The Nuclear Symmetry Energy: constraints from Giant Resonances Xavier Roca-Maza Dipartimento di Fisica, Università degli Studi di Milano and INFN, Sezione di Milano Trobada de Nadal 2012
Table of contents: Introduction The Nuclear Many-Body Problem Nuclear Energy Density Functionals Symmetry energy Motivation Giant Dipole Resonance Giant Quadrupole Resonance Conclusions
INTRODUCTION
The Nuclear Many-Body Problem: Nucleus: from few to more than 200 strongly interacting and self-bound fermions. Underlying interaction is not perturbative at the (low)energies of interest for the study of masses, radii, deformation, giant resonances,... Complex systems: spin, isospin, pairing, deformation,... Many-body calculations based on NN scattering data in the vacuum are not conclusive yet: different nuclear interactions in the medium are found depending on the approach only EoS, no extensive calculations for nuclei are available Based on effective interactions, Nuclear Energy Density Functionals are successful in the description of masses, nuclear sizes, deformations, Giant Resonances,...
Approximate realization of an exact Nuclear Energy Density Functionals: Kohn-Sham iterative scheme (static approximation) Determine a good E[ρ] Initial guess ρ 0 Calculate potential V eff from ρ 0 Solve single particle (Schrödinger) equation and find single particle wave functions φ i Use φ i for calculating new ρ 1 = A i φ i 2 Repeat until convergence Runge-Gross Theorem: dynamic generalization of the static EDFs. dt{ Φ(t) i t Φ(t) E[ρ(t),t]} = 0 Giant Resonances well described within the small amplitude limit (known as RPA approach)
Nuclear Energy Density Functionals: Main types of successful EDFs for the description of masses, deformations, nuclear distributions, Giant Resonances,... Relativistic mean-field models, based on Lagrangians where effective mesons carry the interaction: L int = ΨΓ σ ( Ψ,Ψ)ΨΦ σ + ΨΓ δ ( Ψ,Ψ)τΨΦ δ ΨΓ ω ( Ψ,Ψ)γ µ ΨA (ω)µ ΨΓ ρ ( Ψ,Ψ)γ µ τψa (ρ)µ e ΨˆQγ µ ΨA (γ)µ Non-relativistic mean-field models, based on Hamiltonians where effective interactions are proposed and tested: VNucl eff = Vlong range attractive +V short range repulsive +V SO +V pair Fitted parameters contain (important) correlations beyond the mean-field Nuclear energy functionals are phenomenological not directly connected to any NN (or NNN) interaction
The uncertainties in S(ρ) impacts on many observables and, therefore, on our understanding on the nuclear effective interaction. The Nuclear Equation of State: Infinite System 30 (Independent particle picture: nucleons move in an effective mean field potential generated by all of them) 20 neutron matter e(ρ,δ=1) e ( MeV ) 10 0 Saturation (0.16 fm 3, 16.0 MeV) S(ρ)~ -10 e(ρ,δ=0) symmetric matter -20 0 0.05 0.1 0.15 0.2 0.25 ρ ( fm 3 ) Physical Review C 86 015803 (2012) e(ρ,δ) = e(ρ,δ = 0)+S(ρ)δ 2 +O [ δ 4] with δ = ρn ρp ρ ( ) n+ρ p 2 S(ρ) = J+L ρ ρ 0 3ρ 0 +K ρ ρ0 [ sym 3ρ 0 +O (ρ ρ0 ) 3]
MOTIVATION
Motivation: The symmetry energy and the structure and composition of a neutron star crust Composition 90 80 70 60 50 40 30 20 FSUGold Protons Neutrons Ni Sr Kr N=50 Se Sn N=82 Cd Pd Ru Mo Zr Sr Kr Composition 80 70 60 50 40 30 20 NL3 N=82 N=32 Fe Sr Kr N=50 10-4 10-3 10-2 10-1 10 0 10 1 Se ρ(10 11 g/cm 3 ) Ge Zn Ni Mo Zr SrKr Courtesy of Dany Page Physical Review C 78, 025807 (2008) The faster the symmetry energy increases with density, the more exotic the composition of the outer crust.
Motivation: The symmetry energy and the neutron distribution in nuclei (quantal S: difference between the neutron and proton root mean square radius) quantal S in 208 Pb (fm) 0.3 0.2 NL3, G1 TM1, NL-SV HS NL-SH, PK1 G2 S271, NL3Λ v1 Z271, PKDD NL3Λ v2 FSUGold DD-ME1,D 3 C NL3Λ v3,tw DD-ME2 SkI2 Gs Rs GSkII SkSM* FKVW GSkI, SIV SkM*, D280, SSk SLy5 SkX, SLy4, SLy7 T6 SkP, D250, D300 D1S, SGII SIII, D1, D260 0.1 0 40 80 120 L (MeV) Covariant Covariant DD, PC Skyrme Gogny 1 2 3 L / J NL3Λ v2 FSUGold NL3Λ v3 NL3 HS TM1 NL-SH NL3Λ v1 SkI2 Gs Rs SkM* SkX, SLy4 T6 SkP SGII SIII (a) (b) (c) SkI5 4 8 12 J a sym (A) (MeV) Physical Review Letters 102, 122502 (2009) The faster the symmetry energy increases with density, the largest the size of the neutron distribution in nuclei. [Exp. from strongly interacting probes: 0.18 ± 0.03 fm (Physical Review C 86 015803 (2012))].
