Towards Understanding Incomplete Fusion Reactions at Low Beam Energies: Modified Sum Rule Model

Abstract

We investigated the enhanced production of nuclei formed via incomplete fusion (ICF) reactions near and above the Coulomb barrier energies (5–8 MeV/A). The cross-sections of the evaporation residues formed in the reactions—${}^{11}$B+${}^{124}$Sn, ${}^{10}$B+${}^{124}$Sn and ${}^{11}$B+${}^{122}$Sn—were measured using off-line gamma-ray spectrometry. The sum rule model (SRM) by Wilczyński et al. predicted the cross-section values too low compared to our experimental results. In earlier studies, the same model has been very successful in explaining ICF reactions at high beam energies (>10 MeV/A). We, therefore, modified the SRM, specifically incorporating the energy dependence in the definition of critical angular momentum ${\ell }_{cr}$. The resulting modified SRM gave an improved theoretical estimate for the reactions we studied.

1. Introduction

Nuclear reactions can be categorized by the impact parameters of the projectile [1]. At low beam energies above the Coulomb barrier, complete fusion (CF) dominates. At high beam energies, non-CF processes, such as incomplete fusion (ICF) and direct reactions, become important. Initial investigations into this phenomenon were carried out by Britt and Quinton [2]. In the massive transfer reactions studied by Kaufmann and Wolfgang [3], fast-moving projectile-like fragments (PLF) were observed in the forward direction. In ICF reactions, a part of the projectile fuses with the target, and the remnant flies off in the forward direction with roughly the projectile’s velocity [4].

Several models have been proposed to explain the observed ICF cross-sections and the underlying reaction mechanisms. Some of these models are breakup fusion model [5,6,7], the sum rule model (SRM) [8], the exciton model [9], the hotspot model [10], the promptly emitted particle model [11], the overlap model [12,13,14,15], and multi-step direct reaction theory [16]. Recently, there has been an attempt to understand ICF using a 3D classical stochastic projectile breakup model [17,18]. An earlier study [19] identified that the angular distribution of the residues peaks at a significantly larger angle for ICF than for CF. According to Gerschel et al. [20], the localized angular momentum of the entrance channel depends on the shape of the target nucleus, and it is smaller for spherical targets than deformed targets. The mass asymmetry of the entrance channel also plays a vital role in the breakup of the projectile. The ICF fraction increases with mass asymmetry [21,22] for the same projectile velocity. Moreover, there is an enhancement in the breakup probability with increasing beam energy for $\alpha$-cluster nuclei. The incomplete fusion fraction (relative strength of ICF and CF) was found to be independent of the target charge [23,24]. However, there are conflicting results where the fraction was found proportional to the target charge [25,26,27]. A common inference is that some kind of peripheral collision governs the ICF process. This property can be a powerful spectroscopic tool [28,29,30,31,32] to populate the high-spin states of neutron-rich nuclei close to the $\beta$-stability line, which is a region that is difficult to reach through conventional heavy-ion fusion reactions.

In the present work, we have studied the mechanism of CF and ICF processes in the beam energy range of 5–8 MeV/A (near and above the Coulomb barrier) for the following reactions:${}^{11}$B+${}^{124}$Sn, ${}^{10}$B+${}^{124}$Sn, and ${}^{11}$B+${}^{122}$Sn. We have carried out the off-line measurement of cross-sections for the evaporation residues with half-lives ranging from a few minutes to days.

