Low-lying dipole response in stable and unstable nuclei

Low-lying dipole response in stable and unstable nuclei Marco Brenna Xavier Roca-Maza, Giacomo Pozzi Kazuhito Mizuyama, Gianluca Colò and Pier Francesco Bortignon X. Roca-Maza, G. Pozzi, M.B., K. Mizuyama, G. Colò, Phys. Rev. C 85, 2461 (212) Kazimierz Dolny, 26-3 September

Giant Resonances Motivations Giant Resonances PDS Giant Resonances Nuclear collective excitations that are the macroscopic signature of some many-body correlations inside the nucleus. Constraints on the parameters of the equation of state of nuclear matter Monopole Compressibility K Dipole Symmetry Energy S(ρ =.1fm 3 ) Quadrupole Effective mass m Nuclear interaction in a given channel 6-year-studies on GRs (since 1947), two books P.F. Bortignon, A. Bracco, R.A. Broglia, 1998 M.N. Harakeh, A. van der Woude, 21 Resonances in exotic nuclei (n rich) M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 2/15

Pygmy Dipole Strength (PDS) Motivations Giant Resonances PDS Courtesy: D. Vretenar Low-energy peak in the dipole response of neutron rich nuclei due to shell effects, in some models has a coherent nature (resonance), in others. Connection with the slope of the symmetry energy S(ρ) at saturation: L = 3ρ d dρ S(ρ) ρ=ρ? M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 3/15

Pygmy Dipole Strength (PDS) Motivations Giant Resonances PDS To have a better understanding of the PDS, we need a more exhaustive analysis of its microscopic Courtesy: properties D. Vretenar employing different nuclei and a representative set of nuclear interactions Low-energy peak in the dipole response of neutron rich nuclei X. Roca-Maza, G. Pozzi, M.B., K. Mizuyama, G. Colò, Phys. Rev. C 85, 2461 (212) due to shell effects, in some models has a coherent nature (resonance), in others. Connection with the slope of the symmetry energy S(ρ) at saturation: L = 3ρ d dρ S(ρ) ρ=ρ? M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 3/15

List of ingredients RPA Motivations Self-consistent HF+RPA with Skyrme interactions, with different isovector properties. (L = 37.63 MeV) (L = 48.27 MeV) (L = 1.52 MeV) Continuum is discretized. Large basis due to zero range force. Three nuclei: 68 Ni, Sn, 28 Pb Total Energy and charge radii RPA 68 Ni Sn 28 Pb E [MeV] r c [fm] E [MeV] r c [fm] E [MeV] r c [fm] 611.48 3.88 1136.916 4.727 1667.328 5.512 591.464 3.918 114.18 4.73 1636.336 5.57 64.588 3.88 1121.76 4.71 1654.224 5.48 Exp. 59.376 3.866 19.32 1636.336 5.51 M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 4/15

List of ingredients RPA Motivations RPA Self-consistent HF+RPA with Skyrme interactions, with different isovector properties. (L = 37.63 MeV) (L = 48.27 MeV) (L = 1.52 MeV) Continuum is discretized. Large basis due to zero range force. Three nuclei: 68 Ni, Sn, 28 Pb We focus in particular on RPA and unperturbed dipole strength. the isoscalar or isovector nature. the transition densities. coherence of p-h contributions. M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 4/15

RPA Motivations RPA RPA state: ν = ph X(ν) ph ph 1 +Y (ν) ph hp 1 Reduced transition probabilities B(EJ : ν) = ν ˆF J 2 = ph (X(ν) + Y(ν) ph ) p ˆF J h Strength function S(E) = ν ν ˆF J 2 δ(e E ν) Operators ˆF (IV) 1M = 2 Z N A n=1 rny 1M(ˆr n) 2 N Z A p=1 rpy 1M(ˆr p) ˆF (IS) 1M = A i=1 r 3 i Y 1M (ˆr i ) Transition density: nature of excitation (surface - volume, isoscalar - isovector) δρ ν(r) = 1 2J+1 ph (X(ν) ph + Y(ν) ph ) p Y J h up(r)u h (r) r 2 ph 2 M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 5/15

