6.8 Appendix H: Exercises and Answers
235
Use the decay rate expression of Sect. 4.2 to estimate the “immediate” betaradioactivity so generated; for sake of simplicity, ignore the neutrons released
in the above reaction. If this radioactivity falls out over an area of 10 square
miles, what will be the resulting immediate radioactivity in Curies per square
meter? To put your result in perspective, household smoke detectors use 1 μCi
alpha-emitters as ionization sources to help detect smoke particles.
F.1 In Appendix F, an expression is derived for the probability that the reduced
flight-path-length x (0 < x < 2) for a neutron escaping from within a sphere
will lie between x and x + dx:
P(dx) =
3
4
1 −
x
2
2
dx.
(a) What is the average value of the reduced escape-path length, x =
x P(dx)? (b) Determine also the root-mean-square escape-path length,
x 2
. (c) A convenient measure of the “scatter” of measured x-values is the
quantity x =
x 2
− x
2 . What is x here?
Answers
1.1 (a) 17.59 MeV, (b) 4.78 MeV, (c) 182.0 MeV, (d) –2.642 MeV
1.2 7.34 grams
1.3 4.784 MeV
1.4 (a) E γ = 5.24 MeV (b) E γ = 15.44 MeV (c) Reaction impossible; E γ =
–48 MeV
1.5 (a) Threshold 3.034; Q = −2.642; P & n energies 0.307, 2.051; both forward
(b) No threshold; Q = 17.59; He & n energies 0.99, 19.60; He backward; n
forward
(c) No threshold; Q = 1.674; Ne & H energies 0.068 and 4.356; both forward
(d) Threshold 17.298; Q = −16.143; S & Mg energies 1.304 and 12.55; S
backward; Mg forward
(e) Threshold 51.07; Q = −47.85; Es & D energies 2.816 and 9.331; both
forward
1.6 K
proton
alpha
K
γ
alpha ∼ 120
1.9 Q/Q ~ 0.07; all approximations satisfied
1.10 8.57 meters
1.11 5.47 MeV; fissile
1.12 d/R = 7.16; K > 3.73 MeV; not satisfied
2.1 σ ~ 1.723 bn
2.2 Penetration probability ~0.00863
2.3 8.63 ns
2.4 14.2 kg
2.5 ~8.6 kg
2.6 v(t crit ) ~ 2.38 × 10
5 m s
−1 ; ~ 1.2% neutron speed
235
Use the decay rate expression of Sect. 4.2 to estimate the “immediate” betaradioactivity so generated; for sake of simplicity, ignore the neutrons released
in the above reaction. If this radioactivity falls out over an area of 10 square
miles, what will be the resulting immediate radioactivity in Curies per square
meter? To put your result in perspective, household smoke detectors use 1 μCi
alpha-emitters as ionization sources to help detect smoke particles.
F.1 In Appendix F, an expression is derived for the probability that the reduced
flight-path-length x (0 < x < 2) for a neutron escaping from within a sphere
will lie between x and x + dx:
P(dx) =
3
4
1 −
x
2
2
dx.
(a) What is the average value of the reduced escape-path length, x =
x P(dx)? (b) Determine also the root-mean-square escape-path length,
x 2
. (c) A convenient measure of the “scatter” of measured x-values is the
quantity x =
x 2
− x
2 . What is x here?
Answers
1.1 (a) 17.59 MeV, (b) 4.78 MeV, (c) 182.0 MeV, (d) –2.642 MeV
1.2 7.34 grams
1.3 4.784 MeV
1.4 (a) E γ = 5.24 MeV (b) E γ = 15.44 MeV (c) Reaction impossible; E γ =
–48 MeV
1.5 (a) Threshold 3.034; Q = −2.642; P & n energies 0.307, 2.051; both forward
(b) No threshold; Q = 17.59; He & n energies 0.99, 19.60; He backward; n
forward
(c) No threshold; Q = 1.674; Ne & H energies 0.068 and 4.356; both forward
(d) Threshold 17.298; Q = −16.143; S & Mg energies 1.304 and 12.55; S
backward; Mg forward
(e) Threshold 51.07; Q = −47.85; Es & D energies 2.816 and 9.331; both
forward
1.6 K
proton
alpha
K
γ
alpha ∼ 120
1.9 Q/Q ~ 0.07; all approximations satisfied
1.10 8.57 meters
1.11 5.47 MeV; fissile
1.12 d/R = 7.16; K > 3.73 MeV; not satisfied
2.1 σ ~ 1.723 bn
2.2 Penetration probability ~0.00863
2.3 8.63 ns
2.4 14.2 kg
2.5 ~8.6 kg
2.6 v(t crit ) ~ 2.38 × 10
5 m s
−1 ; ~ 1.2% neutron speed