Signature dependent triaxiality for shape evolution from superdeformation in rapidly rotating $^{40}$Ca and $^{41}$Ca

We investigate the possible occurrence of the highly-elongated shapes near the yrast line in $^{40}$Ca and $^{41}$Ca at high spins on the basis of the nuclear energy-density functional method. Not only the superdeformed (SD) yrast configuration but the yrare configurations on top of the SD band are described by solving the cranked Skyme-Kohn-Sham equation in the three-dimensional coordinate-space representation. It is suggested that some of the excited SD bands undergo band crossings and develop to the hyperdeformation (HD) beyond $J \simeq 25 \hbar$ in $^{40}$Ca. We find that the change of triaxiality in response to rotation plays a decisive role for the shape evolution towards HD, and that this is governed by the signature quantum number of the last occupied orbital at low spins. This mechanism can be verified in an experimental observation of the positive-parity SD yrast signature-partner bands in $^{41}$Ca, one of which ($\alpha=+1/2$) undergoes crossings with the HD band while the other ($\alpha=-1/2$) shows the smooth evolution from the collective rotation at low spins to the non-collective rotation with oblate shape at the termination.


Introduction
Rotational motion, a typical collective mode of excitation in nuclei, emerges to restore the rotational symmetry broken by the nuclear deformation [1]. In most cases, a prolate deformation occurs naturally in the ground state, and then the total spin is generated by the collective rotation about the axis perpendicular to the symmetry axis. On the other hand, the spin is constructed only by the alignment of the single-particle orbitals when the rotation axis coincides with the symmetry axis. Actually, the system may deviate from the axial symmetry due to the Coriolis effect as soon as it rotates, and the direction of the angular momentum vector generated by the single-particle orbitals can be different from either the symmetry axis or the rotation axis. Therefore, the interplay and coupling between the collective and single-particle motions have to be considered selfconsistently to investigate the rotational motions of nuclei [2]. Superdeformed (SD) states exhibiting beautiful rotational spectra are an ideal situation in which the concept of the nuclear deformation is realized, and thus provide an opportune playground for the study of rotational motions. The study of SD bands has been an active field in nuclear physics and the rotational bands have been observed up to high spins in various mass regions [3] since its discovery in 1986 [4]. Recently, the high-spin structures in light N ≃ Z nuclei near the doubly-magic 40 Ca nucleus have been studied experimentally [5][6][7][8][9][10][11][12][13][14][15][16], and SD bands have been observed in such as 36 Ar [5,6], 40 Ar [9], 40 Ca [11,12], 42 Ca [17], and 44 Ti [16]. An interesting feature in this mass region is the coexistence of states with different shapes in low energy, which is caused by the single-particle excitations from the core and the coherent shell effects of neutrons and protons. Complementary theoretical models have been used in attempt to describe microscopically the SD state in 40 Ca by employing the interacting shell model [18,19], the nuclear energy-density functional (EDF) method [20][21][22], and the various cluster models [23][24][25]. The yrast spectroscopy of these light nuclei brings unique opportunity to investigate the mechanism for the occurrence of the SD band and the possibility of the hyperdeformed (HD) band because the single-particle density of state around the Fermi levels is low so that we can study in detail the deformed shell structures responsible for the SD and HD bands.
In quest of HD bands, shape evolution from the SD states associated with an increase in spin has been investigated because the nuclear rotation could stabilize the HD state due to its large moment of inertia. In fact HD bands as well as SD bands with various configurations are predicted in the cranked Skyrme-Kohn-Sham (SKS) calculation [20] and in the cranked relativistic-mean-field (CRMF) calculation [22], in which the observed SD band in 40 Ca is reasonably explained. An interesting feature in these predictions is that the obtained SD states are triaxially deformed: The magnitude of triaxial deformation is almost constant γ ∼ −10 • as a function of spin in the CRMF [11,22] while the SD states obtained in the cranked SKS calculation has a slightly smaller triaxiality γ ∼ 6 -9 • changing with spin [20]. The antisymmetrized molecular dynamics (AMD) calculation gives the similar result to the ones in the mean-field calculations with γ ∼ 15 • , and the K π = 2 + band is predicted to appear due to the triaxial deformation of the SD band [25]. The CRMF calculation predicts also that one of the SD bands exhibits a band termination [22]. We note however that the preceding studies do not clarify how the triaxial deformation and the termination emerge in the SD bands, and how the SD bands change their shape from SD to HD.
