Approximate projection method for the construction of multi-$\alpha$-cluster spaces

An approximate but straightforward projection method to molecular many $\alpha$-particle states is proposed and the overlap to the shell model space is determined. This construction of the space poses an alternative to other techniques which use a more general, larger subset of the shell model space and/or use involved numerical methods. The resulting space is in accordance with the shell model, but still contains states which are not completely symmetric under permutations of the $\alpha$ particles, which is one reason to call the construction approximate. A new contribution is the construction of the six- and seven-$\alpha$-particle spaces. The errors of the method propagate toward larger numbers of $\alpha$- and larger shell excitations. In order to show the practicality of the construction proposed, the obtained spaces are applied, within an algebraic cluster model, to ${}^{20}\mathrm{Ne}, {}^{24}\mathrm{Mg}$, and ${}^{28}\mathrm{Si}$, each treated as a many-$\alpha$-particle system. Former results on ${}^{12}\mathrm{C}$ and ${}^{16}\mathrm{O}$ are re-examined.

Figure 1

Spectrum of ${}^{20}\mathrm{Ne}$, as a five-$\alpha$-particle states. The theoretical spectrum (left panel) is compared to experiment (right panel). The theoretical spectrum was obtained including SO(4) mixing. The dominant $\mathrm{\Delta }{n}_{\pi }(\lambda ,\mu )$ quantum numbers are indicated below each band, where $\mathrm{\Delta }{n}_{\pi }$ refers to the number of shell excitations. Experimental data are retrieved from [40].

Figure 2

Spectrum of ${}^{24}\mathrm{Mg}$, as a six-$\alpha$-particle states. The theoretical spectrum (left panel) is compared to experiment (right panel). The theoretical spectrum was obtained, including SO(4) mixing. The dominant $\mathrm{\Delta }{n}_{\pi }(\lambda ,\mu )$ quantum number are indicated below each band, where $\mathrm{\Delta }{n}_{\pi }$ refers to the number of shell excitations. The dominant $\mathrm{\Delta }{n}_{\pi }(\lambda ,\mu )$ quantum numbers are indicated below each band, where $\mathrm{\Delta }{n}_{\pi }$ refers to the number of shell excitations. Experimental data are retrieved from [40].

Figure 3

Spectrum of ${}^{28}\mathrm{Si}$, as a seven-$\alpha$-particle states. The theoretical spectrum (left panel) is compared to experiment (right panel). On the left hand side the theoretical spectra, obtained including SO(4) mixing, is depicted, compared to experiment (right hand side). The dominant $\mathrm{\Delta }{n}_{\pi }(\lambda ,\mu )$ quantum numbers are indicated below each band, where $\mathrm{\Delta }{n}_{\pi }$ refers to the number of shell excitations. Experimental data are retrieved from [40].