6.1. Slowing-down in graphite

Fission neutrons are emitted with an energy spectrum ranging from ~ 10 to 0.5 MeV.

Collisions with moderator and heavy metal atoms slow down the neutrons until their energy reaches an equilibrium with the thermal motion of the medium. For light nuclei, like carbon, only elastic scattering takes place at energies of interest for nuclear reactors and the collision can be studied as a classical mechanics problem.

The collision can be considered both in the centre of mass frame of reference and in the laboratory frame of reference, the first one being only a means to calculate the data in the laboratory system. The problem of the slowing down by elastic collision is treated in most reactor physics textbooks, so that we only quote here the most significant results. Defining іjj the scattering angle in the centre of mass frame of reference, we can obtain (see, for example, ref. 1, § 7.1.1)

(6.1)

where A = atomic weight of scatterer in units of neutron mass,

v’ = neutron velocity in the laboratory frame of reference before collision, v = neutron velocity in the laboratory frame of reference after collision,

E’, E = neutron energies corresponding to the velocities v’ and v.

The neutron loses most energy when its direction of motion is reversed (cos ф = -1) and thus:

(6.2)

It is obvious that even in the case of isotropic scattering in the centre of mass system, the scattering is anisotropic in the laboratory system, except for heavy nuclei where the two systems practically coincide.

In the case of isotropic scattering in the center of mass system (which is usually the case below 1 MeV) all values of E are equally likely in the interval

aE’^E^E’

tThe present definition of a is the most commonly used in the literature. Unfortunately in the past some authors (e. g. Wigner, Weinberg and Nordheim) defined a as 4AI(A + l)2 so that a is replaced by (1 — a) in every formula.

so that the probability of having a final energy in the small interval dE about E is

when aE’ E E’ and zero when E > E’ or E < aE’.

The scattering kernel is then given by

Z,(E’^E) = Xs(E’)g(E’^E).

The average cosine of the scattering angle in the Laboratory system is given by

cos в = (6.3)

The average logarithmic energy loss per collision is given by

f-lnf-f ln^g(E’^E)dE = +T^—na. (6.4)

.C J aE’ & A OL

The average number of collisions from fission energy Ef to thermal energy Elh is

We can now compare the different moderating properties of materials used in nuclear reactors (Table 6.1).

Table 6.1 H D Be C 0 U A 1 2 9 12 16 238 a 0 0.111 0.640 0.716 0.778 0.983 і 1.000 0.725 0.209 0.158 0.120 0.00838 n (2 MeV -> 0.0253 eV) 18 25 86 114 150 2172

We see that graphite, because of its high atomic weight, is a rather poor moderator, but it has the advantage of a very low absorption cross-section, so that if we take as representative the parameter £(2s/2„), graphite is only inferior to heavy water and beryllium.

As we have seen, the average logarithmic energy loss per collision is independent of energy: this suggests the convenience of using a logarithmic energy scale, called lethargy (и), defined as follows:

и = In Щ (6.6)

where E0 is the highest energy considered, usually 10 MeV.

Considering the slowing down in an infinite homogeneous medium we can write the following balance equation:

Х,(Е)ф(Е) = S(E) + fE Xs(E’^E)4>(E’)dE’ (6.7)

which is the slowing-down equation.

The slowing-down equation is actually a simplification of the Boltzmann equation

(4.2) from which it can be obtained supposing time- and space-independence and integrating over all angles. This equation can always be numerically solved using the multi-group method. For spectrum calculations the Boltzmann equation in a more complete form is normally used (P,, B„ or diffusion approximation), without the need of supposing space independence (see Chapter 8).

Considering only slowing down in the moderator we have for elastic scattering

X,(E)<fi(E) = S(E) + 1-!— [ ‘ ХЛЕ’)ф(Е’)Щ^. (6.8)

1-а JE £J

If %S(E) = const and Xa = 0 at the energies below the fission spectrum where S(E) = 0 we obtain as a solution a 1 IE dependence of the neutron flux. This is a very important result because the scattering cross-section of most moderators is constant over a very broad energy range.