Stopping effects in U+U collisions with a beam energy of 520 MeV/nucleon
Xiao-Feng Luo,1, ∗ Xin Dong,1 Ming Shao,1 Ke-Jun Wu,2 Cheng Li,1
Hong-Fang Chen,1 and Hu-Shan Xu3
University of Science and Technology of China, Hefei, Anhui 230026, China
Institute of Particle Physics, Hua-Zhong Normal University, Wuhan, Hubei 430079, China
Institute of Modern Physics, Chinese Academy of Sciences, LanZhou, Gansu 730000, China
(Dated: November 4, 2018)
A Relativistic Transport Model (ART1.0) is applied to simulate the stopping effects in tip-tip
and body-body U+U collisions, at a beam kinetic energy of 520 MeV/nucleon. Our simulation
results have demonstrated that both central collisions of the two extreme orientations can achieve
full stopping, and also form a bulk of hot, dense nuclear matter with a sufficiently large volume
and long duration, due to the largely deformed uranium nuclei. The nucleon sideward flow in
the tip-tip collisions is nearly 3 times larger than that in body-body ones at normalized impact
parameter b/bmax < 0.5, and that the body-body central collisions have a largest negative nucleon
elliptic flow v2 = −12% in contrast to zero in tip-tip ones. Thus the extreme circumstance and the
novel experimental observables in tip-tip and body-body collisions can provide a good condition and
sensitive probe to study the nuclear EoS, respectively. The Cooling Storage Ring (CSR) External
Target Facility (ETF) to be built at Lanzhou, China, delivering the uranium beam up to 520
MeV/nucleon is expected to make significant contribution to explore the nuclear equation of state
(EoS).
PACS numbers: 24.10.Lx,25.75.Ld,25.75.Nq,24.85.+p
I. INTRODUCTION
In recent years, the ultra-relativistic high energy heavy
ion collisions performed at SPS/CERN and RHIC/BNL
sNN ∼ 10 − 200 GeV) focus on high temperature
and low baryon density region in nuclear matter phase
diagram [1] to search a new form of matter with par-
tonic degree of freedom-the quark-gluon plasma (QGP)
[2, 3, 4, 5]. However, no dramatic changes of experimen-
tal observables, such as jet-quenching, elliptic flow and
strangeness enhancement, have been observed yet [6]. On
the other hand, the heavy ion collisions performed at the
BEVALAC/LBNL and SIS/GSI [7, 8] in last two decades
were used to produce hot and compressed nuclear mat-
ter to learn more about the nuclear equation of state
(EoS) [13, 14] at high baryon density and low temper-
ature region of the phase diagram. Although we have
made great efforts to study the nuclear EoS, theoreti-
cally and experimentally, a solid conclusion can hardly
be made. Then, it is still worthwhile to systematically
study on the collision dynamics as well as the EoS observ-
ables. Recently, for more understanding of the nuclear
matter phase diagram and EoS at high net-baryon den-
sity region, it is proposed to collide uranium on uranium
target at External Target Facility (ETF) of Cooling Stor-
age Ring (CSR) at Lanzhou, China with a beam kinetic
energy of 520 MeV/nucleon. [10].
Uranium is the largest deformed stable nucleus, and
has approximately an ellipsoid shape with the long and
short semi-axis given by Rl = R0(1 + 2δ/3) and Rs =
∗contact author: science@mail.ustc.edu.cn
FIG. 1: (Color online) (a) body-body collisions (b) tip-tip
collisions
R0(1 − δ/3), respectively, where R0 = 7 fm is the effec-
tive spherical radius and δ = 0.27 is the deformation pa-
rameter [9]. Consequently, one has Rl/Rs = 1.3. In our
simulation, we consider two extreme orientations: the so-
called tip-tip and body-body patterns with the long and
short axes of two nuclei are aligned to the beam direc-
tion, respectively [12], see Fig. 1 for illustration. The
two types of orientations can be identified in random ori-
entations of U+U collisions by making proper cutoffs in
experimental data, such as the particle multiplicities, el-
liptic flow and so on [10, 11]. With the two extreme
collision orientations, some novel stopping effects which
are believed responsible for some significant experimental
observables, such as particle production, collective mo-
tion as well as attainable central densities, can be ob-
tained. Due to the large deformation of the uranium
nuclei [11, 12] , it is expected that the tip-tip collisions
can form a higher densities nuclear matter with longer
duration than in body-body or the spherical nuclei colli-
sions, which is considered to be a powerful tool to study
http://arxiv.org/abs/0704.0424v2
the nuclear matter phase transition at high baryon den-
sity [12], and the body-body central collisions may reveal
a largest out-of-plane elliptic flow (negative v2) at high
densities, which can be a sensitive probe to extract the
early EoS of the hot, dense nuclear matter [12, 17]. The
novel experimental observables can be effectively utilized
to study the possible nuclear matter phase transition and
the nuclear EoS [12, 13, 14, 15, 16, 17, 18, 19, 20]. For
comparing with tip-tip and body-body collisions, a type
of gedanken ”sphere-sphere” collisions without deforma-
tions of uranium nuclei are also included in the simula-
tion.
The ART1.0 model [21, 22] derived from Boltzmann-
Uehling-Uhlenbeck (BUU) model [23] has a better treat-
ment of mean field and Pauli-Blocking effects [23] than
cascade models [24]. The fragments production mech-
anism and partonic degree of freedom are not present
in the ART1.0 model. A soft EoS with compressibility
coefficient K = 200 MeV is used throughout the simu-
lation and the beam kinetic energy of uranium nuclei is
set to 520 MeV/nucleon if not specifically indicated. In
the next section, we discuss about the stopping power
ratio and selection of impact parameter b. In Sec. 3,
the evolution of baryon and energy densities as well as
thermalization of central collision systems are studied. In
Sec. 4, some experimental observables, such as nucleon
sideward flow and elliptic flow are also investigated. We
summarize our results in Sec. 5.
II. STOPPING POWER OF TIP-TIP AND
BODY-BODY COLLISIONS
Large stopping power can lead to remarkable pressure
gradient in the compressed dense matter. It is generally
also considered to be responsible for transverse collec-
tive motion [25], the maximum attainable baryon and
energy densities as well as thermalization of collision sys-
tems. Thus, the study of the stopping power in U+U
collisions may provide important information for under-
standing the nuclear EoS and collision dynamics.
A. Selection of impact parameter
The nuclear stopping power and geometric effects in
U+U collisions rely strongly on the impact parameter
b. Considering the conceptual design of the CSR-ETF
detector [10], two methods are invoked here to estimate
the impact parameter. The first one is the multiplicity
of forward neutrons with polar angle θ < 20o in the lab
frame which can be covered by a forward neutron wall.
