素核宇融合による計算基礎物理学の進展
2011.12.05 (Mon.)

光核反応に対する密度汎関数理論の応用
Application of Density Functional Theory to photo-nuclear reaction

江幡 修一郎Ebata Shuichiro
HPCI戦略分野５
研究課題２ 大規模量子多体計算による核物性の解明と応用

東京大学 原子核科学研究センター(Center for Nuclear Study, Univ. Tokyo; CNS)

素核宇融合による計算基礎物理学の進展
江幡 修一郎Working area of models
Mean field theory based on DFT

Shell Model
Cluster Model

Stable : 200
Known : 3175
Total : 7000 ~?

素核宇融合による計算基礎物理学の進展
江幡 修一郎Characteristic structure of Unstable nuclei
Unstable nuclear region is expanding

Neutron-rich nuclei have . . .

KnownabStleStable

Neutron-richCharacteristic structure
New elementary mode ?
(Collective mode ?)Neutron-Skin structure“Pygmy dipole resonance” (PDR)

Neutron-Halo structure

素核宇融合による計算基礎物理学の進展
江幡 修一郎Excited states of unstable nuclei have a important roleon the r-process neucleosynthesis.
Example for 210Pb,
E1 strength function expressed in LorentzianSolar abundance calculatedwith the Lorentzian E1 strengths

S. Goriely, Phys. Lett. B 436, 10 (1998)

PDR is a small part of Total strengthbut, the effect for the abundance is large.

素核宇融合による計算基礎物理学の進展江幡 修一郎

As the first step,
Photo-nuclear reaction (E1 mode)
To describe and understand
Excited modes and Dynamics of
Various nuclei (stable, unstable)

素核宇融合による計算基礎物理学の進展
江幡 修一郎ContentsIntroduction
Models and working area,
Unstable nuclei and E1 mode
Formulation of Cb-TDHFBExample of linear response cal.
for E1 mode (172Yb)
Results of low-lying E1 strengthfor heavy nuclei (A > 100)
Summary and Future work

素核宇融合による計算基礎物理学の進展
江幡 修一郎Important points for nuclear structure
Nucleus have mean-field

Deformation of mean-field

Nucleon in the mean-field→ shell structure (magic number)→ single-particle states

Nilsson diagram

W.Nazarawicz, et. al. Nucl. Phys. A435 (1985) 397.

Pairing correlation
Odd-even mass staggering

A.Bohr and B.R.Mottelson Nuclear structure vol.1 (1969)

素核宇融合による計算基礎物理学の進展
素核宇融合による計算基礎物理学の進展
江幡 修一郎江幡 修一郎Several Mean-field models based on DFT
Statics

Dynamics

No Pairing

Hartree-Fock(HF)

Time-Dependent HF(TDHF, RPA)BCS Pairing

HF+BCS---With Pairing

Hartree-FockBogoliubov(HFB)

TDHFB(QRPA)※ RPA: Random-Phase Approximation※ QRPA: Quasi-particle RPA

素核宇融合による計算基礎物理学の進展
江幡 修一郎Nuclear dynamics described by TDHF theory
H.Flocard, S.E.Koonin and M.S.Weiss Phys. Rev. C17 (1978) 1682

素核宇融合による計算基礎物理学の進展
江幡 修一郎Several Mean-field models based on DFT
Statics

Dynamics

No Pairing

Hartree-Fock(HF)

Time-Dependent HF(TDHF, RPA)BCS Pairing

HF+BCS

Cb-TDHFBWith Pairing

Hartree-FockBogoliubov
(HFB)

TDHFB
(QRPA)※ RPA: Random-Phase Approximation※ QRPA: Quasi-particle RPA

素核宇融合による計算基礎物理学の進展
江幡 修一郎What is Cb-TDHFB ? More detail ... S. Ebata et al., PRC82, 034306
Cb-TDHFB can be derived from TDHFB represented in canonical basis*,with an approximation of pairing potential which is diagonal as like BCS.
*Canonical basis diagonalize density matrix.
: Occupation probability: Pair of k-state: Pair probability(no restriction of time-reversal)

Cb-TDHFB is a time-dependent scheme including
pairing correlations as in the BCS approximation.

Cb-TDHFB equations

Properties of Cb-TDHFB

TDHF
HF+BCS

素核宇融合による計算基礎物理学の進展
江幡 修一郎Linear response calculation with TD scheme (procedure)Calculate HF or HF+BCS ground state

A instantaneous external field
one-body operator

Calculate the time-evolution with TDHF or Cb-TDHFBStrength function S(E;F) is obtained with Fourier transformation.

: Smoothing parameter (~1 MeV)

素核宇融合による計算基礎物理学の進展
江幡 修一郎Linear response calculation with TD scheme ( for 20Ne )If we chose the suitable k ,we can get the results of linear response.

ContentsIntroduction
Models and working area,
Unstable nuclei and E1 mode
Formulation of Cb-TDHFBExample of linear response cal.
for E1 mode (172Yb)
Results of low-lying E1 strengthfor heavy nuclei (A > 100)
Summary and Future work

素核宇融合による計算基礎物理学の進展
江幡 修一郎Example : Photo-absorption cross section of 172Yb
3D Cb-TDHFB
Cb-TDHFB can reproduce
the photo-absorption cross section of 172Yb.

