A theoretical study of Cooper minima in Rydberg atoms

This theoretical study investigates the parameters that affect the location of Cooper minima in photo-excitation spectra of Rydberg atoms. The bound-bound and bound-free transitions are examined. In this study, the wave functions used are obtained from Schwinger's variational principle for a central screened Coulomb potential. These Schwinger wave functions are analytic integral representations which depend explicitly on the quantum defect µ (or the phase shift δ for continuum states). The radial dipole matrix elements are evaluated analytically and reduce to one contour integral on finite limits that is easily evaluated by standard numerical techniques. The wave functions compare favorably with Hartree-Fock wave functions, and the dipole matrix elements compare favorable with with experimental results. The energy Е_CM of the final state for which a Cooper minimum occurs is studied as a function of the difference in quantum defect Δµ = µ_ƒ – µ_i of the initial and final states. It is shown that the major determiner of E_CM is Δµ, while the absolute quantum defects μ of initial and final states are minor corrections. It is seen that the occurence of Cooper minima is qualitatively similar for all l → l + 1 transitions. Likewise, the Cooper minima of all l → l + 1 transitions are also qualitatively similar. However the trends in Cooper minima are qualitatively different for l → l + 1 transitions compared to l → l + 1 transitions. It is seen that Cooper minima that occur far from the ionization threshold are independant of of the initial state principle quantum number (i.e. initial state energy), while Cooper minima that occur close to threshold are shifted to higher energy as the initial state principle quantum number is increased. Results compare favorable with Hartree-Fock calculations and limited experimental evidence.