Investigations of nuclear chirality at iThemba LABS

doi:10.1007/s11467-023-1340-0

Front. Phys.

2024, Vol. 19

Issue (2) : 24302 https://doi.org/10.1007/s11467-023-1340-0

TOPICAL REVIEW

Investigations of nuclear chirality at iThemba LABS

R. A. Bark¹(

), E. A. Lawrie^1,², C. Liu³, S. Y. Wang³

¹. iThemba Laboratory for Accelerator Based Sciences, National Research Foundation, P.O. Box 722, Somerset-West 7129, South Africa
². Department of Physics, University of the Western Cape, Private Bag X17, 7535, Bellville, South Africa
³. Shandong Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, Institute of Space Sciences, Shandong University, Weihai 264209, China

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Abstract

Progress in the studies of chirality in atomic nuclei at iThemba LABS is reviewed. New regions of chirality, around mass 80 and 190 have been discovered using the AFRODITE array, specifically in the nuclei ⁷⁴As, ^78,80,82Br, ⁸¹Kr, and ^193,194,198Tl. Many phenomena have been observed, including multiple chiral bands in the same nucleus, the coexistence of octupole correlations and nuclear chirality, and the coexistence of pseudo spin and nuclear chirality. The best example of chiral degeneracy to date was found in ¹⁹⁴Tl. The level scheme of ¹⁰⁶Ag has been revisited and interpreted in terms of two- and four-quasiparticle bands. Investigations using the particle-rotor model have shown that the fingerprints of chirality in the two-quasiparticle system only can occur in an idealised model description. For systems with a higher number of quasiparticles, the calculations showed that nuclear chirality can persist.

Keywords chirality high-spin states particle-rotor model

Corresponding Author(s): R. A. Bark

Issue Date: 26 September 2023

Cite this article:

R. A. Bark,E. A. Lawrie,C. Liu, et al. Investigations of nuclear chirality at iThemba LABS[J]. Front. Phys. , 2024, 19(2): 24302.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1340-0
https://academic.hep.com.cn/fop/EN/Y2024/V19/I2/24302

Fig.1 Top: Excitation energies calculated with the PRM for a triaxial nucleus with

γ

= 30° and with nucleon configuration

π h 11 / 2 ? ν h 11 / 2 ? 1

[1]. Bottom: A sketch illustrating the left- and right-handed systems formed in angular momentum space by the angular momenta of the valence proton,

j π

, of the valence neutron,

j ν

and of the rotation,

R

Fig.2 Calculated [3] relative excitation energy for the

π h 11 / 2 ? ν h 11 / 2 ? 1

chiral bands,

Δ E = E s i d e ? E y r a s t

, for

ε 2

= 0.15 and for three different cases. The solid red line corresponds to

γ

= 30° and fixed single-particle angular momenta of both valence nucleons, (e.g.,

j π

and

j ν

are fixed along the short and long nuclear axes, respectively). The dashed–dotted blue line corresponds to

γ

= 30°, the same Fermi levels for protons and neutrons, but allowing their configuration spaces to span over 5 single-particle orbitals each. The dashed black line corresponds to

γ

= 20° and the same configuration space of 5 orbitals, for protons and neutrons.

Fig.3 Calculated expectation values for the angles between the angular momenta

j π

j ν

and

R

, for the

π h 11 / 2 ? ν h 11 / 2 ? 1

chiral bands, for

ε 2

= 0.15, and for three cases of chiral bands, as described in Fig.2 and in the text. The data for the yrast and side chiral bands are denoted with open and filled symbols, respectively. The red triangles correspond to

γ

= 30°, and proton and neutron restricted to the lowest- and highest-energy orbitals of the

h 11 / 2

shell only – i.e. where

j π

and

j ν

are fixed along the short and long nuclear axes, respectively. The blue squares correspond to

γ

= 30°, the same Fermi levels for protons and neutrons, but allowing their corresponding configuration spaces to span over 5 single-particle orbitals. The black circles correspond to

γ

= 20° and the same configuration spaces of 5 orbitals for protons and neutrons. Reproduced from Ref. [6].

Fig.4 Calculated probability for vanishing projection of the total angular momentum along the short (s), intermediate (i), and long (l) axes for the

π h 11 / 2 ? ν h 11 / 2 ? 1

chiral partner bands and for restricted (fixed single-particle angular momenta) and non-restricted (free single-particle angular momenta) configurations and for

ε 2

= 0.15. Reproduced from Ref. [6].

