The Nuclear Shell Model

Transcription

1 The Nuclear Shell Model N. A. Smirnova CEN Bordeaux-Gradignan, France Bridging Methods in Nuclear Theory IPHC, Strasbourg, June 26 30, 2017

2 The Nuclear Shell Model Table of contents 1 Introduction to the nuclear many-body problem: variational against diagonalization methods (lecture of Ph. Quentin on self-consistent theories, Wednesday) 2 Shell model Basis construction Solution of the eigenvalue problem Matrix elements of transition operators Effective nucleon-nucleon (NN) interactions (lecture of L. Bonneau on NN interaction, Thursday lecture of H. Molique on group theory, Friday morning) 3 No-Core Shell Model for Light Nuclei (lecture of R. Lazauskas on few-body problems, Tuesday) 4 Shell-model code ANTOINE (E. Caurier, F. Nowacki, IPHC Strasbourg) afternoon session Lecture of A. P. Zuker Beyond the Shell Model on Friday afternoon

3 Structure of complex nuclei Nuclear charge (matter) density distribution ρ ch ( r) (ρ m( r)) with sharp radius R r 0 A 1/3. Empirical evidence on the existence of an average potential and the corresponding shell structure (From masses, nucleon separation energies, low-energy spectra, etc.) Independent-particle motion near Fermi-level. (From nucleon transfer reactions: nuclear mean-field with strong spin-orbit splitting and large shell gaps) Pairing (superfluid behavior) at low excitation energy. (S n versus S 2n ; two-nucleon spectra, comparison of spectra of even-even and even-odd, odd-even nuclei, etc.) Low-lying multipole (quadrupole) modes, vibrational or rotational energy structures. (Coulomb excitation; scattering of charge particles; heavy-ion fusion-evaporation reactions, etc) The aim of the microscopic theory is to describe these motions starting from a NN force.

4 Nuclear many-body problem Limits of nuclear existence r process protons rp process A=10 A=12 50 neutrons A~60 82 Density Functional Theory self consistent Mean Field ab initio few body calc ulations 0 h Shell Model No Core Shell Model Towards a unified description of the nucleus

5 Nuclear many-body problem Non-relativistic Hamiltonian for A nucleons Ĥ = i=1 p 2 i 2m + [ p 2 ] i Ĥ = 2m + U( r i ) + i=1 }{{} Ĥ (0) i<j=1 i<j=1 W ( r i r j ) W ( r i r j ) U( r i ), i=1 } {{ } ˆV Mean-field theories Search for the most optimum mean-field potential starting from a given two-body interaction + correlations Shell model Schematic average potential + residual interaction

6 Variational approach to the nuclear many-body problem Starting point: antisymmetric product wave function φ α1 (1) φ α1 (2)... φ α1 (A) Ψ(1, 2,..., A) = 1 φ α2 (1) φ α2 (2)... φ α2 (A) A! φ αa (1) φ αa (2)... φ αa (A) The best wave function is determined via a variational principle: δ Ψ H Ψ = δψ H Ψ = 0 with φ αi ( r) 2 d r = 1.

7 Self-consistent mean-field potential Hartree-Fock equations ( 2 2m + U H( r) ) φ i ( r) + U F ( r, r )φ i ( r )d r = ε i φ i ( r) Direct (Hartree) term U H ( r) = φ b ( r )W ( r, r )φ b ( r )dr b F Exchange (Fock) term U F ( r, r ) = b F φ b ( r)w ( r, r )φ b ( r ) Iterative solution of Hartree-Fock equations: φ (0) i ( r) φ (1) i ( r), ε (1) i... φ (HF) i ( r), ε (HF) i U (0) H ( r), U(0) F ( r, r ) U (1) H ( r), U(1) F ( r, r )... U (HF ) ( r)

8 Self-consistent mean-field potential Direct (Hartree) term U H ( r) = b F φ b( r )W ( r, r )φ b ( r )d r U H ( r) = ρ( r )W ( r, r )d r ρ( r) = φ b ( r) 2 b F Example: a delta-force W ( r, r ) δ( r r ) U H ( r) ρ( r)

9 Hartree-Fock ground state properties From the product HF wave function, a number of ground-state properties can be calculated: Ψ HF (1, 2,..., A) = 1 A! A A i=1 φ HF α i (i) Ψ HF Ĥ Ψ HF = E 0 Ψ HF A ˆr i 2 Ψ HF = r 2 i=1 Ψ HF A ˆρ( r i ) Ψ HF = ρ( r) i=1 Application to open-shell nuclei and calculation excitation spectra require inclusion of correlations (pairing, etc) beyond mean-field techniques (lecture of Ph. Quentin).

