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A recent comparison of the average fusion cross section, σF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\langle \sigma _\mathrm {F}\right\rangle $$\end{document}, for energies just above the Coulomb barrier for the 12-15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{12-15}$$\end{document}C + 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{12}$$\end{document}C systems found that the behaviour as a function of projectile neutron excess could not be satisfactorily explained by static barrier penetration model calculations and suggested that the neutron dynamics plays an important rôle. In this work we demonstrate that the (15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{15}$$\end{document}C,14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{14}$$\end{document}C) single neutron transfer has a significant influence on the above barrier 15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{15}$$\end{document}C + 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{12}$$\end{document}C total fusion, although not quite in the way expected since it leads to a reduction in the cross section, contrary to the trend in the measured σF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\langle \sigma _\mathrm {F}\right\rangle $$\end{document}. However, this result underlines the danger of ignoring the effect of neutron transfer reactions on fusion in systems involving neutron halo nuclei.

In a recent article deSouza et al. [1] make a systematic comparison of measurements [2,3] of the fusion excitation functions for the 12−15 C + 12 C systems at energies just above the respective Coulomb barriers. They find that the average fusion cross section, σ F , shows a more rapid increase with increasing projectile neutron number than can be accounted for by static barrier penetration models using potentials based on calculated nuclear matter densities. It is suggested that the failure of the static models adequately to describe the dependence of the average fusion cross sections on neutron excess could be interpreted as due to the importance of neutron dynamics. However, time-dependent Hartree-Fock model calculations also failed to reproduce the behaviour of the fusion cross sections. In this brief note we wish to point out that in the 15 C + 12 C system coupling to the ( 15 C, 14 C) single-neutron transfer reaction has a significant influence on the fusion cross section excitation function and that while this coupling does not explain the observed dependence of σ F on neutron excess-it significantly reduces the fusion cross section over the measured range of incident energiesa e-mail: nicholas.keeley@ncbj.gov.pl (corresponding author) its influence is sufficiently important that it must be included in any attempt at understanding the fusion of this neutronrich system. By contrast, the single-nucleon transfer reactions have a negligible effect on the fusion cross sections for the other systems.
The most important structural particularity of 15 C compared to the other carbon isotopes considered by deSouza et al. [1], 12 C, 13 C and 14 C, is the halo nature of its 1/2 + ground state. Not only is this state weakly bound, S n = 1.2 MeV, but its wave function is an almost pure 2s 1/2 neutron coupled to the 14 C core in its 0 + ground state. This suggests that, in addition to the expected significance of breakup effects, stripping of the valence neutron should also play an important rôle in reactions involving 15 C, since the tail of its wave function extends to a considerable distance beyond the 14 C core.
The weak binding of the valence neutron in 15 C is also reflected in the reaction Q value for one-neutron removal compared to the other carbon isotopes considered in Ref. [1]. The relevant Q values for single-nucleon transfers are listed in Table 1.
It can immediately be seen that single-nucleon transfers may be ruled out as a significant influence on the 12 C + 12 C system in the energy region under consideration since the reaction Q values are large and negative. Similar considerations apply to the other systems for both single-neutron addition and single-proton removal from the projectile. Of the other transfer processes listed in Table 1 the 12 C( 13 C, 12 C) 13 C "elastic transfer" (and, by extension, its "inelastic transfer" counterparts) and the 12 C( 14 C, 13 C) 13 C single-neutron removal reaction could possibly influence the fusion cross section at the energies under consideration here, with the 12 C( 15 C, 16 N) 11 B, 12 C( 14 C, 15 N) 11 B and 12 C( 13 C, 14 N) 11 B proton transfer reactions becoming progressively less likely, judging by the reaction Q values. In this work we are not concerned with attempting to fit the measured fusion excitation functions for the systems investigated by deSouza et al., we merely wish to establish the exceptional nature of the influence of the ( 15 C, 14 C) singleneutron transfer reaction on the fusion of the 15 C + 12 C system. Therefore, we only consider couplings to inelastic excitations of the target and projectile nuclei and single-nucleon transfer reactions. Following the kinematic considerations discussed in the preceding paragraph, in addition to 15 C + 12 C we limit our analysis to the 12 C + 13,14 C systems to provide benchmarks for the influence of single-nucleon transfer reactions on the fusion cross section.
