Influence of Prompt Neutron Emission on the Final Distribution of Mass, Kinetic Energy, and Charge of Fragments from Actinide Fission

Abstract

In nuclear fission of actinides samples, the fragments emit prompt neutrons before they reach the detectors. This chapter shows the effects of that emission on the final distribution of mass, charge, kinetic energy, and prompt neutron multiplicity of fragments. Those effects depend on the experimental technique. This chapter shows the curve of the maximal value of total kinetic energy as a function of primary fragment mass for the reactions 233U(nth, f), 235U(nth, f), and 239Pu(nth, f) which reflect the deformation properties of fragments in their ground states and the Coulomb effect on their scission configurations. The odd–even effects on the cold fission region will be also presented.

1. Introduction

In 1939, Otto Hahn and Fritz Strassman discovered the fission of the uranium nucleus induced by neutrons [1], which was experimentally corroborated by Meitner and Frisch [2]. In 1939, von Halban et al. [3] discovered that fission fragments emit neutrons. To interpret the experimental results, Bohr and Wheeler [4] proposed a model in which the potential surface of the fissile nucleus is a function of several variables that represent the deformation of the nucleus, which starts from a well of stability to rise through the potential surface and reach the saddle point. From the saddle point, the system descends through a fission valley (where the attractive nuclear interaction and the repulsive Coulomb interaction compete) to the scission point where finally two separate complementary fragments are formed. The interpretation of the fission process may be based on the potential surface of the fissile system [5].

From the scission point, the two independent fragments are repelled by electrostatic interaction until they reach their final kinetic energy. On the way, the fragments decay mainly by gamma and neutron emission.

In that scheme, to investigate the nature of fission, it results necessary to know at least the characteristics of the final state of the fission process, i.e., the scission configuration. For this reason, the goal of most fission experiments is to see how the process might manifest itself in the distribution of primary fragment mass and kinetic energy distribution.

In this chapter, the effects of the emission of prompt neutrons on the distribution of mass, charge, and kinetic energy of the fragments, compared to the primary quantities will be reported. We will see how the experimental technique affects the results of their measurements.

2. Kinetic energy distribution as a function fragment mass

Let us list some definitions and relations to be used to show the effects of neutron emission on the distribution of mass, charge, kinetic energy, and average prompt neutron multiplicity of the fission fragments.

The fissile nucleus is characterized by its charge and mass $Z_0$ and $A_0$, respectively. The primary complementary fragments are characterized by their masses ($A,A^{\prime }$), charges ($Z,Z^{\prime }$), and kinetic energies ($E,E^{\prime }$), respectively.

The conservation laws of mass and linear momentum are expressed in the relations.

$$A_0=A+A^{\prime },$$E1

and

$$\mathit{AE}=A^{\prime }{E}^{\prime },$$E2

respectively.

The total kinetic energy is.

$$\mathit{TKE}=E+E^{\prime },$$E3

The final complementary fragments masses.

$$m=A-n$$E4

and

$$m^{\prime }=A^{\prime }-n^{\prime },$$E5

where $n$ and $n^{\prime }$ are their corresponding number of emitted prompt neutrons.

For each isobaric fragmentation, the values of $n$ and $n^{\prime }$ are given approximately by the relations.

$$n=\overline{\nu }\left(1-\frac{\mathit{TKE}-\overline{\mathit{TKE}}}{\alpha }+r\right)$$E6

and

$$n^{\prime }={\overline{\nu }}^{\prime }\left(1-\frac{\mathit{TKE}-\overline{\mathit{TKE}}}{\alpha }-r\right),$$E7

where $\overline{\mathit{TKE}}$ is the average total kinetic energy, $\overline{\nu }$ and $\overline{\nu \prime }$ are the average prompt neutron multiplicities, respectively, $\alpha$ is a parameter taken from experimental data, and $r$ is a correlation term.

Due to the emission of neutrons, the values of the final kinetic energy of the fragments will be, approximately,

$$e=E\left(1-\frac{n}{A}\right),$$E8

and

$$e^{\prime }=E^{\prime }\left(1-\frac{n^{\prime }}{A^{\prime }}\right)$$E9

These relations are formulated considering no recoil effect by neutron emission. If one simulates an isotropic prompt neutron emission relative to each fragment center of masses, the result is a kinetic distribution characterized by an average obeying approximately the same relations with standard deviations depending on the emitted neutron kinetic energy. The result is equivalent to a lower resolution in kinetic energy measuring.

