Nonrelativistic Dipole Approximation Research Articles

Theoretical angular dependence of L3 sub-shell X-rays following photoionization

Angular distribution studies are made for L3 sub-shell X-rays of high Z elements; W, Pt, Hg, Pb, and U in the angular span 0°–180° at incident photon energies; near L3 edge, where considerable anisotropy is anticipated and at 60 keV, the energy of experimental interest from standard laboratory source Am-241. Angular distribution of X-rays originating from L3 sub-shell vacancies for each element is derived from theoretical alignment parameter already evaluated under non-relativistic dipole approximation (NDA) in a point Coulomb potential (A20) and in a screened Coulomb potential (A20*) that accommodates the effect of screened nuclear drag on electron wave function, and analytical perturbation theory (A20**). The normalized Lℓ and Lα intensity distributions show more anisotropy at L3 threshold than at 60 keV for all the elements and under all theoretical models. Besides this, the anisotropy for Lℓ is larger than Lα which comprises the unresolved Lα1 and Lα2 lines with opposite anisotropy. The angular distribution trends are almost similar for all the elements. The anisotropy values at L3 threshold energy are assuredly greater than the reported experimental uncertainties (5–8 %) of X-ray intensities. Anisotropy magnitude is found sensitive to the theoretical models. The present anisotropy trends resemble the results of Zhang et al (Journal of Quantitative Spectroscopy and Radiative Transfer, 244 (2020) 106844), however, the values are not quite same.

A plane wave theory of photoionization using the dipole acceleration form: predictions concerning circular dichroism in photoelectron angular distributions at high energies

Photoelectron angular distributions for randomly oriented atoms and molecules are characterized by the linear and circular dichroic parameters β and b 1, respectively. In the region where the photoelectron kinetic energy is several 100 eV, the values of b 1 and β are expected to be constants. In this study, we show that β for all hydrogen atom orbitals other than s orbitals approach values between 0 to 0.5 at the high-energy limit in the nonrelativistic dipole approximation and that the values can be obtained by the plane wave approximation using the dipole acceleration form. These limiting values are smaller than the widely accepted values for molecules. We discuss the implications of this result for the circular dichroism in photoelectron angular distributions of linear molecules.

Inner shell alignment of atoms in photon impact ionization

The ionization of an atom can result in alignment of the residual ion. In the present work alignment studies are made for some of the high Z elements using different approaches. The theoretical values of the alignment parameter (A2) have been calculated by using the non-relativistic dipole approximation in a point Coulomb potential and analytical perturbation theory in a screened Coulomb potential. Empirical alignment evaluations are done by interpolating experimental L x-ray fluorescence cross-sections along with radiative decay rates. The alignment results are further compared with available theoretical and experimental measurements.

Polarization correlations in the elastic Rayleigh scattering of photons by hydrogenlike ions

The (elastic) Rayleigh scattering of hard x rays by hydrogenlike ions has been investigated within the framework of second-order perturbation theory and Dirac's relativistic equation. The focus of this study was, in particular, on two questions: (i) How is the polarization of scattered photons affected if the incident light is itself (linearly) polarized, and (ii) how do the nondipole contributions to the electron-photon interaction and the relativistic contraction of the wave functions influence such a polarization transfer? Detailed calculations were performed for Ne${}^{9+}$, Xe${}^{53+}$, and U${}^{91+}$ targets and for photon energies up to ten times the $1s$ ionization threshold of the ions. From the comparison of these fully relativistic computations with the (nonrelativistic) dipole approximation we conclude that relativistic and higher-multipole effects often lead to a significant or even complete depolarization for heavy targets and at high photon energies.

Differential L-shell radiative recombination rate coefficients for bare uranium ions interacting with low-energy electrons

Results of the calculations of differential L−shell radiative recombination (RR) rate coefficients for bare uranium ions colliding with free electrons using the nonrelativistic dipole approximation and fully relativistic calculations are reported. The rate coefficients were obtained for very low, in the range of meV, relative electron-ion energies. We demonstrate that even for such low relative ion-electron energies the relativistic effects significantly modify the differential RR rate coefficients for the L−subshells and, as a result, the measurements of the relative electron energy dependence of the L-RR rates could be used for studying of the relativistic effects. These effects are strongest for the L 3-subshell, which is discussed here in more details.

K-shell differential radiative recombination rates for bare uranium ions interacting with low-energy electrons

We report on the calculations of the K-shell differential radiative recombination (RR) rate coefficients for very low, in the range of meV, relative electron–ion energies. The rate coefficients were derived for bare uranium ions colliding with free electrons both within the nonrelativistic dipole approximation and using fully relativistic calculations. We show that even for such low relative ion–electron energies, the differential rate coefficient reveals strong relativistic effects. We demonstrate that the measurements of the relative electron energy dependence of the RR rates represent a very sensitive tool for precise studies of the RR process and, in particular, for probing the fine details of the relativistic effects in the RR of ions with electrons. The results are discussed in the context of the first x-ray state-selective RR experiment performed for very low relative electron–ion energies.

