Isospin Projected Antisymmetrized Molecular Dynamics and its Application to ${}^{10}$B

To investigate $pn$ pair correlations in $N=Z=\textrm{odd}$ nuclei, we develop a new framework based on the generator coordinate method of the $\beta\gamma$ constraint antisymmetrized molecular dynamics. In the framework, the isospin projection is performed before the energy variation to obtain the wave function optimized for each isospin. We apply the method to ${}^{10} \textrm{B}$ and show that it works well to describe coexistence of $T=0$ and $T=1$ states in low-energy spectra. Structures of low-lying states and $pn$ correlations are investigated. Strong $M1$($0^+_1\rightarrow 1^+_1$) and $E2$($1^+_1\rightarrow 1^+_2$) transitions are understood by the spin excitation of the $pn$ pair and the rotation of a deformed core, respectively.

To investigate pn pair correlations in N = Z = odd nuclei, we develop a new framework based on the generator coordinate method of the βγ constraint antisymmetrized molecular dynamics. In the framework, the isospin projection is performed before the energy variation to obtain the wave function optimized for each isospin. We apply the method to 10 B and show that it works well to describe coexistence of T = 0 and T = 1 states in low-energy spectra. Structures of low-lying states and pn correlations are investigated. Strong M 1(0 + 1 → 1 + 1 ) and E2(1 + 1 → 1 + 2 ) transitions are understood by the spin excitation of the pn pair and the rotation of a deformed core, respectively.

Introduction
Proton and neutron (pn) correlation is one of the key phenomena to understand properties of nuclei along the N = Z line in the nuclear chart (see Ref. [1] and references therein). Unlike in identical pair correlations, two channels, T = 0 and T = 1, are possible in pn pair correlations, and both channels play an important role in various nuclear structures. Competition between T = 0 and T = 1 pn pairs has been attracting a great interest and discussed to describe level ordering of J π T = 1 + 0 and J π T = 0 + 1 states in N = Z = odd nuclei and neighboring nuclei. In the heavy-mass region, the competition has been investigated with mean-field approaches [2][3][4][5][6][7]. In the light-mass region, properties of a pn pair at the nuclear surface have been studied in detail to understand low-lying spectra of N = Z = odd nuclei [8][9][10].
As discussed in the study with a three-body model calculation [8], a proton and a neutron around a core nucleus form a pn pair in the T = 0 or T = 1 channel because of the S-wave attraction between nucleons. The T = 1 pn pair is the mirror state of the dineutron pair, which is often discussed in neutron-rich nuclei. The appearance of the T = 0 pn pair is peculiar to N = Z = odd nuclei. Unlike the T = 1 pair, the T = 0 pn pair has the finite intrinsic spin S = 1 like a deuteron, and therefore it provides different J states in lowenergy region because of angular momentum coupling. Indeed, it is experimentally known that many T = 0 states coexist along with the J π T = 0 + 1 state in low-energy spectra of N = Z = odd nuclei [1]. Moreover, a high J state with T = 0 comes down to the ground state in many N = Z = odd nuclei in the light-mass region except for those with closed-shell cores. For example, the ground state spin parity of 10 B with Z = N = 5 is 3 + , for which the importance of three-nucleon forces is discussed in the no-core shell model calculations [11,12].
In systematic study of general N = Z = odd nuclei, it is important to describe the competition between the T = 0 and T = 1 pn correlations while taking into account spin configurations as well as nuclear deformation. In the light-mass N = Z region, cluster structure is another important feature which brings rich structures together with the pn correlations. The pn pair feature and its dynamics can be affected by the cluster structure.
To deal with these problems, we developed a new method based on the antisymmetrized molecular dynamics (AMD) [13][14][15][16][17] with constraint on quadrupole deformation parameters called βγ constraint AMD [18,19]. The βγ constraint AMD has been proved to be a useful approach to study structures of stable and unstable nuclei. In the AMD method, existence of a core nucleus, a pn pair, and a cluster structure are not apriori assumed, but degrees of freedom of all nucleons are independently treated in basis wave functions given by Slater determinants of Gaussian wave packets. Nevertheless, if a system favors a structure with a pn pair and cluster structures, such the structure is obtained in the energy variation. In the new method, the isospin projection is performed before the energy variation to obtain the AMD wave function optimized for each isospin state. We call this method, the isospin projected βγ constraint AMD, T βγ-AMD. Energy levels of N = Z = odd nuclei are calculated with the T βγ-AMD method combined with the generator coordinate method (GCM), called T βγ-AMD+GCM. To test the applicability of the new framework, we apply the method to 10 B, in which 2α + pn structures are found in low-lying T = 0 and T = 1 states. We investigate pn correlations around the 2α core, and show the importance of the finite spin of the T = 0 pn pair, which couples with the pn motion and the core rotation, in the low-lying states of 10 B.
The paper is organized as follows. We explain the framework of the T βγ-AMD in the section 2, and the adopted effective nuclear interactions in section 3. We show the calculated results of 10 B in section 4, and give discussion of structures of 10 B focusing on the pn correlation in section 5. Finally, a summary and an outlook are given in section 6.