We initially attempted to understand the experimental cross-sections with theoretical calculations based on the original sum rule model (SRM) [8] but could not find a good agreement. The major discrepancy was the low estimate of cross-sections for the ICF channels corresponding to $\alpha$ and ${}^8$Be breakup, as also pointed out in earlier studies [33,34]. Moreover, contradictory observations were reported regarding the onset of ICF channels. According to SRM, the ICF starts to occur at an angular momentum of $\ell \ge {\ell }_{cr}$ (critical angular momentum for CF). This has been found to be consistent with the multiplicity measurements [4,8,20] at high beam energies. In several studies [19,33,34,35,36,37,38] at beam energies just above the Coulomb barrier, the maximum entrance channel’s angular momentum (${\ell }_{max}$) is quite low. On the other hand, ${\ell }_{cr}$, that is independent of beam energy in the original SRM, remains unaffected. Hence, SRM underestimates the observed ICF cross-sections because of the narrow ℓ-window (${\ell }_{cr}\le \ell \le {\ell }_{max}$). In our present study, we have introduced some modifications to SRM by redefining ${\ell }_{cr}$ to be beam-energy-dependent and reproduced our experimental results in a much better way than possible otherwise. The basic formalism of the model remains the same. Moreover, we used the same set of input parameter values for all three reactions and obtained promising results.

2. Experiments

The experiments were performed at the 14UD Pelletron facility of the Tata Institute of Fundamental Research, Mumbai, India. We irradiated the enriched ${}^{122,124}$Sn foils using ${}^{10,11}$B beams of energy values of 56–78 MeV. The targets with thicknesses in the range of 1.5–2.0 mg/cm${}^2$ were prepared with minimum non-uniformity (less than 2%) by the rolling technique. During each irradiation run, two targets were used together in a target-catcher assembly. The first target facing the beam served as an energy degrader for the second target. In effect, two targets were irradiated at the same time at two different energies to ensure the best use of the beam time. We monitored the beam current continuously and maintained it within 20–40 nA. In particular, the beam current was kept constant within 2–3% variation during the last hour of irradiation to avoid any significant systematic error for the short-lived nuclei. After each irradiation, the data were collected off-line using a coaxial high-purity Germanium (HPGe) ORTEC detector. Figure 1 shows the typical gamma-ray spectra for the three reactions. A standard ${}^{152}$Eu radioactive source was utilized to determine the energy calibration and absolute efficiency of the detector. The estimated error in the measured efficiency was less than 5%.

Table 1. Evaporation residues formed in the reactions: ${}^{11}$B+${}^{124}$Sn, ${}^{10}$B+${}^{124}$Sn, and ${}^{11}$B+${}^{122}$Sn. The abbreviations “m”, “h”, and “d” stand for minute, hour, and day, respectively.

Evaporation Residues	Decay Mode (%)	${\mathit{T}}_{1/2}$	${}^{11}$B${+}^{124}$Sn	${}^{10}$B${+}^{124}$Sn	${}^{11}$B${+}^{122}$Sn	E${}_{\mathit{\gamma }}$ (keV)	${\mathit{a}}_{\mathit{\gamma }}$(%)
${}^{130}$Cs	EC 98.4	29.21 m	5n	-	-	536.1	3.8
${}^{129}$Cs	EC 100	32.06 h	6n	5n	4n	371.9	30.6
${}^{128}$Cs	EC 100	3.66 m	-	6n	5n	442.9	26.8
${}^{127}$Cs	${\beta }^{+}$ 100	6.25 h	-	7n	6n	124.7	11.37
${}^{127}$Xe	EC 100	36.4 d	-	$p6n$	$p5n$	202.9	68.7
${}^{128}$I	${\beta }^{-}$ 93.10	25 m	$\alpha 3n$	$\alpha 2n$	$\alpha n$	442.9	12.62
${}^{126}$I	${\beta }^{-}$ 47.3	12.93 d	$\alpha 5n$	$\alpha 4n$	$\alpha 3n$	388.6	35.6
	EC 52.7	12.93 d	$\alpha 5n$	$\alpha 4n$	$\alpha 3n$	666.3	32.6
${}^{124}$I	EC 100	4.176 d	-	$\alpha 6n$	$\alpha 5n$	602.7	62.9
${}^{123}$I	EC 100	13.22 h	-	-	$\alpha 6n$	158.9	83.3
${}^{126}$Sb	${\beta }^{-}$ 86.4	19.15 m	$\alpha 2p3n$	$\alpha 2p2n$	$\alpha 2pn$	666.1	86
	${\beta }^{-}$ 100	12.35 d	$\alpha 2p3n$	$\alpha 2p2n$	$\alpha 2pn$	666.1	99.6
${}^{122}$Sb	${\beta }^{-}$ 97.59	2.734 d	-	-	${}^8$Be$3n$	564.2	70.67

