Study of neutron emission in nuclear system decay for
1
H, 2H, 4He, and 12C - Pb collisions at 1-2 AGeV
V.I. Yurevich1, R.M. Yakovlev2, V.G. Lyapin2
1
2
JINR, Laboratory of high energies, Dubna, Russia
V.G. Khlopin Radium institute, St. Petersburg, Russia
Neutron production double differential cross sections for the interactions of 2-GeV p
and d, 4-GeV 4He, and 24-GeV 12C ions with Pb nuclei were measured by the TOF method in
angular range 30o-150o. The four-component Moving Source Model was applied for the
neutron data analyses. The temperature and velocity parameters of the sources corresponding
to the decay of highly excited nuclear system are in a good agreement with the results
obtained in charged fragment emission studies. The freeze-out temperature for the thermal
decay by fragmentation was estimated as Tf = 4.70.3 MeV. The mean neutron multiplicities
and beam energy fraction coming to the neutron emission are discussed.
1. Introduction
Neutron emission in intermediate-energy hadron-nucleus and nucleus-nucleus collisions
plays an important role in reaction dynamics and energy balance. A motivation to study
neutrons is
 the neutrons are emitted at all stages of the nuclear system evolution in decay processes;
 the neutron spectra bring the most direct information about excited nuclear matter
properties which is not distorted by the Coulomb interaction as it has place for the
charged fragment emission.
Thus, one can expect that by analysis of the neutron production double differential cross
sections some unique information on the collision dynamics and reaction mechanisms can be
obtained. A basic principle of such experimental data analysis is a selection of main reaction
stages with essential contributions to the neutron emission. For this purpose overall picture of
nuclear system evolution based on results of experiments and theory can be used.
Traditionally, two mechanisms of neutron production, the pre-equilibrium emission and
the evaporation, are considered for low- and medium-energy reactions and next mechanism,
the intra-nuclear cascade, is added at higher energies. Such approach was used in [1] for
development of three-component Moving Source Model (MSM) and it was applied for fitting
of the double differential cross sections of neutron production in the proton-nucleus
interactions at intermediate energies up to 3 GeV [2]. A contribution of the elastic and quasielastic nucleon-nucleon collisions is important at small angles and it is taken into account by
additional Gaussian-like term.
On the other hand, already more than 25 years the MSM is successfully used for analysis
of results on charged particle and fragment emission in the hadron-nucleus and nucleusnucleus collisions at energies from hundred MeV to about 10 GeV per nucleon [3,4]. Basic
approach in this analysis is selection of the emission from the participant nucleon region and
from the target and projectile spectators. The collisions of GeV protons and light nuclei with
heavy nuclei are effective method to prepare equilibrium highly excited nuclear system with
small excitation of collective modes. Information obtained in these experiments is extensively
used in theoretical studies of the reaction dynamics and it supports the idea about existence of
liquid-gas phase transition in the fragmentation process.
1
Two stages of the hot nuclear system decay were observed in [5-9]. At the earlier stage
the moving non-equilibrium hot source is formed in the central interactions following the first
collisions in the participant region. The temperature and lifetime of this source are estimated
as 18-20 MeV [8] and 25-30 fm/c [7,9] respectively. The energetic low-mass fragments are
emitted at this non-equilibrium stage. Further expansion and thermalization up to thermal
break-up of the nuclear system takes additional ~40 fm/c [6,9,10]. According to the
statistical multifragmentation model [11] remnant with excitation energy ~2-3 MeV/nucleon
continues expansion, gets thermal equilibrium and then decays on nucleons and fragments at a
moment characterized by the freeze-out temperature Tf . Experimental study of the charged
fragment emission gave estimation of the Tf  4-5.5 MeV [5-8,12-15]. Multiplicity and masses
of the nuclear fragments depend on the initial excitation, and in a limit of very high excitation
the decay to many IMFs occurs (thermal multifragmentation).
It is naturally to expect that both the hot non-equilibrium source and the thermal
fragmentation of the excited remnant give an essential contribution to the neutron emission.
These sources have to be included to the MSM for the neutron data analysis in accordance
with the new experimental and theoretical results. It was a motivation for our attempt to
modify the MSM. Certainly, one has to expect that the source parameters found in the
analysis of the neutron emission won’t be in contradiction with the values obtained in the
studies of charged fragment emission.
Main aim of our investigation is study of the neutron emission mechanisms in the
interactions of 2-GeV p and d, 4-GeV 4He, and 24-GeV 12C with Pb nucleus. For this purpose
a new approach in the MSM framework was developed. First of all our efforts were focused
on search and study of the neutron components corresponding to disintegration of the hot
nuclear system by the non-equilibrium and thermal fragmentation decays.
