Relativistic Modifications Research Articles

Relativistic modifications of inter-ionic potentials in lead fluoride

The relativistic modifications of the inter-ionic potentials in cubic lead fluoride are calculated by an ab initio fully relativistic method. The relativistic reduction in the uncorrelated short-range Pb2+-Pb2+ interaction by a factor of about 3 is the major feature in the small but significant relativistic stabilisation of the entire crystal. The electron gas method is shown to overestimate the relativistic corrections to the interionic potentials by a factor of between 2 and 3.

Wave dispersion and ray propagation in a weakly relativistic electron plasma: Implications for the generation of auroral kilometric radiation

Relativistic modifications to wave dispersion near the electron gyrofrequency are known to be important for sufficiently energetic electrons. In general, if the characteristic momentum of the energetic particles (po/moc) is greater than the ratio of the plasma frequency to the gyrofrequency (ωp/Ωe), wave propagation can be markedly different than the classical cold plasma limit. We investigate the wave dispersion in a weakly relativistic electron plasma, specifically in the context of the generation of auroral kilometric radiation (AKR). To do this, we have used a simple electron distribution consisting of a ring of energetic electrons at constant perpendicular momentum together with a stationary cold component. It is found that for such an electron distribution, where the two electron components have different gyrofrequencies due to relativistic effects, a new wave mode is introduced. This wave is trapped between the two electron gyrofrequencies. We find that the group velocity for such a wave is small; convective growth lengths for the wave are calculated to be of the order of 1 km. Consequently, ray propagation in a simple model for the density cavity present on auroral zone field lines is studied. We find that the ray paths tend to bring waves to altitudes where the growth rate maximizes; at these altitudes the waves propagate primarily across the ambient magnetic field. This could result in substantial amplification of the wave, since damping is minimized, and the waves may undergo multiple transitions across the auroral arc. However, the wave is trapped and must undergo mode conversion to escape from the auroral arc. Since the wave polarization is found to be very similar to the R‐X mode polarization, we deduce that coupling to the R‐X mode at the edge of the auroral arc may be reasonably efficient.

A simulation study of kilometric radiation generation along an auroral field line

Relativistic modifications to wave dispersion near the electron gyrofrequency are known to be important in a low‐density, energetic plasma. Previous investigations based on simple model distribution functions have suggested that these relativistic dispersion effects may substantially alter the predictions of the cyclotron maser theory as applied to the generation of auroral kilometric radiation (AKR). The present work combines a model for the high‐altitude auroral zone with local particle simulations in order to assess the quantitative implications of relativistic dispersion for the generation of AKR. Electron distribution functions, which include a loss cone and a “hole” at lower energies produced by the parallel electric field, are determined along an auroral field line. Number densities of both energetic auroral electrons and low‐energy backscattered electrons are obtained from a fit to the observed density‐depleted region on an auroral field line. It is found that the hot magnetospheric component is the dominant species at altitudes greater than 1.7 RE geocentric. These auroral zone parameters and distributions are used as input to one‐ and two‐dimensional particle simulations. The simulations show that the electric field results in more free energy in the hot electron population at the lower altitudes. This is counteracted, however, by the increased density of backscattered and secondary electrons at these altitudes, which tends to quench the maser instability. The most intense radiation is found to occur between 1.75 and 2.0 RE, corresponding to AKR frequencies in the range 200‐300 kHz. The peak in the radiation is very close to 90° to the ambient magnetic field, with an offset of a few degrees in the direction of the loss cone. The dominant emission is in the extraordinary mode with a wave frequency ω/Ωe ≈ 0.99. A weak (∼1%) ordinary mode component is produced in association with the extraordinary mode due to a relativistic coupling between perpendicular wave electric fields and parallel currents.

Relativistic eikonal theory of electron capture.

The eikonal theory of electron capture by relativistic projectiles is developed. It is shown that as long as the electron spin is not measured, a density-matrix formalism can greatly simplify the calculation. The exact eikonal cross section is expressed in terms of two-dimensional integrals. For 1s-1s capture an approximate closed-form expression valid for not too high nuclear charges is also given. This form explicitly displays contributions from magnetic capture and from relativistic modifications of the electron orbits. At each stage of the development the corresponding Oppenheimer-Brinkman-Kramers (OBK) result can be retrieved as a special case. It is concluded that the multiple-scattering contributions described by the eikonal approach are crucial in reducing the calculated cross section significantly below the OBK and second-Born-approximation values. In this way, good agreement with experimental data is obtained.

Relativistic modifications of covalent bonding in heavy elements: Calculations for T1H

To determine whether the “orthogonal triplet bond” proposed by Pyper is an important factor in the bonding of hydrogen to a very heavy element with a single p 1 2 valence election, we computed dissociation curves for T1H using fully relativistic quantum methods. Our results and literature information indicate that the orthogonal triplet interaction makes no significant contribution to bonding.

