Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Angular correlations in the two-photon decay of hydrogenlike ions: Relativistic Green’s-function approach

A. Surzhykov, Peter Koval, S. Fritzsche

Published Feb 24, 2005• 39 citations
Publisher

Abstract

The angular correlations in the two-photon decay of hydrogenlike ions are studied within the framework of second-order perturbation theory, based on Dirac's equation. Particular attention has been paid to the effects which arise from the higher (nondipole) terms in the expansion of the electron-photon interaction. It is shown that the photon-photon angular correlation function, which is found symmetric with respect to the angle $\ensuremath{\theta}=90\ifmmode^\circ\else\textdegree\fi{}$ in the electric dipole approximation, becomes asymmetric because of the nondipole contributions, and that this effect is enhanced as the nuclear charge $Z$ increases. Detailed computations on the photon-photon angular distribution have been carried out for the $2{s}_{1∕2}\ensuremath{\rightarrow}1{s}_{1∕2}$ and $3{d}_{5∕2}\ensuremath{\rightarrow}1{s}_{1∕2}$ transitions in neutral hydrogen (H) as well as for hydrogenlike xenon $({\mathrm{Xe}}^{53+})$ and uranium $({\mathrm{U}}^{91+})$ ions, and are compared with previous nonrelativistic results by Au [Phys. Rev. A 14, 531 (1976)]).

DOI
10.1103/physreva.71.022509

Related papers

Angular correlations in the two-photon decay of aligned hydrogenlike ions

Lesya Borowska, A. Surzhykov, Th. Stöhlker +1 more

The two-photon decay of highly charged hydrogenlike ions is studied within the framework of second-order perturbation theory, based on Dirac's equation. Attention has been paid to the angle-angle correlations between the two emitted photons. In particular, the density matrix theory is applied in order to explore how these angle-angle correlations are affected by the alignment of the excited states of the ions. Detailed computations have been carried out for the $3{d}_{5∕2}\ensuremath{\rightarrow}1{s}_{1∕2}$ transition in hydrogenlike uranium ${\mathrm{U}}^{91+}$ ions for different (degrees of) initial alignment. It is shown that the angular correlation function is very sensitive to the alignment of the $3{d}_{5∕2}$ state, an effect which has to be accounted for in the analysis of all (future) two-photon decay experiments, for which the excited state of the ions is produced by relativistic ion-atom collisions.

Physical Review A•Dec 29, 2006•8 citations

Electron angular distributions in the two-photon ionization of hydrogen-like ions: a relativistic description

Peter Koval, S. Fritzsche, A. Surzhykov

The angular distribution of the emitted electrons, following the two-photon ionization of hydrogen-like ions, is studied within the framework of second order perturbation theory and the Dirac equation. Using a density matrix approach, we have investigated the effects which arise from the polarization of the incoming light as well as from the higher multipoles in the expansion of the electron–photon interaction. For medium- and high- Z ions, in particular, the non-dipole contributions give rise to a significant change in the angular distribution of the emitted electrons, if compared with the electric dipole approximation. This includes a strong forward emission while, in the dipole approximation, the electron emission always occurs symmetrically with respect to the plane which is perpendicular to the photon beam. Detailed computations for the dependence of the photoelectron angular distributions on the polarization of the incident light are carried out for the ionization of H, and Xe 53+ and U 91+ (hydrogen-like) ions.

arXiv:0705.0136•Journal of Physics B Atomic Molecular and Optical Physics•Dec 17, 2003•18 citations

Parametrization of the angular correlation and degree of linear polarization in two-photon decays of hydrogenlike ions

