In Chapter 4, we extend the FTBCS1 (FTBCS1 + SCQRPA) theory introduced in Chapter 3 to finite angular momentum to study the pairing properties of hot nuclei, which rotate noncollectively about the symmetry axis. The FTBCS1 theory includes the QNF whereas the FTBCS1 + SCQRPA also takes into account the correction due
to dynamic coupling to SCQRPA vibrations. The proposed extension is tested within the doubly degenerate equidistant model with N = 10 particles as well as some realistic (spherical and deformed) nuclei, 20
O, 22
Ne, and 44
Ca. The numerical calculations were carried out within the FTBCS, FTBCS1, and FTBCS1 + SCQRPA for the pairing gap, total energy, and heat capacity as functions of temperature T, total angular momentum M , and angular velocity . The corrections due to the Lipkin-Nogami particle-number projection were also discussed. The analysis of the numerical results show that the proposed approach is able to reproduce the effect of thermally assisted pairing correlation, which takes place in the schematic model within the FTBCS theory, according to which a finite pairing gap can reappear within a given temperature interval, ( ), while it is zero beyond this interval. However, this phenomenon does not show up in realistic nuclei under consideration.
Under the effect of QNF, the paring gaps obtained within the FTBCS1 at different values M of angular momentum decrease monotonously as T increases, and do not collapse even at hight T in the schematic model as well as realistic nuclei. The effect of thermally assisted pairing correlation is seen in all the cases, but in such a way that the pairing gap reappears at a given and remains finite at , in qualitative agreement with the canonical results of Ref. [31].
The backbending of the moment of inertia is found in the schematic model and in 22Ne in the low temperature region, whereas it is washed out with increasing temperature. This effect does not occur in 20O and 44Ca, in consistent with existing experimental data and results of other theoretical approaches.
The effect caused by the corrections due to the dynamic coupling to SCQRPA vibrations on the pairing gaps, total energies, and heat capacities is found to be significant in the region around the critical temperature of the SN phase transition and/or at large angular momentum M (or angular velocity ). It is larger in lighter systems. As the result, all the signatures of the sharp SN phase transition are smoothed
out in both schematic model and realistic nuclei. The SCQRPA corrections also significantly enhance the reappearance of the thermal gap at finite angular momentum. On the other hand, the effect caused by the corrections due to PNP is important only at temperatures below , and at quite low angular momenta. In particular, it makes backbending less pronounced at T = 0.
Summary and outlook
The present dissertation studies the effects of quantal and thermal fluctuations on the pairing properties of atomic nuclei. As these effects are very important in finite small systems, they need to be taken into account so that the predictions by the well-known theories such as the BCS one and RPA become reliable when applied to light nuclei. Despite many attempts to include the quantal and thermal fluctuations in the BCS and RPA theories, a fully self-consistent microscopic approach has been absent so far.
This issue is resolved for the first time in the present dissertation, where a self- consistent quasiparticle random-phase approximation (SCQRPA) is proposed that works for atomic nuclei and other finite systems not only at zero temperature but also at finite temperature and angular momentum at any given value of the pairing interaction parameter. The SCQRPA also incorporates the particle-number projection within the Lipkin-Nogami method to remove the quantal fluctuations caused by the violation of particle number inherent in the conventional BCS theory. The quantal and thermal fluctuations are included in the SCQRPA in a fully selfconsistent and microscopic way. The proposed approach was tested within the model cases, whose exact solutions are possible, as well as several realistic nuclei ranging from light to heavy systems.
In Chapter 1, the dissertation conducts a systematic comparison for pairing properties of finite systems at non-zero temperature by embedding the exact solutions of the pairing Hamiltonian in three principle thermodynamic ensembles, namely, the grand canonical ensemble (GCE), canonical ensemble (CE) and micro canonical ensemble (MCE). The tests within the Richardson model show the small differences between the results obtained within the exact calculations of the GCE and CE even in systems with small particle numbers. For MCE, there is a ambiguity in the temperature
extracted from the discrete level density, which depends on the shape and parameter of the smoothing function. A novel formula to extract the thermal pairing gap from odd- even mass energy differences is also suggested in Chapter 1. This formula subtracts the uncorrelated single-particle motion from the total energy or binding energy of systems resulting a much better agreement with the canonical gap than the simple extension of the odd-even formula to finite temperature.
Chapter 2 presents a fully derivation of the SCQRPA and SCQRPA with Lipkin-Nogami particle-number projection (LNSCQRPA) at zero temperature for the Richardson model. The results obtained in this chapter show that the use of Lipkin- Nogami method, significantly improves the agreement between the results obtained within the present approach and the exact ones in the overall region of coupling strength G ranging from weak to strong coupling. In the limit of very large G, all the approximations offer the same value as that of the exact one. The SCQRPA proposed in this chapter can be useful to describe the ground state as well as excited states of small systems such as light and unstable nuclei, where the validity of the quasi-boson approximation and that of the conventional BCS are poor.
