Prospects with Extended RPA Theories P. Papakonstantinou Institut f¨ ur Kernphysik, T.U.Darmstadt 1
Overview ■ Introduction ■ From RPA to Second RPA - Formalism and technicalities - Results on Giant Resonances - Issues to be considered ■ Conclusion and Outlook 2 2
Introduction transformed nuclear collective ? realistic interactions excitations ■ 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 3 3
From the textbook ■ RPA - Microscopic theory of small-amplitude density fluctuations - Single-particle excitation operators f L ( r ) Y LM (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 Γ ν - But also : energetically shifts them by ∆ ν dǫ Γ ν ( ǫ ) Dispersion relation: ∆ ν ( E ) = P � 2 π E − ǫ 4 4
Present Work ■ 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 ■ Technical issues to be dealt with Giant Resonances ■ Formalism and consistency ■ Learning about the inter- issues of the present SRPA action and the method! method ☞ In most of what follows a UCOM-transformed Argonne V18 potential is used 5 5
UCOM-HF + PT 0 N max = 12 -2 E/A [MeV] -4 -6 -8 . 6 5 R ch [ fm ] 4 3 2 . 1 4 He 24 O 40 Ca 48 Ni 68 Ni 88 Sr 100 Sn 132 Sn 208 Pb 16 O 34 Si 48 Ca 56 Ni 78 Ni 90 Zr 114 Sn 146 Gd ● HF exp � HF+PT2 � HF+PT2+PT3 6 6
UCOM-HF + PT 0 A Ca -2 E / A [MeV] -4 -6 -8 . 36 38 40 42 44 46 48 50 52 54 0 A Sn -2 E / A [MeV] -4 -6 -8 . 100 104 108 112 116 120 124 128 132 A 7 7
UCOM-HF UCOM UCOM V low-k SIII NL3 EXP. V low-k EXP. SIII NL3 (AV18) (AV18) 20 single particle energy levels [MeV] 1f 5/2 1f 5/2 2p 1/2 2p 1/2 2p 3/2 2p 3/2 0 1f 7/2 1f 7/2 1d 3/2 1d 3/2 2s 1/2 -20 2s 1/2 1d 5/2 1d 5/2 1p 1/2 -40 1p 1/2 1p 3/2 1p 3/2 -60 1s 1/2 -80 1s 1/2 40 Ca -100 protons neutrons 8 8
Standard RPA ■ Vibration creation operator: ph O † Q † ph X ν ph Y ν Q † ν = � ph − � ph O ph ; Q ν | RPA � = 0 ; ν | RPA � = | ν � ■ Standard RPA - the RPA vacuum is approximated by the HF ground state: O † ph → a † � RPA | . . . | RPA � → � HF | . . . | HF � ; p a h ■ RPA equations in ph − space: � � � � � � X ν X ν A B = � ω ν Y ν Y ν − B ∗ − A ∗ A ph,p ′ h ′ = δ pp ′ δ hh ′ ( e p − e h )+ H hp ′ ,ph ′ ; B ph,p ′ h ′ = H hh ′ ,pp ′ ; H = H int = T rel + V UCOM ☞ Self-consistent HF+RPA: spurious state and sum rules 9 9
