We present , a synthesis tool to automatically create correct-by-construction reactive systems and shields
from qualitative or quantitative specifications in probabilistic environments.

A shield is a special type of reactive system used for run-time enforcement; i.e., a shield enforces a given qualitative or quantitative specification of a running system while interfering with its operation as little as possible.
Shields that enforce a qualitative or quantitative specification are called safety-shields
or optimal-shields, respectively. Safety-shields can be implemented as pre-shields or as post-shields,
optimal-shields are implemented as post-shields.
Pre-shields are placed before the system and restrict the choices of the system. Post-shields are implemented after the system and are able to overwrite  the  system’s  output.
is based on the probabilistic model checker , adding model checking algorithms for stochastic games
with safety and mean-payoff objectives.
To the best of our knowledge, is the only synthesis tool able to
solve 2$\sfrac{1}{2}$-player games with mean-payoff objectives without restrictions on the state space.
Furthermore, adds the functionality to synthesize safe and optimal strategies
that implement reactive systems and shields.

§ INTRODUCTION

Reactive synthesis aims to automatically construct correct and efficient systems w.r.t. a formal specification
and has been increasingly used in a wide range of safety-critical applications.

A natural model for reactive synthesis is to model some inputs from the environment probabilistically
and some adverserially.
For adverserial inputs, the synthesized system assumes the worst case, for probabilistic inputs the average case.
The corresponding synthesis problem is mapped to solving a competitive stochastic turn-based game, i.e., a
$2\sfrac{1}{2}$-player game. Qualitative specifications specify the functional requirements of reactive systems.
With a quantitative specification such as mean-payoff objectives, we can measure how well a system satisfies the specification.

Shield synthesis defines a synthesis framework to construct run-time enforcement modules called shields
to guarantee the correctness of running systems. The concept of shielding is very general.

Shields that enforce qualitative objectives are so-called safety-shields [1]. For safety-shields, we
distinguish between pre-shielding and post-shielding as depicted in Fig. <ref>.
In pre-shielding the shield is implemented before the system
and restricts the choices for the system to a list of correct actions.
Pre-shielding is becoming increasingly important in the setting of safe reinforcement learning [9].
In post-shielding, the shield monitors the actions selected by the system and corrects them
if the chosen action could lead to a specification violation.

Shields that enforce quantitative measures are called optimal-shields [2]
and are implemented as post-shields.

is able to synthesize optimal-shields that enforce a mean-payoff objective.
Optimal-shields that enforce multiple quantitative objectives can be obtained via a linear combination to give an approximate solution of a single mean-payoff objective. For instance, the decision whether an optimal-shield should interfere
could be based on first, a performance objective to be minimized by the shield, and second, an interference cost for changing the output of the system. An optimal-shield can then be computed by minimizing a single mean-payoff objective obtained by combining both measures, thus guaranteeing maximal performance with minimal interference.

The decision whether the optimal-shield should interfere is based on two quantitative measures:
first, a performance objective to be minimized by the shield, and second, an interference cost for changing the output of the system.

An optimal-shield can be computed by minimizing a single mean-payoff objective obtained from combining the performance and the interference measure, thus guaranteeing maximal performance with minimal interference.

box = [rectangle, rounded corners, minimum height=0.8cm, text centered, draw=myblack]
arrow = [thick,->,>=stealth]
point = [coordinate]

(env) [box] Environment;
[inner sep=1pt, right of=env, xshift=2.75cm, yshift=1.62cm] (shield) ;
(system) [box,  right of=env, xshift=2.4cm] System;

[arrow, thick] (1.05, 0) – node [anchor=south] (2.8,0);

[thick] (1.2,0) –  (1.2,1.2);
[arrow, thick] (1.2,1.2) |- node[anchor=north]  (3,1.2);
[text width=1cm] at (2,1.5) input;

[arrow, thick] (3.42, 0.8) – node [anchor=east]$\{$actions$\}~~$ (3.42,0.4);

[thick] (3.42, -0.4) –  (3.42,-0.7);

[arrow, thick] (3.42,-0.7) -| node[pos=0.2, above] action (env);

[text width=1cm] at (3.7,1.3) $Sh$;

box = [rectangle, rounded corners, text width=2cm, minimum height=0.8cm, text centered, draw=myblack]
arrow = [thick,->,>=stealth]
point = [coordinate]

(env) [box] Environment;
(system) [box, text width=1cm, right of=env, xshift=2.4cm] System;

[inner sep=1pt, right of=env, xshift=2.73cm, yshift=-1.23cm] (shield) ;

[arrow, thick] (1.05, 0) – node [anchor=south] (2.8,0);

[arrow, thick] (3.42,-2.2) -| node[pos=0.3, above] action' (env);

[arrow, thick] (3.42, -0.4) – node [anchor=west]action (3.42, -0.95);
[thick] (3.42, -2.05) –  (3.42,-2.2);
[text width=1cm] at (1.8,-0.5) input;
[thick] (1.2, 0) –  (1.2,-1.5);
[arrow, thick] (1.2,-1.5) |- node[anchor=north]  (2.9,-1.5);
[text width=1cm] at (3.7,-1.5) $Sh$;

Left: Pre-shielding. Right: Post-shielding.

Tempest capabilities.
The core functionality of is the synthesis of reactive systems and shields in environments that incorporate uncertainty. To the best of our knowledge, is the only tool able to solve 2$\sfrac{1}{2}$-player games with mean-payoff objectives and qualitative objectives given in probabilistic temporal logics, without any restrictions on the state space.
Furthermore, is designed as synthesis tool. Therefore, the computed strategies
can intuitively be used as the synthesized system, which is not the case for many game-solving tools.
is the first tool available for the synthesis of shields
and is able to synthesize pre-safety and post-safety-shields, and optimal-shields.

Implementation and availability.
The tool is written in C++ and
builds upon the code-base of the model checker  [8],
extending existing features to provide the capability of solving stochastic games.
is available under the GPL-3 open source license.
The tool and its source code, along with a docker image and several examples,
are available from the web page[<http://www.tempest-shielding.xyz>].

Connections to other tools.
Probabilistic model checking tools like  [10] and  [8] provide verification of quantitative reward-based  properties and qualitative properties in probabilistic temporal logics.

Many synthesis tools based on games are available and widely used, for example,  [3] is able to solve qualitative timed games,  [4] solves qualitative stochastic games,
and  [5] solves mean-payoff 2-player games.

3.0 [7] is able to solve turn-based stochastic multi-player games
under a variety of properties including long-run average [6].

However, for solving long-run average objectives, needs the game to be a controllable multi-chain, i.e., one of the players needs to be able to reach every end-component from any state with probability one. This is a strong assumption on the structure of the game graph, which many models used in synthesis do not fulfill.
In contrast, does not rely on any assumptions on the structure of the game graph.

Acknowledgements. We would like to thank both Tim Quatmann and Joost-Pieter Katoen for their continuous help on getting acquainted with the source code of .

§ MODEL AND PROPERTY SPECIFICATION

* Quantitative Synthesis (i.e. rewards)
LRA for SMGs

* Probabilistic Synthesis (quantitative w.r.t. probabilities)
F,G,U,X,G<=k (fragment of RPATL)

* Intro: Shields are reactive systems we describe in two different fashions w.r.t. ... post and pre shielding.

* Intro: Everything talks about 2.5 player SMGs

* Intro: specs (objectives, properties)

Model Specification.
supports turn-based stochastic multi-player games (SMGs)
and uses PRISM-games's modelling language to describe the game [7].
The players are divided in two competing coalitions, where the first team is working together to satisfy or maximize/minimize a property
given in rPATL [6].
In each state,  exactly one player chooses an available probabilistic transition to determine the next state.
A strategy for a player determines the choices of transitions made by the player.

Property Specification.
For the synthesis of reactive systems, uses the property specification language of PRISM-games to express properties in
rPATL [6].

We give a few examples that can be used in :

* $\langle\langle 1, 2\rangle\rangle P_{max=?} [F~target]$:
Using the operator $P_{max=?}$, computes a strategy for the  player coalition of player 1 and 2 that guarantees
to reach target with the largest probability.

* $\langle\langle 1, 2\rangle\rangle R^r_{max=?} [ S ]$: Using the operators $R$ and $S$, synthesizes
a strategy that maximizes the expected averaged reward $r$ in the long-run.

Synthesis of safety-shields.
For the synthesis of shields, extends the property specification language.
Let the safety-value of an action in a certain state be the maximal
probability to stay safe within the next $k$ steps when executing this action.
A safety-shield decides whether an action is blocked in a certain state based
on either an absolute threshold $\gamma$, or a relative threshold $\lambda$.
A shield using an absolute threshold blocks all actions with a safety-value smaller than $\gamma$.
By using a relative threshold, actions are blocked with a safety-value smaller than
the best safety-value of an action in the current state times $\lambda$.