The other method is to make use of the parameter Erat
[26], which is the ratio of the total transverse kinetic en-
ergy to the total longitudinal one. The particles are also
required to be within θ < 20o in the lab frame, while
the two qualities are calculated within the center of mass
0 0.2 0.4 0.6 0.8 1
tip-tip
body-body
0 0.2 0.4 0.6 0.8 10
maxb/b
FIG. 2: Upper: Forward neutron multiplicity and Lower:
Erat, as a function of normalized impact parameter b/bmax in
both tip-tip and body-body collisions.
system (c.m.s.).
Erat =
Ezi (1)
The normalized impact parameter b/bmax is used to rep-
resent centralities of tip-tip and body-body collisions and
the bmax of the two cases are quite different from each
other. As shown in Fig. 2, with either method, obvious
linear dependence of the normalized impact parameter
are demonstrated in both tip-tip and body-body near
central collisions. Then, the two methods can be com-
bined to determine the impact parameter to identify the
most central collision events in both tip-tip and body-
body collisions.
B. Stopping power ratio definition and evolution
It is difficult to obtain a universally accepted estimate
of the nuclear stopping power in heavy ion collisions due
to a proliferation of definitions of the concept [27]. The
stopping power ratio R [28] is employed to measure the
degree of stopping and defined as:
|Ptj |/
|Pzj | (2)
, the total nucleon transverse momentum |Ptj | divided by
the total absolute value of nucleon longitudinal momen-
tum |Pzj | in the c.m.s.. The ratio is wildly used to de-
scribe the degree of thermalization and nuclear stopping
by low and intermediate energies heavy ion collisions.
It’s a multi-particle observable on an event-by-event ba-
sis, which for an isotropic distribution is unity.
Fig. 3 shows the time and normalized impact param-
eter dependence of the stopping ratio R for three con-
ditions: tip-tip, body-body and sphere-sphere collisions.
0 10 20 30 40
=0max(a)b/b
tip-tip
body-body
sphere-sphere
0.2 0.4 0.6 0.8 1
(b)Minibias
tip-tip
body-body
sphere-sphere
t(fm/c) maxb/b
FIG. 3: (Color online) (a)The time evolution of the stopping
ratio R in tip-tip, body-body and sphere-sphere central colli-
sions, and (b) the stopping ratio R as a function of b/bmax in
minimum biased collisions.
When the ratio R reaches the value of 1, full stopping of
the collision system is considered to be achieved, and the
momenta is also isotropy, which are not sufficient but nec-
essary for thermal equilibrium of collision systems [28].
For R > 1, it can be explained by preponderance of mo-
mentum flow perpendicular to the beam direction [29]. It
is shown that all of the three conditions can achieve full
stopping when the stopping ratio R=1, the correspond-
ing time for body-body and tip-tip central collisions are
about 15 fm/c and 25 fm/c, respectively. Larger stopping
ratio and faster evolution to full stopping are observed
for body-body central collisions than tip-tip and sphere-
sphere ones at the early stage, which may indicate a more
violent colliding process for body-body central collisions
due to the sizable initial transverse overlap region. Al-
though the stopping ratio of tip-tip central collisions is
lowest than the other two cases at the early time, it raises
sharply later and even exceeding one. So, it means that
longer reaction and passage time can be obtained in tip-
tip central collisions than body-body and sphere-sphere
ones, which may indicate the nucleons in tip-tip colli-
sions can undergo more binary collisions to reach higher
transverse momentum.
In Fig. 3(b), the R of the three conditions are gradu-
ally decrease with the increase of the normalized impact
parameter. When b/bmax < 0.5, the ratio is always larger
for tip-tip collisions than the other two cases, while for
b/bmax > 0.5 all of the three conditions almost have the
same stopping power ratio.
III. BARYON, ENERGY DENSITY AND
THERMAL EQUILIBRIUM
Considering the discrepancy of stopping power be-
tween tip-tip and body-body collisions, it is interesting
4 (a)Baryon Density 
tip-tip
body-body
Au-Au
0 10 20 30 40 50
(b)Energy Density
tip-tip
body-body
Au-Au
t(fm/c)
FIG. 4: The evolution of (a) baryon and (b) energy densities
in tip-tip, body-body and Au+Au central collisions.
to study further about the baryon and energy densities
evolution in both cases. As the full stopping and de-
formation effects in U+U collisions, it is believed higher
local baryon and energy densities system with long du-
ration can be created, which is considered to be a signif-
icant condition to study the nuclear EoS at high bayonic
density region.
A. The evolution of baryon and energy densities
The evolution of baryon and energy densities in the
central zone of tip-tip and body-body as well as Au+Au
central collisions are illustrated in Fig. 4.
In Fig. 4, it is observed the maximum attainable
baryon and energy densities for both tip-tip and body-
body central collisions are about 3.2 ρ0 and 0.8 GeV/fm
respectively, while the Au+Au one are about 2.6 ρ0 and
0.6 GeV/fm3. Both the baryon and energy densities in
U+U collisions are higher than the Au+Au one. Once a
baryon density threshold of ρ > 2.5 ρ0 is required, the
corresponding duration in tip-tip central collisions ∼ 20
fm/c (from ∼ 8 fm/c to ∼ 28 fm/c) is longer than ∼
10 fm/c ( from ∼ 8 fm/c to ∼ 18 fm/c ) of body-body
one, which is as predicted. But the peak densities have
no significant discrepancy between the two cases unlike
those at the energy region of the Alternating Gradient
Synchrotron (AGS) [12], which may be attribute to the
full stopping at the CSR energy.
B. Thermalization of the U+U collision systems
As mentioned before, the stopping ratio R = 1 is a
necessary but not sufficient condition for thermal equi-
librium of the collision system. In order to approach a
thermal equilibrium, a long duration of reaction is needed
for nucleons to undergo sufficient binary collisions. As
0 10 20 30 40 50
tip-tip
body-body
Au-Au
0 10 20 30 40 50
tip-tip
body-body
Au-Au
t(fm/c)
FIG. 5: The evolution of (a) volume with high density (ρ >
2.5ρ0) in tip-tip, body-body and Au+Au central collisions,
and (b) the scaled mean kinetic energy 2
< Ek >, within a
sphere of radius 2fm around the system mass center.
shown in Fig. 4(a), obvious long duration has been ob-
tained in both tip-tip and body-body central collisions.
It is therefore possible thermal equilibrium at the time
of freeze-out can be achieved.
