towards a universal nuclear structure...

Towards a universal

nuclear structure model

Xavier Roca-MazaCongresso del Dipartimento di Fisica

Milano, June 28−29, 2017

1

Table of contents:

Brief presentation of the group Motivation Model and selected results Conclusions

2

Brief presentation of the group:

The group interests focus on theoretical nuclear structure

studies: that is, on the study of i) the strong interaction that

effectively acts between nucleons in the medium; and ii) suit-

able many-body techniques to understand the very diverse

nuclear phenomenology.

P. F. Bortignon (PO)

G. Colò (PO)

X. Roca-Maza (RTD)

E. Vigezzi (Dir. Ric.)

3

Some open lines of research in which the group is

involved:

Density functional theory: successful approach to thestudy of the nuclear matter EoS or of the mass, size andcollective excitations in nuclei

Nuclear field theory: successful approach to the study offragmentation of sp states, resonance widths or half-lives

Parity violating and conserving e-N scattering:

characterize the electric and weak charge distribution innuclei

Superfluidity in nuclei: driven by the pairing interaction

Nucleon transfer reactions: probe spectroscopicproperties and superfluidity

Astrophysical appliactions: neutron stars, electroncapture, β−decay, Gamow-Teller resonances

4

Today...

I will concentrate on some of the basiccomplexities of the nuclear many-body

problem and briefly explain the way along ourgroup is working

5

Motivation: The Nuclear Many-Body Problem Nucleus: from few to more than 200 strongly interacting

and self-bound fermions (neutrons and protons). Complex systems: spin, isospin, pairing, deformation, ... 3 of the 4 fundamental forces in nature are contributing to

the nuclear phenomena (as a whole driven by the strong

interaction).

β−decay: weak process

Nuclei: self-bound systemby the strong interaction[In the binding energyBcoul ∼ −Bstrong/(3 to 10)]

α−decay: interplaybetween the strong andelectromagnetic interaction

6

Motivation: The Nuclear Many-Body Problem Underlying interaction: the “so called” residual strong

interaction = nuclear force, the one acting effectively

between nucleons, has not been derived yet from firstprinciples as QCD is non-perturbative at the low-energiesrelevant for the description of nuclei.

The nuclear force in practice: effective potential fitted tonucleon-nucleon scattering data in the vacuum. 3 Bodyforce are needed. 4 Body?

7

Motivation: The Nuclear Many-Body Problem State-of-the-art many-body calculations based on these

potentials are not conclusive yet:

Which parametrization of the residual strong interactionshould we use?

Which many-body technique is the most suitable? Exp. B/A(0.16 fm−3) = −16 MeV 8

So, two important points to remember:

Working with the exact Hamiltonian and most general

wave function is not possible at the moment

Best attempts so far show too large discrepancies andcan be applied only to nuclei up to mass number 10-20

approx.

9

Motivation: So what to do?

Effective interactions solved at the Hartree-Fock or

Mean-Field but fitted to experimental data in many-body

system have been shown to be successful in thedescription of bulk properties of all nuclei (masses,

nuclear radii, deformations, Giant Resonances...)

These effective models can be understood as anapproximate realization of a nuclear energy density

functional E[ρ].

Density Functional Theory rooted on the

Hohenberg-Kohn theorem ⇒ exact functional exists.

The advantage of these approximate E[ρ] is that theyare quite versatile: nowadays our unique tool to self-

consistently access the ground state and some excited

state properties of ALL atomic nuclei

10

Applicability of nuclear E[ρ] as compared to other

methods

11

Nuclear mean-field models (or EDFs)

〈Ψ|H|Ψ〉 ≈ 〈Φ|Heff |Φ〉 = E[ρ]

where Φ is a Slater determinant and ρ is a one-body densitymatrix ⇒ we expect reliable description for the expectationvalue of one-body operators.Main types of models:

Relativistic based on Lagrangians where effective mesonscarry the interaction (π, σ,ω...).

