Solving Dirac Equation Research Articles

Compact dwarfs made of light-quark nuggets

Utilizing an equiparticle model with both linear confinement and leading-order perturbative interactions, we obtain systematically the properties of strangelets and nonstrange quark matter (udQM) nuggets at various baryon (A) and charge (Z) numbers, where the detailed single-quark-energy levels are fixed by solving Dirac equations in mean-field approximation. We then examine the structures of compact dwarfs made of light strangelets or udQM nuggets forming body-centered cubic lattices in a uniform electron background. Despite that the strangelets and udQM nuggets generally become more stable at larger A, the compact dwarfs are still stable since the fusion reactions between those objects do not take place in the presence of a Coulomb barrier, which is similar to the cases of light nuclei in normal white dwarfs. If udQM dwarfs or strangelet dwarfs are covered with normal matter, their masses and radii become larger but do not exceed those of ordinary white dwarfs. Finally, we investigate the radial oscillation frequencies of udQM dwarfs and strangelet dwarfs and find that their frequencies are typically higher than traditional white dwarfs. The stability of compact dwarfs are then analyzed by examining radial oscillation frequencies of the fundamental mode, where compact dwarfs covered by normal matter are still stable. Published by the American Physical Society 2024

Interface effects of quark matter: Light-quark nuggets and compact stars

The interface effects of quark matter play important roles in the properties of compact stars and small nuggets, such as strangelets and nonstrange quark matter ($ud\mathrm{QM}$) nuggets. By introducing a density derivative term to the Lagrangian density and adopting Thomas-Fermi approximation, we find it is possible to reproduce the results obtained by solving Dirac equations. Adopting certain parameter sets, the energy per baryon of $ud\mathrm{QM}$ nuggets decreases with baryon number $A$ and become more stable than nuclei at $A\ensuremath{\gtrsim}300$. The effects of quark matter symmetry energy are examined, where $ud\mathrm{QM}$ nuggets at $A\ensuremath{\approx}1000$ can be more stable than others if large symmetry energy is adopted. In such cases, larger $ud\mathrm{QM}$ nuggets will decay via fission and the surface of a $ud\mathrm{QM}$ star will fragment into a crust made of $ud\mathrm{QM}$ nuggets and electrons, which resembles the cases of a strange star's crust. The corresponding microscopic structures are then investigated adopting spherical and cylindrical approximations for the Wigner-Seitz cells, where the droplet phase is found to be the most stable configuration with $ud\mathrm{QM}$ stars' crusts and $ud\mathrm{QM}$ dwarfs made of $ud\mathrm{QM}$ nuggets ($A\ensuremath{\approx}1000$) and electrons. For the cases considered here, the crust thickness of $ud\mathrm{QM}$ stars is typically $\ensuremath{\sim}200\text{ }\text{ }\mathrm{m}$, which reaches a few kilometers if we neglect the interface effects and adopt Gibbs construction. The masses and radii of $ud\mathrm{QM}$ dwarfs are smaller than typical white dwarfs, which would increase if the interface effects are neglected.

Quantum tunneling in graphene Corbino disk in a solenoid magnetic potential with wedge disclination

We investigate the wedge disclination effect on quantum tunneling of a Corbino disk in gapped-graphene of inner R1 and outer R2 radii in the presence of magnetic flux Φi. We solve Dirac equation for different regions and obtain the solutions of energy spectrum in terms of Hankel functions. The asymptotic behaviors for large arguments allow us to determine the transmission, Fano factor and conductance. We establish the case where the crystal symmetry is modified locally by replacing a hexagon by pentagon, square, heptagon or octagon. We show that the wedge disclination n modifies the amplitude of transmission oscillations. We find that the period of Fano factor oscillations is of the Aharonov–Bohm type, which strongly depends on n where intense peaks are observed. As another result, n changes the minimum and period of conductance oscillations of the Aharonov–Bohm type. We show that n minimizes the effect of resonance and decreases the amplitude of conductance magnitude ΔG oscillations.

