Deformation of hypernuclei studied with antisymmetirzed molecular dynamics
aa r X i v : . [ nu c l - t h ] A p r APS/123-QED
Deformation of hypernuclei studied with antisymmetirzedmolecular dynamics
M. Isaka
Department of Cosmosciences, Graduate School of Science,Hokkaido University, Sapporo 001-0021, Japan
M. Kimura
Creative Research Institution (CRIS),Hokkaido University, Sapporo 060-0810, Japan
A. Dote
Institute of Particle and Nuclear Studies,KEK, Tsukuba, Ibaraki 305-0801, Japan
A. Ohnishi
Yukawa Institute for Theoretical Physics,Kyoto University, Kyoto 606-8502, Japan (Dated: October 8, 2018)
Abstract
An extended version of the antisymmetrized molecular dynamics to study structure of p - sd shellhypernuclei is developed. By using an effective Λ N interaction, we investigate energy curves of Be,
C and , Ne as function of nuclear quadrupole deformation. Change of nuclear deformationcaused by Λ particle is discussed. It is found that the Λ in p -wave enhances nuclear deformation,while that in s -wave reduces it. This effect is most prominent in C. The possibility of the parityinversion in
Ne is also examined.
PACS numbers: Valid PACS appear here . INTRODUCTION One of the unique and interesting aspects of hypernuclei is the structure change causedby the hyperon as an impurity. Many theoretical works have suggested such phenomenacaused by Λ particle as the change of deformation [1–5], the shrinkage of the inter-clusterdistance [1, 2] and the super-symmetric (genuine) hypernuclear state [2, 6–9]. Owing to theexperimental developments some of them have been observed in light p -shell hypernuclei.As examples, we can refer to the reduction of B( E
2) in Li [10] and the identification of thesuper-symmetric (genuine) hypernuclear state in Be [11–13].Today, we can expect that a new experimental facility of Japan Proton Accelerator Re-search Complex (J-PARC) will reveal the spectral information of p - sd shell and neutron-richhypernuclei. Since these normal nuclei have a variety of structure such as coexistence of shelland cluster structure [14–16] and novel exotic clustering [17–21], there must be many in-teresting phenomena peculiar to hypernuclei to be found. Indeed, several pioneering workspredicted exotic structure in sd -shell hypernuclei such as the parity inversion in Ne [7] andvarious rotational bands in
Ne [22]. These works are based on rather limited knowledgeon the ΛN interaction. Since our knowledge of ΛN interaction has been greatly increased bythe recent theoretical and experimental efforts [13, 23–27], we are now able to perform morequantitative and systematic study of the structure change in Λ hypernuclei. It will revealthe dynamics and many interesting aspects of baryon many-body problem.To perform systematic and quantitative study of sd -shell and neutron-rich Λ hypernu-clei, we develop an extended version of the antisymmetrized molecular dynamics (AMD)[18, 28–31], which we shall call HyperAMD. AMD has been successful to describe variousexotic structure of neutron-rich nuclei and highly excited states of stable nuclei. Therefore,HyperAMD is suitable to describe the structure change and exotic structure of hypernuclei.In this study, we introduce HyperAMD and focus on the change of nuclear quadrupoledeformation caused by a Λ particle. By applying HyperAMD to Be, C, Ne and
Newith YNG-ND ΛN interaction [32], it is found that Λ particle changes nuclear quadrupoledeformation. While the Λ particle in s -wave reduces quadrupole deformation as expected,that in p -orbital increases it. Among the calculated hypernuclei, C has shown the mostdrastic change of the nuclear deformation. It is also found that the binding energy of Λparticle depends on the structure of the core nucleus. Namely, the Λ in s -wave coupled2o the deformed core nucleus has smaller binding than that coupled to spherical core. Onthe contrary, the Λ in p -wave coupled to the deformed core has larger binding than thatcoupled to the spherical core. This contradicts to the preceding study [7] in which the parityinversion of Ne is predicted.This paper is organized as follows. In the next section, we explain the theoretical frame-work of HyperAMD. In the section III, the change of energy curves as function of quadrupoledeformation are presented. The trend of the change and its origin are discussed. The finalsection summarizes this work.
II. THEORETICAL FRAMEWORK
In this section, we introduce the theoretical framework of HyperAMD. Compared to thecoupled channel AMD which describes the multi-strangeness system [33], it has better de-scription of the hyperon single particle wave function, but it does not treat multi-strangenessand is limited to single Λ hypernuclei.
