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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 Gaussianbasis 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
4
methods agree.
  r  r 
k l
kl
• But Eexpt=-28.296 MeV  need for threenucleon interaction.
H. Kamada et al., Phys. Rev. C 63 (2001) 034006
IAEA Workshop on NSDD, Trieste, November 2003
Basic symmetries
• Non-relativistic Schrödinger equation:
A
2
k
A
A
p
H
  W2 k ,l    W3 k , l ,m  
k 1 2m k
kl
k  l m

k
 rk , k ,k ,
pk  i k

• Symmetry or invariance under:
–
–
–
–
Translations  linear momentum P
Rotations  angular momentum J=L+S
Space reflection  parity 
Time reversal
IAEA Workshop on NSDD, Trieste, November 2003
Nuclear shell model
• Separation in mean field + residual interaction:
A
A
p2k
H
  W k , l 
k 1 2m k
k l
A
 pk2
  A


 V k    Wk ,l    V k 


k 1 2mk
 k l
k 1

A
mean fi eld
resi dual interaction
• Independent-particle assumption. Choose V
and neglect residual interaction:
 pk2

H  HIP   
 V k 
k 1 2mk

A
IAEA Workshop on NSDD, Trieste, November 2003
Independent-particle shell model
• Solution for one particle:
 pk2

 V k  i k   Ei i k 

2mk

 k    r , , 
i
i
k
k
• Solution for many particles:
i i
1 2
A
iA
1,2, , A   i k 
HIPi1 i2
k1
k
A



, A   Eik i 1i2
i A 1, 2,
k 1 
iA
1, 2, , A
IAEA Workshop on NSDD, Trieste, November 2003
k
Independent-particle shell model
• Antisymmetric solution for many particles
(Slater determinant):
i i
1 2
iA
1,2, , A 
 i 1 i 2
1 i 1 i 2
1
1
2
2
 i A
i A 
1
2
A!
i 1  i 2
A
i A
A
A
• Example for A=2 particles:


1
i1i2 1,2 
 i1 1i 2 2   i1 2i2 1
2
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:

*

 ii
1 2
*

 ii
1 2
1, 2, , AHi i i 1,2, ,A d1d2 dA
0
, A i i i 1, 2, , Ad1 d2 d A
i 1,2,
iA
1 2
A
1 2
A
A
• Application requires choice of H. Many global
parametrizations (Skyrme, Gogny,…) have
been developed.
IAEA Workshop on NSDD, Trieste, November 2003
Poor man’s Hartree-Fock
• Choose a simple, analytically solvable V that
approximates the microscopic HF potential:
 pk2 1
2 2
2 
HIP     m rk   ls lk  sk   ll lk 
2

k  1 2m
A
• 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.
IAEA Workshop on NSDD, Trieste, November 2003
Energy levels of harmonic oscillator
• Typical parameter
values:
  41 A
1/ 3
MeV
 ls
2
 20 A 2 / 3 MeV
 ll
2
 0.1 MeV
b  1.0 A 1/ 6 fm
• ‘Magic’ numbers at 2, 8,
20, 28, 50, 82, 126,
184,…
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 nn 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.
QuickTi me™ et un décompresseur TI FF (non com pres sé) sont requi s pour vi sionner cet te im age.
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:
 pk2
 A
H     V k   VRI k ,l 
 k l
k 1 2m
A
• 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).
IAEA Workshop on NSDD, Trieste, November 2003
The shell-model problem
• Solve the eigenvalue problem associated with
the matrix (n active nucleons):
n
i1 in  VRI k ,l  i1
kl
in
1
n i1
in  i1
in
1 n
• Methods of solution:
–
–
–
–
Diagonalization (Strasbourg-Madrid): 109
Monte-Carlo (Pasadena-Oak Ridge):
Quantum Monte-Carlo (Tokyo):
Group renormalization (Madrid-Newark): 10120
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 harmonicoscillator 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:
 pk2 1
2 2
2 
HIP     m rk   ls lk  sk   ll lk 
2

k  1 2m
A
A
  VRI k , l 
k l
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:
1


2 j  1g, J  0
2
2
2
j JM J Vpairi ng j JM J  

0,
J0
• Spectrum of 210Pb:
IAEA Workshop on NSDD, Trieste, November 2003
Solution of pairing hamiltonian
• Analytic solution of pairing hamiltonian for
identical nucleons in a single-j shell:
n
j n J  Vpairing k , l  j n J   14 Gn  2j  n    3
k l
• Seniority  (number of nucleons not in pairs
coupled to J=0) is a good quantum number.
• Correlated ground-state solution (cfr. superfluidity in solid-state physics).
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:
n /2

S

o
– Even-even nucleus (=0): 
n /2

a
S
– Odd-mass nucleus (=1): j    o
j
j
• 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.
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:
Ex 21 
2j  1g
• Example of Sn
nuclei:

1
2
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:
S  
1
2
j


  0 
2j  1 a j  a j
0
• 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.
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):


 S10  S10 , S10  l  12 al1 1  al1 1 

22
22
– Isovector (S=0,T=1):
010


 S01  S01 , S01  l  21 al1 1  al1 1 

22
2 2
,
S  S

,
S  S

001
IAEA Workshop on NSDD, Trieste, November 2003
10 

10

01

01 

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:
cos S
10

01


01 n / 4

 S sin  S  S
10

o
•  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.
IAEA Workshop on NSDD, Trieste, November 2003
Wigner’s SU(4) symmetry
• Assume the nuclear hamiltonian is invariant
under spin and isospin rotations:
H
nucl



,S  Hnucl,T   Hnucl,Y  0
A
A
A
k 1
k 1
k 1
S   s k , T   t k , Y   s k t k 
• 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
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
BN, Z   a  bg   a  b  C2SU4
–  decay: Gamow-Teller operator Y,1 is a
generator of SU(4)  selection rule in ().
IAEA Workshop on NSDD, Trieste, November 2003
SU(4) breaking from masses
• Double binding energy difference Vnp
VnpN,Z   14 BN,Z   BN  2,Z   BN,Z  2  BN  2, Z  2
• Vnp in sd-shell nuclei:
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 eveneven 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:
 pk2 1
2 2 
H     m rk   Q Q,
2

k 1 2m
A
Q 
A
A
4 
 rk2 Y2  ˆrk    pk2Y2   ˆpk 

5 k  1
k 1
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 jjcoupling 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.
...
IAEA Workshop on NSDD, Trieste, November 2003