Motivation: The symmetry energy and parity violating electron scattering (A pv: relative difference between the elastic cross sections of right- and left-handed electrons) 10 7 A pv 7.4 7.2 7.0 6.8 MSk7 HFB-8 v090 SGII Sk-T6 SkX SLy4 SLy5 SkM* BCP MSkA MSL0 SIV DD-ME2 SkMP SkSM* DD-ME1 DD-PC1 RHF-PKO3 FSUGold Zenihiro [6] Ska Sk-Rs D1S D1N SkP HFB-17 Hoffmann [4] Klos [8] Sk-Gs PK1.s24 Linear Fit, r = 0.995 Mean Field From strong probes RHF-PKA1 SV SkI2 Sk-T4 NL3.s25 PC-PK1 G2 NL-SH SkI5 NL-SV2 PC-F1 TM1 NL-RA1 PK1 NL3 G1 NL3* NL2 NL1 0.1 0.15 0.2 0.25 0.3 r np (fm) Physical Review Letters 106, 252501 (2011) The largest the size of the neutron distribution in nuclei, the smaller the elastic electron parity violating asymmetry. [Exp. from ew probes: 0.302 ± 0.175 fm (Physical Review C 85, 032501 (2012))].
Motivation: The symmetry energy and the Pygmy Dipole Resonance (Pygmy: low-energy excited state appearing in the dipole response of N Z nuclei) Physical Review C 81, 041301 (2010) The faster the symmetry energy increases with density, the larger is the energy (E) and the larger is the probability (P) of exciting the Pygmy state larger the Energy Weighted Sum Rule (EWSR) E P.
Motivation: The symmetry energy and the Gamow-Teller Resonance (GT transitions change spin and isospin of the initial quantum state and favors n 0) Spin orbit splittings around the Fermi surface Proton and neutron single particle potentials V sym 1g 7/2 1g 9/2 90 Zr 1g 9/2 Gamow-Teller transitions determine: weak interaction rates essential in core-collapse dynamics of massive stars in neutron-rich environment, neutrino-induced nucleosynthesis studies of double-β decay useful in the calibration of detectors used to measure neutrinos protons neutrons Physical Review C 86 031306 (2012)
GIANT RESONANCES
Giant Resonances http://www.majimak.com/wordpress/ T = 1, S = 0 and L = 1 IsoVector GDR T = 1, S = 0 and L = 2 IsoVector GQR
Isovector Giant Resonances In isovector giant resonances neutrons and protons oscillate out of phase e.g. within a classical picture: e-m interacting probes basically excite protons, protons drag neutrons thanks to the nuclear strong interaction, when neutrons approach too much to protons, they are pushed out Isovector resonances will depend on oscillations of the density ρ iv ρ n ρ p S(ρ) will drive such oscillations The excitation energy (E x ) within a Harmonic Oscillator approach is expected to depend on the symmetry energy: 1 d ω = 2 U m δ k E dx 2 x 2 e S(ρ) δβ 2 where β = (ρ n ρ p )/(ρ n +ρ p )
Polarizability, Strength distribution and its moments The linear response or dynamic polarizability of a nuclear system excited from its g.s., 0, to an excited state, ν, due to the action of an external isovector oscillating field (dipolar/quadrupolar in our case) of the form (Fe iwt +F e iwt ): F JM = A i r J Y JM (ˆr)τ z (i) ( L = 1,2 Dipole,Quadrupole) is proportional to the static polarizability for small oscillations α = (8π/9)e 2 m 1 = (8π/9)e 2 ν ν F 0 2 /E where m 1 is the inverse energy weighted moment of the strength function, defined as, S(E) = ν ν F 0 2 δ(e E ν ) Isovector energy weighted sum rules (EWSR) are: m 1 = 2 NZ 2m A (1+κ D) m 1 = 2 50 2m 4π A r2 (1+κ Q ) equal to one half of the HF expectation value of [ˆF,[H, ˆF]] (Thouless theorem) and where κ is the dipole/quadrupole enhancement factor