In the present study, we investigate the near-yrast structures in 40 Ca and 41 Ca at high spins on the basis of the cranked SKS method with paying attention to not only the evolution of shape elongation from the SD to HD states but the change of triaxiality in response to rotation. A binding energy, a functional of densities, is minimized with the configuration being constrained so that the polarization associated with the particle-hole (ph) excitations and the time-odd components in the mean field is taken into account. The possible emergence of the HD state is explored, and then the microscopic mechanism is discussed. For 40 Ca, we find that the excited SD bands undergo crossings at J ≃ 25 , in which the hyperintruder shell is occupied by both a neutron and a proton, forming negative-parity HD bands. For realization of the HD states at high spins, the signature quantum number of the last occupied low-Ω orbital near the Fermi level plays a decisive role in connection with the development of triaxiality. We then propose that this mechanism is verified in observing the SD yrast signature-partner bands in 41 Ca, one of which undergoes crossings with the HD band while the other shows the smooth termination from the collective rotation at low spins to the non-collective rotation at high spins.
The article is organized as follows. Section 2 describes the calculation scheme for the study of the near-yrast high-spin states in the framework of the nuclear EDF method; how to constrain and specify the configurations is explained. Section 3 shows the results of the calculation: The near-yrast structures in 40

Calculation scheme
We define the z-axis as a quantization axis of the intrinsic spin and consider the system rotating uniformly about the z-axis. Then, the cranked SKS equation is given by [26] δ where E[ρ] is a nuclear EDF, ω rot andĴ z mean the rotational frequency and the z-component of angular momentum operator, and the bracket denotes the expectation value with respect to the Slater determinant given by the occupied single-particle KS orbitals for a given ω rot .
To discuss the shape of calculated density distribution, it is convenient to introduce the multipole moments given as where ρ(r) is the particle density,R = 5 3A d 3 r r 2 ρ(r), and X lm are real basis of the spherical harmonics We then define the quadrupole deformation parameter β and the triaxial deformation parameter γ by Since we defined the z-axis as a rotation axis, the spherical harmonics is redefined in the transformation (x, y, z) → (z, x, y), and the expressions needed for describing the quadrupole deformations are given as In the numerical calculations, we impose the reflection symmetry about the (x, y)-, (y, z)and (z, x)-planes. We can therefore construct the simultaneous engenfunctions of the parity transformationP and the −π rotation about the z-axisR z = e iπĵz/ : besides the cranked-KS equationĥ with the single-particle Hamiltonian, or the Routhian for ω rot = 0, namelyĥ ′ = δE δρ − ω rotĵz . The eigenvalues π k (= ±1) and r k (= ±i) are called the parity and z-signature, respectively.
Hereafter, we simply call the latter signature. We can introduce the signature exponent quantum number α (= ±1/2) by r ≡ e iπα . The signature exponent α is useful when comparing with the experimental data through the relation α = I mod 2, where I is the total nuclear spin [2].
Since we chose the quantization axis of the intrinsic spin to coincide with the rotation axis (z-axis), we can determine the single-particle wave functions such that they satisfy the following reflection symmetries [26,27]: with σ being the direction of the intrinsic spin. We solve Eq. (8) by diagonalizing the singleparticle Routhianĥ ′ in the three-dimensional Cartesian-mesh representation with the box boundary condition. Thanks to the reflection symmetries (9)-(11), we have only to consider explicitly the octant region in space with x ≥ 0, y ≥ 0, and z ≥ 0. We use a 3D lattice mesh with a mesh size h = 0.8 fm and 12 points for each direction. The differential operators are represented by use of the 9-point formula of finite difference method. For diagonalization of the Routhian, we use the LAPACK dsyevx subroutine [28]. A modified Broyden's method [29] is utilized to calculate new densities during the selfconsistent iteration.
In the present calculation, we employ the Skyrme EDFs for E[ρ] in Eq. (1). All the timeeven densities are included, while the coupling constants for the derivative of spin density are set to zero as in Ref. [26] to avoid the numerical instability [30,31]. In some EDFs such as the SkI series [32], the center-of-mass correction is considered by subtracting P 2 CM /2mA. We, however, take into account the center-of-mass correction simply by i p 2 i /2mA because we do not discuss the total binding energy but rather the relative energy.
