Nuclear Structure
(I) Single-particle models
P. Van Isacker, GANIL, France
NSDD Workshop, Trieste, April-May 2008
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 parametrisation (+ correlations).
• Phenomenological models: Specific nuclei or
properties with local parametrisation.
NSDD Workshop, Trieste, April-May 2008

Nuclear shell model
• Many-body quantum mechanical problem:
A
A
2
p
Hˆ   k  Vˆ2 rk ,rl 
2mk kl
k1
A

 pk2
 A
 
 Vˆ rk  Vˆ2 rk ,rl   V rk 
 kl

k1 2mk
k1
A
mean field
residual interaction
• Independent-particle assumption. Choose V
and neglect residual interaction:
 pk2

ˆ
ˆ
ˆ
H  HIP  
 V rk 

k1 2mk
A
NSDD Workshop, Trieste, April-May 2008


Independent-particle shell model
• Solution for one particle:
p2

  Vˆ r i r   E ii r 
2m

• Solution for many particles:
i1 i2
iA
r1,r2 ,
A
,rA     ik rk 
k1
Hˆ IPi1 i2
 A

,rA    E ik i1 i2
iA r1,r2 ,
k1 
iA
NSDD Workshop, Trieste, April-May 2008
r1,r2,
,rA 

Independent-particle shell model
• Anti-symmetric solution for many particles
(Slater determinant):
i1 i2
iA
r1,r2 ,
,rA  
 i r1   i r2 
1  i r1   i r2 
A!
1
1
2
2
 i r1   i r2 
A
A
• Example for A=2 particles:
1
i1 i2 r1,r2  
 i1 r1  i2 r2    i1 r2  i2 r1 

2
NSDD Workshop, Trieste, April-May 2008
 i rA 
 i rA 
1
2
 i rA 
A
Hartree-Fock approximation
• Vary i (ie V) to minize the expectation value
of H in a Slater determinant:


*

 i1i2

*
i1 i2
r1,r2 ,
i r1,r2 ,
iA
A
,rA Hˆ i1 i2
,rA i1 i2
r1,r2,
r1,r2 ,
iA
iA
,rA dr1dr2
,rA dr1dr2
drA
drA
0
• Application requires choice of H. Many global
parametrizations (Skyrme, Gogny,…) have
been developed.
NSDD Workshop, Trieste, April-May 2008

Poor man’s Hartree-Fock
• Choose a simple, analytically solvable V that
approximates the microscopic HF potential:
pk2 m 2 2

2
ˆ
HIP   
rk   lk  sk   lk 
2

k1 2m
A
• Contains
– Harmonic oscillator potential with constant .
– Spin-orbit term with strength .
– Orbit-orbit term with strength .
• Adjust ,  and  to best reproduce HF.
NSDD Workshop, Trieste, April-May 2008

Harmonic oscillator solution
• Energy eigenvalues of the harmonic oscillator:
1
1


l
j

l

2
E nlj  N  32    2 ll  1   2 1 2
1
l

1
j

l



2
2
N  2n  l  0,1,2, : oscillatorquantumnumber
n  0,1,2, : radialquantumnumber
l  N,N  2, ,1 or 0 : orbit alangular momentum


j  l  12 : totalangular momentum
m j   j, j  1, , j : z projectionof j
NSDD Workshop, Trieste, April-May 2008
Energy levels of harmonic oscillator
• Typical parameter
values:
  41 A1/ 3 MeV

2
 20 A2 / 3 MeV

2
 0.1 MeV
b  1.0 A1/ 6 fm

• ‘Magic’ numbers at 2, 8,
20, 28, 50, 82, 126,
184,…
NSDD Workshop, Trieste, April-May 2008
Why an orbit-orbit term?
• Nuclear mean field is
close to Woods-Saxon:
VˆWS r 

V0
r  R0
1 exp
a
• 2n+l=N degeneracy is
lifted  El < El-2 < 
NSDD Workshop, Trieste, April-May 2008

Why a spin-orbit term?
• Relativistic origin (ie Dirac equation).
• From general invariance principles:
2
r
0 V
ˆ
VSO   rl  s,  r 
  in HO 
r r
• Spin-orbit term is surface peaked 
diminishes for diffuse potentials.
NSDD Workshop, Trieste, April-May 2008
Evidence for shell structure
• Evidence for nuclear shell structure from
–
–
–
–
2+ in even-even nuclei [Ex, B(E2)].
Nucleon-separation energies & nuclear masses.
Nuclear level densities.
Reaction cross sections.
• Is nuclear shell structure
modified away from the
line of stability?
NSDD Workshop, Trieste, April-May 2008
Ionisation potential in atoms
NSDD Workshop, Trieste, April-May 2008
Neutron separation energies
NSDD Workshop, Trieste, April-May 2008
Proton separation energies
NSDD Workshop, Trieste, April-May 2008

Liquid-drop mass formula
• Binding energy of an atomic nucleus:
2
Z Z 1
N  Z

2/3
BN,Z   av A  as A  ac
 as
1/ 3
 N,Z 
 ap
A1/ 3
A
A
• For 2149 nuclei (N,Z ≥ 8) in AME03:
av16, as18, ac0.71, a's23, ap6
 rms2.93 MeV.
C.F. von Weizsäcker, Z. Phys. 96 (1935) 431
H.A. Bethe & R.F. Bacher, Rev. Mod. Phys. 8 (1936) 82
NSDD Workshop, Trieste, April-May 2008
The nuclear mass surface
NSDD Workshop, Trieste, April-May 2008
The ‘unfolding’ of the mass surface
NSDD Workshop, Trieste, April-May 2008