Isovector Isoscalar Unperturbed Dipole strength functions (IV) ˆF (IV) 1M = 2Z N A n=1 r ny 1M (ˆr n ) 2N A A n=1 r py 1M (ˆr p ) 15 1 5 B(E1;IV) (fm 2 MeV 1 ) 2 3 2 1 9 1 11 12 Exp. 11 MeV [1] 68 Ni 3 3 15 B(E1;IV) (fm 2 MeV 1 )45 6 8 9 1 Exp. 9.8 MeV [2] Sn B(E1;IV) (1 1 fm 2 MeV 1 ) 1 2 8 1 6 4 2 1 15 2 E (MeV) 7 8 9 1 Exp. 7.37 MeV [3] Exp. 13.43 MeV [3] 28 Pb 1 15 2 E (MeV) larger L larger PDS peak A. Carbone et. al., PRC 81, 4131 (21). Experiment: [1] O. Wieland et. al., PRL 12, 9252 (29). [2] P. Adrich et. al., PRL 95, 51 (25). [3] N. Ryezayeva et. al., PRL 89, 27252 (22). 5 1 15 2 E (MeV) M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 6/15

Isovector Isoscalar Unperturbed Dipole strength functions (IS) B(E1;IS) (1 2 fm 6 MeV 1 ) B(E1;IS) (1 3 fm 6 MeV 1 ) 3 2 1 3 2 1 pygmy region 1 2 3 4 E (MeV) pygmy region ˆF (IS) 1M 68 Ni 28 Pb = A i=1 r 3 i Y 1M (ˆr i ) B(E1;IS) (1 3 fm 6 MeV 1 ) 12 8 4 pygmy region 1 2 3 E (MeV) Sn 1 2 3 4 E (MeV) M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 7/15

Isovector Isoscalar Unperturbed RPA versus unperturbed strength B IV (E1) [fm 2 MeV 1 ] 8 7 6 5 4 3 2 1 Unperturbed Sn 8 7 6 5 4 3 2 1 E PDS [MeV] E unp [MeV] 8.52 9.32 9.23 1.81 8.64 11.41 1 2 1 2 Energy [MeV] 1 2 - No low energy peak in the unperturbed response. - Indications that the PDS may show some coherency depending on the model. (RPA peaks do not coincide in energy with the unperturbed peak) M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 8/15

Isoscalar or isovector? A local criterion IS - IV nature Transition densities Collectivity A state is 7% isoscalar in a given radial range if δρ (IS) ν (r) > δρ (IV) ν (r) for at least the 7% of the points in the range [N. Paar et. al., PRL13 (29) 3252] M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 9/15

Isoscalar or isovector? A local criterion IS - IV nature Transition densities Collectivity A state is 7% isoscalar in a given radial range if δρ (IS) ν (r) > δρ (IV) ν (r) for at least the 7% of the points in the range [N. Paar et. al., PRL13 (29) 3252] Three regions: [, R], [, R/2], [R/2, R] M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 9/15

Isoscalar or isovector? A local criterion IS - IV nature Transition densities Collectivity A state is 7% isoscalar in a given radial range if δρ (IS) ν (r) > δρ (IV) ν (r) for at least the 7% of the points in the range [N. Paar et. al., PRL13 (29) 3252] Three regions: [, R], [, R/2], [R/2, R] B IV (E1) = 3 ν ( 2Z A drr 3 δρ n ν(r) 2N A drr 3 δρ p ν(r)) B(E1;IV) [fm 2 MeV 1 ] 3 25 2 15 1 5 IS 7% [,R] IS 7% [,R/2] 68 Ni IS 7% [R/2,R] 8 16 24 8 16 24 E [MeV] 8 16 24 M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 9/15

