An extended liquid drop approach

Transcription

1 An extended liquid drop approach Symmetry energy, charge radii and neutron skins Lex Dieperink 1 Piet van Isacker 2 1 Kernfysisch Versneller Instituut University of Groningen 2 GANIL, Caen, France ECT, Aug 2009 Dieperink (KVI) LDM / 25

2 Motivation 1 Provide nuclear input for test of standard model in apv charge and neutron radii 2 Determine the symmetry energy in nuclear matter Dieperink (KVI) LDM / 25

3 Nuclear structure and Atomic parity violation H = G 2 2 d 3 r[ Nρ n (r) + Z(1 4 sin 2 θ W )ρ p (r)]ψ γ 5 ψ < ψ s γ 5 ψ p >= ic sp Z N [us 1 (r)up 2 (r) us 2 (r)up 1 (r)]/r 2 factorize m.e.: H if G F 2 C 2 if N Q W γ = 1 (αz) 2 electr overlap for point nucleus N = ψ s (0)γ 5 ψ p (0) Rp 2γ 2 Nucleus: Q w = Nq n + Zq p (1 4 sin 2 θ W ) + Q new q i : convolution of electr. wf with finite nucleus; expand in (Zα) n q p = (Zα) q n = (Zα)2 (1 + 5( Rn R p ) 2 ) +... (+ corrections to sharp radius) why large Z? since H pv Z 3 enhancement (+rel. corr) aim: for Ra (Z=88), for 0.1% measurement (to go after Q new ) need: R p 1%, (R n R p )/R p 25% (at present no direct exp info) Let us see whether we can meet these requirements Dieperink (KVI) LDM / 25

4 present status of skin from Brown et al PRC76,2009 based upon spherical Skyrme EDF Neutron Skins for Atomic PNC R depends on Skxs(15..25) idea: PREX@Jlab will fix that Questions Irregularities due to high spin intruder orbitals (i13/2) Effect of deformation? Are slopes really constant? Does not fit exp R c very well (except 208 Pb ) Skxs15 Neutron Skins for Atomic PNC A. Brown claims n skins of APNC isotopes track Pb. PREX constrains them. Skxs25 Dieperink (KVI) LDM / 25

5 content Extended Liquid Drop Model see Pawel Danielewicz talk surface symmetry energy LDM adapted shell corrections Applications symmetry energy for A charge radii and isotope shifts neutron skin prediction atomic parity violation: nuclear corrections to weak charge measurements Dieperink (KVI) LDM / 25

6 LDM Conventional Bethe-von Weizsäcker (liquid drop) formula E A = a B A + a surf A 2/3 + S vol (N Z) 2 /A + a C Z 2 A 1/3 + E pair Incomplete form of symmetry energy Need to introduce volume and surface symmetry energy 11.8 Coulomb need to be refined (R c (N, Z) r 0 A 1/3 ) shell corrections need to be added AME03-LDM Dieperink (KVI) LDM / 25

7 Extended Liquid Drop Model Improved BW: E sym = E(vol, surf ) Decompose asymmetry N Z = N s Z s + N v Z v Evol A (N = a B A + S v Z v ) 2 vol A Esurf A = Esurf 0 + S surf (N s Z s ) 2 /A 2/3 minimize under fixed N Z : N s Z s N Z = 1 y S 1+y 1 A 1/3 vol /S surf E A = a B A + a surf A 2/3 + S v 1+yA 1/3 (N Z) 2 /A +... surface+ volume Dieperink (KVI) LDM / 25

8 Myers-Swiatecki, Ann Phys 55 (1969)395 Danielewicz, NPA 727(2003)233; Steiner et al, Phys. Rep. 411,325 differ in the choice of condition N s + Z s = 0, or Z s = 0 Dieperink (KVI) LDM / 25 Extended Liquid Drop Model Improved BW: E sym = E(vol, surf ) Decompose asymmetry N Z = N s Z s + N v Z v surface+ volume Evol A (N = a B A + S v Z v ) 2 vol A Esurf A = Esurf 0 + S surf (N s Z s ) 2 /A 2/3 minimize under fixed N Z : N s Z s N Z = 1 y S 1+y 1 A 1/3 vol /S surf E A = a B A + a surf A 2/3 S + v (N Z) 2 /A yA 1/3 LDM yields also relation between skin and S s, S v (N s Z s A) N N v = ( Rn R 0 ) 3 Rn R 0 R 0 Ns 3N, R p R 0 R 0 = A(Ns Zs) 6NZ A N Z ac 12 ZA2/3 /S v 6NZ 1+A 1/3 /y R n R p R Zs 3Z, Coulomb term: for N = Z R p > R n

