P. Marevic1,2, R.D. Lasseri , J.-P. Ebran , E. Khan , T ... · P. Marevic1,2, R.D. Lasseri2, J.-P....

P. Marevic1,2, R.D. Lasseri2, J.-P. Ebran1, E. Khan2, T. Niksic3, D.Vretenar3

-Nuclear Clustering in the Energy Density Functional Approach-

1 CEA,DAM,DIF2 IPNO3 University of Zagreb

1

Nuclear systems as a mixture of 4 types of fermion

rn

Introduction

proton

neutron

These features leave fingerprints in finite nuclei

a

a

a

aa

rp Fermi liquid

BCS pairing

Clustering

BEC

saturation density

a

aa

h

th

t

ht

O

O

Mott line

crossover

QPT

T=0 MeV

subsaturation density

2-component Fermi gas

3-component Fermi gas

2

Nuclear EDFs provide a unified and consistent description of these various properties

Emergence of ordered sub-structures inside nuclei

IntroductionN

ucl

ei a

t h

igh

sp

in/e

ner

gy

clustering

skin

halo

molecular orbital

proton

neutrona

deformation

low energy resonance

3

4

Introduction

Outline

Theoretical framework : EDF

Pair correlations

Nuclear clustering

5

- Theoretical framework : EDF-

6

Nuclear many-body problem : strategies

EDF

How to incorporate such correlations starting with a HF product state ?

Many-body approaches as implementation of different strategies to apprehend the many-body problem

Traditional starting point of a many-body approach : HF product state

Z,N,Jp

+ + …

Fermi correlations

𝐸

𝐸𝑅𝐻𝐹

𝐸𝐺𝑆

missing correlations

Vertical Philosophy

Z,N,Jp

Reference vacuum

…

Z,N,Jp

Excit. 1p1h

…

…

Z,N,Jp

Excit. 2p2h

…

Z,N,Jp Z,N,Jp Z,N,Jp

= C0 + Ci1p-1h + Cj

2p-2h +

i j

Z,N,Jp

⦿ conservation of symmetries

⦿ physics accounted through numerous rearrangements

of nucleons within the orbitals given by the reference

vacuum

⦿ non collective excitations


Dynamical correlations

Major contribution

EDF

𝐸

𝐸𝑅𝐻𝐹

𝐸𝐺𝑆

Fermi correlations

in all nuclei

|HF⟩ ∼

7

Vertical Philosophy

Z,N,Jp

Reference vacuum

…

Z,N,Jp

Excit. 1p1h

…

…

Z,N,Jp

Excit. 2p2h

…


= C0 + Ci1p-1h + Cj

2p-2h +

i j

Z,N,Jp




vacuum



⦿ Nondynamical correlations

comparable contributions

EDF


𝐸

𝐸𝑅𝐻𝐹

𝐸𝐺𝑆

in all nuclei

|HF⟩ ∼

open shell

systems

|HF⟩≁

……

8

Fermi correlations

𝒒

𝐴𝑟𝑔(𝒒)


Horizontal philosophy

Heff( , ) dynamical correlations transferred in

EDF

𝐸

𝐸𝐺𝑆

𝐸𝑅𝐻𝐹

𝐸𝑈𝐻𝐹

Vertical Philosophy

Z,N,Jp

Reference vacuum

…

Z,N,Jp

Excit. 1p1h

…

…

Z,N,Jp

Excit. 2p2h

…


= C0 + Ci1p-1h + Cj

2p-2h +

i j

Z,N,Jp




vacuum





𝐸

𝐸𝑅𝐻𝐹

𝐸𝐺𝑆

in all nuclei

|HF⟩ ∼

open shell

systems

|HF⟩≁

……

(0,0 )

⦿ Physics accounted for through spontaneous symmetry breaking

+ fluctuations of the order parameter

Z,N,Jp

(q0,j0 ) (q0,j1 ) (q1,j0 )

⦿ broken symmetries

∫ (|q|, arg(q))= dq f(q)

⦿ Collective excitations

9

Fermi correlations

𝒒

𝐴𝑟𝑔(𝒒)



(0,0 )