The syntax for safety-shielding requires to
specify the type of shielding using the keywords and ,
and to define the used threshold.
We give the following examples:

* $ \langle \text{PreSafety}, \gamma=0.9 \rangle \langle \langle shield \rangle \rangle P_{max=?} [ G^{<=14} ~! crash ] $:
Using this property, synthesizes a pre-safety-shield that allows all actions that do not cause a crash with
a maximal probability of $0.9$ within the next $14$ time steps.

* $ \langle \text{PostSafety}, \lambda=0.95 \rangle \langle \langle shield \rangle \rangle P_{max=?} [ G^{<=14} ~! crash ] $:
synthesizes a post-safety-shield that blocks all actions using the relative threshold $\lambda$.

Synthesis of optimal-shields. The property
starts with the keyword followed by the expression used to compute the long run average. For example:

* $ \langle \text{Optimal} \rangle \langle \langle shield \rangle \rangle R^r_{min=?} [ S ] $: computes an optimal shield that guarantees the long-run average reward of $r$.

§ SYNTHESIS OF STRATEGIES

computes a memoryless deterministic strategy, implementing
a reactive system or a shield, under which
the specified property can be guaranteed. The strategy is computed using value iteration
to solve the coalition game. Fig. <ref> shows sample outputs of the strategies
of the first experiment given in Section <ref>, implementing pre-safety and post-safety-shields.
In the pre-shielding case, the strategy provides for any state a list of allowed actions
with its corresponding safety-value.
The strategy for post-shielding defines for every state and available action, the action to be
forwarded by the shield.

[mathescape,language=Python, label= listing_pre_post, basicstyle=7.58]
|Pre-Safety-Shield with absolute comparison (gamma = 0.8):|
|state_id [label]:  'allowed actions' [<value>: (<action_id label)>]:|

0 [move=0     x1=0     y1=0     x2=4     y2=4]:  1.0:(0 e); 1:(1 s)
3 [move=0     x1=1     y1=0     x2=3     y2=4]:  0.9:(0 e); 1:(2 w)
4 [move=0     x1=1     y1=0     x2=4     y2=4]:  0.9:(1 s); 1:(3 n)

[mathescape,language=Python, label= listing_post, basicstyle=7.58]
|Post-Safety-Shield with relative comparison (lambda = 0.95):|
|state_id [label]: 'forwarded actions' [<action_id> label: <forwarded_action_id> label]:|

0 [move=0     x1=0     y1=0     x2=4	    y2=4]:  0e:0e; 1s:1s
3 [move=0     x1=1     y1=0     x2=3	    y2=4]:  0e:2w; 2w:2w
4 [move=0     x1=1     y1=0     x2=4	    y2=4]:  1s:3n; 3n:3n

Synthesized strategies implementing a pre-safety-shield (top) and a post-safety-shield (bottom).

§ IN ACTION

Left: Warehouse floor plan with $6~\times~3$ shelves. Right: Synthesis-times for
controller synthesis, safety-shield synthesis and optimal-shield synthesis.

High-Level Planning in Robotics.
A classical application of reactive synthesis is the domain of automated high-level
planning in robotics. We consider a warehouse floor plan with several
shelves, see Fig. <ref> (left). To parametrise the experiment, we consider
floor plans with  $n~\times~3$ shelves with $2\leq n \leq 20$.
A robot operates together among other robots within the warehouse.
can be used in this setting to synthesize controllers for the robot that perform certain tasks,
as well as shields used to ensure safe operation of the robot, or to guarantee performance.

Controller synthesis:
Using , we synthesize a controller for the robot that repeatedly picks up packages from one of the entrances and
delivers them to the exits.
We use the mean-payoff criterion to specify that the stochastic shortest paths should be taken.

Safety-shield synthesis:
A safety-shield can be synthesized to enforce collision avoidance with other robots.
In the experiments, we used a finite horizon of $14$ steps and a relative threshold of $\lambda=0.9$.

Optimal-shield synthesis:
During operation, a corridor may be blocked.
A robot should not unnecessarily wait for the corridor to be traversable when alternative paths exist.
We synthesize an optimal-shield that penalizes 'waiting' and is able to enforce a detour when waiting
gets too expensive.

Results. The models, parameters, and properties used for all experiments can be found on the website.
The results for the synthesis-times are depicted in Fig. <ref> (right).
The sizes of state space of the game graphs range from $5184$ states for $n=2$ to $186624$ states for $n=20$.
The results for optimal-shields use the axis on the right hand-side.
The times for creating pre-safety and post-safety-shields are identical.
To compare our results, we tried to compute a strategy for the optimal controller in  3.0
which resulted in an error. By proper modelling, we were able to synthesize safe controllers comparable to the
safety-shield using , resulting in better synthesis-times for .

Left: Synthesis times for safety-shield.
Right: Comparison of synthesis times for optimal-shields: vs Avni et.al.'s implementation [2].

§.§ Safety-Shielding

We perform a more detailed case study for safety-shielding on the warehouse setting from Fig. <ref>.
We parametrise the number of other robots which are modelled with adverserial and probabilistic components,
the finite horizon used for the safety analysis, and the number of shelves.
The results are shown in Fig. <ref> (left).

Optimal-Shielding  in Urban  Traffic  Control.
Avni et al. [2] synthesize optimal-shields
that overwrite the commands of a traffic-light controller modeled in the traffic simulator
The optimal-shield needs to balance the number of waiting cars per incoming road
with the cost for interfering with the traffic light controller.
The example is parametrised with the cut-off parameter $k$ that defines the
maximal modelled number of waiting cars per road.
The comparison of the synthesis-times from Avni et.al.’s  implementation and are shown in
Fig. <ref> (right), showing a difference by orders of magnitudes
in favor of .

Left: Warehouse floor plan with $6~\times~3$ shelves. Right: Synthesis-times for
controller synthesis, safety-shield synthesis and optimal-shield synthesis.

High-Level Planning in Robotics.
A classical application of reactive synthesis is the domain of automated high-level
planning in robotics. We consider a warehouse floor plan with several
shelves, see Fig. <ref> (left). To parametrise the experiment, we consider
floor plans with  $n~\times~3$ shelves with $2\leq n \leq 20$.
A robot operates together among other robots within the warehouse.
can be used in this setting to synthesize controllers for the robot that perform certain tasks,
as well as shields used to ensure safe operation of the robot, or to guarantee performance.

Controller synthesis:
Using , we synthesize a controller for the robot that repeatedly picks up packages from one of the entrances and
delivers them to the exits.
We use the mean-payoff criterion to specify that the stochastic shortest paths should be taken.

Safety-shield synthesis:
A safety-shield can be synthesized to enforce collision avoidance with other robots.
In the experiments, we used a finite horizon of $14$ steps and a relative threshold of $\lambda=0.9$.

Optimal-shield synthesis:
During operation, a corridor may be blocked.
A robot should not unnecessarily wait for the corridor to be traversable when alternative paths exist.
We synthesize an optimal-shield that penalizes 'waiting' and is able to enforce a detour when waiting
gets too expensive.

Results. The models, parameters, and properties used for all experiments can be found on the website.
The results for the synthesis-times are depicted in Fig. <ref> (right).
The sizes of state space of the game graphs range from $5184$ states for $n=2$ to $186624$ states for $n=20$.
The results for optimal-shields use the axis on the right hand-side.
The times for creating pre-safety and post-safety-shields are identical.
To compare our results, we tried to compute a strategy for the optimal controller in  3.0
which resulted in an error. By proper modelling, we were able to synthesize safe controllers comparable to the
safety-shield using , resulting in better synthesis-times for .

Left: Synthesis times for safety-shield.
Right: Comparison of synthesis times for optimal-shields: vs Avni et.al.'s implementation [2].

§.§ Safety-Shielding

We perform a more detailed case study for safety-shielding on the warehouse setting from Fig. <ref>.
We parametrise the number of other robots which are modelled with adverserial and probabilistic components,
the finite horizon used for the safety analysis, and the number of shelves.
The results are shown in Fig. <ref> (left).

Optimal-Shielding  in Urban  Traffic  Control.
Avni et al. [2] synthesize optimal-shields
that overwrite the commands of a traffic-light controller modeled in the traffic simulator
The optimal-shield needs to balance the number of waiting cars per incoming road
with the cost for interfering with the traffic light controller.
The example is parametrised with the cut-off parameter $k$ that defines the
maximal modelled number of waiting cars per road.
The comparison of the synthesis-times from Avni et.al.’s  implementation and are shown in
Fig. <ref> (right), showing a difference by orders of magnitudes
in favor of .

