Optical potentials and knockout reactions from Green functions - - PowerPoint PPT Presentation

Optical potentials and knockout reactions from Green functions treatment

Andrea Idini

“Connecting Bound States to the Continuum” FRIB, 11-22 March 2018 Andrea Idini

15/06/2018 Andrea Idini

Green functions many-body method

!" # + %& = (

)

1 + − -)

./01 ± %&

H(A) = T Tc.m.(A + 1) + V + W

Källén–Lehmann spectral representation Unperturbed case Green function self-consistent methods find spectra of the Hamiltonian operator

!34 # + %& = (

5

6"

7 83 65 79: 65 79: 84 9 6" 7

# − +5

79: + +" 7 + %&

+ (

)

6"

7 83 9 6) 7;: 6) 7;: 84 6" 7

# − +"

7 + +) 7;: − %&

Σ∗

= +

Dyson Equation

‘‘Dressed’’ (with correlation) Particle Propagator ‘‘Bare’’ Propagator

Green functions many-body method

15/06/2018 Andrea Idini

% & + '( = %) & + '( + %) & + '( Σ∗ & + '( %(& + '() H

H(A) = T Tc.m.(A + 1) + V + W

H

Σ∗

= +

Dyson Equation

Interaction between the particle and the system (physical choice) Fragments and changes energy

• f the ‘‘bare’’ state

‘‘Dressed’’ (with correlation) Particle Propagator ‘‘Bare’’ Propagator

Green functions many-body method

15/06/2018 Andrea Idini

% & + '( = %) & + '( + %) & + '( Σ∗ & + '( %(& + '()

Σ,- & + '( = .

/

0,

/0- /

& − 2/ + '(

Escher & Jennings PRC66 034313 (2002)

Σ∗ = +

Σ corresponds to the Feshbach’s generalized optical potential Dyson Equation Equation of motion Corresponding Hamiltonian

15/06/2018 Andrea Idini

% + ℏ' 2) ∇+

'

, -, -/; %, Γ = 2 - − -/ + ∫ 5-′′Σ∗ -, -//; %, Γ ,(-//, -; %, Γ) 9 -, -/ = − ℏ' 2) ∇+

' + Σ∗ -, -/; %, Γ

Green functions many-body method

, : + ;< = ,= : + ;< + ,= : + ;< Σ∗ : + ;< ,(: + ;<)

Σ∗

= +

Particle hole ‘polarization’ propagator (ph-RPA) Particle-particle (pp-RPA) two-body correlation ‘ladder’ propagator

Faddeev RPA ADC(3) n p

HF ADC(1)

15/06/2018 Andrea Idini

Courtesy of C. Barbieri

Hamiltonian method: self consistent Green functions

% & + '( = %) & + '( + %) & + '( Σ∗ & + '( %(& + '()

Independent Particle Collective Phonon mean field Random Phase Approximation

15/06/2018 Andrea Idini

(Non) Hamiltonian method: nuclear field theory ansatz

Σ∗

= +

Σ∗ =

Nuclear Field Theory Self Consistent Green Function

Σ∗ =

coupling from Hamiltonian matrix elements Coupling of physical quantities

Σ∗(&)

Dyson Equation

Vertices Summation (&)

Vertices (ME)

((), &) particle structure self-energy (optical potential) Full single valence space Exploits different truncations

Building Blocks Constitues Central Part

15/06/2018 Andrea Idini

How the imaginary part arises in dissipative systems Σ"# $ + &' = )

*

+"

*($)+# *($)

$ − /* + &' 0"# $ + &' = $ + &' − Σ"# $ + &'

12

Complex roots of the Green function Implemented in NFT

Energy Strength Energy HF 1st Iteration FWHM ∝ '

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How the imaginary part arises in dissipative systems Σ"# $ + &' = )

*

+"

*($)+# *($)

$ − /* + &' 0"# $ + &' = $ + &' − Σ"# $ + &'

12 Energy Strength Energy Energy HF 1st Iteration FWHM ∝ ' FWHM ∝ 4+{Σ} Convergence, ' → 0

Complex roots of the Green function Implemented in NFT

The irreducible self-energy is a nucleon- nucleus optical potential*

Σ"#

∗

% + '( = Σ"#

* + + ,

• "

, -# ,∗

% − /, ± '(

Nu Nucleon elastic scattering

Σ123

E

*Mahaux & Sartor, Adv. Nucl. Phys. 20 (1991), Escher & Jennings PRC66:034313 (2002)

Σ143

/5 correlated mean-field resonances beyond mean-field

Σ*

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Correlations

Courtesy of C. Barbieri

Σ","$

%,&∗ ())

Σ%,&∗(+, +,, ))

Σ∗

= +

• Solve Dyson equation in HO Space, find

+/ 21 2%,& + + ∫ 4+,+,/ Σ%,&∗ +, +,, ) 2%,& +′ = E 2%,&(+)

• diagonalize in full continuum momentum space

Σ","$

%,&∗ ())

Nmax

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!" = $ Φ&

'() Φ*.,. '

• Kn

Knockout Sp Spect ctrosc scopic Factor

• rs

.- 20 12,4 . + ∫ 7.8.8- Σ2,4∗ ., .8, ; 12,4 .′ = E 12,4(.)