β =0.32∆n = 0.76 [MeV]∆p = 0.55 [MeV]

Heavy nucleus
Deformed nucleus
Including pairing

Γ= 1.0 [MeV]

K=0Total cal. cost : 300 CPU hours(with a Single processor; Intel Core i7 3.0 GHz)
Box size : R=15[fm], mesh=1[fm] (3D-Spherical)

Canonical-basis space (HF+BCS g.s.) :146 states for neutron,
98 states for proton
Experimental data:
A.M.Goryachev and G.N.Zalesnyy Vopr. Teor. Yad. Fiz. 5, 42 (1976).

素核宇融合による計算基礎物理学の進展
江幡 修一郎

K=1-

Dimension : HF vs. HF+BCS vs. HFB
HF

Pairing correlation

HF+BCS

Generalize: Canonical basis

HFB: Generalized quasi-particle
state: BCS quasi-particle state*One body density matrix is diagonalized in Canonical basis.

N : nucleon #N' : canonical basis #N = N'

N < N'

M : basis #Dimension

TDHFB: Generalized
density matrix

素核宇融合による計算基礎物理学の進展
江幡 修一郎: Generalized
Hamiltonian: Pair potentialDimension : HF+BCS vs. HFB in coordinate space representation
Wave function in 3D-Cartesian coordinate space
Example )(24/0.8)3 = 27,000 points

HF+BCS

HFB
Difference of matrix elements

Dimension(N' ~ 300 for 238U)

Heavy ion collision (238U to 238U)
Cb-TDHFB : 500 – 600 canonical basis→ Parallelization for canonical basis(Cb-TDHFB can conserve the orthogonality between canonical-basis)
素核宇融合による計算基礎物理学の進展
江幡 修一郎ContentsIntroduction
Models and working area,
Unstable nuclei and E1 mode
Formulation of Cb-TDHFBExample of linear response cal.
for E1 mode (172Yb)
Results of low-lying E1 strengthfor heavy nuclei (A > 100)
Summary and Future work

素核宇融合による計算基礎物理学の進展
江幡 修一郎Low-lying E1 strength : Pygmy dipole resonanceCommon sense of PDR ?
The peak of PDR appears
lower than one of GDR.
PDR peaks appear around separation energy.
Experiments
SNPygmy resonance in experiment

26Ne; J. Gibelin et al., Phys. Rev. Lett. 101, 212503(2008).

68

Ni; O. Wieland et al., Phys. Rev. Lett. 102, 092502(2009).

130,132

Sn; P. Adrich et al., Phys. Rev. Lett. 95, 132501(2005).

208

Pb; N. Ryezayavaet al., Phys. Rev. Lett. 89, 272502(2002).
etc.

P. Adrich, et al.,Phys. Rev Lett. 95, 132501 (2005)
素核宇融合による計算基礎物理学の進展
江幡 修一郎Isovector Dipole

Setup
External filed :Isovector dipole mode (for E1 strength)
Effective Interaction : Skyrme force (SkM*),
Smoothed Pairing strength G (ref. N. Tajima et al. NPA603(1996)23): cutoff function

Neutron and Proton
vibrate in anti-phase.

Nucleus : 14-28O, 18-32Ne, 18-40Mg, 24-46Si, 28-50S, 32-58Ar, 34-64Ca,
56-80

Ni, 78-86Zn, 80-96Ge, 82-100Se, 84-106Kr, 86-124Sr, 88-132Zr, 100-132Mo, 120-134Ru,
122-136
Pd, 124-138Cd, 98-140Sn, 128-142Te, 130-142Xe, etc. ( about 250 kinds of Nucleus )Calculation space (3D-Spherical meshed box):
For heavy nuclei (A > 70),
we use the box has radius 15 [fm] and meshed by 1.0 [fm].
Lattice points
素核宇融合による計算基礎物理学の進展
江幡 修一郎To quantify the low-lying E1 strength systematically,
The ratio of low-lying E1 strengthin Total E1 strength (sum rule).

ex.)

We use the ratio to analyze
the low-lying E1 strength
for all calculated nuclei.

“Low-energy”

26

Ne on a 208Pb target

J. Gibelin, et al., Phys. Rev Lett. 101, 212503 (2008)

素核宇融合による計算基礎物理学の進展
江幡 修一郎Results of Low-lying E1 strength for heavy nuclei (A > 100)

Proton number

Neutron number(Magic #)CalculatedStableKnown(Magic #)素核宇融合による計算基礎物理学の進展
江幡 修一郎r-process ?

N#-dependence of the Low-lying E1 strength for heavy nuclei

N=50

N=82

素核宇融合による計算基礎物理学の進展
江幡 修一郎ContentsIntroduction
Models and working area,
Unstable nuclei and E1 mode
Formulation of Cb-TDHFBExample of linear response cal.
for E1 mode (172Yb)
Results of low-lying E1 strengthfor heavy nuclei (A > 100)
Summary and Future work

素核宇融合による計算基礎物理学の進展
江幡 修一郎Summary & Future work
Development of Cb-TDHFB method in 3D
We develop Cb-TDHFB which can investigate the excited state of heavy nuclei
with small numerical cost.
Cb-TDHFB is suitable to the parallelization for canonical-basis.

Systematic study of E1 mode with Cb-TDHFB
We investigate the low-lying E1 strength systematically (for 250 kinds of nuclei)
including the effects of deformation and pairing.
We can see a very characteristic neutron number dependence of low-lying E1 modes
for heavy nuclei.

Apply the Cb-TDHFB to ...
Pygmy dipole resonance for more heavy nuclei,
Other modes (ISQ, ISM, IVM, scissors, etc.) systematically,Reaction (Heavy ion collision, Fusion, Fission, etc.)
素核宇融合による計算基礎物理学の進展
江幡 修一郎