Sturmian Expansion Research Articles

Gordon Decomposition of the Magnetizability of a Dirac One-Electron Atom in an Arbitrary Discrete Energy State

We present a Gordon decomposition of the magnetizability of a Dirac one-electron atom in an arbitrary discrete energy eigenstate, with a pointlike, spinless, and motionless nucleus of charge Ze. The external magnetic field, by which the atomic state is perturbed, is assumed to be weak, static, and uniform. Using the Sturmian expansion of the generalized Dirac–Coulomb Green function proposed by Szmytkowski in 1997, we derive a closed-form expressions for the diamagnetic (χd) and paramagnetic (χp) contributions to χ. Our calculations are purely analytical; the received formula for χp contains the generalized hypergeometric functions 3F2 of the unit argument, while χd is of an elementary form. For the atomic ground state, both results reduce to the formulas obtained earlier by other author. This work is a prequel to our recent article, where the numerical values of χd and χp for some excited states of selected hydrogenlike ions with 1⩽Z⩽137 were obtained with the use of the general formulas derived here.

Quasirelativistic Coulomb Sturmian basis sets for atomic physics problems: illustration of their applicability to two-photon decay rates in hydrogen-like atoms

The continuum and discrete quasirelativistic one-electron Sturmian basis sets associated with the second-order Dirac–Coulomb equation are constructed by solving appropriate Sturm–Liouville problems. The potential-weighted orthogonality and closure relations obeyed by the members of such sets are discussed. The special case where both quasirelativistic Coulomb Sturmian basis sets are used to obtain solutions to the Gell-Mann–Feynman equation is presented. It is proved that for the continuum quasirelativistic Sturm–Liouville problem, the spectrum of potential strengths is continuous and covers the whole real axis. We have also introduced the complete outgoing continuum quasirelativistic Coulomb wavefunctions and presented the discretization of its radial components. The compact representation of the fully relativistic and quasirelativistic Coulomb Green’s functions is outlined. By making use of the Sturmian expansion of the Coulomb Green’s functions, closed-form expressions of two-photon spontaneous decay rates are derived for arbitrary multipole channels. For the sake of assessing the domain of applicability of quasirelativistic basis sets, an illustrative application is then carried out for two-photon decay rates for hydrogen-like ions. A comparison of present results with reference values previously obtained within a fully relativistic scheme (Bona et al 2014) is undertaken.

Relativistic two-dimensional hydrogen-like atom in a weak magnetic field

A two-dimensional (2D) hydrogen-like atom with a relativistic Dirac electron, placed in a weak, static, uniform magnetic field perpendicular to the atomic plane, is considered. Closed forms of the first- and second-order Zeeman corrections to energy levels are calculated analytically, within the framework of the Rayleigh–Schrödinger perturbation theory, for an arbitrary electronic bound state. The second-order calculations are carried out with the use of the Sturmian expansion of the two-dimensional generalized radial Dirac–Coulomb Green function derived in the paper. It is found that, in contrast to the case of the three-dimensional atom (Stefańska, 2015), in two spatial dimensions atomic magnetizabilities (magnetic susceptibilities) are expressible in terms of elementary algebraic functions of a nuclear charge and electron quantum numbers. The problem considered here is related to the Coulomb impurity problem for graphene in a weak magnetic field.

Static magnetic multipole susceptibilities of the relativistic hydrogenlike atom in the ground state: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

Static magnetic multipole susceptibilities of the relativistic hydrogenlike atom in the ground state: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

Two-dimensional hydrogen-like atom in a weak magnetic field

.We consider a non-relativistic two-dimensional (2D) hydrogen-like atom in a weak, static, uniform magnetic field perpendicular to the atomic plane. Within the framework of the Rayleigh-Schrödinger perturbation theory, using the Sturmian expansion of the generalized radial Coulomb Green function, we derive explicit analytical expressions for corrections to an arbitrary planar hydrogenic bound-state energy level, up to the fourth order in the strength of the perturbing magnetic field. In the case of the ground state, we correct an expression for the fourth-order correction to energy available in the literature.

