Along with the scaling calculation, the monopole vibration can also be studied
with a constrained approach [76, 103, 104, 105, 106]. In the constrained
method, one has to minimize the constrained energy functional:

$\displaystyle\int{dr}\left[{\cal
H}-{\eta}{r}^{2}{\rho}\right]=E(\eta)-\eta\int{dr}{r}^{2}\rho,$ (3.43)

with respect to the variation of densities and meson fields. Here $\eta$ is
the constrained parameter. The densities, field and energy obtained from the
solution of the above constrained equation ( Eq. 3.43 ) are the function of
parameter $\eta$. The rms radius of the nucleus is given in terms of the
parameter $\eta$:

${R_{\eta}}=\bigg{[}\frac{1}{A}\int drr^{2}\rho\bigg{]}^{1/2},$ (3.44)

where A is the mass number of the nucleus. We are assuming that the parameter
$\eta$ is very small, so E($\eta$) can be expanded around $\eta$=0, which
corresponds to the ground state energy of the nucleus and $R_{0}$ gives the
rms radius of the nucleus in ground state. The expansion of $E(\eta)$ around
$\eta=0$ is given by:

$E(\eta)=E(0)+{R_{0}}\frac{\partial
E(\eta)}{\partial\eta}\bigg{|}_{\eta=0}+{R_{0}}\frac{\partial
E(\eta)}{\partial\eta}\bigg{|}_{\eta=0}.$ (3.45)

The second order derivative of the expansion gives the effective
incompressibility of finite nucleus and it is defined as :

${K_{A}}^{c}=\frac{1}{A}{R_{0}}\frac{\partial^{2}E(\eta)}{\partial\eta^{2}}\bigg{|}_{\eta=0}.$
(3.46)

We can calculate the incompressibility using this simple formula. In classical
approximation the average constraint excitation energy is given by :
${E_{x}}^{c}=\sqrt{\frac{A{K_{A}}^{c}}{B_{m}^{c}}}$, where $B_{m}^{c}$ is the
mass parameter. The expression for the mass parameter both in relativistic and
non-relativistic are given in Eq. ( 3.41) and Eq. ( 3.42), respectively.
Again, in the sum rule approach constrained excitation energy can be expressed
as the ratio of moments of strength distribution. Constrained energy is
defined as : $E_{x}^{c}=\sqrt{\frac{m_{1}}{m_{-1}}}$, where $m_{1}$ and
$m_{-1}$ are the energy weighted and inverse energy weighted moments,
respectively. $m_{-1}$ is interpreted as the polarizability of the system ie.
nothing but the measure of energy changed due to excitation of the nucleus.
Expression for the $m_{-1}$ is given by :

$m_{-1}=-\frac{A}{2}\frac{\partial{R_{\eta}}^{2}}{\partial\eta}\bigg{|}_{\eta=0}=\frac{1}{2}\frac{\partial^{2}E(\eta)}{\partial\eta^{2}}\bigg{|}_{\eta=0}.$
(3.47)

One can estimate $m_{-1}$ by calculating first derivative of ${R_{\eta}}^{2}$
with respect to $\eta$ or second derivative of $E(\eta)$ with respect to the
$\eta$. We have calculated $m_{-1}$ using both 5-point and 3-point formula.
$m_{1}$ is proportional to the mass parameter, which can expressed as :
$m_{1}=\frac{2}{m}A<r^{2}>=\frac{2}{m^{2}}{B_{M}}^{nr}$. From the inequality
relations satisfied by the moments of the strength distribution, we get
$\frac{m_{1}}{m_{-1}}\leq\frac{m_{3}}{m_{1}}$. This implies that the scaling
energy gives the upper limit of the resonance, while the constrained energy
gives the lower limit of the resonance spectra. The resonance width [76, 107]
is defined as:

$\displaystyle{\Sigma}$ $\displaystyle=$
$\displaystyle\sqrt{\left({E_{m}}^{s}\right)^{2}-\left({E_{m}}^{c}\right)^{2}}=\sqrt{({\frac{m_{3}}{m_{1}}})^{2}-{(\frac{m_{1}}{m_{-1}}})^{2}}.$
(3.48)

###  3.3 Results and Discussions

Table 3.1: The calculated binding BE and charge radius $r_{ch}$ obtained from relativistic extended Thomas-Fermi (RETF) approximation is compared with relativistic Hartree (with various parameter sets) and experimental results [108, 109].The RETF results are given in the parentheses. The empirical values [110, 19] of nuclear matter saturation density $\rho_{0}$, binding energy per nucleon BE/A, incompressibility modulus $K$, asymmetry parameter $J$, and ratio of effective mass to the nucleon mass $M^{*}/M$ are given in the lower part of the table. Energies are in MeV and radii are in fm.  Nucleus | Set | BE (calc.) | BE (Expt.) |  $r_{ch}$ (calc.) |  $r_{ch}$ (Expt.)
---|---|---|---|---|---
16O | NL1 | 127.2(118.7) | 127.6 | 2.772(2.636) | 2.699
| NL3 | 128.7(120.8) |  | 2.718(2.591) |
| NL3∗ | 128.1(119.5) |  | 2.724(2.603) |
| FSUGold | 127.4(117.8) |  | 2.674(2.572) |
40Ca | NL1 | 342.3(344.783) | 342.0 | 3.501(3.371) | 3.478
| NL3 | 341.6(346.2) |  | 3.470(3.343) |
| NL3∗ | 341.5(344.2) |  | 3.470(3.349) |
| FSUGold | 340.8(342.2) |  | 3.429(3.327) |
48Ca | NL1 | 412.7(419.5) | 416.0 | 3.501(3.445) | 3.477
| NL3 | 414.6(422.6) |  | 3.472(3.426) |
| NL3∗ | 413.5(420.3) |  | 3.469(3.429) |
| FSUGold | 411.2(418.0) |  | 3.456(3.418) |
90Zr | NL1 | 784.3(801.1) | 783.9 | 4.284(4.232) | 4.269
| NL3 | 781.4(801.7) |  | 4.273(4.219) |
| NL3∗ | 781.6(798.7) |  | 4.267(4.219) |
| FSUGold | 778.8(797.3) |  | 4.257(4.214) |
116Sn | NL1 | 989.5(1013.7) | 988.7 | 4.625(4.583) | 4.625
| NL3 | 985.4(1014.6) |  | 4.617(4.571) |
| NL3∗ | 986.4(1011.0) |  | 4.609(4.569) |
| FSUGold | 984.4(1010.7) |  | 4.611(4.569) |
208Pb | NL1 | 1638.1(1653.7) | 1636.4 | 5.536(5.564) | 5.501
| NL3 | 1636.9(1661.2) |  | 5.522(5.541) |
| NL3∗ | 1636.5(1655.2) |  | 5.512(5.538) |
| FSUGold | 1636.2(1661.4) |  | 5.532(5.541) |
Set | NL1 | NL3 | NL3∗ | FSUGold | empirical
$\rho_{0}$ | 0.154 | 0.150 | 0.148 | 0.148 | 0.17
$E/A$ | 16.43 | 16.31 | 16.30 | 16.30 | 15.68
$K$ | 211.7 | 271.76 | 258.27 | 230.0 | $210\pm 30$
$J$ | 43.6 | 38.68 | 37.4 | 32.597 | $32\pm 2$
$M^{*}/M$ | 0.57 | 0.594 | 0.60 | 0.61 | 0.6

####  3.3.1 Force parameters of relativistic mean field formalism

First of all, we examined the predictive power of various parameter sets. In
this context we selected NL1 as a successful past set, and a few recently used
forces like NL3, NL3∗, and FSUGold with varying incompressibilities as shown
in Table 4.1 (lower part of the table). The ground state observable obtained
by these forces are depicted in Table 4.1. Along with the relativistic
extended Thomas-Fermi (RETF) results, the relativistic Hartree values are also
compared with the experimental data [108, 109]. The calculated RMF results
obtained by all the force parameters considered in the present thesis are very
close to the experimental data [108, 109]. A detailed analysis of the binding
energy and charge radius clearly show that NL1 and FSUGold have superior
predictive power for 16O in RMF level. The advantage of FSUGold decreases with
increased mass number of the nucleus. Although the predictive power of the
relatively old NL1 set is very good for binding energy, it has a large
asymmetry coefficient $J$, which may mislead the prediction in unknown
territories, like the neutron drip-line or super heavy regions. The RETF
prediction of binding energy and charge radius (numbers in the parenthesis) is
very poor with the experimental data as compared to the RMF calculations.
However, for relatively heavy masses, the RETF results can be used within
acceptable error. In general, taking into account the binding energy BE and
root mean square charge radius $r_{ch}$, one may prefer to use either of the
NL3 or NL3∗ parametrization.

Table 3.2: The results of isoscalar giant monopole resonance with various parameter sets for some known nuclei are compared with recent experimental data [111]. The calculations are done with relativistic extended Thomas-Fermi (RETF) approximation using both scaling and constrained schemes. The values of $\Delta_{1}$, $\Delta_{2}$, and $\Delta_{3}$ are obtained by subtracting the results of (NL3, NL3∗), (NL3∗, FSUGold), and (NL3, FSUGold), respectively. The monopole excitation energies with scaling $E^{s}$ and constrained $E^{c}$ are in MeV.  Nucleus | NL1 | NL3 | NL3∗ | FSUGold | Expt. | $\Delta_{1}$ | $\Delta_{2}$ | $\Delta_{3}$
---|---|---|---|---|---|---|---|---
| ${E}^{s}$ | ${E}^{c}$ | ${E}^{s}$ | ${E}^{c}$ | ${E}^{s}$ | ${E}^{c}$ | ${E}^{s}$ | ${E}^{c}$ |  |  |  |
${}^{16}{O}$ | 23.31 | 21.75 | 27.83 | 25.97 | 26.86 | 25.20 | 26.97 | 25.17 | 21.13$\pm$0.49 | 0.97 | 0.11 | 0.86
${}^{40}{Ca}$ | 20.61 | 19.77 | 24.01 | 23.16 | 23.32 | 22.48 | 22.98 | 22.30 | 19.20$\pm$0.40 | 0.69 | 0.43 | 1.03
${}^{48}{Ca}$ | 19.51 | 18.67 | 22.69 | 21.73 | 22.01 | 21.11 | 21.72 | 20.88 | 19.90$\pm$0.20 | 0.68 | 0.29 | 0.97
${}^{90}{Zr}$ | 16.91 | 16.41 | 19.53 | 19.03 | 18.97 | 18.50 | 18.60 | 18.21 | $17.89\pm 0.20$ | 0.56 | 0.37 | 0.97
${}^{110}{Sn}$ | 15.97 | 15.50 | 18.42 | 17.94 | 17.90 | 17.44 | 17.52 | 17.13 |  | 0.52 | 0.38 | 0.90
${}^{112}{Sn}$ | 15.87 | 15.39 | 18.29 | 17.81 | 17.78 | 17.32 | 17.42 | 17.02 | 16.1$\pm$0.10 | 0.51 | 0.36 | 0.86
${}^{114}{Sn}$ | 15.76 | 15.28 | 18.16 | 17.67 | 17.65 | 17.18 | 17.31 | 16.90 | 15.9$\pm$0.10 | 0.51 | 0.34 | 0.85
${}^{116}{Sn}$ | 15.63 | 15.19 | 18.02 | 17.52 | 17.51 | 17.04 | 17.19 | 16.77 | 15.80$\pm$0.10 | 0.51 | 0.32 | 0.83
${}^{118}{Sn}$ | 15.51 | 15.03 | 17.87 | 17.36 | 17.37 | 16.89 | 17.07 | 16.63 | 15.6$\pm$0.10 | 0.50 | 0.30 | 0.80
${}^{120}{Sn}$ | 15.38 | 14.90 | 17.72 | 17.20 | 17.22 | 16.73 | 16.94 | 16.49 | 15.4$\pm$0.20 | 0.50 | 0.28 | 0.78
${}^{122}{Sn}$ | 15.24 | 14.76 | 17.56 | 17.03 | 17.07 | 16.57 | 16.81 | 16.34 | 15.0$\pm$0.20 | 0.49 | 0.24 | 0.77
${}^{124}{Sn}$ | 15.11 | 14.61 | 17.40 | 16.85 | 16.91 | 16.40 | 16.67 | 16.19 | 14.80$\pm$0.20 | 0.48 | 0.24 | 0.72
${}^{208}{Pb}$ | 12.69 | 12.11 | 14.58 | 13.91 | 14.18 | 13.55 | 14.04 | 13.44 | 14.17$\pm$0.28 | 0.40 | 0.14 | 0.54
${}^{286}{114}$ | 11.32 | 10.60 | 13.00 | 12.14 | 12.64 | 11.83 | 12.55 | 11.79 |  | 0.36 | 0.09 | 0.45
${}^{298}{114}$ | 11.05 | 10.31 | 12.68 | 11.80 | 12.33 | 11.50 | 12.29 | 11.53 |  | 0.35 | 0.04 | 0.37
${}^{292}{120}$ | 11.28 | 10.53 | 12.96 | 12.07 | 12.60 | 11.76 | 12.48 | 11.69 |  | 0.36 | 0.12 | 0.27
${}^{304}{120}$ | 11.04 | 10.28 | 12.67 | 11.77 | 12.33 | 11.47 | 12.25 | 11.47 |  | 0.34 | 0.08 | 0.42

Before accepting NL3 or NL3∗ as the working parameter set for our further
calculations, in Table 4.2, we have given the excitation energy of some
selective nuclei both in light and super heavy regions with various parameter
sets for some further verification. The isoscalar giant monopole energies
$E^{s}$ and $E^{c}$ are evaluated using both scaling and constraint
calculations, respectively. The forces like NL1, NL3, Nl3∗ and FSUGold have a
wide range of incompressibility $K_{\infty}$ starting from 211.7 to 271.7 MeV
(see Table 4.1). Because of the large variation in $K_{\infty}$ of these sets,
we expect various values of $E^{s}$ and $E^{c}$ with different
parametrization. From Table 4.2, it is noticed that the calculated results for
16O and 40Ca differ substantially from the data. Again this deviation of
calculated results continues decreasing with increasing mass number,
irrespective of the parameter set. This may be due to the use of semi-
classical approximations like Thomas-Fermi and extended Thomas-Fermi. In these
approaches, quantal corrections are averaged out. When we are going from light
to the heavy and then super heavy nuclei, the surface correction decreases
appreciably. Consequently, the contribution to monopole excitation energy
decreases with mass number A. In column 11, 12 and 13 of Table 4.2, the
differences in $E_{x}$ obtained from various parameter sets are given, namely,
$\triangle_{1}$ is the difference in monopole excitation energy obtained by
NL3 and NL3∗. Similarly, $\triangle_{2}$ and $\triangle_{3}$ are the ISGMR
difference with (NL3∗, FSUGold) and (NL3, FSUGold), respectively. The values
of $\triangle_{1}$, $\triangle_{2}$ or $\triangle_{3}$ go on decreasing with
increasing mass number of the nucleus without depending on the parameter used.
In other words, we may reach the same conclusion in the super heavy region
irrespective of the parameter set. However, it is always better to use a
successful parameter set to explore an unknown territory. In this context, it
is safer to choose the NL3 force for our further exploration. The second
observation is also apparent from the Table. It is commonly believed that
mostly the incompressibility of the force parameter affects the excitation
energy of ISGMR of the nucleus. That means, forces having different
$K_{\infty}$ have different excitation energy for the same nucleus. For
example, ${}^{208}{Pb}$ has excitation energy 14.58 and 14.04 MeV with NL3 and
FSUGold, respectively. Although, the ground state binding energy of 208Pb,
either with Hartree (RMF) or REFT approximation matches well with NL3 and
FSUGold parameter sets (see Table 4.1), their ISGMR differ by 0.54 MeV, which
is quite substantial.The reason behind this difference in $E^{s}$ with various
parameter sets is not yet clear. As we have stated, the incompressibility
$K_{\infty}$ is not the only controlling key to tune the monopole excitation
energy. A lot of effort has been devoted to show that the ISGMR excitation
energy can be altered by modifying other variables of the force parameter like
effective mass $M^{*}$ and the symmetry energy coefficient $a_{4}$ [113].
Thus, the relation $E^{M}=\sqrt{\frac{K_{A}}{M<r^{2}>}}$ needs modification
with the inclusion of some other variables including the nuclear matter
incompressibility, where $<r^{2}>$ is the rms matter radius and $M$ the mass
of the nucleon. Actually, it is a long running debate and not yet clear, and
the factors responsible for the ISGMR are invite more work in this direction.

