Page 256 - Dynamics and Control of Nuclear Reactors

APPENDIX C Basic reactor physics 257

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• N¼Density of target nuclei (number of nuclei/cm )
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1
2
• Macroscopic cross section: Σ¼σN (cm /cm or cm )
3
• Reaction rate: R¼Σϕ (number of interactions/cm -s)
• Fission reaction rate: R f ¼Σ f ϕ
• R f ¼[Nσ f ] [n(t)v]
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• R f is the number of fission reactions/cm -s
• σ f is the microscopic fission cross section
The reaction rate in a 1/v absorber is given by
R ¼ N σϕ
or
R ¼ Nc=vÞnv
ð
where c is a constant.
Note that the velocity terms cancel. Therefore, the reaction rate for 1/v absorbers
is independent of the neutron energy.

Example C.1
The energy of neutrons in an experimental reactor is approximately equal to 0.0253 eV. This
corresponds to a speed of about 2200 m/s for neutrons. As an exercise, let us let flux
2
12
ϕ¼2 10 /(cm s). Calculate the neutron density.
12
2 10 cm 2 s 1
6
n ¼ ϕ=v ¼ ¼ 9 10 =cm 3
2200 100m s 1
In the above example, use the microscopic absorption cross section, σ a ¼694 b
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24
for U-235. The density of target nuclei is N¼0.05 10 /cm . Then
Σ a ¼ Nσ a ¼ 34 cm 1
The reaction rate is given by the following:
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13
R ¼ ϕΣ a ¼ 6:8 10 =cm s
R is also the rate at which U-235 nuclei are consumed.
Note that the microscopic fission cross section of U-235 at E¼0.025 eV is,
σ f ¼582 b. The capture cross section is σ c ¼112 b and the absorption cross section
is σ a ¼σ f +σ c ¼694. The ratio σ f /σ a ¼582/694¼0.84. That is, 84% of thermal
neutron absorptions in U-235 result in a fission reaction.
If an absorber cross section decreases slower with increasing neutron energy (and
neutron velocity) than 1/v, then the absorption rate relative to 1/v absorbers increases
as neutron energy increases. The reverse is true for absorbers whose cross section
decreases faster than 1/v. The energy dependence of low energy absorptions plays
an important role in determining the dynamics of power reactors.
The power produced in a reactor is given by the following:
P ¼ NVÞ σ f nv FWatt (C.3)
ð