Mach-like emission from nucleon scattering in proton-nucleus reaction
ABSTRACT The fast-stage nucleon emission of proton-nucleus (pA) reactions from 300A
MeV to 1.8A GeV has been investigated using the quantum molecular-dynamics
model. It is found that the sideward angular spectrum of nucleon emission
presents an interesting Mach-like structure at the early stage of the collision
(tens of fm/c). The sideward angular peak value varies from about 45\circ to
near 73\circ, depending on the bombarding energy. Nucleons emitted from the
vicinity of the sideward peak tend to have a fixed momentum value about 0.5
GeV/c, independent of the bombarding energy as well as the impact parameter.
Additionally, the sideward angular peak value is almost independent of the
equation of state, indicating that binary collision at the early fast stage in
the intermediate energy pA reaction plays an important role in the emergence of
arXiv:1106.3388v1 [nucl-th] 17 Jun 2011
Mach-like emission from nucleon scattering in proton-nucleus reaction
G. Q. Zhang,1,2Y. G. Ma∗,1X. G. Cao,3and C. L. Zhou1,2
1Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China
2Graduate School of the Chinese Academy of Sciences, Beijing 100080, China
3Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
(Dated: June 20, 2011)
The fast-stage nucleon emission of proton-nucleus (pA) reactions from 300A MeV to 1.8A GeV
has been investigated by the quantum molecular dynamics model. It is found that the sideward
angular spectrum of nucleon emission presents an interesting Mach-like structure at the early stage
of the collision (tens of fm/c). The sideward angular peak value varies from about 45◦to near 73◦,
depending on the bombarding energy. Nucleons emitted from the vicinity of the sideward peak tend
to have a fixed momentum value about 0.5 GeV/c, independent of the bombarding energy as well
as the impact parameter. Additionally, the sideward angular peak value is almost independent of
equation of state, indicating that binary collision at the early fast stage in the intermediate energy
pA reaction plays an important role for emergence of Mach-like emission.
PACS numbers: 25.40.-h
Canonical emission is of very interesting phenomenon
occurs in different fields in physics. Recently BNL Rel-
ativistic Heavy Ion Collider data have shown that a hot
and dense quark-gluon plasma (QGP) medium is created
in ultrarelativistic heavy-ion collision (HIC). The QGP
behaves like an almost perfect fluid and to be opaque to
jets created in the initial stage of the collision. The ex-
perimental dihadron correlation function [1–4] exhibits
an interesting double-peak structure at angles opposite
to the trigger jet. It has been suggested [5, 6] that such a
structure could be an evidence for Mach cone. However,
different theoretical interpretations have been provided,
such as Cherenkov-like gluon radiation model , shock
wave model in hydrodynamic equations , jet deflec-
tion  and strong parton cascade mechanism  etc.
Since it is still unclear that the main mechanism for the
emergence of the double peak structure, many experi-
mental and theoretical works suggest that it should de-
pend on the nature of the hot and dense matter created
in the collisions. Very recently, Betz et al. suggested
that the conical emission can also arise due to averaging
over many jet events in a transversally expanding back-
ground in ultrarelativistic heavy-ion collisions. Further-
more, they found that the apparent width of the away-
side shoulder correlation is insensitive to the details of
the energy momentum deposition mechanism as well as
to the system size . Even though much effort on di-
hadron correlation has been done, the mechanism of such
double peak structure is still in debating.
On the other hand, Mach-like structure has been no-
ticed in heavy ion collision at several GeV energy in
∗Corresponding author. E-mail address: firstname.lastname@example.org
earlier time. For an example, Aichelin et al. had used
quantum molecular dynamics (QMD) model to study the
asymmetry reaction system20Ne +197Au at 1050A MeV
in central collisions . A shock wave picture was de-
picted at the fast stage. While the projectile nucleons
punch through the heavy target with a supersonic speed,
a strong compression about 2ρ0 can be reached and a
strong transverse force is put on the projectile surface.