Framework of T βγ constraint AMD
The antisymmetrized molecular dynamics is one of the useful approaches to describe the cluster aspect of light nuclei. However, the conventional AMD method is not sufficient for N = Z = odd nuclei, in which the isospin T = 0 and T = 1 states degenerate in the low-energy region. In this section, we explain the framework of the isospin projected AMD, which is a newly developed method for the study of N = Z = odd nuclei. We also describe the detailed formulation of the present calculation which is based on the isospin projected version of the βγ constraint AMD combined with the GCM.

AMD framework
The AMD framework is based on a variational method. An AMD wave function is a Slater determinant of Gaussian wave packets: where A refers to the antisymmetrization operator and φ i refers to the ith single-particle wave function. φ i is expressed by a product of the spatial(ψ i ), spin(ξ i ), and isospin(n i ) parts as, where The parameters of Gaussian centroids {Z i } i=1,2,...,A and those of spin orientations {ξ i } i=1,2,...,A are optimized in the energy variation, whereas each isospin configuration is fixed to be a isospin eigenstate as |p or |n .
In the AMD wave function expressed by a Slater determinant, some symmetries such as the parity and rotational symmetries are usually broken. To obtain physical wave functions for energy levels, the parity and angular momentum projections are performed to restore the broken symmetries in the AMD framework. In most cases, the parity projection is performed before the energy variation but the angular momentum projection is done after the variation to save computational costs except for the extended AMD calculation with the variation after the parity and angular-momentum projections (AMD+VAP J ) [20]. Namely, energy variation is performed for the AMD wave function projected on to the parity eigenstate with the parity projection operator P π as |Φ π = P π |Φ .
The expectation value of an observable O is calculated by the parity and angular momentum eigenstates Φ πJ M K projected from the obtained AMD wave function as, where P J M K is the angular momentum projection operator. For numerical calculation of the energy variation, we adopt the frictional cooling method, which is a gradient method.
To describe the ground and excited states, we superpose various AMD wave functions based on the GCM with the βγ constraint AMD method [19], which is an efficient method to choose basis AMD wave functions for a multi-configuration calculation.
In the βγ constraint AMD, the energy variation is performed under the constraint on quadrupole deformation parameters β and γ. The method has been proved to be useful to describe spatially developed cluster states as well as deformed states. To perform the energy variation with a constraintC [Φ] = C i , we add a penalty term to the energy expectation value as, where η is an enough large positive value. After the energy variation with the penalty term, we obtain the optimized configuration Φ (C i ) for the energy minimum state in the AMD We superpose the wave functions obtained for different constraint values C i based on the GCM method with the generator coordinateC [Φ]. In the βγ constraint AMD, C i indicates the set of the quadrupole deformation parameters, C i = (β i , γ i ), which specifies the nuclear deformation. Then, the GCM wave function is given by the superposition of the parity and angular momentum projected wave functions obtained by the βγ constraint AMD as, where K refers to the intrinsic magnetic quantum number and i max is the number of superposed wavefunctions. The coefficients g πJ iK are determined by the diagonalization of the Hamiltonian and norm matrices.

