# Illicit object detection in X-ray images using Vision Transformers ††thanks:
Research was possible due to the funding from the European Union’s Horizon
Europe research and innovation programme under grant agreement No 101073876
(Ceasefire)

Jorgen Cani  Department of Informatics and Telematics
Harokopio University of Athens
Athens, Greece
<EMAIL_ADDRESS>Ioannis Mademlis  Department of Informatics and Telematics
Harokopio University of Athens
Athens, Greece
<EMAIL_ADDRESS>Adamantia Anna Rebolledo Chrysochoou  Department of
Informatics and Telematics
Harokopio University of Athens
Athens, Greece
<EMAIL_ADDRESS>Georgios Th. Papadopoulos  Department of Informatics and
Telematics
Harokopio University of Athens
Athens, Greece
<EMAIL_ADDRESS>
###### Abstract

Illicit object detection is a critical task performed at various high-security
locations, including airports, train stations, subways, and ports. The
continuous and tedious work of examining thousands of X-ray images per hour
can be mentally taxing. Thus, Deep Neural Networks (DNNs) can be used to
automate the X-ray image analysis process, improve efficiency and alleviate
the security officers’ inspection burden. The neural architectures typically
utilized in relevant literature are Convolutional Neural Networks (CNNs), with
Vision Transformers (ViTs) rarely employed. In order to address this gap, this
paper conducts a comprehensive evaluation of relevant ViT architectures on
illicit item detection in X-ray images. This study utilizes both Transformer
and hybrid backbones, such as SWIN and NextViT, and detectors, such as DINO
and RT-DETR. The results demonstrate the remarkable accuracy of the DINO
Transformer detector in the low-data regime, the impressive real-time
performance of YOLOv8, and the effectiveness of the hybrid NextViT backbone.

###### Index Terms:

Object Detection, X-rays, Vision Transformers, Deep Neural Networks

##  I Introduction

The detection of concealed illicit items using X-ray images is a common
security procedure [1, 2]. X-ray scanners are a non-invasive way to examine
suspicious containers, such as mailed parcels or luggage, in places where
security is of utmost importance, such as airport terminals, customs offices,
post offices, etc. [3]. The detection process involves X-ray machines
directing beams of high-energy radiation towards the object being scanned. The
varying densities of materials within the object cause the X-rays to be
absorbed at different rates. The resulting high resolution X-ray images, where
denser materials appear lighter and less dense materials appear darker, are
manually inspected for the presence of illicit items (e.g., firearms) by
security officers in real-time [4].

However, X-ray scans present certain shortcomings that perpetrators can
exploit for concealing contraband, such as the occlusion of layered objects, a
cluttered environment, similarity of different objects, as well as certain
material properties which may impact the image appearance [5]. Additionally,
heavy traffic during rush hours may mentally overload the security officers,
causing poor decisions during manual inspection. Hence, efficient and well-
performing automated solutions are necessary for overcoming such issues.

Recent advances in object detection and in the exploitation of multiple
modalities [6] have led to promising X-ray image analysis algorithms [7]. Deep
Neural Network (DNN) architectures such as Convolutional Neural Networks
(CNNs) [8] and Vision Transformers [9] are the state-of-the-art approaches in
general object detection, where the DNN analyzes its input image and outputs
the bounding boxes/Regions-of-Interests (RoIs) of all objects visible on it,
in pixel coordinates, along with the corresponding class labels. However, when
it comes to detecting illicit items in X-ray scan images, the more recent and
powerful Transformer architectures, which rely on the attention mechanism
[10], have not been extensively employed. Relevant literature is still mostly
dominated by CNN-based approaches [11, 12, 13], similarly to visual firearms
recognition in the RGB domain [14, 15].

In order to address these gaps in the literature, this paper attempts to
systematically assess the capabilities of Vision Transformers at the task of
illicit item detection in X-ray images. A hybrid architecture that internally
combines convolutions and the attention mechanism is also being evaluated.
Thus, extensive experimental comparisons are conducted using the SWIN [16] and
Next-ViT [17] deep neural backbones, as well as the DINO [18] and RT-DETR [19]
detection heads. Additionally, YOLOv8 [20] is included as a baseline CNN-based
one-stage object detector. Evaluation is conducted by pretraining on SIXRay
[21], i.e., a relevant, large-scale public dataset, and then finetuning the
object detectors on CFray, i.e., a smaller, custom dataset containing X-ray
images of parcels containing firearms or related objects.

The remainder of this paper is organized in the following manner. Section II
briefly surveys the state-of-the-art regarding illicit item detection in X-ray
images using DNNs. Section III presents the specific deep neural architectures
that are being evaluated in this paper. Subsequently, Section IV discusses the
evaluation datasets, process and results. Finally, Section V draws conclusions
from the conducted study.

##  II Related Work

A wide range of the methods have been employed over the years for detecting
illicit items in X-ray scan images. The method in [22] adopts a complex CNN
pretrained in natural RGB images and finetunes it at the X-ray domain, in an
attempt to address the common issue of DNN overfitting to limited data. This
is a significant problem in relevant literature, since it is difficult to
construct large-scale X-ray image datasets due to the high cost and low
availability of X-ray scanners. In addition, the percentage of “positive”
objects (meaning parcels or luggage that indeed contain illicit items) is
typically very low in such datasets, rendering effective DNN training even
more difficult.

Common DNNs for general object detection have repeatedly been evaluated in
this domain, including Faster R-CNN [23], Mask R-CNN [24], and RetinaNet [25].
However, the standard approach is to use real-time one-stage object detectors
such as YOLO [26] or SSD [27], due to their significantly higher inference
speed. LightRay [13], an alternative lightweight object detector, is a
modification of YOLOv4 designed to detect small illicit items detection in
complex environments. It consists of a MobileNetV3 [28] backbone CNN, a
feature enhancement network and a Lightweight Feature Pyramid Network (LFPN)
[29], that obtains information about objects at different scales, and it also
includes a Convolutional Block Attention Module (CBAM) [30] for refining
feature maps using a spatial attention mechanism.

A different approach is taken in [31], where a new mechanism called Foreground
and Background Separation (FBS) is proposed to separate illicit items from
complex or cluttered backgrounds. This is accomplished by using a feature
extraction DNN combined with Spatial Pyramid Pooling (SPP) and a Path
Aggregation Network to extract high-level features. These feature maps serve
as input to two neural decoders that simultaneously reconstruct the background
and foreground. An attention module then directs the model’s focus to the
foreground objects.

YOLO-CID [11] is a modified YOLOv5 architecture that focuses on real-time
performance. The architecture is enhanced with the Stem [32] and CGhost [33]
neural modules. Thus, it achieves accuracy competitive with respect to the
baseline, but with a smaller model complexity and higher inference speed.

##  III Object Detection Methods

This section details the selected deep neural architectures, starting from the
backbone networks and then moving on to the object detection heads.