Non-relativistic based on effective Hamiltonians (Yukawa,Gaussian or zero-range two body forces)

⇒ Both give similar results

⇒ phenomenological models → difficult to connect to the

residual strong interaction

12

Examples: Binding energies

Relativistic model by Milano and Barcelona groups

0 40 80 120 160 200 240 280Mass number A

-4

-2

0

2

4

6

(Eth

-Eex

p)/

|Eex

p|*

10

0 [

%]

DD-MEδ

(c)

+0.5%

-0.5%

[PHYSICAL REVIEW C 89, 054320 (2014)]

Remarkable accuracy on thousands of measured binding

energies13

Examples: Charge radiiTheory-lines / Experiment - circles

20 40 60 80 100Neutron number N

3.6

3.8

4

4.2

4.4

4.6

4.8

5

Char

ge

rad

ii r c [

fm]

B

(b)

Sr (Z=38)Kr (Z=36)

Se (Z=34)

0 10 20 30 40Neutron number N

1.5

2

2.5

3

3.5

Ch

arg

e r

adii

r c [

fm]

A

(c)

80 100 120 140Neutron number N

5

5.2

5.4

5.6

5.8

6

Char

ge

rad

ii r c [

fm]

C

(d)

Cm (Z=96)

Pu (Z=94)U (Z=92)

Pu (exp)Cm (exp)

Pb (Z=82)Hg (Z=80)

Pt (Z=78)

[PHYSICAL REVIEW C 89, 054320 (2014)]

14

Examples: Giant Monopole and Dipole Resonances

Non-relativistic model by Milano and Aizu (Japan) groups

10 12 14 16 18E

x[MeV]

0

3

6

9

12

10

−2 R

ISG

MR [f

m4 M

eV−1

]

SLy5SAMi 208

Pb

Exp.

8 12 16E

x[MeV]

0

5

10

15

20

25

RIV

GD

R [f

m2 M

eV−1

]

SLy5SAMi

Exp.

208Pb

R = nuclear response function (in dipole resonance is related with theprobability of a photon absorption by the nucleus or σγ)

Good description excitation energy and integrated R but not the

width of the resonance.15

Examples: Gamow Teller Resonance

Gamow-TellerResonance driven bystrong nuclear force(analogous transitions toβ−decay).

0 5 10 15 20 25E

x[MeV]

0

20

40

60

RG

T−

[M

eV−1

]

SAMiExp. PRC 85, 064606 (2012)

Exp. PRC 52, 604 (1995)

208Pb

[Phys.Rev. C86 (2012) 031306]

16

Mean-field approach: overall good descriptionof ground state and excited state properties in

nuclei

It is beyond the MF approach: accuratedescription of the fragmentation of the

single-particle and collective states

17

Examples: Observables beyond the MF approach

Single particle (sp) states:Density of the system is well described within the MF

approach (E[ρ]) while sp are not satisfactorily reproduced.

[J. Phys. G. 44 (2017) 045106]

18


Spectroscopic factor Sis associated to n, j, l:

Removal probability for

valence protons

* From theory: For a boundA− 1 final state

Stheo =

∫d~p|〈ΦA−1|a~p|ΦA〉|2

* Transfer reaction: A → A− 1

dσ

dΩ

∣

∣

∣

exp=

Sexpdσ

dΩ

∣

∣

∣

theo

[Nuclear Physics A 553 (1993) 297-308]

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Possible solution to these problems: PVC

There is NO explicit interplay of collective motion

(e.g. giant resonances) and single-particle motion in the

mean-field approach

Solution: take into account their interplay

For example,

in spherical nuclei: mainly sp+surface vibrations →

Particle Vibration Coupling model (PVC)

while in deformed nuclei: mainly sp+rotations

PVC model in a diagramtic way (e.g. Σ or sp potential):

⇒ “Selection of most relevant diagrams”

⇒ same Veff at all vertices (fitted at Mean-Field level)

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Some selected examples

21


Single particle states:

-24

-20

-16

-12

-8

-4

0

4

8E

[M

eV]

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

3s1/2

1h11/2

2d3/2

1g9/2

2d5/2

1g9/2

2p1/2

2p3/2

ProtonsNeutrons

2f5/2

1h9/2

3p1/2

3p3/2

2f7/2

3s1/2

1h11/2

2d3/2

1g7/2

RMF QVC EXP RMF QVC EXP

120Sn

2d5/2

1g7/2

[Phys. Rev. C 85, 021303(R)]

22

Examples: Observables beyond the MF approachTotal single particlestrength associated ton, j, l: transfer reaction

0

0.2

0.4

0.6

0.8

0 2 4 6 8 10 12

Excitation energy in 41

Ca [MeV]

f7/2 [g.s. of 41

Ca]

