Structures in the energy distribution of the scission neutrons: Finite neutron-number effect

The scission neutron kinetic energy spectrum is calculated for ${}^{236}\mathrm{U}$ in the frame of the dynamical scission model. The bidimensional time-dependent Schrödinger equation with a time-dependent potential is used to propagate each neutron wave function during the scission process, which is supposed to last $1\times {10}^{-22}$ sec. At the end, we separate the unbound parts and continue to propagate them as long as possible (in this case $50\times {10}^{-22}$ sec) in the frozen fragments approximation. At several time intervals, the Fourier transforms of these wave packets are calculated in order to obtain the corresponding momentum distributions, which lead to the kinetic energy distributions. The evolution of these distributions in time provides an interesting insight into the separation of each neutron from the fissioning system and asymptotically gives the kinetic energy spectrum of that particular neutron. We group the results in substates with given projection $\mathrm{\Omega }$ of the angular momentum on the fission axis to study its influence on the spectrum. Finally, the sum over all $\mathrm{\Omega }$ values is compared with a typical evaporation spectrum as well as with recent precise measurements in the reaction ${}^{235}\mathrm{U}(n_{th},f)$. Structures are present both in the scission-neutron spectrum and in the data.

Figure 1

Square modulus of the unbound ${\mathrm{WF}}_{22}$ (left column) and energy distribution (right column) at different times $T$. The wave functions at $T=0$ and $50\times {10}^{-22}$ sec are represented relative to those at $T=20\times {10}^{-22}$ sec. The values on the ordinates of the histograms are $P_{22}\left(E_{\mathrm{kin}}\right)$ probabilities multiplied by 100. $E_{\mathrm{kin}}^{\mathrm{mean}}=\frac{{\sum }_{m,n}{E}_{\mathrm{kin}}P{E}_{\mathrm{kin}}^{1/2}}{{\sum }_{m,n}P{E}_{\mathrm{kin}}^{1/2}}$ where $P=k_{\rho }{\left|F\right|}^{2}d{k}_{\rho }d{k}_z$. $N$ is the probability that the wave function is inside the nucleus at a given time $T$. The equipotential line corresponding to $V_0/2$ is also plotted (left column, as plain blue line) to position the fragments just before scission.

Figure 2

The same as in Fig. 1 but for the unbound ${\mathrm{WF}}_{26}$.

Figure 3

The same as in Fig. 1 but for the unbound ${\mathrm{WF}}_{28}$.

Figure 4

Square modulus of the unbound ${\mathrm{WF}}_{11}^{5/2}$ (left column) and energy distribution (right column) at different times $T$. The projection of the angular momentum on the fission axis of this wave function is $\mathrm{\Omega }=5/2$. The wave functions at $T=0$ and $50\times {10}^{-22}$ sec are represented relative to those at $T=20\times {10}^{-22}$ sec. The values on the ordinates of the histograms are $P_{11}\left(E_{\mathrm{kin}}\right)$ probabilities multiplied by 100. $E_{\mathrm{kin}}^{\mathrm{mean}}=\frac{{\sum }_{m,n}{E}_{\mathrm{kin}}P{E}_{\mathrm{kin}}^{1/2}}{{\sum }_{m,n}P{E}_{\mathrm{kin}}^{1/2}}$ where $P=k_{\rho }{\left|F\right|}^{2}d{k}_{\rho }d{k}_z$. $N$ is the probability that the wave function is inside the nucleus at a given time $T$.

Figure 5

The same as in Fig. 4 but for the unbound ${\mathrm{WF}}_{12}^{5/2}$.

Figure 6

The same as in Fig. 4 but for the unbound ${\mathrm{WF}}_{13}^{5/2}$.

Figure 7

Square modulus of the unbound ${\mathrm{WF}}_{04}^{7/2}$ (left column) and energy distribution (right column) at different times $T$. The projection of the angular momentum on the fission axis of this wave function is $\mathrm{\Omega }=7/2$. The wave functions at $T=0$ and $50\times {10}^{-22}$ sec are represented relative to those at $T=20\times {10}^{-22}$ sec. The values on the ordinates of the histograms are $P_{04}\left(E_{\mathrm{kin}}\right)$ probabilities multiplied by 100. $E_{\mathrm{kin}}^{\mathrm{mean}}=\frac{{\sum }_{m,n}{E}_{\mathrm{kin}}P{E}_{\mathrm{kin}}^{1/2}}{{\sum }_{m,n}P{E}_{\mathrm{kin}}^{1/2}}$ where $P=k_{\rho }{\left|F\right|}^{2}d{k}_{\rho }d{k}_z$. $N$ is the probability that the wave function is inside the nucleus at a given time $T$.

Figure 8

The same as in Fig. 7 but for the unbound ${\mathrm{WF}}_6^{7/2}$.

Figure 9

Kinetic energy distributions at $T=50\times {10}^{-22}$ sec for substates defined by the quantum number $\mathrm{\Omega }$. $\text{Yield}=\sum P_i\left(E_{\mathrm{kin}}\right)\times v_i^2$.

Figure 10

Kinetic energy distributions at $T=50\times {10}^{-22}$ sec calculated with all neutron states together with recent experimental results [20] from the reaction ${}^{235}\mathrm{U}(n_{th},f)$. Two typical evaporation spectra [25] characterized by nuclear temperatures $\text{Temp}=1.0$ and $0.9\mathrm{MeV}$ are also plotted for comparison. The EVN spectra and the SN histogram are normalized to the data. $\text{Yield}=\sum P_i\left(E_{\mathrm{kin}}\right)\times v_i^2$.

Figure 11

The SN spectrum convoluted with Gaussian resolution functions with half-width equal 0.3 and 0.4 MeV.

Figure 12

The time evolution of the energy distribution for scission neutrons with $\mathrm{\Omega }=1/2$.

Figure 13

Square modulus of the total and emitted ${\mathrm{WF}}_{13}$ at $\mathrm{\Omega }=1/2$ immediately after scission (left column) and the corresponding energy distributions (right column). $\mathrm{N}\uparrow$ and $\mathrm{N}\downarrow$ are the square moduli of the spin-up and spin-down components.

Figure 14

The same as in Fig. 13 but for ${\mathrm{WF}}_{14}$ at $\mathrm{\Omega }=1/2$.

Figure 15

The same as in Fig. 13 but for ${\mathrm{WF}}_{28}$ at $\mathrm{\Omega }=1/2$.

Figure 16

Square modulus of the total and emitted ${\mathrm{WF}}_{14}$ at $\mathrm{\Omega }=5/2$ immediately after scission (left column) and the corresponding energy distributions (right column).

Figure 17

The same as in Fig. 16 but for ${\mathrm{WF}}_{15}$ at $\mathrm{\Omega }=5/2.$