Sum of oscillator-strength moments and asymptotic behaviour of thermally induced broadening and shift of energy levels of Rydberg atom circular states

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 = Z²/(n³k_BT ) ≪ 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/n⁶.