Statistical analysis in nuclear DFT: central depression as example P. G. Reinhard 1 1 Institut für Theoretische Physik II, Universität Erlangen-Nürnberg ISNET17, York (UK), 8. November 217 Statistical analysis in nuclear DFT: central depression asisnet17, example York (UK), 8. November 217 1 / 29
Outline 1 Motivation: nuclear bubbles bubble nuclei 2 Framework: nuclear DFT, calibration& statistical analysis Typical structure of nuclear density functionals Optimization of model parameters (χ 2 fits), statistical analysis 3 Central density ρ() and central depression w of nuclear charge density Charge formfactor, charge density and all that 4 Nuclear DFT and central depression: trends, correlations, predictions Averages and variances: trends with size and proton number Correlations (with Coulomb energy, model parameters, groups,...) Inter-correlations: nuclei in different regions 5 Toward estimating a systematic error Confrontation with data, variation of the model, 6 Conclusions Statistical analysis in nuclear DFT: central depression asisnet17, example York (UK), 8. November 217 2 / 29
Acknowledgements (short list) Bastian Schütrumpf (GSI) & Witek Nazarewicz (MSU) central depression [PRC 96 (217) 2436] Jochen Erler and Peter Klüpfel Skyrme parametrizations [PRC 79 (29) 3431] Witek Nazarewicz (MSU) Fayans functionals [PRC 95 (217) 64328] Jörg Friedrich charge formfactors etc [NPA 373 (1982) 192, NPA 459 (1986) 1] Statistical analysis in nuclear DFT: central depression asisnet17, example York (UK), 8. November 217 3 / 29
1) Motivation: nuclear bubbles bubble nuclei Statistical analysis in nuclear DFT: central depression asisnet17, example York (UK), 8. November 217 4 / 29
Super-heavy elements (SHE) bubble nuclei? example:local density distribution ρ(r = ) for SHE just above known region phen. (RMF) other examples in some light nuclei ( 34 Si, 46 Ar) bubble = dip at ρ(), central depression develop measure & test with stat. analysis Statistical analysis in nuclear DFT: central depression asisnet17, example York (UK), 8. November 217 5 / 29
2) Framework: nuclear DFT, calibration & statistical analysis Statistical analysis in nuclear DFT: central depression asisnet17, example York (UK), 8. November 217 6 / 29
A nuclear density functional non-relativistic Skyrme-Hartree-Fock energy: E = E kin + d 3 re pot(ρ p, ρ n, τ p, τ n, J ls,p, J ls,n, χ pair,...time odd...) + E CoM pairing density spin orbit density kinetic-energy density density E pot = 1 [ T = C () T (ρ ) ρ 2 T + CT ρ T ρ T + C (kin) T τ T ρ T + C (ls) T J T ρ T ] + t {p,n} density dependence: C () T (ρ ) = C () T () + C(ρ) T ρα, C (pair) t (ρ ) = V (pair) t isospin recoupling: ρ = ρ n + ρ p, ρ 1 = ρ n ρ p C (pair) t (ρ )χ 2 t ( 1 ρ ρ pair ) universal = same parameters for all systems insufficient ab-initio data model parameters from χ 2 -fit to empirical data in the following using parametrization SV-min from PRC 79 (29) 3431. fit data: E, r rms, R diffr., σ surf, ɛ ls, (3) pair in semi-magic nuclei (255 data points) Statistical analysis in nuclear DFT: central depression asisnet17, example York (UK), 8. November 217 7 / 29
Physical model parameters For the smooth part: liquid-drop model parameters (LDM) E/A eq volume energy isoscalar ρ eq equilibrium density isoscalar K incompressibility isoscalar bulk J symmetry energy isovector L slope of symmetry energy isovector a surf surface energy isoscalar surface a surf,sym surface symmetry energy isovector For shell fluctuations: kinetic and pairing parameters m /m effective mass isoscalar κ TRK TRK sum rule enhancement isovector influence b ls,t = spin-orbit strength isoscalar on spectra b ls,t =1 spin-orbit strength isovector V pair,p proton pairing strength sensitive V pair,n neutron pairing strength to spectra ρ pair pairing density dependence Statistical analysis in nuclear DFT: central depression asisnet17, example York (UK), 8. November 217 8 / 29
