Happy the man who has been able to discern the cause of things

Transcription

1 Happy the man who has been able to discern the cause of things Theories Models Virgil, Georgica A first rate theory predicts A second rate theory forbids A third rate theory explains after the facts Alexander I. Kitaigorodskii

2 Modeling the Atomic Nucleus Theoretical bag of tricks

3 The Nuclear Many-Body Problem H ˆ = T ˆ + V ˆ ˆ T = A 2 p ˆ " i, V ˆ = 2m i i=1 one-body H ˆ " = E" " V ˆ 2b (i, j) + V ˆ 3b (i, j,k) i<i two-body " i<i<k three-body Kinetic energy Potential energy ( ) " = " r 1, r 2,K, r A ;s 1,s 2,K,s A ;t 1,t 2,K,t A 3A nucleon coordinates in r-space nucleon spins: ±1/2 nucleon isospins (p or n): ±1/2 Eigenstate of angular momentum, parity, and ~isospin # A& 2 A A! N!Z! Bottom line: coupled integro-differential equations in 3A dimensions

4 The nuclear many-body problem

5 Weinberg s Laws of Progress in Theoretical Physics From: Asymptotic Realms of Physics (ed. by Guth, Huang, Jaffe, MIT Press, 1983) First Law: The conservation of Information (You will get nowhere by churning equations) Second Law: Do not trust arguments based on the lowest order of perturbation theory Third Law: You may use any degrees of freedom you like to describe a physical system, but if you use the wrong ones, you ll be sorry!

6 LQCD quark models ab initio CI DFT collective models scale separation Resolution Effective Field Theory How are nuclei made? Origin of elements, isotopes Hot and dense quark-gluon matter Hadron structure Hadron-Nuclear interface Nuclear structure Nuclear reactions New standard model Applications of nuclear science To explain, predict, use

7 Interfaces provide crucial clues number of nuclei < number of processors!

8 Theory of nuclei is demanding rooted in QCD insights from EFT many-body interactions in-medium renormalization microscopic functionals low-energy coupling constants optimized to data crucial insights from exotic nuclei Many-body dynamics Input Forces, operators Open channels 11 Li 100 Sn 240 Pu 298 U many-body techniques o direct ab initio schemes o symmetry breaking and restoration high-performance computing interdisciplinary connections nuclear structure impacted by couplings to reaction and decay channels clustering, alpha decay, and fission still remain major challenges for theory unified picture of structure and reactions

9 Ab initio theory for light nuclei and nuclear matter Ab initio: QMC, NCSM, CCM, (nuclei, neutron droplets, nuclear matter) Ab initio input NN+NNN interac+ons Renormaliza+on Many body method Observables Direct comparison with experiment Pseudo-data to inform theory Input: Excellent forces based on the phase shift analysis and few-body data EFT based nonlocal chiral NN and NNN potentials SRG-softened potentials based on bare NN+NNN interactions

10 Few-nucleon systems A=2: many years ago!! 3 H: 1984 (1% accuracy)!!! 3 He: 1987!!! 4 He: 1987!! 5 He: 1994 (n-α resonance)!!! A=6,7,..12: !

11 Green s Function Monte Carlo (imaginary-time method) ψ 0 ψ τ = lime τ ( ) = e ˆ ( H ˆ E 0 )τ ( H E 0 )τ ψ V ψ V ψ( 0) = ψ V, ψ( ) = ψ 0 τ = nδτ ψ τ ( )* ( ) = e ˆ trial wave function ( H E 0 )Δτ Quantum Monte Carlo (GFMC) 12 C No-Core Shell Model 14 F, 14 C Faddeev-Yakubovsky Lattice EFT 12 C (Hoyle) Coupled-Cluster Techniques 17 F, 56 Ni Fermionic Molecular Dynamics +,- n ψv

12 Nucleon-Nucleon Interaction NN, NNN, NNNN,, forces GFMC calculations tell us that: V π / V ~ 70 80% V π ~ 15MeV/pair V R V 3 ~ 5MeV/pair ~ 1MeV/three short-range three-body T ~ 15MeV/nucleon V C ~ 0.66MeV/pair of protons

