172
4 Complicating Factors
were originally emitted in either material and subsequently (1) stayed in their birth
material and caused a fission, possibly after scattering several times; (2) returned to
their birth material after one or more escape/reflection trips into and out of the other
material; or (3) caused a fission in the other material, again possibly after several trips
into and out of each. The model developed here attempts to incorporate the various
possible neutron fates in a plausible way without trying to track multiple generations
of neutron creation and travel. To do so, I simplify the situation by assuming that any
neutrons which migrate from the core into the tamper remain within the tamper, and
that any fissions induced within the tamper arise only from core-escaped neutrons
and not from secondary neutrons released in fissions within the tamper. On this basis,
it is possible to set up a four-factor formula to model the ratio of
238 U to
239 Pu fissions,
238 U fissions
239 Pu fissions
∼
0.3
0.7
∼ 0.43.
(4.44)
Each factor can be estimated from straightforward energy and cross-section
considerations, as follows.
Assume that a total of F fissions occur in the plutonium core; the intent here is
that this number incorporates fissions due to neutrons arising from anywhere within
the core/tamper structure. If the number of secondary neutrons liberated per fission
is ν, then the net number of neutrons liberated will be F(ν − 1); the “–1” appears
because one neutron is consumed in causing each fission.
Some secondary neutrons will escape from the core into the surrounding tamper.
If the probability of escape is p esc , then F(ν − 1)p esc neutrons will enter the tamper.
However, only those neutrons which are more energetic than the fission threshold
of
238 U will have a chance of inducing a fission. Designate the fraction of neutrons
that are sufficiently energetic as f thresh . This will give F(ν − 1)p esc f thresh neutrons
than can potentially induce fissions in the tamper, but not all will do so because of
competing effects, notably inelastic collisions with
238 U nuclei. If the probability
that a neutron will induce a fission is p fiss , then the four-factor analog of (4.44) is
238 U fissions
239 Pu fissions
=
F(ν − 1) p esc f thresh p f iss
F
= (ν − 1) p esc f thresh p f iss .
(4.45)
Each of these factors is analyzed and estimated in the following paragraphs.
(i) Secondary neutrons. This is the most straightforward factor. For
239 Pu, ν ~ 3.2,
so (ν − 1) ~ 2.2.
(ii) Escape probability. The probability that a neutron liberated somewhere within
a sphere of radius R will reach the surface of the sphere and escape without
being consumed in a fission was analyzed in Sect. 4.2. Applying that model to
a 6.3 kg core of
239 Pu under a compression ratio of 2.5 gives p esc = (0.33, 0.48,
0.56, 0.59, 0.61, 0.62) for a maximum number of scatterings S from zero to
4 Complicating Factors
were originally emitted in either material and subsequently (1) stayed in their birth
material and caused a fission, possibly after scattering several times; (2) returned to
their birth material after one or more escape/reflection trips into and out of the other
material; or (3) caused a fission in the other material, again possibly after several trips
into and out of each. The model developed here attempts to incorporate the various
possible neutron fates in a plausible way without trying to track multiple generations
of neutron creation and travel. To do so, I simplify the situation by assuming that any
neutrons which migrate from the core into the tamper remain within the tamper, and
that any fissions induced within the tamper arise only from core-escaped neutrons
and not from secondary neutrons released in fissions within the tamper. On this basis,
it is possible to set up a four-factor formula to model the ratio of
238 U to
239 Pu fissions,
238 U fissions
239 Pu fissions
∼
0.3
0.7
∼ 0.43.
(4.44)
Each factor can be estimated from straightforward energy and cross-section
considerations, as follows.
Assume that a total of F fissions occur in the plutonium core; the intent here is
that this number incorporates fissions due to neutrons arising from anywhere within
the core/tamper structure. If the number of secondary neutrons liberated per fission
is ν, then the net number of neutrons liberated will be F(ν − 1); the “–1” appears
because one neutron is consumed in causing each fission.
Some secondary neutrons will escape from the core into the surrounding tamper.
If the probability of escape is p esc , then F(ν − 1)p esc neutrons will enter the tamper.
However, only those neutrons which are more energetic than the fission threshold
of
238 U will have a chance of inducing a fission. Designate the fraction of neutrons
that are sufficiently energetic as f thresh . This will give F(ν − 1)p esc f thresh neutrons
than can potentially induce fissions in the tamper, but not all will do so because of
competing effects, notably inelastic collisions with
238 U nuclei. If the probability
that a neutron will induce a fission is p fiss , then the four-factor analog of (4.44) is
238 U fissions
239 Pu fissions
=
F(ν − 1) p esc f thresh p f iss
F
= (ν − 1) p esc f thresh p f iss .
(4.45)
Each of these factors is analyzed and estimated in the following paragraphs.
(i) Secondary neutrons. This is the most straightforward factor. For
239 Pu, ν ~ 3.2,
so (ν − 1) ~ 2.2.
(ii) Escape probability. The probability that a neutron liberated somewhere within
a sphere of radius R will reach the surface of the sphere and escape without
being consumed in a fission was analyzed in Sect. 4.2. Applying that model to
a 6.3 kg core of
239 Pu under a compression ratio of 2.5 gives p esc = (0.33, 0.48,
0.56, 0.59, 0.61, 0.62) for a maximum number of scatterings S from zero to