Joint ICTP-IAEA Workshop on Nuclear Structure Decay Data: Theory and Evaluation August Introduction to Nuclear Physics - 1

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1 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

2 Nuclear Structure (I) Single-particle models P. Van Isacker, GANIL, France

3 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.

4 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 ( )

5 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 )

6 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 ( ) [ ]

7 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.

8 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.

9 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 l +1 2 ( ) j = l 1 2 oscillator quantum number radial quantum number orbital angular momentum total angular momentum z projection of j

10 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,

11 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 <

12 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.

13 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?

14 Ionisation potential in atoms

15 Neutron separation energies

16 Proton separation energies

17 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

18 The nuclear mass surface

19 The unfolding of the mass surface

20 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 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 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

21 Shell-corrected LDM

22 Shell structure from E x (2 1 )

23 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

24 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

25 Variable shell structure

26 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.

27 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).

28 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.

29 Schematic short-range interaction Delta interaction in harmonic-oscillator basis: Example of 42 Sc 21 (1 neutron + 1 proton):

30 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 : < Monte Carlo : < Example : 8n + 8p in pfg 9/2 ( 56 Ni). Ψ i1 i 2 i A M. Honma et al., Phys. Rev. C 69 (2004)

31 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 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

32 The three faces of the shell model

33 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

34 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

35 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

36 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.

37 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:

38 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 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