Basic properties - PowerPoint PPT Presentation

���������������������� �� � •Basic properties of nuclei •TDDFT (TDMF) for nuclear collective motion •Re-quantization of collective submanifold •Application to alpha reaction in giant stars ������������������� 2017.12.6-8 @ ��

Saturation properties of nuclear matter • Symmetric nuclear matter w/o Coulomb A – N Z = = 2 • Constant binding energy per nucleon – Constant separation energy B S 16 MeV ≈ ≈ A n ( p ) • Saturation density 3 1 0 . 16 fm k 1 . 35 fm − − ρ ≈ ⇒ ≈ F – Fermi energy 2 2 ! k T F 40 MeV = ≈ F 2 m

����������� (AME2016) Fig. 7. Two-neutron separation energies N = 122 to 145 120 125 130 135 140 145 20 20 Magic number: N =126 213 Pa 217 U 212 Th 18 18 211 Ac 210 Ra 209 Fr 208 Rn S 2n (MeV) 207 At 235 Bk 233 Cm 206 Po 243 Fm 221 Np 241 Es 231 Am 239 Cf 16 16 205 Bi 229 Pu 204 Pb 245 Fm 203 Tl 244 Es 243 Cf 202 Hg 14 14 242 Bk 201 Au 241 Cm 200 Pt 240 Am 239 Pu 12 12 238 Np 237 U 236 Pa 235 Th 234 Ac 10 10 233 Ra 232 Fr 231 Rn 229 At 227 Po 8 8 224 Bi 220 Pb 210 Au 218 Tl 208 Pt 216 Hg 6 6 120 125 130 135 140 145 Neutron number N

����������� (AME2016) Fig. 4. Two-neutron separation energies N = 62 to 85 60 65 70 75 80 85 Magic number: N =82 152 Lu 130 Sm 121 Ce 123 Pr 146 Tm 132 Eu 128 Pm 119 La 135 Gd 144 Er 150 Yb 25 126 Nd 137 Tb 25 142 Ho 140 Dy 118 Ba 155 Hf 117 Cs 116 Xe S 2n (MeV) 115 I 157 Hf 114 Te 113 Sb 20 20 156 Lu 112 Sn 155 Yb 154 Tm 111 In 110 Cd 152 Ho 109 Ag 150 Tb 108 Pd 107 Rh 148 Eu 15 15 106 Ru 146 Pm 105 Tc 104 Mo 144 Pr 103 Nb 143 Ce 102 Zr 142 La 101 Y 141 Ba 10 140 Cs 10 100 Sr 99 Rb 139 Xe 138 I 127 Rh 124 Ru 137 Te 121 Tc 118 Mo 112 Zr 109 Y 115 Nb 136 Sb 103 Rb 107 Sr 129 Pd 135 Sn 134 In 5 5 133 Cd 132 Ag 60 65 70 75 80 85 Neutron number N

Failure of the mean-field models Nakatsukasa et al., RMP 88, 045004 (2016) • In order to explain the nuclear saturation within the mean-field picture, we need an extremely small value of the effective mass. 1 − * m 3 5 B 1 ⎛ ⎞ 0 . 4 ⎜ ⎟ = + ≈ ⎜ ⎟ m 2 2 A T ⎝ ⎠ F – Inconsistent with the experimental data. • A solution [ ] [ ] E h ρ ⇒ ρ φ = ε φ i i i – Energy density functional E δ [ ] h ρ ≡ (Rearrangement terms) δρ

Nuclear energy density functional • Spin & isospin degrees of freedom – Spin-current density is indispensable. • Nuclear superfluidity à Kohn-Sham- Bogoliubov eq. – Pair density (tensor) is necessary for heavy ! nuclei. [ ] E , , J ; ρ τ κ q q q q kinetic pair density spin-current U ( t ) U ( t ) h ( t ) ( t ) − λ Δ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ µ µ E ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ * ⎜ ⎟ ⎜ ⎟ * V ( t ) V ( t ) ( ) µ ( t ) h ( t ) − Δ − − λ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ µ µ

