Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Dirac relativistic electron equation”

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 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Correlated Dirac–Coulomb–Breit multiconfigurational self-consistent-field methods

The fully correlated frequency-independent Dirac–Coulomb–Breit Hamiltonian provides the most accurate description of electron–electron interaction before going to a genuine relativistic quantum electrodynamics theory of many-electron systems. In this work, we introduce a correlated Dirac–Coulomb–Breit multiconfigurational self-consistent-field method within the frameworks of complete active space and density matrix renormalization group. In this approach, the Dirac–Coulomb–Breit Hamiltonian is included variationally in both the mean-field and correlated electron treatment. Here, we also analyze the importance of the Breit operator in electron correlation and the rotation between the positive- and negative-orbital space in the no-virtual-pair approximation. Atomic fine-structure splittings and lanthanide contraction in diatomic fluorides are used as benchmark studies to understand the contribution from the Breit correlation.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS1994

Electron Correlation in 4-Component Relativistic Calculations

The full 4-component Dirac-Coulomb equation can nowadays be used in molecular calculations, The first step in solving this relativistic many-electron equation usually consists of solving the closed or open-shell Diarc-Fock equations. Like in non-relativistic calculations the outcome does not account for the effects of electron correlation. This can in principle be remedied by developing relativistic variants of electron correlation methods like Configuration Interaction or Coupled Cluster. In this talk the differences and similarities of such relativistic approaches as compared to non-relativistic methods will be reviewed. Results of Configuration Interaction calculations on the PtH molecule and on the MeF(sub 6, sup 2-) (Me= Co, Rh, Ir) complexes will be presented to give an impression of the kind of results that currently can be obtained.

Visscher, Luuk↗
DOE OSTI2021

Quantum field theory with the generalized uncertainty principle II: Quantum Electrodynamics

Highlights: • Relativistic Generalized Uncertainty Principle gives Frame independent minimum length. • Quantum Gravity modified Dirac equation from Modified Klein–Gordon Equations. • There exists a Quantum Gravity modified Lagrangian for a spinor field theory. • Feynman vertices contain 2 fermions & up to 5 gauge bosons are allowed. • Minimum length modifies the amplitude of Electrodynamic electron–muon scattering. Continuing our earlier work on the application of the Relativistic Generalized Uncertainty Principle (RGUP) to quantum field theories, in this paper we study Quantum Electrodynamics (QED) with minimum length. We obtain expressions for the Lagrangian, Feynman rules and scattering amplitudes of the theory, and discuss their consequences for current and future high energy physics experiments. We hope this will provide an improved window for testing Quantum Gravity effects in the laboratory.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS2012

Positron Interactions with Atoms and Ions

Dirac, in 1928, combining the ideas of quantum mechanics and the ideas of relativity invented the well-known relativistic wave equation. In his formulation, he predicted an antiparticle of the electron of spin n-bar/2. He thought that this particle must be a proton. Dirac published his interpretation in a paper 'A theory of electrons and protons.' It was shown later by the mathematician Hermann Weyl that the Dirac theory was completely symmetric between negative and positive particles and the positive particle must have the same mass as that of the electron. In his J. Robert Oppenheimer Memorial Prize Acceptance Speech, Dirac notes that 'Blackett was really the first person to obtain hard evidence for the existence of a positron but he was afraid to publish it. He wanted confirmation, he was really over cautious.' Positron, produced by the collision of cosmic rays in a cloud chamber, was detected experimentally by Anderson in 1932. His paper was published in Physical Review in 1933. The concept of the positron and its detection were the important discoveries of the 20th century. I have tried to discuss various processes involving interactions of positrons with atoms and ions. This includes scattering, bound states and resonances. It has not been possible to include the enormous work which has been carried out during the last 40 or 50 years in theory and measurements.

Bhatia, Anand K.↗
NASA NTRS1976

On the equation of state for an electron gas in an intense magnetic field

In this paper we derive the equation of state for a relativistic electron gas imbedded in a static homogeneous magnetic field of arbitrary strength. The derivation is based on the evaluation of the energy-momentum tensor and the use of Dirac's equation for such a problem. Contrary to a derivation presented several years ago, the present derivation is completely gauge-invariant. We also show how to recover, in an exact manner, the perfect gas law for the case of weak magnetic fields.

