Basic properties - - PowerPoint PPT Presentation

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

Saturation properties of nuclear matter

• Symmetric nuclear matter w/o Coulomb

–

• Constant binding energy per nucleon

– Constant separation energy

• Saturation density

– Fermi energy

MeV 16

) (

≈ ≈

p n

S A B

1 3

fm 35 . 1 fm 16 .

− −

≈ ⇒ ≈

F

k ρ

2 A Z N = = MeV 40 2

2 2

≈ = m k T

F F

!

• (AME2016)

Neutron number N S2n (MeV)

• Fig. 7. Two-neutron separation energies N = 122 to 145

120 125 130 135 140 145 120 125 130 135 140 145 6 8 10 12 14 16 18 20 6 8 10 12 14 16 18 20

208Pt 200Pt 210Au 201Au 216Hg 202Hg 218Tl 203Tl 220Pb 204Pb 224Bi 205Bi 227Po 206Po 229At 207At 231Rn 208Rn 232Fr 209Fr 233Ra 210Ra 234Ac 211Ac 235Th 212Th 236Pa 213Pa 237U

217U

238Np 221Np 239Pu 229Pu 240Am 231Am 241Cm 233Cm 242Bk 235Bk 243Cf 239Cf 244Es 241Es 245Fm 243Fm

Magic number: N=126

Neutron number N S2n (MeV)

• Fig. 4. Two-neutron separation energies N = 62 to 85

60 65 70 75 80 85 60 65 70 75 80 85 5 10 15 20 25 5 10 15 20 25

103Rb 99Rb 107Sr 100Sr 109Y

101Y

112Zr 102Zr 115Nb 103Nb 118Mo 104Mo 121Tc 105Tc 124Ru 106Ru 127Rh 107Rh 129Pd 108Pd 132Ag 109Ag 133Cd 110Cd 134In 111In 135Sn 112Sn 136Sb 113Sb 137Te 114Te 138I

115I

139Xe 116Xe 140Cs 117Cs 141Ba 118Ba 142La 119La 143Ce 121Ce 144Pr 123Pr 126Nd 146Pm 128Pm 130Sm 148Eu 132Eu 135Gd 150Tb 137Tb 140Dy 152Ho 142Ho 144Er 154Tm 146Tm 155Yb 150Yb 156Lu 152Lu 157Hf 155Hf

• (AME2016)

Magic number: N=82

Failure of the mean-field models

• In order to explain the nuclear saturation

within the mean-field picture, we need an extremely small value of the effective mass.

– Inconsistent with the experimental data.

• A solution

– Energy density functional (Rearrangement terms)

[ ] [ ] [ ]

δρ δ ρ φ ε φ ρ ρ E h h E

i i i

≡ = ⇒ 4 . 1 2 5 2 3

1 *

≈ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + =

− F

T A B m m

Nakatsukasa et al., RMP 88, 045004 (2016)

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.

[ ]

q q q q

J E κ τ ρ ; , , !

spin-current kinetic pair density

( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − Δ − Δ − ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

* *

t V t U E t V t U t h t t t h

µ µ µ µ µ

λ λ

Nuclear deformation

Ebata, Nakatsukasa, Phys. Scr. 92 (2017) 064005

Deformation landscape

Quadrupole deformation

Nuclear deformation predicted by DFT

Intrinsic Q moment

Deformation landscape

N = 82 Z = 50

Time-dependent density functional theory (TDDFT) for nuclei

• Time-odd densities (current density, spin

density, etc.)

• TD Kohn-Sham-Bogoliubov-de-Gennes eq.

[ ]

) ( ); ( ), ( ), ( ), ( ), ( ), ( t t T t s t j t J t t E

q q q q q q q

κ τ ρ ! ! ! "

( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − Δ − Δ − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

* *

t V t U t h t t t h t V t U t i

µ µ µ µ

λ λ

spin-current current spin spin-kinetic kinetic pair density

Deformation effects for photoabsorption cross section

SkM* functional

Intrinsic Q moment

Yoshida and Nakatsukasa,

• Phys. Rev. C 83, 021404 (2011)

Real-time simulation

“Partial”-space particle-number projection

10-3 10-2 10-1 100 101 102 103 120 135 150 (+4p; Ce)

σ (mb)

10-3 10-2 10-1 100 101 102 103 120 135 150 120 135 150 (+3p; La)

• Expt. Y.X. Watanabe et al.