Z,N,Jp


EDF

𝐸

𝐸𝐺𝑆

𝐸𝑅𝐻𝐹

𝐸𝑈𝐻𝐹

(q0,j0 ) (q0,j1 ) (q1,j0 )


∫ (|q|, arg(q))= dq f(q)


Initial wave function

Optimized wave function with {q(1)}

Vertical Philosophy

Z,N,Jp

Reference vacuum

…

Z,N,Jp

Excit. 1p1h

…

…

Z,N,Jp

Excit. 2p2h

…


= C0 + Ci1p-1h + Cj

2p-2h +

i j

Z,N,Jp




vacuum





𝐸

𝐸𝑅𝐻𝐹

𝐸𝐺𝑆

in all nuclei

|HF⟩ ∼

open shell

systems

|HF⟩≁

……

10

Fermi correlations

𝒒

𝐴𝑟𝑔(𝒒)



(0,0 )



Z,N,Jp


EDF

𝐸

𝐸𝐺𝑆

𝐸𝑅𝐻𝐹

𝐸𝑈𝐻𝐹

(q0,j0 ) (q0,j1 ) (q1,j0 )


∫ (|q|, arg(q))= dq f(q)


Vertical Philosophy

Z,N,Jp

Reference vacuum

…

Z,N,Jp

Excit. 1p1h

…

…

Z,N,Jp

Excit. 2p2h

…


= C0 + Ci1p-1h + Cj

2p-2h +

i j

Z,N,Jp




vacuum





𝐸

𝐸𝑅𝐻𝐹

𝐸𝐺𝑆

in all nuclei

|HF⟩ ∼

open shell

systems

|HF⟩≁

……

11

Fermi correlations

𝒒

𝐴𝑟𝑔(𝒒)



(0,0 )



Z,N,Jp


EDF

𝐸

𝐸𝐺𝑆

𝐸𝑅𝐻𝐹

𝐸𝑈𝐻𝐹

(q0,j0 ) (q0,j1 ) (q1,j0 )


∫ (|q|, arg(q))= dq f(q)


Vertical Philosophy

Z,N,Jp

Reference vacuum

…

Z,N,Jp

Excit. 1p1h

…

…

Z,N,Jp

Excit. 2p2h

…


= C0 + Ci1p-1h + Cj

2p-2h +

i j

Z,N,Jp




vacuum





𝐸

𝐸𝑅𝐻𝐹

𝐸𝐺𝑆

in all nuclei

|HF⟩ ∼

open shell

systems

|HF⟩≁

……

12

Fermi correlations

𝒒

𝐴𝑟𝑔(𝒒)



(0,0 )



Z,N,Jp


EDF

𝐸

𝐸𝐺𝑆

𝐸𝑅𝐻𝐹

𝐸𝑈𝐻𝐹

(q0,j0 ) (q0,j1 ) (q1,j0 )


∫ (|q|, arg(q))= dq f(q)


Vertical Philosophy

Z,N,Jp

Reference vacuum

…

Z,N,Jp

Excit. 1p1h

…

…

Z,N,Jp

Excit. 2p2h

…


= C0 + Ci1p-1h + Cj

2p-2h +

i j

Z,N,Jp




vacuum





𝐸

𝐸𝑅𝐻𝐹

𝐸𝐺𝑆

in all nuclei

|HF⟩ ∼

open shell

systems

|HF⟩≁

……

13

Fermi correlations

EDF

Vertical Philosophy

Z,N,Jp

Reference vacuum

…

Z,N,Jp

Excit. 1p1h

…

…

Z,N,Jp

Excit. 2p2h

…


= C0 + Ci1p-1h + Cj

2p-2h +

i j

Z,N,Jp




vacuum





𝐸

𝐸𝑅𝐻𝐹

𝐸𝐺𝑆

in all nuclei

|HF⟩ ∼

open shell

systems

|HF⟩≁

……

𝒒

𝐴𝑟𝑔(𝒒)



(0,0 )



Z,N,Jp


𝐸

𝐸𝐺𝑆

𝐸𝑅𝐻𝐹

𝐸𝑈𝐻𝐹

(q0,j0 ) (q0,j1 ) (q1,j0 )


∫ (|q|, arg(q))= dq f(q)


𝐸𝑝𝑟𝑜𝑗

Novel many-body approaches combining both philosophies

P. Arthuis, J. Ripoche, T. Duguet, D. Lacroix, J.-P. Ebran

J.Ripoche et al PRC 95, 014326 (2017)

T. Duguet et al EPJA 51, 162 (2015)

14

Fermi correlations

15

- Pair correlations -

How to describe a pair of nucleons in the medium ?