10 Shell model: energy matrix diagonalization Non-relativistic Hamiltonian for A nucleons [ p 2 ] i Ĥ = 2m + U( r i ) + W ( r i r j ) U( r i ) i=1 }{{} i<j=1 i=1 }{{} ĥ i ˆV Construction of a basis from single-particle states ĥφ α ( r) = ε α φ α ( r) {ε α, φ α ( r)} Spherical potential U( r) = U(r): α = {n α, l α, j α, m α } φ nljm ( r) = R nlj(r) r m l ms [ ] (j) Y l (θ, ϕ) χ 1 2 m }{{} (lm l 1 2 ms jm)y lm l (θ,ϕ)χ 12 ms

11 Single-particle wave functions Radial differential equation 2 2m R (r) + 2 2m Normalization condition l(l + 1) R(r) + [U(r)R(r) + a r 2 ls f ls (r)] R(r) = εr(r) φ nlsjm ( r) 2 d r= m l m sm l m s 1 1 (lm l 2 ms jm)(lm l 2 m s jm) χ 1 2 m s χ 1 2 m s R nl (r) 2 dr = Y lml (θ, φ)y lm l (θ, φ)dω R nl (r) 2 dr = Parity ˆPφ nlsjm ( r) = ˆPφ nlsjm ( r) = ( 1) l φ nlsjm ( r)

12 Examples of spherically-symmetric potentials..... Square-well potential + strong spin-orbit term J (kr) l+1/2 U(r) r s d l=0 (100) l=2 (93) ex. N = 4 g l=4 (75.5). Harmonic oscillator potential + orbital + spin-orbit term U(r) e 2 L l +1/2 n-1 ( ν2r) 2 r N : degenerate in n, l ( ν = m ω ) h

13 Harmonic oscillator potential U(r) = mω2 r 2 + α 2 l l + β l s ( ε N = ω 2n + l + 3 ) ( = ω N + 3 ), 2 2 N = 0, 1, 2,..., l = N, N 2,..., 1 or 0 n = (N l)/2. Harmonic oscillator potential possesses many symmetry properties which make it a preferable choice as a basis. N=5 N=4 N=3 N=2 N=1 N=0 M. Mayer (1949) 2p 1f 0h 1d 2s 0g 1p 0f 0d 1s O. Haxel, H. Jensen, H.E. Suess (1949) 0p 0s p 1/2 1f 2p 5/2 3/2 0h 1f 9/2 7/2 0h 1d 11/2 2s 3/2 0g 1/2 7/2 1d 5/2 0g 9/2 1p 1/2 0f 5/2 1p 3/2 0f 7/2 0d 1s 3/2 1/2 0d 5/2 0p 1/2 0p 3/2 0s 1/2

14 Isospin The idea of W.Heisenberg: proton and neutron are considered as two states of a nucleon. ( ) ( ) v ππ v νν v πν 0 1 π =, ν = 1 0 Isospin operators (in analogy with the Pauli matrices): t = 1 2 τ, τ x = ( ) ( 0 i, τ y = i 0 ) ( 1 0, τ z = 0 1 ) Single-particle wave functions ( ) 1 φ ν (r) = φ( r) = φ( r) θ 0 t=1/2,mt =1/2, ( ) 0 φ π (r) = φ( r) = φ( r) θ 1 t=1/2,mt = 1/2.

15 Isospin and classification of nuclear states ˆT = ˆt i, ˆTz = i=1 ˆt zi. i=1 ] Charge independent Hamiltonian: [Ĥ, ˆT = 0. Realistic situation M T = 1 2 (N Z ), 1 2 (N Z ) T A 2. m p m n ; ˆVCoulomb = Z i<j=1 e 2 r i r j ˆV Coulomb = ˆV (T =0) + ˆV (T =1) + ˆV (T =2). E(T, M T ) = a(t ) + b(t )M T + c(t )M 2 T,

