Nuclear physics: a laboratory for many-particle quantum mechanics or From model to theory in nuclear structure physics G.F. Bertsch University of Washington Stockholm University and the Royal Institute Technology Sept. 18, 2014 1. Models vs. theories 2. The nuclear landscape 3. The unitary fermion gas model 4. Light nuclei ab initio 5. The multiconfiguration shell model 6. Models of pairing 7. Self-consistent mean field and density-functional theory 8. Extensions of SCMF
What are models good for? N. Bohr (1937) Models: -capture the essential phenomena -are simple, at least in formulation
Examples of information nuclear models The shell model model H = 2 2 2m V 1 0f(r)+V s r df dr s 4 parameters Explains magic numbers, explains many ground-state spins and parities The liquid drop model B(N,Z) =a v A a s A /3 Z 2 a c A a (N Z) 2 1/3 a A + δ A 1/2 A = N + Z 5 parameters Reproduces binding energies to 3 MeV r.m.s., out totals ranging from 100 to 2000 MeV.
Characteristics of good theories -need only a small set of parameters -have wide predictive power -have intrinsic criteria for limits of validity
Starting point: the nuclear interaction In the two-body sector, the nuclear interaction is just barely able, or not, to produce a bound state. nn channel: r ~ 2 fm; a_nn ~ 18 fm np channel: V ~ -100 MeV; E_d = -2.2 MeV! (deg) 60 40 20 0 Argonne v 18 np Argonne v 18 pp Argonne v 18 nn SAID 7/06 np! (deg) 160 120 80 40 3 S1 Argonne v 18 np SAID 7/06 np -20 1 S0 0-40 0 100 200 300 400 500 600 E lab (MeV) -40 0 100 200 300 400 500 600 E lab (MeV) An informative model: the unitary Fermi gas Interaction is zero-range with a strength that produces a zero-energy bound state
The unitary model in the three-body system--what we learned 1. There are an infinite number of weakly bound 3-body states (Efimov 1971). 2. It is the only nontrivial three-body problem that can be solved exactly. PRL 97, 150401 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending 13 OCTOBER 2006 Unitary Quantum Three-Body Problem in a Harmonic Trap Félix Werner and Yvan Castin Laboratoire Kastler Brossel, École Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, France (Received 18 July 2005; published 10 October 2006) We consider either 3 spinless bosons or 3 equal mass spin-1=2 fermions, interacting via a short-range potential of infinite scattering length and trapped in an isotropic harmonic potential. For a zero-range model, we obtain analytically the exact spectrum and eigenfunctions: for fermions all the states are universal; for bosons there is a coexistence of decoupled universal and efimovian states. All the universal states, even the bosonic ones, have a tiny 3-body loss rate. For a finite range model, we numerically find for bosons a coupling between zero angular momentum universal and efimovian states; the coupling is so weak that, for realistic values of the interaction range, these bosonic universal states remain long-lived and observable. 3. Exact solutions are useful for testing numerical methods (GFMC, hyperspherical harmonics,...)
The uniform gas of fermions with short-ranged interactions H = i 2 i 2m + g i<j δ(r i r j ) Interaction is only defined in a cutoff space; the physical expansion parameter is the scattering length a. Perturbative limit (1960ʼs many-body theory): Now we look at the limit a The Hamiltonian has no internal parameters! E N = ξ 2 kf 2 2m 3 5
