Scattering and One Neutron Pick-Up Reaction on a ¹⁰B Target with Deuterons at an Energy of 14.5 MeV

Abstract

The elastic and inelastic scattering of deuterons on ¹⁰B nuclei and the ¹⁰B(d, t)⁹B reaction were studied at a deuteron energy of 14.5 MeV. In inelastic scattering, differential cross-sections for transitions to ¹⁰B states at excitation energies, E_x, of 0.718 MeV (1⁺), 2.154 MeV (1⁺), and 3.59 MeV (2⁺) were measured. The cross-sections of the (d, t) reaction were measured for the ground (3/2⁻) and excited states of the ⁹B nucleus at E_x = 2.361 MeV (5/2⁻) and 2.79 MeV (5/2⁺). An analysis of the corresponding angular distributions was carried out using the coupled channel method. As a result of the calculations, the values of the quadrupole deformation parameters (β₂ ≈ 0.7 ± 0.1) for various transitions in the ¹⁰B nucleus in inelastic scattering were extracted. From the analysis of the (d, t) reaction, the values of spectroscopic amplitudes (SA = 0.67 and SA = 0.94) for transitions to the states of the ⁹B nucleus were extracted. The results obtained here, taking into account possible measurement errors, are in good agreement with the previously obtained data and the theoretical predictions.

1. Introduction

Single-nucleon transfer reactions induced by light ions have been established as a prominent tool in the field of nuclear spectroscopy. Following pioneering studies [1], a vast amount of research was carried out leading to the publication of a number of papers (see, e.g., [2,3,4] and the references therein). Reactions with light particles (p, d, ³He and α) play a decisive role due to the simple and known structure of these particles.

Direct nuclear reactions with the simultaneous transfer of several particles in the form of a cluster are also widely used to obtain spectroscopic information.

Nuclear clustering is among the most interesting phenomena in nuclear physics, strongly affecting reaction dynamics. An example is the famous Hoyle state in ¹²C, which is formed in stars by the fusion of three alpha particles [5]. Several studies have been dedicated to the investigations of clustering phenomena (see, e.g., [6,7,8] and the references therein). In a previous study [8], the effect of ⁸Be transfer on elastic scattering was studied for the ²⁰Ne+²⁸Si system at energies of about 50 MeV, where both nuclei have a pronounced α-cluster structure. However, the large separation energies of ⁸Be for these nuclei (ε > 12 MeV) would seem to rule out this possibility.

The study of the elastic scattering of deuterons is also of great importance. Analysis of the corresponding angular distributions based on the optical model allows us to extract the values of the interaction potential parameters, knowledge of which is necessary to obtain data on the nuclear structure within the framework of certain reaction mechanisms.

However, it is known that the systematics of optical potentials at heavy nuclei are different from those in the light nuclei region (of the atomic mass of A ≤ 20). This is largely explained by the more pronounced effects of the clustering of light nuclei. As a result, the role of exchange mechanisms with cluster transfer, physically indistinguishable from elastic scattering, increases. This leads to an anomalous increase in scattering cross-sections at large angles, which cannot be explained within the framework of the standard optical model. For the ¹⁰B target, the behavior of the cross-sections can also be significantly influenced by spin–orbit interactions due to the large spin (3⁺) of this nucleus. A systematic analysis of elastic scattering on p-shell nuclei was carried out in Ref. [9], with deuteron incident energies of between 5 and 170 MeV. The elastic scattering of deuterons, specifically on a ¹⁰B target, was studied in Refs. [10,11,12] at energies of 11.8 MeV, 15 MeV, and 13.6 MeV, respectively.

Provided that the direct mechanism dominates, reactions of the (p, d), (d, t) type with the pickup of one neutron are a good tool for studying hole states of nuclei. However, there is only scant information available about reactions starting from ¹⁰B and ending in ⁹B in comparison to those from neighboring nuclei. The (d, t) reaction on a ¹⁰B target was previously studied at deuteron energies in the region of 12–18 MeV [10,11,13,14,15] and at an energy of 33.6 MeV [16].

