New measurements and reanalysis of¹⁴N elastic scattering on¹⁰B target

In elastic scattering betweenp- andsd-shell nuclei, a considerable increase in the differential cross sections at large angles is often observed. This effect, often called anomalous large-angle scattering (ALAS), is particularly observed in the interaction of nuclei with a pronounced α-cluster structure. Such an increase in the cross sections cannot be reproduced within the framework of an optical model with reasonable potential. In a number of works [1-7], the description of ALAS is achieved by including a contribution from the transfer mechanism of nucleon(s) added to the elastic scattering. This addition has led to a significant improvement in the description of the differential cross sections of the elastic scattering in the full angular range obtained in the framework of the coupled reaction channels (CRC) and distorted waves Born approximation (DWBA) methods. This phenomenon has so far been observed in the elastic scattering and reactions between 1p-shell nuclei that differ in mass by one, two, and even three nucleons in addition to that of the α particle.

There are extensive previously published elastic transfer studies including¹⁴N(¹⁰B,¹⁴N)¹⁰B. The elastic scattering for the¹⁴N+¹⁰B system was measured at the energies of 38.1–50.0 MeV [2], 73.9 MeV, and 93.6 MeV [8]. In Ref. [8] the angular distributions of¹⁴N elastically scattered by¹⁰B were measured in the limited angular range of up to 50° in the center of the mass system. In Ref. [2], measurements of angular distributions were extended to large angles (~ 150°) where a significant rise was exhibited. This rise was reproduced by taking into consideration the effect of α-cluster transfer.

In the present work, elastic scattering in the¹⁴N +¹⁰B system was measured atE_lab= 21.0 and 24.5 MeV in order to verify the role of α-cluster transfer between¹⁴N and¹⁰B at energies closer to the Coulomb barrier. The interest in such a study is because the number of reaction channels in the scattering decreases rapidly around the Coulomb barrier. The available data [2,8] for the¹⁴N+¹⁰B system were also reanalyzed using both the optical model (OM) and the DWBA method to obtain an energy-dependent potential that could fairly reproduce the elastic scattering data in the full angular range by taking into account the contribution of the α-transfer mechanism.

This work complements the ongoing program for studying elastic scattering at low energies at the DC-60 cyclotron [5,9], in which scattering in the collision of light heavy ions is studied to take into account the possible contribution of the particle exchange mechanism. The paper is organized as follows. In Section 2, the measurement procedure of¹⁴N +¹⁰B angular distribution is described. Section 3 is devoted to theoretical analysis, results, and discussion. A summary is given in Section 4.

The differential cross sections for elastic scattering of¹⁴N ions on¹⁰B nucleus were measured at the DC-60 cyclotron of Institute of Nuclear Physics (Astana, Kazakhstan).¹⁴N ions were accelerated up to energies of 21.0 and 24.5 MeV and then impinged on a self-supported¹⁰B (with 90% enrichment) target with a thickness of 30 µg/cm². The target was located in the center of the scattering chamber with diameter of 43 cm. Reaction products were recorded with one telescope consisting of two silicon detectors (∆E-Е). ∆Еis a surface barrier detector with a thickness of 10 μm, andEis the complete absorption detector with a thickness of 200 μm. The measurements were performed in the angular range of 31°–153° with an increment of 2° in the center of mass system. A Faraday cup was used to integrate the total incident currents on the target. The ΔE-Espectrum of the reaction products and typical energy spectra of¹⁴N are shown inFig. 1. InFig. 1(a), the ΔE-Espectrum demonstrates excellent Z separation between reaction products (B, C, and N). The energy resolution was about 500 keV and was mainly determined by the energy spread in the cyclotron beam and the target thickness. Such resolution allowed for a clear identification of the several peaks in the spectrum. A typical energy spectrum of¹⁴N scattered by¹⁰B is shown inFig. 1(b).

Figure 1.(color online) (a) ΔE-Espectrum of the reaction products for the¹⁴N+¹⁰B system. (b) The energy spectrum of¹⁴N scattered by the¹⁰B target at the beam energy of 24.5 MeV measured at an angleθ= 42°.

