$\alpha$-cluster excited states in $^{32}$S

$\alpha$-cluster excited states in $^{32}$S are investigated with an extended $^{28}$Si+$\alpha$ cluster model, in which the $^{28}$Si core deformation and rotation, and the $\alpha$-cluster breaking are incorporated. In the GCM calculation with the extended $^{28}$Si+$\alpha$ cluster model, the $\alpha$-cluster excited states are obtained near the $^{28}$Si+$\alpha$ threshold energy. The $^{28}$Si core deformation and rotation effects, and also the $\alpha$-clusters breaking in the $^{28}$Si+$\alpha$ system are discussed. It is found that the rotation of the oblately deformed $^{28}$Si core gives a significant effect to the $\alpha$-cluster excited states whereas the $\alpha$-cluster breaking gives only a minor effect.

Candidates for the α-cluster excited states in 32 S have been reported in the 28 Si( 6 Li,d) 32 S (α transfer) reaction by Tanabe et al. [56] in the 1980s. A couple of states observed in 10 ∼ 15 MeV region may correspond to the α-cluster excited states. Recently, in the experiments of the 28 Si + α elastic-scattering reaction, Lönnroth et al. observed many resonances above the 28 Si + α threshold energy, and interpreted them as fragmentation of α-cluster excited band starting from the bandhead energy E x = 10.9 ± 0.5 MeV, a few MeV higher energy than the 28 Si + α threshold [57]. Another experiment for the α-cluster excited states in 32 S is the inelastic scatterings on 32 S by Itoh et al. [58]. They observed excited states near the 28 Si + α threshold energy are considered to be candidates for α-cluster excited bands with the bandhead energies E x = 6.6 and 7.9 MeV. To understand the α-cluster excited states in 32 S, theoretical studies are now requested.
In a history of theoretical studies of cluster structures in the p-shell and sd-shell regions, multi-α models using the Brink-Bloch α-cluster wave functions [48] have been applied to Z = N = even nuclei. With the multi-α models, systematic calculations of 3-dimensional αcluster configurations were performed from 16 O to 44 Ti [49]. For 28 Si, the 7α-cluster model was used to discuss the shape coexistence of the oblate and prolate states [50]. The multi-α model was also used for 20 Ne to take into account the 16 O core structure change in 16 O+α cluster states in 20 Ne [52]. However, in these studies with the multi-α models, constituent α clusters are assumed to be the ideal 0s-closed configuration, and therefore the contribution of the spin-orbit interaction is completely omitted even though it is significant in mid-shell nuclei. In other words, α clusters in nuclei should be more or less broken from the ideal configuration to gain the spin-orbit interaction. To take into account the α-cluster breaking and the contribution the spin-orbit interaction, an extension of cluster models has been done in the study of the 16 O+α cluster states in 20 Ne [53]. Cluster structures in sd-shell nuclei were also investigated by the antisymmetrized molecular dynamics (AMD) [5], in which the existence of clusters is not assumed but the formation and breaking of cluster structures are automatically described in the model. In the AMD calculation for 28 Si, the oblately deformed state with a 7α-like configuration was obtained for the 28 Si ground state consistently with the 7α-cluster model calculation [50], however, it was shown that the oblate ground state is different from the ideal 7α configuration but it contains the significant cluster breaking because of the spin-orbit interaction [51]. In the systematic studies with the AMD by Taniguchi et al., the α-cluster excited states were suggested in various sd-shell nuclei [54,55]. In these studies, the existence of clusters are not assumed a priori, but core deformation and the α-cluster breaking are taken into account in the AMD framework. However, the rotation of the core in the α-cluster excited states is not sufficiently considered.
Our aim in this paper is to theoretically investigate the α-cluster excited states in 32 S. The question to be answered is whether the α-cluster band appears near the 28 Si+α threshold energy. If the case, we are going to predict its properties such as the bandhead energy, the level spacing (the rotational constant), and the α-decay width. We also intend to clarify the core deformation and rotation effects as well as the α-cluster breaking effect in the α-cluster excited states. In α-cluster excited states in the sd-shell region, the core deformation may occur, and the rotation of the deformed core could play an important role. Moreover, an α cluster at the nuclear surface can be dissociated because of the spin-orbit potential. To incorporate the core deformation and rotation as well as the α-cluster breaking, we construct a new extended cluster model for the 28 Si + α system by extending the conventional cluster model, which relies on the inert cluster assumption. We apply the method and investigate the properties of α-cluster excited states in 32 S.
The contents of this paper are as follows. In Sec. 2, we explain the formulation of the extended 28 Si + α-cluster model. We show the calculated results in Sec. 3, and discuss the 28 Si core structure and the α-cluster breaking effect in the α-cluster excited states in 32 S in Sec. 4. Finally, a summary and an outlook are given in Sec. 5.

