Analytical results for the motion of a Rydberg electron around a polar molecule: effects of a magnetic field of the arbitrary strength

Abstract

We study effects of a magnetic field on the classical bound motion of a highly excited (Rydberg) electron around a polar molecule, the latter being treated as the point-like electric dipole. We consider the magnetic field B to be parallel or antiparallel to the electric dipole, so that the system has the axial symmetry. (The ${B} = 0$ case was studied analytically in one of our previous papers.) We obtain analytical results for the arbitrary strength of the magnetic field. We show that the presence of the magnetic field opens up new ranges of the bound oscillatory-precessional motion of the Rydberg electron, the oscillations being in the meridional direction ($\theta$-direction) and the precession being along parallels of latitude ($\varphi$-direction). In particular, it turns out that in one of the new ranges of the motion, the period of the $\theta$-oscillations has the non-monotonic dependence on primary parameter of the system. This is a counterintuitive result.

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Appendix: Formulas from paper [8] referred to in the main text

$$\begin{aligned} \begin{array}{lr} x_{{2}}(K) = \displaystyle (-1)^{{2/3}}2^{{1/3}}/[(729K^{{2}} - 108)^{{1/2}} - 27K]^{{1/3}} \\ \quad - (-1)^{{1/3}}[(729K^{{2}} - 108)^{{1/2}} - 27K]^{{1/3}}/(2^{{1/3}}3). \quad (10) \\ x_{{3}}(K) =\displaystyle 2^{{1/3}}/[(729K^{{2}} - 108)^{{1/2}} - 27K]^{{1/3}} \\ \quad - [(729K^{{2}} - 108)^{{1/2}} - 27K]^{{1/3}}/(2^{{1/3}}3).\qquad \qquad (11) \\ \tau _{{0}}= 4\tau (1) = 10.488.\qquad \qquad \qquad \qquad \qquad \qquad \quad (20)\\ T = 10.488 [\text {mr}^{{4}}/\text {(2eD})]^{{1/2}}. \qquad \qquad \qquad \qquad \quad \qquad (21) \\ \tau (x, K) = \displaystyle \pm \mathop \int \limits _{\mathrm{x}_{{2}}(K)}^{\mathrm{x}} dz/(- z^{{3}} \quad + z - K)^{{1/2}}, \qquad \qquad (22)\\ T_{{\uptheta }}(K) = \displaystyle 2 \mathop \int \limits _{\mathrm{x}_{{{2}}({K})}}^{\mathrm{x}_{{3}}({K}}) dz/(- z^{{3}} + z - K)^{{1/2}}. \qquad \quad \qquad (26)\\ \varphi (K, x) = \displaystyle K^{{1/2}} \mathop \int \limits _{\mathrm{x}_{{2}}(K)}^\mathrm{x} dz/[(1 - z^{{2}})[(- z^{{3}} + z - K)^{{1/2}}].\\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (29) \end{array} \end{aligned}$$