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NUCLEAR STRUCTURE
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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
From molecules to atomic nuclei
10-10m=1Å
10-15m=1fm
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
Basic physical observables in nuclei
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Electric quadrupole momentum
Angular momentum
Magnetic dipole momentum
Parity
Energy levels
Decay rates
Electric
quadrupole
moment
Magnetic
dipole
moment
Units used in nuclear physics
Length
1 fm =10-15 m
Energy
1 MeV = 106 eV
1 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
Nuclear chart
stability of nuclei
Limits of stable nuclei
exotic nuclei
Nuclear size
from electron scattering experiments
R  r0 A1/3
r0  1.2fm
Binding energy
B  (ZM p  NMn  ZM e - M)c 2
Mass defect
Δ  (M  A)c
2
Example
Binding energy/nucleon: B/A
Liquid drop model
Weizsäcker semiempirical formula (1935)
Symmetry energy
Liquid drop energy versus (Z,N)
Surface vibration and rotation
Deformation parameters
of the nuclear surface
Vibrational states
Rotational states
Total spin I and its projections
to laboratory (M) and intrinsic (K) systems
Ω
Parameters in the intrinsic system
Ω is the rotation angle
β & γ vibrations of a deformed shape
Rotational-vibrational model
Rotational bands
built on top of the
vibrational band head
Sakai-Sheline rule
vibrational states → rotational bands
Nuclear force
Deuteron:
the simplest nuclear system
Deuteron
spin & magnetic moment
Electromagnetic versus strong field
Yukawa potential
Shell model
Nuclear mean field:
the selfconsistent single particle potential
created by all nucleons
Mean field potential
for protons and neutrons
Spin-orbit interaction
Example
Shell model magic numbers
appear due to the spin-orbit interaction
Spherical shell model scheme
The last nucleon of an odd-even (even-odd) nucleus determines
the nuclear properties (spin, quadrupole and magnetic moments)
Schmidt limits for magnetic moments
Schmidt limits for quadupole moments
Second quantisation in the mean field
Each spherical level is filled by 2j+1 nucleons
with different projections
Fermi level
creation/annihilation
operators for
nucleons (fermions)
ψ k ( x) 
 a k 0

k'

k

k
{a k , a }  a k a  a a k  δ kk'
Ground state is a Slater determinant
obeying the Pauli exclusion principle
Ψ gs (x 1 ,..., x F )  det[ψ1 (x 1 )ψ 2 (x 2 )...ψ F (x F )] 
 Ψ gs  a1 a 2 ......a F 0
Particle (croses) and
hole (open circles) states
p-h excitation:
a p a h gs
(p,2p) reaction in the shell model
Residual interaction
among nucleons in the mean field
Multipole expansion
l=0 : pairing
l=2 : quadrupole-quadrupole
V(r1 , r2 )   Vl (r1 , r2 )Pl (cos12 )
l
Particle-particle (p-p) short-range interaction
describes pairing correlations
Hamiltonian

Ĥ  N̂  GP̂ P̂
where
N̂   a a k

k
k
P̂   a a


k
k

k
Quasiparticle
approximation
α k  u k a k  v k a k
Ground state
=BCS vacuum
α k BCS  0
BCS  BCS
π
BCS
ν
Occupation probabilities
Gap parameter
Δ  G  BCS | a k a k | BCS
k
Normal system
Fermi level
Superfluid system
Proton gap versus Z
Particle-hole (p-h) long-range interaction
describes collective excitations:
1) low-lying surface vibrations
2) giant resonance of protons against neutrons
Hamiltonian
p-h excitation

ph
λ
ph

p h λμ
X (a a )
Ψgs
Ĥ  N̂  F(Q̂ λ Q̂ ) 0
p
where
h
N̂   a k a k
k
Q̂   p || r Y || h (a a h ) 

ph

p
Distribution of collective excitations
for various multipolarities versus energy
Giant
resonance
Low-lying
vibrational state
Collective model
Nilsson model of single particle states
in the deformed intrinsic system
Single particle energy versus deformation
Deformed Hamiltonian
Ĥ def  Ĥ sph  δm N ω 2 r 2 Y20
δ  β2
DECAY PROCESSES
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Alpha decay, cluster emission
Beta decay
Gamma decay
Fission and fusion
Nuclear decay modes
Decay law
Decay width
Γ=ħλ
Narrow decaying resonance (Γ is small)
is a quasi-stationary process
Decay rate (activity)
Alpha decay
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 interpretation
of the wave function
Rext
↓
int 
Internal region
 Aext
External region
Quantum penetration explains
Geiger-Nuttall law for α
and cluster decays (C, O, Ne, Mg, Si)
Coulomb parameter
Z1Z 2 e 2
Z1Z 2 e 2
χ

v
Q
Beta decay
Fermi & Gamow-Teller transitions
Gamma decay
Parity rules for gamma transitions
Decay operators
in second quantisation:
gamma transitions
beta transitions
γ̂ 

e

,
f
|
V̂
|

,
i
a
 τ
λμ
τf a τi
τ  p,n
f, i
β̂ -   p, f | V̂λμ | n, i a pf a ni
f, i
β̂    n, f | V̂λμ | p, i a nf a pi
f, i
V̂λμ  r Yλμ
λ
V̂F  1
V̂GT  σ
Fission
&
fusion
Fission - liquid drop model
Energy release for various processes
Strutinsky shell-model correction
The double humped barrier
determines the occurrence of superhevy nuclei
Density of levels
liquid drop
shell model
Superheavy nuclei
are formed by fusion
and detected by alpha decay chains
Fusion energy
The Sun