††thanks: Present address: Physics Division, Argonne National Laboratory,
Argonne, IL 60439, USA††thanks: Deceased††thanks: Present address: Physics
Division, Argonne National Laboratory, Argonne, IL 60439, USA††thanks: Present
address: Department of Radiology, University of Chicago, Chicago, IL 60637,
USA††thanks: Present address: Department of Physical Science, Anne Arundel
Community College, Arnold, MD 21012, USA††thanks: Present address: Physics
Department, Yale University, New Haven, CT 06511, USA

# Evolution of chirality from transverse wobbling in 135Pr

N. Sensharma<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Notre Dame, Notre Dame, IN 46556, USA  Department of Physics and
Astronomy, University of North Carolina Chapel Hill, NC 27599, USA  Triangle
Universities Nuclear Laboratory, Duke University, Durham, NC 27708, USA  U.
Garg<EMAIL_ADDRESS>Department of Physics and Astronomy, University of Notre
Dame, Notre Dame, IN 46556, USA  Q. B. Chen  Department of Physics, East China
Normal University, Shanghai 200241, China  S. Frauendorf  Department of
Physics and Astronomy, University of Notre Dame, Notre Dame, IN 46556, USA  S.
Zhu  National Nuclear Data Center, Brookhaven National Laboratory, Upton, NY
11973, USA  J. Arroyo  Department of Physics and Astronomy, University of
Notre Dame, Notre Dame, IN 46556, USA  A. D. Ayangeakaa  Department of Physics
and Astronomy, University of North Carolina Chapel Hill, NC 27599, USA
Triangle Universities Nuclear Laboratory, Duke University, Durham, NC 27708,
USA  D. P. Burdette  Department of Physics and Astronomy, University of Notre
Dame, Notre Dame, IN 46556, USA  M. P. Carpenter  P. Copp  Physics Division,
Argonne National Laboratory, Argonne, IL 60439, USA  J. L. Cozzi  Department
of Physics and Astronomy, University of Notre Dame, Notre Dame, IN 46556, USA
S. S. Ghugre  UGC-DAE Consortium for Scientific Research, Kolkata 700 064,
India  D. J. Hartley  Department of Physics, United States Naval Academy,
Annapolis, MD 21402, USA  K. B. Howard  Department of Physics and Astronomy,
University of Notre Dame, Notre Dame, IN 46556, USA  R. V. F. Janssens
Department of Physics and Astronomy, University of North Carolina Chapel Hill,
NC 27599, USA  Triangle Universities Nuclear Laboratory, Duke University,
Durham, NC 27708, USA  F. G. Kondev  T. Lauritsen  J. Li  Physics Division,
Argonne National Laboratory, Argonne, IL 60439, USA  R. Palit  Department of
Nuclear and Atomic Physics, Tata Institute of Fundamental Research, Mumbai 400
005, India  A. Saracino  Department of Physics and Astronomy, University of
North Carolina Chapel Hill, NC 27599, USA  Triangle Universities Nuclear
Laboratory, Duke University, Durham, NC 27708, USA  D. Seweryniak  Physics
Division, Argonne National Laboratory, Argonne, IL 60439, USA  S. Weyhmiller
Department of Physics and Astronomy, University of Notre Dame, Notre Dame, IN
46556, USA  J. Wu  Physics Division, Argonne National Laboratory, Argonne, IL
60439, USA

(August 28, 2024)

###### Abstract

Chirality is a distinct signature that characterizes triaxial shapes in
nuclei. We report the first observation of chirality in the nucleus 135Pr
using a high-statistics Gammasphere experiment with the 123Sb(16O,4n)135Pr
reaction. Two chiral-partner bands with the configuration
$\pi(1h_{11/2})^{1}\otimes\nu(1h_{11/2})^{-2}$ have been identified in this
nucleus. Angular distribution analyses of the $\Delta I=1$ connecting
transitions between the two chiral partners have revealed a dominant dipole
character. Quasiparticle triaxial rotor model calculations are in good
agreement with the experiment. This is the first time that both signatures of
triaxiality–chirality and wobbling–have been observed in the same nucleus.

##  I Introduction

Nuclear chiral rotation is a unique signature that helps in the identification
of triaxial nuclear shapes. Chirality appears when the axis of rotation does
not lie in any of the three principal planes of the nucleus Frauendorf and
Meng (1997); Frauendorf (2001); Meng and Zhang (2010); Meng and Zhao (2016).
This arrangement occurs when there is a finite angular momentum component
along all three principal axes of a triaxial rotor. Chirality is
experimentally manifested as two nearly identical $\Delta I=1$ sequences with
the same parity and close excitation energies. A number of nuclei have been
found to exhibit nuclear chiral rotation. Some examples include nuclei in the
regions $A\sim 80$ (80Br Wang  _et al._ (2011)), $A\sim 100$ (104Rh Vaman  _et
al._ (2004)), $A\sim 130$ (135Nd Zhu  _et al._ (2003); Mukhopadhyay  _et al._
(2007),134Pr Starosta  _et al._ (2001); Tonev  _et al._ (2006, 2007), 128Cs
Grodner  _et al._ (2006), 133Ce Ayangeakaa  _et al._ (2013)), and $A\sim 190$
(188Ir Balabanski  _et al._ (2004)). A comprehensive compilation can be found
in Ref. Xiong and Wang (2019). In all cases, two $\Delta I=1$ bands with close
excitation energies were reported. The small energy difference between the two
bands were attributed to tunneling between the left- and right-handed
configurations in the case of static chirality or chiral rotation (CR)
Frauendorf and Meng (1997), and to an oscillation between the two
configurations in the case of chiral vibrations (CV) Starosta  _et al._
(2001).

Figure 1:  Angular momentum geometry for (left) transverse wobbling (TW) and
(right) the chiral vibration (CV) mode in the body fixed frame, where $l$,
$m$, and $s$ correspond to the long, medium, and short axis, respectively.
$\bm{J}$ is the total angular momentum vector. The TW orbit is centered around
the $s$-axis while the CV orbit around the black axis in the $l$-$s$ plane is
the total quasiparticle angular momentum. The quasineutron and quasiproton
angular momenta are shown as the violet and blue arrows aligned with the $l$\-
and $s$\- axes, respectively. With increasing $J$, the quasiparticle arrows
start following the motion of $\bm{J}$. Figure adapted from Ref. Sensharma
(2022). Figure 2:  Partial level scheme of 135Pr developed in the present
work. The lowest level shown is an 11/2- isomeric level with $E_{x}$ = 358.0
keV. The five connecting transitions between dipole bands DB2 and DB1 have
been newly identified in this work. The tentative $\gamma$-ray transitions are
given as dotted lines.

Figure 1 illustrates the case of the 135Pr nucleus. The angular momentum of
the odd $h_{11/2}$ quasiproton aligned with the short ($s$) axis is combined
with a pair of $h_{11/2}$ quasineutrons aligned along the long ($l$) axis. The
resulting particle angular momentum vector lies in the $l$-$s$ plane indicated
by the black arrow. The total angular momentum $\bm{J}$ precesses around this
axis, i.e., it oscillates between the left- and right-handed octants on the
sphere of constant angular momentum. With increasing $J$, the quasiparticle
angular momentum starts to follow the motion of $\bm{J}$, due to the Coriolis
force. The CV evolves into a strongly anharmonic mode, where $\bm{J}$ spends
most of the full period in the chiral positions between the $l$-$s$ and the
$l$-$m$ planes, while rapidly moving over or tunneling through the potential
barriers at the two planes (CR).

