Thermal nuclear pairing within the self-consistent quasiparticle RPA
The self-consistent quasiparticle RPA (SCQRPA) is constructed to study the effects of fluctuations on pairing properties in nuclei at finite temperature and -projection of angular momentum. Particle-number projection (PNP) is taken into account within the Lipkin-Nogami method. Several issues such as the smoothing of superfluid-normal phase transition, thermally assisted pairing in hot rotating nuclei, extraction of the nuclear pairing gap using an improved odd-even mass difference are discussed. A novel approach of embedding the PNP SCQRPA eigenvalues in the canonical and microcanonical ensembles is proposed and applied to describe the recent empirical thermodynamic quantities for iron, molybdenum, dysprosium, and ytterbium isotopes.
Sharp phase transitions such as the superfluid-normal (SN) or shape ones are prominent features of infinite systems such as metal superconductors, ultra-cold gases, liquid helium, etc. They are well described by many-body theories such as the BCS, RPA or quasiparticle RPA (QRPA). The situation changes in finite small systems such as atomic nuclei, where strong quantal and thermal fluctuations strongly or completely smooth out these sharp phase transitions. It is well known that the conventional BCS, RPA or QRPA theories fail in a number of cases in the description of the ground states as well as excited states of these systems. The reason is that strong fluctuations invalidate the assumptions, based on which the main equations of these theories have been derived. Amongst these assumptions are the Cooper pairs, which violate the particle-number conservation, and the closely related quasiboson-approximation (QBA) used in the (Q)RPA, which violates the Pauli principle between the fermion pairs. These assumptions cause the BCS and QRPA to break down at a certain critical value of the pairing interaction parameter , below which the BCS theory only has a trivial solution with zero pairing gap 0. The same is true in the weak coupling region, where the particle-particle RPA is valid but its solution also breaks down at . Meanwhile, the exact solution of the pairing problem exposes no singularity at any . Similarly, at finite temperature 0, the omission of quasiparticle-number fluctuations (QNF) within the BCS theory leads to the collapse of the pairing gap at the critical temperature , corresponding to the temperature of the SN phase transition in infinite systems. Meanwhile, the exact eigenvalues of the pairing problem embedded in the canonical ensemble (CE) shows a smooth decreasing pairing energy with increasing due to thermal fluctuations incorporated in the CE . In rotating nuclei, strong fluctuations also smear out the Mottelson-Valatin effect, according to which the pairing gap, existing at zero angular momentum 0, would collapse at a certain critical angular momentum . This situation means that, in order to be reliable, the BCS, RPA, and/or QRPA theories need to be corrected to include these effects of fluctuations when applied to nuclei, in particular, the light ones. This is done within the framework of the self-consistent QRPA (SCQRPA) presented in this work.
We consider the pairing Hamiltonian , where is the particle-number operator, and are the pairing operators. The operators and are respectively the single-particle creation and destruction operators. This Hamiltonian has been diagonalized exactly in . The exact partition function is constructed by embedding the exact eigenvalues into the CE as , with the degeneracy , inverse temperature , and being the total seniority of the system. Knowing the partition function , one calculates the free energy , entropy , total energy , heat capacity , and pairing gap as , , , , and , where is the single-particle occupation number on the th level obtained by averaging the state-dependent occupation numbers within the CE .
where and are the Bogoliubov’s coefficients,
with the renormalized single-particle energies
, the quasiparticle-pair operators , , and is the QNF on th level. To avoid level-dependent gaps , the level-weighted gap ( is the number of levels) is considered in the numerical results. Because of coupling to collective vibrations beyond the quasiparticle mean field, the quasiparticle occupation number is not given by a Fermi-Dirac distribution of free fermions, but is found from the integral equation 
where the mass operator and the quasiparticle damping are functions of , SCQRPA eigenvalues , SCQRPA and amplitudes, SCQRPA phonon occupations numbers , as well as and . The SCQRPA submatrices and contain the screening factors and so that the set of SCQRPA equations should be solved self-consistently with Eqs. (1), (2) and (5) to simultaneously determine , chemical potential , , , and . To eliminate particle-number fluctuations inherent in the BCS theory, the Lipkin-Nogami (LN) particle-number projection (PNP)  is applied on top of Eqs. (1) and (2). The ensuing theory, called the LNSCQRPA theory, has also been extended to include the finite -projection of angular momentum (noncollective rotation) . The set of obtained equations is formally the same except that now, depending on the single-particle spin projections with being the angular velocity, one has two types of quasiparticle occupation number, , so that . At 0 and 0 the SCQRPA theory reduces to its zero temperature and non-rotating limit, where .