The angular distributions of elastic scattering of¹⁴N on¹⁰B nucleus at the energies of 21.0 MeV and 24.5 MeV are shown inFig. 2. These distributions were normalized to OM cross sections at small angles, where it is relatively independent of the parameters of the nuclear potential. The normalization error was smaller than 10%. As can be seen fromFig. 2, in the front hemisphere, a smooth decrease in the cross sections is observed without a pronounced oscillation structure. With a further increase in the angles, the decrease stops and a weakly expressed diffraction structure appears with cross sections rising at angles greater than 120°. We estimated the systematic error of measured cross sections to be no larger than 10%. The statistical error was 1%–5% during our measurements in the forward hemisphere region. It increased at large angles, but nowhere did it exceed 10%. The error bars inFig. 2are smaller than the size of the experimental points.

Figure 2.Experimental angular distributions of elastic scattering of¹⁴N by¹⁰B nuclei at the beam energies of 21.0 MeV and 24.5 MeV.

It is well known that the extracted values for spectroscopic amplitudes from scattering and transfer reactions are highly dependent on the potential parameters used in the various analyses. Based on this fact, the experimental data for the¹⁴N+¹⁰B nuclear system were first analyzed under the assumption of pure potential scattering. In this case, the differential cross sections were calculated in the framework of the phenomenological optical model and the interaction potential was found by fitting the calculated cross sections to the experimental data. The considered total potential is the sum of three terms:a) ${V_C}(R)$ : Coulomb potential of a uniform charged sphere with radius parameter fixed at 1.3 fm;b) real volume part; andc) an imaginary volume part of nuclear potential with both Woods-Saxon shapes. Thus, the total potential has the following shape:

whereV₀,a_VandW₀,a_Ware the depths and diffusenesses of the real and imaginary parts of the nuclear potential, respectively. Radii in these expressions are defined as ${{R}_{V,W,C}}={{r}_{V,W,C}}{\rm{ x }}\left( {\rm{A}}_{\rm{p}}^{{}^{\rm{1}}\!\!\diagup\!\!{}_{3}\;}+{\rm{A}}_{\rm{t}}^{{}^{\rm{1}}\!\!\diagup\!\!{}_{3}\;} \right)$ . In addition to our measurements for¹⁴N+¹⁰B elastic scattering angular distributions atE_lab=21.0 and 24.5 MeV, we also reanalyzed data at all the available energies: atE_lab=38.1, 42.0, 46.0, and 50.0 MeV, taken from Ref. [2], and atE_lab= 73.92 and 93.6 MeV, taken from Ref. [8], in order to obtain energy-dependent potential parameters for this nuclear system. We performed OM calculations using starting parameters taken from Ref. [8], and only the three parametersV₀,W₀, anda_Wwere varied and the other three,r_V,r_W, anda_V,were fixed at 0.92, 1.29, and 0.77 fm, respectively.

Angular distributions at energies of 24.5–50.0 MeV showed a significant increase in differential cross sections at backward angles, which was assumed to arise from the contribution of α-cluster transfer between¹⁴N and¹⁰B. The experimental data at 21.0 MeV, which is closer to the Coulomb barrier energy for the¹⁴N+¹⁰B nuclear system, only showed a slight increase in cross sections at backward angles. However, a rise is already observed at energy 24.5 MeV, confirming that the transfer phenomenon is highly dependent on the projectile’s energy. The angular distributions at the two higher energies 73.92 and 93.6 MeV were measured in a limited angular range and did not extend beyond ~ 50°, and thus, the transfer phenomenon is not presented.

As mentioned above, experimental data in the energy range 21–50 MeV showed a significant increase in cross sections at backward angles, except atE_lab=21.0 MeV, which showed a slight increase. Such increase in cross sections could not be reproduced by the simple OM, and its origin was investigated in terms of α-cluster transfer implemented through the distorted waves Born approximation (DWBA) method. As the ground state spin-parity of¹⁴N isJ^π=1⁺, an alpha particle coupled to a¹⁰B core withJ^π=3⁺can be found in two differential orbitals with angular momentumL=2 orL=4. The configuration with the higher angular momentum should be favored as the cluster configuration withL=4 has a theoretical spectroscopic factor much larger than that of the one withL=2 (SA₂=0.11 andSA₄=0.83) [10]. However, two possible configurations have been taken into account. The experimental angular distributions at the different energies were analyzed up to angles 90° by the optical model to exclude the effect of cluster transfer and other effects that could affect the cross sections at backward angles using the FRESCO code [11]. The comparisons between the experimental angular distributions in the energy range 21–93.6 MeV and theoretical calculations using both OM and DWBA, incorporated in the Coupled Reactions Channels program (FRESCO) [11], are shown inFigs. 3-5with the potential parameters listed inTable 1.

