MICROSCOPIC DESCRIPTION OF GIANT DIPOLE RESONANCES

Abstract

The ever-intriguing nuclear many-body system can be better understood by micro scopic models, which can relate the microscopic degrees of freedom with the bulk properties which are detectable in the lab setting. Such models become more useful if they can be applied to a large domain of energies, such as temperature extremes (T), spin (I), isospin, and density. Hence, the study of nuclear extremities gains sig nificance, especially in light of recent experimental development. Giant Resonances (GRs) are the fundamental mode of excitations that can be built on any nuclear state and hence are the most versatile tool to study nuclei in all possible domains. Most importantly, the microscopic approach to GDR, which connects the single-particle degrees of freedom with the experimental observable collective transitions, can give a better understanding of the structure of the nucleus itself. Isovector E1 (dipole) transition has the highest probability; thus, giant dipole resonance (GDR), which pri marily refers to isovector GDR, is the most prominent among all the GRs. Goldhaber and Teller (GT) first proposed [1] a theoretical description of GDR as protons and neutrons moving against each other collectively, known as the GT mode with centroid energy of the form E ∝ A−1 6 . Later another explanation was proposed by Steinwedel and Jensen (SJ) that GRs are collective vibrations of proton fluid against neutron f luid [2], with centroid energy of the form E ∝ A−1 3 , known as the SJ mode. It is understood that both modes could coexist with GT mode dominant in lower mass [3]. There are several microscopic models which are developed over the years, such as the phonon damping model (PDM) [4], the time dependant Hartree-Fock (TDHF) [5], the separable random-phase approximation (SRPA) [6], the extended quantum molec ular dynamics (EQMD)model[7], the finite amplitude quasiparticle random phase ap proximation (FAM-QRPA) [8], the QRPAbasedontherelativistic Hartree-Bogoliubov model [9], and the relativistic random phase approximation (RRPA) [10]. Linear re sponse theory (LRT) framework [11–13] is one of the most common RPA-based mi croscopic methods which addresses the collective excitations in the nucleus. There are stochastic [14], and semi-classical [15] approaches as well. Some of them are restricted to spherical or axially deformed nuclei. In a spherical nucleus, the GDR strength man ifests as a single peak, but in a deformed nucleus, it splits depending upon the shape of the nucleus [16, 17]. Most of the microscopic approaches are not extended to hot and rotating nuclei where thermal fluctuations are crucial. The thermal effects include shape fluctuations and fluctuations in the pairing field, which are difficult to incor porate in microscopic approaches. The PDM considers shape fluctuations at higher angular momentum also [18] but is restricted to rotation about the symmetry axis. The thermal fluctuations are well handled while following macroscopic approaches for GDR [19–21], which are still the best available models in explaining the data of hot and rapidly rotating nuclei. With a motive to fill this gap, we revisit the microscopic approaches, which can be easily extended to high spin and temperature. In this work, we discuss a microscopic approach for GDR in deformed nuclei within the framework of linear response theory (LRT) [22] that can be combined with a microscopic-macroscopic approach for nuclear free energy calculations. The equilibrium deformation of the nucleus is identified by minimizing the potential en ergy in deformation space calculated using the Strutinsky method with the triaxial Woods-Saxon (WS) potential. The single-particle wavefunctions resulting from the same calculations are utilized to evaluate the GDR properties. We present a compar ison between the macroscopic and microscopic approaches to GDR by comparing the results with the experimental data reported in Ref. [23]. Importantly, we study the correlations between the single-particle configurations and the GDR cross-sections in 150Nd and 152Sm. These nuclei lie at the critical point between the spherical and axi ally deformed phase transition [24, 25] corresponding to the quasisymmetry X(5) [26, 27] and these shape transitions reflect the underlying modifications of single-particle shell structure [28]. One of the advantages of using a microscopic model is the possi bility of investigating the fine structure present in the GDR cross-section. We employ the technique of continuous wavelet transform (CWT) analysis to extract the fine structure from the microscopic results and present a comparison with the fine struc ture investigation reported in Ref. [29]. After establishing our microscopic model for ground-state GDR, we extend the model to the cases of thermally excited nuclei. Several earlier works into GDR fo cused on the high-J region [30, 31] and recently, the studies have indicated a potential astrophysical implication of GDR [32–35]. On the other hand, the GDR at low T (warm nuclei) is relatively less explored, and only recently has it gained some momen tum [20, 36–39]. Although it is difficult to populate nuclei at low temperatures, recent experimental development has been proven quite promising and made it feasible. The properties of nuclei at low T, such as the existence of pairing phase transitions and the effect of fluctuations in the pairing field (PF), are still a part of continuous research. We have extended the microscopic model to the case of warm nuclei, and the effect of the residual pairing interaction is taken through the quasi-particle wave functions cal culated using the triaxial Woods-Saxon (WS) Hamiltonian within the BCS approach for pairing. The nuclear shape does not directly couple to the GDR in the micro scopic models, but instead, it affects the underlying single-particle spectrum and wave functions, which influence the GDR response function. The same WS Hamiltonian is utilized to obtain the potential energy surface (PES) using the Nilsson-Strutinsky method [19] to get the average GDR observables. The choice of the mean-field plays a crucial role in shell correction, which further affects the average GDR observables in both macroscopic and microscopic models. Thus we discuss the role of the mean f ield by comparing our results with those obtained with Nilsson Hamiltonian [20]. In the present work, we discuss the role of thermal shape fluctuations (TSF) and PF at low T within both the microscopic and macroscopic models for GDR. Wealso present a quantum algorithm to obtain the response of the atomic nucleus to a small external electromagnetic perturbation. The Hamiltonian of the system is presented by a Harmonic Oscillator, and the linear combination of unitaries (LCU) method is utilized to simulate the Hamiltonian on the quantum computer. The out put of the Hamiltonian