Prospects with Extended RPA Theories P. Papakonstantinou Institut - - PowerPoint PPT Presentation

Prospects with Extended RPA Theories

• P. Papakonstantinou

Institut f¨ ur Kernphysik, T.U.Darmstadt

1

Overview

2

■ Introduction ■ From RPA to Second RPA

• Formalism and technicalities
• Results on Giant Resonances
• Issues to be considered

■ Conclusion and Outlook

2

Introduction

3

■ Description based on RPA theories ■ Why extended RPA theories?

• More physics
• Convergence issues with respect to model space

■ What kind of extension is appropriate?

... remains to be seen

• Second RPA to begin with

transformed realistic interactions nuclear collective excitations

?

3

From the textbook

4

■ RPA

• Microscopic theory of small-amplitude density fluctuations
• Single-particle excitation operators fL(r)YLM(ˆ

r) (+isospin)

• GRs: coherent superpositions of ph excitations
• Change in single-particle Hamiltonian treated self-consistently

■ Why beyond RPA

• Damping of GRs due to

coupling of ph state to 2p2h states and higher coupling to surface vibrations increases the width of GRs Γν

• But also: energetically shifts them by ∆ν

Dispersion relation: ∆ν(E) = P

2π

• dǫΓν(ǫ)

E−ǫ

4

Present Work

5

■ Two-body UCOM Hamiltonian

☞Only state-independent, short-range correlations

are treated

■ A Second-order RPA Method

☞Large-scale calculations in closed-shell nuclei

■ Interesting results on

Giant Resonances

■ Learning about the inter-

action and the method!

■ Technical issues to be dealt with ■ Formalism and consistency

issues of the present SRPA method

☞ In most of what follows a UCOM-transformed Argonne V18 potential is used

5

UCOM-HF + PT

6

• 8
• 6
• 4
• 2

. E/A [MeV] Nmax = 12

4He 16O 24O 34Si 40Ca 48Ca 48Ni 56Ni 68Ni 78Ni 88Sr 90Zr 100Sn 114Sn 132Sn 146Gd 208Pb

1 2 3 4 5 6 . Rch [fm]

exp

• HF

HF+PT2 HF+PT2+PT3

6

UCOM-HF + PT

7

36 38 40 42 44 46 48 50 52 54

• 8
• 6
• 4
• 2

. E/A [MeV]

ACa

100 104 108 112 116 120 124 128 132 A

• 8
• 6
• 4
• 2

. E/A [MeV]

ASn

7

UCOM-HF

8

• 100
• 80
• 60
• 40
• 20

20

single particle energy levels [MeV]

UCOM (AV18) SIII NL3 EXP. UCOM (AV18) SIII NL3 EXP.

40Ca

neutrons protons

1s1/2 1p3/2 1p1/2 1d5/2 2s1/2 1d3/2 1s1/2 1p3/2 1p1/2 1d5/2 2s1/2 1d3/2 1f7/2

Vlow-k Vlow-k

1f5/2 1f7/2 1f5/2 2p3/2 2p1/2 2p3/2 2p1/2

8

Standard RPA

9

■ Vibration creation operator:

Q†

ν = phXν phO† ph − phY ν phOph ;

Qν|RPA = 0 ; Q†

ν|RPA = |ν ■ Standard RPA - the RPA vacuum is approximated by the HF ground state:

RPA| . . . |RPA → HF| . . . |HF ; O†

ph → a† pah ■ RPA equations in ph−space:

• A

B −B∗ −A∗ Xν Y ν

• = ων
• Xν

Y ν

• Aph,p′h′ = δpp′δhh′(ep−eh)+Hhp′,ph′ ; Bph,p′h′ = Hhh′,pp′ ; H = Hint = Trel+VUCOM

☞Self-consistent HF+RPA: spurious state and sum rules

9

Second RPA

10

■ Vibration creation operator: Includes 2p2h configurations

Q†

ν = phXν phO† ph − phY ν phOph + p1h1p2h2X ν p1h1p2h2O† p1h1p2h2

−

p1h1p2h2Yν p1h1p2h2Op1h1p2h2 ■ The SRPA vacuum is approximated by the HF ground state:

SRPA| . . . |SRPA → HF| . . . |HF

■ SRPA equations in ph ⊕ 2p2h−space:

      A A12 B A21 A22 −B∗ −A∗ −A∗

12

−A∗

21

−A∗

22

            Xν X ν Y ν Yν       = ων       Xν X ν Y ν Yν       Aph,p′h′ = δpp′δhh′(ep−eh)+Hhp′,ph′ ; Bph,p′h′ = Hhh′,pp′ ; H = Hint = Trel+VUCOM A12: interactions between ph and 2p2h states A22: δp1p′

1δh1h′ 1δp1p′ 1δh1h′ 1(ep1 + ep2 − eh1 − eh2) + interactions among 2p2h states

10

Second RPA

11

■ Large model spaces:

• Number of states up to ≈ 106 for the present cases – can get larger
• But SRPA matrix is sparse and reduction to half the size is always possible

11

Second RPA

11

■ Large model spaces:

• Number of states up to ≈ 106 for the present cases – can get larger
• But SRPA matrix is sparse and reduction to half the size is always possible

■ Use Lanczos

• Find only the lowest eigenvalues |ǫν|
• ... or the ones closest to a set value E0, e.g.

