An intranuclear cascade model for inelastic scattering and breakup reactions involving deuterons and alpha particles

ABSTRACT

The intranuclear cascade model is extended to cluster-induced (deuteron and alpha particle) nuclear reactions involving inelastic scattering and breakup reactions. The proposed model explains the projectile breakup process by describing the projectile cluster as a superposition of several states. The incident cluster and the produced cluster are assumed to be collections of independent particles and may undergo nuclear interaction through nucleon–nucleon interaction with the target nucleus. Trajectory deflections for the projectile and ejecta are incorporated in the model to account for angular distributions. Calculations with the proposed model followed by the generalized evaporation model are performed for validation by comparing with experimental double-differential cross-section spectra produced by bombarding an ²⁷Al target separately with 80-MeV and 99.6-MeV deuterons and 140-MeV alpha particles. The calculation results show good agreement with experimental spectra.

KEYWORDS:

• Intranuclear cascade model
• cluster-induced nuclear reaction
• projectile break up
• double-differential cross section
• alpha particle
• deuteron

1. Introduction

Particle transport codes are widely used in a variety of applications to simulate radiation shielding, isotope burnup, material activation, and radiation dose. The physical processes considered for hadron transport include electromagnetic processes and hadronic processes, the latter of which are simulated using nuclear reaction models. Nowadays, nuclear reactions are modeled in two stages: the fast cascade process followed by the static process. The intranuclear cascade (INC) model, the quantum molecular dynamics (QMD) model, and other models are used to describe the cascade stage that emits fast particles. After the cascade stage, the residual nucleus de-excites via evaporation or fission, for which an evaporation–fission model is used. To describe the fast process, we have developed an INC code [1] that has been incorporated into the widely used Monte Carlo Particle and Heavy Ion Transport code System (PHITS) [2].

Fast-stage reactions have a variety of final channels. Hence, there is strong demand to improve the predictive accuracy of INC models and to expand their application to a variety of nuclear reactions involving many species of incident and emitted particles with a wide energy range. A highly predictive INC model has been realized for nucleon-induced nucleon emission, that is, (N, N’x) reactions [3–6]. Some studies have considered heavy-ion reactions, but were limited to proton or pion production reactions [7–9]. More recently, there has been some study of reactions induced by light clusters [10,11], but with little attention paid to the production of the clusters themselves.

All the final channels of cluster disintegration have similar strengths, so it is important to describe all channels equally (while noting that pickup-like reactions are negligibly weak). However, the predicted spectra of inelastic cluster scattering such as (d, d'x) and (α, α'x) reactions are poor. This is because both INC and QMD models treat nucleons as classical particles, and hence the composite nature of clusters (which is essential in cluster scattering calculations) is not treated. Typical examples are shown in Figures 1 and 2 for the 80-MeV (d, d'x) and 140-MeV (α, α'x) reactions, respectively. In those figures, dashed and solid lines represent results calculated by INCL [11] and JQMD [12], respectively.

Figure 1. INCL (dashed lines) and JQMD (solid lines) model calculations for deuteron bombardment of an ²⁷Al target at 80 MeV in comparison with experimental values (dots) taken from [18 ]. For visualization purposes, each data-set has been multiplied by the factor shown in brackets.

Figure 2. As Figure 1, but for ⁵⁸Ni(α, α'x) reaction at 140 MeV.

In previous work, we expanded our INC model to include reactions for proton-production and proton-induced cluster production at bombardment energies of 200 MeV or higher [6,13]. The model was subsequently extended to lower beam energies (∼50 MeV) by including collective excitations, trajectory deflections, and barrier transmission coefficients [14]. However, such improvements apply to (p, p'x) reactions only. Moreover, the double-differential cross-section (DDX) spectra of 100-Me V ²⁷Al(d, d'x) reactions have been explained successfully by introducing wave-function superposition into the INC model of the deuteron ground state [13]. In the present study, we develop this approach to explain (d, px) reactions, and we expand the model to include alpha-induced reactions, namely (α, α'x), (α, ³Hex), (α, tx), (α, dx), (α, px), and (α, nx). We validate the model by using experimental data for an ²⁷Al target. The dependences on incident energy and target mass are investigated in the future.

