1 Introduction

Nuclear reactions induced by composite particles, such as deuterons, He, Be, and Li nuclei, play a key role in accessing the structural properties of these projectiles, as well as serving as a direct method of generating light particle beams. The study of projectile breakup has garnered strong attention from the nuclear physics community for several decades now. Breakup of the deuteron into proton and neutron has been extensively investigated due to the simplicity of the projectile and its fragments as well as a variety of possible applications, from radioisotope production [1, 2] to modern surrogate reactions strategies [3]. The study of weakly bound projectiles, the so-called halo nuclei, has motivated upgrades in many laboratories, which now provide a great variety of experimental data on such reactions [4, 5]

Many possible outgoing channels can be open in a reaction involving a composite projectile. At intermediate and higher energies, the direct reaction mechanism dominates the reaction dynamics and one can associate much of the particle emission with fast events. The slow process of projectile absorption, compound nucleus formation, and particle evaporation becomes less significant at higher energy, although these reactions are still present and play an important role in emission at backward angles.

Most studies on 3He-induced breakup reactions were conducted in the 1980s, when more refined theoretical models of these reactions were being developed. Although there is great interest at present in this type of reaction, the relatively strong binding and simple structure of 3He have contributed to a general lack of interest in its breakup. Little attention has been given to the excellent experimental data provided by Matsuoka et al. [6], Djaloeis et al. [7], and Aarts et al. [8]. The breakup of 3He into protons and deuterons has thus not been fully explored within a more modern reaction formalism.

In this work, we analyze part of the experimental data provided by Djaloeis et al. [7] by comparing these with calculations of the proton and deuteron inclusive emission spectra and double differential cross sections for reactions induced by 3He at 130 MeV on two heavy targets, 93Nb and 197Au. We use the post-form DWBA framework to calculate elastic and nonelastic breakup cross sections assuming a zero-range interaction between the two fragments. One of our objectives is to determine a good optical potential for the entrance channel, which we hope to extend to a global characterization of inclusive emissions from 3He-induced reactions on different target nuclei.

We have organized this work as follows: we provide a brief overview of the reaction formalism employed in the Section 2. Sections 3 and 4 compare our calculations to the experimental data and give our concluding remarks, respectively.

2 Theoretical Formalism and Optical Potentials

The theoretical study of projectile breakup had its origins in the 1940s with the well-known work of Serber on deuteron breakup [9]. In the 1980s, the breakup models were extended by several groups, using the DWBA to describe nonelastic breakup, also called incomplete fusion [10,11,12,13,14,15]. For the sake of brevity, we will only sketch a basic outline of the inclusive breakup reaction formalism. For a more detailed discussion on the development of the modern models, we point out the following recent references [4, 16,17,18,19].

The energy and angular distributions of a breakup reaction characterize it as a direct process. The inclusive particle emission cross sections consist of two principal components, an elastic (EB) one and a nonelastic breakup (NEB) one. Taking the reaction to be \(^{3}\text {He}+A\rightarrow b + X\), and using the zero-range post-form DWBA, we can write the double differential inclusive cross section as

$$ \begin{array}{@{}rcl@{}} \frac{d^{3}\sigma }{d{k_{b}^{3}}} &=& \frac{d^{3}\sigma^{\text{EB}}}{d{k_{b}^{3}}} + \frac{d^{3}\sigma^{\text{NEB}}}{d{k_{b}^{3}}}, \end{array} $$
(1)

where b represents one of the projectile fragments (b = p or b = d).

If we represent the projectile with the label a, the nonelastic contribution is obtained by calculating the reaction cross section of x + A, using the imaginary part of the x + A optical potential,

$$ \begin{array}{@{}rcl@{}} \frac{d^{3}\sigma^{\text{NEB}}}{d{k^{3}_{b}}}&=& -\frac{ 2(2\pi)^{-3} }{\hbar v_{a}} \langle {\Psi}_{x}|W_{xA}| {\Psi}_{x} \rangle , \end{array} $$
(2)

and the effective wave function for the particle x after breakup,

$$ \begin{array}{@{}rcl@{}} | {\Psi}_{x} \rangle = D_{0}\left (\chi_{b}^{(-)} G_{x}^{(+)} {\Lambda} |\chi_{{a}}^{(+)} \right \rangle , \end{array} $$
(3)

which we calculate in the zero-range approximation

$$ V_{a}(\vec{r})\phi_{a}(\vec{r}) \approx D_{0} \delta (\vec{r}) $$
(4)

with D0 = − 128.75 MeV fm3/2.