The chiral configuration evolves through the alignment of the two
quasineutrons from the single quasiproton configuration illustrated in the
left panel of Fig. 1, which was studied in our previous work Frauendorf and
Dönau (2014); Matta  _et al._ (2015); Sensharma  _et al._ (2019). There, we
demonstrated the appearance of transverse wobbling (TW), which is yet another
manifestation of triaxiality. The present work establishes the existence of
chirality in 135Pr. Although the two identifying signatures of triaxial nuclei
have been separately established in the aforementioned cases, the simultaneous
observation of both in the same nucleus, which proves their mutual relation,
had never been demonstrated before.

Table 1: Initial level energies (Ei), $\gamma$ energies (Eγ), initial and final spins ($I_{\text{i}}^{\pi}$ $\to$ $I_{\text{f}}^{\pi}$), branching ratios (B.R.), mixing ratios ($\delta$), and adopted multipolarities for the transitions presented in Fig. 2. The last two columns present the experimental transition probability ratios for the in-band transitions of DB1 and DB2 as well as for the $\Delta I=1$ transitions connecting the two bands. Ei (keV) | Eγ (keV) |  $I_{\text{i}}^{\pi}$ $\to$ $I_{\text{f}}^{\pi}$ | B.R. | $\delta$ | Mult. |  $\frac{B(M1;\Delta I=1)}{B(E2;\Delta I=2)}$ $(\frac{\mu_{N}^{2}}{e^{2}b^{2}})$ | $\frac{B(E2;\Delta I=1)}{B(E2;\Delta I=2)}$
---|---|---|---|---|---|---|---
730.7(2) | 372.7(4) | $\frac{15}{2}^{-}\to\frac{11}{2}^{-}$ | 1.00 |  | E2 |  |
1390.3(3) | 659.6(4) | $\frac{19}{2}^{-}\to\frac{15}{2}^{-}$ | 1.00 |  | E2 |  |
1477(5) | 746.3(9) | $\frac{17}{2}^{-}\to\frac{15}{2}^{-}$ | 1.00 | -1.40${}_{-0.12}^{+0.14}$ | M1+E2 |  |
2203.1(5) | 811.6(10) | $\frac{21}{2}^{-}\to\frac{19}{2}^{-}$ | 0.55(1) | -1.63${}_{-0.03}^{+0.02}$ | M1+E2 |  |
| 726.1(10) | $\frac{21}{2}^{-}\to\frac{17}{2}^{-}$ | 0.45(1) |  | E2 |  |
2244.4(4) | 854.1(4) | $\frac{23}{2}^{-}\to\frac{19}{2}^{-}$ | 1.00 |  | E2 |  |
2616.4(4) | 1226.1(4) | $\frac{23}{2}^{-}\to\frac{19}{2}^{-}$ | 1.00 |  | E2 |  |
2998.6(4) | 754.2(7) | $\frac{25}{2}^{-}\to\frac{23}{2}^{-}$ | 0.36(1) | -2.16${}_{-0.05}^{+0.06}$ | M1+E2 |  |
| 795.5(6) | $\frac{25}{2}^{-}\to\frac{21}{2}^{-}$ | 0.64(2) |  | E2 |  |
3243.9(4) | 999.5(3) | $\frac{27}{2}^{-}\to\frac{23}{2}^{-}$ | 1.00 |  | E2 |  |
3487.9(4) | 1243.4(4) | $\frac{27}{2}^{-}\to\frac{23}{2}^{-}$ | 0.29(2) |  | E2 |  |
| 871.5(5) | $\frac{27}{2}^{-}\to\frac{23}{2}^{-}$ | 0.71(5) |  | E2 |  |
3863(4) | 375.1(7) | $\frac{29}{2}^{-}\to\frac{27}{2}^{-}$ | 1.00 |  | M1+E2 |  |
3954.1(4) | 710.2(5) | $\frac{29}{2}^{-}\to\frac{27}{2}^{-}$ | 0.18(1) |  | M1+E2 |  |
| 955.5(5) | $\frac{29}{2}^{-}\to\frac{25}{2}^{-}$ | 0.83(5) |  | E2 |  |
4292.8(4) | 1048.4(7) | $\frac{31}{2}^{-}\to\frac{27}{2}^{-}$ | 0.39(5) |  | E2 |  |
| 804.9(5) | $\frac{31}{2}^{-}\to\frac{27}{2}^{-}$ | 0.16(2) |  | E2 |  |
| 430.1(7) | $\frac{31}{2}^{-}\to\frac{29}{2}^{-}$ | 0.45(6) | -0.166(4) | M1+E2 |  |
4319.2(4) | 1075.3(4) | $\frac{31}{2}^{-}\to\frac{27}{2}^{-}$ | 1.00 |  | E2 |  |
4703(4) | 384.8(5) | $\frac{33}{2}^{-}\to\frac{31}{2}^{-}$ | 0.6(1) |  | M1+E2 |  |
| 840.9(6) | $\frac{33}{2}^{-}\to\frac{29}{2}^{-}$ | 0.22(5) |  | E2 |  |
| 410.2(7) | $\frac{33}{2}^{-}\to\frac{31}{2}^{-}$ | 0.23(5) | $-0.165^{+0.001}_{-0.002}$ | M1+E2 | 4.34(4) | 1.01(1)
5028.3(4) | 735.5(5) | $\frac{35}{2}^{-}\to\frac{31}{2}^{-}$ | 0.22(4) |  | E2 |  |
| 324.8(5) | $\frac{35}{2}^{-}\to\frac{33}{2}^{-}$ | 0.8(1) | -0.159(2) | M1+E2 | 15.5(4) | 5.3(1)
5065.6(5) | 1111.5(4) | $\frac{33}{2}^{-}\to\frac{29}{2}^{-}$ | 1.00 |  | E2 |  |
5110.9(4) | 1156.8(4) | $\frac{33}{2}^{-}\to\frac{29}{2}^{-}$ | 1.00 |  | E2 |  |
5162.7(5) | 843.5(5) | $\frac{35}{2}^{-}\to\frac{31}{2}^{-}$ | 1.00 |  | E2 |  |
5374.4(5) | 308.8(5) | $\frac{35}{2}^{-}\to\frac{33}{2}^{-}$ | 0.51(2) |  | M1+E2 |  |
| 263.3(4) | $\frac{35}{2}^{-}\to\frac{33}{2}^{-}$ | 0.49(2) |  | M1+E2 |  |
5451.8(4) | 748.8(5) | $\frac{37}{2}^{-}\to\frac{33}{2}^{-}$ | 0.230(1) |  | E2 |  |
| 423.6(6) | $\frac{37}{2}^{-}\to\frac{35}{2}^{-}$ | 0.770(2) | -0.145(3) | M1+E2 | 7.1(1) | 1.19(2)
5709.1(5) | 598.2(6) | $\frac{37}{2}^{-}\to\frac{33}{2}^{-}$ | 0.16(3) |  | E2 |  |
| 335.3(5) | $\frac{37}{2}^{-}\to\frac{35}{2}^{-}$ | 0.84(2) | -0.049(2) | M1+E2 | 7.4(3) | 0.23(1)
5950.5(5) | 922.2(5) | $\frac{39}{2}^{-}\to\frac{35}{2}^{-}$ | 0.35(2) |  | E2 |  |
| 498.2(6) | $\frac{39}{2}^{-}\to\frac{37}{2}^{-}$ | 0.65(3) | -0.169(2) | M1+E2 | 6.9(3) | 1.14(6)
5996.8(6) | 834.1(4) | $\frac{39}{2}^{-}\to\frac{35}{2}^{-}$ | 1.00 |  | E2 |  |
6093.9(5) | 384.8(6) | $\frac{39}{2}^{-}\to\frac{37}{2}^{-}$ | 0.5(1) | -0.150(3) | M1+E2 | 6.3(2) | 1.36(3)
| 642.2(6) | $\frac{39}{2}^{-}\to\frac{37}{2}^{-}$ | 0.33(8) | -0.10(6) | M1+E2 | 0.9(6) | 0.03(2)
| 719.6(6) | $\frac{39}{2}^{-}\to\frac{35}{2}^{-}$ | 0.18(5) |  | E2 |  |
6504.6(5) | 1052.8(4) | $\frac{41}{2}^{-}\to\frac{37}{2}^{-}$ | 0.235(8) |  | E2 |  |
| 554.2(6) | $\frac{41}{2}^{-}\to\frac{39}{2}^{-}$ | 0.77(3) | -0.055(2) | M1+E2 | 17.2(8) | 0.24(1)
6523.2(5) | 814.1(5) | $\frac{41}{2}^{-}\to\frac{37}{2}^{-}$ | 0.15(2) |  | E2 |  |
| 428.7(8) | $\frac{41}{2}^{-}\to\frac{39}{2}^{-}$ | 0.67(7) | -0.127(5) | M1+E2 | 14.3(6) | 1.80(7)
| 572.9(6) | $\frac{41}{2}^{-}\to\frac{39}{2}^{-}$ | 0.19(2) | -0.15(3) | M1+E2 | 1.7(4) | 0.16(4)
6878.9(6) | 882.1(4) | $\frac{43}{2}^{-}\to\frac{39}{2}^{-}$ | 1.00 |  | E2 |  |
6982.9(5) | 888.9(6) | $\frac{43}{2}^{-}\to\frac{39}{2}^{-}$ | 0.24(2) |  | E2 |  |
| 459.5(6) | $\frac{43}{2}^{-}\to\frac{41}{2}^{-}$ | 0.35(3) | -0.206(7) | M1+E2 | 5.6(2) | 1.61(6)
| 476.9(6) | $\frac{43}{2}^{-}\to\frac{41}{2}^{-}$ | 0.41(4) | -0.11(2) | M1+E2 | 6(1) | 0.46(8)
7027.6(5) | 1077.1(5) | $\frac{43}{2}^{-}\to\frac{39}{2}^{-}$ | 0.49(2) |  | E2 |  |
| 522.1(6) | $\frac{43}{2}^{-}\to\frac{41}{2}^{-}$ | 0.51(2) | -0.138(5) | M1+E2 | 7.4(3) | 0.73(3)
7502.7(5) | 979.5(5) | $\frac{45}{2}^{-}\to\frac{41}{2}^{-}$ | 0.286(4) |  | E2 |  |
| 519.5(7) | $\frac{45}{2}^{-}\to\frac{43}{2}^{-}$ | 0.71(1) | -0.029(5) | M1+E2 | 11(2) | 0.05(1)
| 474.2(6) | $\frac{45}{2}^{-}\to\frac{43}{2}^{-}$ |  |  | – |  |
7594.7(5) | 1090.1(4) | $\frac{45}{2}^{-}\to\frac{41}{2}^{-}$ | 0.329(8) |  | E2 |  |
| 567.5(6) | $\frac{45}{2}^{-}\to\frac{43}{2}^{-}$ | 0.67(2) | -0.059(7) | M1+E2 | 12(2) | 0.18(2)
7801.1(7) | 922.2(4) | $\frac{47}{2}^{-}\to\frac{43}{2}^{-}$ | 1.00 |  | E2 |  |
8033.1(5) | 1050.2(6) | $\frac{47}{2}^{-}\to\frac{43}{2}^{-}$ | 0.436(5) |  | E2 |  |
| 530.3(8) | $\frac{47}{2}^{-}\to\frac{45}{2}^{-}$ | 0.564(7) | -0.076(8) | M1+E2 | 7.7(8) | 0.23(2)
| 436.8(6) | $\frac{47}{2}^{-}\to\frac{45}{2}^{-}$ |  |  | – |  |
8595.5(5) | 1092.8(5) | $\frac{49}{2}^{-}\to\frac{45}{2}^{-}$ | 0.38(2) |  | E2 |  |
| 562.1(6) | $\frac{49}{2}^{-}\to\frac{47}{2}^{-}$ | 0.62(3) | -0.018(2) | M1+E2 | 10(1) | 0.014(2)
8757.9(7) | 956.8(4) | $\frac{51}{2}^{-}\to\frac{47}{2}^{-}$ | 1.00 |  | E2 |  |

