The Nuclear Shell Model

N. A. Smirnova

CEN Bordeaux-Gradignan, France

Bridging Methods in Nuclear Theory IPHC, Strasbourg, June 26–30, 2017

N. A. Smirnova The Nuclear Shell Model 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

N. A. Smirnova The Nuclear Shell Model Structure of complex nuclei

Nuclear charge (matter) density distribution ρch(~r) (ρm(~r)) with sharp 1/3 radius R ' r0A . 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.

(Sn versus S2n; 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.

N. A. Smirnova The Nuclear Shell Model Nuclear many-body problem

126 Limits of nuclear existence 82 ss ce ro r‐p

50 s es c y ro or protons 82 ‐p The ld p al Fie r on n 28 cti ea 20 un t M 50 y F en it ist ens ns D ‐co 8 28 elf 2 20 neutrons s 0 2 8 10 ~6 A= 12 A A=

Toward s a unified ab initio 0 h Shell description of the nucleus few‐body Model calc ulations No ‐Core Shell Model

N. A. Smirnova The Nuclear Shell Model Nuclear many-body problem

Non-relativistic Hamiltonian for A nucleons

A A X ~p2 X Hˆ = i + W (~r −~r ) 2m i j i=1 i

A A A X ~p2 X X Hˆ = i + U(~r ) + W (~r −~r ) − U(~r ) , 2m i i j i i=1 i

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

N. A. Smirnova The Nuclear Shell Model Variational approach to the nuclear many-body problem

Starting point: antisymmetric product wave function

φα1 (1) φα1 (2) . . . φα1 (A)

1 φα2 (1) φα2 (2) . . . φα2 (A) Ψ(1, 2,..., A) = √ . . .. . A! . . . .

φαA (1) φαA (2) . . . φαA (A)

The best wave function is determined via a variational principle: Z ~ 2 ~ δhΨ|H|Ψi = hδΨ|H|Ψi = 0 with |φαi (r)| dr = 1.

N. A. Smirnova The Nuclear Shell Model Self-consistent mean-ﬁeld potential

Hartree-Fock equations

2 ! Z − ~ ∆ + U (~r) φ (~r) + U (~r, r~0)φ (r~0)dr~0 = ε φ (~r) 2m H i F i i i

Direct (Hartree) term Exchange (Fock) term

Z X X ∗ ~0 ~0 ~0 ~0 U (~r, r~0) = φ∗(~r)W (~r, r~0)φ (r~0) UH (~r) = φb (r )W (~r, r )φb(r )dr F b b b∈F b∈F

Iterative solution of Hartree-Fock equations:

(0) ~ (1) ~ (1) (HF) ~ (HF) φi (r) φi (r), εi . . . φi (r), εi ↓ % ↓ % ↓ % (0) ~ (0) ~ ~0 (1) ~ (1) ~ ~0 (HF) ~ UH (r), UF (r, r ) UH (r), UF (r, r ) ... U (r)

N. A. Smirnova The Nuclear Shell Model Self-consistent mean-ﬁeld potential

Direct (Hartree) term

Z X ∗ ~0 ~0 ~0 ~0 UH (~r) = φb(r )W (~r, r )φb(r )dr b∈F

R ~0 ~0 ~0 UH (~r) = ρ(r )W (~r, r )dr

P 2 ρ(~r) = |φb(~r)| b∈F

Example: a delta-force

W (~r, r~0) ∝ δ(~r − r~0)

UH (~r) ∝ ρ(~r)

N. A. Smirnova The Nuclear Shell Model Hartree-Fock ground state properties

From the product HF wave function, a number of ground-state properties can be calculated:

Application to open-shell nuclei and calculation excitation spectra require inclusion of correlations (pairing, etc) → beyond mean-ﬁeld techniques (lecture of Ph. Quentin).