presents the $\beta$-decaying evaporation residues with lifetimes greater than 3 min, which was the minimum time required to start the off-line measurement after the irradiation. Other specifications, e.g., decay mode, half-life (T${}_{1/2}$: min to days), gamma-ray energy (E${}_{\gamma }$) and branching intensity (abundance: $a_{\gamma }$) [39], are also listed in the table. We determined the absolute intensity of the characteristic gamma transitions of daughter nuclei by finding the photopeak area with the gf3 program of the RADWARE package [40]. The data were analyzed using the prescription in our earlier report [37]. In that report, we also described our method of double-channel analysis for the cases of two parents $\beta$-decaying to the same daughter nuclei, e.g., the decays of ${}^{128}$Cs and ${}^{128}$I to ${}^{128}$Xe.

3. Results and Discussion

3.1. Disentangling CF and ICF

Complete fusion (CF) is expected to be the dominant process at the beam energies studied here. We consistently observed high cross-section values for the Cesium isotopes formed through CF via the emission of several neutrons. The statistical model code PACE4 [41] based on the Bass model [42], which describes the decay of an equilibrated compound nucleus following the Monte Carlo procedure, can theoretically estimate the cross-section of evaporation residues for the CF process.

We first adjusted the value of the level density parameter ‘a’ in PACE4 as it is the most significant among the other parameters. After examining how well PACE4 values matched with the experimental results on the total observed cross-section for CF reactions at various beam energies, we fixed $a=A/8$, where A is the mass number of the compound nucleus. We found $\sum {\sigma }_{PACE4}\left(\mathtt{CF}\right)\approx \sum {\sigma }_{Expt.}\left(\mathtt{CF}\right)\approx \sum {\sigma }_{Expt.}(\mathit{xn}$-channel). Other parameters in PACE4 were kept at their default values. For the evaporation residues formed through the emission of $\alpha$ and ${}^8$Be, the measured cross-section values (${\sigma }_{Expt.}(\mathtt{CF}+\mathtt{ICF})$) were too high (at least an order of magnitude) compared to the PACE4 prediction (${\sigma }_{PACE4}\left(\mathtt{CF}\right)$). The implication is that for these ejectiles, the CF contribution was negligibly small. Thus, PACE4 not only estimated the total CF cross-sections reasonably well, it clearly identified the presence of ICF reaction channels whenever the calculated values were too low compared to the experimental results. We finally focused our attention on Cesium isotopes ($\mathit{xn}$-channel) to study CF and iodine isotopes ($\alpha$-channel) and antimony isotopes (${}^8$Be-channel) to understand the ICF reaction mechanism.

3.2. Excitation Functions

Figure 2, Figure 3 and Figure 4 present the excitation function plots (cross-section vs. energy) showing both the experimental and PACE4 results for the studied reactions: ${}^{11}$B+${}^{124}$Sn, ${}^{10}$B+${}^{124}$Sn, and ${}^{11}$B+${}^{122}$Sn, respectively. As mentioned earlier, the CF process predominantly produced Cesium isotopes, whereas ICF produced isotopes of iodine ($\alpha$-channel) and antimony (${}^8$Be-channel). In addition, PACE4 predicted some unobserved stable nuclei, such as ${}^{128}$Xe and ${}^{127}$I. Notably, there remained some mismatch between the experimental results and PACE4 values in these figures for some Cesium isotopes individually. Interestingly, though, the values agreed reasonably well regarding their total production (the sum of all the observed Cs isotopes). In the following sections, we discuss the excitation function plots for each reaction separately.

