Unified description of the fission probability for highly excited nuclei

ABSTRACT

We present a new model to describe the fission probability of the high-energy fission model, as deduced from the intra-nuclear cascade calculation with the Intra-Nuclear Cascade model of Liége (INCL) version 4.6 and Prokofiev’s phenomenological systematics of the proton-induced fission cross-sections. This model is implemented in the de-excitation model of the generalized evaporation model (GEM), and applied to Monte Carlo spallation reaction simulation using the Particle and Heavy Ion Transport code System (PHITS). Comparing with experimental data for subactinide nuclei shows that this model can provide a unified prediction of the proton-, neutron-, and deuteron-induced fission cross-sections with markedly improved accuracy. The calculated fission fragments tend to shift to higher mass numbers. To account for the isotopic distributions of fission fragments within the framework of a coupled INCL/GEM, modification of INCL would be required, especially for description of the highly excited states of residual nuclei.

KEYWORDS:

• Fission cross-section
• fission probability
• fission fragment
• PHITS
• spallation reaction
• high-energy fission model

1. Introduction

Recently, the Particle and Heavy Ion Transport code System, (PHITS) [1], has been successfully used in various studies, e.g. the shielding design of accelerator facilities [2,3], medical radiation [4,5], cosmic-rays [6,7], and accelerator-driven nuclear transmutation of high-level radioactive wastes [8,9]. In the Monte Carlo simulation of particle transport at energies above several tens of mega-electron volts, the spallation model serves as an event generator to provide event-by-event information of secondary particles and the residual nucleus produced from the spallation reaction (i.e. yield, kinetic energy, momentum, isotopic distribution, etc.).

In general, the spallation reaction is divided into two processes: the intra-nuclear cascade (INC) process and the de-excitation process (these are also referred to as the pre-equilibrium process and the equilibrium process, respectively). Once the spallation reaction occurs by bombarding a target nucleus with a high-energy particle, relatively high-energy particles such as nucleons, light charged particles, and/or pions are emitted from the target nucleus in the INC process; this process leads to excitation of the residual nucleus. After an equilibrium state is reached, relatively low energy neutrons, charged particles, and photons are emitted from the highly excited nucleus through the de-excitation process. For subactinide and actinide nuclei, this process is subdivided into two competitive processes: the evaporation process and the fission process.

The current version of PHITS employs a combination of the Intra-Nuclear Cascade model of Liège version 4.6 (INCL4.6) [10,11] and the generalized evaporation model (GEM) [12–14] as defaults for simulating the INC process and the de-excitation process, respectively. A recent benchmark study of the PHITS code [15] has demonstrated that the default spallation model (INCL4.6/GEM) reproduces some experimental data reasonably well; however discrepancies have been also revealed for some physical properties, including the yield of spallation products, as well as the energy and angular distributions of secondary particles. This study focuses on the spallation product yields.

Although spallation products can be produced from both the evaporation process and the fission process, this study exclusively treats the high-energy fission model for subactinide nuclei implemented in GEM. Regarding high-energy fission, Fukahori et al. [16–18] proposed a unified, simple expression that describes proton-, neutron-, and photon-induced fission cross-sections for both subactinide and actinide nuclei. Prokofiev [19] improved the proton-induced fission cross-section systematics using numerous experimental results compiled in the EXchange FORmat (EXFOR) experimental nuclear reaction database [20], and demonstrated that his systematics works relatively well for a wide range of target nuclei from threshold energies to the GeV range. These findings suggest that the fission probability used in the high-energy fission model should be describable using a relatively simple expression, e.g. one based on systematics.

In this paper, we propose a new model to describe the fission probability. The model is deduced from the INC calculation and Prokofiev’s proton-induced fission cross-section systematics, and it is applied to a Monte Carlo spallation reaction simulation by PHITS version 3.02. Here, INCL4.6 was employed for the calculation of the INC process, i.e. the excited state of the residual nuclei, which is used as an input parameter for the GEM, was calculated using INCL4.6. The model calculations are compared with experimental data of proton-, neutron-, and deuteron-induced fission cross-sections and fission-fragment production cross-sections. The cross-sections were calculated within the framework of the PHITS simulation incorporating our model, in which they are normalized to the non-elastic, or reaction, cross-section systematics implemented in PHITS [21]; for proton-induced and neutron-induced reactions, the Pearlstein–Niita cross-section systematics [22] was used; for deuteron-induced reactions, we employed the Minomo–Washiyama–Ogata (MWO) systematics [23] based on the continuum-discretized coupled-channels (CDCC) model calculation [24–26]. In addition, we present the necessary modifications to INCL to account for the final fission fragment yields within the framework of the GEM.

