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Beyond-mean-field study of 37ΛAr based on the Skyrme-Hartree-Fock model

  • We present the hypernuclear states of 37ΛAr obtained using the Skyrme-Hartree-Fock (SHF) model and a beyond-mean-field approach, including angular momentum projection (AMP) and the generator coordinate method (GCM). A comprehensive energy spectrum is given, which includes normally deformed (ND) and super deformed (SD) hypernuclear states with positive or negative parities. Energy levels corresponding to the configurations in which a Λ hyperon occupies the s-, p-, or sd-shell orbitals are discussed. For the s-shell Λ, we pay special attention to the ND and SD states corresponding to the configurations 36ArN sΛ and 36ArS sΛ, where 36ArN and 36ArS denote the ND and SD nuclear cores, respectively. The disagreements between different models over the Λ separation energy of the SD state in previous studies are revisited. For the p-shell Λ, four rotational bands are predicted, and the impurity effects are shown. Furthermore, two energy levels corresponding to the configurations 36ArSΛ[101]32 and 36ArSΛ[101]12 are obtained below the separation threshold of 36Ar+Λ within 0.5 MeV. For the sd-shell Λ, three bound states are found near the separation threshold, and the mechanism behind these states are discussed.
      PCAS:
    • 23.20.Lv(γ-transitions and level energies)
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    [2] M. Bender, H. Flocard, and P. H. Heenen, Phys. Rev. C 68, 044321 (2003) doi: 10.1103/PhysRevC.68.044321
    [3] C. E. Svensson, A. O. Macchiavelli, A. Juodagalvis et al., Phys. Rev. Lett. 85, 2693 (2000) doi: 10.1103/PhysRevLett.85.2693
    [4] B. N. Lu, E. Hiyama, H. Sagawa et al., Phys. Rev. C 89, 044307 (2014) doi: 10.1103/PhysRevC.89.044307
    [5] M. Isaka, M. Kimura, E. Hiyama et al., Prog. Theor. Exp. Phys. 2015, 103D02 (2015) doi: 10.1093/ptep/ptv138
    [6] X. R. Zhou, E. Hiyama, and H. Sagawa, Phys. Rev. C 94, 024331 (2016) doi: 10.1103/PhysRevC.94.024331
    [7] X. Y. Wu, H. Mei, J. M. Yao et al., Phys. Rev. C 95, 034309 (2017) doi: 10.1103/PhysRevC.95.034309
    [8] J. W. Cui, X. R. Zhou, and H. J. Schulze, Phys. Rev. C 91, 054306 (2015)
    [9] J. W. Cui, X. R. Zhou, L. X. Guo et al., Phys. Rev. C 95, 024323 (2017) doi: 10.1103/PhysRevC.95.024323
    [10] W. Y. Li, J. W. Cui, and X. R. Zhou, Phys. Rev. C 97, 034302 (2018) doi: 10.1103/PhysRevC.97.034302
    [11] J. W. Cui and X. R. Zhou, Prog. Theor. Exp. Phys. 2017, 093D04 (2017)
    [12] J. W. Cui and X. R. Zhou, Commun. Theor. Phys. 73, 115301 (2021) doi: 10.1088/1572-9494/ac06bd
    [13] R. H. Dalitz and A. Gal, Phys. Rev. Lett. 36, 362 (1976) doi: 10.1103/PhysRevLett.36.362
    [14] H. Bandō, Nucl. Phys. A 450, 217c (1986) doi: 10.1016/0375-9474(86)90556-7
    [15] H. Mei, K. Hagino, and J. M. Yao, Phys. Rev. C 93, 011301(R) (2016) doi: 10.1103/PhysRevC.93.011301
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    [17] M. Isaka, M. Kimura, A. Doté et al., Phys. Rev. C 87, 021304(R) (2013) doi: 10.1103/PhysRevC.87.021304
