Positions of zeros in the multipole matrix elements of the point Coulomb field.

Abstract

Minima in the photoionization cross sections (Cooper minima) are known to occur for all types of transitions in atoms, even for dipole transitions from the ground state. These minima are caused by zeros in the transition matrix elements which are, in turn, the result of a zero value for the radial integral of a particular energy. For transitions involving the point Coulomb field there are no zeros in the dipole matrix elements, but there are zeros in higher multipole transitions. The zeros in the quadrupole matrix elements involve transitions between low-angular-momentum bound states and free states with two more units of angular momentum. Zeros in octupole matrix elements fall into two categories. The first involves transitions between low-angular-momentum bound states and free states with three more units of angular momentum and the second involves transitions between all s and p states. In this paper the position (energy) of the zeros in the multipole matrix elements of the point Coulomb field are examined. Why the dipole matrix elements of hydrogen have no zeros and what causes the zeros in the higher-multipole matrix elements are then discussed.