Angular Momentum Angular Momentum Projection in TDHF Projection in TDHF

Dynamics :Dynamics :application to Coulomb application to Coulomb excitation and fusionexcitation and fusion

C. Simenel1,2

In collaboration with

M. Bender2, T. Duguet2, F. Nunes2

1) CEA-SPhN, Saclay 2) MSU/NSCL, US

Motivations

Nuclear reactions• elastic, inelastic, deep-inelastic, transfer, break-up, fusion, fission…• Coulomb + nuclear interactions, couplings, multistep process, tunnel effect…

Whole nuclear chart • from light to superheavy elements • from proton to neutron drip lines

Fully microscopic theory

effective interaction (Skyrme, Gogny…) Beyond mean-field

• long range dynamical correlations• mixing of trajectories

Present status

Time dependent mean field

• Time-Dependent Hatree-Fock theory P.A.M. Dirac, Proc. Camb. Phil. Soc. 26, 376 (1930)

• First application to nuclear physics Y.M. Engel et al., NPA 249, 215 (1975) P. Bonche, S. Koonin and J.W. Negele, PRC 13, 1226 (1976)

• 3D calculations of nuclear reactions H. Flocard, S.E. Koonin and M.S. Weiss, PRC 17, 1682 (1978) K.-H. Kim, T. Otsuka and P. Bonche, JPG 23, 1267 (1997) C. S., P. Chomaz and G. de France, PRL 86, 2971 (2001) C. S., P. Chomaz and G. de France, PRL 93, 102701 (2004)

• no pairing (except QRPA)

Present status

Beyond time dependent mean field

• Extended TDHF D. Lacroix, P. Chomaz and S. Ayik, PRC 58, 2154 (1998)

• Time Dependent Density Matrix S.J. Wang and W. Cassing, Ann. Phys. 159, 328 (1985) M. Tohyama, PRC 36, 187 (1987)

• Stochastic TDHF O. Juillet and P. Chomaz, PRL 88, 142503 (1998)

• Time Dependent Generator Coordinate Method P.-G. Reinhard, R.Y. Cusson and K. Goeke, NPA 398, 141 (1983) J.F. Berger, M. Girod and D. Gogny, NPA 428, 23c (1984) H. Goutte, J.F. Berger, P. Casoli and D. Gogny, PRC 71, 024316 (2005)

• Projected TDHF (present work)

Today's objectives: reactions with a deformed projectile

effect of the initial orientation on the reaction many TDHF trajectories

Coulomb excitation (rotation)

• angular momentum projection to calculate the J-population • Interferences between initial orientations

Fusion

• incoherent mixing of TDHF trajectories realistic fusion probability (between 0 and 1)• Effect of Coulomb excitation in the approach

I) Projected TDHF : formalism

A) Projection on angular momentum

• static case angular momentum "projector":

rotated Slater determinant:

• evolution no feed back of the correlations - on the Slater evolution TDHF trajectories - on the superposition functions f() is constant

initial state correlations in the observation only

€

JMΦ =1

NgK

*

K=−J

J

∑ ˆ P MKJ Φ

= dr Ω ∫ f

r Ω ( ) Φ

r Ω ( )

€

Ψ t( ) = ˆ U (t) JMΦ

= dr Ω ∫ f

r Ω , t( ) Φ

r Ω , t( )

€

ˆ P MKJ = JM JK

€

Φ r

( ) = ˆ R r Ω ( ) Φ

€

≈ dr Ω ∫ f

r Ω ( ) ΦTDHF

r Ω , t( )

I) Projected TDHF : formalism

B) Projection on angular momentum

• exact J-population

we use high computationnal cost

• approximated J-population accurate if the vibration and the rotational speed are small

€

PJMex (t) = JM Ψ(t)

2

= Ψ(t) ˆ P MMJ Ψ(t)

= JiM iΦ ˆ U +(t) ˆ P MMJ ˆ U (t) JiM iΦ

€

PJMap (t) = JMΦ Ψ(t)

2

= JMΦ ˆ U (t) JiM iΦ2

€

ˆ P MMJ = JM JM

I) Projected TDHF : formalism

C) The code: symmetries and numerical tests

• initial condition: isotropic distribution of (J=0)

• axial symmetry of small impact parameter or small dynamical deformation of (t) or sudden approximation

• the HF g.s. has an axial symmetry, a time-reversal symmetry and a good parity

• the evolved Slater determinants have a plane of symmetry only one collision partner can be deformed