N. A. Smirnova The Nuclear Shell Model Shell model: energy matrix diagonalization

Non-relativistic Hamiltonian for A nucleons

A A A X ~p2 X X Hˆ = i + U(~r ) + W (~r −~r ) − U(~r ) 2m i i j i i=1 | {z } i

Construction of a basis from single-particle states ˆ hφα(~r) = εαφα(~r) → {εα, φα(~r)}

Spherical potential U(~r) = U(r): α = {nα, lα, jα, mα}

(j) Rnlj (r) h i φnljm(~r) = Yl (θ, ϕ) × χ 1 r 2 m | {z } P 1 (lml ms|jm)Ylm (θ,ϕ)χ 1 2 l ms ml ms 2

N. A. Smirnova The Nuclear Shell Model Single-particle wave functions

Radial differential equation 2 2 l(l + 1) − ~ R00(r) + ~ R(r) + [U(r)R(r) + a f (r)] R(r) = εR(r) 2m 2m r 2 ls ls

Normalization condition

Parity

ˆ ˆ l Pφnlsjm(~r) = Pφnlsjm(−~r) = (−1) φnlsjm(~r)

N. A. Smirnova The Nuclear Shell Model Examples of spherically-symmetric potentials

. Square-well potential + strong spin-orbit term

Jl+1/2 (kr) U(r) s l=0 (100) r d l=2 (93) ex. N = 4 g l=4 (75.5)

. Harmonic oscillator potential + orbital + spin-orbit term

l +1/2 22 U(r) e 2 L n-1 (ν r)

. r

. mω . (ν = ) √ h N : degenerate in n, l

N. A. Smirnova The Nuclear Shell Model Harmonic oscillator potential

N=5 2p1/2 2 2 2p 1f5/2 2p mω r 1f 3/2 ~ ~ ~ 0h9/2 U(r) = + αl · l + βl · ~s 1f7/2 0h 82 2 0h11/2 N=4 2s 1d3/2 2s1/2 1d 0g7/2 1d 3 3 0g 50 5/2 εN = ω 2n + l + = ω N + , 0g9/2 ~ ~ N=3 1p 2 2 1p 0f1/2 1p5/2 0f 28 3/2 0f7/2 N = 0, 1, 2,..., 20 N=2 1s 0d3/2 l = N, N − 2,..., 1 or 0 0d 1s1/2 0d5/2 n = (N − l)/2 . 8 N=1 0p 0p 1/2 0p3/2 2 Harmonic oscillator potential N=0 0s 0s possesses many symmetry 1/2 properties which make it a preferable choice as a basis. M. Mayer (1949) O. Haxel, H. Jensen, H.E. Suess (1949)

N. A. Smirnova The Nuclear Shell Model Isospin

The idea of W.Heisenberg: proton and neutron are considered as two states of a nucleon.

0 1 vππ ≈ vνν ≈ vπν π = , ν = 1 0

Isospin operators (in analogy with the Pauli matrices):

1 0 1 0 −i 1 0 ~t = ~τ, τ = , τ = , τ = 2 x 1 0 y i 0 z 0 −1

Single-particle wave functions 1 φ (r) = φ(~r) = φ(~r) θ , ν 0 t=1/2,mt =1/2 0 φ (r) = φ(~r) = φ(~r) θ . π 1 t=1/2,mt =−1/2

N. A. Smirnova The Nuclear Shell Model Isospin and classiﬁcation of nuclear states

A A ˆ X ˆ X T = ˆti , Tz = ˆtzi . i=1 i=1 h i Charge independent Hamiltonian: Hˆ , Tˆ = 0 .

1 1 A M = (N − Z), (N − Z) ≤ T ≤ . T 2 2 2

Realistic situation

Z 2 ˆ X e mp ≈ mn; VCoulomb = |~ri −~rj | i

N. A. Smirnova The Nuclear Shell Model Two-particle wave function for identical fermions

Angular-momentum coupled state (ja 6= jb)

X (J) Φ (~r ,~r ) = (j m j m |JM)φ (~r )φ (~r ) = φ (~r ) × φ (~r ) ja(1)jb(2);JM 1 2 a a b b jama 1 jbmb 2 ja 1 jb 2 M mamb

ja ≡ (nalaja), J = |ja − jb|, |ja − jb| + 1,..., ja + jb, while M = −J, −J + 1,..., J − 1, J. Not antisymmetric with respect to permutation of two identical fermions !