Figure 3.(color online) Comparison between the experimental data (black circles) and calculations for the¹⁰B(¹⁴N,¹⁴N)¹⁰B elastic scattering atE_lab= 21.0, 24.5, and 38.1 MeV. The dashed black curves denote pure optical model fits to the data for anglesθ_c.m.< 90◦. The dashed red and solid blue curves denote the results of calculations taking into account the process of elastic transfer of the α-particle in the¹⁰B(¹⁴N,¹⁰B)¹⁴N reaction toL= 2 andL= 4 orbitals, respectively.

Figure 4.(color online) Same asFig. 3but at energies 42.0, 46.0, and 50.0 MeV.

Figure 5.Comparison between experimental angular distributions for¹⁰B(¹⁴N,¹⁴N)¹⁰B elastic scattering atE_lab= 73.92 and 93.6 MeV (black circles) and the theoretical calculations (dashed curves) using OM.

The significant increase in differential cross sections at large angles can be explained as arising from α-cluster transfer as shown by DWBA calculations that were performed to explore this possibility. In this case, the exchange of the α-cluster between the two interacting nuclei¹⁴N, to be treated as (¹⁰B – Core) + (α-particle – valence), leads to an exit channel that is physically indistinguishable from the entrance channel. Therefore, the differential cross section is the square of the sum of amplitudes from the pure elastic scattering and the exchange mechanism of the cluster transfer, as follows [12,13]:

where ${f_{el}}(\theta )$ is the elastic scattering amplitude, ${f_{\rm DWBA}}(\pi - \theta )$ is the amplitude calculated using the distorted wave method with the replacement $\theta \to \pi - \theta $ , parameterα=pi(coherence of amplitudes), andSis the product of the two spectroscopic amplitudes (SA) of the transferred particle in the entrance and exit channels, which are the same in the case of elastic transfer.

Calculations of transfer are performed using the optical potential parameters obtained from fitting the data in the forward hemisphere up to 90°. The bound state wave function for the relative motion of theα-particle and¹⁰B in the¹⁴N configuration cluster plus core is defined by a Woods-Saxon potential with fixed radius $R=1.25{\rm{x }} $ $\left( 4_{{}}^{{}^{1}\!\!\diagup\!\!{}_{3}\;}+10_{{}}^{{}^{1}\!\!\diagup\!\!{}_{3}\;} \right)$ fm and the diffusenessa= 0.65 fm. The potential depth was adjusted to reproduce the binding energy of the cluster (11.612 MeV). The quantum numbers of the radial motion of the clusters relative to the core were determined from the following expression of the shell model [13,14]:

whereNis the number of nodes of the radial wave function of relative motion (taking into account the node atr= 0) andLis the corresponding orbital momentum of the cluster; Σ denotes the sums of similar quantities for nucleons entering a cluster in a bound state. Two different sets for cluster quantum numbers for the overlaps $\left\langle {{{}^{{\rm{14}}}{\rm{N}}}} | {{\alpha { + ^{{\rm{10}}}}{\rm{B}}}} \right\rangle $ are used in our calculations, as listed inTable 2,due to the two possibilities of transfer toL=2 andL=4 orbitals. The same potential parameters are taken for the entrance channel (¹⁴N+¹⁰B) and the exit channel (¹⁰B+¹⁴N). The spectroscopic amplitude was taken as a free parameter that is varied to give the best agreement between the theoretical calculations and the experimental data. The variation of the extractedSA₂andSA₄withχ²/NatE_lab= 21–50 MeV is shown inFigs. 6and7,respectively. The values ofSA₂andSA₄at the different energies are listed inTable 1. As can be seen, the average values ofSA₂andSA₄are 0.58 ± 0.10 and 0.81 ± 0.12, respectively. The obtainedSA₂value overestimates the theoretical value calculated by Kurath [10] in the framework of the shell model, whereas the extractedSA₄is in good agreement.