Pedro Manuel Duarte Gonçalves Amaro, F. Fratini, S. Fritzsche +3 more

The spontaneous two-photon emission in hydrogenlike ions is investigated within the framework of second-order perturbation theory and Dirac's equation. Special attention is paid to the angular correlation of the emitted photons as well as to the degree of linear polarization of one of the two photons, if the second is just observed under arbitrary angles. Expressions for the angular correlation and the degree of linear polarization are expanded in powers of cosine functions of the two-photon opening angle, whose coefficients depend on the atomic number and the energy sharing of the emitted photons. The effects of including higher (electric and magnetic) multipoles upon the emitted photon pairs beyond the electric-dipole approximation are also discussed. Calculations of the coefficients are performed for the transitions $2{s}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}$, $3{d}_{3/2}\ensuremath{\rightarrow}1{s}_{1/2}$, and $3{d}_{5/2}\ensuremath{\rightarrow}1{s}_{1/2}$, along the entire hydrogen isoelectronic sequence ($1\ensuremath{\le}Z\ensuremath{\le}100$).

arXiv:1208.3169•Physical Review A•Oct 16, 2012•15 citations

Negative-continuum effects on the two-photon decay rates of hydrogenlike ions

A. Surzhykov, José Paulo Santos, Pedro Manuel Duarte Gonçalves Amaro +1 more

Two-photon decay of hydrogenlike ions is studied within the framework of second-order perturbation theory, based on the relativistic Dirac's equation. Special attention is paid to the effects arising from the summation over the negative-energy (intermediate virtual) states that occur in such a framework. In order to investigate the role of these states, detailed calculations have been carried out for the $2{s}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}$ and $2{p}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}$ transitions in neutral hydrogen H as well as for hydrogenlike xenon ${\text{Xe}}^{53+}$ and uranium ${\text{U}}^{91+}$ ions. We found that for a correct evaluation of the total and energy-differential decay rates, summation over the negative-energy part of Dirac's spectrum should be properly taken into account both for high-$Z$ and low-$Z$ atomic systems.

arXiv:0908.2498•Physical Review A•Nov 30, 2009•18 citations

Angular and polarization correlations in two--photon decay of hyperfine 2s states in hydrogenlike ions

F. Fratini

The angular and polarization properties of the photon pair emitted in the two--photon decay of $2s$ hyperfine states in hydrogenlike ions are investigated within the relativistic Dirac framework and second order perturbation theory. The studied transitions are $2s_{1/2}\,(F=1,0)\to1s_{1/2}\,(F=1,0)$ in Hydrogen atom and $2s_{1/2}\,(F=4,3)\to1s_{1/2}\,(F=4,3)$ in hydrogenlike Uranium ion. Two different emission patterns are found: For non-spin-flip transitions, the angular correlation ($i.e.$ the angular distribution of the emitted photons) is of the type $\sim 1+\cos^2θ$, while the linear polarizations of the emitted photons are approximately parallel one another; For spin-flip transitions, the angular correlation is of the type $\sim 1-1/3\cos^2θ$, while the linear polarizations of the emitted photons are approximately orthogonal one another.

arXiv:1203.6601•Mar 29, 2012•0 citations

Angular and polarization analysis for two-photon decay of 2s hyperfine states of hydrogenlike uranium

L. Safari, Pedro Manuel Duarte Gonçalves Amaro, José Paulo Santos +1 more

The amplitude of two-photon transitions between hyperfine states in hydrogenlike ions is derived based on the relativistic Dirac equation and second-order perturbation theory. We study angular and linear polarization properties of the photon pair emitted in the decay of $2s$ states, where spin-flip and non-spin-flip transitions are highlighted. We pay particular attention to hydrogenlike uranium, since it is an ideal candidate for investigating relativistic and high-multipole effects, such as spin-flip transitions. Two types of emission patterns are identified: (i) non-spin-flip transitions are found to be characterized by an angular distribution of the type $W(\ensuremath{\theta})\ensuremath{\sim}1+{cos}^{2}\ensuremath{\theta}$ while the polarizations of the emitted photons are parallel; and (ii) spin-flip transitions have somewhat smaller decay rates and are found to be characterized by an angular distribution of the type $W(\ensuremath{\theta})\ensuremath{\sim}1\ensuremath{-}1/3{cos}^{2}\ensuremath{\theta}$ while the polarizations of the emitted photons are orthogonal, where $\ensuremath{\theta}$ is the angle between photons directions. Deviations due to nondipole and relativistic contributions are evaluated for both types of transitions. This work is the first step toward exploring the effect of the nucleus over the angular and polarization properties of the photon pairs emitted by two-photon transitions.

