Existence of Exotic Torus Isomer States
and Their Precession Motion
Takatoshi Ichikawa (YITP)
Collaborators:
K. Matsuyanagi, J.A. Maruhn, N. Itagaki

Rotation about the symmetry axis

ZThe equator part is expanded
due to the centrifugal force→Moment of inertia increases
Example: The earth

The rotation about the symmetry axis
is quantum-mechanically forbidden

Spin alignments
Z

I=

I=0

Z

ii(=mi )i+Ω
+Ω’±Ω
= +(m = z +z)

non-collective motion

By spin alignments which break the time-reversal
symmetry, quantum objects can have the angularmomentum about the symmetry axis

Objective
A drastic example:
If many nucleons breaking the time-reversal
symmetry rotates about the symmetry axis
→ torus configuration
Search for stable torus isomers in high-spinstates from 28Si to 56Ni using the cranked
Hartree-Fock (HF) method
How is their excited states?
→ Precession motion→ Describe the precession motion using thetime-dependent Hartree-Fock (TDHF) methodDensity distribution of a stable solution= 1.2 MeV

Cranked 3D Hartree-Fock method→ Code Sky3dJˆz = 0Ĥ

40Ca

varying !
40

Y (fm)

10(Torus)

Ca0–10(a)–100X (fm)10(b)–100X (fm)Ex = 170.45 MeV

10Convergence of K (Jz)
Fission100

Ca (Torus)(SLy6)

80^〈JKz 〉 (!)

A stable solution
exists from 1.2 to
1.7 MeV40

Torus state60

The stable solution
has only K = 60!ω0.5 MeV
1.0 MeV
1.5 MeV
2.0 MeV40
20

Other states

005000
10000
Number of Iteration

15000

quasi stable stateswith 20 and 100

Alignments of orbital angular momentum⇤ ! `zGood quantum number

Neutron| Λ | −Λ ＋ΛGround state76543210(i) K = 20(ii) K = 60(iii) K = 100
2 2 (4 + 5 + 6 ) (3 + 4 + 556 +7 )2 2
2 2insufficient
centrifugal force

Unbound4in the direction perpendicular to the symmetry axis at t = 0,
the torus nuclei start to perform the precession motion. Thisprecession motion is related with a rotation about an axis perpendicular to the symmetry axis, called a perpendicular axis.
In Ref. [5], we already showed that this precession motion isa pure collective motion to restore the broken symmetry anddescribed by the coherent superposition of many 1p-1h states.
We will investigate that other torus nuclei also perform theprecession motion decoupled with other collective modes and36
40
close to the ridged-body one.
Ar its moment of inertia isCa
10 whether
We can also check the stability of the obtained torus nuclei byproviding an impulsive force.
0 Figure 1 shows a schismatic picture of the precession motion taken from Ref. [5]. At t = 0, the torus nucleus is placedon the x-y plane with the angular momentum, K (= Jz ), alongforce–10 the z axis in the laboratory(a) frame. When an impulsive(b)is provided in the negative x direction (the dotted line) att = 0, the vector of the total angular momentum becomes ⃗I
4448line). We call this vector
Ti
Cr the precession axis. Af10 (the dashedter that, the symmetry axis of the density distribution in thebody-fixed flame (the bold solid line) starts to rotate about the0 precession axis with the rotational angle, φ. In the precession
motion, the value K is conserved and its direction is identicalto the bold solid line. The tilting angle, θ, is defined as the–10 angle between the bold(c)solid and the dashed lines.
(d) Then, themoment of inertia for the rotation about a perpendicular axis,
T⊥ , is estimated by T⊥ = I/ωprec , where ωprec denotes the52–10 motion.
0To10
build the first
Fefrequency of the precession
10 rotation
excited states of the precession motion,
we provide an impulX (fm)sive force so as to become the state with I = K + 1.
0 To calculate the precession motion, we determine the time
evolution of the density distribution by solving the TDHF
equation motion given by i! ρ̇ = [ Ĥ, ρ]. To solve the TDHF
code Sky3d and take a Talyor expan–10 equation, we use the(e)sion to the time-development operator up to the 12th orderin the code. The setups of spatial grid points and interac–10
10those of the cranked HF calculations
tions
are the0same asmentionedXabove.
(fm) We start to perform calculations from theinitial density distributions obtained by the cranked HF calculations. The time step of the TDHF calculations is 0.2

TABLE I. Calculated results for obtained stable torus nuclei. The
excitation energy, Eex , is measured from the ground state. The symbols ρ0 , R0 , and d denote the fitted Gaussian parametersfor "obtained!

