30
S. Mallik et al.
considers a collision between < p i > and < p j > and obtain a p for < p i > and
−p for < p j >. This p is added to all p i ’s and −p to all p j ’s. This conserves
both energy and momentum as one progresses in time. For the second event, new
Monte Carlo sampling of A p on A t will be started at time zero, similarly for event 3,
event 4, etc. The calculation of the collision part becomes very time-consuming and,
for this case within each time step, two-body collision is need to be checked between
(A p + A t )N test test particles. Now, to study nuclear liquid–gas phase transition one
needs to simulate collisions between fairly large nuclei. Therefore, it is very difficult to handle this operation with the existing model. Hence, one has to modify the
transport model so that it can be used for fairly large nuclei.
To overcome this problem, the fluctuation added BUU method is modified in
the following way. N test Monte Carlo simulations of A p nucleons with positions
and momenta and N test simulations of A t nucleons with positions and momenta are
to be done as before. As in cascade calculation [24, 26, 28] for nucleon–nucleon
collisions 1 on 1’(event1), 2 on 2’(event2) etc are considered with cross section σ nn .
For event 1, within each time step, nn collisions only between 1 and 1’ (i.e., between
first (A p + A t ) test particles) will be considered. The collision is checked for Pauli
blocking. If a collision between i and j in event 1 is allowed, [27] is to be followed
and N test − 1 test particles closest to i are to be picked and the same momentum
change p of them as ascribed to i is to be given. Similarly, N test − 1 test particles
closest to j are to be selected and these are to be ascribed the momentum change
−p, the same as suffered by j. As a function of time, this is continued till event 1
is over. For Vlasov propagation, all test particles are utilized. For event 2, one has to
return to time t = 0, the original situation (or a new Monte Carlo sampling for the
original nuclei), follow the above procedure but consider nn collisions only between
2 and 2’ {i.e., between (A p + A t ) + 1 to 2(A p + A t ) test particles}. This can be
repeated for as many events as one needs to build up enough statistics. Finally, to
identify fragments, two test particles are considered as the part of the same cluster
if the distance between them is less than or equal to 2 fm [17, 30].
Before applying the modified method in phase transition study, at first, one has
to check whether modified prescription results are comparable with the existing
BUU prescription or not. For this, simulations have to be done by using both the
methods for a central collision reaction of symmetric system
40 Ca+
40 Ca. Figure 3.1
shows the comparison of mass distribution obtained from existing and modified
BUU prescription for beam energies 25 and 50 MeV/nucleon at t = 200 fm/c. The
results obtained from two methods are similar because (a) the number of collisions
in an event is statistically the same. (b) In the original formulation, the objects
that collided were picked from a fine-grain sampling of phase-space density. In the
modified method, these are picked from a coarse grain sampling of the same phasespace density. But many events are needed, so statistically, it should not matter. (c)
Characteristics of scattering are the same and (d) The same Vlasov propagation is
used.
The advantage of this method over the existing method is that here, for one event,
nn collisions need to be considered between (A p + A t ) test particles, whereas in the
existing method, collisions need to be checked between (A p + A t )N test test particles.
S. Mallik et al.
considers a collision between < p i > and < p j > and obtain a p for < p i > and
−p for < p j >. This p is added to all p i ’s and −p to all p j ’s. This conserves
both energy and momentum as one progresses in time. For the second event, new
Monte Carlo sampling of A p on A t will be started at time zero, similarly for event 3,
event 4, etc. The calculation of the collision part becomes very time-consuming and,
for this case within each time step, two-body collision is need to be checked between
(A p + A t )N test test particles. Now, to study nuclear liquid–gas phase transition one
needs to simulate collisions between fairly large nuclei. Therefore, it is very difficult to handle this operation with the existing model. Hence, one has to modify the
transport model so that it can be used for fairly large nuclei.
To overcome this problem, the fluctuation added BUU method is modified in
the following way. N test Monte Carlo simulations of A p nucleons with positions
and momenta and N test simulations of A t nucleons with positions and momenta are
to be done as before. As in cascade calculation [24, 26, 28] for nucleon–nucleon
collisions 1 on 1’(event1), 2 on 2’(event2) etc are considered with cross section σ nn .
For event 1, within each time step, nn collisions only between 1 and 1’ (i.e., between
first (A p + A t ) test particles) will be considered. The collision is checked for Pauli
blocking. If a collision between i and j in event 1 is allowed, [27] is to be followed
and N test − 1 test particles closest to i are to be picked and the same momentum
change p of them as ascribed to i is to be given. Similarly, N test − 1 test particles
closest to j are to be selected and these are to be ascribed the momentum change
−p, the same as suffered by j. As a function of time, this is continued till event 1
is over. For Vlasov propagation, all test particles are utilized. For event 2, one has to
return to time t = 0, the original situation (or a new Monte Carlo sampling for the
original nuclei), follow the above procedure but consider nn collisions only between
2 and 2’ {i.e., between (A p + A t ) + 1 to 2(A p + A t ) test particles}. This can be
repeated for as many events as one needs to build up enough statistics. Finally, to
identify fragments, two test particles are considered as the part of the same cluster
if the distance between them is less than or equal to 2 fm [17, 30].
Before applying the modified method in phase transition study, at first, one has
to check whether modified prescription results are comparable with the existing
BUU prescription or not. For this, simulations have to be done by using both the
methods for a central collision reaction of symmetric system
40 Ca+
40 Ca. Figure 3.1
shows the comparison of mass distribution obtained from existing and modified
BUU prescription for beam energies 25 and 50 MeV/nucleon at t = 200 fm/c. The
results obtained from two methods are similar because (a) the number of collisions
in an event is statistically the same. (b) In the original formulation, the objects
that collided were picked from a fine-grain sampling of phase-space density. In the
modified method, these are picked from a coarse grain sampling of the same phasespace density. But many events are needed, so statistically, it should not matter. (c)
Characteristics of scattering are the same and (d) The same Vlasov propagation is
used.
The advantage of this method over the existing method is that here, for one event,
nn collisions need to be considered between (A p + A t ) test particles, whereas in the
existing method, collisions need to be checked between (A p + A t )N test test particles.