3.2.1. Reaction: ${}^{11}$B+${}^{124}$Sn

In Figure 2a, the observed CF channels are shown. The PACE4 calculation reasonably reproduced the cross-sections for ${}^{129}$Cs (6n), while there was a mismatch for ${}^{130}$Cs (5n). However, the general behavior of the experimental cross-sections followed the trend of the PACE4 results. Moreover, the total cross-section values for ${}^{129}$Cs (6n) and ${}^{130}$Cs (5n) were reasonably close to the PACE4 results. Figure 2b shows the cross-sections for the iodine and antimony isotopes. For the two iodine isotopes (${}^{128}$I ($\alpha$3n) and ${}^{126}$I ($\alpha$5n)), where the experimental values are roughly an order of magnitude higher than the PACE4 values (shown by dashed curves), the values observed indicated the breakup of $\alpha$ particle at the pre-equilibrium stage (ICF). The case for ${}^{126}$Sb produced through ${}^8$Be breakup was similar and was negligibly predicted by the PACE4 calculation. The production of ${}^{126}$Sb could also be due to ICF emitting an $\alpha$ particle followed by the emission of 2p3n.

3.2.2. Reaction: ${}^{10}$B+${}^{124}$Sn

Figure 3a presents the observed cross-sections for ${}^{127,128,129}$Cs (7n, 6n, and 5n).There was an increasing trend with beam energy for ${}^{127}$Cs, agreeing well with the PACE4 calculation. For both ${}^{128}$Cs and ${}^{129}$Cs, although there was a significant difference between the PACE4 and experimental results, the total cross-sections were close. The reaction produced three iodine isotopes, ${}^{124,126,128}$I, with high cross-sections compared to PACE4 results, suggesting the ICF process with the $\alpha$-breakup and emission of neutrons, denoted as $\alpha xn$ with x = 6, 4, and 2, respectively. As mentioned earlier, the contribution of CF to iodine production was very small, as seen by the PACE4 results. The production of ${}^{126}$Sb demands the emission of 8 particles (mass ∼1u each), which could be either two $\alpha$ particles or ${}^8$Be, but we recognized them as the $\alpha 2p2n$ channel. The reason was simply the unlikelihood of the process emitting solely ${}^8$Be at the pre-equilibrium stage without any other particle at the later stage of evaporation residue. Again, for both ${}^{126}$Sb and ${}^{127}$Xe, with cross-sections much higher than the PACE4 values, the major contribution came from the ICF process, as evident from Figure 3b. The nucleus ${}^{127}$Xe ($p6n$), which was produced significantly in our data and was analyzed using the precursor decay analysis method [43], did not interest us in the context of the present study.

3.2.3. Reaction: ${}^{11}$B+${}^{122}$Sn

Figure 4a presents the isotopes ${}^{127,128,129}$Cs produced through the $6n$, $5n$, and $4n$ channels via the CF process, respectively. The overall experimental trend for ${}^{128}$Cs followed the PACE4 results. The experimental cross-sections for ${}^{127}$Cs agreed well with the calculation and showed an increasing trend with beam energy. However, the cross-sections for ${}^{129}$Cs were somewhat higher than PACE4 values, but their behavior matched. Similar to the other two reactions described above, the total cross-sections of all the observed Cesium isotopes were reasonably close to the PACE4 values. We observed many iodine isotopes (${}^{123,124,126,128}$I), as shown in Figure 4b, produced through $\alpha xn$ channels (ICF) with cross-sections much higher than the corresponding PACE4 values. The case for ${}^{122}$Sb, which was produced through ICF process involving ${}^8$Be breakup, was similar. The measured cross-sections for ${}^{127}$Xe ($p5n$) were quite high but irrelevant for the present study.

4. Theoretical Discussion

4.1. Sum Rule Model (SRM)

The sum rule model (SRM), proposed by Wilczyński et al. [8], is an attempt to unify CF and ICF into a single framework. The model relies on the generalized concept of critical angular momentum [44] and the localization of successive ℓ-windows above the critical angular momentum for CF. The absolute cross-section for the $i^{\mathrm{th}}$ reaction channel is given as

$$\sigma \left(i\right)=\pi ƛ{}^2\frac{{\sum }_{\ell =0}^{{\ell }_{\mathit{max}}}(2\ell +1)T_{\ell }\left(i\right)p\left(i\right)}{{\sum }_i{T}_{\ell }\left(i\right)p\left(i\right)},$$

