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
Motivation



Infinite systems
Finite systems
(metal superconductors, ultra-cold gases, liquid helium, etc.)
(atomic nuclei, ultra-small metallic grains, 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



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)
Testing ground: Pairing model
H    j Nˆ j  G  Pˆj Pˆj ’ ,
j, j’
j
j
j
Nˆ j   a jma jm , Pˆj 
1
j
m
 
 
ˆ
ˆ
 a a , Pj  Pj ,
m 1

jm

~
jm
 j  j  12 , Ojm~    Oj  m .
j m
ˆ 

N
j 
Pˆ j , Pˆ   jk 1 
,
 j 


 2 jk Pˆ j , Nˆ j , Pˆk  2 jk Pˆ j .

Nˆ
j
, Pˆk


k



Exact solutions:
A. Volya, B.A. Brown, V. Zelevinsky, PLB 509 (2001) 37
Shortcoming: impracticable at T ≠0 for N > 14
Shortcomings of BCS
T=0
Violation of particle number
 PNF: N = N2 - N2
 Collapse of BCS at G  Gc
G>Gc :
T 0
Omission of QNF:
N =  N   - N 2
 Collapse of BCS gap at T = Tc


Gc
Tc ~ 0.57 (0)
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

Gc
Energy of the first excited state
(For ppRPA: ω= E2 – E1)
1. SCQRPA at T = 0
BCS equations with SCQRPA corrections

 j     j ,

N  2  j D j vk2  12 1  D j
,
j
 j  2Gu j v j N
  G  ju j v j D j ,
j
N
D j  1  2n j ,
2
j
 n j 1  n j  ,
nj  N
2
Dj ,
j
j
(2 j ) .
1   j    2 1   j   
u  1
, v j  1
, E j  ( j    Gv 2j ) 2  2j
2 
E j 
2 
E j 
G
 j   j 
u 2j '  v 2j ' A j A j ' j  A j A j ' .

 j D j j'
2
j



SCQRPA equations
Q    j
1
Dj
X

j A

j
Y j A
… = SCQRPA|…|SCQRPA
j
,
Dj 
A
j
,A

j'
 
jj '
1
2
1
j
 Y 
 2
j
Dj .
.
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
Doubly folded equidistant multilevel pairing model
levels, N particles

Ground-state energy
Energy of first excited state
2. SCQRPA at T  0
FT-BCS equations with QNF

 Tre
1  D ,
H
O  Tr Oe-
Thermal average in the GCE:

 j     j , N  2  j D j vk2  12
- H
j
j
  G  ju jv j D j ,
 j  2Gu j v j N
j
D j  1  2n j ,
 j   j 

N

2
j
2
j
Dj ,
 n j 1  n j .
G
 u 2j '  v 2j ' A j A
 j D j j'

j ' j
 A j A
j'
.
Dynamic coupling to SCQRPA vibrations
G j E  
1
1
,
˜
2 E  E j  M j E 
E˜ j  bj  q jj ,
bj   j   u 2j  v 2j  2Gu j v j  u j'v j'  Gv 4j ,
j'
g j  j'  Gu j v j u 2j'  v 2j'  ,
q jj  Gu 2j v 2j ,

 

 

~
~

n j     E j  
 2 1  n j     E j  

M j     V j


2
2
~
~
2
2

   E j    


E




j




V j   g j  j ' D j ' X

j'



 ,


 j    mM j   i .

Y j '  ,
j'
FTBCS1(FTLN1) + SCQRPA
nj 
1


 j  e   1
1
d
   E˜ j  M j      
2
2
j
N = 10
N=50
Thermally assisted
pairing correlation
(pairing
reentrance effect)


T = 0, M  0
T = 0, M = 0
T  0, M  0

L. G. Moretto, NPA 185 (1972) 145
R. Balian, H. Flocard, M. Vénéroni,
PR 317 (1999) 251
T
Mc

M
normal
T1

 T2
 T1
superfluid

T2M T
SCQRPA at T0 & M0
H '  H   Nˆ   Mˆ ,
Pairing Hamiltonian including z-projection of total
angular momentum:

M   mk (ak ak   ak
ak ) .
k
Bogoliubov transformation + variational procedure:
 k     k ,
1


N  2 vk2  (1  2vk2 )( nk  nk ) 
2

k 
M   mk ( nk  nk )

  G  u k vk D k ,
k
k
 k  Guk vk N
2
Dk ,
k
D k  1  nk  nk ,
QNF:
N
2
k

k


k


k

 k  
1

1

2
Ek

Ek 

k
 n 1 n  n 1 n
FTBCS1:
u k2 

 k   k 
A k A k '  A k A k '  0 ,
G
Dk
 2
 k  
1


,
v

1

 k
2
Ek






( k    Gvk2 ) 2  2k
 u
2
k'

 vk2' A k A k '  A k A k '
k'
nk 
1
.
1  exp[  ( Ek  mk )]
.
Dynamic coupling to SCQRPA vibrations(T0 & M0)
Gk E  
1
1
,

˜
2 E  E k mk  M k E 
E˜ k  bk qkk ,
bk  k   uk2  v k2  2Guk v k  uk v k   Gv k4 ,
k 
gk k   Guk v k uk2  v k2 ,
qkk  Guk2v k2 ,
 
M k E    V


 2
k


1  nk  
nk  


 ,
~
~
 E  Ek  mk    E  Ek  mk    

Vk   g k k  D k  X
k

k
n 
FTBCS1:
1


   E˜ k

k


 k  e

 1
1
d
mk  M k      
1
n 
,
1  exp[  ( Ek  mk )]

k

Y k   ,  k    m M k   i  .
2
2
k
A k A k '  A k A k '
 0.
Thermally assisted pairing
N=10
Thermally assisted pairing
M0
4. Odd-even mass formula at T  0
Uncorrelated s.p energy
Odd-even mass formula at T  0
56Fe
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) = ln(E)
(E) = r(E)(E)
FTSMMC:
Alhassid, Bertsch, Fang
PRC 68 (2003) 044322
Experiments:
PRC 78 (2008) 054321,
74 (2006) 024325
94
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.