Cluster-orbital shell model approach for unstable nuclei and the developments

Transcription

1 CANHP2015 (week 6), Oct. 2015, Kyoto, Japan Cluster-orbital shell model approach for unstable nuclei and the developments Hiroshi MASUI Kitami Institute of Technology

2 Outline of my talk 1. Cluster-orbital shell model approach 2. Comparison with Gamow shell model 3. Radius of oxygen isotopes

3 0. Basic motivation

4 Nuclear structure and the size Halo nuclei I. Tanihata, H. Savajols, R. Kanungo, PPNP68 (2013)

5 The typical halo nuclei: 11 Li ( 9 Li+2n) Loosely bound valence-nucleon above the core nucleus n n 9 Li I. Tanihata, H. Savajols, R. Kanungo, PPNP68 (2013)

6 We want to study these nuclei on the same footing Core Valence Core Valence Cluster-orbital shell model (COSM) Y. Suzuki and K. Ikeda, PRC38 (1988)

7 1. Cluster-orbital shell model

8 Cluster-orbital shell model approach Hamiltonian CoM motion is removed from the Hamiltonian Y. Suzuki and K. Ikeda, PRC38 (1988) Ĥ = ĥ i + i i< j ( p i p j A C + v ij ) r 2 Core-N (1-body) N-N (2-body) r 1 r N An/-symmetriza/on with the nucleons in the core Core nucleus Orthogonality Condi/on Model (OCM) ˆΛ λ F.S. F.S. λ (F.S. : Pauli-forbidden states)

9 Basis function (For N-valence nucleon system) Eigenfunction: Gaussian expansion (GEM) Not eigenfunctions of the core+n sub-system Width parameter: Geometrical progression E. Hiyama, Y. Kino, and M. Kamimura, Prog. Part. Nucl. Phys. 51, 223 (2003). S. Aoyama, T. Myo, K. Kato, and K. Ikeda, Prog. Theor. Phys. 116, 1 (2006). T. Myo, Y. Kikuchi, H. M and K. Kato, Prog. Part. Nucl. Phys. 79, 1 (2014).

10 Combined with the Complex scaling method (CSM) Complex-scaled eigenvalues on the physical Riemann sheet T. Myo, Y. Kikuchi, H. M and K. Kato, Prog. Part. Nucl. Phys. 79, 1 (2014).

11 2. Comparison with Gamow shell model

12 Complex k-plane Im.k Bound states Continua Re. k Anti-bound states (Virtual states) Resonant states

13 Complex scaling method Resolution of the identity

14 Gamow shell model approach N. Michel, W. Nazarewicz, M. Płoszajczak, and K. Bennaceur, Phys. Rev. Lett. 89, (2002) R. Id Betan, R. J. Liotta, N. Sandulescu, and T. Vertse, Phys. Rev. Lett. 89, (2002) N. Michel, W. Nazarewicz, M. Płoszajczak, and J. Okołowicz, Phys. Rev. C 67, (2003) N. Michel, W. Nazarewicz, and M. Płoszajczak, Phys. Rev. C70, (2004). K. Fossez, N. Michel, M. Płoszajczak, Y. Jaganathen, and R. M. Id Betan Phys. Rev. C 91, (2015)

15 Gamow shell model Single-particle states Berggren basis G. Papadimitriou, J. Rotureau, N. Michel, M. Płoszajczak,and B. R. Barrett, Phys. Rev. C 88, (2013).

16 Basis function GEM with CS and GSM GEM with CS GSM

17 2-Steps for the comparison with GSM H.M, K. Kato, K. Ikeda, PRC75(2007) Prepare the single-particle complete set with the complex scaling 2. Expand the many-body state with the complex-scaled single-particle states

18 Comparison with the Gamow shell model Two examples: H.M, K. Kato, K. Ikeda, PRC75(2007) O ( 16 O+2n) : Stable nucleus Bound, narrow resonant states, continua 6 He ( 4 He+2n) : Halo nucleus Resonant states, continua

19 Interac/on Oxygen systems Core N: Semi-microscopic poten/al [3] Adjusted to 17 O (5/2 +, 1/2 +, 3/2 + ) N-N : Volkov No.2 [4] (M=0.58, B=H=0.07) Adjusted to 18 O (0 + ) Size parameter of the core 16 O core (h.o.): fixed size as R rms ( 16 O) = 2.54 fm (b = fm) [3] T. Kaneko, M. LeMere, and Y. C. Tang, PRC44 (1991) [4] A. B. Volkov, NPA74 (1965).