isospin projection
In N = Z = odd nuclei, the isospin projection is necessary to describe different isospin states T = 0, 1, which degenerate in a low-energy region. As done for the parity projection, the isospin projection enables us to obtain the better wave function optimized for each isospin. The isospin projection is also important to obtain proper intrinsic spin structures because the isospin of a pn pair restricts the intrinsic spin of the pair because of the Fermi statistics of two nucleons.
The isospin projection operator is defined as where D T T M T K is the Wigner's D function and R T (Ω) is the SU(2) rotation operator in the isospin space. In the present calculation of N = Z nuclei, we approximately perform the isospin projection with the following operator, where π T = (−1) Z , − (−1) Z for T = 0, 1 respectively. Here P pn is the operator which exchanges the isospin, proton ↔ neutron, of all nucleons in the AMD wave function: where |−p ≡ |n and |−n ≡ |p . For a N = Z = odd system having a proton and a neutron with a core nucleus, the operator P T is a good approximation of the isospin projection operator provided that the core nucleus is an approximately T = 0 state.

isospin projected AMD
In the isospin projected AMD framework, the isospin projection is performed before the energy variation. In the present calculation, the method is used for the βγ constraint AMD. Namely, the energy variation is performed with the β and γ constraints after the isospin and parity projections but before the angular momentum projection as with the penalty term as whereβ andγ are defined as We call this method, the T -projected βγ-constraint AMD (T βγ-AMD). In this method, β and γ are the parameters which control the development of clusters including pn pairs. In the application to 10 B nucleus with 2α + pn structures, γ plays a role of the parameter for spatial development of the pn pair whereas β is related to the α-α distance, as shown later.
After the energy variation, we perform the GCM calculation by superposing the projected AMD wave functions with respect to the discretized generator coordinates β i and γ i as, In the present calculation, we omit the isospin mixing in the wave functions. Note that the isospin mixing by the Coulomb force can be easily taken into account in the present framework by superposing different T states in the GCM calculation.

Hamiltonian and parameters
The Hamiltonian used in the present work is where K, K cm , V c , V ls , and V Coulomb are the kinetic energy, kinetic energy of the center of mass, central force, spin-orbit force, and Coulomb force, respectively. The Coulomb force is approximated by a 7-range Gaussian as done in Ref. [21]. For the central force and the spin-orbit force, the following effective nuclear forces are adopted as done in the previous work [22]. The Volkov No.2 force [23] is used for V c as, where v 1 = −60.65 MeV, v 2 =61.14 MeV, a 1 =1.80 fm and a 2 =1.01 fm. As for the Wigner, Majorana, Bartlett, and Heisenberg parameters, we adopt the same parameters as those used in the previous work [22], W = 1 − M =0.40, M =0.60, and B = H =0.06. These parameters reproduce the αα scattering phase shift. Bartlett and Heisenberg parameters were fitted to the relative energy between T = 0 and T = 1 states of the 10 B spectra in the AMD+VAP J calculation.
For V ls , the spin-orbit term of the Gaussian 3-range soft-core force (G3RS) [24,25] is used as, where b 1 =0.60 fm, b 2 =0.447 fm. The strengths used in the present calculation are u 1 = −u 2 =1300 MeV, which were fit to reproduce the ls splitting between 3/2 − and 1/2 − states in 9 Be in the AMD+VAP J calculation [22]. 7

Results
To test the applicability of the T βγ-AMD, we apply it to 10 B and show advantages of this method. 10 B is an ideal benchmark nucleus for the following reasons. Firstly, 10 B is a N = Z nucleus, in which different isospin states appear in its low-energy region. Indeed, J π T = 3 + 0, 1 + 0 and 0 + 1 states are known in the low-lying 10 B spectra. In second, 10 B is considered to have the 2α + pn structure. It means that 10 B can be a good example of a system having a pn pair around a deformed core. We investigate structures of low-lying T = 0 and T = 1 states in 10 B while focusing on the pn correlation around the 2α core.