These investigations were carried out more than 40 years ago. In these studies, the angular distributions of deuterons and tritons with transitions to the ground and excited states of ¹⁰B and ⁹B nuclei were studied. Based on distored-wave Born approximation (DWBA) analysis, the values of the spectroscopic factors for transitions to the ground and first excited states of ⁹B were extracted [10]. They turned out to be significantly less than the theoretical predictions of the shell model with intermediate coupling [17]. The reason for this may be due to the lack of a reliable optical potential for the interaction of tritons with ⁹B nuclei in the output channel and the low energy of tritons from the (d, t) reaction (Q = −2.18 MeV) with a deuteron energy of 11.8 MeV [10]. It is also noteworthy that the analysis of this reaction did not consider the possible contribution of the exchange mechanism with the transfer of the ⁸Be cluster in the ¹⁰B(d, ¹⁰B)d reaction. Another aspect of the problem is that the¹⁰B nuclei, relating to the nuclei of the middle of the p-shell, have extremely high quadrupole deformation [18]. The deformation of the nucleus is reflected by the value of the nuclear quadrupole moment for ¹⁰B (Q = +84.72 mb). In this case, when calculating cross-sections for the pickup reaction, it is necessary to consider the collective nature of nuclear states. This is supported by the results of studies of reactions (α, t) [19] and (d, t) on ¹¹B nuclei [20], where it was shown that taking into account the quadrupole couplings of the ground and excited states of nuclei strongly influences the cross-sections for reactions (α, t) and (d, t).

In Ref. [21], the data for quasi-elastic scattering in the ⁸Li + ⁹⁰Zr system at an energy of 18.5 MeV were described within the framework of the Coupled Reaction Channels (CRC) method, with the inclusion of couplings with excited states of the ⁸Li and ⁹⁰Zr nuclei and the transfer reactions⁹⁰Zr(⁸Li, ⁷Li)⁹¹Zr and ⁹⁰Zr(⁸Li, ⁹Be)⁸⁹Y. It was also shown that the coupling with the single-neutron stripping reaction has an important influence on the (quasi)elastic scattering of ⁸Li + ⁹⁰Zr at this energy. Therefore, in this case, the CRC method is the most adequate for analysis.

In the current study, the elastic and inelastic scattering of deuterons by ¹⁰B nuclei with excitation of the levels at 0.718 MeV (1⁺), 2.154 MeV (1⁺), and 3. 59 (2⁺) are investigated at an energy of 14.5 MeV. In addition, the reaction ¹⁰B(d, t)⁹B with transitions to the ground (3/2⁻) and excited (2.361 MeV and 2.79 MeV) states of the ⁹B nucleus was studied in more detail than in the previous study [10]. The main purpose was to obtain information on the role of the direct mechanism for a neutron pickup using the CRC method. Another task was to elucidate the connection between transitions in the (d, t) reaction with collective excitations of the¹⁰B nucleus.

2. Experimental Technique and Measurement Results

The measurements were performed on a 14.5 MeV deuteron beam extracted from the U-150M isochronous cyclotron of the Institute of Nuclear Physics (Almaty, Kazakhstan). The energy spread in the beam was about 150 keV.

The target was a self-supporting boron film that was 0.2 mg/cm² thick with a¹⁰B enrichment of 90%. Its thickness was determined from the energy loss of α-particles of a radioactive source with an accuracy of about 8%. The target was located in the center of a scattering chamber (Figure 1). The design of the chamber allowed us to perform measurements in the broad angular range (from 10° to 170° in the laboratory system) using telescopes with the following counters: T₁, T₂ or T₃ (Figure 1). In our experiment, only one telescope of counters for each counter was used. Using telescope T₄, it was possible to carry out measurements in the range of angles of 5–10° in the laboratory system. The distance of the detectors from the target could be varied from 100 to 350 mm. The scattering chamber had a configuration that was specially designed (A in Figure 1) for taking measurements at small angles. A small-angle telescope (T₄) was positioned at a distance of up to 100 cm from the target. Regarding the total acceptance angle relating to the solid angle of the detector, the size of the beam spot and beam position stability was about 1.5° in the center mass system.