16 Two-particle wave function for identical fermions Angular-momentum coupled state (j a j b ) Φ ja(1)j b (2);JM( r 1, r 2 ) = m am b (j a m a j b m b J M)φ jam a ( r 1 )φ jb m b ( r 2 ) = [ φ ja ( r 1 ) φ jb ( r 2 ) ] (J) M j a (n al aj a), J = j a j b, j a j b + 1,..., j a + j b, while M = J, J + 1,..., J 1, J. Not antisymmetric with respect to permutation of two identical fermions! Normalized and antisymmetric state Φ jajb ;JM( r 1, r 2 ) = 1 (j a m a j b m b J M) [ φ jam N a ( r 1 )φ jb m b ( r 2 ) φ jb m b ( r 1 )φ jama ( r 2 ) ] m am b Since the Clebsch-Gordan coefficients have the following property: (j a m a j b m b J M) = ( 1) ja+jb J (j b m b j a m a J M) Φ jajb ;JM( r 1, r 2 ) = 1 [ (ja m a j b m b J M)φ jam N a ( r 1 )φ jb m b ( r 2 ) m am b ] ( 1) ja+jb J (j b m b j a m a J M)φ jb m b ( r 1 )φ jama ( r 2 )

17 Two-particle wave function for identical fermions Normalized and antisymmetric state j a j b Φ jajb ;JM( r 1, r 2 ) = 1 { [φja ( r 1 ) φ jb ( r 2 ) ] (J) 2 M ( 1)ja+j b+j [ φ jb ( r 1 ) φ ja ( r 2 ) ] } (J) M j a (n al aj a), J = j a j b, j a j b + 1,..., j a + j b, while M = J, J + 1,..., J 1, J. j a = j b = j Φ j 2 ;JM ( r 1, r 2 ) = 1 + ( 1)J 2 [ φj ( r 1 ) φ j ( r 2 ) ] (J) M Important consequence: if j a = j b = j, then J = 0, 2, 4,..., 2j 1 (ν0d 5/2 ) 2 : J = 0, 2, 4

18 Two-particle wave function for protons and neutrons j a j b Φ jaj b ;JMTM T ( r 1, r 2 ) = { [φja ( r 1 ) φ jb ( r 2 ) ] (J) M +( 1)ja+j b+j+t [ φ jb ( r 1 ) φ ja ( r 2 ) ] } (J) ΘTMT M 2 Θ 1,1 = θ 1/2,1/2 (1)θ 1/2,1/2 (2), Θ 1, 1 = θ 1/2, 1/2 (1)θ 1/2, 1/2 (2), Θ 1,0 = [ θ 1/2,1/2 (1)θ 1/2, 1/2 (2) + θ 1/2, 1/2 (1)θ 1/2,1/2 (2) ] / 2, Θ 0,0 = [ θ 1/2,1/2 (1)θ 1/2, 1/2 (2) θ 1/2, 1/2 (1)θ 1/2,1/2 (2) ] / 2, j a = j b = j Φ j 2 ;JMTM T ( r 1, r 2 ) = 1 ( 1)J+T 2 [ φj ( r 1 ) φ j ( r 2 ) ] (J) M Θ TM T Remark: if j a = j b = j, then (J + T ) is odd! (ν0d 5/2 ) 2 : J = 0, 2, 4 (ν0d 5/2 π0d 5/2 ) : J = 0, 2, 4(T = 1); J = 1, 3, 5 (T = 0)

19 Many-particle wave function: J(T )-coupled states. J-coupled state Consider N identical fermions in a single-j shell. We construct a totally antisymmetric and coupled to good J N-nucleon wave function from a set of totally antisymmetric (N 1)-nucleon wave functions coupled to all possible J : Φ j(n) χjm ( r 1,..., r N ) = χ J [ j N 1 (χ J )j } ] j N χj Φ j(n 1) χ J M ( r 1,..., r N 1 ) φ jm ( r N ), where [ } ] j N 1 (χ J )j j N χj are one-particle coefficients of fractional parentage (cfp s) Repeat this procedure for N particles in j orbital and so on. Construct thus basis states by consecutive coupling of angular momenta and antisymmetrization.

20 Many-particle wave function: m-scheme basis Slater determinants φ α1 ( r 1 ) φ α1 ( r 2 )... φ α1 ( r A ) Φ α (1, 2,..., A) = 1 φ α2 ( r 1 ) φ α2 ( r 2 )... φ α2 ( r A ) A! φ αa ( r 1 ) φ αa ( r 2 )... φ αa ( r A ) where α i = (n i, l i, j i, m i ) and α stores a set of single-particle configurations {α 1, α 2,..., α A } M = i=1 m i Projection on J can be performed.