Modified liquid-drop formula
• Add surface, Wigner and ‘shell’ corrections:
Z Z 1
 N,Z 
2/3
BN,Z   av A  as A  ac
 ap
1/ 3
1/ 3
Sv

1 y s A1/ 3
A
A
4T T  1
2
 af n  n    aff n  n  
A
• For 2149 nuclei (N,Z ≥ 8) in AME03:
av16, as18, ac0.71, Sv35, ys2.9, ap5.5,
af0.85, aff0.016
 rms1.16 MeV.
NSDD Workshop, Trieste, April-May 2008
Shell-corrected LDM
NSDD Workshop, Trieste, April-May 2008
Shell structure from Ex(21)
NSDD Workshop, Trieste, April-May 2008
Evidence for IP shell model
• Ground-state spins and
parities of nuclei:
j in  nl jm  J 


J
l in  nl jm  l   

j
j
NSDD Workshop, Trieste, April-May 2008
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.
QuickTime™ et un décompresseur TIFF(non compressé) sont requis pour visionner cette image.
J.M. Cavedon et al., Phys. Rev. Lett. 49 (1982) 978
NSDD Workshop, Trieste, April-May 2008
Variable shell structure
NSDD Workshop, Trieste, April-May 2008
Beyond Hartree-Fock
• Hartree-Fock-Bogoliubov (HFB): Includes
pairing correlations in mean-field treatment.
• Tamm-Dancoff approximation (TDA):
– Ground state: closed-shell HF configuration
– Excited states: mixed 1p-1h configurations
• Random-phase approximation (RPA): Correlations in the ground state by treating it on
the same footing as 1p-1h excitations.
NSDD Workshop, Trieste, April-May 2008

Nuclear shell model
• The full shell-model hamiltonian:
pk2
 A
Hˆ    Vˆ rk  VˆRI rk ,rl 
 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).
NSDD Workshop, Trieste, April-May 2008
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.
NSDD Workshop, Trieste, April-May 2008
Schematic short-range interaction
• Delta interaction in harmonic-oscillator basis:
• Example of 42Sc21 (1 neutron + 1 proton):
NSDD Workshop, Trieste, April-May 2008
Large-scale shell model
• Large Hilbert spaces:
i'1 i' 2
n
i' A
Vˆ r ,r  
RI
k
l
i1 i2
iA
kl
– Diagonalisation : ~109.
– Monte Carlo : ~1015.
• Example : 8n + 8p in
pfg9/2 (56Ni).
M. Honma et al., Phys. Rev. C 69 (2004) 034335
NSDD Workshop, Trieste, April-May 2008
The three faces of the shell model
NSDD Workshop, Trieste, April-May 2008
Racah’s SU(2) pairing model
• Assume pairing interaction in a single-j shell:
1


2 j 1g0, J  0
2
2
2
ˆ
j JMJ Vpairingr1,r2  j JMJ  
0,
J 0

• Spectrum 210Pb:

NSDD Workshop, Trieste, April-May 2008

Solution of the pairing hamiltonian
• Analytic solution of pairing hamiltonian for
identical nucleons in a single-j shell:
j nJ
n
1
n
ˆ
V
r
,r
j

J

g
 pairing k l 
0 4 n   2 j  n    3
1kl
• Seniority  (number of nucleons not in pairs
coupled to J=0) is a good quantum number.
• Correlated ground-state solution (cf. BCS).
G. Racah, Phys. Rev. 63 (1943) 367
NSDD Workshop, Trieste, April-May 2008
Nuclear superfluidity
• Ground states of pairing hamiltonian have the
following correlated character:
– Even-even nucleus (=0): Sˆ  o , Sˆ   aˆ aˆ
n /2

– Odd-mass nucleus (=1):

m
aˆ


Ý
Ý
Ý
m m

 
Sˆ 
• Nuclear superfluidity
leads
to

n /2
m
o
+
– Constant energy of first
 2 in even-even nuclei.
– Odd-even staggering in masses.
– Smooth variation of two-nucleon separation
energies with nucleon number.
– Two-particle (2n or 2p) transfer enhancement.
NSDD Workshop, Trieste, April-May 2008
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.
NSDD Workshop, Trieste, April-May 2008
Pairing gap in semi-magic nuclei
• Even-even nuclei:
– Ground state: =0.
– First-excited state: =2.
– Pairing produces
constant energy gap:
Ex 21  12 2 j 1G
• Example of Sn isotopes:

NSDD Workshop, Trieste, April-May 2008

Elliott’s SU(3) model of rotation
• Harmonic oscillator mean field (no spin-orbit)
with residual interaction of quadrupole type:
pk2 1

2 2
ˆ
H    m rk  g2Qˆ  Qˆ ,
2

k1 2m
A
A
Qˆ    rk2Y2 rˆk 
k1
A
  pk2Y2 pˆ k 
k1
J.P. Elliott, Proc. Roy. Soc. A 245 (1958) 128; 562
NSDD Workshop, Trieste, April-May 2008