Dirac-Coulomb Green Function 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.

Delbrück Scattering above the Pair Production Threshold: Going beyond the Born Approximation

AbstractA theoretical method to calculate Delbrück scattering amplitudes for photon energies above the electron‐positron () pair production threshold is presented. The method is based on the application of the relativistic Dirac–Coulomb Green function and describes the interaction of the virtual pair with the Coulomb field of a target to all orders in the coupling strength parameter . To illustrate the application of the developed approach, detailed calculations have been performed for the scattering of 2.754 MeV photons off bare ions with a wide range of nuclear charge numbers. Results of these calculations clearly indicate that the higher‐order terms beyond the Born approximation lead to a strong enhancement of the imaginary part of the Delbrück amplitude and have to be taken into account for the analysis and guidance of gamma‐ray scattering experiments.

All-Order Coulomb Corrections to Delbrück Scattering above the Pair-Production Threshold.

We report calculations of Delbrück scattering that include all-order Coulomb corrections for photon energies above the threshold of electron-positron pair creation. Our approach is based on the application of the Dirac-Coulomb Green's function and accounts for the interaction between the virtual electron-positron pair and the nucleus to all orders in the nuclear binding strength parameter αZ. Practical calculations are performed for the scattering of 2.754MeV photons off plutonium atoms. We find that including the Coulomb corrections enhances the scattering cross section by up to 50% in this case. The obtained results resolve the long-standing discrepancy between experimental data and theoretical predictions and demonstrate that an accurate treatment of the Coulomb corrections is crucial for the interpretation of existing and guidance of future Delbrück scattering experiments on heavy atoms.

Calculations of Delbrück scattering to all orders in αZ

We present a theoretical method to calculate Delbr\"uck scattering amplitudes. Our formalism is based on the exact analytical Dirac-Coulomb Green's function and therefore accounts for the interaction of the virtual electron-positron pair with the nucleus to all orders, including the Coulomb corrections. The numerical convergence of our calculations is accelerated by solving the radial integrals that are involved analytically in the asymptotic region. Numerical results for the collision of photons with energies 102.2 and 255.5 keV with bare neon and lead nuclei are compared with the predictions of the lowest-order Born approximation. We find that our method can produce accurate results within a reasonable computation time and that the Coulomb corrections enhance the absolute value of the Delbr\"uck amplitude by a few percent for the studied photon energies.

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.

Second-order Stark effect and polarizability of a relativistic two-dimensional hydrogenlike atom in the ground state

The second-order Stark effect for a planar Dirac one-electron atom in the ground state is analyzed within the framework of the Rayleigh-Schr\"odinger perturbation theory, with the use of the Sturmian series expansion of the generalized Dirac-Coulomb Green function. A closed-form analytical expression for the static dipole polarizability of that system is found. The formula involves a generalized hypergeometric function ${}_{3}F_{2}$ with the unit argument. Numerical values of the polarizabilities for relativistic planar hydrogenic atoms with atomic numbers $1\leqslant Z\leqslant68$ are provided in a tabular form. A simple formula for the polarizability of a nonrelativistic two-dimensional hydrogenic atom, reported previously by several other authors, is recovered from our result in the nonrelativistic limit.

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

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.

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.

Publisher's Note: Stark-induced magnetic anapole moment in the ground state of the relativistic hydrogenlike atom: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function [Phys. Rev. A73, 022511 (2005)

Publisher's Note: Stark-induced magnetic anapole moment in the ground state of the relativistic hydrogenlike atom: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function [Phys. Rev. A<b>73</b>, 022511 (2005)

