Effects of cluster-shell competition and BCS-like pairing in $^{12}$C

The antisymmetrized quasi-cluster model (AQCM) was proposed to describe {\alpha}-cluster and $jj$-coupling shell models on the same footing. In this model, the cluster-shell transition is characterized by two parameters; $R$ representing the distance between {\alpha} clusters and {\alpha} describing the breaking of {\alpha} clusters, and the contribution of the spin-orbit interaction, very important in the $jj$-coupling shell model, can be taken into account starting with the {\alpha} cluster model wave function. Not only the closure configurations of the major shells, but also the subclosure configurations of the $jj$-coupling shell model can be described starting with the {\alpha}-cluster model wave functions; however, the particle hole excitations of single particles have not been fully established yet. In this study we show that the framework of AQCM can be extended even to the states with the character of single particle excitations. For $^{12}$C, two particle two hole (2p2h) excitations from the subclosure configuration of $0p_{3/2}$ corresponding to BCS-like pairing are described, and these shell model states are coupled with the three {\alpha} cluster model wave functions. The correlation energy from the optimal configuration can be estimated not only in the cluster part but also in the shell model part. We try to pave the way to establish a generalized description of the nuclear structure.

on this idea; however we transform the cluster model wave functions directly to the ones of the jj-coupling shell model and try to pave the way to establish a generalized description of the nuclear structure. Also, antisymmetrized molecular dynamics (AMD) and Fermionic molecular dynamics (FMD) have been successfully introduced to describe both characters of shell and cluster models [17][18][19][20][21]. In these models, central positions of all the nucleons are optimized under some constrains. On the other hand, in our approach, we introduce much fewer and controllable parameters, which allow the description of excited configurations.
In AQCM, we transform Brink-type α cluster model wave function [22] to the jj-coupling shell model wave function by giving imaginary part for the Gaussian center parameters. This procedure has some similarity with the idea of Fock-Bargmann space developed by Filippov et al. [23]. In Ref. [23], they discussed 6 He and the hyperspherical harmonics basis states have been introduced for the description of two valence neutrons outside of the α core, and the matrix elements of the Hamiltonian have been extracted from the expectation value obtained by using a Gaussian wave packet. We also use Gaussian wave packets; however, in our study, we directly transform the wave function to the jj-coupling shell model and the breaking effect of the α cluster part can be discussed.
This paper is organized as follows. We describe our formulation in this work including the review for AQCM in Sec. II. The results and discussion are given in Sec. III. Finally, we present conclusion and outlook in Sec. IV.