A. Wave function
A single Λ hypernucleus consists of A nucleons and a Λ particle is described by the wavefunction that is the eigenstate of the parity,Ψ ± = ˆ P ± Ψ int , (1)where ˆ P ± is the parity projector, and the intrinsic wave function Ψ int is given by the directproduct of the Λ single particle wave function ϕ Λ and the wave function of A nucleons Ψ N ,Ψ int = ϕ Λ ⊗ Ψ N . (2)The nuclear part is described by a Slater determinant of nucleon single particle wave packets,Ψ N = 1 √ A ! det { ψ i ( r j ) } , (3) ψ i ( r j ) = φ i ( r j ) · χ i · η i , (4) φ i ( r ) = Y σ = x,y,z (cid:18) ν σ π (cid:19) / exp (cid:26) − ν σ (cid:0) r − Z i (cid:1) σ (cid:27) , (5) χ i = α i χ ↑ + β i χ ↓ , (6) η i = proton or neutron , (7)3here ψ i is the i -th nucleon single-particle wave packet consists of spatial φ i , spin χ i andisospin η i parts. The spatial part φ i is represented by a deformed Gaussian. Its centroid Z i is a complex valued three-dimensional vector. The width parameters ν σ are real numbersand take independent value for each direction, but are common to all nucleons. The spinpart is parameterized by the complex parameters α i and β i , and the isospin part is fixed toproton or neutron. Z i , ν σ , α i and β i are the variational parameters of the nuclear part.To describe various wave functions of the Λ particle, the Λ single particle wave functionis represented by the superposition of Gaussian wave packets, ϕ Λ ( r ) = M X m =1 c m ϕ m ( r ) , ϕ m ( r ) = φ m ( r ) · χ m , (8) φ m ( r ) = Y σ = x,y,z (cid:18) ν σ ρπ (cid:19) / exp (cid:26) − ν σ ρ (cid:0) r − z m (cid:1) σ (cid:27) , (9) χ m = a m χ ↑ + b m χ ↓ , (10) ρ ≡ m Λ m N . (11)Again, each wave packet is parametrized by the centroid of Gaussian z m , the spin direction a m and b m . z m , a m , b m and c m are the variational parameters of the Λ part. The widthparameter ν σ are equal to those of nuclear part. The number of the superposition M istaken to be large enough to have the energy convergence of the variational calculation. B. Hamiltonian
The Hamiltonian used in this study is given asˆ H = ˆ T N + ˆ V NN + ˆ V Coul + ˆ T Λ + ˆ V Λ N − ˆ T g . (12)Here, ˆ T N , ˆ T Λ and ˆ T g are the kinetic energies of nucleons, Λ particle and the center-of-massmotion. Since we have superposed Gaussian wave packets to describe the Λ single particlewave function, it is rather time consuming to remove the spurious motion of the center-of-mass exactly. To reduce the spurious motion, we keep the center-of-mass of wave packets atthe origin of the coordinate, A X i =1 Z i + M X m =1 √ ρ z m = 0 . (13)4e expect that the spurious energy is not large in sd -shell hypernuclei, since the numberof nucleons is much larger than the hyperon. A similar method is also applied in the otherAMD studies [28].Our model wave function is designed to describe the low-momentum phenomena as in thecase of the conventional shell model and we shall use the low-momentum effective interaction.We have used the Gogny D1S interaction [34] as an effective nucleon-nucleon interaction ˆ V NN ,that has been successfully applied to the stable and unstable normal nuclei. As an effectiveΛN interaction, we have used the central part of the YNG-ND interaction [32]. The YNG-NDinteraction depends on the nuclear Fermi momentum k F through the density-dependence ofthe G-matrix in nuclear medium. In this work, we apply respectively k F =1.14 and 1.17 fm − for Be and
C, that are so determined to approximately reproduce the binding energy of Λin s -wave. For Ne and
Ne, we apply the same value as
C, since there is no experimentaldata. The Coulomb interaction is approximated by the sum of seven Gaussians.
