Spin and Isospin excitations in Nuclei Some general comments on EDFs Motivation and present situation: example GTR Propose new fitting protocols

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1 Towards the improvement of spin-isospin properties in nuclear energy density functionals Xavier Roca-Maza Dipartimento di Fisica, Università degli Studi di Milano and INFN, via Celoria 16, I-2133 Milano, Italy Selected Topics in Nuclear and Atomic Physics Fiera di Primiero (TN), September 3 - October 4, 215 1

2 Table of contents: Spin and Isospin excitations in Nuclei Some general comments on EDFs Motivation and present situation: example GTR Propose new fitting protocols One based on empirical indications One based on the good results found in RHF 2

3 Spin and Isospin excitations in Nuclei Nucleons are fermions characterized by their spin and isospin Nucleons with spin (isospin) may be change their state in phase: spin-scalar S= modes (isospin-scalar T= modes); or out of phase: spin-vector S=1 modes (isospin-vector T=1 modes) They can be excited by strong probes (charge-exhange reactions) and/or weak interaction (axial-vector current couples to the spin and induces β decay processes) One of the most important nuclear excitation modes is the Gamow Teller Resonance which is a pure spin-isospin mode (i.e., from a theoretical picture, it is excited by an operator Ô στ) Spin-isospin modes of excitation (such as the GTR) give direct information on the spin-isospin channel of the effective interaction 3

4 Example: β decay transition Courtesy of Y. Fujita; taken from his lectures colo/lectures/lectures.html 4

5 Example: Gamow Teller transition Courtesy of Y. Fujita; taken from his lectures colo/lectures/lectures.html 5

6 Some comments on the nuclear many-body problem: 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 EoS and (only very recently) few groups in the world are able to perform extensive calculations for light and medium mass nuclei Based on effective interactions, Nuclear Energy Density Functionals are successful in the description of masses, nuclear sizes, deformations, Giant Resonances,... 6

7 Approximate realization of an exact Nuclear Energy Density Functionals: Kohn-Sham iterative scheme (static approximation) Determine a good E[ρ] Initial guess ρ Calculate potential V eff from ρ 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]} = Giant Resonances well described within the small amplitude limit (known as RPA approach) 7

8 Nuclear Energy Density Functionals: Main types of successful EDFs for the description of masses, deformations, nuclear distributions, Giant Resonances,... Relativistic H o HF 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 Ψ ˆQγ µ ΨA (γ)µ Non-relativistic HF models, based on Hamiltonians where effective interactions are proposed and tested: VNucl eff = Vlong range attractive + V short range repulsive + V SO Fitted parameters contain (important) correlations beyond the mean-field Nuclear energy functionals are phenomenological not directly connected to any NN (or NNN) interaction 8

9 Drawbacks on current EDFs??? On the one side, we expect that the H(F)+RPA method based on nuclear effective interactions of the Skyrme, Gogny or Relativistic types enables an effective description of the nuclear many-body problem and can be understood as an approximate realization of an EDF reasonable description of g.s. energy and density of the system On the other side, there are still some open problems at that level on the... accurate determination of spin-isospin properties 9

10 Motivation: Gamow Teller Resonance The E x is not properly described in H(F)+RPA (Neither the strength Beyond 1p-1h RPA effects) SGII a earliest attempt to give a quantitative description of the GTR SkO b accurate in ground state finite nuclear properties and improves the GTR PKO1 c relativistic HF, reasonable GTR still not perfect. Relativistic H d : residual interaction modified ad-hoc R GT [MeV 1 ] R GT [MeV 1 ] R GT [MeV 1 ] Zr Exp. x 3 PRC 55, 299 (1997) PRC 64, 6732 (21) 28 Pb RHF-PKO1 Exp. PRC 85, 6466 (212) SkO SGII 48 Ca Exp. x 3 PRL 13, 1253 (29) SLy E x [MeV] a N. Giai and H. Sagawa, Phys. Lett. B 16, 379 (1981), b P.-G. Reinhard et al., Phys. Rev. C 6, (1999), c H. Liang, N. Van Giai, and J. Meng, Phys. Rev. Lett. 11, (28), d N. Paar, T. Nikšić, D. Vretenar, and P. Ring, Phys. Rev. C 69,

11 Motivation: Gamow Teller Resonance Exchange (Fock) effects on GTR in relativistic models Effect of Migdal term fitted to 28 Pb in RH 11

