Tetrahedral shape and surface density wave of 16 O caused by α -cluster correlations
aa r X i v : . [ nu c l - t h ] A ug Tetrahedral shape and surface density wave of O caused by α -cluster correlations Yoshiko Kanada-En’yo
Department of Physics, Kyoto University, Kyoto 606-8502, Japan
Yoshimasa Hidaka
Theoretical Research Division, Nishina Center, RIKEN, Wako 351-0198, Japan α -cluster correlations in the 0 +1 and 3 − states of C and O are studied using the method ofantisymmetrized molecular dynamics, with which nuclear structures are described from nucleondegrees of freedom without assuming existence of clusters. The intrinsic states of C and Ohave triangle and tetrahedral shapes, respectively, because of the α -cluster correlations. Theseshapes can be understood as spontaneous symmetry breaking of rotational invariance, and theresultant surface density oscillation is associated with density wave (DW) caused by the instabilityof Fermi surface with respect to particle-hole correlations with the wave number λ = 3. O(0 +1 )and O(3 − ) are regarded as a set of parity partners constructed from the rigid tetrahedral intrinsicstate, whereas C(0 +1 ) and C(3 − ) are not good parity partners as they have triangle intrinsicstates of different sizes with significant shape fluctuation because of softness of the 3 α structure. E − to 0 +1 states in C and O are also discussed.
I. INTRODUCTION
Nuclear deformation is one of typical collective motions in nuclear systems. It is known that the ground states ofnuclei often have static deformations in the intrinsic states, which are regarded as spontaneous symmetry breaking ofthe rotational invariance due to many-body correlations. Not only normal deformations of axial symmetric quadrupoledeformations but also triaxial and octupole deformations have been attracting interests.In light nuclear systems, further exotic shapes owing to cluster structures have been suggested. For instance, atriangle shape in C and a tetrahedral one in O have been discussed using cluster models, which a priori assume3 α - and 4 α -cluster structures for C and O. In old days, non-microscopic α -cluster models have been applied inorder to understand the energy spectra of C and O [1, 2]. Wheeler has suggested low-lying 3 − states of C and O as vibration of the triangle and tetrahedral configurations of 3 and 4 α particles, respectively [1]. These statesare now considered to correspond to the lowest negative-parity states C(3 − , 9.64 MeV) and O(3 − , 6.13 MeV)established experimentally. In 1970’s, semi-microscopic cluster models [3–13], a molecular orbital model [14], and alsoa hybrid method of shell model and cluster model [15] have been applied in order to investigate cluster structures of C and O.For C, the ground state is considered to have the triangle shape because of the 3 α -cluster structure. In addition,a further prominent triangle 3 α structure has been suggested in C(3 − , 9.64 MeV). The 0 +1 and 3 − states in C areoften described as partners constructed by the rotation of the equilateral triangle 3 α configuration having the D symmetry even though the cluster structure of the ground state, C(0 +1 ), may not be as prominent as that of the C(3 − ). In cluster models, α clusters are a priori assumed and it is not be able to check cluster formation. Usingthe method of antisymmetrized molecular dynamics (AMD) [16, 17, 17, 19, 20], one of the authors (Y. K-E.) hasconfirmed the 3 α cluster formation in C from nucleon degrees of freedom without assuming existence of clusters forthe first time [21, 22]. The AMD result for C was supported by the calculation of the method of Fermionic moleculardynamics [23], which is a similar method to the AMD. Recently, ab initio calculations using realistic nuclear forceshave been achieved for C and reported the 3 α cluster formation in C [24–26]. In contrast to the triangle shapein the C(0 +1 ) and C(3 − ), a 3 α cluster gas-like state without a specific shape has been suggested for the 0 +2 stateby cluster models [5, 7–10, 12, 27]. In such a cluster gas state, α particles are weakly interacting like a gas and thenormal concept of nuclear deformation may be no longer valid.Let us consider the cluster phenomena in C from the viewpoint of symmetry breaking. Since the Hamiltonian ofnuclear systems has rotational invariance, a nucleus has a spherical shape if the rotational symmetry is not broken,However, in the intrinsic state of C(0 +1 ), the spherical shape changes to the triangle shape via the oblate shapebecause of the α -cluster correlation. It means the symmetry breaking from the rotational symmetry to the axialsymmetry, and to the D symmetry. In the group theory, it corresponds to O(3) → D ∞ h → D . This symmetrybreaking from the continuous group to the discrete (point) group in the triangle shape is characterized by surfacedensity oscillation, namely, a standing wave at the edge of the oblate state, and can be regarded as a kind of densitywave (DW) caused by the particle-hole correlation carrying a finite momentum. This is analogous to the DW ininfinite matter with inhomogeneous periodic density, which has been an attractive subject in various field such asnuclear and hadron physics [28–46] as well as condensed matter physics [47, 48]. Indeed, in our previous work, wehave extended the DW concept to the surface density oscillation of finite systems and connected the triangle shapewith the DW on the oblate state [49].Similarly to the triangle shape with the D symmetry in C, a tetrahedral shape with the T d symmetry in Ohas been suggested based on 4 α -cluster model calculations in order to understand O(3 − , 6.13 MeV) [1, 3, 13], Thetetrahedron shape is supported also by experimental data such as the strong E − → +1 [50] and α -transfer cross sections on C [51]. In 4 α -cluster models, the tetrahedral shape has been suggested also for theground state of O [3, 13]. Moreover, algebraic approaches for the 4 α system have been recently