A system of motors inside the chamber was used to carry out all the necessary movements, i.e., the replacement of the target, the angular and radial displacement of the telescopes, the change in the target plane orientation relative to the beam axis, etc.

The charged products of nuclear reactions were recorded using a counter telescope consisting of thin (ΔE) and thick (E) silicon detectors with thicknesses of about 50 µm and 2 mm, respectively. The separation of deuterons and triton from other charged particles was carried out via a two-dimensional analysis system (ΔE−E) using computer-aided measurement and control CAMAC-standard electronics and a processing program implemented on a personal computer. The overall energy resolution was mainly determined by assessing the energy spread in the beam. The absolute cross-sections were determined based on the known target thickness, the solid angle, and the beam current measured using a Faraday cup (FC in Figure 1). Part of a typical two-dimensional (ΔE−E) spectrum of particles from the ¹⁰B + d interaction is presented in Figure 2, showing the separation of p, d, and t.

The examples of the spectra of deuterons scattered by ¹⁰B and tritons from the reaction ¹⁰B(d, t)⁹B, measured at an energy of 14.5 MeV, are shown in Figure 3 and Figure 4, respectively. In the spectrum of deuterons (Figure 3), in addition to the elastic peak, there are transitions to the states at 0.718 MeV (1⁺), 2.154 MeV (1⁺), 3.59 (2⁺), and 4.77 (3⁺). All of them are connected to the ground 3⁺ state of the ¹⁰B nucleus by strong quadrupole (E2) transitions [22].

In the spectrum of tritons (Figure 4), along with an intense transition to the ground state (3/2⁻) of the ⁹B nucleus, a transition to the 2.361 MeV state (5/2⁻) is observed. In addition, there is a peak corresponding to an excitation energy of 2.79 MeV. Here, the states (5/2⁺) and (1/2⁻) are known [22], which were not separated in our experiment. The 5/2⁺level could not be excited in the reaction of neutron pickup from the 1p_3/2 shell. Its excitation would indicate an admixture of the 2s1d shell in the ground state of ¹⁰B.

The experimental angular distributions of deuterons scattered by ¹⁰B with transitions to the states at 0.718 MeV (1⁺), 2.154 MeV (1⁺), and 3.59 MeV (2⁺), and tritons from the reaction ¹⁰B(d, t)⁹B with transitions to the ground (3/2⁻), and the excited states of ⁹B at E_x = 2.361 MeV (5/2⁻), and E_x = 2.79 MeV (5/2⁺, 1/2⁻) are shown in Figure 5. As can be seen from the figure, the angular distributions of transitions to the ground (J^π = 3/2⁻) and excited state (5/2⁻) of the ⁹B nucleus in the reaction (d, t) have a structure characteristic of neutron transfer with orbital angular momentum l = 1.The angular distribution for the peak corresponding to E_x = 2.79 MeV is very different from the corresponding distributions for the transitions of states 3.2⁻ and 5.2⁻ of the ⁹B nucleus.

The systematic error of the measured cross-sections, according to our estimate, did not exceed 10%. The absolute cross-sections were determined based on the known target thickness (with the error not exceeding 8%), the solid angle of the telescope (error no more than 1%), and a beam current measured with a Faraday cup (with accuracy of 1%). When measuring in the region of the forward hemisphere, the statistical errors were about 1–5%. At larger angles, they increased, but never exceeded 15%.

The error bars shown in Figure 5 are smaller than the size of the experimental points.

3. Analysis of the Experimental Data and Discussion

3.1. Elastic Scattering

Most of the existing systematic deuteron potentials are based on the analysis of the angular distributions of the elastic scattering cross-sections of deuterons from heavy targets with A ≥ 30. It is known that the systematics developed in such heavy-mass regions are different from that of the light-mass regions (A ≤ 20). A systematic analysis of elastic scattering on p-shell nuclei was carried out in Ref. [9]. In the calculations, we used phenomenological Woods–Saxon potentials in the following form:

$$U\left(r\right)=-Vf\left(r\right)-i{W}_{V}g\left(r\right)-{W_S{g}^{\prime }\left(r\right)-V}_{\mathrm{s}\mathrm{o}}h\left(r\right)\left(l\sigma \right)+V_{\mathrm{C}}\left(r\right).$$

(1)

V_C(r) is the Coulomb potential resulting from a uniform charge distribution of radius 1.3A^1/3 fm. The calculations used potentials with both volume (W_V) and surface (W_S) absorption, as well as the spin–orbit (V_so) potential. V denotes the depth of the real potential, l and σ denote the orbital moment and the spin of projectile, respectively.

