Deformation and $\alpha$ clustering in excited states of $^{42}$Ca

The coexistence of various low-lying deformed states in $^{42}$Ca and $\alpha$--$^{38}$Ar correlations in those deformed states have been investigated using deformed-basis antisymmetrized molecular dynamics. Wave functions of the low-lying states are obtained via parity and angular momentum projections and the generator coordinate method (GCM). Basis wave functions of the GCM calculation are obtained via energy variations with constraints on the quadrupole deformation parameter $\beta$ and the distance between $\alpha$ and $^{38}$Ar clusters. The rotational band built on the $J^\pi = 0_2^+$ (1.84 MeV) state as well as the $J^\pi = 0_3^+$ (3.30 MeV) state are both reproduced. The coexistence of two additional $K^\pi = 0^+$ rotational bands is predicted; one band is shown to be built on the $J^\pi = 0_3^+$ state. Members of the ground-state band and the rotational band built on the $J^\pi = 0_3^+$ state contain $\alpha$--$^{38}$Ar cluster structure components.


Introduction
Drastic structural changes initiated by low excitation energies are a significant characteristic of nuclear systems, and the coexistence of deformed states and cluster structures is a typical phenomenon. In the mass number region A ∼ 40, low-lying normal-deformed (ND) and superdeformed (SD) bands with many-particle-many-hole (mp-mh) configurations have been confirmed experimentally in 36,38,40 Ar [4,12,15], 40 Ca [5], 42 Ca [1,9], and 44 Ti [11]. The SD band in 36 Ar, ND and SD bands in 40 Ca, and SD band in 44 Ti are considered to have configurations of 4p8h, 4p4h, 8p8h, and 8p4h, respectively, relative to the sd-shell doubleclosed structure. Coupling of the cluster structure components in deformed states such as the α-cluster structure in the ND band of 40 Ca [10,14,17,18] and the ground-state band in 44 Ti [8,10,18] has also be investigated.
In 42 Ca, deformed states with mp-mh configurations and clustering behavior have been observed experimentally, and the rotational band built on the J π = 0 + 2 (1.84 MeV) state (K π = 0 + 2 band) has been observed [1,9]. This rotational band has a large moment of inertia that is similar to the ones of the SD bands in 36 Ar and 40 Ca, scaled by A 5/3 [9], which is proportional to square of quadrupole deformation parameter β in the liquid-drop model. In contrast to the small level spacings, the in-band E2 transition strengths are rather weak and are of the same order as those of the ND band in 40 Ca. With regard to α-cluster structures, strong population to the J π = 0 + 1 and 0 + 3 (3.30 MeV) states has been observed in α-transfer reactions to 38 Ar, and the ratios of the cross sections of α and 2n transfer reactions suggest that the J π = 0 + 2 and 0 + 3 states have configurations of 6p4h and 4p2h, respectively [3]. Theoretically, the α+ 38 Ar orthogonality condition model (OCM) describes 4p2h states with α-38 Ar cluster structures, but a rotational band with a 6p4h configuration is not obtained in low-lying states [13]. To understand the structures in 42 Ca, various deformation with mp-mh configurations and clustering should be taken into account, but such a study has never been performed. The structures of low-lying states in 42 Ca have not yet been clarified.
This paper aims to clarify the structures of excited deformed bands in positive-parity states of 42 Ca by focusing on the coexistence of rotational bands with mp-mh configurations. The α-38 Ar cluster correlations in low-lying deformed states are also discussed. To discuss coexistence and mixing of deformed and cluster structures, the generator coordinate method (GCM) are used.
This paper is organized as follows; In Sec. 2, the framework of this study is explained briefly. In Sec. 3, results of energy variation to obtain GCM basis are shown. In Sec. 4, coexistence of various deformed states and their structures are discussed. In Sec. 5, relations of E2 transition strengths and particle-hole configurations are discussed. Finally, conclusions are given in Sec. 6.

Framework
In this section, the framework of the study is explained briefly. Details of the framework are provided in Refs. [6,7,16].