Thus, nucleons on the projectile surface will get a trans-
verse velocity and emit in the sideward direction. The
picture is very similar to Mach Cone phenomenon. Re-
cently, Rau et al.  adopted the hydrodynamic model
 and UrQMD  to simulate the Mach-like wave in
the asymmetric system20Ne +238U at 1-20A GeV. They
got clear Mach cone structure as the picture emerged in
hydrodynamic dynamics. While, in the default UrQMD
dynamics, they also got similar sideward peak excitation
function, although UrQMD has a different mechanism
forming the sideward peak and gets a broader width in
the sideward angular distribution.
Considering that proton-nucleus reaction (pA) is rel-
ative simple in reaction mechanism in comparison with
the heavy ion reactions, there is a potential advantage to
give some hints to understand the mechanism of Mach-
like phenomenon in HIC by investigating the fast-stage
nucleon emission. In past several decades, the intermedi-
ate energy proton-induced reactions (pA) play important
roles in the wide applications and fundamental research
fields [15–17]. However, the mechanism of pA collision
is still not well understood, especially on how to form
the sideward peak angular distributions of intermediate
mass fragments (IMF) and light charged particles (LCP)
[18–22]. In early 1970s, Remsberg and Perry  showed
that the angular distribution of IMF dominates 60◦to
70◦which is coincided with the angular distribution of
Mach Shock conical emission predicted by hydrodynam-
ics models. Hirata et al.  used a newly developed
non-equilibrium percolation (NEP) model and concluded
that a doughnut shape structure results in the IMF side-
ward peak angular distribution. Hsi et al. [20–22] per-
formed exclusive experimental studies on pA reactions
and claimed that the sideward peak of IMF originates
in kinematic-focusing effects associated with statistical
and thermal multifragmentation of an expanding residue.
They argued that the sideward peak was not likely to be
the result of shock wave, based on their experimental in-
clusive angular distribution without sideward peak for
LCPs. Using angular correlation analysis, they also de-
nied the sideward peak of IMF angular distribution could
come from the breakup of exotic geometric shapes.
Although different physical hypotheses are employed,
Quantum Molecular Dynamics model could demonstrate
similar phenomenon that hydrodynamical model pre-
dicts. In this Letter, we adopt quantum molecular dy-
namics model to investigate pA reactions. The reaction
system p +208Pb is taken as an example. Various ener-
gies and impact parameters, as well as hard or soft EOS,
are scanned systematically. The focus is concentrated
on nucleon emission at the fast stage of the collision, in-
cluding the kinetic energy distribution and polar angular
ISOSPIN DEPENDENT QUANTUM
MOLECULAR DYNAMICS MODEL
Quantum Molecular Dynamics model bases on an
n-body theory, which simulates heavy ion reactions
at intermediate energies on an event by event basis
[23, 24].The Isospin Dependent QMD (IDQMD) is
an extension version of QMD, which is suitable to
describe from Fermi energy up to 2A GeV with the
isospin effects considered: different density distribution
for neutron and proton, the asymmetry potential term
in mean field, different experimental cross-section for
neutron-proton (np) and proton-proton (pp,nn), Pauli
blocking for neutron and proton separately [25, 26].
Each nucleon is presented in a Wigner distribution
function with a width
√L (here L = 2.16 fm2) centered
around the mean position ? ri(t) and the mean momentum
? pi(t), φi(? r,t) =
The mean field in IDQMD model is:
USky+ UCoul+ UYuk+ Usym, where USky, UCoul,
UYuk, and Usymrepresents the Skyrme potential, the
Coulomb potential, the Yukawa potential and the
symmetry potential interaction, respectively . The
Skyrme potential is:USky
where ρ0 = 0.16 fm−3and ρ is the nuclear density.