§ CONCLUSION AND FUTURE WORK

We have introduced , a tool for the synthesis of reactive systems and shields
with properties given in rPATL capturing qualitative and quantitative objectives
with probabilities.
Currently, supports perfect-information SMGs with full information.
In future work, we will investigate in efficient techniques to deal with partial information.
Furthermore, we will extend 's synthesis algorithm to support strategies with finite memory
with deterministic and stochastic-updates.

[1]
M. Alshiekh, R. Bloem, R. Ehlers, B. Könighofer, S. Niekum, and U. Topcu.
Safe reinforcement learning via shielding.
In Proceedings of the Thirty-Second AAAI Conference on
Artificial Intelligence, (AAAI-18), New Orleans, Louisiana, USA, February
2-7, 2018, pages 2669–2678. AAAI Press, 2018.

[2]
G. Avni, R. Bloem, K. Chatterjee, T. A. Henzinger, B. Könighofer, and
S. Pranger.
Run-time optimization for learned controllers through quantitative
In Computer Aided Verification - 31st Int. Conference, CAV
2019, New York City, NY, USA, July 15-18, 2019, Lecture Notes in Computer
Science, pages 630–649. Springer, 2019.

[3]
G. Behrmann, A. Cougnard, A. David, E. Fleury, K. G. Larsen, and D. Lime.
Uppaal-tiga: Time for playing games!
In W. Damm and H. Hermanns, editors, Computer Aided
Verification, 19th Int. Conference, CAV 2007, Berlin, Germany, July 3-7,
2007, Proceedings, volume 4590 of Lecture Notes in Computer Science,
pages 121–125. Springer, 2007.

[4]
K. Chatterjee, T. A. Henzinger, B. Jobstmann, and A. Radhakrishna.
Gist: A solver for probabilistic games.
In T. Touili, B. Cook, and P. Jackson, editors, Computer Aided
Verification, pages 665–669. Springer Berlin Heidelberg, 2010.

[5]
K. Chatterjee, T. A. Henzinger, B. Jobstmann, and R. Singh.
Quasy: Quantitative synthesis tool.
In P. A. Abdulla and K. R. M. Leino, editors, Tools and
Algorithms for the Construction and Analysis of Systems, pages 267–271,
Berlin, Heidelberg, 2011. Springer Berlin Heidelberg.

[6]
T. Chen, V. Forejt, M. Z. Kwiatkowska, D. Parker, and A. Simaitis.
Automatic verification of competitive stochastic systems.
Formal Methods Syst. Des., (1):61–92, 2013.

[7]
T. Chen, V. Forejt, M. Z. Kwiatkowska, D. Parker, and A. Simaitis.
Prism-games: A model checker for stochastic multi-player games.
In Tools and Algorithms for the Construction and Analysis of
Systems - 19th Int. Conference, TACAS 2013, Rome, Italy, March 16-24, 2013.
Proceedings, Lecture Notes in Computer Science, pages 185–191. Springer,

[8]
C. Dehnert, S. Junges, J. Katoen, and M. Volk.
A storm is coming: A modern probabilistic model checker.
In CAV (2), LNCS, pages 592–600. Springer, 2017.

[9]
N. Jansen, B. Könighofer, S. Junges, A. Serban, and R. Bloem.
Safe reinforcement learning using probabilistic shields (invited
In 31st Int. Conference on Concurrency Theory, CONCUR 2020,
September 1-4, 2020, Vienna, Austria (Virtual Conference), LIPIcs, pages
3:1–3:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.

[10]
M. Z. Kwiatkowska, G. Norman, and D. Parker.
PRISM 4.0: Verification of probabilistic real-time systems.
In G. Gopalakrishnan and S. Qadeer, editors, Computer Aided
Verification - 23rd Int. Conference, CAV 2011, Snowbird, UT, USA, July
14-20, 2011. Proceedings, Lecture Notes in Computer Science, pages 585–591.
Springer, 2011.


# Constraining equation of state of nuclear matter by charge-changing cross
section measurements of mirror nuclei

Jun-Yao Xu 11These authors contributed equally to this work. Zheng-Zheng
Li11These authors contributed equally to this work. Bao-Hua Sun
<EMAIL_ADDRESS>Yi-Fei Niu<EMAIL_ADDRESS>Xavier Roca-Maza
<EMAIL_ADDRESS>Hiroyuki Sagawa  Isao Tanihata  School of Physics,
Beihang University, Beijing 100191, China  School of Nuclear Science and
Technology, Lanzhou University, Lanzhou 730000, China  Dipartimento di Fisica,
Università degli Studi di Milano and INFN, Sezione di Milano, 20133 Milano,
Italy  RIKEN Nishina Center, Hirosawa 2-1, Wako, Saitama 351-0198, Japan

###### Abstract

The nuclear symmetry energy plays a key role in determining the equation of
state (EoS) of dense, neutron-rich matter, which connects the atomic nuclei
with the hot and dense matter in universe, thus has been the subject of
intense investigations in laboratory experiments, astronomy observations and
theories. Various probes have been proposed to constrain the symmetry energy
and its density dependence. Currently, the extensive data yield already a good
and consistent constraint to the symmetry energy ($E_{\text{sym}}(\rho)$) at
saturation density, but do not yet give a consistent result of one critical
EoS parameter, $L$, the density dependence of the symmetry energy. In this
work, we report a new probe of $L$ at saturation density. A good linear
correlation is found between $L$ and the charge changing cross section
difference ($\Delta\sigma_{\text{cc}}$) of mirror nuclei 30Si-30S for both the
Skyrme-Hartree-Fock theory (SHF) and covariant (relativistic) density
functionals (CDF). We found that the pairing effect for this mirror pair is
essential to get a consistent correlation between $L$ and
$\Delta\sigma_{\text{cc}}$ in both the SHF and CDF. Here, the cross sections
are calculated on the same target and at the same energy using the zero-range
optical-limit Glauber model. The linearity is found to be in the same
precision as those found between $L$ and neutron skin thickness or proton
radius difference.

###### keywords:

Symmetry energy , Equation of state of nuclear matter , Charge-changing cross
section , Mirror nuclei

††journal: Physics Letter B

##  1 Introduction

The symmetry energy of nuclear matter, $E_{\text{sym}}(\rho)$, is the energy
required to create an isospin asymmetry between neutrons and protons. It plays
a key role in understanding the equation of state (EoS) of dense, neutron-rich
matter, and has a crucial impact in nucleus-nucleus collisions, in the
structure and stability of atomic nuclei, and also in supernova explosions and
neutron star stability [1, 2, 3, 4, 5, 6, 7, 8]. The investigation of
$E_{\text{sym}}(\rho)$ can be accomplished from normal to light density region
by compressing and heating up nuclear matter in high-energy heavy-ion
collisions [9, 10, 11], by properties of finite nuclei [12, 5], and by
detecting the binary neutron-star merger event [13, 14].

The symmetry energy at saturation density, $\rho_{0}$, has been constrained to
be around 30 MeV [15, 16]. However, the density dependence of symmetry energy,
$L=3\rho{[\partial E_{sym}(\rho)}/{\partial\rho}]|_{\rho=\rho_{0}}$, remains
to have a significant uncertainty. $L$ provides the dominant baryonic
contribution to the pressure in neutron stars [17] and affects the neutron
skin thicknesses in heavy nuclei [18, 19]. Constraints of $L$ have been put
on, e.g., using the neutron-skin thickness of finite nuclei [20, 21], the
dipole polarizability of finite nuclei [22, 23], the radius of neutron stars
[24], proton-elastic scattering [25], charge-exchange reactions [26, 27, 28,
29], the nucleon global optical potentials [30], and also from heavy-ion
collisions [31] and astrophysical heavy pulsar masses [32]. The latest
compilations give $L$ in the range of 40.5-61.9 [23], 42.4-75.4 [33] and
30.6-86.8 MeV [34].

The neutron skin thickness, particularly for 208Pb, is of great contemporary
interest due to the strong correlation between $R_{\text{skin}}^{208}$ and the
slope of the symmetry energy $L$ [35, 36, 37, 38]. The updated neutron skin
thickness of 208Pb ($R_{\text{skin}}^{208}$) from the PREX-1 and -2 combined
data is determined as 0.283$\pm$0.071 fm [39]. Within a specific class of
relativistic energy density functionals, Reed et al. determines $L=(106\pm
37)$ MeV [40], implying a stiff symmetry energy at the typical densities of
$\rho\sim 0.15$ fm-3 in atomic nuclei. It is systematically larger than
previous limits based on both theoretical approaches and experimental
measurements. From the linear correlation between $L$ and neutron skin
thickness of 208Pb within a class of extended Skyrme density functionals, Yue
et al. [41] gives $L=85.5\pm 22.2$ MeV by the constraints from the PREX
experiment and the symmetry energy density slope $L(\rho)$ values at
subsaturation cross density $\rho_{c}$=0.11/0.16 $\rho_{0}$. A reanalysis of
the PREX data based on different types of nuclear energy density functions
(EDF) gives $L=54\pm 8$ MeV [42]. The discrepancy comes from the fact that the
significant one-sigma uncertainty of PREX-2 value of the parity-violating
asymmetry (APV) makes it difficult to use it to constraint current EDFs.