Energy variation with and without the isospin projection
In the framework of the isospin projected AMD, we obtain the state optimized for each isospin T = 0, 1 because the energy variation is performed for the isospin projected AMD wave function as explained previously. The variation after the isospin projection (VAP T ) is essential to clarify the nuclear structure along the N = Z line on the nuclear chart because different isospin states have almost same energy and compete each other in N = Z = odd nuclei. This is an advantage of the T βγ-AMD superior to the usual βγ-AMD, in which the variation is performed without the isospin projection. In this subsection, we compare the energy surface on the βγ plane obtained by the Tβγ-AMD with that of the βγ-AMD and show the availability of the variation after the isospin projection in the present framework, Tβγ-AMD.
First, we show the energy on the βγ plane obtained by the T βγ-AMD. Figure 1 shows the T = 0, 1 π = + energy surfaces of 10 B obtained by the variation after the parity and isospin projections. The minimum point in the T = 1 plane ( Fig. 1(b)) is located at (β cos γ, β sin γ) = (0.4, 0.1) and that in the T = 0 plane ( Fig. 1(a)) is at (0.38, 0). The finite γ value is mainly caused by spatial development of the pn pair as discussed later.
Next, we discuss the results obtained by the βγ-AMD, which is a conventional AMD method with the parity projection without the isospin projection. Figure 2(a) shows the π = + energy surface of 10 B obtained by the βγ-AMD. It is notable that there is a gap along the line from (0.2,0.1) to (0.6,0). This is caused by the drastic change of the intrinsic structure from the small γ region to the relatively large γ region.
We also calculate the energy expectation values of the T = 0, 1 eigenstates projected from the βγ-AMD wave functions obtained by the variation before the isospin projection  Fig. 1(a), whereas the T = 1 energy surface in the large γ region is similar to the T = 1 result of the VAP T in Fig. 1(b). This means that, for the small γ region, the VBP T gives the wave functions almost same as those optimized for the T = 0 component, and for the large γ region, it produces the solutions similar to those optimized for the T = 1 component.
From these results, it is clear that the VBP T method is not applicable to obtain the optimum wave functions on the βγ plane for each isospin. As seen in Fig. 2(b), the VBP T fails to obtain the T = 1 states in the small γ region, that is., the T = 1 wave functions with prolate deformations. As already shown in Fig. 1, the T = 0 and T = 1 states exist in each point on the βγ plane. However, in the VBP T , the T = 0 and T = 1 states can not be optimized separately, but either of T = 0, 1 states or a mixed state of T = 0 and T = 1 components is optimized. The discontinuity on the βγ plane in the VBP T results can be a crucial problem to describe the γ mode, which is related to pn pair motion, and to calculate excited states in each isospin channel.
On the other hand, the VAP T technique enables us to optimize T = 0 and T = 1 components separately even though they almost degenerate at each constraint point. Consequently, the discontinuity on the βγ plane disappears and the smooth energy surfaces are obtained for each isospin states by the T βγ-AMD as shown in Fig. 1. This is one of the advantages of the present method and important to describe the γ mode for pn pair motion in detail.

