Relativistic Atom Research Articles

Hydrogenlike atoms in uniformly charged sphere model of atomic nucleus. I. Reference calculations of energy levels

We have adopted the uniformly charged sphere model of atomic nucleus for one-electron atoms H, Ar17+, Xe53+, Hg79+, and U91+ and calculated several energy levels by solving the matching equations of the inward and the outward radial wave functions for the Schrodinger as well as for the Dirac equations. The finite-size effect (FSE) of the atomic nucleus on the energy levels of nonrelativistic atoms has been found one order of magnitude smaller than the effect on the relativistic atoms. The FSE works mainly on s states for nonrelativistic atoms, whereas for relativistic atoms, it works on s and p1/2 states. The energy levels of p states have been calculated lower than the energies of the corresponding s states when we have included the FSE.

Basis-set expansion method for relativistic atoms in the atomic-nucleus model of a finite sphere of constant electric field

Use of a new atomic-nucleus model is proposed for relativistic atomic and molecular calculaitons; the atomic nucleus has a sphere of finite size and the electric field in the sphere is assumed constant. It is shown that for the basis-set expansion method based on this model, the Slater-type functions of integral principal quantum numbers can be appropriately adopted and the Slater-type function of “ n=l” quantum numbers (e.g., 1p, 2d,3f, etc.) appearing in the point-nucleus model are not needed even if we use the scheme of kinetic balance. Test calculations on one-electron hydrogenic atoms imply the applicability of the proposed atomic-nucleus model to many-electron systems.

Kinetically balanced geometric Gaussian basis sets for relativistic atoms

We present Dirac–Hartree–Fock (DHF) atomic structure results for many-electron atoms (Z=2–86) using kinetically balanced geometric basis sets of Gaussian functions. There is no ‘‘variational collapse’’ or bound failure if we impose the condition of kinetic balance between the large and the small component spinors along with proper boundary conditions for bound state orbitals. Furthermore, with a finite-size nucleus model, the DHF total energy for atoms converges more rapidly to the numerical Dirac–Fock (DF) limit with a smaller number of basis functions than those calculations where a point nucleus is assumed. By systematically extending the geometric basis sets, we have obtained the DHF total energies for all atoms(up to Z=86) within a few millihartrees of the numerical (DF) limit, with smooth convergence from above.

Kinetically balanced geometric Gaussian basis set calculations for relativistic many-electron atoms with finite nuclear size

We have performed Dirac-Fock atomic structure calculations on many-electron atoms using kinetically balanced, geometric Gaussian basis sets. Our results show that “variational collapse” or bound failure does not occur even at Z=86. Furthermore, with a finite-size nuclear model, we obtain results close to the relativistic Hartree-Fock limit, with a smooth convergence from above. This opens up the possibility of optimisation of exponents in Gaussian basis set calculations within the Dirac-Fock approach for many-electron atoms.

Dissolution of relativistic atoms into negative energy states

The problem of atomic dissolution by means of decay to the negative energy continuum is discussed. The derivation of the one-electron central-field Hamiltonian from quantum electrodynamics is made as an example. It is found that the operators that project the Coulomb interaction into positive and negative energy states of the Dirac noninteracting Hamiltonian cause the eigenstates of the atomic Hamiltonian to break up into two sets. One set is expandable in the positive energy noninteracting states, and this set propagates forward in time. The other set is expandable in terms of the negative energy noninteracting states and propagates backward in time. There is, therefore, no danger that transitions will occur from the forward propagating eigenstates into the negative continuum with continued propagation in the forward direction, regardless of the magnitude of the nuclear charge.

Orientation effects for passage of a relativistic positronium atom through crystals

We examine the passage of a relativistic positronium atom through a crystal for a small entrance angle with respect to the crystallographic axes. We show that the probability for an elastic interaction of the positronium atom with the crystal has orientation dependence. The probability of breakup of a positronium atom in a crystal is significantly greater than in an amorphous target.

