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NSDD Workshop, Trieste, February 2006
Nuclear Structure(I) Single-particle models
P. Van Isacker, GANIL, France
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NSDD Workshop, Trieste, February 2006
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.
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NSDD Workshop, Trieste, February 2006
Nuclear shell model• Many-body quantum mechanical problem:
• Independent-particle assumption. Choose V and neglect residual interaction:
ˆ H pk2
2mkk1
A
ˆ V 2 rk,rl kl
A
pk2
2mk
ˆ V rk
k1
A
mean field
ˆ V 2 rk,rl
kl
A
V rk k1
A
residual interaction
ˆ H ˆ H IP pk
2
2mk
ˆ V rk
k1
A
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NSDD Workshop, Trieste, February 2006
Independent-particle shell model• Solution for one particle:
• Solution for many particles:
p2
2m ˆ V r
i r E ii r
i1i2 iAr1,r2,,rA ik
rk k1
A
ˆ H IPi1i2 iAr1,r2,,rA E ik
k1
A
i1i2 iA
r1,r2,,rA
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NSDD Workshop, Trieste, February 2006
Independent-particle shell model• Anti-symmetric solution for many particles
(Slater determinant):
• Example for A=2 particles:
i1i2 iAr1,r2,,rA
1A!
i1r1 i1
r2 i1rA
i2r1 i2
r2 i2rA
iA
r1 iAr2 iA
rA
i1i2r1,r2
12
i1r1 i2
r2 i1r2 i2
r1
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NSDD Workshop, Trieste, February 2006
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.
i1i2 iA
* r1,r2,,rA ˆ H i1i2 iAr1,r2,,rA dr1dr2 drA
i1i2 iA* r1,r2,,rA i1i2 iA
r1,r2,,rA dr1dr2drA0
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NSDD Workshop, Trieste, February 2006
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 .– Orbit-orbit term with strength .
• Adjust , and to best reproduce HF.
ˆ H IP pk
2
2m
m 2
2rk
2 lk sk lk2
k1
A
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NSDD Workshop, Trieste, February 2006
Harmonic oscillator solution • Energy eigenvalues of the harmonic oscillator:
Enlj N 32 2l l 1 2 1
2 l j l 12
12 l 1 j l 1
2
N 2n l 0,1,2, : oscillator quantum numbern 0,1,2, : radial quantum numberl N,N 2,,1or 0 : orbital angular momentum
j l 12 : total angular momentum
m j j, j 1,, j : z projection of j
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NSDD Workshop, Trieste, February 2006
Energy levels of harmonic oscillator• Typical parameter
values:
• ‘Magic’ numbers at 2, 8, 20, 28, 50, 82, 126, 184,…
41 A 1/ 3 MeV
2 20 A 2 / 3 MeV
2 0.1 MeV
b 1.0 A1/ 6 fm
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NSDD Workshop, Trieste, February 2006
Why an orbit-orbit term?• Nuclear mean field is
close to Woods-Saxon:
• 2n+l=N degeneracy is lifted El < El-2 <
ˆ V WS r V0
1 exp r R0
a
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NSDD Workshop, Trieste, February 2006
Why a spin-orbit term?• Relativistic origin (ie Dirac equation).• From general invariance principles:
• Spin-orbit term is surface peaked diminishes for diffuse potentials.
ˆ V SO r l s, r r0
2
rVr
in HO
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NSDD Workshop, Trieste, February 2006
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?
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NSDD Workshop, Trieste, February 2006
Ionisation potential in atoms
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NSDD Workshop, Trieste, February 2006
Neutron separation energies
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NSDD Workshop, Trieste, February 2006
Proton separation energies
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NSDD Workshop, Trieste, February 2006
Liquid-drop mass formula• Binding energy of an atomic nucleus:
• For 2149 nuclei (N,Z ≥ 8) in AME03: avol16, asur18, acou0.71, asym23, apai13
rms2.93 MeV.