A. AQCM wave function
As in many conventional models, the single-particle wave function of AQCM (φ i ) consists of the spatial (ψ i ), spin (χ i ), and isospin (τ i ) parts, The spatial part of the single-particle wave function has a Gaussian shape [22], where ν is the width parameter. From these single-particle wave functions, the Slater determinant of A nucleon system Ψ = A[φ 1 , . . . , φ A ] is constructed, where A is the antisymmetrizer for all nucleons. If we give the same value for the Gaussian center parameter ζ i of four nucleons (spin-up proton, spin-down proton, spin-up neutron, and spin-down neutron) as in the so-called Brink model, they form an α cluster, and the contribution of the spin-orbit interaction vanishes because of the antisymmetrization effect. In Ref. [5], the AQCM wave functions were shown to describe subclosure configurations of the jj-coupling shell model. The Gaussian center parameters {ζ i } are complex vectors, and the imaginary parts are introduced as where e (spin) i is a unit vector for the intrinsic-spin orientation of i-th nucleon, and Λ is an order parameter for the dissolution of the cluster. By introducing Λ, α clusters are transformed to quasi clusters with the spin-orbit contribution.
B. Description of subclosure configuration ( 12 C case) Before extending AQCM to describe single particle excitations, here we review the description of the subclosure configuration for the 12 C case [2]. This part is the mathematical interpretation of AQCM and not needed in the actual calculation; however we have to recall the important parts for further extension of the model. Since the neutron part is introduced in the completely same way, here we concentrate on the proton part. The protons i = 1 and 2 are in a common quasi cluster with spin-up and spin-down. Based on the original idea of Eq. (3), the Gaussian center parameters are introduced as and where e x and e y are unit vectors on the x and y axes, respectively. There are put on the x axis, and imaginary parts are given in the y and −y directions, since their intrinsic spins are quantized along the z axis (z and −z directions). They are introduced as time reversal partners. The squares in the powers of the single-particle wave functions can be expanded as where χ ↑ and χ ↓ stand for spin-up and down, respectively, and τ 1 and τ 2 are isospin wave functions of the protons. In Eq. (6), the cross-term part in the power of the exponential can be Taylor expanded, and by substituting Eq. (4), this factor is described as For Λ = 1, by using the spherical harmonics Y lm (Ω) and introducing the radial part of the spatial wave function R 0l (r), the single-particle wave function of the proton i = 1 can be expressed as where and and s l and t l are the normalization factors of Y ll (Ω) and R 0l (r), respectively. The proton i = 1 has spin-up, and the spherical harmonics Y ll (Ω) with spin-up has j z = l + 1/2, which only couples to j = l + 1/2 (stretched configuration), and the spin-orbit interaction works attractively. Thus the proton i = 1 is described as a linear combination of j-upper orbits with j = l + 1/2 and j z = j, where a j is a coefficient for the r|j, j orbit with a separated factor of R j− 1 2 . The proton i = 2 is the time reversal partner of i = 1 with spin-down, For other protons, i = 3 and 4 are introduced as in the same quasi cluster, and i = 5 and 6 also belong to the same quasi cluster, but this is different from the one for i = 3 and i = 4. Their wave functions are introduced by rotating both the spatial and spin parts of the protons i = 1, 2 about the y axis as where i = 1, 2. The rotation does not change the total angular momentum j, and the resultant single-particle wave functions are also linear combinations of j-upper orbits. Here, α, β, γ are the Euler angles, andR(α, β, γ) is the rotation operator. The parameters θ 1 and θ 2 are rotational angles, and they are introduced as θ 1 = 2π/3 and θ 2 = 4π/3, which give an equilateral triangular shape of the three α clusters when Λ is equal to zero. The r|j, j orbit after the rotation can be expressed aŝ where d j km (β) is Wigner's small d function, Thus the rotated single-particle wave function is expressed aŝ The result shows that when Λ is equal to unity, all the single-particle wave functions are described as the linear combinations of j-upper orbits, and the Slater determinant has only the (0s 1/2 ) 4 (0p 3/2 ) 8 component at the lowest order of R.

C. Extension of AQCM
Here we explain our new model, which is the extension of AQCM.

Total wave function
The total wave function is expressed as a linear combination of different Slater determinants based on the generator coordinate method (GCM) as whereP J MK andP π are the angular momentum and parity projection operators. Here k = 1, 2 . . . is a label for different basis states. The coefficients {c n k } are determined by solving the Hill-Wheeler equation, and n = 1, 2, . . . denotes the n-th excited state obtained after the diagonalization of the Hamiltonian. In this paper, we particularly pay attention to the 0 + states, thus J = M = K = 0 and π = +.
For the basis states, we prepare both the shell and cluster model ones. For the shell model part, we use AQCM, and in addition to the subclosure configuration of 0p 3/2 , we introduce five different two particle two hole (2p2h) configurations. For the cluster model space, we introduce thirty different three α configurations. In total, we superpose 6 + 30 = 36 basis states and diagonalize the Hamiltonian. For the width parameter ν (= 1/2b 2 ) in Eq. (2), we take b = 1.4 fm.
For the cluster model basis states, as schematically shown in Fig. 1, the configurations are introduced with isosceles triangular shapes. The parameters d and h are the base and height of the isosceles triangle, respectively, and they are taken as d = 1, 2, . . . , 5 fm and h = 1, 2, . . . , 6 fm. There are 5 × 6 = 30 basis states for the cluster model side.

