Microscopic cluster model in harmonic oscillator traps

Microscopic cluster model (MCM) is a successful theoretical framework to investigate nuclear clustering phenomena. In this work, we put it in a harmonic oscillator trap (HOT) and propose a new method abbreviated as MCM-HOT to study scattering states of cluster systems in free space. As proofs of concept, we compute the scattering phase shifts for some single-channel two-cluster systems. The numerical results given by MCM-HOT agree well with those from the conventional microscopic $R$-matrix method, implying that our novel method MCM-HOT can be useful in studying scattering processes of light neutron-nucleus systems.

Figure 1

Left: scattering of $\alpha$ particle and neutron in free space. Right: by introducing an artificial potential trap (illustrated here with harmonic oscillator trap), $\alpha$ particle and neutron are confined in it. The scattering properties in an infinite volume on the left can be linked to the discrete energy spectra inside the trap on the right. The trap size $b$ is required to be much larger than the range of short-range interactions between particles.

Figure 2

The phase shifts of $s$-wave ${\frac{1}{2}}^{+}$ state computed with microscopic $R$ matrix and MCM-HOT. The solid line represents results with microscopic $R$-matrix method, the circular, triangular, and square markers correspond to results obtained with MCM-HOT using the first, second, and third eigenenergies, respectively. For these first three eigenstates, the maximum values of $\omega$ adopted are all less than or equal to 1.5 MeV. For the first, second, and third eigenstates, the values of $\omega$ are $\{0.30,0.35,0.40,...,1.50\}\mathrm{MeV}$, $\{0.50,0.55,0.60,...,1.50\}\mathrm{MeV}$, and $\{0.70,0.75,0.80,...,1.50\}\mathrm{MeV}$, respectively.

Figure 3

(a) The phase shifts of ${\frac{1}{2}}^{-}$ state computed with microscopic $R$ matrix and MCM-HOT. For the first, second, third and fourth eigenstates, the values of $\omega$ are $\{0.30,0.35,0.40,...,1.00\}\mathrm{MeV}$, $\{0.50,0.55,0.60,...,1.50\}\mathrm{MeV}$, $\{0.70,0.75,0.80,...,1.50\}\mathrm{MeV}$, and $\{0.90,0.95,1.00,...,1.50\}\mathrm{MeV}$, respectively. (b) The phase shifts of ${\frac{3}{2}}^{-}$ state computed with microscopic $R$ matrix and MCM-HOT. The maximum values of $\omega$ are also all less than or equal to 1.5 MeV for first four eigenstates. In both (a) and (b), the circular, square, triangular, and rhombus markers correspond to the results calculated with MCM-HOT using the first, second, third, and fourth eigenenergies, respectively, the solid lines represent results with microscopic $R$ matrix. For the first, second, third, and fourth eigenstates, the values of $\omega$ are $\{0.20,0.25,0.30,...,0.60\}\mathrm{MeV}$, $\{0.30,0.35,0.40,...,1.50\}\mathrm{MeV}$, $\{0.50,0.55,0.60,...,1.50\}\mathrm{MeV}$, and $\{0.70,0.75,0.80,...,1.50\}\mathrm{MeV}$, respectively.

Figure 4

(a), (b), (c) The phase shifts of $0^{+}$, $2^{+}$ and $4^{+}$ states of $\alpha +\alpha$ system (without Coulomb interaction) computed with microscopic $R$ matrix and MCM-HOT in the GCM framework. For the eigenstates utilized, the maximum values of $\omega$ adopted are all less than or equal to 1.5 MeV. For $0^{+}$ state, ${\omega }_{\min }$ corresponding to the second, third, fourth, and fifth eigenstates are 0.50, 0.70, 0.90, 1.10 MeV, respectively. For $2^{+}$ state, ${\omega }_{\min }$ corresponding to the first, second, third, and fourth eigenstates are 0.50,0.70,0.90,1.10 MeV, respectively. For $4^{+}$ state, ${\omega }_{\min }$ corresponding to the first, second, third, and fourth eigenstates are 0.70, 0.90, 1.10, 1.30 MeV, respectively. The step size for $\omega$ is 0.10 MeV. The 21 generator coordinates used for solving the bound states are uniformly distributed in the range [0.1, 21] fm.

Figure 5

(a), (b), (c) The phase shifts of $0^{+}$, $2^{+}$, and $4^{+}$ states of $\alpha +\alpha$ system (without Coulomb interaction) computed with microscopic $R$ matrix and MCM-HOT in THSR framework. For the eigenstates utilized, the maximum values of $\omega$ adopted are all less than or equal to 1.5 MeV. For $0^{+}$ state, ${\omega }_{\min }$ corresponding to the second, third, fourth, and fifth eigenstates are 0.50,0.70,0.90,1.10 MeV, respectively. For $2^{+}$ state, ${\omega }_{\min }$ corresponding to the first, second, third, and fourth eigenstates are 0.50, 0.70, 0.90, 1.10 MeV, respectively. For the $4^{+}$ state, ${\omega }_{\min }$ corresponding to the first, second, third, and fourth eigenstates are 0.80, 1.00, 1.10, 1.30 MeV, respectively. The step size for $\omega$ is 0.10 MeV. The parameters of the THSR wave functions used for solving the bound states are ${\displaystyle \frac{{\beta }_{\perp }}{b^2}}=0.5,1,2,...,10,{\beta }_z={10}^{-6}$ fm.

Figure 6

(a) The phase shifts of ${\frac{1}{2}}^{-}$ state for $\alpha +n$ system obtained by MCM-HOT and $R$-matrix method with first three eigenstates. For the first, second, and third eigenstates, the solid symbols correspond to the following $\omega$ values: $\{0.3,0.4,0.5,...,1.0\}\mathrm{MeV}$, $\{0.4,0.5,0.6,...,1.5\}\mathrm{MeV}$, and $\{0.5,0.6,0.7,...,1.5\}\mathrm{MeV}$, respectively. The hollow symbols correspond to $\omega =\{1.75,2,2.25,2.5\}$ MeV. (b) The discrepancy between $E_{\mathrm{Virial}}$ and the eigenenergy $E_{\mathrm{c}.\mathrm{m}.}$, The horizontal axis represents the numerical eigenenergy, while the vertical axis displays the percentage difference between the values of $E_{\mathrm{c}.\mathrm{m}.}$ and $E_{\mathrm{Virial}}$ relative to $E_{\mathrm{c}.\mathrm{m}.}$.

Figure 7

(a) The phase shifts of $2^{+}$ state for $\alpha +\alpha$ system (without Coulomb potential) obtained by MCM-HOT and $R$-matrix method with first three eigenstates. For the first, second, and third eigenstates, the solid symbols correspond to the following $\omega$ values: $\{0.3,0.4,0.5,...,1.5\}\mathrm{MeV}$, $\{0.3,0.4,0.5,...,1.5\}\mathrm{MeV}$, and $\{0.4,0.5,0.6,...,1.5\}\mathrm{MeV}$, respectively. The hollow symbols correspond to $\omega =\{1.75,2,2.25,2.5\}$ MeV. (b) The discrepancy between $E_{\mathrm{Virial}}$ and the eigenenergy $E_{\mathrm{c}.\mathrm{m}.}$, The horizontal axis represents the numerical eigenenergy, while the vertical axis displays the percentage difference between the values of $E_{\mathrm{c}.\mathrm{m}.}$ and $E_{\mathrm{Virial}}$ relative to $E_{\mathrm{c}.\mathrm{m}.}$.