Post on 20-Aug-2019
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
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
Examples: Observables beyond the MF approach
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]
19
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)
20
Examples: Observables beyond the MF approach
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
25
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.
26
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
Sec. Order, Λ=1.5 fm-1
Sec. Order, Λ=2 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
Sec. Order, Λ=0.5 fm-1
Sec. Order, Λ=1 fm-1
Sec. Order, Λ=1.5 fm-1
Sec. Order, Λ=2 fm-1
Note: since we do not know the TRUE EoS we chose onecalculation of EoS as a benchmark
28
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)]
29