Wave function
The wave functions in low-lying states are obtained by using the parity and angular momentum projection (AMP) and the GCM with deformed-basis antisymmetrized molecular dynamics (AMD) wave functions. A deformed-basis AMD wave function |Φ is the Slater determinant of Gaussian wave packets that can deform triaxially such that whereÂ denotes the antisymmetrization operator, and |ϕ i denotes a single-particle wave function. The |φ i , |χ i , and |τ i denote the spatial, spin, and isospin components, respectively, of each single-particle wave function |ϕ i . The real 3 × 3 matrix K denotes width of the Gaussian single-particle wave functions that can deform triaxially, which is common to all nucleons. The Z i = (Z ix , Z iy , Z iz ) are complex parameters to denote the centroid of each single-particle wave function in phase space. The complex parameters χ ↑ i and χ ↓ i denote the spin directions. Axial symmetry are not assumed.

Energy variation
Basis wave functions of the GCM are obtained via the energy variation with a constraint potential V cnst after projection onto positive-parity states, whereĤ is Hamiltonian, andP r denotes the parity operator. Variational parameters are K, Z i , and χ ↑,↓ i (i = 1, ..., A). The isospin component of each single-particle wave function is fixed as a proton (π) or a neutron (ν). The Gogny D1S force is used as the effective interaction.
To obtain deformed and cluster structure wave functions, two types of constraints V cnst are used: the quadrupole deformation parameter β of the total system and the distance d between α and 38 Ar clusters, Here β is the matter quadrupole deformation parameter. The distance d between α and 38 Ar clusters are defined as distance between centers of mass of α and 38 Ar clusters, where i ∈ α and 38 Ar mean that ith nucleon is contained in α and 38 Ar clusters, respectively. It should be noted that the σ (= x, y, z) component of the spatial center of the single-particle wave function |ϕ i is ReZiσ √ νσ . Details of the constraint of intercluster distance are provided in Ref. [16]. When sufficiently large values are chosen for v β and v d , the resultant values β and d become β 0 and d 0 , respectively.

Generator coordinate method
After performing the constraint energy variation for |Φ + , we superpose the optimized wave functions by employing the quadrupole deformation parameter β and the distances d between α and 38 Ar clusters, whereP J π M K is the parity and total angular momentum projection operator, and Φ β i and Φ d i are optimized wave functions with β and d constraints for the constrained values β = β  quadrupole deformation parameter β 2p 4p2h 6p4h 8p6h π = + J π = 0 + Fig. 1 The energy curves as functions of the quadrupole deformation parameter β for positive-parity (dashed lines) and J π = 0 + (solid lines) states. Circles, triangles, crosses, and squares indicate the 2p, 4p2h, 6p4h, and 8p6h configurations, respectively (see text).
case of d-constrained wave functions, the z-axis is chosen as the vector which connects the α and 38 Ar clusters. The coefficients f β iK and f d iK are determined by the Hill-Wheeler equation, Then we get the energy spectra and the corresponding wave functions that expressed by the superposition of the optimum wave functions, {|Φ β i } and {|Φ d i }. Figure 1 shows the energy curves as functions of β, which are obtained by the energy variations with the constraint on β. Harmonic-oscillator (HO) quanta of obtained wave functions for protons and neutrons, N π and N ν , respectively, are (N π , N ν ) = (0, 0), (2, 0), (2,2), and (4, 2) on and close to the β-energy surface relative to the lowest allowed state. The N τ (τ = π or ν) are defined as

