Extension of the Böhnel model of multiplicity counting to include space- and angular dependence

Passive methods of nuclear safeguards determine the important parameters of an unknown sample from the statistics of the number of the neutrons emitted from the item. Following Böhnel [1], the methodology of traditional multiplicity counting is based on the first three factorial moments of the number of neutrons, emitted from the sample due to one source event. These “Böhnel moments” were derived in the so-called point model, in which no spatial or angular dependence of the neutron transport is assumed. In this work the model is extended in that these factorial moments are derived in a one-speed space-dependent model, in which the position and velocity direction of the neutrons is accounted for. Coupled backward-type master equations are derived for the generating functions of the number distribution of neutrons leaving the sample by one starting neutron, and by one source event, respectively. Due to the abandoning of 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. However, quantitative solutions can be obtained by a collision-number type expansion (Neumann-series expansion) for all moments. Due to the low number of the collisions, this expansion converges rather fast. Note that the leakage multiplication of the point model, which is a function of the first collision probability, does not appear as an unknown in this model, although at the price that the geometry and material composition of the item must be assumed known. The quantitative solution enables one to compare the point model and the space-angle dependent model and hence to estimate the error of the former. Quantitative results will be given in the paper and at the conference.Reference [1] K. Böhnel, The Effect of Multiplication on the Quantitative Determination of Spontaneously Fissioning Isotopes by Neutron Correlation Analysis. Nucl. Sci. Engng90, 75 (1985)