Prompt emission calculations for ²³⁹Pu(n_th,f) with the DSE model code and a pre-neutron fragment distribution Y(A,TKE) based on the four-dimensional Langevin model

Abstract

The request for new and/or more accurate data of independent fission product yields (FPY) and other distributions of pre- and post-neutron fragments constitutes a priority on an international level. The prompt emission model codes nowadays employed are refined enough to answer this request with the condition of using reliable distributions Y(A,TKE) of pre-neutron fragments. Up to now in the majority of cases such distributions were determined by experimental data. Y(A,TKE) from theoretical calculations can extend the use of prompt emission model codes to fissioning systems for which the experimental information is very scarce or completely missing. Such Y(A,TKE), recently obtained from four-dimensional Langevin model calculations in the frame of multi-modal fission, is tested by its use (as input) in the deterministic prompt emission model code DSE, for the case of ²³⁹Pu(n_th,f). The obtained prompt emission results (e.g. prompt neutron multiplicity distributions ν(A), ν(TKE)) succeed to describe the experimental data and the independent FPY agree with the experimental data, too. Although visible differences between these results and those obtained with a an experimental Y(A,TKE) distribution exist. The present investigations have also emphasized other interesting aspects, e.g. concerning the pre-neutron fragments which contribute to the pronounced peaks and dips in the structure of both the mass yield Y(A_p) and the isotonic yield Y(N_p) of post-neutron fragments, in connection with the role played by the even–odd effect in fragment charge, as well as how the position and magnitude of pronounced peaks and dips are influenced by the Y(A,TKE) distribution. Independent FPY separately for each number “n” of emission sequences leading to the last residual fragment Y(A_p = A-n), reported for the first time, have shown how and which of these components contribute to the structure of total Y(A_p). The crucial role of the energy partition in fission is revealed again by showing that a good choice of the R_T parameterization can lead to very close results to those obtained with a TXE partition based on modeling at scission. On the contrary the use of an unique R_T value for all fragmentations can alter -in some fission cases—the sawtooth shape of ν(A) and its agreement with experimental data, despite the refinement of the prompt emission treatment itself and of a reliable Y(A,TKE) distribution employed in calculations.

Appendices

Appendix A

1.1 Equations of four-dimensional Langevin model

The four-dimensional Langevin model employed to simulate the time evolution of the fissioning nucleus deformation along its fission path from ground state up to scission, consists of solving the following equations on the corresponding potential energy with transport coefficients:

$$ \begin{aligned} \frac{{dq_{\mu } }}{dt} =&\,(m^{ - 1} ){}_{\mu \nu }p_{\nu } \hfill \\ \frac{{dp_{\mu } }}{dt} =&\,\frac{\partial F(q,T)}{{\partial q_{\mu } }} -\frac{1}{2}\,\frac{{\partial (m^{ - 1} )_{\nu \sigma } }}{{\partial q_{\mu } }}\,p_{\nu } p_{\sigma } \\ &- \gamma_{\mu \nu } (m^{ - 1})_{\nu \sigma } p_{\sigma } + \sqrt {T_{\mu }^{eff} } g_{\mu \nu }R_{\nu } (t) \hfill \\ \end{aligned}$$

(1)

in which {q_μ}(with μ from 1 to 4) is a set of collective variables of nuclear shape and {p_μ} the corresponding conjugate momenta. A sum of the repeated indices is taken, excluding μ in the last term of the dp_μ/dt equation.

F(q,T) (taken as F = V–TS where V is the nuclear potential energy, T the nuclear temperature and S the entropy) is the temperature-dependent free energy, which is calculated as a sum of the liquid drop energy and the shell and pairing corrections as described in Ref. [59]. The finite-depth two-center Wood-Saxon potential is employed for the single particle energies used in the calculation of shell and pairing corrections with parameters from Ref. [60].

The nuclear deformation is introduced into the wave function using the shape of the two-center shell model [26] expanded by Cassini ovaloids. Regarding the transport coefficients, a collective inertia tensor m_μν is calculated under the Werner-Wheeler approximation [61]. The friction tensor γ_μν is calculated with the wall-and-window formula [17, 62, 63] with a damping factor of 0.27. The strength of the random force g_μν connects to the friction tensor through the fluctuation–dissipation theorem at T = 1 MeV, namely g_μσg_σμ = γ_μν and R_ν is the stochastic force having a nature of white noise. The effective temperature T_μ^eff is introduced for q_μ not to drop below a certain zero-point energy:

$$ T_{\mu }^{eff} = \frac{{\hbar \omega_{\mu } }}{2}\,\coth \left( {\frac{{\hbar \omega_{\mu } }}{2T}} \right) $$

(2)

in which $\hbar \omega_{\mu } /2$ corresponds to the zero-point energy of the harmonic oscillator.

