$^7$Be and $^7$Li nuclei within the no-core shell model with continuum

TL;DR

This study uses the no-core shell model with continuum to accurately describe $^7$Be and $^7$Li nuclei, predicting known and new resonances, with implications for astrophysics and nuclear physics.

Contribution

It applies the ab initio NCSMC method with chiral interactions to predict properties and resonances of $^7$Be and $^7$Li, including new resonances built on $^6$Li and $^6$He.

Findings

Reproduces all known states of $^7$Be and $^7$Li.

Predicts several new resonances, especially in $^7$Li.

Identifies an $S$-wave resonance near the threshold in $^6$He(p,$eta$)$^7$Li.

Abstract

The production of $^7$Be and $^7$Li nuclei plays an important role in primordial nucleosynthesis, nuclear astrophysics, and fusion energy generation. The $^3\mathrm{He}(\alpha , \gamma) ^7\mathrm{Be}$ and $^3\mathrm{H}(\alpha , \gamma) ^7\mathrm{Li}$ radiative-capture processes are important to determine the $^7$Li abundance in the early universe and to predict the correct fraction of pp-chain branches resulting in $^7$Be versus $^8$B neutrinos. In this work we study the properties of $^7$Be and $^7$Li within the no-core shell model with continuum (NCSMC) method, using chiral nucleon-nucleon interactions as the only input, and analyze all the binary mass partitions involved in the formation of these systems. The NCSMC is an ab initio method applicable to light nuclei that provides a unified description of bound and scattering states and thus is well suited to investigate systems with many resonances and pronounced clustering like $^7$Be and $^7$Li. Our calculations reproduce all the experimentally known states of the two systems and provide predictions for several new resonances of both parities. Some of these new possible resonances are built on the ground states of $^6$Li and $^6$He, and thus represent a robust prediction. We do not find any resonance in the p${+}^6$Li mass partition near the threshold. On the other hand, in the p${+}^6$He mass partition of $^7$Li we observe an $S$-wave resonance near the threshold producing a very pronounced peak in the calculated S factor of the $^6\mathrm{He} (\mathrm{p},\gamma) ^7\mathrm{Li}$ radiative-capture reaction, which could be relevant for astrophysics and its implications should be investigated.

Tables

Table 1. Table 1: Comparison between the NCSMC and the experimental relative ( E B subscript 𝐸 𝐵 E_{B} ) and total ( E 𝐸 E ) energies of the J π = 3 / 2 1 − superscript 𝐽 𝜋 3 superscript subscript 2 1 J^{\pi}=3/2_{1}^{-} and 1 / 2 1 − 1 superscript subscript 2 1 1/2_{1}^{-} bound states of 7 Be produced with the 3 He + 4 He and 6 Li + p reactions. Here, the E B subscript 𝐸 𝐵 E_{B} energy represents the energy of the state with respect the threshold of the reaction. All calculations were performed at N max = 11 subscript 𝑁 max 11 N_{\mathrm{max}}=11 in the HO expansion and using the SRG-evolved NN N 3 LO chiral interaction  Entem and Machleidt ( 2003 ) at the resolution scale of λ SRG = 2.15 subscript 𝜆 SRG 2.15 \lambda_{\mathrm{SRG}}=2.15 fm -1 .

3He + 4He $\leftrightarrow$ 7Be	$J^{\pi }=3/2^{-}$	$J^{\pi }=1/2^{-}$
$E_B$    [MeV]	-1.52	-1.26
Exp. [MeV]	-1.587	-1.157
$E$     [MeV]	-36.98	-36.71
Exp. [MeV]	-37.60	-37.17
6Li + p $\leftrightarrow$ 7Be	$J^{\pi }=3/2^{-}$	$J^{\pi }=1/2^{-}$
$E_B$    [MeV]	-5.73	-5.39
Exp. [MeV]	-5.606	-5.177
$E$     [MeV]	-36.47	-36.13
Exp. [MeV]	-37.60	-37.17

Table 2. Table 2: Properties of the ground state of 7 Be computed with the NCSM and NCSMC approaches using the 3 He + 4 He mass partition and compared with the experimental data. The reduced transition probabilities are from the ground state 3 / 2 1 − 3 superscript subscript 2 1 3/2_{1}^{-} to the first excited state 1 / 2 1 − 1 superscript subscript 2 1 1/2_{1}^{-} . The ANCs C l ​ j subscript 𝐶 𝑙 𝑗 C_{lj} are shown for both the ground state and the 1 / 2 1 − 1 subscript superscript 2 1 1/2^{-}_{1} state.

3He + 4He	NCSM	NCSMC	Exp.	Refs.
$r_{\mathrm{ch}}$ [fm]	2.38	2.62	2.647(17)	Nörtershäuser et al. (2009)
$r_m$ [fm]	2.15	2.41	2.42(4)	Dobrovolsky et al. (2019)
$Q$ [e fm2]	-4.57	-6.14	-
$\mu$ [${\mu }_N$]	-1.14	-1.16	-1.3995(5)	Nörtershäuser et al. (2009)
B(E2) [e2 fm4]	10.90	20.02	26(6)(3)	Henderson et al. (2019)
B(M1) [${\mu }_N^2$]	1.55	1.49	-
${\left(C_{p3/2}\right)}^2$ [fm-1]	-	15.78	${23.3}_{-2.3}^{+1.0}$	Tursunmahatov and Yarmukhamedov (2012)
${\left(C_{p1/2}\right)}^2$ [fm-1]	-	13.22	${15.9}_{-1.5}^{+0.6}$	Tursunmahatov and Yarmukhamedov (2012)

Table 3. Table 3: Energies ( E r subscript 𝐸 𝑟 E_{r} ) and widths ( Γ Γ \Gamma ) in MeV of the resonant states of 7 Be computed with the NCSMC and compared with the existing experimental data  Tilley  et al. ( 2002 ) . Only the resonances with T = 1 / 2 𝑇 1 2 T=1/2 are considered and the resonance energies are given with respect the threshold of the corresponding reaction.