GCM basis obtained by energy variation
where n π and n ν denote proton and neutron numbers, respectively, and N 0π and N 0ν denote HO quanta of the lowest allowed states for protons and neutrons, respectively. The (0, 0), (2, 0), (2,2), and (4, 2) configurations correspond to configurations of [(pf  Through the AMP, largely deformed states gain higher binding energies, and the energy of the local-minimum state for the 6p4h configuration projected onto the J π = 0 + state becomes lower than that for 4p2h configuration. The order of the local-minimum energies are the reverse of that before the AMP. Upper panel of Fig. 3 shows energy curves of α-38 Ar cluster structures as functions of intercluster distance between α and 38 Ar clusters obtained by energy variations with the α-38 Ar intercluster-distance constraint. In the calculations, two types, labeled A and B types, of α-38 Ar cluster structure wave functions are obtained that differ in the orientation of the 38 Ar clusters. An 38 Ar cluster has two proton-hole configuration relative to the sd-shell double-closed structure. Proton holes of 38 Ar clusters in A-and B-type wave functions are in parallel and orthogonal directions to an α cluster, respectively. In small intercluster distance region, the A type has similar energies as the minimum energy on the β-energy surface. But the B type is still excited, and the energies are similar to those at the local minimum of the 4p2h configuration of β-energy curves. Lower panel of Fig. 3 shows HO quanta for protons and neutrons, respectively, relative to the lowest-allowed state in 42 Ca. At small intercluster distance, the A type goes to the lowest-allowed state, whereas proton part of the B type are 2 ω excited, which is the 4p2h configuration. It is because protons on the direction of an α cluster are occupied in sd-shell in the B type 38 Ar clusters. Owing to the Pauli principle, two protons are in the pf -shell even in small α-38 Ar intercluster distance. Density distributions of A-and B-type wave functions with d = 5.0 fm are shown in Figs. 2 (e) and (f), respectively. They have two spatially localized subsystems corresponding α and 38 Ar clusters, which show α-38 Ar cluster structures. Shapes of 38 Ar clusters are distorted due to intercluster interaction though ground state of 38 Ar is almost spherical. Figure 4 shows the level scheme of the positive-parity states in 42 Ca up to the J π = 8 + states obtained via the AMP and the GCM. The GCM bases are deformed-structure wave functions with configurations of 2p, 4p2h, 6p4h, and 8p6h obtained via energy variations with the β constraint and the α-38 Ar cluster structure wave functions obtained via energy 6  variations with the α-38 Ar intercluster distance constraint to a maximum of 9.0 fm. Wave functions that contain more than 0.5 % of the J + K components P J+ KK are adopted to GCM basis for each JK to avoid numerical errors in the AMP. Convergence of the GCM calculation was confirmed by a comparison with a restricted set of basis wave functions (the energies of states listed in Fig. 4 change by less than 0.3 MeV when the number of basis wave functions is halved). Three K π = 0 + rotational bands coexist in the excited states, labeled as ND1, ND2, and SD. Figure 4.1 shows squared overlaps of J π = 0 + states and J π = 0 + components of wave functions obtained by energy variation with the β constraint for the GS, ND1, ND2 and SD bands. The dominant components of the ND1, ND2 and SD states have 6p4h, 4p2h, and 8p6h configurations, respectively, and the quadrupole deformation parameters of their dominant components are β = 0.43, 0.28, and 0.53, respectively. The ground-state (GS) band has a 2p configuration. The theoretical level spacings of the GS band are underestimated although the GS band is considered to have a simple [(f 7/2 ) 2 ] ν structure. This underestimation does not affect qualitative properties of ND1, ND2 and SD bands because particle-hole configurations of their dominant components are much different from those of the GS band.

Coexistence of various rotational bands 4.1. Level scheme
The present results suggest existence of side bands of the ND1, ND2, and SD bands due to triaxial deformation. The ND1S1 and ND1S2 are side bands of the ND1 band whose dominant K components are |K| = 2 and 4, respectively. The ND2S and SDS bands are side bands of the ND2 and the SD bands, respectively, with dominant components of |K| = 2. The ND2 and SD bands, and the side bands of the ND1, ND2, and SD bands, are theoretical predictions; candidate states have not been observed yet.

α-38 Ar cluster correlations
To analyze the α-38 Ar cluster structure correlations in the low-lying rotational bands, squared overlaps of the band head states and α-38 Ar cluster structure components were 7/11 s ua a ua u ati n a a t  calculated for GS, ND1, ND2 and SD bands, as shown in Fig. 6. The J π = 0 + GS and 0 + ND2 states have large amount of A-and B-type α-38 Ar cluster structure components, respectively, at large intercluster distances as well as at small distances. Particle-hole configuration of the A-and B-type α-38 Ar cluster structure wave functions are equal to the the dominant particle-hole configurations of the J π = 0 + GS and 0 + ND2 states, respectively, at small intercluster distance region, which shows particle-hole configuration of cluster wave functions at small intercluster distance are important for coupling to deformed states. The J π = 0 + ND1 and 0 + SD states have small amount of α-38 Ar cluster structure components for any intercluster distance. Distributions of squared overlaps are similar up to high-spin state in each band.

E2 transition strengths
The B(E2) values of the in-band transitions in the theoretical ND1, ND2, and SD bands, and the experimental K π = 0 + 2 band in Weisskopf units are listed in Table 1. The theoretical values for the ND1 band are within error of the experimental values for the K π = 0 + 2 band. The in-band B(E2) values from higher-spin state of the ND2 band are smaller, which is caused by mixing of components other than 4p2h configuration in higher-spin states. The B(E2) values of the SD band are much larger than those of ND1 and ND2 bands.