Within these limits relatively full coupling schemes were included in calculations of the fusion excitation functions using the code fresco [5]. We begin by considering inelastic excitations of both target and projectile nuclei. In all three systems coupling to the 4.44-MeV 2 + excited state of the 12 C target was included, with the B(E2) taken from Raman et al. [6] and the nuclear deformation length, δ 2 = −1.40 fm, from Ref. [7]. For the 15 C + 12 C system, coupling to the 0.74-MeV 5/2 + bound excited state of 15 C was included using the B(E2) of Ref. [8], with the corresponding nuclear deformation length δ 2 = 0.52 fm derived from the B(E2) assuming the rotational model and a radius of 1.2×15 1/3 fm. In the 14 C + 12 C system coupling to the 7.01-MeV 2 + excited state of 14 C was included using the B(E2) of Raman et al. [6] and a nuclear deformation length δ 2 = −0.60 fm, taken from Ref. [10]. We have followed Ref. [10] in assuming a slight oblate deformation for 14 C, although this does not affect the results for the fusion cross section. In the 13 C + 12 C system couplings to the 3.09-MeV 1/2 + , 3.68-MeV 3/2 − and 3.85-MeV 5/2 + states were included. Although these are not considered to be collective states they were treated as such for the sake of simplicity. The respective B(E1), B(E2) and B(E3) values were taken from Ref. [8] and the corresponding nuclear deformation lengths from Ref. [9]. While the use of standard collective model form factors for these states is questionable, Peterson et al. [9] note that for 13 C(α,α ) inelastic scattering the results do not differ greatly from those using more realistic form factors and the inelastic scattering data, including that for the L = 1 transition to the 3.09-MeV 1/2 + level, are rather well reproduced. Thus, collective Table 2 Overlaps plus corresponding spectroscopic amplitudes (S) and shell model level (n j ) included in the full CRC calculations. The signs of the spectroscopic amplitudes are consistent with the phase convention used in fresco [12] model form factors should be adequate for our purposes. No mutual excitation was included in any of the three systems studied. Coulomb multipoles were of the standard form for a deformed charged sphere while nuclear multipoles were calculated by numerically deforming the radius of the diagonal potential and projecting by Gaussian quadrature onto the appropriate multipole.
The following single-neutron transfer couplings were also investigated: 12 C( 15 C, 14 C) 13 C, 12 C( 14 C, 13 C) 13 C, and 12 C( 13 C, 12 C) 13 C. The various overlaps, together with the spectroscopic amplitudes used are listed in Table 2.
Due to the special nature of the 13 C + 12 C system with "elastic transfer" and "inelastic transfer" of the single neutron between the two 12 C cores the various 13 C | 12 C + n overlaps were implemented as both "projectile-like" and "targetlike" and the same inelastic couplings were included in the "exit" partition as for the entrance. Inelastic couplings were also included in the 13 C + 13 C exit partition of the 14 C + 12 C system for both projectile-like and target-like 13 C nuclei.
The real parts of the input optical model potentials were calculated within the double-folding framework with the M3Y nucleon-nucleon effective interaction [16] using the code dfpot [17]. The required 12 C, 14 C and 15 C nuclear matter densities were taken from Dobrovolsky et al. [18], and the 13 C density from Ahmad et al. [19]. The imaginary parts of the optical potentials were of Woods-Saxon squared form, with depth W = 50 MeV, radius fm and diffuseness a W = 0.3 fm, effectively reproducing the incoming wave boundary condition [20]. In this model the total fusion cross section is defined as the sum of the absorption by the imaginary parts of the potentials in all channels and is thus the difference between the reaction cross section and the sum of the integrated cross sections for all channels included in the coupling scheme. While the total fusion calculated in this way is complete within the model space used since it sums over the absorption in all channels explicitly included in the coupling scheme, it should be noted that the measured total fusion cross section may contain contributions from other processes. The calculations are compared with the data in Fig. 1. The results of three calculations are plotted for each system: the bare, no coupling result (denoted by the dotted curves), calculations including coupling to inelastic excitations of the projectile and target only (denoted by the dashed curves) and calculations including couplings to both inelastic excitations and neutron transfers (the solid curves). In all three systems the inelastic excitations have a negligible effect on the fusion excitation function, as do the single-neutron transfer couplings in the 13 C + 12 C and 14 C + 12 C systems. In the 15 C + 12 C system the single-neutron transfer does have a visible effect, reducing the calculated fusion cross section, and while this is only of the order of 10% or so for the incident energies where there are data, the size of the effect is comparable with the differences discussed by deSouza et al. [1], so must be regarded as significant. A similar reduction of the total fusion cross section at energies just above the Coulomb barrier due to coupling to the ( 15 C, 14 C) single-neutron transfer reaction was also predicted for the 15 C + 208 Pb system [21]. Test calculations for the 14 C + 12 C and 13 C + 12 C systems additionally including the 12 C( 14 C, 15 N) 11 B and 12 C( 13 C, 14 N) 11 B single proton transfers respectively found a negligible influence of these reactions on the corresponding above barrier total fusion cross sections. A meaningful test of the effect of coupling to the 12 C( 15 C, 16 N) 11 B single proton transfer on the 15 C + 12 C total fusion cross section could not be performed due to a lack of available spectroscopic amplitudes for the 16 N | 15 C + p overlaps, although the results of the similar calculations for the 14 C + 12 C and 13 C + 12 C systems suggest that it does not play an important rôle.