Angular momentum projected energy surfaces and intrinsic structures
We discuss the angular momentum projected energies of the T βγ-AMD, which are calculated by the angular momentum projection of the wave functions obtained by the T βγ-AMD. Figure 3 shows the energy surfaces on the βγ plane for J π T = 0 + 1, 3 + 0, and 1 + 0 states. Here the K-mixing is taken into account and the lowest energy is shown at each point on the βγ plane except for Fig. 3(c). The J π T = 1 + 0 states on the βγ plane contain two components with different K quanta corresponding to the low-lying 1 + 1 0 and 1 + 2 0 states. Figures 3(b) and (c) show the energy surfaces for the first and the second lowest energies obtained by the K-mixing at each point. The 3 + 0 energy surface is approximately described by the energy of the K = 3 component on the βγ plane, whereas the 1 + 1 0 and 1 + 2 0 energy surfaces are roughly described by the energies of the K = 1 and K = 0 components, respectively, though strictly speaking the K-mixing occurs for γ = 0 states. Compared with the T = 0 and T = 1 energy surfaces before the angular momentum projection in Fig. 1, the energy minimum positions are shifted toward larger deformation β regions by the angular momentum projection meaning that deformed states are more favored after the angular momentum projection. The energy minimum state at (β cos γ, β sin γ) = (0.54, 0.12) for β = 0.55, γ = 12.5 • on the J π T = 0 + 1 energy surface (Fig. 3(a)) is the dominant component of the lowest isovector (T = 1) state, 10 B(0 + 1 ), and that at (0.52, 0.08) for β = 0.53, γ = 8.7 • on the 3 + 0 energy surface (Fig. 3(d)) approximately describes the lowest isoscalar (T = 0) state, the 10 B ground state (3 + 1 0). The energy minimum state at (0.5, 0.21) on the J π T = 1 + 1 0 energy surface and that at (0.29, 0.27) on the J π T = 1 + 2 0 energy surface approximately correspond to the 10 B(1 + 1 ) and 10 B(1 + 2 ), respectively.
Let us discuss intrinsic structures of the states obtained by the T βγ-AMD. The AMD wave functions before the projections at the energy minimums on the J π T = 0 + 1, 3 + 0, 1 + 1 0 and 1 + 2 0 energy surfaces ( Fig. 3(a-d)) are regarded as the intrinsic states of the dominant components of the 0 + 1, 3 + 0, 1 + 1 0, and 1 + 2 0 states of 10 B. In Fig. 4, we show the intrinsic density distribution of the minimum energy states projected onto the xz plane by integrating the density along the y-axis, ρ (x, z) = ρ (x, y, z) dy. The density distribution shows the structure of a 2α core with a pn pair in the low-lying states of 10 B. In the 0 + 1 state, three T = 1 pn pairs with the anti-parallel spin configuration (S = 0) are found in the right side forming an α cluster with a S = 0 pn pair nearby the α cluster. In the 3 + 0 state, the T = 0 (S = 1) pn pair is formed but it shows no spatial development because the pn pair is rotating around the 2α core in the L = 2(D-wave) and strongly attracted by the spin-orbit potential from the core. In the 1 + 1 0 and 1 + 2 0 states, a T = 0 pn pair has the parallel spin configuration indicating the formation of the T = 0, S = 1 pn pair. The pn pair in the 1 + 0 states is farther away from the 2α core compared with those in the 0 + 1 and 3 + 0 states indicating remarkable spatial development of the T = 0, S = 1 pn pair.
From the analysis of intrinsic structures obtained by the T βγ-AMD on the βγ plane, it is found that, the T = 0, S = 1(T = 1, S = 0) pn pair is formed in the T = 0(T = 1) states, and it goes away in the transverse direction from the 2α with increase of γ. In other word, the γ mode on the βγ plane approximately corresponds the pn pair motion relative to the 2α core.
Let us come back to the energy surfaces shown in Fig. 3. The 0 + 1 energy surface shows a plateau from the minimum along the γ parameter up to the γ = 60 • line, which corresponds to a soft mode of the T = 1, S = 0 pn pair motion relative to the 2α core in the 0 + 1 1 state. The 1 + 1 0 energy surface shows a more remarkable plateau for the spatial development of the T = 0, S = 1 pn pair. In contrast, the 3 + 0 energy surface does not show such a plateau toward the γ = 60 • direction meaning that the pn pair in the 3 + 1 0 state is deeply bound near the core because of the spin-orbit potential from the core.

