Nuclear Matter Incompressibility and Giant Monopole Resonances

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1 Nuclear Matter Incompressibility and Giant Monopole Resonances C.A. Bertulani Department of Physics and Astronomy Texas A&M University-Commerce Collaborator: Paolo Avogadro 27th Texas Symposium on Relativistic Astrophysics - Dallas, TX - Dec 8-13,

2 Iron core in core-collapse supernovae The iron core grows and becomes dynamically unstable as it approaches the critical mass. The main reactions are 56 Fe 13 4 He + 4n p + e n + ν e Needs energy (124 MeV) Neutrino escapes Pressure decreases and the core collapses The Equation of State (EoS) has to cover densities of g/cm 3 Nuclei are bound into nuclei until they merge (ρ 0 ~ g/cm 3 ) Pressure is essentially dominated by e - (γ = lnp/ lnρ = 4/3) For ρ > ρ 0, nuclear matter becomes very hard (γ ~ 2) 2

3 EOS Below nuclear density: - Matter is described by nuclei surrounded by a gas of nucleons and alpha particles - Electrons are uniformly distributed in space, inside and outside nuclei. - In equilibrium µ e = µ n µ p - At high densities nuclear matter fills the space uniformly - Techniques for computing EOS: liquid drop, Thomas-Fermi, Hartree-Fock - EOS is obtained by minimizing the free energy of the system for each entropy and Y e 3

4 EOS Above nuclear density: - Maximum density occurs when the inner central core reaches ρ ~ 5ρ 0 - the compressibility has to be enough to allow the existence of neutron stars with M ~ 1.4 M or larger. # K = 9ρ 2 d2 $ E(ρ) / ρ% & dρ 2 ρ fm -3 E" # ρ $ % = E " # ρ 0 $ % K ( * ) ρ ρ 0 ρ 0 + -, 2-16 MeV 4

5 EOS Experimental Observations - Analysis of Giant Isoscalar Resonances ISGMR (T=0, L=0) - Relativistic heavy ion collisions - Neutron masses Giant Resonance: Coherent vibration of nucleons in a nucleus - Resonances related to incompressibility: ISGMR, ISGDR, ISGQR ISGDR (T=0, L=1) ISGQR (T=0, L=2) E ISGMR K A c 1 m r 2 K A = K 1+ ca 1/3 ( ) N Z + K τ & $ % A ' ) ( 2 + K Coul Z 2 A 4/3 - K Coul is basically model independent - Measurements over several isotopes should give Kτ - Kτ critical to understand neutron stars 5

6 EOS Theoretical Methods - Build an energy functional E[ρ] using an mean field calculation Each such a functional characterizes a K - Get excitations such as the ISGMR from a self-consistent QRPA calculation For the nucleon-nucleon interaction V(r i,r j ) = V ij NN + V ij Coul V ij Coul = e2 4 A τ 2 ij + τ ij, i,j=1 r i r j τ = τ + τ ij i j V ijnn = t 0 (1+ x 0 P σ ij )δ(r i r j ) t (1+ x P σ )[ k! ij ij δ(r i r j ) + δ(r i r j ) k! 2 ij ]+ t 2 (1+ x 2 P σ ij ) k! ij δ(r i r jj ) k! ij t (1+ x P σ )ρ α 3 3 ij iw 0! k ij δ(r i r j )(! σ i +! σ j )! k ij, t, x,, W & ( ' r i + r j 2 ) + δ(r r ) + i j * i i α 0 are 10 Skyrme parameters E[ρ] = Φ T + V ij Coul + V ij NN Φ 6

7 + pairing HF + BCS HFB Δ i = 1 2 j G ij Δ j ( ε j λ) Δ j h HF Δ λ h Δ HF u + λ v k k = E k u v k k v NN eff = Skyrme + pairing force, % V = V 0. 1 η' ρ r -. & ρ 0 ( ) 30, "volume"pairing 5 η = 41, "surface"pairing 5 6 1/2, "mixed"pairing ( * ) α / 1 01 δ ( r 1 r 2 ), ρ 0 = 0.16 fm, α =1 7

8 EOS + Pairing Protons and neutrons tend to pair up, much like Cooper pairs of electrons in superconductors. Pairing is important in nuclei and neutron stars, but a clear understanding of the microscopic foundation of the pairing functional is still lacking. ( ) = v 1 η ρ 0 - & % ρ 0 v r,r' Volume pairing + -, $ ' ) ( γ. 0 0 δ r r' / ( ) Skyrme K SLy5 230 SkM* 216 Skxs Surface pairing Blaizot Phys. Rep. 64, 171 (1980) Mixed pairing Shlomo,Youngblood PRC 47, 529 (1993) 8

9 Pairing Measure three-point Δ (3) = # $ ( ) N B(N 1) + B(N +1) 2B(N) % & four-point Δ (4) = # $ ( ) N 3B(N 1) 3B(N) B(N 2) + B(N +1) % & or higher? From experiment: Δ (3) larger for (-1) N = +1 Δ (3) smaller for (-1) N = -1 Δ (4) reflects average of Δ (3) (N) and Δ (3) (N-1) (Δ (4) no additional information) 9

10 Can Microscopic Models do better than LDM? rms for binding energies rms for separation energies Δ o (3) = 12 A 1/ 3 MeV rms = 0.3 MeV experiment

