Derivation of theoretical formula for the third order neutron correlation technique by using importance function

The third order neutron correlation technique is one of the subcriticality measurement techniques. This technique was originally proposed by Furuhashi and utilizes the second and third order neutron correlation factors Y(T) and X(T), which are evaluated from the variance and third order central moment of neutron counts detected during a counting gate width T, respectively. We can obtain the absolute value of subcriticality from the Y∞ and X∞, which are saturation values of Y(T) and X(T) when T goes to infinity. In the previous paper, we derived the generalized theoretical formulas of Y∞ and X∞ that took account of both the spatial and neutron energy effects. Its derivation was based on a heuristic method in which we consider the branching process of neutron family, and we utilized the α-eigenfunction expansion technique. Then, the previous formulas are expressed by multiple sums involving both the α-eigenfunctions and their adjoint functions. In this paper, we derive the new compact expressions of Y∞ and X∞ by using the importance functions related to the neutron detection process. Present derivation is based on the stochastic equation of neutron transport. New expressions have two advantages. We can clarify the physical meanings of Y∞ and X∞. And, without α-eigenfunction expansion, we can calculate Y∞ and X∞ directly by calculating the importance functions which satisfy the adjoint neutron transport equations. Moreover, we prove that new expressions of Y∞ and X∞ are identical with the previous ones, so that we can conclude that our previous derivation based on a heuristic method is correct.