Hamiltonian
The Hamiltonian used in the present calculation iŝ whereT is the total kinetic energy operator andT G is the kinetic energy operator of the center of mass motion. For the central forceV C , we use the Volkov No.2 force [24] given bŷ where V a = −60.65 MeV, V r = 61.14 MeV, α = 1.80 fm, and ρ = 1.01 fm are the original values. We take M = 1 − W = 0.6. Here, B and H denote the Bartlett and Heisenberg terms, which are added to remove the bound state of two neutrons. We take B = H = 0.125. For the spin-orbit forceV LS , we use the spin-orbit part of the G3RS force [25] given byV where η 1 = 0.447 fm and η 2 = 0.6 fm are the original values. The coefficients V LS1 = −V LS2 = 1600 MeV are determined to give a reasonable energy for the ground state in 12 C. Also, the validity of V LS1 = −V LS2 = 1600 MeV is checked in Ref [2]. The operatorV Coulomb is the Coulomb potential for protons.

III. RESULTS AND DISCUSSION
As already seen, the closure configurations of the major shells can be described by conventional α cluster models, and subclosure configurations of the jj-coupling shell model can be described by AQCM. Here we extend AQCM. At first we discuss the AQCM wave functions with general Λ values and next show how to describe particle hole excitations. For 12 C, single particle excitations from 0p 3/2 to 0p 1/2 and 0d 5/2 are introduced for the proton part, neutron part, and proton-neutron part, and the effect of BCS-like pairing is incorporated. Finally these shell-model-like wave functions are coupled with the three α cluster wave functions.

A. AQCM wave functions with general Λ values
We already discussed that Λ = 0 corresponds to α cluster states and Λ = 1 with small R corresponds to the jj-coupling shell model states. However, the discussion for the general Λ values (Λ = 0, 1) is insufficient. In this subsection, we investigate the feature of the AQCM wave functions with general Λ values in 12 C.
Using the relations for the spherical harmonics and Eqs. (8) and (11), the single-particle wave function of the proton i = 1 [Eq. (6)] with general Λ becomes We introduce jj-coupling bases using the Clebsch-Gordan coefficients, Thus the single-particle wave function of proton i = 1 with general Λ becomes where s and p are indexes to distinguish the s and p orbits. Similarly, the single-particle wave function of the proton i = 2, which is time reversal of i = 1, becomes The single-particle wave functions of protons i = 3 − 6 are generated by multiplying the rotational operators R(0, 2π/3, 0) orR(0, 4π/3, 0) for the single-particle wave functions of protons i = 1, 2 as in original AQCM [2], Thus the proton part of the wave function becomes where we omit the isospin part of the wave function. The neutron part is introduced in the completely same way. In Eq. (37), two protons occupy 0s orbits and the others are described by the superposition of four different 0p orbits. We can easily check that Λ = 1 gives the (0s 1/2 ) 2 (0p 3/2 ) 4 configuration with a small enough R value as in original AQCM [2].