20 Results for the oxygen isotopes Energy and R rms of O Energy levels of 18 O GSM: N. Michel, W. Nazarewicz, M. Płoszajczak, and J. Okołowicz, Phys. Rev. C 67, (2003)

21 Energy levels of 19 O Energy levels of 20 O GSM: N. Michel, W. Nazarewicz, M. Płoszajczak, and J. Okołowicz, Phys. Rev. C 67, (2003)

22 Expansion with the single-particle states for 18 O Although the NN-int. are different, COSM and GSM give almost the same result.

23 Interac/on Helium systems Core N: Semi-microscopic poten/al [5] Reproduce the phase-shie of 4 He+n N-N : Minnesota [6] (u=1.0) Effec/ve 3-body force[7] Adjusted to 6 He ground state [5] H. Kanada, T. Kaneko, S. Nagata, andm. Nomoto, Prog. Theor. Phys. 61, 1327 (1979). [6] D. R. Thompson, M. LeMere, and Y. C. Tang, Nucl. Phys. A286, 53 (1977). [7] T. Myo, K. Kato, S. Aoyama, and K. Ikeda, Phys. Rev. C 63, (2001).

24 Results for the Helium isotopes Energy of Helium isotopes R rms of Helium isotopes GSM[14]: N. Michel, W. Nazarewicz, M. Płoszajczak, and J. Okołowicz, Phys. Rev. C 67, (2003) GSM[19]: G. Hagen, M. Hjorth-Jensen, and J. S. Vaagen, Phys. Rev. C 71, (2005). [42] F. Ajzenberg-Selove, NPA490, 1 (1988). [43] I. Tanihata, NPA478, 795c (1998). [44] I. Tanihata et al., PLB289, 261 (1992). [45] G. D. Alkhazov et al., PRL78, 2313 (1997).

25 Expansion with the single-particle states for 6 He (p 3/2 )-contribution for 6-8 He GSM[14]: N. Michel, W. Nazarewicz, M. Płoszajczak, and J. Okołowicz, Phys. Rev. C 67, (2003) GSM[19]: G. Hagen, M. Hjorth-Jensen, and J. S. Vaagen, Phys. Rev. C 71, (2005).

26 (p 3/2 )- and (p 1/2 )-contributions for 6 He (L max =5) (L=1) (L=1) When the model space is the same, results becomes similar GSM[19]: G. Hagen, M. Hjorth-Jensen, and J. S. Vaagen, Phys. Rev. C 71, (2005).

27 C n 2 = (p 3 / 2 ) 2 R rms E C n 2 = (p 1/ 2 ) 2 L max =5 L max =5 ϕ ϕ : L = j ±1/2 Ψ = C n Φ n Although the number of channel increases, contributions of (p 3/2 ) 2 and (p 1/2 ) 2 are almost the same

28 Correlation of valence nucleons Energy of 6 He V-base ECM (T-base) is important S. Aoyama et al. PTP93 (1995) T-base

29 Details of poles and continua 2 C n = Pole + S1+ S2 (p 3/2 ) 2 Pole C n 2 = (p 3 / 2 ) 2 (p 1/2 ) 2 Pole S2 C n 2 = (p 1/ 2 ) 2 S2 S1 S1

30 Precise comparison between GEM with CS and GSM

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32 Precise comparison between two criteria [Gaussian expansion/slater basis] with complex scaling Gamow shell model

33 Hamiltonian One-body part Two-body part KKNN-poten/al ˆΛ i F.S. F.S. i

34 Interac/on 4 He+pp/nn systems Core N: Semi-microscopic poten/al [5] Reproduce the phase-shie of 4 He+n N-N : Minnesota [6] (u=1.0) Effec/ve 3-body force[7] Adjusted to 6 He ground state Core-N Coulomb pot. [5] H. Kanada, T. Kaneko, S. Nagata, andm. Nomoto, Prog. Theor. Phys. 61, 1327 (1979). [6] D. R. Thompson, M. LeMere, and Y. C. Tang, Nucl. Phys. A286, 53 (1977). [7] T. Myo, K. Kato, S. Aoyama, and K. Ikeda, Phys. Rev. C 63, (2001).