C. Frictional cooling method with constraints
Using the frictional cooling method, the variational parameters of the model wave func-tion are so determined that the total energy is minimized under the constraints. We haveimposed two constraints on the variational calculation. The first is on the nuclear quadrupoledeformation parameter β that is achieved by adding the parabolic potential, h ˆ V β i = v β ( β − β ) , (14)to the total energy. Here β denotes the quadrupole deformation of the nuclear wave functionΨ N [28]. The deformation of nuclear part becomes equal to β after the variation. It is notedthat there are no constraint on the nuclear quadrupole deformation γ and the deformationof the Λ single particle wave function. They have the optimum value after the variationalcalculation for each given value of β .Another constraint is on the Λ single particle wave function,ˆ V s = Λ | ϕ s ih ϕ s | , (15) h r | ϕ s i = exp {− ρ ¯ νr } , (16)¯ ν = √ ν x ν y ν z (17)5y using sufficiently large value for Λ, this potential forbids the Λ in s -wave. Therefore, byswitching off and on this potential, we respectively obtain the hypernuclear state in whicha Λ particle dominantly occupies in s - and p -waves.The total energy plus constraint potentials, E ′ = h Ψ ± | ˆ H | Ψ ± ih Ψ ± | Ψ ± i + h ˆ V s i + h ˆ V β i (18)is minimized using the frictional cooling method. The imaginary time development equationsof the variational parameters are given as, dXdt = µ ~ ∂E ′ ∂X ∗ , (19) X = Z i , z m , α i , β i , a m , b m , ν σ , ν Λ i , (20)where µ is arbitrary negative real number. It is easy to proof that E ′ decreases as timedevelops, and after sufficient time steps we obtain the energy minimum under the constraint.By this method, we obtain the hypernuclear wave function for given β , the total parity andthe Λ single particle orbital. In the present work, Λ dominantly occupies s - or p -wave and weshall use the notation Λ s and Λ p for them. Combined with the parity projection, we obtainfour different configurations in which Λ s and Λ p couple to the positive- and negative-paritystates of the core. They are denoted as Ψ + N ⊗ Λ s , Ψ − N ⊗ Λ s , Ψ + N ⊗ Λ p and Ψ − N ⊗ Λ p in thefollowing. III. RESULTS AND DISCUSSIONSA. general trend of the energy curves
We have performed the variational calculation for Be, C, Ne and
Ne. To illustratethe change of nuclear deformation, Figure 1 shows energy curves of hypernuclear stateswith different configurations and corresponding normal nuclei as functions of deformation β . Each energy curve has an energy minimum shown by the open circle, and the bindingenergies, quadrupole deformation and radius at the minimum are listed in Table I. Thebinding energy of Λ is defined as the difference of energy between the ground state of thecore nucleus and the hypernuclear states, B Λ = B ( A +1Λ X ) − B ( A X g.s. ) (21)6 +3.0MeV +12.0MeV +12.0MeV+4.0MeV+4.0MeV +5.0MeV(a) (c)(e) (g)(f)(d) (b)quadrupole deformation β β β β e n e r gy [ M e V ] e n e r gy [ M e V ] FIG. 1: (color online) Energy curves as function of nuclear quadrupole deformation β for (a) Be,(b)(c)
C, (d)(e)
Ne and (f)(g)
Ne. (a),(b),(d) and (f) compare the positive-parity states ofnormal nuclei (Ψ + N ) with the hypernuclear states of Ψ + N ⊗ Λ s and Ψ + N ⊗ Λ p configurations. (c),(e) and(f) compare the negative-parity states (Ψ − N ) with the hypernuclear states of Ψ − N ⊗ Λ s and Ψ − N ⊗ Λ p configurations. Open circle shows the energy minimum on each curve. Energies of hypernuclei areshifted as shown in the figure for the sake of the presentation. Λ s i ng l e p a r ti c l e e n e r gy [ M e V ] quadrupole deformation β FIG. 2: (color online) The single particle energy of Λ s and Λ p of C as function of the quadrupoledeformation of core nucleus C. Solid (dashed) line shows the energy of Λ s coupled to the positive(negative) parity state of C. Dotted line shows the energy of Λ p coupled to the negative-paritystate of C. As the general trend, the shape of energy curve is not strongly modified by Λ particleexcept for
C, and deformation β at the minima are slightly changed. In all cases, Λ s reducesquadrupole deformation. This is consistent with the cluster model calculations [22, 27]and the (relativistic) mean-field calculations [3–5] that demonstrated the reduction of β by7 ABLE I: Calculated total and Λ binding energies B , B Λ in MeV, quadrupole deformation pa-rameters β and γ , and the root mean square radius in fm at the minimum of each energy curve.Central values of observed energies [12, 13, 35, 36] are also listed in parenthesis. The energies forΛ p states of Be and
C are estimated from the observed excitation energies given in Ref. [13].