2. Experiment
The measurements of the neutron production double differential cross sections were
carried out by the TOF method in the target fragmentation region between 30 o and 150o with
the relativistic ion beams of the Dubna synchrophasotron. The experimental setup is
schematically shown in Fig. 1.
Fig.1. A scheme of the TOF spectrometer: C1-C3 – the scintillation beam counters; T – the Pb
target; VC – the scintillation veto counters; D – the neutron detectors.
The neutrons were registered with three types of detectors based on stylben crystals and
large volume plastic scintillators. The detector efficiencies were mainly studied in special
experiments in wide energy range from 100 keV up to hundreds MeV. The gamma-quanta
2
were rejected by the TOF and pulse-shape discrimination (for the stylben detectors). The
charged particles were vetoed by the thin plastic scintillation counters VC placed in front of
the neutron detectors. The system of beam counters C1-C3 with thin plastic scintillators was
applied for beam monitoring and event triggering. Event selection was produced by trigger
used a fast coincidence of pulses from 1) the upstream counters C1 and C2 for p and 4He
beams and 2) the upstream counters and the veto downstream counter C3 for d and 12C beams.
In the last case the veto-signal produced by the downstream counter discriminated interactions
with any charged particles for the deuteron beam and with fragment charge Zf > 5 for the
carbon beam at angles less 2o to the beam axis. Taking into account the trigger selection of the
events the effective reaction cross sections 1.695, 1.77, 2.44, and 3.0 b were used for
determination of the mean neutron multiplicities for the reactions p, d, 4He, and 12C + Pb
respectively. Detail information about the experiment can be found in [16,17].
3. Development of the Moving Source Model
The modified MSM bases on a picture of the collision dynamics shown in Fig.2
corresponding to our knowledge about this phenomenon. Assumption that the pre-equilibrium
emission before the last evaporation stage is the second order process and gives smaller
contribution in comparison with other main sources included in the model and shown in the
figure is used.
Fig.2. A scheme of the nuclear system decay modes in central and peripheral collisions of
GeV hadrons and light nuclei with heavy nuclei used as a base for modification of the MSM.
The source 1 reflects the first nucleon-nucleon interactions as initial stage of the collision.
The hot source 2 is formed in the interactions with the dense core of target nucleus in the
central collisions. Further nuclear system evolution depends on the excitation energy value.
At low, medium and high excitations the formation of excited remnant, the fragmentation to
heavy fragment plus some light fragments and nucleons, and the multifragmentation to some
IMFs are realized respectively. The excited remnant, heavy fragment, and fission fragments
produced in the peripheral and central collisions decay by the evaporation process, and they
are joined to the source 4 due to the common decay process and small velocity in the lab.
frame. The ejection of neutrons in the fragmentation and multifragmentation processes is
reproduced by the source 3. Maximum contribution of the third source (fragmentation) is
expected in a neutron energy range between 5 and 20 MeV taking into account the freezeout temperature Tf ~ 5 MeV found in the charged fragment emission studies.
3
The MSM is based on the traditional assumption that in a source frame the neutron
emission is isotropic with invariant cross section described by the formula
1 d 2
 A exp(  / T ) ,
p dEd
(1)
where E and  – the neutron kinetic energy (MeV) in the lab. frame and the source frame
respectively, the neutron momentum p is calculated as p  ( E 2  2Em)1 / 2 ,  – the solid angle,
T – the temperature parameter (MeV). The Lorenz transformation to the lab. frame gives
the following expression for the four-source model
4
E  m  p i cos 
d 2
  pAi exp{(
 m) / Ti },
dEd i 1
(1   i2 )1 / 2
(2)
where θ – the angle of emitted neutron in the lab. frame. There are three parameters for each
the source: the amplitude Ai, the temperature Ti and the velocity i=Vi/c. In accordance with
the picture of the decay processes (Fig.2), relations T1>T2>T3>T4 and 1>2>3>4 between
the temperature and velocity parameters of the different sources are expected.
4. Application of the MSM for neutron emission analyses
d2/dEd, mb MeV-1 sr-1
The developed MSM was applied for analyses of the neutron production double
differential cross sections obtained at large angles (θ30o) in our measurement and some other
available experiments for p+Pb reaction [18-25]. For this purpose the expression (2) was used
for fitting of the data. The fitting procedure consisted of two steps. At the first step the source
3 was excluded from the fitting process. Here we tried to get a good description of the neutron
spectra in ranges above 30 MeV by the terms 1 and 2 and below 5 MeV by the term 4. It was
found that the best fits corresponded to some underestimation between 5 and 30 MeV. The
next step was the fitting with the term 3 stating from the parameters obtained at the step 1.
Finally, a perfect agreement in all energy range and for any angle above 30 o was reached. The
two examples of the fitting for reactions 4-GeV 4He+Pb and 24-GeV 12C+Pb are shown in
Fig.3 and Fig.4.
