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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
Constrained HFB + Local QRPA Constrained HFB + Local QRPA - - PowerPoint PPT Presentation
Constrained HFB + Local QRPA Constrained HFB + Local QRPA - - PowerPoint PPT Presentation
Constrained HFB + Local QRPA Constrained HFB + Local QRPA ( / ) ( ) ( )
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) , (
rot vib
V T T H
2 2 vib
) , ( 2 1 ) , ( ) , ( 2 1
D D D T
3 1 2 rot
2 1
k k k
T J
5D quadrupole collective Hamiltonian Application to shape coexistence in Se and Kr
- N. Hinohara, et al, Prog. Theor. Phys. 119(2008), 59; PRC 80 (2009),014305.
(Generalized Bohr-Mottelson Hamiltonian) : collective potential vibrational inertial masses rotational moments of inertia 日野原さんのトーク： (1+3)次元のASCC 法 (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
1
q
2
q
68Se
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One can extract the collective degree(s) of freedom the system itself chooses
Adiabatic Self-consistent Collective Coordinate (ASCC)Method
- T. Marumori, T. Maskawa, F. Sakata and A. Kuriyama, PTP 64
(1980), 1294.
By courtesy of Hinohara-san
Matsuo, Nakatsukasa, and Matsuyanagi, PTP 103(2000), 959.
Adiabatic approx. to SCC method
ASCC Basic Equations ASCC Basic Equations
Time-dep. variational principle
) , , , ( ) , , , ( N H t i N p q p q
Adiabatic expansion
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Moving-frame HFB equation (from 0-th order in p) Local harmonic equations (moving-frame QRPA equations) (from 1st-order in p) (from 2nd-order in p)
ASCC Basic Equations ASCC Basic Equations
Not included in QRPA Not included in HFB
moving-frame Hamiltonian
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ASCC Basic Equations ASCC Basic Equations
Collective Hamiltonian Canonical variable conditions
HFB平衡点でのmoving-frame QRPA 方程式 通常のQRPA 方程式 Gauge-fixing Hinohara et al., PTP117(2007) 451.
~
) ( ) (
N qi
i N ) ( ˆ ), ( ˆ ) ( q q q
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q q g T
2 1
q G G q G T
1 2
1 2 ˆ
Classical kinetic energy: Quantized kinetic energy: ) 3 , 2 , 1 ( i q
i i
4
q
5
q
:3 Euler angles
Pauli’s prescription
:2 deformation parameters
Laplacean in a curved space + 5D Hamiltonian
Requantization Requantization
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1
q
2
q
TDHFB phase space
sin sin
x x Collective manifold 2-dimensional Mapping onto the (β,γ) plane 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
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General Bohr-Mottelson Hamiltonian (5D quadrupole collective Hamiltonian): Pauli’s prescription Classical Quadrupole Collective Hamiltonian:
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Collective Schrodinger equation: Collective wave function: Normalization:
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Derivation of vibrational masses Scale transformation B=1
1
q
2
q
) (
D
) ( 2
2
D
collective coordinates Vibrational part of collective Hamiltonian In terms of quadrupole deformation
- ne-to-one
correspondence
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Collective mass can be calculated through Vibrational energy written in terms of quadrupole defromation Without numerical derivatives
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Local QRPA(LQRPA) equations: Constrained HFB(CHFB) equation: Curvature terms(1Dでは寄与小さかった) を落とす Approximations:
) , ( ) , (
2 1 q
q ) , ( ˆ
CHFB
H ) , ( ˆ
2 1 M
q q H
1
q
2
q 最初からfully self-consistentな2D ASCCの計算は大変なので・・・
Constrained HFB + Local QRPA method
Moving-frame Hamiltonian CHFB Hamiltonian
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Constrained HFB + Local QRPA method
Solve the constrained HFB eq. at each point on the (β, γ) plane Solve the LQRPA eqs. on top of each CHFB state Pick up 2 collective modes from the LQRPA modes obtained above Calculate the inertial mass functions Diagonalize the 5D Quadrupole Hamiltonian to calculate the energy spectrum
) , ( ) , ( ˆ
) (
P ) , ( ˆ
) (
Q ) , (
2
) , (
k
J
) , (
D ) , (
D ) , (
D
) , ( ) , ( ) , (
Local QRPA mass
) , ( V
the contribution from the time-odd mean field included
Local Thouless-Valatin eqs.