Stark-induced magnetic anapole moment in the ground state of the relativistic hydrogenlike 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, 825 1997; 30, 2747E1997 is used to derive an analytical formula for the static magnetic anapole toroidal dipole moment induced in the ground state of the relativistic hydrogenlike atom by a weak, spatially uniform, static electric field. An expression for the anapole polarizability for the system in question is found. This expression contains a single generalized hypergeometric series 3 F 2 with the unit argument. In the non- relativistic limit our result agrees with that of Lewis and Blinder Phys. Rev. A 52, 4439 1995. Over the past three decades, in atomic and molecular physics an interest arose in effects associated with magnetic anapole moments. These effects may be divided into two principal categories. The first category comprises those atomic and molecular phenomena in which a nuclear mag- netic anapole moment, resulting from parity nonconservation in nuclear forces, manifests itself cf., e.g., Refs. 1-6. The second category includes those effects which evince the magnetic anapole moment of an atomic or a molecular elec- tronic cloud 7-22. In this paper, we shall be concerned with an effect falling into the second of the aforementioned categories. Specifi- cally, we shall be interested in the anapole moment induced in the ground state of the one-electron Dirac atom by a weak, spatially uniform, static electric field. This problem was al- ready considered a decade ago, both nonrelativistically and relativistically, by Lewis and Blinder 16; the reason for which we have decided to revisit it is that the relativistic calculations carried out in Ref. 16 were approximate in character. Exploiting the Sturmian expansion of the general- ized Dirac-Coulomb Green function 23-27, in the present paper we shall calculate the Stark-induced anapole moment exactly. The structure of the paper is as follows. In Sec. II we define the anapole moment. In Sec. III we consider the one- electron Dirac atom in the weak, spatially uniform, static electric field and apply the first-order Rayleigh-Schrodinger perturbation theory, together with the Green functions tech- nique, to derive an approximate expression for a perturbed electronic wave function. Next, in Sec. IV, this approximate wave function is used to determine an induced electric cur- rent density in the atom. Subsequently, the anapole moment of the resulting i.e., unperturbed plus induced current dis- tribution is considered. It is found that only the induced cur- rent contributes and the form of this contribution is calcu- lated analytically exploiting the Sturmian functions technique. The paper ends with conclusions, constituting Sec. V, and with several appendixes containing supplemen- tary material.

Gordon decomposition of the static dipole polarizability of the relativistic hydrogen-like atom: application of the Sturmian expansion of the first-order Dirac–Coulomb Green function

The Sturmian expansion of the generalized first-order Dirac–Coulomb Green function (Szmytkowski 1997 J. Phys. B: At. Mol. Opt. Phys. 30 825, 2747) is used to derive an analytical formula for the static dipole polarizability of the relativistic hydrogen-like atom in the ground state. The formula contains a generalized hypergeometric series 3F2 with the unit argument. It is identical with one found recently, in a completely different way, by Yakhontov (2003 Phys. Rev. Lett. 91 093001) and is the most compact among all known analytical expressions for the polarizability. Partitioning of the polarizability into convection and spin-polarization components, resulting from the Gordon decomposition of an induced electronic charge density, is carried out. It appears that the spin-polarization component of the polarizability vanishes.

Dynamic polarizability of the relativistic hydrogenlike atom: Application of the Sturmian expansion of the Dirac-Coulomb Green function

We utilize the Sturmian expansion of the Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30, 825 (1997)] to obtain components of the dynamic dipole polarizability tensor of the relativistic hydrogenlike atom in the ground state. It is found that the tensor may be expressed in terms of two independent quantities: a scalar polarizability and a vector polarizability, the latter being of the relativistic origin. In the static and nonrelativistic limits the previously known expressions for the scalar polarizability are recovered.

Relativistic calculation of two-photon bound-bound transition amplitudes in hydrogenic atoms

Using a representation of the (first-order) Dirac Coulomb Green function in terms of an expansion over a Sturmian basis, we have computed two-photon probability amplitudes for transitions in hydrogenic atoms from the ground state towards states in the $L$ and $M$ shells. The treatment takes care of the relativistic structure of the atoms according to the Dirac theory and of the fully retarded interaction with the electromagnetic field. We present in detail the Sturmian-like expansion for the Dirac Coulomb Green function and we discuss the relative importance of relativistic and retardation effects occurring for higher nuclear charge $Z$. Comparison is made also with the nonrelativistic Schr\"odinger treatment, retardation included or not.