Systematics of input parameters for the Los Alamos model with sequential emission

Abstract

The application of the Point-by-Point treatment to numerous fission cases (i.e. 74 cases including many actinides fissioning spontaneously or induced by thermal neutrons and fast neutrons with energies up to the threshold of the second chance fission) has emphasized interesting systematic behaviors of different quantities characterizing the fission fragments. Such systematics can provide values of the input parameters for a new version of the Los Alamos (LA) model including the sequential emission of prompt neutrons, which was recently developed and validated. The prompt neutron spectrum results based on the systematics of input parameters are obtained in good agreement with the experimental data of many actinides. The LA model with sequential emission together with the proposed systematics of its input parameters can predict prompt neutron spectra of fissioning nuclei without any experimental information, thus it becomes an useful tool for prompt neutron spectrum evaluations.

Appendices

Appendix

1.1 A1. Basic relations of the LA model with sequential emission

The main relations of the LA model with sequential emission [1, 2] are synthesized as following.

The center-of-mass energy spectrum of each neutron (indexed k) successively emitted from the light and heavy fragment is calculated as

$$\begin{aligned} \Phi _{k}^{(L,H)} (\varepsilon )= & {} \int \limits _0^{T_{\max \,k}^{(L,H)} } {P_{k}^{(L,H)} (T)\,\phi _{k}^{(L,H)} (\varepsilon )\,dT}\nonumber \\= & {} \varepsilon \,\sigma _{c\,L,H}^{(k)} (\varepsilon )\int \limits _0^{T_{\max \,k}^{(L,H)} } K_{k}^{(L,H)} (T)\,P_{k}^{(L,H)} (T)\exp (-\varepsilon /T)\;dT \nonumber \\ \end{aligned}$$

(a1.1)

with the normalization constant

$$\begin{aligned} K_{k}^{(L,H)} (T)=\left( {\int \limits _0^\infty {\varepsilon \;\sigma _{c\,L,H}^{(k)} (\varepsilon )\;\exp (-\varepsilon /T)\;d\varepsilon } } \right) ^{-1} \end{aligned}$$

(a1.2)

in which the compound nucleus cross-sections of the inverse process $\sigma _{c\,L,H}^{(k)} (\varepsilon )$ refer to each fragment emitting neutrons, i.e. the initial light or heavy fragment for $\hbox {k}=1$, the first light and heavy residual fragment for $\hbox {k}=2$, the second light and heavy residual fragment for $\hbox {k}=3$, etc. [1, 2].

The residual temperature distribution $P_{k}^{(L,H)}(T)$ corresponding to each emission sequence entering Eq. (a1.1) has a triangular form given by the following relation (in which the indexes L,H are omitted):

$$\begin{aligned} P_{k} (T)=\left\{ {\begin{array}{l} {2T} \big / {T_{\max \,k}^{2} \quad T\le T_{\max \,k} } \\ 0\qquad \qquad \quad \quad T>T_{\max \,k} \\ \end{array}} \right. \end{aligned}$$

(a2)

with the maximum temperature [2]

$$\begin{aligned} T_{\max \,k}^{(L,H)} =\frac{3}{2}\;r_{k}^{(L,H)} \;<\!T_{i}\!>_{L,H} \end{aligned}$$

(a3)

in which $r_{k}^{(L,H)}$ are the constant values of the temperature ratios given in Fig. 1 of Ref. [1], i.e. $r_{k=1}^{(L,H)}{=}{} \textit{0.7, r}_{k=2}^{(L,H)}{=}$$\textit{0.5, r}_{k=3}^{(L)}{=}{} \textit{0.42, r}_{k=3}^{(H)}{=}{} \textit{0.36, r}_{k=4}^{(L)}{=}{} \textit{0.38, r}_{k=4}^{(H)}{=}{} \textit{0.28, r}_{k=5}^{(L)}{=}$$\textit{0.33, r}_{k=5}^{(H)}{=}{} \textit{0.20}$ and ${<}\hbox {T}_{\mathrm {i}}{>}_{\mathrm {L,H}}$ are the average initial temperatures of the light and heavy fragment, which are input parameters.

