4.7.3 Attenuation of fast neutrons: neutron
moderation and diffusion
In materials containing atoms of low atomic mass,
neutrons of all energies can lose a significant fraction of their energy in a
single elastic collision and such materials are referred to as moderators. In
heavy nuclei appreciable energy loss in a collision is only possible at high
energies where inelastic scattering can occur. The neutron dose rate from a
point source of fast neutrons falls off with distance r approximately as
exp(−Σremr)/4πr2, where
Σrem depends on the medium where Σ has been defined
earlier. This macroscopic cross-section is called the removal cross-section and
since all interactions tend to remove energy from the beam its value is not too
different from the total macroscopic cross-section
(NσnT) of the material, but is slightly lower.
This exponential fall off is only approximate and holds less well for media in
which hydrogen is the principal fast neutron attenuator. In the table overleaf
the removal cross-section refers to a fission neutron source.
In the slowing-down region the average number of
collisions,
 , to slow a neutron from energy E1 to energy
E2 is equal to
ln(E1/E2)/ξ, where ξ
is the average change per collision in the logarithm of the energy. At energies
below that at which scattering becomes entirely elastic, ξ is
independent of energy and is approximately equal to 2/(A +
 ).
The spatial distribution of neutrons of energy E2 which have
slowed down from a point source of energy E1 is of the form
exp(−r2/4τ)2 where τ is referred to as the Fermi Age
and is the mean square distance a neutron migrates in slowing down from
E1 to E2. It is given by:
|
 |
where D is the diffusion coefficient and equal to
(3ΣnT − 3bΣnn)−
1 and b is the average value of cos Ψ where
Ψ is the angle of scatter of a neutron in a collision. The table
refers to the age of neutrons from a fission source slowing down to an energy
of 1.46 eV. This value, which is just above the thermal region, is appropriate
to the age determined from the measured spatial distribution of the resonance
neutrons detected by indium foils.
The root mean square distance a neutron travels from the
position where it is thermalized to the point where it is absorbed is the
thermal diffusion length, L, and is equal to
(4/π)1/4(Dth/ΣnA)1/2
where Dth is the value of the
diffusion coefficient averaged over the thermal neutron spectrum and
ΣnA is assumed to have a l/υ dependence and is
evaluated at an energy kTn where
Tn is the temperature of the medium.
Properties of moderators and shielding
materials†
(At 20 °C unless stated otherwise)
|
Material |
|
∑rem/m−1 |
ξ |
τ/(103
mm2) |
L/mm |
Dth/mm |
ts/μs |
tth/μs |
|
|
|
|
|
|
|
|
|
|
|
|
|
H2O . . . . . |
1.00 |
9.0a |
0.948 |
2.67c |
27a |
1.4a |
6 |
205 |
20 |
|
D2O
(pure) . . . |
1.10 |
9.1a |
0.570 |
11.7c |
940 |
8.4 |
53 |
~105 |
33 |
|
Diphenyl (C12H10) |
|
|
|
|
|
|
|
|
|
|
85 °C
. . . |
0.99 |
7.1b |
0.812 |
4.6d |
48 |
2.6 |
13 |
354 |
23 |
|
Paraffin Wax |
|
|
|
|
|
|
|
|
|
|
(C30H62) .
. |
0.89 |
10.9b |
0.913 |
1.8 |
21 |
1.1 |
7 |
160 |
21 |
|
Be . . . . . |
1.85 |
13.0a |
0.209 |
7.32e |
208a |
5.0a |
50 |
3 460 |
90 |
|
BeO . . . . . |
3.00 |
14.3b |
0.173 |
9.38e |
290a |
5.0a |
102 |
7 000 |
109 |
|
Graphite . . . . |
1.67 |
8.1a |
0.158 |
29.8c |
520a |
8.5a |
140 |
13 000 |
119 |
|
Concrete‡ . . . |
2.3 |
8.8b |
0.55 |
10.0 |
77 |
6.0 |
30 |
400 |
30 |
|
Al . . . . . . |
2.70 |
7.9a |
0.072 |
430 |
200 |
55 |
900 |
8 800 |
262 |
|
Fe . . . . . . |
7.86 |
16.8a |
0.035 |
33.0 |
12.7 |
3.4 |
360 |
19 |
540 |
| Pb . . . . . . |
11.35 |
11.6a |
0.0096 |
600 |
121.8 |
9.2 |
2720 |
640 |
1960 |
| Bi . . . . . . |
9.75 |
9.8a |
0.0095 |
800 |
320 |
11.2 |
3000 |
3
660 |
1990 |
| U . . . . . . |
18.9 |
1.7a |
0.0084 |
§ |
13.7 |
7.0 |
2040 |
11 |
2250 |
|
|
|
|
|
|
|
|
|
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† The data in this table are obtained from
old sources as they are not needed in modern calculations though they remain
valuable in compilations like this: a, experimental value; b, derived from
components; c, UK Nuclear Data File (see for example Report AEEW-M 1208); d,
taken from ANL 5800; e, ENDF/B data (see BNL 50274). Values not marked are
obtained from approximate formulae.
‡ Composition in 103
kg m−3: H, 0.023; O, 1.22; C, 0.0023; Mg, 0.005; Al, 0.078;
Si, 0.775; K, 0.03; Ca, 0.1; Fe, 0.03; Na, 0.037.
§ Because of its large absorption resonance
integral and its small value of ξ, almost no neutrons slowing down in
uranium reach thermal energies.
The slowing down time, ts, is the time
taken for neutrons to slow down from an energy E to the thermal energy
E0 and is independent of E when E »
E0. The mean thermal neutron lifetime, tth,
refers to thermal diffusion before the neutron is absorbed in the medium. In
the table values of ts and tth have been
obtained from the simple formulae
 and  where the averages are taken
over the slowing-down and thermal regions respectively.
M.G. Sowerby / R.A. Forrest
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