Table 3.3: The predicted proton and neutron drip-lines PDL and NDL for $O$, $Ca$, $Ni$, $Sn$, $Pb$, Z=114, and Z=120 in relativistic mean field formalism (RMF) with various parameter sets are compared with experimental (wherever available) and Finite Range Droplet Model (FRDM) predictions [112].  Nucleus | RMF | FRDM | Expt. [108]
---|---|---|---
| NL1 | NL3 | NL3∗ | FSUGold |  |
| PDL | NDL | PDL | NDL | PDL | NDL | PDL | NDL | PDL | NDL | PDL | NDL
O | 12 | 29 | 13 | 30 | 12 | 30 | 12 | 27 | 12 | 26 | 12 | 28#
Ca | 34 | 69 | 33 | 71 | 34 | 71 | 34 | 66 | 30 | 73 | 35# | 58 #
Ni | 49 | 94 | 50 | 98 | 50 | 98 | 51 | 94 | 46 | 99 | 48 | 79
Sn | 99 | 165 | 100 | 172 | 100 | 172 | 99 | 1 64 | 94 | 169 | 99# | 138#
Pb | 178 | 275 | 180 | 281 | 180 | 280 | 179 | 269 | 175 | 273 | 178 | 220#
114 | 267 | 375 | 271 | 392 | 274 | 390 | 271 | 376 | 269 | 339 | 285# | 289#
120 | 285 | 376 | 288 | 414 | 288 | 410 | 289 | 396 | 287 | 339 | - | -

####  3.3.2 Proton and neutron drip-lines

In Table 4.3 we have shown the proton and neutron drip-lines (PDL and NDL) for
various parameter sets. The neutron (or proton) drip-line of an isotope is
defined when the neutron (or proton) separation energy $S_{n}$ (or $S_{p})\leq
0$, where $S_{n}=BE(N,Z)-BE(N-1,Z)$ or $S_{p}=BE(N,Z)-BE(N,Z-1)$ with BE(N,Z)
is the binding energy of a nucleus with N neutron and Z proton. From the
table, it is seen that all the interactions predict almost similar proton and
neutron drip-lines. If one compares the drip-lines of NL3 and NL3∗, then their
predictions are almost identical, explicitly for lighter mass nuclei. Thus,
the location of the drip-line with various forces does not depend on its
nuclear matter incompressibility or asymmetry coefficient. For example, the
asymmetry coefficient $J=43.6$ MeV and $K_{\infty}=211.7$ MeV for the NL1 set
and these are 38.68 and 271.76 MeV in the NL3 parametrization. The
corresponding proton drip-lines for O isotopes are 12 and 13, and the neutron
drip-lines are 29 and 30, respectively. Similar effects are noticed for other
isotopes of the considered nuclei (see Table 4.3).

Figure 3.1: The excitation energy isoscalar giant monopole resonance (ISGMR)
for O, Ca, Ni, and Sn isotopes from proton to neutron drip-lines as a function
of mass number.  Figure 3.2: The excitation energy isoscalar giant monopole
resonance (ISGMR) for Pb, Z=114, and Z=120 isotopes starting from proton to
neutron drip-lines as a function of mass number.

####  3.3.3 Isoscalar giant monopole resonance

It is well understood that the ISGMR has a direct relation to the
incompressibility of nuclear matter, which decides the softness or stiffness
of an equation of state [19]. This EOS also estimates the structure of neutron
stars, such as mass and radius. Thus, the ISGMR is an intrinsic property of
finite nuclei as well as nuclear equations of state and needed to be
determined to shine some light into nuclear properties. The excitation
energies of ISGMR for O, Ca, Ni, Sn, Pb, Z=114, and Z=120 isotopic series are
given in Figs. 3.1 and 3.2. The results are calculated by using both
constrained and scaling approaches in the isotopic chain, starting from proton
to neutron drip-lines. We use the relation
$E_{m}^{s}=\sqrt{\frac{AK_{A}^{s}}{B_{m}}}$ with the mass parameter
$B_{m}=\int d{\bf r}r^{2}{\cal H}$. The figure shows that excitation energy
obtained from scaling calculation is always greater than the constrained
value. The difference between the monopole excitation of scaling and
constrained calculations, generally gives the resonance width
$\Sigma=\frac{1}{2}\sqrt{E_{3}^{2}-E_{1}^{2}}$, with
$E_{3}=\sqrt{\frac{m_{3}}{m_{1}}}$ and $E_{1}=\sqrt{\frac{m_{1}}{m_{1}}}$ in
terms of the ratios of the integral moments
$m_{k}=\int_{0}^{\infty}d\omega\omega^{K}S(\omega)$ of the random phase
approximation (RPA) strength function $S(\omega)$ [95]. It is also equivalent
to $m_{1}=\frac{2}{m}A<r^{2}>$ and from dielectric theorem, we have
$m_{-1}=-\frac{1}{2}A(\frac{\partial
R_{\eta}^{2}}{\partial\eta}){\bigg{|}_{\eta=0}}$.

Now consider Fig. 3.1, where the excitation energies of giant isoscalar
monopole resonance $E_{x}$ for lighter mass nuclei are plotted. For Z=8 the
excitation energy decreases towards both the proton (A=12, ${E_{x}}^{s}$=
22.51 MeV) and neutron drip-lines (A=26, ${E_{x}}^{s}$ =21.22 MeV). Excitation
energy has a maximum value near N=Z (here it is a double closed isotope with
Z=8, N=8, ${E_{x}}^{s}$= 27.83 MeV). Similar trend is followed in the isotopic
chain of Ca with Z=20. We find the maximum excitation energy at 40Ca
(${E_{x}}^{s}={24.07}$ MeV), whereas ${E_{x}}^{s}$ is found to be smaller both
in the proton (A=34, ${E_{x}}=23.31$ MeV) and neutron drip-lines (A=71,
${E_{x}}^{s}=16.80$ MeV). However, the trends are different for isotopic
chains of higher Z like Z=50, 82, 114, and 120. In these series of nuclei, the
excitation energy monotonically decreases starting from proton drip-line to
neutron drip-line. For example, 180Pb and 280Pb are the proton and neutron
drip nuclei having excitation energy ${E_{x}}=15.63$ and ${E_{x}}=11.45$ MeV,
respectively. Fig. 3.2 shows clearly the monotonic decrease of excitation
energy for super heavy nuclei. This discrepancy between super heavy and light
nuclei may be due to the Coulomb interaction and the large value of isospin
difference. For lighter values of Z, the proton drip-line occurs at a
combination of proton and neutron where the neutron number is less than or
near to the proton number. But for higher Z nuclei, the proton drip-line is
exhibited at a larger isospin. As the excitation energy of a nucleus is a
collective property, it varies smoothly with its mass number, which is also
reflected in the figures. Consider the isotopic chain of Z=50: the drip-line
nucleus (A=100) has excitation energy 18.84 MeV and the neutron drip nucleus
A=171 has $E_{x}=13.39$ MeV. The difference in excitation energy between these
two isotopes is 5.32 MeV. This difference in proton and neutron drip nuclei is
4.31 MeV for Z=82 and is 2.37 MeV in Z=114. In summary, for higher Z nuclei,
the variation of excitation energy in an isotopic chain is less than for a
lighter Z nucleus. Again, by comparing with the empirical formula of
$E_{x}=CA^{-1/3}$, our predictions show similar variation throughout the
isotopic chains. Empirically, the value of $C$ is found to be 80 [114].
However, if we select $C=70-80$ for lighter mass isotopes and $C=80-86$ for
the super heavy region, then that fits well with our results, which are
slightly different than C=80 obtained by fitting the data for stable nuclei
[114].

Table 3.4: The calculated excitation energies in the RETF formalism with FSUGold parameter set are compared with other theoretical formalisms and experimental data [75, 115, 116].  Nucleus | Formalism (parameter) | Excitation energy
---|---|---
16O | RPA(FSUGold) | 23.09
| CRMF(FSUGold) | 22.89
| RETF(FSUGold) | 25.17
| Expt. | 21.13$\pm$0.49
40Ca | RPA(FSUGold) | 20.67
| CRMF(FSUGold) | 20.67
| RETF(FSUGold) | 22.30
| Expt. | 19.18$\pm$0.37
90Zr | RPA(FSUGold) | 17.44
| CRMF(FSUGold) | 17.70
| RETF(FSUGold) | 18.21
| Expt. | 17.89$\pm$0.20
208Pb | RPA(FSUGold) | 13.76
| CRMF(FSUGold) | 13.50
| RETF(FSUGold) | 13.44
| Pairing+MEM | 14.0
| Expt. | 13.5$\pm$0.1
204Pb | RETF(FSUGold) | 13.6
| Pairing+MEM | 13.4
| Expt. | 13.7$\pm$0.1
206Pb | RETF(FSUGold) | 13.51
| Pairing+ MEM | 13.4
| Expt. | 13.6$\pm$0.1
Figure 3.3:  The difference between the monopole excitation energies of
scaling and constrained calculations
$\Sigma=\frac{1}{2}\sqrt{E_{3}^{2}-E_{1}^{2}}$ as a function of mass number A
for O, Ca, Ni, and Sn.  Figure 3.4:  Same as Fig. 3.3, but for Pb, Z=114, and
Z=120.

In table 3.4, we have shown the results obtained in the RETF formalism with
the FSUGold parameter set and compared with other predictions like the random
phase approximation (RPA) and constrained relativistic mean filed (CRMF)[117].
The experimental data are also given, where ever available, for comparison. A
comparative study of these results shows that for light nuclei such as
${}^{16}{O}$, the RPA and CRMF give better results than the semi-classical
RETF approximation. For example, the difference between the excitation
energies of ${}^{16}O$ in the RPA and RETF formalism is $\sim$ 2 MeV and it is
only 0.1 MeV for ${}^{208}Pb$. This implies, for heavy nuclei, that the RETF
gives comparable results with RPA and CRMF. On the other hand, the RETF
results obtained by constrained calculation are within the experimental error
bar. In RETF, the quantal correction is averaged out. Thus, the RETF result of
monopole excitation energy differs from the RPA prediction for light nuclei,
where quantal correction has a significant role in structural properties. In
heavy nuclei, the number of nucleons are larger, so the quantal and semi-
classical approaches are almost similar. This could be a reason for the
accuracy of application of semi-classical calculation to heavy mass nuclei. We
have also compared our results with Pairing+MEM (mutually enhanced magicity)
[118] and experimental data[119] for $Pb$ isotopes. The Pairing+MEM theory
says that a magic nucleus like ${}^{208}Pb$ has a little higher excitation
energy than its neighboring isotopes. But recent experimental data are not in
favor of the manifestation of such an effect. Our theoretical results also
come to the same conclusion.

There is no direct way to calculate $\Sigma$ in the scaling or constrained
method as in RPA. If we compare the excitation energy obtained from scaling
calculations with the non-relativistic RPA result, then it is evident that the
scaling gives the upper limit of the energy response function. On the other
hand, the constrained calculation predicts the lower limit [76]. As a result,
the resonance width $\Sigma$ is obtained from the root mean square difference
of $E_{x}^{s}$ and $E_{x}^{c}$. We have plotted $\Sigma$ for the light nuclei
in Fig. 3.3 and for super heavy in Fig. 3.4. For lighter nuclei, $\Sigma$ is
larger both in the proton and neutron drip-lines. As one proceeds from proton
to neutron drip-line, the value of $\Sigma$ decreases up to a zero isospin
combination (N=Z or double closed) and then increases. For example, $\Sigma$=
10.92, 5.0 and 8.9 MeV for ${}^{12}{O}$, ${}^{16}O$, and ${}^{26}O$,
respectively. Similar trends are also followed by the medium nuclei Z=20, 28,
and 50 isotopic chains. This conclusion can be drawn from the results of the
excitation energy also (see Figs. 3.1 and 3.2), i.e., the difference between
the scaling and constrained excitation energies are more in proton and neutron
drip-lines as compared to the Z=N region. The value of $\Sigma$ in an isotopic
chain depends very much on the proton number. It is clear from the isotopic
chains of $\Sigma$ for O, Ca, Ni, Sn, Pb, and Z=114, 120. All of the
considered series have their own behavior and show various trends. Generally,
for lighter elements, it decreases initially to some extent and again
increases monotonically. On the other hand, for heavier nuclei like Pb, Z=114,
or Z=120 this characteristic of $\Sigma$ differs with different mass number
and can be seen in Fig. 3.4.

####  3.3.4 Incompressibility of finite nuclei

Figure 3.5:  The incompressibility modulus obtained by both scaling and
constrained approaches in the isotopic series of O, Ca, Ni, and Sn.  Figure
3.6:  Same as Fig. 3.5, but for Pb, Z=114, and Z=120.

The nuclear matter incompressibility $K_{\infty}$ is a key quantity in the
study of the equation of state. It is the second derivative of the energy
functional with respect to density at the saturation and is defined as
$K_{\infty}=9\rho_{0}\frac{\partial^{2}\cal
E}{\partial\rho^{2}}|_{\rho=\rho_{0}}$, which has a fixed value for a
particular force parametrization. It is well understood that a larger
$K_{\infty}$ of a parameter set gives a stiff EOS and produces a massive
neutron star. It has also a direct relation to the asymmetry energy
coefficient $J$ of the parameter set [48]. In the limit A approaches
infinitely large, the finite nucleus can be approximated to an infinite
nuclear matter system (N=Z for symmetry and $N\neq Z$ for asymmetry matter).
Thus, it is instructive to study the nature of the incompressibility of a
finite nucleus $K_{A}$ in the isotopic chain of super heavy nuclei. Here, we
calculate the $K_{A}$ as a function of mass number for the light nuclei
considered in the present study (O, Ca, Ni, Sn) and then extend the
calculations to Pb, Z=114 and Z=120 in the super heavy region. Our calculated
results are shown in Figs. 3.5 and 3.6. The incompressibility of finite nuclei
follows the same trend as the excitation energy. For light nuclei, the
incompressibility has a small value for the proton and neutron drip-lines,
whereas it is maximum in the neighborhood of double closed combinations.

It can be easily understood from Fig. 3.5 that, at some particular proton to
neutron combination, the $K_{A}$ is high, i.e., at this combination of N and
Z, the nucleus is more incompressible. In other words, the larger the
incompressibility of a nucleus, the more compressible it will be. Here, it is
worthy to mention that the nuclear system becomes less compressible near both
the neutron and proton drip-lines. This is because of the instability
originating from the repulsive part of the nuclear force, revealing a large
neutron-proton ratio, which progressively increases with the neutron/proton
number in the isotope without much affecting the density [120]. Similar to the
excitation energy, it is found that $K_{A}$ obtained by the scaling method is
always higher than the constrained calculation. The decrease in
incompressibility near the drip-line regions is more prominent in constrained
calculation than the scaling results. From leptodermous expansion [19], we can
get some basic ideas about this decrease in the vicinity of drip-lines. The
expression for finite nucleus incompressibility can be written as

$\displaystyle
K_{A}=K_{\infty}+K_{Sur}A^{-{\frac{1}{3}}}+K_{\tau}{I^{2}}+K_{Coul}{Z}^{2}{A^{-\frac{4}{3}}},$
(3.49)

where $I={\frac{N-Z}{A}}$. The coefficient $K_{\tau}$ is negative, so
incompressibility decreases with ${N-Z}$. For the Ca chain, the
incompressibilities obtained by scaling and constrained calculations are
compared with the Hartree-Fock plus RPA results [19] in Fig. 3.5. From Fig.
3.5, one can see that $K_{A}$ evaluated by the semi-classical approximation
deviates from RPA results for lighter isotopes, contrary to the excellent
matching for the heavier Ca nuclei. This is because of the exclusion of the
quantal correction in the semi-classical formalism. For higher mass nuclei,
this correction becomes negligible and compares to the RPA predictions. This
result is depicted in Fig. 3.6 for Pb and super heavy chain of nuclei. Here
the results show completely different trends than for the lighter series. The
incompressibility has a higher value in the vicinity of the proton drip-line
and decreases monotonically towards the neutron drip-line. Because, for high
Z-series, the proton drip-line appears at a greater value of N in contrast to
the lighter mass region. Again, the incompressibility decreases with neutron
number from the proton to neutron drip-lines.

Figure 3.7:  Compressibility for finite nuclei obtained by scaling calculation
$K_{A}^{s}$ versus nuclear matter incompressibility $K_{\infty}$.

Finally, we would like to see the trend of $K_{A}$ with nuclear matter
incompressibility for various force parameters and also with the size of a
nucleus which can reach the infinite nuclear matter limit. For this we choose
${}^{286,298}114$, ${}^{292,304}120$, and 40Ca as the selected candidates as
shown in Fig. 3.7. Although, the super heavy nuclei approach the nuclear
matter limit, we can not reproduce the $K_{\infty}$ from $K_{A}$. This may be
due to the asymmetry needed to form a bound nucleus, which is the reason for
the deviation. That means the asymmetry $I$ of $K_{A}$ and $K_{\infty}$
differs significantly (where $I=\frac{N-Z}{N+Z}$), which is the main source of
deviation of $K_{A}$ from $K_{\infty}$. Also, this deviation arises due to the
surface contribution of the finite nuclei. For a quantitative estimation, we
have calculated the $K_{A}^{s}$ for different force parameters having various
$K_{\infty}$ at saturation. We find almost a linear variation of $K_{A}^{s}$
with $K_{\infty}$ for the considered nuclei as shown in Fig. 3.7. For Ca
isotopes also we find variation similar in nature, but with smaller $K_{A}$
than the super heavy nuclei.

###  3.4 Application of scaling and constrained formalism to Sn isotopes

In this part of the chapter, we applied the scaling and constrained formalism
to analyse the anamolous nature of Sn Isotopes. In the previous section, we
have discussed the excitation energy and incompressibility of super heavy and
drip line nuclei. The recent experiment on the isotopic chain of Sn isotopes
indicates the mismatch between the theoretical prediction and the experimental
data of the excitation energy of isoscalar giant monopole resonance energy.
This problem is addressed in the name of ” why the tin nucleus is so floppy
?”. So theoretical physicists are now more interested in the medium heavy
region (A$\sim$100). In this section the excitation energy and the
incompressibility isotopes of Cd and Sn nuclei are discussed, within the
framework of relativistic Thomas-Fermi and relativistic extended Thomas-Fermi
approximations. A large number of non-linear relativistic force parameters are
used in the calculations that a parameter set is capable of reproducing the
experimental monopole energy of Sn isotopes, when its nuclear matter
incompressibility lies within $210-230$ MeV, however, it fails to reproduce
the GMR energy of other nuclei. That means, simultaneously a parameter set can
not reproduce the GMR values of Sn and other nuclei.