This paper is organized as follows. In Section 2, we describe the INC model used in this study, as well as how we improve it to include cluster-induced reactions. In Section 3, the present model is validated by comparing calculation results with experimental data. Finally, our conclusions are presented in Section 4.

2. Theoretical model

2.1. INC model for nucleon incidence

Our INC model for nucleon incidence sits within the framework of a two-stage model that consists of the INC for the fast-stage and the generalized evaporation model (GEM) [15] for the static stage. The present calculation procedure is based on our previous model, which make good predictions for nucleon-induced reactions [14,16].

We describe briefly here our INC model of nucleon incidence reactions. The target nucleus is treated as a sphere with distributed nucleon density and momentum. The nucleon density follows a Woods–Saxon-type distribution [17], whereas the momentum follows the degenerate Fermi distribution in the nuclear potential of 45 MeV. Because any nucleon below the Fermi sea is treated as unevolved, these parameters are those associated with an NN collision. The maximum nuclear radius is defined as R_max = r₀ + 5a₀with a₀ = 0.54fm and $r_0=(0.978+0.0206A^{1/\phantom{13}3})A^{1/\phantom{13}3}$, where A is the atomic mass number. The maximum impact parameter for nucleon incidence is taken as R_max. The energetic nucleons travel in straight lines inside the nucleus. If two nucleons approach each other within a distance $r=\sqrt{{\sigma }_{\mathrm{NN}}}/\phantom{\sqrt{{\sigma }_{\mathrm{NN}}}\pi }\pi ,$_, they can undergo NN collision, the occurrence of which judged according to Pauli blocking. When a nucleon hits the target nuclear surface, it either escapes or is reflected depending on the transmission coefficient. Neutrons do not experience a Coulomb barrier, and the transmission coefficient for proton emission with the modified Gamow transmission factor is (1) $$\begin{array}{ccc}& & T_c^G\left(\epsilon \right)=exp\hfill \\ & & \left[-\frac{2{\lambda }_r{R}_{max}}{h/\phantom{h2\pi }2\pi }\frac{\sqrt{2m\epsilon }}{E_{\mathrm{ct}}^{\text{'}2}}\left({cos}^{-1}{E}_{\mathrm{ct}}^{\text{'}}-E_{\mathrm{ct}}^{\text{'}}\sqrt{1-E_{\mathrm{ct}}^{\text{'}2}}\right)\right],\hfill \end{array}$$(1) with (2) $$\begin{array}{c}\hfill E_{\mathrm{ct}}^{\text{'}}=\sqrt{\frac{\epsilon }{{\lambda }_c}{V}_{\mathrm{coul}}}\mathrm{and}{V}_{\mathrm{coul}}=\frac{e^2{Z}_{\mathrm{target}}}{R_{max}}.\end{array}$$(2)

Here, λ_r and λ_c are parameters that depend on the target nucleus. Furthermore, ϵ is the emission energy, V_coul is the Coulomb barrier, and R_max satisfies the range $R_{\mathrm{ct}}^{\text{'}}\le R_{max}\le R_{\mathrm{ct}}^{\text{'}}/\phantom{R_{\mathrm{ct}}^{\text{'}}{E}_{\mathrm{ct}}^{{}^{\text{'}}2}}{E}_{\mathrm{ct}}^{{}^{\text{'}}2}$, with classical turning point outside the potential, R′_ct = λ_rR_max. We assume the reflected particle does not escape during cascade stage. The incident particle and the emitted particle are both deflected at the nuclear surface because of the nuclear potential.

