Deformations of sd and pf shell L hypernuclei

Masahiro Isaka (RIKEN)

Grand challenges of hypernuclear physics

2 body interaction between baryons (nucleon, hyperon)

– hyperon-nucleon (YN)

– hyperon-hyperon (YY)

Structure change caused by hyperon(s)

– No Pauli exclusion between N and Y

– YN interaction is different from NN

A major issue in hypernuclear physics

“Hyperon as an impurity in nuclei”

L hypernucleus Normal nucleus As an impurity

+

Interaction: To understand baryon-baryon interaction

Structure: To understand properties of baryon many-body system

Today’s talk: Deformations of sd- and pf- shell L hypernuclei

Recent achievements in (hyper)nuclear physics Knowledge of LN effective interaction

Study of light (s, p-shell) L hypernuclei

– Accurate solution of few-body problems [1]

– LN G-matrix effective interactions [2]

– Increases of experimental information [3]

Development of theoretical models

Through the study of unstable nuclei

Ex.: Antisymmetrized Molecular Dynamics (AMD)[4]

• AMD can describe dynamical changes of various structure

• No assumption on clustering and deformation

[1] E. Hiyama, NPA 805 (2008), 190c, [2] Y. Yamamoto, et al., PTP Suppl. 117 (1994), 361., [3] O. Hashimoto and H. Tamura, PPNP 57 (2006), 564., [4] Y. Kanada-En’yo et al., PTP 93 (1995), 115.

Recent developments enable us to study structure of L hypernuclei

Toward heavier and exotic L hypernuclei Experiments at J-PARC, JLab and Mainz etc.

Hypernuclear chart will be extended to heavier regions

Taken from O. Hashimoto and H. Tamura, PPNP 57 (2006), 564.

n-rich

Developed cluster structure

Exotic cluster

Coexistence of shell and cluster

“Structure of hypernuclei”

Various deformations • Superdeformation • Coexistence of

deformations • Triaxial deformation

Today’s talk

Superdeformed (SD) states

SD states: 1:2 axis ratio of deformation

Observed by experiments in low-energy regions of 40Ca

Ex.) 40Ca J. R. MacDonald, et al., PRC3, 219(1971), E. Ideguchi, et al., PRL87, 222501(2011) W. Gerace and A. Green, NPA93, 110(1967); NPA123, 241 (1969)

Ground state (b = 0: Spherical)

SD state

Figures: AMD calc. by Y. Taniguchi, et al., PRC 76, 044317 (2007)

1:2

Low-lying positive-parity states exist due to deformation

– 7/2-(g. s.): (almost) spherical

– Very low-lying 3/2+: deformed

SD states also exist in Sc ?

Coexistence of deformations

Ex.) 45Sc and 47Sc I. P. Johnstone, NPA110(1968)429, J. Styczen, et al., NPA262(1976) 317

Ex(3/2+) = 0.01 MeV (Degenerated) Ex(3/2+) = 0.77 MeV

Explained by Nilsson model

Consistent with shell-model picture

BL with different deformations(structures) General trend of L binging energy (BL)

L coupled to the compact state is more deeply bound

E. Hiyama, et al., PRL 85 (2000) 270

BΛ = 7.9 MeV BΛ = 10.3 MeV

12C

13C L

3-

0 2 +

3a cluster

Shell-model like Compact (Hoyle state)

E. Hiyama and Y. Yamamoto, PTP128(2012)105

BL is smaller in SD states? L modifies excitation spectra with coexistence of deformed states?

Triaxial deformation in sd shell nuclei Many nuclei manifests various quadrupole deformation

– Most are prolately or oblately deformed (axially symmetric)

Parameterized by quadrupole deformation parameters b and g

Triaxial deformed nuclei are not many and its (direct) identification is not easy.