Canuto, V.↗
DOE OSTI2020

Dirac's equation and its implications for density functional theory based calculations of materials containing heavy elements

Electronic structure calculations based on density functional theory (DFT) give quantitatively accurate predictions of properties of most materials containing light elements. For heavy materials, and in particular for f -electron systems, DFT based methods can fail both qualitatively and quantitatively for two distinct reasons: their failure to describe confinement effects arising from localized f -electron behavior and their incomplete or approximate treatment of relativity. In addition, different methods for incorporating relativistic effects, which give identical results in most light materials, can give different predictions in heavy elements. In order to develop a quantitative capability for calculating the properties of these materials, it is essential to separate the predictions of the underlying equations from the uncertainty introduced in approximations used in computation. Working toward that goal, here we have developed a code, called dirac-fp , which is based directly on solving the Dirac-Kohn-Sham equations and uses the full-potential linear muffin-tin orbital (FP-LMTO) approach to electronic structure. In order to assess the performance of dirac-fp , we perform calculations on three different face-centered cubic materials using different approximate treatments of relativity: the scalar relativistic (SR) approach commonly used in most solid-state DFT codes, the scalar relativistic plus spin-orbit coupling corrections (SR+SO) approach which includes spin-orbit coupling self-consistently using the SR states inside the muffin tins, and the Dirac-Kohn-Sham (Dirac) approach implemented in dirac-fp . Performing calculations on thorium, in which relativistic effects should be strong, aluminum, in which relativistic effects should be negligible, and gold, in which relativistic effects play an intermediate role, we find that the Dirac approach is able to provide theoretically consistent results in the electronic structure and ground-state properties across all three materials.

3-dimensional systems↗
DOE OSTI2023

New insights into the electronic structure of α-U and δ-Pu

Here, this work presents the results of a theoretical study of the electronic structure of two actinide metals, α -U and δ -Pu. We compare our ab-initio results obtained with the recently developed self-consistent Vertex corrected GW approach with previously published experimental measurements such as photo-electron spectroscopy, for the occupied density of states, and bremsstralung isochromat spectroscopy (BIS) and inverse photo-electron spectroscopy (IPES), for the unoccupied density of states. Our ab-initio approach includes all important relativistic effects (it is based on Dirac’s equation) and it represents the first application of the Vertex corrected GW approach in the physics of actinides. Overall, our theoretical results are in good agreement with the experimental data, which supports the level of approximations which our theoretical method is based upon. By comparing our vertex corrected GW results with our results obtained with less sophisticated approaches (local density approximation and self-consistent GW) we differentiate the strength of correlation effects in Uranium and Plutonium. Also, our theoretical results allow us to elucidate the subtle differences between the previously published experimental BIS and IPES data on the unoccupied density of states in α -U.

36 MATERIALS SCIENCE↗
DOE OSTI2020

A fully implicit, scalable, conservative nonlinear relativistic Fokker–Planck 0D-2P solver for runaway electrons

Upon application of a sufficiently strong electric field, electrons break away from thermal equilibrium and approach relativistic speeds. These highly energetic ‘runaway’ electrons (~ MeV) play a significant role in tokamak disruption physics, and therefore their accurate understanding is essential to develop reliable mitigation strategies. As such, we have developed a fully implicit solver for the 0D-2P (i.e., including two momenta coordinates) relativistic nonlinear Fokker–Planck equation (rFP). As in earlier implicit rFP studies (NORSE, CQL3D), electron–ion interactions are modeled using the Lorentz operator, and synchrotron damping using the Abraham–Lorentz–Dirac reaction term. However, our implementation improves on these earlier studies by (1) ensuring exact conservation properties for electron collisions, (2) strictly preserving positivity, and (3) being scalable algorithmically and in parallel. Key to our proposed approach is an efficient multigrid preconditioner for the linearized rFP equation, a multigrid elliptic solver for the Braams–Karney potentials, and a novel adaptive technique to determine the associated boundary values. We verify the accuracy and efficiency of the proposed scheme with numerical results ranging from small electric-field electrical conductivity measurements to the accurate reproduction of runaway tail dynamics when strong electric fields are applied.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Compton rocket effect due to the action of radiation reaction force in degenerate plasma

A closed set of fluid equations with radiation reaction force (RRF) are constructed from the moments of the appropriate single particle kinetic equation describing a relativistic degenerate (high density) electron plasma. The closure, in analogy with the Maxwellian closure for non-degenerate plasmas, is affected via a parametrized Fermi-Dirac distribution. It is shown that the degeneracy increases RRF just as will be predicted from the so-called “Compton Rocket” effect.