Alignment Studies for Tungsten Near L3 Sub-Shell Threshold Via Theoretical, Experimental and Empirical Methods

Alignment studies are made for tungsten near L3 sub-shell threshold using theoretical, experimental and empirical approaches. Experimentally to measure alignment parameter, the angular distribution of L x-rays of tungsten (W-74) is measured in the angular range 0° to 120°, where maximum anisotropy is expected. The experimental measurements are performed in XRF laboratories of Raja Ramanna Center for Advanced Technology (RRCAT), Indore, India using a three-dimensional double reflection set-up. The weighted average of alignment values with 10% error comes 0.155 ± 0.009. Theoretically, the value of alignment parameter A20 is calculated using non-relativistic dipole approximation in a point Coulomb potential and is found 0.151 at L3 threshold energy (10.676 keV). For empirical A20 evaluations, IGELCS interpolated experimental LXRF cross-section σ*Lg (g = α, l) values of Mann et al with 8% reported errorsare used along with the radiative decay rates. The comparison among theoretical, experimental and empirical values are similar and values being >0.1 at L3 threshold energy are certainly higher than the 5 to 8 percent uncertainties quoted in earlier experimental results.

Two-photon ionization of ground state hydrogen from the photoelectric threshold up to 40 keV

Our study of the ionization of the 1s state of the hydrogen atom through the absorption of two identical photons covers a region of higher energies than those considered up to now; it aims to establish the region of validity of nonrelativistic dipole approximation for the total generalized cross section.

Two-photon above-threshold ionization of hydrogen over the photon energy range from 15 eV to 50 keV

We investigate the absorption of two identical photons from the ground state of hydrogen-like atoms over an energy range that extends beyond that explored up to now. Our approach is based on a hybrid formula, valid in second-order perturbation theory, in which the ${\mathbf{A}}^{2}$ contribution from the nonrelativistic Hamiltonian is treated exactly, while the $\mathbf{A}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{P}$ contribution is calculated in dipole approximation. We find that, at least up to 50 keV, the nonrelativistic dipole approximation, based only on the $\mathbf{A}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{P}$ contribution, determines the values of the total cross section. Our numerical results, covering photon energies from 90 nm (13.7 eV) to 0.0248 nm (50 keV) are in very good agreement with most previous theoretical works. Differences with recent results are discussed.

Study of the ionization of hydrogen atoms in an intense circularly polarized laser field

Based on the non-relativistic theory and dipole approximation, we have developed a new program to study the interaction between a circularly polarized laser and hydrogen atoms. In this paper, we present our calculation of above-threshold ionization (ATI) rate based on the new code and compare the results with that of a pertubation theory calculation. The comparison shows that our non-perturbative calculations can produce ATI peak suppression.

Experimental and theoretical anisotropy studies for tungsten

An attempt has been made to study the angular distribution of L X-rays of tungsten (W-74) both theoretically and experimentally in the angular range 0°–120° at 10.676 keV i. e. at photon energy lying between L3 and L2 edges of W where maximum anisotropy is expected. The experimental measurements are performed in XRF laboratories of Raja Ramanna Center for Advanced Technology (RRCAT), Indore, India using a three-dimensional double reflection geometrical set-up. Theoretically, the value of alignment parameter A 20 used for angular studies is calculated using non-relativistic dipole approximation in a point Coulomb potential. From the normalized angular L X-ray intensities (experimental and theoretical) for L ℓ and L α peaks it has been observed that the experimental anisotropy is more as compared to the theoretical one, furthermore, the anisotropy for L ℓ is more in comparison to L α peaks due to the unresolved L α 1 and L α 2 lines with opposite anisotropic parameters under both procedures.

Calculation of photon angular distribution and polarization for radiative recombination for high-charged hydrogen-like ions

In this paper a systematic study is carried out on the angular distribution and polarization of photons emitted following radiative recombination of H-like ions by a non-relativistic dipole approximation. In order to incorporate the screening effect due to inner-shell electrons, a distorted wave approach is used. The dependences of the calculated angular distribution and polarization on the reduced energy and nuclear charge are fitted by the corresponding empirical formulas respectively.

Calculation of photon angular distribution and polarization for radiative recombination for highly charged helium-like ions

A systematic study is carried out on the angular distribution and polarization of photons emitted following radiative recombination of He-like ions by a non-relativistic dipole approximation. In order to incorporate the screening effect due to inner-shell electrons, a distorted wave approach is used. The dependence of the calculated angular distribution and polarization on the reduced-energy and nuclear charge are fitted by the corresponding empirical formulas, respectively.