FRAMEWORK
To investigate α-cluster excited states in 32 S, we construct the extended cluster model for the 28 Si+α system to take into account the 28 Si core deformation and rotation, and the α-cluster breaking. In this section, we first explain the Brink-Bloch α-cluster model (a conventional cluster model), and then, we describe the formulation of the extended 28 Si+α-cluster model.

Brink-Bloch α-cluster model
In the Brink-Bloch α-cluster model [59], a Z = N = 2n nucleus is composed of nα clusters. Each α cluster is described by the (0s) 4 harmonic oscillator (h.o.) configuration localized around a certain position. The total nα-cluster wave function Φ nα of the A = 4n-body system is written by the following antisymmetrized single-particle wave functions, where A is the antisymmetrizing operator for all nucleons, R i is the center of the ith αcluster (i = 1, · · · , n), χ σ and τ σ are the spin and isospin parts of the single-particle wave function, respectively, and ν is the width parameter.

Extended cluster model for α-cluster breaking
For description of the α-cluster breaking due to the spin-orbit potential from a core, we apply the method proposed by Itagaki et al. [53]. In this method, an α-cluster breaking is incorporated by adding a spin-dependent imaginary part to the Gaussian centers of singleparticle wave functions so as to gain the spin-orbit potential, where the parameter λ α represents the degree of the α-cluster breaking, and e spin,j is the unit vector oriented to the intrinsic spin direction of the jth nucleon (j = 1, · · · , 4). If λ α is zero, this model becomes the conventional cluster model (Brink-Bloch α-cluster model) and describes the intrinsic spin saturated state, where the expectation value of the spin-orbit potential vanishes. When λ α is positive, spin-up and spin-down nucleons in the α cluster obtain finite momenta with opposite directions so as to gain the spin-orbit potential.
2.3. Extended cluster model for 28 Si core Fig. 1 Schematic figures for spatial configurations of Gaussian centers. (a) The 7α-cluster configuration with a pentagon shape for the 28 Si core. (b) The configuration for the present 28 Si + α cluster model.
To describe the 28 Si core structure, we adopt an extended 7α-cluster model where the parameter Λ c for the cluster breaking is incorporated to take into account the spin-orbit interaction effect. The present extended 7α-cluster model is based on the study of 28 Si with the Brink-Bloch 7α-cluster model [50] and that with the method of the AMD [51]. Bauhoff et al. used the 7α-cluster model with a pentagon configuration [50], and succeeded to describe the oblate ground state and the K π = 5 − rotational band with the D 5h symmetry of a pentagon configuration. The 7α-cluster model wave function Φ 7α forms a pentagon configuration as shown in Fig. 1(a), and is described as whereR z is the rotation operator around the z axis, and d 1 and d 2 are the distance parameters for 7α cluster positions. The pentagon configuration of the 7α-cluster structure of 28 Si has been also supported by the AMD calculation where α clusters are not a priori assumed [51]. Differently from the Bauhoff's 7α-cluster model, the 28 Si wave function obtained by the AMD for the ground state is not the ideal 7α-cluster wave function without the cluster breaking but it is a 28-body wave function with a pentagon configuration of 7α clusters having the cluster breaking.
Based on the AMD result for 28 Si, we construct an extended 7α-cluster model for the 28 Si core by respecting the symmetry for the 2π/5 rotation as follows, Here, Φ α ′ (d 2 e x , Λ c ) represents the wave function for a broken α cluster, where ↑ y and ↓ y are the intrinsic spin of the y direction, and nucleon momenta takes the z direction. Λ c is the parameter for the cluster breaking in the 7α-cluster model for the 28 Si core and called the 7α-cluster breaking parameter in this paper. In the case of the d 1 → 0 and d 2 → 0 limit, this extended 7α-cluster model wave function Φ28 Si (d 1 → 0, d 2 → 0, Λ c ) describes the 0d 5/2 subshell closed configuration of the jj-coupling shell model at Λ c = 1 and the oblately deformed state at Λ c = 0. Note that 5α clusters in the 28 Si core are broken α clusters written by the previously mentioned method for the α-cluster breaking proposed by Itagaki et al. The concept of the present model for the 28 Si core is similar to that of an extended 3α cluster model for 12 C proposed by Suhara et al. [60].
In the present calculation, the parameters, d 1 and d 2 , for positions of 7α-clusters are fixed to be the optimized values, d 1 = 0.20 fm, d 2 = 0.27 fm, that give the minimum energy of 28 Si in the 7α-cluster model without the cluster breaking (Λ c = 0). Hereafter, we define the 28 Si wave function with the fixed d 1 and d 2 values as Φ28 Si (Λ c ) ≡ Φ28 Si (d 1 = 0.20 fm, d 2 = 0.27 fm, Λ c ) parameterized by Λ c .

