Quantum defect theory. XIII. Radiative transitions

Absorption of radiation by neutral atoms and molecules, and by positive ions, gives infinite series of spectral lines converging to the first ionisation threshold, and photoionisation with infinite series of resonances converging to higher thresholds. Quantum defect theory has been used in many previous studies of such processes. The authors present a systematic treatment. Results obtained include: (i) formulae which give the dependence of the dipole moment on the effective quantum numbers in closed channels; (ii) formulae, convenient for numerical work, which involve only real quantities; (iii) a parametric formula for the case of two channels, one open and one closed. The parametric formula describes a complete Rydberg series of resonances. It involves a profile factor (x+q) 2 /(1+x 2 ), where x=tan( pi ( nu + alpha ))/tanh( pi beta ) and where, for the closed channel, nu is the effective quantum number and mu =( alpha +i beta ) is the complex quantum defect. For pi beta <<1 the resonances are narrow and one recovers the formula of Fano which has x=(E-E 0 )/ 1 / 2 Gamma where E 0 and Gamma are such that the scattering matrix has a pole at an energy E=E 0 - 1 / 2 i Gamma .