Three-body calculation of the1slevel shift in kaonic deuterium with realisticK¯Npotentials

The $1s$ level shift in kaonic deuterium was calculated using Coulomb Sturmian expansion of Faddeev equations. The convergence of the method yields an $\sim\ 1\ eV$ accuracy for the level shifts. We used three different, realistic, multichannel $\bar{K}N$ interactions reproducing all known experimental two-body $\bar{K}N$ data. The different results suggest, that the level shift should be in the range $\Delta E\sim(800\pm30)-(480\pm20){\rm i}\ \ eV$. The (almost) exact level shifts were compared with values, given by the commonly used approximations.

Dynamics of atoms in strong laser fields I: A quasi analytical model in momentum space based on a Sturmian expansion of the interacting nonlocal Coulomb potential

In this study we present a model that we have formulated in the momentum space to describe atoms interacting with intense laser fields. As a further step, it follows our recent theoretical approach in which the kernel of the reciprocal-space time-dependent Schrödinger equation (TDSE) is replaced by a finite sum of separable potentials, each of them supporting one bound state of atomic hydrogen (Tetchou Nganso et al. 2013). The key point of the model is that the nonlocal interacting Coulomb potential is expanded in a Coulomb Sturmian basis set derived itself from a Sturmian representation of Bessel functions of the first kind in the position space. As a result, this decomposition allows a simple spectral treatment of the TDSE in the momentum space. In order to illustrate the credibility of the model, we have considered the test case of atomic hydrogen driven by a linearly polarized laser pulse, and have evaluated analytically matrix elements of the atomic Hamiltonian and dipole coupling interaction. For various regimes of the laser parameters used in computations our results are in very good agreement with data obtained from other time-dependent calculations.

The nuclear size and mass effects on muonic hydrogen-like atoms embedded in Debye plasma

Effects of finite nuclear size and finite nuclear mass are investigated for muonic atoms and muonic ions embedded in the Debye plasma. Both nuclear charge radii and nuclear masses are taken into account with experimentally determined values. In particular, isotope shifts of bound state energies, radial probability densities, transition energies, and binding energies for several atoms are studied as functions of Debye length. The theoretical model based on semianalytical calculations, the Sturmian expansion method, and the perturbative approach has been constructed, in the nonrelativistic frame. For some limiting cases, the comparison with previous most accurate literature results has been made.

Closed-form expression for the magnetic shielding constant of the relativistic hydrogenlike atom in an arbitrary discrete energy eigenstate: Application of the Sturmian expansion of the generalized Dirac–Coulomb Green function

We present analytical derivation of the closed-form expression for the dipole magnetic shielding constant of a Dirac one-electron atom being in an arbitrary discrete energy eigenstate. The external magnetic field, by which the atomic state is perturbed, is assumed to be weak, uniform, and time independent. With respect to the atomic nucleus we assume that it is pointlike, spinless, motionless, and of charge $Ze$. Calculations are based on the Sturmian expansion of the generalized Dirac--Coulomb Green function [R. Szmytkowski, J. Phys. B 30, 825 (1997); erratum R. Szmytkowski, J. Phys. B 30, 2747(E) (1997)], combined with the theory of hypergeometric functions. The final result is of an elementary form and agrees with corresponding formulas obtained earlier by other authors for some particular states of the atom.

Static electric multipole susceptibilities of the relativistic hydrogenlike atom in the ground state: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

The ground state of the Dirac one-electron atom, placed in a weak, static electric field of definite $2^{L}$-polarity, is studied within the framework of the first-order perturbation theory. The Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30 (1997) 825, erratum: 30 (1997) 2747] is used to derive closed-form analytical expressions for various far-field and near-nucleus static electric multipole susceptibilities of the atom. The far-field multipole susceptibilities --- the polarizabilities $\alpha_{L}$, electric-to-magnetic cross-susceptibilities $\alpha_{\mathrm{E}L\to\mathrm{M}(L\mp1)}$ and electric-to-toroidal-magnetic cross-susceptibilities $\alpha_{\mathrm{E}L\to\mathrm{T}L}$ --- are found to be expressible in terms of one or two non-terminating generalized hypergeometric functions ${}_{3}F_{2}$ with the unit argument. Counterpart formulas for the near-nucleus multipole susceptibilities --- the electric nuclear shielding constants $\sigma_{\mathrm{E}L\to\mathrm{E}L}$, near-nucleus electric-to-magnetic cross-susceptibilities $\sigma_{\mathrm{E}L\to\mathrm{M}(L\mp1)}$ and near-nucleus electric-to-toroidal-magnetic cross-susceptibilities $\sigma_{\mathrm{E}L\to\mathrm{T}L}$ --- involve terminating ${}_{3}F_{2}(1)$ series and for each $L$ may be rewritten in terms of elementary functions. Exact numerical values of the far-field dipole, quadrupole, octupole and hexadecapole susceptibilities are provided for selected hydrogenic ions. Analytical quasi-relativistic approximations, valid to the second order in $\alpha Z$, where $\alpha$ is the fine-structure constant and $Z$ is the nuclear charge number, are derived for all types of the far-field and near-nucleus susceptibilities considered in the paper.