The total kinetic energy of the final fragments is defined as

$$\mathit{tke}=e+e^{\prime }$$E10

Experimentally one has access to the values of final fragment mass and kinetic energy. The fragment masses measured by the double energy method ($2E$), defined as provisional masses of the complementary fragments, $\mu$ and ${\mu }^{\prime }$, are calculated using the conservations relations.

$$A_0=\mu +{\mu }^{\prime },$$E11

and

$$\mathit{\mu e}={\mu }^{\prime }{e}^{\prime }.$$E12

From the Eqs. (8), (9), (11), and (12), one obtains.

$$\mu =A_0\frac{e^{\prime }}{e^{\prime }+e}=A_0\frac{E^{\prime }}{E^{\prime }+E\times F}=A_0\frac{A}{A+A^{\prime }\times F},$$E13

where

$$F=\frac{1-n/A}{1-n^{\prime }/A^{\prime }}.$$E14

The provisional mass $\mu$ may be lower or higher than $A$, depending on $F$. For the reactions ²³³U(n_th, f), ²³⁵U(n_th, f), and ²³⁹Pu(n_th, f), there is a fragment mass region where $\overline{\nu }\gg {\overline{\nu }}^{\prime }$. In this case $\mu >A$. Obviously, $m$ will be equal or lower than $A$.

2.1 Kinetic energy distribution as a function of the final fragment mass

In 1978, using the Monte Carlo method, the experiment to measure the distribution of $e$ as a function of $m$ of fragments from the fission induced by thermal neutrons of ²³⁵U was simulated. The simulation was carried out before the corresponding experiment. The simulated standard deviation of the final light fragment kinetic energy ${\sigma }_e$ showed a peak of 9 MeV for $m=109$ in a mass region where the primary standard deviation is approximately 5 MeV. This peak was the result of the neutron emission combined with the effects of the slopes of the mass yield and the average kinetic energy of the fragments. Experimental results, obtained on the LOHENGRIN mass spectrometer, confirmed that prediction [6]. See Figure 1.

Figure 1.

Standard deviation of the final kinetic energy distribution as a function of the final mass ($m$) of fragments from fission of ²³⁵U induced by thermal neutrons. The output of the Monte Carlo simulation (squares) is to be compared with the experimental result (triangles) [6].

In 1983, Behafaf et al. repeated that experiment extending the domain of measured masses to the heavy region, finding a peak in the standard deviation for $m=125$, which was not reproduced by their Monte Carlo simulation [7]. See Figure 2. In 2007, through another Monte Carlo simulation, Montoya et al. reproduced the mentioned peak [8]. See Figure 3. The origin of the discrepancy between those results is the discontinuity between $m=124$ and $m=125$ on the curve of $\overline{e}$ that Montoya et al. used as input for the simulation while probably was not used in the Belhafaf et al. simulation.

Figure 2.

Standard deviation of the final kinetic energy distribution as a function of the final mass ($m$) of fragments from fission of ²³⁵U induced by thermal neutrons. The output of Monte Carlo simulation (squares) does not reproduce the experimental peak (triangles) at $m$ =125 obtained by Belhafaf et al. [7].

Figure 3.

Standard deviation of the final fragment kinetic energy distribution as a function of the final mass ($m$) of fragments from fission of ²³⁵U induced by thermal neutrons. The output of Monte Carlo simulation (squares) [8] and experimental result (triangles) reproduce the experimental peaks obtained by Belhafaf et al. [7].

2.2 Kinetic energy distribution as a function of provisional fragment mass

In 1978, using the $2E$ technique, Asghar et al. measured the quantities $\overline{e}$, ${\sigma }_e,\overline{\mathit{tke}}$, and ${\sigma }_{\mathit{tke}}$ as a function of provisional mass $\mu$ of fragments from the reactions ²³⁵U(n_th, f) and ²³⁹Pu(n_th, f), respectively [9]. For each of both reactions, those authors obtained a peak near the symmetric mass region in the corresponding ${\sigma }_{\mathit{tke}}$ curves. For the reaction ²³⁵U(n_th, f), the peak on ${\sigma }_e$ around $\mu =109$ reaches the value of approximately 6 MeV. This value must be compared to the 9 MeV found by Brissot et al. for ${\sigma }_e$ around $=109$. These authors used the LOHENGRIN mass spectrometer.

