2 New Signatures of Phase Transition from Models of Nuclear Multifragmentation
15
weights ν / / tot (partial widths). The energy of the emitted particle is then obtained
by another Monte Carlo sampling of its energy spectrum. The energy, mass, and
charge of the nucleus is adjusted after each emission. This procedure is followed for
each of the primary fragment produced at a fixed temperature and then repeated over
a large ensemble and the observables are calculated from the ensemble averages.
The number and type of particles emitted and the final decay product in each event
are registered and are taken into account properly keeping in mind the overall charge
and baryon number conservation.
2.2.3 Lattice Gas Model
The lattice gas model is considerably more complicated than the percolation model
[22] but expositions of the model exist [8, 25, 26] and we refer to [26] for details. Let
A = N + Z be the number of nucleons in the system that dissociates. We consider
D
3 cubic boxes where each cubic box has volume (1.0/0.16) f m
3 . D
3 is larger than
A (they have the same value in the bond percolation model). Here, D
3
/A = V f /V 0
where V 0 is the normal volume of a nucleus with A nucleons and V f is the freezeout volume where partitioning of nucleons into clusters is computed. For nuclear
forces, one adopts nearest neighbor interactions. Following normal practice, we use
neutron–proton interactions v np = − 5.33 MeV and set v nn = v pp = 0.0. Coulomb
interaction between protons is included. Each cube can contain 1 or 0 nucleon. There
is a very large number of configurations that are possible (a configuration designates
which cubes are occupied by neutrons, which by protons and which are empty;
we sometimes call a configuration an event). Each configuration has an energy.
If a temperature is specified, the occupation probability of each configuration is
proportional to its energy: P∝exp(-E/T). This is achieved by Monte Carlo sampling
using the Metropolis algorithm.
Calculation of clusters needs further work. Once an event is chosen we ascribe to
each nucleon a momentum. Momentum of each nucleon is picked by Monte Carlo
sampling of a Maxwell–Boltzmann distribution for the prescribed temperature T.
Two neighboring nucleons are part of the same cluster if P
2
r /2μ + < 0 where is
v np or v nn or v pp . Here, P r is the relative momentum of the two nucleons and μ is
the reduced mass. If nucleon i is bound with nucleon j and j with k then i, j, k are
part of the same cluster. At each temperature, we calculate 50,000 events to obtain
average energy < E > and average multiplicity n a (where a is the mass number of
the cluster) of all clusters. A cluster with one nucleon is a monomer, one with two
nucleons is a dimer, and so on. The total multiplicity is M =
n a and
an a = A,
where A = N + Z is the mass number of the dissociating system.
15
weights ν / / tot (partial widths). The energy of the emitted particle is then obtained
by another Monte Carlo sampling of its energy spectrum. The energy, mass, and
charge of the nucleus is adjusted after each emission. This procedure is followed for
each of the primary fragment produced at a fixed temperature and then repeated over
a large ensemble and the observables are calculated from the ensemble averages.
The number and type of particles emitted and the final decay product in each event
are registered and are taken into account properly keeping in mind the overall charge
and baryon number conservation.
2.2.3 Lattice Gas Model
The lattice gas model is considerably more complicated than the percolation model
[22] but expositions of the model exist [8, 25, 26] and we refer to [26] for details. Let
A = N + Z be the number of nucleons in the system that dissociates. We consider
D
3 cubic boxes where each cubic box has volume (1.0/0.16) f m
3 . D
3 is larger than
A (they have the same value in the bond percolation model). Here, D
3
/A = V f /V 0
where V 0 is the normal volume of a nucleus with A nucleons and V f is the freezeout volume where partitioning of nucleons into clusters is computed. For nuclear
forces, one adopts nearest neighbor interactions. Following normal practice, we use
neutron–proton interactions v np = − 5.33 MeV and set v nn = v pp = 0.0. Coulomb
interaction between protons is included. Each cube can contain 1 or 0 nucleon. There
is a very large number of configurations that are possible (a configuration designates
which cubes are occupied by neutrons, which by protons and which are empty;
we sometimes call a configuration an event). Each configuration has an energy.
If a temperature is specified, the occupation probability of each configuration is
proportional to its energy: P∝exp(-E/T). This is achieved by Monte Carlo sampling
using the Metropolis algorithm.
Calculation of clusters needs further work. Once an event is chosen we ascribe to
each nucleon a momentum. Momentum of each nucleon is picked by Monte Carlo
sampling of a Maxwell–Boltzmann distribution for the prescribed temperature T.
Two neighboring nucleons are part of the same cluster if P
2
r /2μ + < 0 where is
v np or v nn or v pp . Here, P r is the relative momentum of the two nucleons and μ is
the reduced mass. If nucleon i is bound with nucleon j and j with k then i, j, k are
part of the same cluster. At each temperature, we calculate 50,000 events to obtain
average energy < E > and average multiplicity n a (where a is the mass number of
the cluster) of all clusters. A cluster with one nucleon is a monomer, one with two
nucleons is a dimer, and so on. The total multiplicity is M =
n a and
an a = A,
where A = N + Z is the mass number of the dissociating system.