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 E ( ) S = − T F + V • Binding energy in the mean field T F V S ε = + = − F A T i =  2 k i 2 " % T i + V ∑ − B = , $ ' # 2 & 2 m i = 1 ! $ = A 3 5 T F + V # & " 2 % • Saturation property 5 B Inconsistent with S T F V = − = ⇒ A nuclear binding 4

“DFT” (DDHF) model • Mean-field model * E T V m m 0 . 4 = Φ + Φ ⇒ ≈ • Density-dependent Hartree-Fock (DDHF) m * m = 0.7 ~1 [ ] Φ E = Φ T + V ρ ⇒ – Energy density functional [ ] ≡ δ E [ ] ⇒ [ ] ϕ i = ε i ϕ i , E ρ h ρ h ρ δρ – LDA à Skyrme-like EDF

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

Simulation of nuclear reaction � E = E 2 � E 1 < E 2 < E 3 � 58 Ni+ 208 Pb � Skyrme EDF: SLy5 � Inelastic collision � Fusion � E = E 1 � E = E 3 � Quasi-fission �

Fragment mass distribution � Sekizawa and Yabana, Phys. Rev. C 88, 014614 (2013) Phys. Rev. C 90, 064614 (2014); EPJ Web Conf. 86, 00043 (2015) � 58 Ni+ 208 Pb è A 1 + A 2 V P � B 1 + B 2 …… . Particle number projection technique � V T � Simenel, PRL 105, 192701 (2010) � E = 328.4 MeV � (-6p; Ti) (-5p; V) (-4p; Cr) (-3p; Mn) (-2p; Fe) (-1p; Co) (0p; Ni) σ [ mb ] � Mass number of lighter fragments � Experiment (Corradi et al, PRC 66, 024606 (2002)) �

Quasi-fission: 124 Sn+ 238 U � Sekizawa, PhD Thesis (2014, Univ. of Tsukuba) � E lab ~ 6 MeV/ A E lab ~ 6 MeV/ A Very little particle transfer � Sn ß U (6p + 10n) 核子は ほ と ん ど移行 せず 6 陽子 ， 10 中性子の移行 E lab ~ 9 MeV/ A E lab ~ 9 MeV/ A Very little particle transfer � Sn à U (12p + 16n) 核子は ほ と ん ど移行 せず 12 陽子 ， 16 中性子の移行 “Inverse” quasi-fission � Another way to produce Superheavy Elements? �

Fusion hindrance phenomena � Guo and Nakatsukasa, EPJ Web Conf.(2012) 38, 09003 � “extra-push energy” Threshold energy (TDDFT) � 132 Sn + 132 Sn Exp. data � Frozen density (FD) approx. (Coulomb barrier height) �

� Fusion hindrance phenomena � Washiyama, PRC91, 064607 (2015) � Z P Z T =1600 � Z P Z T =1764~1932 � Z P Z T =2000 � Z � TDDFT EXPT � Exp: Schmidt and Morawek, Rep.Prog.Phys. 54 (1991) 949. The threshold energy for fusion: E th � � E extra = E th - V B

� ����������������������� Washiyama, PRC91, 064607 (2015) � 96 Zr+ 124 Sn system � Densities at R=10.0, 12.1, and 14.0 fm for E cm = 250 MeV � 0 � 0.16 (fm -3 ) � Heavy system vs. Light system � 96 Zr + 124 Sn � 40 Ca + 40 Ca � 6/10 � �

Choice of variables ( R,P ) � • R is defined by R = R R − R L R L R R Space “ L ” and “ R ” � • 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 �

���������������������������� Matsuo, Nakatsukasa, Matsuyanagi, PTP 103, 959 (2000) Equa?ons are expanded up to 2 nd order in collec?ve momenta “Moving mean-field equation” H − ( ∂ V ∂ q ) ˆ (0th) δ Ψ ( q ) ˆ H M ( q ) ≡ ˆ ˆ H M ( q ) Ψ ( q ) = 0, Q ( q ) H M ( q ), i ˆ ˆ (− B ( q ) ˆ % ' (1st) δ Ψ ( q ) Q ( q ) P ( q ) Ψ ( q ) = 0 & H M ( q ), ˆ ˆ (− C ( q ) ˆ % ' (2nd) δ Ψ ( q ) P ( q ) i Q ( q ) & 1 % ' H M ( q ),( ∂ V ∂ q ) ˆ ˆ ( , ˆ % ' Q ( q ) Q ( q ) ( Ψ ( q ) = 0 − & & 2 B ( q ) Collec?ve Hamiltonian “Local harmonic equation” (LHE) H Ψ ( q , p ) ≈ 1 2 B ( q ) p 2 + V ( q ) H ( q , p ) = Ψ ( q , p ) ˆ # % V ( q ) = Ψ ( q ) ˆ H , ˆ ˆ & , ˆ # % H Ψ ( q ) , B ( q ) = Ψ ( q ) Q ( q ) Q ( q ) & Ψ ( q ) $ $ This gives the exact total mass, M=Am, for the translational motion.

�������������������������� Finding decoupled canonical variables Assuming the collective variables ( ) → ( q , p ; q i , p i ) ( ) ξ α , π α R = R ξ R is chosen by hand ) = 1 2 B αβ ( ξ ) π α π β + V ( ξ ) ) = 1 ( H ξ , π 2 B αβ ( ξ ) π α π β + V ( ξ ) ( H ξ , π ⇒ 1  2 + V ( R ) B ( R ) P ≈ 1 2 B ( q ) p 2 + V ( q ) + H ( q , p ) R 2 (map from q to R ) ⇒ 1 B ( R ) = ∂ R ∂ R 2 + V ( q ( R ))  ∂ ξ β B αβ ( ξ ) 2 B ( R ) P R ∂ ξ α 2 = B ( R ) + ∂ R ∂ R " % B ( R ) = ∂ R ∂ q j B ij ( q ) B ( q ) $ ' ∂ q i ∂ q # & B ( R ) ≠  The intrinsic d.o.f. affects the B ( R ) mass

����������������������� � ��� çè ����� Wen et al., arXiv:1510.03612 � Poten?al V ( R ) V ( q ) = Ψ ( q ) ˆ H Ψ ( q ) V ( R ) = V ( R ( q )) Iner?al mass " $ H , i ˆ % , i ˆ " ˆ $ B ( q ) = Ψ ( q ) Q ( q ) Q ( q ) % Ψ ( q ) M ( R ) # # m 2 ' * B ( R ) = B ( q ) ∂ R 1 , M ( R ) = ) , ∂ q B ( R ) ( + Model space: 3D grid space 16 x 10 x 10 fm 3 (Δx=Δy=Δz=0.8 fm)

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 8 Be