HXν = ǫνXν ⇐ ⇒ H′Xν = ǫ′

νXν ,

• H′ ≡ H − E0I

ǫ′

ν ≡ ǫν − E0

• 11-a

Second RPA

11

■ Large model spaces:

• Number of states up to ≈ 106 for the present cases – can get larger
• But SRPA matrix is sparse and reduction to half the size is always possible

■ Use Lanczos

• Find only the lowest eigenvalues |ǫν|
• ... or the ones closest to a set value E0, e.g.

HXν = ǫνXν ⇐ ⇒ H′Xν = ǫ′

νXν ,

• H′ ≡ H − E0I

ǫ′

ν ≡ ǫν − E0

• ■ Alternatively, reduce to an ω−dependent problem of RPA size
• ... especially if you ignore interactions within 2p2h space:

Aphp′h′ − → Aphp′h′(ǫ) = Aphp′h′ +

• PHP ′H′

A∗

ph PHP ′H′Ap′h′ PHP ′H′

ǫ − (ǫP + ǫP ′ − ǫH − ǫH′) + iη

11-b

SRPA Eigenstates

12

50 100 150 200 250 300 20 40 60 80 100 BISM(Eν) [fm4] Eν [MeV] O16 eMax06 lMax06 aHO01.80 :: ISM distributions srpa0 srpa rpa 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 100 1000 50 100 150 200 250 Eν [MeV] srpa0 srpa rpa 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 100 1000 srpa srpa0 rpa

SRPA and its diagonal approximation (”srpa0”) vs RPA

12

SRPA Eigenstate Density

13

50 100 150 200 250 300 350 400 Eν [MeV] BrinkBoeker:: He4 eMax08 aHO01.80 JPC210 SRPA0 HF SRPA 1 10 100 1000 50 100 150 200 250 P(Eν) Eν [MeV] av18 E100900:: O16 eMax06 aHO01.80 JPC010 SRPA0 HF SRPA

SRPA vs its diagonal approximation and unperturbed states

13

SRPA - Diagonal approximation

14

0.05 0.1 0.15 0.2 0.25 0.3 0.35 10 20 30 40 50 60 RIVD(E) [fm2/MeV]

16O, Nmax=12

IVD

SRPA full SRPA diag. RPA 50 100 150 200 250 300 350 10 20 30 40 50 60 RISQ(E) [fm4/MeV] E [MeV]

48Ca, Nmax=8

ISQ

SRPA full SRPA diag. RPA

14

Results on GRs

15

UCOM :: RPA and SRPA

16

5 10 15 20 25 30 10 20 30 40 50 60 RISQ(E) [fm4/MeV] E [MeV]

16O

ISQ exp 20 40 60 80 100 120 140 10 20 30 40 50 60 E [MeV]

40Ca

ISQ exp 10 20 30 40 50 60 70 80 RISM(E) [fm4/MeV]

16O

ISM exp 50 100 150 200 250 300 350 400

40Ca

ISM nMax06 lMax06 SRPA RPA exp 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 RIVD(E) [fm2/MeV] IVD

16O

experiment 0.2 0.4 0.6 0.8 1 1.2 IVD

40Ca

experiment 16

UCOM :: RPA and SRPA

16

5 10 15 20 25 30 10 20 30 40 50 60 RISQ(E) [fm4/MeV] E [MeV]

16O

ISQ exp 20 40 60 80 100 120 140 10 20 30 40 50 60 E [MeV]

40Ca

ISQ exp 10 20 30 40 50 60 70 80 RISM(E) [fm4/MeV]

16O

ISM exp 50 100 150 200 250 300 350 400

40Ca

ISM nMax06 lMax06 SRPA RPA exp 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 RIVD(E) [fm2/MeV] IVD