13 GFMC: S. Pieper, ANL 1-2% calculations of A = 6 12 nuclear energies are possible excited states with the same quantum numbers computed

14 dinal form factor F(q) C: ground state and Hoyle state exp ρ 1b ρ 1b+2b state-of-the-art computing Wiringa et al. Phys. Rev. C 89, (2014); A. Lovato et al., Phys. Rev. Lett. 112, (2014) ρ ch (r) r (fm) q (fm -1 ) E [MeV] f pt (k) C(G.S.)! 12 C(0 + 2 ) f tr FORM FACTOR 12 C M(E0) AV18+IL7 one-way orthog. - f pt (k) - 9 May k (fm -1 ) The ADLB (Asynchronous Ψ T O + P2 6 Dynamic Load- GFMC O + G2 GFMC 18 + P2 5 make calculations GFMC O + P2of 12 C with a complete Hamiltonian Experiment (two- and three-nucleon 4 of the Argonne BGP. 3 The computed 6 Z f tr (k) / k 2 (fm 2 ) 0 12 C M(E0) AV18+IL7 one-wa binding energy is 93.5(6) MeV compared 2 to the experimental value of MeV and the point rms radius 1 is 2.35 fm vs Pieper et al., 2.33 QMC from experiment Data from M. Chernykh et al., Phys. Rev. Lett. 105, (2010) (1) Right panel [f tr (k)/k 2 ] proportional to M(E0) at k = (3) (3) Large errors at small k due to large Monte Carlo errors Can get better value at k =0by computing R drr 2 r 2 tr (r) Exp 88(2) Balancing) version of GFMC was used to potential AV18+IL7) on 32,000 processors Results with best wave function in good agreement with data 0 + Epelbaum et al., Phys. Rev. Lett. 109, 92(3) (2012). Lattice EFT Th Lahde et al., Phys. Lett. B 732, 110 (2014) (2) k 2 (fm

15 S2n (MeV) The frontier: neutron-rich calcium isotopes probing nuclear forces and shell structure in a neutron-rich medium 52 Ca mass TITAN@TRIUMF Gallant et al, PRL 109, (2012) AME2003 TITAN K Ca Sc Neutron Number N 54 Ca 2 + S 2n (MeV) Ca: 20 protons, 34 neutrons Experiment ISOLTRAP NN+3N (MBPT) CC (Hagen et al.) KB3G GXPF1A ISOLTRAP@CERN Wienholtz et al, Nature (2013) 54 Ca mass Neutron number N CC theory Hagen et al., PRL109, (2012) RIBF@RIKEN Steppenbeck et al Nature (2013)

16 Anomalous Long Lifetime of 14 C Determine the microscopic origin of the suppressed β-decay rate: 3N force 0.29 Maris et al., PRL 106, (2011) GT matrix element N3LO NN only N3LO + 3NF (c D = -0.2) N3LO + 3NF (c D = -2.0) s p sd pf sdg pfh sdgi pfhj sdgik pfhjl configuration space Dimension of matrix solved for 8 lowest states ~ 10 9 Solution took ~ 6 hours on 215,000 cores on Cray XT5 Jaguar at ORNL

17 Anthropic Principle h9p://en.wikipedia.org/wiki/anthropic_principle The anthropic principle (from Greek anthropos, meaning "human") is the philosophical considera+on that observa+ons of the physical Universe must be compa+ble with the conscious life that observes it. Some proponents of the anthropic principle reason that it explains why the universe has the age and the fundamental physical constants necessary to accommodate conscious life. Anthropic considera+ons in nuclear physics: U. Meissner. h9p://arxiv.org/abs/ The nucleosynthesis of carbon- 12 and Hoyle state Non- anthropic scenario Anthropic scenario (fine- tuned Universe)

18 Dean Lee "Viability of Carbon- Based Life as a Func+on of the Light Quark Mass", Phys. Rev. Le (2013) "Dependence of the triple- alpha process on the fundamental constants of nature", Eur. Phys. J. A 49 (2013) 82 "Varying the light quark mass: impact on the nuclear force and Big Bang nucleosynthesis", Phys. Rev. D 87 (2013)

19 Ab ini+o calcula+on of the neutron- proton mass difference Science 347, 1452 (2015) The result of the neutron- proton mass splifng as a func+on of quark- mass difference and electromagne+c coupling. In combina+on with astrophysical and cosmological arguments, this figure can be used to determine how different values of these parameters would change the content of the universe. This in turn provides an indica+on of the extent to which these constants of nature must be fine- tuned to yield a universe that resembles ours.

