Recent activities in time-dependent density-functional theory - - PowerPoint PPT Presentation

Recent activities in time-dependent density-functional theory

Takashi Nakatsukasa Center for Computational Sciences, University of Tsukuba and RIKEN Nishina Center

Japan-China Symposium on Nuclear Physics (JCNP2015), Suita, Osaka, November 7-12, 2015

Contents

• “Mean-field” model vs “Density-functional”

model

• Time-dependent density-functional theory

(TDDFT) for nuclear reaction

– Quasi-fission (QF)

• Particle number projection
• Inverse QF

– Fusion

• Potential
• Fusion path & inertial mass

“Mean-field” model

• Separation energy
• Binding energy in the mean field
• Saturation property

S = − TF +V

( )

Inconsistent with nuclear binding E

S V TF

F

− = + = ε −B = Ti + V 2 " # $ % & '

i=1 A

∑

, Ti = 2ki

2

2m = A 3 5TF + V 2 ! " # $ % &

⇒ = A B S V TF 4 5 − =

“DFT” (DDHF) model

• Mean-field model
• Density-dependent Hartree-Fock (DDHF)

– Energy density functional – LDA à Skyrme-like EDF 4 .

*

≈ ⇒ Φ + Φ = m m V T E

E = Φ T +V ρ

[ ] Φ

⇒ m* m = 0.7 ~1

E ρ

[ ] ⇒

h ρ

[ ] ϕi =εi ϕi ,

h ρ

[ ] ≡ δE

δρ

• Linear response

– Deformed QRPA calculations – Finite amplitude method

• Superfluid systems

– Full TD-BdG scheme

• Realistic reaction simulation

– “Macroscopic” quantities (potential, friction) – Fragment PNP

• Reaction path and inertial mass

Recent progress in time-dependent density-functonal theory (TDDFT)

Simulation of nuclear reaction

Quasi-fission Fusion Inelastic collision

58Ni+208Pb

E = E1

E1 < E2 < E3

E = E2 E = E3 Skyrme EDF: SLy5

Fragment mass distribution

(-6p; Ti) (-5p; V) (-4p; Cr) (-3p; Mn) (-2p; Fe) (-1p; Co) (0p; Ni)

σ [ mb ] Mass number of lighter fragments VP VT

Particle number projection technique

Experiment (Corradi et al, PRC 66, 024606 (2002))

58Ni+208Pb è A1 + A2

B1 + B2 …….

Sekizawa and Yabana, Phys. Rev. C 88, 014614 (2013) Phys. Rev. C 90, 064614 (2014); EPJ Web Conf. 86, 00043 (2015)

Simenel, PRL 105, 192701 (2010)

E = 328.4 MeV

Quasi-fission: 124Sn+238U

核子はほとんど移行せず 核子はほとんど移行せず

6 陽子，10 中性子の移行 12 陽子，16 中性子の移行

Elab~9 MeV/A Elab~9 MeV/A Elab~6 MeV/A Elab~6 MeV/A

Very little particle transfer Sn ß U (6p + 10n) Sn à U (12p + 16n) Very little particle transfer “Inverse” quasi-fission

Another way to produce Superheavy Elements? Sekizawa, PhD Thesis (2014, Univ. of Tsukuba)

Frozen density (FD) approx. (Coulomb barrier height)

132Sn + 132Sn

Threshold energy (TDDFT)

“extra-push energy”

• Exp. data

Fusion hindrance phenomena

Guo and Nakatsukasa, EPJ Web Conf.(2012) 38, 09003

Fusion hindrance phenomena

• ZPZT=1600

ZPZT=2000 ZPZT=1764~1932 Z

• Exp: Schmidt and Morawek, Rep.Prog.Phys. 54 (1991) 949.

The threshold energy for fusion: Eth Eextra = Eth - VB

Washiyama, PRC91, 064607 (2015) TDDFT EXPT

0.16 (fm-3)

Densities at R=10.0, 12.1, and 14.0 fm for Ecm= 250 MeV

96Zr+124Sn system

Washiyama, PRC91, 064607 (2015)

6/10

• Heavy system vs. Light system
• 96Zr +124Sn

40Ca +40Ca

Choice of variables (R,P)

• R is defined by
• P is calculated from current in L & R
• This definition is questionable after two

nuclei touch each other.

• Need reliable definition of canonical

collective variables

RL RR

R = RR − RL

Space “L” and “R”

• Equa?ons are expanded up to 2nd order in collec?ve

momenta Collec?ve Hamiltonian

(0th) δ Ψ(q) ˆ HM (q) Ψ(q) = 0, ˆ HM (q) ≡ ˆ H −(∂V ∂q) ˆ Q(q) (1st) δ Ψ(q) ˆ HM (q),i ˆ Q(q) % & ' (− B(q) ˆ P(q) Ψ(q) = 0 (2nd) δ Ψ(q) ˆ HM (q), ˆ P(q) i % & ' (−C(q) ˆ Q(q) − 1 2B(q) ˆ HM (q),(∂V ∂q) ˆ Q(q) % & ' (, ˆ Q(q) % & ' ( Ψ(q) = 0

H(q, p) = Ψ(q, p) ˆ H Ψ(q, p) ≈ 1 2 B(q)p2 +V(q) V(q) = Ψ(q) ˆ H Ψ(q) , B(q) = Ψ(q) ˆ H, ˆ Q(q) # $ % &, ˆ Q(q) # $ % & Ψ(q)

“Moving mean-field equation” “Local harmonic equation” (LHE) This gives the exact total mass, M=Am, for the translational motion.

Matsuo, Nakatsukasa, Matsuyanagi, PTP 103, 959 (2000)

• H ξ,π

( ) = 1

2 Bαβ(ξ)παπ β +V(ξ) ≈ 1 2 B(q)p2 +V(q)+ H(q, p) ⇒ 1 2 B(R)P

R 2 +V(q(R))

B(R) = ∂R ∂q " # $ % & '

2

B(q)

H ξ,π

( ) = 1

2 Bαβ(ξ)παπ β +V(ξ) ⇒ 1 2  B(R)P

R 2 +V(R)

 B(R) = ∂R ∂ξ α ∂R ∂ξ β Bαβ(ξ) = B(R)+ ∂R ∂q i ∂R ∂q j Bij(q)

The intrinsic d.o.f. affects the mass

B(R) ≠  B(R)

Finding decoupled canonical variables Assuming the collective variables

ξ α,πα

( ) →(q, p;q i, pi)

R = R ξ

( )

R is chosen by hand

(map from q to R)

çè

Poten?al

V(q) = Ψ(q) ˆ H Ψ(q)

Iner?al mass

B(q) = Ψ(q) ˆ H,i ˆ Q(q) " # $ %,i ˆ Q(q) " # $ % Ψ(q) B(R) = B(q) ∂R ∂q ' ( ) * + ,

2

, M(R) = 1 B(R)

Model space: 3D grid space 16 x 10 x 10 fm3 (Δx=Δy=Δz=0.8 fm)

V(R) =V(R(q)) V(R) M(R) m

Wen et al., arXiv:1510.03612

Summary

• TDDFT for nuclear reaction

– Quasi-fission (QF) – Mass distribution by PNP – Possibility for inverse QF – Fusion hindrance mechanism

• Self-consistent collective coordinate

– Fusion/Fission path – Inertial mass, beyond cranking/GCM-GOA – First successful application to α+α ßà 8Be