Modeling the Atomic Nucleus Theoretical bag of tricks
The nuclear many-body problem
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
Interfaces provide crucial clues number of nuclei < number of processors!
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
Overview Basics Highlights Outlook Matrices ET 3NF RG Soft Applying e (H E T ) to a trial ground state vector Consider a vector var.i and its expansion in eigenstates of the Hamiltonian H: var.i = X k C k k i where H k i = E k k i E.g., var.i is a variational guess for the ground state General: f (H) k i = f (E k ) k i (where f specified by power series) Later: powers of H (Lanczos method) Here, apply imaginary time propagation e iht with = it: (!1)i = lim!1 e (H E T ) var.i!1! C 0 e (E 0 E T ) 0 i We project out the ground state! [MATLAB example available] Note the use of the trial energy E T. Why? How do I get E 0? In practice, we break up the imaginary time into small intervals to be able to calculate: e (H E T ) = e (H E T ) Dick Furnstahl New methods
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
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
GFMC: S. Pieper, ANL 1-2% calculations of A = 6 12 nuclear energies are possible excited states with the same quantum numbers computed
dinal form factor F(q) 10 0 10-1 10-2 10-3 12 C: ground state and Hoyle state exp ρ 1b ρ 1b+2b state-of-the-art computing Wiringa et al. Phys. Rev. C 89, 024305 (2014); A. Lovato et al., Phys. Rev. Lett. 112, 182502 (2014) ρ ch (r) 0.08 0.04 0.00 0 2 4 r (fm) 10-4 0 1 2 3 4 q (fm -1 ) E [MeV] 82 84 86 88 90 92 f pt (k) 10-1 10-2 10-3 12 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 2013 10-4 0 1 2 3 4 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 92.16 MeV and the point rms radius 1 is 2.35 fm vs Pieper et al., 2.33 QMC from experiment. 0 0.5 Data from M. Chernykh et al., Phys. Rev. Lett. 105, 022501 (2010) 2 + 2 + 82.6(1) Right panel [f tr (k)/k 2 ] proportional to M(E0) at k =0 0 + 83(3) 0 + 84.51 85(3) Large errors at small k due to large Monte Carlo errors 2 + 2 + Can get better value at k =0by computing R drr 2 r 2 tr (r) 87.72 0 + 92.16 Exp 88(2) Balancing) version of GFMC was used to potential AV18+IL7) on 32,000 processors Results with best 0 + 2 wave function in good agreement with data 0 + Epelbaum et al., Phys. Rev. Lett. 109, 92(3) 252501 (2012). Lattice EFT Th Lahde et al., Phys. Lett. B 732, 110 (2014). -91.7(2) k 2 (fm
S2n (MeV) 20 18 16 14 12 10 8 6 4 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, 032506 (2012) AME2003 TITAN K Ca Sc 27 28 29 30 31 32 33 34 35 Neutron Number N 54 Ca 2 + S 2n (MeV) 18 16 14 12 10 8 6 4 2 54 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 28 29 30 31 32 33 34 35 36 37 38 Neutron number N CC theory Hagen et al., PRL109, 032502 (2012) RIBF@RIKEN Steppenbeck et al Nature (2013)
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, 202502 (2011) GT matrix element 0.03 0.02 0.01 0-0.01-0.02-0.03 0.3 0.2 0.1 0-0.1 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
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 ( )
ˆ H = t i + 1 2 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 : 2 10 9 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
Overview Basics Highlights Outlook CI QMC CC React SM NM DFT Atoms Unstable proton-dripping fluorine-14 with NCSM Ab initio calculation using soft inverse-scattering potential New: theory preceded recent experimental measurement! P. Maris et al., PRC 81, 021301(R) (2010) V.Z. Goldberg et al., Phys. Lett. B 692, 307 (2010) Matrix dimension 2 10 9, 2.5 hours on 30,000 cores Dick Furnstahl New methods
