${}^6\mathrm{Li}$ in a three-body model with realistic Forces: Separable versus nonseparable approach

Background: Deuteron induced reactions are widely used to probe nuclear structure and astrophysical information. Those ($d,p$) reactions may be viewed as three-body reactions and described with Faddeev techniques.

Purpose: Faddeev equations in momentum space have a long tradition of utilizing separable interactions in order to arrive at sets of coupled integral equations in one variable. However, it needs to be demonstrated that their solution based on separable interactions agrees exactly with solutions based on nonseparable forces.

Methods: Momentum space Faddeev equations are solved with nonseparable and separable forces as coupled integral equations.

Results: The ground state of ${}^6\mathrm{Li}$ is calculated via momentum space Faddeev equations using the CD-Bonn neutron-proton force and a Woods-Saxon type neutron(proton)-${}^4\mathrm{He}$ force. For the latter the Pauli-forbidden $S$-wave bound state is projected out. This result is compared to a calculation in which the interactions in the two-body subsystems are represented by separable interactions derived in the Ernst-Shakin-Thaler (EST) framework.

Conclusions: We find that calculations based on the separable representation of the interactions and the original interactions give results that agree to four significant figures for the binding energy, provided that energy and momentum support points of the EST expansion are chosen independently. The momentum distributions computed in both approaches also fully agree with each other.

Figure 1

The S- and P-wave phase shifts in the $n\alpha$ and $p\alpha$ subsystems as function of the neutron/proton laboratory kinetic energy. The solid (dashed) lines represent the calculations with the modified Bang interaction [39] for the $n\alpha$ and $p\alpha$ systems. The phase-shifts extracted from an R-matrix fit [45] are shown for $n\alpha$ by filled circles and for $p\alpha$ by filled diamonds.

Figure 2

The off-shell $t$ matrix elements $t_{l_{np}^{\prime }{l}_{np}}(p^{\prime },p;E)$ for the $np$ system as function of the off-shell momentum $p$. The center of mass energy is $E_{2b}=-50$ MeV while the total angular momentum and spin are fixed at $J_{np}=S_{np}=1$. The $t$ matrix elements $t_{00}(p^{\prime },p;E)$ are shown in panel (a) while the ones corresponding to $l_p=l_p^{\prime }=2$ are illustrated in panel (b). Panel (c) shows the matrix elements $t_{20}(p^{\prime },p;E)$. The solid and dashed lines depict the $t$ matrix elements computed with the CD-Bonn potential for $p^{\prime }$= 0.3 ${\mathrm{fm}}^{-1}$ and $p^{\prime }$= 0.8 ${\mathrm{fm}}^{-1}$ respectively. The results obtained using a rank-6 separable representation of the CD-Bonn potential are indicated by upward triangles for $p^{\prime }$= 0.3 ${\mathrm{fm}}^{-1}$ and by diamonds for $p^{\prime }$= 0.8 ${\mathrm{fm}}^{-1}$. The six EST support points are located at $\{E_l,p_l\}=\{-60,0.4\},\{-60,1.1\},\{-60,2.5\},\{-5,0.4\},\{-5,1.1\},\{-5,2.5\}$. The energies have units of MeV while the momenta are given in ${\mathrm{fm}}^{-1}$.

Figure 3

Panels (a) and (b) show the momentum distributions $n\left(p\right)$ in the $\left(np\right)\text{−}\alpha$ (solid line), $\left(n\alpha \right)\text{−}p$ (dotted line), and $\left(p\alpha \right)\text{−}n$ (dashed line) arrangement channels of the ${}^6\mathrm{Li}$ ground state calculated with the CD-Bonn [43] $np$ interaction and the modified Bang [44] $n\alpha$ interaction. For the $p\alpha$ interaction the Coulomb interaction given in Eq. (29) is added. Panels (c) and (d) show the momentum distributions $n\left(q\right)$ for the same arrangement channels as panels (a) and (b). The crosses as well as the upward and downward triangles correspond to the same calculations but using separable forces.

Figure 4

The probability $N_{\beta }\left(\lambda \right)$ for the ${}^6\mathrm{Li}$ three-body ground state as a function of the projector strength $\lambda$ calculated according Eq. (34). The solid, dashed, and dot-dashed lines represent the $S_{1/2}$, $P_{1/2}$ and $P_{3/2}$ partial wave states of the $n\alpha$ subsystem. The dashed vertical line indicates the value of $\lambda$ at which the $S_{1/2}$ state becomes unbound.