3He + 4He	NCSMC		Exp.
$J^{\pi }$	$E_r$	$\Gamma$	$E_r$	$E_x$	$\Gamma$
$7/2_1^{-}$	3.61	0.33	2.98	4.57	0.175
$5/2_1^{-}$	4.87	1.00	5.14	6.73	1.2
$7/2_2^{-}$	9.98	0.40	7.68	9.27	-
6Li + p	NCSMC		Exp.
$J^{\pi }$	$E_r$	$\Gamma$	$E_r$	$E_x$	$\Gamma$
$5/2_2^{-}$	1.83	0.66	1.60	7.21	0.40
$7/2_2^{-}$	3.99	0.60	3.66	9.27	-
$3/2_2^{-}$	3.24	2.19	4.29	9.9	1.8

Table 4. Table 4: Energies ( E r subscript 𝐸 𝑟 E_{r} ) and widths ( Γ Γ \Gamma ) in MeV of some of the new predicted resonant states of 7 Be computed with the NCSMC. Only the resonances with T = 1 / 2 𝑇 1 2 T=1/2 are considered and the resonance energies are given with respect the threshold of the 6 Li + p mass partition.

6Li + p	NCSMC
$J^{\pi }$	$E_r$	$\Gamma$
$1/2^{-}$	4.11	2.15
$5/2^{-}$	6.79	4.47
$1/2^{+}$	4.22	0.96
$3/2^{+}$	7.26	1.69

Table 5. Table 5: Comparison between the NCSMC and the experimental relative ( E B subscript 𝐸 𝐵 E_{B} ) and total ( E 𝐸 E ) energies of the J π = 3 / 2 1 − superscript 𝐽 𝜋 3 superscript subscript 2 1 J^{\pi}=3/2_{1}^{-} and 1 / 2 1 − 1 superscript subscript 2 1 1/2_{1}^{-} bound states of 7 Li produced with the 3 H + 4 He, 6 Li + n, and 6 He + p reactions. Here, the E B subscript 𝐸 𝐵 E_{B} energy represents the energy of the state with respect the threshold of the reaction. All calculations were performed at N max = 11 subscript 𝑁 max 11 N_{\mathrm{max}}=11 in the HO expansion and using the SRG-evolved NN N 3 LO chiral interaction  Entem and Machleidt ( 2003 ) at the resolution scale of λ SRG = 2.15 subscript 𝜆 SRG 2.15 \lambda_{\mathrm{SRG}}=2.15 fm -1 .

3H + 4He $\leftrightarrow$ 7Li	$J^{\pi }=3/2^{-}$	$J^{\pi }=1/2^{-}$
$E_B$    [MeV]	-2.43	-2.15
Exp. [MeV]	-2.467	-1.989
$E$     [MeV]	-38.65	-38.37
Exp. [MeV]	-39.25	-38.77
6Li + n $\leftrightarrow$ 7Li	$J^{\pi }=3/2^{-}$	$J^{\pi }=1/2^{-}$
$E_B$    [MeV]	-7.38	-7.05
Exp. [MeV]	-7.250	-6.772
$E$     [MeV]	-38.13	-37.79
Exp. [MeV]	-39.25	-38.77
6He + p $\leftrightarrow$ 7Li	$J^{\pi }=3/2^{-}$	$J^{\pi }=1/2^{-}$
$E_B$    [MeV]	-10.39	-10.06
Exp. [MeV]	-9.975	-9.498
$E$     [MeV]	-38.06	-37.73
Exp. [MeV]	-39.25	-38.77

Table 6. Table 6: Properties of the ground state of 7 Li computed with the NCSM and NCSMC approaches using the 3 H + 4 He mass partition and compared with the experimental data. The reduced transition probabilities are from the ground state 3 / 2 1 − 3 superscript subscript 2 1 3/2_{1}^{-} to the first excited state 1 / 2 1 − 1 superscript subscript 2 1 1/2_{1}^{-} . The ANCs C l ​ j subscript 𝐶 𝑙 𝑗 C_{lj} are shown for both the ground state and the 1 / 2 1 − 1 subscript superscript 2 1 1/2^{-}_{1} state.

3H + 4He	NCSM	NCSMC	Exp.	Refs.
$r_{\mathrm{ch}}$ [fm]	2.21	2.42	2.39(3)	Jager et al. (1974)
$Q$ [e fm2]	-2.67	-3.72	-4.00(3)	Voelk and Fick (1991)
$\mu$ [${\mu }_N$]	3.00	3.02	3.256	Raghavan (1989)
B(E2) [e2 fm4]	3.49	7.12	8.3(5)	Weller et al. (1985)
B(M1) [${\mu }_N^2$]	2.05	2.00	-
${\left(C_{p3/2}\right)}^2$ [fm-1]	-	12.21	12.74(110)	Igamov and Yarmukhamedov (2007)
${\left(C_{p1/2}\right)}^2$ [fm-1]	-	9.99	9.00(90)	Igamov and Yarmukhamedov (2007)

Table 7. Table 7: Energies ( E r subscript 𝐸 𝑟 E_{r} ) and widths ( Γ Γ \Gamma ) in MeV of the resonant states of 7 Li computed with the NCSMC and compared with the existing experimental data  Tilley  et al. ( 2002 ) . The resonance energies are given with respect the threshold of the corresponding mass partition. All the states have isospin T = 1 / 2 𝑇 1 2 T=1/2 except the 3 / 2 3 − 3 superscript subscript 2 3 3/2_{3}^{-} state in 6 He + p which has T = 3 / 2 𝑇 3 2 T=3/2 .