20 Fusion of Light Nuclei Computational nuclear physics enables us to reach into regimes where experiments and analytic theory are not possible, such as the cores of fission reactors or hot and dense evolving environments such as those found in inertial confinement fusion environment. Ab ini+o theory reduces uncertainty due to conflic+ng data NIF The n- 3 H elastic cross section for 14 MeV neutrons, important for NIF, was not known precisely enough. Delivered evaluated data with required 5% uncertainty and successfully compared to measurements using an Inertial Confinement Facility First measurements of the differential cross sections for the elastic n- 2 H and n- 3 H scattering at 14.1 MeV using an Inertial Confinement Facility, by J.A. Frenje et al., Phys. Rev. Lett. 107, (2011)

21 Configuration interaction techniques light and heavy nuclei detailed spectroscopy quantum correlations (lab-system description) Input: configura+on space + forces NN+NNN interac+ons Matrix elements fi9ed to experiment Renormaliza+on Method Diagonaliza+on Trunca+on+diagonaliza+on Monte Carlo Observables Direct comparison with experiment Pseudo-data to inform reaction theory and DFT

22 Average one-body Hamiltonian 120 Sn Unbound! states! Coulomb! barrier! Discrete! (bound)! states! ε F ε F 0! Surface! region! n p A i=1 Flat! bottom! H ˆ 0 = h i, h i = 2 2M 2 i +V i h i φ k ( i) = ε k φ k i ( )

23 ˆ H = t i i Nuclear shell model v ij = (t i +V i ) i, j i i j + $ & & % ' V ) i ) i ( 1 2 v ij i, j i j One-body Hamiltonian Construct basis states with good (J z, T z ) or (J,T) Compute the Hamiltonian matrix Diagonalize Hamiltonian matrix for lowest eigenstates Number of states increases dramatically with particle number Full fp shell for 60 Zn : J z states 5,053,594 J = 0,T = 0 states 81,804, 784 J = 6,T =1 states Can we get around this problem? Effective interactions in truncated spaces (P-included, finite; Q-excluded, infinite) Residual interaction (G-matrix) depends on the configuration space. Effective charges Breaks down around particle drip lines Residual interactioni P + Q =1

24 Microscopic valence-space Shell Model Hamiltonian Energy (MeV) Coupled Cluster Effective Interaction (valence cluster expansion) CCEI Exp. 22 O USD G.R. Jansen et al., Phys. Rev. Lett. 113, (2014) Energy (MeV) In-medium SRG Effective Interaction O MBPT IM-SRG NN+3N-ind IM-SRG NN+3N-full 4 + (4 + ) (2 + ) (0 + ) Expt. S.K. Bogner et al., Phys. Rev. Lett. 113, (2014)

25 Diagonalization Shell Model (medium-mass nuclei reached;dimensions 10 9!) Honma, Otsuka et al., PRC69, (2004) Martinez-Pinedo ENAM 04

26 26

27 Nuclear Density Functional Theory and Extensions NN+NNN interac+ons Input Density dependent interac+ons Density Matrix Expansion Technology to calculate observables Global properties Spectroscopy DFT Solvers Functional form Functional optimization Estimation of theoretical errors Op+miza+on Energy Density Func+onal Fit- observables experiment pseudo data DFT varia+onal principle HF, HFB (self- consistency) Symmetry breaking Symmetry restora+on Mul+- reference DFT (GCM) two fermi liquids Time dependent DFT (TDHFB) self-bound superfluid (ph and pp channels) self-consistent mean-fields broken-symmetry generalized product states Observables Direct comparison with experiment Pseudo- data for reac+ons and astrophysics

28 Mean-Field Theory Density Functional Theory Degrees of freedom: nucleonic densities Nuclear DFT two fermi liquids self-bound superfluid mean-field one-body densities zero-range local densities finite-range gradient terms particle-hole and pairing channels Has been extremely successful. A broken-symmetry generalized product state does surprisingly good job for nuclei.