3 Numerical results and discussions
Shown in Fig. 1 are the energies of the ground state (a) and first excited state (b) obtained at 0 within several approximations as well as by exactly diagonalizing the pairing Hamiltonian for the schematic model, which consists of doubly-folded equidistant levels with the single-particle energies chosen as MeV. The displayed results are for the half-filled case with 10, and plotted as functions of the pairing interaction parameter . It is seen that the LNSCQRPA describes rather well the exact energies of both the ground and first excited states without any discontinuity in the region around , where all other approaches such as the RPA, QRPA, and SCQRPA collapse.
The level-weighted gap, total energy, and heat capacity obtained for the systems with 10 and 50 are shown as functions of temperature in Fig. 2. Beside the predictions by the SCQRPA, LNSCQRPA, as well as by their corresponding limits, FTBCS1 and FTLN1, where coupling to QRPA is omitted (i.e. is described by the Fermi-Dirac distribution for free fermions), and the finite-temperature (FT) BCS results, the exact CE results are also shown. This figure clearly demonstrates how QNF smooth out the sharp SN phase transition in finite systems. The pairing gap never collapses, but decreases monotonously with increasing , whereas the spike at in the heat capacity, which serves as a signature of sharp SN phase transition within the FTBCS, becomes strongly depleted to a broad bump.
At finite angular momentum 0, the FTBCS theory predicts the Mottelson-Valatin effect, according to which, the zero-temperature pairing gap decreases with increasing and collapses at because the angular momentum blocks the levels close to the Fermi surface [Fig. 3 (a) and 3 (c)]. Thermal effects relax the blocking, opening some levels around the Fermi surface for pairing. This leads to the thermally assisted pairing gap (or pairing reentrance), according to which at a certain the pairing gap becomes finite even at [7, 8]. With increasing thermal effects again break the pairs so that the gap disappears at [See Fig. 3 (a) for 1]. In finite systems, the QNF smooth out both the Mottelson-Valatin transition and thermal assisted pairing, for instance, for 10 with 0.5 MeV at 1, the gap only decreases monotonously with increasing but never vanishes [See Fig. 3 (d) for 1], whereas at 3, the pairing gap reappears at but remains finite with further increasing [See Fig. 3 (b) for 3].
The odd-even mass difference contains the admixture with the contribution from uncorrelated single-particle configurations, which increases with . Therefore, the simple extensions of this formula to obtain the three-point and four-point gaps, in principle, do not hold at finite temperature. We propose an improved odd-even mass difference formula at 0, namely
where is the total energy of the system with particles within the grand canonical ensemble (GCE) (GC) or CE (C); with coming from uncorrelated single-particle configurations.
Shown in Fig. 4 are the pairing gaps ( 3 and 4), obtained for 9 and 10 ( 10) by using the simple extension of the odd-even mass formula to 0 as well as the modified gaps from Eq. (6), and the canonical gaps . It is seen in Fig. 4 that the naive extension of the odd-even mass formula to 0 fails to match the temperature-dependence of the canonical gap . The gap even turns negative at 2.4 MeV, suggesting that such simple extension of the odd-even mass difference to finite is invalid. The modified gap is found in much better agreement with the canonical one, whereas the modified four-point gaps practically coincide with the canonical gaps. The comparison in Fig. 4 suggests that formula (6) is a much better candidate for the experimental gap at 0, rather than the simple odd-even mass difference.
