# Analysis and Discussion of Neutron Flux

The analysis results of neutron flux distribution and neutron spectrum are summarized in this chapter with some discussion. Neutron flux distribution and spectrum in the core are shown in Sects. 13.3.1 and 13.3.2, respectively. All analyses are done by a continuous energy Monte Carlo code named MVP-II [2] with JENDL-4.0 [3] library. The code has the function to simulate the experiment not only in eigenvalue mode but also in time-dependent mode, where the necessary time of neutron flight is used to account for the elapsed time after the injection of DT neutrons.

In this chapter, fast and thermal neutrons are in the energy range less than 4 eV and more than 100 keV, respectively.

**Fig. 13.1 Typical core configuration (13 fuel rods case). T-target tritium target**

**Table 13.1 Measured subcriticality in the Kyoto University Critical Assembly (KUCA) experiment**

Number of fuel rods |
Measured subcriticality [$] (standard deviation: 1σ) |

19 |
2.32 (0.02) |

17 |
6.40 (0.08) |

15 |
10.9 (0.2) |

13 |
13.4 (0.2) |

9 |
28.2 (1.1) |

6 |
49.4 (1.0) |

## Neutron Flux Distribution

Neutron flux distribution is drastically changed with the change in subcriticality of the system, because of the change in the number of fuel rods in the core. Figures 13.2, 13.3, and 13.4 show the neutron flux distribution along the central line from T-target to core for 13 fuel rods case in eigenvalue mode and time-dependent mode, respectively. Neutron flux distribution evaluated in time-dependent mode changes as a function of elapsed time after the injection of D-T neutrons, and the shape of neutron flux distribution is almost stable after 1e-4 s (Figs. 13.3 and 13.4). The comparison of neutron flux between two modes shows the discrepancy (Figs. 13.5 and 13.6). Thermal neutron flux distribution of the time-dependent mode is smaller at fuel region, but higher at the reflector region, than those of eigenvalue mode, although fast neutron flux distribution is almost the same between the two modes. Figures 13.5 and 13.6 also show that the neutron spectrum is different between two modes, and the details are discussed in the next section.

## Neutron Spectrum

The neutron spectrum at the fuel region is shown in Figs. 13.7 and 13.8. The neutron spectrum in time-dependent mode changes as a function of elapsed time, although

**Fig. 13.2 Neutron flux distribution evaluated in eigenvalue mode (13 fuel rods)**

**Fig. 13.3 Neutron flux distribution evaluated in time-dependent mode (fast energy range, 13 fuel rods)**

the neutron spectrum in eigenvalue mode is singular. In addition to this, the neutron spectrum at the fuel region evaluated in eigenvalue mode is almost the same among different subcritical states, because there is no change in fuel composition through the experiment. Neutron spectrum at the fuel region evaluated in time-dependent mode changes as a function of elapsed time after the injection of D-T neutrons, but the shape of the spectrum becomes stable after around 1e-4 s, although the

**Fig. 13.4 Neutron flux distribution evaluated in time-dependent mode (thermal energy range, 13 fuel rods)**

**Fig. 13.5 Comparison of neutron flux distribution (fast energy range,13 fuel rods)**

magnitude of neutron flux decreases with elapsed time (Fig. 13.8). The neutron spectrum is compared between two modes, where the shape of the spectrum in timedependent mode is almost stable (at 1e-3 s). The comparison of the neutron spectrum is shown in Fig. 13.9. The neutron spectrum in time-dependent mode depends on the subcriticality of the system, and the spectrum becomes soft as the subcriticality becomes deep. This tendency can be understood by considering the following facts. The magnitude of neutron flux decreases as a function of elapsed time in the subcritical system, and the decreasing speed of neutron flux becomes

**Fig. 13.6 Comparison of neutron flux distribution (thermal energy range, 13 fuel rods)**

**Fig. 13.7 Neutron spectrum at fuel region evaluated in eigenvalue mode**

high for the deep subcritical system. Here the neutrons are slowed down by colliding with the medium in the thermal core, and the time needed in the slowing-down process does not depend on the subcriticality of the system. Therefore, the change in the neutron spectrum is caused by the time delay of decrease in thermal energy range where the neutrons are slowed down compared to that in the fast energy range.

The difference in neutron spectrum will cause the difference in collapsed cross sections widely used in design survey calculations, and the degree of the difference

**Fig. 13.8 Neutron spectrum at fuel region evaluated in time-dependent mode (13 fuel rods)**

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):

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*, ϕα 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

**Fig. 13.9 Comparison of neutron spectrum among several cases**

supercritical state. However, the difference is expected to be not remarkable compared to subcritical state, because excess reactivity is remarkably small compared to the subcriticality, as readily expected.

# Conclusions

The neutron flux distribution evaluated in time-dependent mode changes with elapsed time after the ignition of neutrons into the subcritical system, and the neutron distribution in energy and space becomes almost stable in about 1 e-4 s after the ignition.

There is a remarkable difference in neutron spectrum between two results in

*k*-eigenvalue and time-dependent modes. The neutron spectrum (at 1 μs after the ignition) evaluated in time-dependent mode is softer than that in *k*-eigenvalue mode, and the difference is more remarkable in a deep subcriticality system. This difference is caused by the fact that additional time is necessary to be moderated before decreasing neutron flux in the thermal energy range, and the time is independent of the subcriticality of the system, depending only on the material composition of the system.

The neutron spectrum of a pulsed neutron reactor is to be evaluated in alpha-

eigenvalue mode instead of *k*-eigenvalue mode to match the neutron spectrum during the decrease with elapsed time after the ignition of pulsed neutrons into the subcritical system.