17 Momentum and Density Dependence of the Nuclear Mean Field …
247
where the strength parameters 0 , , γ and ex of SNM are related to the corresponding
“l” and “ul” strengths of ANM as
0 =
l
0 +
ul
0
2
, , γ =
l
γ +
ul
γ
2
and ex =
l
ex +
ul
ex
2
(17.48)
and their expressions in terms of interaction parameters can be obtained from using
(17.43a)–(17.43f). The functions J (k f ) and I (k, k f ) can be written from (17.41)
and (17.45), respectively, by replacing k n( p) by k f . The explicit expressions of the
functions J (k f ) and I (k, k f ) can be written provided the form factor f (r ) of the finite
range interaction is specified. The two conventional forms, Yukawa and Gaussian,
are used widely in the nuclear calculations. For Yukawa form, f (r ) =
e
−r/α
(r/α)
, the
analytical expressions for J (k f ) and I (k, k f ) are given as
J Y (k f ) =
3 6
32k 6
f
+
9 4
8k 4
f
ln
1 +
4k 2
f
2
−
3 4
8k 4
f
+
9 2
4k 2
f
−
3 3
k 3
f
tan
−1
2k f
, (17.49)
I Y (k, k f ) =
3
2
((
2
+ k
2
f − k
2
)
8kk
3
f
ln
2
+ (k + k f )
2
2 + (k − k f ) 2
+
3
2
2k
2
f
−
3
3
2k
3
f
tan
−1
k + k f
− tan
−1
k − k f
.
(17.50)
For Gaussian form, f (r ) = e
−r
2 /α
2 , the corresponding expressions are
J G (k f ) =
⎡
⎣ 3 6
16k 6
f
−
9 4
8k 4
f
+
3 4
8k 4
f
−
3 6
16k 6
f
e
−4k 2
f
2 +
3 3
2k 3
f
2k f
0
e
−t 2 dt
⎤
⎦ , (17.51)
I G (k, k f ) =
3
4
8kk
3
f
ex p
−
k + k f
2
− ex p
−
k − k f
2
+
3
3
4k
3
f
(k+k f )
(k−k f )
e
−t
2 dt.
(17.52)
The momentum scale in these (17.49)–(17.52) is given by =
1
α
for the Yukawa
and =
2
α
for the Gaussian form of the interactions, where α is the corresponding
range of the interactions. It may be noted that at a given density ρ, the functions
I Y (k, k f ) and I G (k, k f ) vanish in the limit of high momentum k. Similarly, the
functions J Y (k f ) and J G (k f ) vanish in the limit of very high density ρ.