In order to construct a feasible description for pairing within the CE, the eigenvalues of the LNBCS and LNSCQRPA, obtained for each total seniority at 0, are embedded into the CE by using the partition function ( LNBCS, LNSCQRPA). The resulting approaches are called the CE-LNBCS and CE-LNSCQRPA, respectively . These solutions are also embedded into the microcanonical ensemble (MCE) by using the Boltzmann’s definition for entropy, , where is the number of accessible states within the energy interval (). The corresponding approaches are called the MCE-LNBCS and MCE-LNSCQRPA, respectively .
The CE heat capacities and MCE entropies for several nuclei are shown in Fig. 5 as functions of and excitation energy , respectively. The single-particle energies are calculated within the deformed Woods-Saxon potentials. In order to have a consistent comparison with the recent experimental data in , we carried out calculations by using the CE-LNBCS and CE-LNSCQRPA for Fe, where pairing is included within the complete shell above the Ca core. For Mo isotopes, pairing is included in the 22 degenerated single-particle levels above the Ca core. For Dy and Yb the same is done on top of the Sn core. It is clear from this figure that the CE-LNSCQRPA results agree quite well with the experimental data , which are also deducted from the CE. The MCE entropies, obtained within the MCE-LNBCS and MCE-LNSCQRPA using 1 MeV, are plotted versus the experimental data. It is seen that the MCE-LNSCQRPA entropy not only offers the best fit to the experimental data but also predicts the results up to higher 10 MeV.
The proposed LNSCQRPA theory can describe without discontinuity the pairing properties of hot noncollectively rotating nuclei at any values of pairing interaction parameter , temperature , and angular momentum . In the limit of zero temperature and zero angular momentum, it offers the best fits to the exact solutions in the weak coupling region with large particle numbers for the energy of the first excited state, whereas the SCQRPA reproduces well the exact one in the strong coupling region. In the limit of very large all the approximations predict nearly the same value as that of the exact one. Under the effect of QNF, the paring gaps obtained at different values of angular momentum decrease monotonously as increases, and do not collapse even at hight . The effect of thermally assisted pairing (pairing reentrance) shows up in such a way that the pairing gap reappears at a given 0 and remains finite at , in qualitative agreement with the results of Ref. . We suggest a novel formula to extract the pairing gap at 0 from the difference of total energies of even and odd systems, where the contribution of uncorrelated single-particle motion is subtracted. Its prediction is found in much better agreement with the canonical gap than the simple extension of the odd-even mass formula to 0. Finally, we embedded the solutions of the LNBCS and LNSCQRPA into the CE and MCE, and found that the CE-LNSCQRPA predictions are in quite good agreements with the exact results as well as the recent experimental data. The present approach can also describe simultaneously and self-consistently the experimentally extracted total energy, heat capacity, and entropy within both CE and MCE treatments. It is simple and feasible for the application to larger finite systems, where the exact matrix diagonalization and/or solving the Richardson equation are impracticable to find all eigenvalues, whereas other methods, such as the quantum Monte-Carlo calculations, are time consuming.
-  Volya A, Brown B A and Zelevinsky V 2001 Phys. Lett. B 509 37.
-  Hung N Q and Dang N D 2009 Phys. Rev. C 79 054328.
-  Hung N Q and Dang N D 2007 Phys. Rev. C 76 054302; 2008 Ibid. 77 029905(E).
-  Dang N D and Hung N Q 2008 Phys. Rev. C 77 064315.
-  Lipkin H J 1960 Ann. Phys. (NY) 9 272; Nogami Y 1965 Phys. Lett. 15 4.
-  Hung N Q and Dang N D 2008 Phys. Rev. C 78 064315.
-  Moretto L G 1971 Nucl. Phys. A 185 145.
-  Balian R, Flocard H and Vénéroni M 1999 Phys. Rep. 317 251.
-  Frauendorf S, Kuzmenko N K, Mikhajlov V M and Sheikh J A 2003 Phys. Rev. B 68 024518.
-  Hung N Q and Dang N D 2010 Phys. Rev. C 81 057302.
-  Melby E et al. 1999 Phys. Rev. Lett. 83 3150; Guttormsen M et al. 2000 Phys. Rev. C 62 024306; Schiller A et al. 2001 Phys. Rev. C 63 021306(R); Algin E et al. 2008 Phys. Rev. C 78 054321.