21 Exercise 1 Basis construction Consider 2 neutrons in 0f 7/2 orbital. Write down the basis in J-coupled form. Write down the basis in m-scheme. How many different J-states exist in this model space?

22 Solution to exercise 1 Basis construction Basis in J-coupled scheme: (0f 7/2 ) 2 ; J, T = 1 J = 0, 2, 4, 6

23 Solution to exercise 1 (continued) Basis construction Basis in m-scheme: m 1 m 2 ; M : 7 2, 5 2 ; 6 7 2, 3 2 ; 5 7 2, 1 2 ; 4 5 2, 3 2 ; 4 7 2, 1 2 ; 3 5 2, 1 2 ; 3 7 2, 3 2 ; 2 5 2, 1 2 ; 2 3 2, 1 2 ; 2 7 2, 5 2 ; 1 5 2, 3 2 ; 1 3 2, 1 2 ; 1 7 2, 7 2 ; 0 5 2, 5 2 ; 0 3 2, 3 2 ; 0 1 2, 1 2 ; 0 7 2, 5 2 ; 1 5 2, 3 2 ; 1 3 2, 1 2 ; 1 7 2, 3 2 ; 2 5 2, 1 2 ; 2 3 2, 1 2 ; 2 7 2, 1 2 ; 3 5 2, 1 2 ; 3 7 2, 1 2 ; 4 5 2, 3 2 ; 4 7 2, 3 2 ; 5 7 2, 5 2 ; 6 There are 4 different J-states.

24 Solution of a many-body Schrödinger equation Construct a basis in the valence space for each J { } (J) Φ (J) k = (j a ) na J a (j b ) n b J b..., Ĥ (0) Φ k = E (0) k Φ k Expand unknown wave function in terms of basis functions n Ψ p = a kp Φ k ĤΨ p = E p Ψ p (Ĥ = Ĥ(0) + ˆV ) k=1 k Multiplying by Φ l, we get a system of equations n H lk a kp = E p a lp k=1 diagonalization of the matrix H lk = Φ l Ĥ Φ k = E (0) k δ lk + V lk

25 Basis dimension and choice of the model space Basis dimension grows quickly ( ) ( ) Ωπ Ων dim N π N ν Model space: a few valence orbitals beyond the closed-shell core. ε m ε 2 1 free orbitals valence space Example: 60 Zn 30 in pf -shell dim( 60 Zn) = ( 20 ) ( = 20! 20! 10!10! 10!10! )

26 Practical shell-model for 18 O in sd-shell Ĥ = Ĥ(0) + ˆV = ĥ(1) + ĥ(2) + ˆV Single-particle energies: ε(0d 5/2 ) = E B ( 17 8 O 9 ) E B( 16 8 O 8 ) = MeV ε(1s 1/2 ) = ε(0d 5/2 ) + E ex( 17 O; 1/2 + 1 ) = MeV ε(0d 3/2 ) = ε(0d 5/2 ) + E ex( 17 O; 3/2 + 1 ) = MeV Basis of states for each (JT ) denoted as j a j b JT : Φ 1 (0 + ) d5/ , T = 1 : Φ 2 (0 + ) s1/ Φ 3 (0 + ) d3/ H 11 H 21 H 12 H 22 H 13 H 23 H 31 H 32 H , T = 1 : Φ 1 (1 + ) d 5/2 d 3/2 11 Φ 2 (1 + ) s 1/2 d 3/2 11 ( ) H11 H 12 H 21 H 22 and so on for J = 2, 3, 4.

