p. marevic1,2, r.d. lasseri , j.-p. ebran , e. khan , t ... · p. marevic1,2, r.d. lasseri2, j.-p....
TRANSCRIPT
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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
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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
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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
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4
Introduction
Outline
Theoretical framework : EDF
Pair correlations
Nuclear clustering
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5
- Theoretical framework : EDF-
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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
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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
Nuclear many-body problem : strategies
Dynamical correlations
Major contribution
EDF
𝐸
𝐸𝑅𝐻𝐹
𝐸𝐺𝑆
Fermi correlations
in all nuclei
|HF⟩ ∼
7
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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
Nuclear many-body problem : strategies
⦿ Nondynamical correlations
comparable contributions
EDF
Dynamical correlations
𝐸
𝐸𝑅𝐻𝐹
𝐸𝐺𝑆
in all nuclei
|HF⟩ ∼
open shell
systems
|HF⟩≁
……
8
Fermi correlations
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𝒒
𝐴𝑟𝑔(𝒒)
Nuclear many-body problem : strategies
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
…
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
⦿ Nondynamical correlations
comparable contributions
Dynamical correlations
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
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𝒒
𝐴𝑟𝑔(𝒒)
Nuclear many-body problem : strategies
Horizontal philosophy
(0,0 )
⦿ Physics accounted for through spontaneous symmetry breaking
+ fluctuations of the order parameter
Z,N,Jp
Heff( , ) dynamical correlations transferred in
EDF
𝐸
𝐸𝐺𝑆
𝐸𝑅𝐻𝐹
𝐸𝑈𝐻𝐹
(q0,j0 ) (q0,j1 ) (q1,j0 )
⦿ broken symmetries
∫ (|q|, arg(q))= dq f(q)
⦿ Collective excitations
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
…
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
⦿ Nondynamical correlations
comparable contributions
Dynamical correlations
𝐸
𝐸𝑅𝐻𝐹
𝐸𝐺𝑆
in all nuclei
|HF⟩ ∼
open shell
systems
|HF⟩≁
……
10
Fermi correlations
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𝒒
𝐴𝑟𝑔(𝒒)
Nuclear many-body problem : strategies
Horizontal philosophy
(0,0 )
⦿ Physics accounted for through spontaneous symmetry breaking
+ fluctuations of the order parameter
Z,N,Jp
Heff( , ) dynamical correlations transferred in
EDF
𝐸
𝐸𝐺𝑆
𝐸𝑅𝐻𝐹
𝐸𝑈𝐻𝐹
(q0,j0 ) (q0,j1 ) (q1,j0 )
⦿ broken symmetries
∫ (|q|, arg(q))= dq f(q)
⦿ Collective excitations
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
⦿ Nondynamical correlations
comparable contributions
Dynamical correlations
𝐸
𝐸𝑅𝐻𝐹
𝐸𝐺𝑆
in all nuclei
|HF⟩ ∼
open shell
systems
|HF⟩≁
……
11
Fermi correlations
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𝒒
𝐴𝑟𝑔(𝒒)
Nuclear many-body problem : strategies
Horizontal philosophy
(0,0 )
⦿ Physics accounted for through spontaneous symmetry breaking
+ fluctuations of the order parameter
Z,N,Jp
Heff( , ) dynamical correlations transferred in
EDF
𝐸
𝐸𝐺𝑆
𝐸𝑅𝐻𝐹
𝐸𝑈𝐻𝐹
(q0,j0 ) (q0,j1 ) (q1,j0 )
⦿ broken symmetries
∫ (|q|, arg(q))= dq f(q)
⦿ Collective excitations
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
⦿ Nondynamical correlations
comparable contributions
Dynamical correlations
𝐸
𝐸𝑅𝐻𝐹
𝐸𝐺𝑆
in all nuclei
|HF⟩ ∼
open shell
systems
|HF⟩≁
……
12
Fermi correlations
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𝒒
𝐴𝑟𝑔(𝒒)
Nuclear many-body problem : strategies
Horizontal philosophy
(0,0 )
⦿ Physics accounted for through spontaneous symmetry breaking
+ fluctuations of the order parameter
Z,N,Jp
Heff( , ) dynamical correlations transferred in
EDF
𝐸
𝐸𝐺𝑆
𝐸𝑅𝐻𝐹
𝐸𝑈𝐻𝐹
(q0,j0 ) (q0,j1 ) (q1,j0 )
⦿ broken symmetries
∫ (|q|, arg(q))= dq f(q)
⦿ Collective excitations
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
⦿ Nondynamical correlations
comparable contributions
Dynamical correlations
𝐸
𝐸𝑅𝐻𝐹
𝐸𝐺𝑆
in all nuclei
|HF⟩ ∼
open shell
systems
|HF⟩≁
……
13
Fermi correlations
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EDF
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
⦿ Nondynamical correlations
comparable contributions
Dynamical correlations
𝐸
𝐸𝑅𝐻𝐹
𝐸𝐺𝑆
in all nuclei
|HF⟩ ∼
open shell
systems
|HF⟩≁
……
𝒒
𝐴𝑟𝑔(𝒒)
Nuclear many-body problem : strategies
Horizontal philosophy
(0,0 )
