Neutron fluctuations: The importance of being delayed

The neutron population in a nuclear reactor is subject to fluctuations in time and in space due to the competition of diffusion by scattering, births by fission events, and deaths by absorptions. As such, fission chains provide a prototype model for the study of spatial clustering phenomena. In order for the reactor to be operated in stationary conditions at the critical point, the population of prompt neutrons instantaneously emitted at fission must be in equilibrium with the much smaller population of delayed neutrons, emitted after a Poissonian time by nuclear decay of the fissioned nuclei. In this work, we will show that the delayed neutrons, although representing a tiny fraction of the total number of neutrons in the reactor, actually have a key impact on the fluctuations, and their contribution is very effective in quenching the spatial clustering.

Figure 1

Monte Carlo simulation of a collection of ${10}^3$ branching Brownian motions at four successive times: (a) $t=0$, (b) $t=10$, (c) $t=50$, (d) $t=100$. Starting from a uniform spatial distribution at the initial time, the population later shows a wild patchiness due to the interplay between diffusion, reproduction, and absorption.

Figure 2

Simplified sketch of fission chains in a water-moderated reactor. A chain begins with a neutron emitted from a fission event (1) in the fuel (F). The neutron diffuses (2) in water (W) and may later come back to the fuel, where it can either undergo an absorption event (3, in which case the fission chain comes to a halt, since no further neutrons are re-emitted), or a new fission event (4). In this latter case, a number of prompt neutrons are emitted instantaneously. At a later time, following the decay of the excited fissioned nuclei, a small fraction of delayed neutrons are also emitted (5) and start diffusing in water (6). Both prompt and delayed neutrons contribute to sustaining the fission chains.

Figure 3

A scheme of the key events involved in the evolution of the neutron and precursor populations. (a) At fission, a random number of $i$ prompt neutrons and $j$ precursors are created with rate ${\alpha }_{i,j}$, and the incident neutron is lost. (b) At absorption, with rate $\mu$, the incident neutron is lost. (c) Upon decay, with rate $\lambda$ a precursor gives rise to a delayed neutron.

Figure 4

Evolution of the average neutron population $\langle n\left(t\right)\rangle$ (a) and the average precursor population $\langle m\left(t\right)\rangle$ (b), starting from a zero-reactivity equilibrium condition. Solid lines are the exact solutions in Eqs. (B2), and dashed lines are the asymptotic solutions provided in Eqs. (6). The parameters are the following: $n_0={10}^3,\eta =8.333\times {10}^{-3},\alpha {\nu }_m=1.2$, and $\lambda ={10}^{-2}$. Red (upper) curves correspond to a supercritical reactor with $\rho =5\times {10}^{-3}$, green (lower) curves correspond to a subcritical reactor with $\rho =-5\times {10}^{-3}$, and blue (central) curves correspond to an exactly critical reactor with $\rho =0$.

Figure 5

Evolution of the normalized and centered second moment of the neutron population $u\left(t\right)$, starting from a zero-reactivity equilibrium condition. Solid lines are the numerical solutions of the exact Eq. (16), and dashed lines are the asymptotic solutions provided in Eqs. (21) and (26). The parameters are the following: $n_0={10}^3,\eta =8.333\times {10}^{-3},\alpha {\nu }_m=1.2,\lambda ={10}^{-2},{\nu }_n^{\left(2\right)}=2,{\nu }_{nm}=2.4\times {10}^{-2}$, and ${\nu }_m^{\left(2\right)}=4\times {10}^{-3}$. Red (lower) curves correspond to a supercritical reactor with $\rho =5\times {10}^{-3}$, green (upper) curves correspond to a subcritical reactor with $\rho =-5\times {10}^{-3}$, and blue (central) curves correspond to an exactly critical reactor with $\rho =0$. The dotted-dashed black line corresponds to the asymptotic value $u_{\infty }$ expected for the supercritical configuration, as given in Eq. (25).

Figure 6

Evolution of the normalized and centered second moments $v\left(t\right)$ (a) and $w\left(t\right)$ (b), starting from a zero-reactivity equilibrium condition. Solid lines are the numerical solutions of the exact Eqs. (17) and (18), respectively, and dashed lines are the asymptotic solutions provided in Eq. (22) [Eq. (27) for the critical case] and Eq. (22) [Eq. (27) for the critical case], respectively, for small reactivities and $\eta \ll 1$. The parameters are the following: $n_0={10}^3,\eta =8.333\times {10}^{-3},\alpha {\nu }_m=1.2,\lambda ={10}^{-2},{\nu }_n^{\left(2\right)}=2,{\nu }_{nm}=2.4\times {10}^{-2}$, and ${\nu }_m^{\left(2\right)}=4\times {10}^{-3}$. Red (lower) curves correspond to a supercritical reactor with $\rho =5\times {10}^{-3}$, green (upper) curves correspond to a subcritical reactor with $\rho =-5\times {10}^{-3}$, and blue (central) curves correspond to an exactly critical reactor with $\rho =0$. The dotted-dashed black line corresponds to the asymptotic value $v_{\infty }=w_{\infty }$ expected for the supercritical configuration, as given in Eq. (25).

Figure 7

Evolution of the normalized and centered pair correlation function $u(r,t)$ for a one-dimensional domain (a), starting from a zero-reactivity equilibrium condition. Solid lines are the numerical solutions of the exact Eq. (45), and dashed lines are the asymptotic solutions provided in Eqs. (21) and (26). The pair correlation functions are displayed at time $t=3\times {10}^4$, with the following parameters: ${\mathcal{N}}_0={10}^3,\eta =8.333\times {10}^{-3},\alpha {\nu }_m=1.2,\lambda ={10}^{-2},{\nu }_n^{\left(2\right)}=2,{\nu }_{nm}=2.4\times {10}^{-2}$, and ${\nu }_m^{\left(2\right)}=4\times {10}^{-3}$. The presence of a peak close to the origin is the signature of spatial clustering. Red (lower) curves correspond to a supercritical reactor with $\rho =5\times {10}^{-3}$, green (upper) curves correspond to a subcritical reactor with $\rho =-5\times {10}^{-3}$, and blue (central) curves correspond to an exactly critical reactor with $\rho =0$. (b) An inset displays the same curves with a logarithmic scale on the ordinate axis.

Figure 8

Evolution of the normalized and centered pair correlation function $v(r,t)$ (a) and $w(r,t)$ (b) for a one-dimensional domain, starting from a zero-reactivity equilibrium condition. Solid lines are the numerical solutions of the exact Eqs. (46) and (47), respectively, and dashed lines are the asymptotic solutions provided in Eqs. (22) [Eq. (27) for the critical case] and (23) [Eq. (28) for the critical case], respectively. The pair correlation functions are displayed at time $t=3\times {10}^4$, with the following parameters: ${\mathcal{N}}_0={10}^3,\eta =8.333\times {10}^{-3},\alpha {\nu }_m=1.2,\lambda ={10}^{-2},{\nu }_n^{\left(2\right)}=2,{\nu }_{nm}=2.4\times {10}^{-2}$, and ${\nu }_m^{\left(2\right)}=4\times {10}^{-3}$. Red (lower) curves correspond to a supercritical reactor with $\rho =5\times {10}^{-3}$, green (upper) curves correspond to a subcritical reactor with $\rho =-5\times {10}^{-3}$, and blue (central) curves correspond to an exactly critical reactor with $\rho =0$.