Neutron Energy Range
Thus far we have not discussed the dependence of cross sections on neutron kinetic energy. To take energy into account we write each of the above cross sections as functions of energy
by letting
and similarly, as a result of Eq. (2.30),
The energy dependence of cross sections is fundamental to neutron behavior in chain reactions and thus warrants detailed consideration. We begin by establishing the upper and lower limits of neutron energies found in fission reactors. Neutrons born in fission are distributed over a spectrum of energy. Defining x(E)dE as the fraction of fission neutrons born with energies between E and E+dE, a reasonable approximation to the fission spectrum is given by
where E is in MeV and x(E) is normalized to one:
The logarithmic energy plot of Fig. 2.2 shows the fission spectrum, x(E). Fission neutrons are born in the MeV energy range with
an average energy of about 2 MeV, and the most probable energy is about 3/4 MeV. The numbers of fission neutrons produced with energies greater than 10MeV is negligible, which sets the upper limit to the energy range of neutrons in reactors. Neutrons born in fission typically undergo a number of scattering collisions before being absorbed.Aneutron scattering froma stationary nucleus will transfer a part of its momentum to that nucleus, thus losing energy. However at any temperature above absolute zero, the scattering nuclei will possess random thermal motions. According to kinetic theory, the mean kinetic energy of such nuclei is
where k is the Boltzmann constant and T is the absolute temperature. For room temperature of T =293.61K the mean energy amounts to 0.0379 eV. Frequently, thermal neutron measurements are recorded at 1.0 kT, which, at room temperature, amounts to 0.0253 eV. In either case these energies are insignificant compared to the MeV energies of fission neutrons. Thus the scatting of neutrons causes them to lose kinetic energy as they collide with nearly stationary nuclei until they are absorbed or are slowed down to the eV range. In the idealized situation where no absorption is present, the neutrons would eventually come to equilibrium with the thermal motions of the surrounding nuclei. The neutrons would then take the form of the famed Maxwell-Boltzmann distribution
where E is in eV, Boltzmann’s constant is k = 8.617065x10-5 eV/K, and M(E) is normalized to one:
Figure 2.2 shows M(E) along with x(E) to indicate the energy
range over which neutrons may exist in a nuclear reactor. Realize,
however, that some absorption will always be present. As a result the
spectrum will be shifted upward somewhat from MðEÞ since the
absorption precludes thermal equilibrium from ever being completely
established. The fraction of neutrons with energies less than 0.001eV
in the room temperature Maxwell-Boltzmann distribution is quite
small, and we thus take it as the lower bound of energies that we
need to consider. In general, we may say the primary range of interest
for neutron in a chain reactor is in the range 0.001 eV<10mev.
Thus the neutron energies range over roughly 10 orders of magnitude!
For descriptions of neutron cross sections important to reactor physics,
it is helpful to divide them into to three energy ranges.We refer to
fast neutrons as being those with energies over the range where significant
numbers of fission neutrons are emitted: 0.1 MeV<10mev.
We call thermal neutrons those with small enough energies that
the thermal motions of the surrounding atoms can significantly affect
their scattering properties: 0.001 eV<1.0>
in between as epithermal or intermediate energy neutrons:
1.0 eV<0.1mev.
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by letting
and similarly, as a result of Eq. (2.30),
The energy dependence of cross sections is fundamental to neutron behavior in chain reactions and thus warrants detailed consideration. We begin by establishing the upper and lower limits of neutron energies found in fission reactors. Neutrons born in fission are distributed over a spectrum of energy. Defining x(E)dE as the fraction of fission neutrons born with energies between E and E+dE, a reasonable approximation to the fission spectrum is given by
where E is in MeV and x(E) is normalized to one:
The logarithmic energy plot of Fig. 2.2 shows the fission spectrum, x(E). Fission neutrons are born in the MeV energy range with
an average energy of about 2 MeV, and the most probable energy is about 3/4 MeV. The numbers of fission neutrons produced with energies greater than 10MeV is negligible, which sets the upper limit to the energy range of neutrons in reactors. Neutrons born in fission typically undergo a number of scattering collisions before being absorbed.Aneutron scattering froma stationary nucleus will transfer a part of its momentum to that nucleus, thus losing energy. However at any temperature above absolute zero, the scattering nuclei will possess random thermal motions. According to kinetic theory, the mean kinetic energy of such nuclei is
where k is the Boltzmann constant and T is the absolute temperature. For room temperature of T =293.61K the mean energy amounts to 0.0379 eV. Frequently, thermal neutron measurements are recorded at 1.0 kT, which, at room temperature, amounts to 0.0253 eV. In either case these energies are insignificant compared to the MeV energies of fission neutrons. Thus the scatting of neutrons causes them to lose kinetic energy as they collide with nearly stationary nuclei until they are absorbed or are slowed down to the eV range. In the idealized situation where no absorption is present, the neutrons would eventually come to equilibrium with the thermal motions of the surrounding nuclei. The neutrons would then take the form of the famed Maxwell-Boltzmann distribution
where E is in eV, Boltzmann’s constant is k = 8.617065x10-5 eV/K, and M(E) is normalized to one:
Figure 2.2 shows M(E) along with x(E) to indicate the energy
range over which neutrons may exist in a nuclear reactor. Realize,
however, that some absorption will always be present. As a result the
spectrum will be shifted upward somewhat from MðEÞ since the
absorption precludes thermal equilibrium from ever being completely
established. The fraction of neutrons with energies less than 0.001eV
in the room temperature Maxwell-Boltzmann distribution is quite
small, and we thus take it as the lower bound of energies that we
need to consider. In general, we may say the primary range of interest
for neutron in a chain reactor is in the range 0.001 eV
Thus the neutron energies range over roughly 10 orders of magnitude!
For descriptions of neutron cross sections important to reactor physics,
it is helpful to divide them into to three energy ranges.We refer to
fast neutrons as being those with energies over the range where significant
numbers of fission neutrons are emitted: 0.1 MeV
We call thermal neutrons those with small enough energies that
the thermal motions of the surrounding atoms can significantly affect
their scattering properties: 0.001 eV
in between as epithermal or intermediate energy neutrons:
1.0 eV
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