A method for cluster recognition from nucleon distributions generated in calculations of relativistic collisions of light particles (protons, α-particles) with nuclei in the framework of the UrQMD model is proposed. The excitation energy of the clusters which is necessary to take into account for the de-excitation of the calculated fragments was estimated from empirical considerations. The approach was applied to calculate mass distributions of fragments in p + Fe collisions for different proton energies and showed a good correspondence to experimental results. The software implementation of the clustering method and a visualization of cluster formation substantially facilitate applications of the proposed method

A Polanski

G A Ososkov

G Musulmanbekov

S V Mitsyn

T I Mikhailova

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A CLUSTERING APPROACH IN THE UrQMD
TRANSPORT MODEL FOR NUCLEAR COLLISIONS
AT RELATIVISTIC ENERGIES
S. V. Mitsyn a, G.Musulmanbekov a, T. I. Mikhailova a,
G. A. Ososkov a, 1, A. Polanski b
a Joint Institute for Nuclear Research, Dubna
b National Centre for Nuclear Research, Swierk, Poland
A method for cluster recognition from nucleon distributions generated in calculations of relativistic
collisions of light particles (protons, α-particles) with nuclei in the framework of the UrQMD model is
proposed. The excitation energy of the clusters which is necessary to take into account for the de-exci-
tation of the calculated fragments was estimated from empirical considerations. The approach was
applied to calculate mass distributions of fragments in p+Fe collisions for different proton energies and
showed a good correspondence to experimental results. The software implementation of the clustering
method and a visualization of cluster formation substantially facilitate applications of the proposed
method.
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PACS: 25.75.-q
INTRODUCTION
Multifragmentation is a physical process occurring in high-energy collisions of nuclei
or in collisions of nucleus with energetic light particles. In such types of reactions a large
number of neutrons is emitted, which explains the interest to this issue in applied sciences [1].
Heavy-ion collisions, besides particle (proton, neutron, meson, hyperon, etc.) production, are
accompanied by nuclear spallation with the formation of nuclear fragments. The consistent
understanding of nuclear spallation over the whole energy range is a challenging task for
theoretical models.
1E-mail: ososkov@jinr.ru
A Clustering Approach in the UrQMD Transport Model for Nuclear Collisions 653
The models that are used to describe this type of reactions can be divided into two main
groups: macroscopic (statistical and hydrodynamical) and microscopic (cascade, transport)
models. The main difference between cascade and transport models is that in the latter
the dynamical evolution of the collision is followed microscopically in time, while in the
former the potentials are taken into account only globally. The Quantum Molecular Dynamic
(QMD) model [2, 3] belongs to the transport class and describes the evolution in time of the
system of nucleons. The interaction is based on a nonrelativistic density-dependent Skyrme-
type [4] equation of state with additional Yukawa and Coulomb potentials. The stochastic
collision term gives rise to large 
uctuations, compared with other transport approaches
like the Boltzmann transport models, which work with the phase-space distribution. This
is a very important feature when simulating the collision process including fragmentation.
QMD was successfully used for calculations of isotopes production for collision energies
less than 0.4 GeV [5]. There exists a version for higher energies, Ultra-relativistic Quantum
Molecular Dynamic, UrQMD [6], which includes resonance excitation of nucleons and string
dynamics. However, this version does not include nuclear fragmentation. Observations of
multifragmentation of nuclei in collisions with protons at relativistic energies have been done
in p + Au collisions by the Karnaukhov group [7] in Dubna; however, they were mainly
interpreted in the terms of the cascade model.
Therefore, in this paper we develop a mathematical algorithm for fragment identiˇca-
