17 Momentum and Density Dependence of the Nuclear Mean Field …
255
Using
(1)
T , the μ
(1)
T is obtained from the evaluation of the density, ρ =
f T (
→
k )d
3 k.
The process is repeated till the desired accuracy is achieved. This procedure simultaneously gives the distribution function f T (
→
k ), the single particle energy T (k, ρ)
and chemical potential μ T (ρ) [38]. The f T (
→
k ) thus determined is used to evaluate
H T (ρ). The pressure P T (ρ) in SNM can be obtained as
P T (ρ) = μ T (ρ)ρ − F T (ρ) = μ T (ρ)ρ − H T (ρ) + T S,
(17.65)
where F T (ρ) is the free energy density and S is the entropy density at density ρ
and temperature T. The entropy density S is expressed in terms of the occupancy
distribution function η T (
→
k ) as
S = −
ξ
(2π) 3
η T (
→
k ) ln η T (
→
k ) + [1 − η T (
→
k )]ln[1 − η T (
→
k )]
d
3 k.
(17.66)
17.3.3 Parameters for ANM
In order to study the properties of ANM, one needs to know the individual parameters
for the like and unlike contributions,
l
0 , ,
ul
0 , ,
l
γ , ,
ul
γ , ,
l
ex and
ul
ex , which are subject
to the constraint given in (17.48). Here, it is assumed implicitly that the ranges for
interaction between like and unlike pairs of nucleons are the same.
The momentum and temperature dependence of the mean field in nuclear matter
are solely simulated through the finite range exchange part. Therefore, the splitting of
the exchange strength ex into
l
ex and
ul
ex should be decided by utilizing constraint
following from the study in the related area. Here, the splitting is decided from
finite temperature calculation of nuclear matter. The splitting of ex into
l
ex and
ul
ex is subject to the constraint
l
ex +
ul
ex = 2 ex . Further, if
l
ex is known, then the
occupancy distribution function in pure neutron matter (PNM) can be calculated for
given ρ and T. Thus, the entropy density in PNM can be obtained from (17.66). The
entropy density in PNM is shown in Fig. 17.9 in panels (a) and (b) as a function
of density for arbitrary splittings of ex into
l
ex and
ul
ex , at T = 40 and 60 MeV,
respectively. The results of SNM at same ρ and T is also shown in the figures
as solid curves. The findings can be summarized as, if 0
l
ex <
2
3
ex , then the
PNM result for entropy density will cross-over the SNM one at some density which
corresponds to a larger value for higher
l
ex value in the range. For
2
3
ex < <
l
ex 2 ex ,
the results for entropy density in PNM lie below that of SNM at all densities. For
l
ex =
2
3
ex , it is found that the PNM results for entropy density approaches that of
SNM result asymptotically in the region of large density. These findings are true at
all temperature T. Once the splitting for ex into
l
ex and
ul
ex is known, then the n- and
p-effective masses in ANM can be studied. The n- and p-effective masses calculated
from (17.5) as a function of Y p for k = k f 0 and ρ = ρ 0 predict the n-effective mass
255
Using
(1)
T , the μ
(1)
T is obtained from the evaluation of the density, ρ =
f T (
→
k )d
3 k.
The process is repeated till the desired accuracy is achieved. This procedure simultaneously gives the distribution function f T (
→
k ), the single particle energy T (k, ρ)
and chemical potential μ T (ρ) [38]. The f T (
→
k ) thus determined is used to evaluate
H T (ρ). The pressure P T (ρ) in SNM can be obtained as
P T (ρ) = μ T (ρ)ρ − F T (ρ) = μ T (ρ)ρ − H T (ρ) + T S,
(17.65)
where F T (ρ) is the free energy density and S is the entropy density at density ρ
and temperature T. The entropy density S is expressed in terms of the occupancy
distribution function η T (
→
k ) as
S = −
ξ
(2π) 3
η T (
→
k ) ln η T (
→
k ) + [1 − η T (
→
k )]ln[1 − η T (
→
k )]
d
3 k.
(17.66)
17.3.3 Parameters for ANM
In order to study the properties of ANM, one needs to know the individual parameters
for the like and unlike contributions,
l
0 , ,
ul
0 , ,
l
γ , ,
ul
γ , ,
l
ex and
ul
ex , which are subject
to the constraint given in (17.48). Here, it is assumed implicitly that the ranges for
interaction between like and unlike pairs of nucleons are the same.
The momentum and temperature dependence of the mean field in nuclear matter
are solely simulated through the finite range exchange part. Therefore, the splitting of
the exchange strength ex into
l
ex and
ul
ex should be decided by utilizing constraint
following from the study in the related area. Here, the splitting is decided from
finite temperature calculation of nuclear matter. The splitting of ex into
l
ex and
ul
ex is subject to the constraint
l
ex +
ul
ex = 2 ex . Further, if
l
ex is known, then the
occupancy distribution function in pure neutron matter (PNM) can be calculated for
given ρ and T. Thus, the entropy density in PNM can be obtained from (17.66). The
entropy density in PNM is shown in Fig. 17.9 in panels (a) and (b) as a function
of density for arbitrary splittings of ex into
l
ex and
ul
ex , at T = 40 and 60 MeV,
respectively. The results of SNM at same ρ and T is also shown in the figures
as solid curves. The findings can be summarized as, if 0
l
ex <
2
3
ex , then the
PNM result for entropy density will cross-over the SNM one at some density which
corresponds to a larger value for higher
l
ex value in the range. For
2
3
ex < <
l
ex 2 ex ,
the results for entropy density in PNM lie below that of SNM at all densities. For
l
ex =
2
3
ex , it is found that the PNM results for entropy density approaches that of
SNM result asymptotically in the region of large density. These findings are true at
all temperature T. Once the splitting for ex into
l
ex and
ul
ex is known, then the n- and
p-effective masses in ANM can be studied. The n- and p-effective masses calculated
from (17.5) as a function of Y p for k = k f 0 and ρ = ρ 0 predict the n-effective mass