∎

e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>e3e-mail:
<EMAIL_ADDRESS>e4e-mail<EMAIL_ADDRESS>e5e-mail<EMAIL_ADDRESS>e6e-mail<EMAIL_ADDRESS>
11institutetext: Department of Physics, Korea University, Seoul 02841,
Republic of Korea22institutetext: Facility for Rare Isotope Beams, Michigan
State University, East Lansing, Michigan 48824, USA33institutetext: Department
of Physics and Astronomy, Michigan State University, East Lansing, Michigan
48824, USA44institutetext: RIKEN Nishina Center, Hirosawa 2-1, Wako, Saitama
351-0198, Japan55institutetext: Department of Physics, Kyoto University, Kita-
shirakawa, Kyoto 606-8502, Japan66institutetext: Department of Physics, Tohoku
University, Sendai 980-8578, Japan77institutetext: Instituto de Física,
Universidade Federal do Rio de Janeiro, Centro de Tecnologia, Bloco A,
21941-909 Rio de Janeiro, Rio de Janeiro, Brazil88institutetext: Departamento
de Engenharia Nuclear, Universidade Federal de Minas Gerais UFMG, Av.
Presidente Antônio Carlos, 6.627, 31270-901 Belo Horizonte, Minas Gerais,
Brazil99institutetext: Institut für Kernphysik, Technische Universität
Darmstadt, D-64289 Darmstadt, Germany1010institutetext: GSI Helmholtzzentrum
für Schwerionenforschung, Planckstrasse 1, 64291 Darmstadt,
Germany1111institutetext: Faculty of Physics, Astronomy and Applied Computer
Science, Jagiellonian University, Kraków, Poland1212institutetext: Division of
Experimental Physics, Rudjer Boskovic Institute, Zagreb,
Croatia1313institutetext: Department of Physics, Rikkyo University, Nishi-
Ikebukuro 3-34-1, Tokyo 171-8501, Japan1414institutetext: Department of Life
and Environmental Agricultural Sciences, Tottori University, Tottori 680-8551,
Japan1515institutetext: Rare Isotope Science Project, Institute for Basic
Science, Daejeon 34047, Republic of Korea1616institutetext: Department of
Physics, Tokyo Institute of Technology, Tokyo 152-8551,
Japan1717institutetext: Institute of Nuclear Physics PAN, ul. Radzikowskiego
152, 31-342 Kraków, Poland1818institutetext: Cyclotron Institute, Texas A&M
University, College Station, Texas 77843, USA1919institutetext: Nikhef
National Institute for Subatomic Physics, Amsterdam,
Netherlands2020institutetext: Department of Physics, Tsinghua University,
Beijing 100084, PR China2121institutetext: Department of Chemistry, Texas A&M
University, College Station, Texas 77843, USA

# Isoscaling in central Sn+Sn collisions at 270 MeV/u

J.W. Leee1,addr1  M.B. Tsange2,addr2,addr3  C.Y. Tsangaddr2,addr3  R.
Wange3,addr2  J. Barneyaddr2,addr3  J. Esteeaddr2,addr3  T. Isobee5,addr4  M.
Kanekoaddr4,addr5  M. Kurata-Nishimuraaddr4  W.G. Lynche4,addr2,addr3  T.
Murakamie6,addr4,addr5  A. Onoaddr6  S.R. Souzaaddr7,addr8  D.S. Ahnaddr4  L.
Ataraddr9,addr10  T. Aumannaddr9,addr10  H. Babaaddr4  K. Boretzkyaddr10  J.
Brzychczykaddr11  G. Cerizzaaddr2  N. Chigaaddr4  N. Fukudaaddr4  I.
Gasparicaddr12,addr4,addr9  B. Hongaddr1  A. Horvataddr9,addr10  K. Iekiaddr13
N. Ikenoaddr14  N. Inabeaddr4  G. Jhangaddr2  Y.J. Kimaddr15  T.
Kobayashiaddr6  Y. Kondoaddr16  P. Laskoaddr17  H.S. Leeaddr15  Y.
Leifelsaddr10  J. Łukasikaddr17  J. Manfrediaddr2,addr3  A.B. McIntoshaddr18
P. Morfouaceaddr2  T. Nakamuraaddr16  N. Nakatsukaaddr4,addr5  S.
Nishimuraaddr4  H. Otsuaddr4  P. Pawłowskiaddr17  K. Pelczaraddr11  D.
Rossiaddr9  H. Sakuraiaddr4  C. Santamariaaddr2  H. Satoaddr4  H. Scheitaddr9
R. Shaneaddr2  Y. Shimizuaddr4  H. Simonaddr10  A. Snochaddr19  A.
Sochockaaddr11  T. Sumikamaaddr4  H. Suzukiaddr4  D. Suzukiaddr4  H.
Takedaaddr4  S. Tangwancharoenaddr2  Y. Toganoaddr13  Z.G. Xiaoaddr20  S.J.
Yennelloaddr18,addr21  Y. Zhangaddr20 (the S$\pi$RIT collaboration)

(Received: date / Revised version: date)

###### Abstract

Experimental information on fragment emissions is important in understanding
the dynamics of nuclear collisions and in the development of transport model
simulating heavy-ion collisions. The composition of complex fragments emitted
in the heavy-ion collisions can be explained by statistical models, which
assume that thermal equilibrium is achieved at collision energies below 100
MeV/u. Our new experimental data together with theoretical analyses for light
particles from Sn+Sn collisions at 270 MeV/u, suggest that the hypothesis of
thermal equilibrium breaks down for particles emitted with high transfer
momentum. To inspect the system’s properties in such limit, the scaling
features of the yield ratios of particles from two systems, a neutron-rich
system of ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ and a nearly symmetric
system of ${}^{108}\mathrm{Sn}+{}^{112}\mathrm{Sn}$, are examined in the
framework of the statistical multifragmentation model and the antisymmetrized
molecular dynamics model. The isoscaling from low energy particles agree with
both models. However the observed breakdown of isoscaling for particles with
high transverse momentum cannot be explained by the antisymmetrized molecular
dynamics model.