3H + 4He	NCSMC		Exp.
$J^{\pi }$	$E_r$	$\Gamma$	$E_r$	$E_x$	$\Gamma$
$7/2_1^{-}$	2.79	0.214	2.18	4.652	0.069
$5/2_1^{-}$	4.04	0.785	4.14	6.604	0.918
$7/2_2^{-}$	9.33	0.435	7.10	9.57	0.437
6Li + n	NCSMC		Exp.
$J^{\pi }$	$E_r$	$\Gamma$	$E_r$	$E_x$	$\Gamma$
$5/2_2^{-}$	0.48	0.21	0.20	7.454	0.080
$3/2_2^{-}$	1.83	1.70	1.50	8.75	4.712
$1/2_2^{-}$	2.60	2.44	1.84	9.09	2.752
$7/2_2^{-}$	2.91	0.039	2.32	9.57	0.437
6He + p	NCSMC		Exp.
$J^{\pi }$	$E_r$	$\Gamma$	$E_r$	$E_x$	$\Gamma$
$3/2_3^{-}$	1.74	0.63	1.26	11.24	0.26

Table 8. Table 8: Energies ( E r subscript 𝐸 𝑟 E_{r} ) and widths ( Γ Γ \Gamma ) in MeV of some of the new predicted resonant states of 7 Li computed within the NCSMC. The resonance energies are given with respect the threshold of the 6 Li + n and 6 He + p mass partitions.

6Li + n	NCSMC
$J^{\pi }T$	$E_r$	$\Gamma$
$5/2^{-}\mathrm{\hspace{0.17em}1}/2$	4.79	7.73
$1/2^{-}\mathrm{\hspace{0.17em}1}/2$	7.71	6.13
$1/2^{+}\mathrm{\hspace{0.17em}1}/2$	3.78	1.11
6He + p	NCSMC
$J^{\pi }T$	$E_r$	$\Gamma$
$1/2^{+}\mathrm{\hspace{0.17em}1}/2$	0.23	0.13
$3/2^{-}\mathrm{\hspace{0.17em}1}/2$	1.94	0.41
$1/2^{-}\mathrm{\hspace{0.17em}3}/2$	3.03	2.65
$5/2^{-}\mathrm{\hspace{0.17em}3}/2$	4.43	2.10
$3/2^{-}\mathrm{\hspace{0.17em}3}/2$	4.55	5.21

Equations

$$\hat{H}=\frac{1}{A}\sum _{i<j=1}^A\frac{({\hat{\vec{p}}}_i-{\hat{\vec{p}}}_j{)}^2}{2m}+\sum _{i<j=1}^A{\hat{V}}_{ij}^{NN},$$

$$|{\Psi }_A^{J^{\pi }T}⟩=\sum _{N=0}^{N_{\max }}\sum _i{c}_{Ni}^{J^{\pi }T}|ANi{J}^{\pi }T⟩.$$

$$|{\Psi }_{A\text{=}7,\frac{1}{2}}^{J^{\pi }T}⟩=$$

$$+\sum _{\nu }\int dr{r}^2\frac{{\gamma }_{\nu }^{J^{\pi }T}(r)}{r}{\mathcal{A}}_{\nu }|{\Phi }_{\nu r,\frac{1}{2}}^{J^{\pi }T}⟩,$$

$$|{\Psi }_{A\text{=}7,-\frac{1}{2}}^{J^{\pi }T}⟩=$$

$$+\sum _{\nu }\int dr{r}^2\frac{{\gamma }_{\nu }^{J^{\pi }T}(r)}{r}{\mathcal{A}}_{\nu }|{\Phi }_{\nu r,-\frac{1}{2}}^{J^{\pi }T}⟩,$$

$$|{\Phi }_{\nu r,\frac{1}{2}}^{J^{\pi }T}⟩=$$

$$\displaystyle\times\,Y_{\ell}(\hat{r}_{4,3})\Big{]}^{(J^{\pi}T)}_{\frac{1}{2}}\;\frac{\delta(r{-}r_{4,3})}{rr_{4,3}}\,,$$

$$|{\Phi }_{\nu r,\frac{1}{2}}^{J^{\pi }T}⟩=$$

$$\times \frac{\delta (r-r_{6,1})}{r{r}_{6,1}},$$

$$|{\Phi }_{\nu r,-\frac{1}{2}}^{J^{\pi }T}⟩=$$

$$\displaystyle\times\,Y_{\ell}(\hat{r}_{4,3})\Big{]}^{(J^{\pi}T)}_{-\frac{1}{2}}\;\frac{\delta(r{-}r_{4,3})}{rr_{4,3}}\,,$$

$$|{\Phi }_{\nu r,-\frac{1}{2}}^{J^{\pi }T}⟩=$$

$$\times \frac{\delta (r-r_{6,1})}{r{r}_{6,1}},$$

$$|{\Phi }_{\nu r,-\frac{1}{2}}^{J^{\pi }T}⟩=$$

$$\times \frac{\delta (r-r_{6,1})}{r{r}_{6,1}}.$$

Full text

Purpose: In this work we study the properties of 7Be and 7Li within the no-core shell model with continuum (NCSMC) method, using chiral nucleon-nucleon interactions as the only input, and analyze all the binary mass partitions involved in the formation of these systems.

Methods: The NCSMC is an ab initio method applicable to light nuclei that provides a unified description of bound and scattering states and thus is well suited to investigate systems with many resonances and pronounced clustering like 7Be and 7Li.

Results: Our calculations reproduce all the experimentally known states of the two systems and provide predictions for several new resonances of both parities. Some of these new possible resonances are built on the ground states of 6Li and 6He, and thus represent a robust prediction. We do not find any resonance in the p${+}^{6}$Li mass partition near the threshold. On the other hand, in the p${+}^{6}$He mass partition of 7Li we observe an $S$-wave resonance near the threshold producing a very pronounced peak in the calculated S factor of the ${}^{6}\mathrm{He}(\mathrm{p},\gamma)^{7}\mathrm{Li}$ radiative-capture reaction.

Conclusions: While we do not find a resonance near the thershold in the p${+}^{6}$Li channel, in the case of 6He${+}$p reaction a resonant $S$-wave state is predicted at a very low energy above the reaction threshold, which could be relevant for astrophysics and its implications should be investigated. We note though that this state lies above the three-body breakup threshold not included in our method and may be influenced by three-body continuum correlations.

I Introduction

The $A{=}7$ systems, in particular 7Be and 7Li, play an important role in primordial nucleosynthesis, nuclear astrophysics, and fusion energy experiments.