29 Nuclear Energy Density Functional isoscalar (T=0) density isovector (T=1) density ( ρ 0 = ρ n + ρ p ) ( ρ 1 = ρ n ρ p ) +isoscalar and isovector densities: spin, current, spin-current tensor, kinetic, and kinetic-spin + pairing densities E = H(r)d 3 r p-h density p-p density (pairing functional) Expansion in densities and their derivatives Constrained by microscopic theory: ab-initio functionals provide quasi-data! Not all terms are equally important. Usually ~12 terms considered Some terms probe specific experimental data Pairing functional poorly determined. Usually 1-2 terms active. Becomes very simple in limiting cases (e.g., unitary limit) Can be extended into multi-reference DFT (GCM) and projected DFT

30 Examples: Nuclear Density Functional Theory Traditional (limited) functionals provide quantitative description BE differences Mass table δm=0.581 MeV Goriely, Chamel, Pearson: HFB-17 Phys. Rev. Lett. 102, (2009) Cwiok et al., Nature, 433, 705 (2005)

31 Description of observables and model-based extrapolation Systematic errors (due to incorrect assumptions/poor modeling) Statistical errors (optimization and numerical errors) S 2n (MeV) S 2p (MeV) proton number FRDM HFB-21 SLy4 UNEDF1 UNEDF0 SV-min exp 4 N= drip line Er experiment neutron number Erler et al., Nature 486, 509 (2012) S 2n (MeV) 2 0 Er neutron number

32 Quantified Nuclear Landscape proton number Z=28 Z=20 stable nuclei known nuclei drip line S 2n = 2 MeV SV-min Z= ~3,000 Z=82 N=82 two-proton drip line N= N=28 N=50 Nuclear Landscape 2012 N=20 neutron number 0 current proton number N=184 neutron number two-neutron drip line How many protons and neutrons can be bound in a nucleus? Erler et al. Nature 486, 509 (2012) from B. Sherrill Literature: 5,000-12,000 Skyrme- DFT: 6,900±500 syst N=258 Asymptotic freedom? FRIB DFT

33 From nuclei to neutron stars (a multiscale problem) Gandolfi et al. PRC85, (2012) J. Erler et al., PRC 87, (2013) M (M solar ) NN NN+NNN 1.97(4) 0.15 fm fm R (km) _ R skin ( 208 Pb)/R skin SV-min C AB =0.82 _ R(1.4 M.)/R The covariance ellipsoid for the neutron skin R skin in 208 Pb and the radius of a 1.4M neutron star. The mean values are: R(1.4M )=12 km and R skin = 0.17 fm. Major uncertainty: density dependence of the symmetry energy. Depends on T=3/2 three-nucleon forces

34 ISNET: Enhancing the interac+on between nuclear experiment and theory through informa+on and sta+s+cs JPG Focus Issue: h9p://iopscience.iop.org/ /page/isnet Around 35 papers (including nuclear structure, reac+ons, nuclear astrophysics, medium energy physics, sta+s+cal methods and fission ) Remember that all models are wrong; the prac+cal ques+on is how wrong do they have to be to not be useful (E.P. Box) Error es+mates of theore+cal models: a guide J. Phys. G (2014)

35 Informa+on Content of New Measurements J. McDonnell et al. Phys. Rev. Le9. 114, (2015) Developed a Bayesian framework to quantify and propagate statistical uncertainties of EDFs. Showed that new precise mass measurements do not impose sufficient constraints to lead to significant changes in the current DFT models (models are not precise enough) Bivariate marginal estimates of the posterior distribution for the 12- dimensional DFT UNEDF 1 parameterization. We can quantify the statement: New data will provide stringent constraints on theory