††journal: Eur. Phys. J. A

##  1 Introduction

During a collision involving heavy-ions at energies well above the Coulomb
barrier, nuclear matter is driven through very different configurations. In
the early stages, violent collisions between its constituents take place,
causing matter to be heated up. If these collisions occur above the production
thresholds, sub-atomic particles such as pions, for instance, may be produced
StockR_PR_1986_135_259 ; SengerPeter_PPNP_2004_53_1 ;
OnoAkira_PPNP_2019_105_139_HIC_dynamics ;
IkenoNatsumi_PRC_2016_93_044612_AMD_SnSn300 ; HongJun_PRC_2014_90_024605 .
Many particles are ejected in this pre-equilibrium stage XuHM_PRC_1994_50_1659
, carrying away an appreciable amount of energy from the system. At the same
time, the relative collective motion between the colliding nuclei leads to the
compression of nuclear matter and densities higher than the saturation density
found in the core of normal nuclei such as lead
IkenoNatsumi_PRC_2016_93_044612_AMD_SnSn300 ;
BorderieB_PPNP_2019_105_82_Phase_Transition_Nuclei .

Dynamical treatments WolterHermann_PPNP_2022_125_103962 have been developed to
study such early stages. The density, the isospin configuration, and the
temperatures attained by the system are sensitive to the physics input
parameters employed by the models WolterHermann_PPNP_2022_125_103962 ;
XuHM_PRC_1994_50_1659 ; LiBaoAn_PRL_1997_78_1644 ; TanWP_PRC_2001_64_051901 ;
JhangG_SpiRIT_PLB_2021_813_136016_pion_ratio . Therefore, important
information on the Nuclear Equation of State (EOS) may be provided by these
dynamical approaches. Determining the input parameters of the models using
experimental observables is of particular interest, as some of them are
closely related to the rate of equilibration of the system, providing deeper
insight into the dynamics of the collisions.

After pre-equilibrium emission, a freeze-out configuration is reached and many
fragments including protons and neutrons are emitted XuHM_PRC_1994_50_1659 .
Its behaviors can be described both by dynamical
OnoAkira_PPNP_2019_105_139_HIC_dynamics ; WolterHermann_PPNP_2022_125_103962
and statistical models BondorfJP_PR_1995_257_133_SMM ; DasCB_PR_2005_406_1 ;
BotvinaAS_EPJA_2006_30_121 ;
BorderieB_PPNP_2019_105_82_Phase_Transition_Nuclei . In the former case, it
appears as a result of the dynamical path taken by the system. In the case of
statistical models, a freeze-out configuration is assumed. In both cases, most
of the excited fragments predicted by the models would decay before detection
and, therefore, a de-excitation treatment BotvinaAS_NPA_1987_475_663 ;
TanWP_PRC_2003_68_034609_ISMM must be applied before comparing theoretical
predictions with the experimental data. Consequently, important vestiges of
the freeze out configuration may be blurred by the de-excitation process.

In this context, the nuclear isoscaling phenomenon
TsangMB_MSU_PRC_2001_64_054615_isoscaling_SMM_EES ;
TsangMB_MSU_PRL_2001_86_5023_isoscaling_experiment , first reported in Ref.
XuHS_MSU_PRL_2000_85_716_isoscaling for Sn+Sn reactions at 50 MeV/u, is a very
useful tool as the ratios of yields from two different reactions (which differ
mainly in the isospin composition) is weakly affected by the fragment de-
excitation process TsangMB_MSU_PRC_2001_64_054615_isoscaling_SMM_EES ,
retaining information on the system’s configuration right after the violent
stages of the reaction. Particularly, under certain conditions
TsangMB_MSU_PRC_2001_64_054615_isoscaling_SMM_EES ;
SouzaSR_PRC_2009_80_044606_isoscaling_symmetryenergy ;
RamiF_PRL_2000_84_1120_FOPI_Isospin_tracing , it may be related to the
symmetry energy, which makes this observable specially relevant to
investigations on the nuclear EOS. The isoscaling analysis considers the
ratio:

$R_{21}(N,Z)=Y_{2}(N,Z)/Y_{1}(N,Z)\;,$ (1)

where $Y_{1}(N,Z)$ and $Y_{2}(N,Z)$ stand for the yields of species $(N,Z)$,
observed in two similar reactions “1” and “2” , where $N$ and $Z$ respectively
denote the neutron and proton numbers of the isotope. It has been found that
this ratio follows a simple scaling law
TsangMB_MSU_PRL_2001_86_5023_isoscaling_experiment ;
TsangMB_MSU_PRC_2001_64_054615_isoscaling_SMM_EES

$R_{21}(N,Z)=C\exp(\alpha N+\beta Z),$ (2)

where $\alpha$ and $\beta$ are the scaling parameters and C is a normalization
constant. By convention, reaction 2 is chosen to have a larger isospin
asymmetry compared with that of reaction 1.

This scaling property can be derived in the framework of the grand-canonical
ensemble TsangMB_MSU_PRC_2001_64_054615_isoscaling_SMM_EES . From it, simple
relationships between the neutron and proton chemical potentials, $\mu_{n}$
and $\mu_{p}$, respectively, and the scaling parameters are obtained:

$\Delta\mu_{n}\equiv\mu_{n}^{(2)}-\mu_{n}^{(1)}=\alpha T\;$ (3)

and

$\Delta\mu_{p}\equiv\mu_{p}^{(2)}-\mu_{p}^{(1)}=\beta T\;.$ (4)

In these expressions, $T$ symbolizes the temperature at the freeze-out
configuration and the superscripts ($i=1,2$) represent reactions “1” or “2”.

