Mack C. Atkinson

Exploring the link between finite nuclei and infinite nuclear matter

Mack C. Atkinson

TRIUMF

Progress in Ab Initio Techniques in Nuclear Physics

arXiv 2001.07231 Mack C. Atkinson TRIUMF 1 / 17

arXiv 2001.07231

Introduction

Fundamental properties of nuclear matter:

1 ρ0 ≈ 0.16 fm−3

2 E0 ≈ −16 MeV

Where do the values for ρ0 and E0 come from?

How does this relate to finite nuclei?

Claim: The value of E0 is not necessarily -16 MeV

Introduction

Fundamental properties of nuclear matter:1 ρ0 ≈ 0.16 fm−3

2 E0 ≈ −16 MeV

Introduction

2 E0 ≈ −16 MeV

Introduction

2 E0 ≈ −16 MeV

Introduction

2 E0 ≈ −16 MeV

Introduction

2 E0 ≈ −16 MeV

Introduction

2 E0 ≈ −16 MeV

Outline

1 More discussion of nuclear matter

2 Green’s function formalism

3 The dispersive optical model (DOM)

4 Binding energy densities and nuclear matter

Outline

Nuclear Saturation

Nuclear saturation is evident from nuclear charge radii: RA ∝ A1/3

Nuclear charge densities extracted from elastic electron-scattering experiments saturate inthe coreThese experiments revealed that this saturation density is ρ0 ≈ 0.16 fm−3

0 2 4 6 8 10

r [fm]

Matter Densities

16O40Ca48Ca58Ni

124Sn208Pb

Nuclear Saturation

Nuclear charge densities extracted from elastic electron-scattering experiments saturate inthe core

These experiments revealed that this saturation density is ρ0 ≈ 0.16 fm−3

0 2 4 6 8 10

r [fm]

Matter Densities

16O40Ca48Ca58Ni

124Sn208Pb

Nuclear Saturation

Nuclear charge densities extracted from elastic electron-scattering experiments saturate inthe core

These experiments revealed that this saturation density is ρ0 ≈ 0.16 fm−3

0 2 4 6 8 10

r [fm]

Matter Densities

16O40Ca48Ca58Ni

124Sn208Pb

Nuclear Saturation

Nuclear charge densities extracted from elastic electron-scattering experiments saturate inthe coreThese experiments revealed that this saturation density is ρ0 ≈ 0.16 fm−3

0 2 4 6 8 10

r [fm]

Matter Densities

16O40Ca48Ca58Ni

124Sn208Pb

Saturation Energy

BE (A,Z ) = aVA− aSA2/3 − 3

Z (Z − 1)

Rc− 1

2aA(A− 2Z )2A−1BE (A,Z ) = aVA− aSA

2/3 −

Z (Z − 1)

Rc− 1

2aA(A− 2Z )2A−1BE (A,Z ) = aVA−XXXXaSA

2/3 −

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

XXXXXXXXX

2aA(A− 2Z )2A−1

Bethe-Weizsacker 1935

Infinite nuclear matter1 No Coulomb2 No surface3 Symmetric NM

=⇒ E0/A = aV ≈ −16 MeV

E0 is determined empirically (model dependent)

ρ0 ≈ 0.16 fm−3 is determined experimentally

W. H. Dickhoff, D. Van Neck MBT Exposed! (2008)

M. Baldo et al., PRC 86, 064001, (2012)

Saturation Energy

BE (A,Z ) = aVA− aSA2/3 − 3

Z (Z − 1)

Rc− 1

2aA(A− 2Z )2A−1

BE (A,Z ) = aVA− aSA2/3 −

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

XXXXXXXXX

2aA(A− 2Z )2A−1

=⇒ E0/A = aV ≈ −16 MeV

M. Baldo et al., PRC 86, 064001, (2012)

Saturation Energy

BE (A,Z ) = aVA− aSA2/3 − 3

Z (Z − 1)