(1)

where $ƛ$ is the reduced deBroglie wavelength of the entrance channel. We followed the earlier prescription for calculating the maximum angular momentum (${\ell }_{max}$) [45,46]. The denominator in Equation (1) normalizes the reaction probabilities. The probability $p\left(i\right)$ for the $i^{\mathrm{th}}$ reaction channel (including CF) was described as proportional to the ground state Q value ($Q_{gg}\left(i\right)$) [11], written as

$$p\left(i\right)=exp\left[\left\{\frac{Q_{\mathit{gg}}\left(i\right)-Q_c\left(i\right)}{T}\right\}\right],$$

(2)

where T is the effective temperature to be treated as a parameter. Both the CF and ICF processes involve the formation of di-nuclear systems with the charges $Z_1^f$ (ejectile), $Z_2^f$ (target+captured fragment), $Z_1^{in}$ (projectile), and $Z_2^{in}$ (target) in the final and initial states. The change in the Coulomb energy ($Q_c\left(i\right)$) due to transfer of charge is given as

$$Q_c=q_c\left(Z_1^f{Z}_2^f-Z_1^{\mathit{in}}{Z}_2^{\mathit{in}}\right)e^2,$$

(3)

where $q_c$ is a parameter. The transmission coefficient $T_{\ell }\left(i\right)$ for the i^th reaction channel is given by

$$T_{\ell }\left(i\right)={\left[1+exp\left\{\frac{\ell -{\ell }_{\mathit{lim}}\left(i\right)}{{\Delta }_{\ell }}\right\}\right]}^{-1},$$

(4)

where ${\Delta }_{\ell }$ is the diffuseness in the $T_{\ell }$ distribution treated as a parameter. The probability distribution was normalized using a factor ($\equiv N_{\ell }$ on the y-axis of Figure 5), the same as in the denominator of Equation (1), by considering all possible reaction channels.

The limiting angular momentum [8] is defined as

$${\ell }_{\mathit{lim}}\left(i\right)=\frac{\mathtt{mass}\mathtt{of}\mathtt{projectile}}{\mathtt{mass}\mathtt{of}\mathtt{captured}\mathtt{fragment}}\times {\ell }_{\mathit{cr}}\left(\mathtt{target}+\mathtt{captured}\mathtt{fragment}\right),$$

(5)

In the original SRM [8], the critical angular momentum (${\ell }_{cr}$) was obtained by balancing forces [47], given as

$${\left({\ell }_{\mathit{cr}}+\frac{1}{2}\right)}^2=\frac{\mu {\left(C_1+C_2\right)}^3}{{\hslash }^2}\left[4\pi \gamma \frac{C_1{C}_2}{C_1+C_2}-\frac{Z_1{Z}_2{e}^2}{{(C_1+C_2)}^2}\right],$$

(6)

where $C_i$ are the half-density radii [48]. The other notations—$Z_i$ for the charge of the i^th di-nuclear system, $\mu$ for the reduced mass, and $\gamma$ for the surface tension coefficient—are the same as in the original SRM [8]. The reaction cross-section for the reaction ${}^{11}$B+${}^{122}$Sn, as an example, can be written as

$$\sigma \left[{}^{122}\mathrm{Sn}\left({}^{11}\mathrm{B},{}^{A}Z\right)\right]=\sum _x\sigma {\left({}^{11}\mathrm{B}{,}^{A}Zxn\right)}^{133-A-x}(55-Z).$$

(7)

The notation ${}^{A}Zxn$ represents sub-channels, where the emission of an ejectile ${}^{A}Z$ is accompanied by x neutrons presumably emitted at a later stage after the formation of the equilibrated system, resulting in the evaporation residues ${}^{133-A-x}(55-Z)$.

Earlier works in the literature demonstrated the success of SRM in reproducing the cross-section for ICF channels in ${}^{12}$C+${}^{160}$Gd [49] and ${}^{14}$N+${}^{159}$Tb [8] reactions at high beam energies. However, it failed at low beam energies [33,34,37,38], as in our present work.