Fig.5 Calculated intra- and inter-band

B (M 1)

transition probabilities for the

π h 11 / 2 ? ν h 11 / 2 ? 1

chiral partner bands, with fixed single-particle angular momenta (restricted configuration space) and free single-particle angular momenta (non-restricted configuration space) at

ε 2

= 0.25. Open and filled symbols denote intra- and inter-band

B (M 1)

rates, respectively. Reproduced from Ref. [6].

Fig.6 Calculated energy staggering parameter

S (I) = [E (I) ? E (I ? 1)] / (2 I)

and difference in the excitation energies

Δ E

of the

π h 11 / 2 ? ν h 11 / 2 ? 1

partner bands for fixed single-particle angular momenta (restricted configuration space) and free single-particle angular momenta (non-restricted configuration space), at

γ

= 30° and

ε 2

= 0.15, 0.25. Reproduced from Ref. [6].

Fig.7 Calculated probability distributions for the projections of the total angular momentum on the short (s), intermediate (i), and long (l) nuclear axes, for the yrast and the side

π h 9 / 2 ? ν i 13 / 2 ? 3

partner bands and at

γ

= 30° and

ε 2

= 0.15. Reproduced from Ref. [11].

Fig.8 Calculated excitation energies

E

and angular momenta of the neutrons

j n

along the nuclear short x-axis for the six lowest-energy bands for realistic configuration descriptions and for the chiral bands at

ε 2

= 0.25 and

γ

= 20°, 30°. The calculated bands are labelled bands 1, 2, 3, 4, 5 and 6 according to their excitation energy at low spin. Reduced from Ref. [13].

Nucleus	Reaction(s)	Beam energy (MeV)	Target thickness (mg/cm²)	Events ( $× 109$ )	Refs.

Mass 80 region
⁷⁴As	⁷⁴Ge( $α$ ,p3n)	59, 63	2.9 + backing	1.9 $γ γ$	Xiao, et al. [19]
⁷⁸Br	⁷⁰Zn(¹²C,p3n)	60, 65	0.85	1.5 $γ γ$	Liu, et al. [20]
				0.16 p $γ γ$	Liu, et al. [20]
⁸⁰Br	⁷⁶Ge(¹¹B, $α$ 3n)	54	1.8 + backing	0.2 $γ γ$	Wang, et al. [21]
	⁷⁶Ge(⁷Li,2n)	35	0.2, 0.6 + backing	1.2 $γ γ$ $a)$	Wang, et al. [21]
⁸¹Kr	⁸²Se ( $α$ ,5n)	65, 68	0.4	1.5 $γ γ$	Mu, et al. [22]
⁸²Br	⁸²Se ( $α$ ,p3n)	65, 68	0.4	1.5 $γ γ$	Liu, et al. [23]
Mass 100 region
¹⁰⁶Ag	⁹⁰Zr (¹⁴N,4n)	71	17	3.4 $γ γ$	Lieder, et al. [24]
			0.7	0.4 $γ γ$	Lieder, et al. [24]
Mass 190 region
¹⁹³Tl	¹⁶⁰Gd(³⁷Cl,4n)	167	1.0		Ndayishimye, et al. [25]
	¹⁸¹Ta(¹⁸O,6n)	105	1.0	10.0 $γ γ$	Ndayishimye, et al. [25]
¹⁹⁴Tl	¹⁸¹Ta(¹⁸O,5n)	91, 93	Stacked 2 or 3 × 0.5	3.0 $γ γ$	Masiteng, et al. [10, 26]
	¹⁸¹Ta(¹⁸O,5n)	91	1.0 + backing	$γ γ$	Masiteng, et al. [27]
¹⁹⁸Tl	¹⁹⁷Au( $α$ ,3n)	40	13.0	$γ γ$	Lawrie, et al. [28, 29]
	¹⁹⁷Au( $α$ ,3n)	40	0.2	$e γ$ $b)$	Lawrie, et al. [28, 29]

Tab.1 Details of experiments covered in this review.

Fig.9 An example of a DSAM line-shape analysis [24]. The 966.6 keV

γ

-ray line is the transition that deexcites the 16^? level of Band 2 in ¹⁰⁶Ag. (a) The 45° spectrum (forward direction) and (b) the 135° spectrum (backward direction).

