A Directed-Graph Branching Treatment of Internal Equilibration Rates and Application to Astromers

The Gupta-Meyer treatment of nuclei with long-lived isomers computes the effective internal equilibration rate by assuming that the higher-lying nuclear levels, through which the ground and isomeric states communicate, are in steady-statel [1]. The effective rate for transition between the ensemble of states associated with the ground state and the ensemble of states associated with the isomeric state then becomes a sum over probabilities of pathways between the ground and isomeric state. These pathways include cycles, and the number of cycles grows dramatically as the temperature increases. Thus, at high temperatures, the number of pathways to sum over becomes large. We have solved for the steady state abundances of the upper-lying levels by means of branchings on a directed graph representing the nuclear levels and the transitions among them. This allows us to factor out the cycles into a factor involving a finite sum, which provides a useful interpretation of the effective transition rates in terms of essential transitions (without cycles) then modified by this factor. The kth best branching algorithm [2] then lets us focus on the most important branchings, which may be relevant for determining the most important transitions for experimental or theoretical study for effective equilibration rates of astromers. 

[1] Gupta, S. S. and Meyer, B. S. (2001) PRC, 64, 025805.

[2] Camerini, P. and Fratta, Luigi and Maffioli, F., (1980) Networks, 10, 91 - 109.