On the determination of radial matrix elements for high-ntransitions in hydrogenic atoms and ions

Abstract

With the aid of the factorization method developed by Schrödinger, and Infeld and Hull, we use ladder operators to derive a set of recurrence relations from which electric multipole transition (radial) matrix elements of the type ⟨n′ℓ′|rk|nℓ⟩ in hydrogenic atoms and ions can be obtained with the knowledge of a few particular integrals. These ‘initial values’ for the recurrence relations are very simply derived by elementary algebra, with the aid of the properties of the associated Laguerre function. As a check of these recurrence relations, special cases of transitions with n = n′ are considered, and thereby some apparently new matrix element relations are found. This method of computation, first proposed by Infeld and Hull, is efficient, highly accurate also for very large values of the principal quantum number, and generates no numerical instability as n and n′ both grow large. As illustrations of the method, electric dipole (k = 1, Δℓ = ±1) oscillator strengths and line strengths can readily be derived for principal quantum numbers of several hundred, and, in addition, electric quadrupole (k = 2, Δℓ = 0, ± 2) and electric octupole (k = 3, Δℓ = ±1) matrix elements are derived and tabulated. While these results are ‘exact’ for non-relativistic, hydrogenic atoms and ions, they clearly provide useful approximate values for Rydberg states of any atomic radiator. As an illustration of the potential usefulness of these results, we refer to Griem's formulation of the Stark broadening for transitions between states of high principal quantum number in hydrogen (Rydberg–Rydberg transitions), as recently generalized by Watson, for application to radio astronomical spectra from the H II regions. The present (recurrence relation) method, which is also compared to some similar methods in the literature, is shown to be very convenient, in that customary simplifications of the atomic matrix elements for large n and n′ can be entirely avoided, with very little price of additional complexity in the treatment. The present results are therefore available to assist in any computations involving Rydberg–Rydberg transitions, e.g. the re-evaluation of Stark broadening theory for such highly excited states, should new astronomical data continue to show the need for such work.

Exact expression of the impact broadening operator for hydrogen Stark broadening

“…The study of recombination transitions requires the evaluation of matrix elements of operator R between states with high principal quantum number n. This is impossible when n 50 (see Hey 2006 for a numerical solution of this problem).…”

“…Now we use the recurrence relations for the radial integrals obtained by Infeld & Hull (1951) (see also Hey (2006)), that are shown here:…”

“…To our knowledge, these discrepancies remain and have even been qualified as "mysterious" (Griem 2005;Watson 2006) 1 . In this context, approximate expressions are required to calculate transition probabilities between states with very high principal quantum numbers (n > 100) (Watson 2006;Hey 2006). This need rises because, although the analytical expressions of those transitions probabilites are known (Gordon 1929;Bethe & Salpeter 1957), they are not computationally manageable due to the large value of the factorials involved.…”

“…This need rises because, although the analytical expressions of those transitions probabilites are known (Gordon 1929;Bethe & Salpeter 1957), they are not computationally manageable due to the large value of the factorials involved. In recent works (Watson 2006;Hey 2006) recurrence relations have been used to obtain transition probabilities that are much easier to evaluate. The investigation can be carried out in this way for high values of n; although, as recognised by one of the authors (Watson 2006), the mystery of the discrepancy with the experiments cannot be solved with those approximate expressions.…”

Ultrafast effective multilevel atom method for primordial hydrogen recombination

“…(2) using an 11-point Newton-Cotes method, where the radial matrix elements g(n, l, κ, l ′ ) were obtained using the recursion relation given by Burgess [32]. Einstein A-coefficients were computed by using the recursion relations obtained by Hey [40] for the radial matrix elements R nl n ′ l ′ . Finally, we obtained the probabilities P i K using a sparse matrix technique similar to that of Ref.…”