6.8 Appendix H: Exercises and Answers
229
Is this satisfied in the case of a 1-MeV neutron striking a nucleus of
235 U?
2.1 Using the empirical nuclear-radius expression of the previous problems, estimate the geometrical cross-sectional area of a
235 U nucleus; give your result
in barns. How does your result compare to the fission cross-section for fast
neutrons for this isotope, σ f = 1.235 bn?
2.2 Because cadmium-113 has an enormous cross-section for capturing thermal
neutrons, strips of cadmium metal are often used in control mechanisms in
reactors. Given ρ = 8.65 gr cm
−3 , A = 112.904 gr mol
−1 and σ capture =
20,600 barns, compute the probability that a neutron will penetrate through a
strip of Cd-113 of thickness 0.05 mm.
2.3 Neutrons with speed v neut corresponding to classical kinetic energy K MeV are
traveling between nuclei in a material of atomic weight A gr mol
−1 , density
ρ gr cm
−3 , and fission cross-section σ f barns. Show that the time between
fissions τ = λ f
v neut can be expressed as
τ =
1.20 A
σ f ρ
√
K
nanoseconds
Compute τ for the case of 2-MeV neutrons in
235 U: A = 235, ρ = 18.71 gr
cm
−3 and σ f = 1.235 bn.
2.4 For
233 U, n = 4.794 × 10
22 cm
−3 , σ f = 1.946 b, σ el = 4.447 b, ν = 2.755, and
ρ = 18.55 g cm
−3 . What does diffusion theory indicate for the critical mass
of this isotope?
233 U would make an excellent nuclear explosive, but it does
not occur naturally.
2.5 You have 5 kg of beryllium-oxide to use as a tamper. What uncompressed
mass of
239 Pu can be rendered just critical, assuming spherical geometry?
2.6 Working from the development of bomb efficiency in Sect. 2.4, show that the
speed of the expanding core at the time of second criticality is given by
v (t crit ) =
α αr
τ
.
Evaluate v (t crit ) for a
235 U core with C = 1.5 bare critical masses using the
values given in Table 2.2. How does v (t crit ) compare to the average neutron
speed?
2.7 Working from the development of bomb efficiency in Sect. 2.5, show that the
pressure within the expanding core at the time of second criticality is given by
P(t crit ) =
α
2
r ρr
3 τ 2
.
Evaluate P(t crit ) in the case of a
235 U core with C = 1.5. Express your result
in atmospheres: 1 atm ~ 101,000 Pa.
2.8 Consider a hypothetical fissile material with ρ = 17.3 gr cm
−3 , A = 250 gr
mol
−1 , σ f = 1.55 bn, σ el = 6 bn, and ν = 2.95. Use CriticalityAnalytic
229
Is this satisfied in the case of a 1-MeV neutron striking a nucleus of
235 U?
2.1 Using the empirical nuclear-radius expression of the previous problems, estimate the geometrical cross-sectional area of a
235 U nucleus; give your result
in barns. How does your result compare to the fission cross-section for fast
neutrons for this isotope, σ f = 1.235 bn?
2.2 Because cadmium-113 has an enormous cross-section for capturing thermal
neutrons, strips of cadmium metal are often used in control mechanisms in
reactors. Given ρ = 8.65 gr cm
−3 , A = 112.904 gr mol
−1 and σ capture =
20,600 barns, compute the probability that a neutron will penetrate through a
strip of Cd-113 of thickness 0.05 mm.
2.3 Neutrons with speed v neut corresponding to classical kinetic energy K MeV are
traveling between nuclei in a material of atomic weight A gr mol
−1 , density
ρ gr cm
−3 , and fission cross-section σ f barns. Show that the time between
fissions τ = λ f
v neut can be expressed as
τ =
1.20 A
σ f ρ
√
K
nanoseconds
Compute τ for the case of 2-MeV neutrons in
235 U: A = 235, ρ = 18.71 gr
cm
−3 and σ f = 1.235 bn.
2.4 For
233 U, n = 4.794 × 10
22 cm
−3 , σ f = 1.946 b, σ el = 4.447 b, ν = 2.755, and
ρ = 18.55 g cm
−3 . What does diffusion theory indicate for the critical mass
of this isotope?
233 U would make an excellent nuclear explosive, but it does
not occur naturally.
2.5 You have 5 kg of beryllium-oxide to use as a tamper. What uncompressed
mass of
239 Pu can be rendered just critical, assuming spherical geometry?
2.6 Working from the development of bomb efficiency in Sect. 2.4, show that the
speed of the expanding core at the time of second criticality is given by
v (t crit ) =
α αr
τ
.
Evaluate v (t crit ) for a
235 U core with C = 1.5 bare critical masses using the
values given in Table 2.2. How does v (t crit ) compare to the average neutron
speed?
2.7 Working from the development of bomb efficiency in Sect. 2.5, show that the
pressure within the expanding core at the time of second criticality is given by
P(t crit ) =
α
2
r ρr
3 τ 2
.
Evaluate P(t crit ) in the case of a
235 U core with C = 1.5. Express your result
in atmospheres: 1 atm ~ 101,000 Pa.
2.8 Consider a hypothetical fissile material with ρ = 17.3 gr cm
−3 , A = 250 gr
mol
−1 , σ f = 1.55 bn, σ el = 6 bn, and ν = 2.95. Use CriticalityAnalytic