1 Introduction

The nuclear fission of actinides begins at the saddle point of the fission barrier when the fissile nucleus is deformed without return and ends at the scission point, from which the fragments separate acquiring their final kinetic energy by Coulomb repulsion. Before and after the scission point neutrons are emitted ^[1].

One of the quantities proper to study fission dynamics is the average prompt neutron multiplicity as a function of the primary mass of fragments ν-. However, because prompt neutron emission, it is not possible to measure the primary fragment mass; and the experimental values associated with ν- as a function of fragment mass depend on the used measurement technique. This has been shown for reactions ²³³U(nth, f), ²³⁵U(nth, f), ²⁵²Cf(sf) ^[2-10], ²³⁹Pu(nth, f) ^[11-15], and spontaneous fission of ²⁵²Cf ^[16], respectively.

Recently, a Monte Carlo simulation of the measurement by the 1V1E technique has been performed and applied to the case of ²³⁹Pu(nth, f) ^[17]. In this work, a simulation of the measurement by the 2E technique is performed and applied to that reaction.

The 2E technique consists of calculating the mass of the complementary fission fragments using the measured values of kinetic energy in the momentum conservation relations. Furthermore, conservation of the number of nucleons in the reaction is assumed.

The uncertainty of the value of the kinetic energy is proportional to the number of neutrons emitted before the fragment reaches the detector. The average number of neutrons depends on the mass of the fragment. Added to this is the resolution of the energy measurement, which depends on the detector. As an example, the total resolution of the mass of the Göök et al. experiment is 4 u ^[18].

2 Input data of the simulation

In the simulation of neutron emission, the state of deformation of the fragments is not considered. Furthermore, the symmetry or asymmetry characteristics of the kinetic energy distribution of the fragments are not distinguished.

The input data are quantities associated with the primary complementary fragments, i.e., the mass numbers (A and A’, respectively), the kinetic energy (E and E’, respectively) and the number of neutrons emitted by each fragment (n and n’, respectively).

As a function of one of the primary complementary fragments mass (A), we use the following quantities: the yield Y (Data taken from Ref. ^[19]. See Fig. 1); a Gaussian distribution of total kinetic energy of complementary fragments (TKE = E + E’) with its average TKE¯ (Data taken from Ref. ^[19]. See Fig. 2), and its standard deviation σTKE (An approximation of the experimental curve taken from Ref. ^[20]. See Fig. 3); the average prompt neutron multiplicity ν-s (A sawtooth approach of data taken from Ref. ^[13]. See Fig. 4); and the slope of the neutron multiplicity against total kinetic energy, dv/d < TKE >(An approach to experimental data taken from Ref. ^[13], See Fig. 5). The prompt neutron multiplicity as a function of total kinetic energy is assumed to be

Figure 1 Simulation of the measurement by the 2E technique of the mass yield of fragments from the ²³⁹Pu(nth,f) reaction. We show the yield of the primary fragment mass (Y(A) diamonds, taken from Ref. ^[19].) and the yield of simulated as measured (by 2E technique) fragment mass ( y(μ), squares), which are the input and output data, respectively, in the simulation. 

Figure 2 Simulation of the measurement by the 2E technique of the average total kinetic energy as a function of the mass of fragments from the ²³⁹Pu(nth,f) reaction. We show the average total kinetic energy as a function of the primary fragment mass (diamonds, values taken from Ref. ^[19]) and the corresponding to the pseudo mass (squares), which are the input and output data, respectively, in the simulation. 

Figure 3 Simulation of the measurement by the 2E technique of standard deviation of the total kinetic energy distribution as a function of the mass of fragments from the ²³⁹Pu(nth, f) reaction. We show the standard deviation of the total kinetic energy distribution as a function of the primary fragment mass (diamonds) and the corresponding to the pseudo mass (squares), which are the input and output data, respectively, in the simulation. For comparison, experimental values (triangles, from Ref. ^[20]) are presented. 

Figure 4 The average neutron multiplicity (measured by the 2E technique), as a function of the mass of fragments from the ²³⁹Pu(nth,f) reaction. We show the simulated average prompt neutron multiplicity as a function of the primary fragment mass ( ν-s, diamonds) and the corresponding to fragment pseudo mass output of simulation ( ν-μ, squares), compared with the corresponding to the experimental data taken from Ref. ^[13] ( ν-, triangles). 

Figure 5 Simulation of the measurement by the 2E technique of the average neutron multiplicity as a function of the mass of fragments from ²³⁹Pu(nth, f) reaction. The slope of neutron multiplicity against total kinetic energy, -dv/d <TKE> (MeV^-1), plotted as a function of primary fragment mass (diamonds) an approach from the experimental data (circles) from Ref. ^[13]. 