Incompressibility of nuclear matter, also knows as compression modulus
$K_{\infty}$ has a special interest in nuclear and astro-nuclear physics,
because of its fundamental role in guiding the equation of state (EOS) of
nuclear matter. The $K_{\infty}$ can not be measured by any experimental
technique directly, rather it depends indirectly on the experimental
measurement of isoscalar giant monopole resonance (ISGMR) for its confirmation
[19]. This fact enriches the demand of correct measurement of excitation
energy of ISGMR. The relativistic parameter with random phase approximation
(RPA) constraints the range of the incompressibility $270\pm{10}$ MeV [49,
121] for nuclear matter. Similarly, the non-relativistic formalism with
Hartree-Fock (HF) plus RPA allows the acceptable range of incompressibility
$210-220$ MeV, which is less than the relativistic one. It is believed that
the part of this discrepancy in the acceptable range of compressional modulus
comes from the diverse behavior of the density dependence of symmetry energy
in relativistic and non-relativistic formalism [122]. Recently, both the
formalism come to a general agreement on the value of nuclear matter
incompressibility i.e., $K_{\infty}=240\pm{10}$ MeV [123, 90, 124]. But the
new experiment on Sn isotopic series i.e., 112Sn$-^{124}$Sn rises the question
”why Tin is so fluffy ?”[125, 126, 127]. This question indicates toward the
correct theoretical investigation of incompressibility. Thus, it is worthy to
investigate the incompressibility in various theoretical formalisms. Most of
the relativistic and non-relativistic models reproduce the strength
distribution very well for medium and heavy nuclei, like 90Zr and 208Pb,
respectively. But at the same time it overestimates the excitation energy of
Sn around 1 MeV. This low value of excitation energy demands lower value of
nuclear matter incompressibility. Till date, lots of effort have been devoted
to solve this problem like inclusion of pairing effect [128, 129, 130, 131,
132, 133, 134], mutually enhanced magicity (MEM) etc. [118]. But the pairing
effect reduces the theoretical excitation energy only by 150 KeV in Sn
isotopic series, which may not be sufficient to overcome the puzzle.
Similarly, new experimental data are not in favor of MEM effect [119].
Measurement on excitation energy of 204,206,208Pb shows that the MEM effect
should rule out in manifestation of stiffness of the Sn isotopes. Here, we use
the relativistic Thomas-Fermi (RTF) and relativistic extended Thomas-Fermi
(RETF) [135, 136, 137, 138, 139] with scaling and constraint approaches in the
frame-work of non-linear $\sigma-\omega$ model [99].

###  3.5 Analysis of Sn results

We calculate the GMR energy using both the scaling and constraint methods in
the frame-work of relativistic extended Thomas-Fermi approximation using
various parameter sets for Cd and Sn nuclei and compared with the excitation
energy with moments ratio $({m_{3}/m_{1}})^{1/2}$ and $({m_{1}/m_{-1}})^{1/2}$
obtained from multipole-decomposition analysis (MDA).

Figure 3.8: The isoscalar giant monopole excitation energy obtained by scaling
method with various parameter sets are compared with experimental data of
$\sqrt{{\frac{m_{3}}{m_{1}}}}$ and $\sqrt{{\frac{m_{1}}{m_{-1}}}}$ for Cd and
Sn isotopes [125, 126]. The upper and lower panels are
$\sqrt{{\frac{m_{3}}{m_{1}}}}$ and $\sqrt{{\frac{m_{1}}{m_{-1}}}}$,
respectively.  Figure 3.9: (a) The variation of giant monopole excitation
energy $E_{x}$ with mass number A for Sn isotopes, (b) the scaling monopole
excitation energy within RETF and RTF formalisms compared with the
experimental moments ratio ${\sqrt{{m_{3}/m_{1}}}}$ [119], (c) variation of
the difference of giant monopole excitation energy obtained from RETF and RTF
($\triangle{E}=RETF-RTF$) formalisms with various parameter sets for Sn
isotopic chain, (d) moments ratio ${\sqrt{{m_{3}/m_{1}}}}$ for Sn isotopes
obtained with NL3+$\Lambda_{V}$ is compared with NL3, QRPA(T5) and
experimental data.

The basic reason to take a number of parameter sets is that the infinite
nuclear matter incompressibility of these forces cover a wide range of values.
For example, NL-SH has incompressibility 399 MeV, while that of NL1 is 210
MeV. From MDA analysis, we get different moments ratio, such as $m_{3}/m_{1}$,
$m_{0}/m_{1}$ and $m_{1}/m_{-1}$. These ratios are connected to scaling,
centroid and constraint energies, respectively. In Fig. 3.8, we have shown the
$(m_{3}/m_{1})^{1/2}$ and $(m_{1}/m_{-1})^{1/2}$ ratio for isotopic chains of
Cd and Sn. The results are also compared with experimental data obtained from
Research Center for Nuclear Physics (RCNP) [125, 126]. From the figures, it is
cleared that the experimental values lie between the results obtained from
FUSG (FSUGold) and NL1 force parameters. If one compares the experimental and
theoretical results for 208Pb, the FSUG set gives better results among all.
For example, the experimental data and theoretical result for monopole
energies are 14.17$\pm$0.1 and 14.04 MeV, respectively. From this, one could
conclude that the infinite nuclear matter incompressibility lies nearer to
that of FSUG (230.28 MeV) parametrization. But experimental result on ISGMR in
RCNP shows that the predictive power of FUSG is not good enough for the
excitation energy of Sn isotopes. This observation is not only confined to
RCNP data, but also persists in the more sophisticated RPA. In Table 3.5, we
have given the results for QRPA(T6), RETF(FSUG) and RETF(NL1). The
experimental data are also given to compare all these theoretical results.

Table 3.5: Moments ratio for Sn isotopes using RETF approximation with FSUGold and NL1 sets are compared with QRPA(T6) predictions [140]. Nucleus |  $({m_{3}/m_{1}})^{1/2}$(MeV) |  $({m_{1}/m_{-1}})^{1/2}$(MeV)
---|---|---
| QRPA | RETF | RETF | Expt. | QRPA | RETF | RETF | Expt.
| (T6) | (FSUG) | (NL1) |  | T6) | (FSUG) | (NL1) |
112Sn | 17.3 | 17.42 | 15.86 | 16.7 | 17.0 | 17.2 | 15.39 | 16.1
114Sn | 17.2 | 17.32 | 15.75 | 16.5 | 16.9 | 16.9 | 15.28 | 15.9
116Sn | 17.1 | 17.19 | 15.63 | 16.3 | 16.8 | 16.77 | 15.15 | 15.7
118Sn | 17.0 | 17.07 | 15.51 | 16.3 | 16.6 | 16.63 | 15.03 | 15.6
120Sn | 16.9 | 16.94 | 15.38 | 16.2 | 16.5 | 16.44 | 14.89 | 15.5
122Sn | 16.8 | 16.81 | 15.24 | 15.9 | 16.4 | 16.34 | 14.75 | 15.2
124Sn | 16.7 | 16.67 | 15.1 | 15.8 | 16.2 | 16.19 | 14.6 | 15.1

The infinite nuclear matter incompressibility $K_{\infty}$ with T6 parameter
set [140] is 236 MeV and that of FSUG is 230.28 MeV. The similarity in
incompressibility (small difference of 6 MeV in $K_{\infty}$) may be a reason
for their prediction in equal value of GMR. The table shows that, there is
only 0.1 MeV difference in QRPA(T6) and RETF(FSUG) results in the GMR values
for 112Sn$-$116Sn isotopes, but the results exactly match for the
118Sn$-$124Sn isotopes. This implies that for relatively higher mass nuclei,
both the QRPA(T6) and RETF(FSUG) results are almost similar. If some one
consider the experimental values of Sn isotopic series, then QRPA(T5) gives
better result. For example, experimental value of $(m_{3}/m_{1})^{1/2}$ for
112Sn is 16.7$\pm$ 0.2 MeV and that for QRPA(T5) is 16.6 MeV. It is shown by
V. Tselyaev et al. [140] that the T5 parameter set with $K_{\infty}=202$ MeV,
better explains the excitation energy of Sn isotopes, but fails to predict the
excitation energy of 208Pb. The experimental data of ISGMR energies for 90Zr
and 114Sn lies in between the calculated values of T5 and T6 forces. In brief,
we can say that the RPA analysis predicts the symmetric nuclear matter
incompressibility within $K_{\infty}=202-236$ MeV and our semi-classical
calculation gives it in the range $210-230$ MeV. These two predictions almost
agree with each other in the acceptable limit.

Table 3.6: Moments ratio ${\sqrt{{m_{1}/m_{-1}}}}$ for Pb isotopes within RETF is compared with pairing+ MEM results and experimental data. Nuclear Mass | (${m_{1}/m_{-1}})^{1/2}$ (MeV) |  $\Sigma$(MeV)
---|---|---
| pairing+MEM | our work | Expt. | our work | Expt.
204Pb | 13.4 | 13.6 | 13.7$\pm$0.1 | 2.02 | 3.3$\pm$0.2
206Pb | 13.4 | 13.51 | 13.6$\pm$0.1 | 2.03 | 2.8$\pm$0.2
208Pb | 14.0 | 13.44 | 13.5$\pm$0.1 | 2.03 | 3.3$\pm$0.2

In Table 3.6, we have displayed the data obtained from a recent experiment
[119] and compared our results. Column two of the table is also devoted to the
result obtained from pairing plus MEM effect [118]. The data show clearly that
our formalism (RETF) predicts the excitation energy more accurately than the
result obtained by Pairing+ MEM prediction for Pb isotopes. For example, the
difference between the pairing+MEM result and experimental observation is 0.3
MeV for 204Pb, which is away from the experimental error. It is only 0.1 MeV
(within the error bar) in RETF calculation. This trend also follows for 206Pb
and 208Pb. As it is mentioned earlier, we have not included the pairing
correlation externally, yet the result is good enough in comparison with
MEM+pairing as pairing has a marginal role for the calculation of collective
excitation. Fig. 3.9(a) shows the variation of excitation energy with mass
number A for Sn isotopes. This results can also be treated as the monopole
excitation energy $E_{x}$ with proton-neutron asymmetry ($I=\frac{N-Z}{N+Z}$).
The graph shows that the variation of monopole excitation energy $E_{x}$ with
both NL3 and FSUGold are following similar pattern as experimental one with a
different magnitude. In Fig. 3.9(b), we have compared our results obtained
from RETF(NL3) and RTF(NL3) with the experimental data for Sn isotopes. The
graph shows that there is only a small difference ($\sim 0.2$ MeV) in scaling
monopole energies obtained with RETF and RTF calculations. Interestingly, the
RETF correction is additive to the RTF result instead of softening the
excitation energy of Sn isotopes. To see the behavior of different
parametrization, we have plotted Fig. 3.9(c), where it has shown the
difference of ${\sqrt{m_{3}/m_{1}}}$ obtained with RETF and RTF results
($\triangle{E}=RETF-RTF$) for various parameter sets. For all sets, except
NL1, we find $\triangle{E}$ as positive. Thus, it is a challenging task to
entangle the term which is the responsible factor to determine the sign of
RETF-RTF. Surprisingly, for most of the parameter sets, RTF is more towards
experimental data. In spite of this, one cannot says anything about the
qualitative behavior of RETF. Because, the variation of the density at the
surface taken care by the RETF formalism, which is essential. One more
interesting observation is that, when one investigate the variation of
$\triangle{E}$ in the isotopic chain of Sn, it remains almost constant for all
the parameter sets, except FSUGold. In this context, FSUGold behaves
differently.

Variation of RETF-RTF with neutron-proton asymmetry for FSUGold set shows some
possible correlation of RETF with the symmetry energy, which is absent in all
other parameter sets. Now it is essential to know, in which respect the
FSUGold parameter set is different from other. The one-to-one interaction
terms for NL3, NL2, NL1 and NL-SH have similar couplings. However, the FSUGold
is different from the above parameters in two aspects, i.e., two new coupling
constants are added. One corresponds to the self-interaction of $\omega$ and
other one corresponds to the isoscalar-isovector meson coupling. It is known
that self-interaction of $\omega$ is responsible for softening the EOS [87,
141] and the isoscalar-isovector coupling takes care the softening of
symmetric nuclear matter[142]. The unique behavior shown by the FSUGold
parametrization may be due to the following three reasons:

  1. 1.

introduction of isoscalar-isovector meson coupling $\Lambda_{V}$,

  2. 2.

introduction of self-coupling of $\omega-$meson,

  3. 3.

or simultaneous introduction of both these two terms with refitting of
parameter set with new constraint.

In order to discuss the first possibility, we plotted NL3+$\Lambda_{V}$(0.03)
in Fig. 3.9(d). The graph shows that there is no difference between NL3 and
NL3+$\Lambda_{V}$, except the later set predicts a more positive RETF-RTF. It
is well known that, the addition of $\Lambda_{V}$ coupling, i.e.,
NL3+$\Lambda_{V}$(0.03) gives a softer symmetry energy [143]. This implies
that, models with softer energy have greater difference in RETF and RTF. At a
particular proton-neutron asymmetry, RETF-RTF has a larger value for a model
with softer symmetry energy. This observation is not conclusive, because all
the parameter sets do not follow this type of behavior. Quantitatively, the
change of RETF-RTF in the Sn isotopic series is about $70\%$ with FSUGold
force, while this is only $20-30\%$ in NL3 and other parameter sets.

In Table 3.7, we have listed the $\rho-$meson contribution to the total
binding energy. From the analysis of our results, we find that only
$\rho-$contribution to the total binding energy change much more than other
quantity, when one goes from RTF to RETF. But this change is more prominent in
FSUGold parameter set than other sets like NL1, NL2, NL3 and NL-SH. Simple
assumption says that the absent of $\Lambda_{V}$ coupling may be the reason
for this behavior in rest of the parameter sets. But we have checked for the
parameter NL3+$\Lambda_{V}$, which does not follow the assumption. This also
shows similar behavior like other sets. In Table 3.8, we have given the
results for FSUGold, NL3+$\Lambda_{V}$ and NL1. The data show a huge
difference of monopole excitation energy in RETF and RTF with FSUGold
parameter set. For example, the $\rho-$meson contribution to the GMR in RETF
for 112Sn is 21.85 MeV, while in RTF it is only 0.00467 MeV.

Table 3.7: Contribution of the $\rho-$meson to the total binding energy in the RTF and RETF approximations with FSUGold, NL1 and NL3+$\Lambda_{V}$ parameter set. RETF represent $\rho$ meson contribution to total energy from $R^{2}$ term and RETF$\Lambda_{V}$ represent contribution from the ${\Lambda_{v}}{R^{2}}{V^{2}}$.  Nucleus | FSUGold | NL3(0.03) | NL1
---|---|---|---
| RETF | RETF$\Lambda_{V}$ | RTF$\Lambda_{V}$ | RTF | RETF | RTF | RETF | RTF
112Sn | 21.85 | -6.66 | -0.00130 | 0.00467 | 20.60 | 20.12 | 17.91 | 16.73
114Sn | 28.72 | -8.64 | -0.00202 | 0.00664 | 27.11 | 26.48 | 23.51 | 22.00
116Sn | 36.42 | -10.83 | -0.00248 | 0.00829 | 34.40 | 33.62 | 29.73 | 27.90
118Sn | 44.87 | -13.23 | -0.00298 | 0.01013 | 42.42 | 41.499 | 36.52 | 34.37
120Sn | 54.04 | -15.82 | -0.00353 | 0.01213 | 51.13 | 50.05 | 43.84 | 41.37
122Sn | 63.88 | -18.58 | -0.00411 | 0.01429 | 60.48 | 59.24 | 51.63 | 48.85
124Sn | 74.33 | -21.49 | -0.00473 | 0.01660 | 70.43 | 69.03 | 59.84 | 56.76

However, this difference is nominal in NL3+$\Lambda_{V}$ parameter set, i.e.,
it is only 0.48 MeV. Similarly, this value is 1.18 MeV in NL1 set. The
contribution of $\rho-$meson to total energy comes from two terms: (i) one
from $\Lambda_{V}{R^{2}}{V^{2}}$ and other (ii) from $\rho^{2}$. We have
explicitly shown that contribution comes from $\Lambda_{V}{R^{2}}{V^{2}}$
makes a huge difference between the GMR obtained from RETF and RTF formalisms.
This type of contribution does not appear from NL3+$\Lambda_{V}$. For example,
in 112Sn the contribution of $\Lambda_{V}{R^{2}}{V^{2}}$ with RETF formalism
is -6.0878 MeV, while with RTF formalism is -5.055 MeV.

The above discussion gives us significant signature that the contribution of
$\Lambda_{V}$ may be responsible for this anomalous behavior. But an immediate
question arises, why NL3+$\Lambda_{V}$ parameter set does not show such type
of effects, inspite of having $\Lambda_{V}{R^{2}}{V^{2}}$ term. This may be
due to the procedure in which $\Lambda_{V}{R^{2}}{V^{2}}$ term is interpreted
in these two parameter sets. In NL3+$\Lambda_{V}$(0.03), the
$\Lambda_{V}{R^{2}}{V^{2}}$ term is not added independently. The $\Lambda_{V}$
and $g_{\rho}$ are interdependent to each other to fix the binding energy and
difference in neutron and proton rms radii $R_{n}$-$R_{p}$. But in FSUGold,
$\Lambda_{V}$ coupling constant is incorporate independently to reproduce the
nuclear observable.