The ${}^{3}\mathrm{He}(\alpha,\gamma)^{7}\mathrm{Be}$ and ${}^{3}\mathrm{H}(\alpha,\gamma)^{7}\mathrm{Li}$ radiative-capture processes are crucial for the determination of the primordial 7Li abundance in the early universe Burles et al. (1999); Nollett and Burles (2000); Nollett (2001) and for predicting the correct fraction of pp-chain branches resulting in 7Be versus 8B neutrinos Adelberger et al. (1998, 2011). Measuring these reactions at the very low solar energies required for astrophysics modeling is extremely challenging due to the suppression of the reaction probability caused by the Coulomb repulsion between the reactants. Consequently, despite the several experimental measurements Singh et al. (2004); Bemmerer et al. (2006); Confortola et al. (2007); Brown et al. (2007); Di Leva et al. (2009); Carmona-Gallardo et al. (2012); Bordeanu et al. (2013); Brune et al. (1994); Kontos et al. (2013), a predictive theoretical description is needed to reliably guide the extrapolation of higher-energy experimental data down to the desired solar values Leva et al. (2016). A summary of the experimental status of production and destruction of 7Be at the relevant energies for astrophysics can be found in Ref. Leva et al. (2016).

A recent experimental investigation of the ${}^{6}\mathrm{Li}(p,\gamma)^{7}\mathrm{Be}$ capture reaction at Lanzhou He et al. (2013) hinted at a possible resonant enhancement of this cross section near the threshold. If real, this enhancement would have consequences for nuclear astrophysics. A new experiment Broggini et al. (2018) is also in progress at the Laboratory of Underground Nuclear Astrophysics lun (LUNA). A theoretical investigation of $S$-wave resonances in 7Be is then called for.

Furthermore, the {}^{6}\mathrm{Li}(n,$${}^{3}\mathrm{H}){}^{4}\mathrm{He} resonant reaction is important for tritium breeding at facilities dedicated to the demonstration of fusion energy generation with deuterium-tritium fuel such as ITER ite .

Starting from the early 60’s several theoretical papers have been devoted to the microscopic description of the ${}^{3}\mathrm{He}(\alpha,\gamma)^{7}\mathrm{Be}$ and ${}^{3}\mathrm{H}(\alpha,\gamma)^{7}\mathrm{Li}$ radiative-capture processes Tombrello and Parker (1963); Kajino (1986); Mertelmeier and Hofmann (1986); Csótó and Langanke (2000); Neff (2011); Gustavino et al. (2016); Dohet-Eraly et al. (2016); Bertulani et al. (2016); Solovyev and Igashov (2017); Tursunov et al. (2018); Takács et al. (2018); Higa et al. (2018). In particular, in Ref. Dohet-Eraly et al. (2016) these reactions were investigated within the ab initio No-Core Shell Model with Continuum (NCSMC) using a renormalized chiral nucleon-nucleon interaction. The calculated astrophysical $S$ factors displayed a reasonable agreement with experimental data.

The scope of the present work is to extend this previous investigation of 7Be and 7Li nuclei published in Ref. Dohet-Eraly et al. (2016) considering all the possible binary mass partitions involved in the formation of such systems and analyzing the spectra in a wider energy range.

The paper is organized as follows: In Sec. II we give the formalism of the NCSMC, while in Sec III we display the results for the 7Be and 7Li. We first show the comparison with the experimentally known states and then we will show our predictions for new possible resonant states. Finally, in Sec. IV we draw our conclusions.

II Theoretical Framework

Since the aim of this paper is to extend the work of Ref. Dohet-Eraly et al. (2016), we adopt the same conditions. The starting point of our approach is the microscopic Hamiltonian

[TABLE]

which describes nuclei as systems of $A$ non-relativistic point-like nucleons interacting through realistic inter-nucleon interactions. For consistency with Ref. Dohet-Eraly et al. (2016), we only use the nucleon-nucleon ($NN$) interaction. Typically, also three-nucleon ($3N$) contributions can be taken into account. In the present work we adopt the $NN$ chiral interaction Entem and Machleidt (2003) developed by Entem and Machleidt up to the fourth order (N3LO) in the chiral expansion. In the framework of chiral effective field theory (EFT) Weinberg (1990, 1991), the Lagrangian is expanded in powers of $(Q/\Lambda_{\chi})^{n}$, where $Q$ is the external momentum and $\Lambda_{\chi}$ represents the hard scale of the theory and it is chosen of the order of 1 GeV. Such an expansion allows a systematic improvement of the interaction and provides a hierarchy of the $NN$ and many-nucleon interactions which naturally arise in a consistent scheme Ordóñez et al. (1994); van Kolck (1994); Epelbaum et al. (2002); Epelbaum (2006).

A faster convergence of the NCSMC calculations is obtained by softening the chiral interaction through the similarity renormalization group (SRG) technique Wegner (1994); Bogner et al. (2007); Roth et al. (2008); Bogner et al. (2010); Jurgenson et al. (2009). The general scheme for such a renormalization procedure is to keep two- and three-body SRG induced terms in all calculations, even in the case when the initial chiral $3N$ force is not included. In the current work, to be consistent with Ref. Dohet-Eraly et al. (2016) and due to the too high computational effort of dealing with a three-nucleon projectile, we also discard the induced $3N$ terms.

The $NN$ potential is softened via the SRG and we evolved the interaction up to $\lambda_{\mathrm{SRG}}=2.15$ fm*-1*, where the parameter $\lambda_{\mathrm{SRG}}$ specifies the resolution scale at which the $NN$ potential is evolved. The value of $\lambda_{\mathrm{SRG}}$ adopted in this work is the same as that one used in Ref. Dohet-Eraly et al. (2016) and it has been chosen as the best value that allows us to reproduce the experimental separation energies of 7Be and 7Li.

