The puzzle of complete fusion suppression in weakly-bound nuclei: a Trojan Horse effect?

TL;DR

This paper introduces a new approach to understanding the suppression of complete fusion in reactions involving weakly-bound nuclei, attributing it mainly to non-elastic breakup modes and explaining large alpha yields via a Trojan Horse mechanism.

Contribution

The paper presents a novel method linking fusion suppression to reaction mechanisms and demonstrates its application to Li+Bi reactions, highlighting the role of non-elastic breakup and Trojan Horse effects.

Findings

Suppression of complete fusion is mainly due to non-elastic breakup modes.

Elastic breakup plays a minor role in fusion suppression.

Large alpha yields are explained by a Trojan Horse mechanism.

Abstract

Experimental studies of nuclear collisions involving light weakly-bound nuclei show a systematic suppression of the complete fusion cross section by $\sim$30\% with respect to the expectation for tightly bound nuclei, at energies above the Coulomb barrier. Although it is widely accepted that the phenomenon is related to the weak binding of these nuclei, the origin of this suppression is not fully understood. In here, we present a novel approach that provides the complete fusion for weakly bound nuclei and relates its suppression to the competition between the different mechanisms contributing to the reaction cross section. The method is applied to the $^{6,7}$Li+$^{209}$Bi reactions, where we find that the suppression of complete fusion is mostly caused by the flux associated with non-elastic breakup modes, such as the partial capture of the projectile (incomplete fusion), whereas the elastic breakup mode is found to play a minor role. Finally, we demonstrate that the large $\alpha$ yields observed in these reactions can be naturally explained as a consequence of a {\it Trojan Horse} mechanism.

Equations

$${\sigma }_R\approx {\sigma }_{\mathrm{CF}}+{\sigma }_{\mathrm{inel}}+{\sigma }_{\mathrm{EBU}}+{\sigma }_{\mathrm{NEB}}^{(b)}+{\sigma }_{\mathrm{NEB}}^{(x)}.$$

$${\frac{d^2\sigma }{d{E}_{b}d{\Omega }_b}|}_{\mathrm{NEB}}=-\frac{2}{\hslash v_a}{\rho }_b(E_b)⟨{\phi }_x({\vec{k}}_b)|\mathrm{Im}[U_{xA}]|{\phi }_x({\vec{k}}_b)⟩,$$

Full text

Introduction.–

Fusion between atomic nuclei constitutes a complicated quantum-mechanical dynamical process, whose outcome is critically dictated by a delicate interplay between the coupling of the relative motion of the colliding partners with their internal degrees of freedom.

Experiments with light weakly-bound stable nuclei (such as 6,7Li and 9Be) have shown that the complete fusion (CF) cross sections (defined as capture of the complete charge of the projectile) are suppressed by $\sim$20-30% compared to the case of tightly bound nuclei Dasgupta et al. (1999); Tripathi et al. (2002); Dasgupta et al. (2002, 2004); Mukherjee et al. (2006); Rath et al. (2009); Canto et al. (2015). The effect has been attributed to the breakup of the weakly bound projectile prior to reaching the fusion barrier, with the subsequent reduction of probability of complete capture. This interpretation is supported by the presence of large $\alpha$ yields as well as target-like residues which are consistent with the capture of one of the fragment constituents of the projectile, a process which is usually termed as incomplete fusion (ICF).

To account for these observations, some authors have proposed a two-step scenario Dasgupta et al. (2002); Diaz-Torres et al. (2007) in which the projectile, due to its loosely bound structure, breaks into two or more fragments, and then one of them is captured by the target. However, dynamical calculations based on a three-dimensional classical dynamical model Diaz-Torres et al. (2007), which implement this two-step breakup-fusion mechanism, can only explain a small fraction of the observed CF suppression for 9Be Cook et al. (2016) and 8Li Cook et al. (2018) reactions. Coupled-channels calculations, including the coupling to low-lying excited states of the projectile and target Dasgupta et al. (1999, 2002); Kumawat et al. (2012); Zhang et al. (2014); Fang et al. (2015) also fail to describe experimental fusion data.