applied to describethe energy spectra of O based on the T d symmetry and its excitation modes [52]. However, the cluster formation northe tetrahedral shape in O have not been confirmed yet. In Hartree-Fock calculations, the spherical p -shell closedstate is usually obtained for the ground state solution except for calculations using particularly strong exchangenuclear interactions [53–55]. Recently, we applied the AMD method to O and found a tetrahedral shape with the4 α -cluster structure with a microscopic calculation from nucleon degrees of freedom without assuming existence ofclusters [56]. More recently, in a first principle calculation using the chiral nuclear effective field theory, the tetrahedralconfiguration of 4 α has been found in the ground state of O[57].The possible tetrahedral shape in O may lead to symmetry breaking from continuous to discrete groups; thebreaking of the O(3) symmetry to the T d symmetry. Problems for O to be solved are as follows: Does the symmetrybreaking occurs to form the tetrahedral shape in the ground state? Can the tetrahedral shape be understood as akind of DW? Whether the 0 +1 and 3 − states can be understood as a set of partners constructed by projection from asingle intrinsic state with the tetrahedral shape? What are analogies with and differences from C?Our aim is to clarify the α -cluster correlations and intrinsic shapes of the 0 +1 and 3 − states in O and comparethem with those in C. To confirm the problem whether the tetrahedron shape is favored in the intrinsic states of O, we perform variation after spin-parity projection (VAP) in the framework of AMD [21]. The AMD+VAP methodhas been proved to be useful to describe structures of light nuclei and succeeded to reproduce properties of the groundand excited states of C [21, 22]. By analyzing the obtained results for the 0 +1 and 3 − states of C and O, we showthat triangle and tetrahedral intrinsic shapes arise because of α -cluster correlations in C and O, respectively. Wealso give a simple cluster model analysis using the Brink-Bloch (BB) α -cluster wave function [59] the appearances ofthe triangle and tetrahedral shapes in C and O, respectively.This paper is organized as follows. In the next section, the framework of the AMD+VAP is explained. The resultsfor C and O obtained using the AMD+VAP are shown in Sec. III. In Sec. IV, we give discussions based on clustermodel analysis and show correspondence of cluster wave functions to surface DWs. A summary is given in Sec. V.
II. VARIATION AFTER PROJECTION WITH AMD WAVE FUNCTION
We explain the AMD+VAP method. For the details of the AMD framework, the reader is refereed to, for instance,Refs. [20, 21]. In the AMD framework, we set a model space of wave functions and perform the energy variation toobtain the optimum solution in the AMD model space. An AMD wave function is given by a Slater determinant ofGaussian wave packets, Φ
AMD ( Z ) = 1 √ A ! A{ ϕ , ϕ , ..., ϕ A } , (1)where the i th single-particle wave function is written by a product of spatial, intrinsic spin, and isospin wave functionsas ϕ i = φ X i χ σi χ τi , (2) φ X i ( r j ) = (cid:18) νπ (cid:19) / exp (cid:8) − ν ( r j − X i √ ν ) (cid:9) , (3) χ σi = ( 12 + ξ i ) χ ↑ + ( 12 − ξ i ) χ ↓ . (4) φ X i and χ σi are the spatial and intrinsic spin functions, and χ τi is the isospin function fixed to be pro-ton or neutron. Accordingly, the AMD wave function is expressed by a set of variational parameters, Z ≡{ X , X , . . . , X A , ξ , ξ , . . . , ξ A } . The width parameter ν relates to the size parameter b as ν = 1 / b and is chosento be ν = 0 .
19 fm − that minimizes energies of C and O. Gaussian center positions X , . . . , X A and intrinsicspin orientations ξ , . . . , ξ A for all single-nucleon wave functions are independently treated as variational parameters.Therefore, in the AMD framework, nuclear structures are described from nucleon degrees of freedom without assumingexistence of clusters. Despite of it, the model wave function can describe various cluster structures owing to the theflexibility of spatial configurations of Gaussian centers and also shell-model structures because of the antisymmetriza-tion. If a cluster structure is favored in a system, the corresponding cluster structure is automatically obtained in theenergy variation.An AMD wave function is regarded as an intrinsic wave function and usually does not have rotational symmetry.To express a J π state, an AMD wave function is projected onto the spin-parity eigenstate,Φ( Z ) = P JπMK Φ AMD ( Z ) , (5)where P JπMK is the spin-parity projection operator. The spin-parity projection can be understood as restoration of thesymmetry. To obtain the wave function for the J π state, variation after projection (VAP) is performed with respect tovariational parameters { Z } of the AMD wave function. Namely, we perform the variation of the energy expectationvalue h Φ( Z ) | H | Φ( Z ) i / h Φ( Z ) | Φ( Z ) i for the J π projected AMD wave function and obtain the optimum parameterset { Z opt J π } for the J π state. This method is called AMD+VAP. The AMD wave function before the projection isexpressed by a single Slater determinant. However, the spin-parity projected AMD wave function is no longer a Slaterdeterminant and contains some kind of correlations beyond the Hartree-Fock approach. III. AMD+VAP RESULTS OF C AND O We perform the AMD+VAP calculation to obtain the lowest positive- and negative-parity states, 0 +1 and 3 − , of Cand O and discuss properties of the obtained states such as intrinsic shapes, cluster structures, and E A. Intrinsic shapes of C and O The density distribution of the intrinsic wave functions Φ
AMD ( Z opt0 + ) Φ AMD ( Z opt3 − ) for C and O obtained using theAMD+VAP are shown in Figs. 1 and 2. C(0 +1 ) and C(3 − ) show triaxial deformations with triangle shapes, while O(0 +1 ) and O(3 − ) show tetrahedral shapes. The quadrupole deformation parameters ( β, γ ) are ( β, γ ) = (0 . , . . , .