9 shell corrections several methods have been proposed, Strutinsky, Koura... here we adapt microscopic mass model of Duflo-Zuker rms dev 500 kev, (14-28 parameters) idea: count number of valence particles (n v, z z ) with respect to closed shells E sh = E(n v, z v ) Example: monopole force in single j-shell with degeneracy D j = 2j + 1, seniority v = 0 E pair (n v ) = g D n v (D n v + 2) g n v. n v + 2g D n v n v D n v =number of holes absorb in core Dieperink (KVI) LDM / 25

10 shell corrections(2) E shell (N, Z) = a 1 S 2 + a 2 (S 2 ) 2 + a 3 S 3 + a np S np with n v D n n v S 2 = n v n v D n + z v z v D z, n v <<D n v + z v S 3 = n v n v (n v n v ) D n + z v z v (z v z v ) D z, S np = n v n v z v z v D n D z, Magic numbers: 6, 14, 28, 50, 82, 126, 184 similar to terms in microscopic mass formula of Duflo and Zuker they refer to S 3 as monopole drift (changes sign midshell) Dieperink (KVI) LDM / 25

11 examples 10.1 AME03-LDM 6.61 AME03-LDMsc without shell corr with shell corr rms deviation 2.4 MeV MeV (mostly due to light A) Dieperink (KVI) LDM / 25

12 Illustration: neutron separation energies Often convenient to consider relative quantities, e.g. neutron separation energies S 2n = E(N, Z) E(N 2, Z) vs Z N= relevance: extrapolation, Z Dieperink (KVI) LDM / 25

13 Determination of Symmetry energy E LDM a B A + a surf A 2/3 + (N Z)2 A aim: determine S v, y = Sv S s 1 2 ( de S v Z(Z 1) + a 1+yA 1/3 C + E A 1/3 pair + E shell in isolation of other LDM parameters dn de dz ) = N Z A S Z A + a c + δe A 1/3 shell µ a = 1 2 (µ n µ p ) = 1 2 [B(N 1, Z) B(N, Z 1)] (isovector chem. pot.) invert (N Z) S A = A N Z (µ a δe C δe shell ) independent of a B, a surf subtlety: SU(2): (N Z) 2 = 4Tz 2 = 4T (T + 1) suggests N Z N Z+1 partially absorbs Wigner energy Symmetrize removal and addition µ a = 4 1 [B(N 1, Z) B(N, Z 1) + B(N, Z + 1) B(N + 1, Z)] Dieperink (KVI) LDM / 25

14 Symmetry energy from exp n-p sep energies S A = A N Z (µ a + δe C ) no shell corr N Note: the regularity of shell effects (up/down sloping) Dieperink (KVI) LDM / 25

15 Symmetry energy from exp n-p sep energies S A = A N Z (µ a + δe C ) S A = A N Z (µ a + δe C + δe shell ) no shell corr N shell corr Note: the regularity of shell effects (up/down sloping) particle-particle: E sh n v + z v [(n v + 1) + (z v 1)] = 0 particle-hole (or h-p) : E sh ±(n v z v [(n z + 1) (z v 1)]) = ±2 problem at N = 40, Z = 38 (midshell in 28-50) Dieperink (KVI) LDM / 25

16 Symmetry energy for nuclear matter S A (N, Z) = S v 1+yA 1/3, y = S v /S s fit in isolation of other parameters Leads to 2-par fit: S 1 A = 1+yA 1/3 S v extrapolation to A fit values S v = 32.5 ± 1.8, y = 2.95 ±.4 (sh.c) S v = 28.5 MeV (no sh corr) compare with Danielewicz using IAS and sh corr from Koura S v = 32.8 y=2.8 with sh corr (Koura) S v = 29 ± 2, y = 2.4 ± 0.4 (without) 1/S v Note correlation between S v and y A 1/3 Dieperink (KVI) LDM / 25

17 Symmetry energy S(ρ) Can be converted to nuclear matter S(ρ) : S(ρ 0 ) S v and S s related to some S(ρ < ρ 0 ) Using Thomas-Fermi: E a = µ2 a 4 dr ρ(r) S S v /S s = 3 r 0 dr ρ(r) ρ 0 ( Sv S(ρ) 1) simplest case: take S(ρ) = S v.(ρ/ρ 0 ) γ note γ = 1 S(ρ) = constant S s = we find γ 0.7 ± 0.1 Danielewicz γ = 0.65 ± 0.1 soft EOS Dieperink (KVI) LDM / 25

18 radii Radii express in terms of iso-scalar/vector R i (N, Z) = R 0 (N, Z) ± N Z 2A R 1(N, Z) (i=n,p) mass radius: R 0 = r 0 A 1/3 + aa 2/3 + c (N Z)2 A 2, in practice c 0 isovector radius R 1 = b R 0,1 depend only weakly on N Z by charge symmetry: R p (N, Z) determines R n however there are Coulomb effects 1 in case of sharp radius: δr c R 0 = ac A 8/3 144S v i.e. for N = Z R p > R n NZ(1+y 1 A 1/3 ) 2 polarization correction (if R is converted to rms radius): since ρ p (r) ρ n (r) in interior p n Dieperink (KVI) LDM / 25