Pair correlation

Reduced density matrix Form of the A-body wave function

⦿ All the information of a many-body system is contained in its Nth-order

density matrix

⦿ Only keep information about p-“cluster” embedded in the medium

composed by the other N-p particles :

⦿ 2-RDM : eigenfunctions provide an in-medium pair wave function

⦿ BCS assumption: all the pairs of fermions occupy the same pair wave

function :

16

2-neutron distribution properties

Pair correlation

Covariant HFB with projection on good particle number prior to variation

R.D. Lasseri et al, in prep

rnrms

dnn

17

2-neutron distribution properties

Pair correlation

R.D. Lasseri et al, in prep18

19

- Nuclear Clustering -

20

First indicator: mass density at the MF level

Nuclear clustering How do es clustering show up in nuclear EDFs ?

5.5

Marevic et al, submitted to PRC

16O

21

Localization measure


Reinhard et al, Phys. Rev. C 83, 034312 (2011)

Conditional probability

0.5 : signals a nearly homogeneous Fermi gas

1 : localized a-like state (in N=Z systems)

r

C

4He 20Ne

22

Beyond MF level



23

Beyond MF level



24

rotation-vibration bands


Bijker (2016)

Work in progressEbran et al, in prep

0+

2+

4+

0+0+

1-

2+

3-

4+

Marevic et al, in prep

Ebran, Khan, Niksic & Vretenar, Nature 487, 341-344 (2012) - PRC 87, 044307 (2013) 25

Guidance : the localization parameter

Nuclear clustering Deepe r understanding about nuclear clustering

quantum liquid

𝛼~1

localizationcrystal

⦿ average inter-particle distance ⇒ density

⦿ depth of the confining potential

⦿ particle number

26

Influence of the particle number


Ebran, Khan, Niksic & Vretenar, PRC 87, 044307 (2013) - PRC 89, 031303(R) (2014)

Clustering is more likely to be found in light systems

⦿ particle number

27Ebran, Khan, Niksic & Vretenar, PRC 89,031303(R) (2014)

Girod & Schuck, PRL 111,132503 (2013)


Influence of the density

⦿ average inter-particle distance ⇒ density

Electrons in quantum dots

Neutral bosons in rotating trap

Nucleons in 40Ca

Yannouleas and Landman, Rep.Prog.Phys. 70, 2067 (2007)

Deeper confining potential

28

Influence of the confining potential



29



30




2

8

20

40

2

4

10

16

28

70

40

2

6

14

26

44

2

4

6

12

18

24

36

48

60

Nazarewicz et. al., AIP Conf. Proc. 259, 30 (1992)

Harmonic oscillator case

31



32


Influence of the neutron excess

2-center cluster in a N=Z system Neutron rich isotope

melting

+

-

+ +-

-

-

+

+

Covalent bonding

p s p*

Be case

s1/2

p3/2

p1/2

d5/2

p3/2

p1/2s1/28

p*3/2

6 N=8 magic number breaking

33


0+1

0+2


10Be 12Be 14Be

34



35



36



Conclusion & Perspectives

Nuclear EDFs frame the various nuclear properties in a unified and consistent way

Clustering in ground and excited states of nuclei : impact of the average density and the depth of

the confining potential

Di-neutron like configuration at the surface of superfluid nuclei

37

Covalent bonding in neutron rich systems

Thank you for your attention

Particle number projection, conditional probability distribution, form factor, quarteting … under

development

Key feature : breaking/restoration of symmetries to efficiently account for nondynamical correlations

Good description of the ground state of ~50 nuclei


EDF

38

Breaking U(1) : good description of the ground state of ~300 nuclei


EDF

39

Breaking U(1) and O(3) : good description of the ground state of all nuclei


EDF

40

P. Marevic1,2, R.D. Lasseri , J.-P. Ebran , E. Khan , T ... · P. Marevic1,2, R.D. Lasseri2, J.-P....

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