Based on the anti-correlation of neutron multiplicity associated with the complementary fission fragments observed by Vorobyev et al. ^[21] in the case of the reaction ²⁵²Cf (sf), a term obeying a Gaussian distribution with an average 0 and a standard deviation ν-s(A)/3 is added to νs(A,TKE) and a similar term with a standard deviation ν-s(240-A)/3 is subtracted from νs(240-A,TKE).

The conservation relations of mass, energy, and linear momentum of the primary fragments with masses A and A’, are the following:

Where A₀ = 240, and

3 Output data of the simulation

The output data are quantities assumed to be measured by the 2E technique associated with the complementary fragments, i.e., the mass ( μ and μ', respectively) and the kinetic energy ( e and e', respectively). For the output data, as a function of the fragment mass calculated by the 2E method (μ), the following quantities are obtained: the yield (y. See Fig. 1); the distribution of the total kinetic energy of final complementary fragments (tke=e+e') with its average ( tke¯. See Fig. 2) and its standard deviation ( σtke. See Fig. 3); and the average prompt neutron multiplicity ( ν-μ. See Fig. 4). Due to the loss in mass by neutron evaporation, the values of the kinetic energy of final complementary fragments are, approximately,

and

With the double energy technique, e and e' are measured. To calculate the “pseudo” mass of complementary fragments (μ and μ', respectively) the conservation relations 2 and 3 are applied to those quantities, i.e.,

and

From (4)-(6), we obtain

where

4 Results

In the fragment mass region around A ≈ 15, referred to the Y(A) curve, the yield curve of the calculated mass y(μ) is shifted to higher mass values. See Fig. 1. This effect occurs because, in that region, in average, n>n' (See Fig. 4) and A ≈ A’, then T < 1. Consequently, from (7), we deduce that

In the fragment mass region around A ≈ 125, referred to the Y(A) curve, the yield of the calculated mass y(μ) is shifted to lower mass values. See Fig. 1. This effect occurs because, in that region, in average, n<n' (See Fig. 4) and A ≈ A’, then T > 1. Consequently, from (7), we deduce that

In general, the average final total kinetic energy (tke¯) is lower than the average primary total kinetic energy (TKE¯). See Fig. 2. This is explained by (4a) and (4b).

Curve TKE¯ shows a discontinuity in A = 140. In the same pseudo mass value, i.e., μ = 140, σtke is higher than the average between the neighboring values corresponding to μ =139,141. See Fig. 2 and Fig. 3, respectively. This result is because primary masses neighboring to 140, with different kinetic energy distributions, converge to μ = 140. Similar behavior occurs in the mass region around A = 122, 123.

Figure 4, we represent the simulated average of prompt neutron multiplicity as a function of the pseudo mass ν-μμ. In the mass regions around A = 115 and A > 150, where μ > A and Y_A is a decreasing function of fragment mass, an overestimation of ν-μ(μ) referred to ν-s is observed. Our results on ν-μ(μ) reproduce the experimental data of Tsuchiya et al.^[13] and Göök et al.^[18], respectively. See Fig. 6. To interpret those results, we use the definition of the average prompt neutron multiplicity as a function of pseudo mass, i.e.,

Figure 6 The average neutron multiplicity (measured by the 2E technique), as a function of the mass of fragments from the ²³⁹Pu(nth, f) reaction. We show the simulated average prompt neutron multiplicity as a function of the primary fragment mass ( ν-s, diamonds) and the corresponding to the fragment pseudo mass output of simulation ( ν-μ, squares), compared with the corresponding to the experimental data of Tsuchiya et al.^[13] and with Göök et al.^[18], respectively. 

where YA=μ-n is the yield of primary fragment mass μ - n that emitted n neutrons. From this equation, we can deduce that if the mass yield Y_A is constant as a function of primary mass A, then ν-μ=ν-A; and, if it is a decreasing (increasing) function of A, then ν-μ>ν-A (ν-μ<ν-A).

5 Discussion

We simulated the measurement, by the 2E technique, of the average prompt neutron multiplicity as a function of fragment mass for the ²³⁹Pu(nth, f) reaction. Compared to input values ν-s(A), an overestimation of the output values ν-μ(μ) in the mass regions A ≈ 115 and A > 150, respectively, is observed. This behavior is because, in those mass regions, there are two conditions: i) Y is a decreasing function of A and ii) μ > A. The result of the simulation suggests that, in those mass regions, the 2E experimental values ν-e, referred to the primary quantities ν-, are overestimated.