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Systematics of different quantities related to sequential prompt emission in fission

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Abstract

A deterministic treatment of sequential neutron emission, based on recursive equations of the residual temperatures, was applied to numerous fission cases (i.e. 49 cases including nuclei fissioning spontaneously or induced by thermal and fast neutrons) for which reliable experimental data of fragment distributions exist. This fact emphasizes systematics and correlations between different average quantities characterizing the initial and residual fission fragments and the sequential prompt neutron emission. The ratios of the average residual temperature following the emission of each neutron to the average temperature of initial fragments (\({<}{T}_{{k}}{>}/{<}{T}_{{i}}{>}\)) are almost constant for all studied fission cases. They do not depend on the prescriptions used for the compound nucleus cross-sections of the inverse process and the level density parameters of initial and residual fragments. This finding allows the determination of a general form of the residual temperature distribution for each emission sequence \({P}_{{k}}(T)\) with the maximum temperature related only to the average temperature of initial fragments. In this way the sequential emission can be included into the Los Alamos model. Prompt neutron spectra in the center-of-mass and laboratory frames can be calculated for each emission sequence. The systematic of the ratios of average residual temperatures and energies to the initial ones together with other linear correlations between different prompt emission and residual quantities allow to obtain indicative values of different average prompt emission quantities in the absence of any prompt emission model.

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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 study are contained in this published article.]

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Acknowledgements

This work was supported by a grant of the Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2016-0014 (contract No. 7/2017), within PNCDI III.

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Correspondence to Anabella Tudora.

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Communicated by P. Capel

Appendices

Appendix

1.1 A1: Constant values of residual temperature and energy ratios

For emission sequences with \(k {>} 2\) the ratios \({{<}E}_{r}^{(k)}{{>} /{<}E^{*}{>}}\) and \({{<}T}_{k}{{>} /{<}T}_{i}{{>}}\) of light fragments are higher than those of heavy fragments. This is due to the magnitude of the difference between the average residual energies of light and heavy fragments \(\Delta \overline{E_{r}^{(k)} } ={<}E_{r}^{(k)} {>}_{L} -{<}E_{r}^{(k)} {>}_{H} \). The average residual energies \({{<}E}_{r}^{(k)}{{>}}\) as a function of \({{<}{} { TXE}{>}}\) are plotted in Fig. 13 using the same symbols and colors as in the main text. For a better visualization of their behavior, the linear fits are plotted with thin solid lines (light fragments) and dashed lines (heavy fragments) using the same color as the respective symbol.

Fig. 13
figure 13

Average residual energies of the light and heavy fragment groups (full and open symbols, respectively) corresponding to the emission sequences with \({k} = 1 \) (red circles), \({k} = 2\) (blue triangles), \({k} = 3\) (green diamonds) and \({k} = 4\) (dark yellow up triangles). The linear fits are plotted with solid lines for light fragments and dashed lines for the heavy fragments using the same color as the respective symbol

Looking at Fig. 13 it can be observed that for the first and second residual fragments (red and blue symbols and lines) the differences \(\Delta \overline{E_{r}^{(k)} } \) are visibly lower (less than 1 MeV) compared to \(\Delta \overline{E_{r}^{(k)} } \) corresponding to the emission sequences with \({k} = 3\) and \({k} = 4\) (green and dark yellow symbols and lines) which are of about 1.5 MeV. These larger differences for \(k {>} 2\) are reflected in higher values of the \({{<}E}_{r}^{(k)}{{>} /{<}E^{*}{>}}\) ratios of light fragments than of heavy fragments.

The temperature ratio \({{<}T}_{k}{{>} /{<}T}_{i}{{>} }\) depends on the square-root of the ratio \({{<}E}_{r}^{(k)}{{>} /{<}E^{*}{>} }\) and    the square-root of the level density parameter ratio \({{<}a}_{i}{{>} /{<}a}_{k}{{>}}\). For all k the level density parameter ratios \({{<}a}_{i}{{>} /{<}a}_{k}{{>} }\) are close to 1 (e.g. they are of about 0.99 (LF) and 0.98 (HF) for \({k} = 2\), 0.96 (LF) and 0.93 (HF) for \({k} = 3\), 0.94 (LF) and 0.88 (HF) for \({k} = 4\)). Consequently, the higher \({{<}E}_{r}^{(k)}{{>} /{<}E^{*}{>} }\) ratios of light fragments for \(k{>} 2\) are reflected in the temperature ratios which are also higher for light fragments than for heavy fragments.

