# Summary and Current Status

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## Abstract

The nuclear matter equation of state (EOS) plays an important role in our understanding of nuclear bulk properties, as well as processes taking place in the stellar environments (i.e. dynamics of supernovae collapse and structure of neutron stars). The EOS also sheds light on the phase transitions taking place in the heavy ion collision reactions. Its exact determination is being pursued extensively, both experimentally and theoretically. Nuclear incompressibility is an important parameter of the nuclear matter EOS. The centroid energy of one of the nuclear compression modes – the isoscalar Giant Monopole Resonance (ISGMR) – is a direct experimental tool to constrain the value of nuclear matter incompressibility.

## Keywords

Symmetry Energy Giant Resonance Centroid Energy Supernova Collapse Nuclear Matter EquationThe nuclear matter equation of state (EOS) plays an important role in our understanding of nuclear bulk properties, as well as processes taking place in the stellar environments (i.e. dynamics of supernovae collapse and structure of neutron stars). The EOS also sheds light on the phase transitions taking place in the heavy ion collision reactions. Its exact determination is being pursued extensively, both experimentally and theoretically. Nuclear incompressibility is an important parameter of the nuclear matter EOS. The centroid energy of one of the nuclear compression modes – the isoscalar Giant Monopole Resonance (ISGMR) – is a direct experimental tool to constrain the value of nuclear matter incompressibility.

We have measured giant resonance (GR) strength distributions in a series of ^{106, 110, 112, 114, 116}Cd and ^{204, 206, 208}Pb isotopes with a view of answering one of the “open” questions in nuclear physics [7, 13]: “Why are Sn isotopes so soft?”. Additionally, with an aim to further extend the work to more neutron-rich nuclei, the last part of this thesis is dedicated to establishing the feasibility of deuterium as a probe for future GR studies in nuclei far from the line of stability and thus explore the density dependence of symmetry energy in more detail.

Experiments were performed at the Research Center for Nuclear Physics (RCNP), Osaka University, Japan. Extremely forward angle inelastic scattering measurements (including 0^{∘}) were made to take advantage of the distinctive angular distributions exhibited by transitions of different multipolarity using the high resolution spectrometer, Grand Raiden. Elastic scattering measurements were performed in order to obtain the optical model parameters. Multipole decomposition analysis (MDA) was performed to extract the strength distributions for the various multipole. Distorted-wave Born approximation (DWBA) angular distributions were calculated in the frame-work of the hybrid model for *α* scattering experiments and the phenomenological model for the deuteron scattering experiment. A hybrid optical model was constructed with the single folding calculation (using density dependent *α*-nucleon interaction) for the real part and Wood-Saxon shaped potential for the imaginary part.

The ISGMR strength distributions were extracted for the Cd isotopes. The ISGMR centroid energies show softness similar to that in the Sn isotopes (observed in a previous study). The relativistic, as well as the non-relativistic calculations including pairing effects, overestimate the ISGMR centroid energies in the Cd isotopes. The asymmetry term in the nuclear incompressibility, K_{ τ }, extracted from the analysis of Cd isotopes was found to be − 555 ± 75 MeV [111].

We investigated an intriguing theoretical conjecture to explain the softness in the Sn and Cd isotopes, viz. consequences of the MEM effect on ISGMR centroid energies. The GR measurements in Pb isotopes were dedicated to testing this MEM effect as predicted to manifest in Pb isotopes. The ISGMR centroid energies measured in the series of Pb isotopes indicated a standard A\(^{-1/3}\) dependence in stark contrast to a sharp increase of 0.6 MeV in the ISGMR centroid energy of ^{208}Pb when compared to the centroid energy of ^{204}Pb that was predicted as resulting from the MEM effect. These results clearly established that MEM effect does not play a measurable role in the energy of the ISGMR, thereby leaving the question of “softness” in the Sn and Cd unanswered still [6].

_{ τ }obtained from the GR experiments and other independent measurements the value of J is found to lie between, 27.7 ≤ J ≤ 35.6 MeV corresponding to K

_{ τ }= −550 ± 100 MeV, as shown in Fig. 6.1 [7].

_{ ∞ }and K

_{ τ }. The “experimental” values thus obtained from the ISGMR for K

_{ ∞ }and K

_{ τ }taken together can provide a means of selecting the most appropriate of the interactions used in nuclear structure and EOS calculations. Figure 6.2 shows K

_{ ∞ }values plotted against different K

_{ τ }values used in different interactions. The constraints put by K

_{ τ }values obtained from the ISGMR in the Sn and Cd isotopes and the currently adopted value of K

_{ ∞ }= 240 ± 20 MeV [7, 13], leaves only a small number of the commonly used interactions as “acceptable”.

To facilitate future GR studies in radioactive isotopes, the feasibility of deuteron inelastic scattering for such studies was tested as a third and the last part of this thesis work. The ISGMR and ISGQR strength distributions in ^{116}Sn and ^{208}Pb were extracted using a high energy (100 MeV/u) deuteron beam. For the first time, the MDA technique was successfully employed to delineate different multipole contributions reliably. Various features of these strength distributions are compared with the previous measurements; they agree within experimental errors. This established the feasibility of using deuteron probe to study GRs in the radioactive nuclei using inverse kinematics [119]. With the new radioactive beam facilities becoming available world wide and the improved detector systems for such measurements, GR studies would be possible in the radioactive nuclei. With these measurements in nuclei far from stability line, one hopes to answer the question of “softness” in some detail and also explore the density dependence of the symmetry energy in the nuclear matter EOS.

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