The elastic breakup contribution is obtained by integrating the elastic breakup differential cross section over the energy and angle of the other fragment:

$$ \begin{array}{@{}rcl@{}} \frac{d^{3}\sigma^{\text{EB}}}{d{k^{3}_{b}}}&=& \frac{(2\pi)^{-5}}{\hbar v_{a}} \int d{k^{3}_{x}} \left |\langle b,x|T|a \rangle \right |^{2} \\ && \qquad \qquad \quad \times \delta (E_{a}+\epsilon - E_{x} -E_{b}). \end{array} $$
(5)

where 𝜖 is the separation energy of the projectile into b and x. We approximate the elastic breakup transition amplitude by its DWBA:

$$ \begin{array}{@{}rcl@{}} \langle b,x|T| a \rangle = \langle \chi_{b}^{(-)} \chi_{x}^{(-)} |V_{a} |\chi_{{a}}^{(+)} \phi_{a}\rangle \end{array} $$
(6)

and simplify this further using the zero-range approximation, to write

$$ \begin{array}{@{}rcl@{}} \langle b,x|T| a \rangle = D_{0}\langle \chi_{b}^{(-)} \chi_{x}^{(-)} {\Lambda} |\chi_{{a}}^{(+)} \rangle , \end{array} $$
(7)

where Λ accounts for finite-range effect corrections [20].

This approach has already been used with a fair degree of success to describe experimental data for the case of reactions induced by deuterons (a = d). Inclusive proton emission cross sections were obtained for different targets and energies and compared to experimental data in [4, 16, 17, 19].

Here, we obtain the proton and deuteron distorted waves by employing the Koning-Delaroche [21] and Han-Shi-Shen [22] optical potentials, respectively. For the 3He-target interaction, we first attempted to use the standard Becchetti-Greenlees [23] optical potential, but without success, due to the high energy of the projectile. The folding-like São Paulo (SP) potential [24, 25] was then employed in the incident channel both nuclei. For the sake of simplicity in its implementation, we performed a best fit of a Wood-Saxon-like function to the São Paulo potential. This provided potential parameters that could be easily used in the breakup calculation.

In Fig. 1, we show the real part of the SP potential, for the 3He + 197Au interaction, in comparison to the best Wood-Saxon fit. Although the shape and strength of the SP potential are well described by a Wood-Saxon function, we found it to have a very large diffusivity aSP ≈ 0.95 fm, for both 93Nb and 197Au. This leads to a too absorptive imaginary potential, when we use the standard prescription of W(r) = NIV (r). We have thus fixed the diffusivity to a more standard value of a0 = 0.65 fm and maintained it fixed throughout the calculations. We still take the strength V0 and the reduced radius r0 (R = r0A1/3) parameters from the fit to the SP potential and take the imaginary part of the interaction to be W0 = 0.78V0, as is usually done with the SP potential. The parameters used for the two targets are given in Table 1. The same geometrical parameters are employed for both the real and imaginary parts of the potentials.

Fig. 1
figure 1

Real part of the São Paulo optical potential for the entrance channel 3He-197Au (markers). The best fit of a Wood-Saxon-like function is also shown (solid line). The adjusted parameters are explained in the text and given in Table 1

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Table 1 Reduced radius and potential depth of the Wood-Saxon potential adjusted to the São Paulo potential for the 3He-induced reactions on 93Nb and 197Au at 130 MeV
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3 Inclusive Cross Sections

Although we have restricted this study to reactions on heavy targets, we have converted the differential cross sections to the laboratory (LAB) frame for comparison to the experimental measurements. Differences of a few percent between the calculated CM and LAB cross sections can be observed. All quantities shown are in the LAB frame, unless said otherwise. The experimental data for deuteron emissions were extracted from Djaloeis et al. [7] and an 3He separation energy of 𝜖 = − 5.49 MeV was employed.

In Fig. 2, we present the deuteron differential spectrum at the fixed angle of θd = 7.5° for 93Nb and at θd = 9.0° for 197Au. The Becchetti-Greenlees potential reaches its limit of application at energies of about 40 MeV [7, 26] and cannot be reliably used to model faster incident projectiles. We have found that the SP potential, on the other hand, provides excellent agreement with the experimental data. The center of the distribution around ∼80 MeV is well reproduced, especially for 93Nb. Small shifts associated with details of the geometry of the nucleus can be seen in the calculations. The low-energy part of the spectra is expected to have contributions from compound nucleus evaporation, a component not calculated in our model. With the success of this first comparison, we have extended our calculations of deuteron emissions to different angles.

