The method of partial distributions is used to calculate the population distribution of three level maser systems in which spontaneous emission and cross relaxation as well as thermal relaxation and induced absorption processes are included. The results obtained are in agreement with the solution of the relevant rate equations. The method has the advantage of speed and suggestiveness. Two examples are presented: (1) A three level optical maser is analyzed in which one spontaneous emission transition probability is included. The solution in this example does not make the linear approximation to the Boltzmann factor. (2) A three level microwave maser is analyzed in which one cross relaxation mechanism is included. The solution to this problem by the method of partial distributions does require use of the linear approximation to the Boltzmann factor.
W. A. Barker and J. D. Keating, "Optical and Cross Relaxation Maser Level Populations by Partial Distribution," Appl. Opt. 1, 335-338 (1962)