11 Blocking Procedure for Odd Nucleons E F V pairing (k) Λ E F for one orbit and its time-conjugate v2 ½, u 0 partner Λ ~ 50 MeV 11

12 Pairing Improves Nuclear Properties Separation energies staggering Sn 12

13 Pairing vs Level Properties Δ (3) = 1 2 N ( 1) [ B( N 1) + B( N + 1) 2B( N) ] Fermi energy (λ= B/ N) s.p. level density (g(e)= dn/ e) 2 B N λ = N N (3) 1 2( 1) Δ = 2 g ( λ) N degenerate shell λ λ does not vary with N! Δ(3) = 0 e n+1 e n λ valence shell full N=2n measure of gap in single-particle spectrum (-1) N = +1, dλ/dn e n+1 e n! Δ(3) = (e n+1 e n )/2 Jahn-Teller mechanism: spherical symmetry spontaneously broken (2j+1)! double-degenerate orbits Δ (3) alternates for (-1) N = + and - 13

14 Pairing vs Level Properties Macroscopic-microscopic model: ( N Z ) B = E sp ~ E sp + E macro 2 (3) 23 E macro =! + ai, ai = 23 MeV Δ = MeV A A E sp contribution : g( λ) = 3a π, a A 2 8 MeV Δ (3) 1 g λ, E sp = A k= 1 e ( ) 25 A MeV k Δ (3) ( N) 1 2 δe 2 Δ (3) even + LDM corrections ( ) Δ (3) ( odd) [ ] e n +1 e n Satula, Dobaczewski, Nazarewicz, PRL 81, 3599 (1998)

15 The unclear Nuclear Pairing UNEDF Collaboration Mass tables for 2,400 nuclei have been analyzed using different forms and methodologies for the pairing mechanism New functional forms for the pairing interaction have been proposed experiment UNEDF theory Δ (3) = 1 2 N ( 1) [ B( N 1) + B( N + 1) 2B( N) ] Bertsch, Bertulani, Nazarewicz, Schunck, Stoitsov, PRC 79, (2009) 15

16 QRPA: The Role of the Rearrangement Term Avogadro, Bertulani, PRC 88, (2013) Fully self consistent EWSR = 99.2% h = δe kin δρ + δe skyrme δρ + δe pair δρ + δe Coul δρ Without Rearrangement in EWSR =116% δh rearr δρ = δ δρ # % $ δe pair δρ & ( ' 0 if E pair depends on density Calculations without rearrangements tend to return higher centroids respect to the fully selfconsistent case. 16

17 Dependence on Functional Skxs20 very well Skm* well! SLy5 bad results S(E) = 0 F 0 j 2 δ( E E ) 0 j A i=1 F 0 = r i 2 m k = 0 E k S(E) de 17

18 Isovector pairing v r,r' + - ( ) = v 1 η ρ 0 - & % ρ 0, $ ' ) ( γ. 0 0 δ r r' / ( ), % MSH (r,r') = v. ρ 0 1 ( 1 δ. )η s ' - & v pair, MSH (r,r') = v. 0 1 η+ η 1 τ 3 δ. - v pair ρ 0 α ( s % * δη n ' ) & Margueron, Sagawa, Hagino, PRC 76, (2007) ρ ρ 0 ( ) ρ ρ η 2 ( δ ρ ' ρ 0 0 & ( * ) ) + * α n 2 / 1 1 δ(r,r') 0 / 1 1 δ(r,r') 0 ρ = ρ n +ρ p δ = ρ n ρ p ρ Yamagami, Shimizu, Nakatsukasa, PRC 80, (2009) 18

19 Isovector pairing Good globabl fits to pairing gaps Bertulani, Liu,, Sagawa, PRC 85, (2012) 19

20 Isovector pairing Reasonable Nuclear Radii MSH IS+IV pairing Bertulani, Liu,, Sagawa, PRC 85, (2012) 20

21 Radii and Skins PWBA 12 C Measuring R n in 208 Pb constrains the pressure of neutron matter at ~ 2/3ρ 0 = 0.1 fm -3. C. Horowitz Δr np Neutron stars ( θ ) dσ dω = σ M F Z 2 ch arge q ( ) 2 Hofstadter,

22 Isovector pairing Reasonable Nuclear Skins MSH IS+IV pairing Radii from spin-dipole resonances Krasznahorkay et al., PRL 82, 3216 (1999) & Antiprotonic atoms Trzcinskaet al., PRL 87, (2001) Bertulani, Liu,, Sagawa, PRC 85, (2012) 22

23 Isovector pairing Improves Centroids of ISGMR 23

24 Isovector pairing ISGMR Comparison to Recent Data Avogadro, Bertulani, PRC 88, (2013) 24

25 Conclusions EOS & Pairing Conundrum Nuclear pairing evidently improves masses, separation energies and staggering effects in microscopic approaches: HF + BCS or HFB. Inclusion of pairing complicates determination of best Skyrme models. E.g., 20% of the nuclei well explained with the SLy5 interaction, and 10% with the SkM* interaction. Isovector pairing improves nuclear properties ß at the cost of additional parameters Detailed studies using HFB + QRPA + isovector pairing with comparison with newest data on ISGMR (RCNP and TAMU) à ISGMR is better reproduced with the soft interaction Skxs20 (K 202 MeV), in contrast with the generally accepted value for K 230 MeV. 25