Solving Klein’s paradox

We figure out the famous Klein’s paradox arising from the reflection problem when a Dirac particle encounters a step potential with infinite width. The key is to piecewise solve Dirac equation in such a way that in the region where the particle’s energy E is greater (less) than the potential V, the solution of the positive (negative) energy branch is adopted. In the case of Klein–Gordon equation with a piecewise constant potential, the equation is decoupled to positive and negative energy equations, and reflection problem is solved in the same way. Both infinitely and finitely wide potentials are considered. The reflection coefficient never exceeds 1. The results are applied to discuss the transmissions of particles with no mass or with very small mass.

Model potential study of Rydberg one-electron spectrum of thallium

Semi-empirical model potential calculations of excitation energies, fine structure splittings, oscillator strengths and line strengths ratios were computed for high Rydberg transitions in principal, sharp and diffuse series of thallium one-electron spectrum with allowance for core–valence electron correlations in core polarization picture. Relativistic effects are fully included through solving Dirac equations and core–valence electron correlation is represented in core polarization picture by appropriate term in model potential optimized to reproduce ionization energies of the system for lowest states of given |nlj〉 symmetry. The dipole-moment operator of transition is modified in accordance. Systematic trends are investigated in line strength ratios and oscillator strengths along the spectral series, the latter showing hydrogen-like behavior for large n* effective quantum number of upper states of transition. The influence of core–valence correlation (core polarization) on line strength ratios and oscillator strengths along the spectral series is studied in detail demonstrating its importance even for transitions to highly excited Rydberg states with n=20. The strong effect of cancellation of positive and negative contributions to transition matrix element was observed for oscillator strengths and line strength ratios in principal series of thallium spectrum.

Semi-empirical model potential study of Rydberg transitions in gold one-electron spectrum

Semi-empirical model potential calculations of oscillator strengths for Rydberg transitions in principal, sharp and diffuse series of gold one-electron spectrum are presented. Relativistic effects are fully included through solving Dirac equations and core–valence electron correlation is represented in core polarization picture by appropriate part in model potential. The dipole-moment operator of transition is modified in accordance. Systematic trends are investigated in oscillator strengths along the spectral series showing hydrogen-like behavior for large n∗ effective quantum number of upper states of transitions. The so-called Cooper minimum in oscillator strengths is observed for principal series transitions. The influence of core–valence correlation (core polarization) on oscillator strengths along the spectral series is studied in detail​ demonstrating its importance even for transitions to rather highly excited Rydberg states with n≈18. In addition, excitation energies for some levels which positions in the neutral gold spectrum is unknown so far, are also predicted.

Energy levels of graphene magnetic circular quantum dot

We study the energy levels of graphene magnetic circular quantum dot surrounded by an infinite graphene sheet in the presence of an electrostatic potential. We solve Dirac equation to derive the solutions of energy spectrum associated with different regions composing our system. Using the continuum model and applying boundary conditions at the interface, we obtain analytical results for the energy levels. The dependence of the energy levels on the quantum dot radius, magnetic field and electrostatic potential is analyzed for the two valleys K and . We show that the energy levels exhibit characteristics of interface states and have an energy gap.

Variational calculations for relativistic two dimensional strongly interacting fermions: Application to 2D liquid 3He

In this paper, a variational method has been used to investigate the properties of two dimensional (2D) strongly interacting fermions. As an application, the results have been applied to 2D liquid H3e as a real fermionic system. We showed that this variational method, known as lowest order constrained variational method, can be used to relativistic 2D fermionic systems with a good accuracy. We proved that different features of a relativistic many body system can be studied without solving Dirac equation. In the case of 2D liquid H3e, our calculations show that at higher temperatures, relativistic corrections are more significant. For this system, as we expect, we show that relativistic effects are not considerable at low temperatures.

Transmission in graphene through time-oscillating linear barrier

Transmission probabilities of Dirac fermions in graphene under linear barrier potential oscillating in time are investigated. Solving Dirac equation we end up with the solutions of the energy spectrum depending on several modes coming from the oscillations. These will be used to obtain a transfer matrix that allows to determine transmission amplitudes of all modes. Due to numerical difficulties in truncating the resulting coupled channel equations, we limit ourselves to low quantum channels, i.e. l = 0, ± 1, and study the three corresponding transmission probabilities.