Y (fm)

Y (fm)

Y (fm)

Systematics of torus isomers

22

2

density distributions, given by ρ(r, z) = ρ0 e − (r − R 0 ) +z /d Thesymbols T⊥ and T ∥ denote the ridged-body moment of inertia forthe rotations about the axis perpendicular to the symmetry axis andabout the symmetry axis, respectively.
System(SLy6)
36 Ar
40 Ca
44 Ti
48 Cr
52 Fe(SkI3)
36 Ar
40 Ca
44 Ti
48 Cr
52 Fe(SkM∗ )
36 Ar
40 Ca
44 Ti
48 Cr
52 Fe

Jz
Eex
ρ0R0dT⊥T∥(!) (MeV) (fm −3 ) (fm) (fm) (!2 /MeV) (!2 /MeV)
3660
4472
52

123.89169.71151.57191.25183.32

0.137
0.129
0.137
0.132
0.138

5.12
6.07
6.30
7.19
7.47

1.62
1.61
1.61
1.60
1.60

14.3
21.1
24.6
33.8
39.1

26.4
39.6
46.5
64.7
75.1

3660
4472
52

125.15173.52153.02
193.66183.70

0.146
0.138
0.146
0.141
0.147

5.01
5.90
6.17
7.00
7.31

1.58
1.58
1.58
1.57
1.57

13.7
19.9
23.6
32.0
37.5

25.3
37.5
44.6
61.3
71.9

3660
44
72
52

124.80167.84152.20192.40187.08

0.131
0.122
0.131
0.125
0.131

5.16
6.17
6.36
7.30
7.55

1.65
1.64
1.64
1.63
1.63

14.6
21.8
25.1
34.9
40.0

26.9
41.0
47.5
66.7
76.7

Skyrme interactions. We ⇢plot ⇠
the density
distributions for the2/3⇢
0 the SLy6 inDensity:
obtained torus nuclei at the z = 0 plane withteraction in Fig. 3. In the figure, the contours correspond tomultiple steps of 0.015 fm −3 . The color is normalized by thelargest density in each plot. The obtained results are tabulated
in Table I. The excitation energies measured from the groundstate are also tabulated in the third column of Table I.

Width: similar to α particle

Macroscopic circulating current of nucleons
40Ca

KJ = 60?
?
T?

How is the moment of inertia for the rotation about anaxis perpendicular to the symmetry axis whenmacroscopic circulating current occurs?

Precession motion☝️Z’Z

The torus isomer has a very large
angular momentum about thesymmetry axis
If we give an impulsive force to thesymmetry axis
→ Precession motion starts
Describe the precession motion
using TDHF method

cf. a top in zero gravity

Time evolution of density distribution
TDHF equation→ Code Sky3di~⇢˙ = [ĥ, ⇢]give an impulsive force at t = 0 by an external field→ I = K + 1 = 61
40Ca

One period: Tprec = 403.0 fm/c2⇡!prec == 3.08 MeV/~T prec

T?TDHF =→I= 19.8 ~2 /MeV!?
rid2T? = 21.1 ~ /MeVRPA2T=
19.6~/MeV→ ?

SummaryWe find stable isomers with the exotic torus configuration in high-spinstates from 36Ar to 52Fe
To build the torus isomer of 40Ca with K = 60 , totally 12 nucleons with Λ= +4, +5, and +6 are aligned with the symmetry axis in both proton andneutron
We also describe the precession motion of torus isomers using the TDHF
method and the obtained moments of inertia for the rotation about anaxis perpendicular to the symmetry axis are almost the rigid-body values
By comparing to the RPA calculation, we found that the precession
motion obtained by the TDHF calculation corresponds to the excitedmode generated by coherent superposition of many 1p-1h excitations
PRL 109, 232503 (2012); PRC 89, 011305(R) (2014)