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Bulletin of the American Physical Society

A systematic analysis of cancellations in atomic-multipole transitions

Reinhard K. Kremer, Constantine E. Theodosiou

Published Dec 1, 1992• 0 citations
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Abstract

The authors have carried out exhaustive calculations of monopole, dipole, and quadrupole transition radial matrix elements as functions of the effective principal quantum numbers of the initial and final electronic state. The general features of the results of these calculations have been displayed in three-dimensional graphs for the general cases. The locations in (n*,n{prime}*)-space of the zeroes of these matrix elements have been found and appear to follow a generic behavior. This behavior has been fitted to a simple analytic form and the parameters of the fits are given for monopole, dipole and quadrupole transitions involving l = 0, 1, 2, and 3 states. These results are expected to be of great value to experimentalists working with atomic Rydberg states, especially of alkali-like systems.

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The authors have carried out exhaustive calculations of monopole, dipole, and quadrupole transition radial matrix elements as functions of the effective principal quantum numbers of the initial and final electronic state. The general features of the results of these calculations have been displayed in three-dimensional graphs for the general cases. The locations in (n*,n{prime}*)-space of the zeroes of these matrix elements have been found and appear to follow a generic behavior. This behavior has been fitted to a simple analytic form and the parameters of the fits are given for monopole, dipole and quadrupole transitions involving l = 0, 1, 2, and 3 states. These results are expected to be of great value to experimentalists working with atomic Rydberg states, especially of alkali-like systems.
The radial electric dipole moments of bound-bound, bound-free, and free-free transitions of high-lying (n>15)(n>15)(n>15) Rydberg states and low-lying continuum states $(\ensuremath{\epsilon}<400$ ${\mathrm{cm}}^{\ensuremath{-}1})$ are calculated for hydrogen and the alkali-metal atoms. All dipole matrix elements for the transitions between low-lll (s(s(s, ppp, ddd, and f)f)f) states of H, Li, and Rb are presented. Results of similar calculations for the other alkali-metal atoms (Na, K, and Cs) are summarized. These numerical results, using quantum defect theory, are compared with semiclassical approximations. The dipole matrix elements cross zero at certain combinations of initial and final states. The matrix elements at these so-called Cooper minima in Li, K, Rb, and Cs are presented in detail.
Microwave electric-field measurements with Rydberg atoms require accurate Rydberg-transition electric-dipole matrix elements. We calculate these matrix elements for Rb and Cs using a Dirac-Fock plus core-polarization method, which core-valence exchange is treated explicitly and core-valence correlation is represented by a cutoff core-polarization potential. A smooth cutoff-radius model fitted to quantum-defect energies reproduces the target energies with MHz-level residuals for states up to n=90n=90n=90. Comparisons with model-potential matrix elements and internal length-velocity consistency show that the sss-ppp transitions are generally the most robust, whereas the ppp-ddd and ddd-fff transitions show stronger atom- and branch-dependent sensitivity to short-range modeling. The largest relative differences arise from radial-integral cancellation, where short-range phase changes are amplified by the small final matrix element. These results provide a practical reliability assessment for Rydberg-transition electric-dipole matrix elements used in field-sensing applications.
Using quantum-defect theory wave functions, we have computed dipole radial matrix elements involving Rydberg states of the singly-ionized alkaline-earth atoms. In order to test the results, we have calculated a set of transition probabilities, absorption oscillator strengths and radiative lifetimes for S, P, D, and F states up to n ∼30. Transition probabilities are in good agreement with available data. We report the lifetimes in terms of scaling relations of type τnl=τ0(n*)α. The variation of oscillator strengths with respect to the effective principal quantum number n* is discussed.
Rates of depopulation by blackbody radiation (BBR) and effective lifetimes of alkali-metal nSnSnS, nPnPnP, and nDnDnD Rydberg states have been calculated in a wide range of principal quantum numbers $n\ensuremath{\le}80$ at the ambient temperatures of 77, 300, and 600 K. Quasiclassical formulas were used to calculate the radial matrix elements of the dipole transitions from Rydberg states. Good agreement of our numerical results with the available theoretical and experimental data has been found. We have also obtained simple analytical formulas for estimates of effective lifetimes and BBR-induced depopulation rates, which well agree with the numerical data.
As shown by Pauli, [Z. Phys. 36, 336 (1926)], the electric dipole operator r\mathbf{r}r can be replaced by the Runge-Lenz vector A\mathbf{A}A when operating within the n2{n}^{2}n2 degenerate manifold of hydrogenic states of principal quantum number nnn. We seek to develop similar rules for higher multipole operators by expressing equivalent operators in terms only of the two vector constants of motion---the orbital angular momentum L\mathbf{L}L and the Runge-Lenz vector A\mathbf{A}A---appropriate to the degenerate hydrogenic shell. Equivalence of two operators means here that they yield identical matrix elements within a subspace of Hilbert space that corresponds to fixed nnn. Such equivalent-operator techniques permit direct algebraic calculation of perturbations of Rydberg atoms by external fields and often exact analytical results for transition probabilities. Explicit expressions for equivalent quadrupole and octupole operators are derived, examples are provided, and general aspects of the problem are discussed.
Minima in cross sections (Cooper minima) are known to occur for all types of transitions in atoms. These minima are caused by zeros in the transition matrix elements which are the result of a zero value for the radial integral of a particular energy. These minima can occur in both the bound-bound and bound-free transition matrix elements. In this paper the approximate position (energy) of the zeros in the bound-bound quadrupole and octupole matrix elements of the point Coulomb field are presented. The positions are found to change in the same manner as those found in the continuum, even to the extent that a zero first occurring in a bound-bound matrix element will move into the corresponding bound-free matrix element.