230
6 Appendices
to determine the bare threshold (spherical) critical radius and mass of this
material. What are the values of α, Δr, efficiency, and yield for a core of C
= 3 critical masses of this material if E f = 185 MeV and if the secondary
neutrons have E = 2 MeV? Take γ = 1/3. If the initial number of neutrons is
taken to be one, what are the fission and criticality-shutdown timescales? (Do
not worry about any refinements to the efficiency calculation as discussed at
the end of Sect. 2.6.) If you have a FORTRAN compiler available, obtain the
numerical-simulation program of Sect. 2.6 and apply it to the same situation;
what yield does the program predict?
2.9 What outer radius and mass of tungsten-carbide tamper will render just
threshold critical a 7-kg spherical core of uncompressed
239 Pu? Hint: Fig. 2.10.
2.10 Consider a mass m of a pure fissile material whose normal density is ρ o , with
m being less than the bare threshold critical mass for the material. Show that
this mass can be made critical by compressing it to a radius given by
r compress ≤
3m
4 π ρ o R o
,
where R o is the threshold critical radius at normal density. Show further that
if C is the number of threshold critical masses represented by m (C < 1), then
the ratio of the density at this compressed radius to the initial density is given
by
ρ compress
ρ o
=
1
C
.
Evaluate numerically for 100 grams of
235 U.
2.11 The diffusion equation for neutrons in a bomb core, (2.18), can be applied in
any coordinate system, provided that the expression for ∇
2 N in that system
is used. To this end, consider a cubical core that extends from 0 < x < L, 0 <
y < L, and 0 < z < L. Solve the diffusion equation in Cartesian coordinates.
Show that if the simplified boundary condition N(L C ) = 0 is used, then the
side length for threshold criticality is given by
L C =
π
√ ν − 1
λ f λ t ,
where the symbols have the same meanings as in Sect. 2.2. Compare this result
to that in Sect. 2.8.1 to show that the critical mass for a cubical bomb core
of a given material is 3
5/2
(4π ) ∼ 1.24 times that of a spherical core of the
same material with this boundary condition. If you are familiar with quantum
mechanics, the solution to this problem is very similar to that of a particle in
a three-dimensional infinite potential box.
6 Appendices
to determine the bare threshold (spherical) critical radius and mass of this
material. What are the values of α, Δr, efficiency, and yield for a core of C
= 3 critical masses of this material if E f = 185 MeV and if the secondary
neutrons have E = 2 MeV? Take γ = 1/3. If the initial number of neutrons is
taken to be one, what are the fission and criticality-shutdown timescales? (Do
not worry about any refinements to the efficiency calculation as discussed at
the end of Sect. 2.6.) If you have a FORTRAN compiler available, obtain the
numerical-simulation program of Sect. 2.6 and apply it to the same situation;
what yield does the program predict?
2.9 What outer radius and mass of tungsten-carbide tamper will render just
threshold critical a 7-kg spherical core of uncompressed
239 Pu? Hint: Fig. 2.10.
2.10 Consider a mass m of a pure fissile material whose normal density is ρ o , with
m being less than the bare threshold critical mass for the material. Show that
this mass can be made critical by compressing it to a radius given by
r compress ≤
3m
4 π ρ o R o
,
where R o is the threshold critical radius at normal density. Show further that
if C is the number of threshold critical masses represented by m (C < 1), then
the ratio of the density at this compressed radius to the initial density is given
by
ρ compress
ρ o
=
1
C
.
Evaluate numerically for 100 grams of
235 U.
2.11 The diffusion equation for neutrons in a bomb core, (2.18), can be applied in
any coordinate system, provided that the expression for ∇
2 N in that system
is used. To this end, consider a cubical core that extends from 0 < x < L, 0 <
y < L, and 0 < z < L. Solve the diffusion equation in Cartesian coordinates.
Show that if the simplified boundary condition N(L C ) = 0 is used, then the
side length for threshold criticality is given by
L C =
π
√ ν − 1
λ f λ t ,
where the symbols have the same meanings as in Sect. 2.2. Compare this result
to that in Sect. 2.8.1 to show that the critical mass for a cubical bomb core
of a given material is 3
5/2
(4π ) ∼ 1.24 times that of a spherical core of the
same material with this boundary condition. If you are familiar with quantum
mechanics, the solution to this problem is very similar to that of a particle in
a three-dimensional infinite potential box.