Mirror nuclei are the nuclei with the same atomic number $A$ but with the
proton number $Z$ and neutron number $N$ interchanged. Assuming a perfect
charge symmetry in atomic nuclei, the point-proton density distribution
$\rho_{p}$ of one nucleus $(Z,N)$, would be the same as the point-neutron
density distribution $\rho_{n}$ of the mirror nuclide ($N,Z$). As a step
further, one expects that the proton distribution radius of one nucleus
$(Z,N)$ would be the same as the neutron distribution radius of the mirror
nuclide ($N,Z$). Then the difference of proton radii between mirror nuclei,
$R_{\text{mirr}}$, should naturally reflect the neutron skin thickness of
($Z,N$), $R_{\text{skin}}(Z,N)$, namely,

$\displaystyle R_{\text{skin}}(Z,N)$ $\displaystyle\equiv
R_{n}(Z,N)-R_{p}(Z,N)$ (1) $\displaystyle\approx R_{p}(N,Z)-R_{p}(Z,N)\equiv
R_{\text{mirr}}(Z,N).$

The linear correlation between the charge radius difference of mirror nuclei,
$R_{\text{mirr}}$, and the slope parameter $L$ was first identified for the
mirror pair 30S and 30Si by Wang et al. [43]. Brown [44] examined by
constructing 48 Skyrme functionals that the charge radius differences of
mirror nuclei are proportional to $L$ at saturation density for a particular
mirror pair, even in the presence of the Coulomb correction. The same findings
were confirmed later in an approach based on the relativistic mean-field model
spanning a large range of the slope of the symmetry energy $L$ [45, 46]. Given
the $R_{\text{mirr}}$ – $L$ correlation at the nuclear saturation density,
Brown [47] deduced $L=5$–$70$ MeV based on charge radii of the 36Ca -36S and
38Ca -38Ar mirror pairs. They found a different slope for the linearity by
comparing the theoretical predictions based on Skyrme and Relativistic mean
field functionals.

Experimentally, the measurement of charge radii of atomic nuclei can be
performed by electron scattering [48, 49, 50], $\mu$ atomic spectrum [51, 52],
laser spectroscopy [53, 54]. However, due to factors such as the yield of
unstable nuclei and the requirement of radioactive isotopic targets, all these
methods have certain limitations when studying peculiar nuclei far away from
the stability line. Recently, a new and effective method has been developed to
extract charge radii of unstable nuclei [55, 56]. It relies on the precise
charge-changing cross section measurements $\sigma_{\text{cc}}$ on a well
known target in combination with Glauber-type model interpretation. This
method is suitable to extract the proton radii of radioactive isotopes, and
has been used in the last years to determine the proton radii of several
isotopic chains, e.g., 12-19C [56, 57], 7,9-12,14B [58], 17-22N [59], 30Ne and
32,33Na [60].

The concrete purpose of this work is essentially twofold: One intention is to
propose a potential new probe to study $L$ from the aspect of charge-changing
cross section measurements of mirror nuclei. This method is easy to evaluate
and has a similar precision to the neutron skin thickness $R_{\text{skin}}$
and the charge radius difference $R_{\text{mirr}}$ of mirror nuclei. The
second purpose is that we will discuss how robust of this new probe regards to
reaction energies and masses of mirror nuclei.

##  2 Charge-changing cross section difference of mirror nuclei

The charge-changing cross section ($\sigma_{\text{cc}}$) reflects the
probability to remove one or more protons from the incident nucleus during a
collision with a target nuclide. Assuming that only the protons in the
projectile will contribute to the charge changing reaction, the experimental
data at a few hundred MeV/nucleon or higher can be reasonably described by the
zero-range optical-limit Glauber model [61, 62, 55]. Glauber model is based on
the individual nucleon-nucleon collisions in the overlap volume of the high
energy colliding nuclei, and the nuclide is modeled as uncorrelated nucleons
sampled from the density distributions. In this framework,
$\sigma_{\text{cc}}$ can be written as:

$\sigma_{\text{cc}}=\int{(1-T^{p}(\boldsymbol{b}))d\boldsymbol{b}}\;,$ (2)

where $\boldsymbol{b}$ and $T^{p}(\boldsymbol{b})$ are the impact parameter
and and the transmission function, respectively.

$T^{p}(\boldsymbol{b})$ reads

$\begin{split}T^{p}(b)=&exp\Big{\\{}-\sigma_{pp}\int{[\rho^{\text{P}}_{p}(\boldsymbol{b-s})\cdot\rho^{\text{T}}_{p}(\boldsymbol{s})}]d\boldsymbol{s}\\\
&-\sigma_{pn}\int{[\rho^{\text{P}}_{p}\boldsymbol{(b-s)}\cdot\rho^{\text{T}}_{n}(\boldsymbol{s})}]d\boldsymbol{s}\Big{\\}}\;.\end{split}$
(3)

Here $\sigma_{pp}$ and $\sigma_{pn}$ represent the free proton-proton and
proton-neutron reaction cross section, respectively. s and (b-s) are the
distances from the centers of the target and projectile nuclei, respectively.
$\rho^{\text{T(P)}}_{p(n)}$ is the proton (neutron) density distribution in
target (projecitle) nuclide integrated along the beam axis. Reaction targets
like carbon or hydrogen are routinely used in the cross section measurements
and have well established density distributions. $\rho^{\text{P}}_{p}$ of
incident nuclei and $\sigma_{\text{cc}}$ are thus correlated. In case of a
proton target, the proton distribution density can be described by a Dirac
density function $\delta$. Eq. (3) can be simplified as:

$\begin{split}T^{p}(b)=&exp\Big{\\{}-\sigma_{pp}\int{[\rho^{\text{P}}_{p}(\boldsymbol{b-s})\cdot\delta^{\text{T}}_{p}(\boldsymbol{s})}]d\boldsymbol{s}\Big{\\}}\\\
=&exp[-\sigma_{pp}\cdot\rho^{\text{P}}_{p}(\boldsymbol{b})]\;.\end{split}$ (4)

Calculations of $\sigma_{\text{cc}}$ depend on the accuracy in the physical
inputs, and moreover how well one can treat the collision process of
projectile protons with target nucleons in terms of nucleon-nucleon
interactions. Effects induced by, e.g., the zero-range approximation and the
role of neutrons in the projectile nuclei have been discussed [63, 64, 65,
66]. However, in the present work, the absolute cross sections are less
significant. Instead, we focus on the difference of the charge-changing cross
section ($\Delta\sigma_{\text{cc}}$) of mirror nuclei on the same target at
the same energy. In this case, all the physical inputs are identical except
the proton density distributions in projectiles. The uncertainties induced by
the common parts, e.g., the density distribution in target nucleon, would be
largely cancelled out. To quantify this, we refer to an empirical formula of
$\sigma_{\text{cc}}$ [67],

$\begin{split}\sigma_{cc}=\pi[R^{\text{mat}}_{\text{targ}}+R^{\text{prot}}_{\text{proj}}+a\frac{R^{\text{mat}}_{\text{targ}}R^{\text{prot}}_{\text{proj}}}{R^{\text{mat}}_{\text{targ}}+R^{\text{prot}}_{\text{proj}}}-c]^{2}\;,\end{split}$
(5)

where $R^{\text{mat}}_{\text{targ}}$ and $R^{\text{prot}}_{\text{proj}}$ are
the matter radius of the target nuclide and the radius of the proton
distribution of projectile nuclide, respectively. $a$ is related to the
surface interaction term and $c$ reflects the effect of surface transparency.
Considering the charge symmetry in mirror nuclei, $\Delta\sigma_{cc}$ can be
approximated as:

$\begin{split}\Delta\sigma_{cc}\approx
2\pi(1+af^{2})[R^{\text{mat}}_{\text{targ}}+afR^{\text{prot}}_{\text{proj}}(Z,N)-c]R_{\text{mirr}}\;,\end{split}$
(6)

where
$f=R^{\text{mat}}_{\text{targ}}/[R^{\text{mat}}_{\text{targ}}+R^{\text{prot}}_{\text{proj}}(Z,N)]$.
To the first order, $\Delta\sigma_{\text{cc}}$ is nearly linearly correlated
to $R_{\text{mirr}}$ and thus the neutron skin thickness. Moreover, the
uncertainty induced by the matter radius of the target nuclide, is much
reduced considering the small $R_{\text{mirr}}$.