Optimization of model parameters and statistical analysis N data ( global quality measure: χ 2 Of (p) O exp ) 2 f (p) =, O f = fit observable, f =1 O 2 f p = (p 1,..., p Np ) = model parameters. O f = adopted error optimal parameters p : χ 2 (p ) = minimal ( ) statistical interpretation: W (p) exp χ 2 (p) probability for parameter set p = average A = dp WA(p), variance 2 A = dp W (A A) 2. Taylor expansions: Covariance matrix C 1 p i p j = pi pj χ = average A = A(p ), variance 2 A = pa Ĉ pa covariance A B = pa Ĉ pb coefficient of determination (CoD): p r 2 AB = A B 2 2 A 2 B multiple correlation coeff. (MCC): RAB 2 = r BAi (r AA ) 1 ij r Aj B ij CoD RAB 2 of a group of obs. A = (A 1,..., A g) with obs. B Statistical analysis in nuclear DFT: central depression asisnet17, example York (UK), 8. November 217 9 / 29
Stages of model development model observable physics explorations modeling range of applicability (RoA) observable in RoA data fit observables in RoA direct or model dep.? statistical analysis averages unresolved trends in fit obs. predictions, extrapolations comparison with data? variances strong vs. soft model params. extrapolation errors CoD, MCC sensitivity analysis (in-)dependent obs./params. superflous parameters redundant or useful data? directions for extensions systematic errors no systematic way trial and error, variations of model = back to physics
3) Central density ρ() and central depression w Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 11 / 29
Density profiles for various nuclear sizes proton density ρ p [fm -3 ] neutron density ρ n [fm -3 ].1.8 SV-min.1.8.6.6.4.2 32 Og 28 Pb 132 Sn 48 Ca 34 Si.4.2-1 -8-6 -4-2 2 4 6 8 1 r [fm] r [fm] develop closely around average density nuclear saturation central density ρ p/n () fluctuates ± about average shell effects = ρ p/n () contains info on both: bulk properties (LDM) & shell effects Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 12 / 29
Safe and unsafe regions of formfactor F(q) = d 3 r e iq r ρ(r) charge density ρ c [fm -3 ] charge formfactor q*f c (q).6 28 Pb 1 exp. SV-min.5.4.3 1.1 region of short-range correlations.2.1 safe mean-field.1 exp. area.1 SV-min 2 4 6 8 1.5 1 1.5 2 2.5 3 3.5 4 r [fm] q [fm -1 ] systematic deviation for F(q) above q > 2.5/fm suppressed by short-range correl. r sensitive to large q? central density ρ() not a perfect mean-field observable = try alternative measure for central depression from F (q) in q < 2.5/fm Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 13 / 29
A safe (?) measure: central depression parameter w liquid drop model (LDM) for density distribution: ) ρ model(r) G σ (1 + w r2 θ(r r 2 d,1 r ), G σ Gaussian folding, surface width σ R d,1 1. diffraction radius from 1. zero in F(q).6 28 box-equivalent radius Pb.5.4.3.2.1 w= w=.1 w=.2 w=.3 1 2 3 4 5 6 7 8 r [fm] compare systematics = ( ) Rd,1 w w = 15.3 1 R d,2 R d,2 2. diffraction radius from 2. zero in F (q) = task: what is the more robust observable ρ() or w? Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 14 / 29
Formfactor zeroes and critical region charge formfactor q*f c (q) 4 Ca charge formfactor q*f c (q) 28 Pb 1 σ surf 1 1.1.1.1 1. zero R diffr safe mean-field area 2. zero w region of short-range correlations.5 1 1.5 2 2.5 3 3.5 4 r [fm] 1.1.1.1 safe mean-field area region of short-range correlations.5 1 1.5 2 2.5 3 3.5 4 q [fm -1 ] 1.&2. zero, 1. maximum diffraction radius R diffr, surface thickn. σ surf, central depr. w small nuclei closer to critical q 2.5/fm = definition of w less robust Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 15 / 29