27 Exercise 2: Energies of 0 + states in 18 O Diagonal Two-body matrix elements (TBMEs) H 11 = 2ε(d 5/2 ) + d 2 5/2 V d 2 5/2 01 }{{} 2.82 MeV H 22 = 2ε(s 1/2 ) + s 2 1/2 V s2 1/2 01 }{{} 2.12 MeV H 33 = 2ε(d 3/2 ) + d 2 3/2 V d 2 3/2 01 }{{} 2.18 MeV Non-diagonal TBMEs H 12 = H 21 = d 2 5/2 V s2 1/2 01 }{{} 1.32 MeV H 23 = H 32 = s 2 1/2 V d 2 3/2 01 }{{} 1.08 MeV H 13 = H 31 = d 2 5/2 V d 2 3/2 01 }{{} 3.19 MeV

28 Practical shell-model for 18 O in sd-shell Eigenvalues (g.s. and two excited 0 + states): E(0 + 1 ) = MeV Egs(0+ 1 ) = 0 MeV E(0 + 2 ) = MeV Eex(0+ 2 ) = MeV E(0 + 3 ) = MeV Eex(0+ 3 ) = MeV Eigenstates: Ψ(0 + 1 ) = a 11 d5/ a 21 s1/ a 31 d3/ Ψ(0 + 2 ) = a 12 d5/ a 22 s1/ a 32 d3/ Ψ(0 + 2 ) = a 13 d5/ a 23 s1/ a 33 d3/ akp 2 = 1 k The full spectrum: Repeat the same procedure for J π = 1 +, 2 +, 3 +, 4 +.

29 Shell-model codes m-scheme codes ANTOINE (E. Caurier) (W. Rae, B. A. Brown) MSHELL (T. Mizusaki) REDSTICK (W. E. Ormand, C. Johnson)... J-coupled codes NATHAN (E.Caurier, F.Nowacki) DUPSM (Novoselsky, Vallières) Ritsschil (Zwarts)... Features Matrix dimension: and beyond Lanczos diagonalization algorithm Calculation of the matrix elements on-the-fly

30 Lanczos algorithm Creation of a tri-diagonal matrix: Ĥ 1 = E E 12 2 Ĥ 2 = E E E Matrix elements: E 11 = 1 Ĥ 1 E 12 2 = (Ĥ E 11) 1 E 21 = E 12, E 22 = 2 Ĥ 2 E 23 3 = (Ĥ E 22) 2 E How to get the lowest states converged: ( ) E11 E 12 E 21 E 22 E 11 E 12 0 E 21 E 22 E 23 0 E 32 E 33 E 11 E E 21 E 22 E E 32 E 33 E E 43 E You may need only a few iterations to get the lowest state of a ( ) matrix converged!

31 Calculation of observables General scheme: Construct the basis: Φ k Expand the wave function: Ψ p = k matrix {H lk }. a kp Φ k and compute the Hamiltonian Solution of the Shrödinger equation by Hamiltonian matrix diagonalization: {H lk } E p, Ψ p (coefficients a kp ) Calculation of matrix elements of the operators T fi Ψ f Ô Ψ i 2 Electroweak operators: Ô(E, LM) = e(k)r L (k)y LM (ˆr(k)) k=1 Ô(M, 1M) = µ n(g s(k) s(k) + g l (k) l(k)) k=1 Ô(F ) = τ ± (k), Ô(GT ) = σ(k)τ ± (k)... k=1 k=1

32 Exercise 3 Electromagnetic transitions in 17 O Calculate the B(E2; 1/ /2+ g.s.) in 17 O modeled as a valence neutron in a 1s0d shell beyond the 16 O closed-shell core. We take: R 1s1/2 (r)r 2 R 0d5/2 (r)dr 12 fm 2. 0 Compare your result to experimental value B exp(e2) = 6.3 e 2.fm 4. What can you conclude?

33 Solution to exercise 3 Electromagnetic transitions in 17 O The single-particle electric multipole operator reads Ô(E, 2M) = er 2 Y 2M (θ, φ), The reduced probability of the EL-transition from the initial to the final state is B(EL; J i J f ) = 1 2J i + 1 J f O(EL) J i 2. In our case, there is one valence particle: J i = (1s 1/2 ), J f = (0d 5/2 ). B(E2; 1/2 + 5/2 + ) = 1 2 0d 5/2 Ô(E2) 1s 1/2 2 = 1 2 n f =0, l f =2, s f = 1 2, j f = 5 2 er 2 Y 2 (θ, φ) n i =1, l i =0, s i = 1 2, j i = = ( ) e2 R0d (r)r 2 R 1s1/2 (r)dr 2 1 5/2 2 ; 5 2 Y 2(θ, φ) ; }{{} r 2 2 B(EL; j i j f ) = e 2 1 { 1 4π r 2 2 (2j f + 1)(2l i + 1)(2L + 1)(l i 0L0 l f 0) 2 l 2 f j f L j i l i } 2

