IAS and Skin Constraints. Symmetry Energy

Transcription

1 IS and Skin Constraints on the Pawel Natl Superconducting Cyclotron Lab, US 32 nd International Workshop on Nuclear July, 2013, East Lansing, Michigan

2 ρ in Nuclear Mass Formula Textbook Bethe-Weizsäcker formula: E = a V + a S 2/3 Z 2 + a C 1/3 + a (N Z ) 2 a + E mic Symmetry energy: charge n p symmetry of interactions nalogy with capacitor: E a = a a (N Z ) 2 (N Z )2 a a?volume Capacitance? E a = independent capacitors r E = Q2 2C (N Z )2 a a (N Z )2 a V a + 2/3 a S a Thomas-Fermi (local density) approximation: C ρ dr a a () = = S(ρ) aa V, for S(ρ) aa V TF breaks in nuclear surface at ρ < ρ 0 /4 PD&Lee NP818(2009)36

3 ρ in Nuclear Mass Formula Textbook Bethe-Weizsäcker formula: E = a V + a S 2/3 Z 2 + a C 1/3 + a (N Z ) 2 a + E mic Symmetry energy: charge n p symmetry of interactions nalogy with capacitor: E a = a a (N Z ) 2 (N Z )2 a a?volume Capacitance? E a = independent capacitors r E = Q2 2C (N Z )2 a a (N Z )2 a V a + 2/3 a S a Thomas-Fermi (local density) approximation: C ρ dr a a () = = S(ρ) aa V, for S(ρ) aa V TF breaks in nuclear surface at ρ < ρ 0 /4 PD&Lee NP818(2009)36

4 ρ in Nuclear Mass Formula Textbook Bethe-Weizsäcker formula: E = a V + a S 2/3 Z 2 + a C 1/3 + a (N Z ) 2 a + E mic Symmetry energy: charge n p symmetry of interactions nalogy with capacitor: E a = a a (N Z ) 2 (N Z )2 a a?volume Capacitance? E a = independent capacitors r E = Q2 2C (N Z )2 a a (N Z )2 a V a + 2/3 a S a Thomas-Fermi (local density) approximation: C ρ dr a a () = = S(ρ) aa V, for S(ρ) aa V TF breaks in nuclear surface at ρ < ρ 0 /4 PD&Lee NP818(2009)36

5 ρ in Nuclear Mass Formula Textbook Bethe-Weizsäcker formula: E = a V + a S 2/3 Z 2 + a C 1/3 + a (N Z ) 2 a + E mic Symmetry energy: charge n p symmetry of interactions nalogy with capacitor: E a = a a (N Z ) 2 (N Z )2 a a?volume Capacitance? E a = independent capacitors r E = Q2 2C (N Z )2 a a (N Z )2 a V a + 2/3 a S a Thomas-Fermi (local density) approximation: C ρ dr a a () = = S(ρ) aa V, for S(ρ) aa V TF breaks in nuclear surface at ρ < ρ 0 /4 PD&Lee NP818(2009)36

6 Mass and Skin Fits : Skin: E a = av a r np = 2 3 (N Z ) av a a S a 1/3 ( r rms a a N Z 1/3 aa S ) Coul PD NP723(2003)233 a S a L

7 Fits in L a V a Plane Lie-Wen Chen et al PRC82(10)024321

8 Charge Invariance?a a ()? Conclusions on sym-energy details, following E-formula fits, interrelated with conclusions on other terms in the formula: asymmetry-dependent Coulomb, Wigner & pairing + asymmetry-independent, due to (N Z )/ - correlations along stability line [PD NP727(03)233]! Best would be to study the symmetry energy in isolation from the rest of E-formula! bsurd?! Charge invariance to rescue: lowest nuclear states characterized by different isospin values (T,T z ), T z = (Z N)/2. Nuclear energy scalar in isospin space: sym energy E a = a a () (N Z )2 = 4 a a () T 2 z E a = 4 a a () T 2 = 4 a a() T (T + 1)

9 Charge Invariance?a a ()? Conclusions on sym-energy details, following E-formula fits, interrelated with conclusions on other terms in the formula: asymmetry-dependent Coulomb, Wigner & pairing + asymmetry-independent, due to (N Z )/ - correlations along stability line [PD NP727(03)233]! Best would be to study the symmetry energy in isolation from the rest of E-formula! bsurd?! Charge invariance to rescue: lowest nuclear states characterized by different isospin values (T,T z ), T z = (Z N)/2. Nuclear energy scalar in isospin space: sym energy E a = a a () (N Z )2 = 4 a a () T 2 z E a = 4 a a () T 2 = 4 a a() T (T + 1)