##  3 RESULTS AND DISCUSSIONS

The ground-state properties of mirror nuclei including the radii, the proton
and neutron density distributions and the slope of the symmetry energy are
computed by the Skyrme-Hartree-Fock theory (SHF) with 39 different Skyrme
forces [68, 69, 70, 71, 72, 73, 74], including SLy, SkI, SV and SIII-SVI
series. We take 30Si-30S pair to study the relation between
$\sigma_{\text{cc}}$ and $L$. The proton skin thickness of 30S is about a
factor of 2 thicker than that of the neutron skin thickness of 30Si in the
mean-field calculations. This interesting phenomenon [75] is interpreted as
the enhanced effect in the neutron deficient nuclei due to both the Coulomb
repulsion and the symmetry energy, which together push protons out to the
surface and thus create a larger proton skin.

Figure 1: (a) Charge-changing cross section for 30Si and 30S on a proton
target as functions of beam energy. (b) Same as (a) but for the difference
$\Delta\sigma_{\text{cc}}$ and the relative difference
$\Delta\sigma_{\text{cc}}/\sigma_{\text{cc}}$. The band represents the
variance when using the different proton density distributions in the SHF
models.

We calculate the $\sigma_{\text{cc}}$ of 30Si and 30S on the 1H target using
the zero-range optical limit Glauber model. A proton target is suggested to be
a better choice to extract the nuclear size [76, 77, 78]. Other targets, like
the commonly used carbon target, would complicate the present study of
interest due to the additional contribution of neutrons in target nuclei. We
would like to mention that all the results shown in this work remain valid
when using a carbon target, although a slightly worse linearity with $L$ is
found.

Figure 2: (Color online) Correlation between $\sigma_{\text{cc}}$ and $R_{p}$
for 30Si and 30S (a), and that between the difference in cross sections
($\Delta\sigma_{\text{cc}}$) and in $rms$ proton radii ($R_{\text{mirr}}$)
(b). The charge-changing cross sections are calculated at 900 MeV/nucleon on a
proton target. The dashed line is the linear fit to the calculated results by
SHF. Numbers in parentheses represent the correlation coefficients. Refer to
the text for details.

The charge-changing cross sections of mirror nuclei and their difference
versus the incident energy are plotted in Fig. 1. The band represents the
variation caused by choosing different proton density distributions in the SHF
models. As expected, the energy evolution follows the same trend as
$\sigma_{pp}$. The cross section of 30S is systematically larger than that of
30Si due to the enhanced collision of additional 2 protons in 30S. The cross
section difference is minimal at an energy of around 250 MeV/nucleon, and keep
almost constant at the energies of 700-1000 MeV/nucleon. The relative
difference, $\Delta\sigma_{\text{cc}}/\sigma_{\text{cc}}$, on the other hand,
reaches the maximum value of about 13% at 250 MeV/nucleon, and has the maximum
dispersion at 700-900 MeV/nucleon. Hereafter we take 900 MeV/nucleon as an
illustration for further discussions. At this energy,
$\Delta\sigma_{\text{cc}}$ is more than 40% larger than 250 MeV/nucleon, thus
is easier experimentally. Furthermore, the maximum dispersion makes it best to
distinguish the impact induced when using various proton density
distributions.

In Fig. 2, we draw the scatter plots of $\sigma_{\text{cc}}$ versus root-mean-
square ($rms$) proton distribution radii ($R_{p}$) for 30Si and 30S. In a
geometric collision model, $\sigma_{\text{cc}}$ should be approximately
correlated to the square of $R_{p}$. To the first order, $\sigma_{\text{cc}}$
versus $R_{p}$ follows well a straight line with a linear correlation
coefficient of about 1.00. This results in the linear correlation between the
cross section difference $\Delta\sigma_{\text{cc}}$ and the proton radius
difference $R_{\text{mirr}}$ for the wide range of Skyrme parametrizations.
The correlation is in a qualitative agreement with the linear fit to the
microscopic calculations and Eq. (6) as shown in Fig. 2(b), namely,
$\Delta\sigma_{\text{cc}}$=102.0(77) $R_{\text{mirr}}$ \+ 26.5(9) (mb), where
$R_{\text{mirr}}$ is in the unit of fm. The corresponding linear correlation
coefficient $r$ is 0.91.

Figure 3:  Slope of the symmetry energy at nuclear saturation density as a
function of $R_{\text{skin}}$ (a), $R_{\text{mirr}}$ (b) and
$\Delta\sigma_{\text{cc}}$ (c) for the 30Si-30S pair. Panels (d)-(f) are the
same as (a)-(c), but for including the pairing correction. The dashed lines
are the relevant linear fits to SHF or/and CDF calculations. Numbers in
parentheses represent the correlation coefficients. Indicated by the color
coded areas are the two recent interpreted $L$ results from the combined PREX
data. Refer to the text for details.

Moreover, we performed the dedicated calculations using the SAMi-ISB forces
[79], an effective Skyrme-like interaction that reconciles standard nuclear
ground-state properties with the current understanding of the density behavior
of the symmetry energy and the reproduction of the isobaric analogue state
(IAS) energy of 208Pb as well as in Sn isotopes. The NOCEX, NOFSIZE, NOISB,
NOPAIR and NOVP refer to the calculations when neglecting Coulomb exchange,
electromagnetic finite size effects, charge-symmetry-breaking (CSB) and
charge-independence-breaking (CIB) effects from nuclear strong force, pairing
and vacuum polarization correction to the Coulomb potential, respectively.
Nicely, the results obtained by switching off these corrections in strong and
Coulomb forces, have little impact on the general linear trend from the wide
range of Skyrme parametrizations.

In Fig. 3, the density dependence of symmetry energy, $L$ are shown as a
function of $R_{\text{skin}}$, $R_{\text{mirr}}$, and
$\Delta\sigma_{\text{cc}}$ for 30Si-30S pair, respectively. Within the Skyrme
functionals, the linear correlation coefficients of $L$ versus
$\Delta\sigma_{\text{cc}}$ reaches 0.95. It is comparable to 0.92 for $L$
versus $R_{\text{skin}}$ and 0.94 for $L$ versus $R_{\text{mirr}}$.
Considering the fact that a direct measurement of its charge radius or the
neutron skin thickness remain a challenge for isotopes like Si,
$\Delta\sigma_{\text{cc}}$ can be used as a new probe to constraint $L$. It
avoids the model uncertainty rooted in the radius determination from the
Glauber-type model. The very different $L$ predictions from Ref. [39, 40] and
Ref. [42] are also indicated by the color-coded areas in this figure. To
constrain $L$, a precision down to 1 mb is required. This is feasible
experimentally because many systematic errors can be reduced in the difference
for mirror nuclei.

We extended the calculations by using the covariant (relativistic) density
functionals (CDF). The widely used parameter sets, nonlinear parameter sets
NL1, 2 [80, 81], density-dependent RMF (DDRMF) parameter sets DD-ME1, 2, PKDD
[82, 83, 84, 85], and density-dependent relativistic Hartree-Fock (DDRHF)
parameter sets PKO1, 2, 3 [86, 87] are shown in Fig. 3. The relativistic
results generally scatter along a linear trend and show a somewhat different
slope from the SHF results. Taking $\Delta\sigma_{\text{cc}}=42.0\pm 0.5$ mb
as an example, the SHF and CDF results would give $L=135\pm 11$ and $79\pm 8$
MeV, respectively. The SHF prediction is about 71% larger than the CDF one.

The different linearities in the relativistic and non-relativistic models in
fact reflect the different structures in prediction of the mirror nuclei, as
noticed for example in Ref. [88]. We have performed a check on the effect of
spin-orbit interactions similar to Ref. [47] on the central bump of the
density for $N$=16 and $Z$=16. Spin-orbit effects in this pair of 30Si-30S has
some impact, but it can not resolve the discrepancy between relativistic and
non-relativistic models. For open shell nuclei, pairing correlations should be
taken into account [89]. We have fitted for each functional the neutron and
proton pairing gaps to the experimental values evaluated by the five-point
mass-difference formulae along the $N$, $Z$=14 and $N$, $Z$=16 isotopic and
isotonic chains. This will allow to fix the pairing strength to a realistic
value and avoid the problems when magic numbers appear in the calculations.
The new results with the inclusion of pairing correlations are shown
accordingly in Fig. 3 (d)-(f). Now the SHF and CDF results are consistent in
describ $L$ versus $R_{\text{skin}}$, $R_{\text{mirr}}$ and
$\Delta\sigma_{\text{cc}}$. The resulting linear correlation coefficient is
somehow in between the values of SHF and CDF without the pairing. The linear
coefficient of $L$ versus $\Delta\sigma_{\text{cc}}$ amounts to be 0.87, which
is the same as those found between $L$ and $R_{\text{skin}}$ or
$R_{\text{mirr}}$. Moreover, we performed the calculations of
$\Delta\sigma_{\text{cc}}$ in the energy range of 100 to 900 MeV/nucleon, and
obtained a similar linearity with $L$.