Rc− 1

2aA(A− 2Z )2A−1

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

XXXXXXXXX

2aA(A− 2Z )2A−1

=⇒ E0/A = aV ≈ −16 MeV

M. Baldo et al., PRC 86, 064001, (2012)

Saturation Energy

BE (A,Z ) = aVA− aSA2/3 − 3

Z (Z − 1)

Rc− 1

2aA(A− 2Z )2A−1

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

XXXXXXXXX

2aA(A− 2Z )2A−1

Infinite nuclear matter

1 No Coulomb2 No surface3 Symmetric NM

=⇒ E0/A = aV ≈ −16 MeV

M. Baldo et al., PRC 86, 064001, (2012)

Saturation Energy

BE (A,Z ) = aVA− aSA2/3 − 3

Z (Z − 1)

Rc− 1

2aA(A− 2Z )2A−1

Z (Z − 1)

Rc− 1

2aA(A− 2Z )2A−1

BE (A,Z ) = aVA−XXXXaSA2/3 −

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

XXXXXXXXX

2aA(A− 2Z )2A−1

Infinite nuclear matter1 No Coulomb

2 No surface3 Symmetric NM

=⇒ E0/A = aV ≈ −16 MeV

M. Baldo et al., PRC 86, 064001, (2012)

Saturation Energy

BE (A,Z ) = aVA− aSA2/3 − 3

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

Rc− 1

2aA(A− 2Z )2A−1

Z (Z − 1)

Rc− 1

2aA(A− 2Z )2A−1

Z (Z − 1)

XXXXXXXXX

2aA(A− 2Z )2A−1

Infinite nuclear matter1 No Coulomb2 No surface

3 Symmetric NM

=⇒ E0/A = aV ≈ −16 MeV

M. Baldo et al., PRC 86, 064001, (2012)

Saturation Energy

BE (A,Z ) = aVA− aSA2/3 − 3

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

Rc− 1

2aA(A− 2Z )2A−1

Z (Z − 1)

XXXXXXXXX

2aA(A− 2Z )2A−1

=⇒ E0/A = aV ≈ −16 MeV

M. Baldo et al., PRC 86, 064001, (2012)

Saturation Energy

BE (A,Z ) = aVA− aSA2/3 − 3

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

Rc− 1

2aA(A− 2Z )2A−1

Z (Z − 1)

XXXXXXXXX

2aA(A− 2Z )2A−1

=⇒ E0/A = aV ≈ −16 MeV

M. Baldo et al., PRC 86, 064001, (2012)

Saturation Energy

BE (A,Z ) = aVA− aSA2/3 − 3

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

Rc− 1

2aA(A− 2Z )2A−1

Z (Z − 1)

XXXXXXXXX

2aA(A− 2Z )2A−1

=⇒ E0/A = aV ≈ −16 MeV

M. Baldo et al., PRC 86, 064001, (2012)

Saturation Energy

BE (A,Z ) = aVA− aSA2/3 − 3

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

Rc− 1

2aA(A− 2Z )2A−1

Z (Z − 1)

XXXXXXXXX

2aA(A− 2Z )2A−1

=⇒ E0/A = aV ≈ −16 MeV

M. Baldo et al., PRC 86, 064001, (2012)

Saturation Energy

BE (A,Z ) = aVA− aSA2/3 − 3

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

Rc− 1

2/3 −

Z (Z − 1)

Rc− 1

2aA(A− 2Z )2A−1

Z (Z − 1)

