2.1 The incident neutron spectrum

Fast neutron generators are used world-wide for all kinds of purposes, such as nuclear data measurement, radiation hardening and radioactive breeding etc. The neutrons from D-T neutron generator are suitable for inducing fission reactions research. Lanzhou University has developed an intense neutron generator based on T(d,n)⁴He reaction with a rotating target, to generate 14.0 MeV mono-energetic neutrons by 300 keV deuteron beams from an accelerator. The fast neutrons produced with a thick target have distinct angular distribution and intrinsic energy spectrum ^[29]. For an incident deuteron energy E_d, the neutron spectrum with a given angle θ and energy E can be calculated by Eq.(1)^[30]:

where, d²Y/(dE_ndΩ_n) is the double differential thick target neutron yields, N is atomic number density of the target, d²σ/(dE_ndΩ_n) is the neutron-produced double differential cross-section, E₀ is the incident deuteron energy, E_d is the deuteron energy in the target, dE_d/dx is the stopping power of the target, and ∑_non(E') is the macroscopic total reaction cross-section. In this work, we calculated the neutron spectrum with incident deuteron beams of 260 keV, as shown in Fig.1.

Fig.1Full-screen View

The outgoing neutron energy spectrum distribution of D-T neutron source

2.2 Sample

Uranium dioxide (100.0% purity) was used as the target. In order to determine optimum thickness of the ²³⁸U sample, we chose ¹²⁸Te and ⁹²Zr as representative nuclides to calculate the different share of fission fragments deposition, as function of the sample thickness. As shown in Fig.2, the deposition share reaches more than 99.90 % at 0.8 mm thickness, where it begins changing slowly with the thickness. Therefore, 1.0 mm target thickness (over 99.95 % fragments deposition share in the sample) was selected.

Fig.2Full-screen View

Simulated deposition share of the fission-fragments, as function of target thickness.

A hollow cylindrical target of Φ20 mm in wall thickness of 1.0 mm was used in the simulation. The target was positioned on the deuteron beam axis. Fission products identified in each step of nuclear reaction inside the sample were considered in Geant4, whereas fission neutrons and decay γ-rays were recorded in the entire 4π steradian.

3.1 Fission yields distribution

The FPY distribution is a key to fission reaction. In the ²³⁸U(n,f) fission process, the fission fragments were simulated in their independent and cumulative yields. The former considered the initial fission-fragment yields only, while the latter took all the fission products including the initial fragments and the β-decayed nuclides from the excited state of fission-fragments.

The independent FPY yields vs. Z and N is shown in Fig.3. The fission fragments spread over upper left of the β-stability line in charge number Z= 35–55 and neutron number N= 60–85, then decay to the bottom of the stability ground state.

Fig.3Full-screen View

Independent yields distribution of the ²³⁸U(n,f) fission reaction induced by D-T neutrons. The color scale refers to the number of events.

The simulated independent FPY distribution was compared with the nuclear data from ENDF/B-VII.I library. As shown in Fig.4, they agree well with each other. According to the nuclear fission theory, the pre-neutron emission mass distribution of neutron-induced antinides can be treated as symmetrical fission which obeys to certain specific rules. As a result, the heavy and light fission-fragment masses have their own symmetrical peak. The heavy-mass peak in fragment-mass distributions is roughly constant and close to A₂=140, and the complementary light-mass peak in fragment-mass distributions can be calculated by A₁=A_f −A₂, where A_f is the fission nucleus mass number. There is a deep valley in symmetrical fission-fragment mass distribution center position A₀=A_f/2. From our simulation, the heavy-mass peak in fragment-mass distributions is at A₂=137, corresponding with the Five-dimensional energy landscapes model ^[31],which gives a less deviation (<5%) than the phenomenological fission potential model ^[32] that assumes A₂=140 for the mass number of heavy fragment. 10⁻⁴

Fig. 4Full-screen View

Comparison of independent FPY from ²³⁸U (n, f) reaction between the calculated and recommended data. (a) The elemental yield versus charge number. (b) The mass chain yield versus mass number.

In Fig. 5, the calculated cumulative FPY distributions for isobaric chain are compared with the experimental data from EXFOR and the evaluated data from ENDF/B-VII.1. In Fig.5(a), the calculated results agree well with the evaluated data, but are higher than the experimental data within error bars. In Fig. 5(b), the three data sets are of good consistency in a wide range, but for A<75 and A>160, our calculation results are lower than the evaluated data.

Fig. 5Full-screen View

Calculated FPY of ²³⁸U (n,f) reaction as function of the charge number (a) and mass number (b), comparing with the recommended data.