Fig. 14
figure 14

Average level density parameter of the light fragments (upper part) and heavy fragments (lower part), initial fragments with black stars and the residual fragments with open symbols (using the same colors as in the main text) as a function of \({<}{ \mathrm{TXE}}{>}\). The total average level density parameters obtained according to Eq. (3) are plotted with full magenta diamonds. The global values (resulting from the linear behaviors of Figs. 4 and 5) are indicated with horizontal black lines

$$\begin{aligned}&\left\langle a \right\rangle =\sum \limits _{\textit{A,Z},\mathrm{TKE}} Y(\textit{A,Z},\mathrm{TKE})\,\frac{1}{n(\textit{A,Z},\mathrm{TKE})}\sum \limits _{k=1}^{n(\textit{A,Z},\mathrm{TKE})} {a_{k} (\textit{A,Z},\mathrm{TKE})} \Bigg / {\sum \limits _{\textit{A,Z,TKE}} {Y(\textit{A,Z},\mathrm{TKE})} }, \end{aligned}$$
(a1)

A2: Global values of level density parameters compared to the calculated ones

The average level density parameters of the initial and residual fragment groups \({{<}a}_{k}{{>} }\) obtained according to Eq. (2) are plotted in Fig. 14 as following: initial fragments (\({k} = 0\)) with black stars, residual fragments (\({k }= 1, 2,\ldots ,5\)) with open symbols using the same colors as in the main text. The total average values of the level density parameter, obtained as are plotted with full magenta diamonds.

As can be seen the global values of the level density parameter (resulting from the slopes of the linear fits given in Figs. 4 and 5) illustrated by the horizontal lines, average well the general trend of the total average level density parameters (full diamonds).

A3: Use of other prescriptions for \(\sigma _{c}(\varepsilon \)) and level density parameters

The excitation energies of initial fragments \(E^{*}\)(A,Z,TKE) resulting from the TXE partition based on modeling at scission obtained with different prescriptions of the level density parameter do not differ significantly [18, 19]. This is due to the partition of available excitation energy at scission, which is made according to the ratio of the level density parameters, i.e. \({E_{\mathrm{sciss}}^{L} } / {E_{\mathrm{sciss}}^{H} }={a_{\mathrm{sciss}}^{L} }/ {a_{\mathrm{sciss}}^{H} }\). As mentioned in Ref. [19], the level density parameter ratio is almost the same for different level density prescriptions.

Contrary, the nuclear temperatures of fragments in the Fermi-gas regime are influenced by the level density parameters. Significant differences exist between the level density parameters provided by the Gilbert–Cameron systematics for spherical nuclei [15] and those given by the Egidy–Bucurescu systematics for the BSFG model [13] (used in Sect. 3). I.e. the level density parameters of the Gilbert–Cameron systematics are higher than those of the Egidy–Bucurescu systematics for BSFG. This fact is reflected in significantly lower temperatures based on the Gilbert–Cameron systematics compared to the temperatures of Sect. 3.

The ratios \({{<}T{>} /{<}T}_{i}{{>} }\) and \({{<}E}_{r}{{>} /{<}E^{*}{>} }\) corresponding to all emission sequences obtained with the prescriptions of a constant \(\upsigma _\mathrm{{c}}\) and level density parameters of the Gilbert–Cameron systematics are plotted with open symbols in Fig. 15. As it can be seen they are very close to (practically the same as) the ratios reported in Sect. 3, plotted here with full symbols.

Fig. 15
figure 15

The ratios \({{<}T{>} /{<}T}_{i}{{>} }\) (blue squares for light fragments and red circles for heavy fragments) and \({<}{E}_{{r}}{>} /{<}{ E}^{*}{>}\) (green squares for light fragments and magenta circles for heavy fragments) corresponding to all emission sequences are plotted as a function of \({<}{ \mathrm{TXE}}{>}\). The ratios based on level density parameters provided by the Gilbert–Cameron (G–C) systematics and constant \(\upsigma _\mathrm{{c}}\) are given with open symbols in comparison with the results of Sect. 3 (based on variable \(\upsigma _\mathrm{{c}}(\varepsilon )\) and level density parameters of the Egidy–Bucurescu (E–B) systematics for BSFG) which are given with full symbols

