Abstract. Successive approximations ae obtained from transport theory for the neutron distribution around a black sphere surrounded by a spherical gap and a non-absorbing medium extending to infinity. Six cases are calculated numerically to show the difference between this treatment and simple diffusion theory. The extrapolation distance and a quantity that is essentially the thermal utilization are calculated in different approximations. It is found that the neutron density in the gap, instead of being a constant as predicted by the simple theory, may vary by a factor of one-third in cases of practical interest.