# Systematic Analysis of Security and Vulnerabilities in Miniapps

Yuyang Han, Xu Ji, Zhiqiang Wang  Beijing Electronic Science and Technology
InstituteBeijingChina and  Jianyi Zhang  Beijing Electronic Science and
Technology InstituteBeijingChina

(2023)

###### Abstract.

The past few years have witnessed a boom of miniapps, as lightweight
applications, miniapps are of great importance in the mobile internet sector.
Consequently, the security of miniapps can directly impact compromising the
integrity of sensitive data, posing a potential threat to user privacy.
However, after a thorough review of the various research efforts in miniapp
security, we found that their actions in researching the safety of miniapp web
interfaces are limited. This paper proposes a triad threat model focusing on
users, servers and attackers to mitigate the security risk of miniapps. By
following the principle of least privilege and the direction of permission
consistency, we design a novel analysis framework for the security risk
assessment of miniapps by this model. Then, we analyzed the correlation
between the security risk assessment and the threat model associated with the
miniapp. This analysis led to identifying potential scopes and categorisations
with security risks. In the case study, we identify nine major categories of
vulnerability issues, such as SQL injection, logical vulnerabilities and
cross-site scripting. We also assessed a total of 50,628 security risk hazards
and provided specific examples.

Miniapps, Security Risk, Vulnerabilities, Least Privilege

††copyright: acmlicensed††journalyear: 2023††conference: Proceedings of the
2023 ACM Workshop on Secure and Trustworthy Superapps; November 26, 2023;
Copenhagen, Denmark.††booktitle: Proceedings of the 2023 ACM Workshop on
Secure and Trustworthy Superapps (SaTS ’23), November 26, 2023, Copenhagen,
Denmark††price: 15.00††isbn: 979-8-4007-0258-7/23/11††doi:
10.1145/3605762.3624432††ccs: Security and privacy Web application security

##  1\. Introduction

According to the Internet Society of China, the miniapp has emerged as a new
genre of application that has attracted widespread attention in recent
years(Yun, 2023). As of 2019, the total number of miniapp users in China has
reached an impressive 1.4 billion, with 780 million new users and 960 million
active users. A miniapp is a sub-application that runs within a mobile
application, serving as a lightweight and agile solution; that provides users
with fast and convenient functions and services. This type of application has
gained remarkable traction on various social networking platforms, especially
on WeChat, one of the most widely used communication applications with 1.3
billion monthly active users(Online, 2023). Although miniapps are essentially
web-based, they provide a seamless and native app-like user experience,
allowing users to access various services without installing separate
applications. Due to their efficiency and ease of use, these miniapps have
gained significant popularity within the mobile Internet ecosystem and have
attracted considerable attention from users and businesses.

However, as miniapps continue to flourish, concerns about their security have
escalated. As with any web application, ensuring security and protecting user
data constitute pivotal aspects that demand careful consideration and
proactive measures. This concern has led the official WeChat miniapps to
introduce a set of security guidelines(Wechat, 2021). Nevertheless, during
their widespread adoption, miniapps have also revealed vulnerabilities and
security issues, attracting the attention of hackers and attackers.
Consequently, security has emerged as a pressing issue that needs to be
addressed within the miniapps ecosystem. According to the Aladdin Research
Institute, the major platforms exert guidance and constraints on miniapps
safety through two different models. Firstly, the product model, embodied by
WeChat and Alipay, employs an official miniapp security testing service to
thwart potential threats proactively. Secondly, the rating model, exemplified
by Tiktok, Kuaishou, and Baidu, uses scoring to intervene in security events.
In addition, third-party markets offer many security products and services to
address and prevent various problems(Institute, 2023).

Meanwhile, security experts have delved into various aspects of miniapps in
recent years to protect user privacy and data integrity and to facilitate a
thriving miniapp ecosystem. Some have performed backward decompilation of
miniapps, while others have examined the differences between miniapps in
Android and iOS clients or explored the variances across different platforms.
Yet, it is noteworthy that research explicitly targeting the security of
miniapp web interfaces and their associated vulnerabilities still needs to be
explored.

In this paper, we focus on a comprehensive analysis of the security challenges
posed by the web interfaces of miniapps. To this end, we propose a powerful
triad threat model for holistically assessing and mitigating security threats.
The model strictly adheres to the principle of least privilege(Krohn et al.,
2005) and the direction of privilege consistency. We also propose a new
analytical framework for analyzing and evaluating miniapps. This model can
identify, diagnose, and mitigate the security risks inherent in miniapps,
strengthening the overall security defences and fostering a more secure and
trustworthy miniapp ecosystem. Our research has systematically focused on the
web interface security of miniapps, particularly emphasising further
investigation and exploration of vulnerabilities in miniapps. During the
experiment, we identified nine categories of vulnerability issues and 50,628
security risks and threats. We provided examples for each vulnerability issue
and recorded an assessment of the security risks.
Contributions. We make the following contributions:

  * •

We presented a new triad threat model, aiming to gain a more comprehensive
insight into the security intricacies of miniapps. Considering the viewpoints
of users, the server side, and potential attackers, and under the guidance of
least privilege and privilege consistency principles, this model evaluates the
security threats and vulnerabilities encountered by miniapps.

  * •

We have constructed a new analysis framework and applied it to traditional
vulnerability detection. For the first time, this framework utilizes a unique
combination of vulnerability detection, involving
AppScan+Proxifier+WeChat(Windows), to detect nearly a thousand well-known
mainstream miniapps. This accomplishment represents an essential milestone in
miniapp security assessment.

  * •

We conducted a comprehensive evaluation of the novel framework on real-world
WeChat miniapps. We meticulously analyzed the detection results, which
resulted in identifying nine distinct categories of significant vulnerability
issues, along with discovering a total of 50,628 security risks and dangers.

##  2\. Related work

Recently, the novel concept of miniapps has been delved into across various
domains. Regarding the miniapps, they can be used on education(Sui et al.,
2020; Chen et al., 2020; Liang and Chang, 2019), online shopping(Rao and Ko,
2021), healthcare(Zhou et al., 2020; Wang et al., 2017), campus services(Jun
et al., 2022),transportation(Cheng et al., 2019), food service(Pai et al.,
2022). For instance, Zhang et al.(Zhang et al., 2021) devised mini-crawlers to
procure miniapps and conducted extensive measurements. Similarly, Zhang et
al.(Zhang et al., 2022) explored the realm of identity obfuscation within
webview-based super apps, while Liu et al.(Liu et al., 2020) fashioned a
dynamic analysis framework tailored for WeChat miniapps. Yang et al.(Yang et
al., 2022) also investigated the vulnerability of cross-miniapp redirection
among miniapps in WeChat and Baidu. Additionally, Lu et al.(Lu et al., 2020)
thoroughly examined the miniapp paradigm, with a specific focus on access
control mechanisms. Recently, Wang et al.(Wang et al., 2023b) delved into the
challenge of concealing APIs in mobile super apps. Meanwhile, Zhang et
al.(Zhang et al., 2023) began an inquiry regarding AppSecret leakage in
miniapps. Furthermore, Wang et al.(Wang et al., 2023a) introduced an
innovative approach named TAINTMINI for detecting sensitive data flow in
miniapps.

OWASP (Open Web Application Security Project)(owasp.org, 2021) and WASC (Web
Application Security Consortium)(hcltechsw.com, 2021) are esteemed
organizations devoted to fortifying the security of Web applications. They
assume a pivotal role in the assessment of web security. In this research
study, we adeptly integrate the identification principles of OWASP Top 10 and
the security guidelines provided by WASC, delving into exploring
vulnerabilities and security evaluation in miniapp security. Our goal is to
assess the security of miniapps systematically, precisely pinpoint potential
vulnerabilities, and lay a foundation for further in-depth analysis.

##  3\. Motivation and Problem Statement

During the thriving expansion of miniapps embedded within mobile applications,
an increasing number of users and enterprises are harnessing these diminutive
wonders to enhance their mobile endeavors. Nonetheless, miniapps face a
substantial spectrum of security threats and vulnerabilities. Previous
research has primarily focused on isolated security issues, resulting in the
need for a holistic perspective and revealing gaps in comprehensive analysis
concerning the harmonious convergence of users, servers, and potential
attackers.

###  3.1. Threat Model

We present a comprehensive triad model for improving the security of miniapp
web interfaces and proactively addressing vulnerabilities. This model
considers the viewpoints of users, servers, and potential attackers, with the
primary goal of providing miniapps with robust resilience to mitigate security
threats effectively. The model integrates proactive measures to prevent
vulnerabilities with passive strategies to mitigate potential attacks. The
methodology used in this model is characterized by its systematic and
meticulous nature. It is an enlightening guide for developers, enabling them
to assess, improve and maintain miniapps’ security. The triad model can be
meticulously designed to strengthen the security of web interfaces and address
the vulnerabilities inherent in miniapps. The model has three primary
dimensions:

  * •

Users: Representing the clientele of miniapps, users are susceptible to
security threats and risks, while their actions can also impact the miniapps
overall security posture.

  * •

Servers: Serving as pivotal communication hubs between miniapps and backend
servers, the security of these nodes directly governs the safeguarding of user
data and system dependability.

  * •

Attackers: Potentially posing security threats, attackers may employ diverse
tactics to assail miniapps, pilfer user data, disrupt servers, tamper with
data, and more.

At the same time, the notion of the ”system side” is introduced. Within the
triad model, the system side encompasses the system layer, which includes the
operating system, the host system, and the miniapp system itself. The dynamic
interaction between the security aspect (including the three components of the
triad model) and the system side, together with the intricate interplay
between the system side, the user side, the server side, and the attacker
side, collectively create the comprehensive security ecosystem of the miniapp
system.

