Relativistic Hydrogenic Wave Functions Research Articles

Calculation of the matrix elements of the Coulomb interaction involving relativistic hydrogenic wave functions

The program MTRDCOUL [1] calculates the matrix elements of the Coulomb interaction between a charged particle and an atomic electron, ∫ψf∗(r)∣R−r∣−1ψi(r)dr. Bound-free transitions are considered, and relativistic hydrogenic wave functions are used. In this revised version a bug discovered in the F3Y CPC Program Library subprogram [2] is fixed. New version program summaryProgram Title: MTRDCOULProgram Files doi: “http://dx.doi.org/10.17632/4cmts2c49b.1 ”Licensing provisions: GNU GPL v3Programming language: Fortran 77Journal reference of previous version: Comput. Phys. Commun. 141 (2001) 73Does the new version supersede the previous version?: YesReasons for the new version: In some applications MTRDCOUL led to unexpected results that were traced back to the erroneous execution of the subprogram F3Y [2]. For example, in some cases F3Y yielded completely different values for the inputs (l,m1,l,m2,l′,m′) and (l,m2,l,m1,l′,m′), while, for symmetry reason, one expects equal results. In the new version this error in F3Y was corrected.Summary of revisions: In the line AAQQ0036 of the original F3Y program [2] the incorrect assignment A2=L1−M1−N2 was replaced by A2=L1−M1−N1. The corrected F3Y was tested by comparing its results with those obtained by a program written for the integral of the product of three spherical harmonics using the 369j program [3]. An agreement within 10−14 was found between the two calculations for all possible arguments of the function belonging to the values of li up to 10.Nature of problem: The theoretical description of the excitation and ionization of atoms by charged particle impact often requires the knowledge of the matrix elements of the Coulomb interaction. Considering that the program can easily be extended to the calculations of matrix elements between wave functions other than the hydrogenic ones, it may find a broad application including the treatment of the electron–electron correlation problems.Solution method: The algorithm is based on the multipole series expansion of the Coulomb potential.Additional comments including Restrictions and Unusual features: The matrix elements are calculated with the following restrictions. The initial bound states are limited to 1s1/2, 2s1/2, 2p1/2, 2p3/2, 3s1/2, 3p1/2, 3p3/2, 3d3/2, 3d5/2. The quantum number l in the final state has a maximum value of 10.AcknowledgmentsThis work was supported by the National Scientific Research Foundation (OTKA, Grant No. K109440). [1]L. Lugosi and L. Sarkadi, Comput. Phys. Commun. 141 (2001) 73.[2]A. Liberato de Brito, Comput. Phys. Commun. 25 (1982) 81.[3]L. Wei, Comput. Phys. Commun. 120 (1999) 222; Erratum: 182 (2011) 1199.AppendixTEST RUN OUTPUT [Display omitted]

K-shell ionization in (0.2–1.75) MeV/u Fe, Co + Cr and Fe + Cu collisions

K-shell ionization cross-sections and mean vacancy sharing probabilities measured in the near-symmetric collision systems Fe, Co, Cu+Cr and Fe+Cu at 0.2–1.75MeV/u energies are reported. Beams of 56Fe, 59Co and 63Cuq+ ions (3⩽q⩽12) delivered by the Van de Graaff tandem accelerator of NIPNE Bucharest were used. The cross-section values have been corrected for multiple ionization in the outer (L,M) shells by using the energy and yield shift method. Mean ionization probabilities per electron in the outer shells have been estimated. Calculations in the Briggs model, using SCA with relativistic hydrogenic wave functions and binding energy corrections, could explain the 2pσ-molecular orbital ionization cross-sections at higher energies (⩾1MeV/u). The vacancy sharing results show that predictions of the two-state coupling Demkov–Meyerhof model are less fulfilled at higher energies (⩾1.5MeV/u).

Calculation of the matrix elements of the Coulomb interaction involving relativistic hydrogenic wave functions

A program is presented for calculation of the matrix elements of the Coulomb interaction between a charged particle and an atomic electron, ∫ψ f †( r)| R− r| −1ψ i( r) d r . Bound-free transitions are considered. Relativistic hydrogenic wave functions are used for the numerical evaluation of the matrix elements. The applied algorithm is based on the multipole series expansion of the Coulomb potential. The radial part of the terms of this series expansion (known as G functions) can also be obtained.