###  III-A Backbone Networks

SWIN [16] is a commonly used hierarchical Vision Transformer that attempts to
enhance scale-invariance in image analysis. It is composed of the Swin
Transformer block, which replaces the standard multi-head self attention (MSA)
block. SWIN is a strong backbone network option for object-detection methods,
as seen in [18] and [34]. Four variants of the SWIN Transformer network exist:
tiny, small, base and large. This paper utilizes the base configuration of
SWIN, consisting of 88 million parameters.

Figure 1: The architecture of the NextViT-s backbone. Image from [17].

NextViT [17] is a backbone network that aims for a latency/accuracy trade-off.
ViTs have great results, but are slow to infer. In contrast, CNNs are fast to
infer but inferior to ViTs in terms of performance. NextViT introduces the
Next Convolution Block (NCB) and Next Transformer Block (NTB), which,
respectively, capture both local and global information. The architecture is
primarily designed for industrial/embedded use-cases in mind, such as security
inspection devices in airport terminals, subways, etc. It relies on the Next
Hybrid Strategy (NHS), which creates four stages of various NCB and NTB
blocks, where each stage’s number of blocks depends on the configuration of
the model. The available configurations are small, base and large.

This paper utilizes the small variant of NextViT, which is depicted in Fig. 1.
The figure shows that NextViT-s consists of four stages, with the final layers
being the closest to the final output. Overall, there are 24 layers:

  1. 1.

Layers 0-4 are stem layers.

  2. 2.

Layers 4-6 are NCB components, named Stage 1.

  3. 3.

Layers 7-10 contain three NCB components, followed by an NTB component, named
Stage 2.

  4. 4.

Layers 11-20 contain four NCB component, followed by an NTB component, which
in turn, is followed by four other NCB components and one final NTB component,
named Stage 3.

  5. 5.

Layers 21-23 contain two NCB components and one NTB component, named Stage 4,
which is the final stage.

###  III-B Detection Head Networks

Detection Transformer (DETR) [35] is a Transformer-based detector, which is
placed after a CNN backbone. It handles object detection as a set prediction
task. DETR is not only anchor-free, but also eliminates the need of
handcrafted components (e.g., the Non-Maximum Suppression algorithm). DETR
with Improved deNoising anchOr boxes (DINO) [18] is a DETR-like method that
boasts impressive performance gains by accumulating various minor enhancements
over the base DETR and by reintroducing the usage of anchor boxes in a
Transformer setting. The goal of DINO is to avoid duplicate bounding box
outputs that correspond to a single ground-truth object, which is achieved by
adding two different types of noise to a single ground-truth RoI during
training. Fig. 2 depicts the DINO architecture.

Real-Time DEtection TRansformer (RT-DETR) [19] is the first real-time end-to-
end DETR-like object detector. The architecture consists of a backbone, the
Efficient Hybrid Encoder, an IoU-aware Query selector, and a Decoder & Head
final component. The object queries in DETR-like models are a set of learnable
embeddings, which are optimized by the decoder component and are mapped to the
classification values and bounding boxes by the prediction head. These queries
however are hard to interpret and optimize. RT-DETR introduces the IoU score
into the objective loss function of the classification branch, which leads to
higher quality encoder features. In common implementations, RT-DETR is
typically combined with the HGNetV2 CNN backbone network, with residual
connections at various stages linking the backbone with the detection head.

You Only Look Once (YOLO) [26] is a fast, anchor-based, single-stage object
detector that performs object localization and classification using a single
CNN. YOLO is an efficient algorithm that has gained popularity due to its
speed and accuracy. Its architecture consists of a backbone network, a neck
network, and a final prediction head. The latest iteration of the
architecture, YOLOv8 [20], is used in this paper.

Many variants of the model exist (nano, small, medium, large, extra-large; n,
s, m, l, x); this paper exploits YOLOv8-l. Vanilla YOLOv8 implementations
cooperate with a variant of the YOLO-specific DarkNet CNN backbone. Residual
connections at various stages are exploited to connect this default CNN
backbone to the detection neck and head. There are a few exceptions in the
literature to the privileged use of DarkNet with YOLO. For instance, AG-YOLO
[36] leverages NextViT for optimal performance in a citrus fruit detection
task, thus better addressing occlusion problems.

Figure 2: The DINO architecture. Image from [18].

##  IV Experimental Evaluation

This section details the experimental evaluation datasets and metrics, the
exact experimental setup and algorithmic configuration, as well as the
obtained results.

###  IV-A Experimental Datasets

This paper has utilized the public SIXray [21] dataset for pretraining the
competing DNNs, before subsequently finetuning them on the smaller, custom
CFray dataset.

The SIXray [21] dataset is a publicly available dataset that serves as an
object detection benchmark for identifying illicit goods on X-ray scan images
of travel luggage. It consists of $1,059,231$ X-ray images taken at subway
stations, where $8,929$ illicit items are annotated. There are $6$ classes
contained in the dataset: “gun”, “knife”, “wrench”, “pliers” and “scissors”.
The final class indicates the absence of illicit objects, named “negative”.
The authors split the dataset into three subsets, named SIXray10, SIXray100
and SIXray1000, where the number indicates the ratio of negative to positive
images. SIXRay-D [37] is a modified version of the dataset, utilizing a
cropping scheme that preserves only the task-related regions of each X-ray
image. This paper exploits the SIXRay-D version of SIXray10.

CFray is a custom dataset that contains X-ray scan images of parcels
containing firearms and firearm components. The images have been captured
using an X-ray scanning machine and annotated using the VoTT
111https://github.com/microsoft/VoTT annotation tool. Parcels have been filled
with both illicit and non-illicit objects to emulate, as closely as possible,
real-life illicit parcels. The dataset contains 1368 annotated images, with
the supported classes being: “SMGs”, “Metal Pistols”, “Components”,
“Revolvers” and “Plastic Pistols”. Fig. 3 depicts indicative reference images.

Figure 3: A sample from the CFray dataset.

###  IV-B Evaluation Metrics

This paper presents evaluation results regarding both the achieved object
detection accuracy and the achieved inference speed.

####  IV-B1 Detection Accuracy Metrics

Object detectors typically output the bounding boxes/RoIs of the detected
objects, the predicted classes, and the corresponding confidence value for
each predicted class. Each input image may contain multiple objects (e.g., gun
components, gun ammunition, and a gun in one parcel), therefore each object
should be separately located and then recognized.

The Intersection over Union (IoU) ratio is calculated by comparing the
predicted bounding box with the ground-truth RoI. Ideally, they should be
totally overlapping:

$\text{IoU}=\frac{\text{size of intersection}}{\text{size of union}}.$ (1)

However, the calculated IoU output is typically not 1. Therefore, a threshold
is used to determine whether the predicted RoI is a sufficiently accurate
result. Thresholds of 0.5 and 0.5-0.95 are used in this paper.