Fig. 2
figure 2

Inclusive differential deuteron spectra at θd = 7.5° for 93Nb (top) and θd = 9.0° for 197Au (bottom) targets at 130 MeV of incident energy. Spectra obtained with the Becchetti-Greenlees optical potential and the modified São Paulo potential for the 3He-target potential are compared (see Table 1 for parameters). The experimental data are taken from [7]

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In Fig. 3, we show deuteron differential spectra at several angles. One observes again a very good correspondence between our calculations and the experimental data. The lower energy part of the experimental spectra, corresponding to compound nucleus emission, becomes more prominent as the scattering angle increases. According to our calculations, the direct breakup component contributes to the deuteron emission at energies higher than about 40 MeV for both 93Nb and 197Au nuclei. At θd = 21°, the compound emission dominates the reaction. Our calculations tend to overestimate the cross sections for the higher energy part of the spectra, a characteristic that is associated with the geometry and diffuseness of the optical potential. The description of these shifts will be addressed in a future work.

Fig. 3
figure 3

Inclusive deuteron spectra from the breakup of 3He on 93Nb (top) and 197Au (bottom) targets for several angles. Optical potential parameters in the entrance channel are given in Table 1. The experimental data are taken from [7]

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Before closing, we present in Fig. 4 the differential spectra of protons for several fixed scattering angles. Although we have no experimental data with which to compare, the calculations serve as a future reference and provide a more complete description of the inclusive breakup emission from 3He. The distributions are centered around ≈ 45 MeV, and the strength of the cross section decreases for larger scattering angles, as in the case of deuterons. One would expect a somewhat larger contribution from compound nucleus emission here, as is usually the case.

Fig. 4
figure 4

Inclusive proton spectra from the breakup of 3He on 93Nb (top) and 197Au (bottom) targets for several angles

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4 Concluding Remarks

In this paper, we have studied the inclusive differential emission spectra of protons and deuterons in reactions induced by 3He at 130 MeV. Breakup data corresponding to two different targets, 93Nb and 197Au, were analyzed. Deuteron differential spectra were compared to the experimental data from [7], while proton emission spectra were presented for completeness. The optical potential of Becchetti-Greenlees was tested, but did not provide a good description of the experimental data due to the incident energy of the experiment. Good agreement to the measurements was obtained by using the São Paulo potential with a modified value of the nuclear diffuseness. The large value for the diffuseness parameter obtained directly from the SPP leads to a strongly absorptive potential, which furnishes only half of the experimental cross section. We decided to adjust a0 in order to keep the simple parameterization NI= 0.78. This choice renormalizes the real part of the interaction by about 1% when compared to the original SPP. The choice of a0 = 0.65 fm follows more closely the values for the diffusivity frequently found in standard optical potentials. This approach facilitates the numerical implementation and provide an easy way of parametrizing the potential for different targets. The zero-range approximation to the DWBA breakup amplitude provided a very good description of the experimental data. The existence of low-energy fragments in the data is understood as being due to compound nucleus emission, a process not included in our model. In general, we have obtained distributions very close in shape and in average values to the reaction measurements. We have thus verified the capability of the model to describe inclusive deuteron emission from 3He-induced reactions.

Although not studied in this work, neutron and diproton (pp) emissions from 3He breakup can also contribute to the reaction dynamics and should be taken into account if one aims to obtain a complete description of the experimental data. We plan to include this emission channel as well, in a forthcoming study. We note that this breakup channel introduces a new level of complexity, as its correct description requires the inclusion of three-body breakup [27]. Compound nucleus emission should also be taken into account in a complete description of differential fragment spectra, as it makes a significant contribution at backward angles.

In the future, we also plan to extend our calculations to heavier projectiles, where the extra degrees of freedom involved in the system can increase the number and variety of emitted fragments. In these cases, the São Paulo potential can be used to represent the ejectile as well as the projectile-target interaction.

We believe that this study helps advance our understanding of the mechanisms for particle emission in reactions induced by weakly bound nuclei. A complete description and understanding of reactions involving two-fragment projectiles will contribute to the extension of the formalism to three-or-more-fragment reactions.