122
3 Producing Fissile Material
The probability of our neutron striking a nucleus of
235 U is given by the ratio of
the total cross-section for
235 U to that for the total cross-section for all processes and
isotopes, weighted by the abundance fraction of that isotope:
Fσ 5
σ total
. Once
the neutron has struck the
235 U nucleus, the probability that it initiates a fission will
be
σ f 5
σ 5
. Hence, by the usual multiplicative process for combining independent
probabilities, the overall probability that the neutron will fission a
235 U nucleus
will be the product of these factors,
Fσ 5
σ total
σ f 5
σ 5
=
Fσ f 5
σ total
. If
fission of a
235 U nucleus liberates on average ν 5 secondary neutrons, then the average
number of neutrons created by one neutron from fissioning a
235 U nucleus will be
ν 5
Fσ f 5
σ total
. Likewise, fissions of
238 U nuclei will give rise, on average, to
ν 8 (1 − F)
σ f 8
σ total
secondary neutrons. The total number of secondary neutrons
created by one initial neutron will then be
k =
F ν 5 σ f 5 + (1 − F)ν 8 σ f 8
σ total
.
(3.4)
In natural uranium, F = 0.0072; we ignore here the very small natural abundance
of
234 U.
We first apply (3.4) to unmoderated (fast) neutrons and natural uranium:
σ total = F
σ f 5 + σ c5
+ (1 − F)
σ f 8 + σ c8
= (0.0072) (1.32407) + (0.9928) (2.9694) = 2.958 bn.
(3.5)
Hence
k f ast =
F ν 5 σ f 5 + (1 − F)ν 8 σ f 8
σ total
=
(0.0072) (2.637) (1.235) + (0.9928) (2.655) (0.3084)
2.958
= 0.283.
(3.6)
Since k fast < 1, a self -sustaining chain reaction using unmoderated neutrons with
natural uranium is impossible. This is why a lump of ordinary uranium of any size
is perfectly safe against a spontaneous chain reaction; a nuclear weapon cannot be
constructed using uranium of natural isotopic composition.
Figure 3.2 shows k fast as a function of F; k fast does not exceed unity until
F ~ 0.53. One must undertake a significant enrichment effort to construct a
uranium bomb. Bomb-grade uranium is usually considered to be 90%
235 U (k ~
2.02).
In the case of moderated neutrons, the story is very different. Here we have
σ total = (0.0072) (584.4 + 98.81) + (0.9928) (0 + 2.717) = 7.617 bn. (3.7)
3 Producing Fissile Material
The probability of our neutron striking a nucleus of
235 U is given by the ratio of
the total cross-section for
235 U to that for the total cross-section for all processes and
isotopes, weighted by the abundance fraction of that isotope:
Fσ 5
σ total
. Once
the neutron has struck the
235 U nucleus, the probability that it initiates a fission will
be
σ f 5
σ 5
. Hence, by the usual multiplicative process for combining independent
probabilities, the overall probability that the neutron will fission a
235 U nucleus
will be the product of these factors,
Fσ 5
σ total
σ f 5
σ 5
=
Fσ f 5
σ total
. If
fission of a
235 U nucleus liberates on average ν 5 secondary neutrons, then the average
number of neutrons created by one neutron from fissioning a
235 U nucleus will be
ν 5
Fσ f 5
σ total
. Likewise, fissions of
238 U nuclei will give rise, on average, to
ν 8 (1 − F)
σ f 8
σ total
secondary neutrons. The total number of secondary neutrons
created by one initial neutron will then be
k =
F ν 5 σ f 5 + (1 − F)ν 8 σ f 8
σ total
.
(3.4)
In natural uranium, F = 0.0072; we ignore here the very small natural abundance
of
234 U.
We first apply (3.4) to unmoderated (fast) neutrons and natural uranium:
σ total = F
σ f 5 + σ c5
+ (1 − F)
σ f 8 + σ c8
= (0.0072) (1.32407) + (0.9928) (2.9694) = 2.958 bn.
(3.5)
Hence
k f ast =
F ν 5 σ f 5 + (1 − F)ν 8 σ f 8
σ total
=
(0.0072) (2.637) (1.235) + (0.9928) (2.655) (0.3084)
2.958
= 0.283.
(3.6)
Since k fast < 1, a self -sustaining chain reaction using unmoderated neutrons with
natural uranium is impossible. This is why a lump of ordinary uranium of any size
is perfectly safe against a spontaneous chain reaction; a nuclear weapon cannot be
constructed using uranium of natural isotopic composition.
Figure 3.2 shows k fast as a function of F; k fast does not exceed unity until
F ~ 0.53. One must undertake a significant enrichment effort to construct a
uranium bomb. Bomb-grade uranium is usually considered to be 90%
235 U (k ~
2.02).
In the case of moderated neutrons, the story is very different. Here we have
σ total = (0.0072) (584.4 + 98.81) + (0.9928) (0 + 2.717) = 7.617 bn. (3.7)