Boundary conditions at thin absorbing shells and plates. I

If two scattering and absorbing media are separated by a thin layer of strongly absorbing material, it may be desirable to know the e€ect of the layer on the thermal neutron distribu- tion. We are concerned chie¯y with spherical shells of very large radii enclosing one medium and surrounded concentrically by another. In the inner and outer media asymptotic density distributions (derivable; except for multiplying constants, from elementary di€usion theory) are attained at distances not greatly exceeding a mean free path from the shell. The purpose of this report is to derive boundary conditions to be imposed upon these solutions to determine the asymptotic distributions correctly. Instead of treating the spherical problem, we determine the boundary conditions at a thin absorbing plate of the same thickness; these may be applied to the spherical problem with an error of the order (thickness of shell)/(radius of shell). This error is investigated in the simple model of elementary di€usion theory. The main problem is treated on the basis of two di€erent approximation methods: (1) based on an expansion in powers of , the ratio of total to capture mean free path in the shell, and (2) based on an expansion in the ratio t/li, where t ˆ=thickness of shell and `i ˆtotal mean free path inside. The ®rst method implies no restriction on t/li , the second none on , provided in each case that only a small portion of incident neutrons are captured in the shell. For aluminium shells separating P-9 from graphite the two methods give almost indistinguishable results for all thicknesses of practical interest. The present report contains the analytic form of the bound- ary conditions under the assumptions of both isotropic and linear anisotropic scattering. It has only been possible up to the present to have numerical work done on the isotropic case. A second report (II) will contain numerical results in the anisotropic case, and applications to the problem of the e€ect of shells on critical sizes of P-9-metal systems with graphite re¯ec- tors. Crown Copyright