XXXXXXXXX

2aA(A− 2Z )2A−1

=⇒ E0/A = aV ≈ −16 MeV

M. Baldo et al., PRC 86, 064001, (2012)

Single-Particle Propagator and the Dyson Equation

G`j(r , r′;E ) =

〈ΨA0 | ar`j |ΨA+1

m 〉〈ΨA+1m | a†r ′`j |ΨA

0 〉E − (EA+1

m − EA0 ) + iη

〈ΨA0 | a†r ′`j |ΨA−1

n 〉〈ΨA−1n | ar`j |ΨA

0 〉E − (EA

0 − EA−1n )− iη

Poles correspond to excitation energies of (A + 1) or (A− 1) nucleus

Numerator like a transition probability to given excitation

Close connection with experimental observables

Perturbation expansion of G leads to the Dyson equation

If the irreducible self-energy (Σ∗) is known, then so is G= + Σ∗

G`j(r , r′;E ) =

m 〉〈ΨA+1m | a†r ′`j |ΨA

0 〉E − (EA+1

m − EA0 ) + iη

〈ΨA0 | a†r ′`j |ΨA−1

0 〉E − (EA

0 − EA−1n )− iη

G`j(r , r′;E ) =

m 〉〈ΨA+1m | a†r ′`j |ΨA

0 〉E − (EA+1

m − EA0 ) + iη

〈ΨA0 | a†r ′`j |ΨA−1

0 〉E − (EA

0 − EA−1n )− iη

G`j(r , r′;E ) =

m 〉〈ΨA+1m | a†r ′`j |ΨA

0 〉E − (EA+1

m − EA0 ) + iη

〈ΨA0 | a†r ′`j |ΨA−1

0 〉E − (EA

0 − EA−1n )− iη

G`j(r , r′;E ) =

m 〉〈ΨA+1m | a†r ′`j |ΨA

0 〉E − (EA+1

m − EA0 ) + iη

〈ΨA0 | a†r ′`j |ΨA−1

0 〉E − (EA

0 − EA−1n )− iη

If the irreducible self-energy (Σ∗) is known, then so is G

= + Σ∗

G`j(r , r′;E ) =

m 〉〈ΨA+1m | a†r ′`j |ΨA

0 〉E − (EA+1

m − EA0 ) + iη

〈ΨA0 | a†r ′`j |ΨA−1

0 〉E − (EA

0 − EA−1n )− iη

The Dispersive Optical Model (DOM)

Irreducible self-energy at positive energies corresponds to an optical potential

Use same functional form as standard optical potentials to parametrize self-energy

Σ∗(r , r ′;E ) is explicitly nonlocal

Dispersion relation connects to negative energies

Dispersive Correction

ReΣ`j(r , r′;E ) = ReΣ`j(r , r

′; εF )− 1

π(εF − E )P

∫ ∞ε+T

dE ′ImΣ`j(r , r′;E ′)[

E − E ′− 1

εF − E ′]

π(εF − E )P

∫ ε−T

−∞dE ′ImΣ`j(r , r

′;E ′)[1

E − E ′− 1

εF − E ′]

This constraint ensures bound and scattering quantities are simultaneously described

′; εF )− 1

π(εF − E )P

∫ ∞ε+T

E − E ′− 1

εF − E ′]

π(εF − E )P

∫ ε−T

′;E ′)[1

E − E ′− 1

εF − E ′]

′; εF )− 1

π(εF − E )P

∫ ∞ε+T

E − E ′− 1

εF − E ′]

π(εF − E )P

∫ ε−T

′;E ′)[1

E − E ′− 1

εF − E ′]

′; εF )− 1

π(εF − E )P

∫ ∞ε+T

E − E ′− 1

εF − E ′]

π(εF − E )P

∫ ε−T

′;E ′)[1

E − E ′− 1

εF − E ′]

′; εF )− 1

π(εF − E )P

∫ ∞ε+T

E − E ′− 1

εF − E ′]

π(εF − E )P

∫ ε−T

′;E ′)[1

E − E ′− 1

εF − E ′]

′; εF )− 1

π(εF − E )P

∫ ∞ε+T

E − E ′− 1

εF − E ′]

π(εF − E )P

∫ ε−T

′;E ′)[1

E − E ′− 1

εF − E ′]

Fitting the Self-energy (40Ca)

Parameters of self-energy varied to minimize χ2

Reproducing the data means self-energy is found

0 30 60 90 120 150 180

Elab >100100>Elab >4040>Elab >2020>Elab >10

p+40Ca

0 30 60 90 120 150 180

n+40Ca

θc.m. [deg] θc.m. [deg]