) , ( ) , ( ) , (
2
D D D
を最小化するペア
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How to choose the two collective modes ?
- 1st. We assume that the mode with lowest-frequency squared
is collective.
) , (
2 1
- 2nd. The other is the mode whose combination with the lowest mode
minimizes the vibrational part of the 5DQH metric:
) , (
2
n
collective
Cranking formula
) , (
2 1
) , (
2 2
We regard the pair which minimizes the vibrational part of the 5DQH metric as collective: Intrinsic vol. element W for α=1 and α=2,3,4・・・ Take the lowest 40 QRPA modes as candidates. Calculate the collective mass for every pair of the candidates
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Comparison with other 5D Quadrupole Hamiltonian approaches
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CHFB＋LQRA法の陽子過剰Se、Krへの適用
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Model space
fitted to the pairing gaps and the quadrupole deformation obtained with Skyrme-HFB by Yamagami et al.
Microscopic Hamiltonian Harmonic oscillator two major shell :Nsh=3, 4 (pf & sdg shells) The s. p. energies: calculated using the modified oscillator potential. Pairing (monopole, quadrupole) + quadrupole p-h interaction P+QQ model:
- M. Yamagami et al.,NPA 693(2001) 579.
Parameters
) (
G
Monopole pairing & quadrupole int. Quadrupole pairing : self-consistent value
Sakamoto et al. PLB245 (1990) 321
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Collective potential Numerical results : absolute minimum Collective path The absolute min. is oblate. The 2nd lowest min. is prolate. The spherical shape is a local maximum.
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LQRPA rotational masses
68Se
Ratio to cranking MoI
3 2 sin ) , ( 4 ) , (
2 2
k Dk
k
J ) , ( / ) , (
(IB) (LQRPA)
k k
J J
1
D
2
D
3
D
(IB) 1 (LQRPA) 1
/ J J
(IB) 2 (LQRPA) 2
/ J J
(IB) 3 (LQRPA) 3
/ J J
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LQRPA vibrational masses
68Se
Ratio to cranking mass
) , (
D
/ ) , ( D
2
/ ) , (
D
β，γに依存して 数十%増加
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Collective wave functions squared
4
for 68Se
Yrast : oblate character, Yrare: prolate character 02 , 23 : β-vibrational
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( ) …B(E2) e2 fm4 effective charge: en =0.4, ep =1.4
Excitation Energies and B(E2)
centrifugal effect Reminiscent of γ-unstable situation B(E2; 62 -> 61 ) >> B(E2; 62 -> 41 ) LQRPA massの方がcranking massより実験とよい一致 e.g.
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LQRPA moments of Inertia
(LQRPA) 1
J
(LQRPA) 2
J
(LQRPA) 3
J
Local QRPA vibrational masses:
72Kr
3 2 sin ) , ( 4 ) , (
2 2
k Dk
k
J
strongly dependent on (β,γ) Extention of Thouless-Valatin MoI to non-equilibrum HFB pts.
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Collective wave functions squared
4
for 72Kr localization
72Kr
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Excitation Energies and B(E2)
EXP：Fischer et al., Phys.Rev.C67 (2003) 064318, Bouchez, et al., Phys.Rev.Lett.90 (2003) 082502. Gade, et al., Phys.Rev.Lett.95 (2005) 022502, 96 (2006) 189901
( ) …B(E2) e2 fm4 effective charge: epol =0.658 The time-odd mean-field lowers the excitation energies. The interband transitions become weaker as angular momentum increases. development of the localization of w.f.
72Kr
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まとめ
展望 現実的な相互作用 Fully self-consistentな2次元ASCC法 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