The NCSMC calculation of the scattering observables requires in input the eigenstates of the two colliding nuclei and the eigenstate of the compound system created during the interaction process. These eigenstates are calculated with the no-core shell model (NCSM)  Navrátil et al. (2000a, b); Barrett et al. (2013), an ab initio method where all nucleons are considered as active degrees of freedom and the many-body wave function is expanded over a complete set of antisymmetric $A$-nucleon harmonic-oscillator (HO) basis states containing up to $N_{\rm max}$ HO excitations above the lowest Pauli-principle-allowed configuration:

[TABLE]

Here, $N$ denotes the total number of HO excitations of all nucleons above the minimum configuration, $J^{\pi}T$ are the total angular momentum, parity and isospin, and $i$ additional quantum numbers. The sum over $N$ is restricted by parity to either an even or odd sequence. The basis is further characterized by the frequency $\Omega$ of the HO well. Square-integrable energy eigenstates expanded over the $N_{\rm max}\hbar\Omega$ basis, $\ket{ANiJ^{\pi}T}$, are obtained by diagonalizing the intrinsic Hamiltonian of Eq.(1). In this work the value of the HO frequency has been chosen to be $\hbar\Omega=20$ MeV, again consistently with Ref. Dohet-Eraly et al. (2016).

The NCSMC wave function is then represented as the generalized cluster expansion. For 7Be we have

[TABLE]

while for 7Li we have

[TABLE]

The first term of Eqs.(3) and (4) consists of an expansion over NCSM eigenstates of the aggregate system (7Be and 7Li) indexed by $\lambda$. These states are well suited to explain the localized correlations of the two 7-body systems, but are inadequate to describe clustering and scattering properties that are better addressed by the second term corresponding to an expansion over the antisymmetrized channel states in the spirit of the resonating group method Wildermuth and Tang (1977); Tang et al. (1978); Fliessbach and Walliser (1982); Langanke and Friedrich (1986); Hofmann and Hale (2008). For 7Be we have

[TABLE]

while for 7Li we have

[TABLE]

The $\nu$ index represents all the quantum numbers on the right-hand side not appearing on the left-hand side and the subscript $\pm\frac{1}{2}$ is the isospin projection, i.e., $(Z-N)/2$. All these terms describe the relative motion of the two colliding nuclei involved in the formation of the composite system during the scattering process. Here, the coordinate $\vec{r}_{4,3}$ in Eq.(5) is the separation distance between the center-of-mass of 4He and 3He, while $\vec{r}_{6,1}$ in Eq.(6) represents the separation distance between 6Li and the proton. The same meaning holds for the separation distances in the channel states of 7Li.

The discrete expansion coefficients $c_{\lambda}^{J^{\pi}T}$ and the continuous relative-motion amplitudes $\gamma_{\nu}^{J^{\pi}T}(r)$ are the solution of the generalized eigenvalue problem derived by representing the Schrödinger equation in the model space of the expansions (3) and (4) Navrátil et al. (2016). The resulting NCSMC equations are solved by the coupled-channel R-matrix method on a Lagrange mesh Descouvemont and Baye (2010); Hesse et al. (1998, 2002). We emphasize that the sums over the index $\nu$ in Eqs.(3) and (4) include all the mass partitions involved in the formation of the compound systems 7Be and 7Li. The NCSMC calculation of these systems with the coupling between the all binary mass partitions is however beyond our present capabilities due to the challenge of dealing with a three-body projectile involved in the 3He + 4He and 3H + 4He reactions. In the present work we will thus consider the different partitions separately. Applications of the NCSMC with a two-body projectile and with the coupling between different mass partitions can be found in Refs. Raimondi et al. (2016); Hupin et al. (2019).

III Results

The NCSMC calculations require in input the NCSM eigenstates of the colliding nuclei and of the composite system. In this work we used eight lowest negative-parity and six lowest positive-parity NCSM eigenstates for 7Be and 7Li with total angular momentum $J\in\{1/2,3/2,5/2,7/2\}$ and isospin $T=1/2$. Concerning the reactants, we used the ground state for 4He $[(J^{\pi}T)=(0^{+}0)]$ and 3He (3H) $[(J^{\pi}T)=1/2^{+}1/2]$, while we used four states for 6Li $[(J^{\pi}T)=(1^{+}0),(3^{+}0),(0^{+}1),(2^{+}1)]$ and two states for 6He $[(J^{\pi}T)=(0^{+}1),(2^{+}1)]$. All the calculations were performed using the SRG evolved chiral N3LO $NN$ interaction Entem and Machleidt (2003) with $\lambda_{\mathrm{SRG}}=2.15$ fm*-1* and with $\hbar\Omega=20$ MeV. In the following subsections we present results obtained at $N_{\mathrm{max}}=11$, the largest space we could reach for technical reasons. However, we performed calculations also at smaller $N_{\mathrm{max}}$ spaces to check convergence.

Every single mass partition was studied separately, neglecting the coupling with the other mass partitions. Depending on the reaction under consideration, the states of the composite system that are below the reaction threshold come out from our calculations as bound states. In the following we will thus show the results for 7Be and 7Li in two separate subsections, presenting first the values of the ground state ($3/2_{1}^{-}$) and the first excited state ($1/2_{1}^{-}$) of both systems obtained from the calculation performed with the different mass partitions. We will then show the results for the resonant states and the comparison with the experimentally known ones, and we will finally show our predictions for new possible resonances.

III.1 The 7Be system

We studied the 7Be nucleus within the NCSMC using the 3He + 4He and 6Li + p reactions, which represent the only two binary mass partitions in this system. All the other mass partitions involve at least three reactants and are not considered in the present work.

The relative ($E_{B}$) and total ($E$) energies of the $J^{\pi}=3/2_{1}^{-}$ and $1/2_{1}^{-}$ states of 7Be computed with the NCSMC are displayed in Tab. 1 and are compared with the corresponding experimental values. Here, the $E_{B}$ energies represent the energy of the state under consideration with respect the threshold of the considered reaction. We can see that in both cases, not only the $E_{B}$ energies, but also the total ones are in a very good agreement with the experimental values. This is particularly true for the 3He + 4He reaction, where the difference between the theoretical and the experimental total energies is less than 1 MeV.