0 30 60 90 120 150 180

p+40Ca

0 30 60 90 120 150 180

n+40Ca

0 1 2 3 4 5 6 7 8

ρ[e·fm

r [fm]

ExperimentDOM

0 50 100 150 200

p+40Ca

Elab [MeV]

05001000150020002500300035004000

0 50 100 150 200

n+40Ca

Elab [MeV]

σtotσreact

Data: J.M. Mueller et al. Phys. Rev. C, 83 064605, 2011

0 30 60 90 120 150 180

p+40Ca

0 30 60 90 120 150 180

n+40Ca

0 30 60 90 120 150 180

p+40Ca

0 30 60 90 120 150 180

n+40Ca

0 1 2 3 4 5 6 7 8

ρ[e·fm

r [fm]

ExperimentDOM

0 50 100 150 200

p+40Ca

Elab [MeV]

05001000150020002500300035004000

0 50 100 150 200

n+40Ca

Elab [MeV]

σtotσreact

0 30 60 90 120 150 180

p+40Ca

0 30 60 90 120 150 180

n+40Ca

0 30 60 90 120 150 180

p+40Ca

0 30 60 90 120 150 180

n+40Ca

0 1 2 3 4 5 6 7 8

ρ[e·fm

r [fm]

ExperimentDOM

0 50 100 150 200

p+40Ca

Elab [MeV]

05001000150020002500300035004000

0 50 100 150 200

n+40Ca

Elab [MeV]

σtotσreact

0 30 60 90 120 150 180

p+40Ca

0 30 60 90 120 150 180

n+40Ca

0 30 60 90 120 150 180

p+40Ca

0 30 60 90 120 150 180

n+40Ca

0 1 2 3 4 5 6 7 8

ρ[e·fm

r [fm]

ExperimentDOM

0 50 100 150 200

p+40Ca

Elab [MeV]

05001000150020002500300035004000

0 50 100 150 200

n+40Ca

Elab [MeV]

σtotσreact

0 30 60 90 120 150 180

p+40Ca

0 30 60 90 120 150 180

n+40Ca

0 30 60 90 120 150 180

p+40Ca

0 30 60 90 120 150 180

n+40Ca

0 1 2 3 4 5 6 7 8

ρ[e·fm

r [fm]

ExperimentDOM

0 50 100 150 200

p+40Ca

Elab [MeV]

05001000150020002500300035004000

0 50 100 150 200

n+40Ca

Elab [MeV]

σtotσreact

0 30 60 90 120 150 180

p+40Ca

0 30 60 90 120 150 180

n+40Ca

0 30 60 90 120 150 180

p+40Ca

0 30 60 90 120 150 180

n+40Ca

0 1 2 3 4 5 6 7 8

ρ[e·fm

r [fm]

ExperimentDOM

0 50 100 150 200

p+40Ca

Elab [MeV]

05001000150020002500300035004000

0 50 100 150 200

n+40Ca

Elab [MeV]

σtotσreact

0 30 60 90 120 150 180

p+48Ca

0 30 60 90 120 150 180

n+48Ca

0 30 60 90 120 150 180

p+48Ca

θc.m. [deg]

−0.01

0 1 2 3 4 5 6 7 8

r [fm]

ExperimentDOM

0 50 100 150 200

p+48Ca

Elab [MeV]

500100015002000250030003500400045005000

0 50 100 150 200

n+48Ca

Elab [MeV]

Fitting the Self-energy (208Pb)

0 30 60 90 120 150 180

p+208Pb

0 30 60 90 120 150 180

n+208Pb

0 30 60 90 120 150 180

p+208Pb

0 30 60 90 120 150 180

n+208Pb

0 2 4 6 8 10

r [fm]

ExperimentDOM

0200400600800

1000120014001600180020002200

0 50 100 150 200

p+208Pb

Elab [MeV]

2000300040005000600070008000900010000

0 50 100 150 200

n+208Pb

Elab [MeV]

Fitting the Self-energy (12C)