The Fig. 5(a) is the evolution of the volume with the
high baryon density(ρ > 2.5 ρ0) for tip-tip, body-body
and Au+Au central collisions, respectively. Both tip-
tip and body-body central collisions have larger volumes
than Au+Au one at the same beam kinetic energy 520
MeV/nucleon. Although the maximum volume attain-
able for body-body central collisions(∼ 220 fm3) is about
two times larger than tip-tip one(∼ 120 fm3), the peak
volume of tip-tip central collisions lasts a much longer
time of ∼ 10 fm/c (from ∼ 15 fm/c to ∼ 25 fm/c) and
much more stable than body-body one. To estimate the
temperature at the freeze-out time, the scaled mean ki-
netic energy of all hadrons in a sphere of radius 2fm
around the system mass center is calculated as 2
< Ek >
[22], which is utilized to reflect the thermalization tem-
perature T of the collision system approximately. As
illustrated in Fig. 5 (b), both tip-tip and body-body
central collisions show a flat region about 75 MeV and
the corresponding time range are about 10 fm/c to 28
fm/c and 10 fm/c to 18 fm/c, respectively. Considering
the time range of the flat region in Fig. 5 (b) associat-
ing with the corresponding range in Fig. 5 (a) and also
looking back to Fig. 4, we obtain a large volume of hot,
dense nuclear matter in both tip-tip and body-body cen-
tral collisions. Consequently, the extreme circumstance
of sufficiently high temperature and density for a signif-
icant large volume and long duration [12, 22] has been
formed in tip-tip and body-body central collisions, which
can provide a good opportunity to study the nuclear EoS
as well as particles in medium properties, especially for
tip-tip case.
The time of freeze out should be cautiously determined
(a)tip-tip π+∆+*N
0 10 20 30 40 50 60
80 (b)Body-Body π+∆+*N
t(fm/c)
FIG. 6: Evolution of the multiplicity of the free pion, N∗ +
∆, N∗ +∆+ π in (a)tip-tip ,and (b)body-body central colli-
sions.
for estimating the thermalization temperature of colli-
sion system. In Fig. 6 the multiplicity evolution of free
pion which are not bounded in baryon resonances and
pion still bounded inside the excited baryon resonances
(∆, N∗) (unborn pion) are displayed. At the Lanzhou
CSR energy region (520 MeV/nucleon), the production
and destruction of the ∆ resonances are mainly through
NN ⇋ N∆ and ∆ → Nπ reactions in which the ∆ decay
rate is always higher than that of the formation of this
resonance and the production of pion is predominated by
the decay of the ∆ resonances (∆ → Nπ) [30]. The total
multiplicity of pion, ∆ and N∗ approaches a saturated
level after a period of evolution, indicating the freeze-
out time about t=28 fm/c and t=18 fm/c for tip-tip
and body-body central collisions, respectively. The larger
maximum attainable total multiplicity of pion, ∆ andN∗
and freeze out earlier indicates a existent of faster evo-
lution and more violently reaction process for the body-
body central collisions than tip-tip case consisting with
the discussing before.
The corresponding temperature about 75 MeV at
freeze-out time can be extracted from the Fig. 5 (b), for
both tip-tip and body-body central collisions. To further
confirm this estimation, both the energy spectrum of the
nucleon and negative-charged pion are studied within the
polar angle range of 900±100 in the c.m.s.. The thermo-
dynamic model [31] predicts that the energy spectra will
be represented by a temperature T which characterizes a
Maxwell-Boltzmann gas
PEdEdΩ
= const× e−Ekin/T (3)
, where P and E are the particle momentum and total
energy in the c.m.s.. Both the energy spectra and the
Boltzmann fit results are shown in Fig. 7. The inverse
slope (e.g. temperature T ) of the nucleons in tip-tip and
body-body central collisions are about 73 MeV and 70
0 0.2 0.4 0.6 0.8
310  
tip-tip
body-body
(a)Nucleon
0 0.2 0.4 0.6 0.8
-π(b)
(GeV)kinE
FIG. 7: (a) Nucleon, and (b) negative-charged pion energy
spectrum at 900 ± 100 in the c.m.s. together with a Maxwell-
Boltzmann fit for both tip-tip and body-body central colli-
sions. The nucleon fit temperature for tip-tip and body-body
are about 73 MeV and 70 MeV, respectively and that of pion
are about 56 MeV and 52 MeV, respectively.
MeV, respectively, which are in good agreement with the
temperature extracted from the Fig. 5(b) at the freeze
out time. The spectra of negative-charged pion show a
different lower temperature than that of nucleon which
may be explained by considering an equilibrated N and
∆ system at thermal freeze out and taking into account
the kinematics of ∆ decay [32]. The nucleon temperature
closely reflects the freeze-out temperature of tip-tip and
body-body central collisions.
In conclusion, thermalization (or near thermalization)
of the collision system corresponding a freeze-out tem-
perature about 75 MeV is likely to be achieved in both
tip-tip and body-body central collisions. However, it’s
also possible that the collision system is still in a non-
equilibrium transport process on its path towards kinetic
equilibration [30].
IV. THE COLLECTIVE FLOW OF U+U
COLLISIONS
Stopping of nuclei in heavy ion collision can lead
to pressure gradient along different directions, result-
ing in collective motion as spectators bounce-off [34]
and participants squeeze-out effects [35]. Since last
two decades, at Bevalac/LBNL and SIS/GSI energies
the so-called ”collective flow” analysis has been estab-
lished [15, 34, 35, 36, 37] to study the collective mo-
tion of the products in heavy ion collisions. The collec-
tive flow resulting from bounce-off and squeeze-out ef-
fects, which can be explained well by the hydrodynamics
model [34, 38], and also be in good agreement with the
experimental data has been observed [39, 40]. Because of
the large deformation of the uranium nuclei, a novel col-
-1 -0.5 0 0.5 1
Soft:tip-tip
Soft:body-body
Cascade:tip-tip
Cascade:body-body
(a)b/b
-1 -0.5 0 0.5 1
(b)b/b
FIG. 8: The mean transverse momentum per nucleon pro-
jected into the reaction plane, < px/A >, as a function
of c.m.s. normalized rapidity is illustrated for tip-tip and
body-body collisions. With normalized impact parameter
cutoff:(a)b/bmax <= 0.5 (b)b/bmax > 0.5.
lective motion is expected [12], to be used to extract the
medium properties and nuclear matter EoS information.
[15, 16, 17, 18, 19, 20].
To perform flow analysis, it is necessary to construct a
imaginary reaction plane defined by direction of the beam
(z) and the impact parameter vector b [43, 45, 46]. In our
simulation, the x− z plane is just defined as the reaction
plane with the beam direction along z positive direction
and the impact parameter vector b along x positive direc-
tion. In last two decades, there are mainly two methods
to study the collective flow at the low and intermediate
energies. One is the sphericity method [28, 34, 41, 42]
which yields the flow angle relative to the beam axis of
the major axis of the best-fit kinetic energy ellipsoid, and
the other is to employ the mean transverse momentum
per nucleon projected into the reaction plane, < px/A >,
to perform nucleon sideward flow analysis [43, 44] which
reflects the spectator bounce-off effects in the reaction
plane. In recent years, it is usual to use an anisotropic
transverse flow analysis method. With a Fourier expan-
sion [47, 48] of the particle azimuthal angle φ distribu-
tion with respect to the reaction plane, different har-
monic coefficients can be extracted, among which the
first harmonic coefficient v1, called directed flow (simi-
lar to sideward flow) and the second harmonic coefficient
v2, called elliptic flow are mostly interested. The ellip-
tic flow reflects the anisotropy of emission particles in
the plane perpendicular to the reaction plane while the
directed flow describes the anisotropy in reaction plane.
