Two-center Dirac Equation Research Articles

Light one-electron molecular ions within the finite-basis-set method for the two-center Dirac equation

During the last decades outstanding results on the precision description of light diatomic molecular compounds have been achieved. The most advanced calculations of electron binding energies have been realized mainly in the framework of the nonrelativistic approach with a consistent account of relativistic and radiative QED corrections. Recently, it has been shown that methods based on the Dirac equation are also suitable for obtaining highly accurate results in simple light molecules. In this paper, we present a completely relativistic method and discuss its application to the description of diatomic systems. In particular, the electronic spectra of the light one-electron quasi-molecular compounds H-H+, He+-He2+ and He+-H+ are analyzed. For this purpose, the two-center Dirac equation is solved by a dual-kinetic balanced finite-basis-set method for axially symmetric systems, called A-DKB. This method allows for a complete relativistic consideration of the electron at fixed inter-nuclear distances. A comparison of the obtained results with the nonrelativistic and relativistic calculations presented in the literature is performed. Without pursuing the goal of high accuracy calculations, the advantages and disadvantages of the approach, as well as possible applications of the method, are discussed in detail.

Light antiproton one-electron quasi-molecular ions within the relativistic A-DKB method.

In the present work, two quasi-molecular compounds each involving one antiproton and one electron (p̄), He+-p̄ and H-p̄, are investigated. Using completely relativistic calculations within the finite-basis method adapted to systems with axial symmetry, the adiabatic potential curves are constructed by numerically solving the two-center Dirac equation. The binding energies of electron are obtained as a function of the inter-nuclear distance and compared with the corresponding nonrelativistic values and relativistic leading-order corrections calculated in the framework of other approaches. A semantic analysis of antiproton quasi-molecular ions with compounds containing a proton (p) instead of an antiproton is given. The advantages of the A-DKB method are demonstrated.

Solution of the two-center Dirac equation with 20-digit precision using the finite-element technique

We present a precise fully relativistic numerical solution of the two-center Coulomb problem. The special case of unit nuclear charges is relevant for the accurate description of the ${\rm H}_2^+$ molecular ion and its isotopologues, systems that are an active experimental topic. The computation utilizes the 2-spinor minmax approach and the finite-element method. The computed total energies have estimated fractional uncertainties of a few times $10^{-20}$ for unit charges and a bond length of 2 atomic units. The fractional uncertainty of the purely relativistic contribution is $1\times10^{-17}$. The result is relevant for future precision experiments, whereas at present the uncertainties arising from the quantum electrodynamic treatment of the rovibrational transition frequencies. are dominant.

Estimating the radiative part of QED effects in superheavy nuclear quasimolecules

A method for calculating the electronic levels in the compact superheavy nuclear quasi-molecules, based on solving the two-center Dirac equation using the multipole expansion of two-center potential, is developed. For the internuclear distances up to $d\sim100$~fm such technique reveals a quite fast convergence and allows for computing the electronic levels in such systems with accuracy $\sim 10^{-6} $. The critical distances $R_{cr}$ between the nuclei for $1\sigma_g$ and $1\sigma_u$ electronic levels in the region $Z\simeq 87-100$ are calculated. By means of the same technique the shifts of electronic levels due to the effective interaction $\Delta U_{AMM}$ of the electron's magnetic anomaly with the Coulomb field of the closely spaced heavy nuclei are evaluated as a function of the internuclear distance and the charge of the nuclei, non-perturbatively both in $Z\alpha$ and (partially) in $\alpha/\pi$. It is shown, that the levels shifts near the lower continuum decrease with the enlarging size of the system of Coulomb sources both in the absolute units and in units of $Z^4 \alpha^5 / \pi n^3$. The last result is generalized to the whole self-energy contribution to the level shifts and so to the possible behavior of radiative part of QED-effects with virtual photon exchange near the lower continuum in the overcritical region.