simulation is utilized in calculating the dipole response with the SWAP test algorithm. The results of the response function computed using the quantum algorithm are compared with the experimental data and provide a good agreement. We show the results for 120Sn and 208Pb to corroborate with the exper imental data in Sn and Pb region and also compare the results with those obtained using the conventional linear response theory. The formalism developed and the results obtained are organized in the following chapters. Chapter 1: In this chapter, we introduce the fundamentals of giant resonances and their importance in understanding several nuclear phenomena. A detailed discus sion is given on the microscopic picture of giant dipole resonances and their historical development. The different theoretical approaches and the experimental efforts for GDR are reviewed with a brief discussion on the inner workings of linear response theory based on the random phase approximation (RPA) used in the present work. Wealso introduce the basics of Hamiltonian simulation on a quantum computer using quantum phase estimation, which is the bedrock of the quantum algorithms used in calculating the response function. With the basics of GDR covered, we lay out the motivation for the present work and give an overall structure of thesis goals. Chapter 2: In chapter 2, we discuss the details of the mean-field used in our models for GDR and potential energy surface calculations. We emphasize the impor tance of a realistic potential, i.e., triaxial Woods-Saxon potential. A brief description of the triaxial Woods-Saxon potential is given with all the individual terms. The method used in the present work to calculate the single-particle energies and wave functions is discussed. We also describe the visualization of WS potential, the WS wave functions, and how it varies as a function of energy and deformation. Chapter 3: This chapter deals with the formalism of the microscopic-macroscopic approach to calculating the energy of the nucleus as a function of deformations. The macroscopic part is referred to the energy obtained with the liquid drop model, and the microscopic part is obtained with triaxial Woods-Saxon potential. We also give details about the issue of plateau conditions while obtaining the shell corrections with the Woods-Saxon potential. We show that the choice of the correct basis can provide a solution to this problem. We also present a brief introduction to the BCS pairing model, which deals with the pairing corrections in the shell corrections in the total free energy of the nucleus. Along with the discussion of shell correction at zero temperature, we also discuss the finite temperature shell correction method to obtain the potential energy surfaces as a function of temperature, which is used in the thermal shape fluctuation model in the later sections of the thesis. Chapter 4: This chapter is dedicated to the models of GDR used in our study, categorized as the macroscopic and microscopic models. We briefly introduce the macroscopic model of GDR, which is used as a benchmark to compare against the random phase approximation (RPA) based linear response theory used in our micro scopic model of GDR. We give a detailed description of the microscopic model where the final form of response function at zero T and finite T. We discuss in detail how the microscopic model connects the single-particle states with the GDR response func tion. We also discuss the form of response function with pairing correlations where BCS wavefunctions are used in the calculations. The fine structure of GDR is ob tained with continuous wavelet transformation (CWT). We introduce the CWT and the importance of scales in the power spectrum, which is obtained after calculating the CWT of the GDR cross-section.Chapter 5: In this chapter, we discuss the results [40] of our microscopic model for ground state GDR in even-even nuclei and compare them with the macroscopic model. We explain the recent experimental data for even-even nuclei 144−152Nd and 152Sm and present a comparison with the macroscopic approach for GDR. We high light the cases where the results from the microscopic approach are sensitive to the change in single-particle configuration despite no change in shape and mass but with a change of two protons in a mid-shell region. We also present the fine structure analysis of the GDR cross-section using the continuous wavelet transform (CWT) framework and elucidate the origin of such fine structures. Chapter 6: In this chapter [41], we discuss the results of our microscopic model for ground state GDR in the odd-even nucleus. We explain the experimental data for odd-even nuclei 175Lu and present a comparison with the macroscopic approach for GDR. Chapter 7: In this chapter, we discuss the results for warm nuclei. The prop erties of nuclei in the low-temperature regime are intriguing and less explored at low temperatures (T < 1 MeV). Giant dipole resonance (GDR) shows there exists a strong interplay between the quantal fluctuations (shell), the thermal fluctuations (shape), and the residual pairing interaction. We present a microscopic model based on random phase approximation (RPA) which is extended to the finite temperature where fluctuations in shape and pairing field are calculated within the thermal shape f luctuation model (TSFM) extended to include the fluctuations in pairing field. We discuss our results for the nuclei 97Tc, 120Sn, 179Au, and 208Pb and corroborate with the available experimental data. We also compare the macroscopic model for GDR and show that despite having fewer parameters than the macroscopic model, the mi croscopic model can reproduce the experimental trend in GDR width in both the Sn and the Pb regions. Our study reveals that the consideration of pairing fluctuations is crucial for a precise match with the experimental data. Chapter 8: In this chapter, we introduce a quantum algorithm for calculating the response function of the nucleus on a quantum computer. We provide the necessary quantum circuits for calculating the full response of the nucleus. We are currently limited to a spherical system due to the finite size of qubit space available in the present era of quantum computing. We show that the complexity of the problem can be simplified by reducing the basis size by selecting the basis states near the Fermi level, which contributes the most to our model. We discuss our results for the nuclear response on the quantum computer, which is corroborated with the experimental results done for the first time in the field. Our study finds that it is possible to calculate a full nuclear response on a quantum computer without resorting to the mean field approximation. Chapter 9: A summary of this thesis is presented in chapter 9 along with the outlook of this work, highlighting the most important conclusions and the future aspects.