Developments In The Calculation Of The Multiplicity Moments In Space-dependent (non-point) Models

At the previous INMM we reported on the space-angle dependent theory of the emission of neutrons from a sample assuming the Böhnel superfission model, and proposed a numerical procedure to calculate the first three factorial moments of the number of the neutrons emitted from a spherical item [1]. Because of the extension of the theory beyond the point model, these are now integral transport equations and not algebraic equations. From these, equations can be derived for the factorial moments of the number of neutrons emitted from the sample by one source event. Unlike in the point model where these equations are algebraic and always linear in the highest order moment, the equations here are transport equations that do not have analytical solutions. In our previous work, we obtained a quantitative solution by solving the inhomogeneous equations directly via a collision-number type expansion (Neumann-series expansion). This method requires the full series expansion method to be applied for all moments separately. The present development is based on the recognition that the homogeneous part of all the moment equations is the same, only the inhomogeneous parts differ. Hence, it is worth to obtain the solutions through a Green’s function technique. Introduction of the Green’s function in the moment equations requires non-conventional methods, and its calculation is more complicated than that of the individual moments, because its dimensionality twice as much in phase space. However, with this method, a Neumann-series expansion is necessary only once, to calculate the Green’s function; once it is available, all order moments, even beyond the third, can be calculated by simple quadrature. Because of this property, solutions are possible to obtain in geometries more complicated than the simple sphere, treated so far; also cylindrical or cubic samples can also be calculated. In the talk both the methodology, as well as concrete quantitative results will be given. These will be compared to the traditional results obtained in the point model, to assess the bias of the latter due to its simplifications, and to assess the role of the geometry in the deviations from the point model.