In Equation (1), f(r), g(r), g′(r), and h(r) are the Woods–Saxon form factors:

$$f\left(r,r_{\mathrm{v}},a_{\mathrm{v}}\right)={\left(1+e^x\right)}^{-1} \mathrm{with} x=(r-r_{\mathrm{v}}{A}^{1/3})/a_{\mathrm{v}},$$

$$g\left(r,r_{\mathrm{w}},a_{\mathrm{w}}\right)={\left(1+e^x\right)}^{-1} \mathrm{with} x=(r-r_{\mathrm{w}}{A}^{1/3})/a_{\mathrm{w}},$$

$$g^{\prime }\left(r,r_{\mathrm{w}},a_{\mathrm{w}}\right)=\left(-4a_{\mathrm{w}}\right){\displaystyle \raisebox{1ex}{$\mathrm{d}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{d}r$}\right.}{\left(1+e^x\right)}^{-1} \mathrm{with} x=(r-r_{\mathrm{ws}}{A}^{1/3})/a_{\mathrm{ws}},$$

$$h\left(r,r_{\mathrm{s}\mathrm{o}},a_{\mathrm{s}\mathrm{o}}\right)={\left({\displaystyle \raisebox{1ex}{$\hslash $}\!\left/ \!\raisebox{-1ex}{$m_{\pi }$}\right.}c\right)}^2{r}^{-1}{\displaystyle \raisebox{1ex}{$\mathrm{d}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{d}r$}\right.}{\left(1+e^x\right)}^{-1} \mathrm{with} x=(r-r_{\mathrm{so}}{A}^{1/3})/a_{\mathrm{so}},$$

where as and rs are the optical model potential (OMP) parameters for the corresponding form factors, c is the speed of light, and m_π represents the pion mass.

The OMP parameters are optimized with the minimization of the χ²-method:

$$\chi 2={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N\left({\left[{\sigma }_i^{\mathrm{e}\mathrm{x}\mathrm{p}}-{\sigma }_i^{\mathrm{t}\mathrm{h}}\right]}^2/\Delta {\sigma }_i^2\right),$$

(2)

where${\sigma }_i^{\mathrm{e}\mathrm{x}\mathrm{p}}$, ${\sigma }_i^{\mathrm{t}\mathrm{h}}$, and $\Delta {\sigma }_i$ are the experimental and theoretical cross-sections and the experimental errors, respectively.

The fitting procedure was carried out with SPI-GENOA [23]. The starting values of the radii of the real and imaginary parts of the OMP for fitting to the optical model are taken from Ref. [24]. The OMPs used in our calculations are listed in

Table 1. Potentials used in the entrance ¹⁰B + d (A1, A2) and in the exit ⁹B + t (B) channels. See text for details.

Pot.	V, MeV	aV, fm	rV, fm	WV(S), MeV	rw, fm	aw, fm	Vso, MeV	rso, fm	aso, fm	rC, fm	σt, mb	χ2	Ref.
A1	71.0	0.95	1.15	13.5	1.34	0.679	6.69	1.25	0.718	1.3	946	2.7	this work
A2	104.22	0.95	0.863	4.17 *	1.59	0.85	12.8	0.863	0.916	1.3	1070	4.9	[10]
B	141.2	0.758	1.156	16.4	1.535	0.99				1.3			[25]

* Surface absorption.

.

For the potential of the exit channel in the ⁹B + t system, we used the potential found in Ref. [25] from the scattering of ³He by ⁹Be at an energy of 13.2 MeV.

A comparison of the calculated cross-sections for elastic scattering of deuterons by ¹⁰B nuclei at an energy of 14.5 MeV is shown in Figure 6.

As can be seen, the calculation with potential A1 from Table 1 reproduces the experimental cross-sections quite well, whereas potential A2 (with surface absorption) gives underestimated cross-sections in the angle range of 45–75°.