    [18] E. Hiyama, M. Kamimura, T. Motoba et al., Phys. Rev. Lett. 85, 270 (2000) doi: 10.1103/PhysRevLett.85.270
    [19] R. Bertini, O. Bing, P. Birien et al., Phys. Lett. B 83, 306 (1979) doi: 10.1016/0370-2693(79)91113-4
    [20] P. H. Pile, S. Bart, R. E. Chrien et al., Phys. Rev. Lett. 66, 2585 (1991) doi: 10.1103/PhysRevLett.66.2585
    [21] T. Hasegawa, O. Hashimoto, S. Homma et al., Phys. Rev. C 53, 1210 (1996) doi: 10.1103/PhysRevC.53.1210
    [22] X. R. Zhou, H. J. Schulze, H. Sagawa et al., Phys. Rev. C 76, 034312 (2007) doi: 10.1103/PhysRevC.76.034312
    [23] M. Bender, K. Rutz, P. G. Reinhard et al., Eur. Phys. J. A 8, 59 (2000) doi: 10.1007/s10050-000-4504-z
    [24] P. Ring and P. Schuck, The Nuclear Many-Body Problem, (New York: Springer-Verlag, Inc., 1980), p.400
    [25] R. Rodriguez-Guzman, J. L. Egido, and L. M. Robledo, Phys. Lett. B 474, 15 (2000) doi: 10.1016/S0370-2693(00)00015-0
    [26] P. Bonche, J. Dobaczewski, H. Flocard et al., Nucl. Phys. A 510, 466 (1990) doi: 10.1016/0375-9474(90)90062-Q
    [27] J. M. Yao, H. Mei, H. Chen et al., Phys. Rev. C 83, 014308 (2011) doi: 10.1103/PhysRevC.83.014308
    [28] H. J. Schulze and E. Hiyama, Phys. Rev. C 90, 047301 (2014) doi: 10.1103/PhysRevC.90.047301
    [29] J. Dobaczewski, W. Satula, B. G. Carlsson et al., Comput. Phys. Commun. 180, 2361 (2009) doi: 10.1016/j.cpc.2009.08.009
    [30] H. Sagawa, X. R. Zhou, X. Z. Zhang et al., Phys. Rev. C 70, 054316 (2004) doi: 10.1103/PhysRevC.70.054316
    [31] J. Terasaki, P. H. Heenen, H. Flocard et al., Nucl. Phys. A 600, 371 (1996) doi: 10.1016/0375-9474(96)00036-X
    [32] M. T. Win, K. Hagino, and T. Koike, Phys. Rev. C 83, 014301 (2011)
    [33] J. M. Yao, Z. P. Li, K. Hagino et al., Nucl. Phys. A 868, 12 (2011)
    [34] C. E. Svensson, A. O. Macchiavelli, A. Juodagalvis et al., Phys. Rev. C 63, 061301(R) (2001) doi: 10.1103/PhysRevC.63.061301
    [35] NationalNuclear DataCenter, Brookhaven National Laboratory, http://www.nndc.bnl.gov/
  • [1] G. L. Long and Y. Sun, Phys. Rev. C 63, 021305(R) (2001) doi: 10.1103/PhysRevC.63.021305
    [2] M. Bender, H. Flocard, and P. H. Heenen, Phys. Rev. C 68, 044321 (2003) doi: 10.1103/PhysRevC.68.044321
    [3] C. E. Svensson, A. O. Macchiavelli, A. Juodagalvis et al., Phys. Rev. Lett. 85, 2693 (2000) doi: 10.1103/PhysRevLett.85.2693
    [4] B. N. Lu, E. Hiyama, H. Sagawa et al., Phys. Rev. C 89, 044307 (2014) doi: 10.1103/PhysRevC.89.044307
    [5] M. Isaka, M. Kimura, E. Hiyama et al., Prog. Theor. Exp. Phys. 2015, 103D02 (2015) doi: 10.1093/ptep/ptv138
    [6] X. R. Zhou, E. Hiyama, and H. Sagawa, Phys. Rev. C 94, 024331 (2016) doi: 10.1103/PhysRevC.94.024331
    [7] X. Y. Wu, H. Mei, J. M. Yao et al., Phys. Rev. C 95, 034309 (2017) doi: 10.1103/PhysRevC.95.034309
    [8] J. W. Cui, X. R. Zhou, and H. J. Schulze, Phys. Rev. C 91, 054306 (2015)
    [9] J. W. Cui, X. R. Zhou, L. X. Guo et al., Phys. Rev. C 95, 024323 (2017) doi: 10.1103/PhysRevC.95.024323
    [10] W. Y. Li, J. W. Cui, and X. R. Zhou, Phys. Rev. C 97, 034302 (2018) doi: 10.1103/PhysRevC.97.034302
    [11] J. W. Cui and X. R. Zhou, Prog. Theor. Exp. Phys. 2017, 093D04 (2017)
