Abstract
The key role in the calculation of post neutron fragment yields (independent fission product yields FPY) is played by (a) the pre-neutron fragment distributions Y(A,Z,TKE) and (b) the prompt neutron multiplicity (calculated in the frame of a prompt emission model). The knowledge of excitation energies of fully accelerated fragments is crucial in any modeling of prompt emission. Consequently in this paper the influence on independent FPY and kinetic energy distributions of post neutron fragments of both the pre-neutron fragment distribution and the partition of total excitation energy (TXE) is investigated. To do that, two reliable experimental Y(A,TKE) distributions of the standard fissioning nucleus 235U(nth,f) and two methods of TXE partition are considered. Nowadays two types of TXE partitions are employed in different prompt emission models, i.e. energy partitions based on different modelings at scission and different parameterizations which allow the TXE sharing by avoiding what is happening at scission. For this reason the methods of TXE partition chosen in this work belong to both types, i.e. (i) the TXE partition based on modeling at scission used in the deterministic prompt emission models PbP and DSE and (ii) the TXE partition according to the temperature ratio RT = TL/TH of complementary fully accelerated fragments (employed in the deterministic HF3D model and the probabilistic Monte-Carlo code CGMF). The independent FPY results (Y(Z,Ap) and Y(Ap)) obtained with both TXE partition methods and both Y(A,TKE) data describe reasonably well the experimental data. The use of different TXE partitions and Y(A,TKE) data does not change the position of the most pronounced peaks and dips in the Y(Ap) structure (e.g. at Ap = 134, 138, 94 and Ap = 136, 141, 97, respectively) and Y(Np) structure (e.g. at Np = 82, 84, 56 and Np = 83, respectively) only their magnitude is influenced by Y(A,TKE) distribution, while the TXE partition influences the position of less pronounced peaks and dips in the structures of Y(Ap) and Y(Np). The TXE partition has an insignificant influence on different kinetic energy distributions of post-neutron fragments, while the differences existing between the kinetic energy distributions of pre-neutron fragments are reflected in the kinetic energy distributions of post-neutron fragments.
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Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: All data generated during this work are included in the published paper.]
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Acknowledgements
A part of this work was done in the frame of the IAEA-CRP “Updating Fission Yield Data for Applications” and the Romanian project ELI-Ro_14.
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Communicated by Cedric Simenel.
Appendix
Appendix
1.1 Influence of Y(A,TKE) data on TXE
The influence of Y(A,TKE) on the average TXE as a function of A and as a function of TKE is illustrated in Fig. 19 where TXE(A) and TXE(TKE) based on the Y(A,TKE) data of Al-Adili and Straede are plotted with the same symbols and colors as the single distributions of pre-neutron fragments given in Fig. 1 of the main text.
As it is expected and it can be seen in the upper part of Fig. 19, the differences between the TXE(A) results based on the Y(A,TKE) data of Al-Adili and Starede reflect the differences between the TKE(A) data, highlighting again the well known inverse correlation between TXE and TKE (i.e. TXE = Q-value + E*CN − TKE, in which E*CN is the excitation energy of the compound nucleus undergoing fission in the case of induced fission and it is zero for spontaneous fission). For instance at AH above 155 (where the TKE(A) data of Straede are lower than TKE(A) of Al-Adili) an inverse behaviour of the TXE(A) results is observed, i.e. TXE(A) based on the data of Straede is higher than TXE(A) based on the data of Al-Adili.
In this figure the TXE(A) result of Okumura et al. [24] is also plotted with open symbols. Differences between TXE(A) of Okumura and our results can be observed. These differences are due to the fact that Okumura et al. have used another fragmentation range and another fragment distribution than those used in this work. However it seems that TXE(A) of Okumura et al. is closer to our TXE(A) based on the Y(A,TKE) data of Straede.
In the lower part of Fig. 19 it can be seen that differences between the TXE(TKE) results based on the Y(A,TKE) of Al-Adili and Straede exist only at TKE values less than 140 MeV.
1.2 Energy components of our TXE partition based on modeling at scission
A result of our method of TXE partition based on modeling at scission (e.g. Refs. [4, 19, 31, 32] is exemplified in Fig. 20: the excitation energy at full acceleration E* (full symbols) and its components the excitation energy at scission E*sc (open symbols) and the extra-deformation energy ΔEdef (stars) are plotted as a function of fragment mass A in the upper part and as a function of fragment charge Z in the lower part. They were obtained by averaging the matrices E*(A,Z,TKE), Esc(A,Z,TKE) and ΔEdef(A,Z,TKE) over the Y(A,Z,TKE) distribution of Eq. (2) based on Y(A,TKE) data of Ref. [37]. The level density parameters of nascent fragments at scission are given by the super-fluid model in which the shell-correction of Möller and Nix [42] and the parameterizations of Ignatyuk [40] for the dumping and asymptotic level density parameter were used.
The results plotted with red symbols are based on the mass excesses of Audi and Wapstra [47] (being already reported, e.g. Ref. [32] and subsequent references) and those plotted with blue symbols on the mass excesses of AME2020 database [39] used in the present work. As it can be seen the differences between the fragment energies based on the two databases of mass excesses are very low, almost insignificant (the blue and red symbols covers each other). Consequently the prompt emission results (such as the prompt neutron multiplicity ν(A,Z,TKE)) obtained by using the mass excesses of AME2020 do not differ significantly from those already reported (which were based on the mass excesses of Ref. [47]).