Single-particle orbitals are labeled by [Nn 3 Λ]Ω(r) with [Nn 3 Λ]Ω being the asymptotic quantum numbers (Nilsson quantum numbers) of the dominant component of the wave function at ω rot = 0 and r the signature of the orbital. To describe various types of rotational bands under the energy variation, the Slater determinantal states are constructed by imposing the configuration of the single-particle KS orbitals. Since the parity and signature are a good quantum number, and the pairing correlations are not included in the present calculation, the intrinsic configurations of interest can be described by the occupation number of particle n for the orbitals specified by the quantum number (π, r); [n (+1,+i) n (+1,−i) n (−1,+i) n (−1,−i) ] q for q = ν (neutron) and π (proton) as in the cranking calculation code hfodd [33].
The procedure of the cranking calculation is as follows: For a specified intrinsic configuration, we first find a selfconsisitent deformed solution at zero rotational frequency ω rot = 0 (or at a finite value of ω rot ) with use of an initial trial state, which is given by a deformed Woods-Saxon potential or another deformed SKS solution. We then increase slightly the rotational frequency by ∆ω rot = 0.1 MeV/ , and obtain the selfconsistent solution. Repeating this with a gradual increase of the rotational frequency, we trace evolution of the rotating deformed state as a function of ω rot . Note that no constraint on shape is imposed, except the reflection symmetry, in performing the cranking calculation.

Superdeformation and hyperdeformation in 40 Ca
The ground state of the 40 Ca nucleus, composed of twenty neutrons and twenty protons, is calculated to be spherical within the present calculation due to the spherical magic number of 20 generated by a gap between the d 3/2 and f 7/2 shells. In the present calculation scheme, the spherical configuration is represented as [7733] ν [7733] π . With an increase in the prolate deformation, the [202]3/2 orbital stemming from the spherical d 3/2 shell grows up in energy, and intersects with the down-sloping [330]1/2 orbital originating from the spherical f 7/2 shell (see Fig. 1 in Ref. [34] for example). After the crossing of these orbitals, one sees an energy gap that corresponds to a normal-deformed (ND) configuration. The ND configuration is thus represented as [6644] ν [6644] π . When the system is further deformed, an extruder [200]1/2 and an intruder [321]3/2 orbitals cross, and an energy gap appears after the crossing of these orbitals, leading to an SD configuration represented as This SD configuration is taken as a reference in the following discussion. The ND and SD configurations may correspond to the four-particle four-hole (4p4h) and eight-particle eighthole (8p8h) configurations with respect to the spherical configuration, respectively. However,  the Nilsson-type deformed wave functions at large deformation are not simply represented by the orbitals within the single major shell, but are expressed by the linear combination of the spherical orbitals with admixture of different major shells of ∆N = 2; the 4p4h or 8p8h configuration in terms the spherical shell model may not directly correspond to the ND or SD configuration in the present model. Furthermore, the low-lying 0 + states located at 3.35 MeV and 5.21 MeV, which are interpreted as the ND and SD states, respectively, in Ref. [11], are actually the mixture of many-particle many-hole configurations as described microscopically by the interacting shell model [19], the generator coordinate method [21], and the AMD [24]. Therefore, we use somewhat loosely the n-particle n-hole configuration below. Note that even the 12p12h configuration, composed of the particles occupying the g 9/2 shell instead of the d 3/2 shell corresponding to the megadeformed (MD) configuration, is also labeled by We show in Fig. 1 the calculated single-particle energies for the reference SD configuration [5555] ν [5555] π obtained by employing the several Skyrme functionals, including SkM* [35], SIII [36], SLy4 [37], SLy6 [37], SkI1 [32], SkI2 [32], SkI3 [32], SkI4 [32], SkI5 [32], UNEDF1-HFB [38], and UNEDF1 [39]. One sees that the SkI series produce the pronounced SD gap energy at a particle number 20. Note that the CRMF calculation in Ref. [22] also gives a high SD-gap energy of ∼ 4 MeV. Excitation energies (a) and kinematic moments of inertia (b) of the reference SD rotational band in 40 Ca as functions of angular momentum obtained by the cranked KS calculation employing the SkM*, SIII, SLy4, SkI4, and UNEDF1-HFB functionals together with the experimental data denoted as band 1 in Ref. [11]. A smooth part AI(I + 1) is subtracted with an inertia parameter A = 0.05 MeV for plotting the excitation energy.