Magnetic-field-induced electric quadrupole moments for relativistic hydrogenlike atoms: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

We consider a Dirac one-electron atom placed in a weak, static, uniform magnetic field. We show that, to the first order in the strength $B$ of the external field, the only electric multipole moments, which are induced by the perturbation in the atom, are those of an even order. Using the Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30, 825 (1997); J. Phys. B 30, 2747 (1997)], We derive a closed-form expression for the electric quadrupole moment induced in the atom in an arbitrary discrete energy eigenstate. The result, which has the form of a double finite sum involving the generalized hypergeometric functions ${}_{3}{F}_{2}$ of the unit argument, agrees with the earlier relativistic formula for that quantity, obtained by us for the ground state of the atom.

Magnetizability of the relativistic hydrogenlike atom in an arbitrary discrete energy eigenstate: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

The Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30, 825 (1997); R. Szmytkowski, J. Phys. B 30, 2747(E) (1997)] is exploited to derive a closed-form expression for the magnetizability of the relativistic one-electron atom in an arbitrary discrete state, with a pointlike, spinless, and motionless nucleus of charge $Ze$. The result has the form of a double finite sum involving the generalized hypergeometric functions ${}_{3}{F}_{2}$ of the unit argument. Our general expression agrees with formulas obtained analytically earlier by other authors for some particular states of the atom. We present also numerical values of the magnetizability for some excited states of selected hydrogenlike ions with $1\ensuremath{\le}Z\ensuremath{\le}137$ and compare them with data available in the literature.

Two-photon decay rates of hydrogenlike ions revisited by using Dirac-Coulomb Sturmian expansions of the first order

A fully relativistic multipole scheme is formulated to study two-photon emission processes in hydrogenlike ions with an infinitely heavy, pointlike, and spinless nucleus of charge up to 100. By making use of the Sturmian expansion of the Dirac-Coulomb Green function of the first order constructed by Szmytkowski, closed-form expressions are derived for arbitrary multipole channels. In the nonrelativistic limit, well-known formulas established previously are retrieved. For the sake of assessing the effectiveness of our approach, numerical applications are then carried out for two-photon decay rates of the selected $2{s}_{1/2}$ and $2{p}_{1/2}$ atomic states. To this end, radial integrals, the most crucial quantities involved in the matrix elements, are treated with great care by means of two suitable techniques that agree with each other quite closely so that very accurate values are obtained regardless of the choice of parameters, such as radial quantum numbers and orders of spherical Bessel functions of the first kind. In addition, the convergence and stability of computations are checked in connection with the intermediate-state summation, which appears within the second-order perturbation theory. As expected, the gauge invariance of our fully relativistic multipole numbers is confirmed. Relativistic effects, and the influence of the negative spectrum of the complete set of Dirac-Coulomb Sturmians of first order and retardation truncations in the transition operator are examined. Finally, a comparison is undertaken of our two-photon relativistic calculations with refined predictions of other authors based on finite basis-set methods widely employed over the past decades.

Electric-field-induced magnetic quadrupole moment in the ground state of the relativistic hydrogenlike atom: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

We consider a Dirac one-electron atom placed in a weak, static, uniform electric field. We show that, to the first order in the strength of the perturbing field, the only magnetic multipole moment induced in the ground state of the atom is the quadrupole one. The tensorial structure of that moment is resolved. Using the Sturmian expansion of the generalized Dirac-Coulomb Green function [Szmytkowski, J. Phys. B 30, 825 (1997); Szmytkowski, J. Phys. B 30, 2747(E) (1997)], we derive a closed-form expression for an $E1\ensuremath{\rightarrow}M2$ cross-susceptibility of the atom in the ground state.