Normalized and antisymmetric state

1 X Φ (~r ,~r ) = (ja ma j m |JM) φ (~r )φ (~r ) − φ (~r )φ (~r ) jajb;JM 1 2 N b b jama 1 jbmb 2 jbmb 1 jama 2 mamb

Since the Clebsch-Gordan coefﬁcients have the following property:

ja+jb−J (ja ma jb mb |JM) = (−1) (jb mb ja ma|JM) 1 X Φ (~r ,~r ) = (ja ma j m |JM)φ (~r )φ (~r ) jajb;JM 1 2 N b b jama 1 jbmb 2 mam b i −(− )ja+jb−J ( | )φ (~ )φ (~ ) 1 jb mb ja ma JM jbmb r1 jama r2

N. A. Smirnova The Nuclear Shell Model Two-particle wave function for identical fermions

Normalized and antisymmetric state

ja 6= jb

1 n o (J) ja+jb+J (J) Φj j ;JM (~r1,~r2) = √ φj (~r1) × φj (~r2) − (−1) φj (~r1) × φj (~r2) a b 2 a b M b a M

ja ≡ (nalaja), J = |ja − jb|, |ja − jb| + 1,..., ja + jb, while M = −J, −J + 1,..., J − 1, J.

ja = jb = j J 1 + (−1) (J) Φ 2 (~r ,~r ) = φ (~r ) × φ (~r ) j ;JM 1 2 2 j 1 j 2 M

Important consequence: if ja = jb = j, then J = 0, 2, 4,..., 2j − 1

2 (ν0d5/2) : J = 0, 2, 4

N. A. Smirnova The Nuclear Shell Model Two-particle wave function for protons and neutrons

ja 6= jb

n o ΘTM (J) ja+jb+J+T (J) T Φj j ;JMTM (~r1,~r2) = φj (~r1)×φj (~r2) +(−1) φj (~r1)×φj (~r2) √ a b T a b M b a M 2

Θ1,1 = θ1/2,1/2(1)θ1/2,1/2(2) , Θ1,−1 = θ1/2,−1/2(1)θ1/2,−1/2(2) , √ Θ1,0 = θ1/2,1/2(1)θ1/2,−1/2(2) + θ1/2,−1/2(1)θ1/2,1/2(2) / 2 , √ Θ0,0 = θ1/2,1/2(1)θ1/2,−1/2(2) − θ1/2,−1/2(1)θ1/2,1/2(2) / 2 ,

ja = jb = j J+T 1 − (−1) (J) Φ 2 (~r1,~r2) = φj (~r1) × φj (~r2) ΘTM j ;JMTMT 2 M T

Remark: if ja = jb = j, then (J + T ) is odd!

2 (ν0d5/2) : J = 0, 2, 4 (ν0d5/2 π0d5/2): J = 0, 2, 4(T = 1); J = 1, 3, 5 (T = 0)

N. A. Smirnova The Nuclear Shell Model 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 JN-nucleon wave function from a set of totally antisymmetric (N − 1)-nucleon wave functions coupled to all possible J0: h o i j(N) ~ ~ X N−1 0 0 N j(N−1) ~ ~ ~ ΦχJM (r1,..., rN ) = j (χ J )j| j χJ Φχ0J0M0 (r1,..., rN−1) φjm(rN ) , χ0J0

where h o i jN−1(χ0J0)j| jN χJ are one-particle coefﬁcients of fractional parentage (cfp’s) Repeat this procedure for N0 particles in j0 orbital and so on. Construct thus basis states by consecutive coupling of angular momenta and antisymmetrization.

N. A. Smirnova The Nuclear Shell Model Many-particle wave function: m-scheme basis

Slater determinants ~ ~ ~ φα1 (r1) φα1 (r2) . . . φα1 (rA) ~ ~ ~ 1 φα2 (r1) φα2 (r2) . . . φα2 (rA) Φ (1, 2,..., A) = √ α . . .. . A! . . . . ~ ~ ~ φαA (r1) φαA (r2) . . . φαA (rA)

where αi = (ni , li , ji , mi ) and α stores a set of single-particle conﬁgurations {α1, α2, . . . , αA}

A X M = mi i=1 Projection on J can be performed.

N. A. Smirnova The Nuclear Shell Model Exercise 1

Basis construction

Consider 2 neutrons in 0f7/2 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?

