Relativistic Dirac Equation Research Articles

Theoretical studies on the scattering of e± off the C2H2 molecule

Abstract By employing previous models [15,18], we report the differential, integrated elastic, inelastic, total (elastic+inelastic), viscosity and momentum transfer cross sections of e±- C2H2 collision dynamics, for the energy range of 1 eV-1 MeV. This report also incorporates Sherman spin polarization function and total ionization cross section. This work provides the first comprehensive study of electron/positron scattering from C2H2 over such a wide energy range. A complex optical model potential (OMP) has been used to represent the projectile-atom interaction. Relativistic Dirac equation has been solved, to get phase-shifts, needed for the calculations of scattering observables, using OMP. Our results are compared with the existing experimental data and theoretical calculations.

Study of electron and positron elastic scattering cross-sections of astro molecule H2S

Abstract The elastic, integrated, momentum transfer, viscosity, energy-dependent, and differential cross-sections and the Sherman function for electron and positron H2S scattering are reported at impact energies ranging from 1 eV to 1 MeV. The average orientations of the polar molecule H2S are considered, and the electron and positron elastic scattering cross-sections of H2S are typically calculated using single scattering-independent atom approximation. The relativistic Dirac equation is solved using the free atom optical potential, which includes the electrostatic interaction potential, exchange potential, correlation polarization potential, and imaginary absorption potential. The present computed cross-section results are compared with the available experimental and theoretical results, and a reasonable agreement is observed.

Study of the electronic structure and electron impact excitation cross section of helium impurities in spherical quantum dots

This manuscript is devoted to explore the atomic structure and electron impact excitation process of atom impurities in quantum dots. To achieve this goal, a method that solves the fully relativistic Dirac equation within the framework of relativistic configuration interaction is proposed. The Gaussian potential is used, which can accurately describe the location of impurities in quantum dots and their local effects on the surrounding electron cloud. The coupled Dirac equation is modified to include a new central potential, providing solutions that include both the continuous and bound state wave functions. The process of electron impact excitation is elucidated using the distorted wave method, all within the framework of relativistic Dirac theory. For illustrative purposes, a detailed investigation of the excitation energies, transition rates, wave functions, and excitation cross sections is carried out for a wide range of confinement strengths of the potential and quantum dot radii, using the helium impurities in spherical quantum dots as an example. Our results reveal that for a given confinement strength of the potential, the bound state wave functions are initially pulled into the inner region by the attractive Gaussian potential well, but eventually reflect the free atom scenario at large quantum dot radii. In contrast, the continuous electron wave functions exhibit monotonic variations as a function of the quantum dot radii. Such behavior of the wave functions gives rise to distinctive phenomena in the variation of excitation energies, transition rates, and excitation cross sections in relation to the potential parameters. Good agreement between the present results and existing data, where available, is obtained. This work holds importance not only for basic research in atomic physics but also for the optical and electronic applications of quantum dots. I.e., in the design and optimization of quantum dot lasers and quantum dot sensors.

Effect of Extreme Variations of Fundamental Constants on the Structure of Atoms and Molecules.

The fundamental constants (FCs) of physics are promoted to dynamic quantities in modern theories. So far most of the literature focused on small fractional variations in the values of FCs. In this paper, we investigate the novel regime of extreme but transient variations of FCs. We focus on the speed of light (c) and show that its variation can dramatically change the electronic structure and chemistry of atoms and molecules. These changes are induced by increased relativistic effects when c is reduced from its nominal value. To model these changes, we solve the fully relativistic Dirac equation at different values of c. We show that at extreme variations of c, the periodic table is truncated, the nominal ground states of atoms can change, water fails to serve as a universal solvent, and the ammonia molecule becomes planar.

Perspective on Coupled-cluster Theory. The evolution toward simplicity in quantum chemistry.

Coupled-cluster theory has revolutionized quantum chemistry. It has provided the framework to effectively solve the problem of electron correlation, the main focus of the field for over 60 years. This has enabled ab initio quantum chemistry to provide predictive quality results for most quantities of interest that are obtainable from first-principle calculations. The best that one can do in a basis is the 'full CI,' the exact solution of the non-relativistic Schrödinger equation or, if need be, the relativistic Dirac equation. With due regard to converging the basis set and adequate consideration of higher clusters and relativity in a calculation, virtually predictive results can be obtained. But in addition to its numerical performance, coupled-cluster theory also offers a conceptually new, many-body foundation for the theory that should be appreciated by all practitioners. The latter is emphasized in this perspective, leading to the 'evolution toward simplicity' in the title. The ultimate theory will benefit from the several features that are uniquely exact in coupled-cluster theory and its equation-of-motion (EOM-CC) extensions.