The presence of chirality has been mainly associated with the appearance of
two closely-spaced $\Delta I=1$ bands. Further evidence for CR is the
relation, $I={\cal J}\omega={\cal J}\left[E(I)-E(I-1)\right]$ Frauendorf and
Meng (1997) being a straight line emanating from the origin, which is often
reformulated as the staggering parameter $S(I)=[E(I)-E(I-1)]/(2I)=1/(2{\cal
J})$ being constant Koike  _et al._ (2003). Selection rules for the
electromagnetic transitions have been derived for the special case of one
proton particle and one neutron hole occupying the same high-$j$ orbital and
coupled to a triaxial rotor with equal minor moments of inertia Koike  _et
al._ (2004).

In this paper, we present new evidence for the existence of chiral geometry in
the 135Pr nucleus by determining the transition-probability ratios for the
$I\rightarrow I-1$ transitions between the chiral-partner bands.

Figure 3: (Color online) The observed coincidence spectrum resulting from the
(top) sum of gates on Eγ = 642, 573, and 477 keV, (middle) sum of all possible
double gates on the $\Delta I$ = 1, M1 transitions within DB1 and DB2, and
(bottom) double gate on 373- and 660-keV transitions. An inset is included in
the bottom panel to magnify and clearly display the three lowest DB2 $\to$ DB1
transitions. The coincident $\gamma$-ray energies are marked against the
respective energy peaks. The peaks marked in red correspond to the $\Delta$I =
1 in-band transitions of DB1 and DB2. Yrast in-band transitions are marked in
black. All other transitions arising from the deexcitation of 135Pr are marked
in blue and an (*) is marked on the transitions that were observed, but are
not displayed, in the level scheme of Fig. 2.

