IAEA Workshop on NSDD, Trieste, November 2003
The nuclear shell model
P. Van Isacker, GANIL, France
Context and assumptions of the model
Symmetries of the shell model:Racah’s SU(2) pairing model
Wigner’s SU(4) symmetry
Elliott’s SU(3) model of rotation
IAEA Workshop on NSDD, Trieste, November 2003
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 parametrization (+ correlations).
• Phenomenological models: Specific nuclei or properties with local parametrization.
IAEA Workshop on NSDD, Trieste, November 2003
Ab initio methods
• Faddeev-Yakubovsky: A≤4
• Coupled-rearrangement-channel Gaussian-basis variational: A≤4 (higher with clusters)
• Stochastic variational: A≤7
• Hyperspherical harmonic variational:
• Green’s function Monte Carlo: A≤7
• No-core shell model: A≤12
• Effective interaction hyperspherical: A≤6
IAEA Workshop on NSDD, Trieste, November 2003
Benchmark calculation for A=4
• Test calculation with realistic interaction: all methods agree.
• But Eexpt=-28.296 MeV need for three-nucleon interaction.
Ψ δ r −rkl( )k<l
4
∑ Ψ
H. Kamada et al., Phys. Rev. C 63 (2001) 034006
IAEA Workshop on NSDD, Trieste, November 2003
Basic symmetries
• Non-relativistic Schrödinger equation:
• Symmetry or invariance under:– Translations linear momentum P– Rotations angular momentum J=L+S– Space reflection parity – Time reversal
H =pk
2
2mkk=1
A
∑ + W2 ξk,ξl( )k<l
A
∑ + W3 ξk,ξl,ξm( )k<l<m
A
∑ +L
ξk ≡r r k,
r σ k,
r τ k{ },
r p k =−ih
r ∇ k[ ]
IAEA Workshop on NSDD, Trieste, November 2003
Nuclear shell model
• Separation in mean field + residual interaction:
• Independent-particle assumption. Choose V and neglect residual interaction:
H =pk
2
2mkk=1
A
∑ + W ξk,ξl( )k<l
A
∑
=pk
2
2mk
+V ξk( )⎡
⎣ ⎢ ⎤
⎦ ⎥ k=1
A
∑mean field
1 2 4 4 4 3 4 4 4 + W ξk,ξl( )
k<l
A
∑ − V ξk( )k=1
A
∑⎡
⎣ ⎢ ⎤
⎦ ⎥
residualinteraction1 2 4 4 4 4 3 4 4 4 4
H ≈HIP =pk
2
2mk
+V ξk( )⎡
⎣ ⎢ ⎤
⎦ ⎥ k=1
A
∑
IAEA Workshop on NSDD, Trieste, November 2003
Independent-particle shell model
• Solution for one particle:
• Solution for many particles:
pk2
2mk
+V ξk( )⎡
⎣ ⎢ ⎤
⎦ ⎥ φi k( ) =Eiφi k( ) φi k( ) ≡φi
r r k,
r σ k,
r τ k( )[ ]
Φi1i2K i A1,2,K ,A( ) = φik
k( )k=1
A
∏
HIPΦi1i2K i A1,2,K ,A( ) = Eik
k=1
A
∑⎛ ⎝ ⎜ ⎞
⎠ Φi 1i2K iA 1,2,K ,A( )
IAEA Workshop on NSDD, Trieste, November 2003
Independent-particle shell model
• Antisymmetric solution for many particles (Slater determinant):
• Example for A=2 particles:
Ψi1i2K i A1,2,K ,A( ) =
1A!
φi1 1( ) φi 12( ) K φi1 A( )
φi 21( ) φi 2
2( ) K φi 2A( )
M M O M
φiA 1( ) φiA 2( ) K φiA A( )
Ψi1i2 1,2( )=12
φi1 1( )φi 22( ) −φi1 2( )φi2 1( )[ ]
IAEA Workshop on NSDD, Trieste, November 2003
Hartree-Fock approximation
• Vary i (ie V) to minize the expectation value of H in a Slater determinant:
• Application requires choice of H. Many global parametrizations (Skyrme, Gogny,…) have been developed.