The IoU is used as a threshold ratio in the mean Average Precision (mAP)
metric, which is one of the evaluation metrics used. To calculate the mAP,
true positives (TP), false positives (FP) and false negatives (FN) are also
required. Precision measures the percentage of correct predictions, while
Recall measures the correct predictions relative to the ground-truth. These
metrics are calculated using the following equations:

$\text{Precision}=\frac{\text{TP}}{\text{TP}+\text{FP}}.$ (2)
$\text{Recall}=\frac{\text{TP}}{\text{TP}+\text{FN}}.$ (3)

The Precision-Recall (PR) curve illustrates the balance between precision and
recall for a given class. Average Precision (AP) is calculated as the area
under the PR curve, ranging from 0 to 1. The formula for AP is as follows:

$\text{AP}=\int_{0}^{1}{p(r)dr}.$ (4)

Finally, the mAP is calculated as the mean value of AP for each class over all
the N classes:

$\text{mAP}=\frac{1}{N}\sum_{k}^{N}{\text{AP}_{k}}.$ (5)

####  IV-B2 Speed Metric

Inference time in milliseconds (ms) is a metric that calculates the time
required for an object detection method to output the bounding boxes,
predicted classes, and confidence value of one input image. It is typically
calculated as an average over multiple images.

###  IV-C Experimental Setup

The methods presented in Section III have been adopted and compared in the
chosen datasets. Various combinations of detectors and backbones have been
exploited, focusing as much as possible on comparing Transformer/hybrid
components. Thus, both DINO and even the YOLOv8 detection head itself have
been combined with the NextViT bacbbone (although YOLO itself is a CNN).

Regarding the connection of RT-DETR and YOLOv8 detectors with the NextViT
backbone, i.e., combinations which have been engineered from scratch in this
paper, a validation process has been utilized to identify the optimal residual
connections. Experiments have shown that the best option for YOLOv8 is to be
connected to backbone layers 7 and 17, while the respective result for RT-DETR
are layers 9 and 19. However, the final performance differences are very small
compared to different connectivity configurations.

All experiments have been conducted on a PC with Ubuntu Linux 22.04, utilizing
a 13th Gen Intel(R) Core(TM) i9-13900K CPU and two NVIDIA GeForce RTX 4070 Ti
GPUs. Development has taken place in Python, PyTorch and Ultralytics. The
training hyperparameters used with the NextViT backbone are the following
ones: AdamW [38] optimizer with 0.01 learning rate, weight decay of 0.0005 and
finally trained for 100 epochs using early-stopping. For SWIN and DINO, the
hyperparameter values given in [18] have been adopted.

###  IV-D Evaluation Results

Table I shows the results of pretraining the chosen methods on SIXray10-D and
finetuning on CFray. All methods are comparable in terms of mAP. However, both
methods using the NextViT backbone require about a tenth of the inference time
of the SWIN-b and DINO methods, while generating results of similar accuracy.
In terms of mAP@50 performance, SWIN+DINO is the most powerful method. In
X-rays, it is important to have good mAP scores at all thresholds, hence the
more stringent mAP@50-95 threshold evaluation. For this threshold, YOLOv8 [20]
combined with NextViT [17] is slightly ahead of SWIN+DINO.

Thus, if real-time performance is needed, the combination of NextViT+YOLOv8 is
without question the best choice. On the other hand, if the most important
priority is to avoid false negatives (as in many security inspection use-
cases) and inference speed is not equally significant, then the SWIN+DINO
combination should be considered.

TABLE I: Experimental evaluation results for the selected method combinations (pretraining on SIXray10-D, downstream finetuning on CFRay). Configuration | Metrics
---|---
Backbone | Detector | mAP@50 | mAP@50-95 | Inf. Time
SWIN-b | DINO | 0.993 | 0.816 | 175.4
NextViT-s | RT-DETR-l | 0.943 | 0.765 | 16
NextViT-s | YOLOv8-l | 0.986 | 0.82 | 11.6
TABLE II: Experimental results of methods trained only on CFray. Configuration | Metrics
---|---
Backbone | Detector | mAP@50 | mAP@50-95 | Inf. Time
SWIN-b | DINO | 0.942 | 0.837 | 175.4
NextViT-s | RT-DETR-l | 0.94 | 0.748 | 16
NextViT-s | YOLOv8-l | 0.984 | 0.76 | 11.7
TABLE III: Experimental evaluation results for the selected method combinations on SIXray10-D. Configuration | Metrics
---|---
Backbone | Detector | mAP@50 | mAP@50-95 | Inf. Time
SWIN-b | DINO | 0.902 | 0.765 | 161.3
NextViT-s | RT-DETR-l | 0.889 | 0.762 | 16
NextViT-s | YOLOv8-l | 0.906 | 0.793 | 12.8

###  IV-E Ablation Studies

Besides the pretraining+downstream finetuning arrangement, additional
experiments have been conducted with single-pass training only on SIXray10-D
or on CFray. The respective results are shown in Tables III and II,
respectively.

Tables I and II demonstrate that pretraining on the SIXray10-D dataset and
then finetuning on CFRay can significantly and clearly improve the achieved
accuracy only for NextViT-s+YOLOv8-l, and in fact only when the stricter IoU
thresholds are considered.

On the other hand, Table III displays the performance of the employed methods
on the SIXray10-D dataset. The NextViT-s+YOLOv8 method outperforms all others
on all metrics, especially on the mAP@50-95 metric. The SWIN-b+DINO
combination, despite its higher model complexity (and, therefore, lower
inference speed) lags in comparison, accuracy-wise. Given that SIXray10-D is a
more reasonably sized dataset compared to the very small CFRay dataset, this
implies that an end-to-end Transformer architecture faces issues in the highly
specific X-ray domain, compared to hybrid solutions that also employ
convolutions, but is surprisingly accurate in the low-data regime as shown by
Table II.

##  V Conclusions

Automated illicit object detection in X-ray images taken at critical
terminals, such as airports, train stations, subways, and ports, is a crucial
task for public safety. This paper evaluates efficient and performant methods
for handling the large volume and throughput of parcels, luggage, and
passengers. The results demonstrate the remarkable accuracy of the DINO
Transformer detector in the low-data regime, the impressive real-time
performance of YOLOv8, and the effectiveness of the NextViT backbone. Future
research directions may involve combining X-ray-specific auxiliary neural
modules with the methods evaluated in this paper to further improve accuracy.

##  VI Acknowledgment

The research leading to these results has received funding from the European
Union’s Horizon Europe research and innovation programme under grant agreement
No 101073876 (Ceasefire). This publication reflects only the authors views.
The European Union is not liable for any use that may be made of the
information contained therein.