11) for C(0 +1 ) and C(3 − ), respectively, and ( β, γ ) = (0 . , .
09) for O(0 +1 ) and O(3 − ). Thetriangle and tetrahedral shapes are caused by α -cluster correlations. Strictly speaking α clusters in the obtainedwave functions do not have ideal (0 s ) configuration but contain some cluster dissociation. Moreover, the intrinsicshapes are somewhat distorted from the regular triangle and tetrahedral shapes as an α cluster is situated slightly farfrom other α s. Nevertheless, these states show surface density oscillation with the wave number λ = 3 as a leadingcomponent as shown later.The deformation mechanism of C and O is interpreted from the viewpoint of symmetry breaking. The highestsymmetry is the sphere, which is realized in the uncorrelated limit; the p / - and p -shell closed configurations of C and O, respectively. Owing to many-body correlations, the symmetry can break into a lower symmetry.Let us consider the intrinsic shape of C. Because of the α -cluster correlation, the rotational symmetry of thespherical state breaks to the axial symmetry of an oblate state and changes to the D symmetry of the regulartriangle 3 α configuration, which breaks into the distorted triangle in the AMD+VAP result. The symmetry change,spherical → oblate → triangle, corresponds to O(3) → D ∞ h → D . Similarly, the intrinsic shape of O is understoodas the symmetry breaking O (3) → T d from the spherical state to the tetrahedral 4 α configuration. Note thatthe continuous group symmetries break to the discrete (point) group ones in the triangle and tetrahedron shapes.The symmetry breaking caused by the α -cluster correlations can be understood as DWs, which cause static densityoscillation at the nuclear surface. As described in the next section, the DWs for the triangle and tetrahedral shapesare characterized by the surface density oscillation with the wave number λ = 3. The order parameter of the DW forthe triangle shape in C is ( Y − − Y +33 ) / √ Y component, and that for the tetrahedralshape in O is ( √ Y + √ Y − − √ Y +33 ) / +1 and 3 − states of C and O obtained using the AMD+VAP,we perform the multipole decomposition of the intrinsic density at r = R as ρ ( R , θ, φ ) = ¯ ρ ( R ) X λµ α λµ Y µλ ( θ, φ ) , (6)and discuss the λ = 3 components. In the present analysis, we take R to be root-mean-square (rms) radii of theintrinsic states. ¯ ρ ( R ) is determined by normalization α = 1. α λµ = ( − µ α ∗ λ − µ because ρ ( r = R , θ, φ ) is real.The density at r = R on the θ - φ plane and that at the θ = π/ C are shown in Fig. 3. As clearly seen, theintrinsic states of C(0 +1 ) and C(3 − ) show surface density oscillation with the wave number λ = 3 on the oblate -4-2 0 2 4 -4 -2 0 2 4 y (f m ) x (fm) -4-2 0 2 4 -4 -2 0 2 4 y (f m ) x (fm) + 12 C(3 ) −12
C(0 ) -4-2 0 2 4 -4 -2 0 2 4 z (f m ) y (fm) -4-2 0 2 4 -4 -2 0 2 4 x (f m ) z (fm) -4-2 0 2 4 -4 -2 0 2 4 x (f m ) z (fm) -4-2 0 2 4 -4 -2 0 2 4 z (f m ) y (fm) FIG. 1: (color on-line) Density distributions for intrinsic states of (left) C(0 +1 ) and (right) C(3 − ) obtained by the AMD+VAPcalculation. The densities integrated on the z , x , and y axes are plotted on the x - y , y - z , and z - x planes, respectively. -4-2 0 2 4 -4 -2 0 2 4 y (f m ) x (fm) -4-2 0 2 4 -4 -2 0 2 4 y (f m ) x (fm) + 16 O(3 ) −16
O(0 ) -4-2 0 2 4 -4 -2 0 2 4 z (f m ) y (fm) -4-2 0 2 4 -4 -2 0 2 4 x (f m ) z (fm) -4-2 0 2 4 -4 -2 0 2 4 x (f m ) z (fm) -4-2 0 2 4 -4 -2 0 2 4 z (f m ) y (fm) FIG. 2: (color on-line) Density distributions for intrinsic states of (left) O(0 +1 ) and (right) O(3 − ) obtained by the AMD+VAPcalculation. The densities integrated on the z , x , and y axes are plotted on the x - y , y - z , and z - x planes, respectively. edge, which comes from the triangle 3 α configuration. Figure 4 shows the amplitudes α λµ of Y µλ component of thesurface density. It is found that the surface density oscillation is characterized by the λ = 3 component reflecting thebreaking of the axial symmetry from the oblate shape to the triangle shape in C.For the intrinsic density of O, we show the θ - φ plot in Fig. 5, and the amplitudes α λµ in Fig. 4, in which thetetrahedral component √ Y / √ Y +33 / −√ Y − / α and α . The open boxesindicate the distortion from the regular tetrahedron. O shows the surface density oscillation with the dominanttetrahedral component reflecting that the rotational symmetry is broken mainly into the T d symmetry.The present result indicates that, the symmetry breaking from the axial symmetry to D symmetry in C andthat from