The spectrum in the laboratory frame of the k-th prompt neutron (emitted isotropically in the center-of-mass frame of the parent fission fragment) is calculated as:

$$\begin{aligned} N_{k}^{(L,H)} (E)=\int \limits _{(\sqrt{E} -\sqrt{E_{f{}L,H} } )^{2}}^{(\sqrt{E} +\sqrt{E_{f{}L,H} } )^{2}} {\frac{\Phi _{k}^{(L,H)} (\varepsilon )\,d\varepsilon }{4\,\sqrt{E_{f\,L,H} \;\varepsilon } }} \end{aligned}$$

(a4)

in which $\Phi _{k}^{(L,H)} (\varepsilon )$ is given by Eqs.(a1) and $E_{f L,H }$ is the kinetic energy per nucleon (the same for all sequences) corresponding to the light and heavy fragment [1, 2].

The spectrum in the center-of-mass and laboratory frames corresponding to all prompt neutrons successively emitted from the light and heavy fragment are obtained by averaging the spectra corresponding to each emitted neutron [Eqs. (a1.1) or (a4)] over the probabilities for emission of each prompt neutron $\textit{Pn}_{k}^{(L,H)}$ which are provided by the systematics reported in the left part of Fig. 3 from Ref. [1].

The total spectrum in the center-of-mass and laboratory frames is calculated as a sum of weighted spectra of all neutrons emitted from the light and heavy fragment, the weights being expressed by the excitation energy ratios $R_{L,H}{=}{{<}{E}^{*}{>}}_{L,H}{{/}{<}{\textit{TXE}}{>}}$.

A2. Guide for the use of the systematics of input parameter for the LA model with sequential emission

Using the the systematics of Sect. 2 the input parameter values for any nucleus (with the charge and mass numbers $\hbox {Z}_{\mathrm {0}}$ and $\hbox {A}_{\mathrm {0}})$, fissioning spontaneously or induced by neutrons (with incident energies up to the threshold of the second chance fission), can be easily calculated as following:

The average total excitation energy of fragments at full acceleration is calculated as:

$$\begin{aligned} {<}{} \textit{TXE}{>}{=}{<}Q{-}{} \textit{TKE}{>}{+}{E}^{*} \end{aligned}$$

(a5)

where $E^*{=}En+Bn$ is the excitation energy of the compound nucleus undergoing fission induced by neutrons (with the incident energy En) and Bn is the neutron binding/separation energy of this nucleus. In the case of spontaneous fission $E^*{=}{} \textit{0}$. ${<}Q$–$\textit{TKE}{>}$ entering Eq. (a5) is the linear fit given in Fig. 4, i.e.:

$$\begin{aligned} {<}Q-\textit{TKE}{>}{=}310.2863\,\textit{XF}-222.8227\quad \,(MeV) \end{aligned}$$

(a6)

in which the fissility parameter is:

$$\begin{aligned} XF=\frac{Z_{0}^{2} /A_{0} }{50.883(1-1.7826\,\eta ^{2})} \end{aligned}$$

(a7)

and $\eta =(N_{0} -Z_{0} )/A_{0}$.

The average total kinetic energy <TKE> of fully-accelerated fragments can be obtained either by subtracting $<Q$–$\textit{TKE}>$ of Eq. (a6) from the average Q-value given by the linear fit of Fig. 5, leading to the following linear dependence on XF (plotted in Fig. 7 with a black line):

$$\begin{aligned} {<}{} \textit{TKE}{>}{=}319.5914\,\textit{XF}-75.3535\quad (MeV) \end{aligned}$$

(a8)

or by using the systematic of Viola et al. [29]:

$$\begin{aligned} {<}{} \textit{TKE}{>}{=}(0.1189\pm 0.0011)\,\frac{Z_{0}^{2} }{A_{0}^{1/3} }+(7.3\pm 1.5) \end{aligned}$$

