If we arbitrarily assume that all neutrons have the same velocity we obtain the energy-independent form of the Boltzmann equation (for simplicity we have here included the fission neutrons in the source term):

The P, approximation consists then in the assumption [see eqns. (4.19) and 4.20)],

(4.24)

(4.25)

where po = ft’ • ft.

Proceeding as in the energy-dependent case we obtain the P, equations

div J(r) + 2гф(г) = 2sod>(r) + S(r),

I grad ф{г) + 2,J(r) = 2si J(r)

introducing the average cosine of the scattering angle [see eqn. (4.17)]

(4.27)

and defining the transport cross-section as

and 2„ = 2, — 2s0 we obtain

div J(r) + Хаф(г) = S(r), з grad ф (r) + J(r)(2a + 2,r) = 0. Defining the diffusion coefficient

D = 3(2,r + 2a) the second eqn. (4.29) takes the form

J(r) = — D grad ф(г),

eqn. (4.31) is called Fick’s law of diffusion.

Substituting (4.31) into the first eqn. (4.29) we have

DV2d>(r)- Хаф(г) + S(r) = 0

which is the neutron diffusion equation.

This equation which is a simplified form of the Boltzmann equation, but can also be directly derived, represents a neutron balance between

DV2d>(r) losses due to leakage,

-Хаф(г) losses due to absorptions,

S(r) source (external or fission source).

Defining the diffusion length

(4.33)

^Ф-1?Ф + о = 0-

The energy-independent diffusion equation is also valid if ф(г, E) is separable in space and energy ф(г, E) = ф(г)(р(Е) in which case the diffusion eqn. (4.32) is valid for ф(г) (one group theory).

The diffusion length is related to the average-distance as the crow flies travelled by a neutron from source to absorption. The solution of eqn. (4.34) in the case of a point source in an infinite homogeneous medium is

ф(г) = А^ (4.35)

where r is the distance from the source and A a coefficient related to the source strength. The number of neutrons absorbed between r and r + dr is 47rr2 drl, uф(r) so that the square of the average r is

[ r2 ■ 4тгг2Хаф(г) dr

4ттг2Хаф(г) dr

Substituting (4.35) in (4.36) and performing the integrations we obtain

r = 6L2, (4.37)

L2 is known as diffusion area.