###  3.2. Key Observations

As shown in Figure 1, we present three potential attack vectors to which the
triad model responds.

Figure 1. There are three ways of potential attacks.

There are three ways of potential attacks.

An attacker directly targets the user: The attacker endeavors to strike
directly at the user, aiming to exploit the user’s vulnerabilities or missteps
to compromise the user’s device or extract sensitive information. This may
involve employing phishing techniques, deploying malicious links, introducing
malware, and more.

The attacker attacks the server side through the user: The attacker adopts an
indirect approach to target the server side by manipulating user behavior. By
inducing a user to visit a malicious website or download a file containing
malevolent code, the attacker manages to infiltrate the server side with the
malicious payload.

The attacker directly attacks the server side: The attacker launches a direct
assault on the server side, striving to take control of it or tamper with
server side data. This type of attack may encompass SQL injection(Halfond et
al., 2006), remote code execution, denial-of-service attacks, and other
malicious activities.

Taken together, the triad model provides an integrated security framework that
requires attackers to consider the principles of least privilege and privilege
consistency, whether they are attacking users directly or exploiting user
behavior to target the server side. This approach helps to improve the
security of systems and prevent the creation and exploitation of
vulnerabilities. In addition, unified attack and defense enable security
organizations to gain a deeper understanding of attacker behavior and
motivations, enabling more effective mitigation of potential threats.

Meanwhile, each facet of the triad framework assumes a pivotal role in
distinct stages of the system side:

  * •

User-centric security: User awareness and authentication mechanisms are
crucial in empowering users to safeguard their data and privacy during
interactions with miniapps and host systems.

  * •

Server fortification: Host system providers must ensure the secure development
and deployment of their applications, including miniapps, to protect user data
and maintain the integrity of the system.

  * •

Attacker mitigation: Preventing potential attackers is crucial for the overall
system security, safeguarding miniapps and other applications from potential
threats.

###  3.3. Problem Statement and Scope

The triad model exhibits the following exquisite characteristics:

  * •

Comprehensiveness: The triad model skilfully encompasses the roles of users,
servers, and attackers, converging with the system side to provide a
comprehensive security research framework. This methodology facilitates the
conduct of multifaceted investigations into the security of miniapps, thereby
giving deep insights into the intricate security aspects of the system.

  * •

Systemic: Unlike a narrow focus on specific security vulnerabilities, the
triad model takes a holistic approach to addressing security concerns
throughout the miniapp lifecycle. It proposes more effective security
solutions by addressing the root causes of security problems.

  * •

Extensibility: Seamlessly integrated with the system side, the triad model
offers remarkable flexibility and scalability. This versatility allows it to
be applied to different types and sizes of miniapps while allowing for
adaptations and enhancements to address evolving technologies and threats.

  * •

Responsive to Diversity: The variety and complexity of security issues within
miniapps makes traditional one-dimensional research methods inadequate. The
introduction of the triad model facilitates a comprehensive exploration of
miniapp security from multiple dimensions and perspectives, leading to
accurate and effective solutions tailored to various scenarios.

In prior work related to miniapps, the security issues inherent in these
applications can be comprehensively categorized from the perspectives of both
attackers and victims.

  * •

Attacker’s perspective: From the perspective of potential attackers, their
intentions revolve around obtaining sensitive user information, stealing data,
disrupting the regular operation of systems, or spreading malicious code. They
use methods such as XSS, CSRF, and SQL injection to target the web interfaces
of miniapps. This method involves manipulating request parameters, forging
requests, and using other mechanisms to carry out their attacks. In addition,
they may use techniques such as fuzz testing and penetration testing to
uncover vulnerabilities in miniapps, including instances such as unauthorized
access and parameter injection, to determine their attack vectors.

  * •

Victim’s perspective: Users could face various risks, ranging from the
potential exposure of personal privacy to the risk of account compromise. When
using miniapps, users should increase their security vigilance, avoid
malicious links and exercise caution when authorizing permissions. Conversely,
if servers are compromised, they could face problems such as exposure to
sensitive data and system downtime. Therefore, it is vital to strengthen
security measures and examine and filter incoming data to prevent attacks.

The triad model provides a new perspective and approach to exploring and
solving security challenges within miniapps. Within this model, a holistic
approach to security is essential, encompassing both offensive and defensive
perspectives to protect user information. Following the principle of least
privilege, granting users only the rights they need can reduce the risk of
breaches and minimize unauthorized operations and data access. At the same
time, maintaining privilege consistency ensures consistent user authentication
and validation across the miniapp system, preventing vulnerabilities due to
inconsistent auditing privileges.

In addition, the security of miniapps web interfaces requires significant
consideration, considering offensive and defensive strategies. Our primary
goal goes beyond simply thwarting the malicious efforts of potential
attackers; it includes proactively identifying and remediating security
vulnerabilities. In addition, we recognize that implementing robust interface
authentication and authorization mechanisms can effectively prevent malicious
users and attackers from directly accessing sensitive data or performing
dangerous operations via interfaces.

To ensure the security of user information, developers must adopt secure
coding practices and perform comprehensive code audits during the development
phase. This proactive approach aims to identify potential logical
vulnerabilities, control discrepancies, instances of SQL injection, and cross-
site scripting attacks. At the same time, a thorough security assessment is
essential when integrating third-party technologies to prevent the
introduction of latent security risks.

Using the triad model in conjunction with a holistic strategy that includes
offensive and defensive measures, the principles of least privilege, and
privilege consistency, a comprehensive improvement in the overarching security
of miniapp web interfaces is achieved. This effort not only preserves user
privacy and data integrity but also strengthens miniapps ability to withstand
adversarial intrusions.

##  4\. Design

This section provides the detailed design of the analysis framework, and
applies it to traditional vulnerability detection. As described in 3.2,
security risks in miniapps occur across three components (i.e., users, servers
and Attackers) at various granularity (i.e., event handlers, miniapp pages,
miniapp programs) in different manners. To address these challenges, the
framework uses a novel combination of vulnerability detection with the pairing
of AppScan+Proxifier+WeChat(Windows) to detect mainstream miniapps. The triad
threat model guides the design and deployment of detection frameworks. By
providing a comprehensive understanding of potential threats and
vulnerabilities from the user, server and attacker perspectives, the triad
model can increase the effectiveness and comprehensiveness of the detection
framework. In particular, for the first time, we employed such a combination
of detection methods. We used the Windows version of the WeChat client to
access the selection miniapp and subsequently directed the data traffic to
AppScan through a Proxifier proxy for the purpose of security risk detection.
The structure of this frame is shown in Figure 2:

Figure 2. Overview of Analyze Framework

Scope. In this study, we focus on systematically studying the security of
miniapps to understand their root causes, and consequences. Given the security
concerns around web interfaces, our goal is to identify vulnerabilities within
miniapps that arise from web interface complications. Given WeChat’s prominent
position as a primary social media platform, coupled with the rapid expansion
of its miniapp ecosystem, this research assumes notable importance. Therefore,
in this paper, we focus on WeChat miniapp in particular due to its popularity
and support for both sensitive data access.

Methodology. As part of the use of the new framework, we are jointly deploying
the following applications in the first instance:

  * •

WeChat platform for Windows: In the previous research focused on miniapps, the
primary research platform was mainly focused on the Android ecosystem. The
primary research methodology included reverse engineering and decompilation
techniques.

  * •

Proxifier: This widely used proxy tool allows applications to connect to the
Internet through intermediary servers, facilitating the mediation and
management of network traffic.

  * •

Appscan: A robust application security scanning tool, Appscan helps to analyze
traffic data relayed through proxies. It identifies critical security risks,
performs categorization assessments, and assists with remediation. Appscan
takes on the task of dissecting the traffic that passes through the Proxifier
proxy and provides a comprehensive evaluation of the underlying security
concerns.

  * •

Burp/Fiddler: While capturing packets for WeChat miniapps, these tools can
intercept network requests sent and received by the miniapps. They allow
systematic analysis of the information encapsulated in these requests.

  * •

Airtest: It, a cross-platform mobile automation testing tool, is used to
automate tests and evaluate the performance of mobile applications. It is well
suited for automating tests and evaluating performance aspects of mobile
applications.

Airtest makes it easy to simulate user actions and data retrieval within
WeChat miniapps. When testing WeChat miniapps: Airtest helps us to efficiently
simulate user interactions and retrieve data from within WeChat miniapps. When
trying WeChat miniapps, Airtest can be used to simulate user actions and, by
creating test scripts, locate and retrieve items within the miniapp. While
simulating user actions, we set up a Proxifier proxy to route the traffic data
of the WeChat miniapp through Proxifier to AppScan. Using AppScan, we analyze
the traffic data passed through the Proxifier proxy. Next, the test
application generates reports and automatically compiles security risks from
the documentation. Finally, we perform a static analysis of various security
risks to consolidate the data further.

AppScan meticulously examines the miniapp’s source code and binary files
during this process, particularly during static analysis. It identifies a
range of vulnerabilities, including but not limited to SQL injection and
cross-site scripting attacks. It also performs a comprehensive dependency
assessment, including third-party libraries and components, to identify
potential vulnerabilities or security issues. At the same time, AppScan can
trace the intricate paths of data flow within miniapps, facilitating early
detection of potential data leakage and identifying insecure data processing
techniques.