Fig. 16
figure 16

The ratios \({{<}T}_{k}{{>} /{<}T}_{i}{{>} }\) as a function of \({<}{\mathrm{TXE}}{>}\) obtained with constant \(\upsigma _\mathrm{{c}}\) and level density parameters provided by the Gilbert–Cameron systematic (open symbols) in comparison with the ratios based on variable \(\upsigma _\mathrm{{c}}(\varepsilon )\) and level density parameters provided by the Egidy–Bucurescu systematics (full symbols)

The temperature and excitation energy ratios corresponding to each emission sequence based on a constant \(\upsigma _\mathrm{{c}}\) and level density parameters of the Gilbert–Cameron systematics do not differ significantly from the ratios of Sect. 3 (which are based on other prescriptions). This fact is exemplified in Fig. 16 for the ratios \({{<}T}_{k}{{>} /{<}T}_{i}{{>}}\). It can be seen that the results based on the present prescriptions plotted with open symbols almost cover the results of Sect. 3 given with full symbols.

The average center-of mass energy \({{<}}\varepsilon {{>} }_{k}\) of each emitted neutron as a function of the average residual temperature \({{<}T}_{k}{{>} }\) based on the present prescriptions is plotted in Fig. 17 with full squares for light fragments and open circles for heavy fragments (the same symbol being used for all sequences). As it can be seen the linear dependence is practically the same for light and heavy fragments (the full squares and open circles almost cover each others), the linear fits (blue and red lines) being almost the same, too. The use of a constant compound cross-section of the inverse process leading to \({<}\varepsilon {>} ={2T}\) is confirmed by these linear fits having the slope values of 2 and very low intercept values of about 10\({}^{{-7}}\).

Fig. 17
figure 17

\({<}\varepsilon {>}_{{k}}\) as a function of \({<}{ T}_{{k}}{>}\) (the same symbol for all k, full squares are for light fragments, open circles for heavy fragments) and the linear fits (a blue line for light fragments and a red line for heavy fragments)

Fig. 18
figure 18

\({<}\varepsilon {>}_{{k}}\) as a function of \({<}{E}_{{r}}^{{(k)}}{>}^{{1/2}}\) corresponding to the light fragments (full squares) and heavy fragments (open circles) using the same symbol for all sequences. Linear fits of data corresponding to each fragment group are also plotted (with a blue line for light fragments and a red line for heavy fragments)

Fig. 19
figure 19

Average level density parameters provided by the Gilbert–Cameron systematics for the initial fragments (black and wine stars for the light and heavy fragments, respectively) and for the residual light and heavy fragments (full and open symbols, respectively) corresponding to the emission sequences \({k} = 1 \) (red circles) and \({k} = 2\) (blue squares). The total average level density parameters are plotted with full black diamonds for light fragments and open black diamonds for heavy fragments. The horizontal solid lines indicate the global values of \({{<}a{>}}\) resulting from the slopes of the linear fits given in Figs. 15 and 16

Table 3 Slopes and intercepts of the linear fits of Fig. 1
Table 4 Slopes and intercepts of the linear fits of Fig. 2

Figure 18 shows \({{<}}\varepsilon {{>} }_{k}\) as a function of \({{<}E}_{r}^{(k)}{{>} }^{1/2}\) based on the present prescriptions (full squares for light fragments and open circles for heavy fragments (the same symbol being used for all sequences). Linear dependences are visible, being fitted by the blue and red line, respectively. The slopes and intercepts of these linear fits are close to the slopes of the linear fits given in Fig. 5 of Sect. 3 where other prescriptions for the level density parameter and \(\upsigma _\mathrm{{c}}(\varepsilon )\) were used.

Following the procedure of Sect. 3, i.e. the use of the slope \(\upalpha = 2\) of \({{<}}\varepsilon {{>} }_{k}\) as a function of \({{<}T}_{k}{{>} }\) and the slopes values of the linear fits given in Fig. 18, the global values of the average level density parameter of about 15 MeV\(^{\mathrm{-1}}\) (for light fragments) and 13 MeV\(^{\mathrm{-1}}\) (for heavy fragments) are obtained. These values are illustrated in Fig. 19 by horizontal solid lines.

Table 5 Slopes and intercepts of the linear fits of Fig. 3 (left part)
Table 6 Slopes and intercepts of the linear fits of Fig 4 (right part)

The average level density parameters of initial fragments (black and wine stars) and of the residual fragments corresponding to the first two emission sequences, calculated according to Eq. (2), are also given in Fig. 17 (with full symbols for light fragments and open symbols for heavy fragments, red circles for \({k} = 1\) and blue squares for \({k} = 2\)). The total average level density parameters obtained according to Eq. (3) are plotted with black diamonds (full for light fragments and open for heavy fragments).