Fig.10 Partial level scheme of ⁸⁰Br showing the chiral partners Band 1 and Band 2. Transitions added by Wang et al. [21] are indicated by stars and red lines.

Fig.11 (a) Excitation energy and (b), (c) staggering parameter

S (I) = [E (I) ? E (I ? 1)] / (2 I)

as a function of spin for the doublet bands in ⁸⁰Br. The filled (open) symbols connected by solid (dotted) curves denote experimental (theoretical) values [21].

Fig.12 Comparisons of the measured and calculated in-band B(M1)/B(E2) ratios for the bands 1 (a) and 2 (b) in ⁸⁰Br [21].

Fig.13 The probability distributions for projection of total angular momentum on the long (

l

-), intermediate (

i

-) and short (

s

-) axis in TPRM for the doublet bands in ⁸⁰Br [21].

Fig.14 Partial level scheme of ⁸²Br showing the chiral partners Band 1 and Band 2. New transitions and levels are indicated in red [23].

Fig.15 The probability distributions for projection of total angular momentum on the long (

l

-), intermediate (

i

-) and short (

s

-) axis in TPRM for the doublet bands in ⁸²Br [23].

Fig.16 Partial level scheme [28] showing the pair of chiral bands in ¹⁹⁸Tl.

Fig.17 Experimental (left panels) and calculated (right panels) excitation energies, staggering

S (I) = [E (I) ? E (I ?

1)] / (2 I)

, and

B (M 1) / B (E 2)

transition probability ratios for the yrast and side bands in ¹⁹⁸Tl at

γ = 44 ∘

. The calculations with and without proton?neutron (

V p n

) interaction are shown. Reproduced from Ref. [29].

Fig.18 Calculated average angles between the angular momentum of the collective rotation and those of the proton

α (p, R)

and neutron

α (n, R)

and also the angle between the angular momenta of the proton and neutron

α (p, n)

for

γ

= 44° (top panels) and

γ

= 30° (bottom panels), and for the yrast and side bands in ¹⁹⁸Tl [29].

Fig.19 Calculated distributions of the projections of the angular momenta of the proton and neutron along the short and long axes, respectively, for

γ = 44 ∘

and for the yrast and side bands in ¹⁹⁸Tl [29].

Fig.20 Partial level scheme showing the negative-parity bands in ¹⁹⁴Tl. The yrast and the side bands from the chiral pair are denoted as Bands 1 and 4, respectively [26].

Fig.21 Excitation energies, alignments, and

B (M 1) / B (E 2)

ratios for the partner bands in ¹⁹⁴Tl. Data involving the 11^? and 12^? levels of the side band are shown with open diamonds. The alignments are calculated with reference parameters of

? 0

= 8

? 2 M e V ? 1

and

? 1

= 40

? 4 M e V ? 3

. Reproduced from Ref. [10].

Fig.22 Potential energy as a function of deformation for the negative parity bands in ¹⁹⁴Tl at (a)

I = 11 ?

and (b)

I

= 21^?. The spacing between the contour lines is 0.25 MeV. Reproduced from Ref. [10].

Fig.23 Excitation energies for the

π h 9 / 2 ? ν i 13 / 2 ? n

, where n = 2,3 bands in ¹⁹⁴Tl (a) and in ¹⁹³Tl (b). Reproduced from Ref. [26].

Fig.24 Alignments for the

π h 9 / 2 ? ν i 13 ? n

bands, where n = 0,1,2,3,4 in ^193,194Tl as a function of the initial spin

I

in panels (a) and (b), and as a function of the rotational frequency in panels (c) and (d). Harris parameters of

? 0 = 8 ? 2

MeV^?1 and

? 1 = 40 ? 4

MeV^?3 were used. The red dashed line in panels (c) and (d) indicates the experimental alignment of the corresponding

ν i 13 / 2 ? n

band in the Hg isotone increased by 2.1

?

, a value that corresponds to the approximate alignment of the

h 9 / 2

proton. Reproduced from Ref. [26].

