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.