Isovector Giant Dipole Resonance: Excitation energy S(ρ = 0.1 fm 1 ) is correlated with the value of the excitation energy (E 1 ) of the IVGDR in spherical nuclei experimental data on E 1 ( 208 Pb) leads to 23.3 MeV < S(ρ = 0.1 fm 3 ) < 24.9 MeV. Physical Review C 77 061304 (2008) Based on the 6.4 hydrodynamical model of 6.2 Ref. [Physics Reports 175 103-261 6 5.8 (1989)], it can be written in 5.6 good approximation within 5.4 EDF [Physical Review Letters 102, 5.2 122502 (2009)] calculations 5 that: E 1 = m1 m 1 4.8 4.6 6 2 m r 2 S(ρ = 0.1)(1+κ 11 12 13 14 15 16 D) E -1 [MeV] f(0.1)={s(0.1)(1+k}} 1/2 [MeV 1/2 ]
Isovector Giant Dipole Resonance: Dipole polarizability: macroscopic approach E the energy within a Liquid Drop based Model; F ext an external field (dipole operator) constrained calculation keeping N and Z fixed: δ{e(ρ,δ)+f ext (Λ,ρ,δ) λ n ρ 0 n (r)dr λ p ρ 0 p (r)dr } = 0 and find: ( m 1 A r2 1/2 48J 1+ 15 J 4 within the same [ model: ] r np = 3 5 J/Q 1+ 9 J 4 t e2 Z 70J t 3r0 2 (I I C ) Q A 1/3 using these expressions: Q A 1/3 ) + rnp surface α D A r2 12J [ 10 2 α D J (MeV fm 3 ) 1+ 5 2 10 9 8 7 6 α D J = 3.1(2) + 18.4(9) r np r = 0.95 EDF MF-Region 20.1(6) x 32(2) 5 0.1 0.15 0.2 0.25 r np (fm) ] 3 e r np+ 2 Z 5 70J rsurface np r 2 1/2 (I I C )
Isovector Giant Quadrupole Resonance: Within the Quantum Harmonic Oscillator approach E IV 2 2m V sym r 2 ( ω 0) 2 r 4 x = 2 ω 0 1+ 5 4 and EDF calculations, one can deduce V sym 8(S(ρ A ) S kin (ρ 0 )) S kin (ρ 0 ) ε F0 /3 and the expression one finds depend on well known quantities from the theory and observable GQR energies. S(ρ A ) = ε F 0 3 B(E2;IV) (10 3 fm 4 MeV -1 ) 6 4 2 0 0 10 20 30 E (MeV) { [ } A (E ) 2/3 IV 2 ( ) 8ε 2 x 2 E IS 2 x ]+1 F 0 S(ρ = 0.1 fm 3 for 208 Pb) = 23.3±0.6 MeV
Isovector Giant Quadrupole Resonance: 0.3 DDMEe Exp. r np (fm) 0.25 0.2 0.15 SAMi-J31 DDMEd DDMEc SAMi-J30 SAMi-J29 DDMEb DDMEa SAMi-J28 SAMi-J27 0.1 6 7 8 9 10 IV 2 IS 2 [(E x ) 2(Ex ) ] / J (MeV) { r np rnp surf = 2 r 2 1/2 3 (I I C) 1 ε F [ 3J 3 I C 7 I I C A2/3 (E IV 24ε F r np 0.14±0.03 fm x ) 2 2(E IS x ) 2 J ]}
Isovector Giant Quadrupole Resonance: Quadrupole polarizability [ m 1 J / <r 4 > 60 55 50 45 40 35 α Q A r4 16πJ EDFs 1+ 7 2 r np+ LNS SkP SAMi-J31 SKM* DDME1 KDE0 SLy4 SLy5 DDME2 DDMEf KDE SAMi-m9 DDMEb SAMi-J29 DDMEg SGIISAMi-m8 SAMi SAMi-m7 SIII DDMEa e 2 Z 70J rsurface np 3 5 r 2 1/2 (I I C ) SkI3 DDMEc Sk255 DDMEd SAMi-J25 r=0.92 30 0.1 0.15 0.2 r np (fm) 0.25 ] NL3 DDMEe m 1 J / <r 4 >= 21.8(1.6) + 109(9) r np
CONCLUSIONS
Conclusions: For heavy nuclei: the macroscopic model presented here contains relevant physics for the description of the different observables. the excitation energy of the IVGDR depend on S(0.1 fm 3 ). within the macroscopic approach S(0.1 fm 3 ) (not observable) or the r np (observable) can be determined by empirical data on the excitation energies of the IS- and IV-GQRs. the dipole and quadrupole polarizabilities depend in an analogous way on the r np. within the macroscopic model presented here, one can build the model independent quantity: 3 5 α D J r 2 4π 7 α Q J r 4 = A 70
Co-workers: G. Colò, P. F. Bortignon and M. Brenna (University of Milan) M. Centelles and X. Viñas (University of Barcelona) N. Paar and D. Vretenar (University of Zagreb, Croatia) B. K. Agrawal (SINP, Kolkata, India) L. Cao (IMP-CAS, Lanzhou, P.R. China)