Overlap	N(Number of nodes)	L	S	J=L+S	Binding energy/MeV
$\left\langle {{{}^{{\rm{14}}}{\rm{N}}}} / {{^{{\rm{10}}}{\rm{B}} + \alpha }} \right\rangle $	2	2	0	2	11.612
	1	4	0	4

Table 2.Cluster quantum numbers for the overlaps used in our calculations.

Figure 6.(color online) Variation ofχ²/Nwith extractedSA₂atE_lab= 21, 24.5, 38.1, 42.0, 46.0, and 50.0 MeV.

Figure 7.(color online) Variation ofχ²/Nwith extractedSA₄atE_lab= 21, 24.5, 38.1, 42.0, 46.0, and 50.0 MeV.

As can be seen fromFigs. 3and4andTable 1, the values of dσ/dσ_RUat the largest angles (dσ/dσ_RU~0.2) and the spectroscopic amplitudes are practically independent of the beam energy. The same result was obtained in Ref. [2].

The contribution of the α-cluster in forming the cross sections at backward angles atE= 46.0 MeV, measured by Takaiet al.[2], was analyzed within the framework of the DWBA method by code LOLA [15]. Their and our analyses are compatible at the forward angles region, but there is a significant difference at the backward angles, as shown inFig. 8. However, in Ref. [2], they did not presentχ²/Nwhen comparing both theoretical calculations. However, it is clearly shown inFig. 8that the current analysis is much better. Our analysis at 46.0 MeV could reproduce the experimental data well, especially at angles > 160°, whereas in the previous analysis the data and the calculations are completely out of phase at large angles (> 160°). It is noteworthy that the extractedSA₄atE= 46.0 MeV from the current work (0.76) is so close to the usedSA₄(0.806) in Ref. [2].

Figure 8.(color online) Comparison between the experimental angular distribution for¹⁴N+¹⁰B elastic transfer at large angles region (black circles) and the current DWBA calculations (solid red curves). The dashed black curve displays the results of Ref. [2].

As mentioned before, the investigation of the¹⁰B(¹⁴N,¹⁰B)¹⁴N α-cluster transfer process toL=2 andL=4 orbitals individually leads to an extractedSA₄value of “0.81 ± 0.12”, which is close to the theoretical value calculated by Kurath [10], whereas the extractedSA₂exceeds the theoretical value. Therefore, further calculations were performed to study the¹⁰B(¹⁴N,¹⁰B)¹⁴N α-cluster transfer process by taking into consideration the transfer toL=2 andL=4 orbitals simultaneously. The potential parameters listed inTable 1are used in the theoretical calculations. The optimalSA₂andSA₄values were searched using the SFRESCO search code [11]. During the search process,SA₂andSA₄were allowed to be changed freely in the range 0.05–0.7 and 0.6–0.95, respectively, till the best agreement between experimental data and calculations was reached. The starting parameters forSA₂andSA₄are 0.1 and 0.83, respectively; these values are adopted from the work of Kurath [10]. The optimal extracted values forSA₂andSA_4,as well as the χ²/Nvalues, at different concerned energies using this technique are listed inTable 3. The comparison between the experimental data and the theoretical calculations using this method are shown inFigs. 9and10. Although in an elastic transfer process the entrance and exit channels are identical and consequently the sameSAshould be taken for both channels, in this case the value ofSA₂was used as the spectroscopic amplitude for the entrance channel and the value ofSA₄was used as the spectroscopic amplitude for the exit channel. Thus, the experimental differential cross sections ${\left( {\dfrac{{{\rm d}\sigma }}{{{\rm d}\Omega }}} \right)_{\exp }}$ are assumed to be described as a coherent sum:

Figure 9.Comparison between the experimental data (black circles) and DWBA calculations including the¹⁰B(¹⁴N,¹⁰B)¹⁴N elastic transfer process toL=2 andL=4 orbitals simultaneously at energies 21.0, 24.5, and 38.1 MeV.

Figure 10.Same asFig. 9but at energies 42.0, 46.0, and 50.0 MeV.