Appendix B

2.1 ν(A) results of TALYS and DSE using the same TXE partition and the same initial fragment distribution

Figure 19 shows the ν(A) result of TALYS reported in Ref. [6] (orange circles) which was obtained with a TXE partition based on R_T = 1.3 and using the initial fragment distributions Y(A) and TKE(A) based on 4-dimenional Langevin model calculations (as described in Sect. 2) in comparison with the ν(A) result of DSE (green diamonds) obtained in the same conditions (i.e. the same TXE partition using R_T = 1.3 and Y(A,TKE) from Langevin model calculations).

Fig. 19

The ν(A) results of TALYS (orange circles) and DSE (green diamonds) obtained with the same TXE partition based on R_T = 1.3 and the same initial fragment distribution (based on Langevin model) together with experimental data (different black and gray symbols) and the DSE result obtained with the TXE partition based on modeling at scission (blue squares)

As it can be seen the ν(A) results of TALYS and DSE (orange circles and green diamonds) are very close to each other. This fact is very interesting and proves the following:

• (1) The crucial role of TXE partition in any treatment of prompt emission.

  The ν(A) results of DSE obtained with two different TXE partitions, one based on modeling at scission (blue squares in Fig. 19) and another based on the unique value R_T = 1.3 for all fragmentations (green diamonds) are very different, only the ν(A) result obtained with the TXE partition based on modeling at scission (blue squares) succeeding to describe well the experimental data.

  In other words the poor agreement of the ν(A) result of TALYS with the experimental data is due to the TXE partition based on an unique value of R_T taken for all fragmentations (R_T = 1.3).

• (2) Two very different treatments of prompt emission in fission (i.e. the statistical Hauser-Feshbach in TALYS and the recursive transcendental equations of residual temperature associated to each emission sequence from each initial configuration in DSE) lead to very similar results.

  Both approaches present convenient and inconvenient aspects. On one side the statistical Hauser-Feshbach treatment of TALYS is more refined. But on the other hand any nuclear reaction code (e.g. TALYS, EMPIRE etc.) needs a lot of input information (e.g. nuclear level schemes, level density prescriptions for the continuum part, data for the treatment of γ-channel like strength functions for s-wave neutrons, transmission coefficients provided by the direct interaction treatment of the inverse reaction of neutron evaporation and so on). And all these data are very scarce in the case of neutron-rich nuclei with a very short life, as the fission fragments are. Moreover, such statistical treatment by TALYS needs many databases and auxiliary input files as well as a longer computational time than the DSE treatment.

  The DSE treatment sequence by sequence consists of solving successive transcendental equations of residual temperature corresponding to each emission sequence from each initial configuration (A,Z,TKE) (covering a large fragmentation and TKE range). But these transcendental equations employ excitation energies of residual fragments taken as an average over their excitation energy spectrum and these equations can be solved only for non-energy dependent level density parameters of fragments. On the other hand the DSE treatment does not need (as input) so much information as TALYS and the computation time is very short.

  The small differences between the ν(A) results of TALYS and DSE which can be observed in Fig. 19 (see comparatively the orange circles and green diamonds) can be also due to

  • the differences between the fragmentation ranges taken into account in TALYS and DSE and

  • a different use of the initial fragment distribution, i.e. as a matrix Y(A,TKE) in DSE and as two single distributions Y(A) and TKE(A) in TALYS, this limiting the number of TKE values taken for each fragmentation in TALYS compared to DSE.

Appendix C

3.1 Multi-modal components of prompt emission quantities and independent FFY

In the case of Y(A,TKE) obtained as the superposition of multi-modal distributions Y_m(A,TKE) weighted with the branching ratios of fission modes (w_m), i.e.

$$ Y(A,{\textit{TKE}}) = \sum\limits_{m} {w_{m} Y_{m} (A,{\textit{TKE}})} $$

(3)

(the index m denoting the fission mode), DSE calculation of the modal components of different quantities characterizing the pre-neutron fragments and the prompt emission as well as of different distributions of post neutron fragments can be performed, too.

The primary results of DSE (consisting of matrices, generically labeled q_k(A,Z,TKE) or q(A,Z,TKE) when they are averaged over the emission sequences) are then averaged over the multiple fission fragment distributions Y_m(A,Z,TKE) of fission modes which are taken as

$$ Y_{m} (A,Z,{\textit{TKE}}) = p(Z,A)\,Y_{m} (A,{\textit{TKE}}) $$

(4)

in which p(Z,A) is the isobaric charge distribution (mentioned in Sects. 3.1.1 and 3.1.2). So that different multi-modal components q_m(A), q_m(TKE), q_m(Z) and the total average values <q_m> are easily obtained. E.g. the modal prompt neutron multiplicity distributions ν_m(A), ν_m(TKE), the total average multiplicity of each mode <ν_m> , the modal components of independent FPY Y_m(A_p) etc.