Extended cluster model for 28 Si + α system
We construct the extended 28 Si+α-cluster model to take into account the 28 Si core deformation and rotation, and the α-cluster breaking. We set the 28 Si core and the α cluster at the inter-cluster distance R, and perform the generator coordinate method (GCM) [61] by treating R as the generator coordinate. The α cluster is parameterized by the α-cluster breaking parameter λ α , whereas the 28 Si core is specified by the 7α-cluster breaking parameter Λ c which changes the 28 Si core deformation from the oblate state to the spherical one. In addition to these parameters, R, λ α , and Λ c , we consider the angle parameter θ to specify the orientation of the oblately deformed 28 Si core. We set the α cluster on the z-axis and define θ for the rotation of the 28 Si core as shown in Fig. 1(b). When θ = 0 • , the symmetric axis of the 28 Si core agrees to the z axis.
Then the 28 Si+α wave function of the extended 28 Si+α-cluster model is written as whereT (R) is the translation operator andR y (θ) is the rotation operator around the y axis. Φ28 Si (R, θ, Λ c ) expresses the extended 7α-cluster model Φ28 Si (Λ c ) rotated by the angle θ and shifted by R. In the extended 28 Si+α-cluster model, the width parameter is chosen to be ν = 0.16fm −2 so as to reproduce the 28 Si radius with the sub-shell closed configuration.

Parity and total-angular-momentum projection
We project the 28 Si+α wave function Φ28 Si+α (R, θ, λ α , Λ c ) to the parity and total angularmomentum eigenstate, whereP ± andP J M K are the parity and the total-angular-momentum projection operators, respectively. In the present paper, we only take the K = 0 component and omit the K-mixing for simplicity.

Generator coordinate method (GCM)
To calculate energy levels of α-cluster states in 32 S, we perform the GCM calculation by superposing the 28 Si + α wave function, where coefficients c (n) i are determined by diagonalizing the norm and Hamiltonian matrices. For the inter-cluster distance R, we superpose the wave functions with R = 1, 2, · · · , 10 fm. For the 7α-cluster breaking parameter Λ c of the 28 Si core, we take two points, Λ c = 0.38 and 0.80, which correspond to oblate and spherical local minimum states of the intrinsic energy of the 28 Si core as described later. For the rotation angle θ of the 28 Si core, we take θ = 0 • , 30 • , 60 • , 90 • for the oblate core (Λ c = 0.38) and take θ = 0 for the spherical core (Λ c = 0.80).
In the present GCM calculation, we omit the α-cluster breaking and fix λ α = 0. More details of the choice of the parameters are described later. We call the calculation with the full diagonalization of the norm and Hamiltonian matrices in the above-mentioned basis wave functions with the parameters (R i , θ i , Λ ci ) "full-GCM" calculation.

Frozen core GCM
In the asymptotic region at a large inter-cluster distance R, the 28 Si core in the lowest 28 Si + α channel should be the ground state of an isolate 28 Si: 28 Si(0 + g.s. ). We also perform the GCM calculation for the 28 Si(0 + g.s. ) + α within the frozen core approximation and compare the result with the previously explained full GCM calculation. We call this calculation "frozen core GCM". In the present work, we express the frozen core wave function by the linear combination of the projected 28 Si(0 + g.s. ) + α wave functions as follows. Let us first consider the adiabatic picture that the 28 Si configuration is optimized at each state of a given inter-cluster distance R. We define the R-fixed 28 Si + α wave function as where parameters (θ k , Λ ck ) = (0 Here coefficients a k (R) are determined by diagonalizing the norm and Hamiltonian matrices for each inter-cluster distance R. By taking an enough large inter-cluster distance R max , we determine the coefficients a k (R max ) in the asymptotic region, which approximately express the ground state configuration of the 28 Si core. We take R max = 10 fm in this paper.
Next, using the coefficients a k (R max ) determined at R max , we define the R-fixed 28 Si(0 + g.s. )+α wave function with the frozen core (the R-fixed frozen core wave function), Then, we perform the frozen core GCM calculation, that is, the GCM calculation of the 28 Si(0 + g.s. ) + α cluster model by superposing the 28 Si(0 + g.s. )+α wave functions with different distance as where coefficients b (n) k are determined by diagonalizing the norm and Hamiltonian matrices.
In order to analyze the α-cluster motion in 32 S states obtained by the full-GCM and those by the frozen core GCM calculations, we calculate the overlap between the 32 S wave functions with the R-fixed 28 Si(0 + g.s. )+α wave function to evaluate the α-cluster component at R,

Hamiltonian
The Hamiltonian operator (Ĥ) iŝ whereT is the kinetic energy andT G is the energy of the center-of-mass motion. As for the effective nuclear forceV nuclear , Volkov No.2 [62] is adopted as the central forceV c and the two-range Gaussian form of the spin-orbit term in the G3SR force [63] is used as the spin-orbit forceV LS . The form of Volkov No.2 is given aŝ where v 1 = −60.65 MeV, v 2 = 61.14 MeV, a 1 = 1.80 fm, a 2 = 1.01 fm. M is the Majorana parameter that is an adjustable parameter. In the present paper, we use M = 0.67. With the Volkov force, reproductions of the binding energy of 32 S and the α-separation energy ( 28 Si + α threshold) are not satisfactory. We also use other M values of the Volkov force to discuss the interaction dependence of the calculated results. 7 The spin-orbit force is given aŝ where b 1 = 0.477 fm, b 2 = 0.600 fm, and P ( 3 O) is the triplet-odd projection operator. We use the strength parameters u 1 = 2000 MeV and u 2 = −2000 MeV which are the same as those used in Ref. [53] for the 16 O + α system. The Coulomb forceV coulomb is approximated by seven Gaussians.

