268
A. Quddus and S. K. Patra
K
N M
0
(x) = −83.4ρ
2/3
0 (x) + 4b 5 ρ
4/3
0 (x) + 10b 6 ρ
5/3
0 (x),
(18.16)
respectively. These quantities are folded in the (18.23)–(18.25) with the weight function to find the corresponding quantities of finite nuclei within the CDFM.
In the CDFM, the one-body density matrix ρ (r, r
) of a nucleus is written as a
coherent superposition of infinite number of one-body density matrices ρ x (r, r
) for
Fluctons (spherical pieces of the nuclear matter) [11, 35, 36],
ρ x (r) = ρ 0 (x))(x − |r|),
(18.17)
with ρ o (x) =
3A
4π x 3 , where x is the spherical radius of a nucleus contained in a uniformly distributed spherical Fermi gas. The one-body density matrix for a finite
nuclear system can be given as [11, 26, 36],
ρ(r, r
) =
∞
0
dx| f (x)|
2
ρ x (r, r
),
(18.18)
where | f (x)|
2 is the weight function (18.22). The term ρ x (r, r
) is the coherent
superposition of the one-body density matrix and defined as
ρ x (r, r
) = 3ρ 0 (x)
J 1
k f (x)|r − r
|
k f (x)|r − r |
×
x −
|r + r
|
2
.
(18.19)
Here, J 1 is the first-order spherical Bessel function. The Fermi momentum of nucleons in the Fluctons with radius x is expressed as k f (x) = (3π
2
/2ρ 0 (x))
1/3
= γ /x,
where γ = (9π A/8)
1/3
≈ 1.52 A
1/3 . The Wigner distribution function of the one
body density matrices in (18.19) is
W (r, k) =
∞
0
dx| f (x)|
2 W x (r, k).
(18.20)
Here, W x (r, k) =
4
8π 3 (x − |r|))(k F (x) − |k|). Similarly, the density ρ (r) within
CDFM can express in terms of the same weight function as
ρ(r ) =
dkW (r, k) =
∞
0
dx| f (x)|
2 3A
4π x 3 (x − |r|)
(18.21)
and it is normalized to the nucleon numbers of the nucleus,
ρ(r)dr = A. The differential equation for the weight function can be obtained in the generator coordinate
by taking the δ-function approximation to the Hill–Wheeler integral equation [36].
The weight function for a given density distribution ρ (r) can be expressed as,
| f (x)|
2
= −
1
ρ 0 (x)
dρ(r )
dr
r =x
,
(18.22)
A. Quddus and S. K. Patra
K
N M
0
(x) = −83.4ρ
2/3
0 (x) + 4b 5 ρ
4/3
0 (x) + 10b 6 ρ
5/3
0 (x),
(18.16)
respectively. These quantities are folded in the (18.23)–(18.25) with the weight function to find the corresponding quantities of finite nuclei within the CDFM.
In the CDFM, the one-body density matrix ρ (r, r
) of a nucleus is written as a
coherent superposition of infinite number of one-body density matrices ρ x (r, r
) for
Fluctons (spherical pieces of the nuclear matter) [11, 35, 36],
ρ x (r) = ρ 0 (x))(x − |r|),
(18.17)
with ρ o (x) =
3A
4π x 3 , where x is the spherical radius of a nucleus contained in a uniformly distributed spherical Fermi gas. The one-body density matrix for a finite
nuclear system can be given as [11, 26, 36],
ρ(r, r
) =
∞
0
dx| f (x)|
2
ρ x (r, r
),
(18.18)
where | f (x)|
2 is the weight function (18.22). The term ρ x (r, r
) is the coherent
superposition of the one-body density matrix and defined as
ρ x (r, r
) = 3ρ 0 (x)
J 1
k f (x)|r − r
|
k f (x)|r − r |
×
x −
|r + r
|
2
.
(18.19)
Here, J 1 is the first-order spherical Bessel function. The Fermi momentum of nucleons in the Fluctons with radius x is expressed as k f (x) = (3π
2
/2ρ 0 (x))
1/3
= γ /x,
where γ = (9π A/8)
1/3
≈ 1.52 A
1/3 . The Wigner distribution function of the one
body density matrices in (18.19) is
W (r, k) =
∞
0
dx| f (x)|
2 W x (r, k).
(18.20)
Here, W x (r, k) =
4
8π 3 (x − |r|))(k F (x) − |k|). Similarly, the density ρ (r) within
CDFM can express in terms of the same weight function as
ρ(r ) =
dkW (r, k) =
∞
0
dx| f (x)|
2 3A
4π x 3 (x − |r|)
(18.21)
and it is normalized to the nucleon numbers of the nucleus,
ρ(r)dr = A. The differential equation for the weight function can be obtained in the generator coordinate
by taking the δ-function approximation to the Hill–Wheeler integral equation [36].
The weight function for a given density distribution ρ (r) can be expressed as,
| f (x)|
2
= −
1
ρ 0 (x)
dρ(r )
dr
r =x
,
(18.22)