Figure 5 shows the probability distribution for the relevant reaction channels formed through ${}^{11}$B+${}^{124}$Sn, as an example. In calculating the denominator of Equation (1), we explicitly considered both CF and ICF channels with the emission of ${}^{4,6}$He, ${}^{6,7}$Li, ${}^{7,8,9,10}$Be, and ${}^{10,11}$B. To avoid any singularity encountered in Equation (5) due to the inclusion of (${}^{11}$B and ${}^{11}$B${}^{{}^{\prime }}$) channels, the corresponding transmission coefficient in Equation (4) was made unity. The CF channel has a broad distribution (Figure 5) because of the largest value of ($Q_{gg}-Q_c$). At high values of ℓ—above the critical angular momentum for CF—the ICF process starts. At first, the $\alpha$-channel and later the ${}^8$Be-channel appear in the successive ℓ-windows. Each distribution peaks roughly at its limiting angular momentum.

It is noticeable from Figure 5a that the maximum input angular momentum (${\ell }_{max}$ at 64 MeV beam energy) could not cover an appreciable portion of the distribution for both $\alpha$- and ${}^8$Be-channels. The effect was greater for the ${}^8$Be-channel. As a result, SRM underestimated the ICF cross-sections, as seen in Figure 6. This is true at other beam energies as well. Interestingly, the discrepancy between the two curves (Figure 6)—one predicted by SRM, and the other the experimental results for the $\alpha$-channel—is rather small for the ${}^{10}$B+${}^{124}$Sn reaction compared to the other two reactions, ${}^{11}$B+${}^{124}$Sn and ${}^{11}$B+${}^{122}$Sn. Such an observation could be attributed to a lower value of $\alpha$-breakup threshold for ${}^{10}$B (4.5 MeV) than for ${}^{11}$B (8.46 MeV), resulting in an early onset of the ICF process. Therefore, the ICF starts comparatively at the lower values of ℓ for ${}^{10}$B+${}^{124}$Sn in comparison to the other two reactions. A similar observation was also presented by Mukherjee et al. [50].

4.2. Modified Sum Rule Model-1 (MSRM1)

A study by Tripathi et al. [51] reported the ICF cross-sections for the reaction ${}^{19}$F+${}^{66}$Zn at low beam energies. They observed a significant contribution from the ICF process at low ℓ-values—less than ${\ell }_{cr}$ for CF—with overlapping channels in ℓ-space. For theoretical understanding, they modified the transmission coefficient in SRM by an exponential factor,

$$T_{\ell }\left(i\right)={\left[1+exp\left(\frac{\ell -{\ell }_{\mathit{lim}}\left(i\right)}{{\Delta }_{\ell }}\right)\right]}^{-1}{\left[\exp (n\ell f_t)\right]}^{-1},$$

(8)

where n is the number of nucleons transferred to the target. The remaining mathematical formulation and parameter values were the same as used by Wilczyński et al. [8] in SRM. The exponential factor in Equation (8) describes the dependence of the transmission coefficient on the number of transferred nucleons. Therefore, with increasing $f_t$, the value of $T_{\ell }\left(i\right)$ decreases steeply for large n, resulting in a rapid increase in the ICF cross-section with increasing mass of the ejectile. Moreover, the low partial-wave contributions to ICF were automatically incorporated. The experimental results on ICF were well reproduced by Tripathi et al. [51]. When we applied this formalism in our study, we obtained a value of ${\ell }_{max}$ covering an appreciable amount of $\alpha$ (${}^4$He) distribution (Figure 5b) compared to the original SRM (Figure 5a). The improvement was due to the suppression of CF distribution, consequently improving the cross-section for the $\alpha$-channel (Figure 6).

