Description of ${}^7\mathrm{Be}$ and ${}^7\mathrm{Li}$ within the Gamow shell model

Background: ${}^7\mathrm{Li}$ and ${}^7\mathrm{Be}$ play an important role in big bang nucleosynthesis and nuclear astrophysics. The ${}^3\mathrm{H}({}^4\mathrm{He},\gamma ){}^7\mathrm{Li}$ radiative capture reaction is crucial for the determination of the primordial ${}^7\mathrm{Li}$ abundance. In nuclear astrophysics, lithium isotopes have attracted great interest because of the puzzling abundance of ${}^6\mathrm{Li}$ and ${}^7\mathrm{Li}$.

Purpose: In this work we study spectra of ${}^7\mathrm{Be}$ and ${}^7\mathrm{Li}$ and elastic scattering cross sections ${}^4\mathrm{He}({}^3\mathrm{He},{}^3\mathrm{He})$ and ${}^4\mathrm{He}({}^3\mathrm{H},{}^3\mathrm{H})$ within the Gamow shell model (GSM) in the coupled-channel formulation (GSM-CC). The evolution of channel amplitudes and spectroscopic factors in the vicinity of the channel threshold is studied for selected states.

Methods: GSM provides the open quantum system formulation of the nuclear shell model. In the representation of GSM-CC, GSM provides the unified theory of nuclear structure and reactions which is suited for the study of resonances in ${}^7\mathrm{Be}$ and ${}^7\mathrm{Li}$ and elastic scattering cross sections involving ${}^3\mathrm{H}$ and ${}^3\mathrm{He}$ projectiles.

Results: GSM-CC in the multimass partition formulation applied to a translationally invariant Hamiltonian with an effective finite-range two-body interaction reproduces well the spectra of ${}^7\mathrm{Be}$ and ${}^7\mathrm{Li}$ and elastic scattering reactions ${}^4\mathrm{He}({}^3\mathrm{He},{}^3\mathrm{He})$ and ${}^4\mathrm{He}({}^3\mathrm{H},{}^3\mathrm{H})$. Detailed analysis of the dependence of reaction channel amplitudes and spectroscopic factors on the distance from the particle decay threshold allowed us to demonstrate the alignment of the wave function in the vicinity of the decay threshold. This analysis also demonstrates the appearance of clustering in the GSM-CC wave function in the vicinity of the cluster decay threshold.

Conclusions: We demonstrated that GSM formulated in the basis of reaction channels including both cluster and proton/neutron channels allows us to describe both the spectra of nuclei with low-energy cluster thresholds and the low-energy elastic scattering reactions with proton, ${}^3\mathrm{H}$, and ${}^3\mathrm{He}$ projectiles. Studying dependence of the reaction channel amplitude and spectroscopic factor in a many-body state on the distance from the threshold, we showed an evolution of the ${}^3\mathrm{He},{}^3\mathrm{H}$ clustering with increasing separation energy from the cluster decay threshold and demonstrated a mechanism of the alignment of many-body wave function with the decay threshold [J. Okołowicz, M. Płoszajczak, and W. Nazarewicz, Prog. Theor. Phys. Suppl. 196, 230 (2012); J. Okołowicz, W. Nazarewicz, and M. Płoszajczak, Fortschr. Phys. 61, 66 (2013)], i.e., the microscopic reorganization of the wave function in the vicinity of the cluster decay threshold which leads to the appearance of clustering in this state.

Figure 1

The calculated energy spectra of ${}^{6,7}\mathrm{Li}$ and ${}^7\mathrm{Be}$, are compared with experimental data [37]. ${}^7\mathrm{Li}$ and ${}^7\mathrm{Be}$ are calculated in GSM-CC using the channel basis with two mass partitions: ${}^4\mathrm{He}+{}^3\mathrm{He}, {}^6\mathrm{Li}+p$ and ${}^4\mathrm{He}+{}^3\mathrm{H}, {}^6\mathrm{Li}+n$, respectively. The spectrum of ${}^6\mathrm{Li}$ is calculated in GSM and wave functions are expanded in the HO basis. Numbers in the brackets indicate the resonance width in keV.

Figure 2

The channel decomposition for selected states of ${}^7\mathrm{Be}$. In different colors, we show the summed GSM-CC probabilities of $p$ and ${}^3\mathrm{He}$ channels. Detailed information about major GSM-CC channel probabilities and spectroscopic factors in individual states of ${}^7\mathrm{Be}$ can be found in Table 3. Information about the number of protons and neutrons in different shells is given in Table 4.

