3.3 Plutonium Production
129
any depletion of them by other reactions. So far as the time-scales go, this is quite
reasonable: Fuel to produce Pu during the Manhattan Project was usually left in
reactors for ~100 days.
To estimate plutonium production, it is helpful to introduce the idea of the reaction
rate expressed in terms of the neutron flux within the reactor. This was touched on
in Sect. 2.1. The neutron flux Φ is the number of neutrons passing through a unit
area (such as 1 cm
2 ) per second. If the neutrons are incident on N target nuclei of
reaction cross-section σ, then the reaction rate will be R = Φ N σ per second; see
the development leading to Eq. (2.2). In a reactor, the number of nuclei involved in
power production will be those of
235 U, and the relevant cross-section will be that
for fission by thermal neutrons of that isotope, σ f 5 . Hence we have
R =
P t
E f
= Φ N 235 σ f 5 ⇒ Φ =
P t
E f N 235 σ f 5
.
(3.20)
To obtain the rate of synthesis of Pu nuclei, use the rate equation again, but with
the number of nuclei of
238 U and its capture cross-section for thermal neutrons, σ c8 :
R Pu = Φ N 238 σ c8 =
P t N 238 σ c8
E f N 235 σ f 5
.
(3.21)
The ratio of the number of nuclei can be expressed in terms of the fractional
abundance F of
235 U in the reactor’s fuel:
R Pu =
P t (1 − F) σ c8
E f F σ f 5
.
(3.22)
Reactor powers are normally quoted in megawatts (MW; P MW ), and fission
energies in MeV. Making these conversions, (3.22) becomes
R Pu =
6.241 × 10
18
P MW (1 − F) σ c8
E
MeV
f
F σ f 5
.
(3.23)
This expression gives the rate of production of Pu in nuclei per second. On
accounting for the mass of
239 Pu nuclei (239.05u = 3.970 × 10
−25 kg) and the
number of seconds in a day, we can transform this into an expression for the number
of grams of Pu produced per day:
R Pu = 214.1
P MW (1 − F) σ c8
E
MeV
f
F σ f 5
(gr day
−1
).
(3.24)
For various reasons, power-producing reactors in the United States use fuel
enriched to F ~ 0.03. For a plant producing electric power at a rate of 1 GW fueled at
F = 0.03 and operating at efficiency η = 0.3 (P MW = 3333), (3.24) gives a production
rate of 593 gr day
−1
= 216 kg yr
−1 , assuming σ f 5 = 584 b, σ c8 = 2.7 b, and E f
129
any depletion of them by other reactions. So far as the time-scales go, this is quite
reasonable: Fuel to produce Pu during the Manhattan Project was usually left in
reactors for ~100 days.
To estimate plutonium production, it is helpful to introduce the idea of the reaction
rate expressed in terms of the neutron flux within the reactor. This was touched on
in Sect. 2.1. The neutron flux Φ is the number of neutrons passing through a unit
area (such as 1 cm
2 ) per second. If the neutrons are incident on N target nuclei of
reaction cross-section σ, then the reaction rate will be R = Φ N σ per second; see
the development leading to Eq. (2.2). In a reactor, the number of nuclei involved in
power production will be those of
235 U, and the relevant cross-section will be that
for fission by thermal neutrons of that isotope, σ f 5 . Hence we have
R =
P t
E f
= Φ N 235 σ f 5 ⇒ Φ =
P t
E f N 235 σ f 5
.
(3.20)
To obtain the rate of synthesis of Pu nuclei, use the rate equation again, but with
the number of nuclei of
238 U and its capture cross-section for thermal neutrons, σ c8 :
R Pu = Φ N 238 σ c8 =
P t N 238 σ c8
E f N 235 σ f 5
.
(3.21)
The ratio of the number of nuclei can be expressed in terms of the fractional
abundance F of
235 U in the reactor’s fuel:
R Pu =
P t (1 − F) σ c8
E f F σ f 5
.
(3.22)
Reactor powers are normally quoted in megawatts (MW; P MW ), and fission
energies in MeV. Making these conversions, (3.22) becomes
R Pu =
6.241 × 10
18
P MW (1 − F) σ c8
E
MeV
f
F σ f 5
.
(3.23)
This expression gives the rate of production of Pu in nuclei per second. On
accounting for the mass of
239 Pu nuclei (239.05u = 3.970 × 10
−25 kg) and the
number of seconds in a day, we can transform this into an expression for the number
of grams of Pu produced per day:
R Pu = 214.1
P MW (1 − F) σ c8
E
MeV
f
F σ f 5
(gr day
−1
).
(3.24)
For various reasons, power-producing reactors in the United States use fuel
enriched to F ~ 0.03. For a plant producing electric power at a rate of 1 GW fueled at
F = 0.03 and operating at efficiency η = 0.3 (P MW = 3333), (3.24) gives a production
rate of 593 gr day
−1
= 216 kg yr
−1 , assuming σ f 5 = 584 b, σ c8 = 2.7 b, and E f