N. A. Smirnova The Nuclear Shell Model Solution to exercise 1

Basis construction Basis in J-coupled scheme: 2 |(0f7/2) ; J, T = 1i J = 0, 2, 4, 6

N. A. Smirnova The Nuclear Shell Model Solution to exercise 1 (continued)

Basis construction

Basis in m-scheme: |m1m2; Mi: 7 5 | 2 , 2 ; 6i 7 3 | 2 , 2 ; 5i 7 1 5 3 | 2 , 2 ; 4i | 2 , 2 ; 4i 7 1 5 1 | 2 , − 2 ; 3i | 2 , 2 ; 3i 7 3 5 1 3 1 | 2 , − 2 ; 2i | 2 , − 2 ; 2i | 2 , 2 ; 2i 7 5 5 3 3 1 | 2 , − 2 ; 1i | 2 , − 2 ; 1i | 2 , − 2 ; 1i 7 7 5 5 3 3 1 1 | 2 , − 2 ; 0i | 2 , − 2 ; 0i | 2 , − 2 ; 0i | 2 , − 2 ; 0i 7 5 5 3 3 1 | − 2 , 2 ; −1i | − 2 , 2 ; −1i | − 2 , 2 ; −1i 7 3 5 1 3 1 | − 2 , 2 ; −2i | − 2 , 2 ; −2i | − 2 , − 2 ; −2i 7 1 5 1 | − 2 , 2 ; −3i | − 2 , 2 ; −3i 7 1 5 3 | − 2 , 2 ; −4i | − 2 , 2 ; −4i 7 3 | − 2 , 2 ; −5i 7 5 | − 2 , 2 ; −6i

There are 4 different J-states.

N. A. Smirnova The Nuclear Shell Model Solution of a many-body Schrödinger equation

Construct a basis in the valence space for each J n o(J) (J) na nb ˆ (0) (0) Φk = (ja)J (jb)J ... , H Φk = Ek Φk a b k

Expand unknown wave function in terms of basis functions n X ˆ ˆ ˆ (0) ˆ Ψp = akpΦk ⇒ HΨp = EpΨp (H = H + V ) k=1

Multiplying by hΦl |, we get a system of equations n X Hlk akp = Epalp k=1 ⇒ diagonalization of the matrix

ˆ (0) Hlk = hΦl |H|Φk i = Ek δlk + Vlk

N. A. Smirnova The Nuclear Shell Model Basis dimension and choice of the model space

Basis dimension grows quickly Ωπ Ων dim ≈ free orbitals Nπ Nν ε m valence space ε2 ε1 Model space: a few valence orbitals beyond the closed-shell core.

60 Example: Zn30 in pf -shell

20 20 dim(60Zn) = 10 10 20! 20! 10 = 10!10! 10!10! ≈ 3.4 × 10 .

N. A. Smirnova The Nuclear Shell Model Practical shell-model for 18O in sd-shell

Hˆ = Hˆ (0) + Vˆ = hˆ(1) + hˆ(2) + Vˆ

Single-particle energies:

17 16 ε(0d5/2) = EB(8 O9 ) − EB(8 O8 ) = −4.143 MeV 17 + ε(1s1/2) = ε(0d5/2) + Eex( O; 1/21 ) = −3.273 MeV 17 + ε(0d3/2) = ε(0d5/2) + Eex( O; 3/21 ) = 0.942 MeV

Basis of states for each (JT ) denoted as |ja jbiJT : |Φ (0+)i ≡ |d2 i   1 5/2 01 H11 H12 H13 + + 2 0 , T = 1 : |Φ2(0 )i ≡ |s1/2i01 ⇒  H21 H22 H23  + 2 H31 H32 H33 |Φ3(0 )i ≡ |d3/2i01

+ + |Φ1(1 )i ≡ |d5/2 d3/2i11 H11 H12 1 , T = 1 : + ⇒ |Φ2(1 )i ≡ |s1/2 d3/2i11 H21 H22 and so on for J = 2, 3, 4.

