Inclusion of sequential emission into the most probable fragmentation approach (Los Alamos model) and its validation

Abstract

A new version of the Los Alamos (LA) model, based on more physical considerations than the previous versions by taking into account the sequential emission of prompt neutrons, is proposed. A residual temperature distribution for each emission sequence P$_{\mathrm {k}}$(T) is considered, so that this new version can provide the prompt neutron spectrum in the center-of-mass and laboratory frames of each neutron successively emitted from the light and heavy fragment. The LA model with sequential emission is applied on 15 fission cases which were not included in the 49 cases used in the elaboration of the systematics of different quantities of residual nuclei, on which the general form of $\hbox {P}_{\mathrm {k}}$(T) is based. The good description of experimental prompt neutron spectrum data of these fission cases by the model results constitutes a relevant validation of the LA model with sequential emission. Systematic behaviours of the average energies (in the center-of-mass and laboratory frames) provided by this model for each neutron sequentially emitted, are also emphasized.

Appendices

Appendix 1

The application of the deterministic modeling with a detailed treatment of sequential emission [14] to numerous fission cases (i.e. 49 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 led to interesting systematic behaviours of different quantities characterizing the residual fragments and the prompt emission, which were reported in Ref. [4].

Fig. 10

(Reproduction of Fig. 1 from Ref. [4]): ratios of the average residual temperature to the initial temperature following the emission of the first (red circles), second (blue squares), third (green diamonds), 4th (dark yellow up triangles) and 5th (orange down triangles) prompt neutron and of all prompt neutrons (black stars) from the light fragments (upper part) and heavy fragments (lower part). The horizontal lines plotted with the same colour as the respective symbol indicate the constant values approximating the respective calculated ratios

One of these systematics, referring to the ratios of average residual temperatures for each emission sequence ${<} \hbox {T}_{\mathrm {k}}{>}$ and for all sequences ${<} \hbox {T}{>}$ to the average temperature of initial fragments ${<}\hbox {T}_{\mathrm {i}}{>}$ , is the basis for the general triangular forms of the residual temperature distributions entering the LA model with and without sequential emission.

This systematic of average temperature ratios is illustrated in Fig. 10 (which reproduces Fig. 1 of Ref. [4]). The ratios ${<}\hbox {T}_{\mathrm {k}}{>} /{<} T_{\mathrm {i}}{>}$ of the 49 fission cases (obtained as first order momentums of the initial and residual temperature distributions provided by the sequential emission modeling of Ref. [14]) are plotted as a function of ${<}\hbox {TXE}{>}$ as following: with red circles for $\hbox {k}=1$ (first emission sequence), blue squares ($\hbox {k}=2$), green diamonds ($\hbox {k}=3$), dark yellow up triangles ($\hbox {k}=4$) and orange down triangles ($\hbox {k}=5$). The ratio ${<}\hbox {T}{>}/{<}T_i{>}$ (corresponding to all emission sequences) is plotted with black stars. The upper and lower part of the figure refers to the sequential emission from the light and heavy fragments, respectively [4]. It is easily seen that all average temperature ratios exhibit a very slow variation with ${<}\hbox {TXE}{>}$ so that they can be considered almost constant. The constant values approximating the mean values of these ratios are represented by horizontal lines plotted with the same colour as the respective symbol [4]. The corresponding values (mentioned in the main text below Eqs. (2.1, 2.2)) are indicated in the right part of each frame. All explanations and aspects concerning this systematic are given in Ref. [4].

We mention only that this systematic of average temperature ratios (illustrated in Fig. 10) is a consequence of the periodicity of nuclear properties, reflected in the behaviour of level densities, neutron separation energies, shell corrections etc. of the multitude of fission fragments (taken into account by the large fragmentation ranges of numerous fissioning nuclei used in the sequential emission model calculations). These fragments (both initial and residuals) cover a large part of the Live Chart of Nuclides (i.e. the medium neutron rich nuclei placed below the beta stability curve).

The simple mathematical procedure to approximate any numerical residual temperature distribution (e.g. the distributions reported by Terrell [5], the distributions provided by the sequential emission modeling reported in Ref. [14]) by a triangular shape, consists of the following:

Using a general expression of a triangular shape:

$$\begin{aligned} P(T)=\left\{ {\begin{array}{l}\ \displaystyle \frac{P_{\max } }{T_{1} }\,T\quad \quad \quad \quad T\le T_{1} \\ \\ \displaystyle \frac{P_{\max } }{T_{2} -T_{1} }(T_{2} -T)\quad T_{1} \le T\le T_{2} \\ \end{array}} \right. \end{aligned}$$

(a1.1)

in which $\hbox {P}_{\mathrm {max}}$ is the maximum value of the triangular P(T), i.e. $\hbox {P}_{\mathrm {max}}=\hbox {P(T}_{1})$ and $\hbox {T}_{2}$ is the maximum temperature.