LATTICE MONTE CARLO CALCULATIONS FOR UNITARY... PHYSICAL REVIEW A 87, 023615(2013) TABLE VI. Historical results for the Bertsch parameter ξ determined experimentally (expt.), by numerical simulation (sim.) and by analytic calculation (anal.), along with publication (pub.) date. Values obtained variationally are upper bounds and are indicated with an asterisk; simulation results without a quoted error bar should be regarded as approximate. Pub. date ξ (expt.) Ref. Pub. date ξ (sim.) Ref. Pub. date ξ (anal.) Ref. 2002-11-07 0.90 (7) [2] 2003-07-31 0.44 (1) [28] 1997-06-01 0.59 [14] 2003-07-02 0.74 (7) [3] 2004-10-05 0.44 (1) [29] 1999-10-01 0.326 [15] 2004-07-27 0.36 (15) [4] 2004-11-10 0.42 (1) [30] 1999-10-01 0.568 [15] 2004-03-23 0.313 10 [5] 2005-08-02 0.42 (1) [31] 2000-10-19 4/9 [16] 2005-01-16 0.51 (4) [6] 2006-01-18 0.07-0.42 [32] 2001-03-14 0.326 [17] 2005-12-14 0.46 (5) [7] 2006-03-10 0.44 [33] 2004-09-03 0.455 [18] 2005-12-16 0.38 (7) [8] 2006-03-17 0.25 (3) [34] 2005-08-30 0.32 [19] 2006-11-30 0.46 +12 5 [9] 2007-06-14 0.449 (9) [35] 2005-08-30 0.24 [19] 2008-11-11 0.39 (2) [10] 2008-03-07 0.31 (1) [36] 2005-08-30 0.5 [19] 2010-04-15 0.41 (1) [11] 2008-03-07 0.306 (1) [36] 2005-10-11 0.42 [20] 2010-02-25 0.415 (10) [12] 2008-08-13 0.292 (12) [37] 2006-08-04 0.475 [21] 2012-01-12 0.376 (4) [13] 2008-08-13 0.329 (5) [37] 2007-02-08 0.36 (1) [22] 2008-08-21 0.37 (5) [38] 2007-04-18 0.279 [23] 2009-05-11 0.292 (24) [39] 2007-04-05 0.300 [24] 2009-11-19 0.4 [40] 2007-04-05 0.367 [24] 2011-04-01 0.383 (1) [41] 2007-04-05 0.359 [24] 2011-06-10 0.383 (1) [42] 2007-04-05 0.376 [24] 2011-08-08 0.398 (3) [43] 2007-06-18 0.391 [25] 2011-12-07 0.372 (5) [44] 2007-06-18 0.364 [25] 2007-06-18 0.378 [25] 2007-07-01 4/9 [26] 2009-01-27 0.377 (14) [27] avoids the multiplicative enhancement in computational cost (4) The many-body ground-state energies (measured in
Getting more realistic Neutron matter equation of state E/N [MeV] 20 15 10 EM 500 MeV EGM 450/500 MeV EGM 450/700 MeV NLO lattice (2009) QMC (2010) APR (1998) GCR (2012) 5 0 0 0.05 0.1 0.15 n [fm -3 ] Tews, et al. PRL 110 032504 (2013)
Getting more realistic on the nuclear interaction About 20 parameters are required to describe the nuclear interaction in sufficient detail to reproduce experimental scattering phase shifts and properties of the deuteron. The Hamiltonian based on that interaction underbinds the 3-nucleon system, overbinds nuclear matter, and produces a spin-orbit field that is too weak. The deficiencies are corrected by including a 3-body interaction arising from pions.
Calculational method of choice for light nuclei: Greenʼs function Monte Carlo S. Pieper et al, Phys. Rev. C70 054325
Excitation energy (MeV) 25 20 15 10 5 0-5 Argonne v 18 with Illinois-7 GFMC Calculations Including IL7 gives correct s.-o. splitting & 10 B g.s. 6 He 0 + 6 Li 1 + 3 + 1 + 7 Li 7/2! 5/2! 5/2! 7/2! 1/2! 3/2! 8 Li 4 + 1 + 3 + 1 + 8 Be 0 + 4 + 3 + 1 + 4 + 0 + 9 Be 7/ 5/ 7/2! 7/2! 3/2! 3/ 5/ 1/2! 5/2! 1/ 3/2! AV18 10 Be AV18 +IL7 Expt. 3, 1 + 0 + 4 + 3 + 0 + 3 + 1 + 1 + 10 B 3 + 4 + 3 + 1 + 1 + 3 + S. Pieper et al, Phys. Rev. C70 054325
Energy (MeV) -20-30 -40-50 -60-70 -80-90 -100 0 + 0 + 4 He 6 He 6 Li 1 + 3 + 1 + 7 Li 7/2! 5/2! 5/2! 7/2! 1/2! 3/2! 8 He 0 + Argonne v 18 with Illinois-7 GFMC Calculations 4 + 0 + 1 + 3 + 1 + 0 + 1 + 4 + 3 + 1 + 4 + 8 Li 9 Li 0 + 8 Be AV18 +IL7 5/2! 1/2! 3/2! 9 Be Expt. 7/ 5/ 7/2! 7/2! 3/2! 3/ 5/ 1/2! 5/2! 1/ 3/2! 1 + 3, 0 + 0 + 3 + 4 + 3 + 1 + 1 + 10 3 Be + 10 B 12 C 0 + 0 +
Multi-configuration shell model See E. Caurier, et al., Rev. Mod. Phys. 77 431 (2005). H = i ε i a i a i + i<j v ij,kl a i a j a la k i =(n,, j, m) labels a shell orbital k<l
Domain of configuration-interaction shell model Matrix size < 10^8
Shell-model Monte Carlo method (an early review: Physics Reports 278 2 (1997). H = i ε i a i a i + i<j v ij,kl a i a j a la k k<l Single-particle energies from the Woods-Saxon potential v ij,kl from a self-consistent separable interaction v(r 1,r 2 )= v L f L (r 1 )f L (r 2 ) Y LM (ˆr 1 )Y LM (ˆr 2 ) L M plus a pairing interaction ( discussed later). L =2, 3, 4 Shell-model Shell-model Monte Carlo Monte method Carlo method
The Monte-Carlo sign problem is avoided by good luck and a good choice of representation. (All v_l are negative and the pairing interaction is expressed as a density-density interaction between like particles.)