2. The model

2.1. Fission probability

Figure 1 gives a schematic of the high-energy fission model implemented in the GEM, together with the INC process. This model, which is based on Atchison’s multi-chance fission model [27,28] implemented in the Los Alamos High-Energy Transport code (LAHET) [29], assumes that fission competes with neutron evaporation at all stages until the excitation energy is fully released or until fission occurs. For each highly excited state, the fission probability is expressed as follows:

(1) $P=\frac{{\mathrm{\Gamma }}_{\mathrm{f}}}{{\sum }_j{\mathrm{\Gamma }}_j},j=\mathrm{f},n,\gamma ,p,d,\alpha ,...$(1)

(2) $\simeq \frac{{\mathrm{\Gamma }}_{\mathrm{f}}}{{\mathrm{\Gamma }}_{\mathrm{f}}+{\mathrm{\Gamma }}_n}=\frac{1}{1+{\mathrm{\Gamma }}_n/{\mathrm{\Gamma }}_{\mathrm{f}}},$(2)

Figure 1. Schematic of the high-energy fission model of the de-excitation process together with the INC process. Open circles represent the fission chances that determine whether fission occurs in accordance with the fission probability $P$. In this example, fission occurs after emitting three neutrons by evaporation. For simplicity, charged-particle evaporation is omitted.

where ${\mathrm{\Gamma }}_j$ $(j=n,\gamma ,p,d,\alpha ,...)$ is the particle evaporation decay width and ${\mathrm{\Gamma }}_{\mathrm{f}}$ is the fission decay width. For subactinide nuclei, Furihata’s original GEM calculates the widths ${\Gamma }_n$ and ${\Gamma }_{\text{f}}$ semi-empirically in the same manner as in Atchison’s model. Baznat et al. [30] refined the GEM model to fit the Prokofiev systematics by adjusting the level density parameters included in the expressions of ${\Gamma }_n$ and ${\Gamma }_{\text{f}}$; this model is embedded in the Monte Carlo N-Particle transport code version 6, MCNP6 [31,32]. It is noted that the original and refined GEM models are based on the Bertini INC model [33] or the ISABEL INC model [34], and on the Cascade-Exciton Model (CEM) [35] and the Los Alamos version of the Quark-Gluon String Model code (LAQGSM) [36], respectively. Therefore, the poor reproducibility of the spallation product yields in the INCL4.6/GEM calculation (see Iwamoto et al. [15]) compared with the ISABEL/GEM (see Furihata and Nakamura [12]) implies that their parametrizations in the GEM would not match those of INCL4.6 implemented in PHITS. Also, as pointed out in Leray et al. [37], the same INC model coupled to different de-excitation models or the same de-excitation model coupled to different INC models gives different results for the spallation product yields. These tendencies suggest that the description of residual nuclei produced from the INC process (i.e. excitation energy and isotopic distribution) remains to be clarified, implying that future possible improvement of INC models would require significant parametrizations and/or theoretical modifications of the de-excitation models.

To avoid the complication of parametrization of the decay widths, we compute neither ${\Gamma }_n$ nor ${\Gamma }_{\text{f}}$; rather, we calculate P directly by deducing from the INC calculation with INCL4.6 and Prokofiev’s fission cross-section systematics. First, we define the total fission probability ${\mathcal{P}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$, which is simply expressed as the ratio of the fission cross-section ${\sigma }_{\mathrm{f}}$ to the non-elastic, or reaction, cross-section ${\sigma }_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{l}}$:

(3) ${\mathcal{P}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=\frac{{\sigma }_{\mathrm{f}}}{{\sigma }_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{l}}}$(3)

where for ${\sigma }_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{l}}$ and ${\sigma }_{\mathrm{f}}$, we employ Pearlstein–Niita’s proton-induced non-elastic cross-section systematics [22] and Prokofiev’s proton-induced fission cross-section systematics, respectively. The Prokofiev systematics is expressed by

(4) $\begin{array}{cc}{\sigma }_{\mathrm{f}}(\mathrm{m}\mathrm{b})=& Q_1\left\{1-exp[-Q_3\left(E_p-Q_2\right)]\right\}\\ \hspace{0.17em}& \times \left(1-Q_{4}ln{E}_p\right),\end{array}$(4)

where $E_p$ is the incident proton energy in MeV and $Q_1$, $Q_2$, $Q_3$, and $Q_4$ are systematics to fit to the experimental data, which are expressed as follows:

(5) $Q_k(X_{\mathrm{C}\mathrm{N}})=exp\left(c_{k,1}+c_{k,2}{X}_{\mathrm{C}\mathrm{N}}+c_{k,3}\sqrt{X_{\mathrm{C}\mathrm{N}}}\right),$(5)

(6) $Q_4(X_{\mathrm{C}\mathrm{N}})=\left\{\begin{array}{cc}0& (X_{\mathrm{C}\mathrm{N}}\le 32.32),\\ c_{4,1}+c_{4,2}{X}_{\mathrm{C}\mathrm{N}}& (X_{\mathrm{C}\mathrm{N}}>32.32),\end{array}\right.$(6)

where $X_{\mathrm{C}\mathrm{N}}$ is the fissility parameter of a compound nucleus ($X_{\mathrm{C}\mathrm{N}}=Z_{\mathrm{C}\mathrm{N}}^2/A_{\mathrm{C}\mathrm{N}}$, $Z_{\mathrm{C}\mathrm{N}}=Z_t+1$, $A_{\mathrm{C}N}=A_t+1$); the indexes $\mathrm{C}\mathrm{N}$ and $t$ are the compound and target nuclei, respectively; and $c_{k,\mathrm{\ell }}$ $(k=1,2,3,4$, $\mathrm{\ell }=1,2,3)$ represent fitting parameters, the values of which are listed in Table 1. The calculated total fission probabilities for the ${ }^{209}$Bi($p,x$), ${ }^{208}$Pb($p,x$), ${ }^{197}$Au($p,x$), ${ }^{181}$Ta($p,x$), and ${ }^{165}$Ho($p,x$) reactions are shown in Figure 2. We see from this figure that the shape of the total fission probabilities is similar to that of the proton-induced fission cross-sections. This is attributable to the fact that the non-elastic cross-sections do not vary significantly with incident proton energies (see Niita et al. [22]).

Table 1. Fitting parameters $c_{k,\mathrm{\ell }}$

	$\mathrm{\ell }=1$	$\mathrm{\ell }=2$	$\mathrm{\ell }=3$
$k=1$	$505.94$	$17.049$	$-185.15$
$k=2$	$11.134$	$-0.1468$	$-0.4187$
$k=3$	$-23.999$	$0.7867$	$-1.210$
$k=4$	$-1.1188$	$0.0352$	–

Figure 2. Total fission probability for ${ }^{209}$Bi($p,x$), ${ }^{208}$Pb($p,x$), ${ }^{197}$Au($p,x$), ${ }^{181}$Ta($p,x$),and ${ }^{165}$Ho($p,x$) reactions as a function of incident proton energy, calculated with Prokofiev’s fission cross-section systematics and Pearlstein–Niita’s non-elastic cross-section systematics.

The INC model provides the event-by-event proton (or charge) number $Z$, mass number $A$, neutron number $N(=A-Z)$, excitation energy $E$, and momentum $p$ of the residual nucleus; as such, the initial properties of the projectile and target nucleus should be already lost at the beginning of the subsequent de-excitation process (See Figure 1.). To apply the Prokofiev systematics to the Monte Carlo spallation reaction simulation, we introduced the corresponding mean incident proton energy $⟨E_p⟩$, mean target charge $⟨Z_t⟩$, and mean target mass $⟨A_t⟩$.

Figure 3 shows the relationship between the mean excitation energy $⟨E⟩$ and the incident proton energy $E_p$ for the calculation of INCL4.6. We found that the incident proton energy can be expressed very well as the following simple form:

(7) $E_p=0.141\times ⟨E{⟩}^{\alpha },$(7)

Figure 3. Relationship between the mean excitation energy after the INC process and incident proton energy for proton-induced reactions for ${ }^{208}$Pb and ${ }^{165}$Ho. Data points and lines indicate the calculation results of INCL4.6 and Equation (7)(7) $E_p=0.141\times ⟨E{⟩}^{\alpha },$(7) , respectively.

where we take $\alpha =1.964-0.000884A_t$. We thus assume that $⟨E_p⟩$ has the following form:

(8) $⟨E_p⟩=0.141\times E^{\alpha }$(8)

with

(9) $\alpha =1.964-0.000884⟨A_t⟩.$(9)