• no charge mixing in the s.p. wave functions

€

Φ(r Ω )

€

Ψ(t)

€

Φ

€

ΦTDHF

r Ω , t( )

I) Projected TDHF : formalism

C) The code: symmetries and numerical tests

• explicit expression of the JM-population

- exact with

and

- approximated€

PJMex (t) = δM 0

2J +1

2π 2NJ2

dβ sinβ d00J β( )

0

π

∫

dβ1dβ 2 sinβ1 sinβ 20

π / 2

∫ dα 1dα 20

π

∫∫∫

Φ β1, t( ) e iα 1ˆ J x e−iβ ˆ J z e iα 2

ˆ J x Φ β 2, t( )

€

PJMap (t) = δM 0

2J +1

πNJ N0

dβ1dβ 2 sinβ1 sinβ 20

π / 2

∫∫

d00J β1( ) dα

0

π

∫ Φ β1, t → −∞( ) e iα ˆ J x Φ β 2, t( )

2

€

NJ = Φ HFˆ P 00

J Φ HF

€

Φ β, t( ) = ˆ U TDHF t( )e−iβ ˆ J z Φ HF

I) Projected TDHF : formalism

C) The code: symmetries and numerical tests

• orthonormalization: convergence with the number of rotational angles

1- |

‹0|0

› |,

| ‹J|

0› |

II) Coulomb excitation : rotational band

A) Classical calculation

• rotation due to Coulomb repulsion

• effects on induced fission 130Xe + 238U (E<B)

f(D)

D

Theoretical calculation Holm et al., PLB 29, 473 (1969)

II) Coulomb excitation : rotational band

A) Classical calculationK.Alder and A. Winther, Electromagnetic Excitation (1978)C. S., P. Chomaz and G. de France, PRL 93, 102701 (2004)

- point like target

- small

- small

- differential equation :

with and

- solution :

- reorientation (=1) :

216

5 βπ

ε =

€

δϕ (ξ ) =3εA2

A1 + A2

sin 2ϕ ∞( ) ξ 1− 1−ξ −1( ) + ln

2

1+ 1−ξ −1

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟−

1

2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

∂∂

ϕ + 2ξ ξ −1( )∂2

∂ξ 2ϕ =

9ε

2ξ

A2

A1 + A2

sin 2ϕ ∞( )

0D

D=

E

ZZeD 21

2

0 =

Z1, A1 Z2, A2

D

€

Δϕ =3εA2

A1 + A2

sin 2ϕ ∞( )1

2+ ln2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

δϕ (t) = ϕ (t) −ϕ ∞

II) Coulomb excitation : rotational band

B) TDHF approach

• self consistent mean field theory

• independant s.p. wave functions

• mean values of one body observables (ex : orientation)

• quantal treatment of inertia

• P. Bonche code K.-H. Kim, T. Otsuka and P. Bonche, JPG 23, 1267 (1997)

deformed projectile + Coulomb potential of the target

• Skyrme forces (SLy4d) T. Skyrme, Phil. Mag. 1 (1956)

( )[ ] ρρρ &ih =,

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sont requis pour visionner cette image.

II) Coulomb excitation : rotational band

B) TDHF approach

• 24Mg (+ 208Pb)

• ECM = 112 MeV (≈B)

• Dinit. = 220fm

• head on collision`

• Rutherford trajectory

• approach phase only

II) Coulomb excitation : rotational band

B) TDHF approach

• 24Mg (+ 208Pb)

• ∞ = 45 deg.

• population of J ?

C. S., P. Chomaz and G. de France, PRL 93, 102701 (2004)

D0

∞ = 45

II) Coulomb excitation : rotational band

C) PTDHF: excitation probability

• set of projected states

- A. Valor, P.-H. Heenen and P. Bonche, NPA 671, 145 (2003) - M. Bender, H. Flocard and P.-H.Heenen, PRC 68, 044321 (2003) et al.,

• 24Mg

€

JMΦ{ }

II) Coulomb excitation : rotational band

C) PTDHF: excitation probability

• time evolution of the J-population

• 24Mg (+ 208Pb)

• ECM = 112 MeV (≈B)

• head on collision

• approaching phase only

Time (fm/c)

PJ(

t)

II) Coulomb excitation : rotational band

C) PTDHF: excitation probability

• 24Mg (+ 208Pb) @ ECM=690 MeV ~ 6B

• angular distribution

• interferences between orientations

• still need nuclear potential (target) and interferences between scattering angles

PJ(

t∞

)

c.m. scattering angle (deg.)