Prolate

Oblate

Triaxial

g = 0◦

g ≈ 30◦

Spherical

g = 60◦

b

g

0 0◦

60◦

Candidate: Mg isotope

Long

Middle

Short

“L in p orbit can be a probe to study nuclear (triaxial) deformation”

Purpose of this study Purpose

To predict the existence of hypernuclear superdeformed states

– Difference of BL, changes of excitation spectra

To predict level structure of the p-states of triaxially deformed L hypernuclei for studying deformation of the core nuclei

Method: Antisymmetrized Molecular Dynamics (AMD)

Extended version of AMD for hypernuclei (HyperAMD)

– No assumption on nuclear deformation

Example) 41LCa, 46

LSc and 48LSc

Example) 25LMg

Theoretical Framework: HyperAMD We extended the AMD to hypernuclei

gNNNN TVTVTH ˆˆˆˆˆˆ -+++= LL

Wave function Nucleon part：Slater determinant

Spatial part of single particle w.f. is described as Gaussian packet

Single particle w.f. of L hyperon: Superposition of Gaussian packets

Total w.f.：

LN：YNG-NF, NSC97f, ESC08c

NN：Gogny D1S

Hamiltonian

HyperAMD (Antisymmetrized Molecular Dynamics for hypernuclei)

( ) ( )=L

m

mm rcr

( ) ( ) m

zyx

mm zrr

--

= ,,

2exp

+= mmm ba

+= ba iii

( ) ( ) ii

zyx

ii Zrr

--

= ,,

2exp

( ) ( ) jiN rA

r

det!

1=

( ) ( ) ( ) ji

m

mm rA

rcr

det!

1 = L

Theoretical Framework: HyperAMD Procedure of the calculation

Energy variation

Cluster Shell

Initial w.f.

nucleons (Gaussian wave packets)

M. Isaka, et al., PRC83, 054304(2011)

Variational Calculation • Imaginary time development method • Variational parameters:

*

i

i

X

H

dt

dX

=

0

iiiiiiiii cbazZX ,,,,,,, ba=

Theoretical Framework: HyperAMD Procedure of the calculation

Energy variation

Cluster Shell

Initial w.f.

nucleons (Gaussian wave packets)

M. Isaka, et al., PRC83, 054304(2011)

Variational Calculation • Imaginary time development method • Variational parameters:

*

i

i

X

H

dt

dX

=

0

iiiiiiiii cbazZX ,,,,,,, ba=

Energy surface Energy variation with constraints on b(and g)

Theoretical Framework: HyperAMD Procedure of the calculation

M. Isaka, et al., PRC83, 054304(2011)

Energy curve

L s.p. energy

Excitation spectra BL: L binding energy

Angular Momentum Projection ( ) ( )

+= sJ

MK

s

K RDdJM *;

Variational Calculation • Imaginary time development method • Variational parameters:

*

i

i

X

H

dt

dX

=

0

iiiiiiiii cbazZX ,,,,,,, ba=

Generator Coordinate Method(GCM) •Superposition of the w.f. with different configuration •Diagonalization of and

MJHMJH s

K

s

K

J

KssK

= ;ˆ;,

MJMJN s

K

s

K

J

KssK

= ;;,

=sK

s

KsK

MJ MJg ;

J

KssKH ,

J

KssKN ,

Discussions

• L in p-orbit as a probe to identify triaxial deformation Example: 25

LMg

• Their existence • L B.E.(BL) and L s. p. energy (eL) with deformation Example: 41

LCa, 46LSc, and 48

LSc

1. Superdeformed states

2. Triaxial deformation of L hypernuclei

M.I., M. Kimura, A. Dote, and A. Ohnishi, PRC87, 021304(R) (2013)

M.I., K. Fukukawa, M. Kimura, E. Hiyama, H. Sagawa, and Y. Yamamoto, PRC89, 024310 (2014)

Superdeformed (SD) states

SD states: 1:2 axis ratio of deformation

Observed by experiments in low-energy regions of 40Ca

Ex.) 40Ca J. R. MacDonald, et al., PRC3, 219(1971), E. Ideguchi, et al., PRL87, 222501(2011) W. Gerace and A. Green, NPA93, 110(1967); NPA123, 241 (1969)