Second RPA ■ Vibration creation operator: Includes 2 p 2 h configurations ph O † p 1 h 1 p 2 h 2 O † ph X ν ph Y ν p 1 h 1 p 2 h 2 X ν Q † ν = � ph − � ph O ph + � p 1 h 1 p 2 h 2 p 1 h 1 p 2 h 2 Y ν − � p 1 h 1 p 2 h 2 O p 1 h 1 p 2 h 2 ■ The SRPA vacuum is approximated by the HF ground state: � SRPA | . . . | SRPA � → � HF | . . . | HF � ■ SRPA equations in ph ⊕ 2 p 2 h − space: X ν X ν A 12 0 A B X ν X ν A 21 A 22 0 0 = � ω ν − B ∗ − A ∗ −A ∗ Y ν Y ν 0 12 Y ν Y ν −A ∗ −A ∗ 0 0 21 22 A ph,p ′ h ′ = δ pp ′ δ hh ′ ( e p − e h )+ H hp ′ ,ph ′ ; B ph,p ′ h ′ = H hh ′ ,pp ′ ; H = H int = T rel + V UCOM A 12 : interactions between ph and 2 p 2 h states A 22 : δ p 1 p ′ 1 δ h 1 h ′ 1 δ p 1 p ′ 1 δ h 1 h ′ 1 ( e p 1 + e p 2 − e h 1 − e h 2 ) + interactions among 2 p 2 h states 10 10
Second RPA ■ Large model spaces: • Number of states up to ≈ 10 6 for the present cases – can get larger • But SRPA matrix is sparse and reduction to half the size is always possible 11 11
Second RPA ■ Large model spaces: • Number of states up to ≈ 10 6 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 | ǫ ν | • ... or the ones closest to a set value E 0 , e.g. H ′ ≡ H − E 0 I � � ⇒ H ′ X ν = ǫ ′ HX ν = ǫ ν X ν ⇐ ν X ν , ǫ ′ ν ≡ ǫ ν − E 0 11 11-a
Second RPA ■ Large model spaces: • Number of states up to ≈ 10 6 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 | ǫ ν | • ... or the ones closest to a set value E 0 , e.g. H ′ ≡ H − E 0 I � � ⇒ H ′ X ν = ǫ ′ HX ν = ǫ ν X ν ⇐ ν X ν , ǫ ′ ν ≡ ǫ ν − E 0 ■ Alternatively, reduce to an ω − dependent problem of RPA size • ... especially if you ignore interactions within 2p2h space: A ∗ ph PHP ′ H ′ A p ′ h ′ PHP ′ H ′ � A php ′ h ′ − → A php ′ h ′ ( ǫ ) = A php ′ h ′ + � ǫ − ( ǫ P + ǫ P ′ − ǫ H − ǫ H ′ ) + i η PHP ′ H ′ 11 11-b
SRPA Eigenstates SRPA and its diagonal approximation (”srpa0”) vs RPA O16 eMax06 lMax06 aHO01.80 :: ISM distributions 300 1000 srpa0 srpa 100 srpa srpa0 10 rpa rpa 250 1 0.1 0.01 0.001 200 0.0001 B ISM (E ν ) [fm 4 ] 1e-05 1e-06 150 1000 100 srpa0 srpa 10 100 rpa 1 0.1 0.01 50 0.001 0.0001 1e-05 0 1e-06 0 20 40 60 80 100 0 50 100 150 200 250 E ν [MeV] E ν [MeV] 12 12
SRPA Eigenstate Density SRPA vs its diagonal approximation and unperturbed states av18 E100900:: O16 eMax06 aHO01.80 JPC010 BrinkBoeker:: He4 eMax08 aHO01.80 JPC210 1000 SRPA0 SRPA0 HF HF SRPA SRPA 100 P(E ν ) 10 1 0 50 100 150 200 250 0 50 100 150 200 250 300 350 400 E ν [MeV] E ν [MeV] 13 13
SRPA - Diagonal approximation 0.35 SRPA full R IVD (E) [fm 2 /MeV] 0.3 16 O, N max =12 SRPA diag. 0.25 RPA IVD 0.2 0.15 0.1 0.05 0 0 10 20 30 40 50 60 350 SRPA full R ISQ (E) [fm 4 /MeV] 300 48 Ca, N max =8 SRPA diag. 250 RPA ISQ 200 150 100 50 0 0 10 20 30 40 50 60 E [MeV] 14 14