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table

# First identification of a doublet wobbling excitation mode in 105Pd

A. Karmakar1,2, P. Datta3, N. Rather${}^{4},$ S. Pal,5 R. Palit,5 A.
Goswami,1,# G.H. Bhat,6 J. A. Sheikh,4,7, S. Jehangir4 ,,,,,,, S.
Chattopadhyay1,2<EMAIL_ADDRESS>S. Frauendorf,8 1Saha
Institute of Nuclear Physics, 1/AF, Bidhan Nagar, Kolkata - 700 064, India
2Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,
Mumnai - 400 094, India  3Ananda Mohan College, Kolkata - 700 009, India
4Department of Physics, Islamic University of Science and Technology,
Awantipora, 192 122, India  5Tata Institute of Fundamental Research, Mumbai-
400005, India  6Department of Physics, GDC Shopian, Jammu and Kashmir, 192
303, India  7Department of Physics, University of Kashmir, Srinagar, 190 006,
India  8Department of Physics, University of Notre Dame, Notre Dame, Indiana
46556, USA  #Deceased

###### Abstract

An experimental investigation of 105Pd has revealed, for the first time, the
existence of two wobbling bands both having one phonon configuration and
originating from the coupling of the wobbling phonon to the ground state band
and to its signature partner. The doublet one-phonon wobbling bands are, in
turn, found to be the signature partner bands. These observations have been
drawn from the measured ratios of the inter-band and intra-band gamma
transition rates. The model calculations based on the triaxial projected shell
model (TPSM) approach have been performed and are found to be in good
agreement with the experimental observations. These calculations provide an
insight into the nature of the observed structures at a microscopic level.

##  I Introduction

The existence of triaxial deformed nuclei has been predicted by numerous mean-
field studies (see e.g. Ref. triaxial ) for an early systematic exploration of
the nuclear chart). A special feature of triaxial nuclei is that they can
exhibit the wobbling motion near the yrast line and was predicted by Bohr and
Mottelson in 1975 bohr . For large angular-momenta, the energy spectrum will
have a harmonic oscillator pattern bohr  with the bands characterized in terms
of the signature quantum number. For these bands, the moments of inertia along
the three principal axes are unequal and the total angular momentum (vector
sum of particle and rotational angular momentum) wobbles about the medium (m)
axis, which has the largest moment of inertia. The nuclear wobbling is
observed as consecutive rotational bands with increasing excitation energy,
corresponding to successive wobbling quanta ($\propto n\hbar\omega_{W}$ for
large I).

Wobbling mode was originally predicted for an even-even system, however, this
special high-spin mode as discussed by Bohr and Mottelson has not been
observed yet. Instead, the first observation of wobbling was reported in 2001
for the odd-mass 163Lu prl_hageman . The authors interpreted their data in the
framework of the particle + triaxial rotor (PTR) model (see e.g. Ref. bohr ).
Two bands, labelled as TSD1 and TSD2, were identified, which were associated
with the triaxial strongly deformed (TSD) minimum with odd-proton occupying
the i13/2 configuration, obtained in the ultimate cranking calculations of
Ref. Bengtsson04 . The lower TSD1 band corresponds to $n$ = 0 and the excited
TSD2 band to the $n=1$ wobbling excitation. The presence of the high-j proton
significantly modifies the wobbling motion. Using the PTR model, Frauendorf
and Dönau dona  classified the collective mode into longitudinal wobbling (LW)
when the angular momentum vector precesses about the medium axis of the
triaxial nuclear shape and transverse wobbling (TW) when it precesses around
an axis transverse to the medium axis. For increasing angular momentum, the
excitation energy of the wobbling quanta increases for LW, and decreases for
TW up to a critical value $I_{c}$, above which the TW changes into the LW
regime. The wobbling mode represents a periodic motion of the three principal
axes of the triaxial charge distribution, which generates strong E2 radiation.
This is observed as collectively enhanced $\Delta I=1$ E2 transitions between
the levels of two consecutive wobbling bands. Recently, TW mode has been
reported in several odd-A nuclei prl_hageman ; garg ; nandi ; 127xe ; 151eu ;
163lu ; 161lu ; 165lu ; 167lu ; 167ta ; 133la ; 135pr ; sensharmaa ; pd_prl .

The TW regime appears when the angular momentum $\mathbf{j}$ of the odd
particle aligns with the short or long axis of the nucleus which is transverse
to the medium axis and for the reported cases it is the short axis. Of course,
the particle alignment is not rigid and there are excitations which correspond
to the reorientation of the odd particle’s angular momentum $\mathbf{j}$, such
that it precesses about the short axis with the projection $j-1$. The bands
based on this configuration have been referred to as signature partner (SP)
bands garg  because it is their traditional nomenclature in high-spin physics,
where the rotational bands group into $\Delta I=2$ signature sequences
connected by strong intra-band E2 transitions. (The signature of the band is
defined by $\alpha=I$ \+ even number.) In contrast to the TW bands, SP bands
are connected by $\Delta I=1$ M1 transitions with the $n=0$ band. In all the
reported cases prl_hageman ; garg ; nandi ; 127xe ; 151eu ; 163lu ; 161lu ;
165lu ; 167lu ; 167ta ; 133la ; 135pr ; sensharmaa ; pd_prl , the SP band has
been observed. Both the TW and signature partner bands have the signature,
$-\alpha$, that is opposite to the signature $\alpha$ of the n=0 band. The
appearance of the collective TW band, in addition to the signature partner
band, represents stringent evidence for triaxial deformation.

If the two distinct modes keep their dominant character, the next bands with
signature $\alpha$ are expected to be the double TW band ($n=2$), the TW based
on the SP band and the particle excitation, where $\mathbf{j}$ precesses about
the short axis with the projection $j-2$. The authors of Ref. 135pr  discussed
the small amplitude limit of the particle triaxial rotor system, where the TW
and SP modes appear as normal vibrational modes, which combine in an additive
way.

Figure 1: The partial level scheme of 105Pd established from the present work.
The levels and transitions marked in Black were reported in pd_prl  while
those marked in Blue were reported pd_conf  by the same group. Our data
confirms all these levels and transitions and we have added three more
transitions, which are marked in red. The spin and parities of Band 2 have
been adapted from Ref. pd_prl

In the case of 135Pr the odd particle is an $h_{11/2}$ proton and accordingly,
the $n=0$ yrast band has $\alpha=-1/2$. The TW and SP bands with $\alpha=1/2$
were identified in Ref. garg . The authors of Ref. sensharmaa  demonstrated
that a fourth band with $\alpha=-1/2$ has the characteristics of the double TW
excitation. The present case study of 105Pd has a structure that is analogous
to 135Pr with the odd proton replaced by the odd $h_{11/2}$ neutron. The
authors of Ref. pd_prl  identified the $\alpha=-1/2$ yrast band (Band 1) and
the $\alpha=1/2$ TW (Band 3) and SP (Band 2) bands depicted in Fig. 1. This
work also observed, at a somewhat larger energy, an $\alpha=-1/2$ band pd_conf
and the authors speculated that it might be the double TW band by analogy to
135Pr. However, they left the nature of the band an open problem because their
data did not allow them to establish such an assignment. Through the detailed
measurements of the mixing ratios of the transition connecting this band with
the TW band, the present Letter provides the experimental data for a
structural assignment. As will be elucidated below, the second $\alpha=1/2$
band is not the double TW band as speculated, but is the first known example
of a TW-on-SP band.