In Tab. 2 we also report the properties of the ground sate of 7Be obtained from the study of the 3He + 4He reaction with the NCSM and NCSMC, and we compare our results with the existing experimental data. Besides the values of the charge radius ($r_{\mathrm{ch}}$), quadrupole moment ($Q$), and magnetic dipole moment ($\mu$), already reported in Ref. Dohet-Eraly et al. (2016), Tab. 2 is now updated with the theoretical predictions of the matter radius and the reduced transition probabilities B(M1; $3/2_{1}^{-}\rightarrow 1/2_{1}^{-}$) and B(E2; $3/2_{1}^{-}\rightarrow 1/2_{1}^{-}$). The experimental measurement of the B(E2) value has been recently performed at the University of Notre Dame Henderson et al. (2019). Our NCSMC result is consistent with this new measurement. Also recently, a new matter radius measurement was published in Ref. Dobrovolsky et al. (2019). Our calculation is in an excellent agreement with the reported value. In addition, we also present the asymptotic normalization coefficients (ANCs) for both the ground state and the $1/2^{-}_{1}$ bound state. The cluster form factors for the two states defined by $r\bra{\Phi^{J^{\pi}T}_{\nu r,\frac{1}{2}}}{\mathcal{A}}_{\nu}\ket{\Psi^{J^{\pi}T}_{A\texttt{=}7,\frac{1}{2}}}$ with the ket and bra from Eqs. (3) and (5), respectively, are shown in Fig. 1. We can see that the NCSMC wave functions extend beyond 10 fm.

In Fig. 2 we show the phase shifts computed with the NCSMC and compared to the experimental spectrum of 7Be. On the left-hand side of the figure we show the results for the two mass partitions. The lower panel on the left contains the phase shifts for the 3He + 4He reaction, while the upper panel contains the results obtained from the 6Li + p calculation. The phase shifts are displayed as functions of the experimental excitation energy and the experimental values of the thresholds are displayed with the two vertical dashed lines. The comparison between the theoretical and the experimental resonance energies are shown in Tab. 3. The column on the right displays the experimental excitation energy spectrum. The solid lines are the known resonant states with their relative energies on the left and their $J^{\pi}$ quantum numbers on the right, while the dashed lines show the thresholds of the two processes, which match with the dashed lines in the left panels.

From Fig. 2 we can clearly see that our results are in a very good agreement with the experimental energy spectrum and all the states are reproduced in the correct order. In the spectrum of the 3He + 4He reaction we have four phase shifts corresponding to the $7/2_{1}^{-},5/2_{1}^{-},5/2_{2}^{-},7/2_{2}^{-}$ states, while for 6Li + p we have the $5/2_{1}^{-},5/2_{2}^{-},7/2_{2}^{-},3/2_{2}^{-}$ states. The $7/2_{1}^{-}$ state is below the 6Li + p threshold and thus it is only showed in the spectrum of the 3He + 4He process. Of course this state comes out from the 6Li + p NCSMC calculation as a bound state, exactly as it happens for the $3/2_{1}^{-}$ and the $1/2_{1}^{-}$ ones. The $3/2_{3}^{-}$ state with $T=3/2$ at $11.01$ MeV is not shown here and it will be discussed in the mirror 7Li system. Finally, the sharp peak in the ${}^{2}\mathrm{F}_{5/2}$ phase shift of the 3He + 4He spectrum and the sharp peak in the ${}^{2}\mathrm{P}_{5/2}$ phase shift of the 6Li + p spectrum deserve a comment. Experimentally, the cross section of the 3He + 4He process only shows a peak in correspondence of the energy of the $5/2_{1}^{-}$ state, while for 6Li + p a peak is only found at the energy of the $5/2_{2}^{-}$ state. Thus, the contribution of the 3He + 4He to the $5/2_{2}^{-}$ state is basically negligible, exactly as it happens for the $5/2_{1}^{-}$ state in the 6Li + p spectrum. Even without the coupling between the two mass partitions, our results correctly reproduce the contributions to the two $5/2^{-}$ states in both spectra.

In Tab. 3 we display the numerical values of the energies and widths of the known states computed with the NCSMC and we compare our results with the experimental values, where we also show the experimental excitation energy, for a better comparison with Fig. 2. Both energies and widths are nicely reproduced even though the agreement with the experimental data is not perfect. For the 3He + 4He mass partition we did not include in the table the $5/2_{2}^{-}$ state, while for 6Li + p we left out the $5/2_{1}^{-}$ state. These two states are indeed too narrow and the widths are negligible in those respective mass partitions.

In Fig. 3 we show our predictions for new possible $T=1/2$ resonances that are experimentally unknown. As seen in the experimental spectrum in Fig. 2, all known states are of negative parity. The interesting result is that we found new possible resonances of both parities. In panel (a) we present the ${}^{4}\mathrm{P}_{1/2}$ and ${}^{6}\mathrm{P}_{5/2}$ phase shifts built on the 6Li ground state and the $3^{+}0$ excited state, respectively. We note that the first ${}^{4}\mathrm{P}_{1/2}$ resonance has been observed experimentally in 7Li, see Sect. III.2. Consequently, we expect that it should be possible to observe it in 7Be as well. The $S$-wave phase shifts built on the four 6Li states included in our calculations are plotted in panel (b). We observe a resonance behavior in particular in the phase shifts built on the $0^{+}$ and $2^{+}$ $T{=}1$ states. The corresponding eigenphase shifts are displayed in the panel (c) and confirm resonances in $1/2^{+},3/2^{+}$ and $5/2^{+}$ channels. The resonance energies and widths of some of these predicted resonances are summarized in Tab. 4. We note that all these predicted resonances are in three-body continuum not included in our calculations. In particular, we anticipate that the $3/2^{+}$ and $5/2^{+}$ will be influenced by the three-body continuum as the $2^{+}1$ states of 6Li is rather broad.

