Int. Symp. Clustering as a Window on the Hierarchical Structures of Quantum Systems (CLUSHIQ2022)
SendaiNov. 2 (Oct. 31-Nov. 3), 2022

Untold story of the Hoyle state- first ab initio calculation of 12C Takaharu Otsuka“Clustering as a window on the hierarchical structure of quantum matter” (2018-2022)
Grant-in-Aid for Science Research on Innovative Areas（新学術領域 MEXT Kakenhi 18H05400)

This work was supported also by MEXT as “Program for Promoting Researches on the Supercomputer Fugaku” (Simulation forbasic science: from fundamental laws of particles to creation of nuclei) and “Priority Issue on post-K computer” (Elucidation ofthe Fundamental Laws and Evolution of the Universe), and by JICFuS.

normal nuclear matter
quantum liquid with a uniform densityof nucleons
α cluster formation - clusters close or apart -Pioneers (before 1960)bond coupling

linear formation

energy

Alpha formation near thethreshold energy

Hoyle
state

Ikeda diagram

The alpha clustering in atomic nuclei wasconsidered to occur near the threshold energy.
This sounds a nice idea.
This picture has been a strong guiding principle
for half a century.
Nevertheless, we re-visit it.

Semi-Hierarchy: Clustering and Hierarchy of MatterExuExHoyle StateΛ(1405)

Λ sd Hadron

α clusterααα3α0Electron Spin

N0 S, T0Color Charge

�
N+𝐾𝐾

HadronMolecule

AtomHadron

QuarkEx

from Nakamura’s talkEx1/ka0Feshbach Molecule+

ExEx
Nucleus
12C(gs)Molecule

 Threshold: Clustering near Threshold  Semi-Hierarchy

This talk  α clustering in nuclei may not need threshold effects
The preformed α particle(=cluster) can come out if the energy is above the threshold.

Hoyle state of 12C

3α

Ex = 7.65 MeV

crucial for the syntheses of carbon and other heavier
12C (Hoyle state) + γ elements in red- and blue-giant stars and further forthe birth of the life, but its structure remains unknown
α threshold

The clustering is one of the fundamental problems
in physics, as is in this project.

8Be(0 +)
1

Foundation from sound underlying bases
Its contemporary versions
Ab initio calculations on clustering aspects• [Green’s Function Monte Carlo (GFMC)]
Variational Monte Carlo (VMC)

8Li(2 +)
1lab. frame density[Wiringa et al. 2000]• No Core Full Configuration (NCFC) :[Cockrel et al. 2012] Not clustering• Lattice EFT : Hoyle state [Epelbaum et al. 2012]
Initial setup••

Besides many microscopic models/formulations,such as FMD, AMD, DFT/MF approaches.

12C (0 +, 2 +)11How to calculate ?
ab initio No-Core Monte Carlo Shell Model(MCSM)

advanced CI method on supercomputers

No inert core, or all nucleons are activatedNucleon-nucleon interactions are fixed prior tothis study, based on fundamental approaches
such as the chiral Effective Field Theory of QCD.

Single-particle states included.
.
.

1p
1s

Presently,
a model space up to Nshell =7 (6 hw) is adopted.
.
.

Nshell=5
Nshell=4
Nshell=3
Nshell=2
Nshell=1

Solve the Schrodinger equation
H Ψ= E Ψ
E : eigenenergyΨ : eigenstate

Nucleon-Nucleon interaction
Be : JISP16+ kinetic energy

fitted to NN scattering + fine tuning

C : Daejeon16

based on chiral EFT with SRG + fine tuning

The interactions are fixed prior to the present calculation.

MCSM eigenstate :NBΨ ( D) = ∑ ci PJ ,Π

Deformed Slater determinantwith three axes of ellipsoidφ (D )(n)n =1

For Jπ projected states, individual orientations are not relevant.

lab.

c1+c2+c3＋・・・

+ C98

・・・
+c99+c100

In order to obtain the snap shot (or intrinsic density profile), all basis vectors are aligned.
CI-calculation values of observables are not changed.

c1intr.

