Partial radiative-recombination cross sections for excited states of hydrogen

The squares of the dipole and quadrupole matrix elements for the free-to-bound transitions of hydrogen up to bound states ‖n=20,l=19〉 are derived in closed analytic form as a function of the kinetic energy of the free electron. Coulomb wave functions are used for the free as well as the bound states and, thus, the results are good for any electron energy. Several interesting effects are found. First, the transition probabilities are maximum for recombination into specific intermediate-angular-momentum states at low energies (w<1 eV) and where the free-state angular momentum is greater than that of the bound state. Further, that specific intermediate-angular-momentum state depends on the kinetic energy of the free electron. This behavior is in contrast to the ‘‘normal’’ behavior of the transition strengths where recombination into s states is greatest and decreases with increasing angular momentum. Second, the quadrupole matrix elements vanish for certain velocities of the free electron. These ‘‘zeros’’ produce minima in the corresponding quadrupole cross sections. Finally, the calculated partial cross sections for recombination into high-angular-momentum states are greater when quadrupole transitions are included.