Figure 4 shows the relationship between $⟨E⟩$ and the mean values of the reduced $Z$ and $A$ numbers ($⟨\mathrm{\Delta }Z⟩$, $⟨\mathrm{\Delta }A⟩$), which can be expressed approximately as

(10) $⟨\mathrm{\Delta }Z⟩=0.0216⟨E⟩-1.037,$(10)

(11) $⟨\mathrm{\Delta }A⟩=0.0457⟨E⟩-1.049,$(11)

Figure 4. Relationship between the mean reduced $Z$ and $A$ numbers ($⟨\mathrm{\Delta }Z⟩$, $⟨\mathrm{\Delta }A⟩$) and the mean excitation energy $⟨E⟩$ after the INC process for proton-induced reactions for ${ }^{209}$Bi, ${ }^{208}$Pb, ${ }^{197}$Au, ${ }^{181}$Ta, and ${ }^{165}$Ho. Data points and lines indicate the calculation results of INCL4.6 and Equations (10)(10) $⟨\mathrm{\Delta }Z⟩=0.0216⟨E⟩-1.037,$(10) and (11(11) $⟨\mathrm{\Delta }A⟩=0.0457⟨E⟩-1.049,$(11) ), respectively.

without dependence on target mass and charge. We, therefore, assume that $⟨Z_t⟩$ and $⟨A_t⟩$ are described as follows:

(12) $⟨Z_t⟩=Z+(0.0216E-1.037),$(12)

(13) $⟨A_t⟩=A+(0.0457E-1.049).$(13)

The fission probability $P$ is deduced from the total probability ${{\rm P}}_{\text{total}}$ calculated with the corresponding mean quantities of $⟨E_p⟩$, $⟨Z_t⟩$, and $⟨A_t⟩$ by introducing a correction factor $g$ as a function of $E$, $⟨Z_t⟩$, and $⟨A_t⟩$:

(14) $P=g(E,⟨Z_t⟩,⟨A_t⟩)\cdot {\mathcal{P}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}(⟨E_p⟩,⟨Z_t⟩,⟨A_t⟩),$(14)

where the factor $g$ is determined so as to fit the Prokofiev systematics and experimental fission cross-sections. We found that the following systematics works relatively well for a wide range of subactinide target nuclei:

(15) $g=max\left\{{\gamma }_0\left(\frac{1}{1+{\mathrm{e}}^{-{\gamma }_1(E-{\gamma }_2)}}-{\gamma }_3\right),0\right\}$(15)

with four adjustable parameters that are written as follows:

(16) ${\gamma }_0=0.46,$(16)

(17) ${\gamma }_1=0.10,$(17)

(18) ${\gamma }_2=57{\left(\frac{X_{{ }^{208}\mathrm{P}\mathrm{b}}}{⟨X_t⟩}\right)}^{5.5},$(18)

(19) ${\gamma }_3=\frac{1}{2}{\left(\frac{X_{{ }^{208}\mathrm{P}\mathrm{b}}}{⟨X_t⟩}\right)}^3,$(19)

where $X_{{ }^{208}\mathrm{P}\mathrm{b}}$ is the fissility parameter of ${ }^{208}$Pb (i.e. $X_{{ }^{208}\mathrm{P}\mathrm{b}}=32.327$) and $⟨X_t⟩$ is the corresponding mean fissility parameter defined as

(20) $⟨X_t⟩=\frac{{⟨Z_t⟩}^2}{⟨A_t⟩}.$(20)

Figure 5 illustrates the calculated $P$ for the corresponding mean of ($Z_t$, $A_t$) for ${ }^{209}$Bi, ${ }^{208}$Pb, ${ }^{197}$Au, ${ }^{181}$Ta, and ${ }^{165}$Ho as a function of excitation energy. Analogous with the fission cross-section, it is assumed that the fission probability should follow a simple trend, having threshold, rapid increase, and plateau. It should also be approximately proportional to the fission cross-section. As such, in Equation (15)(15) $g=max\left\{{\gamma }_0\left(\frac{1}{1+{\mathrm{e}}^{-{\gamma }_1(E-{\gamma }_2)}}-{\gamma }_3\right),0\right\}$(15) , ${\gamma }_0$ denotes an overall suppression parameter to allocate ${{\rm P}}_{\text{total}}$ to each fission chance and ${\gamma }_1$, ${\gamma }_2$, and ${\gamma }_3$ are shape parameters to reproduce the threshold and the curve of the rapid increase.

