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

From molecules to atomic nuclei 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 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 spin 1/2 magnetic moment +2.7928 μN -1.9128 μN parity

Nuclear chart stability of nuclei

Limits of stable nuclei exotic nuclei

Nuclear size from electron scattering experiments

Binding energy Mass defect

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

creation/annihilation Second quantisation in the mean field Each spherical level is filled by 2j+1 nucleons with different projections creation/annihilation operators for nucleons (fermions) Fermi level Ground state is a Slater determinant obeying the Pauli exclusion principle

Particle (croses) and hole (open circles) states p-h excitation:

(p,2p) reaction in the shell model

Residual interaction among nucleons in the mean field Multipole expansion l=0 : pairing l=2 : quadrupole-quadrupole

Quasiparticle Hamiltonian approximation Ground state =BCS vacuum Particle-particle (p-p) short-range interaction describes pairing correlations Quasiparticle approximation Hamiltonian Ground state =BCS vacuum

Occupation probabilities Gap parameter Normal system Superfluid system Fermi level

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 p h

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

DECAY PROCESSES 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

The first probabilistic interpretation 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 ↓ Internal region External region

Quantum penetration explains Geiger-Nuttall law for α and cluster decays (C, O, Ne, Mg, Si) Coulomb parameter

Beta decay

Fermi & Gamow-Teller transitions

Gamma decay

Parity rules for gamma transitions

Decay operators in second quantisation: gamma transitions beta transitions

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