    [12] J. W. Cui and X. R. Zhou, Commun. Theor. Phys. 73, 115301 (2021) doi: 10.1088/1572-9494/ac06bd
    [13] R. H. Dalitz and A. Gal, Phys. Rev. Lett. 36, 362 (1976) doi: 10.1103/PhysRevLett.36.362
    [14] H. Bandō, Nucl. Phys. A 450, 217c (1986) doi: 10.1016/0375-9474(86)90556-7
    [15] H. Mei, K. Hagino, and J. M. Yao, Phys. Rev. C 93, 011301(R) (2016) doi: 10.1103/PhysRevC.93.011301
    [16] M. Isaka, H. Homma, M. Kimura et al., Phys. Rev. C 85, 034303 (2012) doi: 10.1103/PhysRevC.85.034303
    [17] M. Isaka, M. Kimura, A. Doté et al., Phys. Rev. C 87, 021304(R) (2013) doi: 10.1103/PhysRevC.87.021304
    [18] E. Hiyama, M. Kamimura, T. Motoba et al., Phys. Rev. Lett. 85, 270 (2000) doi: 10.1103/PhysRevLett.85.270
    [19] R. Bertini, O. Bing, P. Birien et al., Phys. Lett. B 83, 306 (1979) doi: 10.1016/0370-2693(79)91113-4
    [20] P. H. Pile, S. Bart, R. E. Chrien et al., Phys. Rev. Lett. 66, 2585 (1991) doi: 10.1103/PhysRevLett.66.2585
    [21] T. Hasegawa, O. Hashimoto, S. Homma et al., Phys. Rev. C 53, 1210 (1996) doi: 10.1103/PhysRevC.53.1210
    [22] X. R. Zhou, H. J. Schulze, H. Sagawa et al., Phys. Rev. C 76, 034312 (2007) doi: 10.1103/PhysRevC.76.034312
    [23] M. Bender, K. Rutz, P. G. Reinhard et al., Eur. Phys. J. A 8, 59 (2000) doi: 10.1007/s10050-000-4504-z
    [24] P. Ring and P. Schuck, The Nuclear Many-Body Problem, (New York: Springer-Verlag, Inc., 1980), p.400
    [25] R. Rodriguez-Guzman, J. L. Egido, and L. M. Robledo, Phys. Lett. B 474, 15 (2000) doi: 10.1016/S0370-2693(00)00015-0
    [26] P. Bonche, J. Dobaczewski, H. Flocard et al., Nucl. Phys. A 510, 466 (1990) doi: 10.1016/0375-9474(90)90062-Q
    [27] J. M. Yao, H. Mei, H. Chen et al., Phys. Rev. C 83, 014308 (2011) doi: 10.1103/PhysRevC.83.014308
    [28] H. J. Schulze and E. Hiyama, Phys. Rev. C 90, 047301 (2014) doi: 10.1103/PhysRevC.90.047301
    [29] J. Dobaczewski, W. Satula, B. G. Carlsson et al., Comput. Phys. Commun. 180, 2361 (2009) doi: 10.1016/j.cpc.2009.08.009
    [30] H. Sagawa, X. R. Zhou, X. Z. Zhang et al., Phys. Rev. C 70, 054316 (2004) doi: 10.1103/PhysRevC.70.054316
    [31] J. Terasaki, P. H. Heenen, H. Flocard et al., Nucl. Phys. A 600, 371 (1996) doi: 10.1016/0375-9474(96)00036-X
    [32] M. T. Win, K. Hagino, and T. Koike, Phys. Rev. C 83, 014301 (2011)
    [33] J. M. Yao, Z. P. Li, K. Hagino et al., Nucl. Phys. A 868, 12 (2011)
    [34] C. E. Svensson, A. O. Macchiavelli, A. Juodagalvis et al., Phys. Rev. C 63, 061301(R) (2001) doi: 10.1103/PhysRevC.63.061301
    [35] NationalNuclear DataCenter, Brookhaven National Laboratory, http://www.nndc.bnl.gov/
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Ji-Wei Cui, Ruizhe Wang and Xian-Rong Zhou. Beyond-mean-field study of 37ΛAr based on the Skyrme-Hartree-Fock model[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac6357
Ji-Wei Cui, Ruizhe Wang and Xian-Rong Zhou. Beyond-mean-field study of 37ΛAr based on the Skyrme-Hartree-Fock model[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ac6357 shu
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Beyond-mean-field study of 37ΛAr based on the Skyrme-Hartree-Fock model