C. Description of 1p1h
Next we further improve the AQCM wave function to describe the 1p1h excitations from the subclosure configuration. For this purpose, we generalize the Gaussian center parameter in Eq. (2) as where R is a real number with a dimension of length, and a i , b i , and c i are dimensionless real numbers. Here e x , e y , and e z are unit vectors for the x, y, and z axes, respectively. The spin orientation is no longer fixed along the z axis, and the spin wave function χ i is more generalized as where β i is taken as a real parameter for simplicity. If (a i , b i , c i ) = (1, Λ, 0) and β i = 0 are satisfied, Eq. (41) coincides with original AQCM in Eq. (4). The single-particle wave function is expanded with the jj-coupling shell model bases, {|j, j z }, as in the previous subsections. Using the relation and Eqs. (11), (27)−(30), this generalized single-particle wave function becomes where A i = (a i + b i )/2, B i = (a i − b i )/2, and C i = c i / √ 2, respectively. The single-particle wave function has all the four components of 0p 3/2 orbits and all the two components of 0p 1/2 orbits with different coefficients. Now a proton has all the six components of 0p orbits, and we remove some of them by imposing conditions. If A i sin(β i /2) − √ 2C i cos(β i /2) = 0 is satisfied, we can eliminate the component of |p, 3/2, 1/2 . Similarly, if C i sin(β i /2) − √ 2B i cos(β i /2) = 0 is satisfied, the component of |p, 1/2, −1/2 vanishes. Thus, if A i sin(β i /2) − √ 2C i cos(β i /2) = 0 and C i sin(β i /2) − √ 2B i cos(β i /2) = 0 are simultaneously satisfied, the singleparticle wave function does not have the |p, 3/2, 1/2 and |p, 1/2, −1/2 components. This was for one proton; however if all the protons satisfy the same conditions, the proton part of the wave function also does not have the components of |p, 3/2, 1/2 and |p, 1/2, −1/2 . This is nothing but 1p1h excitation to 0p 1/2 .
In the following part, we simplify the conditions to describe 1p1h. The conditions A i sin(β i /2) − √ 2C i cos(β i /2) = 0 and C i sin(β i /2) − √ 2B i cos(β i /2) = 0 are equivalent to tan( As a i , b i , and c i are real numbers, another condition of |a i | ≥ |b i | is required. As a result, the conditions for the spin part of the wave function become tan(β i /2) = sign(a i c i ) 2(a i − b i )/(a i + b i ), where sign(ξ) = ξ/|ξ|. As an example which realizes the conditions, we show a set for the six protons in Table I. The center of mass of the system is set to the origin. The parameters for protons i = 1 and 2 are equivalent to the ones for the original AQCM wave function [2]. We can confirm that the presence of a real parameter Λ avoids the risk that some of the single particle orbits are not linear independent. For the range of Λ, only 0 < Λ < 1 is allowable. Using these parameters, the wave function of the proton part Ψ p = A[φ 1 , . . . , φ 6 ] describes the (0s 1/2 ) 2 (0p 3/2 ) 3 (0p 1/2 ) 1 configuration at the limit of R → 0.
1. Description of (0s 1/2 ) 2 (0p 3/2 ) 2 (0d 5/2 ) 2 configuration by a linear structure Here we assume a linear shape, and the single-particle wave functions of protons i = 3, 4 are generated by multiplying the rotational operatorR(α = 0, β = π, γ = 0) for the protons i = 1, 2, respectively, and Note that the rotation angle is π so as to generate a linear structure. The Gaussian center parameters of protons i = 5, 6 are set to the origin, and ζ i=5 = ζ i=6 = 0, where i = 5 and 6 are spin-up and spin-down protons, respectively, These six protons are arranged on a straight line, which creates additional nodes owing to the antisymmetrization effect. Thus the proton part of the wave function becomes where the isospin part is omitted. This wave function coincides with the (0s 1/2 ) 2 (0p 3/2 ) 2 (0d 5/2 ) 2 configuration (d 5/2 -2p2h) at the limit of R → 0. The same procedure can be applied to the neutron part in the completely same way.
2. Description of (0s 1/2 ) 2 (0p 3/2 ) 2 (0d 5/2 ) 2 configuration by a regular triangle structure Here we describe the (0s 1/2 ) 2 (0p 3/2 ) 2 (0d 5/2 ) 2 configuration (d 5/2 -2p2h) in a different way; we do not assume a linear shape and the configuration remains with a regular triangular shape. However, in return, the protons i = 3 − 6, are rotated not about the y axis but about the z axis. Then, not only the total angular momentum j, the z component j z is unchanged after the rotation. The single-particle wave functions of protons i = 3, 4 are generated by multiplying the rotational operatorR(α = 2π/3, β = 0, γ = 0) to the protons i = 1, 2, respectively, and The single-particle wave functions of protons i = 5, 6 are generated by multiplying the rotational operatorR(α = 4π/3, β = 0, γ = 0) to the single-particle wave functions of protons i = 1, 2, respectively, and These six protons are arranged in a regular triangular shape and the proton part of the wave function becomes where the isospin part is omitted. This wave function coincides with the (0s 1/2 ) 2 (0p 3/2 ) 2 (0d 5/2 ) 2 configuration (d 5/2 -2p2h) at the limit of R → 0. This is another method to create the d 5/2 -2p2h configuration. The same procedure can be applied to the neutron part.

