence between theory and numerical simulations. Thus, the SnO 2 is confirmed as a natural hyperbolic material with unique photonic properties essential for future applications involving subdiffractional light traffic and detection in the far-infrared range.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Carbon-Based Quantum Information Science with Symmetry Protected Topological States (Final Report, DOE-BES award DE-SC0023105)

This research program established the scientific foundation for the rational, bottom-up design, synthesis, isolation, and investigation of symmetry-protected topological (SPT) electron spin qubits embedded in graphene nanoribbons (GNRs). The work focused on integrating atomically precise low-dimensional carbon nanostructures with emerging quantum logic architectures, providing a pathway toward scalable quantum materials for next-generation computing and sensing technologies. A central component of the program was the elucidation of fundamental relationships between real-space molecular architecture, local spin density distributions, electronic band dispersion, and energy level alignment in atomically precise GNR systems. These correlations define key operational parameters of SPT qubits and were systematically investigated to establish quantitative benchmarks against established molecular and solid-state spin qubit platforms. Attention was given to properties critical for quantum device performance, e.g. decoherence times, spectral sharpness of energy transitions, and tunable exchange interactions between spin states. The research demonstrated that these parameters can be engineered with atomic precision through scalable bottom-up synthetic strategies. Theory-guided design played a central role in identifying candidate structures hosting topologically protected spin states. Experimental validation was performed using both ensemble measurements and single-molecule characterization. In addition to advances in quantum materials synthesis, the program developed and applied spin-sensitive scanning probe microscopy techniques capable of directly probing quantum states and dynamic processes with atomic-scale spatial resolution. These capabilities enabled direct observation and characterization of quantum structures at the single-atom level. While the research activities were primarily hypothesis-driven fundamental investigations, the program adopted a comprehensive materials-by-design framework aimed at translating scientific discoveries into technological concepts compatible with scalable and intelligent manufacturing approaches.

36 MATERIALS SCIENCE↗
DOE OSTI2020

A Radiative Transfer Model for Patchy Landscapes Based on Stochastic Radiative Transfer Theory

The availability of global high-resolution land cover maps provides promising a priori knowledge for characterizing subpixel heterogeneity and improving predictions of directional reflectance of coarse-resolution pixels. Due to mutual shadowing and sheltering effects between the adjacent forest and cropland patches, the spectral nonlinear mixing of patchy ecotones is significant, especially when the sun illuminates the ecotone from the forest side with high solar zenith angle. The spectral linear mixture (SLM) approach leads to overestimation of the bidirectional reflectance factor (BRF) in the red band in the principal plane (PP), with a maximum absolute error (MAE) of 0.0063 and a maximum relative error (MRE) of 52.5%, and to underestimation in the near-infrared band in PP with an MAE of 0.0940 and an MRE of 14.5%. In a scenario with randomly distributed boundary orientations, the overestimation of SLM increases with the degree of fragmentation and the view zenith angle. We propose a Radiative Transfer model for patchy ECotones (RTEC). which improves R² from 0.61 to 0.94 in the red band of Landsat-8 directional reflectance at the validation site. The RTEC model provides an efficient and analytical approach for directional reflectance predictions over heterogeneous patchy landscapes at coarse resolution and will be used for biophysical parameter retrievals [e.g., the leaf area index (LAI)] in future applications.