arXiv:1407.4063•Physical Review A•Jul 29, 2014•9 citations
The angular correlations in the two-photon decay of hydrogenlike ions are studied within the framework of second-order perturbation theory, based on Dirac's equation. Particular attention has been paid to the effects which arise from the higher (nondipole) terms in the expansion of the electron-photon interaction. It is shown that the photon-photon angular correlation function, which is found symmetric with respect to the angle \ensuremathθ=90\ifmmode∘\else\textdegree\fi\ensuremath{\theta}=90\ifmmode^\circ\else\textdegree\fi{}\ensuremathθ=90\ifmmode∘\else\textdegree\fi in the electric dipole approximation, becomes asymmetric because of the nondipole contributions, and that this effect is enhanced as the nuclear charge ZZZ increases. Detailed computations on the photon-photon angular distribution have been carried out for the 2s1∕2\ensuremath→1s1∕22{s}_{1∕2}\ensuremath{\rightarrow}1{s}_{1∕2}2s1∕2​\ensuremath→1s1∕2​ and 3d5∕2\ensuremath→1s1∕23{d}_{5∕2}\ensuremath{\rightarrow}1{s}_{1∕2}3d5∕2​\ensuremath→1s1∕2​ transitions in neutral hydrogen (H) as well as for hydrogenlike xenon (Xe53+)({\mathrm{Xe}}^{53+})(Xe53+) and uranium (U91+)({\mathrm{U}}^{91+})(U91+) ions, and are compared with previous nonrelativistic results by Au [Phys. Rev. A 14, 531 (1976)]).
The two-photon decay of highly charged hydrogenlike ions is studied within the framework of second-order perturbation theory, based on Dirac's equation. Attention has been paid to the angle-angle correlations between the two emitted photons. In particular, the density matrix theory is applied in order to explore how these angle-angle correlations are affected by the alignment of the excited states of the ions. Detailed computations have been carried out for the $3{d}_{5∕2}\ensuremath{\rightarrow}1{s}_{1∕2}$ transition in hydrogenlike uranium U91+{\mathrm{U}}^{91+}U91+ ions for different (degrees of) initial alignment. It is shown that the angular correlation function is very sensitive to the alignment of the 3d5∕23{d}_{5∕2}3d5∕2​ state, an effect which has to be accounted for in the analysis of all (future) two-photon decay experiments, for which the excited state of the ions is produced by relativistic ion-atom collisions.
The angular distribution of the emitted electrons, following the two-photon ionization of hydrogen-like ions, is studied within the framework of second order perturbation theory and the Dirac equation. Using a density matrix approach, we have investigated the effects which arise from the polarization of the incoming light as well as from the higher multipoles in the expansion of the electron–photon interaction. For medium- and high- Z ions, in particular, the non-dipole contributions give rise to a significant change in the angular distribution of the emitted electrons, if compared with the electric dipole approximation. This includes a strong forward emission while, in the dipole approximation, the electron emission always occurs symmetrically with respect to the plane which is perpendicular to the photon beam. Detailed computations for the dependence of the photoelectron angular distributions on the polarization of the incident light are carried out for the ionization of H, and Xe 53+ and U 91+ (hydrogen-like) ions.
The spontaneous two-photon emission in hydrogenlike ions is investigated within the framework of second-order perturbation theory and Dirac's equation. Special attention is paid to the angular correlation of the emitted photons as well as to the degree of linear polarization of one of the two photons, if the second is just observed under arbitrary angles. Expressions for the angular correlation and the degree of linear polarization are expanded in powers of cosine functions of the two-photon opening angle, whose coefficients depend on the atomic number and the energy sharing of the emitted photons. The effects of including higher (electric and magnetic) multipoles upon the emitted photon pairs beyond the electric-dipole approximation are also discussed. Calculations of the coefficients are performed for the transitions $2{s}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}$, $3{d}_{3/2}\ensuremath{\rightarrow}1{s}_{1/2}$, and $3{d}_{5/2}\ensuremath{\rightarrow}1{s}_{1/2}$, along the entire hydrogen isoelectronic sequence ($1\ensuremath{\le}Z\ensuremath{\le}100$).