Results on GRs 15
UCOM :: RPA and SRPA 80 400 nMax06 lMax06 SRPA R ISM (E) [fm 4 /MeV] 70 350 RPA 60 300 50 250 16 O 40 Ca 40 200 ISM ISM 30 150 20 100 exp exp 10 50 0 0 0.45 1.2 experiment experiment R IVD (E) [fm 2 /MeV] 0.4 1 0.35 0.3 0.8 0.25 16 O 40 Ca 0.6 0.2 IVD IVD 0.15 0.4 0.1 0.2 0.05 0 0 30 140 R ISQ (E) [fm 4 /MeV] 120 25 100 20 80 16 O 40 Ca 15 60 ISQ ISQ 10 40 exp exp 5 20 0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 E [MeV] E [MeV] 16 16
UCOM :: RPA and SRPA 80 400 nMax06 lMax06 SRPA R ISM (E) [fm 4 /MeV] 70 350 RPA 60 300 50 250 16 O 40 Ca 40 200 ISM ISM 30 150 20 100 exp exp 10 50 0 0 0.45 1.2 experiment experiment R IVD (E) [fm 2 /MeV] 0.4 1 0.35 0.3 0.8 0.25 16 O 40 Ca 0.6 0.2 IVD IVD 0.15 0.4 0.1 0.2 0.05 0 0 30 140 R ISQ (E) [fm 4 /MeV] 120 25 4 100 20 exp, ( γ ,Xn) R IVD (E) [fm 2 /MeV] 3.5 80 16 O 40 Ca SRPA 15 3 60 ISQ ISQ RPA 2.5 10 40 exp 90 Zr 2 exp 5 20 IVD 1.5 0 0 1 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0.5 E [MeV] E [MeV] 0 0 10 20 30 40 50 60 E [MeV] 16 16-a
Fragmentation of ph states 0.6 350 SRPA SRPA 300 0.5 250 0.4 200 0.3 150 0.2 100 0.1 50 0 0 strength π 1s 1/2 to π 1d 5/2 0.6 RPA RPA 300 0.5 S ISQ (E) [e 2 fm 4 ] 250 0.4 200 0.3 150 0.2 100 0.1 50 0 0 1.2 HF HF 70 1 60 0.8 50 0.6 40 30 0.4 20 0.2 40 Ca 10 0 0 0 10 20 30 40 50 0 10 20 30 40 50 E[MeV] E[MeV] 17 17
Fragmentation of resonances S ISQ (E) [fm 4 ] . 350 RPA 300 SRPA ISQ 250 40 Ca 200 150 100 50 0 1 0.8 0.6 0.4 0.2 0 10 15 20 25 30 E [MeV] 18 18
Fragmentation of resonances S ISQ (E) [fm 4 ] . 350 RPA 300 SRPA ISQ 250 40 Ca 200 150 100 50 0 1 0.8 0.6 0.4 0.2 0 10 15 20 25 30 ten times as many states E [MeV] below 35 MeV 18 18-a
To consider 19
Spurious states 16 O ISD corrected radial operator r 3 − 5 3 � r 2 � r vs r 3 N max = 12 120 0.36 RPA ISD uncorrected R IVD (E) [fm 2 /MeV] 100 0.3 corrected B ISD (E) [fm 6 ] IVD 80 0.24 60 0.18 40 0.12 20 0.06 0 0 0 10 20 30 40 50 60 70 120 0.36 263 1060 SRPA0 ISD uncorrected R IVD (E) [fm 2 /MeV] 100 0.3 corrected B ISD (E) [fm 6 ] IVD 80 0.24 60 0.18 40 0.12 20 0.06 0 0 0 10 20 30 40 50 60 70 E [MeV] 20 20
Low-lying states SRPA0: convergence and stability of low-lying ISQ states 1 750 RPA :: eMax14 lMax10 aHO01.80 SRPA0 :: lMax=10 lMax= 8 0.8 600 48 Ca, IS 2+ lMax= 6 SRPA0 :: E(2 + 1 ), E(2 + R ISQ (E) [fm 4 /MeV] 2 ) vs nMax 0.6 450 ImE [MeV] 0.4 300 0.2 150 0 0 exp 0 2 4 6 8 10 12 14 ReE [MeV] 21 21
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