3.2. The Coupled Channel Analysis

As is known, boron nuclei are highly deformed; thus, low-lying states are of a collective nature. In this case, the coupled channel method is most suitable for describing both scattering and reactions. Calculations were performed employing FRESCO code [26], with the potentials given in Table 1. But for a better agreement between the calculated cross-sections and the experiment, the depths of the imaginary potentials were reduced to values of 10.5 MeV and 3.17 MeV for potentials A1 and A2, respectively. The coupling scheme used in our calculations is shown in Figure 7.

3.2.1. Inelastic Scattering

The inelastic scattering at E_d = 14.5 MeV with excitation of the ¹⁰B states is interpreted within the frame of the collective model.

A comparison of calculated cross-sections with experimental data on elastic and inelastic scattering with transitions to states of ¹⁰B at the excitation energy E_x = 0.718 MeV (1⁺), 2.154 MeV (1⁺) and 3.59 MeV (2⁺) is shown in Figure 8. It can be seen that, taking into account that the channel coupling significantly improves agreement with the experimental data, both potentials (A1 and A2) from Table 1 reproduce the pronounced diffraction in the angular distribution of elastic scattering equally well over the entire range of angles. Calculations with potential A1 (red curves in Figure 8) also describe inelastic scattering quite well. But with potential A2 (blue curves in Figure 8), it is possible to achieve agreement with the experiment only at large angles (larger than 100°), while in the front hemisphere, the shape of the theoretical and measured cross-sections differs greatly.

Therefore, the deformation parameters β₂ were determined, with the potential A1 (red lines in Figure 8) from the condition of the best description of the experimental cross-sections, for the transitions to the 0.718 MeV (1⁺), 2.154 MeV (1⁺), and 3.59 MeV (2⁺) states of ¹⁰B at small angles in the first maximum of the angular distributions. For the above-noted transitions, the following values of quadrupole deformation parameters were obtained: β₂ = 0.76 ± 0.1, β₂ = 0.7 ± 0.1, and β₂ = 0.7 ± 0.1, respectively. The above uncertainties include only statistical and systematic errors. As can be seen, the parameters of the β₂ extracted from the analysis also significantly depend on the choice of optical potential. This introduces additional uncertainty. The values obtained can be compared with the data from a study of inelastic scattering of protons with an energy of 30.3 MeV on ¹⁰B nuclei with excitation of the same states of β₂ = 0.67 ± 0.04, β₂ = 0.49 ± 0.04, and β₂ = 0.45 ± 0.04 [27,28]. Within the errors, the values of the quadrupole deformation parameters found by us are consistent with the results of previous analyses.

3.2.2. The (d, t) Reaction and the Role of Exchange Mechanism with Transfer of ⁸Be

The coupled channel analysis, as shown in Figure 7, included elastic (¹⁰B + d), inelastic scattering with excitation states of 0.718 MeV (1⁺), 2.154 MeV (1⁺), and 3.59MeV (2⁺), one-neutron pickup reactions with transitions to the ground (3/2⁻), and excited states at 2.361 MeV (5/2⁻) and 2.79MeV (5/2⁺). The reaction cross-sections were calculated using the coupled channel Born approximation model (incorporated in the FRESCO code [26]). The calculations were performed with potentials A1 or A2 for the entrance (¹⁰B + d) channel and potential B for the exit (⁹B + t) channel. The parameters of these potentials are given in Table 1.

The wave functions of bound states were calculated with the real Woods–Saxon potential at geometric parameters of r₀ = 1.25fm and a = 0.65 fm. Its depth was chosen so that the required binding energy of the transferred particle was obtained. A comparison of the calculated cross-sections with the experimental data is shown in Figure 9.

The spectroscopic amplitudes (SAs) used in our analysis are given in

Table 2. Spectroscopic amplitudes (SAs) used in calculations of the ¹⁰B(d, t)⁹B reaction.

