2358-19 Joint ICTP-IAEA Workshop on Nuclear Structure Decay Data: Theory and Evaluation 6-17 August 2012 Introduction to Nuclear Physics - 1 P. Van Isacker GANIL, Grand Accelerateur National d'ions Lourds France
Nuclear Structure (I) Single-particle models P. Van Isacker, GANIL, France
Overview of nuclear models Ab initio methods: Description of nuclei starting from the bare nn & nnn interactions. Nuclear shell model: Nuclear average potential + (residual) interaction between nucleons. Mean-field methods: Nuclear average potential with global parametrisation (+ correlations). Phenomenological models: Specific nuclei or properties with local parametrisation.
Nuclear shell model Many-body quantum mechanical problem: ˆ H = A p k 2 A + V ˆ ( 2m 2 r k,r ) l k =1 k k< l A 2 p = k + V ˆ A A ( r ) 2m k + V ˆ ( 2 r k,r ) l V ˆ ( r ) k k =1 k k< l k =1 mean field residual interaction Independent-particle assumption. Choose V and neglect residual interaction: H ˆ H ˆ IP = A k=1 2 p k + V ˆ r k 2m k ( )
Independent-particle shell model Solution for one particle: p 2 2m + V ˆ r ( ) φ i r ( ) = E i φ i ( r) Solution for many particles: Φ i1 i 2 i A ˆ H IP Φ i1 i 2 i A A k=1 ( r 1,r 2,,r A ) = φ ik r k r 1,r 2,,r A A k=1 ( ) ( ) = E ik Φ i1 i 2 i A ( r 1,r 2,,r A )
Independent-particle shell model Anti-symmetric solution for many particles (Slater determinant): Ψ i1 i 2 i A ( r 1,r 2,,r A ) = 1 A! φ i1 r 1 Example for A=2 particles: ( ) = 1 2 φ i 1 r 1 Ψ i1 i 2 r 1,r 2 ( )φ i2 r 2 ( ) φ i1 ( r 2 ) φ i1 ( r A ) ( ) φ i2 ( r 2 ) φ i2 ( r A ) φ i2 r 1 φ ia r 1 ( ) φ ia ( r 2 ) φ ia ( r A ) ( ) φ i1 r 2 ( )φ i2 r 1 ( ) [ ]
Hartree-Fock approximation Vary φ i (ie V) to minize the expectation value of H in a Slater determinant: δ * Ψ i1 i 2 i A * Ψ i1 i 2 i A ( ) ˆ r 1,r 2,,r A H Ψ i1 i 2 i A ( r 1,r 2,,r A )dr 1 dr 2 dr A ( r 1,r 2,,r A )Ψ i1 i 2 i A ( r 1,r 2,,r A )dr 1 dr 2 dr A = 0 Application requires choice of H. Many global parametrizations (Skyrme, Gogny, ) have been developed.
Poor man s Hartree-Fock Choose a simple, analytically solvable V that approximates the microscopic HF potential: ˆ H IP = Contains A k=1 2 p k 2m + mω 2 2 r 2 2 k ζ l k s k κ l k Harmonic oscillator potential with constant ω. Spin-orbit term with strength ζ. Orbit-orbit term with strength κ. Adjust ω, ζ and κ to best reproduce HF.
Harmonic oscillator solution Energy eigenvalues of the harmonic oscillator: E nlj = ( N + 3 2) ω κ 2 l l +1 N = 2n + l = 0,1,2, : n = 0,1,2, : l = N,N 2,,1 or 0 : j = l ± 1 2 : m j = j, j +1,,+ j : 1 ( ) + ζ l j = l + 1 2 2 2 1 l +1 2 ( ) j = l 1 2 oscillator quantum number radial quantum number orbital angular momentum total angular momentum z projection of j
Energy levels of harmonic oscillator Typical parameter values: ω 41 A 1/ 3 MeV ζ 2 20 A 2 / 3 MeV κ 2 0.1MeV b 1.0 A 1/ 6 fm Magic numbers at 2, 8, 20, 28, 50, 82, 126, 184,
Why an orbit-orbit term? Nuclear mean field is close to Woods-Saxon: ˆ V WS V 0 ( r) = 1+ exp r R 0 a 2n+l=N degeneracy is lifted E l < E l-2 <
Why a spin-orbit term? Relativistic origin (ie Dirac equation). From general invariance principles: ˆ V SO = ζ r ( )l s, ζ r ( ) = r 2 0 r V r [ = ζ in HO] Spin-orbit term is surface peaked diminishes for diffuse potentials.
Evidence for shell structure Evidence for nuclear shell structure from 2 + in even-even nuclei [E x, B(E2)]. Nucleon-separation energies & nuclear masses. Nuclear level densities. Reaction cross sections. Is nuclear shell structure modified away from the line of stability?
