nuclear structure phenomenological models from molecules to atomic nuclei. standard model basic...
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NUCLEAR STRUCTURE
• PHENOMENOLOGICAL MODELS• From molecules to atomic nuclei. Standard model• Basic concepts of nuclear physics. Units• Properties of nucleons• Liquid drop model• Surface vibration and rotation• MICROSCOPIC MODELS• Nuclear force • Nuclear mean field• Shell model• Second quantisation in the mean field• Residual interaction. Collective excitations• Collective model. Nilsson model
Standard model
Basic concepts of nuclear physics
nucleon: proton or neutron
nuclide: nucleus uniquely specified by
number of protons (Z) and neutrons (N)
mass number: A=Z+N
isotopes: nuclides with the same Z
ex: 235U and 238U
isotones: nuclides with the same N
ex: 2H, 3He
isobars: nuclides with the same A
atomic mass unit: 1u=1/12 m(12C)
=1.66 10-27kg=931.5 MeV/c2
• Electric quadrupole momentum
• Angular momentum
• Magnetic dipole momentum
• Parity
• Energy levels
• Decay rates
Basic physical observables in nuclei
Units used in nuclear physicsLength
1 fm =10-15 mEnergy
1 MeV = 106 eV1 eV = 1,6 10-19 J
Basic constants
MN=938,90 MeV/c2
ħc=197,33 MeV fm e2=ħc/137=1,44 MeV fm
Properties of nucleons
proton neutron
mass 1.007276=
938.280 MeV/c2
1.008665=
939.573 MeV/c2
charge +1 0
spin 1/2 1/2
magnetic moment +2.7928 μN -1.9128 μN
parity +1 +1
Liquid drop modelWeizsäcker semiempirical formula (1935)
The last nucleon of an odd-even (even-odd) nucleus determinesthe nuclear properties (spin, quadrupole and magnetic moments)
Second quantisation in the mean fieldEach spherical level is filled by 2j+1 nucleons
with different projections
kk'kkkkk'k
kk
δaaaa}a,{a
0a)(ψ
x
creation/annihilationoperators for
nucleons (fermions)Fermi level
Ground state is a Slater determinantobeying the Pauli exclusion principle
0......aaaΨ)](x)...ψ(x)ψ(xdet[ψ)x,...,(xΨ F21gsFF2211F1gs
Residual interaction among nucleons in the mean field
l
12l21l )(cos)Pr,(rV)V( 21 r,r
Multipole expansionl=0 : pairingl=2 : quadrupole-quadrupole
Particle-particle (p-p) short-range interaction describes pairing correlations
QuasiparticleQuasiparticle
approximationapproximation
kkkkk avauα
Ground state Ground state =BCS vacuum=BCS vacuum
νπ
k
BCSBCSBCS
0BCSα
kkk
kkk
aaP
aaN
where
PPGNH
HamiltonianHamiltonian
Particle-hole (p-h) long-range interactiondescribes collective excitations:1) low-lying surface vibrations
2) giant resonance of protons against neutrons
)a(ah||Yr||pQ
aaN
where
)QQF(NH
phhp
kkk
0λ
p
h
HamiltonianHamiltonianp-h excitation
gsλμhpλph
ph
Ψ)a(aX
Distribution of collective excitationsfor various multipolarities versus energy
Giantresonance
Low-lying vibrational state
Nilsson model of single particle statesin the deformed intrinsic system
2
2022
Nsphdef
βδ
YrωδmHH
Single particle energy versus deformation
Deformed Hamiltonian
G. Gamow "Zur Quantentheorie des Atomkernes" (On the quantum theory of the atomic nucleus), Zeitschrift für Physik, vol. 51, 204-212 (1928).
The first probabilistic interpretationof the wave function
int extARext
↓
Internal region External region
Quantum penetration explainsGeiger-Nuttall law for α
and cluster decays (C, O, Ne, Mg, Si)
Coulomb parameter
Q
eZZ
v
eZZχ
221
221
Decay operatorsin second quantisation:
gamma transitions beta transitions
λμλ
λμ YrV
if,
nipfλμ- aain,|V|fp,β
if,
τiτfλμnp,τ
τ aai,|V|f,eγ
σV
1V
GT
F
if,
pinfλμ aaip,|V|fn,β
Strutinsky shell-model correctionThe double humped barrier
determines the occurrence of superhevy nuclei
liquid drop shell model
Density of levels