10
10
4
10
3
a
3
b
4
10
2
10
1
10
0
10
4
3
2
1
-1
10
-1
10
0
10
1
10
2
10
3
10
2
10
1
10
0
10
3
2
1
-1
10
-1
10
0
10
1
10
2
10
3
E, MeV
Fig.3. Contributions of the different sources to the neutron production double differential
cross sections measured for 4-GeV 4He+Pb at 30о (■) и 60о (○) (a) and for 24-GeV 12C+Pb at
53о (b). The curve numbers correspond to the source numbers.
4
d2/dEd, mb MeV-1 sr-1
10
7
10
6
10
30
60
5
o
o
o
10
4
100
10
3
150
10
2
10
1
10
0
10
-1
10
-2
10
-1
3
x 10
a
2
x 10
x 10
o
10
0
10
1
10
2
10
3
10
8
10
7
6
30
o
x 10
10
x 10
10
53
o
5
4
90
o
10
x 10
10
3
150
10
2
10
1
10
0
10
-1
10
-2
10
-1
b
3
2
o
10
0
10
1
10
2
10
3
E, MeV
Fig.4. The fitting of the experimental data on neutron production double differential cross
sections measured for the reactions 4-GeV 4He+Pb (a) and 24-GeV 12C+Pb (b) at various
angles. The points – the experimental data, the curves – the result of the fitting.
The contributions of the sources 2 and 3 to the neutron production cross section increases
with energy and mass of the projectile. In the energy range of incident protons above about
0.5 GeV the temperature parameter values become universal and do not depend on energy.
Our measurements with heavier projectiles gave the same results and showed their
independence on projectile type as it is presented in Fig. 5.
T1=70 MeV
100
T, MeV
T2=21 MeV
10
T3=4.7 MeV
T4=1.6 MeV
1
0,1
0,1
1
10
Ebeam, GeV
Fig.5. Energy dependence of the source temperatures found for our data with different
projectiles p (○), 2H (∆), 4He () and 12C (□) and for some available data for the reaction
p+Pb:  [18],  [19-23], [24],  [25].
The obtained temperature and velocity parameters for the sources 2 and 3 have to be
compared with the corresponding values found for the non-equilibrium hot source and the
thermal equilibrium source in the charged fragment emission studied for similar collisions.
The temperature 212 MeV for the hot source 2 obtained in the neutron emission analysis is
practically reproduces the result [8] for the reaction 4.8-GeV 3He+Au. In the same work the
hot source velocity was measured as ~0.03-0.045 (for Li fragments) that agrees with value
0.04 found for the neutron source 2 for the reaction 4-GeV 4He+Pb.
5
Study of the neutron emission the thermal fragmentation via the liquid-gas phase
transition (the source 3) gives a possibility to estimate the freeze-out temperature Tf.
The found value 4.70.3 MeV is perfectly reproduces the results 4.70.4 MeV of the EOS
collaboration [6] and 5.00.5 MeV of the ISiS collaboration [8] obtained as the isotope ratio
temperature THeDT for the reactions 1-AGeV Au+C and 4.8-GeV 3He+Au respectively.
The slope temperature parameters corresponding to the volume emission of fragments with
Z=2 in the reactions p+Au and 3He+Au at 2 GeV were estimated in [7] as 4.50.3 MeV and
4.60.5 MeV respectively that also supports the neutron result. The neutron source velocity
changes from 0.0025 for protons to 0.007 for carbon projectiles and agrees with the value
0.0060.001 obtained for the thermal equilibrium source of the IMFs formed in the 8.1-GeV
p+Au collisions [26]. The similar results were obtained in [5] for the reaction p+Xe in a range
of proton energies 1-19 GeV and in [8].
There were many theoretical and experimental attempts to estimate the critical
temperature Tc for the nuclear liquid-gas phase transition. The extracted values of Tc for
collisions with Au are from 7.6 MeV [27,28] to 17 MeV [29] and lie between the found
temperatures for the thermal and hot sources.
The made comparison leads us to the apparent conclusion about the same origin of the
hot non-equilibrium and thermal sources for neutrons and charged fragments and shows that
the developed MSM gives adequate fit to the neutron experimental data. Here it is important
to note that physics of the hot source is not clearly understood yet.
The last de-excitation stage by the neutron evaporation described with the source 4 is
characterized by the temperature 1.60.2 MeV that is approximately 50 % higher of the
values observed in nuclear reactions at low energies.
5. Mean neutron multiplicity
The mean neutron multiplicity is important characteristic of the neutron emission process
is the mean neutron multiplicity in a reaction. For our measurements the neutron production
cross sections were calculated by integration of the expression (2) with found parameters over
the neutron energy and solid angle. The mean neutron multiplicities shown in Table 1 were
derived by division of these values by the effective reaction cross sections given in Section 2.