B B Λ β γ p h r i Be Ψ + N Be Ψ + N ⊗ Λ s + N ⊗ Λ p C Ψ + N − N C Ψ + N ⊗ Λ s + N ⊗ Λ p − N ⊗ Λ s Ne Ψ + N − N Ne Ψ + N ⊗ Λ s + N ⊗ Λ p − N ⊗ Λ s − N ⊗ Λ p Ne Ψ + N − N Ne Ψ + N ⊗ Λ s + N ⊗ Λ p − N ⊗ Λ s − N ⊗ Λ p Λ s . On the other hand, it is found that Λ p increases β . The magnitude of the change inquadrupole deformation is strongly dependent on the core nucleus, and the most prominentin C in which Λ s makes C core spherical, while Λ p enhances the core deformation. Thereason of the opposite behavior of Λ s and Λ p and the strong dependence on the core nucleus8s clearly seen in the single particle energy of Λ. Figure 2 shows the single particle energy of Λ( ǫ Λ ( β )) in each parity and Λ single particle state in C. Here ǫ Λ ( β ) is defined as the differencebetween the binding energy of C with the deformation β and that of corresponding stateof C with the same deformation, ǫ Λ ( β ) = B C ( β ) − B C ( β ) . (22)It shows the Nilsson-model-like behavior of the Λ single particle energy. The binding of Λ s becomes shallower as deformation becomes larger. In the case of Λ p , Λ occupies the lowest p -wave that comes down as deformation becomes larger. Therefore, Λ s makes quadrupoledeformation smaller and Λ p in the lowest p -wave makes it larger. The Λ single particle energyvaries within a range of 1 ∼ C (FIG. 1 (b)) manifests the drastic change in quadrupole deformation.Since the positive-parity state of C is quite soft against the quadrupole deformation, smallchange in the Λ single particle energy can result in the large modification in quadrupoledeformation. In other cases, change in Λ single particle energy cannot overcome much largervariation of the core energy and results in minor modification of quadrupole deformation.Therefore, we can conclude that the drastic change of quadrupole deformation by Λ particleoccurs when the core nucleus is quite soft against quadrupole deformation within a range of1 ∼ C may depend on the choice of NN interaction and it willbe investigated in our future work. The behavior of the Λ single particle energy is alsounderstood from the density distribution of the Λ particle and the core nucleus as shown inFigure 3. It shows that as nuclear deformation becomes larger the overlap between the Λ s and the core wave function becomes smaller (for example, compare Ψ + N ⊗ Λ s and Ψ − N ⊗ Λ s of Ne). It leads to the reduction of ΛN attraction. On the contrary, larger deformationmakes the overlap larger in the case of Λ p and increases ΛN attraction (see Ψ + N ⊗ Λ p andΨ − N ⊗ Λ p of Ne). In the case of Λ p , larger deformation reduces the kinetic energy that alsocontribute to the deeper binding of Λ p .Another issue to be mentioned is the reduction of the nuclear radius by Λ particle. Inall cases, the radius of nuclear part is reduced, but the reduction (less than 5%) is muchsmaller than in the case of Li (20%) [10]. More detailed discusstion will be made in our9ext work.
B. discussion on each hypernucleus
The calculated total binding energies of Be and Be underestimate the observed values byabout 5MeV. The underestimation is common to all other hypernuclei. It will be resolved byperforming the angular momentum projection and the generator coordinate method (GCM)that are usually performed in the study of normal nuclei by AMD. Indeed, in the case of Ne, it was shown that AMD calculation [15] reproduces the observed binding energy. Theangular momentum projection and GCM will be performed in our next work. Despite ofthe underestimatin of the total binding energy, B Λ of Λ p is comparable with the observedvalue.The density distribution of Λ p in Be (Ψ + N ⊗ Λ p in FIG. 3) clearly shows that this statecorresponds to the supersymmetric (genuine) hypernuclear state [11–13]. The nuclear parthas the pronouced 2 α cluster structure and the Λ occupies the p orbital parallel to thesymmetry axis. It is also interesting to note that Λ s state reduces the inter-cluster distance,while Λ p state increases it. C manifests the drastic change in the quadrupole deformation. The Λ s makes C corespherical, while Λ p state enhances deformation. It is noted that C has the 0 p / subshellclosure configuration at small deformation and 3 α cluster structure develops as deformationbecomes larger. In other words, the nucleon spin is not satureted at small deformation,while that is alomost zero at larger deformation. The sophisticated AMD calculation [14]has shown that the low-lying states of C have mixed nature between the 0 p / subshellclosure configuration and 3 α cluster structure, and the mixing strength is different for eachstate. Since Λ particle changes the deformation and spin property of C, it will haveinfluence on the ΛN spin-orbit splitting of
C [38–40].Based on the cluster model calculation, the parity inversion in
Ne was suggested bySakuda and Bando [7]. The core nucleus Ne has the α + O cluster state ( J π =1 / − ) 238keV above the ground state ( J π =1 / + ) that has ( sd ) shell structure [41]. They concludedthat Λ s coupled to the J π =1 / − state was more deeply bound than that coupled to the goundstate, and the J π =1 / − ⊗ Λ s configuration becomes the ground state of Ne. They arguedthat the J π =1 / − state has dilute α + O cluster structure and by the reduction of the10nter-cluster distance, Λ s gains larger binding energy than the J π =1 / +1 ⊗ Λ s configuration.Our result shows the opposite trend to their result. Since the positive-parity state is moredeformed than the negative-parity state, the binding of Λ s is weaker when it coupled tothe negative-parity state. This trend is common to other calculations including the clustermodel calculation for Ne [22]. However, AMD fails to reproduce small excitation energy ofthe negative-parity state and it does not have prominent α + O clustering, that are mainlydue to the lack of the angular momentum projection and the GCM calculation. We willneed more sophisticated AMD calculation to settle down this problem.The negative-parity state of Ne has larger deformation than the positive-parity state.Therefore, the Λ s coupled to the positive-parity state is more deeply bound than that coupledto the negative-parity state. It is common to other hypernuclei studied here. On thecontrary, Λ p is more deeply bound to the negative-parity state. Since number of nucleonsin Ne is large enough to bound Λ p , we can expect that the addition of Λ will generatea variety of bound rotational bands in Ne as disscussed by Yamada et al. [22]. We willdiscuss
Ne in detail in the forthcoming paper.