System	Ex, MeV	Jπ	nLj	SA	SA Theory
t→d + n	0.0	1/2+	1S1/2	1.22	1.22 [29]
10B→9B* + n	0.0	3/2−	1P3/2	0.67 ± 0.1 a	−1.09 [17]
	2.361	5/2−	1P3/2	0.94 ± 0.2 a	−0.92 [17]
	2.788	5/2+	1S1/2
10B→8Be + d	0.0	3+	1D3	0.8	0.811 [30]

^a SAs extracted in this analysis.

, along with theoretical predictions. Theoretical values for single-particle hole states in the 1p-shell excited in the pickup of a neutron from the ¹⁰B nucleus were obtained within the framework of the shell model with intermediate coupling [17]. The SA for the t→d + n configuration was calculated in Ref. [29] using the translation-invariant shell model (TISM).

The errors in the extracted spectroscopic amplitudes in addition to statistics were determined by examining the uncertainties of the target thickness (less than 8%) and the uncertainty of normalizing the calculated cross-sections to the experimental data (less than15%). Previously, the ¹⁰B(d, t)⁹B reaction was studied at deuteron energies of 11.8 MeV [10]. In Ref. [10], spectroscopic amplitudes of 0.89 and 0.8 were obtained for transitions to the ground and excited states of the ⁹B nucleus. As can be seen from Table 2, the results found by us, taking into account possible errors, are in good agreement with the previously obtained experimental data and the theoretical predictions.

In addition to the direct neutron pickup, we assessed the role of the exchange mechanism with the transfer of the ⁸Be cluster in the elastic scattering of deuterons on the ¹⁰B nucleus. The calculations of the cross-sections for the exchange mechanism in this case are complicated by the fact that a heavy cluster (⁸Be) can be transferred not only in its ground state but also in its excited states, and spectroscopic amplitudes are not known with sufficient accuracy. This leads to large uncertainties in the calculations. Therefore, for a rough estimate, only the ⁸Betransfer in the ground state (0⁺) was considered. Calculations were carried out with the potential A1 from Table 1, using the theoretical value of the spectroscopic amplitude calculated in Ref. [30] (see Table 2). The results of the calculations are shown in Figure 10.

It can be seen that the cross-sections for the exchange mechanism are more than two orders of magnitude smaller than the experimental values. Thus, one can well assume that the process of excitation does not play a significant role in the elastic scattering of deuterons and the (d, t) reaction.

4. Conclusions

The elastic and inelastic scattering of deuterons on ¹⁰B nuclei, as well as the ¹⁰B(d, t)⁹B reaction, were studied at a deuteron energy of 14.5 MeV. In deuteron scattering, in addition to elastic scattering, differential cross-sections for transitions to ¹⁰B states were measured at excitation energies of 0.718 MeV (1⁺), 2.154 MeV (1⁺), and 3.59 MeV (2⁺). The cross-sections of the (d, t) reaction were measured for the ground (3/2⁻) and excited states of the ⁹B nucleus at E_x = 2.361 MeV (5/2⁻) and 2.79 MeV (5/2⁺).

The optical potentials necessary for the analysis of experimental angular distributions were found from the best description of elastic scattering within the optical model. A good description of scattering over the full angular range was obtained. Further analysis of scattering and reaction (d, t) was carried out using the coupled channel method using the FRESCO program.

Analysis of inelastic scattering based on the collective model made it possible to extract a value of the quadrupole deformation parameter of about β₂ = 0.7±0.1 for transitions to various states of the ¹⁰B nucleus. Within expected errors, the value obtained by us here is consistent with the results of a previous analysis focusing on the inelastic scattering of protons at an energy of 30.3 MeV.

Assuming a direct neutron pickup mechanism, a good enough description of the angular distributions for transitions to the ground (3/2⁻) and first excited (5/2⁻) states of the ⁹B nucleus in the reaction (d, t) was obtained. The spectroscopic amplitudes of these transitions were extracted. The results obtained in the current study, within expected errors, are in quite well agree with the previously obtained experimental data and the theoretical predictions.

An estimate was made of the possible contribution to the elastic scattering of the exchange mechanism of the transfer of the ⁸Be cluster in the ¹⁰B(d, ¹⁰B)d reaction, which was physically indistinguishable from elastic scattering. It is shown that the exchange process is not significant.