Another problem arises in the interpretation of CF of neutron-rich weakly-bound nuclei. In these nuclei, the lowest breakup threshold corresponds to neutron emission. Since CF is operationally defined as capture of the complete charge of the projectile, breakup into one charged fragment and one uncharged one cannot contribute to CF suppression. Still, for the nucleus 8Li, whose lowest breakup threshold is 7Li+$n$ ($S_{n}=2.03$ MeV), a large CF suppression of $\sim$30% has been reported for the 8Li+208Pb Aguilera et al. (2009) and 8Li+209Bi Cook et al. (2018) reactions.

In this Letter, we propose a novel approach to compute CF cross sections of weakly-bound nuclei. Within a unified fully quantum-mechanical framework, the model is able to explain, simultaneously, the large $\alpha$-particle yields, the CF cross sections and the connection of their suppression with the binding energy of the projectile.

Theoretical framework.–

We consider a collision of a weakly-bound two-body projectile (denoted $a=b+x$) with a target nucleus $A$. We are mainly concerned here with the process in which the projectile as a whole fuses with the target nucleus, that is, complete fusion (CF). A realistic evaluation of the CF cross section must take into account the effect of other channels, such as projectile and/or target excitation, transfer and breakup. The explicit inclusion of all these channels in actual calculations is however not possible due to the overwhelming number of processes involved. To overcome this difficulty, the model proposed here takes advantage of the fact that light, weakly-bound nuclei have a marked cluster structure which suggests a natural decomposition of non-elastic channels in terms of the processes undergone by each of the clusters. Furthermore, the sum of the CF plus the other non-elastic channels is a well-constrained quantity since it is given by the reaction cross section ($\sigma_{R}$). Consequently, for a two-body projectile we may write the following approximate decomposition

[TABLE]

In this expression, $\sigma_{\mathrm{inel}}$ corresponds to the excitation of the projectile and/or target without dissociation (i.e., inelastic scattering). The term $\sigma_{\mathrm{EBU}}$ corresponds to elastic breakup, defined as the dissociative processes in which both fragments interact elastically with the target nucleus and hence the three outgoing fragments are emitted in their ground state (i.e., $a+A\rightarrow b+x+A_{\mathrm{gs}}$). Finally, $\sigma^{(b)}_{\mathrm{NEB}}$ and $\sigma^{(x)}_{\mathrm{NEB}}$ account for the so-called non-elastic breakup (NEB) processes, in which one of the two fragments interacts non-elastically with the target nucleus. This includes the ICF described in the introduction but also other processes, such as the projectile dissociation accompanied by target excitation ($a+A\rightarrow b+x+A^{*}$) or the exchange of nucleons between one the projectile fragments and the target. The outlined processes are schematically depicted in Fig. 1 using as example a 6Li+A reaction (modeled as $\alpha+d+A$).

The central idea of the present method is that the quantities $\sigma_{R}$, $\sigma_{\mathrm{inel}}$, $\sigma_{\mathrm{EBU}}$ and $\sigma^{(b,x)}_{\mathrm{NEB}}$ can be reliably calculated with existing reaction formalisms so that the CF section can be inferred from Eq. (1). The pure inelastic scattering cross sections ($\sigma_{\mathrm{inel}}$) are standardly computed by means of coupled-channels calculations including low-lying collective excitations of the projectile and target. The EBU part can be accurately calculated using the continuum-discretized coupled-channels (CDCC) method Austern et al. (1987), which treats the breakup as an excitation to the continuum states. Evaluation of the non-elastic breakup modes is much more challenging because of the large number of processes involved. Here, we propose to use the spectator/participant inclusive breakup model of Ichimura, Austern and Vincent (IAV) Austern and Vincent (1981); Austern et al. (1987); Ichimura et al. (1985), in which the explicit sum over final states arising from the interaction of the participant particle with the target is avoided by using the Feshbach projection formalism, giving rise to a closed-form formula for the the double differential cross section for NEB with respect to the angle and energy of the spectator fragment. For example, if $x$ is the participant particle,

