Damping of giant Damping of giant resonances in extended resonances in extended RPA with ground state RPA with ground state

correlationscorrelations

Kyorin UniversityKyorin University Mitsuru TohyamaMitsuru Tohyama

ContentsContents

Time-dependent density-matrix Time-dependent density-matrix theory (TDDM) and theory (TDDM) and its small-amplitude limit (STDDM)its small-amplitude limit (STDDM) Giant quadrupole resonance in Giant quadrupole resonance in 1616OO SummarySummary

TDDM and STDDMTDDM and STDDM

)(|)1()2()'2()'1(|)():'2'121(

)(|)1()'1(|)():'11(

2 taaaattC

taatt

TDDM is an extended TDHF and gives time evolution of and C2

Time derivatives of and C2

),,(/

),()(|]),1()'1([|)(/

3222

21

CCFtCi

CFtHaatti

BBGKY hierarchy: S. J. Wang and W. Cassing, Ann. Phys. 159(1985)328

),(/

),(/

222

21

CFtCi

CFti

Truncation C3=0 gives TDDM equations

Applications of TDDM GQR: 　 M. Tohyama and A. S. Umar, Phys. Lett. B549 (2002)72 Fusion: Phys. Rev. C65(2002)037601

Ground state: A stationary solution of TDDM eqs.

0)(

0),(/

0'

)(

0),(/

''''''''''

''2''

321'213

''

'1'

213321

321

HPBC

CnFtCi

vCCv

n

CnFtni

　 　

Born term Pair correlations p-h correlation

Time independent form of TDDM

Iterative gradient method

M. Tohyama et al., Eur. Phys. J. A 21, 217(2004)

)(

)(

)(

)(

)1(

)1(

2

1

1

22

11

NF

NF

NC

Nn

NC

Nn

C

F

n

F

C

F

n

F

Neglect of g.s. correlations: STDDM Second RPA

X

x

X

x

db

ca

　

　

STDDM: Linearization of TDDM eqs. for and C2

a, b and d contain n and C : g.s. correlations

Application of STDDM to 2+1 in oxygen isotopes

M. Tohyama et al., Prog. Theor. Phys. 114(2005)1021

Giant quadrupole resonance in Giant quadrupole resonance in 1616OO

Spectrum of electrons scattered from 16O　 　　　　　　　　　　　　　　　　

A. Hotta et al., Phys. Rev. Lett. 33(1974)790

Effective interaction: Skyrme III Effective interaction: Skyrme III Single-particle states:Single-particle states:　 　 nn and and C C : 1p: 1p3/23/2, 1p, 1p1/21/2, 1d, 1d5/25/2

xxαααα’’ : 1s: 1s1/21/2 ~ 1f~ 1f5/25/2

XXαβααβα’’ββ ’ ’ : : 1p1p3/23/2, 1p, 1p1/21/2, 1d, 1d5/25/2

Calculational details

Strength function for r2Y20

RPA

SRPA

STDDM

Reduced strength of SKIII

STDDM and SRPA Only in STDDM

Damping processes

Spectrum of electrons scattered from 16O　 　　　　　　　　　　　　　　　　

A. Hotta et al., Phys. Rev. Lett. 33(1974)790

Strength function for r2Y20

RPA

SRPASTDDM

Original strength of SKIII

SummarySummary

STDDM based on TDDM ground state STDDM based on TDDM ground state was presented. was presented.

STDDM is an ERPA with ground-state STDDM is an ERPA with ground-state correlations.correlations.

STDDM was applied to GQR in STDDM was applied to GQR in 1616O. O. STDDM gives larger fragmentation of STDDM gives larger fragmentation of

22++ states than SRPA. states than SRPA. →→Importance of ground-state Importance of ground-state

correlationscorrelations

Ground state of 16O

Etot(MeV) = EMF + Ecor

= -122.4 -12.5 = -134.9 < EHF= -130.8

1p1p3/23/2 1p1p1/21/2 1d1d5/25/2

nn 0.950.95 0.930.93 0.060.06

QuadrupoleQuadrupole states in states in 1616OO

Experiment*Experiment* STDDMSTDDM

E (MeV)E (MeV) B(E2) (eB(E2) (e22fmfm44)) E (MeV)E (MeV) B(E2)B(E2)6.926.92 3636±4±4 5.05.0 7.87.8

9.859.85 0.670.67±0.27±0.27 9.4, 9.79.4, 9.7 11.811.8

11.5211.52 25.6725.67±2.83±2.83 12.012.0 17.217.2

13.1513.15 13.813.8 12.912.9 17.817.8

15.1515.15 8.18.1±4.1±4.1 15.415.4 1.81.8

16.4616.46 2.72.7±0.9±0.9 16.3, 16.716.3, 16.7 6.86.8

18.518.5 5.15.1±0.5±0.5 18.5, 19.418.5, 19.4 3.03.0

20-3020-30 2020±8±8 20-3020-30 5454

*A. Hotta et al., Phys. Rev. Lett. 33 (1974) 790.*A. Hotta et al., Phys. Rev. Lett. 33 (1974) 790.

Relation to HFB and QRPARelation to HFB and QRPA

*'''' C

M. Tohyama and S. Takahara: Prog. Theor. Phys. 112, 499 (2004)

*'''''' X