4) Nuclear DFT and central depression: trends, correlations, predictions testing set: 3 isotonic chains, fixed neutron number N & varied proton number Z N = 82, 126 examples from normal nuclei and N = 184 chain of SHE Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 16 / 29
Trends & variances for central depression w t & central density ρ t ().1.9.8.7 central proton density ρ p () [fm -3 ] N=82 N=126 N=184 neutrons protons.4.3.2.1 protons neutrons N=82 central depression w N=126 N=184.6.5 5 6 7 8 9 1 11 12 proton number Z -.1 5 6 7 8 9 1 11 12 proton number Z central depression systematically larger for protons Coulomb pressure (?) kink at magic proton number Z = 82 shell effect (in both observables!) no unique trend with proton number; extrapolation errors small Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 17 / 29
Correlations with Coulomb energy CoD: E coul central density ρ t () CoD: E coul central depression w t 1 protons neutrons 1 protons neutrons.8 N=184.8 N=184.6.6.4.4.2 N=82 N=126.2 N=82 N=126 5 6 7 8 9 1 11 12 proton number Z 5 6 7 8 9 1 11 12 proton number Z impact of Coulomb energy increases dramatically for SHE surprisingly little correlation with Coulomb energy for normal nuclei proton and neutron observables show similar correlations Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 18 / 29
Coefficients of determination (CoD) of ρ t () and w t with model params. CoD with central depression w and central density ρ() 1.8.6.4.2 1.8.6.4.2 1.8.6.4.2 E/A ρ eq K L J a surf a surf,sym m * /m κ TRK C ls C ls V pair,p w p w n ρ p ρ n w p w n ρ p ρ n w p w n ρ p ρ n V pair,n ρ pair 32 Og 28 Pb 48 Ca no clearly dominant contributor, influences are spread over many model parameters = look for impact of groups of parameters (LDM versus shell) Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 19 / 29
Trends of multiple correlation coefficients (MCC) MCC with central proton density ρ p () MCC with proton central depression w p 1 1 LDM l*s,pair.8.8.6.6.4 N=82 N=126 N=184.4 N=82.2.2 N=126 N=184 5 6 7 8 9 1 11 12 proton number Z 5 6 7 8 9 1 11 12 proton number Z ρ p(): LDM (=E/A, ρ eq, K, J, L,surf) dominates clearly over shell (=kinetic,ls,pair) ρ() contains shell effects - but you have little influence on them w p: shell -group is as important as LDM (surprise!) Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 2 / 29
Correlations between nuclei in different regions matrix of CoD central density ρ p and central depression w p 1 w p (118,184) w p (18,184) w p ( 98,184) w p ( 88,126).8 w p ( 82,126) w p ( 78,126) w p ( 58, 82).6 w p ( 5, 82) ρ p (118,184) ρ p (18,184) ρ p ( 98,184) ρ p ( 88,126) ρ p ( 82,126) ρ p ( 78,126) ρ p ( 58, 82) ρ p ( 5, 82) Can one improve extrapolations by ρ()&w in existing nuclei? ρ p ( 5, 82) ρ p ( 58, 82) ρ p ( 78,126) ρ p ( 82,126) ρ p ( 88,126) ρ p ( 98,184) ρ p (18,184) ρ p (118,184) w p ( 5, 82) w p ( 58, 82) w p ( 78,126) w p ( 82,126) w p ( 88,126) w p ( 98,184) w p (18,184) w p (118,184).4.2 ρ p( 32 Og) uncorrelated from all other ρ, w other ρ p() more or less inter-correlated w p show generally less inter-correlation ρ p() w p weakly correl. too small inter-correlations Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 21 / 29
5) Toward estimating a systematic error test set for comparison with data: stable, spherical nuclei where F(q)-data exist Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 22 / 29
Comparison with experimental data from ρ charge (r) for stable nuclei central charge density ρ c () [fm -3 ] proton central depression w p.8.7.6 SV-min exp 4 6 8 1 12 14 16 18 2 total nucleon number.5.4.3.2.1 -.1 -.2 SV-min exp 4 6 8 1 12 14 16 18 2 total nucleon number ρ c(): w c: SHE: agrees with data mean-field theory differs from data, situation improves for heavy nuclei in spite of LDM motivation some impact from short-range correlations mean-field predictions increasingly reliable with increasing system size A Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 23 / 29