Figure 5. Fission probability $P$ for the corresponding mean of ($Z_t$, $A_t$) for ${ }^{209}$Bi, ${ }^{208}$Pb, ${ }^{197}$Au, ${ }^{181}$Ta, and ${ }^{165}$Ho as a function of excitation energy.

Thus, $P$ in Equation (14)(14) $P=g(E,⟨Z_t⟩,⟨A_t⟩)\cdot {\mathcal{P}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}(⟨E_p⟩,⟨Z_t⟩,⟨A_t⟩),$(14) is calculated only from the event-by-event information of charge number $Z$, mass number $A$ and excitation energy $E$ of the residual nucleus; therefore, our model is in principle applicable to any projectile, even for deuterons.

2.2. Isotopic distribution

In the fission process, two fission fragments are produced from the fissioning nucleus. The mass and charge of the fission fragments follow the conservation law:

(21) $A=A_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(1)}+A_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(2)},$(21)

(22) $Z=Z_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(1)}+Z_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(2)},$(22)

where $A$ and $Z$ are the mass and charge of the fissioning nucleus, respectively, and $A_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}$ and $Z_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}$ are the mass and charge of the fission fragment, respectively. The mass of one of the fission fragments $A_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(1)}$ is sampled from a Gaussian distribution of mean ${\mu }_{\mathrm{A}}(=A/2)$ and width ${\sigma }_A$. The mass of the other fission fragment $A_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(2)}$ is selected based on Equation (21)(21) $A=A_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(1)}+A_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(2)},$(21) . For ${\sigma }_A$, Furihata’s empirical parametrization is employed:

(23) ${\sigma }_A=C_1{X}^2+C_{2}X+C_3(E-B_{\mathrm{f}})+C_4,$(23)

where $C_1=0.122$, $C_2=-7.77$, $C_3=3.32\times {10}^{-2}$, and $C_4=134.0$ and $B_{\mathrm{f}}$ is the Thomas–Fermi fission barrier formulated by Myers and Świątecki [38]. The charge of a fission fragment $Z_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}$ is selected from a Gaussian distribution of mean ${\mu }_Z$ and width ${\sigma }_Z$ $(=0.75)$, where ${\mu }_Z$ is expressed as

(24) ${\mu }_Z=\frac{Z+Z_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(1)}-Z_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(2)}}{2},$(24)

where

(25) $Z_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(\mathrm{\ell })}=\frac{65.5A_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(\mathrm{\ell })}}{131+{\left(A_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(\mathrm{\ell })}\right)}^{2/3}},\mathrm{\ell }=1,2.$(25)

The total energy $T$ of the fission fragments is written as follows:

(26) $T=E+B_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(1)}+B_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(2)}-B,$(26)

where $E$ and $B$ are the excitation energy and the binding energy of the fissioning nucleus, respectively, and $B_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}$ is the binding energy of the fission fragment. The total kinetic energy (TKE) of the fission fragments follows a Gaussian distribution of mean $ϵ$ and width ${\sigma }_{ϵ}$. For $ϵ$, Rusanov–Itkis–Okolovich’s phenomenological systematics [39] is used:

(27) $ϵ(\mathrm{M}\mathrm{e}\mathrm{V})=\left\{\begin{array}{c}0.131Z^2/A^{1/3}\\ (Z^2/A^{1/3}\le 900),\\ 0.104Z^2/A^{1/3}+24.3\\ (Z^2/A^{1/3}>900).\end{array}\right.$(27)

For ${\sigma }_{ϵ}$, we apply Furihata’s systematics [13]:

(28) ${{\sigma }_{ϵ}}^{{ }^2}(\mathrm{M}\mathrm{e}{\mathrm{V}}^2)=\left\{\begin{array}{c}{C}_1\left(Z^2/A^{1/3}-1000\right)+C_2\\ \left(Z^2/A^{1/3}\le 1000\right),\\ C_2\\ \left(Z^2/A^{1/3}>1000\right),\end{array}\right.$(28)

where $C_1=5.70\times {10}^{-4}$ and $C_2=86.5$. The excitation energies of the fission fragments are given by distributing the total excitation energy, TXE$(=T-\mathrm{T}\mathrm{K}\mathrm{E})$ in accordance with their masses:

(29) $E_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(\mathrm{\ell })}=\mathrm{T}\mathrm{X}\mathrm{E}\times \frac{A_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(\mathrm{\ell })}}{A},\mathrm{\ell }=1,2.$(29)