Zeng, Yelu↗
DOE OSTI2019

Background independent quantum field theory and gravitating vacuum fluctuations

The scale dependent effective average action for quantum gravity complies with the fundamental principle of Background Independence. Ultimately the background metric it formally depends on is selected self-consistently by means of a tadpole condition, a generalization of Einstein’s equation. Self-consistent background spacetimes are scale dependent, and therefore “going on-shell” at the points along a given renormalization group (RG) trajectory requires understanding two types of scale dependencies: the (familiar) direct one carried by the off-shell action functional, and an equally important indirect one related to the continual re-adjustment of the background geometry. This paper is devoted to a careful delineation and analysis of certain general questions concerning the indirect scale dependence, as well as a detailed explicit investigation of the case where the self-consistent metrics are determined predominantly by the RG running of the cosmological constant. Mathematically, the object of interest is the spectral flow induced by the background Laplacian which, on-shell, acquires an explicit scale dependence. Among other things, it encodes the complete information about the specific set of field modes which, at any given scale, are the degrees of freedom constituting the respective effective field theory. For a large class of RG trajectories (Type IIIa) we discover a seemingly paradoxical behavior that differs substantially from the off-shell based expectations: A Background Independent theory of (matter coupled) quantum gravity looses rather than gains degrees of freedom at increasing energies. As an application, we investigate to what extent it is possible to reformulate the exact theory in terms of matter and gravity fluctuations on a rigid flat space. It turns out that, in vacuo, this “rigid picture” breaks down after a very short RG time already (less than one decade of scales) because of a “scale horizon” that forms. Furthermore, we critically reanalyze and refute the frequent claim that the huge energy densities one obtains in standard quantum field theory by summing up to zero-point energies imply a naturalness problem for the observed small value of the cosmological constant.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

Giant Wilson loops and AdS 2 /dCFT 1

The 1/2-BPS Wilson loop in N = 4 supersymmetric Yang-Mills theory is an important and well-studied example of conformal defect. In particular, much work has been done for the correlation functions of operator insertions on the Wilson loop in the fundamental representation. In this paper, we extend such analyses to Wilson loops in the large-rank symmetric and antisymmetric representations, which correspond to probe D3 and D5 branes with AdS 2 × S 2 and AdS 2 × S 4 worldvolume geometries, ending at the AdS 5 boundary along a one-dimensional contour. We first compute the correlation functions of protected scalar insertions from supersymmetric localization, and obtain a representation in terms of multiple integrals that are similar to the eigenvalue integrals of the random matrix, but with important differences. Using ideas from the Fermi Gas formalism and the Clustering method, we evaluate their large N limit exactly as a function of the ’t Hooft coupling. The results are given by simple integrals of polynomials that resemble the Q-functions of the Quantum Spectral Curve, with integration measures depending on the number of insertions. Next, we study the correlation functions of fluctuations on the probe D3 and D5 branes in AdS. We compute a selection of three- and four-point functions from perturbation theory on the D-branes, and show that they agree with the results of localization when restricted to supersymmetric kinematics. We also explain how the difference of the internal geometries of the D3 and D5 branes manifests itself in the localization computation.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2026

Influence of Markovianity and self-consistency on time-resolved spectral functions of driven quantum systems

We present a systematic comparison of the real-time Dyson expansion (RTDE) with established nonequilibrium Green's function (GF) approaches for simulating driven, interacting quantum systems. Focusing on density matrix dynamics, time-off-diagonal GFs, and time-resolved photoemission spectra, we benchmark RTDE against fully self-consistent Kadanoff-Baym equation (KBE) calculations, the generalized Kadanoff-Baym ansatz, and exact diagonalization for small systems using second-order many-body perturbation theory. Using a driven two-band Hubbard model, we show that mean-field single-particle density matrix trajectories provide a reliable baseline for RTDE across a broad range of interaction strengths and excited-carrier populations. Further, RTDE accurately captures correlation effects in the GFs, including long-lived oscillations and revivals that are strongly suppressed by the overdamping inherent to self-consistent KBE schemes. As a consequence, RTDE resolves rich nonequilibrium spectral structure in time-resolved photoemission, such as interaction- and population-dependent quasiparticle splittings and band gap renormalization, which are largely washed out in self-consistent approaches yet are present in exact solutions. Furthermore, our results demonstrate that RTDE bridges the gap between mean-field propagation and full two-time KBE simulations, retaining favorable linear scaling while capturing essential dynamical correlations relevant for ultrafast spectroscopy.