Nonlinear PDE Models in Semi-relativistic Quantum Physics

Abstract We present the self-consistent Pauli equation, a semi-relativistic model for charged spin- 1 / 2 1/2 particles with self-interaction with the electromagnetic field. The Pauli equation arises as the O ⁢ ( 1 / c ) O(1/c) approximation of the relativistic Dirac equation. The fully relativistic self-consistent model is the Dirac–Maxwell equation where the description of spin and the magnetic field arises naturally. In the non-relativistic setting, the correct self-consistent equation is the Schrödinger–Poisson equation which does not describe spin and the magnetic field and where the self-interaction is with the electric field only. The Schrödinger–Poisson equation also arises as the mean field limit of the 𝑁-body Schrödinger equation with Coulomb interaction. We propose that the Pauli–Poisson equation arises as the mean field limit N → ∞ N\to\infty of the linear 𝑁-body Pauli equation with Coulomb interaction where one has to pay extra attention to the fermionic nature of the Pauli equation. We present the semiclassical limit of the Pauli–Poisson equation by the Wigner method to the Vlasov equation with Lorentz force coupled to the Poisson equation which is also consistent with the hierarchy in 1 / c 1/c of the self-consistent Vlasov equation. This is a non-trivial extension of the groundbreaking works by Lions & Paul and Markowich & Mauser, where we need methods like magnetic Lieb–Thirring estimates.

Study of energies spectra and thermodynamic properties of the relativistic Dirac equation using Feynman path integral method

In the current paper, the contribution of a Coulomb-like potential tensor interaction on the solution of the Dirac equation with a new generalized Morse-like potential is investigated using the Feynman path integral method. Relativistic and non-relativistic energy spectra were obtained. It has been established that the Coulomb-like potential eliminates the degeneracy of all pairs of spin doublets. Using the resulting non-relativistic energy eigenvalues equation, three diatomic molecules (H2, LiH, and HCl) were investigated and their thermodynamic properties, including mean energy, free energy, entropy, and specific heat capacity were shown. A comparison with the available literature shows that the thermodynamic plots obtained are consistent with previous work.

Radiative recombination of highly charged ions with polarized electrons

We present a theoretical study of the radiative recombination (RR) of heavy highly charged ions with arbitrary (longitudinally and transversely) polarized electrons. In order to investigate how the spin state of incident electrons affects the linear polarization of emitted photons, we apply the density matrix theory and solutions of the relativistic Dirac equation. Explicit analytical expressions are obtained for the dependencies of the differential cross section and the polarization Stokes parameters of RR photons on the components of the incident electrons polarization vector. Detailed calculations of the degree of linear polarization and the tilt angle that characterizes the orientation of the polarization axis are carried out for radiative recombination with bare ${\mathrm{U}}^{92+}$ and ${\mathrm{Xe}}^{54+}$ ions for different polarization states of the incoming electrons. Based on the results of these calculations, we argue that the linear polarization of RR photons is very sensitive to the spin state of an electron target, and this sensitivity can be easily observed with the help of modern Compton polarization detectors.

Pauli’s Electron in Ehrenfest and Bohm Theories, a Comparative Study

Electrons moving at slow speeds much lower than the speed of light are described by a wave function which is a solution of Pauli's equation. This is a low-velocity limit of the relativistic Dirac equation. Here we compare two approaches, one of which is the more conservative Copenhagen's interpretation denying a trajectory of the electron but allowing a trajectory to the electron expectation value through the Ehrenfest theorem. The said expectation value is of course calculated using a solution of Pauli's equation. A less orthodox approach is championed by Bohm, and attributes a velocity field to the electron also derived from the Pauli wave function. It is thus interesting to compare the trajectory followed by the electron according to Bohm and its expectation value according to Ehrenfest. Both similarities and differences will be considered.

Quantum Transport and Magnetism of Dirac Electrons in Solids.

The relativistic Dirac equation covers the fundamentals of electronic phenomena in solids and as such it effectively describes the electronic states of the topological insulators like Bi_{2}Se_{3} and Bi_{2}Te_{3}. Topological insulators feature gapless surface states and, moreover, magnetic doping and resultant ferromagnetic ordering break time-reversal symmetry to realize quantum anomalous Hall and Chern insulators. Here, we focus on the bulk and investigate the mutual coupling of electronic and magnetic properties of Dirac electrons. Without carrier doping, spiral magnetic orders cause a ferroelectric polarization through the spin-orbit coupling. In a doped metallic state, the anisotropic magnetoresistance arises without uniform magnetization. We find that electric current induces uniform magnetization and conversely an oscillating magnetic order induces electric current. Our model provides a coherent and unified description of all those phenomena. The mutual control of electric and magnetic properties demonstrates implementations of antiferromagnetic spintronics. We also discuss the stoichiometric magnetic topological insulator MnBi_{2}Te_{4}.