##  II Experiment

The experiment was performed using the Gammasphere array at the Argonne
National Laboratory. The details of the experiment are similar to those
described in Ref. Sensharma  _et al._ (2019). An 80-MeV 16O beam impinged on
an enriched 123Sb target to populate the levels of interest in 135Pr. The
target was composed of a 697 $\mu$g/cm2-thick foil of 123Sb with a 15
$\mu$g/cm2 front layer of aluminum. A total of 63 Compton-suppressed High
Purity Germanium detectors of the Gammasphere array were available and data
was acquired in the triple-coincidence mode. Energy and efficiency
calibrations were performed using a standard 152Eu radioactive source. To
increase our statistics, data from the present run were added to data obtained
from the run described in Ref. Sensharma  _et al._ (2019). The combined total
of three- and higher-fold $\gamma$-ray coincidence events was 1.52 $\times$
1010. The RADWARE suite of codes Radford (1995) was utilized to analyze the
combined data by sorting them into $\gamma$-$\gamma$ coincidence matrices and
$\gamma$-$\gamma$-$\gamma$ coincidence cubes. A partial level scheme for 135Pr
relevant for the focus of this work is presented in Fig. 2.

The dipole bands, labeled as DB1 and DB2 in Fig. 2, were first identified by
Refs. Matta  _et al._ (2015) and Matta (2017), respectively. The present work
has confirmed the placement of all the $\gamma$ rays within the two bands, and
has identified five new $\Delta I=1$ transitions connecting them. The top
panel of Fig. 3 presents the spectrum resulting from a coincidence gate on the
sum of the three lowest $\Delta I$ = 1 newly identified DB2 $\to$ DB1
transitions (viz. 642.2, 572.9, and 476.9 keV). The middle panel presents the
spectrum resulting from the sum of all possible coincidence double gates on
the $M1$ in-band transitions within the two dipole bands, and the bottom panel
presents coincidence spectrum resulting from a double gate on the 373- and
660-keV transitions.

Angular distribution measurements have been performed for the three lowest
$\Delta I=1$ interconnecting transitions, and the mixing ratios ($\delta$)
corresponding to the lowest $\chi^{2}$ values were extracted. The details of
the angular distribution measurements are the same as those described in Refs.
Sensharma  _et al._ (2019, 2020). Fig. 4 displays the angular distributions
for the 642.2-, 572.9-, and 476.9-keV transitions connecting the DB2 and DB1
bands with the corresponding $\delta$ value noted on each plot. It is seen
that the values of $\delta$ are small, implying that these transitions are
primarily dipole in character.

Figure 4:  Angular distribution plots for the three lowest $\Delta I=1$
transitions connecting the DB1 and DB2 bands. (Left) The experimental points
are given as black circles, and the solid red lines are fits to the angular
distributions. (Right) Calculated $\chi^{2}$ values between the theoretical
and the corresponding experimental angular distributions presented in the left
panel. The minimum $\chi^{2}$ observed in the plots on the right correspond to
the $\delta$ values obtained from the angular distribution fits.

The experimental energies for the two dipole bands are displayed in Fig. 5
(a). These DB1 and DB2 bands cross each other with a minimal distance of 17
keV at $I$ = $41/2$. In accordance, the rotational frequency $\hbar\omega(I)$
in Fig. 5 (b) indicates that DB1 has a surplus of $\approx$ 2$\hbar$ of
angular momentum at a given $\hbar\omega(I)$ value. It increases below $I$ =
$41/2$ and stays approximately constant above this value. The mixing ratios
extracted from angular distributions were utilized to obtain the transition-
probability ratios, $B(E2,I\rightarrow I-1)/B(E2,I\rightarrow I-2)$ and
$B(M1,I\rightarrow I-1)/B(E2,I\rightarrow I-2)$, which are listed in the last
two columns of Table LABEL:tab:1, and displayed in the panels (c), (d), (e),
and (f) of Fig. 5.

Figure 5:  QTR results in comparison with the experimental data: (a)
excitation energies minus a reference, $E(I)-0.011\times I(I+1)$, (b)
Rotational frequency $\hbar\omega(I-1/2)=[E(I)-E(I-2)]/2$, where the black
straight line shows $\hbar\omega=2AI$ with $A=0.011$MeV/$\hbar^{2}$, (c)
$B(E2;\Delta I=1)/B(E2;\Delta I=2)$ and (d) $B(M1;\Delta I=1)/B(E2;\Delta
I=2)$ ratios for the transitions connecting the DB2 and DB1 bands, (e)
$B(M1;I\to I-1)/B(E2;I\to I-2)$, and (f) $B(E2;I\to I-1)/B(E2;I\to I-2)$
ratios for the DB1 and DB2 bands. Note that the experimental error bars in
(a), (b), (e), and (f) are too small to be visible. Figure 6: The root mean
square angular momentum components along the medium (m-, black squares), short
(s-, red circles), and long (l-, blue triangles) axes of the rotor $R$,
valence proton $j_{\pi}$, valence neutron $j_{\nu}$, and the total angular
momentum $J$ for the DB1 band (top panel) and the DB2 band (bottom panel).

Figure 7:  Distributions of the probability $\mathcal{P}$($\theta$, $\phi$)
for the orientation of the total angular momentum $\bm{J}$ (upper two panels)
and the neutron angular momentum $\bm{j}_{\nu}$ (lower two panels) with
respect to the body-fixed frame (SCS maps) Chen and Frauendorf (2022) for the
DB1 band (top two panels) and the DB2 band (bottom two panels). White, blue,
green, yellow, and red are the colors in increasing order of probability.
Figure 8:  QTR results for the electric quadrupole transition probabilities
for the in-band transitions of DB1 and DB2 (a) B(E2; $\Delta I=2$) and (b)
B(E2; $\Delta I=1$). Panel (c) compares the ratio
$\frac{\left<I||E2||I-1\right>}{\left<I||M1||I-1\right>}$ with the experiment.
For $I^{\pi}$ = 47/2- and 49/2-, the ratio is calculated from the predicted
QTR transition energies. The magnetic dipole transition probabilities as
predicted by QTR are displayed in panel (d). Figure 9:  Same as Fig. 8 but for
the $\Delta I$ = 1 connecting transitions between the DB1 and DB2 bands.

##  III Discussion

To understand the nature of the motion, we have carried out calculations in
the framework of the quasiparticle triaxial rotor model (QTR) Qi  _et al._
(2009); Chen  _et al._ (2018, 2020) for the dipole bands DB1 and DB2, based on
the unpaired nucleon configuration
$\pi(1h_{11/2})^{1}\otimes\nu(1h_{11/2})^{-2}$. In these QTR calculations, the
input deformation parameters $(\beta=0.19,\gamma=22^{\circ})$ were determined
by the self-consistent adiabatic and configuration-fixed constrained covariant
density functional theory (CDFT) Meng  _et al._ (2006); Meng (2016) with the
effective interaction PC-PK1 Zhao  _et al._ (2010). The pairing correlations
in QTR were taken into account by the standard BCS quasiparticle approximation
with the empirical pairing gaps $\Delta_{\pi}=\Delta_{\nu}=12/\sqrt{A}\approx
1.0\leavevmode\nobreak\ \textrm{MeV}$. Setting the energy of the spherical
$h_{11/2}$ shells equal to zero, the proton chemical potential
$\lambda_{\pi}=-2.09\leavevmode\nobreak\ \textrm{MeV}$ was located at the
first level of the $h_{11/2}$ shell, and the neutron chemical potential
$\lambda_{\nu}=0.96\leavevmode\nobreak\ \textrm{MeV}$ was located 0.3 MeV
above the midpoint between the fourth and the fifth levels of the $h_{11/2}$
shell. The ratios between the moments of inertia of the rotor were assumed to
be of the irrotational-flow type
$\mathcal{J}_{k}=\mathcal{J}_{0}\sin^{2}(\gamma-2k\pi/3)$ with
$\mathcal{J}_{0}=25.0\leavevmode\nobreak\ \hbar^{2}/\textrm{MeV}$. In the
calculations of the electromagnetic transitions, the empirical intrinsic
quadrupole moment of $Q_{0}=(3/\sqrt{5\pi})R_{0}^{2}Z\beta$ with
$R_{0}=1.2A^{1/3}$ fm and the gyromagnetic ratios of $g_{R}=Z/A=0.44$,
$g_{\pi}(h_{11/2})=1.21$, and $g_{\nu}(h_{11/2})=-0.21$ were adopted. The
results of the QTR calculations are displayed in Figs. 5 \- 8.