E. Energy levels and principal quantum numbers
We couple all of the 2p2h configurations to the subclosure configuration of 0p 3/2 , and finally the three α cluster wave functions are mixed. Concerning the R and Λ values, for the 1p1h configuration in Sec. III C, R = 0.1 fm and Λ = 0.1 are employed, and for the 2p2h excitation to 0p 1/2 in Sec. III B, we take R = 0.1 fm and (Λ a , Λ b ) = (−3/2, 0). For the 2p2h excitation to 0d 5/2 in Sec. III D 2, we take R = 0.1 fm in Eqs. (4) and (5). We discuss the obtained 0 + energy levels, principal quantum numbers, and E0 transition matrix elements.

Energy levels with the shell model basis states
We start with the shell model configurations introduced in Sec. II C. Figure 2 (a) shows the 0 + energy of 12 C with the subclosure configuration of 0p 3/2 (pn-0p0h), which is −84.5 MeV (the experimental value −92.2 MeV [26]). In Fig. 2 (b), the 0 + levels obtained after coupling with the 2p2h configurations are shown. Here, we mixed five different 2p2h configurations; two nucleons are excited from 0p 3/2 to 0p 1/2 (pp-p 1/2 -2p2h, nn-p 1/2 -2p2h), or they are excited to 0d 5/2 (pp-d 5/2 -2p2h, nn-d 5/2 -2p2h). In addition, we couple a configuration that one proton and one neutron are excited from 0p 3/2 to 0p 1/2 (pn-p 1/2 -2p2h). In this way, we include the effects of BCS-like pairing for the proton part, neutron part, and proton-neutron part. The energy of the ground state becomes −86.9 MeV, and this is lower than that of the subclosure configuration by 2.4 MeV. The reduction is caused by the coherent effects of the three BCS-like pairings. The squared overlap between the ground state of the shell-model basis states and subclosure configuration of 0p 3/2 (pn-0p0h) is 0.91. Finally we couple the shell and cluster basis states. In Table III, the 0 + energies [E (MeV)] and principle quantum numbers (N ) of 12 C are shown. The present interaction gives slightly lower ground state energy for the cluster basis states (−89.1 MeV) compared with the one for the shell model basis states (−86.9 MeV), but this is related to the fine tuning of the interaction parameters. The ground state energy gets lower by 3.5 MeV from the one for the cluster model basis states by mixing both the shell and cluster model basis states (−92.6 MeV), since the spin-orbit interaction was not be taken into account within the cluster model basis states. If we calculate without the 2p2h basis states, namely only within the subclosure configuration of 0p 3/2 and cluster model basis states, the energy is −91.8 MeV. This is higher by 0.8 MeV than the final result, and the mixing of the 2p2h configurations is found to have a certain effect. The principal quantum number for the ground state obtained with the shell model basis states is close to 8, which is the lowest possible value, even though the 2p2h excitations to 0d 5/2 are allowed. On the other hand, the cluster model gives rather large value of 11.22, and this is reduced to 9.15 after coupling with the shell model basis states. The three α configuration shrinks after coupling with the jj-coupling shell model states, as discussed in many preceding works including ours [3,17,18,20]. The 0 + 2 state is the famous Hoyle state, which has the character of weakly interacting three α clusters. Experimentally the state appears at E x = 7.65 MeV, and our final result gives 9.2 MeV. Only within the cluster model basis states, the principal quantum number is 20.01, and this is reduced to 14.00 after coupling with the shell model basis states. Since the ground state wave function is drastically changed after mixing the shell model basis states to the cluster configurations, the 0 + 2 state is also influenced because of the orthogonal condition [27]. The matrix element of the E0 transition between the 0 + 1 and 0 + 2 states is 7.36 e fm 2 , which is 9.22 e fm 2 only within the cluster model basis states (experimental value is 5.52 e fm 2 ). In Table IV, we show the squared overlaps between the 0 + 1,2 states obtained with the full model space and the six shell model basis states introduced in the calculation. The squared overlap between the 0 + 1 state and the pn-0p0h is 0.42, which is reduced from the one only within the shell model basis states (0.91). This is because, using the present interaction parameters, the cluster model basis states gives lower ground state energy compared with the shell model basis states; however this tendency may change when we use slightly different parameter set. Concerning the 2p2h configurations, the squared overlaps between the 0 + 1 state and the pn-, pp-, and nn-p 1/2 -2p2h are about 0.04 − 0.07, which are not negligible. However, the squared overlap between the 0 + 1 state and pp-d 5/2 -2p2h or nn-d 5/2 -2p2h is more than an order of magnitude smaller. For the 0 + 2 state, it has the squared overlap with pn-0p0h by 0.33, but the squared overlaps with the 2p2h configurations are quite small.

