Pairing correlations, orientations and quantum fluctuations in one- and two-nucleon transfer reactions at sub-barrier energies

TL;DR

The paper addresses the challenge of understanding one- and two-nucleon transfer at sub-barrier energies by developing a fully microscopic framework that combines time-dependent covariant density functional theory (TD-CDFT) with dynamical pairing, nuclear deformation orientation sampling, and a generalized time-dependent generator coordinate method (TDGCM) to incorporate quantum fluctuations. It demonstrates that dynamical pairing enhances pair transfer and that transfer probabilities are highly sensitive to the orientation of the oblate $^{96}$Zr nucleus, necessitating orientation averaging to compare with experiment. The study finds that orientation-averaged one-neutron transfer agrees with data, while two-neutron transfer is suppressed below the barrier in TD-CDFT, highlighting the importance of fluctuations captured by TDGCM for accurate sub-barrier dynamics. The generalized TDGCM confirms the essential role of quantum fluctuations in sub-barrier two-neutron transfer but also reveals limitations in the current implementation, pointing to future work to include time evolution of generator states for a fully quantitative description.

Abstract

This work investigates one- and two-neutron transfer in the $^{96}\text{Zr} + {}^{40}\text{Ca}$ reaction at sub-barrier energies using a microscopic framework based on time-dependent covariant density functional theory (TD-CDFT). Pairing correlations are incorporated via the time-dependent BCS approximation, which is shown to significantly enhance pair transfer, as evidenced by an increased two-neutron transfer probability. The oblate deformation of $^{96}$Zr causes the transfer probabilities to vary by orders of magnitude with orientation; a direct comparison with experiment is enabled by averaging results over thirteen systematically chosen orientations. While the orientation-averaged one-neutron transfer probabilities agree well with data, the two-neutron channel is suppressed below the Coulomb barrier. This suppression is attributed to missing quantum fluctuations in the semiclassical TD-CDFT approach. To test this, we employ the generalized time-dependent generator coordinate method (TDGCM), which confirms that quantum fluctuations are essential for an accurate description of sub-barrier two-neutron transfer dynamics.

Figures (4)

• Figure 1: (Color online) One- and two-neutron transfer probabilities as a function of center-of-mass energy for the $^{96}{\rm Zr}+{}^{40}{\rm Ca}$ reaction. The solid squares and circles are experimental values from Ref. Corradi2011PRC. The dotted lines are calculated using TD-CDFT. The solid and dashed lines are obtained by TD-CDFT with time-dependent BCS approximation and with frozen occupation approximation (FOA), respectively. The potential energy surfaces of $^{96}{\rm Zr}$ calculated using CDFT without/with pairing correlations are shown in the left panel, with the equilibrium minimum denoted by the star.
• Figure 2: (Color online) One- and two-neutron transfer probabilities as a function of center-of-mass energy for the reaction $^{96}{\rm Zr}+{}^{40}{\rm Ca}$, calculated using TD-CDFT with dynamical pairing correlations. Results are shown for 16 orientations of the oblate $^{96}{\rm Zr}$ nucleus, comprising 13 distinct and 3 repeated configurations, defined schematically by the azimuthal angle $\phi$ and polar angle $\theta$ on the left: the symmetry axis lies in the (a) $y$-$z$, (b) $x$-$y$, and (c) $x$-$z$ planes, and (d) at a 45$^\circ$ angle relative to both the $z$ axis and the $x$-$y$ plane. Experimental data from Ref. Corradi2011PRC are shown as solid squares and circles. The different lines in each panel correspond to the specific orientations.
• Figure 3: (Color online) One- and two-neutron transfer probabilities for the $^{96}{\rm Zr}+{}^{40}{\rm Ca}$ reaction, averaged over the 13 orientations calculated with TD-CDFT (including dynamical pairing correlations, see Fig. \ref{['fig2']}), plotted as a function of center-of-mass energy.
• Figure 4: (Color online) One- and two-neutron transfer probabilities for the $^{96}{\rm Zr}+{}^{40}{\rm Ca}$ reaction, calculated by TD-CDFT and generalized TDGCM (with dynamic pairing correlations). Both methods start from the initial state with the $^{96}$Zr symmetry axis aligned along the $z$-axis $(\theta, \phi)=(0^\circ, 0^\circ)$. The generalized TDGCM evolution employs five TD-CDFT generator states with initial orientations at $(\theta, \phi)=(0^\circ, 0^\circ),(15^\circ,0^\circ),(30^\circ,0^\circ),(15^\circ,180^\circ),(30^\circ,180^\circ)$.