    Corresponding author: Xian-Rong Zhou, xrzhou@phy.ecnu.edu.cn
  • 1. School of Physics, Xidian University, Xi’an 710071, China
  • 2. FLEETCOR Technologies 3280 Peachtree Rd SUITE 2400, Atlanta, GA 30305, USA
  • 3. Department of Physics, East China Normal University, Shanghai 200241, China

Abstract: We present the hypernuclear states of 37ΛAr obtained using the Skyrme-Hartree-Fock (SHF) model and a beyond-mean-field approach, including angular momentum projection (AMP) and the generator coordinate method (GCM). A comprehensive energy spectrum is given, which includes normally deformed (ND) and super deformed (SD) hypernuclear states with positive or negative parities. Energy levels corresponding to the configurations in which a Λ hyperon occupies the s-, p-, or sd-shell orbitals are discussed. For the s-shell Λ, we pay special attention to the ND and SD states corresponding to the configurations 36ArN sΛ and 36ArS sΛ, where 36ArN and 36ArS denote the ND and SD nuclear cores, respectively. The disagreements between different models over the Λ separation energy of the SD state in previous studies are revisited. For the p-shell Λ, four rotational bands are predicted, and the impurity effects are shown. Furthermore, two energy levels corresponding to the configurations 36ArSΛ[101]32 and 36ArSΛ[101]12 are obtained below the separation threshold of 36Ar+Λ within 0.5 MeV. For the sd-shell Λ, three bound states are found near the separation threshold, and the mechanism behind these states are discussed.

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    I.   INTRODUCTION
    • The normally deformed (ND) and super deformed (SD) states of 36Ar have been the research topics of several theoretical models [1, 2] since the discovery of the SD rotational band [3]. In recent years, properties of the hypernuclear system, 37ΛAr, have often been discussed [47], although this hypernuclear system has not been experimentally observed. The most popular topics associated with 36Ar and 37ΛAr are the SD states; the ground states are ND. For the Λ hyperon occupying the s-shell orbital, the relativistic mean field (RMF) model [4] and beyond-RMF calculations [7] indicate that the SD state of 37ΛAr gives a larger Λ separation energy, SΛ, than the ground state. However, antisymmetrized molecular dynamics (AMD) [5] and the Skyrme-Hartree-Fock (SHF) model [6] show that the SΛ of the SD state is smaller than that of the ground state. Compared to AMD [5] and the beyond-RMF calculations [7], the SHF model [6] merely provides mean-field results, which breaks rotational symmetry and thus hinders further uses of the rotational bands and the transition probabilities observed in laboratories. Since 2015, the beyond-mean-field SHF model for Λ hypernuclei has gradually improved [812]. The angular momentum projection (AMP) technique is used to restore rotational symmetry, and the generator coordinate method (GCM) is employed to deal with shape mixing. Therefore, the beyond-mean-field SHF model is used to research the rotational bands corresponding to the configuration 36ArsΛ, which is the first of the three motivations of this paper.

      Beside the s-shell orbital, the Λ hyperon may be excited onto the p-shell orbitals, and such a Λ hyperon coupled with an ND or SD nuclear core can provide various configurations. These configurations correspond to low-lying energy levels with negative parity, and the beyond-mean-field SHF model can be used to predict them. For example, in a previous study [10], the rotational energy levels corresponding to the configurations 8BepΛ and 8BepΛ [13, 14] were precisely reproduced. Therefore, the second motivation of this paper is to obtain predictions for configurations such as 36ArpΛ, which will help identify the negative-parity energy levels of 37ΛAr in future experiments.

      For several typical hypernuclear systems such as 9ΛBe, 13 ΛC, 21 ΛNe, and 25ΛMg, Λ hyperons on the s-shell orbital or excited onto the p-shell orbitals have been studied using various theoretical models [1012,1518] in recent years. However, the sd-shell Λ hyperon in these hypernuclei does not provide a bound state because they are too light to bind such a Λ hyperon. Some (K,π) and (π+, K+) reactions [1921] reveal that the configuration 39CasdΛ gives bound energy levels (that is, energy levels below the separation threshold of 39Ca+Λ), whereas 31SsdΛ gives unbound energy levels. 37ΛAr is between 32ΛS and 40ΛCa in the hypernuclear chart, which is probably to provide bound states for configurations such as 36ArsdΛ. Hence, the third motivation of this paper is to theoretically investigate the existence of bound states for 36ArsdΛ and obtain the corresponding configurations with Nilsson quantum numbers.

      In this paper, a beyond-mean-field SHF model that considers AMP techniques and the GCM is employed to study the low-lying energy levels corresponding to various configurations of 37ΛAr. These configurations include an ND or SD nuclear core coupled to a Λ hyperon onto the orbitals of s, p, or sd shells. This paper is organized as follows. In Sec. II, the formalism of the beyond-mean-field calculation is introduced. Sec. III presents the results and discussions. In Sec. IV, we draw conclusions about the study.