0

0.2

0.4

0.6

0.8

p1/2

0

0.2

0.4

0.6

0.8

p3/2

0

0.02

0.04

0.06

d5/2

0

0.2

0.4

0.6

0.8

f5/2

0

0.05

0.1

0.15

0.2

0.25

0.3

g9/2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12

Excitation energy in 39

Ca [MeV]

d3/2 [g.s. of 39

Ca]

0

0.2

0.4

0.6

0.8

1

s1/2

0

0.01

0.02

0.03

0.04

ρ lj[

MeV

-1]

-

p3/2

0

0.2

0.4

0.6

0.8

40Ca

SLy5

d5/2

Exp.dataHF+PVC(2

+,3

-,4

+,5

-)

[Phys. Rev. C 86, 034318]

(Remember spectroscopic factor)

Collective strengthphotoabsorption cross sectionin 120Sn

23

Examples: Observables beyond the MF approachOptical potential model based on PVC.Neutron elastic cross section by 16O at 28 MeV

10-4

10-3

10-2

10-1

100

101

0 30 60 90 120 150 180

dσ/

dΩ

[b

arn

/sr]

θ [deg]

E = 28 MeV

Exp.

cPVC

HF

[Phys. Rev. C 86, 041603(R) (2012)]

24

PVC: How to renormalize the effective interacrion? PVC includes many-body correlations not explicitely

included at the Mean-Field level. While Veff fitted at the Mean-Field to experimental data. This implies double counting since parameters contain

correlations beyond Mean-Field

Solution: refit the interaction, that is, renormalize

the theory.

Divergences may also appear in beyond Mean-Fieldcalculations if zero-range Veff are used.

The group is now studiyng different strategies for therenormalization of Veff . Essentially:

In a simplified case (2nd order term is kept, no summationup to infinity). Cutoff and dimensional renormalizationhave been performed

In the full PVC approach by appliyng the subtractionmethod

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Conclusions

Effective interactions solved at Hartree-Fock or

Mean-Field level have been shown to be successful in thedescription of all nuclei (masses, nuclear sizes,

deformations, Giant Resonances...

These effective models can be understood as anapproximate realization of a nuclear energy density

functional E[ρ] ⇒ exact functional exist.

An accurate description of the fragmentation of

single-particle and collective states is reached beyond the

MF approach

Particle Vibration model improve the description of

these observables, although renormalization of theinteraction needs to be investigated.

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Thank you!

27

Simplified PVC: Cutoff renormalization on EoS

Epotential = 〈0|V |0〉+∑ν,0

|〈ν|V |0〉|2

E0 − Eν

where |0〉 is the GS and |ν〉 an excited state [Phys. Rev. C 94, 034311 (2016)]

0 0,05 0,1 0,15 0,2 0,25 0,3

-35

-30

-25

-20

-15

-10

-5

0

E/A

[M

eV]

SLy5 mean field

Sec. Order, Λ=0.5 fm-1

Sec. Order, Λ=1 fm-1



0 0.05 0.1 0.15 0.2 0.25 0.3

ρ [fm-3

]

-30

-25

-20

-15

-10

-5

0

∆E(2

) /A [

MeV

]

(a)

(b)

0 0.05 0.1 0.15 0.2 0.25 0.3

ρ [fm-3

]

-18

-15

-12

-9

-6

-3

0

E/A

[M

eV]

SLy5 mean field





Note: since we do not know the TRUE EoS we chose onecalculation of EoS as a benchmark

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Full PVC: Subtraction method

Subtraction method is based on a simple idea: modify

the theory so that the expectation value of any one-bodyoperator (idealy accurate at the MF) do not change with

respect the MF prediction.

* Its realization is simple(“Σsub(E) = ΣNosub(E) −ΣHF” inducedeff. interaction)* Unfortunatelly corrects only approximatelysome of the studied observables (such asdifferent moments of the response function)

* One (different) subtraction recipe should

be applied for each observable that needs to

be renormalized.

0 5 10 15 20E (MeV)

0

10

20

30

40

10

3xS(Q

uad

rup

ole

) (f

m4 /

MeV

) RPAPVC - Sub. PVC - No Sub.

208Pb

SLy5

[Phys. Rev. C 94, 034311 (2016)]

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towards a universal nuclear structure...

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