The total average values of different quantities are obtained by averaging the average values of fission modes over the modal branching ratios:

$$ <q>= \sum\limits_{m} {w_{m} \, <q_{m}>} $$

(5)

The total distributions of different quantities can be obtained, too, by averaging their multi-modal distributions over the modal probability distributions, i.e.

$$\begin{aligned} q(A) = &\sum\limits_{m} {P_{m} (A)q_{m} (A)},\\ q(\textit{TKE}) =& \,\sum\limits_{m} {P_{m} {(\textit{TKE})}q_{m} {(\textit{TKE})}} ,\\ q(Z)=&\, \sum\limits_{m} {P_{m} (Z)q_{m} (Z)}\end{aligned}$$

(6)

in which the distributions of fission mode probabilities are:

$$ \begin{gathered} P_{m} (A) = w_{m} \,Y_{m} (A)/Y(A) \hfill \\ P_{m} {(\textit {TKE})} = w_{m} \,Y_{m} {(\textit{TKE})}/Y{(\textit{TKE})} \hfill \\ P_{m} (Z) = w_{m} \,Y_{m} (Z)/Y(Z) \hfill \\ \end{gathered} $$

(7)

Note, Eqs. (3–7) can be applied for any fission modes or combinations of fission modes. In the majority of cases the fission modes m = S1, S2 and SL are taken into account. In other cases more than these three modes can be considered (e.g. m = S1, S2, S3, SL and SS) or even combinations of modes, e.g. either two modes an asymmetric and a symmetric one or—as in the case of Langevin calculations for ²³⁹Pu(n_th,f) mentioned in Sect. 2 of the main text—the index m means S1 and S2 + SL, with the multi-modal branching ratios: w_S1 = 19.16% and w_S2+SL = 80.84%.

In the present case, i.e. Y_S1(A,TKE) and Y_S2+SL(A,TKE) (plotted in the upper part of Fig. 1) with their corresponding single distributions given in Fig. 2 (with open green and orange symbols, respectively), the fission mode probabilities given by Eq. (7) are plotted in Fig. 20 with green squares for the S1 mode and orange circles for S2 + SL, as a function of A in the upper part, as a function of TKE in middle and as a function of Z in the lower part.

Fig. 20

Fission mode probabilities as a function of A (upper part), as a function of TKE (middle) and as a function of Z (lower part) plotted with green squares for S1 mode and orange circles for S2 + SL

Multi-modal results of DSE for ²³⁹Pu(n_th,f) are exemplified in Fig. 21 for TXE (in the upper part) and the prompt neutron multiplicity (in the lower part) as a function of A (left parts) and as a function of TKE (right parts). They are plotted with open green symbols for the S1 mode and open orange symbols for S2 + SL. The total single distributions obtained according to Eqs. (6) and (7) are plotted with blue stars. As it can be seen they reproduce the DSE results obtained without fission modes given in the main text (plotted in this figure with full red symbols).

The total average values of each mode, <TXE_m> and <ν_m> , are given in the figure legends, too. The total average values calculated according to Eq. (5) reproduce the total average values obtained without fission modes (differing from these ones with only 0.013% in the case of <TXE> and 0.17% in the case of <ν>).

Fig. 21

The multi-modal components TXE_m(A) and TXE_m(TKE) (upper part) and ν_m(A), ν_m(TKE) (lower part) plotted with open green squares (for S1) and orange diamonds (S2 + SL) and the total distributions based on this modal components (blue stars). The distributions previously obtained without fission mode (full red symbols) are also given for comparison

The multi-modal components of post-neutron fragment yields, obtained by multiplying the DSE result for the respective fission mode, i.e. Y_m(A_p) and Y_m(N_p) respectively, with its branching ratio w_m, are plotted in the upper and lower parts of Fig. 22 with open green squares for S1 and open orange diamonds for S2 + SL. As it can be seen the sum of these components, plotted with blue stars, reproduces the DSE result of Y(A_p) and Y(N_p) respectively, previously obtained without fission modes, which are reported in the main text (plotted in this figure with full red circles).

It can be easily observed that both modes S1 and S2 + SL contribute to the most pronounced peak in the Y(A_p) structure located at A_p = 134 and in the Y(N_p) structure located at N_p = 82, while the structure in the light fragment region of both Y(A_p) and Y(N_p) exhibiting several pronounced peaks is much more due to the contribution of S2 + SL than of S1.

Fig. 22

The multi-modal components of independent FPY (upper part) and isotonic yield of post-neutron fragments (lower part) multiplied with the modal branching ratios are plotted with open green squares (S1) and orange diamonds (S2 + SL), the total Y(A_p) and Y(N_p) as a sum of these components are plotted with blue stars. The Y(A_p) and Y(N_p) results previously obtained without fission modes are also given (full red circles) for comparison