6 D. Chatterjee et al.
fm
5
and calculate the corresponding effective surface energy coefficient a
s ef f
= E
s
A
23
. We then compare it with data from a compilation of Skyrme models
Danielewicz Lee ,
2009 in Fig.
1 . This leads to a
value of C
f in
≈ 75 ± 25.
5 10
15 20
25 30
40 60
80 100
120 140
a
s eff
= E
s
A
23
MeV
C
fin
A = 100
ref ρ
sat
- σ
ρ
sat
+ σ
E
sat
- σ
E
sat
+ σ
K
sat
- σ
K
sat
+ σ
mm- σ
mm+ σ
Figure 1.
Constraint on the finite size parameter using effective surface energy coefficient from a compilation of Skyrme models
black lines.
3.1.2 Method 2 An improved estimate of finite size parameter can be
achieved by comparing the isoscalar surface energy coefficient a
s
= E
IS s
A
23
with the values deduced from systematics of binding energies of finite nuclei
Jodon et al. ,
2016 in Fig.
2 . The value of C
f in
ob- tained using this method is roughly 77.5 ± 12.5.
5 10
15 20
25 30
40 60
80 100
120 140
a
s
= E
s IS
A
23
MeV
C
fin
A = 100
ref ρ
sat
- σ
ρ
sat
+ σ
E
sat
- σ
E
sat
+ σ
K
sat
- σ
K
sat
+ σ
mm- σ
mm+ σ
Figure 2.
Constraint on finite size parameter using surface en- ergy coefficient deduced from systematics of binding energies of
finite nuclei black lines.
3.1.3 Effect of uncertainty of empirical parameters on nuclear surface properties
Using the estimated values of C
f in
determined in the previous section, we study the effect of uncertainty in
the empirical parameters, on the effective surface energy coefficient a
ef f s
Fig. 3
and the diffuseness parameter a Fig.
4 . We vary each empirical parameter one by
one keeping the others fixed. We find that among the isoscalar empirical parameters, uncertainties in the sat-
uration density ρ , finite size parameter C
f in
and the effective mass m ∗ m have the largest effect on the sur-
face energy coefficient a
ef f s
. For the diffuseness param- eter a, the incompressbility K
sat
as well as C
f in
and m ∗ m have the largest influence. The isovector empiri-
cal parameters only have a significant influence at large asymmetry.
3.2 Estimate of finite size parameter using nuclear masses
The estimation of C
f in
in the previous section relies in the uniqueness of the definition of the surface energy.
Unfortunately, the surface energy is not a direct experi- mental observable and the distinction between bulk and
surface requires some modeling. Therefore, we cannot be sure that the functional obtained leads to a reason-
able estimation of the nuclear masses. In an alterna- tive approach, we constrain C
f in
using a fit to exper- imental nuclear masses. For a range of nuclear masses
A, we plot the difference in energy per particle, calcu- lated using ETF model including Coulomb contribu-
tion and experimental values from AME2012 mass ta- ble
Audi et al. ,
2012 ;
Wang et al. ,
2012 .
To adjust the value of C
f in
, we calculate the value of χ
2
= 1
N
N
X
i=1
E
i th
− E
i exp
E
i exp
2
for different ρ
sat
, C
f in
, C
so
. The value corresponding to the minimum of χ
2
at ρ
sat
= 0.154f m
− 3
is found to be C
f in
= 61 corresponding to C
so
= 40, while that corresponding to C
so
= 0 is C
f in
= 59. The corresponding plot for the residuals is displayed in Fig.
5 for the two choices of finite size parameters. It is
evident from the figure that the effect of changing the value of C
so
on the minimum of the energy is negligible. In order to study the sensitivity of the energy per
particle to the uncertainty in the empirical parameters, the effect of variations of the isoscalar empirical
parameters ρ
sat
, E
sat
, K
sat
within error bars on the energy residuals is displayed in Fig.
6 . It is ev-
ident from the figure that apart from the effective mass, C
f in
has the largest effect on the energy residuals.
An empirical NS crust-core description 7
18.5 19
19.5 20
20.5 21
21.5
0.1 0.2
0.3 0.4
0.5 a
s eff
I ρ
sat
ρ
sat
+ σ
ρ
sat
- σ
a Uncertainty in saturation density
17 18
19 20
21 22
23
0.1 0.2
0.3 0.4
0.5 a
s eff
I C
fin
C
fin
+ σ
C
fin
- σ
b Uncertainty in finite size parameter
16 17
18 19
20 21
22 23
0.1 0.2
0.3 0.4
0.5 a
s eff
I m
m+ σ
m- σ
c Uncertainty in effective mass
Figure 3.
Effect of uncertainty in empirical parameters on the variation of effective surface energy coefficient with asymmetry I
0.4 0.42
0.44 0.46
0.48 0.5
0.52 0.54
0.56 0.58
0.1 0.2
0.3 0.4
0.5 a fm
I K
sat
K
sat
+ σ
K
sat
- σ
a Uncertainty in incompressibility
0.35 0.4
0.45 0.5
0.55 0.6
0.1 0.2
0.3 0.4
0.5 a fm
I C
fin
C
fin
+ σ
C
fin
- σ
b Uncertainty in finite size parameter
0.35 0.4
0.45 0.5
0.55 0.6
0.1 0.2
0.3 0.4
0.5 a fm
I m
m+ σ
m- σ
c Uncertainty in effective mass
Figure 4.
Effect of uncertainty in empirical parameters on the variation of diffuseness parameter with asymmetry I
8 D. Chatterjee et al.
-1 -0.5
0.5 1
20 30
40 50
60 70
80 90
100 E
tot
-E
exp
A MeV
A
C
fin
=61,C
so
=40 C
fin
=59,C
so
=0
Figure 5. Difference between theoretical and experimental values
of energy of symmetric nuclei per particle, for the two choices of finite size parameters in Sec.
3.2 .
-2 -1.5
-1 -0.5
0.5 1
1.5 2
20 30
40 50
60 70
80 90
100
E
tot
-E
exp
A MeV
A I = 0
ref ρ
sat+ σ
ρ sat-
σ E
sat+ σ
E sat+
σ K
sat+ σ
K sat+
σ mm+
σ mm-
σ C
fin+ σ
C fin-
σ
Figure 6.
Sensitivity of the difference between theoretical and experimental values of energy of symmetric nuclei per particle, to
the uncertainty in isoscalar empirical parameters.
3.3 Asymmetric nuclei