+c2+c3＋・・・

+ C98

・・・
+c99+c100

Snapshot of density profile
Laboratory-frame

Body-fixed (intrinsic) frame
Q aligned superposed statein the3.7fm

Alignment of MCSM basis vectors (Q aligned)in theEnergy level & transition strength of 12C
ab initio no-core MCSM + Daejeon 16 interaction (Shirokov et al.)based on chiral EFT (Machleidt-Entem, 2011)
charges

protons 1e
neutrons 0e

correlation effects are explicitly treated(no medium correction needed)

Hoyle state

convergence pattern asfunctions of energy variance

Strong deformation (β2~0.6, oblate) in the 0+1 and 2+1
states can now be described from first principles.

Stringent test for the Daejeon 16 interaction and the present No-Core MCSM.

Total and decomposed nucleon-densities in body-fixed framewith proper orthogonalization
groundstatetentatively ~14 MeV

quantum
liquidquantum
liquidclustering
cabac

clustering

linearHoyle
statebground
state

Hoyle
state

MCSM eigenstates are expanded by basis vectors (deformed Slater determinants)
classified by quadrupole moments (T plot by Tsunoda)
Basis vectors are divided into three groups

IIIIIIβ2 < 0.7
triaxialvery prolate

main contributions
Hoyle state

Ground state

triangularoblate(pancake)Rz : Ry : Rx = 2 : 2 : 1

3 α clustes

prolate (rod with balls)
Rz : Ry : Rx = 5 : 1 : 1

Total and decomposed nucleon-densities in body-fixed framewith proper orthogonalization
groundstatetentatively ~14 MeV

quantum
liquidquantum
liquidclustering
cabac

clustering

linear

Hoyle
state

b

ground
state

Hoyle
state

2-dimensionalpresentationca

 ground state

8Be Hoyle state~ 8Be

density profiles of major
MCSM basis vectors(Slater determinants)
in region II for theHoyle state.

c
Most important

α particle
Triangle configurations
with three α clusters
are favored by nuclearforces
Fluctuations emerge
within such configurationsbaA completely different analysis (no physics, data science)classification of MCSM basis vectors by the cluster analysis of through unsupervised
statistical learning
distance :for basis vectors
where parenthesis means a scalar product (overlap integral) with the Jπ =0+ projection

connect basis vectors from the shortest distance to longer up to the threshold leads to almost the same decomposition scheme (the heart of the present picture)~region III~region I~region IITransition from 8Be to 12C, and the crossover in the ground & Hoyle states of 12C

61%

33%

9.8 MeV
7.5 MeV
if pure5.8 MeVCan this be a radial(E0)-type oscillationfrom the groundstate ?

oblate, β 2 ~ 0.66%alpha clustering

94%
quantum liquid

Ground state :the mixing matrix element is ~ -3 MeV (attractive effect) with 6% (ampl. ~ 0.24) alpha clustering.
alpha decay, alpha knockout
Hoyle state
The mixing occurs also due to the orthogonality to the ground state.
The mixing pushes the Hoyle state upwards by ~3 MeV (repulsive effect).

Point-proton radius of the ground state

Matter radius of the Hoyle state
0.36 fm larger than theground-state value
diff. ~ 0.5 fm in experimentby the Ogloblin groupdiff. = 1.1 ~ 1.9 fm in other theories

Abe et al., systematic calculations of ground-state properties

Summary

α clustering from first principles without any assumption for 8,10,12Be and 12C
Perfect ab initio realization of the oblate rotational band in 12C (energy & E2)
Hoyle state shows a novel structure: superposition of quantum liquid and tri-α clustersIt might be interpreted as an “radial” oscillation between compact and cluster structures ?
Some analogue in molecular (trimer) structure ?

