ISOSCALAR GIANT OCTUPOLE RESONANCE ISGOR OF 116 Cd USING SELF-CONSISTENT SKYRME QRPA

Collective models based on the random phase approximation (RPA) are widely used to accurately depict collective modes of response. They can quickly calculate the strength function for the entire nuclear mass range. The quasi-particle random phase approximation (QRPA), which considers the pairing effect, is an enhanced RPA model. It is anticipated that this effect will be significant for open-shell nuclei. In this work, the self-consistent Skyrme Hartree-Fock-Bardeen, Cooper, and Schrieffer (HF-BCS) and QRPA models have been used to study the isoscalar giant octupole resonance (ISGOR) in the 116 Cd isotope. Ten Skyrme-type parameters are utilized in the computations since they may be identified by different values of the incompressibility modulus K MN in nuclear matter. The calculated strength distributions and centroid energy are compared with available experimental data. We saw that the strength distributions varied depending on the type of Skyrme-interaction, and we also observed a definite impact of the K NM values on the centroid energy.


INTRODUCTION
The Bardeen, Cooper, and Schrieffer (BCS) theory [1] provides both the quasi-particle energies and the occupation probabilities of the single particle levels to describe the ground state characteristics of even-even open shell nuclei.This method depends on a set of input single particle states.For each single particle state, the model gives partial occupation probabilities from using a pairing nucleon-nucleon interaction.The pairing effects on different ground state observables are determined using these probabilities.The description of the excited states must go farther because the HF+BCS cannot account for collective effects.The Random Phase Approximation (RPA) theory is frequently extended by the quasiparticle RPA (QRPA) [2,3], which was created to handle pairing and partial occupation probability of the single particle levels.
Giant resonances (GR) [4,5] serve as an example of the collective modes in atomic nuclei that occur at excitation energies between 10 and 30 MeV.These collective modes are related to the nucleons' collective motion inside the nucleus and are divided into different modes [6] based on their multipolarity L, spin S, and isospin T quantum numbers.According to theory, GR is influenced by the response's nucleon participation rate and transition amplitudes.
The E3 response is split into two branches, the 1ħω component has been referred to as the low energy octupole resonance, which is firstly observe by Moss et al. in 1976 utilizing inelastic scattering of alpha particles, while the higher (3ħω) component is referred to as the high energy octupole resonance [7].
The isotope 116 Cd is one of the most promising 2β nuclei thanks to the favorable theoretical estimations of the decay probability [8,9], large energy release Q 2β = 2813.50(13) keV [10], relatively high isotopic abundance δ = 7.49% [11] and a possibility of isotopic enrichment in a large amount.

DESCRIPTION OF CALCULATIONS
The occupation probabilities of the single particle levels and the quasi-particle energies are provided by the Bardeen, Cooper, and Schrieffer (BCS) theory to describe the ground state characteristics of even-even open shell nuclei.When the standard BCS equations, which under spherical symmetry provide particle number n and gap equation  , are coupled with the Hartree-Fock HF equations, the total HF-BCS energy can be determined.
It is necessary to go beyond the HF+BCS, which is unable to account for collective effects, in order to describe the excited states.As a result, the Random Phase Approximation (RPA) theory has been extended to include quasi-particle RPA (QRPA), which is effective in describing collective states of open-shell nuclei [22,23].Here is with, where, here  and  are the particle-particle and hole-hole matrix elements, respectively.The QRPA states |⟩ with matching energy  can be used to determine the strength or response function [24][25][26], The energy moments can be calculate using,