N. A. Smirnova The Nuclear Shell Model Exercise 2: Energies of 0+ states in 18O

Diagonal Two-body matrix elements (TBMEs)

2 2 H11 = 2ε(d5/2) + hd5/2|V |d5/2i01 | {z } −2.82 MeV 2 2 H22 = 2ε(s1/2) + hs1/2|V |s1/2i01 | {z } −2.12 MeV 2 2 H33 = 2ε(d3/2) + hd3/2|V |d3/2i01 | {z } −2.18 MeV

Non-diagonal TBMEs

2 2 H12 = H21 = hd5/2|V |s1/2i01 | {z } −1.32 MeV 2 2 H23 = H32 = hs1/2|V |d3/2i01 | {z } −1.08 MeV 2 2 H13 = H31 = hd5/2|V |d3/2i01 | {z } −3.19 MeV

N. A. Smirnova The Nuclear Shell Model Practical shell-model for 18O in sd-shell

Eigenvalues (g.s. and two excited 0+ states):

+ + E(01 ) = −12.602 MeV Egs(01 ) = 0 MeV + + E(02 ) = −8.097 MeV ⇒ Eex (02 ) = 4.505 MeV + + E(03 ) = 0.622 MeV Eex (03 ) = 13.224 MeV

Eigenstates:

+ 2 2 2 |Ψ(01 )i = a11|d5/2i01 + a21|s1/2i01 + a31|d3/2i01 + 2 2 2 |Ψ(02 )i = a12|d5/2i01 + a22|s1/2i01 + a32|d3/2i01 + 2 2 2 |Ψ(02 )i = a13|d5/2i01 + a23|s1/2i01 + a33|d3/2i01 P 2 akp = 1 k

The full spectrum: Repeat the same procedure for Jπ = 1+, 2+, 3+, 4+.

N. A. Smirnova The Nuclear Shell Model Shell-model codes

m-scheme codes ANTOINE (E. Caurier) http://www.iphc.cnrs.fr/nutheo/code_antoine/menu.html NuShellX@MSU (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: ∼ 1010 and beyond Lanczos diagonalization algorithm Calculation of the matrix elements on-the-ﬂy N. A. Smirnova The Nuclear Shell Model Lanczos algorithm

Matrix elements: Creation of a tri-diagonal matrix: ˆ E11 = h1|H|1i ˆ ˆ H|1i = E11|1i + E12|2i E12|2i = (H − E11)|1i ˆ ˆ H|2i = E21|1i + E22|2i + E23|3i E21 = E12, E22 = h2|H|2i ... ˆ E23|3i = (H − E22)|2i − E21|1i ...

How to get the lowest states converged:

    E11 E12 0 0 E11 E12 0 E11 E12  E21 E22 E23 0  ⇒  E21 E22 E23  ⇒   ⇒ ... E21 E22  0 E32 E33 E34  0 E32 E33 0 0 E43 E44

You may need only a few iterations to get the lowest state of a (103 × 103) matrix converged!

N. A. Smirnova The Nuclear Shell Model Calculation of observables

General scheme:

Construct the basis: |Φk i P Expand the wave function: |Ψpi = akp|Φk i and compute the Hamiltonian k matrix {Hlk }.

Solution of the Shrödinger equation by Hamiltonian matrix diagonalization: {Hlk } ⇒ Ep, |Ψpi (coefﬁcients akp) Calculation of matrix elements of the operators ˆ 2 Tﬁ ∝ |hΨf |O|Ψi i|

Electroweak operators:

A ˆ X L O(E, LM) = e(k)r (k)YLM (ˆr(k)) k=1 A ˆ X ~ O(M, 1M) = µn(gs(k)~s(k) + gl (k)l(k)) k=1 A A ˆ X ˆ X O(F) = τ±(k) , O(GT ) = ~σ(k)τ±(k) ... k=1 k=1

N. A. Smirnova The Nuclear Shell Model Exercise 3

Electromagnetic transitions in 17O

+ + 17 Calculate the B(E2; 1/21 → 5/2g.s.) in O modeled as a valence neutron in a 1s0d shell beyond the 16O closed-shell core. We take: ∞ Z R (r)r 2R (r)dr ≈ 12 fm2. 1s1/2 0d5/2 0 2 4 Compare your result to experimental value Bexp(E2) = 6.3 e .fm . What can you conclude?