[TABLE]

where $\rho_{b}(E_{b})$ is the density of states of the particle $b$, $v_{a}$ is the velocity of the incoming particle, $U_{xA}$ is the optical potential describing $x+A$ elastic scattering, and $\varphi_{x}(\vec{k}_{b},{\vec{r}}_{xA})$ is a projected wave function describing the evolution of the $x$ particle after dissociating from the projectile, when the core is scattered with momentum $\vec{k}_{b}$. This function is obtained from the equation $\varphi_{x}(\vec{k}_{b},{\vec{r}}_{xA})=\int G^{\text{opt}}_{x}({\vec{r}}_{xA},{\vec{r}}^{\prime}_{xA})\langle{\vec{r}}^{\prime}_{xA}\chi_{b}^{(-)}|V_{\text{post}}|\Psi^{3b}\rangle d{\vec{r}}^{\prime}_{xA}$, where $G^{\text{opt}}$ is the optical model Green’s function with potential $U_{xA}$, $\chi_{b}^{(-)}(\vec{k}_{b},{\vec{r}}_{bB})$ is the distorted-wave describing the scattering of the outgoing $b$ fragment with respect to the $B\equiv x+A$ system (obtained with some optical potential $U_{bB}$), $V_{\mathrm{post}}\equiv V_{bx}+U_{bA}-U_{bB}$ is the post-form transition operator and $\Psi^{3b}$ the three-body scattering wave function. Further details can be found in Ref. Lei and Moro (2015). Following our previous works Lei and Moro (2015, 2015, 2017), we approximate $\Psi^{3b}$ by its DWBA form: $\Psi^{3b}({\vec{R}},{\vec{r}})\approx\chi^{(+)}_{a}({\vec{R}})\phi_{a}({\vec{r}})$, where $\chi^{(+)}_{a}({\vec{R}})$ is a distorted wave describing $a+A$ elastic scattering, obtained with some optical potential, and $\phi_{a}({\vec{r}})$ is the projectile ground state wave function. Notice that the expectation value of the imaginary part of the $U_{xA}$ potential in Eq. (2) accounts for all possible non-elastic processes which may take place in $x-A$ scattering (that is, NEB), no matter how diverse or complicated they are. Recent applications of this DWBA version of the IAV model to deuteron Potel et al. (2015); Carlson et al. (2016); Lei and Moro (2015), 6Li Lei and Moro (2015, 2017), and 7Li Lei (2018) induced reactions have shown a very good agreement with existing data. We note that, although a decomposition similar to Eq. (1) has been employed by other authors Parkar et al. (2016), a proper computation of the NEB contributions, using a well founded theory, is a key and novel aspect of the present approach.

Finally, the reaction cross section ($\sigma_{R}$) can be extracted using the elastic S-matrix from the CDCC calculation or from an optical model fit of the elastic data, if available.

Application to the 6,7Li+209Bi reactions.– We apply now the proposed methodology to the reactions 6,7Li+209Bi. CF cross sections for these reactions have been measured by Dasgupta et al. Dasgupta et al. (2002, 2004), at energies below and above the Coulomb barrier ($V_{b}\approx 30$ MeV), and their results are shown in Fig. 2 (yellow circles), with the top and bottom panels corresponding to the 6Li and 7Li cases, respectively. CF suppression is usually measured with respect to the single-barrier penetration model (BPM), which accounts for the quantum tunneling probability through the effective Coulomb plus centrifugal barrier but ignoring the effect of other channels. These BPM calculations (quoted from Ref. Dasgupta et al. (2002)), are shown by magenta dashed lines. The effect of CF suppression is clearly apparent, amounting to $\sim$30% and $\sim$25% for the 6Li and 7Li cases, respectively.

To evaluate the CF cross section in presence of the other non-elastic channels, we make use of Eq. (1). The projectile inelastic scattering and EBU cross sections are obtained from CDCC calculations, using a two-body model ($\alpha+x$, with $x=d$ or $x=t$) for 6,7Li. For the 6Li+209Bi case, these calculations follow closely those performed in Ref. Lei and Moro (2015), so we refer to this work for further details. For the 7Li+209Bi reaction, we employ the $\alpha+t$ model parameters from Ref. Buck and Merchant (1988) and the $t$-target and $\alpha$-target potentials from Refs. F. D. Becchetti and Greenlees (1971) and Barnett and Lilley (1974), respectively. Following our previous works Lei and Moro (2015, 2017), the $d$-target and $t$-target potentials are renormalized to better reproduce the corresponding 6,7Li+209Bi elastic cross sections. Target excitations were not considered, since they have been shown to have a negligible effect on fusion at the above-barrier energies considered here.

