Constrained HFB + Local QRPA Constrained HFB + Local QRPA - PowerPoint PPT Presentation

法による Constrained HFB + Local QRPA 法による Constrained HFB + Local QRPA 大振幅集団運動の記述 大振幅集団運動の記述 佐藤 弘一 ( 京大理 / 理研 ) 日野原 伸生 ( 理研 ) 中務 孝 ( 理研 ) 松尾 正之 ( 新潟理 ) 松柳 研一 ( 理研 / 京大基研 )

Introduction  Formulation  ASCC 法の 2 次元への拡張 Constrained HFB+Local QRPA Application  Oblate-Prolate Shape coexistence in Se &Kr Summary  Hinohara et al., arXiv:1004.5544 KS & Hinohara., arXiv:1006.3694

68 Se 日野原さんのトーク： (1+3) 次元の ASCC 法 Application to shape coexistence in Se and Kr N. Hinohara, et al, Prog. Theor. Phys. 119(2008), 59; PRC 80 (2009),014305. (2+3) 次元への ASCC 法の拡張 ・ 2 次元集団多様体の抽出 & 古典的集団 Hamiltonian の決定 ・集団 Hamitonian を再量子化し集団 Schrödinger 方程式を解く q 2 q 1 5D quadrupole collective Hamiltonian (Generalized Bohr-Mottelson Hamiltonian) :      H T T V ( , ) collective potential vib rot 1 1                  2 2 T D ( , ) D ( , ) D ( , )    vib 2 2 vibrational inertial masses 3 1    2 J T rot k k 2 rotational moments of inertia  k 1

Adiabatic Self-consistent Collective Coordinate (ASCC)Method Matsuo, Nakatsukasa, and Matsuyanagi, PTP 103(2000), 959. Adiabatic approx. to SCC method T. Marumori, T. Maskawa, F. Sakata and A. Kuriyama, PTP 64 (1980), 1294.  One can extract the collective degree(s) of freedom the system itself chooses Time-dep. variational principle          ( , , , N ) i H ( , , , N ) 0 q p q p  t Adiabatic expansion ASCC Basic Equations ASCC Basic Equations By courtesy of Hinohara-san

ASCC Basic Equations ASCC Basic Equations (from 0-th order in p) Moving-frame HFB equation moving-frame Hamiltonian Not included in HFB Local harmonic equations (moving-frame QRPA equations) (from 1st-order in p) Not included in QRPA (from 2nd-order in p)

ASCC Basic Equations ASCC Basic Equations Collective Hamiltonian Canonical variable conditions       ˆ ˆ ( ) ( ), N ( ) i q q q HFB 平衡点での moving-frame QRPA 方程式    ( )  ~   ( ) 0 N Gauge-fixing  q i  Hinohara et al., PTP117(2007) 451. 通常の QRPA 方程式

Requantization Requantization Classical kinetic energy: 1   2   T g q q      (  q i 1 , 2 , 3 ) :3 Euler angles  i i + :2 deformation q q Pauli’s prescription 4 5 parameters Quantized kinetic energy: 5D Hamiltonian     2  1     ˆ 1 T G G    2 q q G    Laplacean in a curved space

TDHFB phase space Mapping onto the ( β , γ ) plane 2-dimensional q 2 Collective manifold q x 1  sin  x  sin  正しい境界条件の下、集団 Schrödinger eq. を解くため、 ( β、γ ) 平面への写像を考える

Classical Quadrupole Collective Hamiltonian: Pauli’s prescription General Bohr-Mottelson Hamiltonian (5D quadrupole collective Hamiltonian):

Collective Schrodinger equation: Collective wave function: Normalization:

Derivation of vibrational masses q Vibrational part of collective Hamiltonian 2 q 1 Scale transformation B=1 collective coordinates one-to-one correspondence   ( ) 2 D 2  ( ) D 0  In terms of quadrupole deformation

Vibrational energy written in terms of quadrupole defromation Collective mass can be calculated through Without numerical derivatives

最初から fully self-consistent な 2D ASCC の計算は大変なので・・・ Approximations: q 2 q Curvature terms(1D では寄与小さかった ) を落とす 1 Moving-frame Hamiltonian CHFB Hamiltonian     ˆ   ˆ H ( q , q ) ( q 1 q , ) ( , ) H ( , ) M 1 2 2 CHFB Constrained HFB(CHFB) equation: Local QRPA(LQRPA) equations: Constrained HFB + Local QRPA method

