240
T. R. Routray et al.
u
n
(k, ρ, Y p ) = ρ n
v
l
d (r )d
3 r + ρ p
v
ul
d (r )d
3 r
+ ρ n
3 j 1 (k n r )
k n r
v
l
ex (r ) j 0 (kr)d
3 r
+ ρ p
3 j 1 (k p r )
k p r
v
ul
ex (r ) j 0 (kr)d
3 r + u R (ρ n , ρ p ),
(17.16)
and
u
p
(k, ρ, Y p ) = ρ p
v
l
d (r )d
3 r + ρ n
v
ul
d (r )d
3 r
+ ρ p
3 j 1 (k p r )
k p r
v
l
ex (r ) j 0 (kr)d
3 r
+ ρ n
3 j 1 (k n r )
k n r
v
ul
ex (r ) j 0 (kr)d
3 r + u R (ρ n , ρ p ),
(17.17)
where j 0 (kr) is the zeroth order Bessel function as a function of the momentum k
of the traversing nucleon and u R (ρ n , ρ p ) is the rearrangement energy arising out of
any explicit dependence of the interactions on the total density, ρ = ρ n + ρ p ,
u R (ρ n , ρ p ) =
ρ
2
n + ρ
2
p
2ρ
ρ
∂v
l
d (r )
∂ρ
d
3 r +
ρ n ρ p
ρ
ρ
∂v
ul
d (r )
∂ρ
d
3 r
+
1
2ρ
ρ n
3 j 1 (k n r )
k n r
2
+
ρ p
3 j 1 (k p r )
k p r
2
ρ
∂v
l
ex (r )
∂ρ
d
3 r
+
ρ n ρ p
ρ
3 j 1 (k n r )
k n r
3 j 1 (k p r )
k p r
ρ
∂v
ul
ex (r )
∂ρ
d
3 r.
(17.18)
In the limit of symmetric nuclear matter, ρ n = ρ p =
ρ
2
and k n = k p = k f , where
k f = (
3π
2 ρ
2
)
1/3 is the Fermi momentum in SNM. The zero-temperature expression
for energy density in ANM given in (17.14) together with (17.15) in the limit of
SNM becomes
H (ρ) =
3
2
10m
k
2
f ρ +
ρ
2
4
(v
l
d (r ) + v
ul
d (r ))d
3 r
+
ρ
2
4
3 j 1 (k f r )
k f r
2
(v
l
ex (r ) + v
ul
ex (r ))d
3 r.
(17.19)
The single particle potentials, the n and p feels in ANM as given in (17.16) and
(17.17), are now the same in the limit of SNM and given as