¹⁴N break-up α emission with ⁵⁹Co, ⁹³Nb, and ¹⁹⁷Au targets at an incident energy of 250 MeV

Abstract

The emission of α particles has been analyzed using a ¹⁴N projectile beam on ⁵⁹Co, ⁹³Nb, and ¹⁹⁷Au targets at an incident energy of 250 MeV, in the angular range from 8° to 40°. The estimated α particles from the break-up channel and the pre-equilibrium mode were compared with the experimentally measured inclusive double differential cross sections at various angles at 250 MeV beam energy. The present results indicate that at such a high incident energy, a significant contribution to the α-particle emission comes from direct reactions, which are dominant at forward angles. At larger emission angles, non-equilibrium and evaporation processes increase and compete with direct reactions.

Introduction

At high incident energies of ~ 10 MeV/A, the break-up analysis of tightly bound projectile nuclei is pivotal in shaping the resulting reaction dynamics and unravelling the mysteries of nuclear structure. In heavy ion nuclear reactions, the compound system processes compete with various direct processes. The major direct processes involve breaking up the incident projectile and/or the transfer of one or more nucleons between the projectile and the target nucleus [1]. The break-up reaction significantly affects the total cross section. The results of an experiment [2] showed that the fission reaction cross section is significantly less than the predicted one. The intricacies of the ¹⁴N break-up may illuminate its implications in nuclear physics.

The earlier studies of the reactions induced by ¹²C and ¹⁴N, revealed the production of large amounts of α particles at ~ 10 MeV/A incident energies. These α particles are primarily emitted in the forward direction, where their spectrum exhibits a broad peak with a mean energy roughly corresponding to the beam velocity. Further experiments have indicated that most of these α particles originate from a projectile break-up [3,4,5,6]. The studies have shown that a significant fraction of the observed α particles originates in the break-up of the projectile into an α-particle and another fragment. One of these fragments may fuse with the target nucleus while the other fragment gets emitted almost undisturbed [7].

In the direct reaction, the projectile ¹⁴N may break into ¹⁰B and an α-particle, or it may break into ¹²C + d, further the ¹²C break into ⁸Be and an α-particle, thus producing three α particles. Secondly, α particles may get emitted after complete and incomplete fusion of the projectile with the target. Also, the contribution from evaporation channels should be considered. Thus, as mentioned above, the energy and angular distributions of the emitted α particles from the different reaction channels would have different values.

At this energy, there is a contribution in the reaction cross section, mainly from a few reactions like: (1) An α-particle produced during the projectile break-up. (2) α particles emitted due to dissociation of another fragment. (3) α particles being a participant fragment and re-emitted with reduced energy. (4) Pre-equilibrium α particles emitted after complete and incomplete fusion. (5) Evaporated α particles. In the past, the contribution of different reaction mechanisms was simulated as described in Ref. [7]. Recently, the contribution of direct processes, pre-equilibrium emission, and evaporation of particles at such a high incident energy has been investigated in the interaction of ¹⁴N with ⁵⁹Co and ⁹³Nb at an incident energy of 250 MeV [8]. A modified PACE code employing Hauser–Feshbach formalism was utilized to calculate emitted α particles cross sections. The modified PACE code takes into account equilibrium and pre-equilibrium processes. More details on the theoretical approach are given in Ref. [9].

In the present work, we have adopted the analysis method described in Ref. [10]. The internal momentum of the α-particle is derived from a Yukawa-type wave function that gives the correct separation energy of α- particle from the ¹⁴N. The primary object of this work is to explore the ¹⁴N break-up α contribution using ⁵⁹Co, ⁹³Nb, and ¹⁹⁷Au targets. In the present break-up study, an α cluster structure of α + ¹⁰B is considered for ¹⁴N nucleus.

The contribution from pre-equilibrium emission and evaporation of particles in the experimentally measured α particle cross sections was already investigated using the same projectile ¹⁴N elsewhere [8]. Therefore, unfolding the ¹⁴N break-up contributions involving medium weight to heavy target masses and comparing them with other possible reaction channels would be interesting.

Experimental details

The experiment was performed at the iThemba LABS, Somerset West, South Africa cyclotron facility. Experimental data were measured for the interaction of ¹⁴N with ¹⁹⁷Au target at an incident energy of 250 MeV. Data were acquired at various scattering angles ranging from 8° to 40° following the method similar to the one for ¹⁴N + ⁵⁹Co, ⁹³Nb systems [8]. More details of the detector arrangement and electronics used in the experiment are described in Ref. [7], and further details of the facility are provided in Ref. [11]. The experimentally measured double differential cross sections for α particles are shown in Figs. 1, 2, 3, 4, 5.