The NEB cross sections are computed with the DWBA version of the IAV model described above. Within our assumed two-body model of 6,7Li, there are two distinct contributions, namely, one in which $x$ interacts non-elastically with the target (with $\alpha$ acting as a spectator) and another in which $\alpha$ interacts non-elastically. The same potentials are used in both calculations, and just the roles of participant and spectator are exchanged in Eq. (2). These $\alpha$ and $x$ yields are displayed, respectively, by squares and diamonds in Fig. 2. In Lei and Moro (2015), we showed that these calculations reproduce very well the inclusive $\alpha$ distributions measured in Ref. Santra et al. (2012) for the 6Li+209Bi reaction.

Finally, the reaction cross sections were evaluated from the elastic S-matrices obtained from the CDCC calculations. These reaction cross sections were found to be very close to those computed with the optical model fit of the elastic cross section from Refs. Santra et al. (2012); Martel et al. (1995).

It is seen in Fig. 2 that the calculated CF cross sections (red solid lines), deduced from Eq. (1), are remarkably close to the data. The separate role of each of the competing channels can be also deduced from this figure. The EBU mechanism ($\alpha+d$ and $\alpha+t$ production) plays a minor role, representing a small fraction of the reaction cross section at the incident energies relevant for this work. Instead, the dominant breakup mechanism in both reactions is the $\alpha$ production due to the (6,7Li,\alpha$$X) NEB. This explains the large $\alpha$ yields observed experimentally in these reactions. This is in fact a rather general feature found independently of the target nucleus Lei and Moro (2017).

The deuteron-production (6Li,d$$X) and triton-production (7Li,t$$X) NEB channels are much smaller than the $\alpha$-production ones. This can be understood as a combination of two effects: (i) the lower Coulomb barrier energy felt by the $d$ and $t$ particles as compared to the $\alpha$ particle and (ii) the smaller reaction cross section for the $\alpha$ particles, owning to its tightly-bound, compact structure.

The fact that the EBU mechanism barely affects the CF cross section explains why classical Cook et al. (2016) and quantum-mechanical calculations Elmahdy et al. (2015), which consider the fusion suppression due to the population of these elastic breakup channels, can only account for a small fraction of this suppression.

Although direct breakup plays a minor role in CF suppression, the degree of suppression has been shown to be closely correlated with the separation energy of the projectile into its cluster constituents Gasques et al. (2009). To investigate this connection within the present framework, we have repeated the calculations varying artificially the separation energy of the 6Li and 7Li nuclei for selected incident energies. The results are shown in Fig. 3 for 6Li + 209Bi at 36 MeV (1.2$V_{b}$) and 7Li + 209Bi at 44 MeV (1.5$V_{b}$). For each case, the BPM limit is indicated by a horizontal line. It is seen that, as the separation energy is increased with respect to its physical value, the reaction cross section decreases monotonically, indicating an overall reduction of non-elastic channels, as expected. The EBU contribution falls very fast, becoming negligible for separation energies of \sim$$3-$4$ MeV. The NEB contributions decrease also with the separation energy, but at a much lower rate, particularly for the $x$-fragment absorption. Interestingly, for large separation energies the difference $\sigma_{R}-\sigma_{\mathrm{EBU}}-\sigma_{\mathrm{NEB}}-\sigma_{\mathrm{inel}}$, that in our model is identified with $\sigma_{\text{CF}}$, tends to the BPM values for both the 6Li and 7Li cases. Thus, in the limit of strong binding, our model predicts no suppression, as expected. This reinforces our interpretation that the CF suppression arises from the flux associated with the transfer/breakup modes due to the weakly-bound structure of the projectile.