tion in the UrQMD nucleon distribution at the ˇnal time and compare the results to the
experimental data.
1. FRAGMENTATION MODEL
The ˇrst step of a UrQMD calculation is to ˇnd the positions of all nucleons at the initial
stage of reaction. Projectile and target are modeled according to the Fermi-gas ansatz: it means
that the initial momenta of the nucleons are randomly chosen between 0 and the local ThomasÄ
Fermi momentum. The nucleons are represented by Gaussian shaped density distributions: its
binding energy should correspond to the value given by the BetheÄWeizséacker formula and
its radius should yield the following mass dependence: R(A) = r0A1/3, where A is the mass
number and the parameter r0 is a function of the nuclear matter ground state density ρ0 [6].
An example of the nucleon distribution for the reaction p + Fe (4 GeV/nucleon) at t = 0
is shown in Fig. 1 (left). Here the collision partners are already boosted to the initial velocity,
such that the nucleus is relativistically contracted. Then the time evolution of the spatial
distribution of nucleons and mesons (pions, kaons, etc.) is traced. For the propagation of the
nucleon coordinates, the set of coupled discretized Hamiltonian equations is solved:
pi(t + Δt) = pi(t) − Δt∇rUi([rj , pj ]),
ri(t + Δt) = ri(t) + Δtpi(t)/m.
(1)
Here Ui([rj , pj]) is the potential acting on particle i, which depends on the coordinates and
momenta of the other particles via the two-body interactions. In between the time steps the
collision term is evaluated stochastically taking into account the two-nucleon cross section.
One can follow the changes of nucleon distributions in time from one time step to another.
654 Mitsyn S. V. et al.
Fig. 1. Example of the time evolution for the reaction p+Fe (4 GeV/nucleon): left Å initial distribution
at t = 0; right Å distribution at time steps t = 5, 20, 35, and 50
The example of time evolution is represented in Fig. 1 (right) for the same reaction. One
can see that, according to the UrQMD results, the energy deposited in the nucleus as a
result of the collision is not distributed uniformly and the nucleus does not have time to
equilibrate. The reaction mechanism looks like a knockout reaction for the fast nucleon and
a fragmentation of the highly excited target residue.
The next step is to identify the nuclear fragments at the ˇnal time of the reaction and
determine their kinetic energies, masses, and binding energies. At the last stage the de-exci-
tation of the excited fragments is calculated.
2. CLUSTER FORMATION
As a result of the UrQMD calculation, we obtain a table with the positions, momenta, and
other characteristics of nucleons and other particles produced in the reaction for the chosen
time steps. To ˇnd the clusters in the ˇnal distribution, we assume that a group of A nucleons
forms a fragment when the following conditions are satisˇed:
1) The nucleons of the group are close to each other, while nucleons which do not belong
to the fragment are away from it.
2) The group radius, i.e., the maximal distance of a particle from the center of mass, is
not greater than Lmaxk = 1.5A
1/3, where A = N(Sk) is the number of nucleons in group k.
In addition, we assume that with condition (2) it is possible to demarcate realistic frag-
ments heuristically. From the many existing clustering algorithms [9] the single linkage
agglomerative [8] was chosen, because it was empirically proved that it was the best in
comparison with the other clustering algorithms.
The mathematical method for cluster recognition, implemented in the UrQMD code,
proceeds as follows. For each nucleon i a vector xi of Euclidean space R3 is deˇned,
which is used as a base for clustering. The clustering result is a set S = {Sj} of disjoint
sets, which are clusters Sj ⊂ V of nucleons V = 1, 2, . . . , n in the current time frame
(where n is the number of fragments to be clusterized). The clustering method is then divided
into 3 steps: (1) application of the single linkage cut to the nucleons; (2) singleton clusters
removal; (3) fragment postˇltering.
A Clustering Approach in the UrQMD Transport Model for Nuclear Collisions 655
1. Single linkage cut. The single linkage cut can be described by the following deˇnition:
a) Let G = 〈V, E〉 be a graph on nucleons V , where E is a set of edges deˇned as
E = {(s, t) |‖ xs − xt ‖< dmax}, s ∈ V, t ∈ V .