35 TBMEs Gaussian/Slater basis (basis functions are analytic function) TBMEs can be calculated analytically Gamow shell model (sinle-particle states are solved numerically) Numerically obtained w.f. : Expanded with analytical func. Re.k Im. k

36 Energy of 6 He (0 + and 2 + ) H. M, K. Kato, N. Michel, M. Płoszajczak, Phys. Rev. C 89 (2014)

37 Energy of 6 Be (0 + and 2 + ) H. M, K. Kato, N. Michel, M. Płoszajczak, Phys. Rev. C 89 (2014)

38 Convergence of the calculation and the related parameters Complex rota/on/berggren basis func/on {α+iβ} {α+iβ} {α-iβ} {α+iβ} Non-Hermi/an (complex symmetric Hamiltonian matrix elements) varia/onal problem Generalized varia/onal problem Ĥ Ĥ Model space Model space

39 Variational parameters GEM + CS Gaussian width parameters: b, N max Complex rotation angle:θ GSM Discretization of continuum and the contour Parameters in H.O. expansion: hω, N max

40 Convergence of resonant poles of 6 He (2 + )

41 Nuclear three-body problem in the complex energy plane: Complex-scaling Slater method A. T. Kruppa, G. Papadimitriou,W. Nazarewicz, and N.Michel,Phys. Rev. C 89, (2014).

42 3. Radius of oxygen isotopes

43 Matter radius of nuclei near the drip-lines A. Ozawa (2001)

44 19 C 11 Li 6 He Relation for the S n and (R rms /A 1/3 ) 11 Be 23 O 26 F 8 He 24 O Large S n Large R rms F-isotopes O-isotopes C-isotopes

45 Reaction cross-section of O-isotopes A. Ozawa et al. NPA693 (2001) R. Kanungo et al. PRC84 (2011) R. Kanungo et al. PRC84 (2011)

46 16 O+XN systems Interac/on H. M, K. Kato and K. Ikeda, PRC73, (2006) Core N: Semi-microscopic poten/al [3] Adjusted to 17 O (5/2 +, 1/2 +, 3/2 + ) N-N : Volkov No.2 [4] (M=0.58, B=H=0.07) Adjusted to 18 O (0 + ) Size parameter of the core 16 O core (h.o.): fixed size as R rms ( 16 O) = 2.54 fm (b = fm) [3] T. Kaneko, M. LeMere, and Y. C. Tang, PRC44 (1991) [4] A. B. Volkov, NPA74 (1965).

47 Core-size dependence 16 O-core Core-N Core configuration: h.o. wave function E(b) = E(b) Core + E(b) Valence Op/mum b might be different in isotopes/isotones

48 16 O-core part Energy of 16 O-core with effective NN-int. [5] Energy of 16 O (additional 3-body force) Volkov No.2 (M=0.58) [5] T. Ando, K. Ikeda, and A. Tohsaki-Suzuki, PTP64 (1980).

49 The core-n part Core-N Hamiltonian: Change of b-parapeter: Strength of the potential

50 Inclusion of the dynamics of the core: R rms are improved H. M, K. Kato and K. Ikeda, PRC73, (2006)

51 Our approaches I) Role of many valence neutrons 16 O+Xn model m-scheme COSM + Gaussian basis II) Role of last one- or two-neutrons Core + n or Core +2n model Coupled-channel model for the core

52 16 O+XN systems Interac/on H. M, K. Kato and K. Ikeda, EPJA42 (2009) Core N: Semi-microscopic Adjusted to 17 O (5/2 +, 1/2 +, 3/2 + ) N-N : Volkov No.2 (M=0.58, B=H=0.07, 0.25) Adjusted to 18 O (0 + ), drip-line at 25 O Size parameter of the core 16 O core (h.o.): fixed size, A 1/6

53 Wave function Φ = m Radial part c m F m (r 1,r 2,...,r N ) (M M T ) m F m (r 1,r 2,...,r N ) = ( g(r 1 ) g(r 2 ) g(r N )) m Product of Gaussian with polynomial Spin-isospin part (M M T ) m Total M and M T are fixed M = m 1 + m m N M T = m T1 + m T m TN We check the expectation value of the total J as <J 2 >