The parameters α = −356 MeV, β = 303 MeV, and
γ = 7/6, correspond to a soft EOS, and α = −124
MeV, β = 70.5 MeV, and γ = 2, correspond to a
hard EOS. UYukis a long-range interaction (surface)
(2πL)3/4exp[−(? r−? ri(t))2
]exp[−i? r·? pi(t)
= α(ρ/ρ0) + β(ρ/ρ0)γ,
-100 10 20
FIG. 1: (Color online) Position correlation 2D histogram Rxy ver-
sus z in a cylindrical coordinate system. The reaction system is 1A
GeV p+208Pb at b = 1 fm with the hard EOS. Left: Evolution
time is 15 fm/c. Right: Evolution time is 25 fm/c. A linear scale
plot is used.
potential, and takes the following form:
rij/√4L) − exp(mrij)erfc(√Lm + rij/√4L)]
Vy = 0.0074GeV, m = 1.25fm−1, L = 2.16 fm2, and
rij is the relative distance between two nucleons. The
strength of symmetry potential is Csym= 32 MeV.
IDQMD treats the many body state explicitly, and
contains correlation effects to all orders and deals with
fragmentation and fluctuation of HICs.
fragments produced in HICs, a simple coalescence rule
is used with the criteria ∆r = 3.5 fm and ∆p = 300
MeV/c between two considered nucleons. Thus, nucle-
ons dominated in Fermi motion will be limited in the
· [exp(mrij)erfc(√Lm −
RESULTS AND DISCUSSIONS
Conical emission structure at the early stage of pA re-
action has been observed in our IDQMD calculation. As
Fig.1 presents for p + Pb at 1A GeV, Mach-like struc-
ture develops at 15 fm/c, while the head of the conical
structure disappears at 25 fm/c.
Fig.2 shows that the yield of nucleons and the average
kinetic energy of nucleons evolve with time. The yield
shows a stable increase for both protons and neutrons.
However, at 15 fm/c, including the protons emitted from
projectile, there are only 2.5 nucleons at each event on
average. This means that at each event, there exists no
Mach-like structure, because, at least, three nucleons are
required to form such structure. In addition, the average
kinetic energy of nucleons presents a dropping situation.
This, therefore, gives a hint that fast nucleons emit at
the early stage and slow nucleons emit at the latter stage.
The trend that neutron yields faster than proton, is con-
sistent with the phenomenon which Ma et al. has found
To show the dynamics process in detail, we select sev-
eral time points during the reaction evolution to see the
kinetic energy (Ek) spectra (Fig. 3) and polar angular
distribution (Fig. 4). For all nucleons, a roughly fixed
FIG. 2: (Color online)Left: The yield of nucleons at different time.
Right: The average kinetic energy at different time for all nucleons
(black square), protons (blue up triangle) and neutrons (red down
triangle). The same reaction system and condition as Fig. 1.
FIG. 3: (Color online): Evolution of the kinetic energy distribution
of nucleons with time. A fixed peak can be found near Ek= 140
MeV before 20 fm/c, then the low energy part comes out and the
former 140 MeV peak is submerged. The same reaction system and
condition as Fig. 1.
Ek peak near 140 MeV can be found before 20 fm/c,
then Ek smears to lower energy. For protons, there ex-
ists a high energy peak at the early stage, and later on
it turns into a high energy tail which stems from those
induced protons, distorted only a little by the mean field.
Hsi et al. had measured the polar angular distri-
bution for “gray proton”(100MeV < Ep <400MeV) and
found no sideward peak. Basing on this consideration,
they denied the possibility of shock-wave-like effects in
pA reaction. However, it is not so easy to select suitable
kinetic energy cut to separate the nucleons emitting at
the early fast stage from the ones evaporating at the lat-
ter stage. In addition, the efficiency for detecting high
energy nucleons should be carefully considered.
In pA reaction, the maximal density is just a little
more than normal nuclear matter density ρ0. Therefore
FIG. 4: (Color online): Evolution of the polar angle distribution
of nucleons with time. A fixed sideward peak can be found near
65◦in laboratory system. The same reaction system and condition
as Fig. 1.