    II.   FORMALISM
    • In a hypernuclear system, the physical quantities we are most concerned with are its eigenstates, eigenenergies, and transition rates. To obtain these physical quantities, the beyond-mean-field SHF model begins with a mean-field state of the Λ hypernucleus in the intrinsic frame of reference [22]

      |Φ(NΛ)(β)=|ΦN(β)|ΦΛ,

      (1)

      where |ΦN(β) represents a nuclear core with the deformation parameter β, and |ΦΛ denotes the single-particle wave function of a Λ hyperon.

      A general eigenstate of the Λ hypernuclear system is given by linear superposition:

      |ΨJMα=βFJα(β)ˆPJMK|Φ(NΛ)(β),

      (2)

      where β plays the role of the generator coordinate, and the AMP operator, ˆPJMK, restores the rotational symmetry of the system. The mean-field state in the intrinsic frame of reference, |Φ(NΛ)(β), is derived by a constrained SHF model [6], and a density-dependent delta interaction (DDDI) [23] is considered for the nucleons as

      G(r)=V0[1ρ(r)ρ0].

      (3)

      The weight function FJα(β) in Eq. (2) and the eigenenergies are determined by the Hill-Wheeler-Griffin (HWG) equation [24],

      β[HJKK(β,β)EJαNJKK(β,β)]FJα(β)=0,

      (4)

      in which the Hamiltonian and norm elements are given by

      HJKK(β,β)=Φ(NΛ)(β)|^HˆPJKK|Φ(NΛ)(β),

      (5)

      NJKK(β,β)=Φ(NΛ)(β)|ˆPJKK|Φ(NΛ)(β).

      (6)

      The corrected Hamiltonians ^H is

      ^H=ˆHλp(ˆNpZ)λn(ˆNnN),

      (7)

      where the final two terms on the right-hand side account for the fact that the projected wave function does not provide the correct number of particles on average [2527], and ˆH is given by the energy density functional (EDF) in Ref. [28].

      For a certain configuration determined by the intrinsic state |Φ(NΛ)(β), the projected energy is derived as

      E(β,J,K)=HJKK(β,β)NJKK(β,β).

      (8)

      Given two general eigenstates, |α;J and |α;J, the reduced E2 transition rate between them is derived as

      B(E2,JαJα)=12J+1|α;J||ˆQ2||α;J|2,

      (9)

      where

      α;J||ˆQ2||α;J=2J+1MμββFJα(β)FJα(β)×CJKJM2μΦ(NΛ)(β)|ˆQ2μˆPJMK|Φ(NΛ)(β).

      (10)

      In the above equation, ˆQ2μ=r2Y2μ(φ,θ) is the electric quadrupole operator [29], and CJKJM2μ denotes the Clebsh-Gordon coefficients.

      Parameters: In this paper, the SLy4 force is used for the NN interaction, and the strength of the pairing force is V0=410 MeV fm3 for both protons and neutrons [30], with a smooth pairing energy cutoff of 5 MeV around the Fermi level [31, 32]. For the NΛ interaction, we use the SLL4 force [28], which offers the best fit for the Λ separation energy in the spherical SHF model [28].

      Notations: Single-particle orbitals of the Λ hyperon are denoted by Nilsson quantum numbers [Nn3ml]Ωπ; the same notation is also adopted in Ref. [4]. Then, a certain configuration of the Λ hypernuclear system is given as A1ZΛ[Nn3ml]Ωπ, in which A1Z is the nuclear core, and Λ[Nn3ml]Ωπ represents a Λ hyperon occupying the orbital [Nn3ml]Ωπ. To emphasize the deformation, 36ArN and 36ArS are used to represent the ND and SD nuclear cores of 37ΛAr, respectively. For example, the configuration 36ArSΛ[110]12 denotes an SD nuclear state of 36Ar coupled with a Λ hyperon occupying the orbital Λ[110]12.

      Model space: Because intrinsic wave functions are kept axially symmetric, the quadrupole deformation parameter β of the nuclear core is the exclusive generator coordinate. If the Λ orbital is specified, the range of β and the number of basis functions determine the model space jointly. In this paper, β is between 1.4 and 2.8, and 90 basis functions are evenly spaced over this range for both 36Ar and 37ΛAr.