N. A. Smirnova The Nuclear Shell Model Solution to exercise 3

Electromagnetic transitions in 17O The single-particle electric multipole operator reads ˆ 2 O(E, 2M) = er Y2M (θ, φ) , The reduced probability of the EL-transition from the initial to the ﬁnal state is

1 2 B(EL; Ji → Jf ) = |hJf ||O(EL)||Ji i| . 2Ji + 1

In our case, there is one valence particle: Ji = (1s1/2), Jf = (0d5/2). 1 B(E2; 1/2+ → 5/2+) = |h0d ||Oˆ (E2)||1s i|2 = 2 5/2 1/2 1 1 5 1 1 |hn =0, l =2, s = , j = ||er 2Y (θ, φ)||n =1, l =0, s = , j = i|2 = 2 f f f 2 f 2 2 i i i 2 i 2 Z 2 1 2 ∗ 2 1 5 1 1 2 e R (r)r R1s (r)dr |h2 ; ||Y2(θ, φ)||0 ; i| . 2 0d5/2 1/2 2 2 2 2 | {z } hr 2i2

1 1 2 2 2 2 2 2 lf jf B(EL; ji → jf ) = e hr i (2jf + 1)(2li + 1)(2L + 1)(li 0L0|lf 0) 4π L ji li

N. A. Smirnova The Nuclear Shell Model Solution to exercise 3

Electromagnetic transitions in 17O (continued)

1/2 2 5/2 2 B(E2; 1/2+ → 5/2+) = e2 1 hr 2i26 × 5(0020|20)2 = 4π 2 1/2 0 = 34.4 e2.fm4.

+ + 2 4 Experimental value Bexp(E2; 1/2 → 5/2 ) = 6.3 e .fm .

This means that the neutron should have an effective charge: e˜n ≈ 0.43 e. because we work in a severely restricted model space (one valence nucleon !).

Standard effective E2 and M1 operators

e˜π ≈ 1.5 e , e˜ν ≈ 0.5 e

g˜s(π) ≈ 0.7 gs(π) , g˜l (π) = gl (π) = 1 | {z } 5.586 g˜s(ν) ≈ 0.7 gs(ν) , g˜l (ν) = gl (ν) = 0 | {z } −3.826

N. A. Smirnova The Nuclear Shell Model Nuclear many-body problem

Non-relativistic Hamiltonian for A nucleons

A A X ~pˆ2 X Hˆ = i + Wˆ (~r −~r ) 2m i j i=1 i

A A A X ~p2 X X Hˆ = i + U(~r ) + W (~r −~r ) − U(~r ) , 2m i i j i i=1 i

A ˆ X ˆ V = V (~ri −~rj ) i

One and two-body Hamitonian

A A ˆ X ˆ X ˆ H = h(~ri ) + V (~ri −~rj ) i=1 i

N. A. Smirnova The Nuclear Shell Model Occupation-number representation (second quantization)

Creation and annihilation operators

† |αi = aα|0i hα| = h0|aα Wave function of a fermion in a quantum state α in coordinate space:

h~r|αi = φα(~r)

(Anti-)commutation relations:

n † o † † aα, aβ = aαaβ + aβ aα = δαβ n † † o aα, aβ = {aα, aβ } = 0

Normalized and antisymmetric A-fermion state:

† † † † |α1α2 . . . αAi = aαA aαA−1 ... aα2 aα1 |0i

N. A. Smirnova The Nuclear Shell Model Operators in the occupation-number formalism

One-body operators

A ˆ X ˆ O = O(~rk ) k=1 Z ˆ ∗ ˆ hα|O|βi = φα(~r)O(~r)φβ (~r)d~r

Second-quantized form of the one-body operator Oˆ :

ˆ X ˆ † O = hα|O|βiaαaβ αβ

For example, the number operator reads

ˆ X ˆ † X † N = hα|1|βiaαaβ = aαaα αβ α

N. A. Smirnova The Nuclear Shell Model Operators in the occupation-number formalism

Symmetric two-body operator acting on an A-fermion system

A ˆ X ˆ T = T (~rk ,~rj ) j

Second-quantized form of the two-body operator Tˆ:

1 X † † Tˆ = hαβ|Tˆ|γδia a a a 4 α β δ γ αβγδ

N. A. Smirnova The Nuclear Shell Model Nuclear many-body Hamiltonian in the occupation-number formalism

Non-relativistic Hamiltonian for A nucleons

A A ˆ X ˆ X ˆ ˆ (0) ˆ H = h(~ri ) + V (~ri −~rj ) = H + V i=1 i

X + 1 X + + Hˆ = ε a a + hαβ|V |γδia a a a , α α α 4 α β δ γ α αβγδ | {z } | {z } one−body term two−body term