Contribution of higher multipole terms of Coulomb interaction to ionization in low-energy heavy-ion collisions

SynopsisThe two-center Dirac equation is solved in spherical coordinates using the multipole decomposition of the electron-nucleus interaction. The total and differential ionization probabilities in collisions of a hydrogenlike ion with a bare nucleus are calculated for different impact parameters. The contribution of the higher multipole terms of the interaction to these probabilities is analyzed. The influence of relativistic effects is examined as well.

Relativistic calculations of the U91+(1s)–U92+ collision using the finite basis set of cubic Hermite splines on a lattice in coordinate space

A new method for solving the time-dependent two-center Dirac equation is developed. The approach is based on the using of the finite basis of cubic Hermite splines on a three-dimensional lattice in the coordinate space. The relativistic calculations of the excitation and charge-transfer probabilities in the U91+(1s)-U92+ collisions in two and three dimensional approaches are performed. The obtained results are compared with our previous calculations employing the Dirac-Sturm basis sets [I.I. Tupitsyn et al., Phys. Rev. A 82, 042701 (2010)]. The role of the negative-energy Dirac spectrum is investigated within the monopole approximation.

Relativistic calculations of charge transfer probabilities in U92+–U91+(1s) collisions using the basis set of cubic Hermite splines

A new approach for solving the time-dependent two-center Dirac equation is presented. The method is based on using the finite basis set of cubic Hermite splines on a two-dimensional lattice. The Dirac equation is treated in the rotating reference frame. The collision of U92+ (as a projectile) and U91+ (as a target) is considered at energy Elab = 6 MeV u−1. The charge transfer probabilities are calculated for different values of the impact parameter. The results obtained are compared with previous calculations (Tupitsyn et al 2010 Phys. Rev. A 82 042701), where a method based on atomic-like Dirac–Sturm orbitals was employed. This work can provide a new tool for the investigation of quantum electrodynamics effects in heavy-ion collisions near the supercritical regime.

Solution of the two-center time-dependent Dirac equation in spherical coordinates: Application of the multipole expansion of the electron-nuclei interaction

A nonperturbative approach to the solution of the time-dependent, two-center Dirac equation is presented with a special emphasis on the proper treatment of the potential of the nuclei. In order to account for the full multipole expansion of this potential, we express eigenfunctions of the two-center Hamiltonian in terms of well-known solutions of the ``monopole'' problem that employ solely the spherically symmetric part of the interaction. When combined with the coupled-channel method, such a wave-function--expansion technique allows for an accurate description of the electron dynamics in the field of moving ions for a wide range of internuclear distances. To illustrate the applicability of the proposed approach, the probabilities of the $K$- as well as $L$-shell ionization of hydrogen-like ions in the course of nuclear $\ensuremath{\alpha}$ decay and slow ion-ion collisions have been calculated.

Relativistic calculations of the charge-transfer probabilities and cross sections for low-energy collisions of H-like ions with bare nuclei

A new method for solving the time-dependent two-center Dirac equation is developed. The time-dependent Dirac wave function is represented as a sum of atomic-like Dirac-Sturm orbitals, localized at the ions. The atomic orbitals are obtained by solving numerically the finite-difference one-center Dirac and Dirac-Sturm equations with the potential which is the sum of the exact reference-nucleus potential and a monopole-approximation potential from the other nucleus. An original procedure to calculate the two-center integrals with these orbitals is proposed. The approach is tested by calculations of the charge transfer and ionization cross sections for the H(1s)--proton collisions at proton energies from 1 keV to 100 keV. The obtained results are compared with related experimental and other theoretical data. To investigate the role of the relativistic effects, the charge transfer cross sections for the Ne^{9+}(1s)--Ne^{10+} (at energies from 0.1 to 10 MeV/u) and U^{91+}(1s)--U^{92+} (at energies from 6 to 10 MeV/u) collisions are calculated in both relativistic and nonrelativistic cases.