    III.   RESULTS AND DISCUSSIONS
    • The potential energy surfaces (PESs) of 36Ar and 37ΛAr derived using the mean-field calculation are shown in Fig. 1, and we can see that there are three energy minima for 36Ar. Two of these minima, located at β=0.16 and β=0.12, both give ND configurations, whereas the third, located at β=0.63, exhibits an SD prolate shape. For 37ΛAr, the addition of one Λ hyperon onto the orbitals of the s, p, or sd shells causes various impurity effects. In our current mean-field calculation, the hyperon Λ[000]12+ makes |β| of the three energy minima slightly smaller, which is already known from early research [33]. Such a reduction in deformation is not clear in Fig. 1 because the nuclear core 36Ar is sufficiently heavy against the NΛ attraction. For the p-shell Λ hyperon, Fig. 1 clearly shows that Λ[110]12 in 37ΛAr makes the prolate minima significantly deeper and nearly eliminates the oblate minimum, whereas Λ[101]32 and Λ[101]12 make the prolate energy minima shallower and enhance the oblate minimum. These opposite phenomena are due to the density distributions of the p-shell Λ orbitals. Fig. 2 shows the density distributions of several p-shell and sd-shell Λs. It must be emphasized that the density distributions of Λ[101]32 and Λ[101]12 are nearly identical and share the same contour map in Fig. 2, and Λ[202]52+ and Λ[202]32+ do the same. In the top-right panel of Fig. 2, we can see that Λ[110]12 is prolately distributed; thus, its coupling to a prolate nuclear core leads to a lower binding energy. However, the top-left panel of Fig. 2 shows that Λ[101]32 and Λ[101]12 are oblately distributed; hence, they prefer oblate nuclear cores. For the sd-shell Λ hyperon, Fig. 1 shows the PESs of configurations 36ArΛ[220]12+, 36ArΛ[202]52+, and 36ArΛ[202]32+ because only these three configurations give bound states, the properties of which will be discussed in detail in the final two paragraphs of this section.

      Figure 1.  (color online) PESs obtained from the mean-field calculations for certain configurations indicated in the legend. The horizontal axis represents the quadrupole deformation parameter β.

      Figure 2.  (color online) Density distributions of the Λ hyperon on the orbitals Λ[101]32 or Λ[101]12 (top-left pannel), Λ[110]12 (top-right panel), Λ[202]52+ or Λ[202]32+ (bottom-left panel), and Λ[220]12+ (bottom-right pannel). The deformation parameter β of each corresponding nuclear core is constrained to the average, ˉβ, of the bandhead given by the GCM calculation.

      Figure 3 gives the projected PESs, E(β,J,K), on each angular momentum J for 36Ar. In this figure, all the ND and SD energy minima of the Jπ=0+ PES are more obvious than those of the mean-field PES, which is due to the energy gained from the restoration of rotational symmetry. The energy levels derived by the GCM are also given in the same figure, and we can clearly see that there are two rotational bands for the ND states and the SD states of 36Ar. In this current calculation, the bandhead of the ND band corresponds to the ground state (g.s.), whereas that of the SD band is the 4th 0+ state. The bandheads of the ND and SD bands are located at –308.02 MeV and –300.96 MeV, respectively, which indicate the binding energy of these two states. Figure 3 also shows that the g.s. and SD bands cross at Jπ=6+, and these calculated results agree with the observed data in Ref. [3]. Besides the ND and SD rotational bands, there are other observed low-lying energy levels such as the 4.4 MeV 2+ level [3]; however, these levels may involve two-quasi-particle (2-qp) excitations [2]. 2-qp excitations are beyond the basis space of this current model; therefore, only the properties of the g.s. and SD bands are discussed hereafter.

      Figure 3.  (color online) Projected PESs, E(β,J), and the GCM energy levels of 36Ar. The angular momentum and parity for each projected PES are given in the legend, and the mean-field PES labeled by 'MF' is also shown for comparison. The solid bullets and horizontal bars indicate the GCM energy levels, which are plotted at their average deformation ˉβ.

      The projected PESs of 36ArΛ[000]12+ are given in Fig. 4, and the GCM energy levels of the g.s. and SD bands are also shown in the same figure. The addition of Λ[000]12+ turns the Jπ=0+ states of 36Ar into those with Jπ=12+ and turns the other states into spin doublets. The band-head energies of the g.s and SD bands are –327.22 MeV and –319.79 MeV, respectively. Therefore, we can deduce that the Λ separation energy, SΛ, for the ground state is 19.20 MeV, whereas SΛ for the Jπ=12+ SD state is 18.83 MeV, which indicates that SΛ of the SD state is smaller than that of the ground state. This is in agreement with the mean-field SHF calculation using the SKI4 parameters [6] but conflicts with the RMF calculations [4, 7]. This conflict stems from the fact that the RMF calculations give an SD state with a localized density, which leads to a larger overlap between Λ and the SD nuclear core. This does not occur in the SHF calculation.