2.2. Extension of INC model

Our model of cluster-induced reactions was developed on the basis of the nucleon reaction model described above. The essential improvement is the inclusion of the breakup of the incident cluster, which is assumed to occur at the initial-state interaction. To begin with, the ground state of the cluster is determined in the same way as for the target nucleus, which is described in Section 2.1. Since the potential depths of V_d for deuteron and V_α for α are sensitive to spectral shapes of emitted protons, we choose their values to fit the experimental data as described below. It is assumed additionally that the initial cluster wave function is given by superposing different states that consist of cluster units. For the alpha particle, the wave function is (3) $$\begin{array}{ccc}\hfill \left|{\alpha }_{\mathrm{init}}\right.⟩& =& c_{\alpha 0}\left|\alpha \right.⟩+c_{\alpha 1}\left|{}^3\mathrm{He}n\right.⟩\hfill \\ & & +c_{\alpha 2}\left|tp\right.⟩+c_{\alpha 3}\left|dd\right.⟩+c_{\alpha 3}\left|nnpp\right.⟩,\hfill \end{array}$$(3) and for the deuteron it is (4) $$\begin{array}{c}\hfill \left|d_{\mathrm{init}}\right.⟩=c_{d0}\left|d\right.⟩+c_{d1}\left|pn\right.⟩,\end{array}$$(4) with the normalization of $$\begin{array}{c}\hfill \sum _{i=\mathrm{all}}{c}_i^2=1\end{array}$$due to orthogonality. The probabilities calculated by these wave functions are used for INC event-by-event calculations. The inclusion of the cluster units allows us to treat their composite nature. The first term on the right-hand side of Equation (3)(3) $$\begin{array}{ccc}\hfill \left|{\alpha }_{\mathrm{init}}\right.⟩& =& c_{\alpha 0}\left|\alpha \right.⟩+c_{\alpha 1}\left|{}^3\mathrm{He}n\right.⟩\hfill \\ & & +c_{\alpha 2}\left|tp\right.⟩+c_{\alpha 3}\left|dd\right.⟩+c_{\alpha 3}\left|nnpp\right.⟩,\hfill \end{array}$$(3) indicates that the incident alpha particle is in a state in which it never disintegrates by interacting with the target nucleus. The second term represents the incident alpha particle breaking up into ³He + 1n as it enters the target nucleus. The resultant ³He particle undergoes no further disintegration in the subsequent reaction process. The momentum ${\stackrel{\rightharpoonup }{\mathrm{P}}}_{{}^{{}^3\mathrm{He}}}$ of the ³He fragment is taken to be (5) $$\begin{array}{c}\hfill {\stackrel{\rightharpoonup }{\mathrm{P}}}_{{}^3\mathrm{He}}=\sum _{N_i=1}^3{\stackrel{\rightharpoonup }{\mathrm{P}}}_{N_i}+\frac{3}{4}{\stackrel{\rightharpoonup }{\mathrm{P}}}_{\alpha },\end{array}$$(5) where ${\stackrel{\rightharpoonup }{\mathrm{P}}}_{N_i}$ is the momentum of the ith nucleon of ³He, and ${\stackrel{\rightharpoonup }{\mathrm{P}}}_{\alpha }$ is the momentum of the alpha-particle projectile. The ³He transport is calculated under assumptions: the three constituent nucleons move in parallel with each other at a speed of the ³He cluster. One of the nucleons may collide with a target nucleon and accordingly receives momentum change, which is treated as the momentum change of the ³He cluster. The third to fifth terms of Equation (3)(3) $$\begin{array}{ccc}\hfill \left|{\alpha }_{\mathrm{init}}\right.⟩& =& c_{\alpha 0}\left|\alpha \right.⟩+c_{\alpha 1}\left|{}^3\mathrm{He}n\right.⟩\hfill \\ & & +c_{\alpha 2}\left|tp\right.⟩+c_{\alpha 3}\left|dd\right.⟩+c_{\alpha 3}\left|nnpp\right.⟩,\hfill \end{array}$$(3) represent the breakup of incident alpha into t + p, d + d, n + n + p + p, respectively, and the disintegration process is the same as that of the second term. Since relative yields of particles emitted from the reactions depend on the coefficients c in Equations (3)(3) $$\begin{array}{ccc}\hfill \left|{\alpha }_{\mathrm{init}}\right.⟩& =& c_{\alpha 0}\left|\alpha \right.⟩+c_{\alpha 1}\left|{}^3\mathrm{He}n\right.⟩\hfill \\ & & +c_{\alpha 2}\left|tp\right.⟩+c_{\alpha 3}\left|dd\right.⟩+c_{\alpha 3}\left|nnpp\right.⟩,\hfill \end{array}$$(3) and (4(4) $$\begin{array}{c}\hfill \left|d_{\mathrm{init}}\right.⟩=c_{d0}\left|d\right.⟩+c_{d1}\left|pn\right.⟩,\end{array}$$(4) ), the values of the coefficients were determined to fit the experimental data as listed in Table 1.