54 B=H=0.07 S n of O-isotopes B=H=0.25 < B=H=0.07 Amrac/on

55 R rms of O-isotopes B=H=0.07 B=H=0.25 Fixed-core Expanded-core b~a 1/6 b: (fm)

56 Comparison with other approaches Expr.: A. Ozawa et al, NPA693 (2001) : m-cosm with b A 1/6 : fixed-b b A 1/6 Fixed-b : H. Nakada, NPA764 (2006) : B. Ab-Ibrahim et al., JPSJ 78 (2009)

57

58 A schematic figure to illustrate the change of the radius of 22 O (fm) 22 O 23 O 24 O R rms [1] 2.88± ± ±0.13 [1] A. Ozawa et al, NPA693 (2001)

59 Inclusion of the core excitation a coupled-channel picture for 16 O H.M, K. Kato, K. Ikeda, NPA895 (2012) 0p-0h 1/ 6 b A 2p-2h ex. b : Fixed Radius : large (0s) 4 (0p) 12 Radius : small [(1s 1/2 ) π (1s 1/2 ) ν ] S=1,T=0 Mean-field-like core [(0 p 1/2 ) π (0p 1/2 ) ν ] S=1,T=0

60 2p-2h ex. b : Fixed TOSM 4P-1 (Myo) Tensor-op/mized shell model T. Myo et al.,ptp117 (2007) Coupling to the higher orbits High-momentum components Radius : small Small b-parameter Op/mized b-parameter for 4 He

61 Inclusion of the core excitation TOSM in 9 Li T. Myo, K. Kato, H. Toki and K. Ikeda, PRC76(2007) 1. Different size for each orbit 2. Specific configura/ons are suppressed due to the Pauli-blocking effect

62 [2p2h excita/ons] 0d 3/2 1s 1/2 0d 5/2 0d 3/2 1s 1/2 0d 5/2 [XpXh excita/ons] 0d 3/2 1s 1/2 0d 5/2

63 A coupled-channel approach Hamiltonian: " $ # T 1 +V 1 Δ 12 T 2 +V 2 + ΔE Δ 21 %" ' $ &# φ 1 φ 2 % " ' = E φ 1 $ & # φ 2 % ' & 1ch (0p0h) 2ch (2p2h) b = 1.1A 1/6 (fm) 1s 1/2 is allowed b = 1.5(fm) : fixed 1s 1/2 is forbidden Coupling term Δ 12 = Δ 21 = -2MeV Energy difference of the core ΔE = 2.5, 5, 7.5, 10 (MeV)

64 17 O (5/2 + ) 0d 3/2 0d 3/2 1s 1/2 1s 1/2 0d 5/2 0d 5/2 19 O (5/2 + ) 16 O-core (0p0h) 16 O-core (2p2h) 0d 3/2 0d 3/2 1s 1/2 1s 1/2 0d 5/2 0d 5/2 18 O-core (0p0h) 18 O-core (2p2h)

65 21 O (5/2 + ) 0d 3/2 0d 3/2 1s 1/2 1s 1/2 0d 5/2 0d 5/2 20 O-core (0p0h) 20 O-core (2p2h) 23 O (1/2 + ) Pauli-forbidden 0d 3/2 0d 3/2 1s 1/2 1s 1/2 0d 5/2 0d 5/2 22 O-core (0p0h) 22 O-core (2p2h)

66 Results for a coupled-channel core+n model approach S n (MeV) Fimed 20 O+n 16 O+n 18 O+n S n are almost reproduced 22 O+n

67 Results for a coupled-channel core+n model approach R rms (fm) A. Ozawa et al. NPA693 (2001) R. Kanungo et al. PRC84 (2011) Mean-field-like core 16 O+n 18 O+n 20 O+n 22 O+n Fixed core

68 Summary Comparison between GEM+CS and GSM Different procedures for preparing the complete set of the basis function They give numerically the same result (at least) for core+2n systems Generalized variational parameters should be optimized carefully Rrms 23 O and 24 O 16 O(fixed size) +Xn : failed to reproduce the experiment Modification of the core is important [An attempt to reproduce the Rrms] Coupled-channel picture (Shrunk core) + (Broad core) is a possible way