In prior research, BurpSuite, Android WeChat, Emulator, and Fiddler were
amalgamated to perform detection tasks. As the WeChat version has been
updated, the effectiveness of the former detection method for miniapp on the
Android platform and within simulators has gradually decreased. Therefore, we
introduce a novel approach, employing a distinct pairing scheme involving
Windows WeChat, Proxifier, AppScan, and BurpSuite. Through this setup, we
analyze the data traffic information of Windows WeChat miniapps using AppScan,
detecting potential vulnerabilities. The results are subsequently cross-
verified and validated through BurpSuite and other indispensable tools during
the testing phase.

##  5\. Evaluation

This section focuses on examining web interface security vulnerabilities
within miniapps. These vulnerabilities are methodically grouped into distinct
focus areas, covering both the frontend and backend. Among the many
vulnerabilities identified are logical inconsistencies, permission management
anomalies, SQL injection susceptibility, cross-site scripting infiltration,
arbitrary file uploads, compromised password security, and other intricate
security nuances. Each classification is accompanied by specific examples that
meticulously illustrate the authentic security challenges faced by miniapps.

###  5.1. Quantifying the Vulnerabilities Risks

Within this section, these vulnerabilities are systematically categorised into
frontend, backend and other relevant dimensions and presented sequentially for
comprehensive analysis.

####  5.1.1. Frontend vulnerabilities

Logic Vulnerabilities: This vulnerability refers to the presence of defects or
errors in the logical flow of an application, resulting in behavior that
deviates from expected results or presents potential security risks.
Typically, these vulnerabilities do not involve syntax errors within the code,
but are related to complications in the program’s business logic or permission
controls. Because of their ability to bypass conventional security measures
and exploit input validation errors to remain undetected, they manipulate the
standard logical flow of a miniapp. As a result, they can cause insecure
permissions, access control issues, unauthorised data retrieval and business
logic errors, among other problems.

Cross-Site Scripting (XSS): This vulnerability attack involves surreptitiously
inserting malicious script code into the input fields of miniapp, forcing it
to execute within the confines of the user’s web browser. It allows malicious
actors to perform harmful actions during a user’s session, including stealing
login credentials, manipulating web page content, and redirecting to malicious
online domains.

####  5.1.2. Backend vulnerabilities

Privilege Escalation: Inadvertent misconfigurations of permissions can give
unauthorised entities access to resources that should be restricted. More
dangerously, these misconfigurations could allow such entities to manipulate
critical system components. Specific fields such as ”id”, ”uid”, and
”UserName”, in conjunction with their associated counterparts, require
scrutiny. Because they can be used as pathways for parameter traversal,
putting sensitive information at significant risk.

SQL Injection: Insufficient user input validation can lead to malicious SQL
query injection to access and modify database data. The SQL injection
vulnerability in miniapps means that user input is not fully validated and
filtered in the backend code of the miniapp. As a result, an attacker can
construct malicious inputs and execute malicious SQL statements to gain
unauthorized access to the database or to modify, delete or leak data from the
database.

Arbitrary file uploads: Owing to the absence of robust file type validation,
malicious actors gain the capability to upload pernicious files that pose a
grave threat to the security of the system.

Weak password security: Within the miniapp, the vulnerability of weak password
security resides in the login authentication or backend management system.
Should users choose passwords that are overly simplistic, readily guessable,
or commonly employed, the security of their accounts diminishes, rendering
them susceptible to password guessing or brute-force cracking attacks.
Furthermore, weak passwords may also engender password-guessing attacks,
brute-force cracking attempts, and instances of multi-account hijacking.

####  5.1.3. Other Security Issues

Leakage of sensitive information: Inadequate management of sensitive
information within miniapps during their design, development or operational
phases can create scenarios where malicious entities can inappropriately
access or disclose user-sensitive data. Furthermore, in the miniapps, improper
handling of user input data within URLs can also expose sensitive information
or facilitate unauthorised access to resources.

Cross-Site Request Forgery: This vulnerability presents a scenario where
malicious actors impersonate authentic users and trick them into unknowingly
executing malicious requests through their web browsers. This manipulation
results in the initiation of unauthorised actions within the miniapp. By
taking advantage of the user’s authenticated session, this vulnerability
exploits the limited ability of miniapps to thoroughly validate the source of
incoming requests, allowing malicious operations to be covertly executed.

Integration of Third-party Technologies: During the development phase of
miniapps, the inclusion of third-party technology components, either developed
by external vendors or by fellow developers (such as plug-ins, libraries,
APIs, etc.), can create security vulnerabilities and potential threats. These
include security breaches, data leakage and the propagation of malicious code.

###  5.2. Case Studies

####  5.2.1. Frontend vulnerabilities

Logic Vulnerabilities: Throughout our experiments, we discovered that a
substantial proportion of the vulnerabilities inherent in the miniapp were
attributed to logic flaws. Among them, we encountered a category known as
”authentication type logic vulnerabilities.” These vulnerabilities encompassed
a range of security concerns, including login bypass (enabling arbitrary user
access), password recovery vulnerabilities, CAPTCHA circumvention,
exploitation of mobile phone CAPTCHAs, SMS-based attacks (commonly referred to
as SMS bombs), and the masking of mobile phone numbers during registration.

During the experiment, we encountered several noteworthy logical anomalies,
which warrant careful attention:

  * •

In our examination of the ”Take goods mall” miniapp, we observed that after
the login authorization, the miniapp omitted the secondary verification of the
user’s mobile phone number to ensure data protection. This oversight resulted
in direct access to the miniapp without verifying the mobile phone number.

  * •

In our evaluation of the miniapp for the ”Haidian Foreign Language Students
Service Platform,” we input ”123,” followed by a single quote during the login
phase and provided a random password. By capturing the package using tools
like BurpSuite, we noticed incorrect account information was returned, yet we
could still directly access the system.

  * •

During our assessment of the ”Yixianjia Jiadele Life Supermarket Online Store”
miniapp, we discovered that entering the verification code ”111111” during the
registration phase resulted in immediate successful registration without
stringent validation of the bound mobile phone number.

  * •

In evaluating the ”To Shoot — Milk Delivery Supermarket Recycling Service”
miniapp, we observed a lack of robust validation for the verification code
during the login phase, allowing unrestricted access using any four-digit
code.

  * •

While examining the ”Panqinkanghua” miniapp, we attempted to replay the
verification code package. We observed multiple SMS messages received on the
phone quickly, indicating a vulnerability to SMS bombing.

  * •

Furthermore, while testing the ”Yonghui Enterprise Purchase” miniapp, we
encountered a four-digit verification code with a five-minute validity period.
And upon investigation, we found that the verification code could be traversed
to obtain the correct sequence.

Cross-Site Scripting (XSS): In the miniapps ”Mom.com Mom’s Good Products,”
”Gourmet Jie Recipe Book,” and ”Jurong Convenience Supermarket Preferred Life
Service Platform,” we have strategically inserted prompt messages into the
input box, comment section, search box, or any other text fields that users
can interact with on the webpage. Our approach strictly adheres to benevolent
experimental testing and forbids using malevolent XSS-constructed statements.
The attack is deemed valid once the prompt messages can be successfully
executed. In the trio mentioned above of miniapps, the prompt statement was
entered into the search field, and subsequently, ”200 OK” was obtained in
BurpSuite.

####  5.2.2. Backend vulnerabilities

Privilege Escalation: An authentication vulnerability through parameters in a
cookie can be exploited by manipulating the UserId value to gain unauthorized
access to user information. When the UserId value is altered to a different
value, the system divulges various essential details about the employee,
including their employee number, name, mobile phone number, and work status
(either ”in” or ”out” of work). For instance, in the case of Shangbiao Brand
Supermarket, by employing Burp Suite to modify the id-data, the corresponding
information for different ids can be accessed, thereby revealing mobile phone
numbers, order numbers, and other sensitive information.

SQL Injection: SQL Injection is a prevalent security vulnerability observed in
miniapps. Its occurrence can largely be attributed to the inadequate handling
of user input data in the backend code, which results in the direct
concatenation of user input into SQL query statements. As a consequence,
attackers can manipulate the execution logic of the SQL statement by modifying
the input, thus gaining unauthorized access to sensitive information and
undermining the integrity of the database.

  * •

During the examination of the ”Jiale Source Fresh Supermarket Corner Store”
miniapp, we encountered an SQL injection vulnerability. By skillfully
constructing and concatenating SQL query statements in the keyword section, we
could access sensitive information from the system, including database
details.

  * •

In the context of the ”Enterprise Management Cloud” miniapp, we identified an
SQL injection vulnerability in the login section. By injecting manipulated SQL
statements into the account section, we were able to trigger error messages
or, at times, inadvertently disclose sensitive information in pop-up windows.

  * •

Similarly, the ”Gourmet Jie Recipe Collection” miniapp displayed SQL injection
vulnerabilities in our examinations. We confirmed this by inputting specific
values into the id section of the SQL query statement.

  * •

Lastly, multiple SQL injections were detected during testing in the ”Water
Sage Technology Campus Direct Drinking Water” miniapp. For instance,
manipulating the uid section allowed access to sensitive information. Should
this miniapp, named ”Campus to provide direct drinking water,” suffer from
information leakage, it could lead to severe consequences.

The Figure 3 shows how an attacker exploits a SQL injection vulnerability to
gain unauthorised access to sensitive information.

Figure 3. An example of the SQL injection process

Arbitrary file uploads: The absence of file type validation permits malevolent
files to be uploaded, thus posing a grave threat to system security.
Primarily, a burp manipulation of the file extension can exploit any location
where an image is uploaded. Secondly, in cases where an ID card is uploaded,
if it is done through a camera, the vulnerability can be circumvented by
modifying the state in the return packet in conjunction with process
validation. For instance, a print miniapp can evade validation and upload an
image by altering the file extension via BurpSuite.