Application of relativistic coupled-cluster theory to electron impact excitation of Mg+ in the plasma environment

A relativistic coupled-cluster (RCC) theory is implemented to study electron impact excitations of atomic species. As a test case, the electron impact excitations of the $3s ~ ^2S_{1/2} - 3p ~ ^2P_{1/2;3/2}$ resonance transitions are investigated in the singly charged magnesium (Mg$^+$) ion using this theory. Accuracies of wave functions of Mg$^+$ are justified by evaluating its attachment energies of the relevant states and compared with the experimental values. The continuum wave function of the projectile electron are obtained by solving Dirac equations assuming distortion potential as static potential of the ground state of Mg$^+$. Comparison of the calculated electron impact excitation differential and total cross-sections with the available measurements are found to be in very good agreements at various incident electron energies. Further, calculations are carried out in the plasma environment in the Debye H\"uckel model framework, which could be useful in the astrophysics. Influence of plasma strength on the cross-sections as well as linear polarization of the photon emission in the $3p ~ ^2P_{3/2} - 3s ~ ^2S_{1/2}$ transition is investigated for different incident electron energies.

Solving Dirac equations on a 3D lattice with inverse Hamiltonian and spectral methods

A new method to solve the Dirac equation on a 3D lattice is proposed, in which the variational collapse problem is avoided by the inverse Hamiltonian method and the fermion doubling problem is avoided by performing spatial derivatives in momentum space with the help of the discrete Fourier transform, i.e., the spectral method. This method is demonstrated in solving the Dirac equation for a given spherical potential in 3D lattice space. In comparison with the results obtained by the shooting method, the differences in single particle energy are smaller than $10^{-4}$~MeV, and the densities are almost identical, which demonstrates the high accuracy of the present method. The results obtained by applying this method without any modification to solve the Dirac equations for an axial deformed, non-axial deformed, and octupole deformed potential are provided and discussed.

Improved analyses forμ−e−→e−e−in muonic atoms by contact interactions

The charged lepton flavor violating (CLFV) processes of ${\ensuremath{\mu}}^{\ensuremath{-}}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}}$ decay by four Fermi contact interactions in a muonic atom for various atoms are investigated. The wave functions of bound and scattering state leptons are properly treated by solving Dirac equations with Coulomb interaction of the finite nuclear charge distributions. This new effect contributes significantly in particular for heavier atoms, where the obtained decay rate is about one order of magnitude larger than the previous estimation for $^{208}\mathrm{Pb}$. We find that, as the atomic number $Z$ increases, the ${\ensuremath{\mu}}^{\ensuremath{-}}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}}$ decay rates increase more rapidly than the result of the previous work of ${Z}^{3}$, suggesting this decay as one of the promising processes to search for CLFV interaction.

Solving Dirac equation using the tridiagonal matrix representation approach

The aim of this work is to find exact solutions of the one dimensional Dirac equation using the tridiagonal matrix representation. We write the spinor wavefunction as a bounded infinite sum in a complete basis set, which is chosen such that the matrix representation of the Dirac wave operator becomes tridiagonal and symmetric. This makes the wave equation equivalent to a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. We solve the recursion relation and obtain the relativistic energy spectrum and corresponding state functions. We are honored to dedicate this work to Prof. Hashim A. Yamani on the occasion of his 70th birthday.

Quantum oscillations and wave packet revival in conical graphene structure

We present analytical expressions for the eigenstates and eigenvalues of electrons confined in a graphene monolayer in the presence of a disclination. The calculations are performed in the continuum limit approximation in the vicinity of the Dirac points, solving Dirac equation by freezing out the carrier radial motion. We include the effect of an external magnetic field and show the appearence of Aharonov-Bohm oscillation and find out the conditions of gapped and gapless states in the spectrum. We show that the gauge field due to a disclination lifts the orbital degeneracy originating from the existence of two valleys. The broken valley degeneracy has a clear signature on quantum oscillations and wave packet dynamics.

