17 Momentum and Density Dependence of the Nuclear Mean Field …
237
In the following section, we shall show that a finite range part in the NN-effective
force is an essential requirement to simulate the k-dependence in the mean field and
the k- and ρ-dependence of the nuclear mean field, can be considered independent
of each other. This we shall work out for the general case of NN-effective force in
the framework of non-relativistic mean field theory.
17.2 Finite Range Effective Interaction and Momentum
Dependence of the Mean Field
One can consider four different effective interactions, namely, the direct and exchange
interaction between pairs of like (l) and unlike (ul) nucleons: v
(l)
d (r ), v
(ul)
d (r ), v
(l)
ex (r )
and v
(ul)
ex (r ). These effective interactions are averaged over angles, spins, and isospins
of the two interacting nucleons and depend on the relative separation, r, between them
as well as on the total nucleon density ρ = ρ n + ρ p of the medium in which they are
interacting. The energy density functional H T (ρ n , ρ p ) in ANM obtained from these
effective interactions can be expressed as
H T (ρ n , ρ p ) =
2
2m
[ f
n
T (
→
k ) + f
p
T (
→
k )]k
2 d
3 k + V
T
(ρ n , ρ p ),
(17.6)
where the first term is the kinetic energy part and second one is the interaction
contribution given by
237
In the following section, we shall show that a finite range part in the NN-effective
force is an essential requirement to simulate the k-dependence in the mean field and
the k- and ρ-dependence of the nuclear mean field, can be considered independent
of each other. This we shall work out for the general case of NN-effective force in
the framework of non-relativistic mean field theory.
17.2 Finite Range Effective Interaction and Momentum
Dependence of the Mean Field
One can consider four different effective interactions, namely, the direct and exchange
interaction between pairs of like (l) and unlike (ul) nucleons: v
(l)
d (r ), v
(ul)
d (r ), v
(l)
ex (r )
and v
(ul)
ex (r ). These effective interactions are averaged over angles, spins, and isospins
of the two interacting nucleons and depend on the relative separation, r, between them
as well as on the total nucleon density ρ = ρ n + ρ p of the medium in which they are
interacting. The energy density functional H T (ρ n , ρ p ) in ANM obtained from these
effective interactions can be expressed as
H T (ρ n , ρ p ) =
2
2m
[ f
n
T (
→
k ) + f
p
T (
→
k )]k
2 d
3 k + V
T
(ρ n , ρ p ),
(17.6)
where the first term is the kinetic energy part and second one is the interaction
contribution given by