δΨi1i2K iA
* 1,2,K ,A( )HΨi1i2K i A1,2,K ,A( )dξ1dξ2K dξA∫
Ψi1i 2K iA* 1,2,K ,A( )Ψi1i2K iA
1,2,K ,A( )dξ1dξ2K dξA∫=0
IAEA Workshop on NSDD, Trieste, November 2003
Poor man’s Hartree-Fock
• Choose a simple, analytically solvable V that approximates the microscopic HF potential:
• Contains– Harmonic oscillator potential with constant .
– Spin-orbit term with strength ls.
– Orbit-orbit term with strength ll.
• Adjust , ls and ll to best reproduce HF.
HIP =
pk2
2m+
12
mω2rk2 −ζls
r l k ⋅
r s k −ζll lk
2⎡
⎣ ⎢ ⎤
⎦ ⎥ k=1
A
∑
IAEA Workshop on NSDD, Trieste, November 2003
Energy levels of harmonic oscillator• Typical parameter
values:
• ‘Magic’ numbers at 2, 8, 20, 28, 50, 82, 126, 184,…
hω ≈41A−1/3 MeV
ζlsh2 ≈20A−2/3 MeV
ζll h2 ≈0.1MeV
∴ b≈1.0A1/6 fm
IAEA Workshop on NSDD, Trieste, November 2003
Evidence for shell structure
• Evidence for nuclear shell structure from– Excitation energies in even-even nuclei– Nucleon-separation energies– Nuclear masses– Nuclear level densities– Reaction cross sections
• Is nuclear shell structure modified away from the line of stability?
IAEA Workshop on NSDD, Trieste, November 2003
Shell structure from Ex(21)
• High Ex(21) indicates stable shell structure:
IAEA Workshop on NSDD, Trieste, November 2003
Shell structure from Sn or Sp
• Change in slope of Sn (Sp) indicates neutron (proton) shell closure (constant N-Z plots):
A. Ozawa et al., Phys. Rev. Lett. 84 (2000) 5493
IAEA Workshop on NSDD, Trieste, November 2003
Shell structure from masses
• Deviations from Weizsäcker mass formula:
IAEA Workshop on NSDD, Trieste, November 2003
Shell structure from masses
• Deviations from improved Weizsäcker mass formula that includes nn and n+n terms:
IAEA Workshop on NSDD, Trieste, November 2003
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 with modified occupation.
QuickTime™ et undécompresseur TIFF (non compressé)sont requis pour visionner cette image.
J.M. Cavedon et al., Phys. Rev. Lett. 49 (1982) 978
IAEA Workshop on NSDD, Trieste, November 2003
Nuclear shell model
• The full shell-model hamiltonian:
• Valence nucleons: Neutrons or protons that are in excess of the last, completely filled shell.
• Usual approximation: Consider the residual interaction VRI among valence nucleons only.
• Sometimes: Include selected core excitations (‘intruder’ states).
H =pk
2
2m+V ξk( )
⎡
⎣ ⎢ ⎤
⎦ ⎥ k=1
A
∑ + VRI ξk,ξl( )k<l
A
∑
IAEA Workshop on NSDD, Trieste, November 2003
The shell-model problem
• Solve the eigenvalue problem associated with the matrix (n active nucleons):
• Methods of solution:– Diagonalization (Strasbourg-Madrid): 109
– Monte-Carlo (Pasadena-Oak Ridge):– Quantum Monte-Carlo (Tokyo):– Group renormalization (Madrid-Newark): 10120
′ i 1K ′ i n VRI ξk,ξl( )
k<l
n
∑ i1K in 1K ni1K in ≡Ψi1K in 1K n( )[ ]
IAEA Workshop on NSDD, Trieste, November 2003
Residual shell-model interaction
• Four 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: interaction.