34 Solution to exercise 3 Electromagnetic transitions in 17 O (continued) { B(E2; 1/2 + 5/2 + ) = e 2 1 1/2 2 5/2 4π r ( ) 2 2 1/2 0 = 34.4 e 2.fm 4. Experimental value B exp(e2; 1/2 + 5/2 + ) = 6.3 e 2.fm 4. } 2 = This means that the neutron should have an effective charge: ẽ n 0.43 e. because we work in a severely restricted model space (one valence nucleon!). Standard effective E2 and M1 operators ẽ π 1.5 e, ẽ ν 0.5 e g s(π) 0.7 g s(π), g l (π) = g l (π) = 1 }{{} g s(ν) 0.7 g s(ν), g l (ν) = g l (ν) = 0 }{{} 3.826

35 Nuclear many-body problem Non-relativistic Hamiltonian for A nucleons Ĥ = i=1 ˆ p 2 i 2m + [ ] p 2 Ĥ = i 2m + U( r i) + i=1 }{{} Ĥ (0) Ŵ ( r i r j ) i<j=1 W ( r i r j ) i<j=1 U( r i ), i=1 } {{ } ˆV The residual interaction is assumed to have a two-body form One and two-body Hamitonian ˆV = i<j=1 ˆV ( r i r j ) Ĥ = ĥ( r i ) + ˆV ( r i r j ) i=1 i<j=1

36 Occupation-number representation (second quantization) Creation and annihilation operators α = a α 0 α = 0 a α Wave function of a fermion in a quantum state α in coordinate space: (Anti-)commutation relations: {a α, a β } { a α, a β r α = φ α( r) = a αa β + a β a α = δ αβ } = {a α, a β } = 0 Normalized and antisymmetric A-fermion state: α 1 α 2... α A = a α A a α A 1... a α 2 a α 1 0

37 Operators in the occupation-number formalism One-body operators Ô = Ô( r k ) k=1 α Ô β = φ α( r)ô( r)φ β( r)d r Second-quantized form of the one-body operator Ô: Ô = α Ô β a αa β αβ For example, the number operator reads ˆN = αβ α ˆ1 β a αa β = α a αa α

38 Operators in the occupation-number formalism Symmetric two-body operator acting on an A-fermion system αβ ˆT γδ = ˆT = j<k=1 φ α( r 1 )φ β( r 2 )ˆT ( r 1, r 2 ) ˆT ( r k, r j ) (1 ˆP 12 ) φ γ( r 1 )φ δ ( r 2 )d r 1 d r 2, Second-quantized form of the two-body operator ˆT : ˆT = 1 αβ ˆT γδ a αa β 4 a δa γ αβγδ

39 Nuclear many-body Hamiltonian in the occupation-number formalism Non-relativistic Hamiltonian for A nucleons Ĥ = ĥ( r i ) + i=1 i<j=1 Ĥ = ε αa αa + α α }{{} one body term ˆV ( r i r j ) = Ĥ(0) + ˆV αβ V γδ a αa + + β a δ a γ, αβγδ } {{ } two body term Two-body term in a JT -coupled form ˆV = 1 j αj β V j γj δ JT (1 + δαβ )(1 + δ γδ ) 4 j αj β j γ j [ δ [ ] ] (JT ) a + j α a + ] (00) (JT ) j β [ãjγ ã jδ

40 Necessary ingredients Single-particle energies from experimental spectra of A core plus a neutron or a proton ε α Two-body matrix elements (TBMEs) from theory? j αj β V j γj δ JT

41 Bare nucleon-nucleon (NN) interaction The NN interaction between two nucleons in the vacuum: NN scattering data, deuteron bound states properties. Elastic scattering in momentum space (Yukawa) ( σ1 q ) ( σ 2 q ) V πnn (1, 2) = g2 πnn 4M 2 q 2 + m 2 π Potential (Fourier transform) in coordinate space V OPEP πnn (1, 2) = g2 πnn ( 4M 2 + m 3 π 12 { σ 1 σ 2 ) ( ) } 3 e m πr (m πr) 2 σ1 r σ 2 r σ 1 σ mπ r 2 m πr high- Meson-exchange theories of NN potential: precision potentials (CD-Bonn, AV18, etc)

42 Concept of effective interaction (operators) Effective in-medium nucleon-nucleon interaction In-medium effects (renormalization of the hard core) Truncated model space ĤΨ = (Ĥ(0) + ˆV )Ψ = EΨ, True wave function: Ψ = a k Φ k. k=1 Wave function in a model space: M Ψ = a k Φ k. k=1 Ψ Ĥeff Ψ = Ψ Ĥ Ψ = E Ψ Ôeff Ψ = Ψ Ô Ψ Approaches to the problem: phenomenological or microscopic.

