Stability of the rod-shaped nuclei
Shoujirou Mizutori, Kansai University of Welfare Sciences
Tsunenori Inakura, Tokyo Institute of Technology

Rod-shaped nuclei :nuclei with elongated shape, nodeless in transverse direction

Main topics of today :Instability/Softness of rod-shaped nuclei against asymmetric deformation
Bending：banana deformation → Two types
On the plane perpendicular to the cranking axis
Boomerang

On the plane including the cranking axis
Asymmetric expansion-contraction：

Yajirobee

Examine TRS(Total Routhian Surface) calculated withMitarashi Dangoconstrained cranked Skyrme density functionalRouthian： Legendre transformed function from Lagrangian only part of variables
In this case
DFT2024,Mizutori&Inakura

Extremely elongated shape of light N=Z nuclei
Exploring α linear-chain states:
Experimentally
Morinaga(1956), Chavallier et al.(1967), Wuasmaa et al. (1992), Rae et al. (1992),
Freer (1995), Curtis et al. (2013)
TheoreticallySuzuki, Horiuchi, Ikeda (1972), Bauhoff et al. (1984), Marsh and Rae (1986)
Itagaki et al. (2001)
Exploring elongated shape inspired by α liner-chain
Nilsson Strutinsky Method : Leander and Larsson (1975)
Skyrme Hartree Fock : Flocard, Heenen, Krieger and Weiss (1984)
Beyond mean field : Bender and Heenen(2003)
Finding out rod-shaped solution in full 3D cranked Skyrme DFT

T. Ichikawa, J.A. Maruhn, N. Itagaki, and S. Okubo,Phys. Rev. Lett. 107, 112501 (2011)
Calculations with modern DFT framework
J.M. Yao, N. Itagaki, and J. Meng. PRC 90, 054307(2014)
J.-P. Erban, E. Khan, T. Niksic, and D. Vretner, PRC90,054329(2014)
P.W. Zhao, N. Itagaki, and J. Meng. Phys. Rev. Lett. 115, 022501(2015)
Y. Iwata, T. Ichikawa, N. Itagaki, J.A. Maruhn, and T. Otsuka, PRC 92, 011303(2015)
Z.X. Ren, S.Q. Zhang, P.W. Zhao, N. Itagaki, J.A. Maruhn and J. Meng,Science China Physics, Mechanics & Astronomy 62 (2019): 1-8
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Rod-shaped solutions of cranked DFT
16O

Ichikawa et. al.
Skyrme density functional.

Solutions w/o constraint

Yao et. al.
covariant density functional.

Solutions for various Skyrme parameters
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Rod-shaped solutions of cranked DFTC isotopes
Zhao et. al.
covariant density functional.

24Mg

Iwata et. al.
Skyrme density functional.

Solutions for various Skyrme parameters
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Rod shaped solutions of N=Z even-even nuclei
Inakura, T and Mizutori, S PRC 98.4(2018):044312
Skyrme density functional

Solutions for various N=Z even-even nuclei

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Aim of this talkStudies so far
Discussions are restricted to the local minima of total Routhian surface(TRS).
In this talkDiscuss softness of TRS in N=Z even-even nuclei
Questions
Are they solid solutions?
Instability/Softness against bending
Instability/Softness against expansion/contraction
Which is responsible for vanishing of the local minima
Instability/Softness against bending
Two way: bend in a plane perpendicular to cranking axis, or including cranking axis
Which is softer?
Instability/Softness against bending
Instability/Softness against antisymmetric expansion-contraction
Which is softer?
Rotational frequency dependence?