Table 1. Values of c coefficients in Equations (3)(3) $$\begin{array}{ccc}\hfill \left|{\alpha }_{\mathrm{init}}\right.⟩& =& c_{\alpha 0}\left|\alpha \right.⟩+c_{\alpha 1}\left|{}^3\mathrm{He}n\right.⟩\hfill \\ & & +c_{\alpha 2}\left|tp\right.⟩+c_{\alpha 3}\left|dd\right.⟩+c_{\alpha 3}\left|nnpp\right.⟩,\hfill \end{array}$$(3) and (4(4) $$\begin{array}{c}\hfill \left|d_{\mathrm{init}}\right.⟩=c_{d0}\left|d\right.⟩+c_{d1}\left|pn\right.⟩,\end{array}$$(4) )

Alpha particle		Deuteron
Cluster unit	c	Cluster unit	c
α	8	d	$\sqrt{70}$
3He + n	$\sqrt{8}$	p + n	$\sqrt{30}$
t + p	$\sqrt{10}$
d + d	$\sqrt{10}$
2p + 2n	$\sqrt{8}$

The impact parameter of the incident cluster is assigned randomly in the range 0–b_max. Here, b_max is the sum of the average radii of the projectile and the target rather than the sum of their cutoff radii (as is the case for nucleon incidence reactions) to fit with the experimental data.

In our INC model, the influence of the nuclear potential causes the incident particle and the outgoing particle to be deflected, which is known as trajectory deflection. For (p, p'x) calculations [14,16], the angular distribution of proton elastic scatterings was used to obtain a comprehensive description of the trajectory deflection. For cluster-induced reactions, trajectory deflection has a strong effect on the angular distributions of the fragments. Therefore, we determine new parameters for the trajectory-deflection angular distribution of each incoming and outgoing cluster. We have shown previously the probability W_def,p of protons being deflected at angle in laboratory system, θ [16]. For each cluster, the parameters were chosen to reproduce the spectral shapes of experimental DDX data for cluster emission reactions on the ²⁷Al target: (6) $$W_{\mathrm{def},d}\left(\epsilon ,\theta ,A\right)=exp\left[-0.001\left(1.3\epsilon +lnA+6\right)\theta \right],$$(6) (7) $$W_{\mathrm{def},t}\left(\epsilon ,\theta ,A\right)=exp\left[-0.001\left(1.2\epsilon +6lnA-5\right)\theta \right],$$(7) (8) $$W_{\mathrm{def},3\mathrm{He}}\left(\epsilon ,\theta ,A\right)=exp\left[-0.001\left(1.2\epsilon +6lnA-5\right)\theta \right],$$(8) (9) $$W_{\mathrm{def},\alpha }\left(\epsilon ,\theta ,A\right)=exp\left[-0.001\left(1.2\epsilon -10lnA+40\right)\theta \right],$$(9) by using the V_d and V_α values determined for proton spectra.

The outgoing particle either travels through or is reflected at the nuclear surface, depending on the transmission coefficient. For clusters, we choose a step function (10) $$T_c\left(\epsilon \right)=\left\{\begin{array}{c}1\epsilon >V_{\mathrm{coul}}\\ 0\epsilon \le V_{\mathrm{coul}}\end{array}\right.$$(10)

Compensation for nuclear recoil and the Q value is applied once the cascade process has finished.