Weak password security: During the testing process, we shall come across
passwords such as ”123456,” ”111111,” ”888888,” ”abc123,” and other feeble,
simplistic account credentials. This situation may arise due to the miniapps’
convenient utilization by employees during the testing phase; however, it
inevitably gives rise to security vulnerabilities.

####  5.2.3. Other Security Issues

Leakage of sensitive information: In the decompiled source code of miniapps,
user information is susceptible to leakage. Even in some miniapps where source
code maintenance is infrequent or has ceased, highly sensitive information,
such as uid and key, might be exposed. Furthermore, in certain registration
and login interfaces, if incorrect or intentional SQL statements are input
consecutively, a prompt box will provide feedback based on the input, allowing
for the extraction of sensitive information through subsequent statement
crafting. During experimentation, we encountered a miniapp that, despite
displaying a prompt box indicating discontinued maintenance, still allowed
access to numerous pages from which a wealth of private information could be
gleaned through decompilation.

Cross-Site Request Forgery: Insufficient CSRF protection could allow an
attacker to execute actions on behalf of an authenticated user
surreptitiously. Once the user logs in and saves their credentials within the
miniapp, the authorized information remains accessible for subsequent logins.
Subsequently, the attacker can employ a malicious link to deceive the user
into clicking on it. Due to the absence of an effective CSRF defense mechanism
in miniapps, the attacker can exploit the user’s active login session to
initiate malevolent requests, all unbeknownst to the user.

Integration of Third-party Technologies: During the meticulous testing phase,
we encountered many miniapps, totaling more than a dozen, which exhibited a
common security flaw -adopting the easily-guessable verification code ”11111.”
Astonishingly, successful registration and login were achieved simply by
inputting this rudimentary code. Remarkably, these miniapps were supported by
the third-party technological expertise of KM Technology. Regrettably, KM
Technology’s services vulnerabilities could potentially jeopardize the
security and integrity of multiple miniapps affiliated with the company.

###  5.3. Evaluation Results

After eliminating extraneous factors, the conclusions of the scan are as Table
1. Total number of severity issues included in the report: 50,628.

Table 1. Security Vulnerabilities in miniapps Severity Level | A Number | Comments
---|---|---
High | 5,456 | Serious, urgent, need to be fixed immediately.
Moderate | 4,042 | General, higher risk, timely attention.
Low | 35,543 | Slight, small risk, timely attention.
Informational | 5,587 | Recommendations, tips, can be optimized.

In miniapps, application data encompasses cookies, JS, parameters, comments,
visited URLs, failed requests, and filtered URLs. Amongst this trove of data,
a myriad of security vulnerabilities lies in wait, constituting a profound
menace to user data’s sanctity and system resources’ stability. These
vulnerabilities can be systematically categorized into risk levels, ranging
from high to medium to low.

  * •

High-risk vulnerabilities pose grave repercussions, encompassing the peril of
breaching the sanctity of administrative privileges, compromising databases,
and executing remote commands on web servers. Moreover, they can potentially
unleash debilitating denial-of-service onslaughts, lay bare delicate
information like source code and user credentials, and even facilitate illicit
transactions.

  * •

Medium-risk vulnerabilities encompass the threat of session hijacking and
manipulation, unauthorized access to sensitive resources, and the exposure of
confidential data transmitted during encryption. In addition, attackers can
subvert authentication mechanisms, infiltrate specific directories, and
extract files or configurations from web servers.

  * •

Low-risk vulnerabilities entail gathering sensitive information, luring users
into divulging crucial data, and extracting server side script source code.
Furthermore, these vulnerabilities may result in information disclosure,
downloading of transient script files, and manipulation of client side
sessions or cookies to impersonate genuine users.

As is shown in Figure 4, the presented data delineates the distribution of
vulnerabilities acquired through framework-based detection, alongside the
proportion of vulnerabilities that underwent our comprehensive classification
and subsequent meticulous verification encompassing nine prominent
vulnerability categories.

Figure 4. These are nine broad categories of vulnerabilities

The data presented in Figure 4 clearly shows that real security threats such
as logical vulnerabilities, cross-site scripting and SQL injection manifest
themselves at a much higher frequency than their proportional representation
in the security risk analysis. This observation highlights the profound
dangers of logical vulnerabilities, XSS and SQL injection to the security of
miniapps. At the same time, while injection vulnerabilities dominate the
landscape of security risk analysis, our study focuses primarily on the
tangible instances of SQL injection vulnerabilities, resulting in their
comparatively lower representation.

Conversely, the incidence of privilege escalation, sensitive information
leakage and arbitrary file upload vulnerabilities in the actual security risk
landscape are significantly lower than their representation in the security
risk analysis. This disparity is because these facets are predominantly
examined through the lens of code audits. This method is separate from our
primary approach within this paper.

##  6\. Discussion

The manuscript above delves into an all-encompassing appraisal of the miniapp,
focusing on the esteemed OWASP Top 10 and WESC. It diligently explores the
security aspect of the miniapps web interface. It conducts thorough research
on vulnerability mining, culminating in the proposition of a triad model that
fosters a more systematic and comprehensive approach to future investigations.
Furthermore, the paper unveils the vulnerabilities unearthed during the
experimentation phase. In light of these security concerns, it is imperative
to persistently enhance and fortify the following aspects of miniapp security
in the times ahead:

  * •

Interface Security: Ensuring the integrity of interfaces is essential to
enable seamless communication between miniapps and backend services. These
interfaces play a crucial role in the miniapp domain, especially in
transferring and managing sensitive user data. Adopting appropriate
authentication and authorisation mechanisms, such as OAuth or token-based
authentication, is paramount in maintaining the security of interfaces. In
addition, implementing rigorous input validation and data filtering is a
fundamental safeguard against malicious intrusion, including threats such as
SQL injection and XSS attacks.

  * •

Platform Security: Platform security, on the other hand, encompasses creating
a safe and sheltered ecosystem for miniapps to thrive within. The safe
encompasses safeguarding the miniapp distribution platform with an unwavering
commitment to data protection. It is imperative to elevate the significance of
passwords that correspond to miniapps and diligently update them promptly.
Additionally, developing robust user login processes and establishing periodic
security reviews and vulnerability remedies serve as steadfast guardians,
ensuring users remain shielded from malicious apps and data breaches.

  * •

Backend Security: Safeguarding the sanctity of the miniapps backend server
stands as an imperative in the realm of backend security. Within this domain,
the backend assumes responsibility for storing and processing invaluable user
data, hence necessitating rigorous access control and robust data encryption
measures. Furthermore, the prohibition of external network access to the
backend management platform remains a steadfast fortification, while the
timely updating of plug-ins, APIs, and other components bolsters defense
walls. Regularly tending to server side scripts and databases by applying
updates and patches mitigates the risks posed by potential attacks.

  * •

Staff Responsibilities: As the custodians of miniapp development and
maintenance, staff have a significant responsibility in ensuring the security
of these miniapps. They must undergo comprehensive security training to
provide a thorough understanding of security threats and mitigation methods.
Adherence to established best practices is a guiding principle in conducting
meticulous security assessments that include source code, interfaces,
platforms and backend systems. This vigilant approach plays a vital role in
the timely identification and remediation of latent security issues, thereby
safeguarding the integrity of the miniapps they manage.

A cohesive defense can be strengthened by carefully managing factors such as
source code security, interface security, platform security, back-end
security, and personnel responsibilities, enabling miniapps to deftly fend off
potential security threats and maintain user privacy and data integrity. As we
diligently address the security challenges inherent in miniapps, future
research could address advanced vulnerability detection techniques, the fusion
of artificial intelligence and security protocols, studies of the miniapp
ecosystem, vulnerability mitigation and management strategies, security
assessment, and authentication mechanisms. These pursuits will foster
collaborative efforts. Through the relentless pursuit of knowledge and in-
depth research, we will continuously improve the security of miniapps, ensure
the protection of user information, and promote the sustainable development of
the miniapp industry.

##  7\. Conclusion

In this paper, we delve into the realm of miniapp security, with a central
focus on discerning and mitigating diverse vulnerabilities that have the
potential to jeopardize the sanctity of user data and the integrity of the
system. To this end, we propose an innovative triad threat model incorporating
users, servers, and potential attackers, upholding the tenets of least
privilege and privilege consistency to safeguard user information. This model
accentuates the significance of comprehensively considering the interplay
between these three entities, thereby creatively designing a novel security
analysis framework for miniapps. Furthermore, our exploration encompasses an
extensive survey and illustrative examples of the manifold security risks
looming in miniapps front-end and back-end spheres. By diligently categorizing
these issues and furnishing specific instances, we illuminate distinct facets
of miniapp security and underscore the criticality of addressing these
vulnerabilities. In culmination, we focus on miniapp security and chart a
course for future research in this domain.

## Ethics and Responsible Disclosure

In adherence to established community practices, we employ our accounts and
computational resources within a controlled environment for thorough analysis
and experimentation aimed at identifying potential security risks. Throughout
our investigative process, we dutifully reported the outcomes of the
vulnerability assessment to the vendor, offering remediation recommendations,
and subsequently obtained acknowledgment of the identified vulnerabilities.
Upholding a commitment to strict confidentiality, we safeguard all information
details before the vendor addresses and provides feedback on the
vulnerabilities, thereby preventing harm to users, miniapp developers, and
platform providers.

###### Acknowledgements.

Supported by the Fundamental Research Funds for Central Universities
(328202204).

