Spectra of a Rydberg Atom in Crossed Electric and Magnetic Fields

Abstract

Contemporary spectroscopic studies of astrophysical and laboratory plasmas frequently deal with extremely large values of principle quantum numbers of atomic systems. These atomic states are very sensitive to electric and magnetic fields of the surrounding medium. While interpreting the spectra of such excited atomic systems, one faces the problem of a huge array of radiative transitions between highly excited atomic levels. Moreover, external electric and magnetic fields significantly complicate the problem because of the absence of standard selection rules typical for the spherical quantization. The analytical expression in the parabolic representation for dipole matrix elements obtained by Gordon contains hyper-geometric series and it has a very complex structure. The matrix elements that involve the presence of electric and magnetic fields are calculated while using a representation closely related to the parabolic quantization on two different axes. This matrix element depends in a complex way on the transition probabilities in the parabolic coordinate system (Gordon’s formulas) and the Wigner d-functions. This circumstance leads to even greater computational difficulties. A method of simplification of these complicated expressions for transition probabilities is demonstrated. The semiclassical approximation for coordinate matrix elements (Gulayev) and recurrence properties of the Wigner d-functions are used. The Hnβ line is under consideration. Specific calculations for the transition 10–8 in the case of parallel and perpendicular fields are presented.

The Coulomb Symmetry and a Universal Representation of Rydberg Spectral Line Shapes in Magnetized Plasmas

“…This leads to the reduction of the expression (11) to double sums. Moreover, as shown in [20,21], these sums are independent.…”

“…Calculation of the transition probabilities in this basis leads one to Formula (11). Using the symmetry properties of the Coulomb field, the Wigner d-functions recurrence relations, and the Rydberg asymptotic formulas for the coordinate matrix elements [9,10], the simple semiclassical approximations for the dipole matrix elements in representation of states (8) are obtained in [20,21] and applied to the spectral line shape calculations. The approximations for the transitions probabilities in this basis are presented in Appendix A (see (A11)-(A16)).…”

“…The detailed derivation of the semiclassical approximation of the dipole matrix elements that is the representation of states (8) is presented in [20,21]. In the Appendix, we briefly discuss the main points of calculating the transition probabilities and present the results in [20,21]. Firstly, it is necessary to consider new quantum number K…”

“…These results and the usage of the specific d-functions properties allow one to simplify the complicated expression (11). The results for the H nα (∆n = n −n = 1) and H nβ (∆n = n −n = 2) series are obtained in [20,21] .The basics of the approach to calculating these matrix elements and the results (see Formulas (A11)-(A16)) are presented in the Appendix A.…”

Review of Rydberg Spectral Line Formation in Plasmas

“…The Gulayev approach [10,11], discussed above, plays a crucial role in this simplification. The details of the derivation of the simplified dipole matrix elements are presented in [24,25]. For the H nα lines:…”