One may question if the light mirror nuclei can be used to constrain EoS,
since they are far from the infinite matter. The infinite matter refers to an
infinite uniform system of nucleons, which is via the strong force without
electromagnetic interactions. In addition, the surface corrections could be
large in light nuclei. Answering this question has to rely on the microscopic
models: can they describe the light, intermediate, and heavy nuclei self-
consistently with a good accuracy? The neutron skin thicknesses of 48Ca and
208Pb are well known to be linearly correlated with $L$ at the nuclear matter
saturation density in e.g., [90, 36, 37, 91, 21]. In Fig. 4, with the same SHF
interaction forces but including the pairing correlations, we examine the
relations between $\Delta\sigma_{\text{cc}}$ of 30Si-30S mirror nuclei and the
neutron-skin thickness of 48Ca and 208Pb. We see a linear correlation for both
48Ca and 208Pb, and a coefficient of 0.84 is reached for the latter. This
indicates that even the light-mass mirror pair, although far away from the
infinite matter, informs us about the density dependence of the symmetry
energy around saturation. However, it should be noted including the pairing
correlations reduces slightly the linearity with $L$.

Having established the existence of a strong correlation between
$\Delta\sigma_{\text{cc}}$ and $L$, we discuss now the experimental
feasibility. Due to the fact that systematic uncertainties associated to the
large set of nuclear models employed will dominate with respect to pairing or
deformation effects, we will neglect such effects in our qualitative
discussion. Specifically, we calculate $\Delta\sigma_{\text{cc}}$ of 16 pairs
of mirror nuclei from 14C-14O to 58Ni-58Zn. Both$\Delta\sigma_{\text{cc}}$ and
$\Delta\sigma_{\text{cc}}/\sigma_{\text{cc}}$ decrease with the increasing
proton number. The relative variance varies from about 29% at $A=20$ to 10% at
$A=52$ for the $T_{Z}$=2 chain, and from about 22% at $A=14$ to about 5% at
$A=58$ for the $Tz=1$ chain. The lighter the mirror nuclei are, the easier the
experiment can be carried out with the present experimental technique.

In the current charge-changing cross section measurements, the typical
uncertainty for charge radius extraction is a few mb and the relative
uncertainty can be as precise as 1% or even lower provided a good statistic.
Therefore, it is experimentally reachable in the current experimental studies.
We noted that a similar high precision in total neutron-removal cross sections
has been proposed to study $L$ in Ref. [63]. As for the isotopic chain, the
slope of the $T_{Z}=2$ chain as a function of mass number $A$ is about a
factor of 2 larger than that for the $T_{Z}=1$ chain. This is simply due to
the enhanced effect from the proton-neutron asymmetry. A particularly
interesting point would be to perform a systematic evaluation for various
mirror nuclei, helping to verify the new $L$ probe.

Figure 4:  Relations between the neutron-skin thickness of 208Pb, 48Ca and the
difference in charge-changing cross sections of 30Si-30S at 900 MeV/nucleon.
Numbers in parentheses represent the correlation coefficients. The linear fits
are also shown.

##  4 Summary

We studies $\sigma_{\text{cc}}$ by using the mean field model SHF and CDF
together with a Glauber model analysis. In this work, we show there is a good
linear correlation between the difference of charge-changing cross sections of
mirror nuclei on the same reaction target at the same energy and the density
dependence of symmetry energy. In the cross section calculations, the proton
density distributions are taken from both the SHF and CDF models. The pairing
correlation is found to be crucial to get linear correlations consistently of
$R_{\text{skin}}$, $R_{\text{mirr}}$, and $\Delta\sigma_{\text{cc}}$ to $L$.
Our results improve the theoretical analysis presented in previous
investigations [44, 45, 47]. Our results also support the recent study [89]
that neglecting pairing would artificially increase the correlation between
$R_{\text{mirr}}$ and $L$.

The present work shows the possibility of using the charge-changing reactions
of mirror nuclei to probe the density dependence of the symmetry energy. A
high precision measurement of $\sigma_{\text{cc}}$ is already possible with
the current facility. Thus, dedicated experiments have been already proposed
at HIRFL-CSR [92] and HIMAC. The energy-, mass-, and isospin-dependent
evaluation are key issues to constrain the symmetry energy. Direct comparisons
with experimental observable are expected soon.

## ACKNOWLEDGMENTS

We would like to thank the reviewers for valuable comments. This work was
supported in part by the National Natural Science Foundation of China (Nos.
U1832211, 11961141004, 11922501 and 12075104). We thank Dr. Andrej Prochazka
for helpful discussions on the Glauber model.