It can be observed that both E*(A) and E*(Z) exhibit a visible sawtooth shape, mainly due to the shell effects, which is well reflected in the well-known sawtooth shape of the prompt neutron multiplicity distribution ν(A) and also in the distribution ν(Z). The visible staggering of E*(Z) in the asymmetric fission region (lower part of Fig. 20) is due to the presence of even–odd effects.
1.3 Basic features of the DSE approach
The DSE treatment consists of solving the following recursive transcendental equations of residual temperature associated to each configuration (A,Z,TKE) (denoting an initial fragment {Z, A} at a TKE value, covering the fragmentation and TKE ranges mentioned in Sect. 2.1). Note, to simplify the notations, in this equation and the next ones, the (A,Z,TKE) configuration is omitted.
in which \(\overline{{E_{r} }}^{(k)}\) is the average excitation energy of the k-th residual fragment {Z, A–k} and \(S_{n}^{(k)}\) is the neutron separation energy from the k-th residual fragment, <ε>k is the average energy in the center-of-mass frame of the k-th emitted prompt neutron, ak and Tk are the level density parameter and the nuclear temperature of the k-th residual fragment, respectively. Note, the sequence index k = 0 means the initial fragment, i.e. \(S_{n}^{(0)}\) is the neutron separation energy from the initial fragment (before prompt neutron emission) and \(\overline{{E_{r} }}^{(0)} = E^*\) is the excitation energy of the initial fragment resulting from the TXE partition (discussed in Sect. 2.3).
Different prescriptions regarding the compound nucleus cross-sections of the inverse process of neutron evaporation from fragments and the level density parameters of initial and residual fragments, entering Eqs. (a1), are discussed in detail in Refs. [20, 23].
Here we mention only that the recursive equations of residual temperatures (a1) (in which the compound nucleus cross-section of initial and residual fragments enter the average center-of-mass energy <ε>k of each neutron successively emitted) can be solved only if an analytical expression for \(\sigma_{c}^{(k)} (\varepsilon )\) is employed. A such expression, proposed by Iwamoto [48], consists of a sum of a constant term σ0 and a s-wave term σs(ε), i.e. \(\sigma_{0} = \pi R^{2}\) (with \(R = r_{0} A^{1/3}\) where r0 is the reduced radius) and \(\sigma_{s} (\varepsilon ) = (\pi /K^{2} )T_{0}\) in which K is the wave number and \(T_{0} = 2\pi \sqrt \varepsilon \,S_{0}\) is the transmission coefficient of the s-wave neutron (S0 being the s-wave neutron strength function of a fission fragment, initial or residual). Consequently the compound nucleus cross-section of the inverse process corresponding to each emission sequence is expressed as:
in which \(\alpha_{k} = (\pi \hbar^{2} /mr_{0}^{2} )\,S_{0}^{(k)} /(A - k)_{{}}^{2/3}\) depends on the mass number and the s-wave neutron strength function of each initial (A, Z) and residual (A-k, Z) fragment of the fragmentation range. Using \(\sigma_{c}^{(k)} (\varepsilon )\) expressed by Eq. (a2), the neutron evaporation spectrum for a given residual temperature Tk becomes
with its first moment given by [20]:
which leads to the following transcendental equations of residual temperatures (corresponding to each emission sequence indexed k associated to an initial (A,Z,TKE) configuration covering the fragmentation and TKE ranges described in Sect. 2.1):
These transcendental equations are numerically solved.
A detailed verification supporting the use of σc(ε) expressed by Eq. (a2) in the successive transcendental equations for residual temperatures can be found in Ref. [20].
The transcendental equations of residual temperature (a5) can be solved only if the level density parameters ak are non-energy dependent. Such level density parameters can be provided by different systematics. To investigate which systematic is appropriate, in Ref. [20] the level density parameters provided by several systematics were compared with the energy-dependent level density parameters of the super-fluid model (with shell corrections of Möller and Nix [42] and the γ and \(\tilde{a}\) a parameterizations of Ignatyuk [40]). This comparison has supported the use in Eq. (a5) of level density parameters for initial and residual fragments provided by the Egidy–Bucurescu systematic for the BSFG model [41].
So that in the present work the DSE approach is used with the following prescriptions: compound nucleus cross-sections of the inverse process \(\sigma_{c}^{(k)} (\varepsilon )\) given by Eq. (a2) (with s-wave neutron strength functions as a function of A based on the <S0> data from RIPL-3 [49]) and level density parameters of initial and residual fragments provided by the Egidy–Bucurescu systematic for the BSFG model [41].
1.4 DSE results of Y(Z,A p) in comparison with other experimental data
The comparison of Y(Z,Ap) results with other data sets of Lang et al. found in EXFOR [45] is illustrated in Fig. 21 for the case of Y(A,TKE) data of Al-Adili et al. [38] and both TXE partitions the method based on modeling at scission (small full circles connected with solid lines to guide the eye) and the procedure using RT = 1.2 (small open circles connected with dashed lines to guide the eye). They are plotted with the same color as the symbols of experimental data for the respective charge number Z.
As it can be seen the Y(Z,Ap) results corresponding to both methods of TXE partition describe reasonably well the experimental data sets of Lang et al.
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Tudora, A. Influence of energy partition in fission and pre-neutron fragment distributions on post-neutron fragment yields, application for 235U(nth,f). Eur. Phys. J. A 58, 126 (2022). https://doi.org/10.1140/epja/s10050-022-00766-y
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DOI: https://doi.org/10.1140/epja/s10050-022-00766-y