Table 3.8: ${}^{S}K_{A}$ and ${}^{C}K_{A}$ are incompressibility for finite nuclei obtained from scaling and constraint methods, respectively are compared with the values obtained from the equation of state (EOS). Nucleus | NL3 | FSUGold
---|---|---
| ${}^{S}K$ | ${}^{C}k$ | ${}^{EOS}K$ | ${}^{S}K$ | ${}^{C}K$ | ${}^{EOS}K$
208Pb | 164.11 | 149.96 | 145 | 147.37 | 134.57 | 138.42
116Sn | 164.64 | 155.39 | 131.57 | 147.11 | 139.71 | 127.64
40P | 136.70 | 110.43 | 105 | 123.40 | 100.36 | 102.53
40Ca | 145.32 | 134.47 | 105 | 130.93 | 123.15 | 102.53

In Table 3.8, we have listed the incompressibility of some of the selected
nuclei in scaling ${}^{S}K_{A}$ and constraint ${}^{C}K_{A}$ calculations.
This results are compared with the computed values obtained from the equation
of state (EOS) model. To evaluate the incompressibility from the EOS, we have
followed the procedure as discussed in Refs. [144, 145]. M. Centelles et al.
[144], parametrized the density for finite nucleus as
$\rho_{A}=\rho_{0}-\rho_{0}/(1+c*{A^{1/3}})$ and obtained the asymmetry
coefficient $a_{sym}$ of the nucleus with mass A from the EOS at this
particular density. Here also, we have used the same parametric form of the
density and obtained the incompressibility for finite nucleus from the EOS.
For example, $\rho_{A}=0.099$ for $A=208$ in FSUGold parameter set, and the
calculated incompressibility is $\sim 145$ MeV at this particular density. We
have also calculated the incompressibility independently in Thomas-Fermi and
extended Thomas-Fermi using scaling and constraint calculations, which are 161
MeV and 146.1 MeV, respectively.

###  3.6 Summary and Conclusions

In summary, we have calculated the isoscalar giant monopole resonance for O,
Ca, Ni, Sn, Pb, Z=114, and Z=120 isotopic series starting from the proton to
the neutron drip-lines. We used four successful parameter sets, NL1, NL3,
NL3*, and FSUGold, with a wide range of incompressibility starting from
$211.7$ MeV to 271.76 MeV to see the dependency of the ISGMR on $K_{\infty}$.
Also, we have analyzed the predictions of ISGMR with these forces, which
originate from various interactions and found that whatever may be their
origin, the differences in ISGMR predicted by them are found to be marginal in
the super heavy region. A recently developed scaling approach in a
relativistic mean field theory is used. A simple, but accurate constrained
approximation is also performed to evaluate the isoscalar giant monopole
excitation energy. From the scaling and constrained ISGMR excitation energies,
we have evaluated the resonance width $\Sigma$ for the whole isotopic series.
This is obtained by taking the root mean square difference of $E_{x}^{s}$ and
$E_{x}^{c}$. The value of $E_{x}^{s}$ is always higher than the constrained
result $E_{x}^{c}$. In a sum rule approach, the $E_{x}^{s}$ can be compared
with the higher and $E_{x}^{c}$ as the lower limit of the resonance width. In
general, we found an increasing trend of $\Sigma$ for both the light and super
heavy regions near the proton and neutron drip-lines. The magnitude of
$\Sigma$ is predicted to be a minimum in the vicinity of N=Z or in the
neighborhood of a double closed nucleus and it is a maximum for highly
asymmetric systems. We have also estimated the incompressibility of finite
nuclei. For some specific cases, the incompressibility modulus is compared
with the nuclear matter incompressibility and found a linear variation among
them. It is also concluded that the nucleus becomes less compressible with the
increase of neutron or proton number in an isotopic chain. Thus neutron-rich
matter, like neutron stars as well as drip-line nuclei, are less compressible
than normal nuclei. In case of finite drip-line nuclei, the nucleus is
incompressible, although it possess a normal density. In brief, we analyzed
the predictive power of various force parameters, like NL1, NL2, NL3, Nl-SH
and FSUGold in the frame-work of relativistic Thomas-Fermi and relativistic
extended Thomas-Fermi approaches for giant monopole excitation energy of Sn-
isotopes. Then the calculation is extended to some other relevant nuclei. The
analysis shows that Thomas-Fermi approximation gives comparable results with
pairing+MEM data. It exactly reproduces the experimental data for Sn isotopes,
when the incompressibility of the force parameter is within $210-230$ MeV,
however, fails to reproduce the GMR data for other nuclei within the same
accuracy.

We have qualitatively analyzed the difference in GMR energies RETF-RTF using
RETF and RTF formalisms in various force parameters. The FSUGold parameter set
shows different behavior from all other forces. Also, extended our
calculations of monopole excitation energy for Sn isotopes with a force
parametrization having softer symmetry energy (NL3+ $\Lambda_{V}$). The
excitation energy decreases with the increase of proton-neutron asymmetry
agreeing with the experimental trend. In conclusion, after all these thorough
analysis, it seems that the softening of Sn isotopes is an open problem for
nuclear theory and more work in this direction are needed.

##  Chapter 4 Perturbative constrained calculation of excitation energy of
ISGMR and IVGDR in semiclassical RMF theory.

In last two chapter, we have extensively discussed the isoscalar giant
monopole resonances in super heavy and medium-heavy nuclei. Semi-classical
approximation like RTF and RETF are used, which are proved very successful to
calculate the collective excitation. We have used constrained and scaling
calculations for the excitation energy of ISGMR. In this chapter, we discuss a
new constrained calculation, which we have developed to calculate the
excitation energy of both ISGMR and IVGDR. From the theoretical point of view
both the old and new constrained calculations are different from each other,
but gives similar results. In old constrained calculation, we minimize the
constrained energy functional with a constrained parameter $\lambda$. We get a
different value of constrained energy with a different value of $\lambda$ and
double derivative of constrained energy with respect to $\lambda$ gives the
constrained excitation energy. The new constrained calculation based on the
perturbative approaches, in which we expand the energy functional in Taylor
series around equilibrium state. We calculated the excitation energy for some
spherical (or double closed ) nuclei, where the experimental data are
available.

###  4.1 Introduction

The nuclear matter incompressibility ($K_{\infty}$) is a measure of stiffness
of the equation of state (EOS). It can be calculated from the curvature of the
EOS around the saturation density ($\rho_{0}$). This quantity is important in
nuclear physics because it is related to many properties of finite nuclei
(such as radii, masses and giant resonances), the dynamics of heavy-ion
collisions, neutron stars and supernovae collapses [19, 146]. As $K_{\infty}$
is not an observable, its value must be deduced from a measurable quantity. In
this respect, an important source of information on $K_{\infty}$ is provided
by the measurements of compression modes, such as the isoscalar giant monopole
resonance (ISGMR), in finite nuclei, in particular in 208Pb. An accepted value
$K_{\infty}=240\pm 20$ MeV, predicted by the majority of mean field models,
extracted from different analysis of experimental data and theoretical
calculation[147].

On the other hand, the existence of strong resonances in the photo-absorption
cross sections was established almost seventy years ago [148] and
theoretically explained few years later [149, 150]. These resonances,
identified with the isovector giant dipole resonance (IVGDR), are correlated
via the electrical polarizability with the neutron skin thickness of neutron
rich nuclei and with the properties of the nuclear symmetry energy around
saturation [151, 152].

The ISGMR and IVGMR can be understood as small amplitude oscillations, which
are the responses of the nucleus to an external field generated by
electromagnetic or hadronic probes. The standard theory to deal with these
oscillations is the random-phase approximation (RPA), which allows to study
microscopically giant resonances. The key output provided by the RPA
calculations is the strength distribution $S_{Q}(E)$ associated to a given
excitation operator ${\bf Q}$. If $S_{Q}(E)$ is concentrated in a narrow
region of the energy spectrum, which is usually the case of the ISGMR and
IVGMR at least for heavy stable nuclei, the knowledge of few low energy
weighted moments of $S_{Q}(E)$ (sum rules) allows to estimate the average
excitation energy of the resonances. In some particular cases it is possible
to express some odd sum rules in a compact form involving only ground-state
properties, which avoids the calculation of the strength distribution [76].
However, the full quantal calculation of these sum rules is still complicated
because the exact ground-state is, in general, unknown. Introducing additional
approximations, some sum rules can be calculated in a rather simple way, as
for instance the cubic energy weighted sum rule, which can be computed by
using the scaling method [19, 76, 153] or the inverse energy weighed sum rule,
which can be obtained through constrained Hartree-Fock (HF) calculations [19,
153, 154, 155].

The theoretical study of giant resonances in the relativistic domain has been
usually done through relativistic RPA (RRPA) calculations. The RRPA
approximation corresponds to the small amplitude limit of the time-dependent
relativistic mean field (RMF) theory [156, 157, 158]. On the other hand the
time-dependent theory has also been directly used to study the isovector
dipole [159] as well as the isoscalar and isovector quadrupole [159] and
monopole [160] oscillations. As difference with the non-relativistic case, the
sum rule theorems, which relate different moments of the RRPA strength with
ground-state properties computed at RMF level, have not been proved in the
relativistic domain [117]. There are few relativistic constrained calculations
available in the literature [117, 103, 104, 105, 158]. In Ref.[158] the
results of constrained RMF calculations of the $m_{-1}$ sum rule were compared
with the corresponding RRPA values and it was suggested that the RRPA $m_{-1}$
moment could be estimated from a RMF calculation. More recently, in Ref.[117],
this suggestion was carefully analyzed for several nuclei along the periodic
table using two different RMF parameter sets. It was found an excellent
agreement between the average energies of ISGMR obtained through RMF
constrained calculation and extracted from RRPA results. In Ref.[81] the
scaling approach to the isoscalar giant monopole and quadrupole resonances in
relativistic Thomas-Fermi (RTF) theory [161, 162] was discussed in detail. It
was found that the semi-classical average excitation energies reproduced well
the RRPA results. In the same reference the excitation energies of the ISGMR
were also calculated performing RTF constrained calculations finding good
agreement with the constrained RMF results of Ref.[105]. As a conclusion,
these results together with the conjecture of Ref.[158] points out that the
scaling method and constrained calculations using the semi-classical RTF
energy density functional may be a very efficient and simplified way to
estimate the average excitation energy of some giant resonances. Although the
constrained RTF approach has been used in [81] to compute the average
excitation energy of the ISGMR, in this chapter we want to use a new
constrained method applied together with the RTF approach, which is able to
study not only the ISGMR but also the IVGDR. This new constrained calculation
is based on a Taylor series expansion of the constrained energy density
functional around the equilibrium. In Section 4.2 we outline the general
formalism to calculate excitation energy of ISGMR and ISVGDR. In Section 4.3
we discussed the application of relativistic Thomas-Fermi approximation to
calculate the energy and particle density. Section 4.4 is dedicated to the
explanation of perturbative approches to calculate the excitation energy of
ISGMR and ISVGDR. Results of our new method with old one are given in section
4.5 and concluded in section 4.6.

###  4.2 Formalism

The strength distribution associated to an excitation operator ${\bf Q}$ is
defined as

$\displaystyle{S_{Q}(E)}={\sum_{n\neq 0}}{|\langle
n|Q|0\rangle|}^{2}\delta(E-E_{n}),$ (4.1)

where $|0\rangle$ and $|n\rangle$ are the ground and excited eigenstates,
respectively, of the exact Hamiltonian $H=T+V$ and $E_{n}$ is the excitation
energy.

The energy weighted moments, also known as sum rules, are defined as

$\displaystyle{m_{k}}=\int_{0}^{\infty}{E^{k}S_{Q}(E)}dE={\sum_{n\neq
0}}{E_{n}}^{K}{|\langle n|Q|0\rangle|}^{2},$ (4.2)

which allow to estimate the average excitation energy as

$\tilde{{E_{K}}}=\sqrt{\frac{m_{k}}{m_{k-2}}}.$ (4.3)

In this work we concentrate in the study of the $\tilde{E}_{1}$ average energy
to estimate the excitation of the ISGMR and the IVGDR. A very useful
approximation arises when the exact energy eigenstates of the Hamiltonian $H$
in (4.1) is replaced by the one-hole-one particle ( 1p1h) RPA ones. In this
case it is possible to show that some sum rules, in particular $m_{1}$ and
$m_{-1}$, can be exactly obtained by replacing the RPA ground-state by the
uncorrelated Hartree-Fock (HF) or by performing constrained HF calculations
(see [76] for more details). In the monopole case, the energy weighted sum
rule $m_{1}$ is almost a model independent quantity and according to the
Thouless theorem it is given by

${m_{1}}={\sum_{n}}{E_{n}}|\langle
n|\sum_{i=1}^{A}|0\rangle|^{2}=2A\frac{{\hbar}^{2}}{m}\langle{r^{2}}\rangle.$
(4.4)

For the isovector dipole, the previous simple approach provided in Eq.(4.4)
does not hold good because the excitation operator ${\bf Q}$ does not commute,
in general, with the potential part $V$ of the Hamiltonian. In the isovector
case the full calculation of the $m_{1}$ sum rule is complicated and usually
this moment is factorized as

${m_{1}}={m_{1}}^{0}(1+\kappa),$ (4.5)

where ${m_{1}}^{0}$ is the sum rule computed assuming $[Q,V]=0$ and $\kappa$
is the so-called enhancement factor. In the dipole case, the ${m_{1}}^{0}$ is
given by the kinetic Thomas-Reiche-Kuhn (TRK) term and therefore, the full
energy weighted sum rule reads as

${m_{1}}=\frac{\hbar^{2}}{2m}\frac{NZ}{A}(1+\kappa).$ (4.6)

For relativistic model the TRK enhancement factor in nuclear matter is given
by [163]

$\kappa(k_{F})=\frac{m}{\sqrt{{m^{*}}^{2}+{k_{F}}^{2}}}-1,$ (4.7)

where $k_{F}$ the Fermi momentum. Notice that in nuclear matter $\kappa$ is a
function of $k_{F}$. In order to estimate the average enhancement factor in
finite nuclei, we use (4.7) together with a local density approximation:

$\kappa=\frac{1}{A}\int{\kappa(k_{F})}\rho(r){\bf dr},$ (4.8)

where the local Fermi momentum $k_{F}$ is related to the finite nucleus
density by $k_{F}(r)=(3\pi^{2}\rho(r))^{1/3}$.

To obtain the inverse energy-weighted sum rule, we take into account the fact
that this sum rule computed at 1p1h RPA level, according to the dielectric
theorem, can also be obtained from a constrained HF calculation performed with
the Hamiltonian $H+\lambda Q$, where $H$ is the Hamiltonian that describe the
nucleus and $Q$ the one-body excitation operator as

$\displaystyle{m_{-1}}={\sum_{n}}\frac{|\langle
n|Q|0\rangle|^{2}}{E_{n}}=\frac{1}{2}\frac{\partial\langle\lambda|Q|\lambda\rangle}{\partial\lambda}\bigg{|}_{\lambda=0}=\frac{1}{2}\frac{\partial^{2}\langle\lambda|H|\lambda\rangle}{\partial\lambda^{2}}\bigg{|}_{\lambda=0},$
(4.9)

where $|\lambda\rangle$ is the HF ground state of the constrained Hamiltonian
$H+\lambda Q$.

