Page 260 - Dynamics and Control of Nuclear Reactors

APPENDIX C Basic reactor physics 261

higher energy of neutrons that interact with the moderator atoms. The energy spec-
trum of moderator atoms, and consequently the energy spectrum of thermalized neu-
trons is given by the Maxwell-Boltzmann distribution. Fig. C.1 shows distributions at
three different temperatures.
In a reactor, the absorption and fission cross sections must be corrected for the
actual temperature of the moderator. The Maxwell-Boltzmann distribution applies
for neutrons in equilibrium with moderator atoms. As shown in elementary reactor
physics books, the “effective” cross section for a material with 1/v dependence in a
moderator at temperature, T is

r ffiffiffiffiffiffiffiffi
1 293
ð
σ TðÞ ¼ σ 0:0253eVÞ (C.9)
1:128 T
The average and most probable energies for neutrons in a Maxwell spectrum are as
follows:
– average neutron energy¼1/2 kT
– most probable neutron energy¼3/2 kT.

where k is the Boltzmann constant and T is the absolute temperature.
Clearly the thermal spectrum shifts to higher energies as moderator temperature
increases. This is called spectral hardening.
Resonances are spikes in a material’s cross section. See Fig. C.2 for the energy-
dependent U-238 cross section. Strong resonances are apparent. Resonances are
important in their effects on steady state and dynamic characteristics of a reactor.
Capture and fission cross sections both exhibit resonant behavior.

× 10 –4
4
293 K
493 K
593 K
Fractional neutron density 2
3

0 1
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
v, Neutron velocity (m/s)
FIG. C.1
Maxwell-Boltzmann distribution of thermal neutrons for three temperatures.