The values of V_d and V_α were determined to reproduce spectral shapes of ²⁷Al (d, px) and ²⁷Al (α, px) reactions, respectively. Figure 3 shows comparison between the results for deuteron potential depths of 10, 15, and 20 MeV (represented by dotted, solid, and dashed lines, respectively) for the ²⁷Al(d, px) reaction at 80 MeV; of these, we choose 15 MeV. We determined the alpha potential depth in the same way by using DDXs of the ²⁷Al(α, px) reaction at 140 MeV. In Figure 4, the results for depths of 30, 40, and 60 MeV are presented as the dash-dotted, solid, and dotted lines, respectively; of these, we chose 40 MeV.

Figure 3. Comparison of the calculation results for proton-production DDXs from 80-MeV deuterons bombarding ²⁷Al at angles of 30°, 45°, and 60° for different values of the deuteron potential depth. The solid circles show the experimental data taken from EXFOR [20]. The dotted, dashed, and line histograms are the INC calculation results for d potential depths of 10, 20, and 15 MeV, respectively. For visualization, the DDXs have been multiplied by the factors indicated.

Figure 4. Comparison of the calculation results for proton-production DDXs from 140-MeV alpha particles bombarding ²⁷Al at angles of 20°, 30°, and 45° for different values of the potential depth for the incident alpha particle. The solid circles show the experimental data taken from EXFOR [20]. The dash-dotted, solid, and dotted line histograms are the INC calculation results for alpha-particle potential depths of 30, 40, and 60 MeV, respectively. For visualization, the DDXs have been multiplied by the factors indicated.

3. Calculation results and discussion

3.1. Deuteron-induced reactions

We tested the validity of our model through comparison with experimental observations for the bombardment of ²⁷Al with deuterons of incident energy 80 or 99.6 MeV. The measured production spectra of deuterons and protons can be compared with the calculation results of our present model in Figures 5–7. Closed circles represent the experimental DDXs and solid-line histograms represent the calculation results. To avoid overlap, some factors indicated in the figures have been multiplied. The contribution of the GEM has been included in each calculation result. In these comparisons, we use the data of Wu et al. [18] and Ridikas et al. [19], and experimental data from EXFOR [20].

Figure 5. INC model calculations coupled with GEM for ²⁷Al(d, d'x) reaction at 80 MeV at angles of 30°–150°. The solid circles show the experimental data taken from EXFOR [20] and the dots are the experimental data taken from [18 ]. For visualization, the DDXs have been multiplied by the factors indicated.

Figure 6. Same as Figure 5, but for ²⁷Al(d, px) reaction at 80 MeV. The solid circles are the experimental data taken from EXFOR [20].

Figure 7. Same as Figure 5, but for ²⁷Al(d, px) reaction at 99.6 MeV. The solid circles are the experimental data taken from EXFOR [20].

Figure 5 allows comparison between experimental DDXs and INC calculation results for ²⁷Al(d, d'x) reactions at 80 MeV for angles of 30°–150°. Besides the experimental data from EXFOR (solid circles), we include numerical values from Wu et al. (dots). The EXFOR database is missing some data around the elastic scattering region, but overall the proposed model accounts successfully for the gross features of the deuteron spectra over the entire energy range.

The proton spectra observed from bombarding ²⁷Al with deuterons at incident energies of 80 and 99.6 MeV are shown in Figures 6 and 7, respectively. In both cases, our INC model succeeds at reproducing the prominent broad peaks at the forwardmost angles at about half the incident energy that are due to breakup of the incident deuterons [18]. Slight inconsistencies can be seen in the high-energy range of each spectrum, where the stripping reaction is dominant. The model calculations show a sharp peak at the highest energy at forward angles, which corresponds to ²⁷Al(d, p)²⁸Al reactions with a positive Q value. In Figure 7, there are discrepancies in discrete-peak regions because of stripping reactions at excitations up to roughly 10 MeV for the 99.6-MeV case. Thanks mainly to the GEM, the low-energy regions (i.e. those that are governed by the evaporation process) are reproduced well. The tendency to underestimate that is observed in the results for spectra at angles of 90° or more is a common shortcoming of INC models.

