An iterative method is proposed for solving the homogeneous (i.e., critical) or inhomogeneous (i.e., source) linear irtegral Boltzmann equation for general geometry. By using successive approximations, these two classes of problems are shown to be mathematically equivalent. For the homogeneous problem, constraints on the algorithm regarding the existence of eigenvalues and the initial approximation are investigated. The algorithm is as the elementary interaction causing the pinning). For higher defect densities, the agreement with the experiments on niobium is better than with the previous theory. This method of correlations seemed suitable for studying the effect of cutting-off'' the small elementary intenactions and for the replacement of the Gauss distribution function by the Poisson distribution function for the number of defects in the elementary volumes. Both these give negative results with respect to the experiments; so far wrte are therefore not able to explain quartitatively the large please delete the above 9 lines======= applied to isotropically scattering slabs and spheres, and is compared with previously published results as well as an independent extrapolation method. For the inhomogeneous problem, an improvement over the normal successive collision method via the use of a Neumann series expansion is used to allow economic parametric studies. Constraintsmore » on the algorithm and methods of efficiently terminating the infinite Neumann series are investigated. The solution via the proposed method as applied to isotropically scattering slabs and spheres is provided in a compact form for a range of multiplication factors and optical dimensions. The shape of the scalar flux distribution is expiained. Extensions of the method to more complex problems are outlined; in particular, the solution to an energy-dependent problem in general geometry is obtained, and the implications of the results are discussed. (7 figures, 2 tables) (auth)« less