As mentioned above, in the work of Takaiet al.[2], the spectroscopic amplitude close to our value was used to reproduce the experimental angular distributions for¹⁰B(¹⁴N,¹⁰B)¹⁴N elastic transfer at all the considered energies. Nevertheless, in the work of Motoboyashi [8], the extractedSAwas 1.3. This discrepancy between the spectroscopic amplitudes predicted by theory and those deduced by a DWBA analysis was reported early by many authors (see, for example, [16,17]) and points to the need for further experimental as well as theoretical exploration of scattering the 1p-shell nuclei.

Figure 11shows the energy dependence of the found by us potentials, which are listed inTable 1. It is seen that a linear function is a fair approximation for the energy dependence for the depths of the real and imaginary parts of the optical potentials in the energy range from 20 to 100 MeV for the¹⁰B+¹⁴N system. With increasing energy, the depth of the real potential decreases and can be approximated by the formulaV₀= 107.2-0.46E,and the depth of the imaginary potential increases and is well described by the formulaW₀= 8.19+0.268E.

Figure 11.(color online) Energy dependence onV₀andW₀for the¹⁰B(¹⁴N,¹⁴N)¹⁰B system.

An important and useful characteristic of optical potentials is the volume integrals overV(r) andW(r),

whereA_pandA_tdenote the masses of projectile (p) and target nucleus (t), respectively.

The volume integrals corresponding to the found potentials, as well as the depths, smoothly vary depending on the energy within 200–300 MeV fm³forJ_Vand 70–210 MeV fm³forJ_W. This is consistent with similar dependencies known from the scattering ofp+¹²С,³He +¹²С, α +¹²С, and¹²С +¹²С [18-20]. Our potentials are physically reasonable, as their volume integrals (J_V) are close to that calculated by microscopic models [18].

The energy dependence of the total reaction cross sectionsσ_Robtained in this work is presented inFig. 12. It can be seen that the rapid growth of cross sections at energies in the range of 20–40 MeV is replaced by a slower one atЕgreater than 50 MeV.

Figure 12.(color online) Energy dependence of the extracted total reaction cross sections for the¹⁰B(¹⁴N,¹⁴N)¹⁰B elastic scattering from the current work.

It is well known that the total reaction cross section calculated in the framework of the optical model is practically independent of the details of the real part of the optical potential and is mainly determined by the parameters of its imaginary part, the depth of which, as can be seen fromTable 1, increases from 14 to 33 MeV with increasing energy up to 93 MeV. The calculated energy dependence of the total cross sections is adequately approximated by the expression:

Angular distributions for¹⁴N+¹⁰B elastic scattering were investigated at energies ofE_lab(¹⁴N) = 21.0 and 24.5 MeV. The measured data obtained at these energies, which are closer to the Coulomb barrier, showed a smooth, structureless decrease in the cross sections in the region of the angles of the forward hemisphere with a slight increase at large angles. Further, data at higher energies 38.1–50.0 MeV showed a Fraunhofer diffraction pattern at forward angles with a remarkable rise in cross sections at backward angles > 90°. This increase is successfully described in terms of the DWBA incorporated in the Coupled Reaction Channels program (FRESCO) by taking into account the contribution ofα-cluster transfer between¹⁴N and¹⁰B. We have extracted the spectroscopic amplitude for the configuration¹⁴N→α+¹⁰B by taking into consideration the two possibilities of transfer toL=2 andL=4 orbitals individually. The extractedSA₂andSA₄values are 0.58±0.10 and 0.81±0.12, respectively. The extractedSA₂value overestimates the known theoretical value, whereas theSA₄value is close to the previously reported one (SA= 0.83). The extracted spectroscopic amplitude for the configuration of¹⁴N as anα-particle coupled to a¹⁰B core suggests that the elastic transfer process is a strong component in the total elastic scattering cross sections. The investigation of the¹⁰B(¹⁴N,¹⁰B)¹⁴N α-cluster transfer process by taking into consideration the transfer toL=2 andL=4 orbitals simultaneously showed that the average extracted values forSA₂andSA₄are 0.5±0.17 and 0.74±0.08, respectively, which agree well with the calculations for transfer process toL=2 andL=4 orbitals individually.

It is shown that the obtained values of the depths of the real and imaginary potentials linearly depend on energy in the energy range from 20 to 100 MeV.

A. A. Ibraheem extends his appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through the research groups program under grant number G.R.P-56-41.