43 Effective Interaction Practical approaches to effective interaction Schematic interaction (parametrized interaction between two nucleons in a nuclear medium) Phenomenological interaction (Fit of TBME s to energy levels of nuclei to be described within the chosen model space) Microscopic interaction (derived from a bare NN-force)

44 Schematic interaction Some examples: V (1, 2) = V 0 exp (µr)/(µr) V (1, 2) = V 0 δ( r 1 r 2 ) V (1, 2) = V 0 δ( r 1 r 2 )(1 + α σ 1 σ 2 ) V (1, 2) = V 0 δ( r 1 r 2 )δ(r 1 R)... V (1, 2) = χq Q (Q = r 2 Y 2µ (Ω r ))... A few parameters (interaction strengths) are fitted to reproduce energy levels in a certain region of (a few) neighboring nuclei local description only!

45 Exercise 4: TBMEs of the δ-force Multipole expansion of the delta-function V (1, 2) = V 0 δ( r 1 r 2 ), δ( r 1 r 2 ) = k δ(r 1 r 2 ) r 1 r 2 2k + 1 4π P k(cos θ 12 ) Diagonal TBMEs between normalized and antisymmetric states ( ) 2 j1 j 2 J 1+( 1) l 1+l 2 +J j 1 j 2 V j 1 j 2 JT = I (2j 1 + 1)(2j 2 + 1) ( ) 2 j 2 V j 2 JT = I (2j + 1) 2 j j J 1 1, if j 0 1 = j 2 = j 2 2 I = 1 1 4π r [R 2 n 1 l 1 (r)r n2 l 2 (r)] 2 dr 0

46 Example 1: 210 Pb in (ν0h 9/2 ) 2 V delta (1, 2) = V 0 δ( r 1 r 2 ) V pairing (1, 2) = GŜ+ Ŝ j 2 a V pairing(1, 2) j 2 b 01 = ( 1) la+l b 1 2 G (2j a + 1)(2j b + 1) E (MeV) Experiment Delta Pairing ,4 +,6 +, Pb128

47 Example 2: 20 Ne and SU(3) model of Elliott Ĥ = i=1 [ p2 i 2m + 1 ] 2 mω2 ri 2 χq Q Q is an algebraic quadrupole operator (Q µ, L µ are SU(3) generators) J.P.Elliott (1958) Group-theoretical classification of nuclear states (analytical solution) see lecture of H. Molique. Experiment SU(3) 20 (80) (42) 6 + E (MeV) Ne Rotational classification of nuclear states as mixing of many spherical configurations

48 Empirical interaction (least-squares-fit method) All TBME s j aj b V j cj d JT are considered as free parameters! Initial interaction parameters Set-up Hamiltonian matrix Eigenvalues Eigenvectors th i i Convergence? yes E,Ψ N data ( Ei Ei 2 ) th exp i= 1 Stop no New interaction parameters New linear system Variation of parameters Examples: 0p-shell: 4 He 16 O (15 TBME s) Cohen, Kurath (1965) 1s0d-shell: 16 O 40 Ca (63 TBME s) Brown, Wildenthal (1988) 1p0f -shell: 40 Ca 80 Zr (195 TBME s) Honma et al (2002, 2004)

49 Microscopic effective interaction A bare NN potential (CD-Bonn, AV18, chiral N3LO, etc) requires regularization and modification to be applied for many-body calculations in a restricted model space. Renormalization schemes (see lecture of L. Bonneau) G matrix followed by the many-body perturbation theory V low k SRG (IM-SRG) Okubo-Lee-Suzuki transformation Successful, but still lack precision of the empirical interactions, mainly due to behavior of centroids. One of possible reasons: absence of 3N forces (A. Poves, A.P. Zuker, 1981; A.P. Zuker, 2003)

50 Modern nuclear shell model and beyond Interacting shell model Oscillator-based shell model with accurate realistic interactions formulated in one or two harmonic-oscillator shells model spaces (large-scale diagonalization). Detailed information on individual states and transitions at low energies Conservation of principal symmetries Numerous applications to nuclear structure, weak interaction and astrophysics E. Caurier et al, Rev. Mod. Phys. 77, 427 (2005)

