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We studied a system consisting of a proton, a muon, and an electron (a μpe system), the muon and the electron being in circular states. We demonstrated that in this case, the muonic motion can represent a rapid subsystem while the electronic motion can represent a slow subsystem – the result that might seem counterintuitive. We used a classical analytical description to find the energy terms for the quasi molecule where the muon rotates around the axis connecting the immobile proton and the immobile electron (i.e., dependence of the energy of the muon on the distance between the proton and electron). We found that there is a double-degenerate energy term. We demonstrated that it corresponds to stable motion. We also conducted an analytical relativistic treatment of the muonic motion and found that the relativistic corrections are relatively small. Then we unfroze the slow subsystem and analysed a slow revolution of the axis connecting the proton and electron. We derived the condition required for the validity of the separation into rapid and slow subsystems. Finally, we showed that the spectral lines, emitted by the muon in the quasi molecule, μpe, experience a red shift compared to the corresponding spectral lines that would have been emitted by the muon in a muonic hydrogen atom (in the μp-subsystem). The relative values of this red shift, which is a “molecular” effect, are significantly greater than the resolution of available spectrometers and thus can be observed. Observing this red shift should be one of the ways to detect the formation of such muonic–electronic negative hydrogen ions.

In the literature, there were studies of Rydberg states of hydrogenic atoms/ions in a high-frequency laser field. It was shown that the motion of the Rydberg electron is analogous to the motion of a satellite around an oblate planet (for a linearly polarized laser field) or around a (fictitious) prolate planet (for a circularly polarized laser field): it exhibits two kinds of precession – one of them is the precession within the orbital plane and another one is the precession of the orbital plane. In this study, we study a helium atom or a helium-like ion with one of the two electrons in a Rydberg state, the system being under a high-frequency laser field. For obtaining analytical results, we use the generalized method of the effective potentials. We find two primary effects of the high-frequency laser field on circular Rydberg states. The first effect is the precession of the orbital plane of the Rydberg electron. We calculate analytically the precession frequency and show that it differs from the case of a hydrogenic atom/ion. In the radiation spectrum, this precession would manifest as satellites separated from the spectral line at the Kepler frequency by multiples of the precession frequency. The second effect is a shift of the energy of the Rydberg electron, also calculated analytically. We find that the absolute value of the shift increases monotonically as the unperturbed binding energy of the Rydberg electron increases. We also find that the shift has a nonmonotonic dependence on the nuclear charge Z: as Z increases, the absolute value of the shift first increases, then reaches a maximum, and then decreases. The nonmonotonic dependence of the laser field-caused energy shift on the nuclear charge is a counterintuitive result.

Charge exchange is one of the most important atomic processes in plasmas. Charge exchange and crossings of corresponding energy levels that enhance charge exchange are strongly connected with problems of energy losses and of diagnostics in high temperature plasmas. Charge exchange was also proposed as an effective mechanism for population inversion in the soft X-ray and vacuum ultraviolet ranges. One of the most fundamental theoretical domains for studying charge exchange is the problem of electron terms in the field of two stationary Coulomb centers (TCC) of charges Z and Z′ separated by a distance R. It presents an intriguing atomic physics: the terms can have crossings and quasi crossings. These intrinsic features of the TCC problem also manifest in different areas of physics, such as plasma spectroscopy: a quasi crossing of the TCC terms, by enhancing charge exchange, can result in an unusual structure (a dip) in the spectral line profile emitted by a Z-ion from a plasma consisting of both Z-and Z′-ions, as was shown theoretically and experimentally. Before the year 2000, the paradigm was that the preceding sophisticated features of the TCC problem and its flourishing applications were inherently quantum phenomena. In 2000, a purely classical description of the crossings of energy terms was presented. In the present paper we study the effect of an electric field (along the internuclear axis) on circular Rydberg states of the TCC system. We provide analytical results for strong fields, as well as numerical results for moderate fields. We show that the electric field has several effects. First, it leads to the appearance of an extra energy term: the fourth classical energy term -in addition to the three classical energy terms at zero field. Second, but more importantly, the electric field creates additional crossings of these energy terms. We show that some of these crossings significantly enhance charge exchange, while other crossings enhance the ionization of the Rydberg quasi molecule. PACS Nos: 32.60.+i, 32.80.Ee, 34.70.+e, 33.80.Be, 31.15.-pRésumé : L'échange de charge est le plus important mécanisme atomique dans les plasmas. L'échange de charge et le croisement des niveaux d'énergie correspondants qui augmente l'échange de charge sont fortement connectés aux problèmes de perte d'énergie et de diagnostique dans les plasmas de haute température. L'échange de charge a aussi été proposé comme un mécanisme efficace pour l'inversion de population dans les domaines VUV et rayon-X mou. Le problème des termes de l'électron dans le champ de deux centres coulombiens stationnaires (TCC) de charges Z et Z′ séparés par une distance R, est central dans l'étude de l'échange de charge. C'est un problème intriguant en physique atomique : les termes peuvent avoir des croisements et des quasi croisements. Ces caractéristiques intrinsèques se manifestent également dans d'autres domaines de la physique, comme la spectroscopie du plasma : un quasi croisement des termes TCC, par augmentation de l'échange de charge, peut donner ...

In our previous paper (Can. J. Phys. 91, 715 (2013) doi: 10.1139/cjp-2013-0077 ) we studied a system consisting of a proton, a muon, and an electron; the muon and the electron being in circular states. The study was motivated by numerous applications of muonic atoms and molecules, where one of the electrons is substituted by the heavier lepton μ–. We demonstrated that in such a μpe quasi molecule, the muonic motion can represent a rapid subsystem while the electronic motion can represent a slow subsystem — a result that may seem counterintuitive. In other words, the muon rapidly revolves in a circular orbit about the axis connecting the proton and electron while this axis slowly rotates following a relatively slow electronic motion. We showed that the spectral lines, emitted by the muon in the quasi molecule, μpe, experience a red shift compared to the corresponding spectral lines that would have been emitted by the muon in a muonic hydrogen atom. In the present paper we generalize this study by replacing the proton in the μpe quasi molecule by a fully stripped ion of nuclear charge Z > 1. We show that in this case, just as in the previously studied case of Z = 1, the muonic motion can represent a rapid subsystem while the electronic motion can represent a slow subsystem. For this to be valid, the ratio of the muonic and electronic angular momenta should be slightly greater than in the case of Z = 1. We demonstrate that the binding energies of the muon for Z > 1 are much greater than for Z = 1 at any finite value of the nucleus–electron distance. Finally we show that the red shift of the spectral lines emitted by the muon (compared to the spectral lines of the corresponding muonic hydrogen-like ion of nuclear charge Z) decreases as Z increases. However, the relative red shift remains within the spectral resolution of available spectrometers at least up to Z = 5. Observing this red shift should be one of the ways to detect the formation of the quasi molecules, μZe.