Multiplicity theory beyond the point model

Passive methods of nuclear safeguards determine the important parameters of an unknown sample from the statistics of the detection of the neutrons emitted from the item. These latter are due to spontaneous fissions and ($\alpha$,n) reactions, enhanced by internal multiplication before leaking out. Based on the original work of Böhnel, 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 space-dependence is accounted for, rather a uniform first collision probability is assumed for each neutron, irrespective of the position of its birth and its velocity direction, and, more important, it is assumed to be the same for all generations in the fission chain as for the source neutrons. The purpose of the present work is to derive the same factorial moments in a one-speed space-dependent model, in which the position and direction of the neutrons is accounted for, but (similarly to the original Böhnel model), no energy dependence is assumed. The integral equations for the moments are solved numerically with a collision number expansion. It is shown that compared to the space-dependent calculations, the unfolding method using the point model underestimates the fissile mass and the underestimation increases with increasing both of fissile mass and the value of $\alpha$.