Fig. 1

(Colour Online) Experimental (open symbols) and calculated (solid curve) energy Spectra of α particles measured at 8° for ¹⁴N with ⁵⁹Co (red circles), ⁹³Nb (blue stars) and ¹⁹⁷Au (green diamonds) at 250

Fig. 2

(Colour Online) Experimental (open symbols) and calculated (solid curve) energy Spectra of α particles measured at 10° for ¹⁴N with ⁵⁹Co (red circles), ⁹³Nb (blue stars) and ¹⁹⁷Au (green diamonds) at 250 MeV

Fig. 3

(Colour Online) Experimental (open symbols) and calculated (solid curve) energy Spectra of α particles measured at 8° (green circles), 10° (red squares), 15° (pink diamonds) and 20° (black stars), for ¹⁴N with ⁵⁹Co at 250 MeV

Fig. 4

(Colour Online) Experimental (open symbols) and calculated (solid curve) energy Spectra of α particles measured at 8° (green circles), 10° (red squares), 15° (pink diamonds) and 20° (black stars), for ¹⁴N with ⁹³Nb at an incident energy of 250 MeV

Fig. 5

(Colour Online) Experimental (open symbols) and calculated (solid curve) energy Spectra of α particles measured at 8° (green circles), 10° (red squares), 12° (blue down-triangle), 15° (pink diamonds) and 20° (black stars), for ¹⁴N with ¹⁹⁷Au at 250 MeV

Theoretical method

We used the Serber approximation [10, 12] to calculate the ¹⁴N break-up into α particle and ¹⁰B fragments. We have considered α and ¹⁰B as spectator and participant fragments, respectively. In this method, the target nucleus plays no role except for the projectile break-up at the collision. The momentum of the emitted α-particle is (p_α) the sum of the momentum due to the center of mass motion of the incident ¹⁴N (p_N) and the momentum due to the internal motion of α-particle in the ¹⁴N at the time of break up (p), that is, $p_{\alpha } = \frac{2}{7} p_N + p$. Also, according to this, approximation, the break-up transition matrix is given by the Fourier transform of the function describing the relative motion of α-particle and ¹⁰B fragment in ¹⁴N. The square of the transition matrix is provided by,

$$ \left| T \right|^2 \propto \left| {\Psi \left( {p_\alpha - \frac{2}{7}p_N } \right)} \right|^2 $$

(1)

where

$$ \Psi \left( p \right) = \frac{1}{{\left( {2\pi \hbar } \right)^{3/2} }}\int {\Psi \left( r \right)\exp \left( { - \frac{i}{\hbar }\left( {p \cdot r} \right)} \right)d^3 r} $$

(2)

is the Fourier transform of the wave function.

$$ \Psi \left( r \right) = C \left( {\frac{\alpha }{2\pi }} \right)^\frac{1}{2} \frac{1}{r}\exp ( - \alpha .r) $$

(3)

Here,

$$ \alpha = \frac{{(2\mu \varepsilon )^\frac{1}{2} }}{2\pi \hbar } $$

(4)

and C is the normalization constant, µ is reduced mass, and ε is the binding energy of α -particle in ¹⁴N, which is 11.61 MeV. So, the obtained transition matrix is given as

$$ |T|^2 \propto \frac{1}{\pi^2 }\frac{{\left( {2\mu \varepsilon } \right)^\frac{1}{2} }}{{[(2\mu \varepsilon + \left( {\frac{2}{7}p_N - p_\alpha } \right)^2 ]^2 }} $$

(5)

The energy distribution of α particles is given by multiplying the transition matrix by the three-body phase space factor [13, 14]. When the target mass number A becomes $\frac{1}{3}A \gg 1$ then Eq. (5) is reduced to the expression.

$$ \frac{d^2 \sigma }{{d\Omega dE}} \propto \frac{4}{\pi }m_\alpha m_B \frac{{p_\alpha p_B \left( {2\mu \varepsilon } \right)^\frac{1}{2} }}{{\left[ {(2\mu \varepsilon + \left( {\frac{2}{7}p_N - p_\alpha } \right)^2 } \right]^2 }} $$

(6)

where m_α and m_B are the masses of α-particle and ¹⁰B, and p_α, p_B, p_N are the momentum of the α-particle, ¹⁰B, and ¹⁴N, respectively. The |T|² term mainly determines the energy spectrum and is only slightly changed by the phase space factor. The peak energy and the FWHM of the bump spectra deduced from the |T|² term only is about $\frac{m_s }{{m_p }}E_N$ and $1.21\sqrt {E_N \varepsilon }$ respectively, at forward angles, which agree with the experimental results.