Study of photon-induced L3 vacancy alignment for elements La to U

Alignment of photon-induced L3 vacancies is studied in rare earth and highZ elements at energies of experimental interest, near thresholds to 60 keV, under nonrelativistic dipole approximation. Numerical calculations of the matrix element are undertaken to produce theoretical data for comparison with the experimental findings. The A2 values being s>0.1 at photoelectron energies <20 keV are certainly higher than 5–8% uncertainties quoted in experimental results. Present findings are from a very basic model, hydrogen-like and can further be treated as reference to observe the impact of screening, relativistic, multipole and retardation corrections to the model

Opacity Bounds

ABSTRACTA bound on the Rosseland mean opacity is derived, assuming that Thomson scattering can be ignored and that radiation can be described by a nonrelativistic dipole approximation. The same method yields an inequality between the Planck mean opacity and the Rosseland mean. Finally, unitarity gives another bound on opacities in the very high temperature regime.

A Calculation of Direct Radiative Recombination Cross Sections and Rate Coefficients

We calculate direct radiative recombination cross sections and ratecoefficients for Be-like O4+, Si10+ and Na-likeFe15+ using nonrelativistic dipole approximation. In order toincorporate the screening effect due to the inner-shell electrons inthe calculation, we use the distorted wave method instead of pureCoulomb approximation with effective charge. The calculated crosssections and rate coefficients are in agreement with other theoreticalcalculations and the experimental data.

CROSS SECTIONS FOR RADIATIVE RECOMBINATION AND THE PHOTOELECTRIC EFFECT IN THE K, L, AND M SHELLS OF ONE-ELECTRON SYSTEMS WITH 1 ≤ Z ≤ 112 CALCULATED WITHIN AN EXACT RELATIVISTIC DESCRIPTION

An extensive tabulation of total cross sections for radiative recombination and, consequently, for the photoelectric effect of hydrogenlike ions with charge numbers 1 ≤ Z ≤ 112 is presented for the K, L, and M shells and electron energies ranging from closely above the threshold to the relativistic regime. The cross sections, accurate to three digits, are based on fully relativistic calculations including the effects of the finite nuclear size and all multipole orders of the photon field. The cross sections for radiative recombination are displayed as a function of the electron energy. By a simple scaling, the dependence on the projectile energy in MeV/u can be derived for accelerator experiments, and, by using elementary formulas, the cross section for the photoelectric effect is obtained as a function of the photon energy. Furthermore, it is pointed out how to scale the cross sections for the capture of heavy Dirac particles. In order to assess the accuracy of cross sections derived from the widely used nonrelativistic dipole approximation, we present a comparison with exact relativistic results for a few representative cases.

Comparison Between the Nonrelativistic Dipole Approximation and the Exact Relativistic Theory for Radiative Recombination

The nonrelativistic dipole approximation (Stobbe theory) for the atomic photoelectric effect and its inverse reaction, radiative recombination, is contrasted with an exact relativistic formulation taking into account relativistic electron motion and all multipole orders for the interaction with the photon. A comparison is performed for total cross sections as a function of energy and nuclear charge as well as for angular distributions. It is found that even for small velocities of the continuum electron, discrepancies between both approaches persist for high nuclear charges.

Azimuthal angular distributions for two-colour three-photon ionization of hydrogen

We study in some detail the photoelectron angular distributions in two-colour three-photon ionization of the ground state hydrogen atom in its dependence on the polarizations of the two electromagnetic fields. We treat the particular case of photons propagating in the same direction and having elliptic polarizations with the same axes. The azimuthal distributions presented refer to the electrons emitted with momenta orthogonal to the photons direction. The calculations are done in nonrelativistic dipole approximation, in velocity gauge, using third-order perturbation theory. We show that the photoelectron azimuthal angular distributions have a fourfold symmetry only for identical linear or circular polarizations. In all the other cases the distribution has a twofold symmetry.

Study of x-ray and L-REC photon emissions from highly ionized swift and projectiles passing through a thin carbon foil

X-rays and L-shell radiative electron capture (L-REC) photons have been detected from highly ionized swift projectiles of and passing through a thin carbon foil at two incident projectile energies. All the K and L peaks show strong shifts towards higher energy as a result of multiple ionization of L- and M-shells. The intensity ratios have been found to be almost half of the value for the singly ionized atoms. The measured positions of various L and K x-ray peaks and the measured intensity ratios have been attributed to certain types of electronic configurations based on the extensive multiconfiguration Dirac-Fock calculations. Generally, it has been found that the lowest-energy configuration plays a rather small role, but in contrast the excited configurations with holes in the L-shell and with occupied 3d, 4s, 4p, 4d or higher subshells seem to be very important. The relative L-REC cross section between higher and lower projectile energies shows excellent agreement with the results obtained under the non-relativistic dipole approximation.