RESULTS
3.1. 28 Si core structure in 28 Si+α system  Fig. 2 The energy expectation value of the isolate 28 Si core. The energy E28 Si (Λ c ) before the parity and total-angular-momentum projection, and the energy E 0 + 28 Si (Λ c ) after the projection are shown by solid and dashed lines, respectively. The width parameter is taken to be ν = 0.16fm −2 .
To discuss effects of the 7α-cluster breaking in the 28 Si core because of the spin-orbit interaction, we show, in Fig. 2, the Λ c dependence of the energy of an isolate 28 Si state before and after the parity and total-angular-momentum projection, In the Λ c = 0.3 ∼ 1.0 region, the 28 Si system gains much energy of the spin-orbit interaction by the 7α-cluster breaking. In the energy curve of E28 Si before the parity and total-angularmomentum projection, there exist two energy minimums at Λ c = 0.38 and Λ c = 0.80 though the energy almost degenerates in this region. We call these two minimums of 28 Si the "oblatetype (Λ c = 0.38)" and "spherical-type (Λ c = 0.80)" states. Here, the oblate-type state is different from the Λ c = 0 state that is the ideal state with the (200) 4 (110) 4 (020) 4 configuration in terms of the (n x , n y , n z ) notation of the h.o. shell-model basis in the sd shell. The energy of the oblate-type state at Λ c = 0.38 is about 18 MeV lower due to the 7α-cluster breaking than that of the Λ c = 0 state having no contribution of the spin-orbit interaction. This result supports the AMD calculation of 28 Si [51] and indicates that the present method of the extended 7α-cluster model is suitable to incorporate the significant energy gain of 28 Si with the 7α-cluster breaking in the oblately deformed 28 Si. In the 0 + projected 28 Si energy, it is found that the oblate-type (Λ c = 0.38) state gains further energy because of the restoration of the rotational symmetry. The present result for the 28 Si core indicates that the rotation of the oblately deformed state can be an important degree of freedom of the 28 Si core structure in the 28 Si + α system as well as the 7α-cluster breaking due to the spin-orbit interaction. Energy expectation value (MeV) ) as a function of the inter-cluster distance R. The parameter Λ c for the 28 Si core structure is fixed to be Λ c = 0.38 (oblate type:solid) and Λ c = 0.80 (spherical type:dashed).
Next, we discuss how the 28 Si core structure in the 28 Si + α system is affected by the existence of an α cluster. The α cluster at the surface of the 28 Si core may affect the feature of the 28 Si core because of the nuclear and Coulomb interactions and also Pauli blocking effect. To discuss features of the 28 Si core with an α cluster at a certain distance R from the core, we fix the parameter λ α = 0 to assume the α cluster without the breaking, and consider the 7α-breaking in the 28 Si core and also the orientation of the oblate-type 28 Si core in the 28 Si + α system. Namely, we analyze the energy expectation value of the parity-projected state before the total-angular-momentum projection, with λ α = 0. Figure 3 shows the 28 Si + α energies for the oblate-type 28 Si core (Λ c = 0.38) and the spherical-type 28 Si core (Λ c = 0.80) set at the orientation θ = 0 • . The energies are plotted as functions of the inter-cluster distance R. It is found that, in the R = 8 fm region, energies of the two cases (Λ c = 0.38 and 0.80) almost degenerate as expected from the energy degeneracy in the isolate 28 Si. In the 2 < R < 5 fm region, the energy for the oblate core is lower than that for the spherical core indicating that, when an α cluster exists at the surface, the oblatetype 28 Si core is energetically favored than the spherical-type because of the smaller overlap, i.e., the weaker Pauli blocking of nucleons between the α cluster and the core for the oblate core at θ = 0 • than in the spherical core case. ig. 4 Energy expectation value of 28 Si + α system (E + 28 Si+α (R, θ, λ α = 0, Λ c = 0.38)) as a function of the inter-cluster distance R. The rotation angle θ of the oblate core is fixed to be θ = 0 • (solid) and θ = 90 • (dashed).
To see the θ dependence of the 28 Si + α energy, we plot the energy expectation value E + 28 Si+α (R, θ, λ α = 0, Λ c = 0.38) of the oblate-type 28 Si core oriented at θ = 0 • and 90 • in Fig. 4. In the small R region (R < 5 fm), the θ = 0 • oriented core is favored because of the weaker Pauli blocking than the θ = 90 • oriented core. On the other hand, the energy does not depend on the core orientation in the large R region, in which the rotational symmetry of the 28 Si core is restored. In the 6 < R < 8 fm region around the barrier, the θ = 90 • oriented core gains slightly larger potential energy than the θ = 0 • core but the energy difference is minor.