4.3. Modified Sum Rule Model-2 (MSRM2)

In our earlier work [37], we proposed a model in an attempt to modify SRM [8]. In the present work, we have further refined the model and named it MSRM2, keeping the basic formalism of the original SRM intact. Considering the classical picture of collision between a projectile and target [52], we redefined ${\ell }_{cr}$ as the angular momentum corresponding to the distance of closest approach (classical turning point). The energy expression for a two-body problem can be written in plane polar coordinates ($r,\theta$) as

$$\frac{1}{2}\mu {\dot{r}}^2+\frac{1}{2}\mu r^2{\dot{\theta }}^2+V\left(r\right)=E,$$

(9)

where $\mu$ is the reduced mass, E is the total energy in center of the mass frame, and $V\left(r\right)$ is the sum of the Coulomb and nuclear potential of the system at the classical turning point corresponding to $r=r_t$, $\dot{r}=0$, resulting in

$$\frac{1}{2}\mu r_t^2{\dot{\theta }}^2+V=E.$$

(10)

Putting ${\ell }_{cr}$ =$\mu r_t^2\dot{\theta }$ in Equation (10) and using the approximation ${\ell }_{cr}^2\gg {\ell }_{cr}+1/4$, we can write

$${\left({\ell }_{\mathit{cr}}+\frac{1}{2}\right)}^2=2\mu r_t^3\left(\frac{E}{r_t}-\frac{V}{r_t}\right),$$

(11)

where $r_t$ is the sum of the half-density radii $C_1$ and $C_2$ of two nuclei, parametrized using $f_c$ and $r_0$ as

$$C_i=f_c{r}_o{A}_i^{\frac{1}{3}}.$$

(12)

By substituting the expression for V (the balance of attractive nuclear and repulsive Coulomb potentials) from Equation (6), we can write Equation (11) as

$${\left({\ell }_{\mathit{cr}}+\frac{1}{2}\right)}^2=\frac{\mu {\left(C_1+C_2\right)}^3}{{\hslash }^2}\left[4\pi \gamma \frac{C_1{C}_2}{C_1+C_2}-\frac{Z_1{Z}_2{e}^2}{{(C_1+C_2)}^2}+\frac{E}{(C_1+C_2)}\right].$$

(13)

In Equation (13), the last term in the square bracket, introduced by us using the classical collision, gave an energy dependence to the critical angular momentum. Incorporating such a term turned out to be significant because it brought in an early onset of ICF reactions at low beam energies. The other two terms, the nuclear part (the first term) and the Coulomb potential (the second term), were the same as in the original SRM. In a different context, Glas and Mosel [53] discussed the energy dependence of ${\ell }_{cr}$.

Figure 5c shows the probability distribution of the relevant channels, including CF, $\alpha$, ${}^8$Be, ${}^{10}$B, and ${}^{11}$B, in the framework of MSRM2 for the reaction ${}^{11}$B+${}^{124}$Sn as an example. The values of the parameters used in our calculation are listed in

Table 2. Parameter values used in the present theoretical calculation using SRM, MSRM1, and MSRM2.

Parameter	SRM	MSRM1	MSRM2
$\gamma$	0.92 MeVfm${}^{-2}$	0.92 MeVfm${}^{-2}$	0.92 MeVfm${}^{-2}$
T	3 MeV	3 MeV	3.5 MeV
$q_c$	0.06 fm${}^{-1}$	0.06 fm${}^{-1}$	0.05 fm${}^{-1}$
${\Delta }_{\ell }$	2 ℏ	2 ℏ	1.4 ℏ
$f_t$	-	0.021	-
$f_c$	-	-	0.71
$r_0$	1.05 fm	1.05 fm	1.08 fm

. The ICF channels have gained height with the corresponding reduction in the CF distribution, in comparison with the original SRM, as seen in Figure 5a. Furthermore, in Figure 5c, the $\alpha$ (${}^4$He) distribution has a tail on the low-ℓ side (sensitive to $q_c$ values), enhancing the cross-section. Most importantly, the ${\ell }_{max}$ now covers a significant portion of the $\alpha$ distribution, bringing the MSRM2 values (dotted curves in Figure 6 closer to the experimental results.