High-resolution study of ion-inducedKα1,2x-ray spectra from high-Zelements

The $K{\ensuremath{\alpha}}_{1,2}$ x rays emitted from La, Tb, Ta, and U targets induced by proton nitrogen, and neon ion bombardment at incident energies between 15 and 28 MeV/nucleon have been measured with a high-resolution crystal spectrometer. The spectra reflect the decay of a K vacancy in the presence of multiple L, M, N,\dots{} shell ``spectator'' vacancies. The L-shell ionization probabilities for impact parameters in the interior of the target's K shell are extracted from the measured spectra and compared with extensive theoretical calculations of ionization probabilities, based on the semiclassical separate-atom perturbation theory [semiclassical approximation (SCA)]. The projectile velocity varies from 60% to 125% of the target L electron velocity, so we can compare experimental data with theoretical predictions as a function of the relative velocity. We find that wave-function effects play an important role for an accurate calculation of the ionization probability. The experimental results agree well with the SCA predictions when Hartree-Fock-type wave functions are used [variationally determined optimized potential (SCA-OPM)]. Relativistic hydrogenic wave functions (SCA-HYD) provide a reliable description of experimental data only in the case of the uranium target. The influence of electron capture and intrashell coupling is also calculated and discussed. An independent comparison of theoretical constructed spectra with the experimental ones confirms the validity of the SCA-OPM. The hole-hole angular momentum coupling between ``active'' and ``spectator'' holes has been observed as predicted.

M-shell X-ray production by C, N and O ions

M-shell X-ray production cross sections for Os, Au and Th targets were measured for C, N and O ions at incident energies from 1.2 MeV to about 36 MeV. Experimental M-shell ionization cross sections, derived by using the effective M-shell fluorescence yields, are compared with the prediction of the SCA calculations with relativistic hydrogenic wave functions. The calculations were performed both in the separated-atom (SA) and united-atom (UA) limits to simulate the binding effect. Generally, the data show better agreement with the SCA-UA cross sections for lower energies and with the SCA-SA predictions for higher energies. For C and O impact on the Au target at about 2 MeV/amu the present data are compared with available experimental results and theoretical predictions of the ECPSSR theory.

M-shell ionization resulting from near-central collisions of mid-Z atoms with 5.5-MeV/amu oxygen ions.

The K\ensuremath{\alpha} and K\ensuremath{\beta} x-ray spectra of palladium and lanthanum bombarded by 5.5-MeV/amu $^{16}\mathrm{O}$ ions were measured with a high-resolution transmission crystal spectrometer. The observed complex structure of the K\ensuremath{\alpha}${\mathit{L}}^{0}$ and K${\mathrm{\ensuremath{\beta}}}_{1,3}$${\mathit{L}}^{0}$ lines consisting of many M satellites was analyzed by means of theoretical line profiles based on multiconfiguration Dirac-Fock calculations. The average M-shell and M-subshell ionization probabilities in near-central collisions are determined. The experimental results are compared to the predictions of the semiclassical-approximation theory. The agreement is much improved by using Dirac-Hartree-Fock (DHF) wave functions instead of screened relativistic hydrogenic wave functions. The importance of DHF wave functions in the calculation of the M-shell ionization probabilities is illustrated with other examples.

Remarks on momentum-space hydrogenic wavefunctions

Two simple expressions are constructed for the momentum representation of the non-relativistic and relativistic hydrogen wavefunctions in terms of Gaussian hypergeometric functions which are either polynomials or absolutely convergent. The usefulness of these results is demonstrated by calculating momentum distributions of the K- and L-shell electrons of 79Au. It is found that unlike K, L1 and L2 subshells, the relativistic effects reduce the momentum distribution of L3 electrons compared with the non-relativistic result.

Wave function effects in inner shell ionization

A method is presented which takes binding effects in inner shell ionization into account in an approximate way. The electron in the initial and final state is described by relativistic Hartree-Fock-Slater or screened relativistic hydrogenic wave functions. By comparing the results of this approach with many different experimental data, it is found that it works very well in most cases.

On the momentum representation of the relativistic Hartree-Fock-Roothaan wavefunctions

Analytical expressions for the momentum representation of the relativistic Hartree-Fock-Roothaan wavefunctions are obtained by means of the Fourier-Bessel transformation. The momentum distributions are calculated by the use of these expressions and the results are compared with the corresponding distributions of the nonrelativistic and relativistic hydrogenic wavefunctions as well as the nonrelativistic Hartree-Fock-Roothaan wavefunctions.

Semiclassical treatment of light-ion induced alignment of theL 3-subshell in heavy atoms

Light ion induced alignment measurements ofL3-subshell in heavy atoms are analyzed within the framework of the semiclassical theory of inner shell ionization. Coulomb deflection effects are exactly taken into account by hyperbolic classical orbits while the electron in the initial and final states is approximated by screened relativistic hydrogenic wave functions. Satisfactory agreement between experiments and theory is obtained for proton- and α-particle induced reactions within the whole region of bombarding energies.