Isoscalar or isovector? IS - IV nature Transition densities Collectivity B ν(r) IV (E1) = 3 ν ( 2Z A drr 3 δρ n ν 2N A drr 3 δρ p ν ν(r)) B(E1;IV) [fm 2 MeV 1 ] 3 25 2 15 1 5 IS 7% [,R] IS 7% [,R/2] 68 Ni IS 7% [R/2,R] 8 16 24 8 16 24 E [MeV] 8 16 24 [N. Paar et. al., PRL13 (29) 3252] IS nature of the PDS due to outermost nucleons (neutrons in a neutron-rich nucleus). M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 1/15

of PDS Transition densities 68 Ni δρ(r) [fm -3 ] 4-4 -8 4-4 -8 4 δρ ν (r) = 1 r p r n IS - IV nature Transition densities Collectivity 2J+1 ph (X(ν) ph + Y(ν) 68 Ni E = 9.77 MeV neutrons protons E = 1.45 MeV -4-8 E = 9.3 MeV 2 4 6 8 1 r [fm] ph ) p Y J h up(r)u h(r) r 2 δρ(r) [fm -3 ] 8-8 8-8 8 r p r n r 2 68 Ni E = 9.77 MeV isovector isoscalar E = 1.45 MeV -8 E = 9.3 MeV 2 4 6 r [fm] 8 1 Around the nuclear surface: all models clearly isoscalar. In the interior: not clear trends. But this part is not quite sensitive to external probes M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 11/15

of PDS Transition densities Sn δρ(r) [fm -3 ] 15 1 5-5 15 1 5-5 15 δρ ν (r) = 1 r p r n IS - IV nature Transition densities Collectivity 2J+1 ph (X(ν) ph + Y(ν) Sn E = 8.52 MeV neutrons protons E = 9.23 MeV 1 5-5 E = 8.64 MeV 2 4 6 8 1 r [fm] ph ) p Y J h up(r)u h(r) r 2 δρ(r) [fm -3 ] 16 8-8 16 8-8 16 r 2 r p r n Sn E = 8.52 MeV isovector isoscalar E = 9.23 MeV 8-8 E = 8.64 MeV 2 4 6 8 1 r [fm] Around the nuclear surface: all models clearly isoscalar. In the interior: not clear trends. But this part is not quite sensitive to external probes M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 12/15

IS - IV nature Transition densities Collectivity Collectivity Relevant p-h excitations in the IS and IV dipole response B(E1 : 1 ) = ph A ph(e1) 2 = ph (X ph n + Yph) p ˆF n 1 h 2 A ph (E1;IV) (fm) 3 2 1-1 2f 7/2 2d 5/2 1i 13/2 1g 9/2 Sn 3s 1/2 3p 3/2 neutrons protons 1g 9/2 1i 13/2 3p 3/2 3s 1/2 1g 9/2 1i 13/2-2 E = 8.52 MeV E = 9.23 MeV E = 8.64 MeV -3 8 9 1 11 1 11 12 9 1 11 12 E ph (MeV) A ph (E1;IS) (fm 3 ) neutrons protons 2 Sn 1i 13/2 3s 1/2 3p 3/2 3s 1/2 3p 3/2 2d 3/2 3p 1/2 1 2d 3/2 3p 3/2 1g 9/2 3s 1/2 1g 9/2 3p 1/2 3s 1/2 3p 3/2 2d 3/2 3p 1/2 1i 13/2 1g 9/2 E = 8.52 MeV E = 9.23 MeV E = 8.64 MeV 8 12 16 8 12 16 8 12 16 E ph (MeV) The largest p-h contributions are the same in IV and IS response: M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 13/15

IS - IV nature Transition densities Collectivity Collectivity Relevant p-h excitations in the IS and IV dipole response B(E1 : 1 ) = ph A ph(e1) 2 = ph (X ph n + Yph) p ˆF n 1 h 2 A ph (E1;IV) (fm) 3 2 1-1 2f 7/2 2d 5/2 1i 13/2 1g 9/2 Sn 3s 1/2 3p 3/2 neutrons protons 1g 9/2 1i 13/2 3p 3/2 3s 1/2 1g 9/2 1i 13/2-2 E = 8.52 MeV E = 9.23 MeV E = 8.64 MeV -3 8 9 1 11 1 11 12 9 1 11 12 E ph (MeV) A ph (E1;IS) (fm 3 ) neutrons protons 2 Sn 1i 13/2 3s 1/2 3p 3/2 3s 1/2 3p 3/2 2d 3/2 3p 1/2 1 2d 3/2 3p 3/2 1g 9/2 3s 1/2 1g 9/2 3p 1/2 3s 1/2 3p 3/2 2d 3/2 3p 1/2 1i 13/2 1g 9/2 E = 8.52 MeV E = 9.23 MeV E = 8.64 MeV 8 12 16 8 12 16 8 12 16 E ph (MeV) The largest p-h contributions are the same in IV and IS response: PDS is one state projected on the two isospin channel. M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 13/15