Chiral Dirac-like fermion in spin-orbit-free antiferromagnetic semimetals

Dirac semimetal is a phase of matter whose elementary excitation is described by the relativistic Dirac equation. In the limit of zero mass, its parity-time symmetry enforces the Dirac fermion in the momentum space, which is composed of two Weyl fermions with opposite chirality, to be non-chiral. Inspired by the flavor symmetry in particle physics, we theoretically propose a massless Dirac-like equation yet linking two Weyl fields with the identical chirality by assuming isospin symmetry, independent of the space-time rotation exchanging the two fields. Dramatically, such symmetry is hidden in certain solid-state spin-1/2 systems with negligible spin-orbit coupling, where the spin degree of freedom is decoupled with the lattice. Therefore, the existence of the corresponding quasiparticle, dubbed as flavor Weyl fermion, cannot be explained by the conventional (magnetic) space group framework. The 4-fold degenerate flavor Weyl fermion manifests linear dispersion and a Chern number of 2, leading to a robust network of topologically protected Fermi arcs throughout the Brillouin zone. For material realization, we show that the transition-metal chalcogenide CoNb3S6 with experimentally confirmed collinear antiferromagnetic order is ideal for flavor Weyl semimetal under the approximation of vanishing spin-orbit coupling. Our work reveals a counterpart of the flavor symmetry in magnetic electronic systems, leading to further possibilities of emergent phenomena in quantum materials.

Wave Packet Dynamics in Synthetic Non-Abelian Gauge Fields.

It is generally admitted that in quantum mechanics, the electromagnetic potentials have physical interpretations otherwise absent in classical physics as illustrated by the Aharonov-Bohm effect. In 1984, Berry interpreted this effect as a geometrical phase factor. The same year, Wilczek and Zee generalized the concept of Berry phases to degenerate levels and showed that a non-Abelian gauge field arises in these systems. In sharp contrast with the Abelian case, spatially uniform non-Abelian gauge fields can induce particle noninertial motion. We explore this intriguing phenomenon with a degenerated Fermionic atomic gas subject to a two-dimensional synthetic SU(2) non-Abelian gauge field. We reveal the spin Hall nature of the noninertial dynamic as well as its anisotropy in amplitude and frequency due to the spin texture of the system. We finally draw the similarities and differences of the observed wave packet dynamic and the celebrated Zitterbewegung effect of the relativistic Dirac equation.

Differential Cross -Section Elastic Scattering Of Electrons By Silver Atoms

The differential cross-section (DCS) was theoretically calculated and measured with constant incident energies ranging from 10 eV to 100eV electron volts as a function of the scattering angle of 10o-150o. Dirac's relativistic equation is used to estimate the cross-sections of electron scattering and the spin polarization of electrons scattering from the silver atom. The obtained results are analyzed and compared with theoretical as well as experimental results of other research. The usefulness of this study is limited to obtaining information about scattering.

Revisiting quantum relativistic effects from phase transition by the catastrophe theory

In this paper we start from the Schrödinger equation to revisit some classical quantum mechanics from the perspective of phase transition process. Here the relativistic effect of particles moving at high speed can be regarded as the phase transition process when the velocity variable increases. Considering that the catastrophe theory could describe qualitatively any phase transition process, we adopt the simplest folding catastrophe type as the potential function in the Schrödinger equation to obtain a revised Schrödinger relativistic equation through the dimensionless analysis first, and then further to derive out the steady-state Klein-Gordon equation and Dirac relativistic equation gradually. These results reveal that the quantum relativistic effect could be considered as the phase transition process, which could be described by adopting the catastrophe models as the potential function in the classical Schrödinger equation.

Beam focus and longitudinal polarization influence on spin dynamics in the Kapitza-Dirac effect

We theoretically investigate the influence of a longitudinal laser polarization component from beam focusing on spin dynamics in Kapitza-Dirac scattering by solving the relativistic Dirac equation with time-dependent perturbation theory. The transverse spacial dependence of the longitudinal beam polarization component is accounted for, by approximating a Gaussian beam with plane-wave components. We find that corrections from a longitudinal laser beam polarization component approximately scale with the second power of the diffraction angle $\epsilon$, from which we conclude that a related influence from beam focusing can be made negligibly small for sufficiently low beam foci.