The Fourier expansion can be expressed as
∼ 1 +
2vncos(nφ) (4)
Fig. 8 shows nucleon sideward flow, < px/A >, for
both tip-tip and body-body minimum biased collisions
(a)Nucleon Flow Parameter
tip-tip
body-body
Au-Au(500MeV/A)
0 0.2 0.4 0.6 0.8 1
10 2(b)Nucleon v
tip-tip
body-body
Au-Au(500MeV/A)
maxb/b
FIG. 9: (a)The nucleon flow parameter F and (b)the c.m.s.
mid-rapidity ( −0.5 < y0 < 0.5 ), nucleon elliptic flow v2 of
three collision conditions as a function of normalized impact
parameter b/bmax with soft EoS.
as a function of normalized rapidity, y(0) = Ycm/ycm, in
which Ycm represents the particle rapidity in c.m.s. and
ycm is the rapidity of the system mass center. To ex-
tract the nuclear EoS information and also demonstrate
the discrepancies of the nucleon sideward flow in tip-tip
and body-body collisions, the cascade events [49], which
neglect the mean field and pauli blocking effects are em-
ployed here to compare with the soft EoS case. In Fig.
8(a),(b), with a soft EoS, it is noted that either tip-tip
or body-body collisions show a spectator bounce-off ef-
fect revealing an obvious ”S” shape [15, 49] at the mid-
rapidity region of −0.5 < y0 < 0.5, while the cascade one
appear a almost vanishing nucleon sideward flow. It can
be understand by the nucleon sideward flow is related
to the mean field, which is mainly responsible for the
pressure gradient of the stopping nuclei, while the mean
field has a strong dependence of the nuclear EoS. There-
fore, the nucleon sideward flow is thought to be a good
indirect probe to extract the nuclear EoS information,
especially tip-tip case for its largely remarkable sideward
flow. A cutoff on normalized impact parameter is also
applied to explore the impact parameter dependence of
nucleon sideward flow. As shown in Fig. 8(b), when
b/bmax > 0.5 the curves of soft EoS and cascade are al-
most superposed with each other, while for b/bmax < 0.5
large discrepancy is observed. The situation is quite sim-
ilar to Fig. 3 (b), almost the same stopping power for
b/bmax > 0.5 and large discrepancy for b/bmax < 0.5 in
tip-tip and body-body minimum biased collisions, which
means there exists a correlation between nuclear stopping
power and sideward flow [33].
The normalized impact parameter dependence of the
collective flow of nucleon is further studied, by analyzing
the ”flow parameter” F [49] and also elliptic flow v2 for
both tip-tip and body-body as well as Au+Au minimum
biased collisions. The flow parameter F is a customarily
used quality to describe the nucleon sideward flow quan-
titatively defined as
d < px/A >
dy(0)
y(0)=0
the slope of the mean transverse momentum per nucleon
projected into the reaction plane at y(0) = 0.
In Fig. 9(a), with b/bmax > 0.5, the nucleon flow pa-
rameter F of tip-tip and body-body collisions are with
similar value. This similarity, along with the almost same
stopping ratioR in Fig. 3(b), indicates a existence of sim-
ilar pressure gradient effects on nucleon sideward flow in
the two collision orientations. While for b/bmax < 0.5,
the flow parameter F of tip-tip collisions is nearly 3 times
larger than that of body-body case. Even the sideward
flow of Au+Au collisions is larger than the body-body
one. It is further confirmed the tip-tip nucleon sideward
flow is a more sensitive probe to extract the information
of nuclear EoS than that of body-body one. The promi-
nence high of the nucleon sideward flow in tip-tip colli-
sions may be resulted from the stronger pressure gradient
between the participants and spectators in the reaction
plane than body-body one, due to the largely deformed
nuclei.
The normalized impact parameter dependent of nu-
cleon elliptic flow v2 at the mid-rapidity region ( −0.5 <
y0 < 0.5 ) is displayed in Fig. 9(b). A significant neg-
ative elliptic flow v2 at this energy region is consistent
with the excitation function of the elliptic flow studied
before [50]. An largest negative v2 about −12% in body-
body central collisions is observed which reflects the large
geometric and squeeze-out effects in the collisions. While
for tip-tip and Au+Au ones the maximum negative v2 are
obtained at mid-centrality. Since both high baryon, en-
ergy densities and large elliptic flow effects, which reflects
an early EoS of the hot dense compression nuclear matter
[17], are available in body-body central collisions. Thus
the body-body nucleon elliptic flow can also be taken as
a sensitive probe of nuclear EoS. The novel behaviors
of nucleon collective flow in tip-tip and body-body col-
lisions are mainly attributed to the large deformation of
the uranium nuclei.
V. SUMMARY
In summary, the CSR-ETF at Lanzhou provide a good
opportunity to systematically study the nuclear EoS at
the high net-baryon density region of nuclear matter
phase diagram. Due to the novel stopping effects in
largely deformed U+U collisions, the simulation based on
ART1.0 demonstrates that full stopping can be achieved
and also a bulk of hot, high densities nuclear matter with
large volume and long duration have been formed in both
tip-tip and body-body collisions. Large nucleon sideward
flow in tip-tip collisions and the significant negative nu-
cleon elliptic flow in body-body central collisions can pro-
vide a sensitive probe to extract nuclear EoS information.
Thus the extreme circumstance and the novel collective
flow in both tip-tip and body-body collisions can provide
a good condition and sensitive probe to study the nu-
clear EoS, respectively. More experimental observables
of U+U collision dynamics should be further studied, due
to the geometric effects.
VI. ACKNOWLEDGEMENT
This work is supported by National Natural Sci-
ence Foundation of China (10575101,10675111) and the
CAS/SAFEA International Partnership Program for
Creative Research Teams under the grant number of
CXTD-J2005-1. We wish to thank Bao-an Li, Feng Liu,
Qun Wang, Zhi-Gang Xiao and Nu Xu for their valuable
comments and suggestions.
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http://arxiv.org/abs/nucl-th/0512009
http://arxiv.org/abs/nucl-ex/9904003

  A Relativistic Transport Model (ART1.0) is applied to simulate the stopping
effects in tip-tip and body-body U+U collisions, at a beam kinetic energy of
520 MeV/nucleon. Our simulation results have demonstrated that both central
collisions of the two extreme orientations can achieve full stopping, and also
form a bulk of hot, dense nuclear matter with a sufficiently large volume and
long duration, due to the largely deformed uranium nuclei. The nucleon sideward
flow in the tip-tip collisions is nearly 3 times larger than that in body-body
ones at normalized impact parameter $b/b_{max}<0.5$, and that the body-body
central collisions have a largest negative nucleon elliptic flow $v_{2}=-12%$
in contrast to zero in tip-tip ones. Thus the extreme circumstance and the
novel experimental observables in tip-tip and body-body collisions can provide
a good condition and sensitive probe to study the nuclear EoS, respectively.
