Correlated-Gaussian approach to linear-chain states: Case of four $\alpha$ particles

We show that correlated Gaussians with good angular momentum and parity provide flexible basis functions for specific elongated shape. As an application we study linear-chain states of four $\alpha$ particles in variation-after-projection calculations in which all the matrix elements are evaluated analytically. We find possible chain states for $J^{\pi }=0^{+},2^{+},4^{+}$ and perhaps $6^{+}$ with the bandhead energy being about 33 MeV from the ground state of ${}^{16}\mathrm{O}$. No chain states with $J\ge 8$ are found. The nature of the rotational sequence of the chain states is clarified in contrast to a rigid-body rotation. The quadrupole deformation parameters estimated from the chain states increase from 0.59 to 1.07 for $2^{+}$ to $6^{+}$. This work suggests undeveloped fields for the correlated Gaussians beyond those problems which have hitherto been solved successfully.

Figure 1

Probability of finding the component with partial wave $l$ in the Gaussian wave packet characterized by $\eta$. See Eq. (8).

Figure 2

Comparison of the radial functions with $l=0$ and 6 between the SG, ${\varphi }_{sl}^{\nu }\left(r\right)$, and the LPG, ${\phi }_{kl}^a\left(r\right)$. The value of $s$ is set to 8 fm. Both $\nu$ and $a$ are given in ${\mathrm{fm}}^{-2}$. Case (a) $\nu =0.163:(a,k)=(0.0796,2)$ for $l=0$ and $(0.1076,1)$ for $l=6$. Case (b) $\nu =0.521:(a,k)=(0.2630,8)$ for $l=0$ and $(0.2605,5)$ for $l=6$. The overlap integral between the LPG and the SG is larger than 0.998 in all the cases.

Figure 3

Energies of LC configurations with the angular momentum $L$ and positive parity. Panel (a) shows the lowest energy calculated in the SS model, whereas (b) shows the lowest energy in the MS model. Although it slightly depends on $L$, $r_{\mathrm{rms}}$ increases from about 2.28 for $H=20$ to 5.62 fm for $H=130$.

Figure 4

$H$-dependence of the contributions of the kinetic energy, the nuclear potential, and the Coulomb potential to the lowest energy LC states obtained in the MS model. The nuclear contribution consists of two-body ($V_{2\mathrm{B}}$) and three-body ($V_{3\mathrm{B}}$) potential energies. Panels (a), (b), (c), (d), and (e) show $L^{\pi }=0^{+}$, $2^{+}$, $4^{+}$, $6^{+}$, and $8^{+}$ cases, respectively.

Figure 5

Comparison of the excitation energies of the $4\alpha$ LC states as a function of $L(L+1)$. Open circles and crosses denote the lowest and the second lowest LC states obtained in the MH model. Data are taken from Yao [25], Suhara [26], Bender [49], and Bauhoff [48].

Figure 6

Plots of $f_z\left(l\right)$ as a function of $l$ for some values of $z$.