3. Results and discussions

3.1. Fission cross-sections

Figure 6 shows the proton-induced fission cross-sections of ${ }^{209}$Bi, ${ }^{208}$Pb, ${ }^{197}$Au, ${ }^{181}$Ta, and ${ }^{165}$Ho, as calculated with the original GEM model and the modified GEM model. These are compared with experimental data from Refs [40–60] and an evaluation of FISCAL, which was developed by Fukahori et al [18]. FISCAL is a code for describing a unified systematics of proton-, neutron-, and photon-induced fission cross-sections; it was utilized for evaluating the JENDL high-energy file, JENDL/HE-2007 [61]. Overall, the modified model calculation agrees well with the experimental data and the Prokofiev systematics. In contrast, these are underestimated by the original GEM model. Whereas FISCAL partially fits the experimental cross-sections, disagreement is seen for the relatively light subactinides. Figure 7 shows the neutron-induced fission cross-sections from ${ }^{209}$Bi, ${ }^{208}$Pb, ${ }^{\mathrm{n}\mathrm{a}\mathrm{t}}$Pb, ${ }^{197}$Au, and ${ }^{181}$Ta calculated with the original and modified models, as compared with experimental data from Refs [62–65]. and the FISCAL evaluation. As was the case for proton-induced fission, both the original GEM model and FISCAL fail to reproduce the experimental results. However, despite the fact that the systematics of the fission probability is constructed from the proton-induced cross-sections, the modified model successfully accounts for the general trends and its calculated values are in agreement with the experimental values.

Figure 6. Proton-induced fission cross-sections of ${ }^{209}$Bi, ${ }^{208}$Pb, ${ }^{197}$Au, ${ }^{181}$Ta, and ${ }^{165}$Ho, as compared with experimental data from Refs [40–60].

Figure 7. Neutron-induced fission cross-sections of ${ }^{209}$Bi, ${ }^{208}$Pb, ${ }^{\mathrm{n}\mathrm{a}\mathrm{t}}$Pb, ${ }^{197}$Au, and ${ }^{181}$Ta, as compared with experimental data from Refs [62–65].

Figure 8 shows the deuteron-induced fission cross-sections of ${ }^{209}$Bi and ${ }^{208}$Pb calculated with the original and modified models, as compared with experimental data from Refs [42,66–68]. Although available experimental data are scarce, it appears that the modified model coupled to the MWO systematics works well for the deuteron-induced fission cross-sections.

Figure 8. Deuteron-induced fission cross section of ${ }^{209}$Bi and ${ }^{208}$Pb, compared with experimental data from Refs [42,66–68].

Figure 9 shows the cross-section ratios between proton- and neutron-induced fissions for ${ }^{209}$Bi, ${ }^{208}$Pb, ${ }^{197}$Au, and ${ }^{181}$Ta calculated with the modified model, as compared with experimental data from Refs [62,69]. It is seen that incident protons are more likely to cause fission than incident neutrons; this trend is linked to the expression of Equation (14)(14) $P=g(E,⟨Z_t⟩,⟨A_t⟩)\cdot {\mathcal{P}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}(⟨E_p⟩,⟨Z_t⟩,⟨A_t⟩),$(14) and related equations, in which the fission probability increases with increases in the fissility parameter. More specifically, although the target $Z$ and $A$ numbers decrease through the INC process unless the incident particle is captured, the incident proton ($Z=1$, $A=1$) contributes to increase both $Z$ and $A$; on the other hand, the incident neutron ($Z=0$, $A=1$) only involves the increase of $A$. Therefore, the fissility parameter $(X=Z^2/A)$ of residual nuclei induced by protons is expected to be larger than that induced by neutrons with the same energy. Furthermore, it is observed in this figure that the ratios decrease as incident energy increases. This is because the relative difference in the fissility parameters becomes small with the increase in incident energy.

Figure 9. Fission cross-section ratios between proton- and neutron-induced fissions for ${ }^{209}$Bi, ${ }^{208}$Pb, ${ }^{197}$Au, and ${ }^{181}$Ta targets calculated with the modified model, as compared with experimental data from [62,69].