α-cluster breaking
We analyze the λ α dependence of the energy expectation value of the 28 Si + α system to see the α-cluster breaking effect on the 28 Si + α system. Figure 5 shows, the energy E28 Si+α (R, θ = 0 • , λ α , Λ c = 0.38) with the α-cluster breaking, namely, λ α optimized at each distance R, compared with the energy for λ α = 0 without the α-cluster breaking. The energy gain by the α-cluster breaking is very small except for the R < 3 fm region. This results indicates that the α-cluster breaking in the 28 Si + α system is minor in the α-cluster excited states having large amplitudes of the α cluster at the surface region (4 < R < 6 fm). Therefore, we ignore the α-cluster breaking effect in the GCM calculation discussed in the next section for simplicity.
In the R < 2 fm region, the finite λ α gives some energy gain to the 28 Si + α system, but it is not appropriate to regard it as the α-cluster breaking because the α-cluster gets into in the inner region of the core and the 28 Si + α picture breaks down in this region. More details of the α-cluster breaking in the 28 Si + α system are discussed later.

GCM calculation
We superpose 28 Si + α wave functions and obtain the ground and excited states of 32 S with the full-GCM calculation described in Sec. 2.6.
The calculated value of the 32 S binding energy is 205.71 MeV which underestimates the experimental binding energy (271.78 MeV), whereas that of the α-separation energy is 13.8 MeV which overestimates the experimental value (6.95 MeV). We can adjust the interaction parameter M of the Volkov force to reproduce either the binding energy or the α-separation energy, but it is difficult to reproduce both data within the present two-body effective interaction. At the end of this section, we show energy levels calculated by using modified interaction parameters to see the interaction dependence of the result. Figure 6 shows the energy levels 32 S obtained by the full-GCM with the default interaction parameters. Energies measured from the 28 Si + α threshold energy are plotted as functions of J(J + 1). In the energy region near the 28 Si + α threshold, we obtain J ± = 0 + , 2 + , 4 + , and 6 + states having a remarkably developed α-cluster structure. We assign these states as α-cluster excited states belonging to an α-cluster band. In Fig. 6, the corresponding α-cluster excited states are shown by circles connected by dashed lines. The bandhead 0 + state starts from E r = 1.58 MeV above the 28 Si + α threshold and the rotational energy approximately follows the expression of the rigid rotor model: with the rotational constant k = 2 /2J = 145 keV up to the 6 + state. We do not obtain an α-cluster excited state with J ± = 8 + . We also obtain other excited states lower than the α-cluster excited states, but their energies change with the increase the number of bases and we can not obtain converged energies. This means that the present model space of the extended 28 Si + α cluster model is not sufficient to describe non-cluster states of 32 S in the low energy region. On the other hand, we obtain good convergence for the energies of the ground state and the α-cluster excited states with respect to the increase of the number of bases.
We show the overlap f J ± n 28 Si+α (R) defined in Eq. (16) between the full-GCM wave function Ψ J ± n 28 Si+α and the R-fixed frozen core wave function Φ J ± 28 Si(0 + g.s. )+α (R) for the α-cluster excited states (J ± = 0 + , 2 + , 4 + , 6 + ) in Fig. 7. We also show the overlap for the ground state. The overlap f J ± n 28 Si+α (R) indicates the α-cluster amplitude at R in the L = J orbit around the 28 Si ground state. It is found that the ground state has no developed α cluster in the large R region. In contrast to the ground state, the α-cluster excited states show the developed α-cluster in the large R region: the 0 + , 2 + , and 4 + states have large amplitudes in the R ∼ 5 fm region whereas the 6 + state has the peak at R = 4 fm with a long tail in the large R region.
We estimate the α-decay widths of the α-cluster excited states using the overlap f J ± n 28 Si+α (R) defined in Eq. (16) with the approximation method in Ref. [64]. Following the method in Ref. [64], the (dimensionless) reduced α width θ 2 α (a) at the channel radius a is approximately 12 where A, A 1 , and A 2 are the mass numbers of 32 S, 28 Si, and α cluster, respectively. Using θ 2 α (a), we calculated the partial α-decay width Γ α of the 28 Si(0 + g.s. ) + α channel in the L-wave (L = J) as, where F L and G L are the regular and irregular Coulomb functions, respectively, γ 2 w is the Wigner limit of the reduced α-width γ 2 w = 3 2 /2µa 2 , µ is the reduced mass, and k = √ 2µE r / . The calculated θ 2 α (a) and Γ α of the α-cluster band in 32 S are shown in Table  1. At a = 6 fm, the reduced α widths are significant as θ 2 α (a) = 0.26 ∼ 0.32 reflecting the spatially developed cluster structure in this band. For the α-decay widths, we calculate Γ α in two cases of the bandhead energy considering the ambiguity of the predicted bandhead energy because the α-decay width is quite sensitive to the α-decay energy. In the first case, we use the energies obtained in the present calculation, in which the bandhead energy is Table 1 The reduced α-decay widths θ 2 α (a) at the channel radii a = 6 and a = 7 fm and the partial α-decay widths Γ α for the 28 Si(0 + g.s. ) + α channel in the l = J-wave. The results calculated with M = 0.67 are shown. The α-decay energies used in the calculation of Γ α are the calculated values starting from E r = 1.6 MeV, and those shifted by 2.3 MeV to adjust the bandhead energy to the experimental value E r = 3.9 MeV reported in Ref. [57].  [57]. Let us discuss comparison with the experimental reports of the α-cluster excited states. In the experiment of elastic 28 Si + α scattering, Lönnroth et al. reported the α-cluster excited band starting from the bandhead energy E r = 3.9 ± 0.5 MeV measured from the 28 Si + α threshold [57]. They evaluated the rotational constant k = 122 ∼ 152 keV from the averaged energies of the fragmented states. In the experiment of α inelastic scattering on 32  In contrast to the strong interaction dependence of the bandhead energy, the rotational constant is not sensitive to the interaction parameter in the present calculation. Although it is difficult to quantitatively predict the handhead energy in the present calculation, we can say that the α-cluster excited states appear near the 28 Si+α threshold and construct the rotational band up to the J π = 6 + state with the rotational constant k = 140 ∼ 150 keV.