In 2020, simulating a primary distribution with any peak on the ${\sigma }_{\mathit{TKE}}$ curve for fragments from the reaction ²³⁹Pu(n_th, f), Montoya reproduced the values of ${\sigma }_{\mathit{tke}}$ as a function of $\mu$ [10], found by Asghar et al. [9]. See Figure 4.

Figure 4.

Standard deviation of the total kinetic energy distribution as a function of the primary fragment mass (diamonds) and the corresponding to the provisional mass (squares), which are the input and output data, respectively, in a Monte Carlo simulation [10], to be compared with experimental data (triangles) obtained by Asghar et al. [9].

The Monte Carlo simulations of fission experiments show that the structures on the standard deviation of the fragment kinetic energy distribution, as a function of mass, are the result of prompt neutron emission and the combined effects of mass yield and average kinetic energy curves. The shape of those effects depends on the measurement technique.

A representative case is the region of light fragment near the symmetry of fission. In this region for both cases $\overline{E}\left(A\right)$,$Y\left(A\right)$ are decreasing functions, and the final mass, due to neutron emission, is lower than the primary mass, i.e., $m<A$. As a consequence from that, for a given $m$, the final kinetic energy distribution is wider than the corresponding to the primary mass $A=m.$

In the same mass region, $n>n^{\prime }$. Then, from the conservation relations, one can easily show that $\mu >A$. As a result of that, the distribution of $e$ values for a given $\mu$ is different compared to the corresponding to the final mass $m$ [11].

3. Average prompt neutron multiplicity

One of the quantities related to the distribution of fragment excitation energy in fission is the average prompt neutron multiplicity $\overline{\nu }\left(A\right)$ as a function of the mass. In 2014, using the $2E$ method, Göök et al. measured the average prompt neutron multiplicity as a function of the mass of fragments from the spontaneous fission of ²⁵²Cf [12]. They found the well-known sawtooth approach, with a noticeable enhancement for $\mu =121$. In 2019, applying a Monte Carlo simulation with input data representing a pre-neutron sawtooth approach (${\overline{\nu }}_s$) without any enhancement for the average prompt neutron multiplicity, Montoya and Romero reproduced [11], as post-neutron, the Göök et al. result [12]. See Figure 5.

Figure 5.

Experimental average prompt neutron multiplicity as a function of mass (triangles) of fragments from the spontaneous fission of ²⁵²Cf obtained by A. Göök et al. using the $2E$ method [12], its sawtooth approach of primary distribution (diamonds), and the corresponding to the calculated by a Monte Carlo simulation (circles) [11].

Those authors explain that the notorious peak in $\overline{\nu }\left(\mu \right)$ at $\mu =121$ is an effect of the yield, the kinetic energy distribution, and the prompt neutron multiplicity as a function of primary fragment mass. In 2019, Montoya applied the Monte Carlo Method to reproduce the results of similar experiments for the ²³³U(n_th, f) and ²³⁵U(n_th, f) reactions [13].

Those studies show that to calculate the quantities distribution associated with the primary fragments it is necessary to apply the Monte Carlo technique to simulate the experiment and reproduce its results.

4. Yield of charge as a function of fragment kinetic energy

Several authors have measured the yield of charge as a function of the kinetic energy of the final light fragments from isobaric fragmentations. Quade et al. studied the reaction ²³³U(n_th, f). The results show that the yield of the lower charge of light fragment mass increases as a function of the final kinetic energy [14].

Using the Monte Carlo method, Montoya and Rivera simulated a constant charge distribution as a function of the light fragment kinetic energy, the prompt neutron emission depending on the kinetic energy, and the yield of mass of primary fragments. Their results show that the increase in charge asymmetry with final kinetic energy is due to the decreasing neutron multiplicity as a function of the primary fragment kinetic energy [15]. See Figure 6.

Figure 6.

Reaction ²³³U (n_th, f). (a) The yield of charge as a function of final fragment kinetic energy for final mass 82 measured by Quade et al. [14]. (b) Results from Monte Carlo simulation of the experiment. Yields of charge lower than 0.1% are neglected [15].