Band assignment
The amount of α-38 Ar cluster components (Fig. 6), particle-hole configurations of dominant components and the in-band transition B(E2) values (Tab. 1) indicate that the ND1 band and the ND2 band head correspond to the experimental K π = 0 + 2 band and J π = 0 + 3 state, respectively. The large amount of α-38 Ar cluster components in the J π = 0 + ND2 state reveals that this state corresponds to the experimental J π = 0 + 3 state because of strong populations to the J π = 0 + 3 state by α-transfer reactions to 38 Ar [3], which are sensitive to α-38 Ar cluster structure components. The ND1 states have small amount of α-38 Ar cluster structure 9/11 components and similar in-band B(E2) values to those of the experimental K π = 0 + 2 band, which indicates that the ND1 band corresponds to the experimental K π = 0 + 2 band. The particle-hole configurations of the ND1 (6p4h) and ND2 (4p2h) bands are consistent with those of the J π = 0 + 2 and 0 + 3 states, respectively, suggested by the results of an α-transfer experiment [3]. The members of the ND2 band apart from the band head and those of the SD band have not been observed yet. The members of the ND2 band could possibly be observed by a combination of α-transfer reactions and γ-spectroscopy experiments because the ND2 band contains large amount of α-38 Ar cluster structure components and has large in-band B(E2) values.
This full-microscopic model reveals the coexistence of three low-lying rotational K π = 0 + bands with 6p4h, 4p2h, and 8p6h configurations in 42 Ca, but the α + 38 Ar OCM, which is a semi-microscopic model, produces only one K π = 0 + rotational band with a 4p2h configuration in the low-lying states. Full-microscopic models treating clustering and various deformations with mp-mh configurations are required to understand the low-lying structures in 42 Ca. A unified treatment of clustering and deformation is important for studying nuclear structures.

Particle-hole configurations and E2 transitions
The in-band B(E2) values, deformations, and particle-hole configurations in the ND1 and ND2 bands found here indicate that the in-band B(E2) values are more sensitive to the proton particle-hole configurations than to deformations, which are not necessarily related to the B(E2) values. Indeed, for ND1 and ND2 bands with the same proton particle-hole configurations, (sd) −2 (pf ) 2 , the calculated in-band B(E2) values are similar although their dominant components have different quadrupole deformations of β = 0.40 and 0.28, respectively. This is inconsistent with a simple collective model in which the B(E2) values are proportional to β 2 . Experimental B(E2) values of the in-band transitions in the ND band in 40 Ca, whose proton particle-hole configurations are also (sd) −2 (pf ) 2 , are similar to the theoretical B(E2) values of the ND1 and ND2 bands in 42 Ca. As the Nilsson orbits show, particle-hole configurations are strongly related to deformation, but a careful consideration both of the particle-hole configurations and deformation are required for understanding the structures of deformed states.

Conclusions
In conclusion, the structures of the deformed states in 42 Ca have been investigated using deformed-basis AMD and the GCM by focusing on the coexistence of various rotational bands with mp-mh configurations and α-38 Ar clustering. In the excited states, three K π = 0 + bands, ND1, ND2, and SD, are obtained, which have dominant 6p4h, 4p2h, and 8p6h configurations, respectively. The ND1 band corresponds to the experimental K π = 0 + 2 band, while the ND2 and SD bands have not yet been observed. The band head of the ND2 band corresponds to the experimental J π = 0 + 3 state. The B(E2) values of the in-band transitions of the ND1 band are consistent with experimental data. The members of the GS and ND2 bands contain large amount of α-38 Ar cluster structure components, which is consistent with results that show the J π = 0 + 1 and 0 + 3 states are strongly populated by 38 Ar( 6 Li, d) reactions. Particle-hole configurations of the dominant components of the GS and ND2 bands are consistent with the suggestions of the α-transfer to 38 Ar experiment. E2 transitions are 10/11 more sensitive to proton particle-hole configurations than to deformation. It is necessary to employ full-microscopic calculations and consider both clustering and various deformations with mp-mh configurations for understanding the low-lying states in 42 Ca.