A Universal Damping Mechanism of Quantum Vibrations in Deep Sub-Barrier Fusion Reactions

We demonstrate the damping of quantum octupole vibrations near the touching point when two colliding nuclei approach each other in the mass-asymmetric $^{208}$Pb + $^{16}$O system, for which the strong fusion hindrance was clearly observed. We, for the first time, apply the random-phase approximation method to the heavy-mass asymmetric di-nuclear system to calculate the transition strength $B$(E3) as a function of the center-of-mass distance. The obtained $B$(E3) strengths are substantially damped near the touching point, because the single-particle wave functions of the two nuclei strongly mix with each other and a neck is formed. The energy-weighted sums of $B$(E3) are also strongly correlated with the damping factor which is phenomenologically introduced in the standard coupled-channel calculations to reproduce the fusion hindrance. This strongly indicates that the damping of the quantum vibrations universally occurs in the deep sub-barrier fusion reactions.

We demonstrate the damping of quantum octupole vibrations near the touching point when two colliding nuclei approach each other in the mass-asymmetric 208 Pb + 16 O system, for which the strong fusion hindrance was clearly observed. We, for the first time, apply the random-phase approximation method to the heavy-mass asymmetric di-nuclear system to calculate the transition strength B(E3) as a function of the center-of-mass distance. The obtained B(E3) strengths are substantially damped near the touching point, because the single-particle wave functions of the two nuclei strongly mix with each other and a neck is formed. The energy-weighted sums of B(E3) are also strongly correlated with the damping factor which is phenomenologically introduced in the standard coupled-channel calculations to reproduce the fusion hindrance. This strongly indicates that the damping of the quantum vibrations universally occurs in the deep sub-barrier fusion reactions. Heavy-ion fusion reactions are an excellent probe to investigate the fundamental features of the dynamics for many-body quantum systems. When a projectile approaches a target, the Coulomb barrier is formed, because of the strong cancellation between the Coulomb repulsion and the nuclear attractive force. Nuclear fusion takes place when the projectile penetrates through this Coulomb barrier. At incident energies in the vicinity of the Coulomb barrier height, called the sub-barrier fusion, the strong enhancements of fusion cross sections, compared to the estimations of a simple one-dimensional potential model, have been observed in many systems. These enhancements are well accounted for in terms of the couplings between the relative motion of the colliding nuclei and the intrinsic degrees of freedom such as collective vibrations of the target and the projectile [1]. The coupled channel (CC) model, which takes into account this mechanism, has been successful in describing such enhancements [2,3].
Recent experiments at extremely low incident energies, called the deep sub-barrier energies, revealed, however, that steep falloffs of the fusion cross sections, compared to the estimations of the standard CC model, emerge in a wide range of mass systems [4,5] (see Ref. [6] for details). These steep falloff phenomena are often called the fusion hindrance. An important quantity for understanding this fusion hindrance is the potential energy at the touching point of the colliding nuclei, which is strongly correlated with the threshold incident energy for the emergence of the fusion hindrance. That is, the fusion hindrance would be associated with the dynamics in the overlap region of the two colliding nuclei (see Fig. 1 in Ref. [7]).
A theoretical challenge is how to extend the standard CC model to describe these fusion hindrance phenomena in the overlap region. Two different models based on assumptions opposite to each other have been proposed [6]. One is the sudden approach proposed by Mişicu and Esbensen [8,9].
They constructed a heavy ion-ion potential with a shallow potential pocket considering the Pauli principle effect acting when two colliding nuclei overlap with each other. The other is the adiabatic approach proposed by Ichikawa et al. [10]. In this approach, neck formations between the colliding nuclei are taken into account in the overlap region. Based on this picture, the sudden and adiabatic processes were smoothly jointed by phenomenologically introducing the damping factor in the coupling form factor [11]. Later, we showed that the physical origin of the damping factor is the damping of quantum vibrations of the target and the projectile near the touching point using the random-phase approximation (RPA) method for the light mass-symmetric 16 O + 16 O and 40 Ca + 40 Ca systems [12].