the rotational symmetry to T d symmetry in O occur because of the α -cluster correlations with triangleand tetrahedral configurations, respectively. As a result, the intrinsic states of the 0 +1 and 3 − states in C ( O)show the surface density oscillation with the dominant λ = 3 components, which are interpreted as the DW on theoblate (spherical) shape. (fm -3 ) C(0 + ) θ ( π rad) φ ( π r a d ) D e n s it y (f m - ) φ ( π rad) C(0 + ) (fm -3 ) C(3 - ) θ ( π rad) φ ( π r a d ) D e n s it y (f m - ) φ ( π rad) C(3 - ) FIG. 3: (color on-line) Surface density at r = R for C(0 +1 ) and C(3 − ) calculated using the AMD+VAP. R is 2.45 fm for0 +1 and 3.03 fm for 3 − . (top) Density plotted on the θ - φ plane. (bottom) Density at θ = π/ D symmetry is plotted for a eye guide by dashed lines. B. Properties of +1 and − states TABLE I: Binding energies (MeV), excitation energies (MeV) for the 3 − states, rms radii (fm), and the E B ( E
3; 3 − → +1 ). The experimental rms radii of the ground states are the rms point-proton radii that are reduced from thecharge radii [62]. B.E. Ex (3 − ) rmsr(0 +1 ) rmsr(3 − ) B(E3) C exp 92.16 9.64 2.309(2) 103(17) C 1-base 86.6 11.8 2.41 2.98 5.3 C 2-base 87.6 11.9 2.51 2.90 41 C 23-base 88.0 10.8 2.53 3.13 61 O exp 127.62 6.13 2.554(5j 205(106) O 1-base 122.9 9.4 2.69 2.70 151 O 2-base 122.9 8.9 2.69 2.71 163
Let us discuss the observable properties of the 0 +1 and 3 − states such as rms radii and E − → +1 )compared with the experimental data. As shown previously, the intrinsic states of C and O show the surfacedensity oscillation with the dominant λ = 3 component. If the 0 +1 and 3 − states are constructed from an intrinsicstate with the λ = 3 component, they can be regarded as a set of parity partners and have a strong E O(0 +1 ) and O(3 − ) are quite similar to each other.However, those of C(0 +1 ) and C(3 − ) are not so similar, but they show a difference in the development of the 3 α cluster. It means softness of the 3 α structure, and therefore, shape fluctuation is expected in realistic C(0 +1 ) and -0.4-0.2 0 0.2 0.4 α | α | α | α | | α | α | α | | α | | α | α l m C(0 + ) -0.6-0.4-0.2 0 0.2 0.4 0.6 α | α | α | α | | α | α | α | | α | | α | α l m C(3 - ) -0.4-0.2 0 0.2 0.4 α | α | α | α | | α | α | α | | α | | α | α l m O(0 + ) -0.4-0.2 0 0.2 0.4 α | α | α | α | | α | α | α | | α | | α | α l m O(3 - ) FIG. 4: Y µλ components ( α λµ ) of the intrinsic surface density at r = R for C(0 +1 ), C(3 − ), O(0 +1 ), and O(3 − ) calculatedusing the AMD+VAP. The hatched areas for O indicate the tetrahedron component √ Y / √ Y +33 / − √ Y − / α hatch30 ≡ p / α and α hatch33 ≡ α . C(3 − ). In order to improve the wave functions for J π states by taking into account the possible shape fluctuation, wesuperpose the basis wave functions obtained by the AMD+VAP for different J π states as is done in usual AMD+VAPcalculations as Ψ( J π ) = X ( J π ) ′ ,K c ( J π ) ′ ,K P JπMK Φ AMD ( Z opt( J π ) ′ ) , (7)where coefficients c ( J π ) ′ ,K are determined by diagonalization of Hamiltonian and norm matrices. We call the calcu-lation using one base with the summation ( J π ) ′ = J π and K = − J, . . . , + J “1-base calculation” and that using twobases with the summation ( J π ) ′ = 0 + , − and K = − J, . . . , + J “2-base calculation”. Here K -mixing is considered inΨ( J π ). In the previous works [21, 22], we have performed further superposition of 23 wave functions, which we call“23-base calculation” in this paper.In Table I, we show energies, rms radii, and E C and O. The experimental excitationenergies of C(3 − ) and O(3 − ) are reasonably reproduced by the calculations. The fact that 1-base and 2-basecalculations give almost the same result for O(0 +1 ) and O(3 − ) indicates that these two states have small shapefluctuation and can be understood as a set of parity partners constructed from the rigid intrinsic state with thetetrahedral shape. The E O obtained by 1-base and 2-base calculations reproduce well the large B ( E
3) of the experimental data. This also supports the tetrahedral shape of the intrinsic state in O. By contrast,the experimental B ( E
3) for C is much underestimated by 1-base calculation. The B ( E
3) is largely enhancedby 2-base calculation mainly because of the shape fluctuation in C(0 +1 ). Considering that both Φ AMD ( Z opt(0 + ) ) and (fm -3 ) O(3 - ) θ ( π rad) φ ( π r a d ) (fm -3 ) O(0 + ) θ ( π rad) φ ( π r a d ) FIG. 5: (color on-line) Surface density at r = R for O(0 +1 ) and O(3 − ) calculated using the AMD+VAP. R = 2 .