Electronic structure↗
DOE OSTI2021

Directing Energy Transfer in Halide Perovskite–Chromophore Hybrid Assemblies

Directing the flow of energy and the nature of the excited states that are produced in nanocrystal–chromophore hybrid assemblies is crucial for realizing their photocatalytic and optoelectronic applications. Using a combination of steady-state and time-resolved absorption and photoluminescence (PL) experiments, we have probed the excited-state interactions in the CsPbBr 3 –Rhodamine B (RhB) hybrid assembly. PL studies reveal quenching of the CsPbBr 3 emission with a concomitant enhancement of the fluorescence of RhB, indicating a singlet-energy-transfer mechanism. Transient absorption spectroscopy shows that this energy transfer occurs on the ~200 ps time scale. To understand whether the energy transfer occurs through a Förster or Dexter mechanism, we leveraged facile halide-exchange reactions to tune the optical properties of the donor CsPbBr 3 by alloying with chloride. This allowed us to tune the spectral overlap between the donor CsPb(Br 1–x Cl x ) 3 emission and acceptor RhB absorption. For CsPbBr 3 -RhB, the rate constant for energy transfer (k ET ) agrees well with Förster theory, whereas alloying with chloride to produce chloride-rich CsPb(Br 1–x Cl x ) 3 favors a Dexter mechanism. These results highlight the importance of optimizing both the donor and acceptor properties to design light-harvesting assemblies that employ energy transfer. Furthermore, the ease of tuning optical properties through halide exchange of the nanocrystal donor provides a unique platform for studying and tailoring excited-state interactions in perovskite–chromophore assemblies.

14 SOLAR ENERGY↗
DOE OSTI2025

DESI 2024 V: Full-Shape galaxy clustering from galaxies and quasars

We present the measurements and cosmological implications of the galaxy two-point clustering using over 4.7 million unique galaxy and quasar redshifts in the range 0.1 < z < 2.1 divided into six redshift bins over a ∼ 7,500 square degree footprint, from the first year of observations with the Dark Energy Spectroscopic Instrument (DESI Data Release 1). By fitting the full power spectrum, we extend previous DESI DR1 baryon acoustic oscillation (BAO) measurements to include redshift-space distortions and signals from the matter-radiation equality scale. For the first time, this Full-Shape analysis is blinded at the catalogue-level to avoid confirmation bias and the systematic errors are accounted for at the two-point clustering level, which automatically propagates them into any cosmological parameter. When analyzing the data in terms of compressed model-agnostic variables, we obtain a combined precision of 4.7% on the amplitude of the redshift space distortion (RSD) signal reaching a similar precision with just one year of DESI data than with twenty years of observation from the previous generation survey. We also analyze the data to directly constrain the cosmological parameters within the ΛCDM model using perturbation theory and combine this information with the reconstructed DESI DR1 galaxy BAO. Using a Big Bang Nucleosynthesis Gaussian prior on the baryon density parameter, ω b , and a weak Gaussian prior on the spectral index, n s , we constrain the matter density is Ω m = 0.296±0.010 and the Hubble constant H 0 = (68.63 ± 0.79)[km s -1 Mpc -1 ]. Additionally, we measure the amplitude of clustering σ 8 = 0.841±0.034. The DESI DR1 galaxy clustering results are in agreement with the ΛCDM model based on general relativity with parameters consistent with those from Planck. The cosmological interpretation of these results in combination with DESI DR1 Ly-α forest data and external datasets are presented in the companion paper [1].

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Good and proper: self-similarity of N -body simulations with proper force softening

ABSTRACT Analysis of self-similarity in scale-free N-body simulations reveals the spatial and temporal scales for which statistics measured in cosmological simulations are converged to the physical continuum limit. We examine how the range of scales in which the two-point correlation function is converged depends on the force softening length and whether it is held constant in comoving or proper coordinates. We find that a proper softening that reaches roughly 1/30th of the inter-particle spacing by the end of the simulation resolves the same spatial and temporal scales as a comoving softening of the same length while using a third fewer time-steps, for a range of scale factors typical to Lambda cold dark matter (ΛCDM) simulations. We additionally infer an inherent resolution limit, set by the particle mass and scaling as a−1/2, beyond which reducing the softening does not improve the resolution. We postulate a mapping of these results with spectral index n = −2 to ΛCDM simulations.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Designs from Local Random Quantum Circuits with SU ( d ) Symmetry