The SCQRPA (LNSCQRPA) at zero temperature proposed in Chapter 2 is extended to finite temperature in Chapter 3. The resulting approaches, FTBCS1(FTLN1) + SCQRPA include the effect of quasiparticle-number fluctuation as well as dynamic coupling to the pairing vibrations. The results obtained within the Richardson model as well as two realistic nuclei, Fe and Sn, show that the quasiparticle-number fluctuation (QNF) is the microscopic origin which smoothes out the superfluid-normal (SN) phase transition in nuclei. As the result, the pairing gaps obtained within the FTBCS1(FTLN1) with or without coupling to the pair vibration do not collapse at critical temperature as predicted by the conventional FTBCS theory but has a tail, which extends to high temperature. The results obtained also agree well with the exact solutions of the model case as well as the finite-temperature quantum Monte
Carlo method for realistic Fe nucleus. The corrections due to the dynamic coupling to the SCQRPA vibrations, which causes the deviation of the quasiparticle occupation number from the Fermi-Dirac distribution for non-interacting fermion, significantly improves the agreement between the FTBCS1(FTLN1)+SCQRPA results and the exact and Quantum Monte-Carlo calculations. These corrections are found to be negligible in realistic heavy nucleus such as Sn.
Chapter 4 proposes a extended version of finite-temperature SCQRPA to finite angular momentum to describe the pairing properties in a hot rotating system. The proposed aprroach is tested within the schematic multilevel model as well as some realistic spherical and deformed nuclei, O, Ne, and Ca. The results obtained again show that the quasiparticle-number fluctuation is the microscopic origin, which smooths out the superfluid-normal phase transition in not only hot but also hot rotating systems. We also found that in this hot rotating systems, because of quasiparticle- number fluctuation, a tiny system in the normal state at zero temperature can turn superconducting at finite temperature. The backbending effect can be seen in the schematic model and in realistic well-deformed Ne nucleus in the region of low temperature, which is consistent with the experimental finding and the results of other theoretical predictions.
The present approach disentangle the origin of the quantal and thermal fluctuations in finite small systems, which could not be revealed in the exact solutions, where all fluctuations are automatically included. Compared to existing methods, the merit of present approach lies in its fully microscopic derivation and simplicity when it is applied to heavy systems, where exact solutions are impracticable because of the huge size of the matrix to be diagonalized.
The present dissertation considers a simple pairing model with the constant monopole pairing interaction. Depending on specific studies, other realistic interactions and/or multipole deformations can also be included in a straightforward manner.
The SCQRPA proposed in the present dissertation has a number of promising applications. To mention a few of them, let us recall that the test of this theory conducted in several neutron-rich isotopes in the present dissertation shows that the SCQRPA can serve as a good tool to study neutron-rich and unstable nuclei, especially the light ones, where the conventional QRPA fails or its validity is questionable.
The study of nuclear level densities, where the BCS overestimates the experimental data [64], shall be another prospective application of the SCQRPA, where the ground-state correlations beyond the QRPA are self-consistently and microscopically taken into account. The study of the SN phase transition in nuclei as isolated self-sustained micro-systems is still an open question from both theoretical and experimental points of view. It is therefore highly desirable to explore the application of the SCQRPA in the study of thermodynamic properties of atomic nuclei within the microcanonical ensemble. This is the subject of our forthcoming study.
Last but not least, relating to the recent experimental observation of the Bose- Einstein condensation (BEC) in ultra-cold atomic Fermi gases [56], the encouraging results, obtained within the SCQRPA presented in Chapters 2 and 3, show a promising perspective for the application of this theory in the study of the crossover region of the BCS-BEC transition in atomic nuclei. This work has already been recently started and is now underway [20].
List of publications
I) Peer-reviewed journals
A) Published:
1) N. Quang Hung and N. Dinh Dang (2007), “Self-consistent quasiparticle random-phase approximation for a multilevel pairing model”, Phys. Rev. C 76, 054302; ibid 77, 029905 (E).
2) N. Dinh Dang and N. Quang Hung (2008), “Pairing within the self-consistent quasiparticle random-phase approximation at finite temperature”, Phys. Rev. C 77, 064315.
3) N. Quang Hung and N. Dinh Dang (2008), “Pairing in hot rotating nuclei”,
Phys. Rev. C 78, 064315.
4) N. Quang Hung and N. Dinh Dang (2009), “Exact and approximate ensemble treatments of thermal pairing in a multilevel model”, Phys. Rev. C 79, 054328.