###  4.3 Relativistic Thomas-Fermi approximation

In this work we describe the average excitation energy of ISGMR and IVGDR by
using Thomas-Fermi approximation to the RMF theory discussed in early
literature [161, 162, 80, 81] . A reason for using semi-classical technique is
to study giant resonances is the fact that these oscillations are strong
collective motion where shell effects have only a marginal impact [81, 164].
In the linear RMF model nucleons interact via the exchange of the effective
$\sigma$, $\omega$ and $\rho$ mesons [8]. However, due to the high
incompressibility $K_{\infty}$ predicted by this model, self-interactions of
the $\sigma$ meson, through non-linear terms, have been introduced in the
formalism to reduce $K_{\infty}$ to a more realistic value [99]. The starting
point of the semi-classical RMF theory is the nucleon-nucleon interacting
effective Lagrangian density, which can be written as

$\displaystyle{\cal H}={\cal E}+g_{v}V\rho+g_{\rho}R\rho_{3}+e{\cal
A}\rho_{p}+{\cal H}_{f},$ (4.10)

where $\rho=\rho_{n}+\rho_{p}$ is the baryon density,
$\rho_{3}=(\rho_{p}-\rho_{n})/2$ is the isovector density and ${\cal E}$ the
nucleon energy density, which at RTF level can be written as [161, 162, 80,
81]

${\cal
E}=\sum_{q}\frac{1}{8\pi^{2}}[k_{F_{q}}\epsilon_{F_{q}}^{3}+k_{F_{q}}^{3}\epsilon_{F_{q}}-{m^{*}}^{4}\frac{k_{F_{q}}+\epsilon_{F_{q}}}{m^{*}}].$
(4.11)

For each kind of nucleon $(q=n,p)$, the local Fermi momentum $k_{F_{q}}$ and
energy $\epsilon_{F_{q}}$ are defined as

$k_{F_{q}}=(3\pi^{2}\rho_{q})^{1/3}\quad;\quad\epsilon_{F_{q}}=\sqrt{k_{F_{q}}^{2}+{m^{*}}^{2}}.$
(4.12)

In Eq. (4.10), ${\cal H}_{f}$ stands for the free contribution of the meson
fields $\phi$, $V$ and $R$, associated to the $\sigma$, $\omega$ and $\rho$
mesons, and for the Coulomb field ${\cal A}$. This contribution reads

$\displaystyle{\cal H}_{f}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[(\vec{\nabla}\phi)^{2}+m_{s}^{2}\phi^{2}\right]+\frac{1}{3}b\phi^{3}+\frac{1}{4}c\phi^{4}-\frac{1}{2}\left[(\vec{\nabla}V)^{2}+m_{v}^{2}V^{2}\right]$
(4.13) $\displaystyle-$
$\displaystyle\frac{1}{2}\left[(\vec{\nabla}R)^{2}+m_{\rho}^{2}R^{2}\right]-\frac{1}{2}\left(\vec{\nabla}{\cal
A}\right)^{2},$

where $m_{s}$, $m_{v}$ and $m_{\rho}$ are the masses of the mesons and
$m^{*}$, entering in Eqs.(4.11) and (4.12, is the nucleon effective mass
defined by $m^{*}=m-g_{s}\phi$. In these equations $g_{s}$, $g_{v}$,
$g_{\rho}$ and $e$ are the coupling constants for the ${\sigma}$, $\omega$ and
${\rho}$ mesons and for the photon, respectively. In Eq.(4.13) $b$ and $c$
stand for the coupling constants associated to the ${\sigma}$ meson non-linear
terms. These self-interactions of the $\sigma-$ meson generates analogous
effect of three body interaction due to its off-shell meson couplings. These
self-interactions are also essential for the saturation properties of infinite
nuclear matter [165, 166, 167].

The semi-classical ground-state densities and meson fields are obtained by
solving the variational equations derived from the energy density (4.10)
constrained by the condition of fixed neutron ($N$) and proton ($Z$) numbers,
which read

$\epsilon_{F_{n}}+g_{v}V-\frac{1}{2}g_{\rho}R-\mu_{n}=0,$ (4.14)
$\epsilon_{F_{p}}+g_{v}V+\frac{1}{2}g_{\rho}R+e{\cal A}-\mu_{p}=0,$ (4.15)
$\Delta V-{m_{v}}^{2}V+{g_{v}}\rho=0,$ (4.16) $\Delta
R-{m_{\rho}}^{2}{R}+{g_{\rho}}\rho_{3}=0,$ (4.17) $\Delta\phi-
m_{s}^{2}{\phi}+g_{s}\rho_{s}-b\phi^{2}-c\phi^{3}=0,$ (4.18)

and

$\Delta{\cal A}+e\rho_{p}=0.$ (4.19)

In Eqs.(4.14) and (4.15) $\mu_{n}$ and $\mu_{p}$ are the neutron and proton
chemical potentials, respectively, introduced to ensure the right $N$ and $Z$
values. In Eq.(4.18) $\rho_{s}$ is the semi-classical scalar density given by

$\rho_{s}=\frac{\partial{\cal E}}{\partial
m^{*}}=\sum_{q}\frac{m^{*}}{2\pi^{2}}[k_{F_{q}}\epsilon_{F_{q}}-{m^{*}}^{2}\frac{k_{F_{q}}+\epsilon_{F_{q}}}{m^{*}}].$
(4.20)

###  4.4 The constrained perturbative approach

We apply now the constrained method to estimate the excitation energy of the
ISGMR and IVGDR. In the semi-classical context one has to minimize the
constrained energy density functional [81]

$\int[{\cal H}-\mu_{n}\rho_{n}-\mu_{p}\rho_{p}-\lambda Q]d{\bf
r}=E(\lambda)-\lambda<Q>.$ (4.21)

In previous studies of the ISGMR using the constrained method [81, 117, 164]
one minimizes Eq. (4.21) for several values of $\lambda$ and then computes
numerically the derivatives respect to $\lambda$, which define the $m_{-1}$
sum rule (see Eq.(4.9)) , using three- or five-point formulas. This technique
may also be applied, in principle, to estimate the excitation energy of the
IVGDR using a deformed code. We present here an alternative route which avoids
the explicit use of deformation. Keeping in mind that the derivatives respect
to the parameter $\lambda$ in Eq.(4.9) are computed at $\lambda=0$, we expand
the constrained energy density functional (4.21) around equilibrium in powers
of $\lambda$ up to quadratic terms, which according to Eq.(4.9), is needed to
compute the $m_{-1}$ sum rule. To this end, we write the nuclear densities and
meson fields as the sum of the unperturbed solutions, obtained by solving
self-consistently Eqs.(4.14)-(4.20), plus a perturbative contribution, which
is linear in $\lambda$ and has the same angular dependence as the excitation
operator $Q$. Notice that, as far as the right number of particles is obtained
by the unperturbed density, the additional constraint $\int\delta\rho_{n}({\bf
r})d{\bf r}=\int\delta\rho_{p}({\bf r})d{\bf r}=0$ must be imposed. This can
be achieved by introducing additional contributions to the neutron and proton
chemical potentials, $\delta\mu_{n}$ and $\delta\mu_{p}$, respectively, which
are also linear in the parameter $\lambda$.

The independent term of the constrained energy functional (4.21) in powers of
$\lambda$ is just the unperturbed energy $E_{0}$ of the ground-state of the
nucleus. The linear term in $\lambda$ are proportional to the equation of
motion of the the unperturbed problem Eqs.(4.14)-(4.20), while the quadratic
contribution provides the set of equations of motion corresponding to the
pertubative part to the nuclear densities and meson fields by applying the
variations principle. According to Eq. 4.9, $m_{-1}$ sum rule is given by one
half of the second derivative of the constrained energy with respect to the
parameter $\lambda$. At this point two important comments are in order. First,
the equations of motion of the perturbative part of the nuclear densities and
fields can also be obtained by performing a second order variation with
respect to the densities and fields in the equations of motion of the
unperturbed problem, Eqs. (4.14)-(4.20). Second, by replacing the solutions of
the perturbative contribution to densities and fields in the second derivative
of the energy density functional respect to the parameter $\lambda$, one can
recast this expression as one half of the first derivative of the excitation
operator $Q$ respect to $\lambda$, in agreement with Eq.(4.9). These two
comments point out that this perturbative approach to the calculation of the
$m_{-1}$ sum rule is properly formulated. We will now apply this perturbative
constrained method to estimate the excitation energies of the ISGMR and IVGDR.

####  4.4.1 Isoscalar giant monopole resonance

The ISGMR is a collective excitation in which proton and neutron vibrate in a
phase to each other. It corresponds to the radial oscillation of nucleus
around its equilibrium radius. In order to calculate excitation energy we
write the excitation operator as $Q=r^{2}-<r^{2}>_{0}$, where
${<r^{2}>_{0}}^{1/2}$ is the rms radius computed using the unperturbed
equilibrium density $\rho_{0}$. We apply the constrained perturbative method
by writing the perturbative nuclear densities and fields as
$\lambda\delta\rho_{n}$, $\lambda\delta\rho_{p}$, $\lambda\delta V_{0}$ ,
$\lambda\delta\phi$, $\lambda\delta R$ and $\lambda\delta{\cal A}$. Due to the
spherical symmetry of the problem, we shall also introduce the corrections
$\lambda\delta\mu_{n}$ and $\lambda\delta\mu_{p}$ to the neutron and proton
chemical potentials. With all these ingredients we expand Eq. (4.21) up to
order $\lambda^{2}$ obtaining

$\displaystyle{\tilde{E}}$ $\displaystyle=$ $\displaystyle\int dr\bigg{[}{\cal
H}^{0}+\lambda\bigg{[}\left({\epsilon^{0}_{F_{n}}}+{g_{v}}V^{0}-\frac{1}{2}g_{\rho}R^{0}\right)\delta\rho_{n}$
(4.22) $\displaystyle+$
$\displaystyle\left({\epsilon^{0}_{F_{p}}}+{g_{v}}{V^{0}}+\frac{1}{2}g_{\rho}R^{0}+e{\cal
A}^{0}\right)\delta\rho_{p}+\left(\Delta{\cal
A}^{0}+e\rho^{0}_{p}\right)\delta{\cal A}$ $\displaystyle+$
$\displaystyle\bigg{(}\Delta{V^{0}}-{m_{v}}^{2}{V^{0}}+g_{v}{\rho^{0}}\bigg{)}\delta{V}+\bigg{(}\Delta
R^{0}-{m_{\rho}}^{2}R^{0}+g_{\rho}{\rho^{0}_{3}}\bigg{)}\delta R$
$\displaystyle-$
$\displaystyle\bigg{(}\Delta\phi-{m_{s}}^{2}\phi^{0}+g_{s}{\rho^{0}_{s}}-b{\phi^{0}}^{2}-c{\phi^{0}}^{3}\bigg{)}\delta\phi\bigg{]}$
$\displaystyle+$
$\displaystyle\frac{\lambda^{2}}{2}\bigg{[}\bigg{(}\frac{\partial{\epsilon^{0}_{F_{n}}}}{\partial\rho^{0}_{n}}\delta\rho_{n}-{g_{s}}\frac{\partial{\epsilon^{0}_{F_{n}}}}{\partial
m_{0}^{*}}\delta\phi+g_{v}\delta V-\frac{1}{2}g_{\rho}\delta
R\bigg{)}\delta\rho_{n}$ $\displaystyle+$
$\displaystyle\bigg{(}\frac{\partial{\epsilon^{0}_{F_{p}}}}{\partial\rho^{0}_{p}}\delta\rho_{p}-{g_{s}}\frac{\partial{\epsilon^{0}_{F_{p}}}}{\partial
m_{0}^{*}}\delta\phi+e\delta{\cal A}+g_{v}\delta V+\frac{1}{2}g_{\rho}\delta
R\bigg{)}\delta\rho_{p}$ $\displaystyle+$
$\displaystyle\bigg{(}g_{v}\delta\rho+\Delta\delta V-m_{v}^{2}\delta
V\bigg{)}\delta V+\bigg{(}g_{\rho}\delta\rho_{3}+\Delta\delta
R-m_{\rho}^{2}\delta R\bigg{)}\delta R$ $\displaystyle+$
$\displaystyle\bigg{(}e\delta\rho_{p}+\Delta\delta{\cal A}\bigg{)}\delta{\cal
A}+\bigg{(}\bigg{(}{g_{s}}^{2}\frac{\partial\rho^{0}_{s}}{\partial{m^{*}}^{0}}+2b\phi^{0}+3c{\phi^{0}}^{2}\bigg{)}\delta\phi$
$\displaystyle-$
$\displaystyle{g_{s}}\frac{\partial{\epsilon^{0}_{F_{n}}}}{\partial{m^{*}}^{0}}\delta\rho_{n}-{g_{s}}\frac{\partial{\epsilon^{0}_{F_{p}}}}{\partial{m^{*}}^{0}}\delta\rho_{p}-\Delta\delta\phi+m_{s}^{2}\delta\phi\bigg{)}\delta\phi\bigg{]}$
$\displaystyle-$
$\displaystyle\mu^{0}_{n}\rho^{0}_{n}-\mu^{0}_{p}\rho^{0}_{p}-\bigg{(}\mu^{0}_{n}+\lambda\delta\mu_{n}\bigg{)}\lambda\delta\rho_{n}-\bigg{(}\mu^{0}_{p}+\lambda\delta\mu_{p}\bigg{)}\lambda\delta\rho_{p}$
$\displaystyle-$
$\displaystyle\lambda(r^{2}-<r^{2}>^{0})(\delta\rho_{n}+\delta\rho_{p})\bigg{]}.$

In the above equation the superscript 0 indicates quantities which are
calculated using the unperturbed densities and fields. The linear
contributions of the perturbative components of densities and fields in
Eq.(4.22) vanish, because their prefactors are just the motion equations of
the unperturbed densities and fields Eqs.(4.14)-(4.20), which are zero for the
self-consistent solutions. Therefore, the constrained energy Eq.(4.21) becomes
a quadratic function of the perturbative corrections to the nuclear densities
and meson fields, from where the motion equations for $\delta\rho_{n}$,
$\delta\rho_{p}$,$\delta V$, $\delta R$, $\delta{\cal A}$ and $\delta\phi$ are
easily obtained:

$\frac{\partial{\epsilon^{0}_{F_{n}}}}{\partial\rho^{0}_{n}}\delta\rho_{n}-{g_{s}}\frac{\partial{\epsilon^{0}_{F_{n}}}}{\partial
m_{0}^{*}}\delta\phi+g_{v}\delta V-\frac{1}{2}g_{\rho}\delta
R-\delta{\tilde{\mu}}_{n}-r^{2}=0,$ (4.23)

$\frac{\partial{\epsilon^{0}_{F_{p}}}}{\partial\rho^{0}_{p}}\delta\rho_{p}-{g_{s}}\frac{\partial{\epsilon^{0}_{F_{p}}}}{\partial
m_{0}^{*}}\delta\phi+e\delta{\cal A}+g_{v}\delta V+\frac{1}{2}g_{\rho}\delta
R-\delta{\tilde{\mu}}_{p}-r^{2}=0,$ (4.24)

$g_{v}\delta\rho+\Delta\delta V-m_{v}^{2}\delta V=0,$ (4.25)

$g_{\rho}\delta\rho_{3}+\Delta\delta R-m_{\rho}^{2}\delta R=0,$ (4.26)

$(e\delta\rho_{p}+\Delta\delta{\cal A}=0,$ (4.27)

and

$\displaystyle\bigg{(}{g_{s}}^{2}\frac{\partial\rho^{0}_{s}}{\partial{m^{*}}^{0}}+2b\phi^{0}+3c{\phi^{0}}^{2}\bigg{)}\delta\phi-{g_{s}}\frac{\partial{\epsilon^{0}_{F_{n}}}}{\partial{m^{*}}^{0}}\delta\rho_{n}-{g_{s}}\frac{\partial{\epsilon^{0}_{F_{p}}}}{\partial{m^{*}}^{0}}\delta\rho_{p}-\Delta\delta\phi+m_{s}^{2}\delta\phi=0.$
(4.28)

In Eqs.(4.23) and (4.24) $\delta{\tilde{\mu}}_{q}=\delta\mu_{q}-<r^{2}>^{0}$
($q=n,p$). The self-consistent solution of the set of equations
Eqs.(4.23)-(4.24) gives the perturbed neutron and proton densities,
$\delta\rho_{n}$ and $\delta\rho_{p}$, respectively, as well as the
corrections to the fields $\delta V$ ,$\delta\phi$, $\delta R$ and
$\delta{\cal A}$, where $\delta_{n}$ and $\delta_{p}$ are the transition
densities. The $m_{-1}$ sum rule is given by the second order derivative of
the semi-classical energy $E(\lambda)$ (see Eq.(4.21)) with respect to the
parameter $\lambda$. Combining Eq.(4.22) with the motion equations
(4.23)-(4.24) for the perturbative nuclear densities and meson and photon
fields are easily obtained

$m_{-1}=\int\bigg{(}\delta{\tilde{\mu}}_{n}\delta\rho_{n}+\delta{\tilde{\mu}}_{p}\delta\rho_{p}+r^{2}\delta\rho\bigg{)}d{\bf
r},$ (4.29)

where $\delta\rho=\delta\rho_{n}+\delta\rho_{p}$. Taking into account the
constraint $\int\delta\rho_{q}d{\bf r}=0$, we get the final form of the
$m_{-1}$,

${m_{-1}}=\int r^{2}\delta\rho d{\bf r}.$ (4.30)

In order to solve the motion equations (4.23)-(4.24) for the perturbative
corrections to the nuclear densities and meson fields we need to know, in the
ISGMR case, the corrections to the neutron and proton chemical potentials. To
this end we first isolate these in equations $\delta\rho_{n}$ and
$\delta\rho_{p}$ and next, using explicitly the constraints
$\int\delta\rho_{n}d{\bf r}=\int\delta\rho_{p}d{\bf r}=0$, we obtain after
little algebra the corrections $\delta{\tilde{\mu}}_{n}$ and
$\delta{\tilde{\mu}}_{p}$ as

$\delta{\tilde{\mu}}_{n}=\frac{\int{k^{0}_{F_{n}}}\bigg{[}{\epsilon^{0}}_{F_{n}}\bigg{(}g_{v}\delta
V-\frac{1}{2}g_{\rho}\delta
R-r^{2}\bigg{)}-{m^{0}}^{*}g_{s}{\delta\phi}\bigg{]}d{\bf
r}}{\int{k^{0}_{F_{n}}}{\epsilon^{0}}_{F_{n}}d{\bf r}},$ (4.31)

and

$\delta{\tilde{\mu}}_{p}=\frac{\int{k^{0}_{F_{p}}}\bigg{[}{\epsilon^{0}}_{F_{p}}\bigg{(}g_{v}\delta
V+\frac{1}{2}g_{\rho}\delta R+e\delta
A-r^{2}\bigg{)}-{m^{0}}^{*}g_{s}{\delta\phi}\bigg{]}d{\bf
r}}{\int{k^{0}_{F_{p}}}{\epsilon^{0}}_{F_{p}}d{\bf r}}.$ (4.32)

The self-consistent solution of the coupled equations (4.23)-(4.24) allows to
obtain the perturbative contribution to the nuclear densities and fields. From
Eq.(4.30) one can obtain the inverse energy weighted sum rule ${m_{-1}}$,
which combined with the $m_{1}$ sum rule given by Eq.(4.4) provides an
estimate of the excitation energy of the ISGMR given by
$E_{1}=\sqrt{m_{1}/m_{-1}}$.

