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![Page 1: John Daoutidis October 5 th 2009 Technical University Munich Title Continuum Relativistic Random Phase Approximation in Spherical Nuclei.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649d0d5503460f949e1ac7/html5/thumbnails/1.jpg)
John Daoutidis
October 5th 2009
Technical University Munich
TitleContinuum Relativistic Random Phase Approximation in Spherical
Nuclei.
Continuum Relativistic Random Phase Approximation in Spherical
Nuclei.
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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
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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!
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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
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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?
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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:
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Exact Coupling to Continuum
How do we solve RPA ?How do we solve RPA ?
S
Linear Response Formalism
Configuration Space Formalism
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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:
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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)
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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
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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
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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
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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
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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
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Isovector Giant Pygmy: IV-GPRIsovector Giant Pygmy: IV-GPR
• Depends on the coupling to the continuum!
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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
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Quasiparticle CRPAQuasiparticle CRPA
Applications: Isovector Giant Dipole Resonances
QcRPA 1-
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• 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
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