The ${}^{6}\mathrm{Li}(p,\gamma)^{7}\mathrm{Be}$ radiative capture reaction

As discussed in the Introduction, recently the ${}^{6}\mathrm{Li}(p,\gamma)^{7}\mathrm{Be}$ capture reaction has been investigated at Lanzhou He et al. (2013) and suggested a possible new structure just above the threshold that has not been observed in the previous measurements Switkowski et al. (1979); Tingwell et al. (1987); Cecil et al. (1992); Prior et al. (2004). In our analysis, we also computed the capture reaction. We do not find any resonance near the 6Li${+}$p threshold as shown in Figs. 2 and  3. Consequently, the shape of our calculated $S$ factor reproduce the trend of the earlier experimental data with no evidence of a new structure at very low energy. Recently, the same reaction has been also investigated in Ref. Dong et al. (2017); Gnech and Marcucci (2019) with similar findings. As in our calculation the coupling between the 6Li${+}$p and 3He${+}^{4}$He mass partitions is not included, our calculated $S$ factor overestimates the data. The extension of our formalism to include the coupling is under way. It is clear that it will improve the description of the magnitude of the $S$ factor but it will not change its shape.

The ${}^{3}\mathrm{He}(\alpha,\gamma)^{7}\mathrm{Be}$ radiative capture reaction

The S factor of the ${}^{3}\mathrm{He}(\alpha,\gamma)^{7}\mathrm{Be}$ radiative capture calculated with the same Hamiltonian and approach as employed in this paper was presented for kinetic energy range up to 3.8 MeV in the center of mass by black solid line in the top panel of Fig. 5 of Ref. Dohet-Eraly et al. (2016). Recently, motivated by the above discussed Lanzhou experiment a new ${}^{3}\mathrm{He}(\alpha,\gamma)^{7}\mathrm{Be}$ measurement has been performed in the energy range between 4 and 4.5 MeV Szücs et al. (2019), i.e., in the region just above the threshold of the ${}^{6}\mathrm{Li}(p,\gamma)^{7}\mathrm{Be}$ capture. This first experiment above 3.1 MeV in the center of mass found a flat structureless S factor. For example, at 4.42 MeV, the $S_{34}=0.55(4)$ keV b has been reported Szücs et al. (2019). Our calculated NCSMC S factor beyond the range shown in Ref. Dohet-Eraly et al. (2016) is monotonically increasing; at 4.42 MeV, we find $S_{34}=0.48$ keV b, which is slightly below the experiment Szücs et al. (2019).

III.2 The 7Li system

We present now the results for 7Li that we obtained analyzing the 3H + 4He, 6Li + n, and 6He + p reactions. In this case there is one more reaction than in the 7Be case and also here these are the all possible binary mass partitions involved in the formation of the 7Li system. Also in this case, the three processes were studied separately.

In Tab. 5 we summarize the values of the relative and total energies of the two $3/2_{1}^{-}$ and $1/2_{1}^{-}$ bound states. Since the 3H + 4He reaction has the lowest energy threshold, the experimental values are well reproduced and the difference between the total energies is less than 1 MeV, similarly as for 3He + 4He mass partition in 7Be. The agreement is also good in the other two cases even if the differences are a bit larger.

In Tab. 6 we display the properties of the ground state obtained from the study of the 3H + 4He reaction with the NCSM and NCSMC. The theoretical results for $r_{\mathrm{ch}}$, $Q$, and $\mu$ are compared with the experimental values, and the theoretical predictions of B(M1; $3/2_{1}^{-}\rightarrow 1/2_{1}^{-}$) and B(E2; $3/2_{1}^{-}\rightarrow 1/2_{1}^{-}$) are reported. As in the 7Be case, the $r_{\mathrm{ch}}$, $Q$, and B(E2) values increase substantially when NCSMC is applied due to the proper physical tail of the NCSMC wave functions. We also present the NCSMC ANCs for the ground state and the $1/2^{-}_{1}$ state that compare quite well with the values extracted from experimental data. The cluster form factors for the two states resamble closely the 7Be ones shown in Fig. 1. We therefore do not show them.

In Fig. 4 we show the results for the phase shifts. The figure is basically organized as Fig. 2, here the difference is that we have three mass partitions instead of two. On the left-hand side there are the three panels displaying the phase shifts of the corresponding process, while on the right-hand side there is the experimental spectrum. Again, the solid lines represent the energy levels while the dashed lines show the reaction thresholds. The theoretical phase shifts in three panels on the left are adjusted to the experimental thresholds displayed with dashed lines. As in the previous case, also for 7Li our method is able to reproduce all the energy levels in the correct order. Two differences must be addressed with respect the previous case. The first difference concerns the $T=3/2$ state at $11.24$ MeV, that we now discuss and demonstrate that it is theoretically well reproduced in particular in the 6He + p scattering. The second difference concerns the two $5/2^{-}$ states at the energies of $6.604$ and $7.454$ MeV, respectively. In this case the threshold of the 6Li + n reaction is exactly in between these two states and thus only the $5/2_{2}^{-}$ state appears in the spectrum of this mass partition. The $5/2_{1}^{-}$ resonance is shown in the spectrum of the 3H + 4He scattering, and once again we find a situation similar to the previous one. The experimental cross section for 3H + 4He has a peak in correspondence of the $5/2_{1}^{-}$ state, while for 6Li + n the peak is found at the energy of the $5/2_{2}^{-}$ state. This experimental observation is reproduced by our calculation and the very small contribution to the $5/2_{2}^{-}$ state from the 3H + 4He process can be seen in its spectrum at the excitation energy of $\sim 9.5$ MeV. The last comment concerns the ${}^{8}\mathrm{P}_{7/2}$ phase shifts in 6Li + n scattering, that is built on the $3^{+}$ state of 6Li and in the figure seems to appear at the threshold of 6He + p. This is purely accidental and simply due to the shift of the theoretical results to the experimental thresholds.

In Tab. 7 we report the energies and widths of the resonant states of 7Li computed with the NCSMC and compared to the experimental values. For 3H + 4He reaction we do not include the $5/2_{2}^{-}$ state. The general agreement with the experimental values is good, and the resonant centroids are basically reproduced by our calculations. In this case, the energy of the $7/2_{2}^{-}$ state is better reproduced by the 6Li + n calculation, even though its contribution to the width is very small.

