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
Top Related