The Cooling Storage Ring (CSR) External Target Facility (ETF) to be built at
Lanzhou, China, delivering the uranium beam up to 520 MeV/nucleon is expected
to make significant contribution to explore the nuclear equation of state
(EoS).

Stopping effects in U+U collisions with a beam energy of 520 MeV/nucleon
Xiao-Feng Luo,1, ∗ Xin Dong,1 Ming Shao,1 Ke-Jun Wu,2 Cheng Li,1
Hong-Fang Chen,1 and Hu-Shan Xu3
University of Science and Technology of China, Hefei, Anhui 230026, China
Institute of Particle Physics, Hua-Zhong Normal University, Wuhan, Hubei 430079, China
Institute of Modern Physics, Chinese Academy of Sciences, LanZhou, Gansu 730000, China
(Dated: November 4, 2018)
A Relativistic Transport Model (ART1.0) is applied to simulate the stopping effects in tip-tip
and body-body U+U collisions, at a beam kinetic energy of 520 MeV/nucleon. Our simulation
results have demonstrated that both central collisions of the two extreme orientations can achieve
full stopping, and also form a bulk of hot, dense nuclear matter with a sufficiently large volume
and long duration, due to the largely deformed uranium nuclei. The nucleon sideward flow in
the tip-tip collisions is nearly 3 times larger than that in body-body ones at normalized impact
parameter b/bmax < 0.5, and that the body-body central collisions have a largest negative nucleon
elliptic flow v2 = −12% in contrast to zero in tip-tip ones. Thus the extreme circumstance and the
novel experimental observables in tip-tip and body-body collisions can provide a good condition and
sensitive probe to study the nuclear EoS, respectively. The Cooling Storage Ring (CSR) External
Target Facility (ETF) to be built at Lanzhou, China, delivering the uranium beam up to 520
MeV/nucleon is expected to make significant contribution to explore the nuclear equation of state
(EoS).
PACS numbers: 24.10.Lx,25.75.Ld,25.75.Nq,24.85.+p
I. INTRODUCTION
In recent years, the ultra-relativistic high energy heavy
ion collisions performed at SPS/CERN and RHIC/BNL
sNN ∼ 10 − 200 GeV) focus on high temperature
and low baryon density region in nuclear matter phase
diagram [1] to search a new form of matter with par-
tonic degree of freedom-the quark-gluon plasma (QGP)
[2, 3, 4, 5]. However, no dramatic changes of experimen-
tal observables, such as jet-quenching, elliptic flow and
strangeness enhancement, have been observed yet [6]. On
the other hand, the heavy ion collisions performed at the
BEVALAC/LBNL and SIS/GSI [7, 8] in last two decades
were used to produce hot and compressed nuclear mat-
ter to learn more about the nuclear equation of state
(EoS) [13, 14] at high baryon density and low temper-
ature region of the phase diagram. Although we have
made great efforts to study the nuclear EoS, theoreti-
cally and experimentally, a solid conclusion can hardly
be made. Then, it is still worthwhile to systematically
study on the collision dynamics as well as the EoS observ-
ables. Recently, for more understanding of the nuclear
matter phase diagram and EoS at high net-baryon den-
sity region, it is proposed to collide uranium on uranium
target at External Target Facility (ETF) of Cooling Stor-
age Ring (CSR) at Lanzhou, China with a beam kinetic
energy of 520 MeV/nucleon. [10].
Uranium is the largest deformed stable nucleus, and
has approximately an ellipsoid shape with the long and
short semi-axis given by Rl = R0(1 + 2δ/3) and Rs =
∗contact author: science@mail.ustc.edu.cn
FIG. 1: (Color online) (a) body-body collisions (b) tip-tip
collisions
R0(1 − δ/3), respectively, where R0 = 7 fm is the effec-
tive spherical radius and δ = 0.27 is the deformation pa-
rameter [9]. Consequently, one has Rl/Rs = 1.3. In our
simulation, we consider two extreme orientations: the so-
called tip-tip and body-body patterns with the long and
short axes of two nuclei are aligned to the beam direc-
tion, respectively [12], see Fig. 1 for illustration. The
two types of orientations can be identified in random ori-
entations of U+U collisions by making proper cutoffs in
experimental data, such as the particle multiplicities, el-
liptic flow and so on [10, 11]. With the two extreme
collision orientations, some novel stopping effects which
are believed responsible for some significant experimental
observables, such as particle production, collective mo-
tion as well as attainable central densities, can be ob-
tained. Due to the large deformation of the uranium
nuclei [11, 12] , it is expected that the tip-tip collisions
can form a higher densities nuclear matter with longer
duration than in body-body or the spherical nuclei colli-
sions, which is considered to be a powerful tool to study
http://arxiv.org/abs/0704.0424v2
the nuclear matter phase transition at high baryon den-
sity [12], and the body-body central collisions may reveal
a largest out-of-plane elliptic flow (negative v2) at high
densities, which can be a sensitive probe to extract the
early EoS of the hot, dense nuclear matter [12, 17]. The
novel experimental observables can be effectively utilized
to study the possible nuclear matter phase transition and
the nuclear EoS [12, 13, 14, 15, 16, 17, 18, 19, 20]. For
comparing with tip-tip and body-body collisions, a type
of gedanken ”sphere-sphere” collisions without deforma-
tions of uranium nuclei are also included in the simula-
tion.
The ART1.0 model [21, 22] derived from Boltzmann-
Uehling-Uhlenbeck (BUU) model [23] has a better treat-
ment of mean field and Pauli-Blocking effects [23] than
cascade models [24]. The fragments production mech-
anism and partonic degree of freedom are not present
in the ART1.0 model. A soft EoS with compressibility
coefficient K = 200 MeV is used throughout the simu-
lation and the beam kinetic energy of uranium nuclei is
set to 520 MeV/nucleon if not specifically indicated. In
the next section, we discuss about the stopping power
ratio and selection of impact parameter b. In Sec. 3,
the evolution of baryon and energy densities as well as
thermalization of central collision systems are studied. In
Sec. 4, some experimental observables, such as nucleon
sideward flow and elliptic flow are also investigated. We
summarize our results in Sec. 5.
II. STOPPING POWER OF TIP-TIP AND
BODY-BODY COLLISIONS
Large stopping power can lead to remarkable pressure
gradient in the compressed dense matter. It is generally
also considered to be responsible for transverse collec-
tive motion [25], the maximum attainable baryon and
energy densities as well as thermalization of collision sys-
tems. Thus, the study of the stopping power in U+U
collisions may provide important information for under-
standing the nuclear EoS and collision dynamics.
