6.8 Appendix H: Exercises and Answers
231
2.12 If you have a FORTRAN compiler available, obtain the numerical-simulation
program of Sect. 2.6 and apply it to the following situation. A fission bomb is
to be made of a pure
239 Pu core (normal density) of mass 30 kg plus a
238 U
tamper of mass 200 kg (A = 238.05 gr mol
−1 ; ρ = 18.95 gr cm
−3 ; σ el = 4.804
bn; ignore any fissility of the tamper). What is the yield? Take γ = 1/3, the
average neutron energy to be 2 MeV, the energy per fission to be 180 MeV,
and assume one initiating neutron.
2.13 Repeat the derivation of Sect. 2.3 for threshold criticality of a tamped core (α =
δ = 0), but take the “outer” boundary condition to be simplified to N tamp (R tamp )
= 0. Show that the criticality condition simplifies to
1 −
R thresh
d core
cot
R thresh
d core
=
λ
tamp
trans
λ
core
trans
1 − R thresh
R tamp
.
In this approximation, what is the threshold critical radius for
235 U for a
tungsten-carbide tamper of outer radius 16 cm? Compare to the result for
an untamped core with the same boundary condition, Sect. 2.1.8.
2.14 It is remarked in Sect. 2.5 that in the case of a gas of uranium nuclei of
normal density of that metal (18.95 gr cm
−3 ), radiation pressure dominates
gas pressure for per-particle energies greater than about 2 keV. This problem
investigates this issue.
For a “gas” of photons, thermodynamics provides the following expression
for the pressure:
P rad =
8π
5 k
4
B
45c 3 h 3
T
4
,
where k B is Boltzmann’s constant, c is the speed of light, h is Planck’s constant,
and T is the absolute temperature. For a gas of “classical” particles, the ideal
gas law can be cast as
P classical = 10
6
ρ N A k B
A
T,
where ρ and A are the density and atomic weight of the material in gr cm
−3 and
gr mol
−1 , respectively; N A is Avogadro’s number as usual. Given that in the
classical case the per-particle energy is 3k B T /2, show that radiation pressure
will dominate over gas pressure for classical per-particle energies satisfying
E >
3
2
45 × 10
6
8π 5
1/3
ch
e
ρ N A
A
1/3
∼
4.908 × 10
−5
ρ N A
A
1/3
eV,
231
2.12 If you have a FORTRAN compiler available, obtain the numerical-simulation
program of Sect. 2.6 and apply it to the following situation. A fission bomb is
to be made of a pure
239 Pu core (normal density) of mass 30 kg plus a
238 U
tamper of mass 200 kg (A = 238.05 gr mol
−1 ; ρ = 18.95 gr cm
−3 ; σ el = 4.804
bn; ignore any fissility of the tamper). What is the yield? Take γ = 1/3, the
average neutron energy to be 2 MeV, the energy per fission to be 180 MeV,
and assume one initiating neutron.
2.13 Repeat the derivation of Sect. 2.3 for threshold criticality of a tamped core (α =
δ = 0), but take the “outer” boundary condition to be simplified to N tamp (R tamp )
= 0. Show that the criticality condition simplifies to
1 −
R thresh
d core
cot
R thresh
d core
=
λ
tamp
trans
λ
core
trans
1 − R thresh
R tamp
.
In this approximation, what is the threshold critical radius for
235 U for a
tungsten-carbide tamper of outer radius 16 cm? Compare to the result for
an untamped core with the same boundary condition, Sect. 2.1.8.
2.14 It is remarked in Sect. 2.5 that in the case of a gas of uranium nuclei of
normal density of that metal (18.95 gr cm
−3 ), radiation pressure dominates
gas pressure for per-particle energies greater than about 2 keV. This problem
investigates this issue.
For a “gas” of photons, thermodynamics provides the following expression
for the pressure:
P rad =
8π
5 k
4
B
45c 3 h 3
T
4
,
where k B is Boltzmann’s constant, c is the speed of light, h is Planck’s constant,
and T is the absolute temperature. For a gas of “classical” particles, the ideal
gas law can be cast as
P classical = 10
6
ρ N A k B
A
T,
where ρ and A are the density and atomic weight of the material in gr cm
−3 and
gr mol
−1 , respectively; N A is Avogadro’s number as usual. Given that in the
classical case the per-particle energy is 3k B T /2, show that radiation pressure
will dominate over gas pressure for classical per-particle energies satisfying
E >
3
2
45 × 10
6
8π 5
1/3
ch
e
ρ N A
A
1/3
∼
4.908 × 10
−5
ρ N A
A
1/3
eV,