Ionisation potential in atoms
Neutron separation energies
Proton separation energies
Liquid-drop mass formula Binding energy of an atomic nucleus: Z( Z 1) ( ) = a v A a s A 2 / 3 a c B N,Z Δ( N,Z) + a p A 1/ 3 For 2149 nuclei (N,Z 8) in AME03: a v 16, a s 18, a c 0.71, a' s 23, a p 6 σ rms 2.93 MeV. A 1/ 3 a ʹ s ( N Z) 2 A C.F. von Weizsäcker, Z. Phys. 96 (1935) 431 H.A. Bethe & R.F. Bacher, Rev. Mod. Phys. 8 (1936) 82
The nuclear mass surface
The unfolding of the mass surface
Modified liquid-drop formula Add surface, Wigner and shell corrections: Z( Z 1) ( ) = a v A a s A 2 / 3 a c B N,Z S v 4T T +1 1+ y s A 1/ 3 A For 2149 nuclei (N,Z 8) in AME03: a v 16, a s 18, a c 0.71, S v 35, y s 2.9, a p 5.5, a f 0.85, a ff 0.016 σ rms 1.16 MeV. ( ) A 1/ 3 a f Δ( N,Z) + a p A 1/ 3 ( n ν + n π ) + a ff ( n ν + n π ) 2
Shell-corrected LDM
Shell structure from E x (2 1 )
Evidence for IP shell model Ground-state spins and parities of nuclei: j in φ nljmj J l in φ nljmj ( ) l = π J π
Validity of SM wave functions Example: Elastic electron scattering on 206Pb and 205Tl, differing by a 3s proton. Measured ratio agrees with shell-model prediction for 3s orbit. J.M. Cavedon et al., Phys. Rev. Lett. 49 (1982) 978
Variable shell structure
Beyond Hartree-Fock Hartree-Fock-Bogoliubov (HFB): Includes pairing correlations in mean-field treatment. Tamm-Dancoff approximation (TDA): Ground state: closed-shell HF configuration Excited states: mixed 1p-1h configurations Random-phase approximation (RPA): Correlations in the ground state by treating it on the same footing as 1p-1h excitations.
Nuclear shell model The full shell-model hamiltonian: ˆ H = A 2 p k 2m + V ˆ A ( r k ) + V ˆ RI r k,r l k=1 k<l ( ) Valence nucleons: Neutrons or protons that are in excess of the last, completely filled shell. Usual approximation: Consider the residual interaction V RI among valence nucleons only. Sometimes: Include selected core excitations ( intruder states).
Residual shell-model interaction Several approaches: Effective: Derive from free nn interaction taking account of the nuclear medium. Empirical: Adjust matrix elements of residual interaction to data. Examples: p, sd and pf shells. Effective-empirical: Effective interaction with some adjusted (monopole) matrix elements. Schematic: Assume a simple spatial form and calculate its matrix elements in a harmonic-oscillator basis. Example: δ interaction.
Schematic short-range interaction Delta interaction in harmonic-oscillator basis: Example of 42 Sc 21 (1 neutron + 1 proton):
Large-scale shell model Large Hilbert spaces: Ψ i'1 i' 2 i' A n k<l V ˆ RI ( r k,r l ) Diagonalization : <10 10. Monte Carlo : <10 15. Example : 8n + 8p in pfg 9/2 ( 56 Ni). Ψ i1 i 2 i A M. Honma et al., Phys. Rev. C 69 (2004) 034335
Symmetries of the shell model Three bench-mark solutions: No residual interaction IP shell model. Pairing (in jj coupling) Racah s SU(2). Quadrupole (in LS coupling) Elliott s SU(3). Symmetry triangle: ˆ H = A k =1 A 1 k< l 2 p k 2m + 1 2 mω 2 r 2 k ζ ˆ ls l k s ˆ k ζ ˆ ll ( ) + V ˆ RI r k,r l l k 2
The three faces of the shell model
Racah s SU(2) pairing model Assume pairing interaction in a single-j shell: j 2 JM J ˆ V pairing r 1,r 2 Spectrum 210 Pb: ( ) j 2 JM J = 1 2 j +1 2 ( )g 0, J = 0 0, J 0
Solution of the pairing hamiltonian Analytic solution of pairing hamiltonian for identical nucleons in a single-j shell: j n υj n V ˆ pairing ( r k,r l ) j n 1 υj = g 0 n υ 4 1 k<l ( )( 2 j n υ + 3) Seniority υ (number of nucleons not in pairs coupled to J=0) is a good quantum number. Correlated ground-state solution (cf. BCS). G. Racah, Phys. Rev. 63 (1943) 367
Nuclear superfluidity Ground states of pairing hamiltonian have the following correlated character: Even-even nucleus (υ=0): Odd-mass nucleus (υ=1): Nuclear superfluidity leads to ( S ˆ ) n / 2 + o, S ˆ + = + a ˆ ˆ m S + ( ) n / 2 o Constant energy of first 2 + in even-even nuclei. Odd-even staggering in masses. Smooth variation of two-nucleon separation energies with nucleon number. Two-particle (2n or 2p) transfer enhancement. m a ˆ + + m a ˆ m
Two-nucleon separation energies Two-nucleon separation energies S 2n : (a) Shell splitting dominates over interaction. (b) Interaction dominates over shell splitting. (c) S 2n in tin isotopes.
Pairing gap in semi-magic nuclei Even-even nuclei: Ground state: υ=0. First-excited state: υ=2. Pairing produces constant energy gap: E ( + x 2 ) 1 = 1 2 j +1 2 ( )G Example of Sn isotopes:
Elliott s SU(3) model of rotation Harmonic oscillator mean field (no spin-orbit) with residual interaction of quadrupole type: ˆ H = A k=1 A k=1 2 p k 2m + 1 2 mω 2 2 r k Q ˆ µ r 2 k Y 2µ ˆ A k=1 ( ) r k ( ) + p k 2 Y 2µ ˆ p k g 2 ˆ Q ˆ Q, J.P. Elliott, Proc. Roy. Soc. A 245 (1958) 128; 562