Norm of overlap wavefunctions But also the shape of the overlap wavefunction!

Collaboration with C. Bertulani

Con Conclusion

• ns (1

(1)

• The non-local generalized optical potential corresponding to

nuclear self energy can be calculated in several, different, ways.

• Imaginary part can arise spontaneously in non-hamiltonian

cases.

• Reaction properties calculated from bound state description

might differ from effective pure single-particle description.

15/06/2018 Andrea Idini

Energy FWHM ∝ "#{Σ} Convergence, ' → 0 Strength

Navràtil, Roth, Quaglioni, PRC82, 034609 (2010)

Σ"

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SRG-N3LO, Λ = 2.66 fm*+ , + +.O 0. 1.

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0.0 2.5 5.0 7.5 10.0 12.5 15.0

Ec.m. (MeV)

−100 100 200

δ (deg)

3/2+ 1/2+ 5/2+ 2 4 6 8 10 12 14 16

Ec.m. (MeV)

−100 100 200

δ (deg)

3/2− 5/2− 1/2− 7/2−

NNLOsat ! + #$O &. (. +)*+ n p

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Using the ab initio optical potential for neutron elastic scattering on Oxygen

20 40 60 80 100 120 140 160 180 0.001 0.01 0.1 1 10 Lister and Sayres, Phys Rev 143, 745

15/06/2018 Andrea Idini 20 40 60 80 100 120 140 160 180 0.01 0.1 1 10

!" = 2.76 MeV

Ψ" # = %∫ '#

( … '# *Φ *,(

• (#

(, … , # *,()Φ *

• (#

(, … , # *)

#

"

#

"

Overlap function

EM results from A. Cipollone PRC92, 014306 (2015)

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Proton particle-hole gap

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• pen circles neutrons, closed protons

!" = $ Φ&

'() Φ*.,. '

• Kn

Knockout Sp Spect ctrosc scopic Factor

• rs

.- 20 12,4 . + ∫ 7.8.8- Σ2,4∗ ., .8, ; 12,4 .′ = E 12,4(.)

Calculated from overlap wavefunctions

20 40 60 80 100

5 10 15 20 25 30

Spectroscopic Factor (%) Separation Energy (MeV) Oxygen Chain

O14 O16 O22 O24

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Collaboration with C. Bertulani

Ov Overlap wa wavefunctions

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Collaboration with C. Bertulani Deviation of quasifree !, !# cross section calculation for different wavefunctions (%&' − %)*)/%)*

Nucleus EB ⌦ r2↵1/2

W S

⌦ r2↵1/2

GF

CW S CGF σW S

qf

σGF

qf

σW S

kn

σGF

kn

C2SGF (state) [MeV] [fm] [fm] [fm−1/2] [fm−1/2] [mb] [mb] [mb] [mb]

14O (π1p3/2) 8.877

2.856 2.961 6.785 7.172 27.38 28.60 27.19 27.42 0.548

5% <1%

Con Conclusion

• ns an

and Pe Perspectives

• We are developing an interesting tool to study nuclear

reactions effectively. We have defined a non-local generalized optical potential corresponding to nuclear self energy.

• Spectroscopic Factors from ab-initio overlap wavefunctions

differ from effective wood saxon. These do not seem to depend much on proton-neutron asymmetry

15/06/2018 Andrea Idini

20 40 60 80 100 120 140 160 180 0.001 0.01 0.1 1 10 Lister and Sayres, Phys Rev 143, 745

Wh Why op

• ptical po

potentials tials?

Koning, Delaroche, NPA713, 231 (2002)

• Optical potentials reduce many-body

complexity decoupling structure contribution and reactions dynamics.

• Often fitted on elastic scattering data

(locally or globally)

• A microscopic model is difficult but

worth it

15/06/2018 Andrea Idini

• Fig. 2. Comparison of predicted neutron total cross sections and experimental data, for nuclides in the Mg–Ca mass region, for the energy range 10 keV–250 MeV. For

50 100 150

[mb/sr] Ω /d σ d

5

10

12

10

19

10

26

10

33

10 Ca

40

n+

< 10

0 < E < 20

10 < E < 40

20 < E < 100

40 < E > 100

E

50 100 150 Ca

40

p+

[deg]

cm

θ

Dickhoff, Charity, Mahzoon, JPG44, 033001 (2017)