DOM fit of 12C can be compared with ab-initioresults

0 30 60 90 120 150 180

0 1 2 3 4 5 6

r [fm]

ExperimentDOM

0 50 100 150 200

Elab [MeV]

200400600800100012001400160018002000

0 50 100 150 200

Elab [MeV]

Data in DOM fit: J.M. Mueller et al. Phys. Rev. C, 83 064605, 2011

DOM Binding Energy Density

0 1 2 3 4 5 6 7 8

4πr2E/A

[MeV·fm

4πr2ρ

[fm−3]

r [fm]

ρ2ρnTEV

4πr2E/A

[MeV·fm

4πr2ρ

[fm−3]

r [fm]

CD-BonnAV18N3LO

Shape of E(r) is consistent with NM calculations

Core of nucleus is NM-like (ρ ≈ ρ0)

Core of nucleus minimally contributes to EA0

=⇒ NM is not well-constrained by EA0

∫d3r

∫d3r ′

[〈r |T |r ′〉〈r ′|ρ|r〉+ δ3(r − r ′)

∫ ε−f

−∞dEESh(r ;E )

]= 4π

∫ ∞0

drr2E(r)p

N Z DOM EA0 Exp. EA

012C 5.80 5.90 -93.5 -92.2

40Ca 19.9 19.8 -340 -34248Ca 27.9 19.9 -418 -416

208Pb 126 81.7 -1629 -1637

Nuclear matter points: A. Rios et al., PRC 89, 044303, (2014)

0 1 2 3 4 5 6 7 8

4πr2E/A

[MeV·fm

4πr2ρ

[fm−3]

r [fm]

ρ2ρnTEV

4πr2E/A

[MeV·fm

4πr2ρ

[fm−3]

r [fm]

CD-BonnAV18N3LO

∫d3r

∫d3r ′

[〈r |T |r ′〉〈r ′|ρ|r〉+ δ3(r − r ′)

∫ ε−f

−∞dEESh(r ;E )

]= 4π

∫ ∞0

drr2E(r)p

N Z DOM EA0 Exp. EA

012C 5.80 5.90 -93.5 -92.2

40Ca 19.9 19.8 -340 -34248Ca 27.9 19.9 -418 -416

208Pb 126 81.7 -1629 -1637

0 1 2 3 4 5 6 7 8

4πr2E/A

[MeV·fm

4πr2ρ

[fm−3]

r [fm]

ρ2ρnTEV

4πr2E/A

[MeV·fm

4πr2ρ

[fm−3]

r [fm]

CD-BonnAV18N3LO

∫d3r

∫d3r ′

[〈r |T |r ′〉〈r ′|ρ|r〉+ δ3(r − r ′)

∫ ε−f

−∞dEESh(r ;E )

]= 4π

∫ ∞0

drr2E(r)

N Z DOM EA0 Exp. EA

012C 5.80 5.90 -93.5 -92.2

40Ca 19.9 19.8 -340 -34248Ca 27.9 19.9 -418 -416

208Pb 126 81.7 -1629 -1637

0 1 2 3 4 5 6 7 8

4πr2E/A

[MeV·fm

4πr2ρ

[fm−3]

r [fm]

ρ2ρnTEV

4πr2E/A

[MeV·fm

4πr2ρ

[fm−3]

r [fm]

CD-BonnAV18N3LO

∫d3r

∫d3r ′

[〈r |T |r ′〉〈r ′|ρ|r〉+ δ3(r − r ′)

∫ ε−f

−∞dEESh(r ;E )

]= 4π

∫ ∞0

drr2E(r)

N Z DOM EA0 Exp. EA

012C 5.80 5.90 -93.5 -92.2

40Ca 19.9 19.8 -340 -34248Ca 27.9 19.9 -418 -416

208Pb 126 81.7 -1629 -1637

0 1 2 3 4 5 6 7 8

4πr2E/A

[MeV·fm

4πr2ρ

[fm−3]

r [fm]