Physics↗
DOE OSTI2022

Strong Relativistic Effects in Lanthanide-Based Single-Molecule Magnets

Lanthanide-based single-molecule magnets (SMMs) are promising building blocks for quantum memory and spintronic devices. Designing lanthanide-based SMMs with long spin relaxation time requires a detailed understanding of their electronic structure, including the crucial role of the spin-orbit coupling (SOC). While traditional calculations of SOC using the perturbation theory applied to a solution of the non-relativistic Schrödinger equation are valid for light atoms, this approach is questionable for systems containing heavy elements such as lanthanides. We investigate the accuracy of the perturbation estimates of SOC by variationally solving the Dirac equation for the [DyO] + molecule, a prototype of a lanthanide-based SMM. We show that the energy splittings between the M J states involved in spin relaxation depend on the interplay between strong SOC and dynamic electron correlations. Here, we demonstrate that this interplay affects the resonances between the spin and vibrational transitions, and, therefore, the spin relaxation time.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS1991

Synchrotron radiation with radiation reaction

A rigorous discussion is presented of the classical motion of a relativistic electron in a magnetic field and the resulting electromagnetic radiation when radiation reaction is important. In particular, for an electron injected with initial energy gamma(0), a systematic perturbative solution to the Lorentz-Dirac equation of motion is developed for field strengths satisfying gamma(0) B much less than 6 x 10 to the 15th G. A particularly accurate solution to the electron orbital motion in this regime is found and it is demonstrated how lowest-order corrections can be calculated. It is shown that the total energy-loss rate corresponds to what would be found using the exact Larmor power formula without including radiation reaction. Provided that the particle energy and field strength satisfy the same contraint, it is explicitly demonstrated that the intuitive prescription for calculating the time-integrated radiation spectrum described above is correct.

Nelson, Robert W.↗
DOE OSTI2024

Anomalous conductivity due to relativistic Landau quantization

Here we use a recently developed kinetic model derived from the Dirac equation, in order to study electromagnetic wave propagation in superstrong magnetic fields, such as in magnetars, where relativistic Landau quantization is prominent. The leading contribution to the conductivity tensor in such a plasma is calculated. It is found that the electron Hall current has an anomalous contribution, in the quantum relativistic regime, where the effective particle energy has a significant contribution from the diamagnetic and Zeeman energy. As a result, a new quantum resonance frequency appears, and the dispersion relation for the left- and right-hand polarized modes are strongly modified for long and moderate wavelengths. The implications for magnetar physics are discussed.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Time inversion symmetry in the Dirac and Schrödinger-Pauli theories

The Schrödinger-Pauli theory is generally believed to give a faithful representation of the nonrelativistic and weakly relativistic limit of the Dirac theory. However, the Schrödinger-Pauli theory is fundamentally incomplete in its account of broken time inversion symmetry, e.g., in magnetically ordered systems. Here, in the Dirac theory of the electron, magnetic order breaks time inversion symmetry even in the nonrelativistic limit, whereas time inversion symmetry is effectively preserved in the Schrödinger-Pauli theory in the absence of spin-orbit coupling. In the Dirac theory, the Berry curvature $1/(2m^2 c^2)$ is thus an intrinsic property of nonrelativistic electrons similar to the well-known spin magnetic moment $e\hbar/(2m)$, while this result is missed by the nonrelativistic or weakly relativistic Schrödinger-Pauli equation. In ferromagnetically ordered systems, the intrinsic Berry curvature yields a contribution to the anomalous Hall conductivity independent of spin-orbit coupling.

Winkler, R. [Northern Illinois Univ., DeKalb, IL (↗
DOE OSTI2026

Gaunt and Breit two-electron contributions to mean-field transformations and fine structure splitting

Materials utilized by novel energy systems are often studied using weakly correlated mean-field theories. However, if these systems incorporate heavy elements, relativistic effects must be included. Therefore, a Kramers unrestricted coupled cluster with singles and doubles excitation formalism within a molecular mean-field exact two-component framework (X2C mmf ) using a four-component Dirac–Hartree–Fock (DHF) reference state is presented. The exact X2C mmf transformed normal-order Hamiltonian incorporates all one-electron and two-electron (2e) contributions from the Coulomb, Gaunt, and Breit operators and is used with the equation of motion method to calculate the excitation energies of the alkali group of elements. Using this framework, the effects of 2e Gaunt and Breit integrals are studied. Results demonstrate growing contributions from these integrals to the generated X2C mmf mean-fields and electronic fine structure calculations with increasing atomic number. Overall, this paper outlines the method, its effect within the X2C mmf approach, and lays the foundation for future theoretical development of relativistic calculations within this framework.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Pushing the limits of the periodic table — A review on atomic relativistic electronic structure theory and calculations for the superheavy elements