Fig.25 Line shape analysis of two transitions de-exciting the 28⁺ level of Band 2. Analysis of the 483-keV peak at (a) forward and (b) backward, angles, and (c) analysis of the 931-keV peak in the spectrum that is a sum of the forward and backward spectra. The fit of the Doppler broadened peak of interest is shown with red solid line. Peaks without Doppler broadening are fitted with apparatus line shapes, shown in blue. The black solid line shows the fit for all peaks. In panel (d) the

χ 2

functions from the analysis of the spectra shown in (b) and (c) are plotted. Reproduced from Ref. [27].

Fig.26 Partial level scheme of the high-energy part of four bands of ¹⁹⁴Tl. The lifetimes (shown in red) are measured in ps. Reproduced from Ref. [27].

Fig.27 Experimental and calculated (a)

B (M 1)

, and (b)

B (E 2)

, transition probabilities in the negative-parity bands of ¹⁹⁴Tl. Measured excitation energies are shown in (c). The calculated energies for the four lowest-energy bands and for

γ

= 40° and

γ

= 30° are plotted in (d) and (e), respectively. The calculated bands are labeled with A, B, C and D according to their excitation energy. Reproduced from Ref. [27].

Fig.28 Partial level scheme of ¹⁹³Tl. New transitions are shown in red. Transitions with revised placement are shown in blue. Reproduced from Ref. [25].

Fig.29 Potential energy surfaces calculated with the cranked Nilsson?Strutinsky codes for the negative-parity bands in ¹⁹³Tl. Reproduced from Ref. [25].

Fig.30 Partial level scheme of ¹⁰⁶Ag [24]. The widths of the arrows represent the relative intensities of the transitions. The lifetimes, in ps, are indicated in red; their uncertainties are, on average, 20%.

Fig.31 Comparison of experimental (symbols) and calculated (lines) excitation energies and electromagnetic transition properties of (a) Band 1 and (b) Bands 2 and 3 [24]. The open squares for Band 1 indicate results from Ref. [80]. The theoretical results for

π g 9 / 2 ? 1 ? ν g 7 / 2 2 ν h 11 / 2

configurations of Bands 2 and 3 are displayed as dashed and dotted lines, respectively. The

B M 1

values of bands 2 and 3 are also compared with calculations using a configuration, shown as dash?dot?dotted and dash?dotted lines, respectively.

Fig.32 Quasiparticle alignment i as function of rotational frequency for (a) Band 1 and (b) Bands 2 and 3 in ¹⁰⁶Ag. The experimental points are shown as full symbols. The alignments deduced from the neighbouring nuclei are indicated as straight lines. Results of theoretical calculations are displayed as dashed and dotted lines. The Harris parameters are

? 0 = 8.9 ? 2

MeV^?1 and

? 1 = 15.7 ? 4

MeV^?3. Reproduced from Ref. [24].

Fig.33 The level scheme of ⁷⁸Br (left panel) and the experimental excitation energies, energy staggering parameters

S (I) = [E (I) ? E (I ? 1)] / 2 I

, and reduced transition probability ratios

B (M 1) / B (E 2)

for M

χ

D in ⁷⁸Br in comparison with the TPRM calculations (right panel). Reproduced from Ref. [20].

Fig.34 The experimental B(E1)/B(E2) values (a) and energy displacement

δ E

(b) between Bands 1 and 3 as functions of spin in ⁷⁸Br, together with those in ¹²⁵Ba [84] and ²²⁴Th [83].

Fig.35 The level scheme of ⁷⁴As (left panel) and the experimental excitation energies, energy staggering parameters

S (I) = [E (I) ? E (I ? 1)] / 2 I

, and reduced transition probability ratios

B (M 1) / B (E 2)

for chiral doublet bands in ⁷⁴As in comparison with the TPRM calculations (right panel). Reproduced from Ref. [19].

Fig.36 The experimental B(E1)/B(E2) values (a) and energy displacement

δ E

(b) between Bands 1 and 3 in ⁷⁴As, as functions of spin, together with those in ⁷⁸Br and ²²⁴Th [83].

Fig.37 Level scheme of ⁸¹Kr [22].

Fig.38 The experimental

E (I)

and

B (M 1) / B (E 2)

for bands 2, 3 with the

π g 9 / 2 2 ? ν g 9 / 2 ? 1

configuration and bands 5, 6, 7 with the

π g 9 / 2 (p 3 / 2, f 5 / 2) ? ν g 9 / 2 ? 1

configuration in ⁸¹Kr in comparison with the MPRM results [22].

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