Solution of multi-dimensional neutron transport equation of the spherical harmonics method using the finite Fourier transformation and quadrature formula

A solution of the neutron transport equation in multi-dimensions of the spherical harmonics method is given by applying the finite Fourier transformation method. A quadrature formula of the form of the Gauss-Legendre one is derived using the associated Legendre functions. Using this quadrature formula for the integration over the polar angle and the finite Fourier transformation, equations for the spherical harmonics components of the angular flux at the region boundary of the constant cross sections are derived which are simpler than the previous ones. It is found that the characteristic roots of the differential equations which are satisfied by the spherical harmonics components of higher order are simply given by the roots of the associated Legendre functions.