In this Letter, we show that the damping of the quantum vibrations near the touching point is a universal mechanism in the deep sub-barrier fusions and is responsible for the fusion hindrance. A typical example optimally suited for this purpose is the recent precise data for the 208 Pb + 16 O fusion [5]. The performances of both the sudden and adiabatic models have been well tested in this system [9,11]. The adiabatic model can reproduce well the experimental data rather than the sudden model for the fusion hindrance. To discriminate which model is a better description, we here show the physical origin of the damping factor introduced in Ref. [11] in the heavy-mass asymmetric 208 Pb + 16 In the standard CC model (and the sudden model), the vibrational modes of the individual colliding nuclei are assumed not to change, even when the two nuclei strongly overlap with each other. However, as shown in Ref. [12], the single-particle wave functions are drastically changed by level repulsions, which are associated with the neck formations. We apply the RPA method to the heavymass asymmetric system, 208 Pb + 16 O, and show that these mechanisms lead to damping of quantum vibrations in the colliding nuclei near the touching point. This is exhibited by a drastic decreases of the B(E3) strengths carried by low-energy RPA excitation modes.
To illustrate our main idea, we first discuss the Nilsson diagram for protons as a function of the center-ofmass distance, R, in the 208 Pb + 16 O system. We calculate the mean-field potential for the 208 Pb + 16 O system using the folding procedure with the single Yukawa function [13]. Before the touching point, we assume the spherical shape for both nuclei. After the touching point, we describe the nuclear shapes with the reflection-asymmetric lemniscatoids parametrization [14]. (The parametrization dependence is negligible, because in this Letter we do not discuss the strongly overlapping region.) Based on these densities, we also calculate the Coulomb potential. We use the radius for the proton and neutron potentials, R 0 , with R 0 = 1.27A 1/3 fm, where A is the total nucleon number. The depths of the neutron and proton potentials for individual 16 O and 208 Pb nuclei, V T and V P , are taken from Ref. [15]. In the folding procedure, we smoothly joint the two different depth parameters of the mean-field potentials for 18 O and 208 Pb by the function where z c denotes the center position between the two surfaces of the colliding nuclei and µ denotes the smoothing parameter. We take µ = 0.8 fm, which is the same as the diffuseness parameter of the single-particle potential. In the calculations, the origin is located at the center-of-mass position of the two nuclei.
Using the obtained mean-field potentials, we solve the axially-symmetric Schrödinger equation with the spinorbit force. The details of the model and the parameters are similar to Refs. [13,15]. Then, the z component of the total angular momentum, Ω, is the good quantum number. Note that the parity is not a good quantum number because the mean-field potential for the whole system breaks the space-reflection symmetry. We expand the single-particle wave functions in terms of the deformed harmonic-oscillator bases in the cylindrical coordinate representation. The deformation parameter of the basis functions is determined so as to cover the target and the projectile. The basis functions with energies lower than 26 ω are taken into account. Figure 1 shows the Nilsson diagram as a function of the center-of-mass distance R. In the figure, we can see extremely strong Coulomb effect of 208 Pb on 16 O. The singleparticle p 1/2 and p 3/2 states in 16 O are shown by the (red) thick solid lines. Even at the large separation distance R = 20 fm, the energies of these two states are higher than the Fermi energy of the s 1/2 state in 208 Pb. The miss-match of the two Fermi levels between 16 O and 208 Pb occurs due to the strong Coulomb effect. At an infinite separation distance, the energies of the p 1/2 and p 3/2 states for 16 O are −5.88 and −10.7 MeV, respectively. Thus, at R = 20 fm, the depth of the mean-field potential for 16 O becomes shallow by about 5 MeV due to the Coulomb effect from 208 Pb. The single-particle energies of the p 1/2 and p 3/2 states in 16 O remarkably increase with decreasing R due to the increasing Coulomb effect from 208 Pb. Then, many level crossings and repulsions between the energy levels of 16 O and 208 Pb occur. With decreasing R, the energy of the p 1/2 state becomes positive around R = 18 fm, that is, it changes into a resonance state, but there is still a sufficiently high Coulomb barrier. After that, it goes across the f 5/2 and p 3/2 states of 208 Pb around R = 16 fm and the p 1/2 state of 208 Pb around R = 13 fm. Below R = 13 fm, the Coulomb barrier becomes lower due to the attractive nuclear mean-field potential. Then, the strong mixture of the single-particle states between 16 O and 208 Pb starts in many levels, which causes many level splittings seen in the Nilsson diagram.
We now solve the RPA equation at each R for the massasymmetric 208 Pb + 16 O system. We calculate the first excited 3 − (octupole vibrational) states of 16 O and 208 Pb, which give the main contributions in the standard CC calculations. We can easily apply the RPA method to the dinuclear system, because its the wave function is described with a one-center Slater determinant. We take the singleparticle levels for each neutron and proton up to 200th and the coherent superposition of all one-particle one-hole states with excitation energies below 30 MeV. We follow the diabatic single-particle configuration corresponding to the ground state of 16 O. The occupied p 1/2 and p 3/2 states in 16 O are represented by the light gray (red) thick curves in Fig. 1. We carry out the RPA calculation avoiding immediate vicinities of the level-crossing points. We use the density-dependent residual interaction taken from Ref. [16] and tune it so that the energy of the spurious center-ofmass motion becomes zero. We calculate B(E3) values for the RPA solutions with Ω = 0 in individual nuclei using the shifted octupole operator, Q 30 (R − R ′ 0 ), where R ′ 0 is the center-of-mass position of the projectile or target nucleus.