The calculations just presented rule out the direct breakup (6Li$\to\alpha+d$ and 7Li$\to\alpha+t$) and point toward the $\alpha$-production NEB mechanisms as the main responsible mechanism for the CF suppression in 6,7Li-induced reactions. As noted earlier, these channels are associated, respectively, with deuteron and triton reactions with the target nucleus. This includes particle transfer, target excitation and ICF. This may seem unexpected if one notes that the average deuteron and triton kinetic energies in the incident 6Li and 7Li projectiles are of the order, or even smaller, than their respective Coulomb barrier energies for the $d$+209Bi and $t$+209Bi systems ($\sim$10-11 MeV). For such low incident energies, the free $d$+209Bi and $t$+209Bi reaction cross sections are very small, in spite of which, the three-body 209Bi(6,7Li, \alpha$$X) cross sections are remarkably large. This phenomenon is not new and was first pointed out by Baur Baur (1986), who explained it invoking a “Trojan Horse mechanism”. The idea is that, for a three-body reaction of the form $a+A$, with $a=b+x$, a particular channel of the form $a+A\to b+c+C$ will be enhanced with respect to the free, two-body reaction $x+A\to c+C$ due to the fact that the $a+A$ system is above its Coulomb barrier. Loosely speaking, the $x$ particle is brought inside its Coulomb barrier by the heavier particle $a$. The method has become a standard tool in nuclear astrophysics as an indirect way of obtaining information of low-energy charged-particle induced reactions by means of three-body reactions (see e.g. Spitaleri et al. (2011)) and its formal aspects can be found elsewhere Typel and Baur (2003). We illustrate here the phenomenon for the two reactions under study. For that, in Fig. 4 we compare the reaction cross sections for the two-body reactions $d$+209Bi and $t$+209Bi, as a function of the center-of-mass energy for each system, with the three-body cross sections 209Bi(6Li, \alpha$$X) (top) and 209Bi(7Li, \alpha$$X) (bottom) for several 6,7Li incident energies. The vertical arrow in each panel denotes the position of the Coulomb barrier for the $d/t$+209Bi system. As expected, the reaction cross section for the two-body reactions drops very quickly as the energy approaches the Coulomb barrier. By contrast, the three-body cross sections remain very large, even at energies well below their nominal barrier. These results provide a natural explanation of the large $\alpha$ yields observed experimentally and confirmed by the IAV model.

The picture that emerges from these calculations is the following. The weakly bound projectile $a$ overcomes the $a+A$ Coulomb barrier, bringing also the $x$ fragment inside its Coulomb barrier via the just described Trojan Horse mechanism. This triggers the non-elastic processes between $x$ and $A$ which give rise to the large variety of emerging fragments observed experimentally and, in turn, to the suppression of CF. The present results add numerical support to the suggestion put forward by Cook et al. Cook et al. (2018), who conjectured that it is clustering and weak-binding, but not breakup in the usual sense, that is responsible for the CF suppression.

Summary and conclusions.–

In summary, we have proposed a new method to compute CF cross sections in collisions of light, weakly-bound nuclei. The method links these cross sections with the reaction and the transfer/breakup cross sections. These quantities can be reliably evaluated with state-of-the-art reaction frameworks, namely, the CDCC method for the EBU part, and the inclusive breakup model of IAV for the NEB. Application to the 6,7Li + 209Bi reactions, shows an excellent agreement with the CF data for these systems, and shows that the CF suppression originates from the flux associated with non-elastic breakup modes, most notably the $\alpha$ production channels. The large yields observed for these channels can be naturally explained as due to a Trojan Horse mechanism. Contrary to the assumption made in some works, we find that the direct breakup channels (${}^{6}\text{Li}\to\alpha+d$ and ${}^{7}\text{Li}\to\alpha+t$), which can be identified with our EBU contribution, play a very small role for these systems.

Although the calculations presented here have been restricted to the 6,7Li projectiles, we expect the conclusions to be valid for other weakly bound nuclei for which CF suppression have been also reported, such as 9Be or 8Li. An interesting question that arises is how the relative importance of the different competing mechanisms evolve as the separation energy of the projectile decreases, such as in the extreme cases of the halo nuclei 11Li, 6Li or 11Be. We note that, although the methodology proposed here is in principle applicable to these more exotic systems, its application may require (i) going beyond the DWBA approximation adopted here for the NEB cross sections and (ii), in the case of 11Li and 6He, a description of the projectile in terms of a three-body cluster model.