$\alpha$ and triton clustering in ${}^{35}\mathrm{Cl}$

Background: The coupling of clusters and deformed structures is important for the dynamics of a nuclear structure. Threshold energy is discussed to explain cluster structure coupling to deformed states but the relation between threshold energy and excitation energy involves open problems. Negative-parity superdeformed (SD) states were observed via a $\gamma$-spectroscopy experiment in ${}^{35}\mathrm{Cl}$. However, their detailed structure is unclear.

Purpose: An analysis of the coupling of cluster structures in deformed states and high-lying cluster states in ${}^{35}\mathrm{Cl}$ is used to investigate cluster structures coupling to deformed states and excitation energy of high-lying cluster states.

Method: Antisymmetrized molecular dynamics (AMD) and the generator coordinate method (GCM) are used. An AMD wave function corresponds to a Slater determinant of Gaussian wave packets. Energy variational calculations with constraints on deformation and clustering are used to obtain wave functions of deformed structures and $\alpha \text{−}{}^{31}\mathrm{P}$ and $t\text{−}{}^{32}\mathrm{S}$ cluster structures. The wave functions are adopted as the GCM basis, and wave functions of ground and excited states are calculated.

Results: Various deformed bands are obtained and predicted. A $K^{\pi }={\frac{1}{2}}^{-}$ deformed band (corresponding to the observed SD band) dominates the deformed structure and compact $\alpha \text{−}{}^{31}\mathrm{P}$ and $t\text{−}{}^{32}\mathrm{S}$ cluster structure components. Particle-hole configurations of dominant components with deformed and cluster structures are similar. In high-lying states, almost pure $\alpha \text{−}{}^{31}\mathrm{P}$ and $t\text{−}{}^{32}\mathrm{S}$ cluster states are obtained in negative-parity states, and excitation energies of the $t\text{−}{}^{32}\mathrm{S}$ cluster states exceed those of $\alpha \text{−}{}^{31}\mathrm{P}$ cluster states.

Conclusions: Particle-hole configurations of cluster structures with a small intercluster distance are important for coupling to low-energy deformed states. Threshold energies reflect excitation energies of almost pure high-lying cluster states.

Figure 1

Energy curves as functions of the quadrupole deformation parameter $\beta$ are obtained by energy variational calculation with parity projection on (a) positive and (b) negative parity. Specifically, ${\left(pf\right)}_{\tau }^n$ ($\tau =\pi$ or $\nu$) exhibit particle-hole configurations (see text). With respect to the positive-parity states, squares, triangles, and circles denote totally $0\hslash \omega$, $2\hslash \omega$, and $4\hslash \omega$ excited configurations, respectively, from the lowest allowed state. With respect to the negative-parity states, squares, triangles, circles, and crosses denote totally $1\hslash \omega$, $3\hslash \omega$, $5\hslash \omega$, and $7\hslash \omega$ excited configurations, respectively, from the lowest allowed state.

Figure 2

Energies of cluster structure as functions of intercluster distance for positive parity (a) $\alpha \text{−}{}^{31}\mathrm{P}$ and (b) $t\text{−}{}^{32}\mathrm{S}$ and negative parity (c) $\alpha \text{−}{}^{31}\mathrm{P}$ and (d) $t\text{−}{}^{32}\mathrm{S}$ cluster structures. Circles and triangles exhibit energies of S and L types, respectively (see text).

Figure 3

Density distributions of $\alpha \text{−}{}^{31}\mathrm{P}$ cluster states in negative-parity states with $d_{\alpha \text{−}{}^{31}\mathrm{P}}=4.8$ fm for (a) S and (b) L types. Left and right peaks correspond to $\alpha$ and ${}^{31}\mathrm{P}$ clusters, respectively, in each panel.

Figure 4

Relative harmonic oscillator quanta $\mathrm{\Delta }{N}^{\tau }$ of positive-parity (a) $\alpha \text{−}{}^{31}\mathrm{P}$ and (b) $t\text{−}{}^{32}\mathrm{S}$ cluster structure and negative-parity (c) $\alpha \text{−}{}^{31}\mathrm{P}$ and (d) $t\text{−}{}^{32}\mathrm{S}$ cluster structure as functions of the intercluster distance. Circles and triangles denote S-type protons and neutrons, respectively. Squares and inverted triangles denote L-type protons and neutrons, respectively.

Figure 5

Level scheme and of ${}^{35}\mathrm{Cl}$. Left and right parts correspond to the experimental and theoretical levels. Experimental data are based on Refs. [37, 45]. Labels of bands are shown in the squares.

Figure 6

In-band $B(E2;J\to J-2)$ values are plotted as functions of spin $J$ of an initial state for theoretical $K^{\pi }={\frac{1}{2}}^{\pm }$, ${\frac{5}{2}}^{+}$, and ${\frac{13}{2}}^{-}$ bands and the experimental negative-parity SD band ($\mathrm{SD}-$) and its candidate positive-parity partner band (SD$+$). Left and right vertical axes are located in the $e^2{\mathrm{fm}}^4$ and Weisskopf unit, respectively.

Figure 7

Excitation energies of obtained negative-parity states are plotted. The horizontal axis shows the spin of each state. Triangles, squares, circles, and inverted triangles denote members of the $K^{\pi }={\frac{1}{2}}^{-}$, hn-L, “$\alpha$”, and “$t$” bands, respectively (see text). Crosses denote the other states. Dotted, chain, solid, and dashed lines denote trends of the excitation energies of the $K^{\pi }={\frac{1}{2}}^{-}$, hn-L, “$\alpha$”, and “$t$” bands, respectively.

Figure 8

Cluster structure components of the $K^{\pi }={\frac{3}{2}}^{-}$ states in the (a) $K={\frac{1}{2}}^{-}$, (b) hn-L, (c) “$\alpha$”, and (d) “$t$” bands. Solid lines with circles and dashed lines with triangles denoting $\alpha \text{−}{}^{31}\mathrm{P}$ and $t\text{−}{}^{32}\mathrm{S}$ cluster structure components, respectively. Open and closed symbols denote S and L types, respectively.