SkM*, SIII, SLy4, SkI4, and UNEDF1-HFB together with the experimental data [11]. The cranking calculations were carried out up to about ω rot = 2.0 MeV/ with an interval of ∆ω rot = 0.1 MeV/ . As usual, the angular momentum is evaluated as I = J z . A smooth rigid-body part AI(I + 1) is subtracted with an inertia parameter A = 0.05 MeV to make the difference of the results visible. The calculation with the Skyrme functionals tends to overestimate the observed excitation energy. And the calculated kinematic moments of inertia, J = I /ω rot , are large as shown in Fig 2(b), which leads to a gentle slope in the energy vs. angular-momentum plot. Among the functionals we employed, the SkM* functional reproduces reasonably the observed excitation energy of the SD band in 40 Ca. Thus we are going to discuss the high-spin states in 40,41 Ca obtained mainly by using the SkM* functional. The SkI4 functional reproduces reasonably the observed kinematic moment of inertia as shown in Fig. 2(b). Thus, the calculation employing the SkI4 functional will be used to complement the discussion.  is subtracted with an inertia parameter A = 0.05 MeV as in Fig. 2. The experimental data for the normal deformation [11,13] are also shown.
configuration [6644] ν [6644] π , corresponding to the configuration [2,2] calculated in Ref. [22], terminates around I = 16 with the structure ν( where the deformation reaches a weakly-deformed oblate shape as shown in Fig. 4(b). Here, an oblate shape is indicated by γ = ±60 • , and in our definition of the triaxial deformation parameter γ and the choice of rotation axis, γ = −60 • represents a non-collective rotation, where the rotation axis is parallel to the symmetry axis. Thus, the sign of γ in the present definition is different from that in the Lund convention [40] as well as in Ref. [22]. Furthermore, let us mention the notation of the configuration defined in Ref. [22]. The configuration is labeled One sees that the reference SD configuration [5555] ν [5555] π (π = +1, r = +1) or the configuration [4,4] representing the four-neutron and four-proton excitation into the pf shell appears as an SD-yrast band below I ≃ 15, the deformation of which is β ≃ 0.55 and γ ≃ 3 • .
Next, we explore possibility of strongly-deformed states with other configurations. As shown in Fig. 1, the single-particle orbitals just below and above the SD gap at N, Z = 20 are [321]3/2 and [200]1/2, respectively, for both neutrons and protons. As candidates which may appear near the yrast, we calculate all the possible "1p1h" and "2p2h" excitations from the reference SD configuration associated with these two orbitals.  [3,4] following Ref. [22].
Calculated results for these configurations are shown in Figs. 3(a), 4(a), and 4(b). For all the four negative-parity configurations we obtain solutions which commonly have large deformation β ∼ 0.5 at low spins I 12, and hence can be regarded as SD bands. At higher spins, however, a difference grows up. Nevertheless we observe the following systematic trends. An interesting feature in 40 Ca at high spin is that the negative-parity SD bands with the configuration [6554] ν [5555] π (π = −1, r = −1) and the configuration [6545] ν [5555] π (π = −1, r = +1) are predicted to extend to even higher spins I 20 and appear near the yrast line beyond I ≃ 25. Furthermore, they exhibit stepwise increase of deformation reaching β ≃ 0.6 -0.65. We show in Fig. 5 the rotational frequency dependence of the single-particle energy levels (Routhian) for the configuration [6554] ν [5555] π . Due to the polarization associated with the time-odd mean field, the signature splitting is seen even at We here remark relation to the preceding cranked EDF studies. It is predicted in Ref. [22] that negative-parity excited bands with neutron "1p1h" configuration [3,4] as well as proton "1p1h" configuration [4,3] have appreciable triaxiality with positive γ ∼ 10 • (negative γ ∼ −10 • in the convention adopted in Ref. [22]), however the sign and the development of triaxiality are not deeply investigated. We note also that in the present study we obtained proton ph-excited states [4,3], i.e, [5555] ν [6554] π etc., which appear closely to the neutron ph-excited states because the system under consideration is a light N = Z nucleus though we do not show these states in the present article. The random-phase approximation (RPA) in the rotating mean field is more appropriate to describe the excited bands [41][42][43][44][45][46][47], however it is beyond the scope of the present work. The low-lying K π = 1 − state on the SD I π = 0 + state predicted in an RPA description [34,48] [3,3] calculated in Ref. [22]. The deformation of these states is not very large and the deformation parameter is around β ∼ 0.4, which is in between the deformations of ND and SD configurations. This is because a "2p2h" excitation on the reference SD configuration can be also regarded as a "2p2h" excitation on the ND configuration. We found that the triaxial deformation develops with positive-and negative-γ for the configurations [6545] ν [6545] π and [5645] ν [5645] π , respectively [see Fig. 4(b)]. The latter terminates quickly at I ≃ 18. The former configuration develops to the HD state with a band crossing where both a neutron and a proton are promoted to the [440]1/2(r = +i) orbital, corresponding to the configuration [31,31] representing the one-neutron and one-proton excitation into the g 9/2 shell. These behaviors can be explained by the mechanism discussed above, i.e. the occupation of the [200]1/2(r = +i) or [200]1/2(r = −i) orbital favoring the positive-and negative-γ deformations, respectively, as increasing the rotational frequency.

