Nucleur shell model of atom by P. Van isacker

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

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.

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

Benchmark calculation for A=4
• Test calculation with realistic interaction: all
methods agree.
• But Eexpt=-28.296 MeV  need for three-
nucleon interaction.
H. Kamada et al., Phys. Rev. C 63 (2001) 034006

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

Nuclear shell model
• Separation in mean field + residual interaction:
• Independent-particle assumption. Choose V
and neglect residual interaction:

Independent-particle shell model
• Solution for one particle:
• Solution for many particles:

Independent-particle shell model
• Antisymmetric solution for many particles
(Slater determinant):
• Example for A=2 particles:

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.

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.

Energy levels of harmonic oscillator
• Typical parameter
values:
• ‘Magic’ numbers at 2, 8,
20, 28, 50, 82, 126,
184,…

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?

Shell structure from Ex(21)
• High Ex(21) indicates stable shell structure:

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

Shell structure from masses
• Deviations from Weizsäcker mass formula:

Shell structure from masses
• Deviations from improved Weizsäcker mass
formula that includes nn and n+n terms:

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.
J.M. Cavedon et al., Phys. Rev. Lett. 49 (1982) 978

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).

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

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.

Schematic short-range interaction
• Delta interaction in harmonic-oscillator basis.
• Example of 42Sc21 (1 active neutron + 1 active
proton):

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:

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:

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).
G. Racah, Phys. Rev. 63 (1943) 367

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.

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:

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.

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.

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):

Superfluidity of N=Z nuclei
• Ground state of a T=1 pairing hamiltonian for
identical nucleons is superfluid, (S+)n/2o.
• 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.

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.
E.P. Wigner, Phys. Rev. 51 (1937) 106
F. Hund, Z. Phys. 105 (1937) 202

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.

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 ().

SU(4) breaking from masses
• Double binding energy difference Vnp
• Vnp in sd-shell nuclei:
P. Van Isacker et al., Phys. Rev. Lett. 74 (1995) 4607

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

Elliott’s SU(3) model of rotation
• Harmonic oscillator mean field (no spin-orbit)
with residual interaction of quadrupole type:
J.P. Elliott, Proc. Roy. Soc. A 245 (1958) 128; 562

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.

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.
– ...

Nucleur shell model of atom by P. Van isacker

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