Two-body term in a JT -coupled form 1 X p Vˆ = − hj j |V |j j i (1 + δ )(1 + δ ) 4 α β γ δ JT αβ γδ jαjβ jγ jδ h i(JT ) (00) a+ a+ a˜ a˜ (JT ) jα jβ jγ jδ

N. A. Smirnova The Nuclear Shell Model Necessary ingredients

Single-particle energies

εα

from experimental spectra of Acore plus a neutron or a proton

Two-body matrix elements (TBMEs)

hjαjβ |V |jγ jδiJT from theory ?

N. A. Smirnova The Nuclear Shell Model 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)

2 g ~σ1 · ~q ~σ2 · ~q V (1, 2) = πNN πNN 2 2 2 4M ~q + mπ

Potential (Fourier transform) in coordinate space

2 3 OPEP ( , ) = gπNN mπ { ~σ · ~σ VπNN 1 2 4M2 12 1 2 3 3 o e−mπ r + 1 + + 2 3 ~σ1 ·~r ~σ2 ·~r − ~σ1 · ~σ2 mπ r (mπ r) mπ r

Meson-exchange theories of NN potential: high- precision potentials (CD-Bonn, AV18, etc)

N. A. Smirnova The Nuclear Shell Model Concept of effective interaction (operators)

Effective in-medium nucleon-nucleon interaction In-medium effects (renormalization of the hard core) Truncated model space Hˆ Ψ = (Hˆ (0) + Vˆ )Ψ = EΨ , True wave function: ∞ X Ψ = ak Φk . k=1 Wave function in a model space:

Approaches to the problem: phenomenological or microscopic.

N. A. Smirnova The Nuclear Shell Model 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)

N. A. Smirnova The Nuclear Shell Model Schematic interaction

Some examples:

V (1, 2) = −V0 exp (µr)/(µr)

V (1, 2) = −V0δ(r~1 − r~2)

V (1, 2) = −V0δ(r~1 − r~2)(1 + α ~σ1 · ~σ2)

V (1, 2) = −V0δ(r~1 − r~2)δ(r1 − R) ...

2 V (1, 2) = χQ · Q (Q = r Y2µ(Ωr )) ...

A few parameters (interaction strengths) are ﬁtted to reproduce energy levels in a certain region of (a few) neighboring nuclei ⇒ local description only!

N. A. Smirnova The Nuclear Shell Model Exercise 4: TBMEs of the δ-force

Multipole expansion of the delta-function

V (1, 2) = −V0δ(r~1 − r~2) ,

X δ(r1 − r2) 2k + 1 δ(r~1 − r~2) = Pk (cos θ12) r1r2 4π k

Diagonal TBMEs between normalized and antisymmetric states

2 l1+l2+J j1 j2 J 1+(−1) hj1j2|V |j1j2iJT = I (2j1 + 1)(2j2 + 1) 1 1 2 − 2 0 2 2 2 2 2 j j J hj |V |j iJT = I (2j + 1) 1 1 , if j1 = j2 = j 2 − 2 0 ∞ 1 Z 1 I = [R (r)R (r)]2 dr 4π r 2 n1l1 n2l2 0

N. A. Smirnova The Nuclear Shell Model 210 2 Example 1: Pb in (ν0h9/2)

Vdelta(1, 2) = −V0δ(r~1 − r~2) ˆ ˆ Vpairing (1, 2) = −GS+ · S− 1 2 2 la+l p h j | V (1, 2) | j i = −(−1) b G (2ja + 1)(2j + 1) a pairing b 01 2 b

1.5 Experiment Delta Pairing

8+ 6+ + 4+ 8 2+,4+,6+,8+ 6+ 1 4+ 2+ 2+

E (MeV) 0.5

0 0+ 0+ 0+ 210 Pb128

N. A. Smirnova The Nuclear Shell Model Example 2: 20Ne and SU(3) model of Elliott

A X p2 1 Hˆ = − i + mω2r 2 − χQ · Q 2m 2 i i=1

Q is an algebraic quadrupole operator (Qµ, Lµ 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+ + 15 5 + + + 4 4 8 + 8+ 3 2+ 2+ 10 0+ + E (MeV) 6 + - 6 4 0+ 3- 5 2- 4+ 4+ + 2 2+ 0 0+ 0+ 20Ne

Rotational classiﬁcation of nuclear states as mixing of many spherical conﬁgurations

N. A. Smirnova The Nuclear Shell Model Empirical interaction (least-squares-ﬁt method)

All TBME’s hjajb|V |jc jd iJT are considered as free parameters !