Superdeformation and hyperdeformation in 41 Ca: role of the [200]1/2 orbital
For the occurrence of the HD states at high spins in 40 Ca, we found that the occupation of the [200]1/2(r = +i) orbital is necessary because the configurations involving this orbital favor the positive-γ deformation at ω rot > 0, while the configurations involving the   We show in Figs. 4(c) and 4(d) the evolution of deformation for these configurations.
Let us look at the positive-parity configuration [6555] ν [5555] π (π = +1, r = +i) corresponding to the configuration [4,4]. At ω rot = 0, this configuration is nothing but the one neutron added in the [200]1/2(r = +i) orbital to the reference SD configuration   momentum of ∼ 4 whose structure corresponds to the configuration [41,41]. Both a neutron and a proton occupy the g 9/2 shell. On the other hand, its signature partner, the configuration [5655] ν [5555] π (π = +1, r = −i) also corresponding to the configuration [4,4], terminates around I = 20. This is because the occupation of the ν[200]1/2(r = −i) orbital leads to the negative-γ deformation at ω rot > 0. The signature-partner SD bands reveal an opposite evolution of triaxiality depending on the signature quantum number of a neutron which is added to the reference SD state in 40 Ca. Note that we found the solutions corresponding to the MD configuration as shown by the thin lines in Fig. 3(b), and the signature-partner MD bands may appear as the yrast bands above I ≃ 34.
The configuration [6555] ν [6554] π with quantum numbers(π = −1, r = −i), denoted by open square in Fig. 3(b), appears closely to the yrast band at high spins I 30. Figure 6 shows the Routhians for this configuration. Both a neuron and a proton occupy the

Case for the SkI4 functional
One may doubt the above finding can be a specific feature of the EDF employed. We try to dispel the suspicion by investigating the functional dependence of the high-spin structures in 40 Ca and 41 Ca. We employ here the SkI4 functional as mentioned previously. As one can see in Figs. 8(b) and 8(d), the SkI4 functional gives non-zero triaxiality at ω rot = 0 for most of the configurations under study. Therefore, we expect to see clearly the role of signature-dependent triaxiality in shape evolution of the SD states.
Before discussion, let us mention how to choose the sign of γ at ω rot = 0. At ω rot = 0, the sign of γ gives no difference, which means that we have two minima, with the same energy, in the potential energy surface in terms of the quadrupole deformations in the region of −60 • < γ < 120 • . At ω rot > 0, the asymmetry shows up in the total Routhian surface and we can choose the configuration that produces the lowest energy. We thus chose the sign of γ at ω rot = 0, whose configuration gives lower energy at ω rot > 0 continuously.
Then, we are going to investigate the near-yrast structures in 40 Ca at first. Figures 7(a) and 8(a), 8(b) show the excitation energies and deformation. It is seen from comparison of the excitation energies [ Fig. 3(a) vs. Fig. 7(a)] that the relative ordering of the near-yrast SD bands and the ND band are essentially the same as that for SkM*. On the other hand, a difference from SkM* is seen in the absolute magnitude of the excitation energies. For example, the negative-parity excited SD bands are well separated from the reference SD  band at low spins since the SD gap energy at the particle number 20 is higher than that calculated with SkM*. This is similar to the results obtained by the CRMF calculation [22].
The deformation of the obtained SD and ND states in 40  β-deformation at I ≃ 23 and ≃ 20 from β ≃ 0.4 and ≃ 0.5, respectively, to β ≃ 0.6, which is larger than the β-deformation of the reference SD configuration [5555] ν [5555] π . In fact these highly-deformed states correspond to the HD configuration, where the hyperintruder g 9/2 shell is occupied by both a neutron and a proton. As we discussed in the case of SkM*, the occupation of [200]1/2(r = +i) orbital in these bands at low spins induces the shape evolution toward the HD configuration via the level crossing between the [200]1/2(r = +i) and [440]1/2(r = +i) orbitals. At even higher spins I 27, we see a difference between SkM* and SkI4, for which we did not obtain the expected HD state with an additional occupation of the π[440]1/2(r = +i) orbital. Another difference is seen in the MD band, which, in the case of SkI4, crosses with the SD bands at spins lower than the crossings between the SD and HD bands.