Although the formal derivation is based on the grand-canonical ensemble, the
isoscaling property is also found in nearly all statistical models
TsangMB_MSU_PRC_2001_64_041603_isoscaling_condition , including versions of
the Statistical Multifragmentation Model (SMM) which employ the micro-
canonical and canonical ensembles
SouzaSR_PRC_2009_80_044606_isoscaling_symmetryenergy . It has also been
observed in the Antisymmetrized Molecular Dynamics (AMD) model
OnoAkira_SpiRIT_PRC_2003_68_051601_isoscaling_AMD . Calculations based on the
molecular dynamics approach DorsoCO_PRC_2006_73_044601 seem to indicate that
the isoscaling may be observed even when the system has not yet attained
thermal equilibrium. Furthermore, recent experimental results suggest that the
parameters $\alpha$ and $\beta$ are sensitive to the mechanisms responsible
for fragment production. These results call for further investigations on the
nuclear isoscaling property.

Many experimental studies GeraciE_LNS_NPA_2004_732_173_isoscaling_experiment ;
TrautmannW_2006_nuclex_0603027_isoscaling_ALADIN_INDRA ;
FableQ_Arxiv_2022_2202_13850_CaCa_INDRA ;
WuenschelS_TAMU_PRC_2009_79_061602_isoscaling_experiment ;
YoungsM_TAMU_NPA_2017_962_61_isoscaling_experiment have investigated the
isoscaling property in different systems at different bombarding energies. A
close relationship between the parameter $\alpha$ and the symmetry energy
coefficient $C_{\textrm{sym}}$ has been pointed out in Ref.
TsangMB_MSU_PRC_2001_64_054615_isoscaling_SMM_EES . More specifically,
denoting by $Z_{i}$ and $A_{i}$ the atomic and mass numbers of the $i$-th
source, one has:

$\alpha\approx
4C_{\textrm{sym}}[({Z_{1}}/{A_{1}})^{2}-({Z_{2}}/{A_{2}})^{2}]/T\;.$ (5)

This property has been examined in several theoretical
SouzaSR_PRC_2009_80_044606_isoscaling_symmetryenergy ;
BotvinaAS_JINR_PRC_2002_65_044610_isoscaling_ion_induce and experimental
TrautmannW_2006_nuclex_0603027_isoscaling_ALADIN_INDRA ;
FableQ_Arxiv_2022_2202_13850_CaCa_INDRA ;
WuenschelS_TAMU_PRC_2009_79_061602_isoscaling_experiment ;
YoungsM_TAMU_NPA_2017_962_61_isoscaling_experiment works. Studies found that
the isoscaling parameter $\alpha$ and the $(Z/A)^{2}$ difference between the
two sources are related
WuenschelS_TAMU_PRC_2009_79_061602_isoscaling_experiment , and different
isoscaling tendencies between the projectile-like fragments and the emitted
fragments exist YoungsM_TAMU_NPA_2017_962_61_isoscaling_experiment . The
INDRA-GSI collaboration extended this investigation to relativistic collision
energies for carbon induced reactions on Sn isotopes
LeFevreA_PRL_2005_94_162701_INDRA_Alladin .

In this work, we extend earlier investigations at low incident energies of
Sn+Sn reactions to a collision energy (270 MeV/u). We concentrate on central
mid-rapidity events, with impact parameters $b<1.5$ fm. The scaling properties
are studied as a function of the transverse momentum.

##  2 Experiment

The S$\pi$RIT experiment was performed at the Radioactive Isotope Beam Factory
(RIBF) at RIKEN. The primary beams of 238U and 132Xe impinged on the Be target
to produce secondary beams of ${}^{132,124}\textrm{Sn}$ and
${}^{112,108}\textrm{Sn}$ respectively, at 270 MeV/u. The beams bombarded on
the isotopically enriched ${}^{124}\textrm{Sn}$ and ${}^{112}\textrm{Sn}$
targets with areal density of $608\,\textrm{mg/cm}^{2}$ and
$561\,\textrm{mg/cm}^{2}$, respectively. Four reactions with different
neutron-to-proton ratios, $N/Z$, of the total system were measured:
${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ $(N/Z=1.56)$,
${}^{108}\mathrm{Sn}+{}^{112}\mathrm{Sn}$ $(N/Z=1.2)$,
${}^{112}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ $(N/Z=1.36)$, and
${}^{124}\mathrm{Sn}+{}^{112}\mathrm{Sn}$ $(N/Z=1.36)$. In this work, we focus
mainly on the most $(N/Z=1.56)$ and least $(N/Z=1.2)$ neutron rich systems.

Charged particles produced in the reactions were detected with the SAMURAI
Pion Reconstruction and Ion-Tracker Time Projection Chamber (S$\pi$RIT-TPC)
ShaneR_SpiRIT_NIMA_2015_784_513_spirittpc ;
TangwancharoenS_SpiRIT_NIMA_2017_853_44_TPCGatingGrid ;
BarneyJ_SpiRIT_RSI_2021_92_063302_spirittpc installed inside the SAMRURAI
dipole magnet OtsuH_NIMB_2016_376_175_SAMURAIMagnet with a magnetic field of
0.5 T. The effective volume of the TPC is 1344 mm $\times$ 864 mm $\times$ 530
mm, and the target was placed at 15 mm upstream of the entrance window,
resulting in an angular coverage of $\theta<80^{\circ}$ with respect to the
beam axis in the laboratory frame. Description of the associated trigger
arrays located on the side and downstream of the TPC used to select central
events can be found in Refs. LaskoP_NIMA_2017_856_92_KATANA ;
KanekoM_NIMA_2022_1039_167010_KyotoArray . The Generic Electronics for TPCs
(GET) was employed to read out the track signals. The analysis software
S$\pi$RITROOT JhangGenie_SpiRIT_JKPS_2016_69_144_SpiRITROOT ;
LeeJW_SpiRIT_NIMA_2020_965_163840_SpiRITROOT was developed for the track
reconstruction of the charged particles. Detailed performance of the TPC and
GET electronics as well as the software analysis codes have been published in
BarneyJ_SpiRIT_RSI_2021_92_063302_spirittpc ;
JhangGenie_SpiRIT_JKPS_2016_69_144_SpiRITROOT ;
IsobeT_SpiRIT_NIMA_2018_899_43_GET_electronics ;
LeeJW_SpiRIT_NIMA_2020_965_163840_SpiRITROOT . Other technical issues
including space charge correction and extending the dynamic range of the TPC
electronics can be found in TsangCY_SpiRIT_NIMA_2020_959_163477_space_charge ;
EsteeJ_SpiRIT_NIMA_2019_944_162509_tpcdynamicrange .