####  4.4.2 Isovector giant dipole resonance

In case of IVGDR protons and neutrons vibrate in opposite phase to each other
in such a way that the center of mass of the whole system remains unchange.
This constraint is introduced through the excitation operator
$Q=z-<z>=rY_{10}(\Omega)$, where $<z>$ is the z-coordinate of the center of
mass of the nucleus. As in the case of ISGMR, we write the neutron and proton
densities as the sum of unperturbed solution plus a perturbative term, which
follows the geometry of the excitation operator, i.e. $\rho_{n}({\bf
r})=\rho^{0}_{n}(r)+\lambda\delta\rho_{n}(r)Y_{10}(\Omega)$ and $\rho_{p}({\bf
r})=\rho^{0}_{p}(r)+\lambda\delta\rho_{p}(r)Y_{10}(\Omega)$. With this choice
of the nuclear densities the right normalization of neutrons and protons is
ensured and, consequently, the corrections $\delta\mu_{n}$ and $\delta\mu_{p}$
to the corresponding chemical potentials is not needed in the case of the
IVGDR. Using this parametrization of the nuclear densities we compute now the
expectation values of the dipole operator $Q$ defined before. It is easy to
show that

$\displaystyle<z>=\int\big{(}{\rho_{n}({\bf r})+\rho_{p}({\bf
r})\big{)}rY_{10}(\Omega)d{\bf r}}$ (4.33) $\displaystyle=$
$\displaystyle\frac{1}{A}\int{r^{3}(\delta\rho_{n}(r)+\delta\rho_{p}(r))dr}=\frac{N}{A}<z>_{n}+\frac{Z}{A}<z>_{p},$

and

$\displaystyle<Q>=\int{\big{(}\rho_{p}({\bf r})-\rho_{n}({\bf
r})\big{)}\big{(}rY_{10}(\Omega)-<z>\big{)})d{\bf r}}$ (4.34) $\displaystyle=$
$\displaystyle
2\bigg{[}\frac{N}{A}\int{r^{3}\delta\rho_{p}(r)dr}-\frac{Z}{A}\int{r^{3}\delta\rho_{n}(r)dr}\bigg{]}$
$\displaystyle=$ $\displaystyle\frac{2NZ}{A}\big{(}<z>_{p}-<z>_{n}\big{)}.$

We assume, as in the ISGMR case, that the fields also split into an
unperturbed part plus a corrective contribution that follows the excitation
field, i.e. ${V}={V}^{0}+\lambda\delta{V}Y_{10}$,
${R}={R}^{0}+\lambda\delta{R}Y_{10}$, ${\cal A}={\cal
A}^{0}+\lambda\delta{\cal A}Y_{10}$ and
${\phi}={\phi}^{0}+\lambda\delta{\phi}Y_{10}$. Therefore, one can write the
energy density functional (4.21) in powers of $\lambda$ and derive from this
expansion the motion equations of the nuclear densities and fields, as we have
done in the case of ISGMR starting from Eq.(4.22). However, as we have pointed
out before, it is also possible derive the variational equations for
$\delta\rho_{n}$, $\delta\rho_{p}$, $\delta V$, $\delta R$, $\delta\phi$ and
$\delta{\cal A}$ expanding the motion equations associated to the constrained
energy density functional (4.21). In case of the IVGDR, these equations for
the meson and photon fields are formally given by Eqs.(4.16)-(4.19), keeping
in mind, however, that in this case densities and fields are, the ones
associated to the constrained problem. The variations equations for neutron
and proton are

$\epsilon_{F_{n}}+g_{v}V-\frac{1}{2}g_{\rho}R-\mu_{n}+\lambda
rY_{10}\frac{N}{A}=0,$ (4.35)

and

$\epsilon_{F_{p}}+g_{v}V+\frac{1}{2}g_{\rho}R+e{\cal A}-\mu_{p}-\lambda
rY_{10}\frac{N}{A}=0,$ (4.36)

which correspond to Eqs.(4.14) and (4.15) modified by the contribution of the
excitation field. Expanding the densities and fields in these variations
equations in powers of $\lambda$, the independent term gives the motion
equations of unperturbed densities and fields and the linear terms in $\lambda
Y_{10}$ correspond to the variations equations for the perturbations of
densities and fields, which read

$\frac{\partial{\epsilon^{0}_{F_{n}}}}{\partial\rho^{0}_{n}}\delta\rho_{n}-{g_{s}}\frac{\partial{\epsilon^{0}_{F_{n}}}}{\partial
m_{0}^{*}}\delta\phi+g_{v}\delta V-\frac{1}{2}g_{\rho}\delta
R+\frac{Z}{A}r=0,$ (4.37)

$\frac{\partial{\epsilon^{0}_{F_{p}}}}{\partial\rho^{0}_{p}}\delta\rho_{p}-{g_{s}}\frac{\partial{\epsilon^{0}_{F_{p}}}}{\partial
m_{0}^{*}}\delta\phi+e\delta{\cal A}+g_{v}\delta V+\frac{1}{2}g_{\rho}\delta
R-\frac{N}{A}r=0,$ (4.38)

$g_{v}\delta\rho Y_{10}+\Delta(\delta VY_{10})-m_{v}^{2}\delta VY_{10}=0,$
(4.39)

$g_{\rho}\delta\rho_{3}Y{10}+\Delta(\delta RY_{10})-m_{\rho}^{2}\delta
RY_{10}=0,$ (4.40)

$(e\delta\rho_{p}Y_{10}+\Delta(\delta{\cal A}Y_{10})=0,$ (4.41)

and

$\displaystyle\bigg{(}{g_{s}}^{2}\frac{\partial\rho^{0}_{s}}{\partial{m^{*}}^{0}}+2b\phi^{0}+3c{\phi^{0}}^{2}\bigg{)}\delta\phi
Y_{10}-{g_{s}}\frac{\partial{\epsilon^{0}_{F_{n}}}}{\partial{m^{*}}^{0}}\delta\rho_{n}Y_{10}$
(4.42) $\displaystyle-$
$\displaystyle{g_{s}}\frac{\partial{\epsilon^{0}_{F_{p}}}}{\partial{m^{*}}^{0}}\delta\rho_{p}Y_{10}-\Delta(\delta\phi
Y_{10})+m_{s}^{2}\delta\phi Y_{10}=0.$

As in the case of ISGMR, we can also calculate the $m_{-1}$ sum rule for the
IVGDR from the second order derivative of the constrained energy with respect
to the $\lambda$, which after little algebra and taking into account
Eqs.(4.37)-(4.42) become

$m_{-1}=\frac{\partial^{2}E}{\partial\lambda^{2}}.\int
r^{3}dr\bigg{[}\frac{N}{A}\delta\rho_{p}-\frac{Z}{A}\delta\rho_{n}\bigg{]}.$
(4.43)

###  4.5 Results and Discussions

Before going to discussions of the predictions of our model and compare with
the results obtained applying another theoretical models as well as with the
experimental data, we outline briefly our numerical procedure to estimate
semi-classically the ISGMR and IVGDR excitation energies in finite nuclei.
First, and assuming spherical symmetry, we compute for each nucleus the
unperturbed densities and fields by solving self-consistently the set of
equations (4.14)-(4.20). using the so-called imaginary time-step method [168,
169, 170]. and these unperturbed values we can be use to compute the $m_{1}$
sum rules given by Eqs.(4.4) and (4.6), for the ISGMR and IVGDR, respectively.
Next, we solve iteratively the sets of coupled linear equations (4.23)-(4.27)
for the ISGMR and (4.37)-(4.42) for the IVGDR, which allow to obtain the
perturbative corrections to the nuclear densities and meson and photon fields.
Finally the $m_{-1}$ sum rules can be obtained through Eqs.(4.30) and (4.43)
for the ISGMR and IVGDR, respectively. All these calculations are performed at
RTF level paying special attention to the convergence of the $m_{-1}$ sum
rule. The convergence of this quantity actually depends on the size of the box
where the calculations are performed as well as on the number of iterations
used in the unperturbed calculation.

Table 4.1:  Excitation energy of the ISGMR in MeV of some spherical nuclei calculated as $\sqrt{m_{1}/m_{-1}}$ using the NL3 model. ${E_{c}}(RTF-P)$ is the estimte of the present work, ${E_{c}}(RETF)$ is the RETF result computed as in Ref.[81],${E_{c}}(RMF)$ and ${E_{c}}(RPA)$ are the constrained Hartree and RPA results, respectively, reported in [117]. Nucleus | ${E_{c}}(RTF-P)$ | ${E_{c}}(RETF)$ | ${E_{c}}(RMF)$ | ${E_{c}}(RPA)$ | Expt.
---|---|---|---|---|---
16O | 26.26 | 25.98 | 23.34 | 23.35 | 21.13$\pm$0.49
40Ca | 22.89 | 23.20 | 21.55 | 21.57 | 19.18$\pm$0.37
90Zr | 18.74 | 19.08 | 18.58 | 18.55 | 17.89$\pm$0.20
116Sn | 17.29 | 17.57 | 16.98 | 17.06 | 16.07$\pm$0.12
144Sm | 16.05 | 16.33 | 16.08 | 16.16 | 15.39$\pm$0.28
208Pb | 13.84 | 13.91 | 14.07 | 14.10 | 14.17$\pm$0.28
Table 4.2:  Excitation energy of the IVGDR in MeV of some spherical nuclei calculated as $\sqrt{m_{1}/m_{-1}}$ using the NL3 model. ${E_{c}}(RTF-P)$ is the estimte of the present work and ${E}(RPA)$ are the RPA results reported in [173]. Nucleus | ${E_{c}}(RTF-P)$ | ${E_{c}}(RPA)$ | Expt.[174]
---|---|---|---
16O | 22.05 | 21.1 | 22.3-24
40Ca | 19.90 | 19.57 | 19.8$\pm$0.5
90Zr | 17.18 | 17.19 | 16.5$\pm$0.2
116Sn | 16.25 | 15.77 | 15.7$\pm$0.2
208Pb | 13.54 | 13.16 | 13.3$\pm$0.1
Table 4.3:  Semi-classical $m_{-1}$ sum rule in fm2 MeV-1 of some spherical nuclei computed as explained in this work $(RTF-P)$ and using the DM approach (4.44) ($(RTF-LDM)$ input). All these calculation are performed with the NL3 parameter set.  Nucleus | $m_{-1}(RTF-P)$ | $m_{-1}(RTF-DM)$
---|---|---
16O | 0.219 | 0.192
40Ca | 0.699 | 0.657
68Ni | 1.499 | 1.403
90Zr | 2.130 | 2.076
116Sn | 3.061 | 3.012
120Sn | 3.250 | 3.173
208Pb | 7.685 | 7.256

Experimental information about the excitation energy of the ISGMR in medium
and heavy nuclei is obtained from inelastic scattering of $\alpha$ particles
measured at forward angles [75, 115, 171, 172]. As a first application of our
perturbative constrained approach developed in previous sections, we compute
the excitation energy of the ISGMR for several nuclei which value is
experimentally known [75, 115, 171, 172] using the RMF NL3 parametrization
[49]. Although the sum rule approach, which assumes a well defined peak, for
light nuclei may be questionable, we display in table 4.1 our semi-classical
RTF estimate of the excitation energy of ISGMR of the nuclei 16O, 40Ca, 90Zr,
116Sn, 144Sm and 208Pb calculated with our perturbative constrained approach.
In the same table we also show the constrained values obtained using the
Relativistic Extended Thomas-Fermi Approach (RETF), which includes $\hbar^{2}$
corrections, computed as described in Ref.[81] together with the corresponding
experimental values. For a sake of completeness, we also display in the same
table the constrained Hartree and RPA values of these excitation energies
reported in Ref.[117]. From this table some comments are in order. First, as
we mentioned previously, the constrained Hartree predictions of the excitation
energy of the ISGMR practically concides with the values extracted from RRPA
calculations, which numerically proves the conjecture of Ref.[158]. Second, as
it was pointed out in [81], the importance of the $\hbar^{2}$ corrections in
the excitation energy of the ISGMR is actually small as it can be seen from
the comparison between the RTF and RETF results. Third, the excitation
energies of the ISGMR decreases with increasing mass number, as expected, in
both semi-classical and quantal (constrained Hartree and RRPA) predictions.
However, the slope is larger for the semi-classical than for the quaintly
estimates. For light and medium nuclei the semi-classical values overestimate
the quantal ones while the opposite is true for 208Pb, which is the heaviest
nucleus considered in this work.

The experimental values of the excitation energies of the IVGDR basically come
from the analysis of the measured photo-absorption cross sections as mentioned
before [175]. From a theoretical point of view there are many studies of the
IVGDR at RPA and RRPA levels using different mean field models in both non-
relativistic and relativistic frames [176]. However, from long ago the IVGDR
has also been analyzed from a semi-classical point of view. Let us mention in
this respect the estimate of the $m_{-1}$ sum rule on the basis of the Droplet
Model (DM) [177] reported more than thirty years ago. More recently, the
excitation energies of the IVGDR has been estimated using the scaling method
[178, 179] and performing constrained calculations [179]. In table 4.2 we
display our constrained perturbative estimate of the excitation energy of some
selected spherical even-even nuclei computed with the NL3 parameter set [49]
as well as the RRPA predictions obtained with the same NL3 model [180] and the
experimental values extracted from Ref.[174]. From this table it can be seen
that our semi-classical RTF estimate reproduce remarkably well the RRPA values
and the experimental data. This result point out, as discussed in earlier
literature [177, 178, 179], that semi-classical approaches, as the constrained
perturbative RTF method discussed in this work, are well suited for describing
also the average excitation energy of the IVGDR due to the fact that shell
corrections in this collective oscillation are, actually, small [177].

As mentioned before, the $m_{-1}$ sum rule can also be estimated in the
framework of the LDM as [177]

$m_{-1}=\frac{A<r^{2}>}{48J}\big{(}1+\frac{15}{4A^{1/3}}\frac{J}{Q}\big{)},$
(4.44)

where $\sqrt{<r^{2}>}$ stands for the mass rms radius, $J$ for the symmetry
energy and $Q$ is the so-called surface stiffness coefficient. This
coefficient $Q$ is obtained from a semi-classical RTF calculation in semi-
infinite nuclear matter as explained in Ref.[97] and its corresponding value
is $J/Q$=1.46. The DM estimates of the $m_{-1}$ sum rule at RTF level for
several nuclei are reported in table 4.3. To obtain these values, in addition
to the $J/Q$ ratio and the symmetry energy of the NL3 model ($J=$37.40 MeV),
we also use in Eq.(4.44) the $<r^{2}>$ value obtained from a self-consistent
RTF calculation for each considered nucleus. From table 4.3 we see that the
semi-classical DM and the perturbative constrained RTF estimates of the
$m_{-1}$ sum rule are in a good agreement. Our RTF predictions of $m_{-1}$ are
also in good agreement with the values, computed semi-classically with the
Skyrme Skm∗ force, displayed in Table V of [179]. For example, $m_{-1}$ values
of 0.215, 0.682, 2.054 and 7.047 fm2 MeV-1 are reported in this reference for
the nuclei 16O, 40Ca,90Zr and 208Pb, respectively. In the same reference the
experimental values of the $m_{-1}$ sum rule, obtained as the integral up to
the pion threshold of the measured total nuclear photoabsortion cross section
[177], are also reported in the same Table V. These experimental values,
0.215$\pm$0.004, 0.682$\pm$0.016 and 7.35$\pm$0.51 fm2MeV-1 for 16O, 40Ca and
208Pb, respectively, are in harmony with our semi-classical predictions
reported in table 4.3. However, I should be pointed out that recent
measurements of the electric polarizability in 208Pb, 120Sn and 68Ni give a
smaller experimental value for the $m_{-1}$ sum rule [152].

###  4.6 Summary and outlook

We have estimated the average excitation energy of the isoscalar monopole and
isovector dipole giant resonances for several nuclei within a semiclassical
sum rule approach in the relativistic mean field framework with the NL3
parameter set. We are aware that a sum rule approach only gives information
about some selected moments of the RRPA strength function and that in this
approach a precise prediction of the excitation energy of the reasonance is
only possible if the strength is concentrated in a single peak, which is the
usual scenario for medium and heavy nuclei but not for light nuclei where the
resonance broadens and fragments. The $m_{1}$ sum rules are evaluated at RPA
level using the semi-classical expectation values calculated with the
relativistic Thomas-Fermi approximation . The $m_{-1}$ sum rule is obtained
through a new constrained perturbative calculation also within the
relativistic Thomas-Fermi theory. In this calculation the nuclear densities
and meson and photon fields are splitted into a part corresponding to the
unperturbed solution plus a perturbative correction, chosen in such a way that
it follows the excitation field. The energy density corresponding to the
constrained problem is expanded up to quadratic terms in these perturbations.
The application of the variational principle to this energy density allows to
obtain the equations of motion of the perturbative corrections to nuclear
densities and fields. With this simple semi-classical method the average
excitation energies of the isoscalar monopole and isovector dipole giant
resonances is estimated reasonably well as compared with the more fundamental
but also more cumbersome relativistic RPA calculations. The differences
between both calculations are basically due to shell effects and are less than
5% in medium and hevay nuclei where the sum rule can be considered more
confidently. This result is in agreement with previous findings in earlier
literature pointing out that shell effects have little impact on the average
excitation energies of the giant resonances studied in this chapter. Our
constrained perturbative approach has been obtained at relativistic Thomas-
Fermi level, therefore, it seems reasonable, to improve it by including the
$\hbar^{2}$ corrections which give a more realistic calculation at the nuclear
surface. As far as our method is able to describe collective oscillations with
excitation operators without spherical symmetry, as the isovector dipole, it
seems also appealing to apply our method to another isoscalar and isovector
oscillations of higher multipolarities.

