NSDD Workshop, Trieste, April-May 2008 Nuclear Structure (I) Single-particle models P. Van Isacker,...
-
Upload
marylou-chambers -
Category
Documents
-
view
212 -
download
0
Transcript of NSDD Workshop, Trieste, April-May 2008 Nuclear Structure (I) Single-particle models P. Van Isacker,...
NSDD Workshop, Trieste, April-May 2008
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:
• Independent-particle assumption. Choose V and neglect residual interaction:
€
ˆ H =pk
2
2mkk=1
A
∑ + ˆ V 2 rk,rl( )k<l
A
∑
=pk
2
2mk
+ ˆ V rk( ) ⎡
⎣ ⎢
⎤
⎦ ⎥
k=1
A
∑mean field
1 2 4 4 3 4 4
+ ˆ V 2 rk,rl( )k<l
A
∑ − V rk( )k=1
A
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
residual interaction1 2 4 4 4 3 4 4 4
€
ˆ H ≈ ˆ H IP =pk
2
2mk
+ ˆ V rk( ) ⎡
⎣ ⎢
⎤
⎦ ⎥
k=1
A
∑
NSDD Workshop, Trieste, April-May 2008
Independent-particle shell model
• Solution for one particle:
• Solution for many particles:
€
p2
2m+ ˆ V r( )
⎡
⎣ ⎢
⎤
⎦ ⎥φi r( ) = E iφi r( )
€
Φi1i2K iAr1,r2,K ,rA( ) = φik
rk( )k=1
A
∏
ˆ H IPΦ i1i2K iAr1,r2,K ,rA( ) = E ik
k=1
A
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟Φ i1i2K iA
r1,r2,K ,rA( )
NSDD Workshop, Trieste, April-May 2008
Independent-particle shell model
• Anti-symmetric solution for many particles (Slater determinant):
• Example for A=2 particles:
€
Ψi1i2K iAr1,r2,K ,rA( ) =
1
A!
φi1r1( ) φi1
r2( ) K φi1rA( )
φi2r1( ) φi2
r2( ) K φi2rA( )
M M O M
φiAr1( ) φiA
r2( ) K φiArA( )
€
Ψi1i2r1,r2( ) =
1
2φi1
r1( )φi2r2( ) − φi1
r2( )φi2r1( )[ ]
NSDD Workshop, Trieste, April-May 2008
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
* r1,r2,K ,rA( ) ˆ H Ψi1i2K iAr1,r2,K ,rA( )dr1dr2K drA∫
Ψi1i2K iA
* r1,r2,K ,rA( )Ψi1i2K iAr1,r2,K ,rA( )dr1dr2K drA∫
= 0
NSDD Workshop, Trieste, April-May 2008
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+
mω2
2rk
2 −ζ lk ⋅sk −κ lk2
⎡
⎣ ⎢
⎤
⎦ ⎥
k=1
A
∑
NSDD Workshop, Trieste, April-May 2008
Harmonic oscillator solution
• Energy eigenvalues of the harmonic oscillator:
€
Enlj = N + 3
2( )hω −κ h2l l +1( ) + ζ h2 − 1
2l j = l + 1
21
2l +1( ) j = l − 1
2
⎧ ⎨ ⎩
N = 2n + l = 0,1,2,K : oscillator quantum number
n = 0,1,2,K : radial quantum number
l = N,N − 2,K ,1or 0 : orbital angular momentum
j = l ± 1
2: total angular momentum
m j = − j,− j +1,K ,+ j : z projection of j
NSDD Workshop, Trieste, April-May 2008
Energy levels of harmonic oscillator• Typical parameter
values:
• ‘Magic’ numbers at 2, 8, 20, 28, 50, 82, 126, 184,…
€
h ≈41 A−1/ 3 MeV
ζ h2 ≈ 20 A−2 / 3 MeV
κ h2 ≈ 0.1 MeV
∴ b ≈1.0 A1/ 6 fm
NSDD Workshop, Trieste, April-May 2008
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+ expr − R0
a
NSDD Workshop, Trieste, April-May 2008
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 = ζ r( ) l ⋅s, ζ r( ) =r0
2
r
∂V
∂r= ζ in HO[ ]
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:
• For 2149 nuclei (N,Z ≥ 8) in AME03:
av16, as18, ac0.71, a's23, ap6
rms2.93 MeV.€
B N,Z( ) = av A − asA2 / 3 − ac
Z Z −1( )A1/ 3
− ′ a sN − Z( )
2
A
+ ap
Δ N,Z( )A1/ 3
C.F. von Weizsäcker, Z. Phys. 96 (1935) 431H.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:
• For 2149 nuclei (N,Z ≥ 8) in AME03:
av16, as18, ac0.71, Sv35, ys2.9, ap5.5, af0.85, aff0.016
rms1.16 MeV.
€
B N,Z( ) = av A − asA2 / 3 − ac
Z Z −1( )A1/ 3
+ ap
Δ N,Z( )A1/ 3
−Sv
1+ ysA−1/ 3
4T T +1( )A
− af nν + nπ( ) + aff nν + nπ( )2
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 inφnljmj ⇒ J
l inφnljmj ⇒ −( )l =π⎫ ⎬ ⎭
⇒ J π
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 undécompresseur 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): Cor-relations 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:
• 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 =pk
2
2m+ ˆ V rk( )
⎡
⎣ ⎢
⎤
⎦ ⎥
k=1
A
∑ + ˆ V RI rk,rl( )k<l
A
∑
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 harmonic-oscillator basis. Example: δ 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:
– Diagonalisation : ~109.
– Monte Carlo : ~1015.
• Example : 8n + 8p in pfg9/2 (56Ni).
M. Honma et al., Phys. Rev. C 69 (2004) 034335
€
Ψi'1 i'2K i'Aˆ V RI rk,rl( )
k<l
n
∑ Ψi1i2K iA
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:
• Spectrum 210Pb:
€
j 2JMJˆ V pairing r1,r2( ) j 2JMJ =
− 1
22 j +1( )g0, J = 0
0, J ≠ 0
⎧ ⎨ ⎩
NSDD Workshop, Trieste, April-May 2008
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 nυ J ˆ V pairing rk,rl( )1≤k<l
n
∑ j nυJ = −g01
4n −υ( ) 2 j − n −υ + 3( )
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):– 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 / 2
o , ˆ S + = ˆ a m↓+ ˆ a ̇ ̇ ̇ m ↑
+
m
∑
€
ˆ a mb+ ˆ S +( )
n / 2o
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:
• Example of Sn isotopes:
€
Ex 21+
( ) = 1
22 j +1( )G
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:
€
ˆ H =pk
2
2m+
1
2mω2rk
2 ⎡
⎣ ⎢
⎤
⎦ ⎥
k=1
A
∑ − g2ˆ Q ⋅ ˆ Q ,
ˆ Q μ ∝ 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