Exact solutions of models for continuous and pulsed irradiation and implications for stability and fluctuations

Abstract

The behaviour of materials during irradiation is determined by the point defect concentrations which develop and the manner in which they evolve in time. The controlling factors are the production, recombination and loss to sinks and these are usually modelled with the help of first order (“bimolecular”) non-linear equations. In this paper we present exact solutions which are obtained by transforming them to a Riccati equation. Five cases have been considered and the corresponding models are pertinent to different irradiation conditions which exist at very low or high temperatures or arise due to self ion irradiation etc. The Lyapounov stability of the solutions are discussed together with the phase diagram plots. A general formalism has been given for the case of pulsed irradiation based on the structure of solution of the Riccati equation. The solution enables us to calculate the instantaneous and average value of the concentrations during any pulse cycle. The formalism also enables us to explicitly identify two time scales corresponding to the irradiation and annealing pulses from which the behaviour of the system can be obtained for a sequence of pulses. The formalism is applied to one of the models as an example. The paper concludes with a few general comments on the implication of stability due to diffusion and fluctuations. In particular it is shown that one of the models applicable to situations at very low temperature where recombination dominates and loss to sinks is unimportant the fluctuations will drive the mean square of the concentrations linearly in time.