Core rotation and shape mixing effects
In the full-GCM calculation, we take into account the core rotation and the oblate-spherical mixing as well as the inter-cluster motion by superposing the parity and total-angularmomentum projected 28 Si+α wave functions with R, θ, and Λ c . For the inter-cluster distance, R = 1, 2, · · · , 10 fm are used. For the rotation angle of the 28 Si core, θ = 0 • , 30 • , 60 • , 90 • are used for the oblate core (Λ c = 0.38), and θ is fixed to be θ = 0 • for the spherical core (Λ c = 0.80). Here, we perform GCM calculations with reduced basis wave functions to discuss how the core rotation and the oblate-spherical mixing affect the α-cluster excited states. We discuss the core rotation effect on the energy spectra. In Fig. 9, we compare the energy spectra obtained by the GCM calculations of the oblate core with and without the core rotation. The former is calculated by superposing 28 Si+α wave functions with R = 1, · · · , 10 fm for the Λ c = 0.38 core at θ = (0 • , 30 • , 60 • , 90 • ), and the latter is calculated by those with R = 1, · · · , 10 fm for the Λ c = 0.38 core at the fixed angle θ = 0 • . The mixing of the   spherical core (Λ c = 0.80) is omitted in this analysis for simplicity. As the result, the energy reduction by the core rotation is remarkable for the α-cluster excited states. The band energy is reduced by about 5 MeV, which is almost consistent with 4.4 MeV reduction of the 28 Si(Λ c = 0.38) + α threshold caused by the 0 + projection of 28 Si. It indicates that, in the α-cluster excited states, the α cluster spatially develops and does not disturb the oblate core rotation.
In Fig. 10, we show the energy spectra obtained by the GCM calculation with full base wave functions and that without the spherical core (Λ c = 0.80) wave functions to see the effect of the oblate-spherical mixing. The result shows that the spherical core mixing effect is minor. Energy expectation value (MeV) R (fm) 28 Si+α system 28 Si(0 + g.s. )+α system