We review the progress in atomic structure theory with a focus on superheavy elements and their predicted ground state configurations important for an element’s placement in the periodic table. To understand the electronic structure and correlations in the regime of large atomic numbers, it is essential to correctly solve the Dirac equation in strong Coulomb fields, and to take into account quantum electrodynamic effects. We specifically focus on the fundamental difficulties encountered when dealing with the many-particle Dirac equation. We further discuss the possibility for future many-electron atomic structure calculations going beyond the critical nuclear charge $Z_{crit}≈170$, where levels such as the $1s$ shell dive into the negative energy continuum ($E_{nκ} < –m_{e}c^2$). The nature of the resulting Gamow states within a rigged Hilbert space formalism is highlighted.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1997

Interfacing Relativistic and Nonrelativistic Methods: A Systematic Sequence of Approximations

A systematic sequence of approximations for the introduction of relativistic effects into nonrelativistic molecular finite-basis set calculations is described. The theoretical basis for the approximations is the normalized elimination of the small component (ESC) within the matrix representation of the modified Dirac equation. The key features of the normalized method are the retention of the relativistic metric and the ability to define a single matrix U relating the pseudo-large and large component coefficient matrices. This matrix is used to define a modified set of one- and two-electron integrals which have the same appearance as the integrals of the Breit-Pauli Hamiltonian. The first approximation fixes the ratios of the large and pseudo-large components to their atomic values, producing an expansion in atomic 4-spinors. The second approximation defines a local fine-structure constant on each atomic centre, which has the physical value for centres considered to be relativistic and zero for nonrelativistic centres. In the latter case, the 4-spinors are the positive-energy kinetic al ly-balanced solutions of the Levy-Leblond equation, and the integrals involving pseudo-large component basis functions on these centres, are set to zero. Some results are presented for test systems to illustrate the various approximations.

Dyall, Ken↗
DOE OSTI2025

Effect of interparticle fields and radiation reaction on beam dynamics

The dynamics of relativistic particles in an intense electromagnetic field can be described by the Landau-Lifshitz (LL) equation, where the radiation reaction (RR) is accounted for via a self-force, and interparticle fields are often neglected as an approximation. However, the inclusion of interparticle fields is necessary to ensure energy-momentum conservation, particularly during coherent emission. Here we present (i) an analytical proof showing that the energy-momentum conservation law of the Hamilton-Rohrlich-Dirac action, which is divergence free and describes a generic system of interacting charges, respects causality and provides physically sensible results; (ii) a simple generalization of the LL equation for many particles evaluated as a function of the total field, i.e., the sum of the external and interparticle fields. By performing first-principles numerical simulations of a neutral, relativistic bunch of electrons and positrons (e − /e + ) colliding with a laser pulse, this theory is shown to satisfy energy-momentum conservation when interparticle fields and RR are simultaneously taken into account; and (iii) the combined effect of interparticle fields and RR primarily affects the tail of the particle energy distribution. Additionally, our first-principles simulations show that the effect of interparticle fields on beam energy loss becomes smaller when most of the radiated energy is incoherent.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

Elimination of the linearization error in APW/LAPW basis set: Dirac-Kohn-Sham equations

Here, a detailed account of the implementation of equations of the relativistic density functional theory (RDFT) using basis sets of APW/LAPW type with flexible extensions provided by local orbitals is given. Earlier discoveries of the importance of the high derivative local orbital (HDLO) extension of the APW/LAPW basis set for enhancing the accuracy of DFT calculations are confirmed using a fully relativistic approach and α – U as an example. High energy local orbitals (HELO's), however indispensable for GW calculations, are considerably less efficient in enhancing the accuracy of DFT applications. It is shown that a simplified approach to the relativistic effects, namely considering them only inside the muffin-tin (MT) spheres, produces basically identical results (as compared to fully relativistic approach) for the electronic free energy of the five materials considered in this work. By comparing the effect of the simplified approach on the electronic free energy with its effect on the electronic kinetic energy we conclude that the insensitivity of the free energy to the way we describe the relativistic effects in the interstitial region is related to the variational property of this quantity.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