##  II Experimental Results

The high-spin states of 105Pd were populated through the fusion-evaporation
reaction, where the 63 MeV 13C beam delivered by the 14-UD Pelletron at the
Tata Institute of Fundamental Research (TIFR) was used to bombard a 1 mg/cm2
enriched 96Zr target with Pb backing of 9 mg/cm2 thickness. The de-exciting
$\gamma$ rays were detected by using the Indian National Gamma Array (INGA)
marcos . During the time of the experiment, the array consisted of 18 Compton-
suppressed clover detectors arranged in six rings with three at 40∘, two at
65∘, four at 90∘, three at 115∘, three at 140∘ and three at 157∘ with respect
to the beam direction. The two and higher-fold coincidence data were recorded
by a fast-digital data acquisition system based on Pixie-16 modules marcos .
The corresponding time-stamped data were sorted in a $\gamma$-$\gamma$
symmetric matrix using the multiparameter time-stamped-based coincidence
search (MARCOS) program, developed at TIFR marcos . The matrix was used to
establish the low-lying negative parity levels of 105Pd with the help of the
RADWARE program LEVIT8R rad . The partial level scheme of 105Pd has been shown
in Fig. 1, where the widths of the transitions are proportional to their
relative intensities. The newly placed 254 keV (23/2- $\xrightarrow{}$ 21/2-)
and 300 keV (27/2- $\xrightarrow{}$ 25/2-) transitions between Band 4 and Band
2 are shown in the coincidence spectrum with their respective top transitions,
in Fig. 2. The relative intensities of the gamma transitions obtained from the
present data agree well with the evaluated intensities from Ref. pd_old .
However, the intensities of the newly observed gamma transitions in Ref.
pd_prl  and pd_conf  were not reported. These have been evaluated from the
present data and tabulated in Table 1.

Table 1: Energies, relative intensities, DCO ratios using the detectors at 40∘ and 157∘ ring along with the detectors at 90∘ ring, linear polarizations, mixing ratios and deduced multipolarities of transitions of 105Pd shown in Fig. 1.  Eγ |  Iγ (rel.) |  RDCO |  RDCO | P | $\delta$ | Mult.
---|---|---|---|---|---|---
(keV) | (keV) |  (40∘) |  (157∘) |  |  |
772 | 47.93(22) | 1.04(2) | - | 0.59(3) | 0.1(4) | E2
958 | 29.51(29) | 0.98(2) | - | 0.46(4) | 0.05(4) | E2
939 | 1.01(10) | 0.96(10) | - | 1.15(64) | 0.0(1) | E2
1158 | 1.08(17) | - | 0.65(10) | - | - | M1
604 | 0.64(11) | 0.97(6) | - | - | 0.05(9) | E2
991 | 4.75(16) | 1.20(3) | 1.27(5) | -0.43(27) | 1.8(3) | Mixed
198 | 0.42(11) | 0.60(24) | - | - | 0.02(15) | M1
582 | 32.75(br) | 1.06(7) | - | 0.34(27) | 0.1(1) | E2
1375 | 0.51(34) | 1.09(11) | - | - | 0.10(10) | E2
603 | 1.93(31) | 0.84(4) | 0.78(11) | -0.37(14) | -0.60(5) | Mixed
384 | 19.97(br) | 0.66(5) | - | - | 0.11(4) | M1
814 | 1.98(27) | 0.98(5) | - | 0.56(45) | 0.0(5) | E2
1034 | 3.11(16) | 1.12(6) | 1.15(11) | -0.33(16) | 2.2(5) | Mixed
430 | 0.28(6) | 0.65(7) | - | - | 0.08(7) | M1
808 | 3.31(27) | 0.96(6) | - | 0.33(27) | -0.1(1) | E2
1411 | 0.51(32) | 0.98(14) | - | - | 0.0(1) | E2
453 | 1.69(35) | 0.84(5) | 0.83(12) | - | -0.62(6) | Mixed
378 | 1.04(23) | 0.61(13) | - | - | 0.03(13) | M1
254 | 0.27(9) | 0.94(7) | 0.97(15) | - | 0.35(7) | Mixed
918 | 2.18(29) | 1.00(5) | - | 0.69(50) | 0.0(1) | E2
994 | 2.16(15) | 1.08(5) | 1.23(12) | -0.35(21) | 2.4(7) | Mixed
540 | 0.66(24) | 0.65(8) | - | - | 0.07(7) | M1
1100 | 15.48(47) | 0.99(3) | - | 0.18(8) | 0.1(4) | E2
959 | 1.29(34) | - | - | - | - | -
1084 | 0.48(16) | 1.03(15) | - | 0.89(74) | 0.1(1) | E2
1159 | 0.43(17) | - | - | - | - | -
1006 | 2.90(48) | 1.05(9) | - | 0.75(34) | 0.1(7) | E2
1459 | 0.60(35) | 0.98(13) | - | - | 0.0(15) | E2
300 | 0.12(6) | 0.89(13) | 0.85(20) | - | 0.29(13) | Mixed
1089 | 1.38(32) | 1.05(7) | - | 0.89(45) | 0.1(8) | E2
1152 | 8.96(91) | 1.00(3) | - | 0.25(17) | 0.0(5) | E2
1097 | 1.25(33) | - | - | - | - | -
1262 | 0.18(9) | - | - | - | - | -
1046 | 1.93(41) | 1.03(6) | - | 0.64(43) | 0.1(6) | E2
Figure 2: The gamma spectra in coincidence with (a) 1046 keV (31/2-
$\xrightarrow{}$ 27/2-) keV and (b) 1006 (27/2- $\xrightarrow{}$ 23/2-) keV
transitions of Band 4. The newly placed gamma transitions from Band 4 to Band
2 have been marked in red. The transitions marked in green belong to 102Ru.
This contamination is due to the presence of 1047 keV (15- $\xrightarrow{}$
13-) and 1005 keV (14- $\xrightarrow{}$ 12-) transitions in 102Ru. Figure 3:
Variation of the calculated RDCO as a function of mixing ratio ($\delta$) for
the $\Delta$I = 1 transitions from Band 3 to Band 1 ((a) 991, (b) 1034 keV)
and from Band 4 to Band 2 ((c) 300, (b) 254 keV). The horizontal lines
correspond to the RDCO values at 40∘ along with their uncertainties.