12 Motivation: Gamow Teller Resonance Quenching of the strength Experimentally, the GTR exhausts 6 7% of the Ikeda sum rule: [R GT (E) R GT +(E)]dE = 3(N Z) To explain the problem, two possibilities that go beyond (1p 1h) RPA correlations have been drawn: the effects of the second-order configuration mixing: 2p-2h correlations within the quark model, a n(p) can become a p(n) or a + ( ++ ) under the action of the GT operator and since there is no Pauli blocking for h excitations it may contribute to the GTR. The experimental analysis of 9 Zr quenching ( 2/3) has to be mainly attributed to 2p-2h coupling and not to isobar effects much smaller [T. Wakasa et. al., Phys. Rev. C 55, 299 (1997)]. E x GTR in nuclei mainly in the region of several tens of MeV and the h states are hundreds of MeV above the GT hard to excite the in the nuclear medium. 12

13 Motivation: which gs properties are important for describing the E GTR x? The study a of the GTR and the spin-isospin Landau-Migdal parameter G using several Skyrme sets, concluded that G is not the only important quantity in determining the excitation energy of the GTR spin-orbit splittings also influences the GTR Empirical indications b suggest that G > G > Not a very common feature within available Skyrme forces c a M. Bender, J. Dobaczewski, J. Engel, and W. Nazarewicz, Phys. Rev. C 65, (22); b T. Wakasa, M. Ichimura, and H. Sakai, Phys. Rev. C 72, 6733 (25); T. Suzuki and H. Sakai, Phys. Lett. B 455, 25 (1999), c Li-Gang Cao, G. Colo, and H. Sagawa, Phys. Rev. C 81, 4432 (21) 13

14 Why spin-orbit splittings are important in E GTR x? Schematic picture of single-particle transitions involved in the Gamow Teller Resonance of 9 Zr. Transitions excited by στ operator. E 1 x ɛ π1g7/2 ɛ ν1g9/2 + ɛ 1 ph E 2 x ɛ π1g9/2 ɛ ν1g9/2 + ɛ 2 ph E x ɛ π1g + ɛ ph F. Osterfeld, Rev. Mod. Phys. 64, 491 (1992) 14

15 We propose a new fitting protocol that help improving spin-isospin properties: example with a Skyrme interaction 15

16 (Standard) Skyrme Model [... have a quick look!] Includes central tensor terms (J 2 terms) due to the coupling of tensor and spin and gradients terms and two spin-orbit parameters (same as SkO and some SkI forces) H = K + H + H 3 + H eff + H fin + H SO + H sg + H Coul K H H 3 = h 2 τ/2m = (1/4)t [(2 + x )ρ 2 (2x + 1)(ρ 2 n + ρ 2 p)] (CENTRAL) = (1/24)t 3 ρ α [(2 + x 3 )ρ 2 (2x 3 + 1)(ρ 2 n + ρ 2 p)] (DENSITY DEP.) H eff = (1/8)[t 1 (2 + x 1 ) + t 2 (2 + x 2 )]τρ + (1/8)[t 2 (2x 2 + 1) t 1 (2x 1 + 1)](τ n ρ n + τ p ρ p ) (EFF. MASS) H fin = (1/32)[3t 1 (2 + x 1 ) t 2 (2 + x 2 )]( ρ) 2 (1/32)[3t 1 (2x 1 + 1) + t 2 (2x 2 + 1)][( ρ n ) 2 + ( ρ p ) 2 ] (FIN RANGE) H SO = (1/2)W J ρ + (1/2)W (J n ρ n + J p ρ p ) H sg = (1/16)(t 1 x 1 + t 2 x 2 )J 2 + (1/16)(t 1 t 2 )(J n 2 + J p 2 ) 16

17 Fitting Protocol: Inspired on SLy5 χ 2 definition: χ 2 = 1 Ndata N data i (O theo. i O data i ) 2 ( O data i ) 2 Landau-Migdal parameters in infinite nuclear matter G and G fixed to.15 and.35, respectively, at ρ. Table: Data and pseudo-data O i, adopted errors for the fit O i and selected finite nuclei and EoS. O i O i B 1. MeV 4,48 Ca, 9 Zr, 132 Sn and 28 Pb r c.1 fm 4,48 Ca, 9 Zr and 28 Pb E SO.4 O i π1g in 9 Zr and π2f in 28 Pb e n (ρ).2 O i R. B. Wiringa et al., PRC 38, 11 (1988) 17