A. Selection of impact parameter
The nuclear stopping power and geometric effects in
U+U collisions rely strongly on the impact parameter
b. Considering the conceptual design of the CSR-ETF
detector [10], two methods are invoked here to estimate
the impact parameter. The first one is the multiplicity
of forward neutrons with polar angle θ < 20o in the lab
frame which can be covered by a forward neutron wall.
The other method is to make use of the parameter Erat
[26], which is the ratio of the total transverse kinetic en-
ergy to the total longitudinal one. The particles are also
required to be within θ < 20o in the lab frame, while
the two qualities are calculated within the center of mass
0 0.2 0.4 0.6 0.8 1
tip-tip
body-body
0 0.2 0.4 0.6 0.8 10
maxb/b
FIG. 2: Upper: Forward neutron multiplicity and Lower:
Erat, as a function of normalized impact parameter b/bmax in
both tip-tip and body-body collisions.
system (c.m.s.).
Erat =
Ezi (1)
The normalized impact parameter b/bmax is used to rep-
resent centralities of tip-tip and body-body collisions and
the bmax of the two cases are quite different from each
other. As shown in Fig. 2, with either method, obvious
linear dependence of the normalized impact parameter
are demonstrated in both tip-tip and body-body near
central collisions. Then, the two methods can be com-
bined to determine the impact parameter to identify the
most central collision events in both tip-tip and body-
body collisions.
B. Stopping power ratio definition and evolution
It is difficult to obtain a universally accepted estimate
of the nuclear stopping power in heavy ion collisions due
to a proliferation of definitions of the concept [27]. The
stopping power ratio R [28] is employed to measure the
degree of stopping and defined as:
|Ptj |/
|Pzj | (2)
, the total nucleon transverse momentum |Ptj | divided by
the total absolute value of nucleon longitudinal momen-
tum |Pzj | in the c.m.s.. The ratio is wildly used to de-
scribe the degree of thermalization and nuclear stopping
by low and intermediate energies heavy ion collisions.
It’s a multi-particle observable on an event-by-event ba-
sis, which for an isotropic distribution is unity.
Fig. 3 shows the time and normalized impact param-
eter dependence of the stopping ratio R for three con-
ditions: tip-tip, body-body and sphere-sphere collisions.
0 10 20 30 40
=0max(a)b/b
tip-tip
body-body
sphere-sphere
0.2 0.4 0.6 0.8 1
(b)Minibias
tip-tip
body-body
sphere-sphere
t(fm/c) maxb/b
FIG. 3: (Color online) (a)The time evolution of the stopping
ratio R in tip-tip, body-body and sphere-sphere central colli-
sions, and (b) the stopping ratio R as a function of b/bmax in
minimum biased collisions.
When the ratio R reaches the value of 1, full stopping of
the collision system is considered to be achieved, and the
momenta is also isotropy, which are not sufficient but nec-
essary for thermal equilibrium of collision systems [28].
For R > 1, it can be explained by preponderance of mo-
mentum flow perpendicular to the beam direction [29]. It
is shown that all of the three conditions can achieve full
stopping when the stopping ratio R=1, the correspond-
ing time for body-body and tip-tip central collisions are
about 15 fm/c and 25 fm/c, respectively. Larger stopping
ratio and faster evolution to full stopping are observed
for body-body central collisions than tip-tip and sphere-
sphere ones at the early stage, which may indicate a more
violent colliding process for body-body central collisions
due to the sizable initial transverse overlap region. Al-
though the stopping ratio of tip-tip central collisions is
lowest than the other two cases at the early time, it raises
sharply later and even exceeding one. So, it means that
longer reaction and passage time can be obtained in tip-
tip central collisions than body-body and sphere-sphere
ones, which may indicate the nucleons in tip-tip colli-
sions can undergo more binary collisions to reach higher
transverse momentum.
In Fig. 3(b), the R of the three conditions are gradu-
ally decrease with the increase of the normalized impact
parameter. When b/bmax < 0.5, the ratio is always larger
for tip-tip collisions than the other two cases, while for
b/bmax > 0.5 all of the three conditions almost have the
same stopping power ratio.
III. BARYON, ENERGY DENSITY AND
THERMAL EQUILIBRIUM
Considering the discrepancy of stopping power be-
tween tip-tip and body-body collisions, it is interesting
4 (a)Baryon Density 
tip-tip
body-body
Au-Au
0 10 20 30 40 50
(b)Energy Density
tip-tip
body-body
Au-Au
t(fm/c)
FIG. 4: The evolution of (a) baryon and (b) energy densities
in tip-tip, body-body and Au+Au central collisions.
to study further about the baryon and energy densities
evolution in both cases. As the full stopping and de-
formation effects in U+U collisions, it is believed higher
local baryon and energy densities system with long du-
ration can be created, which is considered to be a signif-
icant condition to study the nuclear EoS at high bayonic
density region.
A. The evolution of baryon and energy densities
The evolution of baryon and energy densities in the
central zone of tip-tip and body-body as well as Au+Au
central collisions are illustrated in Fig. 4.
In Fig. 4, it is observed the maximum attainable
baryon and energy densities for both tip-tip and body-
body central collisions are about 3.2 ρ0 and 0.8 GeV/fm
respectively, while the Au+Au one are about 2.6 ρ0 and
0.6 GeV/fm3. Both the baryon and energy densities in
U+U collisions are higher than the Au+Au one. Once a
baryon density threshold of ρ > 2.5 ρ0 is required, the
corresponding duration in tip-tip central collisions ∼ 20
fm/c (from ∼ 8 fm/c to ∼ 28 fm/c) is longer than ∼
10 fm/c ( from ∼ 8 fm/c to ∼ 18 fm/c ) of body-body
one, which is as predicted. But the peak densities have
no significant discrepancy between the two cases unlike
those at the energy region of the Alternating Gradient
Synchrotron (AGS) [12], which may be attribute to the
full stopping at the CSR energy.
B. Thermalization of the U+U collision systems
As mentioned before, the stopping ratio R = 1 is a
necessary but not sufficient condition for thermal equi-
librium of the collision system. In order to approach a
thermal equilibrium, a long duration of reaction is needed
for nucleons to undergo sufficient binary collisions. As
0 10 20 30 40 50
tip-tip
body-body
Au-Au
0 10 20 30 40 50
tip-tip
body-body
Au-Au
t(fm/c)
FIG. 5: The evolution of (a) volume with high density (ρ >
2.5ρ0) in tip-tip, body-body and Au+Au central collisions,
and (b) the scaled mean kinetic energy 2
< Ek >, within a
sphere of radius 2fm around the system mass center.
shown in Fig. 4(a), obvious long duration has been ob-
tained in both tip-tip and body-body central collisions.
It is therefore possible thermal equilibrium at the time
of freeze-out can be achieved.