The $\gamma$ \- ray multipolarities were determined from the Ratio of
Directional Correlations from Oriented states (RDCO) method dco . For this
measurement, an asymmetric matrix was constructed with the $\gamma$ -ray
energy detected at 90∘ along one axis while the coincident $\gamma$ -ray
energy at 40∘ and 140∘ on the other axis. For the mixed transitions, the
measured RDCO values have been validated by re-measuring the value using the
“157∘” and “90∘” matrix. For the RDCO measurements, the stretched E2 gating
transitions were used. The value for the attenuation coefficient of complete
alignment ($\sigma$/J) of 0.31(3) was estimated from the measured RDCO values
for 1100 keV E2 and 1331 keV E1 transitions, which were assumed to be pure
stretched transitions. The RDCO values for 40∘ (157∘) were calculated using
the computer code ANGCOR ang  (for $\sigma$/J = 0.31(3)) and were found to be
1.0 and 0.58 (1.0 and 0.51) for pure $\Delta$I = 2 and $\Delta$I = 1
transitions, respectively. The linear polarization (P) measurements were also
performed to extract the electromagnetic character of the de-exciting $\gamma$
rays. The results obtained from the present measurements are tabulated in
Table 1 and are found to be consistent with previously known spin and parity
assignments pd_prl ; pd_conf . It may be observed from Table. 1 that the
experimental RDCO values show a significant departure from the calculated
values for the $\Delta$I = 1 transitions between Band 3 $\xrightarrow{}$ Band
1 and Band 4 $\xrightarrow{}$ Band 2 and also for the $\Delta$I = 0
transitions between Band 4 $\xrightarrow{}$ Band 1. Only these transitions
(labelled as mixed in Table. 1) have substantial M1/E2 mixing. The mixing
ratios ($\delta$) of 991 (Ex = 1961 keV), 603 (Ex = 2345 keV), 1034 (Ex = 2775
keV) and 994 (Ex = 3694 keV) keV transitions have been unambiguously
determined through the combined measurement of RDCO and P. The mixing ratios
for the 991, 1034 and 994 keV transitions between the wobbling bands were
reported to be 1.8(5), 2.3(3) and 2.7(6), respectively pd_prl . The values
obtained from the present measurements are 1.8(3), 2.2(5) and 2.4(7)
respectively, which are in good agreement with the previous measurements.
However, for the low energy transitions of 254, 453 (Ex = 3153 keV) and 300
keV (Ex = 4159 keV), the P measurements were not possible due to inadequate
statistics. In this case, the mixing ratios have been estimated by comparing
the experimental RDCO value with the corresponding calculated values for the
different values of $\delta$ using $\sigma$/J = 0.31(3). These plots for the
two $\Delta$I = 1 transitions from Band 3 to Band 1 namely, 991 and 1034 keV
and two of 254 and 300 keV from Band 4 to Band 2, are shown in Fig. 3. For
Band 3 $\xrightarrow{}$ 1 transitions, the higher evaluated value of $\delta$
is consistent with the values obtained from the combined measurement of RDCO
and P. For 254 and 300 keV transitions, the estimated $\delta$ values are
0.35(7) and 0.29(13), respectively. The DCO ratios for all the observed
$\Delta$I = 1 transitions between Band 3 and 4 agree well with the calculated
value for a pure dipole transition. This was also found to be true for the
1158 keV transition from Band 2 ( Ex = 2900 keV ) to Band 1. Thus, for these
transitions, the E2 mixing has been found to be negligible (E2 fraction $\leq$
0.1$\%$). It is worth mentioning that the 17/2-, 21/2- and 25/2- levels of
Band 3 decay by two $\Delta$I = 1 transitions to Band 1 and Band 4. For the
three transitions to Band 1, the E2 fraction
($\frac{\delta^{2}}{1+\delta^{2}}$) is $\approx$ 85$\%$, while for the
transitions to Band 4, the E2 fraction is $\leq$ 0.1$\%$ as obtained from
Table 1. It may also be noted that the E2 fraction is $\approx$ 10$\%$ for the
254 and 300 keV transitions from Band 4 to Band 2, but the lower transition
energies lead to large electric quadrupole transition rates (B(E2)) for these
$\Delta$I = 1 transitions.

Figure 4: The ratio of the rates of the out-band $\Delta$I = 1 transitions and
the in-band E2 transitions of 105Pd are plotted as a function of spin, I. The
E2 and M1 rates for the $\Delta$I = 1 transitions have been estimated using
the evaluated values of the mixing ratios (Table 1) and are shown on the left
and right panels, respectively. The values for the Band 3 $\xrightarrow{}$
Band 1 transitions are shown in (a) and (d), the values for Band 4
$\xrightarrow{}$ Band 2 transitions in (b) and (e), and the values for the
Band 3 $\xrightarrow{}$ 4 transition in (c) and (f). Fig. 4(f) also includes
the value for the Band 2 $\xrightarrow{}$ Band 1 transition (shown in green),
assuming the E2 mixing to be negligible. In Fig. 4(c), the upper bounds
correspond to the upper limit of the uncertainties of the mixing ratio values
given in Table 1. The dotted line on each panel represents the calculated
values from TPSM.

The ratios of the inter-band to intra-band transition rates have been plotted
in Fig. 4, which show two distinct classes of inter-band $\Delta$I = 1
transitions. In one case, the B(E2) values for these transitions are of the
same order as the intra-band B(E2) rates. Thus, these $\Delta$I = 1
transitions from Band 3 to Band 1 and Band 4 to Band 2 ( Fig. 4(a) and Fig.
4(b) ) have substantial collective enhancement in the B(E2) rates. On the
other hand, the $\Delta$I = 1 transitions for Band 3 $\xrightarrow[]{}$ Band 4
and Band 2 $\xrightarrow[]{}$ Band 1 are predominantly M1 in nature. The
B(M1)out/B(E2)in values for the $\Delta$I = 1 transitions are plotted in Fig.
4(d), (e) and (f). The ratios for the out-band and in-band transition rates
for Band 2 to Band 3 and Band 4 to Band 1 have been plotted in Fig. 5, which
shows an increasing trend in the B(M1)out/B(E2)in values for the $\Delta$I = 0
transitions from Band 4 to Band 1 as a function of spin (Fig. 5(b)).

The present analysis establishes that the inter-band $\Delta$I = 1 transitions
for Band 2 $\xrightarrow{}$ Band 1 and Band 4 $\longleftrightarrow$ Band 3 are
predominantly M1, while the $\Delta$I = 1 transitions for Band 3
$\xrightarrow{}$ Band 1 and Band 4 $\xrightarrow{}$ Band 2 shows large
enhancement in B(E2) rates.

