becomes remarkable as the subcriticality of the system becomes deep. It should be
noted that collapsed cross sections depend on the subcriticality of target system
because of the difference in neutron spectrum.
There is one recommendation to evaluate the proper neutron spectrum for
collapsing. In eigenvalue mode, the k-eigenvalue mode expressed as
Eq. (13.13.1) is usually used because the eigenvalue is an unbiased index to
recognize the criticality, but there is another eigenvalue mode, named the alphaeigenvalue mode, expressed as Eq. (13.13.2):
k-eigenvalue mode Lϕ k ¼
1
k
Mϕ k ,
ð13:1Þ
alpha-eigenvalue mode L þ
α
v
h
i
ϕ α ¼ Mϕ α ,
ð13:2Þ
where L is the destruction operator including leakage and absorption reactions, M is
the production operator including fission reactions, k is the k-eigenvalue called the
effective multiplication factor, α is the alpha-eigenvalue, v is the neutron speed, and
ϕ k , ϕ α are the neutron fluxes for each mode. Equation (13.1) is derived from a timedependent equation by eliminating the term of time derivative, but Eq. (13.13.2) is
derived by considering the exponential change of neutron flux in time. Usually a
subcritical system such as the ADSR is operated not in stable but in transient
conditions.
In the subcritical system, the alpha-eigenvalue is negative, and the impact of the
negative alpha-eigenvalue on neutron flux is remarkable at thermal energy range
where the neutron speed is small. Therefore, the neutron spectrum evaluated in
alpha-eigenvalue mode is softer than that in k-eigenvalue mode. Similar to this
consideration, the difference in neutron spectrum could be observed in the
1E-4
1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
1E+3
1E+4
1E+5
1E+6
1E+7
1E+8
1E-3
1E-1
1E+1
1E+3
1E+5
1E+7
Neutron spectrum [arb. unit]
Neutron Energy [eV]
1e-8 [sec]
1e-7 [sec]
1e-6 [sec]
1e-5 [sec]
1e-4 [sec]
1e-3 [sec]
2e-3 [sec]
3e-3 [sec]
Fig. 13.8 Neutron spectrum at fuel region evaluated in time-dependent mode (13 fuel rods)
13 Study on Neutron Spectrum of Pulsed Neutron Reactor
135
noted that collapsed cross sections depend on the subcriticality of target system
because of the difference in neutron spectrum.
There is one recommendation to evaluate the proper neutron spectrum for
collapsing. In eigenvalue mode, the k-eigenvalue mode expressed as
Eq. (13.13.1) is usually used because the eigenvalue is an unbiased index to
recognize the criticality, but there is another eigenvalue mode, named the alphaeigenvalue mode, expressed as Eq. (13.13.2):
k-eigenvalue mode Lϕ k ¼
1
k
Mϕ k ,
ð13:1Þ
alpha-eigenvalue mode L þ
α
v
h
i
ϕ α ¼ Mϕ α ,
ð13:2Þ
where L is the destruction operator including leakage and absorption reactions, M is
the production operator including fission reactions, k is the k-eigenvalue called the
effective multiplication factor, α is the alpha-eigenvalue, v is the neutron speed, and
ϕ k , ϕ α are the neutron fluxes for each mode. Equation (13.1) is derived from a timedependent equation by eliminating the term of time derivative, but Eq. (13.13.2) is
derived by considering the exponential change of neutron flux in time. Usually a
subcritical system such as the ADSR is operated not in stable but in transient
conditions.
In the subcritical system, the alpha-eigenvalue is negative, and the impact of the
negative alpha-eigenvalue on neutron flux is remarkable at thermal energy range
where the neutron speed is small. Therefore, the neutron spectrum evaluated in
alpha-eigenvalue mode is softer than that in k-eigenvalue mode. Similar to this
consideration, the difference in neutron spectrum could be observed in the
1E-4
1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
1E+3
1E+4
1E+5
1E+6
1E+7
1E+8
1E-3
1E-1
1E+1
1E+3
1E+5
1E+7
Neutron spectrum [arb. unit]
Neutron Energy [eV]
1e-8 [sec]
1e-7 [sec]
1e-6 [sec]
1e-5 [sec]
1e-4 [sec]
1e-3 [sec]
2e-3 [sec]
3e-3 [sec]
Fig. 13.8 Neutron spectrum at fuel region evaluated in time-dependent mode (13 fuel rods)
13 Study on Neutron Spectrum of Pulsed Neutron Reactor
135