Gamow shell model description of proton scattering on ${}^{18}\mathrm{Ne}$

Background: The structure of weakly bound/unbound nuclei close to particle drip lines is different from that around the valley of beta stability. A comprehensive description of these systems goes beyond the standard shell model (SM) and demands an open quantum system description of the nuclear many-body system.

Purpose: For that purpose, we are using the Gamow shell model (GSM), which provides a fully microscopic description of bound and unbound nuclear states, nuclear decays, and reactions. We formulate the GSM in coupled-channel (GSM-CC) representation to describe low-energy elastic and inelastic scattering of protons on ${}^{18}\mathrm{Ne}$.

Method: The GSM-CC formalism is applied to a translationally invariant Hamiltonian with an effective finite-range two-body interaction. We discuss in detail the GSM-CC formalism in coordinate space and give the description of the novel equivalent potential method for solving the GSM-CC system of integrodifferential equations. This method is then applied for the description of $(p,p^{\prime })$ reaction cross-sections. The reactions channels are built by GSM wave functions for the ground state $0^{+}$ and the first excited $2^{+}$ of ${}^{18}\mathrm{Ne}$ and a proton wave function expanded in different partial waves. The completeness of this basis is verified by comparing GSM and GSM-CC energies of low-energy resonant states in ${}^{19}\mathrm{Na}$. The differences between the two calculations provide a measure of the missing configurations in the GSM-CC calculation of low-energy states of ${}^{19}\mathrm{Na}$ due to the restriction on the number of excited states of ${}^{18}\mathrm{Ne}$.

Results: We present the first application of the GSM-CC formalism for the calculation of excited states of ${}^{18}\mathrm{Ne}$ and ${}^{19}\mathrm{Na}$, the excitation function, and the elastic/inelastic differential cross-sections in the ${}^{18}\mathrm{Ne}$$(p,p^{\prime })$ reaction at different energies. This is the first unified description of the spectra and reaction cross-sections in the GSM formalism. The method is shown to be both feasible and accurate. The approximate equivalence of GSM and GSM-CC in describing spectra of ${}^{19}\mathrm{Na}$ has been demonstrated numerically.

Conclusions: The GSM in the coupled-channel representation opens a possibility for the unified description of low-energy nuclear structure and reactions using the same Hamiltonian. While both GSM and GSM-CC can describe energies, widths, and wave functions of the many-body states, the GSM-CC can in addition yield reaction cross-sections. The combined application of GSM and GSM-CC to describe energies of resonant states allows to test the exactitude of calculated cross-sections for a given many-body Hamiltonian.

Figure 1

Real (left) and imaginary (right) parts of the channel wave functions for $J_A^{\pi }=5/2^{+}$ at $E_{\mathrm{c}.\mathrm{m}.}=$ 1 MeV. Dashed lines show the channel functions at the first iteration. The final solution for channel wave functions is shown with the solid lines. The dotted lines depict the final solutions for the rescaled channel wave function $u_c$ in Eq. (26). Panels (a) and (b) represent the entrance channel: $(0^{+}\times d_{5/2})$. Panels (c) and (d) correspond to the channel $(2^{+}\times d_{5/2})$, and panels (e) and (f) to the channel $(2^{+}\times s_{1/2})$.

Figure 2

The same as in Fig. 1 but for $E_{\mathrm{c}.\mathrm{m}.}=$ 5 MeV.

Figure 3

The channel wave functions for $J_A^{\pi }=1/2^{+}$ at $E_{\mathrm{c}.\mathrm{m}.}=$ 1 MeV. Panels (a) and (b) show the entrance channel $(0^{+}\times s_{1/2)}$. Panels (c) and (d) correspond to the channel $(2^{+}\times d_{5/2})$, and panels (e) and (f) to the channel $(2^{+}\times s_{1/2})$. For further information, see the caption of Fig. 1.

Figure 4

The same as in Fig. 3 but for $E_{\mathrm{c}.\mathrm{m}.}=$ 5 MeV.

