Angular distribution of gamma rays

The theory of angular distributions of γ-rays is developed systematically, aiming at a phase consistent derivation of angular distribution formulas for gamma rays emitted in the decay of an aligned initial state. The development starts from first principles, that is, the angular distribution formulas are derived directly from perturbation theory and all quantities introduced are carefully and explicitly defined. In particular the mixing ratios are phase consistently related to reduced matrix elements of interaction multipole operators which again are well defined in phase. Hence the mixing ratios become physical quantities which can be extracted from angular distribution measurements and then compared in both magnitude and sign with the predictions of nuclear models (especially the independent particle model). Critical stages in the theoretical development at which either a choice of phase convention has to be made or transformation properties enter are emphasized.

As a first step, the transition probability for emission of gamma radiation with wave vector k and polarization ε from an initial state ¦λ > to a final state ¦μ > is calculated using time-dependent perturbation theory. This step makes no specification of the angular momentum of the initial and final states and no multipole expansion of the interaction. Particular attention is paid to the relation between emission and absorption. In the second step of the calculation the angular momentum of the initial and final states is specified, the interaction is expanded in a series of multipoles and the final angular distribution formula is derived. In order to describe emission of gamma radiation a definite and well-defined part of the interaction Hamiltonian must be expanded. This part of the Hamiltonian is determined by the order in which the initial and final states are written in transition matrix elements. The expansion yields a set of interaction multipole operators which are well defined in phase and transformation properties. There is no uncertainty in the relative phase of the electric and magnetic interaction multipole operators due to an arbitrariness in the phases of the vector potentials. These interaction multipole operators must be used to define the mixing ratios appearing in angular distribution formulas in this paper. It is shown that care must be taken when using Siegert's theorum and “effective” operators.

In the last section, reduced matrix elements of the interaction multipole operators are given explicitly for one- and two-particle states and one- and two-hole states of the independent-particle model.