(a9)

The average excitation energies of complementary fragments ${<}{E}^{*}{>}_{L,H}$ are obtained from the linear fits given in Fig. 9, i.e.:

$$\begin{aligned} \begin{array}{l} R_{L} ={<}{E}^{*}{>}_{L} {/}{<}{} \textit{TXE}{>}{=}0.6323\,\textit{XF}{+}0.0646 \\ R_{H} ={<}{E}^{*}{>}_{H} {/}{<}{} \textit{TXE}{>}{=}-0.6323\,\textit{XF}{+}0.9354 \\ \end{array}\nonumber \\ \end{aligned}$$

(a10)

using the $\textit{<TXE> }$ value calculated according to Eqs. (a5–a7).

Table 2 Input parameters of the LA model with sequential emission provided by the present systematics

Finally the average temperatures of initial light and heavy fragment are given by the linear fits plotted in Fig. 10:

$$\begin{aligned} \begin{array}{l} {<}{T}_{i}{>}_{L} {=}0.0357\,{<}{E}^{*}{>}_{L} {+}0.6111\quad (MeV) \\ {<}{T}_{i}{>}_{H} {=}0.0382\,{<}{E}^{*}{>}_{H} {+}0.6227\quad (MeV) \\ \end{array} \end{aligned}$$

(a11)

Note, for neutron-induced fission at higher En (above of about 3.5 MeV) the excitation energy ratios R$_{\mathrm {L,H}}$ can be slightly modified according to their dependence on En (using the linear fits given in Fig. 8). In this case the values of initial temperatures are determined by using the level density parameter values given by the lines plotted in Fig. 15.

Fig. 15

Average level density parameter ${<}\hbox {a}{>} _{\mathrm {L}}$ (full symbols) and ${<}\hbox {a}{>}_{\mathrm {H}}$ (symbols with a cross inside) obtained from the PbP treatment as a function of En (same symbols and colours as in Fig. 8). Both the constant values approximating the average level density parameters which exhibit a slow variation with En (indicated by horizontal lines and values mentioned in the right part of the frame) and the linear fits are plotted with solid and dashed lines, respectively, in the same colour as the respective symbol

The input parameter values provided by the present systematics for the fissioning nuclei used in the present validation are given in Table 2.

Note, for fissioning nuclei without any experimental information the most probable fragmentation can be easily established by assuming that the mass number of the most probable heavy fragment is 140 (taking into account that the heavy fragment group does not differ significantly from a fissioning actinide to another and in the majority of cases the heavy fragment mass distributions are centered around 140). The most probable charge corresponding to $\hbox {A}_{\mathrm {H}}=140$ is then obtained as the UCD distribution corrected with the charge polarization taken as − 0.5. And the charge number of the most probable heavy fragment is taken as the nearest integer of this most probable charge.

The influence played by the shape of the compound nucleus cross-section of the inverse process $\sigma _{\mathrm {c}}(\varepsilon )$ on the PFNS shape is well known, too (see e.g. Ref. [19] and references therein). Compound nucleus cross-section $\sigma _{\mathrm {c}}(\varepsilon )$ provided by optical model calculations with phenomenological potential parameterizations adequate for the nuclei which are fission fragments (e.g. Becchetti-Greenlees, Koning-Delaroche taken from RIPL-3 [35]) are currently used in refined prompt emission models including many fragmentations (either with a deterministic or a probabilistic Monte-Carlo treatment) and also in all versions of the LA model.

Taking into account that the shape exhibited by the majority of PFNS experimental data can be described by different model results in which $\sigma _{\mathrm {c}}(\varepsilon )$ is provided by optical model calculations using the potential parameterization of Becchetti-Greenlees, for fissioning nuclei without any experimental information, the predicted results of the LA model with sequential emission can be also based on this optical model parameterization.