Ground state (b = 0: Spherical)

SD state

Figures: AMD calc. by Y. Taniguchi, et al., PRC 76, 044317 (2007)

1:2

ND and SD states of 40Ca Ground, ND and SD states

Y. Taniguchi, et al., PRC 76, 044317 (2007)

pf

sd ND

pf

sd SD

pf

sd

proton neutron

GS

Configuration Core nucleus 40Ca: basically same calculation as

W. Gerace and A. Green, NPA93, 110(1967); NPA123, 241 (1969)

Consistent with empirical assignment of the nucleon configurations

Energy curves of 45Sc and 47Sc Several (local) energy minima for each nuclei/parity

Energy curves of 45Sc and 47Sc Several (local) energy minima for each nuclei/parity

Energy curves of 41LCa as a function of b

40Ca(Pos)⊗L(s)

40Ca(Pos)

GS

ND SD

40Ca

41Ca L

41Ca L “GS⊗L”, “ND ⊗L ” and “SD ⊗L ” curves are obtained

SD states will appear in 41LCa

spherical superdeformed

40Ca 40Ca

41Ca L

spherical

41Ca L

superdeformed

Energy surface as a function of b Ground, normal deformed and superdeformed states are obtained

45Sc(Neg)⊗L(s)

45Sc(Pos)⊗L(s)

45Sc(Pos)

45Sc(Neg)

40Ca(Pos)⊗L(s)

40Ca(Pos)

GS

ND SD

GS: sd-shell closed ND: 4p-4h SD: 8p-8h

Consistent with the preceding study

40Ca

GS

ND ND

ND

SD

40Ca 45Sc

45Sc

Y. Taniguchi, et al., PRC 76, 044317 (2007)

Definition:

General trend: eL changes within 1 - 2 MeV as b increases

L single particle energy

L single particle energy

46Sc L 45Sc(Neg)⊗L(s)

45Sc(Pos)⊗L(s)

GS ND SD

41Ca L

40Ca(Pos)⊗L(s)

40Ca(Pos)⊗L(s)

40Ca(Pos)

GS

ND SD

40Ca

Energy surface

eL

Definition:

General trend: eL changes within 1 - 2 MeV as b increases

L single particle energy

Similar to the p shell L hypernuclei

L single particle energy

46Sc L 45Sc(Neg)⊗L(s)

45Sc(Pos)⊗L(s)

GS ND SD

41Ca L

40Ca(Pos)⊗L(s) 13C L

13C(Pos)⊗L(s)

p-shell hypernucleus

Difference of L binding energy BL

ND and SD states are predicted in 41LCa

BL is different among ground, ND and SD states

40Ca

41Ca L

Largest

Superdeformed states in Sc hypernuclei

Various deformations coexist in the g.s. regions We predict ND and SD states with mp-mh configuration

Examples: 46LSc, 48

LSc

Core nuclei (45Sc, 47Sc)

Difference of BL depending on deformation by adding a L

Excitation spectra of 46LSc and 48

LSc Difference of BL leads to the energy shift up of the deformed states

Similar phenomena in 48LSc

We hope these states in Sc L hypernuclei are observed at JLab

Shifted up

Shifted up

Shifted up

Short summary

AMD + GCM framework has been applied to 45Sc and 47Sc, and 41LCa,

46LSc and 48

LSc to investigate deformed excited states.