The right-most panels of Fig. 6 display the expectation values
$J_{i}=\sqrt{\langle J_{i}^{2}\rangle}$ of the total angular momentum
components with respect to the body-fixed axes. For $I\geq 41/2$, the spin
where DB1 and DB2 come closest, $J_{s}\approx 10\hbar$ and $J_{l}\approx
10\hbar$, while $J_{m}$ continuously increases. This is the angular momentum
geometry that characterizes chiral rotation as sketched in Fig. 1. The other
panels illustrate how the geometry comes about. The $l$-component is generated
by the $h_{11/2}$ quasineutron pair. The $s$-component is the sum of the
$h_{11/2}$ quasiproton and quasineutron contributions. The $m$-component is
mainly generated by the collective angular momentum $R$, with an increasing
contribution from the $h_{11/2}$ quasineutron pair. The real angular momentum
composition deviates from the sketch in Fig. 1, because the quasineutron pair
does not have the pure hole character of the schematic figure and reacts to
the Coriolis force by reorienting toward the $m$-axis.

Figure 7 complements the display of the mean values of the angular momentum
components in Fig. 6. It displays the probability distributions of the total
angular momentum $\bm{J}$ vector and of the quasineutron angular momentum
vector $\bm{j}_{\nu}$ as functions of their angles with respect to the body-
fixed frame, where $\theta$ is the angle with the $l$-axis and $\phi$ is the
angle of the projection on the $s-m$-plane with the $s$-axis. The
distributions represent the classical orbits, as sketched in Fig. 1, where
their fuzziness represents the quantum mechanical limit of the concept of
classical orbits. The strict definition of these Spin Coherent State (SCS)
maps can be found in Ref. Chen and Frauendorf (2022).

These figures present the chiral angular momentum arrangement, which is
schematically illustrated in the right panel of Fig. 1. For $I$ = $29/2$ and
$31/2$, the dipole bands have the character of anharmonic chiral vibrations.
The DB1 band has a distribution that corresponds to a squeezed version of the
ellipse in Fig. 1 (right) and its reflection through the $\theta=90^{\circ}$
plane. The distribution of the DB2 partner corresponds to a squeezed orbit
with a larger $\phi$ amplitude, as is expected for an excited CV. For $I$ =
$33/2$, $35/2$ and $39/2$, the distributions of DB1 and DB2 become centered at
the left- and right-handed angular momentum arrangements, and gradually move
towards the $m$-axis. The probability distribution of the angular momentum of
the one-quasiproton configuration (not shown) stays well centered at the
$s$-axis. The angular momentum distribution of the two-quasineutron
configuration, on the other hand, has a constant $l$-component, as indicated
in Fig. 1 (right). Additionally, it has $s$\- and $m$-components, which
increase with $I$.

Figure 5 (a) compares the QTR energies with the experiment. Although there is
a good overall agreement, the QTR energies exhibit a signature staggering that
is not seen in the experiment. Remarkably, the QTR model reproduces the very
small distance of less than 20 keV between the $I$ = $41/2$ levels. The
distributions of DB1 and DB2 in Fig. 7 are very similar. The small difference
indicates that the two states must have opposite intrinsic symmetry with
respect to the chirality-changing transformation. The chiral-partner bands
based on the $\pi h_{11/2}\otimes\nu h_{11/2}$ configuration in the
neighboring isotope 134Pr show the same crossing with a distance of about 30
keV between the $I$ = $14$ levels.

The analog situation for octupole vibrations has been discussed in Ref.
Frauendorf (2008). The one-phonon band has a rotationally aligned angular
momentum of about 3$\hbar$ and odd $I$, which leads to the crossing of the
even-$I$ zero-phonon band. Mixing cannot appear because the bands have
different spin and parity.

As seen in Fig. 5 (b), DB1 has a surplus angular momentum of about 2$\hbar$,
which generates the band crossing. The very small difference in the distance
between the $I$ = $41/2$ levels indicates an opposite symmetry with respect to
an intrinsic transformation for the two states with the same parity and spin.
This can only be the change between the left- and right-handed arrangements of
the three angular momentum constituents. The details of the band-crossing
phenomenon and their relation to the avoided crossing seen in the TW
configuration of 135Pr (see Ref. Chen and Frauendorf (2022)) are yet to be
investigated.

The experimental values of $\omega(I)$ presented in Fig. 5(b) approach the
relation $\omega=AI$ at large angular momenta, which corresponds to a stable
angular momentum component of the quasiparticles in the $s$-$l$ plane and a
small component along the $m$-axis, as displayed in Fig. 1 (right). The
calculations do not reproduce the change of slope, which is consistent with
the continuous increase of the quasineutron $m$-component seen in Fig. 7. The
calculated values of $\omega(I)$ reproduce the difference of about 2$\hbar$ of
aligned angular momentum, which are characteristic for chiral vibrations Chen
and Meng (2018); Chen  _et al._ (2019). However, they show irregularities near
the band crossing, which are not seen in the experiment.

The QTR values for the $B(E2)$ and $B(M1)$ probabilities for the DB1 and DB2
bands displayed in Fig. 8 are very similar, which is expected for chiral-
partner bands. The respective values for interband transitions are presented
in Fig. 9. With the adopted association of the states within DB1 and DB2, the
intraband $B(E2,I\rightarrow I-2)$ values for $I$ = $41/2$ and $43/2$ are
strongly reduced, while the interband values are strongly enhanced, such that
their sum smoothly increases with $I$. The $B(E2,I\rightarrow I-1)$ values
behave in the same way, while their sum smoothly decreases with $I$. The
intraband and interband $B(M1,I\rightarrow I-1)$ values (see Figs. 8 (d) and 9
(d)) have, respectively, a minimum and maximum at $I$ = $43/2$, while their
sum slightly decreases with $I$.

The QTR mixing ratios for the intraband transitions in Fig. 8 (c) follow the
experimental ones fairly well. The experimental differences between DB1 and
DB2 are larger and opposite compared to the QTR calculations. The QTR values
for the interband transitions of $\left<E2\right>/\left<M1\right>\simeq-0.25$
(Fig. 9 (c)) are close to the experimental ones as well.