Also for 7Li our calculations predict new resonant states that are shown in Fig. 5 for 6Li + n, and in Fig. 6 for 6He + p, respectively. The resonance energies and widths of some of these predicted resonances are summarized in Tab. 8. As for 7Be, all currently experimentally known 7Li states have negative parity, while our NCSMC calculations predict new resonances of both parities. Panel (a) of  5 shows the ${}^{4}\mathrm{P}_{1/2}$ and ${}^{6}\mathrm{P}_{5/2}$ phase shifts built on the 6Li ground state and the $3^{+}0$ excited state, respectively. We note that the first resonance in ${}^{4}\mathrm{P}_{1/2}$ partial wave corresponds to the experimentally known state at 9.02 MeV and the phase shift is also included in Fig. 4. The narrow resonances in the ${}^{2}\mathrm{P}_{1/2}$ partial wave built on the 6He ground state presented in Fig. 6 will mix with the much broader ${}^{4}\mathrm{P}_{1/2}(1^{+}0)$ partial waves in a calculation that couples different mass partitions as well as in experiment. Consequently, their widths obtained in the present calculations are unrealistically small and we do not include these resonances in Tab.  8. We note that for $T{=}3/2$, we predict a new $1/2^{-}$ resonance built on the 6He ground state as well as a $3/2^{-}$ and $5/2^{-}$ resonances built in the 6He $2^{+}1$ state, see panel (b) of Fig. 6. As the latter state is unbound, our predictions for the $3/2^{-}$ and $5/2^{-}$ resonances are less robust than that of the $1/2^{-}$ resonance. The $S$-wave phase shifts built on the four 6Li and two 6He states included in our calculations are plotted in panel (b) of Fig. 5 and panel (c) of Fig. 6, respectively. Contrary to the situation in 7Be (Fig. 3), the find resonant behavior in the 6Li+n scattering in the partial waves built on the $T{=}0$ $1^{+}$ and $3^{+}$ 6Li states with the resonances appearing below the $T{=}1$ $0^{+}$ and $2^{+}$ states thresholds. In the 6He+p scattering, the $1/2^{+}$ resonace appear just above the threshold, also below the 6Li $0^{+}1$ state not coupled in the present calculations. Consequently, the prediction of the positive-parity $1/2^{+}S$-wave resonance in 6He+p appears robust. Still, we have to keep in mind that this state is in the three-body continuum (4He+d+n) that is not included in our calculations and, of course, this can affect its properties.

The ${}^{6}\mathrm{He}(p,\gamma)^{7}\mathrm{Li}$ radiative capture reaction

The sharp resonance near the threshold of the 6He + p reaction suggests a resonant $S$ factor for the ${}^{6}\mathrm{He}(p,\gamma)^{7}\mathrm{Li}$ radiative-capture reaction. Indeed, our calculated $S$ factor predicts a very pronounced and sharp peak just above the threshold. Its possible implications for astrophysics, if any, remain to be investigated. As pointed out in the previous subsection, the three-body continuum not included in our calculations could affect this resonance and the $S$ factor. Since we do not include the coupling between the different mass partitions, the magnitude of our calculated S factor is unrealistically large. Consequently, wedo not present the calculated S factor until the coupling of the 7Li mass partitions is implemented in our formalism. Experimental investigation of the ${}^{6}\mathrm{He}(p,\gamma)^{7}\mathrm{Li}$ radiative capture has been performed only at energies well above the threshold, e.g., at $E_{\rm{}^{6}He}{=}40$ MeV/A Sauvan et al. (2001).

IV Conclusions

In Ref. Dohet-Eraly et al. (2016) the 7Be and 7Li systems have been studied within the NCSMC approach investigating the 3He + 4He and the 3H + 4He reactions. In the present work we extended the work of Ref. Dohet-Eraly et al. (2016) and studied these systems in a wider energy range considering all the binary-mass partitions. In addition to the two previous ones, here we also studied the 6Li + p, 6Li + n, and 6He + p reactions and investigated 7Be and 7Li bound states as well as resonances and scattering states. Our results provide a very good description of the experimental energy spectrum of both systems. Not only the bound state energies but also the resonant states are nicely reproduced in the correct order. The widths of the known resonances are also well reproduced. Besides these known states we found several new resonances of both parities, some of them built on the ground state of 6Li and 6He. Finally, we also investigated the ${}^{6}\mathrm{Li}(p,\gamma)^{7}\mathrm{Be}$ and the ${}^{6}\mathrm{He}(p,\gamma)^{7}\mathrm{Li}$ radiative-capture processes. Contrary to the Lanzhou experiment, we did not find any resonance in the $S$-wave near the 6Li + p threshold. Our predicted ${}^{6}\mathrm{Li}(p,\gamma)^{7}\mathrm{Be}$ $S$ factor is non-resonant and smooth at low energies. On the other hand, we predict a pronounced $S$-wave resonance in 7Li near the 6He + p threshold that results in a sharp peak in our predicted $S$ factor for the ${}^{6}\mathrm{He}(p,\gamma)^{7}\mathrm{Li}$ reaction. Its possible implications for astrophysics, if any, remain to be investigated. It also must be noted that this state is already in three-body continuum that we did not include in our calculations and can thus affect our results.

The presented calculations can be improved in three ways. First, the coupling between the different mass partitions needs to be introduced, which would then allow to study transfer reactions such as {}^{6}\mathrm{Li}(\mathrm{n},$${}^{3}\mathrm{H}){}^{4}\mathrm{He}. Second, the chiral three-nucleon interaction shoud be included. Third, the three-body continuum should be considered. The work on the first two points is in progress. The third one is the most challenging and will require more technical development.

Acknowledgements.

This work was supported by the NSERC Grant No. SAPIN-2016-00033 and by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under Work Proposals No. SCW1158 and SCW0498. TRIUMF receives federal funding via a contribution agreement with the National Research Council of Canada. This work was prepared in part by LLNL under Contract No. DE-AC52-07NA27344. Computing support came from an INCITE Award on the Titan supercomputer of the Oak Ridge Leadership Computing Facility (OLCF) at ORNL, from Westgrid and Compute Canada, and from the LLNL institutional Computing Grand Challenge Program.