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As is known, light nuclei are well described by the shell model. However, the shell model does not take into account residual interactions between nucleons. It is the residual forces of proton-proton and neutron-neutron pairing that cause the zero spin of a nucleus whose shells are filled. We propose to apply a three-particle model for the $^{14}$A nucleus, which we will consider as a system consisting of a $^{12}$C core nucleus and two nucleons. The most acceptable approach to solving three-particle problems in nuclear physics is solving the Faddeev equation [1]. However, in the general case, one has to solve a system of two-dimensional differential or integral equations, which become more complicated when taking into account the Coulomb interaction between charged particles. From the point of view of simplicity of solving the problem of bound states of three particles, the method of hyperspherical functions [2,3] is the most convenient.
As an initial stage, we consider the $^{14}$C=$^{12}$C+2n nucleus. We expand the desired three-particle wave function into a system of hyperspherical functions and additionally use the Rayleigh-Ritz variational principle:
$\vert \Psi^{J;\,J_z}\rangle=\sum_{\mu}c_{\mu}\vert \Psi_{\mu}^{J;\,J_z}\rangle\qquad (1)$,
$\langle \delta\Psi^{J;\,J_z}\vert H-E \vert \Psi^{J;\,J_z} \rangle\qquad (2)$,
where $\delta\Psi^{J;\,J_z}$ indicates the variation of $\Psi^{J;\,J_z}$ for arbitrary infinitesimal changes of the linear coefficients $c_{\mu}$, $\mu$ is the index set. The problem of determining $c_{\mu}$ and the energy $E$ is then reduced to a generalized eigenvalue and eigenvector problem of the matrix.The expansion states $\vert \Psi_{\mu}^{J;\,J_z}\rangle$ of Eq. (1) are then given by
$\vert \Psi_{\mu}^{J;\,J_z}\rangle=\rho^{\mu}Y_{\{G\}}(\Omega_5),\qquad (3)$
where $\rho$ and $Y_{\{G\}}(\Omega_5)$ are hyperradius and hyperspherical function, respectively. $\Omega_5$ is a five-dimensional solid angle. As a result, we obtain a system of linear equations for finding the energy and expansion coefficients:
$\sum_{K^{\prime}L^{\prime}S^{\prime};l_{x_1}^{\prime}
\,l_{y_1}^{\prime}\nu^{\prime}}\Bigl[\langle KLS$ $l_{x_1}\,l_{y_1}\nu\vert T-\kappa^2\vert K^{\prime}L^{\prime}S^{\prime} l_{x_1}^{\prime}\,l_{y_1}^{\prime}\nu^{\prime}\rangle\delta_{KK^\prime}\delta_{LL^\prime}\delta_{SS^\prime}\delta_{l_xl_x^\prime}\delta_{l_yl_y^\prime}-$ $\frac{2m}{\hbar^2}\langle KLS l_{x_1}\,l_{y_1}\nu\vert V_1+V_2+V_3\vert K^{\prime}L^{\prime}S^{\prime} l_{x_1}^{\prime}
\,l_{y_1}^{\prime}\nu^{\prime}\rangle\Bigr]c_{\nu^{\prime}K^{\prime}L^{\prime}S^{\prime}}^{l_{x_1}^{\prime}\,l_{y_1}^{\prime}}=0. \qquad (4)$
$T=\frac{d^2}{d\rho^2}-\frac{K(K+4)+15/4}{\rho^2}$ is the hyperradial kinetic energy operator, $\kappa=\sqrt{\frac{2m}{\hbar^2}\epsilon}$ is a wave number for the bound state and $V_1$, $V_2$, $V_3$ are the interaction potentials between particles.
The calculations use the n-n potential from Ref.[4] and the $^{12}$C-n potential (the Woods-Saxon) from Ref.[5], adjusted to describe low-energy data. We calculate the linear system of Eq. (4) and receive the following results for the ground state energy $(\epsilon_{^{12}C-2n} )$ of $^{14}$C=$^{12}$C+2n
| $\quad$ Set of$\;\qquad\;$ |$\qquad\;\;$ Our result for $\qquad$ | $\;\;$ Experimental value of $\;\;$ |
|$\;\; nn\;$ potentials$\;\;$| the binding energy (MeV) | the binding energy (MeV) |
|1 (Yukawa ) $\quad\;$ |$\qquad\qquad$ 14.32 $\qquad\quad\;\;\;$ | $\qquad\qquad$ 13.12 $\qquad\quad\;\;$|
|2 (Gaussian) $\quad\;$ | $\qquad \quad\;\;$ 14.10 $\qquad\quad\;\;\;$ |$\qquad\qquad\qquad\qquad\qquad\;$ |
1. L.D. Faddeev, S.P. Merkuriev, Quantum Scattering Theory for Several Particle Systems, Springer, August 31, 1993.
2. Delves, L. M.: Nucl. Phys. 9, 391 (1959); 20, 275 (1960).
3. Smith, F. T.: Phys. Rev. 120, 1058 (1960); J. Math. Phys. 3, 735 (1962).
4. B. F. Irgaziev, V. B. Belyaev, Jameel-Un Nabi, Phys., Rev.C 87, 035804 (2013).
5. B. F. Irgaziev, Abdul Kabir, Jameel-Un Nabi, Can. J. Phys. 99, 176-184 (2021).
| Section | Nuclear physics (Section 1) |
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