Halo Model Reaction Framework

• Halo Model Reaction Framework is a combined method that represents weakly bound nuclei as cluster systems while discretizing the continuum for reaction dynamics.
• It integrates the Cluster Model and the CDCC technique to quantitatively predict elastic scattering, breakup, fusion, and transfer processes.
• The framework’s predictive power is validated against experimental data on medium-mass targets, highlighting the role of coupled channels in nuclear reactions.

The Halo Model Reaction Framework describes the reaction dynamics of weakly bound and exotic halo nuclei with medium-mass targets by combining the clustering structure of the projectile with advanced reaction methods that explicitly account for coupling to the continuum. This approach is exemplified by the use of the Cluster Model for the projectile description in conjunction with the Continuum-Discretized Coupled-Channel (CDCC) technique, enabling quantitative predictions of elastic scattering, breakup, fusion, and nucleon transfer channels for nuclei such as $^6$Li, $^7$Li, $^9$Be, $^6$He, $^8$He, $^8$B, and $^{11}$Be on targets including $^{58}$Ni, $^{59}$Co, and $^{64}$Zn (Beck et al., 2010).

1. Cluster Model for Halo and Weakly Bound Nuclei

The Cluster Model represents a weakly bound or halo nucleus as a two-body (or few-body) system. For instance, $^6$Li is treated as an $\alpha + d$ cluster, while a halo nucleus such as $^8$B is modeled as a tightly bound core plus a loosely bound proton. The low binding energy of the outer nucleon leads to an extended spatial configuration and facilitates competing reaction pathways: the projectile may remain intact (elastic scattering), dissociate (breakup), partially or completely fuse with the target, or undergo transfer reactions. These processes are strongly coupled through the reaction Hamiltonian, necessitating an explicit inclusion of couplings between the bound and continuum states.

2. Continuum-Discretized Coupled-Channel (CDCC) Methodology

The CDCC framework is specifically designed to incorporate breakup (continuum) effects by discretizing the projectile's continuum spectrum into a set of finite energy bins and treating them on the same footing as bound states. The total three-body wave function is expanded as

$$\Psi(\mathbf{R}, \mathbf{r}) = \sum_i \phi_i(\mathbf{r}) ~ \chi_i(\mathbf{R}),$$

where $\mathbf{r}$ parameterizes the internal cluster coordinates, $\mathbf{R}$ is the projectile–target separation, $\phi_i$ are eigenstates (bound or discretized continuum), and $\chi_i$ are the channel radial wavefunctions. The coupled equations for the channel functions involve coupling potentials,

$$V_{ij}(R) = \langle\phi_i(\mathbf{r}) | V(\mathbf{R}, \mathbf{r}) | \phi_j(\mathbf{r}) \rangle.$$

Solving these equations yields the $S$-matrix elements $S_l$, from which reaction observables are extracted. The breakup is further classified into sequential breakup (SBU) and direct breakup (DBU); in the CDCC context, both mechanisms are included via the continuum basis.

3. Reaction Observables: Formal Definitions

Within the CDCC-based Halo Model Reaction Framework, several key observables are defined in terms of the $S$-matrix:

• Elastic Scattering:

$\frac{d\sigma}{d\Omega} = |f(\theta)|^2$

where $f(\theta)$ is the scattering amplitude related to the $S_l$ by a partial-wave decomposition.

• Total Reaction Cross Section:

$$\sigma_R = \frac{\pi}{k^2} \sum_l (2l+1)[1 - |S_l|^2]$$

where $k$ is the projectile–target wavenumber.

• Breakup Cross Section:

$$\sigma_{\text{break}} = \sum_{i \in \text{cont.}} \frac{\pi}{k^2} (2l_i+1)[1 - |S_i|^2]$$

• Fusion Cross Section:

Fusion, particularly sub-barrier, can be estimated using Wong's formula,

$$\sigma_F(E) = \frac{\hbar\omega R_B^2}{2E} \ln\left[1 + \exp\frac{2\pi(E - V_B)}{\hbar\omega}\right],$$

where $R_B$ and $V_B$ are the barrier radius and height, and $\hbar\omega$ is the curvature parameter.

