In the energy dependent case the diffusion equation cannot be simply obtained from the Pі eqns. (4.22) since the energy dependent equivalent of Fick’s law is

5grad <Mr, J5) + 2,(J5)J(r, J5)= f 2„(E’ -> E)J(r, E’) dE’

in which the flux ф(г, E) does not only depend on the current at energy E, but through the Sji terms is related to the current at all other energies E’ from which scattering into energy E is possible.

This fact can also be easily seen using the multi-group formulation. The multi-group form of the P, equations can be obtained integrating (4.22) on the energy range of each group і (see § 4.7).

div Ji(r) + 2Ііфі(г) = ^ ‘2*о. к~іфк(г)+-г — X‘ 2 п2іифк(г),

к = 1 Keff к = 1

[grad <fr(r) + 2!.,JKr) = X 2si. k~Jk(r).

Here 21 і is weighted by the flux while 2!.,- is weighted by the current. It is convenient to write these equations in matrix form.<9) We then obtain

div J(r) + 2,ф(г) = у- р2іФ(г) + 2s0Ф(г),

Keff

J(r) = — Dgrad ф(г),

where

v°	0 •	1— о
* о	v° . At 2	* о
1—— о •	• О	• 0 2°,n_

is a diagonal matrix

Xi2/i	X^n • •	• дгі 2/n
Xz^fi	^2/2 ‘ ‘	• X^IN
Xstn	A"n2/2 ‘ ‘	■ *n2/n_
2,0, 1 — 1	2,0. ^1 ‘	■ ‘ 2,0, N
2,0. 1—2	2,0.2—2 ‘	■ ‘ 2,0. N
2,0, 1—N	2,0,2—N ‘	■ ‘ 2,0. N

211 2,i, i-i	1 M Ї			1— T z w 1
1 M г	2І2 — 2, 1,2-2			2S 1, N-*2
_ 2,1.1 —N	— 2s 1,2-N			StN — 2S 1, N—N_

The second equation of (4.38a) gives a matrix form of Fick’s law. In order to have the ordinary diffusion equation D should be a diagonal matrix.

Off-diagonal terms obviously disappear for isotropic scattering, in which case 2,i = 0. In general we have in analogy to (4.16) and (4.17)

f X.,(E-*E’) dE’

&o{E) = ————————

I 2,о(E-*E’)dE’

so that

J 2,i(E -*E’) dE’ = jio(E) I Xs0(E->E’) dE’ = Д„(Е)2,(Е).

Inelastic scattering tends to be isotropic because it involves compound nucleus formation and one can assume that the neutron emitted has forgotten the direction of the incident neutron. In the case of elastic scattering its degree of anisotropy depends only on the mass of the scattering nucleus (see §6.1).

COS So — fJLо — — . .

ЗА

Without the need of going as far as assuming isotropic scattering, the energy dependent diffusion equation can be obtained supposing that

f X„(E’ -* E)J(r, E’) dE’ = f X„(E -» E’)J(r, E) dE’

in that case the term

can be written in the form

J(r, E) I Z„(E->E’) dE’ = J(r, E)Zs(E)jlo(E)

and Fick’s law in its usual form can be obtained

J(r, E) = — D(E) grad ф(г, E)

with

1

3[2.(E) + 2«,(E)]’

2«,(Е) = 2.(Я)(1-До).

The energy-dependent diffusion equation takes then the form — D(E)VV(r, E) + 2,(E)0(r, E)

= 12s0(Е’->Е)ф(г, Е’)<ІЕ’ + у-х(Е) J v(E’)Sf(E^(r, E’)dE’ + S(r, E).

(4.39)

In the same way for the multi-group formulation it is possible to diagonalize the matrix D thus obtaining the ordinary multi-group diffusion equations

— DiV20,(r) + 2«0i(r)

2 2s0л~іфк(г) + — г-Хі2,

k=1 Keff к = 1

Let us summarize the conditions under which the diffusion equation is valid:

(a) Using the Pi approximation we have neglected in the spherical harmonics expansion the terms corresponding to / > 1. This implies that one must be sufficiently far from surfaces where the angular flux distribution is greatly anisotropic (e. g. external boundaries and strong absorbers).

(b) In order to obtain the multi-group Fick’s law we have assumed

J 2.,(E’->E)J(r, E’)dE’ = Js.,(E->E’)J(r, E)dE’.

This means that the scattering collision densities are slowly varying functions of energy over the maximum allowable energy change per collision. This cannot be easily fulfilled by light moderators like hydrogen where the energy loss per collision can be very big, but is reasonable for heavier moderators like graphite or beryllium.