The top portion in each part of Figure 6 presents the total CF cross-sections as a function of beam energy for the three reactions. Since the SRM estimated the total CF cross-section values and not the individual evaporation residues, we needed only ${\sigma }_{\mathtt{total}}\left(\mathtt{CF}\right)$ from our experimental measurement for comparison. Thus, the discrepancies noted earlier (Section 3) between our experimental and PACE4 results for some evaporation residues were rather unimportant. The figure clearly demonstrates the improvement we achieved through MSRM2 over SRM and MSRM1 for both CF and ICF processes in all three parts of Figure 6. As expected, the experimental results were mostly low in comparison to theoretical values for all three cases because of some unobserved channels in our measurements. Focusing on the ICF process (lower portion in each part of Figure 6), MSRM2 gives a kind of upper limit. The best match was found for the case of ${}^{11}$B+${}^{122}$Sn with many observed channels, including $\alpha n$, $\alpha 3n$, $\alpha 5n$, $\alpha 6n$, and $\alpha 2pn$, listed in Table 1. Furthermore, MSRM2 ensured the validity of our proposed classical picture of collision by considering ${\ell }_{cr}$ at the turning point.

4.4. 3D Stochastic Breakup Model

Recently, Diaz-Torres et al. [17,18] attempted to model the ICF mechanism based on a 3D classical picture of stochastic breakup. It turned out from the measurements and continuum-discretized coupled channel (CDCC) calculations [54] that the integral of the breakup probability is an exponential function of the distance of the closest approach. The exponent contains a parameter $\alpha$ (in fm${}^{-1}$) that can be varied.

We calculated the cross-sections for the $\alpha$-channel through the ICF process using the FORTRAN program PLATYPUS [55] for the ${}^{11}$B+${}^{124}$Sn, ${}^{10}$B+${}^{124}$Sn, and ${}^{11}$B+${}^{122}$Sn reactions. Figure 7 compares the results of our calculations and experiments. The value of the parameter $\alpha$ in the exponent was chosen to be 0.922 fm${}^{-1}$. The discrepancy between the theoretical and experimental results decreases from part (a) through part (c). We attributed this discrepancy to the number of channels observed in each reaction. For instance, we could only observe two $\alpha$-channels, $\alpha$3n and $\alpha$5n, in the ${}^{11}$B+${}^{124}$Sn reaction (Figure 7a), while there were four such channels observed in ${}^{10}$B+${}^{124}$Sn (Figure 7b) and five in ${}^{11}$B+${}^{122}$Sn (Figure 7c).

5. Conclusions

We have studied complete (CF) and incomplete fusion (ICF) reaction cross-sections in the low-beam-energy range of 5–8 MeV/nucleon. The standard experimental technique of off-line $\gamma$-ray spectrometry was utilized. Among the three reactions studied here, we found a greater ICF contribution in ${}^{10}$B+${}^{124}$Sn and ${}^{11}$B+${}^{122}$Sn than in ${}^{11}$B+${}^{124}$Sn. The low $\alpha$-breakup threshold of ${}^{10}$B possibly played a role in enhancing the cross-section. The reaction ${}^{11}$B+${}^{122}$Sn was unique among the three cases in providing the maximum accuracy for the measurement of ICF cross-sections because it produced a large number of evaporation residues with lifetimes within our measurable range. Hence, ${}^{11}$B+${}^{122}$Sn turned out to be the best case for our experimental study.

We first tried to understand the experimental results in the framework of the well-known theoretical sum rule model (SRM) and found enormous discrepancies. The SRM always underestimated the cross-sections. To improve the theoretical description, we modified the SRM by redefining its critical angular momentum as the distance of closest approach using a classical picture, which brought in the early onset of the ICF process. We labelled our approach as the modified sum rule model-2 (MSRM2), which reproduced the experimental results for both CF and ICF reasonably well. However, the discrepancies existed because of some unobserved channels for which off-line measurements were lacking either due to short lifetimes or stable parent nuclei. Finally, the original SRM estimated very low cross-sections for the ICF process at the beam energy range of 5-8 MeV/nucleon, even for our benchmark reaction ${}^{11}$B+${}^{122}$Sn. The modified SRM (MSRM2) not only visualized the process classically, bringing in the beam energy dependence to the critical angular momentum, it improved the results profoundly.