thermal nuclear pairing within the self-consistent qrpa
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
Thermal nuclear pairing within the self-consistent
QRPA
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, Japan2) Institute for Nuclear Science & Technique, Hanoi – Vietnam
3) Institute of Physics, Hanoi - Vietnam
MotivationInfinite 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 T0, 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)
. ,
, ˆˆ , ˆ , ˆ
, ˆˆˆ
~21
1~
1
,
mjmj
mjj
jjm
mjjmjm
jmjmj
jjj
jjj
j
j
PPaaPaaN
PPGNH
j
j
j
OO
’’
Testing ground: Pairing model
Testing ground: Pairing model
Exact solutions: A. Volya, B.A. Brown, V. Zelevinsky, PLB 509 (2001) 37Shortcoming: impracticable at T ≠0 for N > 14
. ˆ2ˆ , ˆ , ˆ2ˆ , ˆ
, ˆ
1ˆ , ˆ
jjkkjjjkkj
j
jjkkj
PPNPPN
NPP
Tc ~ 0.57 (0)
G>Gc : T 0
Exact
BC
S
T=0
Gc
BCS
Exact
Shortcomings of BCS
Violation of particle number
PNF: N = N2 - N2 Collapse of BCS at G
Gc
Omission of QNF: N = N - N 2
Collapse of BCS gap at T = Tc
ppRPA
QRPA
Gc
Exact
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
Energy of the first excited state(For ppRPA: ω= E2 – E1)
1. SCQRPA at T = 01. SCQRPA at T = 0
. '''
2'
2' jjjjj
jjj
jj
jj vuG
AAAAD
. )2( , 1 , 21
, 2 ,
, 12 ,
2
2
212
jjjjjjjj
jjjjjj
jjjj
jjkjjjj
nnnn
vGuvuG
vN
NND
DND
DD
22222 )( , 12
1 ,1
2
1jjjj
j
jj
j
jj GvE
Ev
Eu
BCS equations with SCQRPA corrections
SCQRPA equations
.
21
1
2
jj
j
YD… = SCQRPA|…|SCQRPA
.,, ''1
jjjjjjjj jjj
DAAAYAXQD
SCQRPA at T = 0 (continued)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 02. SCQRPA at T 0
. '''
2'
2' jjjjj
jjj
jj
jj vuG
AAAAD
. 1 , 21
, 2 ,
, 12 ,
2
2
212
jjjjj
jjjjjj
jjjj
jjkjjjj
nnn
vGuvuG
vN
ND
DND
DD
FT-BCS equations with QNF
Thermal average in the GCE: H-H- eOeO TrTr
Dynamic coupling to SCQRPA vibrations
Dynamic coupling to SCQRPA vibrations
G j E 1
21
E ˜ E j M j E ,
˜ E j b j q jj ,
b j j u j2 v j
2 2Gu jv j u j 'j ' v j ' Gv j
4 ,
q jj Gu j2v j
2 , g j j ' Gu jv j u j '2 v j '
2 ,
n j 1
j e 1 1
˜ E j M j 2 j2
d
. , '
, ~
~
~
~1
''''
2222
2
iMmjgV
E
En
E
EnVM
jjjjjj
jj
j
jj
j
jjjj
YXD
FTBCS1(FTLN1) + SCQRPA
N = 10
N=50
T = 0, M = 0
T = 0, M 0
T 0, M 0
M
T
M
T1
T2
T1 T2
T
L. G. Moretto, NPA 185 (1972) 145R. Balian, H. Flocard, M. Vénéroni,
PR 317 (1999) 251superfluid
normal
Mc
Thermally assisted pairing correlation(pairing reentrance
effect)
SCQRPA at T0 & M0SCQRPA at T0 & M0
QNF:
kkkkk nnnn 112N . '''
2'
2' kkkk
kkk
kkk vu
GAAAA
D
FTBCS1: . )](exp[1
1 , 0''
kkkkkkk mE
n
AAAA
Pairing Hamiltonian including z-projection of total angular momentum:
Bogoliubov transformation + variational procedure:
, 1
,
,
,
2
kkk
kkkkk
kkkk
kk
nn
vGu
vuG
D
DN
D
,ˆˆ' MNHH
M mk (ak ak ak
ak ) .k
222
22
22
)(
12
1 ,1
2
1
)(
))(21(2
12
kkkk
k
kk
k
kk
kkk
k
kkkkk
GvE
Ev
Eu
nnmM
nnvvN
Dynamic coupling to SCQRPA vibrations
(T0 & M0)Dynamic coupling to SCQRPA vibrations
(T0 & M0)
Gk E 1
21
E ˜ E k mk Mk E
,
˜ E k b k qkk ,
b k k uk2 vk
2 2Gukvk u k k v k Gvk
4 ,
qkk Guk2vk
2 , gk k Gukvk u k 2 v k
2 ,
nk
1
k e 1 1
˜ E k mk Mk 2 k2
d
. ,
, ~~12
iMmkgV
mEE
n
mEE
nVEM
kkkkkk
kk
kk
k
kk
kkk
YXD
FTBCS1: . 0 , )](exp[1
1''
kkkkkk
k mEn AAAA
N=10 M 0
Thermally assisted pairing
Thermally assisted pairing
4. Odd-even mass formula at T 04. Odd-even mass formula at T 0
Uncorrelated s.p energy
Odd-even mass formula at T 0Odd-even mass formula at T 0
56Fe 56Fe Pairing is included for
pf+g9/2 major shellabove the 40Ca core
S(E) = ln(E)(E) = (E)(E)
94,98Mo 94,98Mo Pairing is included for 22 levels above
the 48Ca core
FTSMMC: Alhassid, Bertsch, FangPRC 68 (2003) 044322
Experiments: PRC 78 (2008) 054321,
74 (2006) 024325
94
ConclusionsConclusions
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 T0.
A modified formula is suggested for extracting the pairing gap from the differences of total energies of odd and even systems at T0. 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.