From Fig. 5 (e), it is observed that the experimental and theoretical
intraband $B(M1,I\rightarrow I-1)/B(E2,I\rightarrow I-2)$ ratios for the DB1
and DB2 bands are similar, supporting the chiral-partner band interpretation.
With the above-mentioned association of DB1 and DB2 with the lowest QTR
states, the calculated ratios reproduce the experimental ones fairly well,
which also holds for the intraband $B(E2,I\rightarrow I-1)/B(E2,I\rightarrow
I-2)$ values, displayed in Fig. 5 (f). The fact that the $B(E2,I\rightarrow
I-1)$ values are comparable with the $B(E2,I\rightarrow I-2)$ ones is another
indication of chiral geometry. The experimental ratios of DB1 and DB2 differ
more from each other than predicted by the QTR. The discrepancies reflect the
differences of the $\left<E2\right>/\left<M1\right>$ ratios in Figs. 8 (c) and
9 (c). Further, they may be related to differences of the $B(E2,I\rightarrow
I-2)$ values between the chiral-partner bands such as those seen in the
neighboring nucleus 134Pr, which have been attributed to a coupling of the
orientation degrees of freedom to the shape ones Tonev  _et al._ (2007).

Figure 5 (c) compares the experimental and theoretical values of the
$B(E2,I\rightarrow I-1)_{\textrm{out}}/B(E2,I\rightarrow I-2)_{\textrm{in}}$
probability ratios for the $\Delta I$ = 1 transitions connecting DB2 with DB1.
Fig. 5 (d) presents the same comparison for the $B(M1,I\rightarrow
I-1)_{\textrm{out}}/B(E2,I\rightarrow I-2)_{\textrm{in}}$ ratios. The QTR
energy distance between DB2 and DB1 is less than 150 keV for $I$ = $39/2$,
$41/2$, and $43/2$. A possible non-diagonal matrix element of the order of 100
keV missed by the QTR model Hamiltonian will cause a superposition of the QTR
states in DB1 and DB2 that will redistribute the $B(E2)_{\textrm{in}}$ and
$B(E2)_{\textrm{out}}$ values such that their sum remains the same as seen in
Figs. 8 (a) and (b). The same holds for the $B(M1)_{\textrm{in}}$ and
$B(M1)_{\textrm{out}}$ values as seen in Figs. 8 (d) and 9 (d), respectively.
Such a superposition could account for the differences observed in Figs. 5 (c)
and (d). However, it would not change the QTR
$\left<E2\right>/\left<M1\right>$ ratios much, which are similar for the
interband and intraband transitions (see Figs. 8 (c) and 9 (c)). Clearly,
lifetime measurements would be essential to establish the absolute reduced
transition probabilities, and to remove the present ambiguities of being
restricted to only ratios.

As observed in Fig. 5, the QTR calculations predict the same reduced
transition probabilities for the DB1 $\rightarrow$ DB2 transitions as for the
DB2 $\rightarrow$ DB1 ones, which is expected for chiral partners. The absence
of DB1 $\rightarrow$ DB2 transitions in the experiment seems to suggest that
the two bands exhibit chirality that is more akin to a zero- and one-quantum
vibrational state than the static chirality predicted by the QTR model.

The neighboring nucleus 134Pr has been studied in detail in Refs. Starosta
_et al._ (2001); Tonev  _et al._ (2006, 2007). Compared to 135Pr, it has only
one hole-like quasineutron combined with the particle-like quasiproton. Its
angular momentum geometry is closer to the “ideal” arrangement of a proton
particle, a neutron hole and a $\gamma=30^{\circ}$ rotor core, for which the
authors of Ref. Koike  _et al._ (2004) derived selection rules, which generate
the characteristic staggering of the intra and inter band $M1$ transitions. In
Ref. Zhang  _et al._ (2007), the authors demonstrated that the staggering is
quickly quenched when the deformation deviates from $\gamma=30^{\circ}$. In
Ref. Tonev  _et al._ (2007), the authors interpreted their data for 134Pr in
terms of a model that couples the proton particle and the neutron hole with an
IBM core and compared it with the QTR model. The fluctuations of the triaxial
shape, which the IBM core takes into account, removed the staggering pattern
and accounted for the difference between the intra band $B(E2)$ values of the
two partner bands. Nevertheless, the analysis of the angular momentum geometry
pointed to the presence of “weak dynamical chirality”.

With one proton particle and two neutron holes, 135Pr is asymmetric, which
violates prerequisites for the selection rules derived in Ref. Koike  _et al._
(2004). As seen in Figs. 8 and 9, the decay pattern changes to strong intra
band and weak inter band E2 and M1 transitions away from the region where the
partner bands cross. In the crossing region, strong inter band transitions
arise, where the sum of the inter band and intra band values for $B(E2)$ as
well as $B(M1)$ changes smoothly with $I$. The same pattern has been found in
the analog asymmetric case of 135Nd, with two proton particles and one neutron
hole Zhu  _et al._ (2003); Mukhopadhyay  _et al._ (2007). The data were well
reproduced by the QTR calculations in Ref. Qi  _et al._ (2009), where the
analysis of the angular momentum geometry demonstrated the chiral nature of
the two bands. Lifetime measurements Mukhopadhyay  _et al._ (2007)
demonstrated that in contrast to 134Pr, the two bands in 135Nd have very
similar stretched intra band E2 transitions, indicating a more stable shape.
One may speculate that this is the case for the three quasiparticle
configuration in 135Pr as well.

Figure 10: (Color online) Calculated $\chi^{2}$ values between the theoretical
and experimental angular distributions of the 746-, 812-, and 754-keV
transitions in 135Pr (Ref. Matta  _et al._ (2015); Sensharma  _et al._ (2019))
as a function of the mixing ratio $\delta$; the ratio $\sigma/I$ adopted is
0.2. (Insets) Experimental angular distributions of the 746-, 812-, and
754-keV transitions (black full circles), and calculated distributions with
the adopted $\delta$ (red solid line) and the other $\delta$ corresponding to
a higher $\chi^{2}$ (blue dashed line).

In a recent paper, Lv et al. Lv  _et al._ (2022) questioned our earlier work
Matta  _et al._ (2015); Sensharma  _et al._ (2019), which reported on the
observation of TW in 135Pr. As an important result of the present study – the
simultaneous observation of transverse wobbling and chirality – relies on the
angular distribution results, in the following we refute their criticisms –
both experimental and theoretical – and present additional evidence to further
strengthen our arguments.