## References

  * [1] A. W. Steiner, M. Prakash, J. M. Lattimer, P. J. Ellis, Phys. Rep. 411 (25) (2005) 325.
  * [2] B. A. Li, L. W. Chen, C. M. Ko, Phys. Rep. 464 (4-6) (2008) 113–281.
  * [3] J. M. Lattimer, M. Prakash, Phys. Rep. 442 (1-6) (2007) 109–165.
  * [4] H. Shen, F. Ji, J. Hu, K. Sumiyoshi, Astrophys. J 891 (2) (2020) 148.
  * [5] X. Roca-Maza, N. Paar, Prog. Part. Nucl. Phys. 101 (2018) 96–176.
  * [6] G. Burgio, H.-J. Schulze, I. Vidaña, J.-B. Wei, Prog. Part. Nucl. Phys. 120 (2021) 103879.
  * [7] M. Baldo, G. Burgio, Prog. Part. Nucl. Phys. 91 (2016) 203–258.
  * [8] J. M. Lattimer, M. Prakash, Phys. Rep. 621 (2016) 127–164.
  * [9] Y.-X. Zhang, N. Wang, Q.-F. Li, L. Ou, J.-L. Tian, M. Liu, K. Zhao, X.-Z. Wu, Z.-X. Li, Front. Phys. 15 (2020) 54301.
  * [10] H. Li, H. J. Xu, Y. Zhou, X. B. Wang, J. Zhao, L. W. Chen, F. Q. Wang, Phys. Rev. Lett. 125 (2020) 222301.
  * [11] H. jie Xu, H. Li, X. Wang, C. Shen, F. Wang, Phys. Lett. B 819 (2021) 136453.
  * [12] N. Wan, C. Xu, Z. Ren, Phys. Rev. C 94 (4) (2016) 044322.
  * [13] Y. Zhou, L.-W. Chen, Astrophys. J 886 (1) (2019) 52.
  * [14] L. Baiotti, Prog. Part. Nucl. Phys. 109 (2019) 103714.
  * [15] W. D. Myers, W. J. Swiatecki, Nucl. Phys. 81 (1) (1966) 1–60.
  * [16] K. Pomorski, J. Dudek, Phys. Rev. C 67 (2003) 044316.
  * [17] C. J. Horowitz, J. Piekarewicz, Phys. Rev. Lett. 86 (2001) 5647–5650.
  * [18] M. B. Tsang, J. R. Stone, F. Camera, P. Danielewicz, S. Gandolfi, K. Hebeler, C. J. Horowitz, J. Lee, W. G. Lynch, Z. Kohley, R. Lemmon, P. Möller, T. Murakami, S. Riordan, X. Roca-Maza, F. Sammarruca, A. W. Steiner, I. Vidaña, S. J. Yennello, Phys. Rev. C 86 (2012) 015803.
  * [19] C. J. Horowitz, E. F. Brown, Y. Kim, W. G. Lynch, R. Michaels, A. Ono, J. Piekarewicz, M. B. Tsang, H. H. Wolter, J. Phys. G: Nucl. Part. Phys. 41 (9) (2014) 093001.
  * [20] S. Typel, B. A. Brown, Phys. Rev. C 64 (2001) 027302.
  * [21] P.-G. Reinhard, W. Nazarewicz, Phys. Rev. C 93 (2016) 051303.
  * [22] A. Tamii, I. Poltoratska, P. von Neumann-Cosel, Y. Fujita, T. Adachi, C. A. Bertulani, J. Carter, M. Dozono, H. Fujita, K. Fujita, K. Hatanaka, D. Ishikawa, M. Itoh, T. Kawabata, Y. Kalmykov, A. M. Krumbholz, E. Litvinova, H. Matsubara, K. Nakanishi, R. Neveling, H. Okamura, H. J. Ong, B. Özel-Tashenov, V. Y. Ponomarev, A. Richter, B. Rubio, H. Sakaguchi, Y. Sakemi, Y. Sasamoto, Y. Shimbara, Y. Shimizu, F. D. Smit, T. Suzuki, Y. Tameshige, J. Wambach, R. Yamada, M. Yosoi, J. Zenihiro, Phys. Rev. Lett. 107 (2011) 062502.
  * [23] J. M. Lattimer, Y. Lim, Astrophys. J 771 (1) (2013) 51.
  * [24] A. W. Steiner, S. Gandolfi, Phys. Rev. Lett. 108 (2012) 081102.
  * [25] K. Iida, K. Oyamatsu, B. Abu-Ibrahim, Phys. Lett. B 576 (3) (2003) 273–280.
  * [26] A. Krasznahorkay, M. Fujiwara, P. van Aarle, H. Akimune, I. Daito, H. Fujimura, Y. Fujita, M. N. Harakeh, T. Inomata, J. Jänecke, S. Nakayama, A. Tamii, M. Tanaka, H. Toyokawa, W. Uijen, M. Yosoi, Phys. Rev. Lett. 82 (1999) 3216–3219.
  * [27] A. Krasznahorkay, N. Paar, D. Vretenar, M. N. Harakeh, Phys. Lett. B 720 (2013) 428–432.
  * [28] L.-G. Cao, X. Roca-Maza, G. Colò, H. Sagawa, Phys. Rev. C 92 (3) (2015) 034308\.
  * [29] X. Roca-Maza, L.-G. Cao, G. Colò, H. Sagawa, Phys. Rev. C 94 (4) (2016) 044313\.
  * [30] C. Xu, B.-A. Li, L.-W. Chen, Phys. Rev. C 82 (2010) 054607.
  * [31] Y. Wang, Q. Li, Y. Leifels, A. Le Fèvre, Phys. Lett. B 802 (2020) 135249.
  * [32] R. Essick, I. Tews, P. Landry, A. Schwenk, Phys. Rev. Lett. 127 (2021) 192701.
  * [33] B. A. Li, X. Han, Phys. Lett. B 727 (2013) 276–281.
  * [34] M. Oertel, M. Hempel, T. Klähn, S. Typel, Rev. Mod. Phys. 89 (2017) 015007.
  * [35] B. Alex Brown, Phys. Rev. Lett. 85 (2000) 5296–5299.
  * [36] L.-W. Chen, C. M. Ko, B.-A. Li, Phys. Rev. C 72 (2005) 064309.
  * [37] M. Centelles, X. Roca-Maza, X. Vi$\widetilde{n}$as, M. Warda, Phys. Rev. Lett. 102 (12) (2009) 122502.
  * [38] X. Roca-Maza, M. Centelles, X. Vi$\widetilde{n}$as, M. Warda, Phys. Rev. Lett. 106 (2011) 252501.
  * [39] D. Adhikari, H. Albataineh, D. Androic, et al., Phys. Rev. Lett. 126 (2021) 172502\.
  * [40] B. T. Reed, F. Fattoyev, C. Horowitz, J. Piekarewicz, Phys. Rev. Lett. 126 (17) (2021) 172503.
  * [41] T.-G. Yue, L.-W. Chen, Z. Zhang, Y. Zhou, Constraints on the symmetry energy from prex-ii in the multimessenger era (2021).
  * [42] P.-G. Reinhard, X. Roca-Maza, W. Nazarewicz, Phys. Rev. Lett. 127 (2021) 232501\.
  * [43] N. Wang, T. Li, Phys. Rev. C 88 (1) (2013) 83–93.
  * [44] B. A. Brown, Phys. Rev. Lett. 119 (12) (2017) 122502.
  * [45] J. Yang, J. Piekarewicz, Phys. Rev. C 97 (1) (2018) 014314.
  * [46] M. Gaidarov, I. Moumene, A. Antonov, D. Kadrev, P. Sarriguren, E. Moya de Guerra, Nucl. Phys. A 1004 (2020) 122061.
  * [47] B. A. Brown, K. Minamisono, J. Piekarewicz, H. Hergert, D. Garand, A. Klose, K. König, J. D. Lantis, Y. Liu, B. Maaß, A. J. Miller, W. Nörtershäuser, S. V. Pineda, R. C. Powel, D. M. Rossi, F. Sommer, C. Sumithrarachchi, A. Teigelhöfer, J. Watkins, R. Wirth, Phys. Rev. Research 2 (2020) 022035.
  * [48] S. Abrahamyan, A. Acha, A. Afanasev, et al., Phys. Rev. Lett. 109 (2012) 192501\.
  * [49] C. De Jager, H. De Vries, C. De Vries, At. Data Nucl. Data Tables 14 (5) (1974) 479–508.
  * [50] T. Suda, H. Simon, Prog. Part. Nucl. Phys. 96 (2017) 1–31.
  * [51] R. Engfer, H. Schneuwly, J. Vuilleumier, H. Walter, A. Zehnder, At. Data Nucl. Data Tables 14 (5) (1974) 509–597.
  * [52] G. Fricke, C. Bernhardt, K. Heilig, L. Schaller, L. Schellenberg, E. Shera, C. Dejager, At. Data Nucl. Data Tables 60 (2) (1995) 177–285.
  * [53] Z.-T. Lu, P. Mueller, G. W. F. Drake, W. Nörtershäuser, S. C. Pieper, Z.-C. Yan, Rev. Mod. Phys. 85 (2013) 1383–1400.
  * [54] P. Campbell, I. Moore, M. Pearson, Prog. Part. Nucl. Phys. 86 (2016) 127–180.
  * [55] T. Yamaguchi, M. Fukuda, S. Fukuda, G. W. Fan, I. Hachiuma, M. Kanazawa, A. Kitagawa, T. Kuboki, M. Lantz, M. Mihara, M. Nagashima, K. Namihira, D. Nishimura, Y. Okuma, T. Ohtsubo, S. Sato, T. Suzuki, M. Takechi, W. Xu, Phys. Rev. C 82 (2010) 014609.
  * [56] T. Yamaguchi, I. Hachiuma, A. Kitagawa, K. Namihira, S. Sato, T. Suzuki, I. Tanihata, M. Fukuda, Phys. Rev. Lett. 107 (2011) 032502.
  * [57] R. Kanungo, W. Horiuchi, G. Hagen, G. R. Jansen, P. Navratil, F. Ameil, J. Atkinson, Y. Ayyad, D. Cortina-Gil, I. Dillmann, A. Estradé, A. Evdokimov, F. Farinon, H. Geissel, G. Guastalla, R. Janik, M. Kimura, R. Knöbel, J. Kurcewicz, Y. A. Litvinov, M. Marta, M. Mostazo, I. Mukha, C. Nociforo, H. J. Ong, S. Pietri, A. Prochazka, C. Scheidenberger, B. Sitar, P. Strmen, Y. Suzuki, M. Takechi, J. Tanaka, I. Tanihata, S. Terashima, J. Vargas, H. Weick, J. S. Winfield, Phys. Rev. Lett. 117 (2016) 102501.