Fig. 11
Energy expectation values of the R-fixed 28 Si + α system and the R-fixed 28 Si(0 + g.s. ) + α system are plotted. The solid line is the 28 Si + α system and the dashed line is the 28 Si(0 + g.s. ) + α system. overlap R (fm) Fig. 12 The wave function overlap f (R) between the R-fixed 28 Si + α system and the R-fixed 28 Si(0 + g.s. ) + α system defined in Eq. (33).
In the asymptotic region at a large inter-cluster distance R, the 28 Si core should be the ground state of the isolate 28 Si(0 + g.s. ). As discussed previously, the α-cluster excited states contain dominantly the 28 Si(0 + g.s. )+α component. Therefore, it is expected that the frozen core GCM calculation with the inert 28 Si(0 + g.s. ) core assumption can be a leading order 17 approximation at least for the α-cluster excited states. The frozen core GCM calculation is the extreme case of the weak coupling and it is different from the adiabatic picture of the strong coupling. In the previous section, we start from the strong coupling picture, in which the deformed 28 Si core is located at a fixed orientation, and then consider the rotation and shape mixing effects on the α-cluster excited states obtained by the full GCM calculation. In this section, we discuss the features of the α-cluster excited states from the weak coupling picture. Namely, we start from the frozen core 28 Si(0 + g.s. )+α states, and then consider the effect of the core excitations, in particular, the rotational excitation from the 28 Si(0 + g.s. ). Note that the core excitations taken into account in the present model are the rotational excitation such as 28 Si(2 + ) and also the change of the oblate-spherical mixing (shape mixing) from the 28 Si(0 + g.s. ). After comparing the properties of the R-fixed 28 Si+α wave function between the optimized 28 Si core and the inert 28 Si(0 + g.s. ) core cases, we compare the result of the frozen core GCM calculation with that of the full GCM calculation containing the rotational and shape-mixing excitations from the 28 Si(0 + g.s. ) core. For a certain inter-cluster distance R, we define the R-fixed frozen core wave function Φ J ± 28 Si(0 + g.s. )+α (R) in Eq. (14), and also the R-fixed 28 Si + α wave function Φ J ± 28 Si ′ +α (R) in Eq. (13), where the 28 Si core wave function is optimized so as to minimize the energy expectation value of the R-fixed 28 Si + α wave function. We here consider 0 + projected wave functions. In the asymptotic region at a large inter-cluster distance R, Φ J ± 28 Si ′ +α (R) equals to Φ J ± 28 Si(0 + g.s. )+α (R). On the other hand Φ J ± 28 Si ′ +α (R) may deviate from Φ J ± 28 Si(0 + g.s. )+α (R), in the short inter-cluster distance region, in which the core excitation from the 28 Si(0 + g.s. ) occurs because of the existence of the α cluster to gain the total energy.
We plot the energy expectation values of the R-fixed frozen core wave function and the Rfixed 28 Si + α wave function in Fig. 11. In Fig. 12, we show the overlap between the R-fixed  28 Si + α wave function and the frozen core wave function, which is reduced from 1 by the core excitation. It is found that the core excitation from the 28 Si(0 + g.s. ) occurs in the R < 6 fm region and it reduces the energy of the total system 32 S in R ≤ 5 fm. These results indicate that the R > 6 fm region is understood as the ideal weak coupling regime of 28 Si(0 + g.s. )+α, whereas the rotational and shape-mixing excitations of the 28 Si core occur in the R < 6 fm region.
Next, we compare the frozen core GCM calculation given by Eq. (15) with the full-GCM calculation to see the core excitation effects in particular on the α-cluster band. Figure 13 shows the energy spectra obtained by the full-GCM and the frozen core GCM calculations. The energy of the ground state decreases by about 1 MeV from the frozen core GCM to the full-GCM calculation. The energy of the α-cluster band also shifts down slightly because of the core excitation effect.