Figure 10 shows the neutron-induced fission cross-section ratio between ${ }^{\mathrm{n}\mathrm{a}\mathrm{t}}$Pb and ${ }^{209}$Bi, as compared with experimental data of Ref [62]. In this figure, it is seen that ${ }^{209}$Bi is more likely to cause fission than ${ }^{\mathrm{n}\mathrm{a}\mathrm{t}}$Pb owing to the larger fissility parameter of ${ }^{209}$Bi. The modified model successfully reproduces this trend.

Figure 10. Neutron-induced fission cross section ratio between ${ }^{\mathrm{n}\mathrm{a}\mathrm{t}}$Pb and ${ }^{209}$Bi, as compared with experimental data from Ref [62].

3.2. Isotopic distributions

The calculated fission fragment distributions were compared with three experimental values measured using the inverse-kinematics technique at GSI Darmstadt: $p$ + ${ }^{208}$Pb (1000$A$ MeV) [52], $p$ + ${ }^{197}$Au (800$A$ MeV) [49], and $p$ + ${ }^{208}$Pb (500$A$ MeV) [70]. Figure 11 shows the fission fragment mass distributions for each element (${ }_{27}$Co, ${ }_{31}$Ga, ${ }_{35}$Br, ${ }_{39}$Y, ${ }_{43}$Tc, ${ }_{47}$Ag, and ${ }_{51}$Sb) for the three experiments. Due to the underestimation of the fission cross-sections or the fission probabilities, the original model generally underestimates the production cross-sections of the fission fragments. On the other hand, although the modified model accounts for the fission cross-sections, it fails to reproduce the peak positions of the mass distribution, which appear to uniformly shift to higher mass numbers. To clarify this trend, Figure 12 compares the ratio of the mean neutron numbers to each element ($⟨N⟩/Z$) of the fission fragments between the calculations and the experimental data. While the original GEM underestimates both the fission cross-sections and the production cross-sections of the fission fragments, it reproduces the $⟨N⟩/Z$ values fairly well. In contrast, the modified model generally overestimates the $⟨N⟩/Z$ values. This trend is explained as resulting from the alteration of the fission probability. In the modified model, the rate of neutron evaporation in the pre-fission process is suppressed compared with the original model because of the higher fission probability. However, even though the fission probability is modified, the fact remains that both models underestimate the total number of evaporation neutrons; this is likely attributable to the limited excitation energy given by INCL4.6.

Figure 11. Mass distribution for $p$ + ${ }^{208}$Pb (1000$A$ MeV), $p$ + ${ }^{197}$Au (800$A$ MeV), and $p$ + ${ }^{208}$Pb (500$A$ MeV) reactions, as compared with experimental data from Refs [49,52,70].

To investigate how many neutrons should be additionally evaporated within the framework of the modified GEM model when coupled to INCL4.6, we introduced a free parameter $\nu$ and changed Equation (21)(21) $A=A_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(1)}+A_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(2)},$(21) to

(30) $A=A_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(1)}+A_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(2)}+\nu ,$(30)

where $\nu$ represents a nonphysical parameter that corresponds to the number of neutrons forcibly removed from the fissioning system without giving excitation energy. The mass of one of the fission fragments $A_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(1)}$ is sampled from a Gaussian distribution of mean ${\mu }_{\mathrm{A}}(=(A-\nu )/2)$ and width ${\sigma }_A$. The remainder of the de-excitation calculation is the same as that given in the previous section. Figure 13 compares the $⟨N⟩/Z$ distributions between $\nu =0$ and $\nu =3$ together with the experimental data. This figure indicates that, to account for the fission fragment yields with the INCL4.6/GEM, approximately three neutrons should be additionally evaporated in post-fission.

To examine the excitation energy required to evaporate the additional neutrons from the residual nucleus, we introduced a nonphysical free parameter $\eta$ and changed Equation (29)(29) $E_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(\mathrm{\ell })}=\mathrm{T}\mathrm{X}\mathrm{E}\times \frac{A_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(\mathrm{\ell })}}{A},\mathrm{\ell }=1,2.$(29) to

