An equivalence relation for a doubly heterogeneous lattice

This paper considers a lattice consisting of an infinite uniform distribution of clusters in an external moderator and a cluster consisting of a uniform distribution of absorber lumps in an internal moderator. The lattice is characterized by these parameters: cluster mean chord length; probability of neutrons leaving a cluster to collide in the external moderator prior to crossing a cluster; adjustable Bell factor for the clusters; lump mean chord length; probability of neutrons leaving a lump to collide in the internal moderator prior to crossing a lump; adjustable Bell factor for the lumps; internal moderator volume fraction in the cluster; internal moderator macroscopic cross section. The flux in (or resonance integral of) the absorber lump is equivalent to the flux in an infinite medium consisting of the lump material homogeneously mixed with a moderator of a calculated cross section which is dependent on a factor termed the ''double heterogeneity'' factor abbreviated ''doublet''. In the limit of an infinite single cluster, this factor approaches 1, yielding the correct single heterogeneity expression for the moderator cross section. This expression is quite general; the only restriction it places on the lattice structure is that a cluster contain many lumps