Application to the level density of Dy-162 All the shell orbitals between Z=50-90 and N=82-136 are included. Dimension of space: D(M=0) = 6 10^21 PRL 101, 082501 (2008) PHYSICAL R 10 10 10 8 ρ (MeV -1 ) 10 5 10 0 E M E (MeV) -370-372 -374-376 4 8 12 16 20 β (MeV -1 ) 0 5 10 15 E X (MeV) ρ (MeV -1 ) 10 4 10 0 E M N(E x ) lnρ 600 300 0 2 4 6 8 E x (MeV) 0 6 4 2 0 1 2 E x (MeV)
Pairing in nuclei: the phenomenon Definition of pairing gap: (3) o,z (N) =1 2 (E b(z, N + 1) 2E b (Z, N)+E b (Z, N 1)) 3 2.5 Odd N 2 (3) (MeV) 1.5 1 0.5 0-0.5 3 2.5 20 40 60 80 100 120 140 Neutron number N Even N 2 (3) e (MeV) 1.5 1 0.5 0-0.5 20 40 60 80 100 120 140 Neutron number N
The seniority model Define an SU(2) algebra of operators in the Fock space of a j-shell. P + = m>0 a ma m P = P + P 0 = m a ma m The Hamiltonian H = GP + P E v (N) = 1 G(N v)(2ω N v + 2) 4 Ca isotopes in the f_7/2 shell Interaction energy 0-1 -2-3 -4-5 -6-7 -8 0 1 2 3 4 5 6 7 8 neutron number N
The Richardson model See Dukelsky, et al., Rev. Mod. Phys. 76 643 (2004). The model has been widely used to assess the accuracy of BCS theory in various contexts.
0.4 FIG. 7. Neutron average variances σ between HFB and experimental (3) o as a function of the calculated quadrupole deformation β 2.Thedeformationrange 0.2 β 2 0.4 was divided into bins with β 2 = 0.1, and for each bin the variance σ was obtained by averaging the residuals over all nuclei available. Pairing gaps calculated by self-consistent mean-field theory below another mechanism that could simulate the observed trend in A, namely,anisospindependenceoftheeffective pairing interaction. 1.6 Neutrons exp HFB (3) oz 1.2 0.8 0.4 ο (MeV) (3) 1.6 20 40 60 80 100 120 140 Neutron number N 2.0 Protons Bertsch, et al., Phys. Rev. C 79 034306 (2009). 1.2 0.8
Nuclear deformation H.B.G. Casimir (1937):.One is led to the conclusion that the quadrupole moment cannot possibly be due to one proton... It might even been that the nucleus as a whole has a prolate shape and that the nucleus as a whole is rotating about its major axis. Energy gap in the excitation spectrum of even-even nuclei, scaled to 2 Delta(3). 1 0.8 0.6 Ratio 0.4 0.2 0 50 100 150 200 250 Mass number A
The Nilsson model H = P 2 2M + 1 2 M ω 2 zz ω 2 (x y 2 ) + c( 2 2 )+d s δ ω ω z The nucleus is most stable where there are gaps in the single-particle spectrum.
Self-consistent mean field and density-functional theory H = ij 2 2m ij a i a j + v ij,kl a i a j a la k + 3body i<j,k<l Minimize <H> in the space of Slater determinants a i a j = ρ ij a i a j a a k = ρ ik ρ j ρ i ρ jk or in the space of Bogoliubov vacua a i a j = κ ij a i a j a a k = ρ ik ρ j ρ i ρ jk + κ jiκ k Basis of states can be coordinate space mesh, momentum space mesh, or 3-d harmonic oscillator states. There are two popular families of interactions, the Skyrme family based on contact interactions and derivatives and the Gogny family based on short-ranged Gaussian functions. Both have a density-dependent contact interaction for the 3-body term. See Bender, Heenen and Reihard, Rev. Mod. Phys. 75 121 (2003).