Figure 3

The GSM-CC elastic differential cross sections of the reaction ${}^4\mathrm{He}({}^3\mathrm{H},{}^3\mathrm{H}){}^4\mathrm{He}$, calculated at four different c.m. angles, are compared with the experimental data (in dots) [44]. ${}^3\mathrm{H}$ bombarding energy is given in the laboratory frame.

Figure 4

The same as in Fig. 3 but for the reaction ${}^4\mathrm{He}({}^3\mathrm{He},{}^3\mathrm{He}){}^4\mathrm{He}$. Experimental data (in dots) are from Ref. [44].

Figure 5

The GSM-CC elastic differential cross sections of the reaction ${}^6\mathrm{Li}(p,p){}^6\mathrm{Li}$ at three different c.m. angles are plotted as a function of proton energy and compared with experimental data (in dots) [45]. Proton bombarding energy is given in the c.m. frame. Both GSM-CC and experimental cross sections are divided by the Rutherford cross section.

Figure 6

From top to bottom: the real and imaginary parts of the squared amplitudes $b_c^2\equiv {\langle {\tilde{w}}_c|w_c\rangle }^2$, and the partial widths of each reaction channel in the $5/2_2^{-}$ state of ${}^7\mathrm{Li}$. Contributions of all partial waves $\ell j$ in reaction channels ${[{}^6\mathrm{Li}\left(1_1^{+}\right)\otimes n\left(\ell j\right)]}^{5/2^{-}}$ are summed and everything is plotted as a function of the distance of the state to the neutron threshold. The meaning of curves for different channels is explained in the adjacent key. Each curve represents the sum of all reaction channels built on the same many-body state of ${}^6\mathrm{Li}$.

Figure 7

Same as Fig. 6 but for the mirror system ${}^7\mathrm{Be}(5/2_2^{-})$. Everything is plotted as a function of the distance of the state to the proton threshold.

Figure 8

The real and imaginary parts of the channel probabilities $b_c^2$ and the partial widths of each reaction channel in the $7/2_1^{-}$ state of ${}^7\mathrm{Li}$ are shown as a function of the distance with respect to the lowest decay threshold ${[{}^4\mathrm{He}\left(0_1^{+}\right)\otimes {}^3\mathrm{H}\left(L_{\mathrm{c}.\mathrm{m}.}{J}_{\mathrm{int}}{J}_{\mathrm{P}}\right)]}^{J^{\pi }}$. The dashed vertical line gives the GSM-CC energy of the $7/2_1^{-}$ resonance. For more details, see the caption of Fig. 6 and discussion in the text.

Figure 9

Same as Fig. 8 but for the mirror system ${}^7\mathrm{Be}(7/2_1^{-})$. Everything is plotted as a function of the distance of the state to the ${}^3\mathrm{He}$ threshold.

Figure 10

The spectroscopic factors and the channel weights in $5/2_2^{-}$ state of ${}^7\mathrm{Li}$ are shown as a function of the distance with respect to the neutron emission threshold ${[{}^6\mathrm{Li}\left(1_1^{+}\right)\otimes n\left(\ell j\right)]}^{J^{\pi }}$. From top to bottom: (i) real part of the spectroscopic factors $\mathrm{Re}\left[{\mathcal{S}}^2\right]$, (ii) imaginary part of the spectroscopic factors $\mathrm{Im}\left[{\mathcal{S}}^2\right]$, and (iii) real part of the channel weights $\mathrm{Re}\left[b_c^2\right]$.

Figure 11

Same as Fig. 10 but for the mirror system ${}^7\mathrm{Be}(5/2_2^{-})$. the spectroscopic factors and the channel weights $\mathrm{Re}\left[b_c^2\right]$ are shown as a function of the distance with respect to the one-proton decay threshold ${}^6\mathrm{Li}\left(1_1^{+}\right)\otimes p\left(\ell j\right){]}^{5/2^{-}}$.

Figure 12

The spectroscopic factors and the channel weights in $7/2_1^{-}$ state of ${}^7\mathrm{Li}$ are shown as a function of the distance with respect to the triton emission threshold ${[{}^4\mathrm{He}\left(0_1^{+}\right)\otimes {}^3\mathrm{H}\left(L_{\mathrm{c}.\mathrm{m}.}{J}_{\mathrm{int}}{J}_{\mathrm{P}}\right)]}^{J^{\pi }}$. From top to bottom: (i) real part of the spectroscopic factors $\mathrm{Re}\left[{\mathcal{S}}^2\right]$, (ii) imaginary part of the spectroscopic factors $\mathrm{Im}\left[{\mathcal{S}}^2\right]$, and (iii) real part of the channel weights $\mathrm{Re}\left[b_c^2\right]$. The dashed vertical line gives the GSM-CC energy of the $7/2_1^{-}$ resonance.