## References

  * (1)
  * Chen et al. (2020) Xin Chen, Xi Zhou, Huan Li, Jinlan Li, and Hua Jiang. 2020.  The value of WeChat as a source of information on the COVID-19 in China.  _Preprint]. Bull World Health Organ_ 30 (2020).
  * Cheng et al. (2019) Ao Cheng, Gang Ren, Taeho Hong, Kichan Nam, and Chulmo Koo. 2019.  An exploratory analysis of travel-related WeChat mini program usage: affordance theory perspective. In _Information and Communication Technologies in Tourism 2019: Proceedings of the International Conference in Nicosia, Cyprus, January 30–February 1, 2019_. Springer, 333–343.
  * Halfond et al. (2006) William G Halfond, Jeremy Viegas, Alessandro Orso, et al. 2006\.  A classification of SQL-injection attacks and countermeasures. In _Proceedings of the IEEE international symposium on secure software engineering_ , Vol. 1. IEEE, 13–15.
  * hcltechsw.com (2021) hcltechsw.com. 2021.  WASC Threat Classification v2.0 report.  Retrieved May 25, 2025 from https://help.hcltechsw.com/appscan/Enterprise/10.0.1/topics/r_wasc_threat_classifications_report.html
  * Institute (2023) Aladdin Institute. 2023.  Mini Program Security Construction and Development Insights.  Retrieved July 28, 2023 from https://www.aldzs.com/viewpointarticle?id=16623
  * Jun et al. (2022) Wu Jun, Jiajun Li, Fengning Liu, Zesen Yuan, Yu Han, and Jin Zhang. 2022.  A Research on the Construction of Campus Errand Running Service Platform Based on WeChat Mini App.  _World Scientific Research Journal_ 8, 9 (2022), 501–507.
  * Krohn et al. (2005) Maxwell N Krohn, Petros Efstathopoulos, Cliff Frey, M Frans Kaashoek, Eddie Kohler, David Mazieres, Robert Tappan Morris, Michelle Osborne, Steve VanDeBogart, and David Ziegler. 2005.  Make Least Privilege a Right (Not a Privilege).. In _HotOS_.
  * Liang and Chang (2019) Qinzhen Liang and Chengyang Chang. 2019.  Construction of teaching model based on WeChat Mini-Program.  _International Journal of Science_ 16, 1 (2019), 54–59.
  * Liu et al. (2020) Yi Liu, Jinhui Xie, Jianbo Yang, Shiyu Guo, Yuetang Deng, Shuqing Li, Yechang Wu, and Yepang Liu. 2020.  Industry practice of javascript dynamic analysis on wechat mini-programs. In _Proceedings of the 35th IEEE/ACM International Conference on Automated Software Engineering_. 1189–1193.
  * Lu et al. (2020) Haoran Lu, Luyi Xing, Yue Xiao, Yifan Zhang, Xiaojing Liao, XiaoFeng Wang, and Xueqiang Wang. 2020.  Demystifying resource management risks in emerging mobile app-in-app ecosystems. In _Proceedings of the 2020 ACM SIGSAC conference on computer and communications Security_. 569–585.
  * Online (2023) AZhongguancun Online. 2023.  WeChat’s 1.3 Billion Monthly Activities Steadily Sitting on the Throne of the First National APP.  Retrieved July 26, 2023 from https://new.qq.com/rain/a/20230326A028AX00
  * owasp.org (2021) owasp.org. 2021.  OWASP Top 10 - 2021.  Retrieved May 25, 2022 from https://owasp.org/Top10/
  * Pai et al. (2022) Chen-Kuo Pai, Ze-Tian Wu, Seunghwan Lee, Jaeseok Lee, and Sangguk Kang. 2022.  Service Quality of Social Media-Based Self-Service Technology in the Food Service Context.  _Sustainability_ 14, 20 (2022).  https://doi.org/10.3390/su142013483
  * Rao and Ko (2021) Qianhui Rao and Eunju Ko. 2021.  Impulsive purchasing and luxury brand loyalty in WeChat Mini Program.  _Asia Pacific Journal of Marketing and Logistics_ 33, 10 (2021), 2054–2071.
  * Sui et al. (2020) Yiling Sui, Tian Wang, and Xiaochun Wang. 2020.  The impact of WeChat app-based education and rehabilitation program on anxiety, depression, quality of life, loss of follow-up and survival in non-small cell lung cancer patients who underwent surgical resection.  _European Journal of Oncology Nursing_ 45 (2020), 101707.  https://doi.org/10.1016/j.ejon.2019.101707
  * Wang et al. (2023a) Chao Wang, Ronny Ko, Yue Zhang, Yuqing Yang, and Zhiqiang Lin. 2023a.  Taintmini: Detecting Flow of Sensitive Data in Mini-Programs with Static Taint Analysis. In _2023 IEEE/ACM 45th International Conference on Software Engineering (ICSE)_. 932–944.  https://doi.org/10.1109/ICSE48619.2023.00086
  * Wang et al. (2023b) Chao Wang, Yue Zhang, and Zhiqiang Lin. 2023b.  Uncovering and Exploiting Hidden APIs in Mobile Super Apps.  arXiv:2306.08134 [cs.CR]
  * Wang et al. (2017) Feilong Wang, Lily Dongxia Xiao, Kaifa Wang, Min Li, and Yanni Yang. 2017.  Evaluation of a WeChat-based dementia-specific training program for nurses in primary care settings: A randomized controlled trial.  _Applied Nursing Research_ 38 (2017), 51–59.
  * Wechat (2021) Wechat. 2021.  WeChat mini programs’ Safety guidelines Development Principles and Considerations.  Retrieved July 22, 2023 from https://developers.weixin.qq.com/miniprogram/dev/framework/security.html
  * Yang et al. (2022) Yuqing Yang, Yue Zhang, and Zhiqiang Lin. 2022.  Cross miniapp request forgery: Root causes, attacks, and vulnerability detection. In _Proceedings of the 2022 ACM SIGSAC Conference on Computer and Communications Security_. 3079–3092.
  * Yun (2023) Xian Yun. 2023.  Analysis of the development status and development trend of China’s small program industry in 2023.  Retrieved July 25, 2023 from https://www.sgpjbg.com/info/4a34cfc1716174fa600c82a4662c02a4.html
  * Zhang et al. (2022) Lei Zhang, Zhibo Zhang, Ancong Liu, Yinzhi Cao, Xiaohan Zhang, Yanjun Chen, Yuan Zhang, Guangliang Yang, and Min Yang. 2022.  Identity Confusion in WebView-based Mobile App-in-app Ecosystems. In _31st USENIX Security Symposium (USENIX Security 22)_. USENIX Association, Boston, MA, 1597–1613.  https://www.usenix.org/conference/usenixsecurity22/presentation/zhang-lei
  * Zhang et al. (2021) Yue Zhang, Bayan Turkistani, Yuqing Yang, Chaoshun Zuo, and Zhiqiang Lin. 2021.  A measurement study of wechat mini-apps.  _SIGMETRICS21: ACM SIGMETRICS / International Conference on Measurement and Modeling of Computer Systems_ 5, 2, Article 14 (2021), 25 pages.  https://doi.org/10.1145/3460081
  * Zhang et al. (2023) Yue Zhang, Yuqing Yang, and Zhiqiang Lin. 2023.  Don’t Leak Your Keys: Understanding, Measuring, and Exploiting the AppSecret Leaks in Mini-Programs.  arXiv:2306.08151 [cs.CR]
  * Zhou et al. (2020) Kaina Zhou, Wen Wang, Wenqian Zhao, Lulu Li, Mengyue Zhang, Pingli Guo, Can Zhou, Minjie Li, Jinghua An, Jin Li, et al. 2020\.  Benefits of a WeChat-based multimodal nursing program on early rehabilitation in postoperative women with breast cancer: a clinical randomized controlled trial.  _International journal of nursing studies_ 106 (2020), 103565.


# Hyperdeformation in the Cd isotopes: a microscopic analysis

H. Abusara and A. V. Afanasjev  Department of Physics and Astronomy,
Mississippi State University, MS 39762, USA

###### Abstract

A systematics search for the nuclei in which the observation of discrete
hyperdeformed (HD) bands may be feasible with existing detector facilities has
been performed in the Cd isotopes within the framework of cranked relativistic
mean field theory. It was found that the 96Cd nucleus is a doubly magic HD
nucleus due to large proton $Z=48$ and neutron $N=48$ HD shell gaps. The best
candidate for experimental search of discrete HD bands is 107Cd nucleus
characterized by the large energy gap between the yrast and excited HD bands,
the size of which is only 15% smaller than the one in doubly magic HD 96Cd
nucleus.

###### pacs:

21.60.Jz,27.60.+j,21.10.Ma

##  I Introduction

Hyperdeformation (HD) is one of critical phenomena in nuclear structure, the
study of which will considerably advance our knowledge of nuclei at extreme
conditions of very large deformation and fast rotation CRMF-HD ; Dudek . The
studies of HD will also contribute into understanding of the crust of neutron
stars, where extremely deformed nuclear structures are expected (see Ref.
stars  and references therein). Although some experimental evidences of the
existence of HD at low K.98 ; C12  and high spin 152Dy-exp-1 ; 152Dy-exp-2 ;
Hetal.06 ; 60Zn-1  exist, the current experimental knowledge of HD is very
limited. New generation of detectors such as GRETA Greta  and AGATA Agata
will definitely allow to study this phenomenon in more details. However, these
detectors will become functional only in the middle of next decade. Thus, it
is very important to understand whether new experimental information on HD can
be obtained with existing detectors such as GAMMASPHERE Gamma .