Table 7 Maximum temperatures of residual temperature distributions corresponding to each emission sequence and probabilities for emission of each prompt neutron for \({}^{{235}}\)U(n,f) at \({E}_{{n}} = 0.5 \,\mathrm{MeV}\)

As can be seen in the case of light fragments the global value of about 15 MeV\(^{\mathrm{-1}}\) (based on the slopes of the linear fits of Figs. 17 and 18) averages well the calculated total average level density parameters (full black diamonds). In the case of heavy fragments the global value of about 13 MeV\(^{\mathrm{-1}}\) slightly underestimates the general trend of the calculated total average level density parameters (open black diamonds), which is better approximated by a global value of about 13.5 MeV\(^{\mathrm{-1}}\) (illustrated by a horizontal dashed line).

A4: Uncertainties of fit results

The uncertainties of fit results, which are not written in the figure legends (because of lack of space) are given in Tables 3, 4, 5 and 6.

Details about the uncertainties given in Table 2of the main text

The uncertainties of \({<}\varepsilon {>}_{{L,H}}\) and \({<}\varepsilon {>}\) given in Table 2 of the main text are due to the uncertainties of the linear fit results of \({<}\varepsilon {>}_{{k}}\) as a function of \({<}{T}_{{k}}{>}\) given in Fig. 4 and to the uncertainties of the linear fit results of \({Pn}_{\mathrm{k}}\) as a function of \({<}{{ E}^{*}}_{{k}}{>}\) (plotted in the left part of Fig. 3) which are given in Table 5 (because of the lack of space inside the frames of Fig. 3).

The total \({<}\varepsilon {>}_{{L,H}}\) (corresponding to all emitted neutrons from the light and heavy fragments, respectively) are calculated according to Eq. (3) particularized here for the average prompt neutron energy in the center-of-mass frame.

$$\begin{aligned} {<}\varepsilon {>}_{L,H} ={\sum \limits _k {{<}\varepsilon {>}_{k}^{(L,H)} \,Pn_{k}^{(L,H)} } } / {\sum \limits _k {Pn_{k}^{(L,H)} } }, \end{aligned}$$
(a2)

in which the uncertainties of the linear fits mentioned above are propagated, according to the well-known formula of error propagation. The total \({<}\varepsilon {>}\) corresponding to the emission of all prompt neutrons from all fragments is calculated as

$$\begin{aligned} {<}\varepsilon {>}=w_{L} {<}\varepsilon {>}_{L} +w_{H} {<}\varepsilon {>}_{H}, \end{aligned}$$
(a3)

in which \({<}\varepsilon {>}_{{L,H}}\) are given by Eq. (a2) and \(w_{L,H}\) are the weights of the light and heavy fragment groups \(w_{L,H} ={{<}E^*{>}_{L,H} } / {{<}\mathrm{TXE}{>}}\) (with the values of \({<}{E}^{*}{>}_{{L,H}}\) and \({<}\mathrm{TXE}{>}\) mentioned in Sect. 4). The uncertainties of \({<}\varepsilon {>}\) given in Table 2 are obtained by the propagation in Eq. (2) of the uncertainties in \({<}\varepsilon {>}_{{L,H}}\).

A5: Maximum temperatures of \({P}_{k}(T)\) distributions and probabilities for emission of each neutron for\({}^{235}\) U(n,f) at \({E}_{n} =\) 0.5 MeV

For the fission case \({}^{{235}}\)U(\({n}_{\mathrm{0.5MeV}}\),f) the maximum temperatures of the residual temperature distributions corresponding to each emission sequence are obtained according to Eq. (7) using the average initial temperatures \({<}{T}_{{i}}{>}_{{L,H}}\) obtained form the average values \({<}{{ E}^{*}}_{{L,H}}{>}\) and \({<}{ a}_{{L,H}}{>}\) given in Section 4 and the constant values \({r}_{\mathrm{k}}\) of the temperature ratios from Fig. 1. They are given in the first two columns of Table 7.

The probabilities for emission of each prompt neutron, given in the last two columns of Table 7, are obtained from the linear fits given in Fig. 3 (left part) using the average residual temperature values \({<}{ T}_{{k}}{>}_{{L,H}}\) obtained from the systematics of Fig. 1.

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Tudora, A. Systematics of different quantities related to sequential prompt emission in fission. Eur. Phys. J. A 56, 84 (2020). https://doi.org/10.1140/epja/s10050-020-00059-2

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