      Figure 4.  (color online) Same as Fig. 3, but for 37ΛsAr. The notation Λs denotes a Λ hyperon on the s-shell orbital, that is, Λ[000]12+, and only the GCM energy levels of the g.s. and SD bands are preserved for simplification.

      Figures 5 and 6 give the collective wave functions gJα(β) of the ND-band and SD-band states for 36Ar and 37ΛAr, respectively. These two figures reproduce the observed features of the ND and SD bands qualitatively. For each state of the ND band, it is shown that the maximum of gJα(β) corresponds to β within the ND region; for the SD-band states, all the collective wave functions reach their maxima at β0.65, which indicates an SD shape. It is also shown that, compared to those of 36Ar, the s-shell Λ slightly pulls the gJα(β) of the hypernuclear states in 37ΛAr toward the spherical shape, which leads to a reduction in deformation.

      Figure 5.  (color online) Collective wave functions gJα(β) for ND-band states in 36Ar and 37ΛAr. The angular momenta, J, are shown in the legend of each panel.

      Figure 6.  (color online) Same as Fig. 5, but for the SD-band states in 36Ar and 37ΛAr.

      To discuss the reduction in deformation more comprehensively, the g.s. and SD bands of 36Ar and 37ΛAr in Figs. 3 and 4 are extracted and shown in Fig. 7, where the observed data [34, 35] on 36Ar are also given for comparison. Until Jπ=4+, the current model reproduces the observed g.s. band of 36Ar very well. Moreover, the intervals between the energy levels of the calculated SD band are in good agreement with the observed data. However, the current model gives a significantly higher Jπ=6+ energy level for the g.s. band than the observed value, and the predicted SD band is nearly 3 MeV greater than the experimental data. This disagreement with the observed data likely stems from the fact that quasi-particle excitations are absent in the GCM space of our model. In Fig. 7, it is also shown that the addition of Λ[000]12+ shifts the excited energy levels of 37ΛAr slightly upward compared to those of 36Ar because the moment of inertia of the nuclear core is reduced owing to the reduction in defromation. The reduction in deformation is also indicated by the intra-band B(E2) values of 37ΛAr in comparison with those of 36Ar. As shown in Fig. 7, B(E2,52+12+) in the g.s. band of 37ΛAr is reduced by 9% compared to B(E2,2+0+) in the g.s. band of 36Ar, whereas for the SD band of 37ΛAr,B(E2,52+12+) is reduced by approximately 5%. The contrast between the reductions in the B(E2) values of the g.s. band and those of the SD band indicates that deformation of the SD band is more stable than that of the g.s. band. For the g.s. band, the stability of deformation is also indicated by the PESs given in Fig. 1, which show that there are two ND energy minima and the barrier between them is less than 1.5 MeV. Furthermore, the collective wave functions in Fig. 5 show that the two ND energy minima compete with each other. Finally, it is deduced that shape deformation in the g.s. band is relatively soft and unstable. Note that the beyond-mean-field RMF calculation provides a clearer reduction in the B(E2) values in 37ΛAr [7]. This is because the relativistic NΛ interaction is stronger than the nonrelativistic interaction, as verified in Refs. [6, 9].

      Figure 7.  (color online) ND and SD rotational bands of 36Ar and 37ΛsAr compared to the experimental data [34, 35]. The reduced electric quadrupole transition strengths, B(E2), (in units of e2fm4) are provided on the arrows.

      Besides Λ[000]12+, the ND and SD nuclear cores, denoted as 36ArN and 36ArS in this paper, may couple with a Λ hyperon on the p- or sd-shell orbitals, and each specific combination corresponds to a certain configuration of 37ΛAr. Energy levels corresponding to these configurations, which are lower than the separation threshold of 36Ar+Λ, are given in Fig. 8. It is shown that the configurations 36ArNΛ[110]12, 36ArNΛ[101]32, and 36ArNΛ[101]12 give three rotational bands with bandheads at 11.13 MeV, 9.02 MeV, and 8.80 MeV, respectively. Because the ground state of 36Ar is oblate, the configurations 36ArNΛ[101]32 and 36ArNΛ[101]12 give lower energies than the configuration 36ArNΛ[110]12. However, for the SD state, the configuration 36ArSΛ[110]12 gives a rotational band with the bandhead at 13.70 MeV, which is approximately 5 MeV lower than the energy levels given by the configurations 36ArSΛ[101]32 and 36ArSΛ[101]12. This is because the SD state of 36Ar is prolately deformed, and its coupling with Λ[110]12 produces lower energies. Figure 8 also shows that, although the configurations 36ArSΛ[101]32 and 36ArSΛ[101]12 both give bound states, their energy levels are near the separation threshold of 36Ar+Λ. In our earlier research [10], the beyond-mean-field SHF model successfully reproduced the negative-parity energy levels of 9ΛBe, which were denoted as a genuine hypernuclear state and 9Be-analogue. Furthermore, the same model predicted the negative-parity levels of 13 ΛC, which was in good agreement with the observed values [11]. Therefore, the beyond-mean-field SHF model is powerful, and the predicted energy levels in Fig. 8 for the p-shell Λ are reliable and can help identify the negative-parity levels of 37ΛAr in future experiments.