Theoretical efforts to study HD at high spin both in macroscopic+microscopic
(MM) method and in self-consistent approaches were reviewed in Ref. CRMF-HD .
Our recent study of HD within the framework of the cranked relativistic mean
field (CRMF) theory in the Z=40-58 part of nuclear chart CRMF-HD  represents
the first ever systematic investigation of HD within the self-consistent
theory. The general features of the HD bands at high spin have been analysed
in Ref. CRMF-HD . In particular, it was concluded that the density of the HD
states in the vicinity of the yrast line is the major factor which decides
whether or not discrete HD bands can be observed. The high density of near-
yrast HD states will lead to a situation in which the feeding intensity will
be redistributed among many low-lying bands, thus drastically reducing the
intensity with which each individual band is populated. For such densities,
the feeding intensity of an individual band will most likely drop below the
observational limit of the modern experimental facilities. On the contrary,
the large energy gap between the yrast and excited HD configurations will lead
to an increased population of the yrast HD band, thus increasing the chances
of its observation.

The analysis of Ref. CRMF-HD , based on the energy gap between the last
occupied and first unoccupied routhians in the yrast HD configurations,
suggests that the density of the HD bands in the spin range where they are
yrast is high in the majority of the cases. It also indicates the Cd isotopes
as the best candidates for a search of discrete HD bands. However, one has to
remember that this type of analysis may be too simplistic because the
polarization effects induced by particle-hole excitations are neglected. In
particular, it can overestimate the size of the energy gap between the yrast
and excited HD configurations. Realistic analysis of the density of the HD
bands should include significant number of the HD configurations calculated in
a fully self-consistent manner with all polarization effects included. Such
analysis is time-consuming in computational sense and has been performed only
for 124Xe in Ref. CRMF-HD , but its extension to other nuclei is needed. Thus,
the goals of the current manuscript are (i) to perform a fully self-consistent
analysis of the density of the HD bands in the Cd isotopes and (ii) to find
the best nuclei in which experimental study of discrete HD bands can be
feasible with existing experimental facilities.

##  II Theoretical framework and the details of the calculations

The calculations in the present manuscript are performed in the framework of
the CRMF theory without pairing KR.89 ; A150  using numerical scheme of Ref.
CRMF-HD . The CRMF equations for the HD states are solved in the basis of an
anisotropic three-dimensional harmonic oscillator in Cartesian coordinates
characterized by the deformation parameters $\beta_{0}=1.0$ and
$\gamma=0^{\circ}$ and oscillator frequency $\hbar\omega_{0}=41A^{-1/3}$ MeV
(see Ref. CRMF-HD  for details). The truncation of basis is performed in such
a way that all states belonging to the shells up to fermionic $N_{F}$=14 and
bosonic $N_{B}$=20 are taken into account; this truncation scheme provides
sufficient numerical accuracy CRMF-HD . The NL1 parametrization of the RMF
Lagrangian RRM.86  is used in most of our calculations since it provides a
good description of the moments of inertia of the rotational bands in unpaired
regime in the SD and ND minima A150 ; ALR.98 ; A60 ; VRAL.05 , the single-
particle energies for the nuclei around the valley of $\beta$ stability ALR.98
; A250  and the excitation energies of the SD minima LR.98 . Other
parametrizations such as NL3 NL3 , NLSH NLSH , NLZ NLZ  and NL3* NL3*  are
used only to check the size of the HD gaps in the nuclei of interest.

Figure 1: Proton (top panel) and neutron (bottom panel) single-particle
energies (routhians) in the self-consistent rotating potential as a function
of the rotational frequency $\Omega_{x}$. They are given along the deformation
path of the yrast HD configuration in 107Cd and obtained in the calculations
with the NL1 parametrization of the RMF Lagrangian. Long-dashed, solid, dot-
dashed and dotted lines indicate $(\pi=+,r=+i)$, $(\pi=+,r=-i)$,
$(\pi=-,r=+i)$ and $(\pi=-,r=-i)$ orbitals, respectively. Solid (open) circles
indicate the orbitals occupied (emptied). The dashed box indicates the
frequency range corresponding to the spin-range $I=55-80\hbar$ in this
configuration. The arrows indicate the particle-hole excitations leading to
excited HD configurations.

Single-particle orbitals are labeled by $[Nn_{z}\Lambda]\Omega^{sign}$.
$[Nn_{z}\Lambda]\Omega$ are the asymptotic quantum numbers (Nilsson quantum
numbers) of the dominant component of the wave function at $\Omega_{x}=0.0$
MeV. The superscripts sign to the orbital labels are used sometimes to
indicate the sign of the signature $r$ for that orbital $(r=\pm i)$.

The excited HD configurations were built from the yrast HD configurations
obtained in the previous study CRMF-HD  by exciting either one proton or one
neutron or both together. Proton and neutron configurations generated in this
way are labeled by $\pi$i and $\nu$j, where $i=0,1,2,...$ and $j=0,1,2,...$
are integers indicating the corresponding configurations. $\pi$0 $\otimes$
$\nu$0 represents the yrast HD configuration. Total excited configurations
$\pi$i $\otimes$ $\nu$j are constructed from all possible combinations of
proton $\pi$i and neutron $\nu$j configurations excluding the one with $i=0$
and $j=0$. The selection of excited configurations is also constrained by the
condition that the energy gap between the orbital from which the particle is
excited and the orbital into which it is excited do not exceed 2.5 MeV in the
routhian diagram for the yrast HD configuration. All configurations are
calculated in a fully self-consistent manner so that their total energies are
defined as a function of spin.

Fig. 1 illustrates the selection of excited configurations. It shows the
occupation of the proton and neutron orbitals in the yrast HD configuration in
107Cd. According to our criteria only three proton excitations across the
$Z=48$ HD gap are considered. On the contrary, more neutron $ph$-excitations
are allowed across the $N=59$ HD shell gap. Table 1 shows their detailed
structure. For example, the $\nu$1 configuration is created by exciting one
neutron from the [770]1/2+ into [413]7/2+ orbitals. One can notice that we
only consider the $ph$-excitations between the states which do not have the
same combination ($\pi$,$r$) of parity $\pi$ and signature $r$. The computer
code in general can handle the excitations between the states with the same
($\pi$, $r$), but the configurations based on such excitations are less
numerically stable and require more computational time. Because of this reason
and the fact that they do not alter significantly the results for the density
of the HD states, it was decided to neglect them in the calculations. However,
in the cases of large energy gaps between the yrast and excited HD
configurations, they are taken into account.

Table 1: Neutron particle-hole excitations in 107Cd shown in Fig. 1. label | Excitation
---|---
$\nu$1 | [770]1/2+ $\rightarrow$ [413]7/2+
$\nu$2 | [770]1/2+ $\rightarrow$ [413]7/2-
$\nu$3 | [532]3/2- $\rightarrow$ [413]7/2+
$\nu$4 | [532]3/2- $\rightarrow$ [413]7/2-
$\nu$5 | [651]3/2- $\rightarrow$ [413]7/2+
$\nu$6 | [651]3/2+ $\rightarrow$ [413]7/2-

Figure 2: (Color online) Energies of the calculated HD configurations in even-
even 96-106Cd nuclei relative to a smooth liquid drop reference $AI(I+1)$,
with the inertia parameter A=0.01. In each nucleus, the yrast and lowest
excited proton configurations are shown by solid lines. Dot-dashed and dotted
lines represent the yrast lines at low spin built from normal-deformed (ND)
and superdeformed (SD) states, respectively.

##  III Discussion

Figs. 2 and 3 show the density of the HD states in even-even 96-108Cd and odd
mass 107,109Cd nuclei studied using above outlined procedure. The energy gap
between the yrast HD configuration and lowest excited HD configurations is
around 1.5 MeV in 96Cd (Fig. 2a). It is comparable with the energy gap between
the yrast and excited SD configurations in doubly magic SD nucleus 152Dy (Fig.
7 in Ref. A150 ). This energy gap in 96Cd is due to large energy cost of
particle-hole excitations across the $Z=48$ and $N=48$ HD shell gaps which
have similar size (see Fig. 1 and Table 2). All that together indicates that
the 96Cd is a doubly magic HD nucleus. Only proton excitations to the
$[420]1/2^{-}$ orbital above the $Z=48$ HD shell gap result in bound excited
proton configurations, the excitations to other orbitals located above the
$Z=48$ HD shell gap produce the proton-emitting states. The doubly magic
nature of 96Cd nucleus is confirmed also in the calculations with other RMF
parametrizations (Table 2). It is interesting to mention that the RMF
parametrizations aimed at the description of the nuclei far from stability
such as NL3, NL3*, NLSH show larger $Z=48$ and $N=48$ HD shells gaps in
96,107-109Cd than the parametrizations NL1 and NLZ fitted predominantly to
$\beta$-stability nuclei (Table 2).

Figure 3:  (Color online) The same as in Fig. 2 but for 107,108,109Cd. The
yrast HD line in 108Cd is built from two signature-degenerated configurations.

With increasing neutron number the energy gap between the yrast and excited HD
configurations disappears (Fig. 2). This is due to relatively high density of
the neutron states above the $N=48$ HD shell gap (Fig. 1). Indeed, many
excited neutron configurations are located below the lowest excited proton
configurations (Fig. 2). One can also see that even-even 100-104Cd nuclei are
characterized by appreciable density of the HD states in the vicinity of the
yrast HD line (Fig. 2). The analysis of the single-particle structure in these
nuclei indicates that similar density of the HD bands is expected also in odd
mass nuclei 99-105Cd. In no way these nuclei have to be considered as good
candidates for a search of discrete HD bands since the feeding intensity will
be redistributed among many low-lying HD bands. As a result, the feeding
intensity of an individual HD band will most likely drop below the
observational limit of modern experimental facilities. Although there is some
energy gap between the lowest four HD configurations and other excited
configurations in 106Cd, this nucleus does not appear to be a good candidate
for a search of discrete HD bands because the presence of four low-lying HD
configurations will lead to a fragmentation of feeding intensity. This is one
of possible reasons why the HD bands have not been observed in this nucleus
F.05 .