Analytical Calculation of Energy levels of mono- and bilayer Graphene Quantum Dots Used as Light Absorber in Solar Cells

In this paper by solving Dirac equation, we present an analytical solution to calculate energy levels and wave functions of mono- and bilayer graphene quantum dots. By supposing circular quantum dots, we solve Dirac equation and obtain energy levels and band gap with relations in a new closed and practical form. The energy levels are correlated with a radial quantum number and radius of quantum dots. In addition to monolayer quantum dots, AA- and AB-stacked bilayer quantum dots are investigated and their energy levels and band gap are calculated as well. Also, we analyze the influence of the quantum dots size on their energy spectrum. It can be observed that the band gap decreases as quantum dots’ radius increases. On the other hand, increase in the band gap is more in AB-stacked bilayer quantum dots. Using the obtained relations, the band gap is obtained in each state. Comparing the energy spectra obtained from the tight-binding approximation with those of our obtained relations shows that the behavior of the energies as function of the dot size is qualitatively similar, but in some cases, quantitative differences can be seen. As quantum dots radius increases, the analytical results approach to the tight-binding method results.

Supersymmetric analysis of the Dirac-Weyl operator within $\mathcal{PT}$PT symmetry

Two-dimensional effective Hamiltonian for a massless Dirac electron interacting with a hyperbolic magnetic field is discussed within \documentclass[12pt]{minimal}\begin{document}$\mathcal{PT}$\end{document}PT symmetry. Factorization method and polynomial procedures are used to solve Dirac equation for the constant Fermi velocity and the effective potential which is complex Scarf II potential. The more general effective Scarf II potential models are also obtained within pseudo-supersymmetry. Finally, an extension of Panella and Roy's work [Phys. Lett. A 376, 2580–2583 (2012)] to the both \documentclass[12pt]{minimal}\begin{document}$\mathcal{PT}$\end{document}PT symmetric and real Scarf II partner potentials is given using the position dependent Fermi velocity.

Tunnelling of relativistic particles from new type black hole in new massive gravity

In the framework of the three dimensional New MassiveGravity theory introduced by Bergshoeff, Hohm and Townsend, weanalyze the behavior of relativistic spin-1/2 and spin-0 particlesin the New-type Black Hole backgroud, solution of the New MassiveGravity.We solve Dirac equation for spin-1/2 and Klein-Gordonequation for spin-0. Using Hamilton-Jacobi method, we discuss tunnellingprobability and Hawking temperature of the spin-1/2 and spin-0particles for the black hole. We observe that the tunnellingprobability and Hawking temperature are same for the spin-1/2 and spin-0.

TUNNELING OF DIRAC PARTICLES FROM PHANTOM REISSNER NORDSTROM ADS BLACK HOLE

We study the Hawking radiation of Dirac particles from Phantom Reissner Nordstrom ADS black hole by tunneling method. We use charge relativistic Dirac equation to study the emission of such particles. To solve Dirac equation we use WKB approximation and find the tunneling probability of outgoing particles. Finally we find the Hawking temperature for such type of black holes.

Discreteness of Curved Spacetime from GUP

Diverse theories of quantum gravity expect modifications of the Heisenberg's uncertainty principle near the Planck scale to a so-called Generalized uncertainty principle (GUP). It was shown by some authors that the GUP gives rise to corrections to the Schrodinger , Klein-Gordon, and Dirac equations. By solving the GUP corrected equations, the authors arrived at quantization not only of energy but also of box length, area, and volume. In this paper, we extend the above results to the case of curved spacetime (Schwarzschild metric). We showed that we arrived at the quantization of space by solving Dirac equation with GUP in this metric.

PROPERTIES OF $b\bar{b}$ MESONS IN CONSTITUENT QUARK MODELS

The properties of bottomonia are investigated in detail in the constituent quark model. The wave functions of bottomonia are obtained by solving Dirac equation and Schrödinger equation. The potentials between quark and antiquark include color confinement (linear and quadratic) and one-gluon-exchange. Based on the obtained wave functions, the electromagnetic and two-photon decay, electric dipole transition, and hadronic width of bottomonia are calculated. The calculated results are in a good agreement with the experimental data. The results also show that the nonrelativistic and relativistic version of quark model can all describe the properties of bottomonia.