120 135 150 120 135 150 (+2p; Ba) TDHF TDHF+GEMINI 120 135 150 120 135 150 (+1p; Cs)

136Xe+198Pt (Ec.m.≈ 644.98 MeV)

GRAZING w/ evap. 120 135 150 120 135 150 120 135 150 (0p; Xe) 120 135 150 (-1p; I) 120 135 150 120 135 150 (-2p; Te)

MASS NUMBER

120 135 150 120 135 150 (-3p; Sb) 120 135 150 10-3 10-2 10-1 100 101 102 103 120 135 150 (-4p; Sn)

σ (mb)

10-3 10-2 10-1 100 101 102 103 120 135 150

Sekizawa, Phys. Rev. C 96, 014615 (2017)

Reaction above the Coulomb barrier

Simenel, C., 2010, Phys. Rev. Lett. 105, 192701.

136Xe + 198Pt

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.

• 12C

– 12C + & → 16O + * – 16O + & → 20Ne + * – 20Ne + & → 24Mg + * – …

S-factor

– 1 2 =

454676 8

– –

9(;) = exp(−2@A) A = BCBDED ℏG

H ; = 1 ; 9 ; ×J(;) Astrophysical S-factor

Classical Hamiltons form

The TDDFT can be described by the classical form. The canonical variables Number of variables = Number of ph degrees of freedom

 ξ ph = ∂H ∂π ph  π ph = − ∂H ∂ξ ph

Blaizot, Ripka, Quantum Theory of Finite Systems (1986) Yamamura, Kuriyama, Prog. Theor. Phys. Suppl. 93 (1987)

K L, @ = ; N(L, @) NOOP = L + Q@ L + Q@ R

OOP

LOS,@OS NSSP = 1 − L + Q@ R L + Q@

SSP

NOS = L + Q@ 1 − L + Q@ R L + Q@

OS

i ∂ ∂t ρ(t) = ⇥h[ρ(t)], ρ(t)⇤ , hkl ⌘ ∂E[ρ] ∂ρlk . ) are identical to

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 XYZ L @Y@Z + 1(L) expanded up to 2nd order in @ [α, \ = (Uℎ)]

• Point transformation LY, @Y → T^, U^

U^ =

_`a _bc @Y, @Y = _bc _`a U^

• Hamiltonian

K d = K d T, U ≈ 1 2 X e^f T U^Uf + 1(T)

Decoupled submanifold

• Collective canonical variables (T, U)

– LY, @Y → T, U; Th , Uh; i = 2, ⋯ , kOS – Decoupled collective subspace Σ – Σ defined by (Th, Uh) = (0,0)

• Decoupled eq. of motion

U̇h = −

_n _bo − C D _p e _bo UD = 0

Ṫ h = X ehCU = 0 [ K d = K d T, U ≈

C D X

e^f T U^Uf + 1(T) ]

P(q) = i d/dq q1 q2 P2(q) = i d/dq2 P1(q) = i d/dq1 (a) (b)

Symplectic formulation

• Collective canonical variables (T, U)

– LY, @Y → T, U; Th , Uh; i = 2, ⋯ , kOS

• Equations for a decoupled submanifold

_n _`a − _n _b _b _`a = 0

→ LY on Σ XZs t

s _n _`a _b _`u = vD _b _`a

→

_b _`a

t

s _n _`a ≡ _6n _`x_`a − Γ Ys Z _n _`u

Covariant derivative with Γ

Ys Z = C DzZ{ zY{,s + zs{,Y − zYs,{

using the metric zYZ ≡ ∑

_bc _`a _bc _`u ^

Riemannian formulation

• Rewriting a curvature term

_b _`a_`u in Γ Zs Y by

using the decoupled condition

• Equations for a decoupled submanifold

_n _`a − _n _b _b _`a = 0

Moving mean-field eq. XZs t

s _n _`a _b _`u = vD _b _`a

Moving RPA eq.

t

s _n _`a ≡ _6n _`x_`a − Γ Ys Z _n _`u

Covariant derivative with Γ

Ys Z = C DzZ{ zY{,s + zs{,Y − zYs,{

using the metric zYZ ≡ XYZ , XYsXsZ = δY

Z

Numerical procedure

L T,Y = ~T ~LY

_n _`a − _n _b _b _`a = 0

Moving mean-field eq. XZs t

s _n _`a _b _`u = vD _b _`a

Moving RPA eq. Move to the next point

LY + LY = LY + ÄL,b

Y

Moving MF eq. to determine the point: LY

L,b

Y = ~LY

~T

Tangent vectors (Generators)

Canonical variables and quantization

• Solution

– 1-dimensional state: ξ T – Tangent vectors:

_b _`a and _`a _b

– Fix the scale of T by making the inertial mass X e =

_b _`a XYZ _b _`a = 1

• Collective Hamiltonian

– K dÇÉÑÑ T, U =

C D UD + 1

e(T), 1 e T = 1(ξ T ) – Quantization T, U = Qℏ

3D real space representation

X [ fm ] y [ fm ]

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)

• 3D space discretized in lattice
• BKN functional
• Moving mean-field eq.: Imaginary-time method
• Moving RPA eq. Finite amplitude method (PRC

76, 024318 (2007) )

At a moment, no pairing

1-dimensional reaction path extracted from the Hilbert space of dimension of 104 ~105.