TWO-PHOTON TRANSITIONS IN ATOMIC INNER-SHELLS FOR Xe——RELETIVISTIC EFFECT AND ATOMIC SCREENING EFFECT

There is recent interest in atomic inner-shell two-photon decay processes because state-of-art experimental techniques (such as electronic coincident measurement etc.) have made it possible to measure the two-photon decay rate in the X-ray region. We perform relativistic self-consistent field calculations on the two-photon decay rates of (ls)-1→(ns)-1 transitions (n = 2,3,4) and (ls)-1→(nd)-1 transitions (n = 3,4) for hydrogen, hydrogen-like Xe ion and Xe atoms. Comparing the hydrogemc and the hydrogen-like Xe ion rates, we can clearly demonstrate the relativistic effect. Comparing our previous non-relativistic and our present relativistic Xe atom rates, we can also demonstrate the relativistic effect in the atomic inner-shell two photon decays. Comparing the hydrogen-like Xe ion and the Xe atom rates, we can explain the atomic screening effect. After elucidating the relativistic effect and the atomic screening effect, we are convinced that we have obtained the reliable relativistic rates, which can provide a basis to analyze th effect of electron correlations in the atomic inner-shell two-photon decays. We also compare our results with the experimental measurement and other theoretical results.

Multiple ionization and capture in relativistic heavy-ion atom collisions

We show that in relativistic heavy-ion collisions the independent electron model can be used to predict cross sections for multiple inner-shell ionization and capture in a single collision. Charge distributions of 82- to 200-MeV/amu Xe and 105- to 955-MeV/amu U ion beams emerging from thin solid targets were used to obtain single- and multiple-electron stripping and capture cross sections. The probabilities of stripping electrons from the K, L, or M shells were calculated using the semiclassical approximation and Dirac hydrogenic wave functions. For capture, a simplified model for electron capture was used. The data generally agree with theory.

Relativistic many-electron atoms in perturbation and dirac-fock theory

Our previous work on the problem of relativistic many-electron atoms is extended and more rigorously treated than before. The present paper is restricted to the formal consideration of energy levels in the relativistic theory. Both perturbation theory in 1/Z (where − eZ is the nuclear charge) for high Z ions and the Dirac-Fock approximation are considered. The present treatment applies to atoms with Z > 2, and N a closed shell or subshell (where e(N ± 1) is the lepton charge: N ± 1 are the total number of electrons in the non-relativistic limit). Renormalization, radiative corrections, and correlations are explicitly considered. All contributions of leading order in αZ through 1/Z2 in perturbation theory, as well as leading order correlation and radiative correction terms in Dirac-Fock theory are formally exhibited and their relative size is estimated. Besides the usual Lamb shift corrections, additional types of radiative correction are shown to occur. A detailed derivation of the explicit, finite integrals which remain in these additional terms after renormalization, and which must be numerically evaluated, is left to a subsequent paper.

Interaction of relativistic elementary atoms with matter. I. General formulas.

The problem of the interaction of relativistic elementary atoms (Coulomb bound states of elementary particles such as positronium, pionium, etc.) with matter is studied in the reference frame where the atom is initially at rest. An atom of matter is treated as a spinless structureless fast particle. The amplitudes of elementary-atom interaction are derived in the Born approximation under the assumption that a momentum transfer to the atom does not significantly exceed an inverse Bohr radius of the atom. The elementary-atom excitation and ionization processes are considered. The transitions where the spin projection of the atom component is reversed are also studied. In particular the matrix elements for para-ortho and ortho-para transitions are given. The spin structure of the amplitudes is discussed in detail. The sum rules, which allow the calculation of the cross sections summed over atom final states are found. Finally the formulas of the atom interaction cross sections are presented.

Approximation methods for many-electron atoms from a field theory point of view

A summary of a field-theoretic approach to the study of the energies and radiative transition matrix elements of many-electron atoms is presented. The approach can deal with both non-relativistic and relativistic atoms. The method provides a unified description of a number of well-known approximations for calculating energy eigenvalues, and can easily be generalized to include new types of approximations and higher order correlations. Insistence on gauge invariance of transition amplitudes at every level of approximation leads to a generalization of the previously used Hartree-Fock results for transition amplitudes. It involves non-local effective currents.