The Fig. 5(a) is the evolution of the volume with the
high baryon density(ρ > 2.5 ρ0) for tip-tip, body-body
and Au+Au central collisions, respectively. Both tip-
tip and body-body central collisions have larger volumes
than Au+Au one at the same beam kinetic energy 520
MeV/nucleon. Although the maximum volume attain-
able for body-body central collisions(∼ 220 fm3) is about
two times larger than tip-tip one(∼ 120 fm3), the peak
volume of tip-tip central collisions lasts a much longer
time of ∼ 10 fm/c (from ∼ 15 fm/c to ∼ 25 fm/c) and
much more stable than body-body one. To estimate the
temperature at the freeze-out time, the scaled mean ki-
netic energy of all hadrons in a sphere of radius 2fm
around the system mass center is calculated as 2
< Ek >
[22], which is utilized to reflect the thermalization tem-
perature T of the collision system approximately. As
illustrated in Fig. 5 (b), both tip-tip and body-body
central collisions show a flat region about 75 MeV and
the corresponding time range are about 10 fm/c to 28
fm/c and 10 fm/c to 18 fm/c, respectively. Considering
the time range of the flat region in Fig. 5 (b) associat-
ing with the corresponding range in Fig. 5 (a) and also
looking back to Fig. 4, we obtain a large volume of hot,
dense nuclear matter in both tip-tip and body-body cen-
tral collisions. Consequently, the extreme circumstance
of sufficiently high temperature and density for a signif-
icant large volume and long duration [12, 22] has been
formed in tip-tip and body-body central collisions, which
can provide a good opportunity to study the nuclear EoS
as well as particles in medium properties, especially for
tip-tip case.
The time of freeze out should be cautiously determined
(a)tip-tip π+∆+*N
0 10 20 30 40 50 60
80 (b)Body-Body π+∆+*N
t(fm/c)
FIG. 6: Evolution of the multiplicity of the free pion, N∗ +
∆, N∗ +∆+ π in (a)tip-tip ,and (b)body-body central colli-
sions.
for estimating the thermalization temperature of colli-
sion system. In Fig. 6 the multiplicity evolution of free
pion which are not bounded in baryon resonances and
pion still bounded inside the excited baryon resonances
(∆, N∗) (unborn pion) are displayed. At the Lanzhou
CSR energy region (520 MeV/nucleon), the production
and destruction of the ∆ resonances are mainly through
NN ⇋ N∆ and ∆ → Nπ reactions in which the ∆ decay
rate is always higher than that of the formation of this
resonance and the production of pion is predominated by
the decay of the ∆ resonances (∆ → Nπ) [30]. The total
multiplicity of pion, ∆ and N∗ approaches a saturated
level after a period of evolution, indicating the freeze-
out time about t=28 fm/c and t=18 fm/c for tip-tip
and body-body central collisions, respectively. The larger
maximum attainable total multiplicity of pion, ∆ andN∗
and freeze out earlier indicates a existent of faster evo-
lution and more violently reaction process for the body-
body central collisions than tip-tip case consisting with
the discussing before.
The corresponding temperature about 75 MeV at
freeze-out time can be extracted from the Fig. 5 (b), for
both tip-tip and body-body central collisions. To further
confirm this estimation, both the energy spectrum of the
nucleon and negative-charged pion are studied within the
polar angle range of 900±100 in the c.m.s.. The thermo-
dynamic model [31] predicts that the energy spectra will
be represented by a temperature T which characterizes a
Maxwell-Boltzmann gas
PEdEdΩ
= const× e−Ekin/T (3)
, where P and E are the particle momentum and total
energy in the c.m.s.. Both the energy spectra and the
Boltzmann fit results are shown in Fig. 7. The inverse
slope (e.g. temperature T ) of the nucleons in tip-tip and
body-body central collisions are about 73 MeV and 70
0 0.2 0.4 0.6 0.8
310  
tip-tip
body-body
(a)Nucleon
0 0.2 0.4 0.6 0.8
-π(b)
(GeV)kinE
FIG. 7: (a) Nucleon, and (b) negative-charged pion energy
spectrum at 900 ± 100 in the c.m.s. together with a Maxwell-
Boltzmann fit for both tip-tip and body-body central colli-
sions. The nucleon fit temperature for tip-tip and body-body
are about 73 MeV and 70 MeV, respectively and that of pion
are about 56 MeV and 52 MeV, respectively.
MeV, respectively, which are in good agreement with the
temperature extracted from the Fig. 5(b) at the freeze
out time. The spectra of negative-charged pion show a
different lower temperature than that of nucleon which
may be explained by considering an equilibrated N and
∆ system at thermal freeze out and taking into account
the kinematics of ∆ decay [32]. The nucleon temperature
closely reflects the freeze-out temperature of tip-tip and
body-body central collisions.
In conclusion, thermalization (or near thermalization)
of the collision system corresponding a freeze-out tem-
perature about 75 MeV is likely to be achieved in both
tip-tip and body-body central collisions. However, it’s
also possible that the collision system is still in a non-
equilibrium transport process on its path towards kinetic
equilibration [30].
IV. THE COLLECTIVE FLOW OF U+U
COLLISIONS
Stopping of nuclei in heavy ion collision can lead
to pressure gradient along different directions, result-
ing in collective motion as spectators bounce-off [34]
and participants squeeze-out effects [35]. Since last
two decades, at Bevalac/LBNL and SIS/GSI energies
the so-called ”collective flow” analysis has been estab-
lished [15, 34, 35, 36, 37] to study the collective mo-
tion of the products in heavy ion collisions. The collec-
tive flow resulting from bounce-off and squeeze-out ef-
fects, which can be explained well by the hydrodynamics
model [34, 38], and also be in good agreement with the
experimental data has been observed [39, 40]. Because of
the large deformation of the uranium nuclei, a novel col-
-1 -0.5 0 0.5 1
Soft:tip-tip
Soft:body-body
Cascade:tip-tip
Cascade:body-body
(a)b/b
-1 -0.5 0 0.5 1
(b)b/b
FIG. 8: The mean transverse momentum per nucleon pro-
jected into the reaction plane, < px/A >, as a function
of c.m.s. normalized rapidity is illustrated for tip-tip and
body-body collisions. With normalized impact parameter
cutoff:(a)b/bmax <= 0.5 (b)b/bmax > 0.5.
lective motion is expected [12], to be used to extract the
medium properties and nuclear matter EoS information.
[15, 16, 17, 18, 19, 20].