##  Chapter 5 Effects of self interacting $\omega$-meson on finite and
infinite nuclear system.

The most mysterious and the debatable subject in nuclear physics is the
nucleon-nucleon interaction. A lots of theories have been proposed for the
nucleon-nucleon interaction but still it is not clear. The nucleon-nucleon
interaction affects the nuclear structure in a more prominent way. So without
the study of the nucleon -nucleon interaction nuclear structure will remain
incomplete. In this chapter, a detailed study is made for the nucleon-nucleon
interaction based on relativistic mean field theory in which the potential is
explicitly expressed in terms of masses and the coupling constants of the
meson fields. An unified treatment for self-coupling of isoscalar-scalar
$\sigma-$, isoscalar-vector $\omega$-mesons and their coupling constants are
given in an analytic form. The present investigation is focused on the effects
of self-interacting higher order $\sigma$ and $\omega$ fields on nuclear
properties. An attempt is made to explain the many-body effects by higher
order couplings $\sigma-$ and $\omega-$fields, which generally occurs in the
high-density region. Both infinite nuclear matter and the finite nuclear
properties are included in the present study to observe the behavior and
sensitivity of this self-interacting terms.

###  5.1 Introduction

The Nucleon-Nucleon (NN) interaction has been investigated for over half a
century [2]. Probably this is a long standing question in history of nuclear
physics. In fact, describing the nuclear properties in terms of the
interactions between the nucleon pair is indeed the main goal for nuclear
physicists. The NN-interaction in terms of mediated mesons was put forwarded
by Yukawa [2] in 1935. Although the meson theory is not fundamental from the
QCD point of view, it has improved our understanding of the nuclear forces as
well as highlighted some good quantitative results [181, 182]. The modern
theory of NN potential in term of particle exchanges was made possible by the
development of quantum field theory [182]. However, at low-energy, one can
assume that the interactions are instantaneous and therefore the concept of
interaction potential becomes useful. The derivation of a potential through
particle exchange is important to understand the nuclear force as well as
structural properties.

Now-a-days, there are number of developments in the nuclear theory by
introducing quark and gluon in connection with the NN-potential [183, 184].
These models give the fundamental understanding of NN interaction at present.
Here, we are not addressing all these rich and long standing problems about
NN-potential. Our aim is to highlight some basic features of the NN-
interactions arising from relativistic mean field (RMF) Lagrangian [185, 186,
187, 188]. The behavior of this potential gives an idea about the saturation
properties of nuclear force at high density limit.

This chapter is organized as follows. In Section 5.2, we briefly discuss the
theoretical formalism of NN-interaction based on relativistic mean field (RMF)
theory. The general forms of the NN potentials are expressed in the coordinate
space (r-space) in term of masses and coupling constants of the force
parameters. In Section 5.3, we review the effects of modified term in the
Lagrangian and their effects on the finite nuclei and in infinite nuclear
matter. In Section 5.4, we make few comments about the current form of the NN-
interaction to saturation condition of nuclear systems.

###  5.2 Theoretical frameworks

The nuclear potential in relativistic mean field (RMF) is possible via various
mesons interaction with nucleons. The linear relativistic mean field (RMF)
Lagrangian density for a nucleon-meson many-body system [7, 8, 189, 190] is
given as:

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\overline{\psi_{i}}\\{i\gamma^{\mu}\partial_{\mu}-M\\}\psi_{i}+{\frac{1}{2}}\partial^{\mu}\sigma\partial_{\mu}\sigma-{\frac{1}{2}}m_{\sigma}^{2}\sigma^{2}-g_{s}\overline{\psi_{i}}\psi_{i}\sigma$
(5.1)
$\displaystyle-{\frac{1}{4}}\Omega^{\mu\nu}\Omega_{\mu\nu}+{\frac{1}{2}}m_{w}^{2}V^{\mu}V_{\mu}-g_{w}\overline{\psi}_{i}\gamma^{\mu}\psi_{i}V_{\mu}-{\frac{1}{4}}\vec{B}^{\mu\nu}.\vec{B}_{\mu\nu}$
$\displaystyle+{\frac{1}{2}}m_{\rho}^{2}{\vec{R}^{\mu}}.{\vec{R}_{\mu}}-g_{\rho}\overline{\psi}_{i}\gamma^{\mu}\vec{\tau}\psi_{i}.\vec{R^{\mu}}-{\frac{1}{4}}F^{\mu\nu}F_{\mu\nu}$
$\displaystyle-e\overline{\psi}_{i}\gamma^{\mu}\frac{\left(1-\tau_{3i}\right)}{2}\psi_{i}A_{\mu}.$

If, we neglect the $\rho-$ meson, it corresponds to the Walecka model in its
original form [8, 189]. From the above relativistic Lagrangian, we obtain the
field equations for the nucleons and mesons as,

$\displaystyle\Bigl{(}-i\alpha.\bigtriangledown+\beta(M+g_{\sigma}\sigma)+g_{\omega}\omega+g_{\rho}{\tau}_{3}{\rho}_{3}){\psi}_{i}={\epsilon}_{i}{\psi}_{i},$
(5.2)
$\displaystyle(-\bigtriangledown^{2}+m_{\sigma}^{2})\sigma(r)=-g_{\sigma}{\rho}_{s}(r),$
(5.3)
$\displaystyle(-\bigtriangledown^{2}+m_{\omega}^{2})V(r)=g_{\omega}{\rho}(r),$
(5.4)
$\displaystyle(-\bigtriangledown^{2}+m_{\rho}^{2})\rho(r)=g_{\rho}{\rho}_{3}(r).$
(5.5)

In the limit of one-meson exchange and mean-field (the fields are replaced by
their expectation values or c-number), for a heavy and static baryonic medium,
the solution of single nucleon-nucleon potential for scalar ($\sigma$) and
vector ($\omega$, $\rho$) fields are given by [190, 191],

$\displaystyle V_{\sigma}(r)$ $\displaystyle=$
$\displaystyle-\frac{g_{\sigma}^{2}}{4{\pi}}\frac{e^{-m_{\sigma}r}}{r},$ (5.6)

and

$\displaystyle V_{\omega}(r)$ $\displaystyle=$
$\displaystyle+\frac{g_{\omega}^{2}}{4{\pi}}\frac{e^{-m_{\omega}r}}{r},\quad
V_{\rho}(r)=+\frac{g_{\rho}^{2}}{4{\pi}}\frac{e^{-m_{\rho}r}}{r}.$ (5.7)

The total effective nucleon-nucleon potential is obtained from the scalar and
vector parts of the meson fields. This can be expressed as [185],

$\displaystyle v_{eff}(r)$ $\displaystyle=$ $\displaystyle
V_{\omega}+V_{\rho}+V_{\sigma}=\frac{g_{\omega}^{2}}{4{\pi}}\frac{e^{-m_{\omega}r}}{r}+\frac{g_{\rho}^{2}}{4{\pi}}\frac{e^{-m_{\rho}r}}{r}$
(5.8) $\displaystyle-\frac{g_{\sigma}^{2}}{4{\pi}}\frac{e^{-m_{\sigma}r}}{r}.$

####  5.2.1 Non-linear case

The Lagrangian density in the above Eq. 5.1 contains only linear coupling
terms, which is able to give a qualitative description of the nuclear system
[190, 191]. The essential nuclear matter properties like incompressibility and
the surface properties of finite nuclei cannot be reproduced quantitatively
within this linear model. The suppression of the two-body interactions within
a nucleus in favor of the interaction of each nucleon with the average nucleon
density, means that the non-linearity acts as a smoothing mechanism and hence
leads in the direction of the one-body potential and shell structure [99, 141,
192, 193]. The replacement of mass term $\frac{1}{2}m_{\sigma}^{2}\sigma^{2}$
of $\sigma$ field by $U(\sigma)$ and $\frac{1}{2}m_{\omega}^{2}V^{\mu}V_{\mu}$
of $\omega$ field by $U(\omega)$. This can be expressed as

$\displaystyle
U(\sigma)=\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\frac{1}{3}g_{2}\sigma^{3}+\frac{1}{4}g_{3}\sigma^{4},$
(5.9) $\displaystyle
U(\omega)=\frac{1}{2}m_{\omega}^{2}V_{\mu}V^{\mu}+\frac{1}{4}c_{3}(V_{\mu}V^{\mu})^{2}.$
(5.10)

The terms on the right hand side of Eqs.( 5.9)- (5.10), except the first term,
from the non-linear self coupling of the $\sigma$ and $\omega$ mesons,
respectively [99, 141]. Here, the non-linear parameter $g_{2}$ and $g_{3}$ due
to $\sigma-$ fields are adjusted to the surface properties of finite nuclei
[194, 195]. In general most of the successful fits like NL1 and NL3 sets
yield, the $+ve$ and $-ve$ signs for $g_{2}$ and $g_{3}$, respectively. The
negative value of $g_{3}$ is a serious problem in quantum field theory and
responsible for the divergence of a solution in the lighter mass region of
periodic table i.e. for higher density region. As, we are dealing within the
mean field level and with normal nuclear matter density, the corresponding
$\sigma$ field is very small and the $-ve$ value of $g_{3}$ is still allowed
[194, 196]. Again, $c_{3}=\frac{1}{6}\zeta_{0}$ is the non-linear coupling
constant for self-interacting $\omega$-mesons. With the addition of the non-
linear terms in the Eqs. ( 5.9)- (5.10) to the Lagrangian, the field equations
for $\sigma$ and $\omega$\- fields (in Eq.5.6-5.7 ) are modified as:

$\displaystyle(-\bigtriangledown^{2}+m_{\sigma}^{2})\sigma(r)=-g_{\sigma}{\rho}_{s}(r)-g_{2}r\sigma^{2}(r)-g_{3}\sigma^{3}(r),$
$\displaystyle(-\bigtriangledown^{2}+m_{\omega}^{2})V(r)=g_{\omega}{\rho}(r)-c_{3}W^{3}(r).$
(5.11)

Here, $W(r)=g_{\omega}V_{0}(r)$ is the non-linear self-interacting
$\omega$-fields. Because of the great difficulty in solving the above
nonlinear differential equations, it is essential to have a variational
principle for the estimation of the energies [194, 195]. In the static case,
the negative sign of the third term in the Lagrangian is computed with the
correct source function and an arbitrary trial wave function. The limit on the
energy has a stationary value equal to the proper energy when the trial wave-
function is in the infinitesimal neighborhood of the correct one. Now, the
solution for the modified $\sigma$ and $\omega$ fields are given as [194]

$\displaystyle
V_{\sigma}=-\frac{g_{\sigma}^{2}}{4{\pi}}\frac{e^{-m_{\sigma}r}}{r}+\frac{g_{2}^{2}}{4{\pi}}r\frac{e^{-2m_{\sigma}r}}{r}+\frac{g_{3}^{2}}{4{\pi}}\frac{e^{-3m_{\sigma}r}}{r},$
$\displaystyle
V_{\omega}=\frac{g_{\omega}^{2}}{4{\pi}}\frac{e^{-m_{\omega}r}}{r}-\frac{c_{3}^{2}}{4{\pi}}\frac{e^{-3m_{\omega}r}}{r^{2}}.$
(5.12)

The new NN-interaction analogous to $M3Y$ form and is able to improve the
incompressibility and deformation of the finite nuclei results [196]. In
addition to this, the non-linear self coupling of the $\sigma$ and
$\omega$-mesons help to generate the repulsive and attractive part of the NN
potential at $long$ and at $short$ distance, respectively to satisfy the
saturation properties (Coester-band problem) [197]. We are dealing with two
type of mesons, one is scalar ($\sigma$) and other is vector $(\omega)$. The
range of their interactions are also different due to their different masses.
Consider the case of $\sigma-$meson, where the range of interaction is
$\sim\frac{\hbar}{m_{\sigma}c}$ fm. In this range the attractive part of the
potential comes from the exchange of the $\sigma-$meson. The density dependent
many-body effect demands a repulsive part in this region. This is given by the
self interacting terms like $\sigma^{3}$ and $\sigma^{4}$ [165]. A suitable
adjustment of the parameter able to reproduces the proper potential satisfying
the Coester band problem. Generally, the exchange of $\omega-$meson gives the
repulsive potential in the short-range part of the hard core region.
Diametrically, opposite phenomenon is also occurs in case of $\omega-$meson
coupling. Contrary to the non-linearity of $\sigma-$meson nature, the self-
coupling of $\omega-$meson gives an attractive component at very short
distance ($\sim 0.2$ fm) of the nuclear potential. The non-linear terms also
generate the most discussed $3-body$ interaction [165]. The modified effective
nucleon-nucleon interaction is defined as [185]:

$\displaystyle v_{eff}(r)$ $\displaystyle=$ $\displaystyle
V_{\omega}+V_{\rho}+V_{\sigma}$ (5.13)
$\displaystyle=\frac{g_{\omega}^{2}}{4{\pi}}\frac{e^{-m_{\omega}r}}{r}+\frac{g_{\rho}^{2}}{4{\pi}}\frac{e^{-m_{\rho}r}}{r}-\frac{g_{\sigma}^{2}}{4{\pi}}\frac{e^{-m_{\sigma}r}}{r}$
$\displaystyle+\frac{g_{2}^{2}}{4{\pi}}r\frac{e^{-2m_{\sigma}r}}{r}+\frac{g_{3}^{2}}{4{\pi}}\frac{e^{-3m_{\sigma}r}}{r}-\frac{c_{3}^{2}}{4{\pi}}\frac{e^{-3m_{\omega}r}}{r}.$

###  5.3 Results and Discussions

The Eq. ( 5.13) shows that the effective NN-potential in terms of the well
known inbuilt RMF theory parameters of $\sigma$, $\omega$ and $\rho$ meson
fields. Here, we have used RMF (NL3) force parameter along with varying
$c_{3}$ for $\omega$-self interactions to determine the nuclear properties.
The values of the parameters for NL3-force are listed in Table 5.1. Although,
the $\omega^{4}$ term is already there in the FSU-Gold parameter [198, 90],
here we are interested to see the effect of non-linear self coupling of
$\omega$ meson. Thus, we have added the self-interaction of $\omega$ with
coupling constant $c_{3}$ on top of NL3 set and observed the possible effects.

Figure 5.1: The effective NN interaction potentials as a function of distance
$r$ from Eq. 5.6\- 5.12 for NL3 parameter set.

First of all, we have calculated the NN-potential for linear and non-linear
cases using Eq. 5.8 and 5.13, respectively. The obtained results for each
cases are shown in Fig. 5.1. From the figure, it is clear that without taking
the non-linear coupling for RMF (NL3), one cannot reproduce a better NN-
potential. In other word, the depth of the potential for linear and non-linear
are $\sim$ 150 MeV and 50 MeV, respectively. Thus, the magnitude of the depth
for linear case is not reasonable to fit the NN-data. Again, considering the
values of $c_{3}$, there is no significant change in the total nucleon-nucleon
potential. For example, the NN-potential does not change at all for $c_{3}$
$\simeq$ $\pm$ 0.6, which can be seen from Fig. 5.1.

Figure 5.2:  The contribution of $\sigma$-potential from linear, non-linear
and total as a function of distance $r$ for NL3 parameter set. Figure 5.3:
The contribution of $\omega$-potential from linear, non-linear and total as a
function of distance $r$ for for NL3 parameter set.

Further, we have calculated the individual contribution of meson fields to the
NN-potential in particular case of $\sigma$ and $\omega$-mesons. In case of
$\sigma$-field, we have calculated the linear and non-linear contributions
separately, and combined to get the total $\sigma$-potential as shown in Fig.
5.2. From the figure, one can find the non-linear self-interacting terms in
the $\sigma$-field play an important role (contributing a repulsive
interaction) in the attractive part of the $\sigma-$meson domain, giving rise
to a repulsive potential complementing the 3-body effect of the nuclear force
in the total NN-potential [194]. The linear and non-linear contribution of the
$\omega$-field at various $c_{3}$ are shown in Fig. 5.3. The important feature
in this figure is that the linear term give an infinitely large repulsive
barrier at $\sim 0.5$ fm, in which range, the influence of the non-linear term
of the $\omega-$meson is zero. However, this non-linear terms is extremely
active at very short distance ($\sim 0.2$ fm), which can be seen from the
figure.

Figure 5.4: The energy per particle of symmetric nuclear matter as a function
of baryon density for various values of $c_{3}$.