GCM results
We superpose i max = 91 wave functions on the βγ plane for each isospin and calculate energy spectra, moments, and transition strengths. The parity, isospin, and the angular momentum projections with K-mixing are taken into account in the GCM calculation as described in Eq. (21). The generator coordinates, β and γ, effectively describe the pn pair motion relative to the 2α core as well as the α-α motion.
The T = 0 and T = 1 energy spectra of 10 B obtained by the GCM calculation are shown in Fig. 5. Compared with the minimum energies of the J π T energy surfaces on the βγ plane, about 4 MeV energy gain is obtained for the lowest J π T states by the superposition in the GCM calculation mainly because of the quantum mixing of spatial and spin configurations of the pn pair. In particular, the 1 + 1 0 states gains a larger energy than the 3 + 1 0 because of the spatial motion of the pn pair along the plateau on the energy surface. As a result, the excitation energy of the 1 + 1 0 state decreases in the GCM calculation. Moreover, we obtain excited J π T states as a result of configuration mixing as well as K-mixing in the GCM calculation. The energy spectra of the GCM calculation reasonably reproduce the x (fm) x (fm) experimental spectra of T = 0 and T = 1 states. Note that the relative energy between the T = 1 states and T = 0 states strongly depends on the interaction parameters, B and H. Table 1 shows the calculated values of nuclear properties compared with the experimental data. The present T βγ-AMD+GCM calculation reproduces well the ground state properties such as the point-proton radius r p , the electric quadrupole moment Q, and the magnetic moment µ, and also describes reasonably the experimental data of E2 and M1 transition strengths.
In Fig. 6, we show the overlap amplitudes of the 0 + 1, 1 + 1 0, 1 + 2 0 and 3 + 0 states obtained by the GCM calculation with the basis wavefunctions Φ (β i γ i ) on the βγ plane. The overlap amplitudes are calculated by the projection to the subspace composed of J π -projected states, ..,J . For the 10 B(1 + 1 0) and 10 B(0 + 1 ) states, the overlaps are distributed widely along the plateaus indicating the spatial development of the T = 0, S = 1 and T = 1, S = 0 pn pairs, respectively. For the 10 B(3 + 1 0), the overlap is distributed in the small γ region corresponding to the flat region around the energy minimum toward the γ = 0 line on the J π T = 3 + 0-projected energy surface (Fig. 3 (d)). This result indicates that the Spectra of 10 B calculated by the T βγ-AMD+GCM and those of the experimental data taken from Ref. [26]. The minimum energies in the J π -projected energy surfaces of the T βγ-AMD measured from the 3 + 1 0 energy of the T βγ-AMD+GCM are also shown. 10 B(3 + 1 0) has less spatial development of the pn pair and is regarded as the almost prolately deformed state with γ fluctuation.

Discussion
In this section, we discuss the structures of the low-lying states in 10 B and describe the spin and spatial configurations of the pn pair.
As discussed previously, the T = 0 and T = 1 states of 10 B have dominantly the S = 1 and S = 0 pn pairs around the 2α core, respectively. The finite spin (S = 1) of the T = 0 pair couples with the orbital angular momentum L to the total angular momentum J. In the 2α + pn structures, the orbital angular momentum (L pn ) for the pn pair motion relative to the 2α core and that (L 2α ) for the 2α core rotation contribute to the total orbital angular momentum L. Table. 2 shows the expectation values L 2 and S 2 of the squared total orbital angular momentum and total spin angular momentum of the 3 + 1 0, 1 + 1 0, 1 + 2 0, and 0 + 1 1 states obtained by the T βγ-AMD+GCM. S 2 = 2.0 of the T = 0 states indicates the almost pure S = 1 pn pair in the 3 + 1 0, 1 + 1 0, and 1 + 2 0 states, and S 2 = 0.3 of the T = 1 Table 1 Nuclear properties and transition strengths of 10 B are shown. The calculated values obtained by the T βγ-AMD+GCM and these obtained for the minimum energy states of the J π -projected energy surfaces of the T βγ-AMD in Fig. 3 are shown. The results of the AMD+VAP J calculation in Ref. [22] are also shown. The experimental proton radii are derived from the charge radii in [27]. Other experimental data are taken from [26,28].
observable T βγ-AMD T βγ-AMD+GCM AMD+VAP J Exp   Table 1. The 1 + 2 0 state appears from the excitation of the orbital angular momentum from L = 0 to L = 2 mainly  contributed by the rotation of the 2α core (L 2α = 2). The strong E2 transition from the 1 + 2 0 state to the 1 + 1 0 state (see Table 1) shows the feature of the L 2α = 2 excitation in the 1 + 2 0 state. The 3 + 1 0 state has the dominant L = 2 component, which is mainly contributed by the orbital angular momentum L pn = 2 of the S = 0 pn pair. From the cluster point of view, the S = 1 pn pair aligned to L pn = 2 feels strong spin-orbit potential from the 2α core. The strong attraction of the spin-orbit potential for the S = 1 pn pair in the D-wave is the origin of the level inversion in 10 B between the 1 + 1 0 and 3 + 1 0 states with the S-wave and D-wave pn pairs, respectively as pointed out in Ref. [22]. Note that, it corresponds to the p 2 3/2 configuration in the jj-coupling scheme.