Are there any difference between A=8N nuclei and A=8N+4 nuclei?
DFT2024,Mizutori&InakuraConstraints
Induce deformations by multipole operators

α: signature; symmetry about the rotation around the x-axis
α=0 : match with itself with 180 degree rotation
α=1 : reflect with 180 degree rotation

Choice of the axis
Symmetry : z-axis
Cranking : x-axis

Octupole operators for reflection asymmetric shapes

Bending in the plane perpendicular to cranking axis (boomerang)(K=1(-) )Bending in the plane including cranking axis (yajirobee)(K=1(+) )Reflection asymmetric expansion-contraction(K=0)

Quadrupole operator for expansion/contraction

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Our model
Model(Cranked) Constrained Density Functional Theory

with center of mass conditions and principal axis conditions
Only one kind of constraints at a timeother deformations are automatically chosen by the system
Detail of the calculation

SKI3 interaction
Full 3D mesh
Rmax=20fm
ΔR=0.6fm
Constraint operator is dumped outside nucleus

K. Rutz, J.A. Maruhn, P.-G. Reinhard, W. Greiner, Nuclear Phys. A590 (1995) 680

Cranking term is dumped outside nucleus in a different manner

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Kind of deformations induced by octupole operators

Density

K=0K=1(-)

K=1(+)

12C

Peak Position

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Total Routhian Surface vs Potential Energy Surface
Potential energy surface:comparison at equal I (actually <Jx>)-49.6<Jx>=8-49.7<H>(MeV)-49.8

comparison of the minima-49.9-35-50
-50.1

-40

-50.2

minimum
TRS minima
PES minima

-50.3
1

1.02

1.04

1.06

1.08

1.1

1.12

β2

Total routhian surface : comparison at equal ωrot
-70.4

-45

1.14

E-50.4-50hωrot=2.6-70.5<H-ωrotJx>(MeV)-70.6-55-70.7
-70.8

-60

-70.9
-71

3 4 5 6 7 8 9 10 11 12<Jx>-71.1-71.2
11.02

1.04

1.06

1.08
β2

1.1

1.12

1.14

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Bending vs Expansion-Contractionωrot=1.0MeV:16O
K=0K=1(+)
K=1(-)-84.6

yE-ωrot<Jx>

asymmetric
expantion-contraction-85

420-2-4-10

16O

-84.6

-84.8
E-ωrot<Jx>ωrot=1.0MeV:16O-50z

510-85.2-84.8-85-85.2

y-85.404
20-2-4-85.4
-10

0.02

-500.04
z5

βBending in the plane perpendicular 3
to cranking axis

100.060.08

0.1 1.21.22

Bending in the plane including
cranking axis

1.24

1.26

1.28

1.3

β2

Softer in the order of K=1(-), K=1(+), K=0
Extremely soft at low rotational frequency to K=1 deformations
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Bending vs Expansion-Contractionωrot=2.2MeV:16O-98.6

y-99.40-98.8-99420-2-4-10-99.2

asymmetric
expantion-contraction
E-ωrot<Jx>-99
yE-ωrot<Jx>

-98.6

K=0
K=1(+)
K=1(-)

-98.8

-99.6

ωrot=2.2MeV:16O-50z

5104
20-2-4-10

0.02-50z50.04

Bending in the plane perpendicular
to cranking axis

16O

-99.2

-99.4
Bending in the plane including
cranking axis
-99.6
100.060.08
0.11.24
1.26
β3

1.28

1.3

1.32

1.34

β2

Softer in the order of K=1(+), K=1(-), K=0
softer at high rotational frequency to K=0 deformations
relatively not soft at high rotational frequency to K=1 deformations
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Rotational dependence of TRS
16O

0.16

K=1(-)

ωrot dependence:K=1+

ωrot dependence:K=10.1

ωrot = 1.0
ωrot = 1.2
ωrot = 1.4
ωrot = 1.6
ωrot = 1.8
ωrot = 2.0
ωrot = 2.2

0.12
0.1

0.08
0.07

0.08
0.06
0.04

0.06
0.05
0.04
0.03
0.02

0.02

0.01

0-0.02

ωrot = 1.0
ωrot = 1.2
ωrot = 1.4
ωrot = 1.6
ωrot = 1.8
ωrot = 2.0
ωrot = 2.2

0.09

R-R0(MeV)

0.14

R-R0(MeV)

K=1(+)