That means, mostly the (i) linear term of the $\sigma-$ meson is responsible
for the attractive part of the nuclear force (nuclear binding energy) (ii) the
non-linear terms are responsible for the repulsive part of the nuclear force
at long distance, which simulate the 3-body interaction of the nuclear force
[194], which also help to explain the Coester band problem. (iii) similarly,
the linear term of the $\omega-$ meson is restraint for the repulsive part of
the nuclear force (known as hard core) and (iv) the non-linear self-coupling
of the $\omega-$ meson ($\frac{1}{4}c_{3}V_{\mu}V^{\mu}$) is responsible for
the attractive part in the very shortest ($\sim 0.2fm$) region of the NN-
potential. It is worthy to mention that the values of these constants are
different for different forces of RMF theory. Hence, the NN-potential somewhat
change a little bit in magnitude by taking different forces, but the nature of
the potential remains unchanged.

####  5.3.1 Energy density and Pressure density

In the present work, we study the effect of the additional term on top of the
$NL3$ force parameter to the Lagrangian [49], which comes from the self-
interaction of the vector fields with $c_{3}$ as it is done in the Refs. [198,
90, 199, 200]. The inclusion of this term is not new, it is already taken into
account for different forces of RMF and effective field theory motivated
relativistic mean field theory (E-RMF). Here, our aims is to see the effect of
$c_{3}$ to the nuclear system and the contribution to the attractive part of
the hard core of NN-potential. We have solved the mean field equations self-
consistently and estimated the energy and pressure density as a function of
baryon density. The NL3 parameter set along with the additional $c_{3}$ is
used in the calculations[199, 200]. The obtained results for different values
of $c_{3}$ are shown in Figs. 5.4 and 5.5, respectively. From the figure, it
is clearly identify that the $-ve$ value of $c_{3}$ gives the stiff equation
of state (EOS), meanwhile the $+ve$ value shows the soft EOS. It is to be
noted that mass and radius of the neutron star depends on the softness and
stiffness of EOS. Here, in our investigation, we observed that the softening
of the EOS depends on the non-linear coupling of the $\omega-$ meson [193,
90]. The recent measurement of Demorest et. al. [201] put a new direction that
the NL3 force needs slightly softer EOS. However, when we deals with G2
(E-RMF) model, the results of Ref. [202] demands a slightly stiffer EOS. This
implies that, the value of $c_{3}$ should be fixed according to solve the
above discussed problem.

Figure 5.5: The pressure density of symmetric nuclear matter as a function of baryon density for various values of $c_{3}$. Figure 5.6: The binding energy of 40Ca and 208Pb in their ground state for different values of $c_{3}$. Figure 5.7: The excitation energy as a function of $c_{3}$ for 40Ca and 208Pb. (b) The incompressibility of 40Ca and 208Pb as a function of $c_{3}$. Table 5.1: The values of $m_{\sigma}$, $m_{\omega}$, $m_{\rho}$ (in MeV) and $g_{\sigma}$, $g_{\omega}$, $g_{\rho}$ for RMF (NL3) force, along with the self-interacting $\omega-$ field with coupling constant $c_{3}$ . set | ${m}_{\sigma}$ | ${m}_{\omega}$ | $m_{\rho}$ | $g_{\sigma}$ | $g_{\omega}$ | $g_{\rho}$ | $g_{2}(fm^{-1})$ | $g_{3}$ | $c_{3}$
---|---|---|---|---|---|---|---|---|---
NL3 | 508.194 | 782.5 | 763.0 | 08.31 | 13.18 | 6.37 | -10.4307 | -28.8851 | 0.0$\pm$0.6

####  5.3.2 Binding energy, Excitation energy and Compressibility

To see the sensitivity of $c_{3}$ on the finite nuclei, we calculated the
binding energy (BE), giant monopole excitation energy ($E_{x}$) for 40Ca and
208Pb nuclei as representative cases as a function of $c_{3}$. The excitation
energy and incompressibility are calculated by scaling method within the
framework of extended Thomas-Fermi approximation. The obtained results are
shown in upper and lower panel of the Fig. 5.6 and 5.7. The experimental and
empirical data are also displaced for comparison.

From the figure, one observes a systematic variation of binding energy by
employing the isoscalar-vector selfcoupling parameter $c_{3}$. For example,
the binding energy monotonically changes for all values of $c_{3}$. When
$c_{3}=0$, the NL3 set reproduces the original binding energies for both
${}^{40}Ca$ and ${}^{208}Pb$, which are fitted with the data while
constructing the force parameters. As soon as the self-coupling constant is
non-zero, the calculated BE deviates from the data, because of the influence
of $c_{3}$. Again, analyzing the excitation energy, we find reasonable match
of $E_{x}$ with the observation (lower panel of Fig. 5.7). These values of
$E_{x}$ remain almost constant for a wide range of $c_{3}$ ($\sim$ −2 to
$\sim$1), beyond which $E_{x}$ increases drastically for positive value only.
Further, we analyze the variation of compressibility modulus with c3 for 40Ca
and 208Pb (upper panel of Fig. 5.7). We include both positive and negative
values of $c_{3}$ (−2 to +2) to know the effects on the sign of $c_{3}$.
Similar to the monopole excitation energy, we find that the compressibility
modulus does not change with the increase of c3 up to some optimum value. The
empirical (nuclear matter compressibility modulus $K_{\infty}$) data of
${K_{\infty}}^{emp}=210\pm 30$ MeV and the nuclear matter incompressibility
for NL3 set (271.76 MeV) are given in the figure to have an idea about the
bridge between the nuclear matter limit for finite nuclei. In general, these
two values along with the finite nuclei compressibility modulus gives an
overall estimation about a possible link among them in finite nuclei and
nuclear matter limit. It is interesting to notice that although we get a stiff
equation of state with negative value of $c_{3}$ for infinite nuclear matter
system, this behavior does not appear in finite nuclei, i.e. the $K_{A}$ and
$E_{x}$ do not change with sign of $c_{3}$. May be the density with which we
deal in the finite nucleus is responsible for this discrepancy. However,
$K_{A}$ and $E_{x}$ increase substantially after certain value of $c_{3}$,
i.e. the finite nucleus becomes too much softer at about $c_{3}$ $\sim$1.0
resulting a larger incompressibility.

###  5.4 Summary and Conclusions

In summary, we analyzed the effects of the non-linear self-coupling of the
$\sigma-$scalar and $\omega-$vector mesons. At long range, i.e., more than 0.5
fm, the self coupling of the $\sigma-$meson gives a repulsive component
contrary to the attractive part of the linear term. This repulsive nature of
the nuclear potential originated from the nonlinear terms of the
$\sigma-$meson couplings simulate the 3-body force. This 3-body force is
mostly responsible to solve the Coester band problem in RMF formalism. On the
other hand, at extremely short distance, the nonlinear term of the
$\omega-$meson coupling gives a strongly attractive potential for both
$positive$ and $negative$ value of $c_{3}$. This short range distance is about
$0.2fm$, beyond (more than 0.2 fm) which the $vector-$meson interaction itself
shows a strong repulsion due to its linear interaction of the $\omega-$meson
and responsible for the saturation of nuclear force. Thus, one conclude that
the effects of the vector self-coupling is crucial for the attractive nature
of the nuclear force at the extremly short range region and should be taken in
equal footing while constructing the force parameter in relativistic field
theory.

##  Chapter 6 Effects of $NN$ potentials on $p$ nuclides in the A$\sim$100-120
region

In the previous chapter, we discussed the effects of self-interacting $\omega$
-meson ($\omega^{4}$) coupling on various properties starting from the binding
energy of finite nucleus to equation of state of the infinite nuclear matter.
A special attention was given on the effect of $\omega^{4}$ coupling on newly
proposed R3Y nucleon-nucleon potential. In this chapter, we will discuss some
more applications of our proposed nucleon-nucleon interaction, R3Y in the
calculation of astrophysical S-factor. Microscopic optical potentials for low
energy proton reactions have been obtained by folding density dependent M3Y
and R3Y interaction derived from nuclear matter calculation with densities
from mean field approach to study astrophysically important proton-rich nuclei
in mass 100-120 region. We compare S factors for low-energy $(p,\gamma)$
reactions with available experimental data and further calculate astrophysical
reaction rates for $(p,\gamma)$ and $(p,n)$ reactions using both R3Y and M3Y
interaction. Along with the linear R3Y interaction, we choose some nonlinear
R3Y (NR3Y) interactions from RMF calculation and folded them with
corresponding RMF densities to reproduce experimental S factor values in this
mass region. Impact of the non-linearity of NR3Y interaction on S-factor of
proton rich nuclei is discussed in detail.

Figure 6.1:  Comparison of charge density from our calculation with Fourier-
Bessel analysis of experimental electron scattering data[203]

###  6.1 Introduction

In nature 35 nuclei, commonly termed as $p$ nuclei, can be found on the
proton-rich side of the nuclear landscape ranging between 74Se to 196Hg. As
they are neutron deficit, the astrophysical reactions involved in the
synthesis of these elements do not correspond to the slow($s$) or fast($r$)
neutron capture processes. It mainly includes reactions such as proton
capture, charge exchange and photo-disintegration. One can find a detailed
study related to the $p$ process in standard text books [for example,
Illiadis[204]] and reviews[205].

The natural abundances for $p$ nuclei are very low in the order of 0.01% to
1%. In general, the calculation of isotopic abundances require a network
calculation typically involving 2000 nuclei and approximately 20000 reaction
and decay channels and one major problem with this $p$ network is that most of
the nuclei involved in the reaction network are very shortly lived. As a
consequence, it is very difficult to track the $p$ process nucleosynthesis
network experimentally. However, recent radioactive ion beam facilities are
giving new prospects, still we are far away from measuring astrophysical
reaction rates for the main reactions involved in the $p$ process. Thus, one
often has to depend on theoretical models to study these reactions. These type
of calculations acutely exploit the Hauser-Feshbach formalism where the
optical model potential, in a local or a global form, is a key ingredient.
Rauscher et.al. substantially calculated astrophysical reaction rates and
cross sections in a global approach[206]. They further made a comment that the
statistical model calculations may be improved by using locally tuned
parameterization.

Here, we perform a fully microscopic calculation. The framework is based on
microscopic optical model utilizing the theoretical density profile of a
nucleus. In presence of a stable target, electron scattering experiment can be
performed to avail nuclear charge density distribution data. However, in
absence of a stable target, theory remains a sole guide to describe the
density. Therefore, in this work we employ relativistic mean field (RMF)
approach to extract the density information of a nucleus. This has the
advantage of extending it to unknown mass regions. In some earlier works[207,
208, 209, 210, 211], this method has been used to study low energy proton
reactions in the A $\sim$ 55-100 region. Therefore we use this method in A
$\sim$ 100-120 region as an extension of previous works.

The non-linearity in the scalar field[212, 186] in a RMF theory has been
proved very successful in reproducing various observable like nuclear ground
state including nuclear matter properties and the surface phenomena like
proton radioactivity etc. In this chapter, we intend to study the effects of
microscopic optical potentials obtained from nonlinear $NN$ interactions also
in addition to the conventional linear $NN$ interactions in the A$\sim$
100-120 region. We concentrate mainly on the region relevant to the $p$
network and therefore mainly proton rich and stability region of the nuclear
landscape is our main concern.

###  6.2 Procedure

The RMF approach has successfully explained various features of stable and
exotic nuclei like ground state binding energy, radius, deformation, spin-
orbit splitting, neutron halo etc. [8, 213, 163, 214, 7]. The RMF theory is
nothing but the relativistic generalization of the non-relativistic effective
theory like Skyrme and Gogny. This theory does the same job, what the non-
relativistic theory can do, with an additional guarantee that it works in a
better way in high density region[215]. We have used the RMF formalism in both
direct and indirect way. Directly we have calculated the nuclear density,
which is an essential quantity to calculate the optical potential. Indirectly
we used RMF Lagrangian to derive $NN$ interactions also along with the
phenomenologically availed $NN$ interaction model. Here we have used different
types of $NN$ interactions, namely the density dependent M3Y interaction
(DDM3Y) and nonlinear R3Y interactions(NR3Y). The concept of the NR3Y was
originally developed from basic idea of the RMF formalism[216] and will be
discussed later in this section.

In order to calculate the nuclear density, different forms of Lagrangian
densities can be used from RMF approach. The chosen form of the interaction
Lagrangian density is given by

$\displaystyle{\cal L}_{int}$ $\displaystyle=$
$\displaystyle\bar{{\psi}}\bigg{[}{g_{\sigma}}{\phi}-\bigg{(}{g_{\omega}}{V_{\mu}}+\frac{g_{\rho}}{2}{\tau}.{\mathbf{b}_{\mu}}+\frac{e}{2}(1+\tau_{3}){A_{\mu}}\bigg{)}\gamma_{\mu}\bigg{]}{\psi}-\frac{g_{2}}{3}{\phi^{3}}-\frac{g_{3}}{4}\phi^{4}+\frac{\xi}{4}({V_{\mu}}{V^{\mu}})^{2}$
(6.1) $\displaystyle+$
$\displaystyle\Lambda(\mathbf{R}_{\mu}.\mathbf{R}^{\mu})(V_{\mu}V^{\mu}).$

The values of $g_{\sigma}$, $g_{\rho}$ and $g_{\omega}$ are the coupling
constants for sigma, rho, and omega mesons respectively, given in Table 6.1.
The coupling constants for nonlinear terms of sigma are $g_{2}$ and $g_{3}$,
that for omega meson is given by $\xi$ and $\Lambda$ denotes the cross
coupling strength between rho and omega meson.

For example, in case of FSUGold parameter set[90], one can see that, apart
from the usual nucleon-meson interaction terms, it contains two additional
nonlinear meson-meson self interaction terms including isoscalar ($\omega$)
meson self interactions, and mixed isoscalar-isovector ($\omega^{2}R^{2}$)
coupling, whose main aim is to softening the equation of state (EOS) of
symmetric nuclear matter. As a result, the new parameterization becomes more
effective in reproducing quite a few nuclear collective modes, namely the
breathing modes in 90Zr and 208Pb, and the isovector giant dipole resonance in
208Pb[90].

Again there are many other parameter sets in RMF which are different from each
other in various ways like inclusion of new interaction or different value of
masses and coupling constants of the mesons etc. For the comparison and better
analysis we have included different parameter sets (NL3, TM1) as it is a
matter of great concern to check their credibility in astrophysical
prediction. Therefore, for the astrophysical calculations we have used nuclear
densities from different sets of parameters like NL3 and TM1 and folded them
with corresponding $NN$ interactions respectively. In case of DDM3Y
interaction, which is not obtained from the RMF theory, we folded it with RMF
density obtained from FSUGold. This FSUGold folded DDM3Y interaction have been
used in earlier works[207, 208, 209, 210, 211] and successfully reproduced
some astrophysically important cross sections and reaction rates in A $\sim$
55-100 region. Therefore, it will be interesting to see the behavior of such
potential in A $\sim$ 100-120 region.

Typically, a microscopic optical model potential is obtained by folding an
effective interaction, derived either from the nuclear matter calculation, in
the local density approximation, i.e. by substituting the nuclear matter
density with the density distribution of the finite nucleus (for example
DDM3Y), or directly by folding different R3Y interactions using different sets
of parameters from RMF with corresponding density distributions. The folded
potential therefore takes the form

$V(E,\vec{R})=\int\rho(\vec{r^{\prime}})v_{eff}(r,\rho,E)\vec{dr^{\prime}},$
(6.2)

with $\vec{r}=\vec{r^{\prime}}-\vec{R}$ in fm. These effective interactions
($v_{eff}(r,\rho,E)$) are described below in more details.

The density dependent M3Y (DDM3Y) interaction[217] is obtained from a finite
range energy independent G-matrix elements of the Reid potential by adding a
zero range energy dependent pseudo-potential and introducing a density
dependent factor. The interaction is given by

$v_{eff}(r)=t^{M3Y}(r,E)g(\rho).$ (6.3)

Here $v_{eff}(r)$ is a function of r, $\rho$ and $E$, where $E$ is the
incident energy and $\rho$, the nuclear density. The $t^{M3Y}$ interaction is
defines as

$t^{M3Y}=7999\frac{e^{-4r}}{4r}-2134\frac{e^{-2.5r}}{2.5r}+J_{00}(E)\delta(r)$
(6.4)

with the zero range pseudo potential $J_{00}(E)$ given by,

$J_{00}(E)=-276\left(1-0.005\frac{E}{A}\right){\rm MeV}fm^{3}$ (6.5)

and $g(\rho)$ is the density dependent factor expressed as,

$g(\rho)=C(1-b\rho^{2/3}),$ (6.6)

with $C=2.07$ and $b=1.624$ $fm^{2}$ [217].

In 2014, Sahu et.al.[186] introduced a simple form of nonlinear self-coupling
of the scalar meson field and suggested a new $NN$ potential from relativistic
mean field theory (RMFT) analogous to the M3Y interaction. Rather than using
usual phenomenological prescriptions, the authors derived the microscopic $NN$
interaction from the RMF Lagrangian. Starting with the nonlinear relativistic
mean field Lagrangian density for a nucleon-meson many-body system they solved
the nuclear system under the mean-field approximation using the Lagrangian and
obtained the field equations for the nucleons and mesons. It is necessary here
to mention that the authors [186] had taken the nonlinear part of the scalar
meson $\sigma$ proportional to $\sigma^{3}$ and $\sigma^{4}$ in account.
Finally for a normal nuclear medium the resultant effective nucleon-nucleon
interaction, obtained from the summation of the scalar and vector meson fields
takes the form. 111There is a typographical error in the expression of
$v_{eff}$ in Sahu et.al [186] and the corrected form is given in this thesis.

$\displaystyle