[Project Name] Post-Mortem

1
Thermal nuclear pairing within the
self-consistent QRPA

• N. Dinh Dang1,2 and 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 gt 14

4
Shortcomings of BCS
T0
GgtGc T? 0

• Violation of particle number
• ? PNF ?N? ?N2? - ?N?2
• ? Collapse of BCS at G ? Gc

Omission of QNF ?N 2 ? N 2 ? - ?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 ?
E2 E1)
6
1. SCQRPA at T 0
BCS equations with SCQRPA corrections
SCQRPA equations
?? ?SCQRPASCQRPA?
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
N50
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
?
T1
superfluid
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
N10
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 pfg9/2 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 gt Mc and T0) 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?gt14) at T?0.