00

0.02

0.04

0.06
β3

0.08

0.1

-0.01
00.02

0.04

0.06

0.08

0.1

β3

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16O

K=0
ωrot dependence:K=00.6

ωrot dependence of curvature
0.6

ωrot = 1.0
ωrot = 1.2
ωrot = 1.4
ωrot = 1.6
ωrot = 1.8
ωrot = 2.0
ωrot = 2.2

0.5
0.4
0.3
0.2
0.1

0.4
0.3
0.2
0.1

00-0.1

-0.1

0

0.02

β30
β31+
β31-

0.5
R(β3=0.1)-R(β3=0.1)

R-R0(MeV)

Rotational dependence of TRS

0.04

0.06
β3

0.08

0.1

1

1.2

1.4

1.6
1.8
ωrot(MeV)

2

2.2

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12C

TRS
TRS on asymmetric shape0.45

ωrot = 1.0MeV:K=0
ωrot = 1.0MeV:K=1(+)
ωrot = 1.0MeV:K=1(-)
ωrot = 3.0MeV:K=0
ωrot = 3.0MeV:K=1(+)
ωrot = 3.0MeV:K=1(-)

0.35

R-R0(MeV)

0.3

0.4
R(β3=0.10)-R(β3=0.00)

0.4

β30
β31+
β31-

0.25
0.2
0.15
0.1

asymmetric
expantion-contraction

0.3

0.2

Bending in the plane including
cranking axis

0.1

0.05
0-0.05

00

0.02

0.04

0.06
β30.080.1

Bending in the plane perpendicular
to cranking axis
11.52
ωrot(MeV)

2.5

3

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20NeTRS

ωrot dependence of curvature

TRS on asymmetric deformation
0.7

0.5
0.4

y0.3
0.2

0.6
R(β3=0.1)-R(β3=0.1)

0.6

420-2-4-10-50z

5100.1

0.5

0.2
0.1

-0.1

-0.1
0.02

0.04

0.06

0.08

β30.1

420-2-4-10-505

asymmetric
expantion-contraction
Bending in the plane including
cranking axis

0.3

00

β30
β31+
β31-

0.4

0y

R-R0(MeV)

0.7

ωrot = .8MeV:K=0
ωrot = .8MeV:K=(+1)ωrot = .8MeV:K=(-1)ωrot = 1.6MeV:K=0
ωrot = 1.6MeV:K=(+1)ωrot = 1.6MeV:K=(-1)

Bending in the plane perpendicular
to cranking axis
0.8 0.91

1.1 1.2 1.3 1.4 1.5 1.6ωrot(MeV)

10z

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24Mg

TRS
TRS to asymmetric deformation
0.80.8ωrot = .8MeV:K=0
ωrot = .8MeV:K=(+1)ωrot = .8MeV:K=(-1)ωrot = 1.4MeV:K=0
ωrot = 1.4MeV:K=(+1)ωrot = 1.4MeV:K=(-1)

0.6
0.5
0.4

0.7
R(β3=0.10)-R(β3=0.00)

0.7420.3 -20
y0.2-4-10-50z

5100.1
0-0.1

0.6

β30
β31+
β31-

asymmetric
expantion-contraction

0.5
0.4
0.3

Bending in the plane including
cranking axis

0.2
0.1
00

0.02

0.04

0.06
β3

0.08

-0.1

0.1

42
yR-R0(MeV)

ωrot dependence of curvature

Bending in the plane perpendicular
to cranking axis
0.8

0.9

1

1.1
1.2
ωrot(MeV)

1.3

1.4

0-2
-4
-10-50z

510

So far, no essential differences among 12C, 16O, 20Ne, 24Mg are observed.
No signifimant difference between A=8N nuclei and N=8N+4 nulei
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Conclusion
In this work, we check the softness of the rod nuclei against bending,asymmetric expansion-contraction, and expansion/contraction.
• Extremely soft against bending.
• Bending in the plane perpendicular to the cranking axis is softer than thebending in the plane including the cranking axis.
• Softer at lower rotational frequency.
• Expansion is responsible for vanishing second minima at high rotationalfrequency• Mode responsible for vanishing second minima at low rotational frequency hasnot been identified• Relatively not soft against asymmetric expansion-contraction• Softer at high rotational frequency• No significant difference between A=8N nuclei and A=8N+4 nuclei
Future problem
・Microscopic understanding of the rotational frequency dependence
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