The cuboid-shaped S$\pi$RIT TPC lacks the azimuthal symmetry. In this work, we
mainly analyze the data at the azimuthal angles, $\phi$,
$-30^{\circ}<\phi<20^{\circ}$ and $160^{\circ}<\phi<210^{\circ}$ where the
tracks are longest and the track quality is generally much better. The TPC
efficiencies arising from the detector performance are determined by the track
embedding method AndersonM_NIMA_2003_499_659_STARTPC using events generated
from Monte Carlo simulations and GEANT-4. The reconstructed rigidity
momentum/charge, $p/Z$ and the mean energy loss per unit length
$\left<dE/dx\right>$ were provided for each track. The resolution of the
reconstructed momentum and $\left<dE/dx\right>$ for single tracks is 1.6% and
4.6 %, respectively EsteeJustinBrian_phdthesis_MichiganStateUniversity_2020 .
The particle identification (PID) is obtained using the correlation plot of
magnetic rigidity, $p/Z$, and energy loss, $\left<dE/dx\right>$, of the
detected particles, as shown in Fig. 1. Isotopes of $p$, $d$, $t$,
${}^{3}\mathrm{He}$, 4He, ${}^{6}\mathrm{He}$, ${}^{6}\mathrm{Li}$, and
${}^{7}\mathrm{Li}$ can be clearly identified. However, below $p/Z<$ 600
MeV/$c$, the $t$ and ${}^{3}\mathrm{He}$ PID lines merge. In this study, we
mainly focus on particles in the mid-rapidity range $y_{0}=$ 0 - 0.4 with
$y_{0}=y/y_{NN}^{c.m.}-1$ where $y$ is the rapidity of the particle and
$y_{NN}^{c.m.}$ is center-of-mass rapidity of the nucleon-nucleon system.
Since the mid-rapidity gate eliminates tritons below $p/Z<$ 1000 MeV/$c$,
contamination from ${}^{3}\mathrm{He}$ in triton spectra is minimal. On the
other hand, contamination from triton in the 3He spectra has to be determined.

Some physics results regarding the symmetry energy constraints from the
S$\pi$RIT experiments have been published. Charged pion multiplicities and
ratios were published in Ref. JhangG_SpiRIT_PLB_2021_813_136016_pion_ratio and
Ref. EsteeJ_SpiRIT_PRL_2021_126_162701_pion_ratio . Ref.
KanekoM_SpiRIT_PLB_2021_822_136681_Z1particles_AMD focused on $Z$=1 particles
and comparisons of the rapidity distributions with the AMD
modelsOnoAkira_PTP_1992_87_1185_AMD ; OnoAkira_PPNP_2019_105_139_HIC_dynamics
. This paper provides a more comprehensive study than Ref.
KanekoM_SpiRIT_PLB_2021_822_136681_Z1particles_AMD and focuses on the
transverse momentum spectra and yield ratios obtained from light charged
fragments, specifically $p$, $d$, $t$, 3He and 4He.

Figure 1: Particle identification plot of energy loss, $\left<dE/dx\right>$
vs. magnetic rigidity, $p/Z$. The particles emitted from the
${}^{132}\textrm{Sn}+^{124}\textrm{Sn}$ collisions at 270 MeV/u are detected
with the S$\pi$RIT-TPC.

To select central collisions, we assume the impact parameters increase
monotonically with the charged particle multiplicity
CavataC_PRC_1990_42_1760_reduced_impact_parameter ;
BarneyJonathanElijah_phdthesis_MichiganStateUniversity_2019 . Very central
collision events ($b<$ 1.5 fm) similar to those analyzed in Ref.
KanekoM_SpiRIT_PLB_2021_822_136681_Z1particles_AMD are chosen for this work.
The difference between this work and Ref.
KanekoM_SpiRIT_PLB_2021_822_136681_Z1particles_AMD for the rapidity spectra in
$y_{0}=$ 0 - 0.4 (Fig. 2) is within 3 %. The charge multiplicity selection in
both works is different. Indeed, events with multiplicities equal or more than
57 and 56 charged particles, for 132Sn+124Sn and 108Sn+112Sn respectively, are
chosen in this work, while 56 and 55 are used in Ref.
KanekoM_SpiRIT_PLB_2021_822_136681_Z1particles_AMD . In both cases, the impact
parameter gates of ($b<$ 1.5 fm) is chosen. In this work, $b_{\mathrm{max}}$
is defined to be the experimental $b_{\mathrm{max}}$ of 7.52 fm in
${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ and 7.13 fm in
${}^{108}\mathrm{Sn}+{}^{112}\mathrm{Sn}$ while in Ref.
KanekoM_SpiRIT_PLB_2021_822_136681_Z1particles_AMD , $b_{\mathrm{max}}$=10 fm
is defined by the sum of the radii for the projectile and target. These slight
differences in the multiplicity gates do not affect the results of both works.
The total statistics is increased as the tracks from both the left and right
side of the TPC is used in the present work, whereas only the tracks in the
right side of the TPC is used in Ref.
KanekoM_SpiRIT_PLB_2021_822_136681_Z1particles_AMD . These are the main
reasons for the 3 % discrepancy. In any case, the isocaling ratios are
unaffected by the slight discrepancies in the spectra.