10 Charge Invariance?a a ()? Conclusions on sym-energy details, following E-formula fits, interrelated with conclusions on other terms in the formula: asymmetry-dependent Coulomb, Wigner & pairing + asymmetry-independent, due to (N Z )/ - correlations along stability line [PD NP727(03)233]! Best would be to study the symmetry energy in isolation from the rest of E-formula! bsurd?! Charge invariance to rescue: lowest nuclear states characterized by different isospin values (T,T z ), T z = (Z N)/2. Nuclear energy scalar in isospin space: sym energy E a = a a () (N Z )2 = 4 a a () T 2 z E a = 4 a a () T 2 = 4 a a() T (T + 1)

11 a a () Nucleus-by-Nucleus T (T + 1) E a = 4 a a () In the ground state T takes on the lowest possible value T = T z = N Z /2. Through +1 most of the Wigner term absorbed. Formula generalized to the lowest state of a given T (e.g. Jänecke et al., NP728(03)23).?Lowest state of a given T : isobaric analogue state (IS) of some neighboring nucleus ground-state. T z =-1 T z =0 T z =1 Study of changes in the symmetry term possible nucleus by nucleus T=1 T=0 PD&Lee arxiv:

12 Queries in the Context of Data re expansions valid? Coefficient values?? EIS = E? 4 a a () IS E gs = [ T (T + 1) ] + E mic Is the excitation energy linear in the isospin squared?? or a 1 a a a ()? = a V a + 2/3 a S a? = (a V a ) 1 + (a S a ) 1 1/3 Is the volume-surface separation valid? From an a V a -a S a fit can one learn about a V a and L for uniform matter?

13 Insight into IS nalysis IS data: ntony et al. DNDT66(97)1 Shell corrections: Koura et al. ProTheoPhys113(05)305 Excitation energies to IS, EIS, for different, lumped together in narrow regions of. Is E a T (T + 1)? YES

14 Insight into IS nalysis IS data: ntony et al. DNDT66(97)1 Shell corrections: Koura et al. ProTheoPhys113(05)305 Excitation energies to IS, EIS, for different, lumped together in narrow regions of. Is E a T (T + 1)? YES

15 a a () = 4 a a () with Shell Corrections E IS E mic T 2 E mic from: Koura et al. ProTheoPhys113(05)305 Heavy nuclei a a 22 MeV, light a a 10 MeV (more surface + low ρ)

16 Z -Dependence of Symmetry Coefficients? surface less dominant, higher average ρ surface more dominant, lower average ρ Symmetry coefficients on a nucleus-by-nucleus basis

17 Sensitivity to Shell Corrections Fit to raw data ( > 30) in the middle, but: Moller et al. fit: aa V = MeV, aa S = 8.48 MeV von Groote et al.: aa V = MeV, aa S = MeV

18 Comparisons to Skyrme-Hartree-Fock Issues in data-theory comparisons (codes by P.-G. Reinhard): 1. No isospin invariance in SHF - impossible to follow the procedure for data 2. Shell corrections not feasible at such scrutiny as for data 3. Coulomb effects. Solution: Procedure that yields the same results as the energy, in the bulk limit, but is weakly affected by shell effects: (N Z ) r<rc N Z = a a a V a = C r<r c C r<r c ρ S(ρ)

19 a a () from Mean-Field Calculations Skyrme-Hartree- Fock theory (codes by P.-G. Reinhard) Similar behavior with as for IS

20 a a () from Different Mean Fields?Slope L in ρ slope in?? Less impact of the slope L at ρ 0 than expected!??difficulty for L determination??

21 Model-Independent Large- Expansion?? Symbols: results of spherical no-coulomb SHF calcs Lines: volume-surface decomposition - expectation vs fit Symmetric matter energy f/sample Skyrmes Works Symmetry coefficient Not... Expectations from half- matter.

22 Pearson correlation coefficient Can S(ρ) Be Constrained??! r XY = r 1 - strong correlation r 0 - no correlation (X X )(Y Y ) (X X ) 2 (Y Y ) 2 X a a () Y S(ρ) Ensemble of Skyrmes Nearly no information about S(ρ 0 )!

23 Symmetry-Energy Correlations When Strong NO S(ρ) a a! Vehicle: phenomenological mean-field theory

24 Constraints on S(ρ) Demand that Skyrme approximates IS results at > 30 produces a constraint area for S(ρ):?Slope constraint??

25 Correlation at ρ 0?Slope constraint??

26 symmetry Skin & Energy Stiffness Pearson coef of r np = rn rms rp rms & stiffness of S γ(ρ) := ρ S d S d ρ f/different at fixed η = (N Z )/

27 symmetry Skin & Energy Stiffness Pearson coef of r np & stiffness γ(ρ) := ρ S d S d ρ f/different η = (N Z )/ at fixed

28 symmetry Skins from Measurements

29 S(ρ) from Combined Constraints

30 Comparison to Microscopic Calculations PR: V18 + UVIX BHF: V18 + UVIX DBHF: Bonn BHF: Bonn B + micro 3N BHF: V18 + micro 3N IS + r np constraints extrapolation

