Nucleus-nucleus potential from identical-particle interference

Based on the quantum interference between two-identical-nucleus scattering at energies around the Coulomb barrier, the barrier positions for $^{58}$Ni+$^{58}$Ni and $^{16}$O+$^{16}$O are extracted from Mott oscillations in the angular distributions around 90$^{\circ}$ for the first time. The angle separation of pairs of Mott scattering valleys around 90$^{\circ}$ has a direct relationship with the closest distance between two nuclei in elastic scattering. Together with the barrier height from fusion excitation function, the extracted barrier position provides a sensitive probe to constrain the model predictions for the nucleus-nucleus potential barrier.

The study of nucleus-nucleus potential [1][2][3][4][5][6][7][8][9][10][11][12] is one of the most important topics in nuclear physics. The basic features of the nucleus-nucleus potential are commonly described in terms of an interaction that is a function of the center-to-center distance between the projectile and target nuclei and consists of a repulsive Coulomb term and a short-ranged attractive nuclear component. The total potential possesses a maximum at a distance where the repulsive and attractive forces balance each other. This is referred to as the Coulomb barrier [13][14][15][16]. For heavy-ion fusion and scattering reactions, the Coulomb barrier directly influences the behavior of fusion and scattering cross sections, and an accurate extraction of the barrier not only the height but also the position and curvature from fusion cross sections and angular distributions of elastic/inelastic scattering attracted therefore a lot of attentions in many decades. Based on some empirical or realistic nuclear interactions [1][2][3][4][5][6] together with barrier penetration concept [17,18], coupled-channel methods [19,20], optical models [21][22][23] or microscopic dynamics equations [24][25][26][27][28], one attends to obtain the information of the nucleus-nucleus potential barrier through reproducing the measured cross sections. Unfortunately, it is found that the data can be reproduced reasonably well with different potentials combining different theoretical models [21,29]. Because of the ambiguity of optical model potential (commonly known as the Igo ambiguity [30]) due to the complicated parameter space and internal structure of the reaction partners, it is therefore of great significance to directly extract the barrier from measured data or to reduce the model dependence as much as possible. The height of the Coulomb barrier (or the distribution of the barrier height) could be extracted from the precisely measured fusion cross sections [31][32][33][34] or the back-angle quasi-elastic scattering cross sections [35,36]  data from Ref. [39] for the same reaction. It is therefore required to further investigate the barrier position from heavy-ion fusion/scattering cross sections.
For microscopic particles, such as photons, nucleons, heavy nuclei and so on, the quantum effect especially quantum interference between two-identical-particle have been evidently observed, which verifies the wave properties of microscopic particles, such as the positions of the sources of waves and de Broglie wavelength. In the double slit interference experiment of light as shown in Fig.1(a), the slit separation d has a relationship with the linear separation ∆y between fringes on screen (detector): where λ is the wavelength of light, L is the distance between the slit and the screen. It is known that the slit separation should be comparable with the wavelength of light in order to form the evident fringes on the screen. Although the direct double slit interference experiment for heavy nuclei is difficult since the slit separation should be at the femtometer scale, the quantum interference was clearly observed from the angular distribution of elastic scattering between identical projectile and target nuclei as shown in Fig.1(b), which is known as the Mott scattering [40][41][42]. Mott proposed an analytical formula for describing the differential cross section in the center-of-mass system for pure Coulomb scattering of identical particles [40]: where Z is the charge number of nuclei, θ c.m. is the center-of-mass scattering angle, E c.m. is the center-of-mass energy and I is the intrinsic spin of particles. η = Z 2 e 2 v is the Sommerfeld number which is 1 2 the ratio of the characteristic distance of closest approach given by Z 2 e 2 /E c.m. and the reduced wavelength λ [41]. According to Eq.  [42] which is generally at the Coulomb barrier. Considering that the elas-  (TDHF) theory [24], which are in good agreement with the extracted values.
In the proposed method, one needs to know the barrier height beforehand and then to measure the angle separation in Mott oscillations around 90 • at energies around the barrier.
It is therefore necessary to check the sensitivity of the extracted values of barrier position with the change of barrier height. The uncertainty of the extracted barrier height based on a precisely measured fusion excitation function is generally smaller than one MeV. Through adopting different optical potentials, we vary the barrier height within one MeV (but remain the position of potential barrier fixed) and find that the calculated change of the separation ∆θ is smaller than 1% for both 58 Ni+ 58 Ni and 16 O+ 16 O at energies around the barrier. We also note that the measured ∆θ for 16 O+ 16 O at E c.m. = 11.92 MeV is almost the same as that at 10.91 MeV [39]. In addition, it is known that at energies around the Coulomb barrier, the channel coupling effects lead to a distribution of fusion barrier, with which one obtains a mean value of the barrier height. Based on the elastic scattering between two-identical nuclei, the obtained barrier position could be a mean value of the positions of potential barriers considering the channel coupling effects.
To summarized, the quantum interference between two-identical-particle can be clearly observed not only for photons but also for heavy nuclei.