The 13 C + 12 C fusion excitation function is reasonably well described for energies up to approximately E c.m. = 13 MeV; thereafter it is increasingly over-predicted by the calculations. The 14 C + 12 C fusion excitation function is well described over the whole range of available data, the slight over-prediction at the lowest E c.m. value being within the effect of different choices of nuclear matter density as input to the calculation of the double-folded potential. By contrast, the 15 C + 12 C fusion excitation function is well described by the no-coupling barrier penetration calculation and while the introduction of target and projectile inelastic couplings does not affect this good agreement, the single-neutron transfer coupling worsens it, leading to significant under-prediction of the cross section at the two lowest available E c.m. values.
A more global presentation of the dependence of the above barrier fusion on neutron excess is provided by the average fusion cross section, σ F . This is illustrated in Fig. 2 where we compare the σ F values of deSouza et al. [1], obtained by averaging Wong model fits to the experimental fusion cross sections over the interval 10 ≤ E c.m ≤ 18 MeV, with the equivalent quantities derived from the present calculations.
Also plotted on Fig. 2 are the σ F values extracted by deSouza et al. [1] from their barrier penetration model calculations with potentials calculated using nuclear matter densities from relativistic mean field (RMF) calculations employing the NL3 interaction and the Sao Paulo potential as effective interaction. The slope of the present no-coupling results is almost identical, although the σ F values are approximately 5% larger. However, this is within the spread of values obtained with different choices of nuclear matter densities. The behaviour of σ F as a function of neutron excess is not affected by the inclusion of inelastic couplings, neither do the ( 13 C, 12 C) and ( 14 C, 13 C) neutron transfer couplings have a significant influence. By contrast, the ( 15 C, 14 C) singleneutron transfer coupling acts in the opposite sense to the trend of the measured σ F values as a function of neutron excess, i.e. it reduces σ F for 15 C + 12 C to the extent that it is slightly smaller than for 14 C + 12 C. It is thus apparent that while the ( 15 C, 14 C) single-neutron transfer reaction has a significant effect on the total fusion in the 15 C + 12 C system, it does not explain the discrepancy between the trend of the measured and calculated values of σ F as a function of neutron excess, rather, it increases it. However, it is equally clear that any attempt at understanding the observed behaviour must include this process. Some other process, not included in the calculations presented here, must act to compensate for the effect of the single-neutron transfer coupling if the observed increase in σ F for the 15 C + 12 C system beyond the trend as a function of neutron excess expected from "geometric" considerations is to be explained. The most likely candidate is coupling to the 15 C → 14 C + n breakup, although this is difficult to demonstrate in a consistent manner with currently available models. However, the coupled discretised continuum channels (CDCC) calculations presented in Ref. [22] found that for the 6 He + 208 Pb system coupling to the 6 He → α + 2 n breakup did increase the total fusion cross section at similar incident energies with respect to the Coulomb barrier, and test CDCC calculations for the 15 C + 12 C system including coupling to the 15 C → 14 C + n breakup show a similar effect. Whatever the additional coupling may be, without taking into account the ( 15 C, 14 C) single-neutron transfer the good agreement of the no coupling calculation with the data would have led to the-erroneous-conclusion that the influence of coupling in general was negligible in this energy regime.
In summary, the present work provides evidence for the suggested importance of neutron dynamics in explaining the evolution of σ F as a function of neutron excess in the 12−15 C + 12 C systems [1], although perhaps not quite in the way expected. The CRC calculations presented here suggest that some other process, most likely the 15 C → 14 C + n breakup, must affect the total fusion cross section in such a way as to compensate for, or even somewhat exceed, the influence of the ( 15 C, 14 C) single-neutron transfer reaction in the 15 C + 12 C system in the energy regime discussed here. There is also some evidence for a slight, ∼ 5%, reduction of the above-barrier fusion cross section in the 13 C + 12 C system compared to our CRC calculations, possibly due to the formation of molecular orbitals by the single neutron between the two 12 C cores, see, e.g., Ref. [23]. The main conclusion is therefore that in systems involving neutron-rich and neutron halo nuclei the influence of neutron transfer reactions on the total fusion cross section should not be ignored.

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: The data are taken from Refs. [2][3][4].] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/.