Fig. 1 shows a typical PID plot of the charged particles obtained in the
132Sn+124Sn collisions. The heavier clusters with A $>$ 5 nuclei such as
${}^{6}\textrm{He}$, ${}^{6}\textrm{Li}$, ${}^{7}\textrm{Li}$ are not included
in the current analysis due to lack of statistics. To standardize the PID
conditions, the center and width of $\left<dE/dx\right>$ of the PID line for
each particle are fitted with the empirical Bethe-Bloch formula with Gaussian
widths. Only tracks having PID probability larger than $70\%$ ($50\%$ for
${}^{3}\textrm{He}$) and $2.2\sigma$ width of $\left<dE/dx\right>$ are
selected for the analysis.

The overall systematic uncertainties are estimated from the variations of
track multiplicity, track quality LeeJW_SpiRIT_NIMA_2020_965_163840_SpiRITROOT
and PID quality cuts TsangChunYuen_phdthesis_MichiganStateUniversity_2022 . In
the region $p_{T}/A<$ 400 MeV/c, the statistical errors are smaller than
systematic errors which are about 5% and 2% for the absolute yield and yield
ratios, respectively. Outside of this region the statistic and systematic
uncertainties increase with $p_{T}/A$ to exceed 15%. The error bars shown in
this work include both the systematic and statistical errors.

##  3 Particle Spectra

Figure 2: Differential multiplicity as a function of rapidity $y_{0}$ for $p$,
$d$, $t$, 3He, and 4He from top to bottom panels from the collisions of
${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ (black solid circles) and
${}^{108}\mathrm{Sn}+{}^{112}\mathrm{Sn}$ (red open squares) reactions. The
data points are compared to the AMD${}^{(\textrm{S})}$ (solid lines) on the
left panels and AMD${}^{(\textrm{F})}$ (dotted lines) on the right panels.
Figure 3: Differential multiplicity as a function of $p_{T}/A$ for $p$, $d$,
$t$, 3He, and 4He from top to bottom panels. Data points are compared to the
AMD${}^{(\textrm{S})}$ (solid lines) on the left panels and
AMD${}^{(\textrm{F})}$ (dotted lines) on the right panels for
${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ (black) and
${}^{108}\mathrm{Sn}+{}^{112}\mathrm{Sn}$ (red) reactions.

The experimental rapidity and transverse momentum spectra for $Z$=1 and 2
particles are shown in Fig. 2 and Fig. 3, respectively, for the neutron rich
system ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ $(N/Z=1.56)$ (black circles)
and the nearly symmetric system, ${}^{108}\mathrm{Sn}+{}^{112}\mathrm{Sn}$
$(N/Z=1.20)$ (red squares). The data in both left and right panels in each
figure are the same. Comparison of the data to the two different parameter
sets of the AMD models, shown by solid (left panels) and dashed (right panels)
lines, will be discussed in details later.

In all cases, the rapidity spectra show peaking near or at $y_{0}=0$
suggesting high degree of stopping. The slight asymmetry observed around
$y_{0}=0$ for all particles is due to the inefficiencies in the TPC to detect
target rapidity particles which are generally low in energy and emitted at
backward angles in the laboratory frame. In the case of ${}^{3}\mathrm{He}$,
the PID contamination from tritons is significant and the spectra peak is
located slightly off $y_{0}=0$. For the analysis, we assume the spectra is
symmetric at $y_{0}=0$ and only include data in $y_{0}=0$ \- $0.4$.

At mid rapidity, except for proton and ${}^{3}\mathrm{He}$ isotopes, more
particles, $d$, $t$ and ${}^{4}\mathrm{He}$ are produced in the neutron-rich
systems. The same observation is also seen in Fig. 3 for particles with
$p_{T}/A<280$ MeV/$c$. On the contrary, more high energy particles including
the neutron rich tritons, with $p_{T}/A>400$ MeV/$c$, are produced from the
nearly symmetric ${}^{108}\textrm{Sn}+{}^{112}\textrm{Sn}$ system than the
neutron-rich one. This is surprising as one would expect that neutron-rich
systems would produce more neutron-rich isotopes at all energies. One
explanation could be that high energy particles are produced in a dynamical
and non-equilibrium environment and that the hot participant zone is rather
neutron deficient.

##  4 Yield Ratios and Isoscaling

It has been shown that isoscaling occurs only when the two systems have nearly
the same temperature TsangMB_MSU_PRC_2001_64_041603_isoscaling_condition .
Since the absolute temperature cannot be directly measured, we use the
established isotope thermometers based on the double ratio of hydrogen and
helium species

$R_{\textrm{H}-\textrm{He}}=[Y(d)Y(^{4}\textrm{He})]/[Y(t)Y(^{3}\textrm{He})]$
(6)

to examine the relative temperature increase with particle energy in both
reactions:

$T_{\textrm{H}-\textrm{He}}\,\,\textrm{(MeV)}=\frac{14.29}{\log\left(1.59\,R_{\textrm{H}-\textrm{He}}\right)}\;.$
(7)

We note that the isotope temperature can be derived in the grand-canonical
ensemble AlbergoS_INCA_1985_89_1 and it is found to be independent of $N/Z$
ratio of the source in low energy collisions
KundeGJ_PLB_1998_416_56_HHeTemperature .

The upper panel of Fig. 4 displays the H-He temperature evaluated from Eq.
(7). The $T_{\textrm{H-He}}$ for both neutron rich (solid circles) and the
near-symmetric (open squares) systems are nearly the same for low energy
particles. The temperatures increases slowly from around 8 MeV to 10 MeV with
increasing $p_{T}/A$. Above 280 MeV/$c$, temperatures start to increase
dramatically. Furthermore, the temperatures of the two systems begin to
differ. Above 400 MeV/$c$ the temperature of the near-symmetric system is
higher than that of the neutron rich system. The increasing differences in
temperatures with $p_{T}/A$ indicates that the Sn+Sn collisions studied here
(with incident energy of 270 MeV/u) do not form fully equilibrated systems.

