Unified ab initio approach to bound and unbound states: No-core shell model with continuum and its application to ${}^7$He

We introduce a unified approach to nuclear bound and continuum states based on the coupling of the no-core shell model (NCSM), a bound-state technique, with the no-core shell model/resonating group method (NCSM/RGM), a nuclear scattering technique. This new ab initio method, no-core shell model with continuum (NCSMC), leads to convergence properties superior to either NCSM or NCSM/RGM while providing a balanced approach to different classes of states. In the NCSMC, the ansatz for the many-nucleon wave function includes (i) a square-integrable $A$-nucleon component expanded in a complete harmonic oscillator basis and (ii) a binary-cluster component with asymptotic boundary conditions that can properly describe weakly bound states, resonances, and scattering. The Schrödinger equation is transformed into a system of coupled-channel integral-differential equations that we solve using a modified microscopic $R$-matrix formalism within a Lagrange mesh basis. We demonstrate the usefulness of the approach by investigating the unbound ${}^7$He nucleus.

Figure 1

Ground-state energy of ${}^6$He calculated within the NCSM using the SRG-N${}^3$LO $NN$ potential with $\Lambda =2.02$ fm${}^{-1}$. The dependence on the HO frequency for different $N_{\max }$ basis sizes is shown. The points with error bars represent the results of the exponential extrapolation.

Figure 2

Basis size $N_{\text{max}}$ dependence of ${}^6$He and ${}^7$He g.s. energies calculated within the NCSM. The SRG-N${}^3$LO $NN$ potential with $\Lambda =2.02$ fm${}^{-1}$ and the HO frequency of $\hslash \Omega =16$ MeV were used. Exponential extrapolations from the last three $N_{\max }$ points is shown.

Figure 3

Dependence of ${}^6$He excitation energies on the size of the basis $N_{\text{max}}$. NCSM calculations were performed with the SRG-N${}^3$LO $NN$ potential with $\Lambda =2.02$ fm${}^{-1}$ and the HO frequency of $\hslash \Omega =16$ MeV.

Figure 4

Dependence of the NCSM/RGM (a) and NCSMC (b) ${}^6\text{He}+n$ phase shifts of the ${}^7$He $3/2^{-}$ g.s. on the number of ${}^6$He states included in the binary-cluster basis. The short-dashed green curve, the dashed blue curve, and the solid red curve correspond to calculations with ${}^6\text{He}$ $0^{+}$ g.s. only, $0^{+},2^{+}$ states, and $0^{+},2^{+},2^{+}$ states, respectively. The SRG-N${}^3$LO $NN$ potential with $\Lambda =2.02$ fm${}^{-1}$, the $N_{\max }=12$ basis size and the HO frequency of $\hslash \Omega =16$ MeV were used. See text for further details.

Figure 5

Dependence of the NCSM/RGM (a) and NCSMC (b) ${}^6\text{He}+n$ phase shifts of the ${}^7$He $1/2^{-}$ excited state on the number of ${}^6$He states included in the binary-cluster basis. The short-dashed green curve, the dashed blue curve, and the solid red curve correspond to calculations with ${}^6\text{He}$ $0^{+}$ g.s. only, $0^{+},2^{+}$ states, and $0^{+},2^{+},2^{+}$ states, respectively. See Fig. 4 for further details.

Figure 6

Dependence of the NCSMC ${}^6\text{He}+n$ phase shifts of ${}^7$He $3/2^{-}$ (a), $1/2^{-}$ (a), and $5/2^{-}$ (b) states on the size of the HO expansion $N_{\text{max}}$. The ${}^6\text{He}$ $0^{+},2^{+},2^{+}$ states were included in the binary-cluster basis. The SRG-N${}^3$LO $NN$ potential with $\Lambda =2.02$ fm${}^{-1}$ and the HO frequency of $\hslash \Omega =16$ MeV were used.

Figure 7

NCSM/RGM (a) and NCSMC (b) ${}^6\text{He}+n$ diagonal phase shifts (except ${}^6{P}_{3/2}$, which are eigenphase shifts) as a function of the kinetic energy in the c.m. The dashed vertical area centered at $0.43\text{MeV}$ indicates the experimental centroid and width of the ${}^7\text{He}$ g.s. [38, 39]. In all calculations the lowest three ${}^6\text{He}$ states have been included in the binary-cluster basis. The SRG-N${}^3$LO $NN$ potential with $\Lambda =2.02$ fm${}^{-1}$ within the $N_{\max }=12$ basis size and the HO frequency of $\hslash \Omega =16$ MeV were used. See text for further details.

Figure 8

Experimental and theoretical centroid energies for ${}^7\text{He}$ resonances, with the ${}^6\text{He}+n$ threshold as the energy reference. The experimental energy of the $1/2^{-}$ resonance is taken from Ref. [48]. The theoretical values for NCSM/RGM and NCSMC correspond to the $n{-}^6$He kinetic energy in the c.m. when the derivative of the phase shift is maximal; see text for details. The information on the width of the states is given in Table . The calculations are carried out as described in Table , Fig. 7, and in the text.

Figure 9

NCSMC ${}^6\text{He}+n$ $3/2^{-}$ $P$-wave eigenphase shifts as a function of the kinetic energy in the c.m. Calculations are carried out as described in Fig. 7b. See text for further details.

Figure 10

NCSMC ${}^6\text{He}+n$ $3/2_2^{-}$ and $5/2^{-}$ diagonal $P$-wave phase shifts as a function of the kinetic energy in the c.m. The calculation is carried out as described in Fig. 7b. See text for further details.