it is hard to explain the transverse emission by the way
the Ref.  does, which needs high compression gradi-
ent. In IDQMD, with the reaction process going on, the
small value of angular peak, composed by mostly high en-
ergy induced protons, gets more and more feeble, while a
fixed sideward peak shows up at about 65◦for all nucle-
ons, then a thermal part comes out with a peak at ∼90◦,
which will dominate at the latter isotropic stage ( see
Fig.4). The incident protons would also get the chance
of collision with other nucleons in the target. The chance
of collision is determined by experimental proton-proton
and proton-neutron cross-section, including the elastic
and the inelastic channels. After the binary collision,
the projectile proton shares its kinetic energy and mo-
mentum with its collision partner, which results in the
sideward angular distribution peak at the early stage of
The correlation between the momentum and polar an-
gular of nucleons is also studied. Nucleons coming from
the the vicinity of the angular peak tend to get a fixed
momentum value 0.53GeV ( 140MeV) at the early stage
(15fm/c). At the following stage (25fm/c), the sideward
emission nucleons move their momentum peak to lower
value and extend their angular distribution width at the
same time (Figure 5). The situation is very similar to
Mach shock structure that ideal hydrodynamical models
Additionally, different impact parameters, energies and
EOS are investigated systematically. Nucleons emitting
within momentum range 0.3GeV/c < P < 0.6GeV/c are
selected in the following study. After the head of projec-
tile protons punches through the center of208Pb target,
0 0.51 1.52
FIG. 5: (Color online): Correlation between the momentum and
polar angular of nucleons at 15fm/c (left) and 25fm/c (right). Nu-
cleons near 65◦sideward peak have a fixed momentum value about
0.53 GeV/c (140MeV) in Laboratory system. The same reaction
system and condition as Fig. 1.
FIG. 6: (Color online): Excitation function of the sideward angu-
lar peak value of nucleons. The system is p +208Pb from energy
330A MeV to 1.81A GeV with impact parameter from b = 1 fm
to b = 5 fm and hard EOS. The error bars are smaller than the
five angular peak values are sampled continuously, with
a time step 1 fm/c. The average angular peak value is
then calculated from these five peak values as the final
result. Excitation function of the sideward angular peak
value for p +208Pb is presented for the hard EOS (Fig.
6). The sideward angular peak values increase with the
beam energy from about 45◦at 330MeV up to and a
limited 73◦at 1.81GeV. Soft EOS gives the same values
of angular peak values and bombarding energy depen-
dence as the hard EOS case, and also shows insensitivity
to impact parameter (not shown here due to the limited
space). Overall, the results are independent of the impact
parameter and EOS.
CONCLUSION AND SUMMARY
In summary, we apply quantum molecular dynamics
model to revisit the early stage nucleon emission in pA
reaction. It is found, for the first time, that the peak
of sideward angular distribution for the early emitting
nucleons is energy dependent. With the increasing of
beam energy, the sideward angular peak value increases
to a limit value 73◦, while the momentum of the emit-
ting protons tend to get a fixed value 0.53 GeV/c (140
MeV). However, the peak of sideward angular distribu-
tions are independent of impact parameters and EOS.
This Mach-like phenomenon is similar to the result that
hydrodynamical models have predicted, although differ-
ent physical interpretation are employed.
nucleon collision plays the essential role, which results
in the sideward angular emission. In event-level, there
are only 2-3 nucleons emitting at the early stage (before
20 fm/c), so there exists no Mach-like structure in the
event basis. In this case, we are unable to apply the an-
gular correlation method to extract the Mach-like struc-
ture information event by event. In the present model,
the Mach-like structure can be regarded as the scattering
effect mostly by the bombarding proton with the target
nucleons summing over each event, which is very differ-
ent from bulk collective hydrodynamical behavior. This
mechanism may also give some hints to the Mach-like
cone structure in ultra-relativistic energy [5, 6, 8–10, 27].
As far as the experiment is concerned, the key is to sep-
arate the nucleons emitting at the early stage from those
coming from the later stage and to improve the detecting
efficiency of the forward and sideward emitting nucleons,
especially for those “grey nucleons”.
This work was supported in part by the National
Natural Science Foundation of China under Grant No.
11035009, 10979074, 10905085, 10875159, and the Shang-
hai Development Foundation for Science and Technology
under contract No. 09JC1416800, and the Knowledge
Innovation Project of the Chinese Academy of Sciences
under Grant No. KJCX2-EW-N01.
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