The Cosmic Black Hole as a Solution of the Relativistic Quantum Mechanical DIRAC Equation

For many physicists, Albert Einstein’s Theory of General Relativity is the top of Physics. A whole new concept, based on a flexible Space-Time Continuum. A wonderful New and Original insight in the Origins of Space and Time. But nowadays physics requires more than a fundamental theory about Space and Time.
 The Mathematical foundation for a “Quantum Mechanical Model of the Black Hole” is based on a 10-dimensional Space-Time Continuum. This article has been written in projections of a 10-Dimensional Space-Time Continuum within an easier to understand 4-Dimensional Space-Time Continuum. For that reason, this theory will not start with “Einstein’s famous Field Equations”, but the start will be at a very fundamental concept in Physics. Isaac Newton’s 3rd law as a fundament in Classical- and Quantum Mechanics.
 To make the theory of the “Quantum Mechanical Model of the Black Hole” as much understandable as possible, this article starts with a short comprehension of the theory.

Theoretical analysis of the long-distance limit of NMR chemical shieldings.

After some years of controversy, it was recently demonstrated how to obtain the correct long-distance limit [point-dipole approximation (PDA)] of pseudo-contact nuclear magnetic resonance chemical shifts from rigorous first-principles quantum mechanics [Lang et al., J. Phys. Chem. Lett. 11, 8735 (2020)]. This result confirmed the classical Kurland-McGarvey theory. In the present contribution, we elaborate on these results. In particular, we provide a detailed derivation of the PDA both from the Van den Heuvel-Soncini equation for the chemical shielding tensor and from a spin Hamiltonian approximation. Furthermore, we discuss in detail the PDA within the approximate density functional theory and Hartree-Fock theories. In our previous work, we assumed a relatively crude effective nuclear charge approximation for the spin-orbit coupling operator. Here, we overcome this assumption by demonstrating that the derivation is also possible within the fully relativistic Dirac equation and even without the assumption of a specific form for the Hamiltonian. Crucial ingredients for the general derivation are a Hamiltonian that respects gauge invariance, the multipolar gauge, and functional derivatives of the Hamiltonian, where it is possible to identify the first functional derivative with the electron number current density operator. The present work forms an important foundation for future extensions of the Kurland-McGarvey theory beyond the PDA, including induced magnetic quadrupole and higher moments to describe the magnetic hyperfine field.

THERMOELECTRIC EFFECTS IN SELF-SIMILAR MULTIBARRIER STRUCTURE BASED ON MONOLAYER GRAPHENE

Thermoelectric effects have attracted wide attention in recent years from physicists and engineers. In this work, we explore the self-similar patterns in the thermoelectric effects of monolayer graphene-based structures, by using the quantum relativistic Dirac equation. The transfer matrix method has been used to calculate the transmission coefficient. The Landauer–Büttiker formalism and the Cutler–Mott formula were used to calculate the conductance, the Seebeck coefficient, and the power factor. We find self-similar behavior and the scale factors between generations in transport and thermoelectric properties. Furthermore, we implement these scale invariances as general scaling rules. We present a new analytical demonstration of self-similarity in the Seebeck coefficient. These findings can open outstanding perspectives for experimentalists to develop thermoelectric devices.

A Theoretical Study of Scattering of Electrons and Positrons by CO2 Molecule

This article presents a theoretical investigation of the differential, integrated, elastic, inelastic, total, momentum-transfer, and viscosity cross-sections, along with the total ionization cross-section, for elastically scattered electrons and positrons from a carbon dioxide (CO2) molecule in the incident energy range of 1 eV ≤Ei≤ 1 MeV. In addition, for the first time, we report the spin polarization of e±−CO2 scattering systems. The independent atom model (IAM) with screening correction (IAMS) using a complex optical potential was employed to solve the Dirac relativistic equation in partial-wave analysis. The comparison of our results with the available experimental data and other theoretical predictions shows a reasonable agreement in the intermediate- and high-energy regions.

Accurate calculations for the Dirac electron in the field of two-center Coulomb field: Application to heavy ions

The relativistic Dirac equation for a bound electron in the field of two fixed positive charges is revisited. In contrast to the one center case this three dimensional equation is separable only partially around the azimuthal angle φ, because of the commutation of the Dirac Hamiltonian only with the z component of the total angular momentum Jz. In this work we determine the variational exact solution of this two center problem using a basis constructed by linear combinations of relativistic Slater type spinor wave functions with non integer powers of the radii r1 and r2 on the two centers. We present in some detail the determination of the two center integrations involved. The solutions are obtained by a minimax procedure, that we have developed with a new iterative scheme. We use independent large and small components of the Dirac spinor. This permits us to take control of the spurious solutions, and gives us the possibility to avoid them by the appropriate choice of the wave function parameters. We investigate the behavior of the electron in its 1sσg level of the diatomic homo-nuclear systems A2(2Z-1)+, where A represents the heavy element and Z its atomic number. In the case of heavy ions we study the dependance of the electron energy on the internuclear distance, this gives us an indication for the conditions of atomic collapse, which can induce electron-positron pair production. Our approach has the advantage of needing small basis sets for a relative error of the of 10-7-10-8. It can also be extended easily to the excited level such as the 1sσu level.