234
6 Appendices
3.6 A news story indicates that a country that is attempting to develop nuclear
weapons has succeeded in enriching 1000 kg of uranium hexafluoride to 5%,
that is, 5% of the uranium nuclei in the sample are
235 U while the other 95%
are
238 U. How many grams of
235 U are contained in the sample?
4.1. The half-life for spontaneous fission of
242
96 Cm is 7.0 × 10
6 yr. What is the
corresponding rate of spontaneous fissions per kg per second?
4.2. In the study of thermodynamic properties of materials, the following simple
differential equation is used to model the change in volume dV of a sample of
material of initial volume V when it is subjected to a change in pressure dP:
dV
d P
= −
V
B
.
B is the bulk modulus of the material, a measure of its compressibility; a higherB material is more difficult to compress than one of lower B. Assume that over
the range of some compressive event, B averages 30 GPa. If an implosion bomb
subjects a plutonium core to a pressure increase of one million atmospheres
(1 atm ~ 10
5 Pa), integrate the differential equation to estimate the ratio of the
final volume of the plutonium to its initial volume.
4.3 A rogue militia organization in an unstable country claims to have acquired
20 kg of
239 Pu of normal density and to have developed a crude gun-type
bomb. The core contains 3% Pu 240. If they are to have a 50–50 chance of
non-predetonation, what is the maximum tolerable assembly time? Take S max
= 0.
4.4 A 5-kg
239 Pu core is compressed to a density of 20 gr cm
−3 at the time it reaches
first criticality. If it is contaminated with 1.8%
240 Pu, what is the probability
of achieving 70% of the design yield? Take (τ , F, α O , t O ) = (10
−8 s, 45, 1,
10
−5 s) as in Sect. 4.3. What value of t init corresponds to these conditions?
4.5 Prove that Eq. (4.21) reduces to Eq. (4.18) for t init → 0.
4.6 According to the publication of West & Sherwood cited in Sect. 4.4, the yield
of 5.2 MeV alphas on
27 Al is 4.25 × 10
−7 . If the number-density ratio of
aluminum to the fissile material in a bomb core is held to 10
−5 , what maximum
rate of alpha-decays can be tolerated if the production of neutrons is to be kept
to no more than 10
4 per second?
5.1 The purpose of this problem is to make a very crude estimate of the radioactivity
produced by a fission weapon.
Suppose that fission of
235 U happens exclusively by the reaction
235
92 U +
1
0 n →
141
56 Ba +
92
36 Kr + 3
1
0 n
.
Assume that 1 kg of
235 U is fissioned in this way.
141 Ba and
92 Kr both subsequently decay by beta-decay with half-lives of 18 min and 1.8 s, respectively.
6 Appendices
3.6 A news story indicates that a country that is attempting to develop nuclear
weapons has succeeded in enriching 1000 kg of uranium hexafluoride to 5%,
that is, 5% of the uranium nuclei in the sample are
235 U while the other 95%
are
238 U. How many grams of
235 U are contained in the sample?
4.1. The half-life for spontaneous fission of
242
96 Cm is 7.0 × 10
6 yr. What is the
corresponding rate of spontaneous fissions per kg per second?
4.2. In the study of thermodynamic properties of materials, the following simple
differential equation is used to model the change in volume dV of a sample of
material of initial volume V when it is subjected to a change in pressure dP:
dV
d P
= −
V
B
.
B is the bulk modulus of the material, a measure of its compressibility; a higherB material is more difficult to compress than one of lower B. Assume that over
the range of some compressive event, B averages 30 GPa. If an implosion bomb
subjects a plutonium core to a pressure increase of one million atmospheres
(1 atm ~ 10
5 Pa), integrate the differential equation to estimate the ratio of the
final volume of the plutonium to its initial volume.
4.3 A rogue militia organization in an unstable country claims to have acquired
20 kg of
239 Pu of normal density and to have developed a crude gun-type
bomb. The core contains 3% Pu 240. If they are to have a 50–50 chance of
non-predetonation, what is the maximum tolerable assembly time? Take S max
= 0.
4.4 A 5-kg
239 Pu core is compressed to a density of 20 gr cm
−3 at the time it reaches
first criticality. If it is contaminated with 1.8%
240 Pu, what is the probability
of achieving 70% of the design yield? Take (τ , F, α O , t O ) = (10
−8 s, 45, 1,
10
−5 s) as in Sect. 4.3. What value of t init corresponds to these conditions?
4.5 Prove that Eq. (4.21) reduces to Eq. (4.18) for t init → 0.
4.6 According to the publication of West & Sherwood cited in Sect. 4.4, the yield
of 5.2 MeV alphas on
27 Al is 4.25 × 10
−7 . If the number-density ratio of
aluminum to the fissile material in a bomb core is held to 10
−5 , what maximum
rate of alpha-decays can be tolerated if the production of neutrons is to be kept
to no more than 10
4 per second?
5.1 The purpose of this problem is to make a very crude estimate of the radioactivity
produced by a fission weapon.
Suppose that fission of
235 U happens exclusively by the reaction
235
92 U +
1
0 n →
141
56 Ba +
92
36 Kr + 3
1
0 n
.
Assume that 1 kg of
235 U is fissioned in this way.
141 Ba and
92 Kr both subsequently decay by beta-decay with half-lives of 18 min and 1.8 s, respectively.