Results and discussion

The energy spectra of the α particles emitted from the break-up of ¹⁴N at the collision with ⁵⁹Co, ⁹³Nb and ¹⁹⁷Au targets at 250 MeV have been examined. Our analysis includes investigating α + ¹⁰B cluster structure for breaking up ¹⁴N projectile nucleus at various angles, using the Serber Model. The experimentally measured double differential cross section of α particles at different angles is compared with the present results on break-up calculations. Figures 1 and 2 illustrate the measured energy spectra of emitted α particles at 8° and 10°, respectively, in comparison with the corresponding break-up calculations, for reactions involving ¹⁴N with ⁵⁹Co, ⁹³Nb, and ¹⁹⁷Au at an incident energy of 250 MeV. Remarkably, at smaller angles, the reaction cross section remains relatively constant. Furthermore, a conspicuous, broad peak is observed in the energy spectrum, centered around $\frac{2}{7}E_N \sim 70 {\text{MeV}}$. This peak corresponds to α particles originating from incomplete fusion and pre-equilibrium emission. An extended tail is visible, attributed to spectator α particles originating from the direct break-up process.

In the present method, the experimental data were not fitted using any free parameters or distribution models throughout the entire study, except for incorporating a normalization constant. The identical normalization constant was applied across various angles to calculate the break-up cross sections for the specific target under investigation [12]. Direct break-up reactions are predominant at very forward angles. Hence, the normalization constant was chosen to align with measurements at 8°, ensuring uniformity across all angles. There is a notable correspondence between the present calculations and experimental results for all systems, including ⁵⁹Co, ⁹³Nb, and ¹⁹⁷Au at both 8° and 10°. However, for the medium weight target nucleus ⁵⁹Co at 10°, the present calculations underestimate the cross sections, possibly because the same normalization constant was used for both 8° and 10°.

Figures 3, 4, 5 show the measured energy spectra of emitted α particles along with the results of break-up calculations at various angles for ¹⁴N with ⁵⁹Co, ⁹³Nb and ¹⁹⁷Au. The figures clearly indicate that as we move towards more backward angles, the peak gradually diminishes, vanishing entirely after 20°.

Figures 6 and 7 compare measured double differential cross sections with the calculated break-up and pre-equilibrium α particles contributions [8]. These comparisons aim to understand the behaviour of emitted α particles during the reaction across various angles. It is evident that at forward angles (8°–15°), direct reactions dominate and closely match break-up calculations. However, deviations between the calculated break-up spectra and experimental data become apparent as the scattering angle increases towards larger or backward angles (25°–40°), where the non-break-up channels such as pre-equilibrium and equilibrium processes come into play and align more closely with pre-equilibrium calculations. This discrepancy is attributed to the presence and contribution of other reaction channels.

Fig. 6

(Colour Online) The α particles energy spectra at angles 10° (red squares) A, 15° (pink diamonds) B, 25° (blue left-triangle) C, and 40° (brown hexagon) D, for ¹⁴N with ⁵⁹Co at 250 MeV. The filled curve (green) shows the break-up calculation, and the filled curve (pink) shows the contribution of pre-equilibrium α particles

Fig. 7

(Colour Online) The α particles energy spectra at angles 10° (red squares) A, 15° (pink diamonds) B, 30° (indigo triangles) C, and 40° (brown hexagon) D, for ¹⁴N with ⁹³Nb at 250 MeV. The filled curve (green) shows the break-up calculation, and the filled curve (pink) shows the contribution of pre-equilibrium α particles

Conclusion

¹⁴N break-up α contributions have been studied at various angles using ⁵⁹Co, ⁹³Nb and ¹⁹⁷Au targets at an incident energy of 250 MeV. The present work focused on calculating the ¹⁴N breaking up into an α-particle and ¹⁰B nucleus utilizes the Serber model’s asymptotic wave function. The experimentally measured double differential cross section of α particles at various angles is compared with the current results obtained from break-up calculations. The results of these calculations exhibit good agreement with available experimentally measured data, offering a clear explanation for the substantial α-particle yields resulting from the break-up of ¹⁴N and also reflecting the difference in their origins. As the incident energy increases, the projectile becomes less efficient in transferring the kinetic energy to the target in thermal energy. To account for this intriguing experimental observation, it has become necessary to postulate not only the growing significance of incomplete fusion reactions at higher incident energies but also that the rapid dissipation of energy imparted to the ¹⁴N nucleus will not warm the nuclei and lead this reaction to the pre-equilibrium emission of α particles. Break-up calculations based on the α-particle momentum distribution derived from the asymptotic relative wave function of successfully replicate peak energies in the analyzed energy range. Figs. 6 and 7 visually demonstrate the contributions from incomplete fusion, complete fusion, and pre-equilibrium emission of α particles, particularly emphasizing the significance of projectile break-up at very forward angles. However, the Serber model falls short in providing information regarding excited state or resonant break-up phenomena. Furthermore, the model confines us to direct break-up within the target field, neglecting other possible reaction pathways. Hence, for more accurate and comprehensive results, employing DWBA (Distorted Wave Born Approximation) can yield more reliable outcomes [10].