Nuclear forces favor both quantum liquid and α clustering, with more binding for the former.
The transition between them is not a phase transition but a crossover involving mixing.
α cluster emerges even in the well-bound ground state, α decay, α knockout, etc.
“Threshold” is not needed. Soft interactions may not suffice to reproduce this feature.

13, 2234 (2022)

open accessCollaborators
Takashi Abe (RIKEN)
Tooru Yoshida (RIST, CNS, U. Tokyo)
Yusuke Tsunoda (CNS, U. TokyoTsukuba)
N. Shimizu (CNS, U. TokyoTsukuba)
N. Itagaki (KyotoOsaka Metropolitan)
Y. Utsuno (JAERI)
H. Ueno (RIKEN)James Vary (Iowa State U.)
Pieter Maris (Iowa State U.)

END
Thank you for your attention

4He(0+;gs)

Nshell = 2
Nshell = 3

DM ~ 100DM ~ 3 x 103

Nshell = 4

JISP16 NN int.
w/ optimum hw
w/o Coulomb force

DM ~ 4 x 104
DM ~ 3 x 105

Nshell = 5

Extrapolation by the variance
N. Shimizu et al., Phys. Rev. C82, 061305 (2010)
based on T. Mizusaki & M. Imada, Phys. Rev. C65, 064319 (2002)

MCSM eigenstate :NB

Ψ ( D) = ∑ ci PJ ,Π

Deformed Slater determinantwith three axes of ellipsoidφ (D )(n)n =1

For Jπ projected states, individual orientations are not relevant.
c1+c2+c3＋・・・+ C98・・・+c99+c100

For “intrinsic state”, all basis states are aligned so that three axes of theellipsoid are placed on the given directions, e.g. the longest one on the z axis.

c1intr.

+c2+c3＋・・・+ C98

・・・
+c99

+c100

α cluster formation - intuitive image -Pioneers (before 1960)bond coupling

linear formation

The snapshot state in the body-fixed frame is needed,as this snapshot state gives the snapshot of density profile.
( The snapshot state is nothing but the intrinsic state in most literatures. )(The corresponding states in the lab. frame are obtained by rotating it.)
It is difficult (or impossible) to observe it experimentally.

Semi-Hierarchy: Clustering and Hierarchy of MatterExuHoyle StateΛ(1405)

Λ sd Hadron

ααα3α0Electron Spin0 S, T0Color ChargeNα cluster

Ex1/ka0Feshbach Molecule+ExEx
Nucleus
12C(gs)Molecule

 Threshold: Clustering near Threshold  Semi-Hierarchy Degree of Freedom：Neutralization of Charge, Spin(S), Isospin(T)Closed Quantum System

�
N+𝐾𝐾

HadronMolecule

ExOpen Quantum SystemExAtomHadron

QuarkT-plot : visualization of MCSM eigenvector on Potential Energy Surface
stochastically deformed Slater determinant intrinsic shapeeigenstate
amplitude

projection onto Jπ• PES is calculatedby CHF for the shell-model
Hamiltonian• Location of circle : quadrupole

0+1 state of68Ni
triaxial

deformation of unprojected
MCSM basis vectors• Area of circle :overlap probability
between eachprojected basis andeigen wave function

spherical

prolateY. Tsunoda, et al.
PRC 89, 031301 (R) (2014)

Y. Tsunoda

T-plot analysis of 0+ states applied to Be isotopes

T-plot2 excess neutrons
8Be

10Be

4 excess neutrons
12Be

Levels and B(E2)’s of Be isotopes

calculated with
hw=15MeV, Nshell=6
With JISP16 interaction

B(E2) Exp:
8Be Datar et al. 2013 + estimate
by GFMC
10Be McCutchan et al. 2009
12Be Imai et al. 2009

excitation level patterns indicate
the occurance of the rotationalmotion of a deformed “object”→ nucleus seen in the body-fixed(intrinsic) frame