Reduced matrix elements of a tensorial product of n tensor operators in the basis of N-particle angular-momentum eigenstates are expanded into a sum of products of n 1-particle reduced matrix elements and a single (N + n)-particle recoupling coefficient. Application of the formula is illustrated by specific examples. The method leaves the coupling schemes of N-particle states undisturbed, which allows summation over intermediate states in a product of matrix elements to be made for any number of factors. A formula is given for the sum of products of two matrix elements and the extension to a greater number of matrix elements is illustrated by an example which is reduced to a form suitable for numerical evaluation. Such summations over products of matrix elements occur in the perturbation theory of configuration interaction and have hitherto been discussed in terms of effective operators. The connection of the method used here with the effective-operator approach is demonstrated.
Minima in cross sections (Cooper minima) are known to occur for all types of transitions in atoms. These minima are caused by zeros in the transition matrix elements which are the result of a zero value for the radial integral of a particular energy. These minima can occur in both the bound-bound and bound-free transition matrix elements. In this paper the approximate position (energy) of the zeros in the bound-bound quadrupole and octupole matrix elements of the point Coulomb field are presented. The positions are found to change in the same manner as those found in the continuum, even to the extent that a zero first occurring in a bound-bound matrix element will move into the corresponding bound-free matrix element.
We discuss the possibility of zeros in the nonrelativistic radiative continuum-continuum matrix element for electron-atom (inverse) bremsstrahlung. As demonstrated earlier for upward transitions from bound states, the occurrence of different signs for the free-free matrix element in limiting cases, plus the requirement of continuity, implies the existence of zeros. Using knowledge of the sign of the dipole matrix element in the soft- and hard-photon limits with one continuum electron energy held fixed, we show that zeros can occur in the
s-
p matrix element. We discuss the connection of our results to elastic scattering and to Ramsauer-Townsend minima. We consider the observability of zeros in this
(s-
p) matrix element manifested as minima in the cross sections.