Spectra of baryons containing two heavy quarks

Doubly heavy baryons, i.e., the baryons containing two heavy quarks are treated in the adiabatic approximation, considering the motion of the light quark as a relativistic motion. The binding energy and mass spectra of doubly heavy baryons are calculated solving the two-center Dirac equation the one-centre Schrodinger equation for Coulomb plus linear potential. The binding energy of the light quark as a function of the distance between heavy quarks is found.

Crossing symmetry in the high-energy limit and pair production in ultrarelativistic heavy-ion collisions

We briefly review the description of electron propagation in the field of ultrarelativistic colliding nuclei. The crossing symmetry of the resulting amplitude is discussed. We show that the amplitude loses crossing invariance in the high-energy limit so that the conventional formalism to derive the pair production amplitude cannot be applied. Consequently, former calculations of the pair production amplitude using the solution of the two-center Dirac equation are incorrect. On the basis of bremsstrahlung and ${e}^{+}{e}^{\ensuremath{-}}$ pair production, conclusions about the Coulomb corrections in electron scattering and pair production in ultrarelativistic heavy-ion collisions are drawn.

ExactZ2scaling of pair production in the high-energy limit of heavy-ion collisions

The two-center Dirac equation for an electron in the external electromagnetic field of two colliding heavy ions in the limit in which the ions are moving at the speed of light is exactly solved and nonperturbative amplitudes for free electron-positron pair production are obtained. We find the condition for the applicability of this solution for large but finite collision energy, and use it to explain recent experimental results. The observed scaling of positron yields as the square of the projectile and target charges is a result of an exact cancellation of a nonperturbative charge dependence and holds as well for large coupling. Other observables would be sensitive to nonperturbative phases.

Relativistic Collisions of Highly-Charged Ions

The physics of elementary atomic processes in relativistic collisions between highly-charged ions and atoms or other ions is briefly discussed, and some recent theoretical and experimental results in this field are summarized. They include excitation, capture, ionization, and electron-positron pair creation. The numerical solution of the two-center Dirac equation in momentum space is shown to be a powerful nonperturbative method for describing atomic processes in relativistic collisions involving heavy and highly-charged ions. By propagating negative-energy wave packets in time the evolution of the QED vacuum around heavy ions in relativistic motion is investigated. Recent results obtained from numerical calculations using massively parallel processing on the Cray-T3E supercomputer of the National Energy Research Scientific Computer Center (NERSC) at Berkeley National Laboratory are presented.

Two-center Dirac equation with separable potentials

It is shown that the two-center Dirac equation with a separable potential is analytically solvable. As a numerical example we construct a two-center separable potential whose energy eigenvalues resemble the two-center nuclear Coulomb potential.

Relativistic two-center continuum.

A method is presented to define unique continuum states for the two-center Dirac Hamiltonian. In the spherical limit these states become the familiar angular-momentum eigenstates of the radial Coulomb potential. The different states for a fixed total energy \ensuremath{\Vert}E\ensuremath{\Vert}gm may be distinguished by considering the asymptotic spin-angular distribution of states with unique scattering phases. The first numerical solutions of the two-center Dirac equation for continuum states are presented.

Electrons in superheavy quasimolecules

Binding energies and wave functions of inner-shell electronic states in superheavy quasimolecules with $({Z}_{p}+{Z}_{t})\ensuremath{\alpha}>1$ are calculated. Ionization during a collision of very heavy ions is investigated within a molecular basis generated by the solutions of the two-center Dirac equation. Transitions to vacant bound states as well as direct excitation to the continuum are taken into account. We present theoretical values for the ionization probability as a function of impact parameter, bombarding energy, and combined nuclear charge. Our computed results are compared with recent experimental data. It is suggested that relativistic binding energies of electrons in superheavy quasimolecules can be determined experimentally via the impact-parameter dependence of ionization and the anisotropy of quasimolecular radiation.