Electronic relativistic effects onL-shell ionization of atoms by light-ion impact

The $L$-shell ionization cross sections by charged-particle impact have been calculated in the plane-wave Born approximation, using relativistic hydrogenic wave functions for the target electrons. The effects of the binding-energy increase due to penetration of the projectile and the Coulomb deflection of the projectile are taken into account. The calculated values are compared with the corresponding values obtained by other theoretical models as well as the experimental data. It is found that the present theoretical predictions are in satisfactory agreement with the experimental results for low-energy protons.

Electronic relativistic effect on the ratios of deuteron to alpha-particle induced K-shell ionization cross sections

The ratios of the K-shell ionization cross sections induced by deuterons to those induced by α-particles with the same velocities have been calculated in the plane wave Born approximation using relativistic hydrogenic wave functions for the target electrons. The numerical results are compared with the recent experimental valus of Chang et al. The use of relativistic wave functios considerably improves the agreement with the experimental data.

Internal ionization accompanyingKconversion: One-step theory

A one-step theory of internal ionization accompanying $K$ conversion has been developed. The ionization probability and the energy spectrum of the ejected electrons have been calculated using screened relativistic hydrogenic wave functions. It is shown that the probabilities calculated using the present model are in good agreement with the recent experimental values for the $K$-shell internal ionization accompanying $K$ conversion. The calculated results predict that the ionization probability is dependent on the nuclear transition energy and on the multipolarity of the transition. The multipolarity dependence of the spectral shape of the ejected electrons has also been studied.RADIOACTIVITY $^{57}\mathrm{Fe}$, $^{95,\phantom{\rule{0ex}{0ex}}96}\mathrm{Mo}$, $^{97}\mathrm{Tc}$, $^{109}\mathrm{Ag}$, $^{113,\phantom{\rule{0ex}{0ex}}114}\mathrm{In}$, $^{131}\mathrm{Xe}$, $^{137}\mathrm{Ba}$, $^{141}\mathrm{Pr}$, $^{150}\mathrm{Sm}$, $^{169}\mathrm{Tm}$, $^{203}\mathrm{Tl}$; calculated internal ionization probability accompanying $K$ conversion, ejected-electron spectrum.

Electron shakeoff accompanying internal conversion

The electron shakeoff probabilities accompanying the internal conversion process have been calculated in the sudden approximation, using screened relativistic hydrogenic wave functions. The screening constants have been determined from the relativistic self-consistent-field calculations and the presence of a vacancy resulting from internal conversion has been taken into account. The present results indicate that relativistic effects cause an appreciable increase in the probabilities. It is also shown that the prediction of Carlson et al. is a good approximation to the shakeoff probability accompanying internal conversion. The calculated results have been compared with available experimental data. There is fairly good agreement between the calculated and the recently measured values. The need for new data is emphasized.RADIOACTIVITY $^{57}\mathrm{Fe}$, $^{109}\mathrm{Ag}$, $^{113,114}\mathrm{In}$, $^{131}\mathrm{Xe}$, $^{137}\mathrm{Ba}$, $^{141}\mathrm{Pr}$, $^{150}\mathrm{Sm}$, $^{203}\mathrm{Tl}$; calculated shakeoff probability accompanying internal conversion.

L-shell contributions to internal ionization accompanying electron capture

The $L$-shell internal-ionization probability accompanying $K$ capture, $K$-shell internal-ionization probability accompanying $L$ capture, and the ejected-electron spectrum during these processes have been calculated using screened relativistic hydrogenic wave functions. The screening constant for the daughter atom is determined taking into account the presence of a vacancy resulting from electron capture. The results show that $L$-shell contributions are important for the low-energy region of the electron spectrum and for high -$Z$ elements. It is also concluded that the presence of a vacancy in the inner shell of the daughter atom plays an important role.[RADIOACTIVITY $^{37}\mathrm{Ar}$, $^{55}\mathrm{Fe}$, $^{71}\mathrm{Ge}$, $^{131}\mathrm{Cs}$, $^{165}\mathrm{Er}$; calculated internal-ionization probability accompanying EC, ejected-electron spectrum.]

Differential cross sections for the elastic scattering of γ-rays for energies from 145 keV To 750 keV

Differential cross sections for the elastic scattering of γ-rays for energies between 145 keV and 750 keV and for charge numbers from 30 to 82 are presented for scattering angles ranging from 45° to 135°. These differential cross sections have been calculated using form factors and “corrected form factors” obtained from relativistic HFS wave functions and using a semi- empirical procedure for obtaining accurate scattering amplitudes from the form factors. For an easy extension of the tabulation, exact analytical expressions based on relativistic hydrogen wave functions are presented for both form factors and all subshells of the atom. Effective charge numbers are given providing close agreement between form factors calculated from relativistic hydrogen wave functions and from relativistic HFS wave functions.