Theory of Multiple Scattering of Slow Neutron

Abstract

The formula for the cross section for the multiple scattering of slow neutron derived in a previous paper is improved so as to satisfy the condition of detailed balance. The cross section is expressed in terms of the time-dependent propagators of the neutron and the many­ particle space-time correlation functions for the target system. The well-defined prescription for the classical approximation is presented in the last section. In a previous paperl) (referred to as 1), a formula for multiple scattering of a slow neutron has been derived. The formula is written in terms of the time-independent propagators of the neutron and the many-particle space-time correlation functions for the target system, and is a direct extension of the Van Hove formula 2 ) for the ease of single scattering to that of multiple scattering. In the derivation of the formula, however, use has been made of the assumption that the recoil effect of the motions of the target atoms to the intermediate neutron can be discarded. On account of this crude approximation, the cross section in I violates the condition of detailed balance. The purpose of the present paper is to present an improved formula for the cross section which remedies the defect mentioned above. In the present formulation, the recoil effect of the motions of the target atoms on the inter­ mediate neutron is taken into account. By virtue of the improvement of ap­ proximation, the new formula for the cross section proposed in the present paper does satisfy the condition of detailed balance, and the separation of the variables relating to the target system and the incident neutron 'which has been a merit of the Van Hove formalism is still retained. The formula for the cross section is written in terms of the time-dependent propagators of the neutron and the many-particle space-time correlation functions. The time-dependence of the neu­ tron propagator reflects the motions of the target atoms during the time that