16O

experiment 0.2 0.4 0.6 0.8 1 1.2 IVD

40Ca

experiment

0.5 1 1.5 2 2.5 3 3.5 4 10 20 30 40 50 60 RIVD(E) [fm2/MeV] E [MeV] IVD

90Zr

exp, (γ,Xn) SRPA RPA

16-a

Fragmentation of ph states

17

40Ca

0.2 0.4 0.6 0.8 1 1.2 10 20 30 40 50 E[MeV] HF 0.1 0.2 0.3 0.4 0.5 0.6 strength π1s1/2 to π1d5/2 RPA 0.1 0.2 0.3 0.4 0.5 0.6 SRPA 10 20 30 40 50 60 70 10 20 30 40 50 E[MeV] HF 50 100 150 200 250 300 SISQ(E) [e2fm4] RPA 50 100 150 200 250 300 350 SRPA

17

Fragmentation of resonances

18

0.2 0.4 0.6 0.8 1 10 15 20 25 30 E [MeV] 50 100 150 200 250 300 350 SISQ(E) [fm4] .

ISQ

40Ca

RPA SRPA

18

Fragmentation of resonances

18

0.2 0.4 0.6 0.8 1 10 15 20 25 30 E [MeV] 50 100 150 200 250 300 350 SISQ(E) [fm4] .

ISQ

40Ca

RPA SRPA

ten times as many states below 35 MeV

18-a

To consider

19

Spurious states

20

20 40 60 80 100 120 10 20 30 40 50 60 70 0.06 0.12 0.18 0.24 0.3 0.36 BISD(E) [fm6] RIVD(E) [fm2/MeV] RPA ISD uncorrected corrected IVD 20 40 60 80 100 120 10 20 30 40 50 60 70 0.06 0.12 0.18 0.24 0.3 0.36 BISD(E) [fm6] RIVD(E) [fm2/MeV] E [MeV]

263 1060

SRPA0 ISD uncorrected corrected IVD

ISD corrected radial operator r3 − 5

3r2r vs r3 16O

Nmax = 12

20

Low-lying states

21

0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14 150 300 450 600 750 ImE [MeV] RISQ(E) [fm4/MeV] ReE [MeV]

48Ca, IS 2+

exp RPA :: eMax14 lMax10 aHO01.80 SRPA0 :: lMax=10 lMax= 8 lMax= 6 SRPA0 :: E(2+

1), E(2+ 2) vs nMax

SRPA0: convergence and stability of low-lying ISQ states

21

Renormalized RPA

22

Centroid energies — RPA RRPA ....

. exp

10 15 20 25 30

208Pb 100Sn 90Zr 40Ca 16O

E [MeV] IS GMR centroid

208Pb 100Sn 90Zr 40Ca 16O

IS GQR centroid 2 4 6 8 10 12

208Pb 100Sn 90Zr 40Ca 16O

collective ISO state

Fermi-sea depletion: 2.6-5.0%

22

Renormalized RPA

23

Centroid energies — RPA RRPA ....

. exp

10 15 20 25 30

208Pb 100Sn 90Zr 40Ca 16O

E [MeV] IS GMR centroid

208Pb 100Sn 90Zr 40Ca 16O

IS GQR centroid

2 4 6 8 10 12

208Pb 100Sn 90Zr 40Ca 16O

collective ISO state

Fermi-sea depletion: 2.6-5.0%

23

RPA, SRPA, and extensions

24

Nucl.Phys.A477(88)205 etc

RPA SRPA

additional 2nd-order diagrams

24

Summary and Outlook

25

Extended-RPA calculations using correlated realistic interactions

Effective interactions for extended RPA?

✔ Avoiding conceptual problems ✔ More fundamental treatment of nucleon self energy, m∗ (ISQ, IVD) ✘ Two-body UCOM: Soft nuclei due to residual three body effects?

■ Second RPA:

✔ Great improvement over RPA results ✔ model space should be flexible enough to describe residual LRC ✘ Instabilities and inconsistencies

■ Extensions of the present simple SRPA method

25

Summary and Outlook

26

Extended-RPA calculations using correlated realistic interactions

Effective interactions for extended RPA?

✔ Avoiding conceptual problems ✔ More fundamental treatment of nucleon self energy, m∗ (ISQ, IVD) ✘ Two-body UCOM: Soft nuclei due to residual three body effects?

■ Second RPA:

✔ Improvement over RPA results ✔ Model space should be flexible enough to describe residual LRC ✘ Instabilities and inconsistencies

■ Extensions of the present simple SRPA method

Thank you!

26

Thank you!

27

Work in collaboration with:

■ A.G¨

unther, H.Hergert, S. Reinhardt, R.Roth J.Wambach, ...