The increase in H-He temperature as a function of the surface velocity of the
particles, has been observed in Refs. WangJ_PRC_2005_72_024603 ;
BougaultR_JPG_47_025103 . In previous work, a drop of the temperature is also
observed at very high velocity. We do not see the drop. This could be a
consequence of the dramatic decrease of the cross-sections for $p_{T}/A$ above
280 MeV/$c$.

Figure 4: Upper panel: The H-He temperature evaluated using the isotope ratios
of Eq. 6 & 7. Lower panel: The isoscaling yield ratio $R_{21}$ as a function
of transverse momentum for different particles. The break line at $p_{T}/A$ =
280 MeV/$c$ show different trend of isoscaling for particles with low (left
side of the line) and high (right side of the line) $p_{T}/A$. The horizontal
solid lines for $p_{T}/A<$ 280 MeV/$c$ correspond to the average fitted values
of $R_{21}$ using Eq. 2. The dashed lines are used to guide the eye
highlighting the trends of the data for different particles.

Next, we focus on the spectral ratio of isotope yields $R_{21}(N,Z)$ of the
$Z$=1 and 2 particles, observed in the
${{}^{132}\textrm{Sn}}+{{}^{124}\textrm{Sn}}$ and
${{}^{108}\textrm{Sn}}+{{}^{112}\textrm{Sn}}$ systems. These are shown in the
lower panel of Fig. 4 as a function of $p_{T}/A$. The dotted lines connect
data points providing visual guidance to the trends of the data for each
particle. The vertical dashed line at $p_{T}/A=280$ MeV/$c$ marks the
approximate region when the temperatures of the two systems start to differ.

The isotope ratios plotted to the left of the dashed line show isoscaling
characteristics. In each ($N-Z=-1$, 0 and 1) group, $R_{21}(N,Z)$ values are
nearly constant as a function of $p_{T}/A$. Moreover, 3He behave like protons
($N-Z=-1$) with ratio value smaller than 1, and 4He behave similarly to
deuterons ($N-Z=0$) with higher ratio value than protons. The tritons
($N-Z=1$) have the highest ratio values as expected from isoscaling.

The $R_{21}$ ratios for the five particles with $p_{T}/A<$ 280 MeV/$c$ are
plotted as a function of $N$ and $Z$, in the left panels of Fig 5. The three
parameters $\alpha$, $\beta$ and $C$ are simultaneously fitted and the
resulting isoscaling fits are shown as lines in the figure. The two fitted
lines of the data for the isotopes with $Z$ = 1 and 2 are shown separately in
the top left panel, where the slopes of the lines represent the fit parameter
$\alpha$ = 0.29. The three fitted lines for the isotones with $N$ = 0, 1 and 2
are shown on the bottom left panel as a function of $Z$ and the slopes
represent the fit parameter $\beta$ = -0.23. In the absence of Coulomb,
protons and neutrons should behave similarly and one would expect $\alpha$ and
$\beta$ to have similar values but opposite signs. This has been observed in
most of the previous studies of isoscaling. In this case, the magnitude of the
$\alpha$ value is larger than the $\beta$ value. The isoscaling ratios
obtained from these fits are plotted as solid horizontal lines in the bottom
panel of Fig. 4 to the left of the vertical dashed line.

In contrast, except for protons, $R_{21}(N,Z)$ for high energy isotopes shown
on the right side of the vertical dashed line decreases with $p_{T}/A$.
Furthermore, the lines from different isotopes cross over each other. In the
case of $N=2$ particles (triton and 4He), $R_{21}$ falls off suddenly above
$p_{T}/A=$ 280 MeV/$c$ with the largest drop exhibited by tritons. This
reflects the sharper drop in the $t$ and 4He particle spectra at the high
energy. The right panels of Fig. 5 show the ratios $R_{21}$ plotted as a
function of $N$ and $Z$ in the range of $p_{T}/A\geq 400$ MeV/$c$. $R_{21}$
values for t, ${}^{3}\mathrm{He}$ and 4He have nearly the same values leading
to the breakdown of isoscaling. The lines connecting the isotope (upper right
panel) and isotone (lower right panel) data points serve only to guide the eye
and provide a contrast trends of the $R_{21}$ values between fragments with
low (left panels) and high (right panels) transverse momentum.

It is also interesting to note that $R_{21}$ is less than 1 for all high
energy isotopes. For isotopes with $p_{T}/A\approx 500$ MeV/$c$,
$R_{21}\approx$ 0.5 for t, 3He and 4He, i.e. 50% less tritons are produced
from the neutron-rich system of ${{}^{132}\textrm{Sn}}+{{}^{124}\textrm{Sn}}$
than that from the ${{}^{108}\textrm{Sn}}+{{}^{112}\textrm{Sn}}$ system. So
far there is no explanation for the surprising result.

Figure 5:  In each panel, $R_{21}$ values of Z=1 and 2 particles are plotted
as a function of $N$ and $Z$ in the top and bottom panels, respectively. In
the left panels, the lines are fits with Eq. (2) performed for particles with
$p_{T}/A<$ 280 MeV/$c$. The values of $\alpha$ and $\beta$ noted on the top
corners of the panels correspond to the slope of the fitted lines. In the
right panels, the data shown for $pT/A\geq$ 400 MeV/$c$ do not follow Eq. (2),
and the lines joining the data points are mainly used to guide the eyes.

##  5 Model Comparisons

The isoscaling ratios $R_{21}(N,Z)$ of the $Z$ = 1 and 2 particles, obtained
from the experimental yields measured in the
${{}^{132}\textrm{Sn}}+{{}^{124}\textrm{Sn}}$ and
${{}^{108}\textrm{Sn}}+{{}^{112}\textrm{Sn}}$ systems, are shown in Fig. 6. In
each panel, data are compared to the models; SMM (horizontal lines in the top
panel), AMD${}^{(\textrm{S})}$ and AMD${}^{(\textrm{F})}$ (hatched bands in
the middle and bottom pannels, respectively).