IAEA Workshop on NSDD, Trieste, November 2003
Schematic short-range interaction
• Delta interaction in harmonic-oscillator basis.
• Example of 42Sc21 (1 active neutron + 1 active proton):
IAEA Workshop on NSDD, Trieste, November 2003
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:
HIP =pk
2
2m+
12
mω2rk2 −ζls
r l k ⋅
r s k −ζll lk
2⎡
⎣ ⎢ ⎤
⎦ ⎥ k=1
A
∑
+ VRI ξk,ξl( )k<l
A
∑
IAEA Workshop on NSDD, Trieste, November 2003
Racah’s SU(2) pairing model
• Assume large spin-orbit splitting ls which implies a jj coupling scheme.
• Assume pairing interaction in a single-j shell:
• Spectrum of 210Pb:
j2JM J Vpairing j2JM J =−1
22j +1( )g, J =0
0, J ≠0
⎧ ⎨ ⎩
IAEA Workshop on NSDD, Trieste, November 2003
Solution of pairing hamiltonian
• Analytic solution of pairing hamiltonian for identical nucleons in a single-j shell:
• Seniority (number of nucleons not in pairs coupled to J=0) is a good quantum number.
• Correlated ground-state solution (cfr. super-fluidity in solid-state physics).
jnυJ Vpairingξk,ξl( )k<l
n
∑ jnυJ =−14 G n−υ( ) 2j −n−υ +3( )
G. Racah, Phys. Rev. 63 (1943) 367
IAEA Workshop on NSDD, Trieste, November 2003
Pairing and superfluidity
• Ground states of a pairing hamiltonian have superfluid character:– Even-even nucleus (=0):– Odd-mass nucleus (=1):
• Nuclear superfluidity leads to– Constant energy of first 2+ in even-even nuclei.– Odd-even staggering in masses.– Two-particle (2n or 2p) transfer enhancement.
S+( )n j /2
o
aj+ S+( )
nj /2o
IAEA Workshop on NSDD, Trieste, November 2003
Superfluidity in semi-magic nuclei• Even-even nuclei:
– Ground state has =0.
– First-excited state has =2.
– Pairing produces constant energy gap:
• Example of Sn nuclei:
Ex 21+
( )=12 2j +1( )g
IAEA Workshop on NSDD, Trieste, November 2003
Two-nucleon separation energies
• Two-nucleon separation energies S2n:
(a) Shell splitting dominates over interaction.
(b) Interaction dominates over shell splitting.
(c) S2n in tin isotopes.
IAEA Workshop on NSDD, Trieste, November 2003
Generalized pairing models
• Trivial generalization from a single-j shell to several degenerate j shells:
• Pairing with neutrons and protons:– T=1 pairing: SO(5).– T=0 & T=1 pairing: SO(8).
• Non-degenerate shells:– Talmi’s generalized seniority.– Richardson’s integrable pairing model.
S+ ∝ 12 2j+1 aj
+×aj+
( )0
0( )
j∑
IAEA Workshop on NSDD, Trieste, November 2003
Pairing with neutrons and protons
• For neutrons and protons two pairs and hence two pairing interactions are possible:– Isoscalar (S=1,T=0):
– Isovector (S=0,T=1):−S+
01⋅S−01, S+
01= l +12 a
l 12
12
+ ×al 12
12
+⎛ ⎝
⎞ ⎠
001( )
, S−01 = S+
01( )
+
−S+10⋅S−
10, S+10 = l +1
2 al 1
212
+ ×al 1
212
+⎛ ⎝
⎞ ⎠
010( )
, S−10 = S+
10( )
+
IAEA Workshop on NSDD, Trieste, November 2003
Superfluidity of N=Z nuclei• Ground state of a T=1 pairing hamiltonian for
identical nucleons is superfluid, (S+)n/2o.• Ground state of a T=0 & T=1 pairing
hamiltonian with equal number of neutrons and protons has different superfluid character:
Condensate of ’s ( depends on g0/g1).• Observations:
– Isoscalar component in condensate survives only in N~Z nuclei, if anywhere at all.