Figure 5

The equivalent local potentials $V_{c{c}^{\prime }}^{\left(\text{eq}\right)}$ [Sec. 3a] for $J_A^{\pi }=5/2^{+}$ at $E_{\mathrm{c}.\mathrm{m}.}=1$ MeV. The channel indices 0, 1, 2 denote the entrance channel $(0^{+}\times d_{5/2})$, and channels $(2^{+}\times d_{5/2})$, $(2^{+}\times s_{1/2})$, respectively. Panels (a) and (b) show the real and imaginary parts of the diagonal equivalent potential $V_{00}^{\left(\text{eq}\right)}$ in the entrance channel. Panels (c), (d) and (e), (f) depict the off-diagonal equivalent local potentials $V_{01}^{\left(\text{eq}\right)}$ and $V_{02}^{\left(\text{eq}\right)}$, respectively. Panels (g), (h) show the diagonal equivalent potential $V_{22}^{\left(\text{eq}\right)}$. The dotted and solid lines represent the equivalent local potentials corresponding to the first iteration and to the final solution of the CC equations, respectively.

Figure 6

The same as in Fig. 5 but for $E_{\mathrm{c}.\mathrm{m}.}=$ 5 MeV.

Figure 7

The equivalent local potentials $V_{c{c}^{\prime }}^{\left(\text{eq}\right)}$ [Sec. 3a] for $J_A^{\pi }=1/2^{+}$ at $E_{\mathrm{c}.\mathrm{m}.}=1$ MeV. The channel indices 0, 1, 2 denote the entrance channel $(0^{+}\times s_{1/2})$, and channels $(2^{+}\times d_{5/2})$, $(2^{+}\times s_{1/2})$, respectively. Panels (a) and (b) show the real and imaginary parts of the diagonal equivalent potential $V_{00}^{\left(\text{eq}\right)}$ in the entrance channel. Panels (c), (d) and (e), (f) depict the off-diagonal equivalent local potentials $V_{01}^{\left(\text{eq}\right)}$ and $V_{02}^{\text{(eq)}}$, respectively. Panels (g), (h) show the diagonal equivalent potential $V_{22}^{\left(\text{eq}\right)}$. The dotted and solid lines represent the equivalent local potentials corresponding to the first iteration and to the final solution of the CC equations, respectively.

Figure 8

The same as in Fig. 7 but for $E_{\mathrm{c}.\mathrm{m}.}=$ 5 MeV.

Figure 9

$p+{}^{18}$Ne excitation function at different c.m. angles ${\Theta }_{\text{c.m}.}$ in the range from 105 to 180${}^{\circ }$. GSM-CC results are drawn with the solid line. The dotted line corresponds to the GSM-CC calculation using only Coulomb interaction. Panels (a)–(f) correspond to the excitation functions at ${\Theta }_{\text{c.m}.}$ = 105, 120.2, 135, 156.6, 165, and 180${}^{\circ }$, respectively. The experimental data at ${\Theta }_{\text{c.m.}}$ = 105, 135, and 165${}^{\circ }$ are taken from Ref. [30]. The data at ${\Theta }_{\text{c.m.}}$ = 120.2, 156.6${}^{\circ }$ are taken from Ref. [31] and at ${\Theta }_{\text{c.m}.}$ = 180${}^{\circ }$ from Ref. [32].

Figure 10

Elastic angular cross sections for the reaction $p+{}^{18}$Ne at different c.m. energies: $E_{\text{c.m}.}$ = (a) 0.1, (b) 1.0, (c) 1.3, and (d) 1.5 MeV below the inelastic channel threshold. Solid and dashed lines show the GSM-CC results in $\left(sd\text{−}p\right)$ and $\left(sd\right)$ model spaces, respectively. Dotted lines have been obtained using the Coulomb interaction only.

Figure 11

Elastic (left) and inelastic (right) angular cross sections for the reaction $p{+}^{18}$Ne at different c.m. energies $E_{\text{c.m}.}$ = (a), (b) 3 and (c), (d) 5 MeV above the inelastic channel threshold. Solid and dashed lines show the GSM-CC results in $\left(sd\text{−}p\right)$ and $\left(sd\right)$ model spaces, respectively. Dotted lines show results obtained using the Coulomb interaction only.