###  5.1 Statistical Multifragmentation Model

In order to check the extent to which the isoscaling properties observed
experimentally may be understood in a scenario in which a thermal equilibrated
source is formed and undergoes a prompt breakup, we employ the canonical
version of the SMM model described in Refs. SouzaSR_NPA_2019_989_69_ISMM ;
TanWP_PRC_2003_68_034609_ISMM . Many different sources contribute to the
actual data whereas a single source is employed in the calculation due to the
computational effort needed to generate a source distribution. The model
assumes a breakup volume three times larger than that of the source at normal
density, and breakup temperature $T=8$ MeV. The mass and atomic numbers of the
decaying source associated with the 132Sn+124Sn system are $N_{2}=93$ and
$Z_{2}=79$, whereas $N_{1}=71$ and $Z_{1}=71$ are used in the case of the
108Sn+112Sn system. These values have been selected in order to obtain a good
agreement with the measured $R_{21}$ ratios. Different source compositions
lead to slightly different $R_{21}$ values and, therefore, those adopted in
this work should be seen as average values. The predicted isotope ratios
$R_{21}$ are shown as horizontal solid lines in the top panel of Fig. 6.

Figure 6: Isotope yield ratios compared with the models: SMM (top),
AMD${}^{(\textrm{S})}$ (middle) and AMD${}^{(\textrm{F})}$ (bottom). The data
are the same in all panels. The break line at $p_{T}/A$ = 280 MeV/$c$ show
different trend of isoscaling for particles with low (left side of the line)
and high (right side of the line) $p_{T}/A$.

###  5.2 Asymmetrized Molecular Dynamics Model

In Ref. KanekoM_SpiRIT_PLB_2021_822_136681_Z1particles_AMD , the AMD model
OnoAkira_PTP_1992_87_1185_AMD ; OnoAkira_PPNP_2019_105_139_HIC_dynamics has
been employed to describe the rapidity distributions of $Z$ = 1 particles (top
six panels in Fig. 2). The time evolution of the system is calculated by AMD
until $t=300$ fm/$c$. The productions of light charged particles and their
properties are almost determined at this primary stage. The employed version
of AMD considers the process of cluster formation in the final state of every
two-nucleon collision as $N_{1}+N_{2}+B_{1}+B_{2}\to C_{1}+C_{2}$, where each
scattered nucleon $N_{i}$ ($i=1,2$) may form a cluster $C_{i}$ with a
surrounding particle(s) $B_{i}$. Clusters in the $(0s)^{A_{\text{c}}}$
configuration are considered for $1\leq A_{\text{c}}\leq 4$. A formed cluster
may be broken later, e.g., when a nucleon in the cluster is scattered by some
other particle. The important model parameters include the medium effects on
two-nucleon collision cross sections $\sigma_{NN}$ and cluster correlations. A
set of parameters was chosen in Ref.
OnoAkira_JPSCP_2020_32_010076_AMD_IMSigmaNN for a reasonable reproduction of
the FOPI data of central Xe + CsI collisions at 250 MeV/nucleon
ReisdorfW_FOPI_NPA_2010_848_366_HICsystematics ; AndronicA_PRC_2003_67_034907
, paying attention to the yields of light charged particles and heavier
fragments, and a kind of stopping observable of various particle species. (The
stopping observable is expressed as $\frac{1}{2}\langle
p_{T}^{2}\rangle/\langle p_{z}^{2}\rangle$ with transverse and longitudinal
momentum components, $p_{T}$ and $p_{z}$, in the center-of-mass frame.) This
parametrization of Ref. OnoAkira_JPSCP_2020_32_010076_AMD_IMSigmaNN is called
AMD${}^{(\textrm{F})}$ here.

For the present systems of Sn + Sn collisions, the rapidity distributions
predicted by AMD${}^{(\textrm{F})}$ are shown in the right panels of Fig. 2.
The calculated distribution looks more transparent than the data and
underestimates the yields of deuterons and especially that of tritons in the
mid-rapidity region. To reduce this discrepancy in the rapidity distributions
in Ref. KanekoM_SpiRIT_PLB_2021_822_136681_Z1particles_AMD , another
parametrization by adopting larger in-medium $NN$ cross sections $\sigma_{NN}$
is chosen, so that the rapidity distributions become much narrower as shown in
the left panels of Fig. 2. In addition, agreement with the triton multiplicity
was improved by modifying the phase space so that the bound phase space for
the relative coordinate between a two-nucleon pair and another nucleon becomes
approximately $(2\pi\hbar)^{3}$
KanekoM_SpiRIT_PLB_2021_822_136681_Z1particles_AMD . We label this parameter
set as AMD${}^{(\textrm{S})}$. A persistent observed problem is that the
$t/{}^{3}\mathrm{He}$ yield ratio is always underestimated, either by
AMD${}^{(\textrm{F})}$ or AMD${}^{(\textrm{S})}$.

In the present work, we adopt the Skyrme SLy4 effective interaction, which
corresponds to a soft symmetry energy with the slope parameter $L=46$ MeV. The
calculation of stiff symmetry energy with $L=108$ MeV was also analyzed in
Ref. KanekoM_SpiRIT_PLB_2021_822_136681_Z1particles_AMD . However, only the
soft symmetry energy case is shown here since the data seems to have a better
agreement with the soft symmetry energy case. The interaction with soft
symmetry energy overestimates the deuteron binding energy and consequently may
overestimate the deuteron yield PiantelliS_PRC_2019_99_064616 . However, such
an effect of the binding energy or the bound phase space is largely canceled
out when the double ratio is taken between two systems
KanekoM_SpiRIT_PLB_2021_822_136681_Z1particles_AMD .