IV. SUMMARY
An extended version of AMD named HyperAMD has been introduced to investigate struc-ture of p - sd shell hypernuclei. The energy curves of Be, C, Ne and
Ne as functions ofquadrupole deformation are studied. It has been found that Λ s reduces nuclear deformation,while Λ p increases it. It is due to the variation of the single particle energy of Λ as functionof quadrupole deformation. The binding of Λ s decreases as deformation becomes larger,while that of Λ p increases. The variation of Λ single particle energy is within a range of1 ∼ C is very soft against quadrupoledeformation, it manfests the most prominent change of quadrupole deformation. This trendof the deformation change caused by Λ s and Λ p contradicts to the cluster model calculationfor Ne [7], but is consistent with other calculations. More sphisticated AMD calculation11ill be needed to resolve this disagreement. [1] T. Motoba, H. Band¯o and K. Ikeda, Prog. Theor. Phys. , 189 (1983); , 222 (1984).[2] T. Motoba, H. Band¯o, K. Ikeda, and T. Yamada, Prog. Theor. Phys. Suppl. , Chap. 3(1985).[3] Xian-Rong Zhou, H.-J. Schulze, H. Sagawa, Chen-Xu Wu and En-Guang Zhao, Phys. Rev.C , 034312 (2007).[4] M.T. Win and K. Hagino, Phys. Rev. C , 054311 (2008).[5] H.-J. Schulze, M.T. Win, K. Hagino and S. Sagawa, Prog. Theor. Phys. , 569 (2010).[6] R.H. Dalitz and A. Gal, Phys. Rev. Lett. , 362 (1976).[7] T. Sakuda and H. Band¯o, Prog. Theor. Phys. , 241(1988).[9] T. Yamada, K. Ikeda, H. Band¯o and T. Motoba, Phys. Rev. C , 854 (1988).[10] K. Tanida et al ., Phys. Rev. Lett. , 1982 (2001).[11] P. H. Pile et al ., Phys. Rev. Lett. , 2585 (1991).[12] O. Hashimoto et al ., Nucl. Phys. A , 93c (1998).[13] O. Hashimoto and H. Tamura, Prog. Part. Nucl. Phys. , 564 (2006), and references therein.[14] Y. Kanada-En’yo, Phys. Rev. Lett. , 5291 (1998).[15] M. Kimura, Phys. Rev. C , 044319 (2004).[16] M. Kimura and H. Horiuchi, Nucl. Phys. A , 58 (2006).[17] W. von Oertzen, Z. Phys. A , 37 (1996).[18] Y. Kanada-En’yo, H. Horiuchi and A. Ono, Phys. Rev. C , 628 (1995).[19] N. Itagaki and S. Okabe, Phys. Rev. C , 044306 (2000).[20] P. Descouvemomnt, Nucl. Phys. A , 463 (2002).[21] Y. Kanada-En’yo, Phys. Rev. C , 011303 (2002).[22] T. Yamada, K. Ikeda, H. Band¯o and T. Motoba, Prog. Theor. Phys. , 985 (1984).[23] A. Reuver, K. Holinde and J. Speth, Nucl. Phys. A , 543 (1994).[24] T.H. Rijken and Y. Yamamoto, Phys. Rev. C , 044008 (2006).[25] Y. Fujiwara, Y. Suzuki and C. Nakamoto, Prog. Part. Nucl. Phys. , 439 (2007).
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