The parameters $\hbox {P}_{\mathrm {max}}$, $\hbox {T}_{1}$ and $\hbox {T}_{2}$ can be interrelated by taking into account the physical conditions i) and ii) mentioned in the main text.

The condition i) $\int \nolimits _0^{T\max } {P(T)\,dT} =1$ (in which P(T) is given by Eq. (a1.1)) leads to $P_{\max } =2/T_{2}$ and the condition ii) $\int \nolimits _0^{T\max } {T\,P(T)\,dT} ={<}T{>}$ (in which P(T) is given by Eq. (a1.1) and ${<} \hbox {T}{>}$ is the average value provided by the sequential emission modeling [14] or, the same, the first order momentum of a numerical distribution) leads to $T_{1} +T_{2} =3{<}T{>}$.

Consequently the triangular P(T) (with a moderately broad cut-off at high temperatures) which approximates a numerical distribution with the first order momentum value ${<}T{>}$ becomes:

$$\begin{aligned} P(T){=}\left\{ {\begin{array}{l} \displaystyle \frac{2T}{T_{1} (3{<}T{>}-T_{1} )}\quad \quad \quad \quad \quad T\le T_{1} \\ \\ \displaystyle \frac{2}{3{<}T{>}-T_{1} }\,\frac{3{<}T{>}-T_{1} -T}{3{<}T{>}-2T_{1} }\quad T_{1} \le T\le 3{<}T{>}-T_{1} \\ \end{array}} \right. \end{aligned}$$

(a1.2)

The diffuse high temperature cut-off of P(T) given by Eq. (a1.2) can be replaced by a sharp cut-off so that P(T) becomes:

$$\begin{aligned} P(T)=\left\{ {\begin{array}{l} \displaystyle \frac{8\,T}{9{<}T{>}^{2}}\quad T\le \frac{3}{2}{<}T{>} \\ \\ 0\quad \quad \quad \quad T>\displaystyle \frac{3}{2}{<}T{>}\quad \\ \end{array}} \right. \end{aligned}$$

(a1.3)

which is Eqs. (3) or (4) of the main text with $T_{\max } ={3{<}T{>}} \big / 2$ (in which ${<}\hbox {T}{>}$ is the value of the first order momentum of the numerical residual temperature distribution which is approximated by a triangular form).

Appendix 2

The average values of the input parameters given in Table 2 are based on the PbP treatment and the experimental fragment distributions Y(A,TKE) (mentioned in the last column)

Regarding the PbP treatment:

• (a) The fragmentation ranges of all studied fissioning nuclei are constructed as usually (Ref. [2] and references therein), i.e.: a large fragment mass range going from symmetric fission up to a very asymmetric split is considered. For each mass number A of this range, three charge numbers Z are taken as the nearest integer values above and below the most probable charge taken as the unchanged charge distribution (UCD) corrected with the charge polarization:

  $$\begin{aligned} Zp(A)=Z_{UCD} (A)+\Delta Z(A) \end{aligned}$$

  (a2.1)

  The charge polarization $\Delta$Z(A) is taken either dependent on A (provided by the Zp model of Wahl [33]) or the same value is considered for all A, i.e. the average value $\Delta Z =|0.5|$ (with the sign plus for light fragment and minus for the heavy ones). For each fragment {A, Z} of the fragmentation range mentioned above the calculations are done at TKE values covering a large range (e.g. from 100 to 200 MeV) with a step size of 2 MeV.

• (b) The excitation energies of fully accelerated fragments E*(A,Z,TKE) are obtained by sharing TXE of each fragmentation at each TKE value according to the TXE partition method based on modeling at scission usually employed in the PbP treatment [2, 10, 34, 35].

• (c) The level density parameters of fully accelerated fragments a(A,Z,TKE) are provided by the super-fluid model in which the shell-corrections of Möller and Nix from RIPL [36] and the parameterizations of Ignatiuk [19] for the dumping and asymptotic level density parameter were used.

Table 2 Total average values of the input parameters of LA model (without or with sequential emission)

Regarding the average values (given in Table 2).