In Fig. 3, the $p_{T}/A$ spectra predicted by the AMD${}^{(\textrm{S})}$ and
AMD${}^{(\textrm{F})}$ are depicted by the solid lines (left panels) and
dotted lines (right panels) respectively. For proton and deuteron, the overall
shapes and absolute yields are well explained with both AMD${}^{(\textrm{S})}$
and AMD${}^{(\textrm{F})}$. On the other hand, for clusters with $A\geq 3$,
the shapes of the $p_{T}/A$ spectra change significantly when the in-medium
cross sections $\sigma_{NN}$ are increased from AMD${}^{(\textrm{F})}$ to
AMD${}^{(\textrm{S})}$, and the spectra extend to high $p_{T}/A$ in
AMD${}^{(\textrm{S})}$ compared to the experimental data. The change from
AMD${}^{(\textrm{F})}$ to AMD${}^{(\textrm{S})}$ is understood as a natural
consequence of the increased $\sigma_{NN}$ to reduce $\langle
p_{z}^{2}\rangle$ and increase $\langle p_{T}^{2}\rangle$. To reproduce the
experimental data of both the rapidity and $p_{T}/A$ distributions, both
$\langle p_{z}^{2}\rangle$ and $\langle p_{T}^{2}\rangle$ of clusters need to
be reduced, in particular for tritons.This is not possible by only changing
$\sigma_{NN}$ in AMD. (The enhanced production of $Z=2$ isotope yields from
the AMD${}^{(\textrm{S})}$ parameterizations in Fig. 2 suggests that adjusting
the in-medium cluster correlations may also be needed.)

For the low energy particles with $p_{T}/A<280$ MeV/$c$, the behaviour of
$R_{21}$ is qualitatively explained by both AMD${}^{(\textrm{F})}$ and
AMD${}^{(\textrm{S})}$. The predicted $R_{21}$ values are shown as bands in
the middle and bottom panels of Fig. 6. The widths of the bands represent
statistical uncertainties. The predicted $R_{21}$ for all particles except
proton show slightly lower values. Similar to SMM, the $R_{21}$ of tritons
shows the largest difference in the model comparisons. Furthermore, the
$R_{21}$ ratios for tritons from AMD(S) underestimate the data more than the
results from AMD(F).

An emergence of isoscaling in the AMD calculation is not trivial, because AMD
does not assume any equilibrium state, and the fragments are produced within a
rapidly evolving system OnoAkira_SpiRIT_PRC_2003_68_051601_isoscaling_AMD .
However, during the dynamical evolution, clusters are repeatedly created and
broken by the microscopic processes explained above, and therefore a situation
similar to a chemical equilibrium may be realized before particles stop
interacting. The isoscaling observed in AMD is consistent with such a picture,
at least qualitatively. The precise values of $R_{21}$ may depend on the
details of the model ingredients. In the present calculation, all clusters
with the $(0s)^{A_{\text{c}}}$ configuration ($1\leq A_{\text{c}}\leq 4$) are
considered, including the dineutron and diproton correlations. When the
strength of dinucleon correlation is varied, a test calculation shows that the
composition of nucleons and clusters is affected. However, the steep drop of
$R_{21}$ of clusters at high $p_{T}$ down to $R_{21}<1$ in the experimental
data seems to be difficult to explain by a minor modification of the present
AMD model. Direct information on free neutrons can be an important clue to
solve this puzzle. It is available in the calculated results but is not shown
in the present paper.

##  6 Summary and Conclusion

In summary, the isoscaling phenomenon of hydrogen and helium isotopes in
${}^{132}\textrm{Sn}+^{124}\textrm{Sn}$ and
${}^{108}\textrm{Sn}+^{112}\textrm{Sn}$ reactions at beam energy of 270 MeV/u
is presented as a function of $p_{T}/A$. Isoscaling phenomenon up to
$p_{T}/A<280$ MeV/$c$ is found but breaks down for cluster particles with
$p_{T}/A>280$ MeV/$c$. The systems are found to form a thermal equilibrium not
throughout the system but locally, which is evident from the increasing trend
of H-He isotope ratio temperature with increasing $p_{T}/A$. The isoscaling
can be qualitatively explained by both the SMM and AMD models. When the yield
spectra and isoscaling are compared to the predictions from the dynamical
model AMD with two different parameter sets, we do not find any preference in
increasing the default values of $\sigma_{NN}$ as observed in earlier study of
Z=1 particles. While isoscaling breaks down for $p_{T}/A>280$ MeV/$c$
particles in the data, the isoscaling trend in AMD persist. Most intriguely,
the high-momentum clusters are suppressed in the neutron-rich system compared
to the more symmetric system suggesting the non-equilibrium nature of the
emission process especially for the high energy particles.

## Acknowledgement

The authors would like to thank Prof. Pawel Danielewicz for many fruitful
discussions. This work was supported by the U.S. Department of Energy, USA
under Grant Nos. DE-SC0021235, DE-NA0003908, DE-FG02-93ER40773, DE-
FG02-93ER40773, DE-SC0019209, DE-SC0015266, DE-AC02-05CH11231, U.S. National
Science Foundation Grant No. PHY-1565546, the Robert A. Welch Foundation
(A-1266 and A-1358), the Japanese MEXT, Japan KAKENHI (Grant-in-Aid for
Scientific Research on Innovative Areas) grant No. 24105004, JSPS KAKENHI
Grants Nos. JP17K05432, JP19K14709 and JP21K03528, the National Research
Foundation of Korea under grant Nos. 2018R1A5A1025563 and 2013M7A1A1075764,
the Polish National Science Center(NCN) under contract Nos.
UMO-2013/09/B/ST2/04064, UMO-2013/-10/M/ST2/00624, Computing resources were
provided by FRIB, the HOKUSAI-Great Wave system at RIKEN, and the Institute
for Cyber-Enabled Research at Michigan State University. S.R. Souza
acknowledges partial support from CNPq, CAPES, FAPERJ and the use of the
supercomputer Lobo Carneiro, where part of the calculations have been carried
out. This work has been done as part of the project INCT-Física Nuclear e
aplicações, projeto No. 464898/2014-5.

## Data Availability Statement

This manuscript has no associated data or the data will not be deposited.
[Authors’ comment: The data can be available on request sent to the
corresponding author.]

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