Real-time evolution method and its application to the $3\alpha$ cluster system

A theoretical method is proposed to describe the ground and excited cluster states of atomic nuclei. The method makes use of the equation of motion of the Gaussian wave packets to generate the basis wave functions incorporating various cluster configurations. The generated basis wave functions are superposed to diagonalize the Hamiltonian. This method is regarded as the generator coordinate method which uses the real time as the generator coordinate. As a application of our proposed method, we present the benchmark calculation for the $3\alpha$ system. We show that our proposed method works efficiently and yields the results consistent with or better than the other cluster models. We have also discussed briefly the structure of the excited $0^{+}$ and $1^{-}$ states.

Figure 1

The rebound of $\alpha$ clusters described by Eqs. (17, 18, 19). Open circles represent the real part of $\mathit{Z}$, and the solid (dotted) arrows represent the imaginary part of $\mathit{Z}$ before (after) the rebound. The rebound does not change the real parts of ${\mathit{Z}}_i$, ${\mathit{Z}}_j$, and ${\mathit{Z}}_k$ (positions of $\alpha$ clusters), but changes the imaginary parts (momenta). It reverts the momentum of ${\mathit{Z}}_i$ and the center-of-mass momentum between ${\mathit{Z}}_j$ and ${\mathit{Z}}_k$ but conserves the relative momentum between ${\mathit{Z}}_j$ and ${\mathit{Z}}_k$.

Figure 2

Intrinsic density snapshots at the propagation time $t=0$, 500, 1200, 2000, and $4000\mathrm{fm}/c$. Upper (lower) panels show the ensemble of the wave functions obtained by the EOM starting from the wave function at $t=0\mathrm{fm}/c$.

Figure 3

(a) Radius of the intrinsic wave function as function of time. (b) Overlap between the wave functions at $t=0$ and $t$ after the projection to the $J^{\pi }=0^{+}$.

Figure 4

The number of basis wave functions selected by the overlap condition (21). The black dotted line shows the number of the basis wave functions without the selection.

Figure 5

The eigenenergies of the $0^{+}$ states measured from the $3\alpha$ threshold as a function of the propagation time $T$ obtained by the REM without the $r^2$ constraint. Open and filled symbols show the results obtained by using the different initial wave functions at $t=0$ fm/$c$.

Figure 6

Same as Fig. 5 but for radii.

Figure 7

The number of the basis wave functions (top), the energies (middle) and radius (bottom) of the excited $0^{+}$ states as the functions of the cutoff radius $r_{\mathrm{cut}}$. THSR results are taken from Ref. [18].

Figure 8

Electric monopole transition matrix element from the ground state to the excited $0^{+}$ states obtained by the $r^2$ constraint.

Figure 9

Excitation energies measured from the $3\alpha$ threshold. Theoretical results by the REM, THSR [18, 23] and RGM [2, 3] are compared with the experiments [13, 14, 15, 16, 19, 72, 73, 74, 75, 76]. The calculated $4_1^{+}$, $4_2^{+}$, $5_1^{-}$, and $5_2^{-}$ states are labeled according to the observed counterparts.

Figure 10

Excitation modes of the Hoyle state are schematically shown. Arrows show the monopole and dipole transitions.