Two-photon decay of hydrogenlike ions is studied within the framework of second-order perturbation theory, based on the relativistic Dirac's equation. Special attention is paid to the effects arising from the summation over the negative-energy (intermediate virtual) states that occur in such a framework. In order to investigate the role of these states, detailed calculations have been carried out for the $2{s}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}$ and $2{p}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}$ transitions in neutral hydrogen H as well as for hydrogenlike xenon Xe53+{\text{Xe}}^{53+}Xe53+ and uranium U91+{\text{U}}^{91+}U91+ ions. We found that for a correct evaluation of the total and energy-differential decay rates, summation over the negative-energy part of Dirac's spectrum should be properly taken into account both for high-ZZZ and low-ZZZ atomic systems.
The angular and polarization properties of the photon pair emitted in the two--photon decay of 2s2s2s hyperfine states in hydrogenlike ions are investigated within the relativistic Dirac framework and second order perturbation theory. The studied transitions are 2s1/2 (F=1,0)→1s1/2 (F=1,0)2s_{1/2}\,(F=1,0)\to1s_{1/2}\,(F=1,0)2s1/2​(F=1,0)→1s1/2​(F=1,0) in Hydrogen atom and 2s1/2 (F=4,3)→1s1/2 (F=4,3)2s_{1/2}\,(F=4,3)\to1s_{1/2}\,(F=4,3)2s1/2​(F=4,3)→1s1/2​(F=4,3) in hydrogenlike Uranium ion. Two different emission patterns are found: For non-spin-flip transitions, the angular correlation (i.e.i.e.i.e. the angular distribution of the emitted photons) is of the type ∼1+cos⁡2θ\sim 1+\cos^2θ∼1+cos2θ, while the linear polarizations of the emitted photons are approximately parallel one another; For spin-flip transitions, the angular correlation is of the type ∼1−1/3cos⁡2θ\sim 1-1/3\cos^2θ∼1−1/3cos2θ, while the linear polarizations of the emitted photons are approximately orthogonal one another.
The amplitude of two-photon transitions between hyperfine states in hydrogenlike ions is derived based on the relativistic Dirac equation and second-order perturbation theory. We study angular and linear polarization properties of the photon pair emitted in the decay of 2s2s2s states, where spin-flip and non-spin-flip transitions are highlighted. We pay particular attention to hydrogenlike uranium, since it is an ideal candidate for investigating relativistic and high-multipole effects, such as spin-flip transitions. Two types of emission patterns are identified: (i) non-spin-flip transitions are found to be characterized by an angular distribution of the type $W(\ensuremath{\theta})\ensuremath{\sim}1+{cos}^{2}\ensuremath{\theta}$ while the polarizations of the emitted photons are parallel; and (ii) spin-flip transitions have somewhat smaller decay rates and are found to be characterized by an angular distribution of the type $W(\ensuremath{\theta})\ensuremath{\sim}1\ensuremath{-}1/3{cos}^{2}\ensuremath{\theta}$ while the polarizations of the emitted photons are orthogonal, where $\ensuremath{\theta}$ is the angle between photons directions. Deviations due to nondipole and relativistic contributions are evaluated for both types of transitions. This work is the first step toward exploring the effect of the nucleus over the angular and polarization properties of the photon pairs emitted by two-photon transitions.