Institut f¨ ur Kernphysik, TU Darmstadt, Germany ■ H. Feldmeier, K. Langanke, G. Martinez-Pinedo, T. Neff, ... GSI, Darmstadt, Germany

Some related references

■ P

. P ., R. Roth, PLB671, 356 (2009)

■ P

. P ., R. Roth, N.Paar, Phys. Rev. C75, 014310 (2007)

■ N. Paar, P

. P ., H. Hergert, R. Roth, Phys. Rev. C74, 014318 (2006)

■ and many more: http://crunch.ikp.physik.tu-darmstadt.de/tnp/

27

Reduction of the RPA problem

28

■ 2Nx2N RPA problem, with A and B NxN symmetric:

• A

B −B −A Xν Yν

• = ǫν
• Xν

Yν

• ■ Reduction to NxN is always possible in various ways

... even when A ± B are not positive definite

■ Simplest way:

[(A − B)(A + B)]Rν = ǫ2

νRν , with Rν = ǫ−1/2 ν

(Xν + Yν) For real, positive solutions, Xν = 1

2[ǫ1/2 ν I + ǫ−1/2 ν

(A + B)]Rν Yν = 1

2[ǫ1/2 ν I − ǫ−1/2 ν

(A + B)]Rν

[P .P .,EPL78(07)12001]

28

Second RPA with 2p2h coupling

29

0.1 0.2 0.3 0.4 0.5 10 20 30 40 50 RIVD [fm2/MeV] E [MeV] UCOM

16O eMax12 lMax08 aHO01.80

srpa srpa0 rpa 10 20 30 40 50 E [MeV] SRG srpa srpa0 rpa

29

The ISQ resonance of 40Ca - Width?

30

100 200 300 400 500 600 700 5 10 15 20 25 30 35 40 RISQ(E) [fm4/MeV] E[MeV] Continuum RPA 40Ca - SkM*

30

The ISQ resonance of 40Ca - Width?

30

100 200 300 400 500 600 700 5 10 15 20 25 30 35 40 RISQ(E) [fm4/MeV] E[MeV] Continuum RPA 40Ca - SkM* 500 1000 1500 2000 5 10 15 20 25 30 35 40 BISQ(E) [fm4] E[MeV] 40Ca - Brink-Boeker RPA

30-a

The ISQ resonance of 40Ca - Width?

30

100 200 300 400 500 600 700 5 10 15 20 25 30 35 40 RISQ(E) [fm4/MeV] E[MeV] Continuum RPA 40Ca - SkM* 500 1000 1500 2000 5 10 15 20 25 30 35 40 BISQ(E) [fm4] E[MeV] 40Ca - Brink-Boeker RPA 50 100 150 200 250 300 350 400 5 10 15 20 25 30 35 40 BISQ(E) [fm4] E[MeV] 40Ca - UCOM-AV18 RPA

30-b

Wavelet transform – Morlet

31

■ Consider spectrum Ei, B(Ei); fold with, e.g., a Gaussian

S(E) = 1 √ 2πγ

• i

B(Ei) exp−(E−Ei)2/2γ2

■ mother wavelet: Morlet

ψ(x) = π−1/4 cos kx exp−x2/2 ; k = 5

■ Wavelet coefficients: (δE the scale)

C(Ex, δE) = 1 √ δE

• S(E)ψ(Ex − E

δE )dE

■ Next we plot |C(Ex, δE)| (γ = 50keV)

31

The ISQ resonance of 40Ca - Wavelet transform

32

10 20 30 40 500 1000 1500 2000 10 20 30 40 200 400 600 800 1000 1200

20 25 30 35 0.5 1.0 1.5 2.0 2.5 3.0 2 4 6 8 10 12 14 0.5 1.0 1.5 2.0 2.5 3.0 10 15 20 25 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 5 6 7 0.5 1.0 1.5 2.0 2.5 3.0

RPA SRPA

SISQ(E) [fm4/MeV] SISQ(E) [fm4/MeV] Morlet scale [MeV] Morlet scale [MeV] E [MeV] E [MeV] Ex [MeV] power [a.u.]

32

Fragmentation of ph states

33

0.2 0.4 0.6 0.8 1 1.2 5 10 15 20 25 30 35 40 E[MeV] HF 0.2 0.4 0.6 0.8 1 1.2 strength ν0d3/2 to ν1p3/2 RPA 0.2 0.4 0.6 0.8 1 1.2 SRPA 0.4 0.8 1.2 1.6 5 10 15 20 25 30 35 40 E[MeV] HF 0.5 1 1.5 2 2.5 SIVD(E) [e2fm2] RPA 1 2 3 4 5 SRPA

33

Fragmentation of resonances

34

0.2 0.4 0.6 0.8 1 5 10 15 20 25 30 35 40 E [MeV] 200 400 600 800 1000 1200 SISM(E) [fm4] .

ISM

40Ca

RPA SRPA

34