ρ2ρnTEV

4πr2E/A

[MeV·fm

4πr2ρ

[fm−3]

r [fm]

CD-BonnAV18N3LO

∫d3r

∫d3r ′

[〈r |T |r ′〉〈r ′|ρ|r〉+ δ3(r − r ′)

∫ ε−f

−∞dEESh(r ;E )

]= 4π

∫ ∞0

drr2E(r)

N Z DOM EA0 Exp. EA

012C 5.80 5.90 -93.5 -92.2

40Ca 19.9 19.8 -340 -34248Ca 27.9 19.9 -418 -416

208Pb 126 81.7 -1629 -1637

0 1 2 3 4 5 6 7 8

4πr2E/A

[MeV·fm

4πr2ρ

[fm−3]

r [fm]

ρ2ρnTEV

4πr2E/A

[MeV·fm

4πr2ρ

[fm−3]

r [fm]

CD-BonnAV18N3LO

∫d3r

∫d3r ′

[〈r |T |r ′〉〈r ′|ρ|r〉+ δ3(r − r ′)

∫ ε−f

−∞dEESh(r ;E )

]= 4π

∫ ∞0

drr2E(r)

N Z DOM EA0 Exp. EA

012C 5.80 5.90 -93.5 -92.2

40Ca 19.9 19.8 -340 -34248Ca 27.9 19.9 -418 -416

208Pb 126 81.7 -1629 -1637

0 1 2 3 4 5 6 7 8

4πr2E/A

[MeV·fm

4πr2ρ

[fm−3]

r [fm]

ρ2ρnTEV

4πr2E/A

[MeV·fm

4πr2ρ

[fm−3]

r [fm]

CD-BonnAV18N3LO

∫d3r

∫d3r ′

[〈r |T |r ′〉〈r ′|ρ|r〉+ δ3(r − r ′)

∫ ε−f

−∞dEESh(r ;E )

]= 4π

∫ ∞0

drr2E(r)

N Z DOM EA0 Exp. EA

012C 5.80 5.90 -93.5 -92.2

40Ca 19.9 19.8 -340 -34248Ca 27.9 19.9 -418 -416

208Pb 126 81.7 -1629 -1637

0 1 2 3 4 5 6 7 8

4πr2E/A

[MeV·fm

4πr2ρ

[fm−3]

r [fm]

ρ2ρnTEV

4πr2E/A

[MeV·fm

4πr2ρ

[fm−3]

r [fm]

CD-BonnAV18N3LO

∫d3r

∫d3r ′

[〈r |T |r ′〉〈r ′|ρ|r〉+ δ3(r − r ′)

∫ ε−f

−∞dEESh(r ;E )

]= 4π

∫ ∞0

drr2E(r)

N Z DOM EA0 Exp. EA

012C 5.80 5.90 -93.5 -92.2

40Ca 19.9 19.8 -340 -34248Ca 27.9 19.9 -418 -416

208Pb 126 81.7 -1629 -1637

3N Contribution

Energy sum-rule used in DOM assumes no 3-body force

Variational Monte Carlo (VMC) calculations show small 3N contribution

Verifies that the lack of a 3N force in DOM calculations does not affect conclusions

−25−20−15−10−505

10152025

0 1 2 3 4 5

4πr2E/A

[MeV·fm

r [fm]

VMC using AV18+UX

ρETVU

0 1 2 3 4 5

r [fm]

3N Potential Densities

Urbana-XNV3-Ib*NV3-Ia*

NV3-IIb*NV3-IIa*

0 1 2 3 4 5

r [fm]

Energy Densities

Chiral Interactions: M. Piarulli et al., PRL 120, 052503, (2018)

3N Contribution

−25−20−15−10−505

10152025

0 1 2 3 4 5

4πr2E/A

[MeV·fm

r [fm]

VMC using AV18+UX

ρETVU

0 1 2 3 4 5

r [fm]

NV3-IIb*NV3-IIa*

0 1 2 3 4 5

r [fm]

