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Thermal nuclear pairing within the self-consistent QRPA

Thermal nuclear pairing within the self-consistent QRPA. N. Dinh Dang 1,2 and N. Quang Hung 1,3 1) Nishina Center for Accelerator-Based Science, RIKEN, Wako city, Japan 2) Institute for Nuclear Science & Technique, Hanoi – Vietnam 3) Institute of Physics, Hanoi - Vietnam. Motivation.

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Thermal nuclear pairing within the self-consistent QRPA

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  1. Thermal nuclear pairing within the self-consistent QRPA N.Dinh Dang1,2and N. Quang Hung1,3 1) Nishina Center for Accelerator-Based Science, RIKEN, Wako city, Japan 2) Institute for Nuclear Science & Technique, Hanoi – Vietnam 3) Institute of Physics, Hanoi - Vietnam

  2. Motivation Infinite systems (metal superconductors, ultra-cold gases, liquid helium, etc.) • Fluctuations are absent or negligible • Superfuild-normal, liquid-gas, shape phase transitions, etc. • Described well by many-body theories such as BCS, RPA or QRPA Finite systems (atomic nuclei, ultra-small metallic grains, etc.) • Strong quantal and thermal fluctuations • Phase transitions are smoothed out • The conventional BCS, RPA or QRPA fail in a number of cases (collapsing points, in light systems, at T0, at strong or weak interaction, etc. ) When applied to finite small systems, to be reliable, the BCS, RPA and/or QRPA need to be corrected to take into account the effects due to quantal and thermal fluctuations. THE SELF-CONSISTENT QRPA (SCQRPA)

  3. Testing ground: Pairing model Exact solutions: A. Volya, B.A. Brown, V. Zelevinsky, PLB 509 (2001) 37 Shortcoming: impracticable at T ≠0 for N > 14

  4. Shortcomings of BCS T=0 G>Gc : T 0 Violation of particle number  PNF: N= N2 - N2  Collapse of BCS at G  Gc Omission of QNF: N2=  N2  - N 2  Collapse of BCS gap at T = Tc   BCS BCS Exact Exact Gc Tc ~ 0.57 (0)

  5. Shortcomings of (pp)RPA and QRPA: • QBA: Violation of Pauli principle  Collapse of RPA at G  Gc • QRPA is valid only when BCS is valid: Collapse of QRPA at G  Gc  ppRPA Exact QRPA Gc Energy of the first excited state (For ppRPA: ω= E2 – E1)

  6. 1. SCQRPA at T = 0 BCS equations with SCQRPA corrections SCQRPA equations … = SCQRPA|…|SCQRPA

  7. SCQRPA at T = 0 (continued) SCQRPA = BCS + QRPA+ Corrections Due To Quantal Fluctuations GSC beyond the QRPA PNP  SCQRPA + Lipkin Nogami Coupling to pair vibrations

  8. Doubly folded equidistant multilevel pairing model levels, N particles  Ground-state energy Energy of first excited state

  9. 2. SCQRPA at T  0 FT-BCS equations with QNF Thermal average in the GCE:

  10. Dynamic coupling to SCQRPA vibrations FTBCS1(FTLN1) +SCQRPA

  11. N = 10 N=50

  12. Thermally assisted pairing correlation (pairing reentrance effect)    T = 0, M = 0 T = 0, M  0 T  0, M  0   T normal T2  L. G. Moretto,NPA 185 (1972) 145 R. Balian, H. Flocard, M. Vénéroni, PR 317 (1999) 251  superfluid T1 Mc T1 T2 T M M   

  13. SCQRPA at T0 & M0 Pairing Hamiltonian including z-projection of total angular momentum: Bogoliubov transformation + variational procedure: QNF: FTBCS1:

  14. Dynamic coupling to SCQRPA vibrations(T0 & M0) FTBCS1:

  15. Thermally assisted pairing Thermally assisted pairing N=10 M  0

  16. 4. Odd-even mass formula at T  0 Uncorrelated s.p energy

  17. Odd-even mass formula at T  0

  18. 56Fe 94 Pairing is included for pf+g9/2 major shell above the 40Ca core 94,98Mo Pairing is included for 22 levels above the 48Ca core S(E) = lnW(E) W(E) = r(E)D(E) FTSMMC: Alhassid, Bertsch, Fang PRC 68 (2003) 044322 Experiments: PRC 78 (2008) 054321, 74 (2006) 024325

  19. Conclusions • A microscopic self-consistent approach to pairing called the SCQRPA is developed. It includes the effects of QNF and dynamic coupling to pair vibrations. It works for any values of G, N, T and M, even at large N. • Because of QNF: - The sharp SN phase transition is smoothed out in finite systems; - A tiny rotating system in the normal state (at M > Mc and T=0) can turn superconducting at T0. • A modified formula is suggested for extracting the pairing gap from the differences of total energies of odd and even systems at T0. By subtracting the uncorrelated single-particle motion, the new formula produces a pairing gap in reasonable agreement with the exact results. • A novel approach called CE(MCE)-LNSCQRPA is proposed, which embeds the LNSCQRPA eigenvalues into the CE (MCE). The results obtained are very close to the exact solutions, the FTQMC ones, and experimental data. It is simple and workable for a wider range of mass (N>14) at T≠0.

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