b) Then the set of sets S = {Sj} is a single linkage cut, if:
i) there is a one-to-one mapping between the connected components in G and S;
ii) each set Sj is a set of nucleons in a corresponding connected component.
2. Singleton clusters removal. The algorithm puts nucleons that are away from any other
cluster into distinct clusters which are called singletons. These singletons should be removed
(as they are not fragments). The removal is done in the following manner: S <= {Sj :
N(Sj) > 1}.
Postˇltering. It is clear that the algorithm can form clusters which are stretched in
coordinate space, while fragments should be compact in space according to the condition (2).
In order to meet this condition, we perform postˇltering on such fragments, removing distant
nucleons. The postˇltering algorithm is as follows. For each cluster Sk:
a) ˇnd a center of mass μk = M [xj ], j ∈ Sk;
b) erase from Sk nucleons which are farther from the center than Lmaxk : Sk <= {i |‖
xi − μk ‖ Lmaxk }.
Thus, Lmaxk has the meaning of the fragment radius.
3. RESULTS
The cluster formation method discussed above was applied to calculate nuclear fragmen-
tation in the reaction of protons with iron, p + Fe, at proton energies of 1.5 and 1.0 GeV
per nucleon. UrQMD calculations were performed, and the clusterization program allowed
us to obtain the distribution of masses of the produced fragments at different time steps.
It was observed that the smallest time step for the fragmentation process is about 50 fm/c.
The produced fragments are still in excited states. The excitation energies were estimated
Fig. 2. Mass distributions of fragments in p + Fe for different proton energies: a) 1.5 GeV;
b) 1.0 GeV [10]
656 Mitsyn S. V. et al.
from the energy conservation law and divided between fragments in proportion to their mass.
For comparison calculations were also done in the cascade+ percolation model that allows
one to take into account the excitation energy of produced clusters (see [10] and references
therein). The obtained mass distributions for both methods are shown in Fig. 2 and compared
with the experimental results [11]. One can see that the results of both calculations are much
more different from the experimental data, if we deal with primary excited nuclei, but ˇt
experimental data reasonably well after evaporation is taken into account.
CONCLUDING REMARKS
The method describing multifragmentation reactions for energies greater than 0.4 GeV/nu-
cleon with the UrQMD transport model is developed. It uses the single linkage clustering
approach with special postˇltering, which allows one to identify fragments in the ˇnal nu-
cleon distribution. The comparison with experimental data shows that the approach is quite
promising for physical applications. The software implementation of the clustering method
and the visualization of the cluster formations are done in a user-friendly framework, which
substantially facilitates the applications of this method.
REFERENCES
1. Li Ou // J. Phys. G: Nucl. Part. Phys. 2009. V. 36. P. 125104.
2. Aichelin J. // Phys. Rep. 1991. V. 202. P. 233 and references therein.
3. Gossiaux P. B. et al. // Nucl. Phys. A. 1997. V. 619. P. 379.
4. Skyrme T.H. R. Nonlinear Field Theory // Proc. Roy. Soc. Ser. A. 1961. V. 260. P. 127.
5. Kumar S., Puri R. K., Aichelin J. // Phys. Rev. C. 1998. V. 58. P. 1618.
6. Bleicher M. et al. // J. Phys. G. 1999. V. 25. P. 1859; hep-ph/9909407.
7. Avdeyev S. P. et al. // Nucl. Phys. A. 2002. V. 709. P. 392.
8. Agglomerative Hierarchical Cluster Tree Å MATLAB Linkage:
http://www.mathworks.com/help/stats/linkage.html. 2013.
9. Duran B. S., Odell P. L. Cluster Analysis. New York: Springer Verlag, 1947.
10. Musulmanbekov G., Polanski A. Development of Dynamical Model for Simulation of Nuclear
Spallation // XXI Intern. Baldin Seminar on High Energy Physics Problems, Dubna, Sept. 10Ä15,
2012. PoS (Baldin ISHEPP XXI) 096.
11. Villagrasa-Canton C. et al. // Phys. Rev. C. 2007. V. 75. P. 044603.


A CLUSTERING APPROACH IN THE UrQMD TRANSPORT MODEL FOR NUCLEAR COLLISIONS AT RELATIVISTIC ENERGIES

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A CLUSTERING APPROACH IN THE UrQMD TRANSPORT MODEL FOR NUCLEAR COLLISIONS AT RELATIVISTIC ENERGIES

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