At the large separation distance R = 20 fm, we obtain the first 3 − excited states of individual nuclei. The obtained energies and B(E3, 3 − 1 → 0 + 1 ) values are 2.86 MeV and 7.13 × 10 4 e 2 · fm 6 for 208 Pb and 4.64 MeV and 124 e 2 · fm 6 for 16 O. The obtained transition densities and currents for the first 3 − states of 16 O and 208 Pb are depicted in Figs. 2 (a) and (c). At R = 20 fm, these modes are isolated. When the two nuclei approach each other, however, these modes start to fragment into several states. To evaluate the octupole collective strengths carried by low-energy excitations, we then calculate the energy-weighted sum of B(E3) strengths. By checking the spectrum of all obtained RPA modes as a function of R, we determined to take the sum for octupole excitations with E ≤ 4 MeV and E ≤ 6 MeV for 208 Pb and 16 O, respectively.  Fig. 2 (b) and (d). These figures indicate that the octupole collectivities of both 16 O and 208 Pb are considerably diminished by each colliding partner.
The microscopic origin of the damping of these vibrations is easily seen as follows. At R = 20 fm, the main proton components of the 3 − modes are the excitations p 1/2 → d 5/2 and p 3/2 → d 5/2 for 16 O, and the excitations d 3/2 → h 9/2 and s 1/2 → f 7/2 for 208 Pb [see the (red and blue) arrows around R = 19 fm in Fig. 1]. The density distributions of the p 1/2 and d 5/2 states in 16 O are displayed in (a) and (b) of Fig. 4. Their wave functions suffer major modifications near the touching point at R = 11.65 fm, as depicted in (c) and (d) of Fig. 4 [see also the (red) arrow at R = 11.65 fm in Fig. 1]. We can clearly see the neck formations in (d). Also for 208 Pb, similar drastic changes of single-particle wave functions occur for both protons and neutrons near the Fermi surface, causing the damping of the collectivity of the 3 − vibration [see the (blue) arrows around 10.6 fm in Fig. 1].
Finally, to see the correlation with the damping factor phenomenologically introduced in the CC calculation, we compare the calculated results with the damping factor that well reproduced the experimental data of the fusion cross section for 208 Pb + 16 O [11]. The damping factor is given by Φ(r, λ α ) = e −(r−R d −λ α ) 2 /2a 2 d for r < R d + λ α (otherwise ing factor with these parameters normalized at R = 20 fm. We can see that the damping factor strongly correlates with the calculated energy-weighted sums of B(E3) in the lowenergy region, which clearly indicates that the damping of the quantum vibrations indeed occurs when the colliding nuclei approach each other.
In summary, we have demonstrated the damping of the quantum octupole vibrations of both 16 O and 208 Pb, when they approach each other. To show this, we, for the first time, applied the RPA method to the heavy massasymmetric 208 Pb + 16 O system. We have discussed the Nilsson diagram as a function of the center-of-mass distance R and have shown that the single-particle energies in 16 O are largely sifted to the positive-energy direction by the strong Coulomb effects from the heavy-mass 208 Pb in a colliding process. We calculated the B(E3) strengths for 16 O and 208 Pb as a function of R. The obtained B(E3) strengths are substantially damped near the touching point of the colliding nuclei. The obtained energy-weighted sum of B(E3) in the low-energy region exhibits a strong correlation with the damping factor that reproduces well the experimental data of the fusion cross section for 208 Pb + 16 O. This is a clear evidence that the damping of the quantum octupole vibrations indeed occur near the touching point in the deep sub-barrier fusion reactions. The drastic change of single-particle wave functions consitituting the low-energy collective excitations discussed in this paper would commonly occur in all deep sub-barrier reactions. Therefore, the damping of quantum vibrations in both the target and the projectile near the touching point seems to be a universal mechanism causing the fusion hindrance, which should be taken into account in the standard CC model.
A part of this research was funded by the MEXT HPCI STRATEGIC PROGRAM. This work was supported by JPSJ KAKENHI Grant Number 15K05078.