On the other hand, the high density of the HD bands in above discussed nuclei
will most likely favor the observation of the rotational patterns in the form
of ridge structures in three-dimensional rotational mapped spectra Hetal.06 .
The study of these patterns as a function of neutron number can provide a
valuable information about HD at high spin.

Figure 4:  (Color online) Dymanic moments of inertia $J^{(2)}$ (panel (a)),
transition quadrupole moments (panel (b)), and mass hexadecapole moments
$Q_{40}$ (panel (c)) of the yrast HD bands in the nuclei under study. All
these bands have the proton $\pi 6^{2}$ configuration, their neutron
configurations are shown in the right panel. The configuration of the yrast HD
band in 106Cd is shown in panel (a).

Further increase of the neutron number brings the neutron Fermi level to the
region of low density of the neutron states characterized by the large $N=59$
and $N=61$ HD shell gaps (Fig. 1) with the combined size of these two gaps
being around 2.5 MeV (Table 2). As a result, the 107-109Cd nuclei show
appreciable energy gap between the yrast and lowest excited HD configurations
(Fig. 3). This gap is especially pronounced in the case of 107Cd for which it
is around 1.3 MeV. Note that the size of this gap is defined by the size of
the $Z=48$ HD shell gap, since the lowest excited configuration is based on
proton excitation (Fig. 3a). Similar or even larger energy gap between the
yrast and excited HD configurations is expected in the NLZ, NL3, NL3* and NLSH
parametrizations for which the size of the $Z=48$ and $N=59$ HD shell gaps is
at least 1.7 MeV in 107Cd (Table 2). The energy gaps between the yrast and
excited HD configurations at the spins where the HD configurations become
yrast are somewhat lower in 108,109Cd being around 0.9 and 1.1 MeV. This
energy gap in 108Cd is dictated by the size of the $N=61$ HD shell gap since
lowest excited HD configurations are based on neutron excitations. Thus, in
108Cd it will be smaller (similar) in the case of the NLZ (NL3, NL3*)
parametrizations and larger in the NLSH parametrization as compared with the
one obtained in the NL1 parametrization (Table 2). In the case of 109Cd, the
energy gap between the yrast and excited configurations will be larger
(smaller) in the NL3, NL3* and NLSH (NLZ) parametrizations (Table 2).

Table 2: The size of the Z=48, N=59, and N=61 HD shell gaps (in MeV) obtained
with different parametrizations of the RMF Lagrangian for the yrast HD
configurations in 96,107,108,109Cd. They are given at rotational frequency
$\Omega_{x}=1.00$ MeV approximately corresponding to the spin at which the HD
bands become yrast. The lowest (among the different parametrizations) value of
the shell gap is shown by bold style. ’59+61’ line shows the combined size of
the $N=59$ and $N=61$ HD shell gaps. RMF Parametrizations
---
Nucleus | Gap | NL1 | NLZ | NL3 | NL3* | NLSH
96Cd | Z=48 | 1.75 | 1.93 | 2.43 | 2.27 | 2.71
| N=48 | 2.00 | 2.07 | 2.59 | 2.44 | 3.03
108Cd | Z=48 | 1.62 | 1.66 | 2.23 | 1.99 | 2.06
| N=59 | 1.30 | 1.70 | 1.50 | 1.46 | 1.20
| N=61 | 1.20 | 0.74 | 1.20 | 1.19 | 1.50
| 59+61 | 2.50 | 2.44 | 2.70 | 2.65 | 2.70
107Cd | Z=48 | 1.70 | 1.73 | 2.22 | 2.18 | 2.27
| N=59 | 1.89 | 2.16 | 2.08 | 2.04 | 1.74
109Cd | Z=48 | 1.52 | 1.61 | 1.89 | 1.84 | 1.54
| N=61 | 1.37 | 1.16 | 1.83 | 1.74 | 2.16

Two factors make the observation of discrete HD bands in 108Cd 111Two bands
with very extended shapes observed in 108Cd in Refs. Cd108-1 ; Cd108-2  were
assigned as superdeformed in Ref. Cd108 . with existing facilities less
probable than in odd-mass 107,109Cd nuclei. First, the yrast HD line in this
nucleus is built from two signature degenerate configurations (Fig. 3b) in
which the last neutron is placed into one of the signatures of the $[413]7/2$
orbital (see Fig. 1 and Ref. Cd108 ). This reduces the feeding intensity of
each of these bands by factor of 2 as compared with the case when the yrast HD
line is built from single configuration. Second, the energy gap between the
yrast and excited HD configurations decreases with increasing spin (Fig. 2b).
As a result, further reduction of feeding intensity of the yrast HD bands is
expected if the bands are populated at spins higher than the spin at which
they become yrast. On the contrary, the energy gap between the yrast and
excited HD configurations is more constant as a function of spin in 109Cd and
especially in 107Cd. All these results strongly suggest that the 107Cd nucleus
is the best candidate for the experimental search of the discrete HD bands.
This conclusion is also supported by detailed analysis of the single-particle
routhians in the yrast HD configurations of even-even nuclei studied in Ref.
CRMF-HD ; this analysis does not suggest any alternative case which would
provide similar or larger gap between the yrast and excited HD configurations
in even-even, odd and odd-odd nuclei of the $Z=40-58$ part of the nuclear
chart.

The calculated properties of the yrast HD bands in studied nuclei are shown in
Fig. 4. The HD shapes undergo a centrifugal stretching that result in an
increase of the transition quadrupole moments $Q_{t}$ with increasing
rotational frequency. This process also reveals itself in the dynamic moments
of inertia: they increase with increasing rotational frequency in the
frequency range of interest. On the other hand, the mass hexadecapole moments
$Q_{40}$ do not show a clear trend as a function of rotational frequency and
stay nearly constant in the majority of the HD bands. Unpaired band crossings
due to interaction of different single-particle orbitals are seen in the
configurations of the yrast HD bands in 100,102,106Cd nuclei. For example, the
interaction between the $(r=+i)$ signatures of the $\nu[770]1/2$ and
$\nu[532]5/2$ orbitals is responsible for the crossing seen at $\Omega_{x}\sim
1.05$ MeV in the yrast HD band in 106Cd. This crossing may be an extra factor
(in addition to the density of the near-yrast HD bands) which complicates the
observation of the HD bands in 106Cd: such bands have not been observed in
experiment of Ref. F.05 .

The current study clearly shows that the polarization effects in time-even and
time-odd mean fields have an important impact on the density of the HD states
and especially on the energy gap between the yrast and excited HD states. The
latter quantity is appreciably smaller (by up to $\sim 0.5$ MeV; compare Figs.
2 and 3 with Table 2) than the respective HD shell gap in the routhian
diagram.

The role of time-odd mean fields in the definition of the energy gap between
the yrast and excited HD configurations is quite complicated. This is
illustrated by the fact that the energy gap between the yrast HD and the
lowest excited proton and neutron HD configurations is larger by $\approx 0.2$
MeV in the calculations without NM than in the ones with NM at spins where the
HD configurations become yrast $(I\approx 67\hbar)$. This fact reflects two
different mechanisms by which the time-odd mean fields affect the relative
energies of different rotational bands. In the first mechanism, the angular
momentum content of the single-particle orbitals is modified in the presence
of time-odd mean fields, see Ref. AR.00  for details. There are two important
consequences of this mechanism. First, the same total angular momentum of the
system is built at rotational frequency which is by $\sim 25\%$ lower in the
calculations with NM than in the calculations without NM. Second, the changes
of the single-particle angular momenta of the single-particle orbitals
surrounding the HD gaps of interest (the $\pi[420]1/2$ and $\pi[541]1/2$
orbitals for proton subsystem and $\nu[413]7/2$ and $\nu[651]3/2$ for neutron
subsystem (Fig. 1)) induced by NM modify the single-particle energies of these
orbitals. As a result, these gaps are smaller by $\sim 0.12$ MeV in the
calculations with NM at $I=67\hbar$. The second mechanism is related to
additional binding due to time-odd mean fields. The time-odd mean fields are
stronger in the excited HD configuration than in the yrast HD configuration.
Thus, additional binding due to NM is stronger in excited HD configuration
than in the yrast HD configuration. This also leads to the decrease of the
energy gap between the yrast and excited HD configurations in the calculations
with NM as compared with the ones without NM.

The presence of time-odd mean fields reveals itself also in the energy
splitting of the opposite signatures of the $\nu[770]1/2$ orbital visible at
$\Omega_{x}=0.0$ MeV (Fig. 1); the occupied orbital is more bound than
unoccupied one in the RMF theory (Ref. A150 ).