B N,Z avolA asur A2 / 3 acou
Z Z 1 A1/ 3 asym
N Z 2
A
apai
N,Z A1/ 2
C.F. von Weizsäcker, Z. Phys. 96 (1935) 431H.A. Bethe & R.F. Bacher, Rev. Mod. Phys. 8 (1936) 82
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NSDD Workshop, Trieste, February 2006
Deviations from LDM
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NSDD Workshop, Trieste, February 2006
Modified liquid-drop formula• Add surface, Wigner and ‘shell’ corrections:
• For 2149 nuclei (N,Z ≥ 8) in AME03: avol16, asur18, acou0.72, avsym32,
assym79, apai12, af0.14, aff0.0049, r2.5
rms1.28 MeV.
B N,Z avolA asur A2 / 3 acou
Z Z 1 A1/ 3 avsym
4T T r A
assym
4T T r A4 / 3 apai
N,Z A1/ 2 afFmax aff Fmax
2
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NSDD Workshop, Trieste, February 2006
Deviations from modified LDM
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NSDD Workshop, Trieste, February 2006
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NSDD Workshop, Trieste, February 2006
Shell structure from Ex(21)
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NSDD Workshop, Trieste, February 2006
Evidence for IP shell model• Ground-state spins and
parities of nuclei:j in nljmj
Jl in nljmj
l
J
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NSDD Workshop, Trieste, February 2006
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
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NSDD Workshop, Trieste, February 2006
Variable shell structure
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NSDD Workshop, Trieste, February 2006
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): Cor-relations in the ground state by treating it on the same footing as 1p-1h excitations.
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NSDD Workshop, Trieste, February 2006
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).
ˆ H pk2
2m ˆ V rk
k1
A
ˆ V RI rk,rl kl
A
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NSDD Workshop, Trieste, February 2006
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.
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NSDD Workshop, Trieste, February 2006
Schematic short-range interaction• Delta interaction in harmonic-oscillator basis:• Example of 42Sc21 (1 neutron + 1 proton):
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NSDD Workshop, Trieste, February 2006
Large-scale shell model• Large Hilbert spaces:
– Diagonalisation : ~109.– Monte Carlo : ~1015.– DMRG : ~10120 (?).
• Example : 8n + 8p in pfg9/2 (56Ni).
M. Honma et al., Phys. Rev. C 69 (2004) 034335
i'1 i' 2 i' A
ˆ V RI rk,rl kl
n
i1i2 iA
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NSDD Workshop, Trieste, February 2006
The three faces of the shell model
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NSDD Workshop, Trieste, February 2006
Racah’s SU(2) pairing model• Assume pairing interaction in a single-j shell:
• Spectrum 210Pb:
j 2JMJˆ V pairing r1,r2 j 2JMJ
12 2 j 1 g0, J 0
0, J 0
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NSDD Workshop, Trieste, February 2006
Solution of the 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 (cf. BCS).
j nJ ˆ V pairing rk,rl 1kl
n
j nJ g014 n 2 j n 3
G. Racah, Phys. Rev. 63 (1943) 367
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NSDD Workshop, Trieste, February 2006
Nuclear superfluidity• Ground states of pairing hamiltonian have the
following correlated 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.– Smooth variation of two-nucleon separation
energies with nucleon number.– Two-particle (2n or 2p) transfer enhancement.
ˆ S n / 2o , ˆ S ˆ a m
ˆ a Ý Ý Ý m
m
ˆ a m ˆ S n / 2
o
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NSDD Workshop, Trieste, February 2006
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.
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NSDD Workshop, Trieste, February 2006
Pairing gap in semi-magic nuclei• Even-even nuclei:
– Ground state: =0.– First-excited state: =2.– Pairing produces
constant energy gap:
• Example of Sn isotopes:
Ex 21 1
2 2 j 1 G
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NSDD Workshop, Trieste, February 2006
Elliott’s SU(3) model of rotation• Harmonic oscillator mean field (no spin-orbit)
with residual interaction of quadrupole type:
ˆ H pk2
2m 1
2m 2rk
2
k1
A
g2ˆ Q ˆ Q ,
ˆ Q rk2Y2 ˆ r k
k1
A
pk2Y2 ˆ p k
k1
A
J.P. Elliott, Proc. Roy. Soc. A 245 (1958) 128; 562