Variation of model add Fayans functionals Fayans functional: major news: very similar to Skyrem-Hartree-Fock slighlty different density dependence (rational fct. instead of ρ α ) additional ρ terms in surface parameter C ρ T = more flexible modeling of surface and pairing C (pair) t Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 24 / 29
Variation of model add Fayans functionals central proton density ρ p () [fm -3 ] proton central depression w p.7.6.5.4 SV-min Fy-std Fy( r) N=82 N=126 N=184 span in 1. figure SHF phen. RMF 5 6 7 8 9 1 11 12 proton number Z proton number Z ρ p(): error bars even smaller for Fy functional - in spite of more surface terms predictions significantly different (outside error bars) no clear trend w p: differences insignificant (inside error bars) trends different from ρ p() difference Fy SV-min smaller than for other (& older) models (RMF,folded Yukawa).3.2.1 SV-min Fy-std Fy( r) N=82 N=126 N=184 5 6 7 8 9 1 11 12 Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 24 / 29
4) Conclusions central density ρ() & central depression w: from formfactor F(q), unclear impact of high q (short-range correl., shell fluct.) risky observable testing ground for statistical analysis impact of model parameters and Coulomb pressure: E coul only relevant in SHE influences spread over many model parameters no dominant influencer LDM group : decisive for ρ(), still important for w shell group : considerable impact on w, only weak impact on ρ() correlations between different A, model depedence, data: ρ()&w in SHE independent from normal nuclei variation of model (Fayans funct.) yields only small differences theory compatible with data on ρ c(), systematic deviations for w c in small nuclei = : at variance with expectation: ρ() is a better mean-field observable than w both observables can be safely extrapolated to SHE Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 25 / 29
Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 26 / 29
Trends of multiple correlation coefficients (MCC) 1.8 MCC with central proton density ρ p () 1.8 MCC with proton central depression w p J,L LDM l*s,pair.6.6.4 N=82 N=126 N=184.4 N=82.2.2 N=126 N=184 5 6 7 8 9 1 11 12 proton number Z 5 6 7 8 9 1 11 12 proton number Z ρ p(): w p: LDM (=E/A, ρ eq, K, J, L,surf) dominates clearly over shell (=kinetic,ls,pair) already isovector LDM (=J,L) determines a large fraction shell -group is as important as LDM, J,L has much less impact Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 27 / 29
Matrix of CoD for 48 Ca and 32 Og (E/A) sym ρ,sym K J L a surf a surf,sym m /m κ E Coul w p w n ρ p,c /ρ av ρ n,c /ρ av ρ p,c ρ n,c ρ,c ρ 1,c (E/A)sym ρ,sym K J L a surf asurf,sym m /m κ E Coul 32 Og wp wn ρp,c/ρav ρn,c/ρav ρp,c ρn,c ρ,c ρ1,c 48 Ca 1..8.6.4.2 Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 28 / 29
Comparison with experimental data from ρ charge (r) for stable nuclei central charge density ρ c () [fm -3 ] proton central depression w p.8.7 SV-min Fy(std) Fy( r) exp.5.4.3.2.1.6 -.1 -.2 SV-min Fy( r) exp 4 6 8 1 12 14 16 18 2 4 6 8 1 12 14 16 18 2 total nucleon number total nucleon number ρ c(): agrees with data, SV-min generally closest (but insignificant effect) deviations change sign shell fluctuations in mean-field results w c: mean-field theory differs from data, situation improves for heavy nuclei in spite of LDM motivation some impact from high q (short-range correl.?) SHE: mean-field predictions increasingly reliable with increasing system size A Statistical analysis in nuclear DFT: central depression as ISNET17, exampleyork (UK), 8. November 217 29 / 29