(31) $E_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(\mathrm{\ell })}=(\mathrm{T}\mathrm{X}\mathrm{E}+\eta )\frac{A_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(\mathrm{\ell })}}{A},\mathrm{\ell }=1,2,$(31)

where $\eta$ represents an artificially given excitation energy. The remainder of the de-excitation calculation is the same as that given in the previous section. Figure 14 shows the reduced ${\chi }^2$ values for fission fragment yields from the three reactions as a function of $\eta$. Here, the values are scaled so that the minimum values of the fitting functions are normalized to unity. The reduced ${\chi }^2$ values take a minimum when values of several tens of mega-electron volts are given to $\eta$ in Equation (31)(31) $E_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(\mathrm{\ell })}=(\mathrm{T}\mathrm{X}\mathrm{E}+\eta )\frac{A_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{g}}^{(\mathrm{\ell })}}{A},\mathrm{\ell }=1,2,$(31) . Although further analysis is needed, this value seems to be consistent with a value estimated roughly from a neutron separation energy of the fission fragments; for example, assuming that two fission fragments having a neutron separation energy of 8 MeV are produced from the high-energy fission, the minimum excitation energy required for removal of three neutrons from the two fission fragments is calculated as 24 MeV. Thus, our phenomenological approach using the GEM suggests that the calculated excitation energy would be rather low, and an additional excitation energy (i.e. several tens of MeV) would be required to account for the high-energy fission including the fission fragment distributions. Although increasing the excitation energy, which needs to be yielded in some physical way, would worsen the reproducibility of the fission cross-sections, our approach can easily amend them by adjusting the fission probability parameters.

Figure 12. $⟨N⟩/Z$ distribution as a function of $Z$ number for $p$ + ${ }^{208}$Pb (1000$A$ MeV), $p$ + ${ }^{197}$Au (800$A$ MeV), and $p$ + ${ }^{208}$Pb (500$A$ MeV) reactions.

Figure 13. $⟨N⟩/Z$ distribution as a function of $Z$ number for $p$ + ${ }^{208}$Pb (1000$A$ MeV), $p$ + ${ }^{197}$Au (800$A$ MeV), and $p$ + ${ }^{208}$Pb (500$A$ MeV) reactions.

Figure 14. Reduced ${\chi }^2$ values as a function of $\eta$ for $p$ + ${ }^{208}$Pb (1000$A$ MeV), $p$ + ${ }^{197}$Au (800$A$ MeV), and $p$ + ${ }^{208}$Pb (500$A$ MeV) reactions. The values are scaled so that the minimum values of the fitting functions are normalized to unity.

4. Conclusions

We have presented a new model to describe the fission probability of the high-energy fission model by deducing from Prokofiev’s phenomenological systematics of proton-induced fission cross-sections. Although this model treats the fission probability in an entirely phenomenological manner, we do not need to perform hefty parametrizations of theoretical formula, such as the Bohr–Wheeler formula [71]; it is also flexible in its implementation. This model has been implemented in the de-excitation model of the GEM and applied to the Monte Carlo spallation reaction simulation by PHITS. Comparing with experimental data shows that our model is in fairly good agreement with the proton-, neutron-, and deuteron-induced fission cross-sections. Furthermore, our phenomenological approach suggests that an additional excitation energy (i.e. several tens of MeV) would be necessary to account for the isotopic distributions of fission fragments within the framework of INCL4.6/GEM.

According to Boudard et al. [11], the INCL model determines the final state of the INC process using a so-called ‘stopping time’ parameter, which has negative sensitivity to the excitation energy. Although its parametrization has been performed by rather elaborate analyses within the framework of INCL, our results imply that the stopping time value might need to be shortened. It is also pointed out in Mancusi et al. [72] that, to account for the isotopic distributions near the target nucleus, quantum-mechanical features of the nuclear surface should be considered. Thus, the current version of INCL still leaves room for improvement in its theoretical description; therefore, the calculated relationship among $E_p$, $⟨E⟩$, and $(⟨\mathrm{\Delta }Z⟩,⟨\mathrm{\Delta }A⟩)$ could be altered by a possible future improvement of the INCL model. Our phenomenological approach can easily deal with such an alteration.

Future work will focus on investigating how fission fragments are distributed in the de-excitation process, and further modification will be made to the modified GEM model and/or the INCL4.6 implemented in PHITS. The modified model will be utilized in designing the accelerator-driven system for nuclear transmutation and its experimental facility at J-PARC [73].

Acknowledgments

One of the authors (H. Iwamoto) would like to acknowledge many helpful discussions with Drs K. Nishio, T. Sato, T. Ogawa, S. Hashimoto, and Y. Iwamoto of the Japan Atomic Energy Agency (JAEA), and Dr T. Kawano of Los Alamos National Laboratory. The authors are grateful to Dr T. Fukahori of JAEA for providing the FISCAL code.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Japan Society for the Promotion of Science (JSPS) Grants-in-Aid for Young Scientists B [grant number 17K14916].