Nuclear deformation Ebata, Nakatsukasa, Phys. Scr. 92 (2017) 064005 Quadrupole deformation Deformation landscape

Nuclear deformation predicted by DFT Intrinsic Q moment Deformation landscape Z = 50 N = 82

Time-dependent density functional theory (TDDFT) for nuclei • Time-odd densities (current density, spin density, etc.) " ! ! ! [ ] E ( t ), ( t ), J ( t ), j ( t ), s ( t ), T ( t ); ( t ) ρ τ κ q q q q q q q spin-kinetic kinetic current spin-current spin pair density • TD Kohn-Sham-Bogoliubov-de-Gennes eq. U ( t ) U ( t ) h ( t ) ( t ) − λ Δ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ∂ µ µ i ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ * ⎜ ⎟ ⎜ ⎟ V ( t ) * V ( t ) ( ) t ( t ) h ( t ) ∂ − Δ − − λ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ µ µ

Deformation effects for photoabsorption cross section Yoshida and Nakatsukasa, SkM* functional Phys. Rev. C 83, 021404 (2011) Intrinsic Q moment

Reaction above the Coulomb barrier “Partial”-space particle-number projection Simenel, C., 2010, Phys. Rev. Lett. 105, 192701. Expt. Y.X. Watanabe et al. TDHF Real-time simulation TDHF+GEMINI GRAZING w/ evap. 136 Xe+ 198 Pt ( E c.m. ≈ 644.98 MeV) 136 Xe + 198 Pt 10 3 10 3 (+4p; Ce) (+3p; La) (+2p; Ba) (+1p; Cs) 10 2 10 2 σ (mb) 10 1 10 1 10 0 10 0 10 -1 10 -1 10 -2 10 -2 10 -3 10 -3 120 135 150 120 135 150 120 135 150 120 135 150 120 135 150 120 135 150 120 135 150 120 135 150 10 3 10 3 (-4p; Sn) (-3p; Sb) (-2p; Te) (-1p; I) (0p; Xe) 10 2 10 2 σ (mb) 10 1 10 1 10 0 10 0 10 -1 10 -1 10 -2 10 -2 10 -3 10 -3 120 135 150 120 135 150 120 135 150 120 135 150 120 135 150 120 135 150 120 135 150 120 135 150 120 135 150 120 135 150 MASS NUMBER Sekizawa, Phys. Rev. C 96 , 014615 (2017)

Large amplitude collective motion • Decay modes – Spontaneous fission – Alpha decay • Low-energy reaction – Sub-barrier fusion reaction – Alpha capture reaction (element synthesis in the stars) Missing quantum fluctuation, tunneling, etc.

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���� S-factor • ������ 4 5 4 6 7 6 1 2 = – 8 – ������ – ��� 9(;) = exp(−2@A) A = B C B D E D ℏG ����������� H ; = 1 • ��� ; 9 ; ×J(;) Astrophysical S -factor ��������

Classical Hamilton � s form Blaizot, Ripka, � Quantum Theory of Finite Systems � (1986) Yamamura, Kuriyama, Prog. Theor. Phys. Suppl. 93 (1987) The TDDFT can be described by the classical form. ξ ph = ∂ H  i ∂ ∂ t ρ ( t ) = ⇥ h [ ρ ( t ) ] , ρ ( t ) ⇤ , ∂ π ph π ph = − ∂ H h kl ⌘ ∂ E [ ρ ]  . ∂ ξ ph ∂ρ lk ) are identical to K L, @ = ; N(L, @) L OS ,@ OS The canonical variables N SSP = 1 − L + Q@ R L + Q@ L + Q@ R N OOP = L + Q@ OOP SSP 1 − L + Q@ R L + Q@ N OS = L + Q@ OS Number of variables = Number of ph degrees of freedom

