Relativistic effects on information measures for hydrogen-like atoms

The $M$-shell ionization in high-$Z$ atoms by low-energy light $_{1}^{1}\mathrm{H}$, $_{1}^{2}\mathrm{H}$, $_{2}^{3}\mathrm{He}$, and $_{2}^{4}\mathrm{He}$ ions have been studied systematically in the energy range $0.1--1.0\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}∕\mathrm{amu}$ in order to verify the available theoretical approaches describing the $M$-shell ionization by charged particles in asymmetric collisions. The present low-energy data, combined with our earlier results reported for $M$-shell ionization by hydrogen and helium ions for higher energies, form a systematic experimental basis to test the theoretical predictions of $M$-shell ionization based on the plane-wave Born approximation (PWBA), the semiclassical approximation (SCA), and the binary-encounter approximation (BEA). In the PWBA based approaches the energy loss (E), Coulomb deflection (C), perturbed stationary state (PSS), and relativistic (R) effects were considered within the ECPSSR theory and its recent modification, called the ECUSAR theory, in which a description of the PSS effect was corrected to account for the united- and separated-atom (USA) electron binding energy limits. In the SCA calculations with relativistic wave functions the binding effect was included only in the limiting cases of separated-atom and united-atom limits. Possible contribution of the electron capture, multiple ionization, and recoil ionization to the $M$-shell vacancy production, which is dominated for light ions impact by direct single ionization process, are also discussed. The universal scaling of measured $M$-shell x-ray production and ionization cross sections was investigated in detail. Using the present data the isotopic effect has been studied by comparing the measured $M$-shell ionization cross-section ratios for equal-velocity hydrogen $_{1}^{1}\mathrm{H}$ and $_{1}^{2}\mathrm{H}$ as well as helium $_{2}^{3}\mathrm{He}$ and $_{2}^{4}\mathrm{He}$ isotopes. In addition, the ratios of measured ionization cross sections for $_{1}^{2}\mathrm{H}$ and $_{2}^{4}\mathrm{He}$ were used to investigate the role of the binding effect. The present results are of practical importance for the application of particle-induced x-ray emission technique in trace element studies.

M-shell ionization of heavy elements by0.1–1.0MeV∕amuH1,2andHe3,4ions

We discuss, on a phenomenological level, the possible appearance of resonances ine
++e
− scattering at energies in the MeV region. The expected resonance cross section and angular distribution are examined. The observability depends crucially on the attainable energy resolution which is limit by the momentum distribution of the electrons contained in the target.

Resonant Bhabha scattering at MeV energies

Position and momentum information measures are evaluated for the ground state of the \emph{relativistic} hydrogen-like atoms. Consequences of the fact that the radial momentum operator is not self-adjoint are explicitly studied, exhibiting fundamental shortcomings of the conventional uncertainty measures in terms of the radial position and momentum variances. The Shannon and R\'enyi entropies, the Fisher information measure, as well as several related information measures, are considered as viable alternatives. Detailed results on the onset of relativistic effects for low nuclear charges, and on the extreme relativistic limit, are presented. The relativistic position density decays exponentially at large $r$, but is singular at the origin. Correspondingly, the momentum density decays as an inverse power of $p$. Both features yield divergent R\'enyi entropies away from a finite vicinity of the Shannon entropy. While the position space information measures can be evaluated analytically for both the nonrelativistic and the relativistic hydrogen atom, this is not the case for the relativistic momentum space. Some of the results allow interesting insight into the significance of recently evaluated Dirac-Fock vs. Hartree-Fock complexity measures for many-electron neutral atoms.

An approach to calculate both the Fourier transforms of singular spinor components of relativistic eigenfunctions of hydrogen-like atoms and those of their multiple products is developed. The method simplifies calculations of the momentum representation for the eigenfunctions belonging to the discrete spectrum. Besides, it reduces the procedure of the Fourier transform for eigenfunctions of the continuous spectrum to the evaluation of only one one-dimensional integral, allowing the explicit separation of its singular part representing a generalized function. The proposed algorithm is especially convenient for large-scale numerical calculations.

On the fourier transforms of the spinor components of hydrogen-like atom relativistic eigenfunctions

A momentum representation of the Dirac relativistic wave functions for a hydrogenlike atom is obtained in a simple form by means of Fourier transformation. The expressions given are convenient for analytic as well as numerical applications.

Momentum representation of Dirac relativistic wave functions

Relativistic effects on a modified version of Rutherford's scattering cross section are considered up to first-order in the Born approximation for relativistic velocities in the binary-encounter approximation (BEA). The predicted cross sections with protons as projectile are lower than the previous theoretical values at low energies and are seen to be in better agreement with measurements. An approximate relativistic correction factor which accounts for orbital electrons only is compared with exact Dirac corrections, within the BEA model.

C. V. Sheth

Papers

Momentum representation of Dirac relativistic wave functions

Relativistic corrections in K -shell ionization cross sections