Figure 5: The ratio of the rates of the out-band $\Delta$I = 0 and $\Delta$I =
2 transitions from Band 4 to Band 1, Band 2 to Band 3 and the in-band E2
transition of Band 4 as a function of spin, I. The M1 and E2 rates have been
estimated using the evaluated values of the mixing ratios (Table. 1) and
plotted in (a) and (b), respectively. The dotted line on each panel represents
the calculated values from TPSM. Table 2: Transition rates of different bands
from TPSM Calculations used for the comparison plots with experimental values.

|  |  Band 2 |  |
---|---|---|---|---
Spin | B(E2)in | B(E2)out | B(M1)out | Decay to
($\hbar$) | (e2b2) | (e2b2) | ($\mu_{n}^{2}$) |
10.5 | 0.54 | 0.008 | 0.13 | Band 1
|  |  | 0.11 | Band 3
12.5 | 0.82 | 0.008 | 0.18 | Band 1
|  |  | 0.15 | Band 3
14.5 | 0.89 | 0.001 | 0.19 | Band 1
|  |  | 0.09 | Band 3

|  |  Band 3 |  |
---|---|---|---|---
Spin | B(E2)in | B(E2)out | B(M1)out | Decay to
($\hbar$) | (e2b2) | (e2b2) | ($\mu_{n}^{2}$) |
8.5 | 0.72 | 0.33 | 0.14 | Band 1
|  | 0.01 | 0.41 | Band 4
10.5 | 0.80 | 0.32 | 0.12 | Band 1
|  | 0.01 | 0.46 | Band 4
12.5 | 0.72 | 0.23 | 0.11 | Band 1
|  | 0.001 | 0.46 | Band 4
14.5 | 0.70 | 0.18 | 0.11 | Band 1

|  |  | Band 4 |
---|---|---|---|---
Spin | B(E2)in | B(E2)out | B(M1)out | Decay to
($\hbar$) | (e2b2) | (e2b2) | ($\mu_{n}^{2}$) |
7.5 | 0.63 | 0.02 | 0.51 | Band 3
|  | 0.23 | 0.08 | Band 1 ($\Delta$I = 0)
|  | 0.02 | - | Band 1 ($\Delta$I = 2)
9.5 | 0.69 | 0.01 | 0.42 | Band 3
|  | 0.53 | 0.35 | Band 2
|  | 0.20 | 0.17 | Band 1 ($\Delta$I = 0)
|  | 0.02 | - | Band 1 ($\Delta$I = 2)
11.5 | 0.72 | 0.01 | 0.38 | Band 3
|  | 0.60 | 0.37 | Band 2
|  | 0.28 | 0.79 | Band 1 ($\Delta$I = 0)
|  | 0.01 | - | Band 1 ($\Delta$I = 2)
13.5 | 0.74 | 0.004 | 0.35 | Band 3
|  | 0.55 | 0.38 | Band 2
|  | 0.27 | 0.65 | Band 1 ($\Delta$I = 0)
|  | 0.01 | - | Band 1 ($\Delta$I = 2)

##  III Discussion

To provide an insight into the nature of the band structures observed in
105Pd, triaxial projected shell model (TPSM) tpsm  calculations have been
performed. This model has been shown to provide a unified description of the
interplay between collective and single-particle degrees of freedom in
deformed odd-mass nuclei SJ21 ; SB10 ; JM17 ; 151eu . In the present study of
105Pd, the deformed basis states have been obtained by solving the triaxial
Nilsson potential with $\epsilon=0.257$ and $\epsilon^{\prime}=0.12$ pd_prl .
The monopole pairing interaction is employed by solving standard Bardeen-
Cooper-Schriefer (BCS) equations and the resulting Nilsson + BCS states are
then projected onto good angular-momentum states rs82 . For odd-neutron
nuclei, the basis states are composed of one-neutron and one-neutron coupled
to two-proton states, which are sufficient for the discussion of the observed
low-lying band structures in 105Pd. We have also evaluated the transition
probabilities with the effective charges of 1.6e for protons and 0.6e for
neutrons GH14 ; bh14 ; bh14a .

Figure 6: Energies of the projected K configurations, where the short axis is
chosen as the quantization axis 3 to which K refers. The curves are labelled
by three quantities: quasiparticle character, K quantum number and energy of
the quasiparticle state. For instance, [1,1/2,1.31] designates the one
quasineutron state with K= 1/2 having intrinsic energy of 1.31 MeV. Figure 7:
Experimental and calculated values from TPSM for the rotational frequency
(upper panel) and the level energies minus the rotor contribution (lower
panel) as functions of the spin I for Bands 1, 2, 3 and 4. Figure 8:
Probabilities of various projected K configurations in the orthonormal basis
shown in Fig. 6. The curves are labelled by three quantities: quasiparticle
character, K quantum number and energy of the quasiparticle state. For
instance, [1,1/2,1.31] designates one quasineutron state with K= 1/2 having an
intrinsic energy of 1.31 MeV. The symbols and line types agree with Fig.6.

The experimental energies $E(I)$ and the derived rotational frequencies
$\hbar\omega(I)$ are compared with the TPSM values in Fig. 7. The calculations
reproduce the experimental values for all four bands quite well. The levelling
of Band 1 at $\hbar\omega=0.58$ MeV (“upbend” of the function $I(\omega)$)
reflects the rotational alignment of an additional pair of $h_{11/2}$
quasineutrons, which is seen in Fig. 6 as the bands originating from the
three-quasineutron configuration at 4.17 MeV become yrast. A similar levelling
is found in the TPSM results for Bands 2, 3, and 4 and reflects the crossing
of the three-quasineutron configurations. The TPSM levelling is seen in the
experimental frequencies of Bands 3 and 4 as well. However, the two
experimental frequency points of Band 2 do not confirm the TPSM prediction.

The TPSM wavefunctions have been used to calculate the M1 and E2 transition
probabilities between the excited levels of 105Pd, which are listed in Table
2. Figs. 4 and 5 compare the pertaining ratios with the observed inter-band to
intra-band ratios of the transition rates. For all the transitions, the
agreement was found to be quite well, which substantiates the suggested
interpretation.

The E2 probability for Band 3$\rightarrow$Band 1 is enhanced, as expected for
collective transition connecting the $n=1$ TW state with the $n=0$ yrast
state. The transitions Band 2$\rightarrow$Band 1 are weak, expected for the SP
band, which has a different orientation of the $h_{11/2}$ quasineutron than
the yrast band. The transitions Band 4$\rightarrow$Band 2 are enhanced, as
speculated for the collective E2 transition connecting the $n=1$ TW excitation
on top of the SP states with the SP states. The transitions Band
4$\rightarrow$Band 3 are small because the orientation of the $h_{11/2}$
quasineutron differs in the two bands. For the same reason, the transitions
Band 4$\rightarrow$Band 1 are weak.

