Let us consider the time-independent Boltzmann equation (4.4) with isotropic sources (again the fission source has been for simplicity included in S).<l0)

,ft)= I ls(E’^E, Cl’

+ 4l~S(r, E)

С1)ф(г, E’, ft’) dE’ dil’

expanding the scattering cross-section of scatterer і in Legendre polynomials

<tAE’^E, CI’^CI) = 2 s„(E’-»E)Pi(#io) (4-42)

і

where p0 = ft ■ Cl’ — cos во and defining

XAE’ -*• E) = 4тг 2 NME’ -*■ E)

і

where N, is the atomic density of scatterer i, we obtain

a • viKr, e, a) + х, ф(г, в, п) = — J — 2 f s„(E’-*• е)р,(р0)ф(г, e, ft) dE’ dcr

4тг i J

+ 7— S(r, E). (4.43)

47Г

The one-dimensional form of this equation is sufficient for all interesting applications of the Bn method.

If z is the space variable, instead of the variable ft it is sufficient to introduce the variable p = cos в where в is the polar angle between the direction ft and the z-coordinate.

Because of symmetry reasons the flux is then independent of the azimuthal angle <p and we can use the variable

1A(z, E, p) = j^ 1jt(z, E, ft) d<p

so that we have

Making use of the addition theorem of Lengendre polynomials it is possible to express Pi(po) in terms of p and p’

P,(po) = Pl(p)Pl(p’) + 2’Z,(j РГ (P)РГ (p’) cos [m(<p — <p’)]. (4.45)

m=l 1» ~T tTl )l

Carrying out the integration with respect to <p’ from 0 to 27t the second term of (4.45) disappears so that (4.44) becomes

+ ^S(z,£).

We multiply this equation by e‘Bz and integrate over all z obtaining an equation for the Fourier transform of the flux densityt

Equation (4.46) becomes

The Fourier transform of the flux density can be expanded in Legendre polynomials

Ф(В, Е, р) = їф,(В, E)P,{p) (4.49)

l

dividing both sides of the equation by

1 — iyp, where у =

multiplication by Pj(p) and integration over p, because of the orthogonality of the Legendre polynomials, gives

In the Bn approximation the expansion in Legendre polynomials of the scattering cross-section (4.42) is truncated at the nth term. The very important peculiarity of the Bn method is that after having made this truncation no truncation in the expansion of ф

tThe reader is reminded that not all texts use the same definition of a Fourier transform. The definition (4.47) implies an inverse transform:

ip(z, E, ju.) = J e, Bz ф(В, E, ці) dB. Sometimes the following definitions are used:

ф(В, E, ці) = J e IB‘ ip(z, E, ці) dz, ф(г, E, ft) = 2~ J е>Вг Ф(В, E, ці) dB.

A symmetric definition is also possible

ф(В, Е,р)=уІe~,B‘ ф(г, E, p) dz, іl/(z, E, fi)= j е‘Вг ф(В, E, ці) dB

is necessary. The precision with which the components фі of the neutron flux density are calculated is only influenced by this truncation. In general the B„ method gives more exact values for the components ф; than the P, method because neutrons emerging from collisions are more isotropic than the flux (or neutrons entering a collision). If the scattering, for example, is linear in the laboratory system the В, method is exact, and if scattering is isotropic in laboratory system, B0 is exact. The B„ approximation starts from the angular distribution that neutrons would have with isotropic scattering (in the case n = 0) and calculates deviations from this distribution. Besides the anisotropy of the scattering cross-section is a known quantity so that it is easier to assess the value of the approximation which is being made with the B„ method.

The difficulty of the B„ method consists in the fact that in order to obtain the flux it is in principle necessary to perform a Fourier inversion. The only way of avoiding this inversion is to assume separability of the space and energy dependence of the neutron flux

ф(г, Е, B) = WB, E, (l)eiBr

in this case the energy dependence of the Fourier transform is the same as the one of the flux.

While this approximation would be too crude for complete reactor calculations, it may still be quite useful in the case of spectrum calculations where it is often used to obtain within-group fluxes (see §8.1).