RESULTS AND DISCUSSION
In this work, the response in 116 Cd isotope has been studied in the framework of self-consistent QRPA+HFBCS method with Skyrme-type interactions.It is noteworthy to mention that 240 Skyrme interactions that were previously published in the literature underwent analysis by a separate team [27,28] to determine their ability to describe experimental data on nuclear matter, nuclei's properties and observational data of neutron stars, such as the binding energies, radii, effective mass, incompressibility coefficient, symmetry energy density, and fission barriers.The following 10 were chosen to be studied in this work: SkP [12], eMSL09 [13], MSL0 [14], T44 [15], BSK20 [16], Ska [17], SV [18], QMC2 [19], SII [20], and SGOI [17]  The E3 resonance is divided into low and high energy octupole resonance, as was indicated in the introduction section.The low energy octupole resonance includes 25% of the energy-weighted sum rules for electric E3 and the high energy octupole resonance contains 75% of the energy-weighted sum rules, according to the Harmonic Oscillator Shell Model's explanation of the Giant Resonances.When these modes are connected with the octupole residual reaction, a low energy octupole resonance with around 35% of the energy-weighted sum rules and a high energy octupole resonance with 65% of the energy-weighted sum rules are generated [29].
Our calculated fraction energy-weighted sum rules (EWSR/MeV) for ISGOR in 116 Cd are displayed in Fig. 1.The low energy octupole resonance strength is obtained in the energy range from 5 and 15 MeV, and the high energy octupole resonance strength is located between 15 and 35 MeV.Our results were compared with available experimental data using two Lorenzian smearing widths of 3 and 5 MeV.The calculations utilizing emSL09 and QMC2 Skyrme interactions with = 5 MeV was the best in describing the practical results experimental data, as shown in Fig. 1.
Most of interactions work best and agree with data concerning centroid energy (m 1 /m 0 ), widths and profiles of strength.The form of the calculated strength distribution for 116 Cd are in good agreement with the experimental, but the calculated strength distribution peaks were 1-4 MeV higher than the experimental.In 116 Cd, the form of the strength distribution is like Gaussian distribution in the low excitation region but with a large tailing on the high energy extending to 40 MeV.
It is not completely clear, from the experimental point of view, whether the constrained, centroid or scaling energies are more suitable to be compared with the experimental data.However, from what we have just concluded, it can be stated that the reasonable values of nuclear incompressibility that can be extracted from the present 116 Cd data are either the one of SkP or emSL09, namely 200.97 or 229.6 MeV.
The formula of the centroid energy E cen = m 1 /m 0 is measured in a particular energy band close to the resonance peak.Our theoretical calculations of m 1 /m 0 (MeV) for ISGOR in 116 Cd and experimental values [30] are listed in Table 1.
The m 1 /m 0 of the ISGOR as a function of the nuclear matter incompressibility coefficient K NM is presented in Table 1 and depicted in Fig. 2. The calculated centroid energies of the most Skyrme interactions are near and below of the experimental value.We found that emSL09 interaction with K NM = 229.6MeV agree with data.In ISGOR, a high K NM causes the peak to shift to a lower energy while a low K NM causes the peak to shift to a higher energy.For example, an SGOI interaction with a high K NM of 361.59 MeV causes the centroid energy to be 14.964MeV and a low K NM of 200.97 MeV causes the centroid energy to be 16.800MeV.The experimental value of m 1 /m 0 = 18.28 .
. MeV [30] for 116 Cd as shown in Table 2. Fig. 3, shows the significant change in the strength distribution with changing values of K MN .
Whereas we can clearly confirm from the present results that the value of the nuclear matter incompressibility does play a key role in dictating the location of the ISGOR centroid energy, it is also true that the pairing interaction lowers the energy of the ISGOR to some extent, typically few hundreds keV.This qualitative conclusion is the same that was first found in Ref. [31].Thus, the pairing interaction cannot be neglected if one aims to reproduce not only the ISGOR centroid energies in Cd isotopes, but also, more generally, in other open-shell nuclei.
For the low energy octupole resonance, the strength distributions are concentrated in a single peak around 10 MeV and for the high energy octupole resonance, the strength distributions are concentrated in a single peak around 25 MeV.However, the location of the peak found with each Skyrme interaction is slightly different.The SkP interaction predicts lowest peaks while the SGOI interaction gives peaks at the highest energies.As it is known from previous studies, the relative position of the peaks is governed by the nuclear matter incompressibility associated with each effective interaction.

Figure 2 .
Figure 2. Our calculated ISGOR centroid energies (color symbols) as a function of KNM in 116 Cd based on 10 Skyrme-type interaction in comparison with the experimental data [30] (shown as the regions between the dashed blue lines).

Figure 3 .
Figure 3. Using 10 Skyrme-type interaction and 3 MeV wide Lorenzian smearing, we estimated the fraction energy-weighted sum rules (EWSR/MeV) for ISGOR in116 Cd and compared it to the experimental results[30].
The strength function of ISGOR has been subjected to self-consistent QRPA based on HF-BCS calculations.The low energy octupole resonance strength is identified between 5 and 15 MeV, and the high energy octupole resonance strength is focused between 15 and 40 MeV.The low energy octupole resonance strength is discovered between 5 and 15 MeV, while the high energy octupole resonance strength is concentrated between 15 and 40 MeV, putting the isoscalar E3 strength in the range of 5 to 40 MeV in 116 Cd.By using emSL09 with K NM = 229.6MeV.The centroid energy is pushed to a lower energy by high K NM , whereas a higher energy is reached by low K NM .For instance, the eMSL09 interaction with low K NM = 229.60MeV produces the energy equivalent to 18.25 MeV in 116 Cd while the SGOI interaction with high K NM = 361.59MeV does the same for 116 Cd.