75 fm for0 +1 and R =2.77 fm for 3 − . Φ AMD ( Z opt(3 − ) ) shows triangle shapes but they have different amplitudes ( α ) of Y ± component, the shape fluctuationis regarded as amplitude fluctuation of the triangle shape that is taken into account by superposing Φ AMD ( Z opt0 +1 ) andΦ AMD ( Z opt3 − ) in 2-base calculation. A further enhanced B ( E
3) is obtained by 23-base calculation, which reasonablyreproduces the experimental data.In terms of harmonic-oscillator (ho) shell-model basis expansion, the shape fluctuation causes mixing of highershell components. In order to quantitatively discuss the higher shell mixing, we calculate occupation probability of a N shell -shell in the ho shell-model expansion for the obtained wave functions Ψ( J π ) as was done in Refs. [60, 61]. Here,we choose the ho width to be b used for the AMD wave function and define N shell = 0 for the lowest 0 ~ ω configuration.Figure 6 shows the occupation probability for C. For C(3 − ), the probability is distributed widely in the highershell region. However, C(0 +1 ) still has the dominant N shell = 0 component as 90% in 1-base calculation and 80% in2-base and 23-base calculations. Non-negligible N shell ≥ C(0 +1 ) are found in 2-base and 23-basecalculations and they contribute to the enhancement of the E C(0 +1 ) and C(3 − ) indicate significant shapefluctuation in C originating in softness of the triangle 3 α structure. Figure 7 shows the occupation probability for O. The occupation probability distributions for O(0 +1 ) and O(3 − ) are similar to each other. Moreover, they arenot sensitive to the number of base wave functions. These results indicate that these two states in O can be regardedas a set of parity partners constructed from the rigid tetrahedral intrinsic state. Both states contain significant mixingof higher-shell components as approximately 50% indicating the significant ground state correlation because of thetetrahedral 4 α structure. IV. DISCUSSIONS BASED ON CLUSTER MODEL ANALYSIS
In the previous section, we found the triangle and tetrahedral shapes in the intrinsic states of C and O calculatedusing the AMD+VAP, in which we treat nucleon degrees of freedom without assuming existence of clusters. In thissection, we give more fundamental discussions on the triangle 3 α and tetrahedral 4 α structures based on simpleanalyses using a cluster model. We consider the 3 α - and 4 α -cluster wave functions given by the BB α -cluster modeland show that the triangle 3 α and tetrahedral 4 α states correspond to the surface DWs with the wave number λ = 3.We also discuss the role of Pauli blocking between clusters in the triangle and tetrahedral shapes. O cc up a ti on N shell C, 1 base C(0 + ) C(3 - ) 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 O cc up a ti on N shell C, 2 bases C(0 + ) C(3 - ) 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 O cc up a ti on N shell C, 23 bases C(0 + ) C(3 - ) FIG. 6: Occupation probability of the N shell -shell in the harmonic oscillator expansion for C obtained by 1-base, 2-base, and23-base calculations.
A. Brink-Bloch α -cluster wave function The BB α -cluster wave function Φ BB nα [59] for an even-even Z = N = 2 n ( A = 4 n ) nucleus is described by thefollowing nα -cluster wave function consisting of (0 s ) α clusters asΦ BB nα ( S , . . . , S n ) = n A { Φ α ( S )Φ α ( S ) . . . Φ α ( S n ) } , (8)Φ α ( S k ) = ψ S k ,p ↑ (4 k + 1) ψ S k ,p ↓ (4 k + 2) ψ S k ,n ↑ (4 k + 3) ψ S k ,n ↓ (4 k + 4) , (9) ψ S k ,τσ ( j ) = (cid:18) νπ (cid:19) / exp (cid:8) − ν ( r j − S k ) (cid:9) X στ ( j ) , (10)where X τσ is the spin-isospin wave function with τ = { p, n } and σ = {↑ , ↓} . Φ BB nα is specified by the spatial configu-ration { S , . . . , S n } , which indicate center positions of α clusters.Note that the BB wave function is included in the AMD model space. Namely, when the parameters for the AMDwave function are chosen as X k − = X k − = X k − = X k = S k √ ν, (11) χ σ k − χ τ k − = X p ↑ , (12) χ σ k − χ τ k − = X p ↓ , (13) χ σ k − χ τ k − = X n ↑ , (14) χ σ k χ τ k = X n ↓ , (15)with k = 1 , . . . , n , the AMD wave function is equivalent to the BB α -cluster wave function. O cc up a ti on N shell O, 1 base O(0 + ) O(3 - ) 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 O cc up a ti on N shell O, 2 bases O(0 + ) O(3 - ) FIG. 7: Occupation probability of the N shell -shell in the harmonic oscillator expansion for O obtained by 1-base and 2-basecalculations.
