Spherical Harmonic Calculations for a Cylindrical Cell of Finite Height

The spherical harmonic (P₃) approximation to the Boltzmann equation is applied to the case of a finite cylinder, with symmetry about the axis of the cylinder. Solutions are obtained for the case of a neutron source proportional to cos B_zz where z is measured along the axis of the cylinder and B_z² is the axial buckling. These solutions are then expanded in terms of B_z and only terms of order B_z² or less are retained. The approximate solutions are then used to calculate the thermal utilization of a cell of finite height composed of a natural uranium rod surrounded by a D₂O moderator as a function of the axial buckling. The resultant expression for the utilization has the form where f(0) is the utilization of the cell of infinite height and the constant L² corresponds to the thermal diffusion area in two-group theory. Results are obtained for several cells and compared with those obtained using other calculational methods.