• Transfer Channels:

The transfer amplitude involves explicit coupling form factors between breakup and specific bound or resonant states in the target–residual system, typically included as extra couplings in the CDCC equations.

4. Application to Medium-Mass Targets and Reaction Mechanism Interplay

The application of this framework to medium-mass targets (e.g., $^{58}$Ni, $^{59}$Co, $^{64}$Zn) reveals several systematics:

• The structure of both projectile and target (e.g., deformation, collective excitations) significantly affects reaction coupling.
• For $^6$Li/$^7$Li+$^{59}$Co, the coupling to breakup channels—despite modest breakup cross sections—is essential to accurately reproduce elastic angular distributions.
• Enhanced fusion cross sections due to breakup couplings are modest; in many instances, flux is diverted from complete fusion into incomplete fusion or transfer.
• Halo nuclei ($^8$B, $^6$He, etc.) can yield large predicted breakup cross sections, but the influence of continuum couplings on near-barrier elastic scattering is often weaker than anticipated.

This underscores that the low binding energy and spatial extension of a halo do not guarantee strong reaction channel coupling; rather, the net effect depends on the interplay of structure, binding, and dynamical matching.

5. Quantitative and Experimental Validation

The Halo Model Reaction Framework, as implemented via the Cluster Model + CDCC method, enables faithful reproduction and interpretation of a diverse set of experimental observables:

• Elastic scattering angular distributions are highly sensitive to continuum coupling and can be used as stringent tests of microscopic structure inputs.
• Breakup angular distributions, including SBU and DBU, reveal the underlying cluster structure and are accurately described by the CDCC built upon the chosen cluster model.
• Fusion excitation functions on various targets demonstrate that the effect of projectile breakup on complete fusion is largely a redistribution of total flux among stacking reaction channels.

Comparison with experiment across different projectiles and targets affirms the adequacy and adaptability of the framework to diverse systems, provided the cluster representation and reaction couplings are properly encoded.

6. Theoretical Synthesis and Framework Summary

The key workflow in the Halo Model Reaction Framework is:

1. Projectile Structure Input:
   • Model the projectile as a two-body (or few-body) cluster system.
   • Choose the appropriate bound and continuum representations.
2. Continuum Discretization:
   • Discretize the breakup continuum into energy bins for inclusion in the reaction dynamics.
3. CDCC Equation Solution:
   • Compute all relevant couplings and solve the coupled-channel equations numerically.
4. Extraction of Observables:
   • Use the computed $S$-matrix to calculate cross sections for elastic scattering, fusion, breakup, and transfer via well-established formulas.

This framework integrates experimental and theoretical advances, enabling a direct and detailed comparison of models with observables in reactions involving weakly bound and halo nuclei on medium-mass targets. The employment of key formulas such as

$$\frac{d\sigma}{d\Omega} = |f(\theta)|^2, \qquad \sigma_R = \frac{\pi}{k^2} \sum_l (2l+1)[1 - |S_l|^2], \qquad \sigma_F(E) = \frac{\hbar\omega R_B^2}{2E} \ln\left[1 + \exp\frac{2\pi(E - V_B)}{\hbar\omega}\right], \qquad V_{ij}(R) = \langle\phi_i(\mathbf{r})| V(\mathbf{R}, \mathbf{r}) |\phi_j(\mathbf{r})\rangle$$

guarantees the formal consistency and predictive value of the approach.

7. Implications and Outlook

The Halo Model Reaction Framework, anchored by the Cluster Model and CDCC methodologies, provides a systematic and robust paradigm for studying the reaction dynamics of weakly bound and halo nuclei. Its success in accounting for the interplay between reaction mechanisms (elastic, fusion, breakup, transfer) has established it as a standard reference for interpreting and guiding experimental investigations in heavy-ion and exotic nuclear reaction physics. The framework remains the foundation for ongoing developments, including the extension to more complex (three-body or higher) clusterizations, refined couplings (e.g., continuum-continuum), and the inclusion of target structure effects for broader applicability.