IV. CONCLUSION
We have developed the framework of AQCM to describe not only the subclosure configuration of the jj-coupling shell model but also the 2p2h configurations, in addition to the cluster model wave functions. In 12 C, it was shown that the 2p2h excitations from 0p 3/2 to 0p 1/2 and that to 0d 5/2 were successfully described, which enables us to include the effects of BCS-like pairing for the proton part, neutron part, and proton-neutron part. The correlation energy from the optimal configuration can be estimated not only in the cluster part but also in the shell model part.
For the ground 0 + state of 12 C, the interaction of the present calculation gives slightly lower energy for the cluster model basis states (−89.1 MeV) compared with the one for the shell model basis states (−86.9 MeV), and the ground state energy gets lower by 3.5 MeV by mixing both the shell and cluster model basis states (−92.6 MeV). This is because the spin-orbit interaction is not be taken into account within the cluster model basis states. If we calculate without the 2p2h basis states, the energy becomes −91.8 MeV, about 0.8 MeV higher, and the mixing of the 2p2h configurations is found to have a certain effect. Only within the cluster model basis states, the principal quantum number is rather large, 11.22, and this is reduced to 9.15 after coupling with the shell model basis states. The three α configuration shrinks after coupling with the jj-coupling shell model states. The squared overlap between the ground 0 + state and the 0p0h configuration of the jj-coupling shell model is 0.42, and the overlaps with some of the 2p2h configurations are 0.04 − 0.07, which are not negligible.
The 0 + 2 state is the famous Hoyle state, and the present model gives E x = 9.2 MeV. The cluster model basis states give the principal quantum number of 20.01, and this is reduced to 14.00 after coupling with the shell model basis states. Since the ground state wave function is drastically changed after mixing the shell model basis states to the cluster configurations, the 0 + 2 state is also influenced because of the orthogonal condition. The method of describing particle hole excitations is considered to be applied to other light or even heavier nuclei. As an example, description of four particle for hole configurations such as (0s 1/2 ) 4 (0p 3/2 ) 8 (0d 5/2 ) 4 and the coupling with the cluster model wave functions ( 12 C+α, four α's) are going on for 16 O; the understanding of "the mysterious 0 + state" is a long-standing problem [28]. The systematic description of competition between particle hole excitations and cluster states is a challenging subject to be performed in near future.