To perform flow analysis, it is necessary to construct a
imaginary reaction plane defined by direction of the beam
(z) and the impact parameter vector b [43, 45, 46]. In our
simulation, the x− z plane is just defined as the reaction
plane with the beam direction along z positive direction
and the impact parameter vector b along x positive direc-
tion. In last two decades, there are mainly two methods
to study the collective flow at the low and intermediate
energies. One is the sphericity method [28, 34, 41, 42]
which yields the flow angle relative to the beam axis of
the major axis of the best-fit kinetic energy ellipsoid, and
the other is to employ the mean transverse momentum
per nucleon projected into the reaction plane, < px/A >,
to perform nucleon sideward flow analysis [43, 44] which
reflects the spectator bounce-off effects in the reaction
plane. In recent years, it is usual to use an anisotropic
transverse flow analysis method. With a Fourier expan-
sion [47, 48] of the particle azimuthal angle φ distribu-
tion with respect to the reaction plane, different har-
monic coefficients can be extracted, among which the
first harmonic coefficient v1, called directed flow (simi-
lar to sideward flow) and the second harmonic coefficient
v2, called elliptic flow are mostly interested. The ellip-
tic flow reflects the anisotropy of emission particles in
the plane perpendicular to the reaction plane while the
directed flow describes the anisotropy in reaction plane.
The Fourier expansion can be expressed as
∼ 1 +
2vncos(nφ) (4)
Fig. 8 shows nucleon sideward flow, < px/A >, for
both tip-tip and body-body minimum biased collisions
(a)Nucleon Flow Parameter
tip-tip
body-body
Au-Au(500MeV/A)
0 0.2 0.4 0.6 0.8 1
10 2(b)Nucleon v
tip-tip
body-body
Au-Au(500MeV/A)
maxb/b
FIG. 9: (a)The nucleon flow parameter F and (b)the c.m.s.
mid-rapidity ( −0.5 < y0 < 0.5 ), nucleon elliptic flow v2 of
three collision conditions as a function of normalized impact
parameter b/bmax with soft EoS.
as a function of normalized rapidity, y(0) = Ycm/ycm, in
which Ycm represents the particle rapidity in c.m.s. and
ycm is the rapidity of the system mass center. To ex-
tract the nuclear EoS information and also demonstrate
the discrepancies of the nucleon sideward flow in tip-tip
and body-body collisions, the cascade events [49], which
neglect the mean field and pauli blocking effects are em-
ployed here to compare with the soft EoS case. In Fig.
8(a),(b), with a soft EoS, it is noted that either tip-tip
or body-body collisions show a spectator bounce-off ef-
fect revealing an obvious ”S” shape [15, 49] at the mid-
rapidity region of −0.5 < y0 < 0.5, while the cascade one
appear a almost vanishing nucleon sideward flow. It can
be understand by the nucleon sideward flow is related
to the mean field, which is mainly responsible for the
pressure gradient of the stopping nuclei, while the mean
field has a strong dependence of the nuclear EoS. There-
fore, the nucleon sideward flow is thought to be a good
indirect probe to extract the nuclear EoS information,
especially tip-tip case for its largely remarkable sideward
flow. A cutoff on normalized impact parameter is also
applied to explore the impact parameter dependence of
nucleon sideward flow. As shown in Fig. 8(b), when
b/bmax > 0.5 the curves of soft EoS and cascade are al-
most superposed with each other, while for b/bmax < 0.5
large discrepancy is observed. The situation is quite sim-
ilar to Fig. 3 (b), almost the same stopping power for
b/bmax > 0.5 and large discrepancy for b/bmax < 0.5 in
tip-tip and body-body minimum biased collisions, which
means there exists a correlation between nuclear stopping
power and sideward flow [33].
The normalized impact parameter dependence of the
collective flow of nucleon is further studied, by analyzing
the ”flow parameter” F [49] and also elliptic flow v2 for
both tip-tip and body-body as well as Au+Au minimum
biased collisions. The flow parameter F is a customarily
used quality to describe the nucleon sideward flow quan-
titatively defined as
d < px/A >
dy(0)
y(0)=0
the slope of the mean transverse momentum per nucleon
projected into the reaction plane at y(0) = 0.
In Fig. 9(a), with b/bmax > 0.5, the nucleon flow pa-
rameter F of tip-tip and body-body collisions are with
similar value. This similarity, along with the almost same
stopping ratioR in Fig. 3(b), indicates a existence of sim-
ilar pressure gradient effects on nucleon sideward flow in
the two collision orientations. While for b/bmax < 0.5,
the flow parameter F of tip-tip collisions is nearly 3 times
larger than that of body-body case. Even the sideward
flow of Au+Au collisions is larger than the body-body
one. It is further confirmed the tip-tip nucleon sideward
flow is a more sensitive probe to extract the information
of nuclear EoS than that of body-body one. The promi-
nence high of the nucleon sideward flow in tip-tip colli-
sions may be resulted from the stronger pressure gradient
between the participants and spectators in the reaction
plane than body-body one, due to the largely deformed
nuclei.
The normalized impact parameter dependent of nu-
cleon elliptic flow v2 at the mid-rapidity region ( −0.5 <
y0 < 0.5 ) is displayed in Fig. 9(b). A significant neg-
ative elliptic flow v2 at this energy region is consistent
with the excitation function of the elliptic flow studied
before [50]. An largest negative v2 about −12% in body-
body central collisions is observed which reflects the large
geometric and squeeze-out effects in the collisions. While
for tip-tip and Au+Au ones the maximum negative v2 are
obtained at mid-centrality. Since both high baryon, en-
ergy densities and large elliptic flow effects, which reflects
an early EoS of the hot dense compression nuclear matter
[17], are available in body-body central collisions. Thus
the body-body nucleon elliptic flow can also be taken as
a sensitive probe of nuclear EoS. The novel behaviors
of nucleon collective flow in tip-tip and body-body col-
lisions are mainly attributed to the large deformation of
the uranium nuclei.
V. SUMMARY
In summary, the CSR-ETF at Lanzhou provide a good
opportunity to systematically study the nuclear EoS at
the high net-baryon density region of nuclear matter
phase diagram. Due to the novel stopping effects in
largely deformed U+U collisions, the simulation based on
ART1.0 demonstrates that full stopping can be achieved
and also a bulk of hot, high densities nuclear matter with
large volume and long duration have been formed in both
tip-tip and body-body collisions. Large nucleon sideward
flow in tip-tip collisions and the significant negative nu-
cleon elliptic flow in body-body central collisions can pro-
vide a sensitive probe to extract nuclear EoS information.
Thus the extreme circumstance and the novel collective
flow in both tip-tip and body-body collisions can provide
a good condition and sensitive probe to study the nu-
clear EoS, respectively. More experimental observables
of U+U collision dynamics should be further studied, due
to the geometric effects.
VI. ACKNOWLEDGEMENT
This work is supported by National Natural Sci-
ence Foundation of China (10575101,10675111) and the
CAS/SAFEA International Partnership Program for
Creative Research Teams under the grant number of
CXTD-J2005-1. We wish to thank Bao-an Li, Feng Liu,
Qun Wang, Zhi-Gang Xiao and Nu Xu for their valuable
comments and suggestions.
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