Reference Lv  _et al._ (2022) claims that the mixing ratios, $|\delta|$, for
the $\Delta I=1$ transitions connecting the wobbling bands could also be $<$
1, thereby questioning the $E2$ nature of these transitions and, hence, the
wobbling interpretation of the level sequences. Their argument is that the
angular distributions provide two $\delta$ values, both of which are equally
likely. This observation is in direct contrast to the mixing ratios reported
in Refs. Matta  _et al._ (2015); Sensharma  _et al._ (2019). We assert that
their claim is erroneous and unfounded. In our measurements, performed with
the Gammasphere array, angular distribution information could be obtained at
sufficient angles so as to reliably extract both the $A_{2}$ and $A_{4}$
coefficients of the standard angular distribution expression. This allowed for
the mixing ratios to be determined unambiguously by differentiating between
the two possible local $\chi^{2}$ minima, with one of them consistently
providing the better fit as determined by the standard $\chi^{2}$-minimization
techniques. The calculated $\chi^{2}$ values between theoretical and
experimental angular distribution for the 746-, 812-, and 754-keV transitions
connecting the wobbling bands as a function of the mixing ratios $\delta$ are
presented in Fig. 10. Clearly, the data gives only one true minimum in each
case, corresponding to the $|\delta|>1$ value (also reported in Ref. Matta
_et al._ (2015)) for these transitions. In the insets, we have presented the
experimental angular distributions for these transitions along with the
calculated distributions corresponding to the two $\delta$ values obtained
from solving the quadratic equation (See Eq. 2.46 in Ref. Sensharma (2022)).
Just looking at this figure could lead to an erroneous conclusion, as Lv et
al. Lv  _et al._ (2022) appear to have reached, that the two fits are
“similar”; however, the $\chi^{2}$ minimization process unambiguously gives
only one value. The results provided in Refs. Matta  _et al._ (2015);
Sensharma  _et al._ (2019), thus, clearly establish the predominantly $E2$
character of the transitions connecting the wobbling bands, rendering firm
evidence for the occurrence of wobbling motion in this nucleus.

Similar critical comments have been made in Lv  _et al._ (2022) about the
polarization asymmetries reported in our paper. We wish to point out that the
polarization asymmetry measurements were provided merely as a further
confirmation of the high $E2$ admixtures established by the angular
distribution data, and not as a way to determine the mixing ratios therefrom.
For the two transitions where it was possible to extract asymmetry parameters,
those are positive even beyond the limits of the uncertainties. As reported in
our paper, the asymmetry parameters for the transitions identified as
“predominantly $E2$” are all positive; are closer to those for the known pure
$E2$ transitions from the yrast band; and have a sign opposite to that
obtained for the 594-keV transition from the signature-partner band to the
yrast sequence, which has been designated as “primarily $M1$”. Indeed, our
procedure, and the associated arguments, are identical to those used in the
first reports of wobbling motion in nuclei Ødegård  _et al._ (2001); Jensen
_et al._ (2002); Amro  _et al._ (2003): in the case of the Lu isotopes as
well, large negative mixing ratios were found and limited polarization results
were employed to confirm the “electric” nature of the wobbling transitions.

Figure 11: Wobbling energies, Ewobb, as a function of spin for the TW1 and TW2
bands in 135Pr. The experimental values are given as black circles (TW1) and
blue squares (TW2). Also shown for comparison are the QTR values from Ref.
Sensharma  _et al._ (2019) (black and blue dotted lines) and that from Ref. Lv
_et al._ (2022) (black and blue dashed-dotted lines) in the upper panel, and
the corresponding TPSM values (dotted and dashed-dotted red lines) in the
lower panel.

The authors of Ref. Lv  _et al._ (2022) also wrongly claim that, in the
theoretical analyses of Refs. Frauendorf and Dönau (2014); Matta  _et al._
(2015); Sensharma  _et al._ (2019), the frozen alignment approximation was
applied. We want to underline that the experimental results were compared with
the QTR calculations, which fully account for the re-orientation of the
particle angular momentum. The main difference between the QTR calculations of
Ref. Lv  _et al._ (2022) and Refs. Frauendorf and Dönau (2014); Matta  _et
al._ (2015); Sensharma  _et al._ (2019) consists in the ratios between the
moments of inertia. The calculations in Ref. Lv  _et al._ (2022) assume
irrotational-flow ratios. The large ratio of four between the $m$\- and
$s$-axes generates an early realignment of the total angular momentum with the
$m$-axis. As a consequence, the energy differences between the lowest band and
the next two bands increase with angular momentum in contrast to the
experimental energies, as seen in the upper panel of Fig. 11.

In the calculations of Refs. Matta  _et al._ (2015); Sensharma  _et al._
(2019), a ratio of about two is assumed, which leads to a later reorientation
of the total angular momentum. As a consequence, the relative energies first
decrease and then increase, which is the signature of transverse wobbling. As
seen in the upper panel of Fig. 11, the angular momentum dependence of the
relative energies agrees qualitatively with the experimental ones. The ratio
of 2 between the moments of inertia was adjusted such that the QTR energies
come as close as possible to the experimental ones.

The TSPM calculations in the lower panel of Fig. 11 evaluated the rotational
response microscopically. The similarity between the QTR and TSPM results
lends additional credibility to the adjusted ratio of two.

##  IV Conclusion

In summary, a new pair of chiral-partner bands based on the
$\pi(1h_{11/2})^{1}\otimes\nu(1h_{11/2})^{-2}$ configuration has been
established in 135Pr using a high-statistics Gammasphere experiment with the
123Sb(16O,4n)135Pr reaction. The predominant magnetic dipole character of the
$\Delta I=1$ transitions between the chiral-doublet bands has been
demonstrated by the precise measurement of angular distributions. Results from
QTR calculations are in good agreement with the experimental observations, and
demonstrate the chiral partnership. With this observation, both signatures of
triaxiality in nuclei–chirality and wobbling–have been evidenced in the same
nucleus, putting the triaxial nature of nuclear shapes on a firmer than ever
footing.

###### Acknowledgements.

This work has been supported in part by the U.S. National Science Foundation
[Grants No. PHY-1713857 (UND), No. PHY-2011890 (UND), No. PHY-2310059 (UND)
and No. PHY-190740 (USNA)]; by the U. S. Department of Energy, Office of
Science, Office of Nuclear Physics [Contract No. DE-AC02-06CH11357 (ANL),
Grants No. DE-FG02-95ER40934 (UND), No. DE-FG02-97ER41033 (UNC), and No. DE-
FG02-97ER41041 (TUNL)]; and by the National Natural Science Foundation of
China under Grant No. 12205103. This research used resources of ANL’s ATLAS
facility, which is a DOE Office of Science User Facility.