3.2. Alpha-particle-induced reactions

We tested the present INC model for all available channels of the alpha-induced reactions of ²⁷Al at 140 MeV, namely (α, α'x), (α, ³Hex), (α, tx), (α, dx), (α, px), and (α, nx). Figures 8–13 allow comparisons between the calculated DDX spectra (solid lines) and the experimental data (closed circles); for visualization purposes, the DDXs have been multiplied by the factors shown. In these comparisons, we have used the data of Wu et al. [21] and experimental data taken from EXFOR [20].

Figure 8. Same as Figure 5, but for ²⁷Al(α, α'x) reaction at 140 MeV.

Figure 9. Same as Figure 5, but for ²⁷Al(α, nx) reaction at 140 MeV.

Figure 10. Same as Figure 5, but for ²⁷Al(α, ³Hex) reaction at 140 MeV.

Figure 11. Same as Figure 5, but for ²⁷Al(α, tx) reaction at 140 MeV.

Figure 12. Same as Figure 5, but for ²⁷Al(α, dx) reaction at 140 MeV.

Figure 13. Same as Figure 5, but for ²⁷Al(α, px) reaction at 140 MeV.

Figure 8 shows the case for (α, α'x) at laboratory angles of 20°–120°. The experimental data were taken from EXFOR [20]. The highest energy peak of the alpha spectra, which corresponds to elastic scattering, is captured well by our calculations. The calculated alpha-production DDX spectra account well for the experimental results at all angles and over the entire energy range.

Figure 9 allows a comparison between experimental and calculated DDXs of neutrons at 20°, 30°, 45°, 60°, and 120° resulting from bombarding ²⁷Al with 140-MeV alpha particles [20]. The present model reproduces the features of the neutron spectra successfully. The discrepancy observed at 60° can be attributed to experimental uncertainty.

Figures 10 and 11 show energy spectra for (α, ³Hex) and (α, tx) reactions, respectively, with ²⁷Al at 140 MeV for laboratory angles of 20°–120°. The gross features of the ³He energy spectra are reproduced well by the present INC model, except in the high-energy region where the ²⁷Al(α, ³He)²⁸Al reaction is dominant. The overall features of the triton spectra from ²⁷Al are reproduced well by our INC model, except at the high-energy end as in the case of ³He production. The one-neutron stripping-reaction mechanism is not considered in the present study. We believe the inclusion of the stripping-reaction mechanism could improve the accuracy of the model as we added the collective excitation process [14,16] for (p, p’x) reactions.

Deuteron spectra from ²⁷Al for alpha particles at 140 MeV are shown in Figure 12 for laboratory angles of 20°–120°. The INC model accounts well for the forward-angle broad peaks at half the incident energy, which have been ascribed to the breakup of alpha particles [22]. Good agreement with the experimental data is observed over the entire range of the deuteron spectra. The only exception is observed at the high-energy region of 30°.

Figure 13 shows comparison between the experimental and calculated proton energy spectra from 140-MeV alpha particles with ²⁷Al. The present model accounts well for the gross features of the proton spectra. However, at small angles, a difference is observed in the high-energy region that is governed by the stripping reaction, and the present model results in overestimation at large angles (i.e. 90° and 120°).

4. Conclusion

We investigated our previously developed INC model to widen its applicable range to include deuteron- and alpha-induced reactions. We introduced the superposition of ground-state wave functions to explain a variety of final channels. Because the angular distributions are sensitive to the deflection of fragments, we incorporated trajectory deflection for both the cluster projectile and the ejecta. Considering the incident cluster as a collection of independent nucleons, we performed calculations within the framework of our INC model together with a GEM. The calculation results indicated that the proposed model has high predictive power for (d, d'x), (d, px), and all channels of alpha-induced reactions. The proposed model underestimated the high-energy end of spectra, which are occupied by transitions to discrete levels. We believe that the inclusion of the stripping reaction mechanism improves the model accuracy.

Acknowledgement

This work is supported partly by JSPS KAKENHI Grant Numbers 17H03522.

Disclosure statement

No potential conflict of interest was reported by the authors.