α-cluster breaking at the nuclear surface
As mentioned in Sec. 3.2, the α-cluster breaking around the 28 Si core is minor in the surface region. We here discuss details of the α-cluster breaking around the 28 Si core in the 28 Si+α system in comparison with that around the 16 O core in the 16 O+α system to clarify the core dependence of the α-cluster breaking. λ α R (fm) 16 O+α system 28 Si+α (Λ c =0.38) system 28 Si+α (Λ c =0.80) system Fig. 17 The α-cluster breaking parameter λ α optimized to minimize the energies of the 16 O + α, 28 Si + α (Λ c = 0.38), and 28 Si + α (Λ c = 0.80) systems.
We perform a similar analysis of the α-cluster breaking for the 16 O+α system by using the following 16 O+α model wave function, where the 16 O core wave function Φ16 O is written by a tetrahedron formed 4α-clusters with the α-α distance 0.5 fm which is almost equivalent to the double closed p-shell configuration. The width parameter is taken to be ν = 0.195 fm −2 . The λ α is optimized to minimize the energy expectation value of the parity-projected 16 O + α wave function, Figure 15 shows the energy of the R-fixed 16 O+α wave function with the optimized λ α (with the α-cluster breaking) and that with the fixed λ α = 0 (without the α-cluster breaking). As discussed previously, for the 28 Si core case, the energy reduction by the α-cluster breaking is found only in the very short distance region (see Fig. 5), whereas there is almost no energy reduction in the R ≥ 3 fm region where the α-cluster excited states have the α-cluster amplitudes. Differently from the 28 Si+α system, in the 16 O + α system, the significant energy reduction by the α-cluster breaking is found in a relatively wide R region. This energy reduction by the α-cluster breaking shifts the energy minimum position to the short distance region, and it may give a significant effect to α-cluster structure in the ground band of 20 Ne as discussed in Ref. [53]. In Fig. 16, we show energy reductions by the α breaking, i.e., the energy difference between the optimized λ α and the fixed λ α = 0 cases for the 16 O + α and 28 Si + α (Λ c = 0.38) systems. We also show the energy reduction for the spherical 28 Si core (Λ c = 0.80) case. It is found that, the energy reduction of 28 Si+α (Λ c = 0. 38) system is about a half of that of 16 O + α system in the R = 2 ∼ 3 fm region, and that of 28 Si+α (Λ c = 0.80) system is quite small. Thus the α-cluster breaking gives energetically less important effects to the 28 Si+α system than to the 16 O + α system. In Fig. 17, we compare the optimized values of the α-breaking parameter λ α for each system. In both cases of the oblate and spherical 28 Si cores, λ α of the 28 Si + α system is smaller than that of the 16 O + α system at least in the R < 4 fm region. This indicates that, compared with the 16 O + α system, the α-cluster breaking is relatively suppressed in 28 Si + α, in particular, for the case of the spherical-type 28 Si core (Λ c = 0.80).
The α-cluster breaking at the nuclear surface is caused mainly by the spin-orbit potential from the core nucleus, and therefore, it is naively expected that the α-cluster breaking is likely to occur in heavier core systems because of the stronger core potential than light core systems. The present result is opposite to this expectation. The reason is understood by the Pauli blocking effect from the 28 Si core as follows. In general, in the α-cluster breaking mechanism at the nuclear surface, 4 nucleons in the broken α cluster favor to occupy the ls-favored orbits to gain the spin-orbit potential from the core rather than to form the ideal (0s) 4 α-cluster. However, in the 28 Si + α system, the ls-favored 0d 5/2 orbits are occupied by nucleons in the 28 Si core, which block the α-cluster breaking. The 0d 5/2 orbits are fully blocked, in the jj-coupling limit Λ c = 1 for the sub-shell 0d 5/2 -closed 28 Si core. Even though the 28 Si core in the 28 Si+α system is not in this limit, it has a finite Λ c and partially blocks the 0d 5/2 orbits. This picture can describe the suppression of the α-cluster breaking at the surface of the 28 Si core compared with that of the 16 O core where 0d 5/2 orbits are empty, and also the larger suppression for the spherical-type (Λ c = 0.80) 28 Si than that for the oblate-type (Λ c = 0.38) 28 Si core.

CONCLUSION
We investigated the α-cluster excited states in 32 S. We proposed an extended model of the 28 Si+α cluster model by taking into account the 28 Si core deformation and rotation as well as the α-cluster breaking. The 28 Si core is described by the extended 7α-cluster model with the cluster breaking due to the spin-orbit interaction.
Applying the extended 28 Si+α cluster model, we performed the GCM calculation and obtain the α-cluster excited states near the 28 Si+α threshold energy. These states construct the rotational band up to the 6 + state with the rotational constant k = 140 ∼ 150 keV. We can not quantitatively predict the bandhead energy because of the ambiguity of the interaction parameters. The α-cluster excited band obtained in the present work may correspond to one of the experimentally reported bands [57,58]. The calculated rotational constant reasonably agrees to the value of the experimental band reported in Ref. [57]. Although the fragmentation of the α-cluster excited states was observed in the experiment of Ref. [57], no fragmentation is found in the present calculation, maybe, because of the insufficient model space.
From the point of view of the strong coupling picture, we discussed the 28 Si core deformation and rotation effects as well as the α-cluster breaking one in the α-cluster excited states. It is found that the rotation of the oblately deformed 28 Si core significantly reduces the excitation energies of the α-cluster excited states, whereas the α-cluster breaking gives only a minor effect. We also analyzed the feature of the α-cluster excited band from the weak coupling picture using the frozen core 28 Si(0 + g.s. ) + α wave functions. The α-cluster excited states are found to have the dominant 28 Si(0 + g.s. ) + α components. The dimensionless reduced α widths estimated by the 28 Si(0 + g.s. ) + α components are significantly large 22 as θ 2 α (a) = 0.26 ∼ 0.32 at a = 6 fm. We evaluated the partial α-decay widths from the calculated values of θ 2 α (a). We also compared the result of the frozen core GCM calculation with that of the full GCM calculation, and found that the rotational excitation from the 28 Si(0 + g.s. ) plays an role to stabilize the α-cluster excited states. The present model is the extended 28 Si+α cluster model, in which the cluster breaking due to the spin-orbit interaction and also the rotation of the deformed core are taken into account. The cluster breaking effect of the 28 Si core part gives the large energy reduction (18 MeV) of the isolate 28 Si from the 7α-cluster model without the cluster breaking. This is an advantage over conventional cluster models using the Brink-Bloch α-cluster model. Moreover, the rotation effect of the deformed core in 32 S gives about 5 MeV reduction of the α-cluster band energy from that obtained with the fixed core orientation. This indicates the importance of the angular momentum projection of the subsystem in the α-cluster excited states having the deformed core.

Acknowledgements
The numerical calculations were carried out on SR16000 at YITP in Kyoto University.