Initial Set-up Eigenvalues Convergence ? yes N data 2 interaction Hamiltonian Eigenvectors th exp Stop th EE− E,Ψ ∑ ( ii) parameters matrix ii i1=

New New linear Variation of interaction system parameters parameters

Examples:

0p-shell: 4He–16O (15 TBME’s) Cohen, Kurath (1965) 1s0d-shell: 16O–40Ca (63 TBME’s) Brown, Wildenthal (1988) 1p0f -shell: 40Ca–80Zr (195 TBME’s) Honma et al (2002, 2004)

N. A. Smirnova The Nuclear Shell Model 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

Vlow−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)

N. A. Smirnova The Nuclear Shell Model 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)

N. A. Smirnova The Nuclear Shell Model State-of-the-art calculations: backbending in 48Cr

KB3 (semi-empirical interaction in pf-shell model space) Strasbourg-Madrid

For J<10 :

EJ ~ J(J + 1) (J + 1)(2J + 3) Q = Q (J), K ≠ 1 0 3K2 − J(J + 1) spec 5 2 B(E2;J → J − 2) = e2 (JK20 | J − 2,K) Q2 16π 0

J<10: collective rotation J=10-12 : backbending phenomenon (competition between rotation and alignment of 0f7/2 particles) J>12 : spherical states E. Caurier et al, Rev. Mod. Phys. 77 (2005) 427

N. A. Smirnova The Nuclear Shell Model State-of-the-art calculations: superdeformation in 36Ar

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, 042502 (2005)

N. A. Smirnova The Nuclear Shell Model State-of-the-art calculations: mirror bands in A = 51

S. Lenzi, A. Zuker, E. Caurier et al

N. A. Smirnova The Nuclear Shell Model No-Core Shell Model

Non-relativistic Hamiltonian for A nucleons in many N~Ω harmonic oscillator space

A A X ~p2 X Hˆ = i + W (~r −~r ) 2m i j i=1 i

Problem: excitation of the center-of-mass of the system.

Center-of-mass coordinates

A A 1 X X R~ = ~r ; P~ = ~p A i i i=1 i=1

Translational-invariant Hamiltonian

A A X ~p2 P~ 2 X Hˆ = i − + W (~r −~r ) 2m 2mA i j i=1 i

N. A. Smirnova The Nuclear Shell Model No-Core Shell Model

Separation of the harmonic oscillator Hamiltonian into center-of-mass and intrinsic Hamiltonians

A X ~p2 mΩ2~r 2 H = i + i ho 2m 2 i=1 A 2 A ~ 2 2 ~ 2 1 X 2 mΩ X 2 P mAΩ R = ~p − ~p + ~r −~r + + 2mA i j 2A i j 2mA 2 i

N. A. Smirnova The Nuclear Shell Model No-Core Shell Model

Translational-invariant Hamiltonian

A A A X ~p2 mΩ2~r 2 P~ 2 X X mΩ2~r 2 Hˆ = i + i − + W (~r −~r ) − i 2m 2 2mA i j 2 i=1 i

A 2 2 2 A 2 A X ~p mΩ ~r X mΩ X 2 Hˆ = i + i + W (~r −~r )− ~r −~r −Hˆ 2m 2 i j 2A i j CoM i=1 i

Separation of the Center-of-Mass motion 3 Hˆ Ω = Hˆ + β Hˆ − Ω CoM 2~

N. A. Smirnova The Nuclear Shell Model

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.

N. A. Smirnova The Nuclear Shell Model 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..).

N. A. Smirnova The Nuclear Shell Model 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

N. A. Smirnova The Nuclear Shell Model