B. Surface density oscillation of BB wave functions
As explained in Ref. [49], the BB 3 α -cluster wave function for the regular triangle configuration with a smallinter-cluster distance ( d ) can be rewritten asΦ BB3 α, small- d ( ǫ ) ≈ Y τσ (cid:8) ψ ho00 X τσ ( ψ ho1 − + ǫψ ho2+2 ) X τσ ( ψ ho1+1 − ǫψ ho2 − ) X τσ (cid:9) , (16)where ψ ho lm is the harmonic oscillator single-particle orbit, and ǫ is a small real value of the order O ( d ). Here, O ( ǫ )and higher terms in the spatial part for X τσ are omitted. ǫ is regarded as the order parameter for breaking of theaxial symmetry. The density of the Φ BB3 α, small- d ( ǫ ) state at r = ( r, θ, φ ) is given as ρ ( r ) = 4 π / b e − r b (cid:26) r b sin ( θ ) + ǫ √ r b sin ( θ )( e − iφ − e iφ ) + O ( ǫ ) (cid:27) , (17)and its multipole decomposition at r = R is ρ ( R , θ, φ ) = 8 π / b e − R b (cid:26) (1 + 43 R b ) Y ( θ, φ ) − √ R b Y ( θ, φ ) + ǫ √ R b (cid:18) Y − ( θ, φ ) √ − Y +33 ( θ, φ ) √ (cid:19) + O ( ǫ ) (cid:27) . (18)In a similar way, the BB 4 α -cluster wave function for the regular tetrahedral configuration with a small distancecan be rewritten as Φ BB4 α,T d ( ǫ ) ≈ Y τσ (cid:8) ψ ho00 X τσ ψ ho10 X τσ ( ψ ho1 − + ǫψ ho2+1 ) X τσ ( ψ ho1+1 + ǫψ ho2 − ) X τσ (cid:9) . (19)The density of the Φ BB4 α, small- d state is ρ ( r ) = 4 π / b e − r b (cid:26) r b sin ( θ ) + ǫ √ r b sin ( θ )( e iφ + e − iφ ) + O ( ǫ ) (cid:27) , (20)and its multipole decomposition at r = R is ρ ( R , θ, φ ) = 8 π / b e − R b ( (1 + 2 R b ) Y ( θ, φ ) + ǫ r R b ( Y − ( θ, φ ) √ Y +23 ( θ, φ ) √ O ( ǫ ) ) . (21)0Equations (21) and (21) indicate the surface density oscillation with the wave number λ = 3. Note that the Y ± terms in (21) can be transformed to Y − ( θ ′ , φ ′ ) √ Y +23 ( θ ′ , φ ′ ) √ √ Y ( θ, φ ) + √ Y − ( θ, φ ) − √ Y +33 ( θ, φ ) (22)by a Ω rotation ( θ, φ ) → R Ω ( θ, φ ) = ( θ ′ , φ ′ ). C. DW-type correlation at Fermi surface of BB wave functions
In particle-hole representation on the Fermi surface defined by the ǫ = 0 case, the Φ BB3 α, small- d ( ǫ ) and Φ BB4 α, small- d ( ǫ )states can be expressed as | Φ BB3 α, small- d ( ǫ ) i = Y χ (1 + ǫa † ,χ b † ,χ )(1 − ǫa † − ,χ b † − ,χ ) | i oblate F , (23) | i oblate F ≡ Y χ (cid:16) a † ,χ a † − ,χ a † ,χ (cid:17) |−i , (24)and | Φ BB4 α, small- d ( ǫ ) i = Y χ (1 + ǫa † ,χ b † ,χ )(1 + ǫa † − ,χ b † − ,χ ) | i spherical F , (25) | i spherical F ≡ Y χ (cid:16) a † ,χ a † ,χ a † − ,χ a † ,χ (cid:17) |−i , (26)where χ = τ σ , a † lm,χ and b † lm,χ are the particle and hole operators for the single-particle φ lm X τσ state, respectively,and | i oblate F and | i spherical F are the oblate and spherical states in the p -shell, which are the d → α and 4 α systems, respectively. Here, b † lm,χ is defined using the annihilation operator a l − m,χ as b † lm,χ = a l − m,χ . Equations (23)and (25) indicate that the 3 α - and 4 α -cluster wave functions contain the DW-type particle-hole correlations carryingthe finite angular momenta of λ | µ | = 33 and 32, which are consistent with the angular momenta of Y ± and Y ± components contained in the surface density in (18) and (21), respectively. D. Role of Pauli blocking in triangle and tetrahedral shapes
By contrast to C(0 +1 ) with the triangle configuration, the second 0 + state of C is considered to be a clustergas state where 3 α clusters are freely moving in dilute density like a gas without any geometric configurations [7–10, 12, 27]. It means that two kinds of 3 α -cluster states appear in C; the triangle state with localized α clustersand the cluster gas state with nonlocalized α clusters. From the viewpoint of symmetry breaking, the symmetry isbroken to the D in the 0 +1 state, and seems to be restored in the 0 +2 state.The origin of the symmetry breaking and that of the restoration in the 3 α states can be understood by Pauliblocking between α clusters as follows. Let us here discuss the Pauli blocking effect on the α -cluster motion, inparticular, its angular motion. The BB 3 α -cluster wave function Φ BB3 α ( S , S , S ) expresses the localized cluster state,in which α clusters are located around positions S , S , and S . We assume that 2 α clusters placed at S = (0 , d/ , S = (0 , − d/ ,
0) form a 2 α core and the third α is placed at S = ( x, y,
0) = ( r cos ϕ, r sin ϕ,