Summary and outlook
We developed a new framework of the isospin projected AMD with the βγ constraint and the GCM called T βγ-AMD+GCM for study of N = Z = odd nuclei. To test the applicability of the method we applied it to 10 B. The formation of the S = 1 and S = 0 pn pairs around the 2α core in the low-lying T = 0 and T = 1 states of 10 B is described in the T βγ-AMD+GCM calculation. The spatial development of the T = 0 and T = 1 pn pairs as well as the core deformation is controlled by the βγ constraint in the isospin projected AMD. By Excitation Energy (MeV) Exp Fig. 7 The schematic figure of spin configurations of the pn pair and coupling with the orbital angular momentum in the low-lying states of 10 B. The experimental spectra are also shown. L, L αα and L pn denote to the total orbital angular momentum, the orbital angular momentum of the 2α core and that of the center of mass motion of the pn pair. S means to the intrinsic spin of the pn pair.
superposition of the optimized wave functions on the βγ plane with the GCM, the spatial motion of the pn pair as well as the core rotation is taken into account.
The T βγ-AMD+GCM calculation reproduces reasonably the properties of low-lying states of 10 B. The structures of the lowest four states (3 + 1 0, 1 + 1 0, 1 + 2 0 and 0 + 1 1) are understood by the angular momentum coupling of the pn pair internal spin (S), its spatial motion, and the core rotation. The 3 + 1 0(1 + 1 0) state is described by the 2α + pn structure having the T = 0, S = 1 pn pair in the D-wave(S-wave) around the 2α core. Because of the spin-orbit attraction from the core for the S = 1 pn pair in the D-wave, the 3 + 1 0 state comes down to the ground state. The 1 + 2 0 appears from the excitation of the 2α core rotation and has the strong E2 transition to the 1 + 1 0 state. The 0 + 1 1 state, which is the isobaric analogue state of Table 2 The expectation values L 2 and S 2 of the squared total orbital angular momentum and total spin angular momentum. The values obtained by the T βγ-AMD+GCM and those calculated for the energy minimum states of the J π -projected energy surfaces in Fig. 3(T βγ-AMD).
state L 2 S 2 T βγ-AMD T βγ-AMD+GCM T βγ-AMD T βγ-AMD+GCM the 10 Be ground state, is regarded as the spin partner of the 1 + 1 0 state, and it has the strong M1 transition from the 1 + 1 0 state. It was found that the T βγ-AMD+GCM can describe spin configurations and spatial development of the pn pair in T = 0 and T = 1 states as well as the core deformation and rotation. The pn pair motion relative to the core and also the shape fluctuation of the deformed core are taken into account by the βγ constraint with the GCM. This is one of the advantages supplier to the AMD+VAP J which is based on the the J π -projected wave function of a single Slater determinant. It was also shown that the isospin projection before the energy variation is necessary to obtain the optimum solution for each isospin state in Z = N = odd nuclei, in which different isospin states almost degenerate in the low-energy region.
The present method is expected to be useful to investigate structure with a pn pair around a deformed core in other N = Z = odd such as 22 Na and 26 Al, in which various J π T states appear in low-lying spectra. In principle, in the T βγ-AMD+GCM, existence of a pn pair nor clusters is not apriori assumed. Therefore, it is applicable to general N = Z = odd nuclei, and might enable us to make systematic study of pn pair correlations in nuclei along the N = Z line.