2.8 Criticality and Yield: Approximate Methods
103
ultimately be created. All νN secondary neutrons will either escape or be consumed
in causing fissions: νN = N escape + N fission . If we take the condition to have a critical
mass to be that each nucleus does in fact fission, then N of the secondary neutrons
must be consumed in causing fissions. Under this assumption we can write νN =
N escape + N, or N escape = N(ν – 1). This is the number of neutrons that we can permit
to escape. However, we know from the development in Sect. 2.1 that Eq. (2.7), N esc
= N o e
-σ nx , dictates how many must escape, when modified to represent a spherical
core in place of a linear structure of thickness x. The condition for a critical mass
then becomes demanding that the number that must escape be less than or at most
equal to the number that can be permitted to escape:
Nmust
escape
≤ (ν − 1)N .
(2.126)
The issue of modifying N esc for a spherical core is examined in detail in Sect. 6.6
and Exercise F.1, where it is shown that the average distance from any point within a
sphere or radius R to its surface is 3R/4. If we assume that fission is the only reaction
that can happen when a nucleus is struck by a fleeing neutron, then we have
N escape = ν N exp
−
3
4
σ f n R
.
(2.127)
Combining this with (2.126) and solving for the radius R gives
R crit ∼
4
3 σ f iss n
ln
ν
ν − 1
,
(2.128)
which is written as an approximation to emphasize that scattering has been neglected.
This expression gives critical radii of 10.74 and 7.14 cm for
235 U and
239 Pu,
respectively, corresponding to masses of about 97 and 24 kg. These results are closer
to the formal diffusion theory results than those derived in the preceding section,
presumably because they allow for some neutron escape. Sometimes, a simpler
approximation turns out to be a better one!
2.8.3 Estimating the Yield of the Trinity Test by Examining
the Rate of Growth of the Fireball
In this section, the order of magnitude of the explosive yield of the Trinity test is
estimated through a high-school-level conservation of energy analysis of the growth
of its fireball. This analysis is adapted from Reed (2020b), and is somewhat akin to
the treatment of bomb efficiency in Sect. 2.4. Figure 2.21 shows the Trinity fireball
at 25 ms after the explosion. A series of time-lapse images of the explosion can be
found at https://www.trinityremembered.com/photos/test/.
103
ultimately be created. All νN secondary neutrons will either escape or be consumed
in causing fissions: νN = N escape + N fission . If we take the condition to have a critical
mass to be that each nucleus does in fact fission, then N of the secondary neutrons
must be consumed in causing fissions. Under this assumption we can write νN =
N escape + N, or N escape = N(ν – 1). This is the number of neutrons that we can permit
to escape. However, we know from the development in Sect. 2.1 that Eq. (2.7), N esc
= N o e
-σ nx , dictates how many must escape, when modified to represent a spherical
core in place of a linear structure of thickness x. The condition for a critical mass
then becomes demanding that the number that must escape be less than or at most
equal to the number that can be permitted to escape:
Nmust
escape
≤ (ν − 1)N .
(2.126)
The issue of modifying N esc for a spherical core is examined in detail in Sect. 6.6
and Exercise F.1, where it is shown that the average distance from any point within a
sphere or radius R to its surface is 3R/4. If we assume that fission is the only reaction
that can happen when a nucleus is struck by a fleeing neutron, then we have
N escape = ν N exp
−
3
4
σ f n R
.
(2.127)
Combining this with (2.126) and solving for the radius R gives
R crit ∼
4
3 σ f iss n
ln
ν
ν − 1
,
(2.128)
which is written as an approximation to emphasize that scattering has been neglected.
This expression gives critical radii of 10.74 and 7.14 cm for
235 U and
239 Pu,
respectively, corresponding to masses of about 97 and 24 kg. These results are closer
to the formal diffusion theory results than those derived in the preceding section,
presumably because they allow for some neutron escape. Sometimes, a simpler
approximation turns out to be a better one!
2.8.3 Estimating the Yield of the Trinity Test by Examining
the Rate of Growth of the Fireball
In this section, the order of magnitude of the explosive yield of the Trinity test is
estimated through a high-school-level conservation of energy analysis of the growth
of its fireball. This analysis is adapted from Reed (2020b), and is somewhat akin to
the treatment of bomb efficiency in Sect. 2.4. Figure 2.21 shows the Trinity fireball
at 25 ms after the explosion. A series of time-lapse images of the explosion can be
found at https://www.trinityremembered.com/photos/test/.