• Reaction path
• After touching

– No bound state, but – a resonance state in 8Be

Simple case: α + α scattering

α particle4He α particle4He

8Be: Tangent vectors (generators)

δρ at G.S.

• 6
• 3

3 6

y [fm]

• 6
• 4
• 2

2 4 6

z [fm]

• 0.021
• 0.014
• 0.007

0.007 0.014 0.021

Density [fm-3]

ρ at G.S.

• 6
• 3

3 6

y [fm]

• 6
• 4
• 2

2 4 6

z [fm]

0.04 0.08 0.12 0.16

Density [fm-3]

δρ at R = 7.2 fm

• 6
• 3

3 6

y [fm]

• 6
• 4
• 2

2 4 6

z [fm]

• 0.06
• 0.03

0.03 0.06

Density [fm-3]

ρ at R = 7.2 fm

• 6
• 3

3 6

y [fm]

• 6
• 4
• 2

2 4 6

z [fm]

0.04 0.08 0.12 0.16

Density [fm-3]

N(2 ⃗) N(2 ⃗) Tangent vectors (Generators)

8Be: Collective potential

• 40
• 38
• 36
• 34
• 32
• 30

3 3.5 4 4.5 5 5.5 6 6.5 7

V(R) [MeV] R [fm]

ASCC 4e2/R + 2Eα

Represented by the relative distance R Transformation: T → Ü

1 Ü = 1(T Ü )

R

8Be: Collective inertial mass

Ground (resonance) state

2 2.5 3 3.5 4 3 3.5 4 4.5 5 5.5 6 6.5 7 M(R)/m R [fm]

Reduced mass X e(Ü) = ~Ü ~T X e ~Ü ~T = ~Ü ~T

D

á d(Ü) = 1 X e(Ü)

Transformation: T → Ü

á d(Ü) → 2à

α + α scattering (phase shift)

• 150
• 100
• 50

50 100 150 200 250 300 5 10 15 20 25 30 35

δ [deg] E [MeV]

L = 0 L = 2 L = 4 L = 6

Nuclear phase shift

Effect of dynamical change of the inertial mass Dashed line: Constant reduced mass ( á Ü → 2à)

16O + α scattering

• Important reaction to synthesize heavy

elements in giant stars

– Alpha reaction

16O 4He 20Ne

16O + α to/from 20Ne

Reaction path

20Ne

20Ne: Inertial mass

Reduced mass á Ü → 3.2à

Strong increase in the mass near the ground state of 20Ne

T Ü á(Ü) á T = 1

Ü

20Ne: Collective potential

• 146
• 144
• 142
• 140
• 138
• 136
• 134
• 132
• 130

4 4.5 5 5.5 6 6.5 7 7.5 8

V(R) [MeV] R [fm]

16O + α

Quantum tunneling

Alpha reaction16O + α

Synthesis of 20Ne

0.5 1 2 3 4 5 0.5 1 1.5 2 2.5 3

S factor (107MeV b)

16O + α

with reduced mass with ASCC mass

E [ MeV ]

H ; = 1 ; 9 ; ×J(;)

Fusion reaction: Astrophysical S-factor

Effect of dynamical change of the inertial mass Dashed line: Constant reduced mass ( á Ü → 3.2à) With á Ü With ã

16O+16O → 32S: Reaction path

Starting from two 16O configuration

• 250
• 240
• 230
• 220

4 5 6 7 8 9

E [MeV] R [fm]

16O + 16O

16O+16O → 32S: Collective potential

Superdeformed state

Ground state

16O+16O → 32S: Collective mass

Reduced mass áå8

çé(Ü)

áå8

é (Ü)

á(Ü)

• 250
• 240
• 230
• 220

200 400 600 800 1000 1200 1400

E [MeV] 〈 r2Y20 〉

16O + 16O

1022 1023 1024 1025 1026 2 4 6 8 10

S factor [MeV b]

• Ec. m. [MeV]

16O + 16O

with reduced mass with ASCC mass

Fusion reaction16O + 16O

Effect of dynamical change of the inertial mass hinders the fusion cross section by 2 orders of magnitude.

With á Ü With ã

Summary

• Missing correlations in nuclear density

functional

– Correlations associated with low-energy collective motion

• Re-quantize a specific mode of collective

motion

– Derive the slow collective motion – Quantize the collective Hamiltonian – Applicable to nuclear structure and reaction

– – –

• –

–