When considering theoretical predictions one has to keep in mind that they are
subject of the errors in the description of the energies of the single-
particle states, which exist in the RMF theory at spherical shape RBRMG.98 ,
normal deformation A250  and quite likely at superdeformation Cd108 . The
extrapolation from spherical and normal deformation towards HD is itself a
potential source of errors since it is not know how well the response of the
mean field (or the single-particle potential and liquid drop in the MM method)
to the extreme elongation of the nucleus is reproduced in model calculations.
Such errors are not restricted to the self-consistent models; they are also
expected in the phenomenogical potentials (used in the MM method) which
describe single-particle energies at normal deformation better than self-
consistent models. However, several facts support the results and
interpretations given above. First, all RMF parametrizations used in this
study lead to the same HD configurations in 96,107-109Cd nuclei which become
yrast at similar spins (see Ref. CRMF-HD  for comparison of the results
obtained with NL1 and NL3) and to similar sizes of the proton and neutron HD
shell gaps (Table 2). Second, the large size of the $Z=48$ and $N=59$ (and
especially of combined neutron $59+61$ gap) HD shell gaps reduces the
importance of the errors in the description of the energies of specific
single-particle states. Third, the MM results of Ref. A100  suggest similar
conclusions for the nuclei around 108Cd. Indeed, large $Z=48$ shell gap and
low density of the single-particle states in the vicinity of the $N=59$ and
$N=61$ HD shell gaps is clearly visible in Figs. 4 and 5 of Ref. A100 . The
$N=59$ and $N=61$ shell gaps are separated by the signature-degenerated
$7/2^{+}$ state (Fig. 5 in Ref. A100 ). Thus, similar to our case, the yrast
HD line in 108Cd will be formed from two signature degenerated configurations
in the MM calculations.

##  IV Conclusions

In summary, a systematic analysis of hyperdeformation in the Cd isotopes has
been performed in the cranked relativistic mean field theory. The density of
the HD states has been analysed with the goal to find the best cases for
experimental search of the discrete HD bands. Our analysis indicates 96Cd as a
doubly magic HD nucleus in this part of nuclear chart; its magicity is due to
large $Z=48$ and $N=48$ HD shell gaps. However, experimental study of HD in
this nucleus is problematic with existing facilities due to its $N=Z$ status.
The low density of the neutron single-particle states in the vicinity of the
$N=59$ and 61 HD shell gaps and sizable $Z=48$ HD shell gap lead to
appreciable gaps between the yrast and excited HD bands in 107-109Cd nuclei,
thus offering better opportunities to observe discrete HD bands. Among these
three nuclei, the best candidate for observing the discrete HD bands with
existing facilities is 107Cd nucleus. The MM calculations of Refs. A100 ;
SDH.07  indicate that the fission barriers are sufficiently large in the
nuclei around 108Cd so that the HD minimum could survive fission for a
significant range of angular momentum. The stability of the HD minimum is
defined by its depth, the fission barrier height and the height of the barrier
between the HD and normal-deformed/superdeformed minima Dudek ; SDH.07 . Our
study clearly indicates that the HD minimum is localized in the potential
energy surface. However, future studies of the HD in this mass region have to
provide more quantitative answers on these properties of the HD minima in a
fully self-consistent framework.

The work was supported by the U.S. Department of Energy under grant DE-
FG02-07ER41459.

## References

  * (1) A. V. Afanasjev and H. Abusara, Phys. Rev. C 78, 014315 (2008)
  * (2) J. Dudek, K. Pomorski, N. Schunck and N. Dubray, Eur. Phys. J A20, 15 (2004).
  * (3) N. Chamel, Nucl. Phys. A747, 109 (2005).
  * (4) A. Krasznahorkay, M. Hunyadi, M. N. Harakeh, M. Csatlós, T. Faestermann, A. Gollwitzer, G. Graw, J. Gulyás, D. Habs, R. Hertenberger, H. J. Maier, Z. Máté,D. Rudolph, P. Thirolf, J. Timár and B. D. Valnion, Phys. Rev. Lett. 80, 2073 (1998).
  * (5) C. M. Brink, H. Friedrich, A. Weiguny and C. W. Wong, Phys. Lett. B33, 143 (1970).
  * (6) A. Galindo-Uribarri, H. R. Andrews, G. C. Ball, T. E. Drake, V. P. Janzen, J. A. Kuehner, S. M. Mullins, L. Persson, D. Prévost, D. C. Radford, J. C. Waddington, D. Ward, and R. Wyss, Phys. Rev. Lett. 71, 231 (1993).
  * (7) G. Viesti, M. Lunardon, D. Bazzacco, R. Burch, D. Fabris, S. Lunardi, N. H. Medina, G. Nebbia, C. Rossi-Alvarez, G. de Angelis, M. De Poli, E. Fioretto, G. Prete, J. Rico, P. Spolaore, G. Vedovato, A. Brondi, G. La Rana, R. Moro, and E. Vardaci, Phys. Rev. C51, 2385 (1995).
  * (8) B. Herskind, G. B. Hagemann, G. Sletten, Th. Døssing, C. Rønn Hansen, N. Schunck, S. Ødegård, H. Hübel, P. Bringel, A. Bürger, A. Neusser, A. K. Singh, A. Al-Khatib, S. B. Patel, A. Bracco, S. Leoni, F. Camera, G. Benzoni, P. Mason, A. Paleni, B. Million, O. Wieland, P. Bednarczyk, F. Azaiez, Th. Byrski, D. Curien, O. Dakov, G. Duchene, F. Khalfallah, B. Gall, L. Piqeras, J. Robin, J. Dudek, N. Rowley, B. M Nyakó, A. Algora, Z. Dombradi, J. Gal, G. Kalinka, D. Sohler, J. Molnár, J. Timár, L. Zolnai, K. Juhász, N. Redon, F. Hannachi, J. N. Scheurer, J. N. Wilson, A. Lopez-Martens, A. Korichi, K. Hauschild, J. Roccaz, S. Siem, P. Fallon, I. Y. Lee, A. Görgen, A. Maj, M. Kmiecik, M. Brekiesz, J. Styczen, K. Zuber, J. C. Lisle, B. Cederwall, K. Lagergren, A. O. Evans, G. Rainovski, G. De Angelis, G. LaRana, R. Moro, W. Gast, R. M. Lieder, E. Podsvirova, H. Jäger, C. M. Petrache, and D. Petrache, Phys. Scr. T125, 108 (2006).
  * (9) W. von Oertzen, V. Zherebchevsky, B. Gebauer, Ch. Schulz, S. Thummerer, D. Kamanin,G. Royer,and Th. Wilpert, Phys. Rev. C78, 044615 (2008).
  * (10) I-Y. Lee, AIP Conf. Proc 656, 343 (2003).
  * (11) D. Bazzacco, AIP Conf. Proc. 701, 265 (2004).
  * (12) I-Y. Lee, Nucl. Phys. A520, 641c (1990).
  * (13) W. Koepf and P. Ring, Nucl. Phys. A493, 61 (1989).
  * (14) A. V. Afanasjev, J. König and P. Ring, Nucl. Phys. A608, 107 (1996).
  * (15) P.-G. Reinhard, M. Rufa, J. Maruhn, W. Greiner and J. Friedrich, Z. Phys. A323, 13 (1986).
  * (16) A. V. Afanasjev, G. Lalazissis, and P. Ring, Nucl. Phys. A 634, 395 (1998).
  * (17) A. V. Afanasjev, I. Ragnarsson and P. Ring, Phys. Rev. C 59, 3166 (1999).
  * (18) D. Vretenar, A. V. Afanasjev, G. Lalazissis, and P. Ring, Phys. Rep. 409, 101 (2005).
  * (19) A. V. Afanasjev, T. L. Khoo, S. Frauendorf, G. A. Lalazissis, and I. Ahmad, Phys. Rev. C 67, 024309 (2003).
  * (20) G. A. Lalazissis and P. Ring, Phys. Lett. B427, 225 (1998) 225.
  * (21) G. A. Lalazissis, J. König and P. Ring, Phys. Rev. C 55, 540 (1997).
  * (22) M. M. Sharma, M. A. Nagarajan and P. Ring, Phys. Lett. B312, 377 (1993).
  * (23) M. Rufa, P.-G. Reinhard, J. A. Maruhn, W. Greiner, and M. R. Strayer, Phys. Rev. C38, 390 (1988).
  * (24) G. A. Lalazissis, S. Karatzikos, R. Fossion, D. Pena Arteaga, A. V. Afanasjev, P. Ring, Phys. Lett. B671, 36 (2009).
  * (25) P. Fallon, Nucl. Phys. A752, 231c (2005).
  * (26) R. M. Clark, P. Fallon, A. Görgen, M. Cromaz, M. A. Deleplanque, R. M. Diamond, G. J. Lane, I. Y. Lee, A. O. Macchiavelli, R. G. Ramos, F. S. Stephens, C. E. Svensson, K. Vetter, D. Ward, M. P. Carpenter, R. V. F. Janssens, and R. Wadsworth, Phys. Rev. Lett. 87, 202502 (2001).
  * (27) A. Görgen, R. M. Clark, P. Fallon, M. Cromaz, M. A. Deleplanque, R. M. Diamond, G. J. Lane, I. Y. Lee, A. O. Macchiavelli, R. G. Ramos, F. S. Stephens, C. E. Svensson, K. Vetter, D. Ward, M. P. Carpenter, R. V. F. Janssens, and R. Wadsworth, Phys. Rev. C65, 027302 (2002).
  * (28) A. V. Afanasjev and S. Frauendorf, Phys. Rev. C 72, 031301(R) (2005).
  * (29) K. Rutz, M. Bender, P.-G. Reinhard, J. A. Maruhn, and W. Greiner, Nucl. Phys. A634, 67 (1998).
  * (30) A. V. Afanasjev and P. Ring, Phys. Rev. C 62, 031302(R) (2000).
  * (31) R. R. Chasman, Phys. Rev. C64 024311 (2001).
  * (32) N. Schunck, J. Dudek, and B. Herskind, Phys. Rev. C75, 054304 (2007).