The solution of the time-independent multi-group neutron transport equation using spherical harmonics

Abstract

The time-independent multi-group neutron transport equation is solved by expressing the neutron flux for each energy group in the form Φ(R,Ω)= ∑ l=0 N ∑ m=0 l (2l+1)P m l( cosθ)(ψ lmr) cos(mφ)+γ lm(r sin(mθ)) where ф( r, Ω) is the flux at r in direction Ω. The moments ψ lm ( r) and y lm ( r) are functions of the position r and P m l (cos θ) is the associated Legendre polynomial of order lm. θ and ∅ are the spherical co-ordinates of Ω i.e. θ is the angle to the z-axis and ∅ the angle the projection of Ω on to the x− y plane makes with the x-axis. N is the order of the approximation. The resulting moment equations obtained after equating coefficients of P m l (cos θ) cos( m∅) and P m l (cos θ) sin ( m∅) are transformed, by eliminating ψ lm ( r) and y lm ( r) with odd l, to linked second order differential equations for those with even l. These are solved by the familiar algorithms of diffusion theory.