Lang et al. obtained similar results for the reaction ²³⁵U(n_th, f) [16] which were interpreted by Montoya and Rivera with similar arguments [17]. See Figure 7. The fragments with low kinetic energy have high excitation energy so they emit a high number of prompt neutrons. Thus, for a given final mass $m$, the corresponding primary mass and charge would be high. This reasoning is valid in a region where the yield is an increasing function of mass.

Figure 7.

Reaction ²³⁵U(n_th, f). The yield of charge as a function of kinetic energy for mass number 85 of final fragments (after neutron emission). (a) Results measured by Lang et al. [16]; (b) results from Monte Carlo simulation of the experiment [17].

5. Cold fission

5.1 Maximum values of the kinetic energy of the fragments

Due to the emission of neutrons, the fragments reach the detectors with final values of mass and kinetic energy lower than the primary values. In 1981, Signarbieux et al., discovered the phenomenon of cold fission in the reactions ²³³U(n_th, f), ²³⁵U(n_th, f), and ²³⁹Pu(n_th, f) [18]. They detected fragments with high values of kinetic energy and, therefore, with low values of excitation energy, too low to emit neutrons. Using the time-of-flight difference technique, these authors separated neighboring masses, which permitted them to measure the maximal total kinetic energy curves as a function of the primary fragment mass (${\mathit{TKE}}_{\max }\left(A\right)$) [19]. See Figures 8–10, respectively.

Figure 8.

Fragment mass and total kinetic energy distribution in cold fission region from ²³³U(n_th, f). The thick dashed line is the maximal value of the available energy ($Q$), as a function of light fragment mass and its corresponding charge. The thin dashed line is the total kinetic energy as a function of fragment mas, for the light fragment kinetic energy $E=112$ MeV. The thin full lines are equal probability lines: The numbers of detected fission events per a.m.u., whose corresponding total kinetic energy values between two consecutive lines, are indicated in the amplified circle. Those numbers (10 in this case) are chosen such that the separation between consecutive lines is of the order of 1 MeV. Figure is taken from [19].

Figure 9.

Similar to Figure 6 for the reaction ²³⁵U(n_th, f). Figure from [19].

Figure 10.

Similar to Figure 7 for the reaction ²³⁹Pu(n_th, f). Figure from [19].

In 1986, Trochon et al., using twin ionization chambers, obtained experimental values of ${\mathit{TKE}}_{\max }\left(A\right)$ for the reaction ²³⁵U(n_th, f) [20]. See Figure 11. The trends of ${\mathit{TKE}}_{\max }\left(A\right)$ are similar to the $\overline{\mathit{tke}}\left(\mu \right)$ curve, but approximately 20 MeV higher than those values [21, 22, 23]. The ${\mathit{TKE}}_{\max }\left(A\right)$ curves show oscillations that were interpreted as effects of the change of fragment charges that reach these total kinetic energy values. These so-called Coulomb effects produce ledges in the increasing part of ${\mathit{TKE}}_{\max }\left(A\right)$ curve, corresponding to the change of charges that maximize the total kinetic energy [22].

Figure 11.

Maximal value of total kinetic energy as a function of mass and its corresponding charge of fragments from thermal neutron-induced fission of ²³⁵U. Figure taken from [20].

The Coulomb effect is defined as the preference for a more asymmetric charge isobaric fragmentations corresponding to similar $Q$-values at the highest values of total kinetic energy [22]. Lang et al. [16] measured the charge and the mass distribution for $E=108$ MeV. This energy window corresponds to the $\mathit{TKE}$-line 10 MeV lower than the average than the maximal Q-line. For the mass fragmentation 91/135, the charges $Z=$ 36 and 37 correspond to the same $Q$-value (187.5 MeV), but the charge $Z=$ 36 has a yield (75.9 ± 3.3%) higher than the corresponding to $Z=$37 (19.7 ± 3.1%). For the same mass fragmentation, Trochon et al. [20] have deduced that the surviving charge at very high TKE-values is $Z=$ 36. The same holds for the even-even fragmentations ($Z/Z^{\prime },N/N^{\prime })=\left(36/56,56/88\right)$ and ($38/54,54/90)$, whose $Q$ values are 189.8 and 189.3, respectively.