The generation of k -designs (pseudorandom distributions that emulate the Haar measure up to k moments) with local quantum circuit ensembles is a problem of fundamental importance in quantum information and physics. Despite the extensive understanding of this problem for ordinary random circuits, the crucial situations in which symmetries or conservation laws are in play are known to pose fundamental challenges and remain little understood. Here, we construct explicit local unitary ensembles that can achieve high-order unitary k -designs under transversal continuous symmetry, in the particularly important SU ( d ) case. Specifically, we define the convolutional quantum alternating (CQA) group generated by 4-local SU ( d ) -symmetric Hamiltonians as well as associated 4-local SU ( d ) -symmetric random unitary circuit ensembles and prove that they form and converge to SU ( d ) -symmetric k -designs, respectively, for all k < n ( n − 3 ) / 2 , with n being the number of qudits. A key technique that we employ to obtain the results is the Okounkov-Vershik approach to S n representation theory. To study the convergence time of the CQA ensemble, we develop a numerical method using the Young orthogonal form and the S n branching rule. We provide strong evidence for a subconstant spectral gap and certain convergence time scales of various important circuit architectures, which contrast with the symmetry-free case. We also provide comprehensive explanations of the difficulties and limitations in rigorously analyzing the convergence time using methods that have been effective for cases without symmetries, including Knabe’s local gap threshold and Nachtergaele’s martingale methods. This suggests that a novel approach is likely necessary for understanding the convergence time of SU ( d ) -symmetric local random circuits. Published by the American Physical Society 2024

Li, Zimu (ORCID:0000000314736492)↗
DOE OSTI2023

Large-Scale Materials Modeling at Quantum Accuracy: Ab Initio Simulations of Quasicrystals and Interacting Extended Defects in Metallic Alloys

Ab initio electronic-structure has remained dichotomous between achievable accuracy and length-scale. Quantum many-body (QMB) methods realize quantum accuracy but fail to scale. Density functional theory (DFT) scales favorably but remains far from quantum accuracy. We present a framework that breaks this dichotomy by use of three interconnected modules: (i) invDFT: a methodological advance in inverse DFT linking QMB methods to DFT; (ii) MLXC: a machine-learned density functional trained with invDFT data, commensurate with quantum accuracy; (iii) DFT-FE-MLXC: an adaptive higher-order spectral finite-element (FE) based DFT implementation that integrates MLXC with efficient solver strategies and HPC innovations in FE-specific dense linear algebra, mixed-precision algorithms, and asynchronous compute-communication. Furthermore, we demonstrate a paradigm shift in DFT that not only provides an accuracy commensurate with QMB methods in ground-state energies, but also attains an unprecedented performance of 659.7 PFLOPS (43.1% peak FP64 performance) on 619,124 electrons using 8,000 GPU nodes of Frontier supercomputer.

density functional theory↗
DOE OSTI2024

Charge self-consistent density functional theory plus ghost rotationally invariant slave-boson theory for correlated materials

We present a charge self-consistent density functional theory combined with the ghost rotationally invariant slave-boson (DFT+gRISB) formalism for studying correlated materials. Here, this method is applied to SrVO 3 and NiO, representing prototypical correlated metals and charge-transfer insulators. For SrVO 3 , we demonstrate that DFT+gRISB yields an accurate equilibrium volume and effective mass close to experimentally observed values. Regarding NiO, DFT+gRISB enables the simultaneous description of charge-transfer and Mott-Hubbard bands, significantly enhancing the accuracy of the original DFT+RISB approach. Furthermore, the calculated equilibrium volume and spectral function reasonably agree with experimental observations.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2020

Comparison of total emitted solar-induced chlorophyll fluorescence (SIF) and top-of-canopy (TOC) SIF in estimating photosynthesis