0) (Fig. 8(b)). Herewe define the intrinsic frame so that the x - y plane contains S , S , and S .In order to discuss Pauli blocking effect for the angular motion of the third α cluster around the 2 α core, we showin Fig. 8(a) the norm N α of the BB 3 α -cluster wave function defined as N α ( x, y ) ≡ ˜ N α ( x, y )˜ N α ( p x + y , , (27)˜ N α ( x, y ) ≡ h ˜Φ BB3 α ( S , S , S ) | ˜Φ BB3 α ( S , S , S ) i , (28)˜Φ BB3 α ( S , S , S ) = A { Φ α ( S )Φ α ( S )Φ α ( S ) } . (29)The norm N α is normalized by the value at ϕ = 0 on the x -axis for each r , and N α ∼ N α ∼ α cluster. In the small r region, N α is much1suppressed in the ϕ = 0 region because of the antisymmetrization effect. It means that, in the small r region, the third α cluster feels the strong Pauli blocking from the 2 α core on the y -axis, which blocks the angular motion of the third α and localize it well at ϕ = 0 on the x -axis (see Fig. 8(c)). This corresponds to the localized 3 α -cluster state witha triangle configuration. Namely, in a compact 3 α state, the triangle configuration is favored because of the strongPauli blocking effect between α clusters. By contrast, in the large r region, N α is nearly equal to 1 independentlyfrom φ indicating that the third α cluster is almost free from the Pauli blocking (Fig. 8(d)). It corresponds to thenonlocalized cluster state. These are the reasons for appearances of the localized and nonlocalized cluster states.The C(0 +1 ) state contains dominantly the compact 3 α component in the strong Pauli blocking regime, in whichthe triangle configuration is favored because of the Pauli blocking between α clusters. In the C(0 +2 ), α clustersspatially develop and can move freely like a gas in the weak Pauli blocking regime. In the case of O, the tetrahedralconfiguration is favored in a compact 4 α state owing to the same mechanism of the Pauli blocking effect between α clusters. -2 -1 0 1 2 x ν -2-1 0 1 2 y ν / φ (r, )y x(a) (b)free motionPauli blocking(c) small r (d) large r FIG. 8: (a) shows the norm N α for the BB 3 α -cluster wave function plotted on the x - y plane for the third α position. The N α ∼ N α ∼ α position around the 2 α on the y -axis. (c) shows a schematic for the small r case corresponding to a compact3 α state, and (d) shows a schematic for the large r case. V. SUMMARY
We investigated intrinsic shapes of the 0 +1 and 3 − states of C and O. The intrinsic states of C and Oobtained using the AMD+VAP method show the triangle and tetrahedral shapes, respectively, because of the α -cluster correlations. The formation of α clusters in these states was confirmed in the AMD framework, in which wetreated nucleon degrees of freedom without a priori assuming existence of clusters. The surface density shows the λ = 3 oscillation as the leading component, which is associated with the D and T d symmetry.Comparing the intrinsic structures between the 0 +1 and 3 − states, we discussed whether these two states can beunderstood as a set of parity partners. O(0 +1 ) and O(3 − ) have tetrahedral intrinsic shapes similar to each otherand can be understood as a set of parity partners constructed from the rigid intrinsic state with the tetrahedral shape.Because of the tetrahedral intrinsic shape, the B ( E
3; 3 − → +1 ) in O is significantly large. By contrast, C(0 +1 )and C(3 − ) can not be understood as ideal parity partners as C(3 − ) has the triangle shape with a much largersize than C(0 +1 ). Moreover, we found the large shape fluctuation, mainly, the amplitude fluctuation of the triangleshape in C(0 +1 ) and C(3 − ) originating in softness of the triangle 3 α structure. The B ( E
3) for C is enhancedbecause of the amplitude fluctuation in C(0 +1 ).Based on simple analyses using the BB 3 α - and 4 α -cluster model wave functions, we showed the connection oftriangle 3 α and tetrahedral 4 α states with surface DWs caused by the particle-hole correlations carrying the wavenumber λ = 3 on the Fermi surface. It means that the oscillating surface density in the triangle and tetrahedral shapesis associated with the instability of Fermi surface and is related to the spontaneous symmetry breaking because of the2many-body correlation. Pauli blocking between α clusters plays an important role in the appearances of the triangleand tetrahedral configurations in 3 α and 4 α systems, respectively. Acknowledgments
The authors thank to nuclear theory group members of department of physics of Kyoto University for valuablediscussions. Discussions during the YIPQS long-term workshop ”DCEN2011” held at YITP are helpful to advancethis work. The computational calculations of this work were performed by using the supercomputers at YITP. Thiswork was supported by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science (JSPS)Grant Number [Nos.23340067, 24740184, 26400270]. It was also supported by the Grant-in-Aid for the Global COEProgram “The Next Generation of Physics, Spun from Universality and Emergence” from the Ministry of Education,Culture, Sports, Science and Technology (MEXT) of Japan.