51 State-of-the-art calculations: backbending in 48 Cr E ~ J(J + 1) J KB3 (semi-empirical interaction in pf-shell model space) Strasbourg-Madrid For J<10 : (J + 1)(2J + 3) Q0 = Q 2 3K J(J + 1) 5 B E2;J 16π spec (J), K 1 ( J 2) = e ( JK20 J 2,K ) Q 0 J<10: collective rotation J=10-12 : backbending phenomenon (competition between rotation and alignment of 0f 7/2 particles) J>12 : spherical states E. Caurier et al, Rev. Mod. Phys. 77 (2005) 427

52 State-of-the-art calculations: superdeformation in 36 Ar Intruder np-nh configurations can lead even to superdeformation! [sd] 16 [pf] 0-0p0h spherical configuration [sd] 12 [pf] 4-4p4h deformed configuration E. Caurier et al, Phys.Rev.Lett. 95, (2005)

53 State-of-the-art calculations: mirror bands in A = 51 S. Lenzi, A. Zuker, E. Caurier et al

54 No-Core Shell Model Non-relativistic Hamiltonian for A nucleons in many N Ω harmonic oscillator space Ĥ = i=1 p 2 i 2m + i<j=1 W ( r i r j ) Problem: excitation of the center-of-mass of the system. Center-of-mass coordinates R = 1 A r i ; P = p i i=1 i=1 Translational-invariant Hamiltonian Ĥ = i=1 p 2 i 2m P 2 2mA + i<j=1 W ( r i r j )

55 No-Core Shell Model Separation of the harmonic oscillator Hamiltonian into center-of-mass and intrinsic Hamiltonians [ pi 2 H ho = 2m + mω2 r 2 ] i 2 i=1 = 1 ( ) 2 mω p i p 2 ( ) j + 2 P r i r j + 2 2mA 2A 2mA + maω2 R 2 i<j=1 i<j=1 }{{ 2 }}{{} Ĥ int Ĥ CoM

56 No-Core Shell Model Translational-invariant Hamiltonian Ĥ = Ĥ = [ pi 2 2m + mω2 r i 2 ] P 2 2 2mA + i=1 [ pi 2 2m + mω2 r i 2 ] + 2 i=1 i<j=1 i<j=1 W ( r i r j ) mω2 2A W ( r i r j ) i<j=1 i=1 mω 2 r 2 i 2 ( r i r j ) 2 Ĥ CoM Separation of the Center-of-Mass motion Ĥ Ω = Ĥ + β ( Ĥ CoM 3 2 Ω )

57 B.R. Barrett, P. Navratil, J. P. Vary, Ab-initio no core shell model, Prog. Part. Nucl. Phys. 69 (2013) 131 and refs. therein. See lecture of R. Lazauskas for alternative methods.

58 Summary The Nuclear Shell Model The Nuclear Shell Model is a powerful microscopic approach to nuclear structure Very good description of energies and transitions at low energies High predictive power. Important applications : structure of nuclei far from stability (proton rich or neutron-rich nuclei) calculation of weak interaction processes on nuclei for the tests of the Standard Model calculation of the nuclear structure input (masses, half-lives, reaction rates, etc) relevant for astrophysical simulations (r-processes, rp-process, etc..).

59 Some references on the Shell model theory Books P. J. Brussaard, P. W. M. Glaudemans, Shell-Model Applications in Nuclear Spectroscopy (North-Holland, Amsterdam, 1977). K. Heyde, The Nuclear Shell Model (Springer-Verlag, Heidelberg, 2004) A. DeShalit, I. Talmi, Nuclear Shell Theory (Academic Press, New York and London, 1963) I. Talmi, Simple Models of Complex Nuclei, The Shell Model and the Interacting Boson Model (Harwood Academic Publishers, New York, 1993) R. D. Lawson, Theory of the Nuclear Shell Model (Clarendon Press, 1980). Reviews E. Caurier et al, The shell model as a unified view of nuclear structure, Rev. Mod. Phys. 77 (2005) 427. B. R. Barrett, P. Navratil, J. P. Vary, Ab-initio no core shell model, Prog. Part. Nucl. Phys. 69 (2013) 131