At energies far below the Coulomb barrier, measured heavy-ion fusion cross sections fall off more steeply than coupled-channels calculations predict (the fusion-hindrance phenomenon), and the astrophysical S factor consequently develops a maximum at a threshold energy EthrE_{\mathrm{thr}}. Hagino, Balantekin, Lwin, and Thein (Phys. Rev. C 97, 034623, 2018) showed that this maximum follows naturally if the excitation function is fitted with two Woods-Saxon ion-ion potentials, one for the subbarrier region and a distinctly shallower, much more diffuse one for the deep-subbarrier region, but left open what microscopic many-body dynamics after the two nuclei touch produces this abrupt steepening and how large it is system by system. The open problem is to establish, microscopically and quantitatively, the origin of the change in the logarithmic slope L(E)L(E) at EthrE_{\mathrm{thr}} — equivalently, of the large increase of the effective surface diffuseness in the two-potential phenomenology — and to disentangle such dynamical post-touching effects from the static kinematic effect of a negative reaction QQ value. The 2021 field review states that the question of which physical model captures the effect is not settled.

Background

In heavy-ion fusion, two nuclei tunnel through the Coulomb barrier and merge into a compound nucleus. Near the barrier, measured fusion excitation functions σfus(E)\sigma_{\mathrm{fus}}(E) are well reproduced by coupled-channels calculations, which include couplings of the relative motion to low-lying collective excitations and to nucleon-transfer channels of the colliding nuclei; these couplings generate a distribution of effective barrier heights. In 2002, Jiang et al. measured 60^{60}Ni+89^{89}Y down to cross sections of 100 nb and discovered that at deep subbarrier energies the cross sections fall off far more steeply than any coupled-channels extrapolation — the fusion-hindrance phenomenon, since confirmed in many medium-heavy systems and actively searched for in light astrophysical systems.

Two standard analysis tools are used at these energies. The astrophysical S factor, S(E)=Eσfus(E)e2πηS(E)=E\sigma_{\mathrm{fus}}(E)e^{2\pi\eta} with η=ZPZTe2/ℏv\eta=Z_P Z_T e^2/\hbar v the Sommerfeld parameter (the dimensionless strength of the Coulomb interaction; ZPZ_P, ZTZ_T are the projectile and target atomic numbers and vv the relative velocity), removes the trivial Coulomb-penetration energy dependence that dominates light-ion reactions. The logarithmic slope, L(E)=dln⁡(Eσfus)/dEL(E)=d\ln(E\sigma_{\mathrm{fus}})/dE, characterizes the steepness of the excitation function; for a point-charge system with energy-independent S(E)S(E) one has Lcs(E)=πη/EL_{\mathrm{cs}}(E)=\pi\eta/E. Because dln⁡S/dE=L(E)−πη/Ed\ln S/dE=L(E)-\pi\eta/E, the S factor has a maximum exactly where L(E)L(E) crosses Lcs(E)L_{\mathrm{cs}}(E), and this crossing energy has been widely adopted as the hindrance threshold EthrE_{\mathrm{thr}} (denoted EsE_s in the systematics literature). For a system with negative fusion QQ value a maximum is kinematically mandatory, since S(E)→0S(E)\to 0 as E→−QE\to -Q; for positive QQ it is not required, yet maxima are observed in several positive-QQ systems.

Hagino, Balantekin, Lwin, and Thein (2018) asked why the maximum appears. Following the two-potential fit of Thein, Lwin, and Hagino (Phys. Rev. C 85, 057602, 2012), they fitted measured excitation functions (e.g. 64^{64}Ni+64^{64}Ni, 28^{28}Si+64^{64}Ni) with two different Woods-Saxon ion-ion potentials (the standard empirical form V(r)=−V0/[1+exp⁡((r−R0)/a)]V(r)=-V_0/[1+\exp((r-R_0)/a)] for the nuclear part of the interaction, characterized by a radius R0R_0 and a surface-diffuseness parameter aa), one for the subbarrier region (cross sections 10−210^{-2}–10010^{0} mb) and one for the deep-subbarrier region (below 10−310^{-3} mb), defining EthrE_{\mathrm{thr}} where the two fitted excitation functions cross. The deep-subbarrier potential requires a much larger surface diffuseness aa (about 11 fm versus the usual 0.650.65 fm). They showed that the S-factor maximum is then a natural consequence: at E<EthrE<E_{\mathrm{thr}} the steep excitation function makes L(E)L(E) exceed πη/E\pi\eta/E, so S(E)S(E) rises as EE decreases, while above EthrE_{\mathrm{thr}} the normal slope leaves dS/dE<0dS/dE<0.