  * [58] S. Terashima, I. Tanihata, R. Kanungo, A. Estradé, W. Horiuchi, F. Ameil, J. Atkinson, Y. Ayyad, D. Cortina-Gil, I. Dillmann, A. Evdokimov, F. Farinon, H. Geissel, G. Guastalla, R. Janik, M. Kimura, R. Knoebel, J. Kurcewicz, Y. A. Litvinov, M. Marta, M. Mostazo, I. Mukha, T. Neff, C. Nociforo, H. J. Ong, S. Pietri, A. Prochazka, C. Scheidenberger, B. Sitar, Y. Suzuki, M. Takechi, J. Tanaka, J. Vargas, J. S. Winfield, H. Weick, Prog. Theor. Exp. Phys. 2014 (2014) 101D02.
  * [59] S. Bagchi, R. Kanungo, W. Horiuchi, G. Hagen, T. D. Morris, S. R. Stroberg, T. Suzuki, F. Ameil, J. Atkinson, Y. Ayyad, D. Cortina-Gil, I. Dillmann, A. Estradé, A. Evdokimov, F. Farinon, H. Geissel, G. Guastalla, R. Janik, S. Kaur, R. Knöbel, J. Kurcewicz, Y. A. Litvinov, M. Marta, M. Mostazo, I. Mukha, C. Nociforo, H. J. Ong, S. Pietri, A. Prochazka, C. Scheidenberger, B. Sitar, P. Strmen, M. Takechi, J. Tanaka, Y. Tanaka, I. Tanihata, S. Terashima, J. Vargas, H. Weick, J. S. Winfield, Phys. Lett. B 790 (2019) 251–256.
  * [60] A. Ozawa, T. Moriguchi, T. Ohtsubo, N. Aoi, D. Q. Fang, N. Fukuda, M. Fukuda, H. Geissel, I. Hachiuma, N. Inabe, Y. Ishibashi, S. Ishimoto, Y. Ito, T. Izumikawa, D. Kameda, T. Kubo, T. Kuboki, K. Kusaka, M. Lantz, Y. G. Ma, M. Mihara, Y. Miyashita, S. Momota, D. Nagae, K. Namihira, D. Nishimura, H. Ooishi, Y. Ohkuma, T. Ohnishi, M. Ohtake, K. Ogawa, Y. Shimbara, T. Suda, T. Sumikama, H. Suzuki, S. Suzuki, T. Suzuki, M. Takechi, H. Takeda, K. Tanaka, R. Watanabe, M. Winkler, T. Yamaguchi, Y. Yanagisawa, Y. Yasuda, K. Yoshinaga, A. Yoshida, K. Yoshida, Phys. Rev. C 89 (2014) 044602.
  * [61] J. Meng, S.-G. Zhou, I. Tanihata, Phys. Lett. B 532 (3-4) (2002) 209–214.
  * [62] A. Bhagwat, Y. K. Gambhir, Phys. Rev. C 69 (1) (2004) 014315.
  * [63] T. Aumann, C. A. Bertulani, F. Schindler, S. Typel, Phys. Rev. Lett. 119 (2017) 262501\.
  * [64] I. Abdul-Magead, B. Abu-Ibrahim, Nucl. Phys. A 1000 (2020) 121804.
  * [65] Y. Suzuki, W. Horiuchi, S. Terashima, R. Kanungo, F. Ameil, J. Atkinson, Y. Ayyad, D. Cortina-Gil, I. Dillmann, A. Estradé, A. Evdokimov, F. Farinon, H. Geissel, G. Guastalla, R. Janik, R. Knoebel, J. Kurcewicz, Y. A. Litvinov, M. Marta, M. Mostazo, I. Mukha, C. Nociforo, H. J. Ong, S. Pietri, A. Prochazka, C. Scheidenberger, B. Sitar, P. Strmen, M. Takechi, J. Tanaka, I. Tanihata, J. Vargas, H. Weick, J. S. Winfield, Phys. Rev. C 94 (2016) 011602.
  * [66] M. Tanaka, M. Takechi, A. Homma, A. Prochazka, M. Fukuda, D. Nishimura, T. Suzuki, T. Moriguchi, D. S. Ahn, A. Aimaganbetov, M. Amano, H. Arakawa, S. Bagchi, K. H. Behr, N. Burtebayev, K. Chikaato, H. Du, T. Fujii, N. Fukuda, H. Geissel, T. Hori, S. Hoshino, R. Igosawa, A. Ikeda, N. Inabe, K. Inomata, K. Itahashi, T. Izumikawa, D. Kamioka, N. Kanda, I. Kato, I. Kenzhina, Z. Korkulu, Y. Kuk, K. Kusaka, K. Matsuta, M. Mihara, E. Miyata, D. Nagae, S. Nakamura, M. Nassurlla, K. Nishimuro, K. Nishizuka, K. Ohnishi, M. Ohtake, T. Ohtsubo, S. Omika, H. J. Ong, A. Ozawa, H. Sakurai, C. Scheidenberger, Y. Shimizu, T. Sugihara, T. Sumikama, H. Suzuki, S. Suzuki, H. Takeda, Y. Tanaka, Y. K. Tanaka, I. Tanihata, T. Wada, K. Wakayama, S. Yagi, T. Yamaguchi, R. Yanagihara, Y. Yanagisawa, K. Yoshida, T. K. Zholdybayev, Charge-changing cross sections for ${}^{42\textrm{--}51}$ca and effect of charged-particle evaporation induced by neutron removal reaction (2021).
  * [67] L. Chulkov, O. Bochkarev, D. Cortina-Gil, H. Geissel, M. Hellström, M. Ivanov, R. Janik, K. Kimura, T. Kobayashi, A. Korsheninnikov, G. Münzenberg, F. Nickel, A. Ogloblin, A. Ozawa, M. Pfützner, V. Pribora, M. Rozhkov, H. Simon, B. Sitár, P. Strmen, K. Sümmerer, T. Suzuki, I. Tanihata, M. Winkler, K. Yoshida, Nucl. Phys. A 674 (3) (2000) 330–342.
  * [68] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, R. Schaeffer, Nucl. Phys. A 635 (1-2) (1997) 231–256.
  * [69] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, R. Schaeffer, Nucl. Phys. A 643 (4) (1998) 441–441.
  * [70] P.-G. Reinhard, H. Flocard, Nucl. Phys. A 584 (3) (1995) 467–488.
  * [71] P. Klüpfel, P.-G. Reinhard, T. J. Bürvenich, J. A. Maruhn, Phys. Rev. C 79 (2009) 034310.
  * [72] J. Bartel, P. Quentin, M. Brack, C. Guet, H.-B. Håkansson, Nucl. Phys. A 386 (1) (1982) 79–100.
  * [73] M. Beiner, H. Flocard, N. Van Giai, P. Quentin, Nucl. Phys. A 238 (1) (1975) 29–69.
  * [74] H. Krivine, J. Treiner, O. Bohigas, Nucl. Phys. A 336 (2) (1980) 155–184.
  * [75] J. Yang, J. A. Hernandez, J. Piekarewicz, Phys. Rev. C 100 (2019) 054301.
  * [76] W. Horiuchi, Y. Suzuki, T. Inakura, Phys. Rev. C 89 (2014) 011601.
  * [77] W. Horiuchi, S. Hatakeyama, S. Ebata, Y. Suzuki, Phys. Rev. C 93 (2016) 044611.
  * [78] W. Horiuchi, Y. Suzuki, T. Uesaka, M. Miwa, Phys. Rev. C 102 (2020) 054601.
  * [79] X. Roca-Maza, G. Colò, H. Sagawa, Phys. Rev. Lett. 120 (2018) 202501.
  * [80] G. A. Lalazissis, J. König, P. Ring, Phys. Rev. C 55 (1997) 540–543.
  * [81] Y. Sugahara, H. Toki, Nucl. Phys. A 579 (3) (1994) 557–572.
  * [82] T. Nikšić, D. Vretenar, P. Finelli, P. Ring, Phys. Rev. C 66 (2002) 024306.
  * [83] G. A. Lalazissis, T. Nikšić, D. Vretenar, P. Ring, Phys. Rev. C 71 (2005) 024312.
  * [84] W. H. Long, J. Meng, N. V. Giai, S.-G. Zhou, Phys. Rev. C 69 (2004) 034319.
  * [85] S. Typel, H. Wolter, Nucl. Phys. A 656 (3) (1999) 331–364.
  * [86] W. H. Long, H. Sagawa, N. V. Giai, J. Meng, Phys. Rev. C 76 (2007) 034314.
  * [87] W. H. Long, H. Sagawa, J. Meng, N. V. Giai, EPL (Europhysics Letters) 82 (1) (2008) 12001.
  * [88] H. Kurasawa, T. Suda, T. Suzuki, Prog. Theor. Exp. Phys. 2021 (1) (2020) 013D02.
  * [89] P.-G. Reinhard, W. Nazarewicz, Phys. Rev. C 105 (2022) L021301.
  * [90] R. Furnstahl, Nucl. Phys. A 706 (1) (2002) 85–110.
  * [91] M. Kortelainen, J. Erler, W. Nazarewicz, N. Birge, Y. Gao, E. Olsen, Phys. Rev. C 88 (2013) 031305.
  * [92] B.-H. Sun, J.-W. Zhao, X.-H. Zhang, L.-N. Sheng, Z.-Y. Sun, I. Tanihata, S. Terashima, Y. Zheng, L.-H. Zhu, L.-M. Duan, L.-C. He, R.-J. Hu, G.-S. Li, W.-J. Lin, W.-P. Lin, C.-Y. Liu, Z. Liu, C.-G. Lu, X.-W. Ma, L.-J. Mao, Y. Tian, F. Wang, M. Wang, S.-T. Wang, J.-W. Xia, X.-D. Xu, H.-S. Xu, Z.-G. Xu, J.-C. Yang, D.-Y. Yin, Y.-J. Yuan, W.-L. Zhan, Y.-H. Zhang, X.-H. Zhou, Sci. Bull. 63 (2) (2018) 78–80.