Next, we discuss the near-yrast structure of 41 Ca. Figure 7( with an additional proton ph-excitation appear above the positive-parity SD bands. The energy ordering of these bands is essentially the same as that of SkM* [ Fig. 3(b)] as far as the low spin region I 6 is concerned. However, the pair of the positive-parity SD bands [6555] ν [5555] π and [5655] ν [5555] π (8p8h+n) is well separated from the negative-parity SD bands, and stay as the lowest-energy SD bands for a longer spin interval up to I ≃ 12 as compared with the case of SkM*. The trend continues further at higher spins I ≃ 12 -27.
These are due to the larger SD gap at Z = 20 in the SkI4 functional. This indicates that the positive-parity SD/HD bands [6555] ν [5555] π and [5655] ν [5555] π might be populated more strongly in fusion reactions if the reality is close to the results of SkI4 than those of SkM*.
The predicted shape deformation and its evolution with ω rot , including the band termination and hyperdeformation, shown in Figs. 8(a) and 8(b), are also consistent with the results of SkM* except the following differences: All the configurations have non-zero and sizable triaxiality with |γ| 10 • at ω rot = 0, as mentioned above. Correspondingly ω rot -dependence of the triaxiality is weak at small rotational frequency although the triaxiality itself is larger than those in SkM*. This indicates that the effect of the occupation in the orbitals [200]1/2(r = ±i) inducing the positive or negative γ deformation is stronger than that in SkM*. We observe also that the negative-parity bands

Summary
We have investigated the shape evolution of the doubly-magic 40 Ca and its neighboring 41 Ca in response to rotation in the framework of the nuclear energy-density functional method. The cranked Skyrme-Kohn-Sham equation was solved in the 3D lattice to describe various types of configurations including the negative-parity excited bands. We found that the hyperdeformed (HD) states appear above I ≃ 25. The occupation of both a neutron and a proton in the rotation-aligned [440]1/2 orbital brings about the occurrence of the HD states. Here, we found that the development of triaxial deformation is important to understand the property of the near-yrast bands. The configurations in which the [200]1/2(r = −i) orbital is occupied by a neutron and/or a proton undergo the evolution of triaxial deformation with negative value with an increase in spin, and terminate at the intermediate spins without the band crossing. The occupation of the [200]1/2(r = +i) orbital at low spins leads to the realization of the HD states at high spins after the band crossings. We predicted that this mechanism can be verified in an experimental observation of the positive-parity superdeformed (SD) signature-partner bands in 41 Ca which has an 8p8h+n configuration with the last neutron occupying the [200]1/2(r = +i) or [200]1/2(r = −i) orbital on top of the reference SD configuration [5555] ν [5555] π in 40 Ca: One of the signature-partner bands with α = 1/2 (I = 1/2, 5/2, · · · ) undergoes band crossings leading to the HD band, while the other partner band with α = −1/2 (I = 3/2, 7/2, · · · ) terminates below I ≃ 20.

Acknowledgment
We thank T. Inakura and T. Nakatsukasa for valuable discussions, and E. Ideguchi for communications on the present experimental status. This work was supported by the JSPS KAKENHI (Grants No. JP16K17687, No. JP17K05436, and No. JP19K03824) and the JSPS-NSFC Bilateral Program for Joint Research Project on "Nuclear mass and life for unraveling mysteries of the r-process". The numerical calculations were performed on CRAY XC40 at the Yukawa Institute for Theoretical Physics, Kyoto University, and on COMA (PACS-IX) at the Center for Computational Sciences, University of Tsukuba.

A Calculation of the ground-state energy for an odd-A nucleus
For an odd-A nucleus with spherical symmetry, the solutions of the deformed Skyrme-Kohn-Sham equation are not converged generally due to the degeneracy of the single-particle orbitals with different magnetic quantum numbers; (2j + 1)-fold degeneracy. We encountered the oscillation of the calculated J z , expectation value of the z-component of total angular momentum operator. This is because one can not define the symmetry axis and thus the J z can not be defined uniquely. To resolve the degeneracy of the single-particle orbitals, the infinitesimal cranking is introduced: with m being the magnetic quantum number of the total angular momentum. Then, we take the extrapolation to the limit of ω rot → 0.