Energy Densities

3N Contribution

−25−20−15−10−505

10152025

0 1 2 3 4 5

4πr2E/A

[MeV·fm

r [fm]

VMC using AV18+UX

ρETVU

0 1 2 3 4 5

r [fm]

NV3-IIb*NV3-IIa*

0 1 2 3 4 5

r [fm]

Energy Densities

3N Contribution

−25−20−15−10−505

10152025

0 1 2 3 4 5

4πr2E/A

[MeV·fm

r [fm]

VMC using AV18+UX

ρETVU

0 1 2 3 4 5

r [fm]

NV3-IIb*NV3-IIa*

0 1 2 3 4 5

r [fm]

Energy Densities

3N Contribution

−25−20−15−10−505

10152025

0 1 2 3 4 5

4πr2E/A

[MeV·fm

r [fm]

VMC using AV18+UX

ρETVU

0 1 2 3 4 5

r [fm]

NV3-IIb*NV3-IIa*

0 1 2 3 4 5

r [fm]

Energy Densities

Energy in 208Pb

Large core of 208Pb makes it ideal to study

Must remove effect of Coulomb

Account for asymmetry:BE = aVA− 1

2aA(A− 2Z )2A−1

Energy does not match canonical value −18

0 1 2 3 4 5 6

r [fm]

DOMAV18

∫d3rρ(r) =

∫d3rE(r) =⇒ E(r) =

)ρ(r) =⇒ E (r) ≈ A

Energy in 208Pb

2aA(A− 2Z )2A−1

Energy does not match canonical value

0 1 2 3 4 5 6

]r [fm]

DOMAV18

∫d3rρ(r) =

∫d3rE(r) =⇒ E(r) =

)ρ(r) =⇒ E (r) ≈ A

Energy in 208Pb

2aA(A− 2Z )2A−1

0 1 2 3 4 5 6

]r [fm]

DOMAV18

∫d3rρ(r) =

∫d3rE(r) =⇒ E(r) =

)ρ(r) =⇒ E (r) ≈ A

Energy in 208Pb

2aA(A− 2Z )2A−1

0 1 2 3 4 5 6

]r [fm]

DOMAV18

∫d3rρ(r) =

∫d3rE(r) =⇒ E(r) =

)ρ(r) =⇒ E (r) ≈ A

Energy in 208Pb

2aA(A− 2Z )2A−1

Energy does not match canonical value −18

0 1 2 3 4 5 6

]r [fm]

DOMAV18

∫d3rρ(r) =

∫d3rE(r) =⇒ E(r) =

)ρ(r) =⇒ E (r) ≈ A

Energy in the Interior

Energy of each nucleus consistently above -16 MeV

Consistent with SCGF calculation in NM

0 0.2 0.4 0.6 0.8 1 1.2 1.4

r [fm]

12C40Ca48Ca

r [fm]

208PbAV18

M. Baldo et al., PRC 86, 064001, (2012)

0 0.2 0.4 0.6 0.8 1 1.2 1.4

r [fm]

12C40Ca48Ca

r [fm]

208PbAV18

M. Baldo et al., PRC 86, 064001, (2012)

0 0.2 0.4 0.6 0.8 1 1.2 1.4

r [fm]

12C40Ca48Ca

r [fm]

208PbAV18

M. Baldo et al., PRC 86, 064001, (2012)

Summary

Canonical value of saturation energy is determined empirically from the mass formula

Minimal contribution to the binding energy from the core of the nucleus=⇒ Empirical mass formula is not a great constraint for NM

Results are consistent with SCGF calculation of NM using AV18

The value of E0 is not necessarily -16 MeV

Summary

Minimal contribution to the binding energy from the core of the nucleus

=⇒ Empirical mass formula is not a great constraint for NM

Summary

Thanks

Willem Dickhoff

Maria Piarulli

Robert Charity

Hossein Mahzoon

Lee Sobotka

Cole Pruitt

Natalie Calleya

Bob Wiringa

Arnau Rios

Mack C. Atkinson

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Transcript of Mack C. Atkinson