18 Skyrme Aizu Milano interaction: SAMi Parameter set and nuclear matter properties: Table: SAMi parameter set and saturation properties with the estimated standard deviations inside parenthesis value(σ) value(σ) t (75) MeV fm 3 ρ.159(1) fm 3 t (1.4) MeV fm 5 e 15.93(9) MeV t (1.) MeV fm 5 m IS.6752(3) t (7.6) MeV fm 3+3α m IV.664(13) x.32(16) J 28(1) MeV x 1.532(7) L 44(7) MeV x 2.14(15) K 245(1) MeV x 3.688(3) G.15 (fixed) W 137(11) G.35 (fixed) W 42(22) α.25614(37) 18

19 SAMi: spin and spin-isospin instabilities Imposing that spin and isospin d.o.f. at the Fermi surface are stable under generalized deformations [Bäckman et al., Nucl. Phys. A 321, 1 (1979)] 1 + G > 1 + G > Sold lines: SLy5 Dashed lines: SAMi ~2ρ 2-3ρ ~4ρ ~5ρ G G ρ (fm 3 ) Unstable Stable max. densities in NS where only NM present 19

20 Results Equation of State: SAMi vs ab initio calculations e(ρ) ( MeV ) Variational-Wiringa 1988 BHF--Vidana 212 BHF--Li 28 BHF--Baldo 24 SLy5 SAMi ρ =.152 fm 3 E / A = MeV K = MeV J = MeV L = MeV K τ = MeV ρ ( fm 3 ) Figure: Neutron and symmetric matter EoS as predicted by the HF SAMi (dashed line) and SLy5 (solid line) interactions and by the benchmark microscopic calculations of R. B. Wiringa et al., PRC 38, 11 (1988) (circles). State-of-the-art BHF calculations are shown by diamonds I. Vidaña, private communication, triangles Z. H. Li et al., Phys. Rev. C 77, (28) and squares M. Baldo et al., Nucl. Phys. A 736, 241 (24). 2

21 Results Giant Monopole and Dipole Resonances in 28 Pb 1 2 R ISGMR [fm 4 MeV 1 ] SLy5 SAMi Exp. 28 Pb SLy5 SAMi Exp. 28 Pb R IVGDR [fm 2 MeV 1 ] E x [MeV] E x [MeV] Figure: Strength function at the relevant excitation energies in 28 Pb as predicted by SLy5 and the SAMi interaction for GMR and GDR. A Lorentzian smearing parameter equal to 1 MeV is used. Experimental data for the centroid energies are also shown: E c(gmr) = ±.11 MeV [D. H. Youngblood, et al., Phys. Rev. Lett. 82, 691 (1999)] and E c(gdr) = ±.1 MeV [N. Ryezayeva et al., Phys. Rev. Lett. 89, (22)]. 21

22 Results Gamow Teller Resonance in 48 Ca, 9 Zr and 28 Pb A i=1 σ(i)τ ±(i) R GT [MeV 1 ] SkO SGII SAMi Exp. x 3 48 Ca PRL 13, 1253 (29) Figure:Gamow Teller strength distributions in 48 Ca (upper panel), 9 Zr (middle panel) and 28 Pb (lower panel) as measured in the experiment [T. Wakasa et al., R GT [MeV 1 ] Zr Exp. x 3 PRC 55, 299 (1997) PRC 64, 6732 (21) SLy5 Phys. Rev. C 55, 299 (1997), K. Yako et al., Phys. Rev. Lett. 13, 1253 (29), A. Krasznaborkay et al., Phys. Rev. C 64, 6732 (21), H. Akimune et al., Phys. Rev. C 52, 64 (1995) and T. Wakasa et al., Phys. Rev. C 85, 6466 (212)] and predicted by SLy5, SkO, SGII and SAMi forces. R GT [MeV 1 ] Pb RHF-PKO1 Exp. PRC 85, 6466 (212) E x [MeV] 22

23 Results Spin Dipole Resonances in 9 Zr and 28 Pb Operator: Ai=1 M τ ±(i)r L i [Y L(ˆr i ) σ(i)] JM Sum Rule: [RSD (E) R SD + (E)]dE = 9 4π (N r2 n Z r 2 p ) R SD - [fm 2 MeV 1 ] R SD SAMi 9 Zr J π = J π = 1 J π = 2 Total Exp E x [MeV] 5 6 Experiment: K. Yako et al., Phys. Rev. C 74, 5133(R) (26). A Lorentzian smearing parameter 2 MeV is used. R SD - [fm 2 MeV 1 ] SAMi Total 28 Pb Exp J π = J π = 1 J π = E x [MeV] Experiment: T. Wakasa et al., Phys. Rev. C 85, 6466 (212). A Lorentzian smearing parameter 2 MeV is used. 23