Overview Basics Highlights Outlook CI QMC CC React SM NM DFT Atoms Asides on Fluorine-14 calculation 14 F decays by proton emission to 13 O =) proton drip line What if 2 10 9 dimension matrix full? > 10 19 bytes storage? =) obviously many matrix elements must be zero! (Sparse) Only about 20 out of 2 billion eigenvalues needed =) Lanczos method! How to scale to 30,000 cores? =) work with computer scientists =) SciDAC! How do you extrapolate N max!1? (ask me later!) Dick Furnstahl New methods
Overview Basics Highlights Outlook CI QMC CC React SM NM DFT Atoms Size and sparsity of Hamiltonian matrices [from P. Maris] M-scheme basis space dimension Hamiltonian matrices grow rapidly with basis size (N max ) and A = N + Z from combinatorics: 10 10 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 4He 6Li 8Be 10B 12C 16O 19F 23Na 27Al 10 0 0 2 4 6 8 10 12 14 N max Dick Furnstahl New methods
Overview Basics Highlights Outlook CI QMC CC React SM NM DFT Atoms Size and sparsity of Hamiltonian matrices [from P. Maris] But fortunately there are many zero elements so storage is large but feasible. How can we take advantage of sparsity? Dick Furnstahl New methods
Overview Basics Highlights Outlook CI QMC CC React SM NM DFT Atoms Size and sparsity of Hamiltonian matrices [from P. Maris] But fortunately there are many zero elements so storage is large but feasible. How can we take advantage of sparsity? 10 18 11Li (Z= 3, N= 8) 10 15 10 12 dimension # nonzero s NN # nonzero s NNN # nonzero s NNNN # nonzero s NNNNN # matrix elements 10 9 10 6 10 3 10 0 0 2 4 6 8 10 12 basis space truncation N max Dick Furnstahl New methods
Overview Basics Highlights Outlook CI QMC CC React SM NM DFT Atoms Lanczos method in short Consider an arbitrary vector i and its expansion in eigenstates of H, where H k i = E k k i. Then H m i = P k C ke m k ki If m large enough, largest E k will dominate the sum =) project out the corresponding eigenvector To get lowest eigenvalue, use (H I) m with >0 large enough so that E 0 > E max More efficient to diagonalize H in the basis spanned by H k i, H 2 k i,...,h m k i Called the Krylov space Lanczos: orthogonalize basis states as you go, generating H in tri-diagonal form, which is efficiently diagonalized Re-orthonormalization for numerical stability Many computational advantages to treating sparse matrices with Lanczos [see J. Vary et al., arxiv:0907.0209] Dick Furnstahl New methods
Overview Basics Highlights Outlook CI QMC CC React SM NM DFT Atoms!"#$%&'()%*+,-'./01%234(%)'%2'.567%!"4,'(2CX*-"&(1*4+#0+)*'(*BJKWQD*.+"4*/"#52#05+K*,%7*'7/+4*#01/7*(%$#+0*%(-+41'*)060#"4*,+7"*-+$".* @07/*/"#52#0>+)*#+))*7/"(*"*-".Y* *Z+6,+4)*'5*[9;<\*$'##",'4"&'(*6"-+*60$4')$'30$* ***(%$#+"4*)74%$7%4+*$"#$%#"&'()*7'*)'#>+*7/+*3%]]#+* *[)+-*).)7+6"&$*$/04"#*^"60#7'(0"(*54'6*#'@2+(+41.* ***+_+$&>+*`+#-*7/+'4.*'5*a!<* *b+.*5+"7%4+a*$'()0)7+(7*!"#$%&'(#)0(7+4"$&'()* Q2(%$#+'(*5'4$+)*)%334+))*$40&$"#*$'63'(+(7*$'63"4+-*7'*I2(%$#+'(*5'4$+)*'(#.* *F'#%&'()*'5*CX!*"(-*CX9*7/4'%1/*^"60#7'(0"(*-0"1'("#0]"&'(* *CDD25'#-*4+-%$&'(*0(*c"6'@2:+##+4*74"()0&'(*6"740=*+#+6+(7**!"#$%#"&'()*+(",#+-*,.*/01/23+45'46"($+* $'63%&(1*7/4'%1/*89!8:;*34'14"6*! <06+()0'(*'5*6"740=*)'#>+-*5'4*?*#'@+)7* )7"7+)A**B*C=CDE*! F'#%&'(*7''G*B*H*/'%4)*'(*ICJKDDD*$'4+)* '(*!4".*L:J*M"1%"4*"7*NO9P* (+7*-+$".*4"7+** 0)*>+4.*)6"##* Dick Furnstahl F$0+($+*4+5RA**S/.)0$"#*O+>0+@*P+T+4)*089K*IDIJDI*UIDCCV*!'63%7"&'("#*4+5RA**S4'$+-0"*!'63%7+4*F$0+($+*0K*EW*UIDCDV* New methods
Overview Basics Highlights Outlook CI QMC CC React SM NM DFT Atoms Asides on Carbon-14 decay calculation Atomic masses [1 amu = 1/12 mass of 12 C] 14 O: 14.0085953 ± 0.0000001 amu 14 N: 14.0030740 ± 0.0000000 amu 14 C: 14.0032420 ± 0.0000000 amu (from online table of nuclides ) How does each decay? Compare lifetimes: 14 C lives long! Calculation with NCSM using chiral EFT potentials and operator for decay ( 14 6 C! 14 7 N + e + e) Scaling enabled by CS/AM collaborations Role of 3NF is key Determining the contribution of one part of Hamiltonian =) Hellmann-Feynman Dick Furnstahl New methods