In the ²³⁸U(n, f) reaction induced by 14 MeV neutrons, the yields of light fission fragment around mass number of 94 and the heavy fission fragment around mass number 140 are found to be higher because of the nuclear structure effects and suitable N/Z values. Meanwhile, the FPY distribution does not have a third peak around symmetric mass region, due to the lower excitation energy of ²³⁸U(n,f) reaction induced by 14 MeV neutrons, leading to asymmetric fission of fission-fragment with mass number. So, the physical models in Geant4 code are suitable for simulating the neutron-induced fission process for actinide nuclide.

3.2 Fission-fragment kinetic energy distribution

In the fission reaction process of ²³⁸U induced by fast neutrons, the ²³⁸U nucleus splits into two fission fragments with high kinetic energies. They separate and fly away under the action of Coulomb repulsion. Fig.6 shows the relationship between the kinetic energy and mass number of the fission fragments. The nuclides with magic number of nucleons are much more than other fission fragments. The primary kinetic energy of each heavy fission fragment is in the order of 50–110 MeV, which means that the primary fission fragments can be in velocities of 10⁷ m/s, before they slow down via elastic-scatting inside the uranium sample. The huge kinetic energy of fission fragment transforms into the thermal energy or emits a light particle to ground-state in de-excitation process and releases its kinetic energy in elastic-scatting process, which is important for reactor design especially in selecting reactor materials.

Fig.6Full-screen View

The primary FPY kinetic energy distribution from ²³⁸U(n,f) reaction induced by D-T neutrons. The color scale refers to the number of events.

Figure 7 shows the average and total fission-fragment kinetic energies as function of the mass number. According to the momentum conservation law, the light fission-fragments have higher kinetic energy and the heavy have lower kinetic energy. The total kinetic energy is the sum of kinetic energies of a complementary fragment pair. From Fig.7 (b), the total kinetic energy of heavy fission-fragment reaches to the maximum value at mass number of 132 due to the nuclear shell structure effects. The structure of fission fragment with mass number of 132 (Z=50, A=82) approximates a sphere that help to minimize the center-to-center distance of two fission fragments contributes to bigger fission fragment kinetic energy.

Fig.7.Full-screen View

The relationship between the fission-fragment kinetic energy distribution and the fission-fragment mass number. (a) The average kinetic energy distribution. (b) The total kinetic energy distribution.

3.3 Fission neutron spectrum

The fission neutron spectrum, including prompt and delayed neutrons, is of importance in basic research and reactor design. The prompt neutrons are mainly evaporated by the primary neutron-rich fission fragments far from the β-stability line, with high excited energy. They account for over 99% of the total fission neutrons. The delayed neutrons are produced from the β-ray decayed of fission fragments. Fig.8 shows the neutron spectrum from the ²³⁸U(n,f) reaction induced by D-T neutrons.

Fig.8Full-screen View

The fission neutron spectrum of ²³⁸U(n,f) reaction induced by D-T neutrons.

The total fission neutron energy distribution obeys Maxwell-Boltzmann distribution, which can be calculated by:

where, f(E) is the fission neutron energy spectrum, E is the neuron energy, parameter kT = 1.38 is the Maxwell temperature, and n₀ = 2.11×10⁻⁶ is the proportional coefficient of the function.

We also calculated the total nubar (the average fission-neutron count) of ²³⁸U(n,f) reaction at different energies of the incident neutron beams. The results are compared with experimental data in Ref. [33] (Fig.9). The average fission-neutron count (νn) changes mainly with the cross-section of ²³⁸U(n,f) reaction, in a positive correlation with incident neutron energy. The excitation energy of ²³⁸U fission system increases with the incident neutron energy as more neutrons are evaporated immediately in the order of 10^-15s.

Fig.9.Full-screen View

The total nubar of ²³⁸U (n,f) reaction as function of the incident neutron energy.

3.4 Decay γ-ray spectrum of ²³⁸U (n, f) reaction

In the neutron-induced fission reaction, the residual nucleus emits characteristic γ-ray in its de-excitation process, and can be thus identifies by analyzing the decay γg-ray spectrum. The cumulative γ-ray spectra of fission products from ²³⁸U(n,f) reaction induced by D-T neutron source, of different cooling times, are shown in Fig.10.

Fig.10Full-screen View

The cumulative γ-ray spectrum from ²³⁸U fission reaction with different cooling time.

The precise fission-fragment γ-ray data of fission productions ^[34] for analyzing fission yield is partly presented in Table.1. It is possible to distinguish over 34 fission products (Fig.10) of different half-lives using correlative calculation process ^[35]. This is helpful for our future experimental measurements for the cumulative fission yields of ²³⁸U(n,f) reaction.