9 PHQMD—A Microscopic Transport Approach for Heavy-Ion …
109
V (r i0 , t) =
j, j =i
d
3 r i d
3 r j d
3 p i d
3 p j V (r i , r j , r i0 , r j0 )
× f (r i , p i , r i0 , p i0 , t) f (r j , p j , r j0 , p j0 , t),
(9.7)
with f being the Wigner density of the particles
f (r i , p i , r i0 , p i0 , t) =
1
π 3 3 e
−
2
L (r i −r i0 (t))
2 e
−
L
2 2 (p i −p i0 (t))
2 .
(9.8)
Numerical test have shown that the time evolution of the system does not change
if we replace 1/2(ρ int (r i0 , t) + ρ int (r j0 , t)) by ρ int (r i0 , t) or by ρ int (r j0 , t). For the
Skyrme potential we can therefore use the analytical form
V Skyrme (r i0 , t) = α
ρ int (r i0 , t)
ρ 0
+ β
ρ int (r i0 , t)
ρ 0
γ
.
(9.9)
The expectation value of the Coulomb interaction can also be calculated analytically.
The expectation value of the Hamiltonian which enters in (9.2) is finally given by
H = =T + +V
(9.10)
=
i
p
2
i0 + m 2 − m
+
i
V Skyrme (r i0 , t).
p-A experiments have shown that the potential energy of nuclear matter does not
only depend on the density but also on the relative momentum between proton and
heavy ion. This momentum dependence is quite often neglected because it is difficult to implement in transport approaches. It is, however, important if one wants to
describe the data quantitatively because momentum dependent interactions influence
observables in a different way than static interactions. For example, the maximal density which is reached in heavy-ion reactions varies considerably as a function of the
centrality and so does the static interaction. The momentum dependent interaction,
on the contrary, remains constant. Here we introduce a potential which presents a fit
to the most recent data [19] which are presented in Fig. 9.1. We see that they changed
quite a bit as compared to the old data which are represented by a fit. This old fit
has been used up to now in all QMD and IQMD calculations [20]. The momentum
dependent part of the potential we parametrize by
V mom (p i0 , p j0 ) = exp
−c
(p i0 − p j0 ) 2
a(p i0 − p j0 )
2
+ b(p i0 − p j0 )
4
,
ρ
ρ 0
(9.11)
109
V (r i0 , t) =
j, j =i
d
3 r i d
3 r j d
3 p i d
3 p j V (r i , r j , r i0 , r j0 )
× f (r i , p i , r i0 , p i0 , t) f (r j , p j , r j0 , p j0 , t),
(9.7)
with f being the Wigner density of the particles
f (r i , p i , r i0 , p i0 , t) =
1
π 3 3 e
−
2
L (r i −r i0 (t))
2 e
−
L
2 2 (p i −p i0 (t))
2 .
(9.8)
Numerical test have shown that the time evolution of the system does not change
if we replace 1/2(ρ int (r i0 , t) + ρ int (r j0 , t)) by ρ int (r i0 , t) or by ρ int (r j0 , t). For the
Skyrme potential we can therefore use the analytical form
V Skyrme (r i0 , t) = α
ρ int (r i0 , t)
ρ 0
+ β
ρ int (r i0 , t)
ρ 0
γ
.
(9.9)
The expectation value of the Coulomb interaction can also be calculated analytically.
The expectation value of the Hamiltonian which enters in (9.2) is finally given by
H = =T + +V
(9.10)
=
i
p
2
i0 + m 2 − m
+
i
V Skyrme (r i0 , t).
p-A experiments have shown that the potential energy of nuclear matter does not
only depend on the density but also on the relative momentum between proton and
heavy ion. This momentum dependence is quite often neglected because it is difficult to implement in transport approaches. It is, however, important if one wants to
describe the data quantitatively because momentum dependent interactions influence
observables in a different way than static interactions. For example, the maximal density which is reached in heavy-ion reactions varies considerably as a function of the
centrality and so does the static interaction. The momentum dependent interaction,
on the contrary, remains constant. Here we introduce a potential which presents a fit
to the most recent data [19] which are presented in Fig. 9.1. We see that they changed
quite a bit as compared to the old data which are represented by a fit. This old fit
has been used up to now in all QMD and IQMD calculations [20]. The momentum
dependent part of the potential we parametrize by
V mom (p i0 , p j0 ) = exp
−c
(p i0 − p j0 ) 2
a(p i0 − p j0 )
2
+ b(p i0 − p j0 )
4
,
ρ
ρ 0
(9.11)