      Figure 8.  (color online) Comprehensive energy spectra for 37ΛAr. Configurations corresponding to each rotational band are indicated by the solid box and dashed arrow. Energy levels higher than the separation threshold of 36Ar + Λ are neglected for simplification.

      As deduced in Sec. I, 37ΛAr may provide bound states for configurations such as 36ArsdΛ. Thus, in future experiments, theoretical predictions for the energy levels of 36ArsdΛ are crucial for identifying the bound state near the separation threshold of 36Ar+Λ. The current calculation finds bound states for configurations 36ArΛ[220]12+, 36ArΛ[202]52+, and 36ArΛ[202]32+, the energy levels of which are shown in the top right corner of Fig. 8. For 36ArNΛ[220]12+ and 36ArNΛ[202]52+, only the bandheads with Jπ=12+ and 52+ are lower than the separation threshold, and their energies are 18.45 MeV and 17.91 MeV, respectively. For 36ArNΛ[202]32+, the bandhead is with Jπ=32+, and its energy is 17.94 MeV. For the same configuration, there is another bound state with Jπ=52+; however, the energy level is only 0.24 MeV lower than the separation threshold.

      The mean-field PESs of the three configurations discussed above are shown in Fig. 1. It is found that Λ[220]12+ enhances the energy minima on the prolate side of the PES and weakens the oblate one, whereas Λ[202]52+ and Λ[202]32+ exhibit opposite effects. Similar to cases of the p-shell Λ, impurity effects of those in the sd-shell mainly stem from their density distributions, which are shown in Fig. 2. For Λ[220]12+, the main component of its wave function is Y2,0(θ,φ) and it is elongated along the symmetry axis (shown in the bottom-right panel of Fig. 2); hence, it prefers a prolate nuclear core. Conversely, for Λ[202]52+ and Λ[202]32+, the main components of their wave functions are Y2,±2(θ,φ); therefore, they are both compressed in the direction of the symmetry axis (shown in the bottom-left panel of Fig. 2) and prefer oblate nuclear cores. As shown in Fig. 8, this type of effect also causes the GCM energy levels of the configurations 36ArNΛ[202]52+ and 36ArNΛ[202]32+ to be approximately 0.5 MeV lower than that of 36ArNΛ[220]12+. This is because, for all three configurations, the nuclear core 36ArN is oblately distributed, and its coupling with Λ[202]52+ or Λ[202]32+ leads to relatively lower energies.

    IV.   CONCLUSIONS
    • In summary, a beyond-mean-field SHF model that includes AMP techniques and the GCM is introduced in this paper. Based on this model, a comprehensive energy spectrum of 37ΛAr is given and researched in detail. Attention is mainly paid to the bound states that are formed by an ND or SD nuclear core coupled with a Λ hyperon occupying one of the s-, p-, or sd-shell orbitals. For the s-shell Λ hyperon, ND and SD bands corresponding to the configurations 36ArNΛ[000]12+ and 36ArSΛ[000]12+ are given. The intra-band B(E2) values of these two bands decreased compared to those of 36Ar, which is due to the reduction in deformation. For the p-shell Λ hyperon, three ND bands and one SD band are shown, which correspond to the configurations 36ArNΛ[110]12, 36ArNΛ[101]32, 36ArNΛ[101]12, and 36ArSΛ[110]12, respectively. Moreover, the configurations 36ArSΛ[101]32 and 36ArSΛ[101]12 give two bound states, the energy levels of which are below the separation threshold of 36Ar+Λ within 0.5 MeV. For the sd-shell Λ hyperon, three configurations 36ArNΛ[220]12+, 36ArNΛ[202]52+, and 36ArNΛ[202]32+ give bound states with energy levels lying at 18.45 MeV, 17.91 MeV, and 17.94 MeV above the ground state, respectively. Finally, it is expected that the predictions in this paper will aid future experiments in identifying the observed states of 37ΛAr.

    ACKNOWLEDGMENTS
    • One of the authors thanks Xian-Ye Wu for informative discussions.

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