Collectivity Coherence of the different contributions IS - IV nature Transition densities Collectivity B(E1 : 1 ) = ph A ph(e1) 2 = ph (X ph n + Yph) p ˆF n 1 h 2 3 cumulative sum 1 cumulative sum (E1) [e fm] A IV ph 2 1-1 -2 π ν π ν π 1 2 3 n ph Sn 1 2 3 n ph ν 1 2 3 4 n ph A IS ph (E1) [e fm3 ] 8 6 4 2 1 2 3 4 n ph Sn π ν π ν π 1 2 3 4 n ph ν 1 2 3 4 5 n ph The largest p-h contributions are: coherent in the IS channel, less coherent in the IV channel. M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 14/15

Conclusions Motivations Microscopic study of low energy dipole strength in 68 Ni, Sn, 28 Pb using Skyrme-HF+RPA framework. Low-energy peak in the IS and IV strength for all nuclei and models. IV (and IS) peak increases in magnitude with increasing values of L [in agreement with Carbone et al., PRC 81, 4131(R) (21) and Vretenar et al., PRC 85, 44317 (212)]. Systematically more collectivity in the IS than in the IV transitions The low-energy IS response is basically due to the outermost neutrons. Therefore, IS probes seem to be more suitable for the study of the low-energy dipole response. M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei 15/15

Extra material M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei

Correlations χ 2 (p) = ( m i=1 O th. i ) 2 (p) Oi ref. O i ref. χ 2 (p) χ 2 (p ) n i,j=1 (p i p i )M ij (p j p j ) M curvature matrix ( pi pj χ 2 ) Given two observables A and B, we define the covariance as A B = i,j p i A(M 1 ) ij pj B Pearson product-moment correlation coefficient c AB = A B A 2 B 2 c AB = 1 fully correlated c AB = totally uncorrelated M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei

Spurious state Back to Conclusions B(E1;IS) (1 3 fm 6 MeV 1 ) 4 3 2 1 with spurious state (sps) sps correc. dipole operator sps correc. as in App. A New set of states ν orthogonal to the spurious state, i.e. drr 2 r(δρ ñ ν +δρ p ν ) = Equal IV strength with new states drr 2 r(δ Z A ρñ ν N A δρp ν ) = drr 2 r(δ Z A ρn ν N A δρp ν) 1 2 3 4 E (MeV) Assumption (PLB 485, 362 (2)) δρ n,p ν = δρ n,p ν α n,p dρ n,p HF (r) dr M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei

Single particle units Back to Collectivity B(E1;IV) [s.p. units] 7 6 5 4 3 2 1 8.52 MeV 8 9 1 9.23 MeV 8 9 1 E [MeV] Sn 8.64 MeV 8 9 1 B(E1;IS) [s.p. units] 2 16 8.52 MeV 12 8 4 8 9 1 9.23 MeV 8 9 1 E [MeV] 8.64 MeV Sn 8 9 1 If different p-h states are contributing coherently to the PDS, the most prominent peaks should be clearly larger than one. M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei

RPA versus unperturbed strength Back to Unperturbed strength E [MeV] 16 14 12 1 unperturbed PDS unperturbed PDS unperturbed PDS 8 6 68 Ni Sn 28 Pb M. Brenna, X. Roca-Maza, G. Pozzi, K. Mizuyama, G. Colò, P.F. Bortignon Low-lying dipole response in stable and unstable nuclei