31 at ρ 0? CEFT - Hebeler IS+ r np QMC - Gandolfi PDR - Carbone HI - Tsang Masses - Korteleinen Skins - Chen NS - Steiner S(ρ) = a V a + L 3 ρ ρ 0 ρ

32 Conclusions Symmetry-energy term weakens as nuclear mass number decreases: from a a 23 Mev to a a 9 MeV for 8. For 25, a a () may be fitted with aa 1 = (aa V ) 1 + (aa S ) 1 1/3, where aa V 35 MeV and aa S 10 MeV. Weakening of the symmetry term can be tied to the weakening of S(ρ) in uniform matter, with the fall of ρ. Including skin sizes, significant, ±1.0 MeV, constraints on S(ρ) at densities ρ =( ) fm 3. round ρ 0 : strongly correlated a V a = ( ) MeV and L = (35 70) MeV. To do: Dedicated Skyrme interactions. PD&Lee arxiv: PHY

33 Conclusions Symmetry-energy term weakens as nuclear mass number decreases: from a a 23 Mev to a a 9 MeV for 8. For 25, a a () may be fitted with aa 1 = (aa V ) 1 + (aa S ) 1 1/3, where aa V 35 MeV and aa S 10 MeV. Weakening of the symmetry term can be tied to the weakening of S(ρ) in uniform matter, with the fall of ρ. Including skin sizes, significant, ±1.0 MeV, constraints on S(ρ) at densities ρ =( ) fm 3. round ρ 0 : strongly correlated a V a = ( ) MeV and L = (35 70) MeV. To do: Dedicated Skyrme interactions. PD&Lee arxiv: PHY

34 Conclusions Symmetry-energy term weakens as nuclear mass number decreases: from a a 23 Mev to a a 9 MeV for 8. For 25, a a () may be fitted with aa 1 = (aa V ) 1 + (aa S ) 1 1/3, where aa V 35 MeV and aa S 10 MeV. Weakening of the symmetry term can be tied to the weakening of S(ρ) in uniform matter, with the fall of ρ. Including skin sizes, significant, ±1.0 MeV, constraints on S(ρ) at densities ρ =( ) fm 3. round ρ 0 : strongly correlated a V a = ( ) MeV and L = (35 70) MeV. To do: Dedicated Skyrme interactions. PD&Lee arxiv: PHY

35 Conclusions Symmetry-energy term weakens as nuclear mass number decreases: from a a 23 Mev to a a 9 MeV for 8. For 25, a a () may be fitted with aa 1 = (aa V ) 1 + (aa S ) 1 1/3, where aa V 35 MeV and aa S 10 MeV. Weakening of the symmetry term can be tied to the weakening of S(ρ) in uniform matter, with the fall of ρ. Including skin sizes, significant, ±1.0 MeV, constraints on S(ρ) at densities ρ =( ) fm 3. round ρ 0 : strongly correlated a V a = ( ) MeV and L = (35 70) MeV. To do: Dedicated Skyrme interactions. PD&Lee arxiv: PHY

36 Conclusions Symmetry-energy term weakens as nuclear mass number decreases: from a a 23 Mev to a a 9 MeV for 8. For 25, a a () may be fitted with aa 1 = (aa V ) 1 + (aa S ) 1 1/3, where aa V 35 MeV and aa S 10 MeV. Weakening of the symmetry term can be tied to the weakening of S(ρ) in uniform matter, with the fall of ρ. Including skin sizes, significant, ±1.0 MeV, constraints on S(ρ) at densities ρ =( ) fm 3. round ρ 0 : strongly correlated a V a = ( ) MeV and L = (35 70) MeV. To do: Dedicated Skyrme interactions. PD&Lee arxiv: PHY

37 Conclusions Symmetry-energy term weakens as nuclear mass number decreases: from a a 23 Mev to a a 9 MeV for 8. For 25, a a () may be fitted with aa 1 = (aa V ) 1 + (aa S ) 1 1/3, where aa V 35 MeV and aa S 10 MeV. Weakening of the symmetry term can be tied to the weakening of S(ρ) in uniform matter, with the fall of ρ. Including skin sizes, significant, ±1.0 MeV, constraints on S(ρ) at densities ρ =( ) fm 3. round ρ 0 : strongly correlated a V a = ( ) MeV and L = (35 70) MeV. To do: Dedicated Skyrme interactions. PD&Lee arxiv: PHY

38 IS nalysis Skyrme Hartree-Fock a a () = 4 a a () without Shell Corrections E IS T 2 IS data: ntony et al. DNDT66(97)1 Lines: fits to a a () assuming volume-surface competition analogous to that for E 1.??Fundamental knowledge??

39 IS nalysis Skyrme Hartree-Fock Stiffness of the S ρ γ

40 IS nalysis Skyrme Hartree-Fock Robustness of Macroscopic Description?

41 IS nalysis Skyrme Hartree-Fock Constraints at ρ 0