## References

  * Frauendorf and Meng (1997) S. Frauendorf and J. Meng, Nucl. Phys. A 617, 131 (1997).
  * Frauendorf (2001) S. Frauendorf, Rev. Mod. Phys. 73, 463 (2001).
  * Meng and Zhang (2010) J. Meng and S. Q. Zhang, J. Phys. G: Nucl. Part. Phys. 37, 064025 (2010).
  * Meng and Zhao (2016) J. Meng and P. W. Zhao, Phys. Scr. 91, 053008 (2016).
  * Wang  _et al._ (2011) S. Y. Wang, B. Qi, L. Liu, S. Q. Zhang, H. Hua, X. Q. Li, Y. Y. Chen, L. H. Zhu, J. Meng, S. M. Wyngaardt, _et al._ , Phys. Lett. B 703, 40 (2011).
  * Vaman  _et al._ (2004) C. Vaman, D. B. Fossan, T. Koike, K. Starosta, I. Y. Lee, and A. O. Macchiavelli, Phys. Rev. Lett. 92, 032501 (2004).
  * Zhu  _et al._ (2003) S. Zhu, U. Garg, B. K. Nayak, S. S. Ghugre, N. S. Pattabiraman, D. B. Fossan, T. Koike, K. Starosta, C. Vaman, R. V. F. Janssens, _et al._ , Phys. Rev. Lett. 91, 132501 (2003).
  * Mukhopadhyay  _et al._ (2007) S. Mukhopadhyay, D. Almehed, U. Garg, S. Frauendorf, T. Li, P. V. M. Rao, X. Wang, S. S. Ghugre, M. P. Carpenter, S. Gros, _et al._ , Phys. Rev. Lett. 99, 172501 (2007).
  * Starosta  _et al._ (2001) K. Starosta, T. Koike, C. J. Chiara, D. B. Fossan, D. R. LaFosse, A. A. Hecht, C. W. Beausang, M. A. Caprio, J. R. Cooper, R. Krücken, _et al._ , Phys. Rev. Lett. 86, 971 (2001).
  * Tonev  _et al._ (2006) D. Tonev, G. de Angelis, P. Petkov, A. Dewald, S. Brant, S. Frauendorf, D. L. Balabanski, P. Pejovic, D. Bazzacco, P. Bednarczyk, _et al._ , Phys. Rev. Lett. 96, 052501 (2006).
  * Tonev  _et al._ (2007) D. Tonev, G. de Angelis, S. Brant, S. Frauendorf, P. Petkov, A. Dewald, F. Doenau, D. L. Balabanski, Q. Zhong, P. Pejovic, _et al._ , Phys. Rev. C 76, 044313 (2007).
  * Grodner  _et al._ (2006) E. Grodner, J. Srebrny, A. A. Pasternak, I. Zalewska, T. Morek, C. Droste, J. Mierzejewski, M. Kowalczyk, J. Kownacki, M. Kisielinski, _et al._ , Phys. Rev. Lett. 97, 172501 (2006).
  * Ayangeakaa  _et al._ (2013) A. D. Ayangeakaa, U. Garg, M. D. Anthony, S. Frauendorf, J. T. Matta, B. K. Nayak, D. Patel, Q. B. Chen, S. Q. Zhang, P. W. Zhao, _et al._ , Phys. Rev. Lett. 110, 172504 (2013).
  * Balabanski  _et al._ (2004) D. L. Balabanski, M. Danchev, D. J. Hartley, L. L. Riedinger, O. Zeidan, J. ye Zhang, C. J. Barton, C. W. Beausang, M. A. Caprio, R. F. Casten, _et al._ , Phys. Rev. C 70, 044305 (2004).
  * Xiong and Wang (2019) B. W. Xiong and Y. Y. Wang, Atom. Data Nucl. Data Tables 125, 193 (2019).
  * Sensharma (2022) N. Sensharma,  _Wobbling Motion in Nuclei: Transverse, Longitudinal, and Chiral_ (Springer Cham, 2022).
  * Frauendorf and Dönau (2014) S. Frauendorf and F. Dönau, Phys. Rev. C 89, 014322 (2014).
  * Matta  _et al._ (2015) J. T. Matta, U. Garg, W. Li, S. Frauendorf, A. D. Ayangeakaa, D. Patel, K. W. Schlax, R. Palit, S. Saha, J. Sethi, _et al._ , Phys. Rev. Lett. 114, 082501 (2015).
  * Sensharma  _et al._ (2019) N. Sensharma, U. Garg, S. Zhu, A. D. Ayangeakaa, S. Frauendorf, W. Li, G. H. Bhat, J. A. Sheikh, M. P. Carpenter, Q. B. Chen, _et al._ , Phys. Lett. B 792, 170 (2019).
  * Koike  _et al._ (2003) T. Koike, K. Starosta, C. Vaman, T. Ahn, D. B. Fossan, R. M. Clark, M. Cromaz, I. Y. Lee, and A. O. Macchiavelli, AIP Conference Proceedings 656, 160 (2003).
  * Koike  _et al._ (2004) T. Koike, K. Starosta, and I. Hamamoto, Phys. Rev. Lett. 93, 172502 (2004).
  * Radford (1995) D. C. Radford, Nucl. Instrum. Methods Phys. Res., Sect. A 361, 297 (1995).
  * Matta (2017) J. T. Matta,  _Exotic Nuclear Excitations: The Transverse wobbling mode in 135Pr_ (Springer International Publishing, 2017).
  * Sensharma  _et al._ (2020) N. Sensharma, U. Garg, Q. B. Chen, S. Frauendorf, D. P. Burdette, J. L. Cozzi, K. B. Howard, S. Zhu, M. P. Carpenter, P. Copp, _et al._ , Phys. Rev. Lett. 124, 052501 (2020).
  * Chen and Frauendorf (2022) Q. B. Chen and S. Frauendorf, Eur. J. Phys. A 58, 75 (2022).
  * Qi  _et al._ (2009) B. Qi, S. Q. Zhang, J. Meng, S. Y. Wang, and S. Frauendorf, Phys. Lett. B 675, 175 (2009).
  * Chen  _et al._ (2018) Q. B. Chen, B. F. Lv, C. M. Petrache, and J. Meng, Phys. Lett. B 782, 744 (2018).
  * Chen  _et al._ (2020) Q. B. Chen, N. Kaiser, U.-G. Meißner, and J. Meng, Phys. Lett. B 807, 135568 (2020).
  * Meng  _et al._ (2006) J. Meng, J. Peng, S. Q. Zhang, and S.-G. Zhou, Phys. Rev. C 73, 037303 (2006).
  * Meng (2016) J. Meng, ed.,  _Relativistic density functional for nuclear structure_, International Review of Nuclear Physics, Vol. 10 (World Scientific, Singapore, 2016).
  * Zhao  _et al._ (2010) P. W. Zhao, Z. P. Li, J. M. Yao, and J. Meng, Phys. Rev. C 82, 054319 (2010).
  * Frauendorf (2008) S. Frauendorf, Phys. Rev. C 77, 021304 (R) (2008).
  * Chen and Meng (2018) Q. B. Chen and J. Meng, Phys. Rev. C 98, 031303(R) (2018).
  * Chen  _et al._ (2019) Q. B. Chen, N. Kaiser, U.-G. Meißner, and J. Meng, Phys. Rev. C 99, 064326 (2019).
  * Zhang  _et al._ (2007) S. Q. Zhang, B. Qi, S. Y. Wang, and J. Meng, Phys. Rev. C 75, 044307 (2007).
  * Lv  _et al._ (2022) B. Lv, C. Petrache, E. Lawrie, S. Guo, A. Astier, K. Zheng, H. Ong, J. Wang, X. Zhou, Z. Sun, _et al._ , Physics Letters B 824, 136840 (2022).
  * Ødegård  _et al._ (2001) S. W. Ødegård, G. B. Hagemann, D. R. Jensen, M. Bergström, B. Herskind, G. Sletten, S. Törmänen, J. N. Wilson, P. O. Tjøm, I. Hamamoto, _et al._ , Phys. Rev. Lett. 86, 5866 (2001).
  * Jensen  _et al._ (2002) D. R. Jensen, G. B. Hagemann, I. Hamamoto, S. W. Ødegård, B. Herskind, G. Sletten, J. N. Wilson, K. Spohr, H. Hübel, P. Bringel, _et al._ , Phys. Rev. Lett. 89, 142503 (2002).
  * Amro  _et al._ (2003) H. Amro, W. C. Ma, G. B. Hagemann, R. M. Diamond, J. Domscheit, P. Fallon, A. Görgen, B. Herskind, H. Hübel, and D. R. Jensen, Phys. Lett. B 553, 197 (2003).