In order to further investigate the structure of the observed bands in 105Pd,
Fig. 6 shows the energies of the bands projected from the quasiparticle
configurations before the TPSM Hamiltonian is diagonalized. It is important to
point out that the short principal axis is chosen as the 3-axis of
quantization, and “K” denotes the angular moment projection on the short axis.
This is different from the TPSM calculations published so far where the long
principal axis is chosen as the 3-axis. This choice simplifies the
interpretation because, in the TW regime, the odd quasineutron tends to align
its angular momentum with the short axis. This has the consequence that the
final states are much less mixed with respect to K. Of course, such a change
of the quantization axis leaves the observables invariant. Technically, it is
achieved by changing the triaxiality parameter $\gamma$ to an equivalent value
in another sector. The TPSM parameters $\epsilon=0.257$ and
$\epsilon^{\prime}=0.12$ correspond to $\epsilon=\epsilon_{0}\cos\gamma$ and
$\epsilon^{\prime}=\epsilon_{0}\sin\gamma$ with $\epsilon_{0}=0.2795$ and
$\gamma=23.17^{\circ}$ and the 3-axis being long principal axis. Changing
$\gamma\rightarrow-120^{\circ}-\gamma=-143.172^{\circ}$ leaves the results
invariant but moves the 3-axis to the short principal axis and the
corresponding parameters for which the TPSM calculations were performed are
$\epsilon=0.2237$ and $\epsilon^{\prime}=-0.1675$.

Fig. 8 displays the diagonal components of the $K$-distribution of the
wavefunctions for various angular momentum states for the four observed bands.
The lowest sequences of basis states in Fig. 6 are projections from the one-
quasineutron configuration at 1.307 MeV onto different values of $K$. As seen
in Fig. 8, for the states $I$=11/2, 15/2 19/2, … of Band 1 the largest
components are the $K$=11/2 states. Decomposing these basis states into their
quasineutron factors and collective factors (defined as the projections from
the quasiparticle vacuum) shows that the main quasineutron component is
$h_{11/2}$ with the projection of the triaxial Nilsson state along the s-axis,
$k$=11/2. For the $I$= 13/2, 17/2, 21/2, … states of Band 3 band, the same
$K$=11/2 basis states appear with the largest probability. The higher energies
of the $I$= 13/2, 17/2, 21/2, … states compared to the $I$=11/2, 15/2 19/2, as
seen in Fig. 6, are due to a larger fraction of the collective angular
momentum. This is the expected structure for the TW mode with the angular
momentum of the $h_{11/2}$ quasineutron being aligned with the s-axis.

For $I$=17/2, 21/2, …. of band Band 2, the dominating basis state is $K$=7/2
and decomposing it into the quasineutron and collective factors shows that the
most probable component is $h_{11/2}$ with the $k=$7/2 projection. It has the
characteristic signature partner structure, with the angular momentum of the
quasineutron being tilted away from the s-axis. The reorientation of the
quasineutron’s angular momentum suppresses the collective E2 transitions Band
2$\rightarrow$Band 1 and Band 2$\rightarrow$Band 3, and the transitions
acquire the M1 character. For the $I$=15/2, 19/2, 23/2, … of Band 4, the
largest component is $K$ =7/2. The energies of the states are larger as
compared to the 13/2, 17/2, 21/2, …, because of the additional collective
angular momentum of the wobbling motion. The similar orientation of the
quasineutron’s angular momentum allows enhanced E2 Band 4$\rightarrow$Band 2
transitions. For Band 4$\rightarrow$Band 3 transitions, the change of the
quasineutron’s angular momentum alignment with the s-axis suppresses the
collective E2 radiation and generates the M1 part.

The above interpretation in terms of the leading basis states loses relevance
with increasing $I$ because of the mixing of different $K$ states projected
from the quasineutron configuration at 1.307 MeV as well from the
configurations at 1.074 MeV and 1.792 MeV. This is reflected by the decrease
in the probability of the configuration at 1.307 MeV with increasing $I$.

The nucleus 105Pd has a structure that is analogous to 135Pr, where the odd
$h_{11/2}$ quasineutron replaces the odd quasiproton. The latter has been
studied in Refs. dona ; garg ; sensharmaa  using the TPSM and the PTR model.
Both approaches predict that the second band with the signature 11/2, 15/2,
19/2, … has the character of the $n$=2 double TW excitation mode, which is in
agreement with the strong E2 transition to the $n$=1 single TW band observed
in 135Pr sensharmaa . As shown in the present work, this band has a different
nature in 105Pd. The TPSM accounts for the different nature of these bands in
105Pd and 135Pr. At variance, the PTR model predicts an $n$ = 2 double TW
excitations for Band 4 in 105Pd as well Chen_PC . The PTR model predicts the
SP band in both nuclei at twice the energy above the yrast band than observed
pd_prl ; sensharmaa , while the TPSM approach reproduces the experimental
energy difference quite well. The TPSM takes the anti-symmetrization between
the odd nucleon and the nucleons of the collective rotor core into account,
while the PTR model neglects it. This could be the reason for the discrepancy
between the two models.

In summary, the multipolarities and the electromagnetic characters of the
gamma transitions from the excited levels of the second band with angular
momentum states, 15/2, 19/2, 23/2, … (Band 4) have been determined through the
RDCO and polarization measurements. The transitions connecting it with the
previously known single transverse wobbling band (Band 3) are dominated by the
M1 component with only weak E2 admixtures. This excludes the interpretation as
a double transverse wobbling excitation, which has been suggested for the
analogue band in 135Pr. Instead, the band has been interpreted as a transverse
wobbling excitation built on the signature partner band (Band 2). Accordingly,
the inter-band $\Delta$I = 1 transitions show large enhancement in the B(E2)
rates. Triaxial projected shell model calculations account well for the
observed energies and ratios of the transition probabilities. Analyzing the
calculated eigenstates, the doublet wobbling nature of the bands observed in
105Pd has been demonstrated.

In comparison to axial nuclei, triaxial systems have the possibility to rotate
about the third axis as an additional collective degree of freedom, which
generates the wobbling mode as a set of collective excitations. In general,
the collective excitations are expected to appear not only based on the yrast
band, but also on quasiparticle excitations. The present work identified the
first example of such a wobbling mode of a quasineutron excitation - the
signature partner band with a transverse wobbling excitation built on it,
which is an essential indication of its collective nature.

Private communications on the particle-rotor model results for 105Pd with
Prof. Q. B. Chen are gratefully acknowledged. GHB, JAS and NR acknowledge the
Science and Engineering Research Board (SERB), Department of Science and
Technology, Govt. of India for providing financial support under Project no.
CRG/2019/004960 to carry out a part of this research work. AK is thankful to
the Council of Scientific $\&$ Industrial Research (CSIR), India, for the
Senior Research Fellowship vide file no. 09/489(0121)/2019-EMR-I.

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