⦿ Physics accounted for through spontaneous symmetry breaking
+ fluctuations of the order parameter
Z,N,Jp
Heff( , ) dynamical correlations transferred in
𝐸
𝐸𝐺𝑆
𝐸𝑅𝐻𝐹
𝐸𝑈𝐻𝐹
(q0,j0 ) (q0,j1 ) (q1,j0 )
⦿ broken symmetries
∫ (|q|, arg(q))= dq f(q)
⦿ Collective excitations
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
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- Pair correlations -
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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
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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
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2-neutron distribution properties
Pair correlation
R.D. Lasseri et al, in prep18
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19
- Nuclear Clustering -
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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
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16O
21
Localization measure
Nuclear clustering How do es clustering show up in nuclear EDFs ?
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
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22
Beyond MF level
Nuclear clustering How do es clustering show up in nuclear EDFs ?
Marevic et al, submitted to PRC
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23
Beyond MF level
Nuclear clustering How do es clustering show up in nuclear EDFs ?
Marevic et al, submitted to PRC
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24
rotation-vibration bands
Nuclear clustering How do es clustering show up in nuclear EDFs ?
Bijker (2016)
Work in progressEbran et al, in prep
0+
2+
4+
0+0+
1-
2+
3-
4+
Marevic et al, in prep
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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
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Influence of the particle number
Nuclear clustering Deepe r understanding about nuclear clustering
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
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27Ebran, Khan, Niksic & Vretenar, PRC 89,031303(R) (2014)
Girod & Schuck, PRL 111,132503 (2013)
Nuclear clustering Deepe r understanding about nuclear clustering
Influence of the density
⦿ average inter-particle distance ⇒ density
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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
Nuclear clustering Deepe r understanding about nuclear clustering
⦿ depth of the confining potential
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Influence of the confining potential
Nuclear clustering Deepe r understanding about nuclear clustering
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Influence of the confining potential
Nuclear clustering Deepe r understanding about nuclear clustering
⦿ depth of the confining potential
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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
Influence of the confining potential
Nuclear clustering Deepe r understanding about nuclear clustering
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Nuclear clustering Deepe r understanding about nuclear clustering
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
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Nuclear clustering Deepe r understanding about nuclear clustering
0+1
0+2
Influence of the neutron excess
10Be 12Be 14Be
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Nuclear clustering Deepe r understanding about nuclear clustering
Influence of the neutron excess
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Nuclear clustering Deepe r understanding about nuclear clustering
Influence of the neutron excess
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Nuclear clustering Deepe r understanding about nuclear clustering
Influence of the neutron excess
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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
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Good description of the ground state of ~50 nuclei
Nuclear many-body problem : strategies
EDF
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Breaking U(1) : good description of the ground state of ~300 nuclei
Nuclear many-body problem : strategies
EDF
39
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Breaking U(1) and O(3) : good description of the ground state of all nuclei
Nuclear many-body problem : strategies
EDF
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