References [1] J. A. Wheeler, Phys. Rev. , 1083 (1937); ibid. , 1107 (1937).[2] D. M. Dennison, Phys. Rev. , 378 (1954).[3] D. M. Brink, H. Friedrich, A. Weiguny and C. W. Wong, Phys. Lett. B33 , 143 (1970).[4] K. Ikeda et al. , Prog. Theor. Phys. Suppl. , 1 (1972).[5] H. Horiuchi, Prog. Theor. Phys. , 1266 (1974); , 447 (1975).[6] Yu. F. Smirnov, I. T. Obukhovsky. Yu. M. Tchuvil’sky and V. G. Neudatchin, Nucl. Phys. A 235, 289 (1974).[7] E. Uegaki, S. Okabe, Y. Abe and H. Tanaka, Prog. Theor. Phys. , 1262 (1977).[8] E. Uegaki, Y. Abe, S. Okabe and H. Tanaka, Prog. Theor. Phys. , 1621 (1979).[9] Y. Fukushima and M. Kamimura, Proc. Int. Conf. on Nuclear Structure, Tokyo, 1977, edited by T. Marumori [J. Phys.Soc. Jpn. , 225 (1978).[10] M. Kamimura, Nucl. Phys. A351 , 456 (1981).[11] Y. Suzuki, Prog. Theor. Phys. , 1751 (1976).[12] Y. Fujiwara et al. , Prog. Theor. Phys. Suppl. , 29 (1980).[13] W. Bauhoff, H. Schultheis, R. Schultheis Phys. Rev. C 29 , 1046 (1984).[14] Y. Abe, J. Hiura and H. Tanaka, Prog. Theor. Phys. , 352 (1971); ibid. , 800 (1972).[15] N. Takigawa and A. Arima, Nucl. Phys. A , 593 (1971).[16] Y. Kanada-En’yo and H. Horiuchi, Prog. Theor. Phys. , 115 (1995);[17] Y. Kanada-En’yo, H. Horiuchi and A. Ono, Phys. Rev. C , 628 (1995);[18] Y. Kanada-En’yo and H. Horiuchi, Phys. Rev. C , 647 (1995).[19] Y. Kanada-En’yo and H. Horiuchi, Prog. Theor. Phys. Suppl. , 205 (2001).[20] Y. Kanada-En’yo M. Kimura and H. Horiuchi, C. R. Physique
497 (2003).[21] Y. Kanada-En’yo, Phys. Rev. Lett. , 5291 (1998).[22] Y. Kanada-En’yo, Prog. Theor. Phys. , 655 (2007) [Erratum-ibid. , 895 (2009)].[23] M.Chernykh, H.Feldmeier, T.Neff, P.von Neumann-Cosel and A.Richter, Phys. Rev. Lett. , 032501 (2007).[24] E. Epelbaum, H. Krebs, T. A. L´’ahde, D. Lee and Ulf-G. Meissner, Phys. Rev. Lett. , 252501 (2012).[25] A. C. Dreyfuss, K. D. Launey, T. Dytrych, J. P. Draayer and C. Bahri, Phys. Lett. B , 511 (2013).[26] J. Carlson, S. Gandolfi, F. Pederiva, S. C. Pieper, R. Schiavilla, K. E. Schmidt and R. B. Wiringa, Rev. Mod. Phys. ,1067 (2015).[27] A. Tohsaki, H. Horiuchi, P. Schuck, and G. R¨opke, Phys. Rev. Lett. , 192501 (2001).[28] A. W. Overhauser, Phys. Rev. Lett. , 415 (1960).[29] D. M. Brink and J. J. Castro, Nucl. Phys. A216 , 109 (1973).[30] M. de Llano, Nucl. Phys. A , 183 (1979).[31] H. Ui and Y. Kawazore, Z. Phys. A , 125 (1981).[32] R. Tamagaki and T. Takatsuka, Prog. Theor. Phys. ,1340 (1976).[33] T. Takatsuka, K. Tamiya, T. Tatsumi and R. Tamagaki, Prog. Theor. Phys. , 1933 (1978).[34] A. B. Migdal, Rev. Mod. Phys. , 107 (1978).[35] F. Dautry and E. M. Nyman, Nucl. Phys. A , 323 (1979).[36] D. V. Deryagin, D. Y. Grigoriev and V. A. Rubakov, Int. J. Mod. Phys. A , 659 (1992).[37] E. Shuster and D. T. Son, Nucl. Phys. B , 434 (2000).[38] B. Y. Park, M. Rho, A. Wirzba and I. Zahed, Phys. Rev. D , 034015 (2000). [39] M. G. Alford, J. A. Bowers and K. Rajagopal, Phys. Rev. D , 074016 (2001).[40] E. Nakano and T. Tatsumi, Phys. Rev. D , 114006 (2005).[41] I. Giannakis and H. C. Ren, Phys. Lett. B , 137 (2005).[42] K. Fukushima, Phys. Rev. D , 094016 (2006).[43] D. Nickel, Phys. Rev. Lett. , 072301 (2009); Phys. Rev. D , 074025 (2009).[44] T. Kojo, Y. Hidaka, L. McLerran and R. D. Pisarski, Nucl. Phys. A , 37 (2010).[45] S. Carignano, D. Nickel and M. Buballa, Phys. Rev. D , 054009 (2010).[46] K. Fukushima, T. Hatsuda, Rept. Prog. Phys. , 014001 (2011).[47] G. Gruner, Rev. Mod. Phys. 60, 1129 (1988).[48] G. Gruner, Rev. Mod. Phys. 66, 1 (1994).[49] Y. Kanada-En’yo and Y. Hidaka, Phys. Rev. C , 014313 (2011).[50] D. Robson, Phys. Rev. Lett. , 876 (1979).[51] J.P.Elliott, J.A.Evans, E.E.Maqueda Nucl.Phys. A437, 208 (1985).[52] R. Bijker and F. Iachello, Phys. Rev. Lett. , no. 15, 152501 (2014) doi:10.1103/PhysRevLett.112.152501[arXiv:1403.6773 [nucl-th]].[53] J. Eichler, A. Faessler Nucl. Phys., A157,166 (1970).[54] N. Onishi, R.K. Sheline Nucl. Phys. A165, 180 (1971).[55] S. Takami, K. Yabamna, K. Ikeda, Prog. Theor. Phys. , 407 (1996).[56] Y. Kanada-En’yo and Y. Hidaka, arXiv:1208.3275 [nucl-th].[57] E. Epelbaum, H. Krebs, T. A. L´’ahde, D. Lee, Ulf-G. Meissner and G. Rupak, Phys. Rev. Lett. , no. 10, 102501 (2014).[58] R. K. Sheline and K. Wildermuth, Nucl. Phys. 21, 196 (1960).[59] D. M. Brink, International School of Physics “Enrico Fermi”, XXXVI, p. 247 (1966).[60] Y. Suzuki, K. Arai, Y. Ogawa and K. Varga, Phys. Rev. C , 2073 (1996).[61] Y. Kanada-En’yo and T. Suhara, Phys. Rev. C , no. 4, 044313 (2014) doi:10.1103/PhysRevC.89.044313 [arXiv:1401.5517[nucl-th]].[62] I. Angeli, At. Data Nucl. Data Tables87