Various deformed states in 45Sc and 47Sc

– Prediction of the SD states in 45Sc

In 41LCa, 46

LSc and 48LSc,

– Prediction of deformed states corresponding to the core nuclei

– BL is different among the ground, deformed and SD states

Changes of excitation spectra by L

Discussions

• L in p-orbit as a probe to identify triaxial deformation Example: 25

LMg

• Their existence • L B.E.(BL) and L s. p. energy (eL) with deformation Example: 41

LCa, 46LSc, and 48

LSc

1. Superdeformed states

2. Triaxial deformation of L hypernuclei

M.I., M. Kimura, A. Dote, and A. Ohnishi, PRC87, 021304(R) (2013)

M.I., K. Fukukawa, M. Kimura, E. Hiyama, H. Sagawa, and Y. Yamamoto, PRC89, 024310 (2014)

Deformation of nuclei Triaxial deformed nuclei are not many

Its identification is not easy

Prolate

Oblate

Triaxial

g = 0◦

g ≈ 30◦

Spherical

g = 60◦

b

g

0 0◦

60◦

Candidate: Mg isotope

Long

Middle

Short

“L in p orbit can be a probe to study nuclear (triaxial) deformation”

Deformation of 24Mg

24Mg: candidate of triaxially deformed nuclei [1,2]

[1,2] R. Batchelor, et al., Nucl. Phys. 16, 38 (1969)., A. Cohen and J. Cookson, Nucl. Phys. 29, 604 (1962).

Lp in deformed hypernuclei

Large overlap leads deep binding

Middle

Small overlap leads shallow binding

Triaxial deformation Prolate deformation

Observing the 3 different p-states is strong evidence of triaxial deformation

Our (first) task: To predict the level structure of the p-states in 25LMg

parallel perpendicular

9Be L

R.H. Dalitz, A. Gal, PRL 36 (1976) 362. H. Bando, et al., PTP 66 (1981) 2118.; H. Bando, et al., IJMP 21 (1990) 4021.

If 24Mg is triaxially deformed nuclei, 3 different p-states could appear

Purpose Purpose and problem

To reveal triaxial deformation of 24Mg, we will predict the level structure of the p states in 25

LMg

25

LMg

– p-states will split into 3 different states, if 24Mg is triaxially deformed

Method

HyperAMD

(Antisymmetrized Molecular Dynamics for hypernuclei)

– No assumption on symmetry of nuclei

YNG-NSC97f interaction

Results： Single particle energy of L hyperon L single particle energy on (b, g) plane

Single particle energy of L hyperon is different from each p state – This is due to the difference of overlap between L and nucleons

( ) ( ) ( ) ( )iiNiiii VT gbgbgbe ,ˆˆ,, += LLL

25Mg (AMD, L in p orbit) L

Lowest p state 2nd lowest p state 3rd lowest p state

Results： Single particle energy of L hyperon eL

L s. p. energy is different from each other with triaxial deformation

25Mg (AMD) L

Lowest p state (Parallel to long axis)

2nd lowest p state (Parallel to middle axis)

3rd lowest p state (Parallel to short axis)

( ) ( ) ( ) ( )iiNiiii VT gbgbgbe ,ˆˆ,, += LLL

I

II III

I

II III

I

II III

Lowest 2nd Lowest 3rd Lowest

3 p-states with different spatial distributions of L in p-orbit

I II III

Results: Excitation spectra 3 bands are obtained by L hyperon in p-orbit

– 24Mg⊗Lp(lowest), 24Mg⊗Lp(2nd lowest), 24Mg⊗Lp(3rd lowest)

Lowest threshold : in between 8.3 and 12.5 MeV Ne + 21 L L

Splitting of the p states

Summary Summary

HyperAMD + GCM was used to investigate deformations of sd- and pf- shell L hypernuclei

Superdeformed (SD) states

Prediction of the SD states in 41LCa, 46

LSc and 48LSc

BL is different among the ground, deformed and SD states

Triaxial deformation

In 25LMg, three different p-states will appear due to triaxial deformation

Future plan

To predict the production cross sections

Other sd- and pf-shell L hypernuclei: 27LMg (triaxial)

Comparison of BL with cluster states: 13LC (Hoyle analog)

Changes of excitation spectra by L

L can be a probe to identify triaxial deformation of nuclei