– Spin-orbit term reduces isoscalar component.
cosθS+10⋅S+
10 −sinθS+01⋅S+
01( )
n/4o
IAEA Workshop on NSDD, Trieste, November 2003
Wigner’s SU(4) symmetry• Assume the nuclear hamiltonian is invariant
under spin and isospin rotations:
• Since {S,T,Y} form an SU(4) algebra:– Hnucl has SU(4) symmetry.– Total spin S, total orbital angular momentum L,
total isospin T and SU(4) labels () are conserved quantum numbers.
Hnucl,Sμ[ ]= Hnucl,Tν[ ] = Hnucl,Yμν[ ] =0
Sμ = sμ k( ),k=1
A
∑ Tν = tν k( )k=1
A
∑ , Yμν = sμ k( )k=1
A
∑ tν k( )
E.P. Wigner, Phys. Rev. 51 (1937) 106F. Hund, Z. Phys. 105 (1937) 202
IAEA Workshop on NSDD, Trieste, November 2003
Physical origin of SU(4) symmetry
• SU(4) labels specify the separate spatial and spin-isospin symmetry of the wavefunction:
• Nuclear interaction is short-range attractive and hence favours maximal spatial symmetry.
IAEA Workshop on NSDD, Trieste, November 2003
Breaking of SU(4) symmetry
• Breaking of SU(4) symmetry as a consequence of– Spin-orbit term in nuclear mean field.– Coulomb interaction.– Spin-dependence of residual interaction.
• Evidence for SU(4) symmetry breaking from– Masses: rough estimate of nuclear BE from
decay: Gamow-Teller operator Y,1 is a generator of SU(4) selection rule in ().
B N,Z( )∝ a+bgλμν( ) =a+b λμν C2 SU 4( )[ ]λμν
IAEA Workshop on NSDD, Trieste, November 2003
SU(4) breaking from masses
• Double binding energy difference Vnp
Vnp in sd-shell nuclei:
δVnp N,Z( )=14 B N,Z( )−B N −2,Z( )−B N,Z−2( )+B N −2,Z−2( )[ ]
P. Van Isacker et al., Phys. Rev. Lett. 74 (1995) 4607
IAEA Workshop on NSDD, Trieste, November 2003
SU(4) breaking from decay
• Gamow-Teller decay into odd-odd or even-even N=Z nuclei:
P. Halse & B.R. Barrett, Ann. Phys. (NY) 192 (1989) 204
IAEA Workshop on NSDD, Trieste, November 2003
Elliott’s SU(3) model of rotation
• Harmonic oscillator mean field (no spin-orbit) with residual interaction of quadrupole type:
H =pk
2
2m+
12
mω2rk2⎡
⎣ ⎢ ⎤
⎦ ⎥ k=1
A
∑ −κQ⋅Q,
Qμ =4π5
rk2Y2μ ˆ r k( )
k=1
A
∑ + pk2Y2μ ˆ p k( )
k=1
A
∑⎛ ⎝ ⎜ ⎞
⎠
J.P. Elliott, Proc. Roy. Soc. A 245 (1958) 128; 562
IAEA Workshop on NSDD, Trieste, November 2003
Importance and limitations of SU(3)
• Historical importance:– Bridge between the spherical shell model and the
liquid droplet model through mixing of orbits.– Spectrum generating algebra of Wigner’s SU(4)
supermultiplet.
• Limitations:– LS (Russell-Saunders) coupling, not jj coupling
(zero spin-orbit splitting) beginning of sd shell.– Q is the algebraic quadrupole operator no
major-shell mixing.
IAEA Workshop on NSDD, Trieste, November 2003
Generalized SU(3) models
• How to obtain rotational features in a jj-coupling limit of the nuclear shell model?
• Several efforts since Elliott:– Pseudo-spin symmetry.– Quasi-SU(3) symmetry (Zuker). – Effective symmetries (Rowe).– FDSM: fermion dynamical symmetry model.– ...