17 Momentum and Density Dependence of the Nuclear Mean Field …
243
u(k, ρ) = e(ρ) + ρ
de(ρ)
dρ
−
2 k
2
f
2m
+ u
ex
(k, ρ),
(17.30)
where u
ex
(k, ρ) is the functional that contains the k-dependence of the mean field in
SNM and is given as
u
ex
(k, ρ) =
ρ
2
[ j 0 (kr) − j 0 (k f r )]
3 j 1 (k f r )
k f r
v
l
ex (r ) + v
ul
ex (r )
d
3 r.
(17.31)
The saturation property of the iso-vector part of the mean field u τ (k = k f , ρ) is
related to the NSE, E s (ρ). Under the Taylor series expansion of energy density in
ANM H (ρ, β) around β = 0,
H (ρ, β) ∼ = H (ρ) +
1
2
β
2 ∂
2 H (ρ, β)
∂β 2
| β=0 .
(17.32)
The NSE is defined as
E s (ρ) =
1
2ρ
∂
2 H (ρ, β)
∂β 2
| β=0 .
(17.33)
Using (17.14) and (17.15) for H (ρ, β) in (17.33), the expression for E s (ρ) becomes
E s (ρ) =
2 k
2
f
6m
+
ρ
4
v
l
d − v
ul
d
d
3 r +
ρ
4
j
2
0 (k f r )
v
l
ex − v
ul
ex
d
3 r
−
ρ
4
j
2
1 (k f r )
v
l
ex + v
ul
ex
d
3 r.
(17.34)
Further, on evaluating the effective mass in SNM in the limit k → k f ,
m∗
m
=
1 +
m
2 k
∂u
∂k
−1
k=k f
, by using the mean field given in (17.20), one obtains
2 k
2
f
6m
m
m∗
(k = k f , ρ) − 1
= −
ρ
4
j
2
1 (k f r )(v
l
ex + v
ul
ex )d
3 r.
(17.35)
The iso-vector part of the mean field given in (17.27) at k = k f can be expressed, in
terms of NSE, E s (ρ) and effective mass in SNM by using (17.34) and (17.35). Thus
the expression for the iso-vector part u τ (k, ρ) in (17.29), where the k-dependence
has been separated out by subtracting the saturation contribution, becomes
u τ (k, ρ) = 2E s (ρ) −
2 k
2
f
3m ∗ (k = k f , ρ)
+ u
ex
τ (k, ρ),
(17.36)