Relativistic modifications of covalent bonding in heavy elements. the role of partial ionic character and π electrons on ligands

The bonding between an element containing a single valence electron occupying a Dirac—Fock p̄ or p orbital and (i) a group of different electronegativity having no π electrons, (ii) a group containing both σ and π electrons having a large π → σ excitation energy, (iii) a halogen and (iv) another element having a single valence p̄ or p electron, is examined.

Relativistic modifications of covalent bonding in heavy elements : the kappa valence method

It is shown by using the relativistic generalization of spin-valence theory that the j- j coupled ground state of an element having a single valence electron in a Dirac-Fock p̄ or p orbital can form a stable covalent bond to a hydrogen atom. The bond is shown to be a mixture of a normal covalent σ bond and the triplet bond formed by the interaction of two electrons occupying orthogonal orbitals.

Relativistic many-body approach to hyperfine interaction in rare earths: Explanation of experimental result in europium

A first-principles relativistic many-body treatment is carried out for the hyperfine constant in europium atom. This investigation has allowed us to sort out for the first time all the contributing physical mechanisms, core polarization, correlation, relativistic modifications of these, and the purely relativistic Casimir and breakdown of $\mathrm{LS}$ coupling effects. It is found that all these competing contributions have to be accurately evaluated to successfully explain the observed small hyperfine constant.

Relativistic effects in magnetohydrodynamic wave propagation

Relativistic modifications of Alfv́en waves are discussed for cases in which magnetic and relativistic terms are comparable. For atmospheric waves there are corrections to velocities, amplitudes, and polarizations. The mass current of the wave itself causes dispersion.

Relativistic Corrections to Nonrelativistic Two-Particle Dynamical Calculations: Demonstration of the Validity of the Drell-Hearn-Gerasimov Sum Rule for Weakly Bound Composite Particles

The problem of the first-order relativistic corrections to the conventional nonrelativistic perturbation treatment of electromagnetic interactions of bound states is discussed. It is shown how the Thomas precession effect is usually mishandled for small excitation energies acting on a bound state. These results are demonstrated by the explicit calculation of the single-particle matrix elements of the charge density and the low-energy Compton scattering amplitude in a simple bound-state model. The amplitudes, which are known by virtue of relativistic invariance and current conservation, are also explicitly calculated, but agreement is obtained only by applying the relativistic modifications of the c.m. variables found in a previous paper. At the same time, the Drell-Hearn-Gerasimov sum rule is shown to be consistent for this model and also for simple models of the hydrogen atom and the deuteron when the constituent particles have arbitrary masses.

Vacuum Polarization Effects in Proton-Proton Scattering

The problem of vacuum polarization scattering of protons by protons is treated to first order in the vacuum polarization interaction. The phase shifts caused by the interaction are calculated, and the corresponding addition to the $p\ensuremath{-}p$ scattering matrix is constructed. The phase shifts are calculated using Coulomb wave functions as the unperturbed wave functions. The corresponding addition to the $p\ensuremath{-}p$ scattering matrix is then constructed, including exactly the Coulomb phase shift factors $\mathrm{exp}[2i({\ensuremath{\sigma}}_{L}\ensuremath{-}{\ensuremath{\sigma}}_{0})]$ appearing in the series representation of the scattering amplitudes. Other electromagnetic and relativistic modifications of the Coulomb scattering amplitudes are also examined in the limit of low energies. Numerical results are given for the vacuum polarization contributions to the $p\ensuremath{-}p$ scattering cross section over the energy range 1.4-4.2 Mev. It is found that vacuum polarization scattering may easily be confused with nuclear $P$-wave scattering.

AUGER TRANSITIONS IN SILVER

Twenty of the K Auger lines of silver have been resolved and estimates of their relative intensities obtained. Comparison with the theoretical intensities calculated by Burhop indicates that the existing theory is not quantitatively correct and that relativistic modifications offer little hope of improvement. Special attention is directed to the K → LL lines where disagreement with theory is most marked. Comparison with similar observations previously reported for gold and bismuth shows an interesting agreement in the observed ratio of the intensity of the K → LIILIII line to that of the K → LIIILIII line, and in the ratio of the intensity of the K → LILIII line to that of the K → LILI line.

The intensity of X-ray spectrum lines of heavy elements

The intensities of X-ray spectral lines resulting from transitions between the K, L and M shells have been calculated using screened relativistic wave functions. The relativistic modifications are shown to decrease the Lβ3β4 doublet ratio but do not alter the Kα1α2 and Kβ1β3 ratios, in agreement with experiment. The calculated intensities of the forbidden lines arising from quadrupole transitions are in good agreement with experiment. Lines due to transitions arising from the magnetic dipole moment of the electron are found to be very weak, but the LI → K transition might just be observable for heavy elements.