Strategy • Purpose – Recover quantum fluctuation effect associated with “slow” collective motion • Difficulty – Non-trivial collective variables • Procedure 1. Identify the collective subspace of such slow motion, with canonical variables (T, U) 2. Quantize on the subspace T, U = Qℏ

Expansion for “slow” motion • Hamiltonian K = K L, @ ≈ 1 2 X YZ L @ Y @ Z + 1(L) expanded up to 2 nd order in @ [ α, \ = (Uℎ )] • Point transformation L Y , @ Y → T ^ , U ^ _` a _b c U ^ = _b c @ Y , @ Y = _` a U ^ • Hamiltonian d T, U ≈ 1 e ^f T U ^ U f + 1(T) d = K K 2 X

Decoupled submanifold • Collective canonical variables (T, U) – L Y , @ Y → T, U; T h , U h ; i = 2, ⋯ , k OS – Decoupled collective subspace Σ – Σ defined by (T h , U h ) = (0,0) P(q) = i d/dq (a) • Decoupled eq. of motion e _b o U D = 0 _n C _p P 2 (q) = i d/dq 2 U̇ h = − _b o − (b) D P 1 (q) = i d/dq 1 Ṫ h = X e hC U = 0 q 2 q 1 e ^f T U ^ U f + 1(T) ] C d = K d T, U ≈ [ K D X

Symplectic formulation • Collective canonical variables (T, U) – L Y , @ Y → T, U; T h , U h ; i = 2, ⋯ , k OS • Equations for a decoupled submanifold _n _n _b L Y on Σ _` a − _` a = 0 → _b X Zs t _n _` u = v D _b _b _b → s _` a _` a _` a _ 6 n _n _n Z t _` a ≡ _` x _` a − Γ Covariant derivative s Ys _` u Z = C D z Z{ z Y{,s + z s{,Y − z Ys,{ with Γ Ys _b c _b c using the metric z YZ ≡ ∑ ^ _` a _` u

Riemannian formulation Y by _b _` a _` u in Γ • Rewriting a curvature term Zs using the decoupled condition • Equations for a decoupled submanifold _n _n _b _` a − _` a = 0 Moving mean-field eq. _b X Zs t _n _` u = v D _b _b Moving RPA eq. s _` a _` a _ 6 n _n _n Z t _` a ≡ _` x _` a − Γ Covariant derivative s Ys _` u Z = C D z Z{ z Y{,s + z s{,Y − z Ys,{ with Γ Ys using the metric z YZ ≡ X YZ , X Ys X sZ = δ Y Z

Numerical procedure _n _n _b _` a − _` a = 0 Moving mean-field eq. _b X Zs t _n _` u = v D _b _b Moving RPA eq. s _` a _` a Tangent vectors (Generators) Y = ~L Y T ,Y = ~T L L ,b ~L Y ~T Moving MF eq. to determine the point: L Y Move to the next point L Y + L Y = L Y + Ä L ,b Y

Canonical variables and quantization • Solution – 1-dimensional state: ξ T _` a _b – Tangent vectors: _` a and _b – Fix the scale of T by making the inertial mass _` a X YZ _b _b e = X _` a = 1 • Collective Hamiltonian D U D + 1 C d ÇÉÑÑ T, U = e(T) , 1 e T = 1(ξ T ) – K – Quantization T, U = Qℏ

3D real space representation • 3D space discretized in lattice • BKN functional • Moving mean-field eq.: Imaginary-time method • Moving RPA eq. � Finite amplitude method (PRC 76, 024318 (2007) ) Wen, T.N., 96 , 014610 (2017) Wen, T.N., PRC 94, 054618 (2016). Wen, Washiyama, Ni, T.N., Acta Phys. Pol. B Proc. Suppl. 8, 637 (2015) y [ fm ] At a moment, no pairing 1-dimensional reaction path extracted from the Hilbert space of dimension of 10 4 ~10 5 . X [ fm ]