19 Radii charge radii, shell corrections Closed shells: Strong binding small R near closed shells: decrease of binding, increase of R midshell: deformation: increase of binding, increase R rms dev= fm N Dieperink (KVI) LDM / 25

20 Radii charge radii, shell corrections Closed shells: Strong binding small R near closed shells: decrease of binding, increase of R midshell: deformation: increase of binding, increase R N rms dev= fm Try: δr shell /R = a 2 (n v n v + z v z v ) + a pn (n v n v.z v z v ) R c (N, Z) = R 0 (A) + N Z A R 1 + δr c + δr shell Dieperink (KVI) LDM / 25

21 Radii charge radii, shell corrections Closed shells: Strong binding small R near closed shells: decrease of binding, increase of R midshell: deformation: increase of binding, increase R N rms dev= fm fm (only 5 par s!) Try: δr shell /R = a 2 (n v n v + z v z v ) + a pn (n v n v.z v z v ) R c (N, Z) = R 0 (A) + N Z A R 1 + δr c + δr shell some data (Angeli) have large error bars (mix of stat, syst, theor errors) Dieperink (KVI) LDM / 25

22 isotope shifts for Cs Radii isotope shifts r 2 N r 2 N=82 Dieperink (KVI) LDM / 25

23 isotope shifts for Cs Radii isotope shifts r 2 N r 2 N=82 Dieperink (KVI) LDM / 25

24 charge radii, deformation Radii Away from closed shells with say n v, z v 4 deformation of the ground state occurs: increase of binding, increase of radius R(θ) = R 0 (1 + β 2 Y 20 (θ) + β 4 Y 40 (θ)) then R 2 = R 2 spher ( π β ) Simplest: take β 2 from exp BE(2) or Q-moments We are working on a dynamic approach using a collective H to describe excited states. Dieperink (KVI) LDM / 25

25 isotope shifts for Ra Radii isotope shifts r 2 N r 2 N=126 Dieperink (KVI) LDM / 25

26 isotope shifts for Ra Radii isotope shifts r 2 N r 2 N=126 Deformation effect is substantial, absent in spherical EDF Dieperink (KVI) LDM / 25

27 neutron skin Radii options: 1 from isovector term in R: R = N Z A R 1 + δr C e.g. 208 Pb: R = 0.18 ± 0.03 fm reduction of skin by 30% 2 from exp µ a (i) Rn Rp R = A(N S Z S ) 6NZ A N Z a cza 2/3 /S v 6NZ 1+A 1/3 /y (ii) µ a (N, Z) = 2(N Z) A from (i)+(ii) R R = µa A 5/3 0 12S s NZ + 5ac A 4/3 72S s N S sa 1/3 1+A 1/3 /y 5ac 6 Z A 1/3 take µ a from exp, shell effects implicitly included overall uncertainty in R from error in S s (15%) in ratio R N R N S s drops out Dieperink (KVI) LDM / 25

28 Results for R Radii using µ a from exp wiggles mostly occur at magic numbers related to loss of binding not an effect of deformation! 208 Pb exp: R = 0.20 ± 0.04 ± 0.05 fm (anti-protonic atoms) present: method(1): 0.18 fm; method(2): 0.20 fm Brown: fm, Piekarewicz 0.22 fm, vangiai 0.21 fm Dieperink (KVI) LDM / 25

29 Radii Results for R in Radium comparison with Brown et al. uncertainty in R solely due to error in S s, 15%? Dieperink (KVI) LDM / 25

30 isotopic chains Radii Nuclear effects for single atom real nucleus S = constant density model = 1 + (Zα)2 ( 1 2 δ c δ np +...) δ c = R 2 c /R 2 0 1, δ pn = R 2 n/r 2 c 1... To eliminate the atomic uncertainties use isotopic ratio ratio R = E PNC E PNC = Q W Q W ( R p R p ) 2γ 2 N N ( R p R p ) 2γ 2 (1 + f n ( Rn R p ) f n ( R n R )) p S S N = (Zα) 2 N N ( 1 2 δ c δ pn +...) Dieperink (KVI) LDM / 25

31 Summary Radii Extended LDM for masses and radii with simple shell corrections Symmetry energy S v = 32 ± 2MeV, S v /S s = 3.0 ± 0.3 Agrees with Danielewicz Radii R c can be described by similar shell corrections; for a quantitative fit need deformation neutron skin predicted from n-p separation energies, weak shell effects weak disagreement with Brown et al Dieperink (KVI) LDM / 25