II) International conference proceedings
A) Published:
1) N. Quang Hung and N. Dinh Dang (2007), “Self-consistent quasiparticle RPA for a multilevel pairing model”, Proceedings of the 9th Int'l Spring Seminar on
Nuclear Physics, World Scientific, pp. 503 – 510.
2) N. Dinh Dang and N. Quang Hung (2008), “Nuclear pairing at finite temperature and finite angular momentum”, Int. Jour. Mod. Phys. E 17, 2160.
3) N. Dinh Dang and N. Quang Hung (2009), “Nuclear pairing at finite temperature and angular momentum”, Proceedings of the 13th Int'l Symp. on Capture
conference proceedings 1090, pp. 179 – 183.
III) Annual review of RIKEN Accelerator Progress Report
A) Published:
1) N. Quang Hung and N. Dinh Dang (2007), “Self-consistent quasiparticle RPA for a multilevel pairing model”, RIKEN Accel. Prog. Rep. 40, 59.
2) N. Dinh Dang and N. Quang Hung (2008), “Thermal pairing assisted by quasiparticle-number fluctuations”, RIKEN Accel. Prog. Rep. 41, 37.
3) N. Quang Hung and N. Dinh Dang (2008), “Effects of thermal fluctuations and angular momentum on nuclear pairing properties”, RIKEN Accel. Prog. Rep. 41, 38.
B) Submitted:
1) N. Quang Hung and N. Dinh Dang (2009), “Thermodynamical ensemble treatments of nuclear pairing in a multilevel model”, submitted to RIKEN Accel. Prog.
Rep. 42.
2) N. Dinh Dang, N. Quang Hung and P. Schuck (2009), “BCS-BEC transition in finite systems”, submitted to RIKEN Accel. Prog. Rep. 42.
IV) Reports at International Meetings and Seminars
1) N. Quang Hung and N. Dinh Dang (2007), Self-consistent quasiparticle RPA
for a multilevel pairing model, Poster presentation at the 9th Int'l Spring Seminar on
Nuclear Physics, Vico Equense, Italy.
2) N. Quang Hung and N. Dinh Dang (2007), Self-consistent quasiparticle RPA
for a multilevel pairing model, Talk at the seminar for students of RIKEN Asia
Program Associate, RIKEN, Wako, Japan.
for a multilevel pairing model, Talk at the meeting of Heavy-Ion Nuclear Physics
laboratory RIKEN, Wako, Japan.
4) N. Quang Hung and N. Dinh Dang (2008), Effects of thermal fluctuations and
angular momentum on nuclear pairing properties, Talk at the progress report of
research activity in the fiscal year 2007, Heavy-Ion Nuclear Physics laboratory, RIKEN, Wako, Japan.
5) N. Quang Hung and N. Dinh Dang (2008), Effects of thermal fluctuations and
angular momentum on nuclear pairing properties, Poster presentation at CNS-RIKEN
Int'l Symp. on Frontier of Gamma-Ray Spectroscopy and Perspectives for Nuclear Structure Studies (Gamma08), RIKEN, Wako, Japan.
6) N. Dinh Dang and N. Quang Hung (2008), Nuclear pairing at finite
temperature and angular momentum, Contributed talk at the First Int'l Workshop on
State of The Art in Nuclear Cluster Physics, Strasbourg, France.
7) N. Dinh Dang and N. Quang Hung (2008), Nuclear pairing at finite
temperature and angular momentum, Contributed talk at the 13th Int'l Symp. on
Capture Gamma-Ray Spectroscopy and related topics, Cologne, Germany.
8) N. Quang Hung and N. Dinh Dang (2008), Thermodynamic ensemble
treatments of nuclear pairing, Talk at the meeting of Heavy-Ion Nuclear Physics
laboratory, RIKEN, Wako, Japan.
9) N. Quang Hung (2008), Pairing in hot rotating nuclei, Invited seminar at Department of Physics, University of Notre Dame, Indiana, USA.
10) N. Quang Hung and N. Dinh Dang (2008), Pairing within the self-consistent
quasiparticle random-phase approximation at finite temperature and angular momentum, Poster presentation at Int'l Conf. on Interfacing Structure and Reactions at
the Center of the Atom (Kernz08), Queenstown, New Zealand.
11) N. Quang Hung (2009), Pairing in hot rotating nuclei, Invited seminar at Center for theoretical physics, Institute of physics, Hanoi, Vietnam.
12) N. Quang Hung and N. Dinh Dang (2009), Effects of thermal and quantal
fluctuations on superfluid pairing in nuclei, Talk at annual research meeting of Heavy-
Ion Nuclear Physics Laboratory, RIKEN, Wako, Japan.
13) N. Quang Hung (2009), Pairing within the self-consistent quasiparticle
random-phase approximation at finite temperature and angular momentum, Invited
seminar at Theoretical nuclear physics laboratory, RIKEN Nishina Center, RIKEN, Wako, Japan.
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