The average values of the excitation energy ${<}E^*{>}_{\mathrm {L,H}}$ and level density parameter ${<}\hbox {a}{>}_{\mathrm {L,H}}$ of fully accelerated fragments are obtained by averaging the multi-parametric matrices E*(A,Z,TKE) and a(A,Z,TKE) provided by the PbP treatment over the multiple fragment distributions:

$$\begin{aligned} Y(A{,}Z{,}{} \textit{TKE})=p(Z{,}A)\,Y_{\exp } (A{,}{} \textit{TKE}) \end{aligned}$$

(a2.2)

in which the isobaric charge distribution p(Z, A) is a Gaussian function centered on Zp(A) of Eq. (a2.1), with the root-mean-square rms(A) either provided by the Zp model [33] or the same average value rms $=$ 0.6 is taken for all A. $\hbox {Y}_{\mathrm {exp}}$(A,TKE) are the data mentioned in the last column of Table 2.

The average total kinetic energy ${<}\hbox {TKE}{>}$ given in the second column of Table 2 are resulting from the Y(A,TKE) data and the average initial temperatures (4th column) are calculated according to Eq. (11).

The most probable fragmentation (5th column), i.e. the charge and mass numbers of the light and heavy fragment, are the nearest integer values of the average masses ${<}\hbox {A}_{\mathrm {L,H}}{>}$ and charges ${<}\hbox {Z}_{\mathrm {L,H}}{>}$ resulting from the Y(A,Z,TKE) distributions.

Appendix 3

1.1 A3.1 Representation of prompt neutron spectra as ratio to a Maxwellian spectrum

The graphical representation of prompt neutron spectra as ratios to a Maxwellian spectrum is currently used in order to assure a better visualization of the spectrum over the entire energy range of prompt neutrons. Otherwise, for a good visualization of the high and low energy parts of the spectrum two separate frames are needed (one in logarithmic scale and another in linear scale).

In principle in such graphical representations any value of the temperature parameter $\hbox {T}_{\mathrm {M}}$ entering the Maxwellian spectrum

$$\begin{aligned} N_{Maxw} (E)=\frac{2}{\sqrt{\pi }\,T_{M}^{3/2}}\exp (-E/T_{M} ) \end{aligned}$$

(a3.1)

can be used.

In order to avoid graphical representations in which the plotted prompt neutron spectrum (as ratio to a Maxwellian spectrum) is placed too high or too low compared to unity (usually indicated by a horizontal line) the $\hbox {T}_{\mathrm {M}}$ value is chosen so that the first order momentum of the respective prompt neutron spectrum is equal or close to the first order momentum of the Maxwellian spectrum, i.e.

$$\begin{aligned} <E>=\int \limits _0^\infty {E\,N(E)\,dE} \cong \frac{3}{2}T_{M} \end{aligned}$$

(a3.2)

The representation as ratio to a Maxwellian spectrum with $\hbox {T}_{\mathrm {M}}$ obtained from Eq. (a3.2) highlights the differences in shape between the studied prompt neutron spectrum and the so called “equivalent Maxwellian spectrum”.

Note, in the graphical representation of PFNS as ratio to a Maxwellian spectrum, the value used for $\hbox {T}_{\mathrm {M}}$ is usually indicated either in the title of the vertical axis or in the figure legend.

1.2 A3.2 Normalization of experimental data to a calculated prompt neutron spectrum

The experimental prompt neutron spectrum data are usually given in arbitrary units (e.g. counts, part/fission etc., being not normalized to unity) and the data do not cover the entire energy range of prompt neutrons. Consequently each time when a calculated spectrum is compared with the experimental data, the normalization of experimental data to the respective calculated spectrum must be done.

Note, when a calculated/theoretical prompt neutron spectrum (which is, obviously, normalized to unity) is compared with experimental data (normalized to this spectrum), this means a comparison of spectrum shapes.

The well-known simple procedure for normalization of experimental data to a calculated spectrum (mentioned by Madland and Nix in Ref. [1], too) consists of the multiplication of the experimental spectrum data by the normalization factor:

$$\begin{aligned} f=A_{calc} /A{ }_{\exp } \end{aligned}$$

(a3.3)

in which $\hbox {A}_{\mathrm {calc}}$ and $\hbox {A}_{\mathrm {exp}}$ are the surfaces of the calculated spectrum $\hbox {N}_{\mathrm {calc}}$(E) and experimental spectrum $\hbox {N}_{\mathrm {exp}}$(E), both between the lower ($\hbox {E}_{1})$ and upper ($\hbox {E}_{2})$ limits of the experimental data, i.e.

$$\begin{aligned} A_{calc}= & {} \int \limits _{E1}^{E2} {N_{calc} (E)\,dE} \end{aligned}$$

(a3.4)

$$\begin{aligned} A_{\exp }= & {} \int \limits _{E1}^{E2} {N_{\exp } (E)\,dE} \end{aligned}$$

(a3.5)

Note, sometime the word “re-normalization” is used instead of “normalization” because if more calculated spectra are compared with the same experimental data, then these data must be normalized to each calculated spectrum, which means a re-normalization of data each time.