Two phenomenological pictures of the hindrance itself exist, both invoking dynamics after the nuclei touch. In the sudden approximation of Mişicu and Esbensen, nuclear incompressibility and Pauli blocking of overlapping single-particle states make the ion-ion potential develop a shallow pocket and short-range repulsion. In the adiabatic model of Ichikawa, Hagino, and Iwamoto, the coupling strengths of collective excitations are damped once the nuclei overlap, the system deforms into a dinuclear shape with neck formation, and tunneling sees an effectively one-body barrier. Time-dependent Hartree-Fock (TDHF) calculations — mean-field evolution of the full many-nucleon wave function — indicate that Pauli blocking in the overlap region and nucleon exchange through the neck reduce the fusion probability. Neither picture, however, derives the fitted two-potential parameters quantitatively from the underlying many-body dynamics, and the 2018 paper explicitly defers this to future work.

Problem Statement

Establish the microscopic origin of the deep-subbarrier steepening of heavy-ion fusion excitation functions, as posed at the close of Hagino, Balantekin, Lwin, and Thein (2018): identify what detailed many-body dynamics after the two nuclei touch — dissipation, neck formation, nucleon-transfer dynamics, changes of collective inertia, or Pauli blocking — produces the abrupt change of the logarithmic slope L(E)=dln⁡(Eσfus)/dEL(E)=d\ln(E\sigma_{\mathrm{fus}})/dE at the hindrance threshold EthrE_{\mathrm{thr}}, and in particular the much larger effective surface diffuseness (a≈1a\approx 1 fm versus ≈0.65\approx 0.65 fm) that the two-potential fits require below EthrE_{\mathrm{thr}}; determine quantitatively how much L(E)L(E) changes at EthrE_{\mathrm{thr}}, at minimum for one representative system for which the two-potential fit has been performed (e.g. 64^{64}Ni+64^{64}Ni or 28^{28}Si+64^{64}Ni) and, where data permit, for the wider systematics. This includes deciding between the competing sudden (short-range repulsion from nuclear incompressibility and Pauli blocking) and adiabatic (damping of channel couplings during neck formation) interpretations, or showing that neither suffices. It further requires disentangling such dynamical post-touching effects from the static effect of a negative fusion QQ value — establishing under what conditions the kinematic requirement S(E)→0S(E)\to 0 as E→−QE\to -Q, rather than post-touching dynamics, controls the threshold energy — and validating from underlying nuclear dynamics, rather than by phenomenological parametrization, the effective potential-parameter changes obtained in the two-potential fits. These deliverables jointly serve one objective: a physically grounded, quantitatively validated account of why the astrophysical S factor develops a maximum at deep subbarrier energies. The deliverables above are not all jointly mandatory. The core — mandatory and sufficient on its own to resolve the objective — is a microscopic identification of the many-body mechanism behind the change of slope and of the associated increase of the effective surface diffuseness, together with its quantitative determination of the change of L(E)L(E) at EthrE_{\mathrm{thr}} for at least one representative system for which the two-potential fit has been performed. The adjudication between the sudden and adiabatic interpretations, the separation of the negative-QQ kinematic requirement from post-touching dynamics, and the microscopic validation of the two-potential parameters are supporting requirements that a resolution must address explicitly but need not close: a solution that establishes the core for one representative system, states which interpretation the evidence supports, and reports what remains undetermined resolves the objective, whereas a mechanism advanced without the quantitative slope change, or a purely phenomenological parametrization, does not.

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