Also, the relative contributions of the sources 2 and 3 are given there.
Table 1. The mean neutron multiplicities and the contributions of the sources 2 and 3 for the
studied reactions.
Reaction
Mn
(n/interaction)
p+Pb
21.8±3.4
Source 2
contribution
(%)
16
d+Pb
17.1±3.4
14
20
He+Pb
22.5±3.5
19
23
29.1±4.5
16
26
4
12
C+Pb
Source 3
contribution
(%)
22
The mean neutron multiplicity a bit grows coming from protons to helium and then to
carbon ions. The deuterons give the smaller value than the same energy protons but it is
needed to note that the difference lies inside the experimental errors. The contributions of the
sources 2 and 3 are essential 14-19 % and 20-26 % respectively. Also, one can obtain
contributions of these two modes for the neutron emission in central collisions assuming that
such collisions take about a half of the reaction cross sections. A simple calculation shows
6
that about 80 % of the emitted neutrons in the central collisions come from these two sources.
Here one has to keep in mind that the neutron production at small angles in the elastic, quasielastic and projectile fragmentation processes is not taken into account.
Energy dependence of the mean neutron multiplicity was studied on a base of some
available results of TOF [19-25,30] and integral measurements [31-33] for the reaction p+Pb
in a range from 0.1 to 3 GeV. The TOF data were integrated over the neutron energy and solid
angle where for determination of the total mean neutron multiplicity some extrapolations of
he experimental results to low- and high-energy ranges were made. The data [31] obtained by
the moderation technique were corrected on high-energy neutron leakage. The fitting to these
data gave the following expression for the mean neutron multiplicity in the reaction p+Pb and
energy range 0.1-3 GeV
M n  39(1  exp( 0.22E p )) 0.55 ,
(3)
where Ep is the proton energy in GeV. This dependence together with the experimental results
are shown in Fig. 6. The curve (3) is close to the curve produced by the formula from [34]
recommended for incident protons with energy up to 2 GeV. These two dependences are
extrapolated to higher energy range for a comparison with our data found for the reactions
He+Pb and C+Pb. Adding of the neutron production in the elastic, quasi-elastic and projectile
fragmentation processes will raise the points corresponding to our results for the reactions
He+Pb and C+Pb, and agreement with the extrapolated dependences will become much
better.
50
Mn, neutron/reaction
40
30
20
10
0
0,1
1
10
E, GeV
Ebeam, GeV
Fig. 6. A dependence of the mean neutron multiplicities on the beam energy for the reactions
with Pb: ○, ∆, , □ – our results for p, d, He, and C beams respectively; and the results of
other experiments with proton beam: ■ [19-23],  [24], ▲ [25], ● [30],  [31] (corrected on
high-energy neutron leakage), [32,33], the solid curve – our fit (3), the dashed curve – the
Cugnon’s formula [34].
6. Energy coming to neutron emission
The neutron emission takes an essential part of the projectile energy in the reactions.
Estimation of the kinetic energy of the emitted neutrons was performed using the
experimental results [19-24,30] for the reaction p+Pb. It was found that the kinetic energy of
the neutrons takes on average about 31 % of the incident proton energy in GeV range as it is
7
shown in Fig. 7. The main contribution ~27% is given by the high-energy neutrons with
energies above 20 MeV. Estimation of the mean energy coming to the neutron production in
the reaction is obtained by adding of the neutron separation energy ~7 MeV and it gives about
40% of the incident proton energy.
Ekin, MeV
Ekin/Ep, %
Ep , GeV
Fig. 7. Projectile energy dependences of the mean energy coming to the kinetic energy of
neutrons Ekin (left) and the ratio Ekin/Ep (right) for the reaction p+Pb: □ [19-23], ○ [24], ∆ [30],
the curves  the result of fitting.
7. Conclusion
The performed analysis of the neutron production double differential cross sections for
the collisions of relativistic light nuclei with Pb show important role of the neutron emission
in the reaction dynamics and de-excitation process. The neutron emission takes the essential
part of the projectile energy. The developed MSM reflects the real decay processes and gives
the interpretation of the experimental results adequate to our knowledge about main reaction
mechanisms in collisions of light nuclei with heavy nuclei at intermediate-energies. Study of
the neutron emission provides the important information on the reaction and decay
mechanisms including estimation of the temperature at different stages of the excited nuclear
system evolution. The “neutron thermometer” is unique method for direct determination of
the freeze-out temperature on the thermal fragmentation stage. The obtained value of the
freeze-out temperature is 4.7±0.3 MeV and it agrees with results of recent studies of the IMFs
emission in similar collisions [6-8].
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