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Journal of Physics B Atomic Molecular and Optical Physics

The use of oscillator-strength sum rules for evaluating the angular-momentum dependence of the blackbody-radiation-induced broadening and shift of Rydberg-state energy levels

Igor L. Glukhov, Aleksandr A. Kamenski, Vitalii D Ovsiannikov

Published Aug 18, 2020• 5 citations
Publisher

Abstract

Abstract First evaluations of the blackbody-radiation (BBR)-induced shift and broadening of highly excited Rydberg states were based on the account of only the first term of the expansion for the Planck’s distribution in powers of the ratio of the BBR-photon energy to the environmental thermal energy. The corresponding asymptotic expression for the shift is proportional to a sum rule for the oscillator strengths, which is identical for all states, independently of their principal and angular-momentum quantum numbers. The broadening appeared proportional to the sum of the oscillator-strength first moments, which is dependent on only the principal quantum number, being independent of the state’s angular momentum. In this article, we demonstrate that the account of higher-order terms of the Planck’s distribution expansion provides corrections revealing the dependence of asymptotic shift and broadening on the Rydberg-state principal and angular-momentum quantum numbers. The strongest corrections are observed for states with low angular momenta, in agreement with earlier performed numerical calculations. Sum rules for higher moments of oscillator strengths are used for evaluating the corrections for the BBR-induced broadening and shift of Rydberg-state energy levels in neutral atoms and ions of arbitrary nuclear charge Z .

DOI
10.1088/1361-6455/abb032

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Abstract First evaluations of the blackbody-radiation (BBR)-induced shift and broadening of highly excited Rydberg states were based on the account of only the first term of the expansion for the Planck’s distribution in powers of the ratio of the BBR-photon energy to the environmental thermal energy. The corresponding asymptotic expression for the shift is proportional to a sum rule for the oscillator strengths, which is identical for all states, independently of their principal and angular-momentum quantum numbers. The broadening appeared proportional to the sum of the oscillator-strength first moments, which is dependent on only the principal quantum number, being independent of the state’s angular momentum. In this article, we demonstrate that the account of higher-order terms of the Planck’s distribution expansion provides corrections revealing the dependence of asymptotic shift and broadening on the Rydberg-state principal and angular-momentum quantum numbers. The strongest corrections are observed for states with low angular momenta, in agreement with earlier performed numerical calculations. Sum rules for higher moments of oscillator strengths are used for evaluating the corrections for the BBR-induced broadening and shift of Rydberg-state energy levels in neutral atoms and ions of arbitrary nuclear charge Z .
Abstract Based on the rules for summing oscillator-strength moments, we derive analytical expressions for environmentally (thermally) induced energy-level shifts and broadenings of circular Rydberg states with maximum orbital (l) and magnetic (m) quantum numbers, l = |m| = n – 1. The formulae for the energy of interaction with blackbody radiation are presented in the form of series expansion in even powers of the small parameter η = Z 2/(n 3 k B T) ≪ 1 in the region of high temperatures T and principal quantum numbers n (Z is the charge of the residual ion). The expression for the thermally induced width contains a negative temperature-independent term coinciding in absolute value with the spontaneous width, so that the temperature-independent term is absent in the expression for the sum of the spontaneous and thermally induced widths. A similar expansion in powers of η for the thermally induced shift also contains a temperature-independent term proportional to 1/n 6.
Abstract The rates of blackbody-radiation(BBR)-induced transitions from excited n S, n P, n D and n F states into bound states and into continua of the group IIa ions Ca + , Sr + , Ba + at temperatures from T = 100 K to T = 3000 K are calculated in consecutive comparison of the Fues model potential (FMP) with quantum defect method (QDM). The temperature and binding-energy dependencies are determined and analytical asymptotic approximations are proposed for evaluating numerically the contributions of the bound states and continua to the Rydberg-level widths and positions. The general properties of n -dependence at the threshold and regularities of above-threshold ionization cross sections for states with high principal quantum numbers n are used for determining the rates of BBR-induced ionization. Relations are determined between deviations from general regularities for the sum rules of oscillator- strengths and specific non-local dependence of the FMP on optical-electron angular momentum. The QDM appears preferable in comparison with the FMP approach for calculating the amplitudes of radiation transitions from Rydberg states, and specifically sums of matrix-element-dependent terms over very large (in particular, infinite) numbers of discrete states. The fractional (related to natural decay) rates of BBR-induced decay into lower bound states, excitation into upper bound states and ionization transitions into the continuum are presented asymptotically with parameters tabulated for the S, P, D and F Rydberg states.
Abstract We examine the possibilities of refining an asymptotic description and quantitative calculations of the effects induced by thermal blackbody radiation (BBR) of the environment on the Rydberg states of atoms. Numerical values are calculated and asymptotic expressions are proposed for simplified estimates of natural lifetimes and threshold photoionisation cross sections for Rydberg states of rubidium and caesium atoms with large values of the principal quantum number, n ⩾ 20, and small orbital momenta, l = 0, 1, 2, 3. Based on analytical expressions, we present numerical estimates for the contributions of photoionisation probabilities to the BBR-induced broadening of the Rydberg energy level, as well as the contributions of continuum integrals to thermally induced shifts in the Rydberg-state energy levels.
Abstract We present the derivation of simple analytical expressions for calculating the widths of Rydberg states with small orbital angular momenta in Zn+, Cd+, Hg+ ions of group IIb of the periodic system of elements induced by blackbody radiation (BBR) at ambient temperatures from 100 to 3000 K. The probabilities of radiative transitions from excited nS, nP, nD and nF states to all dipole-accessible states of ions are calculated. The wave functions of the quantum defect (QD) method for the initial and final states of the Rydberg electron are used to calculate the amplitudes of transitions between bound states. The dependences of the probabilities of induced decays and excitations on the BBR temperature, the principal and orbital quantum numbers of the Rydberg ion are determined. Analytical expressions are derived for numerical estimates of the contributions of the probabilities of thermally induced decays and excitations to the width of the Rydberg energy level. The numerical values of the coefficients of interpolation polynomials representing asymptotic expansions in powers of the principal quantum number n for the relative probabilities of decays and excitations of Rydberg states with large values of n and small orbital angular momenta l = 0, 1, 2, 3 are calculated.
We consider the Zeeman shift in Rydberg atoms induced by room-temperature blackbody radiation (BBR). BBR shifts to the Rydberg levels are dominated by the familiar BBR Stark shift. However, the BBR Stark shift and the BBR Zeeman shift exhibit different behaviors with respect to the principal quantum number of the Rydberg electron. Namely, the BBR Stark shift asymptotically approaches a constant value given by a universal expression, whereas the BBR Zeeman shift grows steeply with principal quantum number due to the diamagnetic contribution. We show that for transitions between Rydberg states, where only the differential shift between levels is of concern, the BBR Zeeman shift can surpass the BBR Stark shift. We exemplify this in the context of a proposed experiment targeting a precise determination of the Rydberg constant.
Abstract Analytical expressions for spontaneous and induced by blackbody radiation energy shifts and widths of atomic circular states are derived. The features of asymptotic behavior of the total width and energy shifts for highly excited Rydberg circular states in the blackbody radiation field are described. Approximate equations are proposed for simple evaluations of energy characteristics and an absence of temperature-independent spontaneous contribution to the total energy width in a range of high temperatures and principal quantum numbers is demonstrated.