Performance of SCMF on nuclear binding energies For comparision, the 5 parameter liquid drop model reproduces binding energies with a 3 MeV RMS error. -SCMF improvement is a factor 2 -SCMF with phenomenology terms for correlation effects improvement is a factor of 5 -same or better improvement can be obtained with the liquid drop model plus shell corrections plus phenomenological correlation terms
Correlations and the restoration of broken symmetries SCMF of nuclei breaks translational invariance. Depending on the nucleus, it may also breaks rotational symmetry. In the Bogoliubov theory, it further breaks particle number conservation. PHYSICAL REVIEW C 79, 014311(2009) The paradox of self-consistent mean-field theory: the physical states of course respect the symmetries of the Hamiltonion, but one often does better by breaking them.
Extensions of self-consistent mean-field theory for spectroscopy RPA,quasiparticle RPA, time-dependent HF, HFB, DFT Generator Coordinate Methods Collective Hamiltonian Discrete-basis Hill-Wheeler
Progress on a global theory for even-even nuclei HFB mapped to the 5-dimensional Bohr Hamiltonian Delaroche et al., PRC 81 (2010) 13 parameters in the energy functional 100 (a) 80 Z 60 40 20 sign[cos(3 )] HFB 0.40 0.30 0.20 0.10 0.00-0.10-0.20 20 40 60 80 100 120 140 160 180 200 N
STRUCTURE OF EVEN-EVEN NUCLEI USING A MAPPED... PHYSICAL REVIEW C 81,014303(2010) E (MeV) 2 1 6 + 4 + 2796 2468 0 + Expt 1533 2409 19 76 Kr 4 + 6 + Th (3 + 2 ) + 19 861 2993 4 + 3029 2325 0 + 809 2338 4 3 + + 1482 1727 1127 253 0 0 + 1443 0 + 1169 FIG. 1. Experimental and theoretical spectra (MeV) and transition strengths (e 2 fm 4 )of 76 Kr showing the excitations that we examine in the present global study. The experimental spectrum, on the left, is from Ref. [23] as well as from data repository for the 3 + and 4 + members of the γ band [24]. Calculated values are those from Ref. [23].
Global assessment of accuracy: first excited J=2 state a) Excitation Energies 10 b) B(E2,0 +! ) 184 Hg Theoretical (MeV) 1 0.1 Theoretical (e 2 b 2 ) 1 0.1 0.01 18 Ne 0.1 1 Experimental (MeV) 0.01 0.1 1 10 Experimental (e 2 b 2 ) E(exp) = E(theory)+/-40% over 2 order of magnitude Many other observables
The CEA/DAM global survey, on the N=Z line 2+ Excitation energies 2.5 Energy (MeV) 2 1.5 1 0.5 0 SLy4 Exp 5dch 32 34 36 38 40 42 44 46 48 Z 12
Nature 469 68 (2011) 2.5 Energy (MeV) 2 1.5 1 0.5 _... 0 32 34 36 38 40 42 44 46 48 Z 13
A near-term goal: odd-a nuclei No collective Hamiltonian to map onto Restoring angular momentum more difficult Projection technique Onishi Formula (1966) Ψ 1 Ψ 0 = det(u 0 U 1 + V 0 V 1) Robledo s Formula (2009) Ψ 1 Ψ 0 =pf (V1 U 1 1 ) 1 1 (V 0 U 1 0 )
4 FIG. 3: (Color online) Comparison of the calculated lowlying levels grouped into rotational bands with data taken from Ref. [22]. FIG. 4: (Color online) Excitation energies of states in the ground-state band of 25 Mg and B(E2) and B(M1) values for transitions between them. 3/ level in panel (b) even 8.4 MeV. These values are Bender, Heenen, et al., arxiv: 1406.5983 (2014).
Charge radii-- HFB Experimental data from Angeli, ADNDT 87 (2004) 0.03 0.02 a) Charge radius HFB Cm Relative Error 0.01 0-0.01-0.02-0.03 50 100 150 200 250 Mass Number A U 0.03 0.02 a) Charge radius HFB Cm Relative Error 0.01 0-0.01-0.02-0.03 50 100 150 200 250 Mass Number A U
Error assessment INT09-1: Effective field theories and the many-body problem Example: intrinsic error from shell truncation in CI fits Estimated error of 0.2 MeV is close to actual value.