Relationship between a quasipotential equation and a Schr�dinger equation

The quasipotential method, proposed in connection with the theory of the strong interaction, has proved to be a convenient tool for studying relativistic particles when it is possible to use perturbation theory, in particular, in the theory of a relativistic many-electron atom or ion. It is shown that a linear nonunitary transformation of the wave function transforms a quasipotential equation into a Schroedinger equation. The usual conditions of orthonormality of the wave functions and the cluster properties of the potential are recovered.

Interaction of relativistic elementary atoms with matter. II. Numerical results.

The formulas of the cross sections of elementary-atom interaction with atoms of matter given in part I of this study are simplified to the form appropriate for numerical calculations. The cross sections are split into electric, magnetic, and spin parts. The numerical values of each part of the total and excitation cross sections of eight elementary atoms (${A}_{2e}$,${A}_{e\ensuremath{\mu}}$,${A}_{e\ensuremath{\pi}}$, ${A}_{2\ensuremath{\mu}}$, ${A}_{\ensuremath{\mu}\ensuremath{\pi}}$, ${A}_{2\ensuremath{\pi}}$, ${A}_{\ensuremath{\pi}K}$, ${A}_{2K}$) interacting with five typical targets (C, Al, Cu, Ag, Pb) are given. The electric parts are found to give the dominant contribution to all cross sections except those of para-ortho and ortho-para transitions of the elementary atoms composed of two spin-1/2 particles. The results are compared to those published earlier. The points concerning the elementary-atom interaction with matter which are not studied by us are briefly reviewed. The magnetic form factors of elementary atoms are discussed.

Relativistic many-electron atoms in perturbation and Dirac-Fock theory

Relativistic many-electron atoms in perturbation and Dirac-Fock theory

Vacuum-induced friedel-type oscillations in relativistic atoms

It is pointed out that generalized Atiyah-Patodi-Singer η-invariants naturally appear in the thermodynamic potential describing the equilibrium properties of a localized system of relativistic interacting fermions. Under circumstances to be specified, externalU(1) gauge fields induce oscillating charge distributions which should be experimentally detectable for strong Coulomb fields produced by highly stripped large-Z atoms. Casimir energy shifts result be the same mechanism.

Kinetically balanced calculations on relativistic many-electron atoms

The kinetically balanced calculations proposed by Ishikawa, Grant, Stanton et al. have been performed on Be, C, Ni, Ag, and Hg. The results show that variational collapse does not occur.

A comparative study of nonrelativistic and relativistic 5d atoms as impurities in silicon

A comparison is made between calculations performed nonrelativistically and relativistically for W, Re, Os, Ir, Pt, Au and Hg as substitutional impurities in silicon. The calculations were carried out using the relativistic extended Hückel method. The direct and indirect relativistic effects upon the 5 d-like levels and band-gap levels are analysed.

A method of polarizing relativistic proton beams by laser radiation

The method of polarization of protons in the process of optical orientation of hydrogen atoms, obtained in neutralizing negative hydrogen ions, accelerated up to an energy of 200–600 MeV is discussed. The method uses the Doppler shift of a transition frequency for relativistic atoms ( ν c ⩾ 0.5 ) into the region accessible for lasers. The implementation of the method will allow a beam of polarized protons with a pulse intensity of up to 50 mA to be obtained. This value exceeds the intensity achieved so far by three orders of magnitude. The kinetics of the polarization process is calculated. The possible scheme of a laser polarizer is discussed.

Energy level shifts and transition probabilities in a relativistic atom

Energy level shifts and transition probabilities in a relativistic atom

Broken Symmetry in the Local-Spin-Density Approximation and Its Application to Relativistic Atoms

The ground state of the (${d}^{4}{s}^{2}$) configuration shifts by 1.725 eV relative to that of the (${d}^{5}s$) on going from Mo to W. This shift, which arises mainly from relativistic contributions to the W energy, is calculated with 2% accuracy by use of the authors' relativistic pseudopotential and the local-spin-density approximation. That the single determinantal broken-symmetry local-spin-density approximation wave function accounts for the spin-orbit energy, which otherwise could be obtained from the spin-orbit mixing of several multideterminantal terms, may be surprising.