24 Advantages and disadvantages of a RHF theory 24

25 Covariant density functional theory RHF theory achieved quantitative description of B(N, Z) and r ch (PLB 64, 15 (26); PRC 76, (27); EPL 82, 121 (28); PRC 81, 2438 (21)) effective mass splitting in ANM can be described naturally (PLB 64, 15 (26)) nuclear spin-isospin resonances can be described in a fully self-consistent way (PRL 11, (28); PRC 79, (29); PRC 85, 6432 (212)) improvement on the descriptions of nuclear shell structures and their evolutions (PRC 76, (27); EPL 82, 121 (28); PLB 68, 428 (29)) However... * RHF includes non-local potentials v HF (r, r ) * RHF is much more complicated than RH theory. * non negligible computational cost when improving the calculations and/or going beyond the mean-field 25

26 To construct RH functionals from RHF scheme Therefore, it is desirable to find a covariant density functional based on only local potentials, yet keeping the merits of the exchange terms Possible solution: construct RH functionals from RHF scheme Fierz transformation allow to map Fock terms into local Hartree terms (for contact interactions) but, masses of mesons are heavy zero-range approximation is reasonable in nuclei (Skyrme, Relativistic point-coupling approaches,... ) 26

27 Fierz transformation: from α HF to α H α H S = αhf S α H ts = 1 8 αhf S 4 8 αhf V 4 8 αhf V 12 8 αhf tv αhf tv α H V = 1 8 αhf S αhf V αhf tv α H tv = 1 8 αhf S α H T = 1 16 αhf S α H tt = 1 16 αhf S α H PS = 1 8 αhf S α H tps = 1 8 αhf S α H PV = αhf S αhf V αhf V αhf V αhf V αhf tv αhf tv 4 8 αhf tv αhf tv α H tpv = αhf S αhf V 2 8 αhf tv 27

28 Check how good is the mapping in a practical case start with an available RHF parametrization: PKO2 (which is based on a finte-range interaction) perform the zero-range reduction (if needed) perform the Fierz transformation Compare observables sensitive to the Fock terms between the original model and the localized model, such as: Gamow-Teller Effective mass splitting Resonance between neutrons and protons R - ( 1 / M e V ) R - ( 1 / M e V ) e x p t. 4 8 C a G T R u n p e r. P K O 2 P K O 2 - H E ( M e V ) 2 8 P b G T R u n p e r. P K O 2 P K O 2 - H e x p t E ( M e V ) R - ( 1 / M e V ) Z r G T R u n p e r. P K O 2 P K O 2 - H e x p t E ( M e V ) M * D /M ( M * D, p - M * D, n ) / M.1 D B H F. 5 P K O 2 -H P K O ( f m -3 ) This opens the possibility for the development of new nuclear local covariant density functionals 28

29 Preliminary test to build a localized model from RHF 29

30 Test to build a localized model: work in progress Project: Build a local CDF including all terms in the Lagarngian allowed by the symmetries (S, V, ts, tv, T, tt, PS, tps, PV, tpv terms) consider as free parameters the ones corresponding to the S, V and tv channels the rest of the channels will be determined by the Fierz transformations, within the same Hartee scheme, and in a fully self-consistent manner. 4 2 (B exp. B teo. ) / B exp. ) / R ch exp. R ch teo. ( R ch exp rms = 3 MeV rms =.2 fm A e(ρ,δ) (MeV) ρ (fm ) LRHF-DD-PC BHF (Marcello Baldo) ρ (fm 3 ) DBHF (C. Fuchs) Fierz T. α ts = α ts (α S, α V, α tv ) m p * - m n * (MeV) 3

31 Conclusions: We have remainded some of the problems in the spin-isospin channels in Skyrme and RH models using as an example the GTR We have briefly presented the benefits of the new proposed fitting protocol that cure part of the previous problems test the new protocol and show some results when applied with a Skyrme interaction the benefits of using a RHF model proposed a new method to determine a RH model keeping the benefits of a RHF 31

32 Thank you! Work in collaboration with: G. Colò, H. Sagawa, Li-Gang Cao, H. Liang, J. Meng, P. Ring and P. Zhao 32