We report many studies have shown that solar-induced fluorescence (SIF) has a good potential to predict gross primary production (GPP) of vegetation. What we measured by remote sensing or near-surface platforms is top-of-canopy (TOC) SIF (SIF toc ), which is not necessarily equal or proportional to total emitted SIF (SIF tot ) from the entire canopy due to the (re)absorption and scattering effects. However, photosynthesis, the process that plants use to fix carbon from the atmosphere, occurs at the entire vertical canopy. Here, by using the recollision theory, we calculated SIF tot at 760 nm from the measured SIF toc , hyperspectral reflectance (R), canopy interception (i) and leaf albedo (ω l ). Among them, both SIF toc and R can be obtained from concurrent TOC spectral measurements; i and ω l in the near-infrared region can be estimated from the open access datasets with a good accuracy. Our result confirms that the measured SIF toc only accounts for a small fraction of SIF tot : the above-the-canopy sensor can only “see” on average 22.9% of SIF tot at Harvard Forest. SIF tot has the following advantages over SIF toc in estimating GPP: (1) SIF tot improves the diurnal estimate of canopy GPP, especially capable to capture the midday depression of photosynthesis which may cause the large discrepancies between SIF toc and GPP on a diurnal basis, (2) SIF tot produces a stronger correlation with GPP from plants with complex canopy structure or under sky conditions with more diffuse irradiance, and (3) the SIF tot -GPP relationship shows a stronger resilience to environmental stresses. The fluorescence escape ratio (f esc ), the ratio between SIF toc multiplied by π and SIF tot , is mostly determined by the sun-canopy-sensor geometry and leaf inclination distribution. The effect of LAI and the leaf chlorophyll concentration on f esc is marginal at the 760 nm wavelength. Our results suggest that converting SIF toc into SIF tot provides a better tool to understand and estimate GPP.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2020

Unification of nonequilibrium molecular dynamics and the mode-resolved phonon Boltzmann equation for thermal transport simulations

Nano-size confinement induces many intriguing non-Fourier heat conduction phenomena, such as nonlinear temperature gradients, temperature jumps near the contacts, and size-dependent thermal conductivity. Over the past decades, these effects have been studied and interpreted by nonequilibrium molecular dynamics (NEMD) and phonon Boltzmann transport equation (BTE) simulations separately, but no theory that unifies these two methods has ever been established. In this work, we unify these methods using a quantitative mode-level comparison and demonstrate that they are equivalent for various thermostats. We show that different thermostats result in different non-Fourier thermal transport characteristics due to the different mode-level phonon excitations inside the thermostats, which explains the different size-dependent thermal conductivities calculated using different reservoirs, even though they give the same bulk thermal conductivity. Specifically, the Langevin thermostat behaves like a thermalizing boundary in phonon BTE and provides mode-level thermal-equilibrium phonon outlets, while the Nose-Hoover chain thermostat and velocity rescaling method behave like biased reservoirs, which provide a spatially uniform heat generation and mode-level nonequilibrium phonon outlets. These findings explain why different experimental measurement methods can yield different size-dependent thermal conductivity. They also indicate that the thermal conductivity of materials can be tuned for various applications by specifically designing thermostats. Finally, the unification of NEMD and phonon BTE will largely facilitate the study of thermal transport in complex systems in the future by, e.g., replacing computationally unaffordable first-principles NEMD simulations with computationally less expensive spectral BTE simulations.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2023

Effects of anisotropy and disorder on the superconducting properties of niobium

We report results for the superconducting transition temperature and anisotropic energy gap for pure niobium based on Eliashberg’s equations and electron and phonon band structures computed from density functional theory. The electronic band structure is used to construct the Fermi surface and calculate the Fermi velocity at each point on the Fermi surface. The phonon bands are in excellent agreement with inelastic neutron scattering data. The corresponding phonon density of states and electron–phonon coupling define the electron–phonon spectral function, α 2 F(p, p'; ω), and the corresponding electron–phonon pairing interaction, which is the basis for computing the superconducting properties. The electron–phonon spectral function is in good agreement with existing tunneling spectroscopy data except for the spectral weight of the longitudinal phonon peak at ℏω LO = 23 meV. We obtain an electron–phonon coupling constant of λ = 1.057, renormalized Coulomb interaction μ* = 0.218, and transition temperature T c = 9.33 K. The corresponding strong-coupling gap at T = 0 is modestly enhanced, Δ 0 = 1.55 meV, compared to the weak-coupling BCS value $Δ^{\text {wc}}_{0}$ = 1.78 k B T c = 1.43 meV. The superconducting gap function exhibits substantial anisotropy on the Fermi surface. We analyze the distribution of gap anisotropy and compute the suppression of the superconducting transition temperature using a self-consistent T-matrix theory for quasiparticle-impurity scattering to describe niobium doped with non-magnetic impurities. We compare these results with experimental results on niobium SRF cavities doped with nitrogen impurities.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