J=2

J=0

€

PJ t → ∞( ) ≈ 2PJ t = 0( )

Semi-classical

PTDHF"improved" TDHF

III) Fusion of deformed nuclei at the barrier

• 24Mg+208Pb @ 94 MeV

• head-on collision

• initial distance: 20 fm

• the fusion probability depends on the initial orientation

QuickTime™ et undécompresseur GIF

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QuickTime™ et undécompresseur GIF

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III) Fusion of deformed nuclei at the barrier

• 24Mg+208Pb

• initial distance: 20 fm no long range Coulomb excitation

• Isotropic model : red line B0 ~ 97 MeV ε ~ 0.06 ~ εSLy4d/2

• barrier ~ 10% lower than expected (collision term ?)

CM

Ene

rgy

(MeV

)

Orientation β at 20fm

€

B β( ) ≈ B0 1−εAp

1/ 3

Ap1/ 3 + AT

1/ 32 − 3sin2 β( )

⎡

⎣ ⎢

⎤

⎦ ⎥

FUSION

SCATTERING

0 π/4 π/2

III) Fusion of deformed nuclei at the barrier

• Fusion probability - Isotropic distribution at D=20fm : blue line - Isotropic distribution at D=220fm : red points

• reduction of the fusion due to Coulomb excitation

• no concluding effect of nuclear excitation on the fusion probability

Pfu

sP

fus/P

0

ECM (MeV)

Conclusions and perspectives

PTDHF • approximated angular momentum projection on TDHF trajectories • beyond mean field for the observation, ex: PJ(t)• Coulomb excitation strong effect of the interferences• fusion reduction of the fusion due to Coulomb excitation

exact projection interferences between scattering angles effect of interferences (orientations) on fusion ?

feed back of the correlations on the evolution TDGCM pairing (TDHF)

annexe

Plan

I) Projected TDHF: Formalism• Time Dependent Generator Coordinate Method• Projection on angular momentum • The code: symmetries and numerical tests

II) Coulomb excitation: 24Mg rotationnal band• Classical calculation• TDHF approach• PTDHF: excitation probability

III) Fusion of deformed nuclei at the barrier• Rotationnal couplings in the entrance channel• Beyond mean field results

IV) Conclusions and perspectives

I) Projected TDHF : formalism

A) TDGCM

• wave function q: collective variable f : superposition function Φ: Slater determinant

P.-G. Reinhard et al.

Φ(t) ΦTDHF(t) J.F. Berger et al. + H. Goutte et al.

Φ(t) ΦHFB

• time dependent Griffin-Hill-Wheeler equation

€

Ψ t( ) = dq∫ f q, t( ) Φq t( )

€

dq'∫ Φq t( ) H − i∂

∂tΦq ' t( ) f q', t( )

⎧ ⎨ ⎩

−i Φq t( ) Φq ' t( )∂f q', t( )

∂t

⎫ ⎬ ⎭

= 0

I) Projected TDHF : formalism

C) The code: symmetries and numerical tests

• explicit expression of the JM-population

- exact with

and

- approximated

- without interferences

€

PJMex (t) = δM 0

2J +1

2π 2NJ2

dβ sinβ d00J β( )

0

π

∫

dβ1dβ 2 sinβ1 sinβ 20

π / 2

∫ dα 1dα 20

π

∫∫∫

Φ β1, t( ) e iα 1ˆ J x e−iβ ˆ J z e iα 2

ˆ J x Φ β 2, t( )

€

PJMap (t) = δM 0

2J +1

πNJ N0

dβ1dβ 2 sinβ1 sinβ 20

π / 2

∫∫

d00J β1( ) dα

0

π

∫ Φ β1, t → −∞( ) e iα ˆ J x Φ β 2, t( )

2

€

PJMap' (t) = δM 0

2J +1

4dβ sinβ d00

J β( )0

π

∫

Φ β, t → −∞( ) Φ β , t( )2

€

NJ = Φ HFˆ P 00

J Φ HF

€

Φ β, t( ) = ˆ U TDHF t( )e−iβ ˆ J z Φ HF

I) Projected TDHF : formalism

C) The code: symmetries and numerical tests

• test of the overlap between Slater determinants

€

ΦHFˆ U TDHF t( ) Φ HF

€

E =2πh

T= 340.4MeV

≈ ei

i

∑ = 341.6MeV

• 24Mg (+ 208Pb) @ ECM=690 MeV ~ 6B