Muonic–electronic negative hydrogen ion: circular states

Abstract

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.

Oscillatory-Precessional Motion of a Rydberg Electron Around a Polar Molecule

“…This polynomial has a negative minimum equal to −K − 2/3 3/2 at x = − 1/3 1/2 and a maximum equal to −K + 2/3 3/2 at x = 1/3 1/2 . Obviously, for the range of the bound motion to exist, the maximum should be positive, leading to the following requirement (Fox [6]): K < 2/3 3/2 = K max (7) or equivalently (see Equation (4)…”

“…In 2013, Kryukov and Oks [4] started by analytically considering circular orbits of a negative charge in the field of a finite dipole, when the orbital plane was perpendicular to the dipole axis; they used the cylindrical coordinates. (We note that as the application, the authors of paper [4] chose the finite dipole to be made by stationary proton and electron, while the negative charge moving in the field of this dipole was a muon; however, their analytical results are totally applicable to the motion of an electron in the field of any finite dipole.) They showed that stable circular orbits exist if the size of the dipole is greater or equal to some finite value (this value turned out to be the same as in many quantal studies of this system-see, e.g., paper [2] and references therein).…”

“…They showed that stable circular orbits exist if the size of the dipole is greater or equal to some finite value (this value turned out to be the same as in many quantal studies of this system-see, e.g., paper [2] and references therein). Further, the authors of paper [4] went beyond the circular orbits and presented analytically a stable conic-helical motion of the electron in the field of the finite dipole: the helical orbit is confined to the surface of a right frustum of a cone coaxial with the dipole.…”

“…It is worth mentioning that obtaining the analytical results in papers [3,4] was possible because the general problem of the motion of a charged particle in the field of two stationary Coulomb centers (the problem, of which the motion of a charge in the field of a finite dipole is a particular case) is characterized by higher than geometrical (i.e., algebraic) symmetry. The symmetry is manifested by the presence of an additional conserved quantity: the projection of the super-generalized Runge-Lenz vector on the axis connecting the stationary charges [5].…”

Classical Dynamics of Muonic-Electronic Helium and Heliumlike Ions: The Allowance for the Eccentricity of the Muon and Nucleus Orbits

“…Muonic atoms and molecules have numerous applications-see, e.g., papers [1] [2] [3] [4] [5] and references therein. These were the inspiration for studying Rydberg quasimolecules μZe, consisting of a nucleus of charge Z, a muon, and an electron [6] [7] [8].…”