Constrained HFB + Local QRPA method Solve the constrained HFB eq. at each point on the ( β , γ ) plane      V ( , )       ( , ) ( , ) ( , )    Solve the LQRPA eqs. on top of each CHFB state ( , ) Local Thouless-Valatin       2     ˆ ˆ ( , ) ( ) ( ) Q ( , ) P ( , ) eqs. Pick up 2 collective modes from the LQRPA modes obtained above        2 D ( , ) D ( , ) D ( , ) を最小化するペア    Local QRPA mass Calculate the inertial mass functions         J ( , ) D ( , ) D ( , ) D ( , )    k the contribution from the time-odd mean Diagonalize the 5D Quadrupole Hamiltonian to field included calculate the energy spectrum

How to choose the two collective modes ? Take the lowest 40 QRPA modes as candidates.      2 ( , ) 1 st . We assume that the mode with lowest-frequency squared 1 Calculate the collective mass for every pair of the candidates is collective. We regard the pair which minimizes the vibrational part of the 5DQH metric as collective : 2 nd . The other is the mode whose combination with the lowest mode minimizes the vibrational part of the 5DQH metric: Intrinsic vol. element  n   2 ( , )      2 ( , ) 2 collective      2 ( , ) Cranking formula 1 W for α =1 and α =2,3,4 ・・・

Comparison with other 5D Quadrupole Hamiltonian approaches

CHFB ＋ LQRA 法の陽子過剰 Se 、 Kr への適用

Microscopic Hamiltonian P+QQ model: Pairing (monopole, quadrupole) + quadrupole p-h interaction Parameters    ( ) G Monopole pairing & quadrupole int. 0 fitted to the pairing gaps and the quadrupole deformation obtained with Skyrme-HFB by Yamagami et al. Quadrupole pairing : self-consistent value Sakamoto et al. PLB245 (1990) 321 Model space Harmonic oscillator two major shell :Nsh=3, 4 (pf & sdg shells) The s. p. energies: calculated using the modified oscillator potential. M. Yamagami et al.,NPA 693(2001) 579.

Numerical results Collective potential Collective path The absolute min. is oblate. : absolute minimum The 2 nd lowest min. is prolate. The spherical shape is a local maximum.

68Se LQRPA rotational masses            2 2 J ( , ) 4 D k ( , ) sin 2 k 3 k D D D 2 3 1     (LQRPA) (IB) J J ( , ) / ( , ) Ratio to cranking MoI k k (LQRPA) (IB) (LQRPA) (IB) (LQRPA) (IB) J / J J / J J / J 3 3 1 1 2 2

68Se LQRPA vibrational masses         2 D ( , ) D ( , ) / D ( , ) /    Ratio to cranking mass β，γに依存して 数十 % 増加

 4 Collective wave functions squared for 68Se Yrast : oblate character, Yrare: prolate character : β -vibrational 0 2 , 2 3

( ) …B(E2) e 2 fm 4 Excitation Energies and B(E2) effective charge: e n =0.4, e p =1.4 centrifugal effect Reminiscent of γ -unstable situation e.g. B(E2; 6 2 -> 6 1 ) >> B(E2; 6 2 -> 4 1 ) LQRPA mass の方が cranking mass より実験とよい一致

72 Kr LQRPA moments of Inertia Extention of Thouless-Valatin MoI to non-equilibrum HFB pts. (LQRPA) (LQRPA) J J (LQRPA) J 1 2 3            2 2 J ( , ) 4 ( , ) sin 2 3 D k k k strongly dependent on ( β , γ ) Local QRPA vibrational masses:

72 Kr localization 4  for 72Kr Collective wave functions squared

EXP ： Fischer et al., Phys.Rev. C67 (2003) 064318, Excitation Energies and B(E2) Bouchez, et al., Phys.Rev.Lett.90 (2003) 082502. Gade, et al., Phys.Rev.Lett. 95 (2005) 022502, 96 (2006) 189901 ( ) …B(E2) e 2 fm 4 72 Kr effective charge: e pol =0.658 The interband transitions become weaker as angular momentum increases. development of the localization of w.f. The time-odd mean-field lowers the excitation energies.

まとめ 5 次元四重極 Hamiltonian を微視的に決定する方法として、 2 次元 ASCC 法の近 似である CHFB+LQPRA 法を開発し、 Se および Kr の低励起状態に適用した CHFB+LQPRA 法によって求めた慣性質量は、平均場の time-odd 項から の寄与を含んでいる。 Cranking 質量との比較で、平均場の time-odd 項に よって、数十 % 程度の慣性質量の増大が見られた。この増大はβ、γに依 存する。 Se および Kr の計算結果は、それらの低励起状態が、理想的な変形共存とγ - unstable な状態との中間的な状況にある。それらの状態はγ方向の大振幅の shape fluctuation, β振動的な励起、回転運動の兼ね合いによって決まる 回転運動は振動波動関数の ( β , γ ) 平面での局在化の発達に重要な役割を果たす 展望 現実的な相互作用 Fully self-consistent な 2 次元 ASCC 法