John Daoutidis

October 5th 2009

Technical University Munich

TitleContinuum Relativistic Random Phase Approximation in Spherical

Nuclei.

Continuum Relativistic Random Phase Approximation in Spherical

Nuclei.

Contents of the talkContents of the talk

1.1. Density Functional TheoryDensity Functional Theory in relativistic static phenomena, in relativistic static phenomena,

1.1. Method to describe nuclear collective phenomena (RPA),Method to describe nuclear collective phenomena (RPA),

1.1. Exact treatment of the coupling to the continuum,Exact treatment of the coupling to the continuum,

1.1. Results in spherical nuclei and comparison with experiment,Results in spherical nuclei and comparison with experiment,

1.1. Conclusions.Conclusions.

Contents

Density functional

theory

density matrix ρ r ,r'=∑i=1

A

∣ϕ i r ⟩ ⟨ϕ ir' ∣

h=δEδ iiih ˆ

Mean field: Eigenfunctions:

V = δ 2 Eδ δ ρ ρ

Interaction:

HE

Density functional theoryDensity functional theory

exact!

Point-coupling model

RELATIVISTIC POINT-COUPLING RELATIVISTIC POINT-COUPLING INTERACTIONSINTERACTIONS RELATIVISTIC POINT-COUPLING RELATIVISTIC POINT-COUPLING INTERACTIONSINTERACTIONS

+ gradients (finite range)

σ ω ρ

J=0, T=0 J=1, T=0 J=1, T=1

Covariant DFT Dirac s.p. equation

+ density dependent couplings

Contents

Static properties Static properties (binding energies, nuclear radii, deformations, etc). Collective excitations Collective excitations

(surface oscillations, rotations, etc.)

Static DFTStatic DFT

How can we explain reactions that lead to collective phenomena, from individual motion?

TD-DFT Random Phase ApproximationTD-DFT Random Phase Approximationsmall ampl. limit

RPA

2 13

2

8 ( 1)

[(2 1)!!]

LL

SL L c

2

ˆ0 | |

LF

†0 | | .vi tat a a e c c

0t t

0 : ground state density

Photoabsorbtion cross section

Strength function

Time Dependent DFT:

Exact Coupling to Continuum

How do we solve RPA ?How do we solve RPA ?

S

Linear Response Formalism

Configuration Space Formalism

Method #1: Configuration space formalism

Method #1: Configuration space formalism

RPA matrix equation:

Dimension determined by the size of the 1p-1h configuration.

AB matrix

min min* *min min

0j j ph

j j hp

min

min

( )j m i mn ij mnij

j mnij

ˆ| |mnij im V jn

2

ˆˆ ˆ

EV

Interaction:

Limitations of conf. space formalismLimitations of conf. space formalism

1) The p-h configuration space can be very large in the case of medium or heavy nuclei

2) Relativistic RPA requires also transitions to Dirac sea (antiparticles)

Large dimension of RRPA matrix (>7000),Large numerical effort

discrete spectrum

Artificial width:Lorentzian with smearing parameter 2Δ

ω [MeV]

S(

ω)

0

3) Approximate treatment of the continuum ( put the nucleus in a box)

2

ˆ0 | |LS F

2

2 2

1ˆ0 | |LF

4)

Method #2: Linear Response FormalsimMethod #2: Linear Response Formalsim

Linearized Bethe-Salpeter equation:

Simple matrix equation or rank 350 (7 meson channels, 50 r-mesh points)Simple matrix equation or rank 350 (7 meson channels, 50 r-mesh points)

Can have a continuous spectrum (resonance width) if RCan have a continuous spectrum (resonance width) if R00cc’cc’ is exact. is exact.

Response

____

0 0' ' ' ' ' ' ' ' ' '

' '

phaba b aba b abcd cdc d c d a b

cdc d

R R R V R 10

1ph

RR V

†' '

'

1Im

c cc c

cc

S F R F

' '' '

΄ ΄R F

sum of separable terms:

( )ph c ph cc

c

V Q r Q

{ , , }c c c c 0cD T LQ

0 † 0 ''

'

c ccc

cc

R Q R Q

' 10

'

1

cc ph

cc

RR

Full Response function:

Free Response functionFree Response function

J. Daoutidis and P. Ring PRC 80 (2009) 024309

continuum

1. Full continuum and Dirac sea are included (no truncation),

2. Escape width is automatically reproduced,3. One order of magnitude faster numerical calculations.

-u(r) and w(r) are the exact scattering wave functions solutions of Dirac equation for arbitrary energies.

'

'0

0 | | | | 0. .c c

ccv v

Q v v QR b g

E E

'0

'

| | | |. .c c

ccph p h

h Q p p Q hR b g

0'

1| | . .

ˆcc c ch h

R h Q Q h b gh

| ( ) | . .c h ch

h Q G Q h b g

*

*

( ; ) ( '; '( , '; )

( ; ) ( '; '

w r E u r E r rG r r

u r E w r E r r

Free Response function:

neglect υph

OverviewOverview

•Density Functional TheoryDensity Functional Theory in relativistic static phenomena, in relativistic static phenomena,

•Method to describe nuclear collective phenomena (RPA),Method to describe nuclear collective phenomena (RPA),

•Exact treatment of the coupling to the continuum,Exact treatment of the coupling to the continuum,

•Results in spherical nuclei and comparison with experiment,Results in spherical nuclei and comparison with experiment,

•Conclusions.Conclusions.

Contents

Isoscalar Giant Monopole Resonance (breathing mode)Isoscalar Giant Monopole Resonance (breathing mode)

ResultsResults

ISGMR

Continuum RRPA with PC-F1 force: Continuum RRPA with PC-F1 force: J. Daoutidis, P. Ring,J. Daoutidis, P. Ring, PRC 80 (2009) 024309 Discrerte RRPA with PC-F1 force: Discrerte RRPA with PC-F1 force: Niksic et. al. PRC 72 (2005) 014312 Niksic et. al. PRC 72 (2005) 014312

Isovector Giant Dipole ResonanceIsovector Giant Dipole Resonance

Giant isovector dipole oscillations -> neutrons oscillate against protons.

tCRPA =1/20 tDRPA !

. 13.30 0.10ExpE MeV

13.28CRPAE MeV

Isovector Giant Pygmy: IV-GPRIsovector Giant Pygmy: IV-GPR

• Depends on the coupling to the continuum!

Quasiparticle CRPAQuasiparticle CRPA

Pairing correlations play a crucial role for open shell nuclei.RMF+BCS : - Simple - Successful in nuclei when F not close to the continuum (drip lines)

pp

Quasiparticle CRPAQuasiparticle CRPA

Applications: Isovector Giant Dipole Resonances

QcRPA 1-

• We have formulated the continuum QRPA based on the Point Coupling Relativistic mean field theory (PC-F1).

• We applied the continuum QRPA to multipole giant resonances in double magic nuclei as well as spherical open shell Tin isotopes.

• There are quantitative improvements, compared to the discrete RPA (escape widths, minimizing numerical effort, etc.)

Summary and outlookSummary and outlook

•Extend the model to meson exchange forces for better qualitative comparison.

•Include Relativistic Hartree Bogoliubov (RHB) theory in the static problem in order to treat pairing correlations at the cases where BCS fails (drip lines, halo nuclei).

•Extend to include deformed nuclei.

SUMMARY