Ionization of atomic hydrogen with up to four excess photons by circularly and linearly polarized light

The above threshold ionization (ATI) is investigated for H in ns states. The total generalized cross sections are calculated for up to 4 excess photons using perturbation theory and modified Coulomb Green's function Sturmian expansion.

Magnetic-field-induced electric quadrupole moment in the ground state of the relativistic hydrogenlike atom: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

We consider a Dirac one-electron atom placed in a weak, static, uniform magnetic field. We show that, to the first order in the strength $B$ of the perturbing field, the only electric multipole moment induced by the field in the ground state of the atom is the quadrupole one. Using the Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30 (1997) 825; erratum 30 (1997) 2747], we derive a closed-form expression for the induced electric quadrupole moment. The result contains the generalized hypergeometric function $_{3}F_{2}$ of the unit argument. Earlier calculations by other authors, based on the non-relativistic model of the atom, predicted in the low-field region the quadratic dependence of the induced electric quadrupole moment on $B$.

Electric and magnetic dipole shielding constants for the ground state of the relativistic hydrogen-like atom: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

The Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30 (1997) 825; erratum 30 (1997) 2747] is exploited to derive closed-form expressions for electric ($\sigma_{\mathrm{E}}$) and magnetic ($\sigma_{\mathrm{M}}$) dipole shielding constants for the ground state of the relativistic hydrogen-like atom with a point-like and spinless nucleus of charge $Ze$. It is found that $\sigma_{\mathrm{E}}=Z^{-1}$ (as it should be) and $$\sigma_{\mathrm{M}}=-(2Z\alpha^{2}/27)(4\gamma_{1}^{3}+6\gamma_{1}^{2}-7\gamma_{1}-12) /[\gamma_{1}(\gamma_{1}+1)(2\gamma_{1}-1)],$$ where $\gamma_{1}=\sqrt{1-(Z\alpha)^{2}}$ ($\alpha$ is the fine-structure constant). This expression for $\sigma_{\mathrm{M}}$ agrees with earlier findings of several other authors, obtained with the use of other analytical techniques, and is elementary compared to an alternative one presented recently by Cheng \emph{et al.} [J. Chem. Phys. 130 (2009) 144102], which involves an infinite series of ratios of the Euler's gamma functions.

Sturmian expansions for two-electron atomic systems: Singly and doubly excited states

We present a configuration interaction (CI) method based on the Sturmian expansion for bound states of a two-electron atomic system. These Sturmian functions are solutions of one-electron quantum mechanical problems, where the eigenvalue is the magnitude of a short-range potential. Also, they fulfill the long-range boundary conditions of Coulomb potentials. We choose to expand the Sturmians of the CI basis using ${L}^{2}$ Laguerre-type functions. We compute ground and single-excited states energies for He and ${\mathrm{H}}^{\ensuremath{-}}$. Moreover, we are able to obtain energies and widths of double excited states of He, using a Sturmian basis with outgoing boundary conditions. In all cases, our ansatz outperforms other CI calculations, for similar basis size.

Above threshold ionization of atomic hydrogen in ns states with up to four excess photons

In a high-intensity laser field an atom can absorb more photons than the minimum necessary for ionization. It is known as above threshold ionization (ATI). Theoretically it is the most difficult case to handle as we have to consider transitions in continuum. To study ATI we use the perturbation theory and Green's function formalism. We have derived the modified two-term Coulomb Green's function (CGF) Sturmian expansion. In each term explicit summation over all intermediate states is carried out. The transition amplitude may be obtained in a closed form. The generalized cross sections are evaluated for the photoionization of atomic hydrogen in ns states with up to four excess photons. Calculations are performed in a wide range of wavelengths for linear and circular polarization. In the cases for which data are available, our results agree very well with the previous ones.

The second Born approximation in electron-helium scattering in a Nd-YAG laser field

The dynamics of laser-assisted elastic collisions in helium is studied using the second-order Born approximation. Detailed calculations of the scattering amplitudes are performed by using the Sturmian basis expansion. Differential cross sections for elastic scattering with the net absorption/emission of up to two photons are calculated for collision energies of 5 eV, 10 eV, and 20 eV. We discuss the influence of the low-energy electrons on the differential cross section (DCS) as a function of the scattering angle for selected choices of the laser frequency and the number of photons exchanged between the external field and electron-helium system.