5.2 Spherical-prolate configurations

The highest values of ${\mathit{TKE}}_{\max }\left(A\right)$ are reached for the configurations corresponding to the double magic spherical heavy fragments $Z=50$ and $N=82$ and the prolate and soft transitional light nuclei $Z=40-42$ and $N=60-64$ . These conditions allow that

$${\mathit{TKE}}_{\max }\left(A\right)\cong \mathit{CE}\cong Q,$$E15

Where $\mathit{CE}$ is the Coulomb energy at the scission point, and $Q$ is the energy available from the corresponding fragmentation.

Because in the symmetric mass region $\mathit{CE}\gg Q$, in the scission point, the fragments must be highly deformed to constitute a probable configuration. That implies a high deformation energy and therefore a low kinetic energy. This hypothesis is verified in the curve ${\mathit{TKE}}_{\max }\left(A\right)$: There is a steep drop in ${\mathit{TKE}}_{\max }\left(A\right)$ in symmetric fission [22].

5.3 Even-odd effects in cold fission

The curve of ${\mathit{TKE}}_{\max }\left(A\right)$ does not present clear even-odd effects as a function of the mass, contrary to what might have been expected in the case of a superfluid fission process [23]. This is not contradictory to the existence of even-odd effects in the number of protons $\mathit{\delta Z}$ and $\mathit{\delta N}$. If there is no more than one pair of nucleons (protons or neutrons), the relationship between the three even-odd effects is [24, 25]:

$$1+\mathit{\delta A}=\mathit{\delta Z}+\mathit{\delta N}.$$E16

In 1991 Gönnenwein and Börsig [26] found a negative odd–even effect on the minimum excitation energy (${\mathit{TXE}}_{\min }$) as a function of $Z$. In 1993, Hambsch et al. [27] questions the positive odd even effect on that quantity. In 2013, Gönnenwein confirms the negative odd-even effect [28]. In 2016, Mirea proposes a microscopic model to explain the negative odd-even effects on excitation energy in cold fission [29]. In 2017, Montoya and Collin show that there is no contradiction between the positive odd-even effect on ${\mathit{TKE}}_{\max }$ and negative odd-even effect on ${\mathit{TXE}}_{\min }$ as a function of $Z$ [30].

6. Conclusion

The emission of prompt neutrons from fragments from actinides fission does not permit to obtain the pre-neutron mass ($A$) and kinetic energy ($E$) distribution. It is only possible to measure the post-neutron mass ($m$) and kinetic energy ($e$). The difference between the post-neutron and pre-neutron distributions depends on the prompt neutron multiplicity ($\nu$) associated with each complementary fragment. Using a Monte Carlo simulation of neutron emission and the experimental technique, several surprising unexpected results of the fission experiment are explained.

The curve of the standard deviation of the kinetic energy as a function of the final mass${\sigma }_e\left(m\right)$ of fragments from the reaction ²³⁵U(n_th, f) shows a peak around $m=109$. The curve of the average prompt neutron multiplicity as a function of the provisional mass of fragments, $\overline{\nu }\left(\mu \right)$, from the spontaneous fission of ²⁵²Cf, presents a peak around $\mu =122$. The results of Monte Carlo simulations show that both cases may be explained by the interplay of the prompt neutron emission and the rapid decreasing of $Y\left(A\right)$ and $\overline{E}\left(A\right)$ in their respective mass region.

For isobaric fragmentation in the asymmetric region, the preference for the lower charge of the light fragment increases with the final kinetic energy. This result is explained by the fact that fragments with higher kinetic energy have emitted a lower number of neutrons, which correspond to a lower charge nuclei.

In the cold fission region, the fragment excitation energy is so low that there is no emission of neutron emission. Thus, by a mass spectrometer, one can separate neighboring masses. So, we can obtain the curve of the maximal value of the primary total kinetic energy (${\mathit{TKE}}_{\max }$) as a function of the fragment mass ($A$). The curve of the ${\mathit{TKE}}_{\max }\left(A\right)$ shows odd-even, shell, and Coulomb effects.

As a conclusion one may say that, in order to relate primary and final fragment distributions, it is necessary to simulate the primary fragment distribution, the emission of neutrons, and the experimental technique. Two different experimental techniques may produce contradictory results.