Department of Physics and Astronomy

Lecture materials: Physics 8140

1. Introduction

1.0 Slides: Practical applications of nuclear physics (PDF)
1.1 Slides: Phys-8140 course overview, low-energy nuclear physics (PDF)
1.2 Slides: nuclear accelerators (PDF)        1.2 Notes (PDF)
1.3 Notes: nuclear units and constants of nature, with examples (PDF)
1.4 Notes: de Broglie wavelength of electron and heavy-ion beams (PDF)

2. Basic experimental facts and theoretical concepts

2.0 From quarks to nuclei
2.0a Notes: the four fundamental interactions, leptons and hadrons, quarks, standard model (PDF)
QCD and lattice gauge theory: masses of hadrons (Frank Wilczek, Physics Today, 2000)
2.0b Slides: from quarks to nuclei (PDF)
2.0c Notes: the nuclear quantum many-body problem (PDF)

2.1 Basic experimental facts
2.1a Slides: nuclear chart and decay modes (PDF)         2.1a Notes (PDF)
        Nuclear chart, proton and neutron driplines. Alpha, beta, and gamma decay. Radioactive
        decay law: decay rate, half-life, mean life, and level width. Gamov's explanation
        (1928) of alpha emission via potential barrier tunneling. Spontaneous fission:
        charge and mass distribution of fission fragments, fission half-lives. The frontier
        of superheavy elements, exp. evidence for Z=117. Proton and neutron radioactivity
        at the driplines, cluster emission. The "terra incognita" of neutron-rich nuclei.
        Location of neutron dripline for light isotopes.
2.1b Slides: nuclear densities and binding energies (PDF)         2.1b Notes (PDF)
        Nuclear charge densities and rms radii (elastic electron scattering). Nuclear
        binding energies. Liquid drop model and Bethe-Weizsaecker semi-empirical
        mass formula. NOTE: nuclei are not classical liquids! Reason: pair
        correlation function for fermions ("Pauli repulsion"), see Section 4.2 for details.
2.1c Slides: nuclear astrophysics (PDF)         2.1c Notes (PDF)
        Type II Supernovae, iron core collapse, production of neutrons and neutrinos
        via electron capture by protons. Supernova explosion driven by high-energy
        neutrinos. Synthesis of heavier nuclei via the rapid neutron capture process
        (r-process). Neutron stars. The rapid proton capture process (rp-process)
        in nova explosions.
2.1d Slides: RIB facilities (PDF)

2.2 Basic theoretical concepts
2.2 Notes: Binding energies and Q-values (PDF)
        Q-value formalism for nuclear reactions (using energy conservation).
        Applications: energy release in nuclear fission and fusion,
        neutron separation energies and location of neutron dripline (homework),
        threshold energies for heavy-ion fusion reactions (homework).
2.2.1 Notes (PDF)
        Structure of nuclear many-body Hamiltonian: one-body, two-body, and three-body operators.
        Many-particle Schroedinger equation. Quantum systems of identical particles,
        symmetric and anti-symmetric many-particle wave functions. Fermions, bosons and
        Pauli's spin-statistic theorem. Phenomenological one-body Hamiltonians (e.g. nuclear
        shell model): total energy and many-body wave function (Slater determinant). Spin and isospin
        formalism (Pauli matrices and state vectors). Free nuclear Fermi gas model
        (Fermi momentum and Fermi energy as function of density, relation between Fermi energy
        and total energy, explanation of symmetry energy in liquid drop model).

3. Phenomenological models (single-particle and collective)

3.1 Single-particle motion: shell models
3.1a Notes: spherical shell model (PDF)        3.1b Slides (PDF)
        Exp. evidence for nuclear shell structure: "magic numbers" (discontinuities
        in binding energies, first excited 2+ states). Spherical shell model
        with spin-orbit force due to strong interaction. Calculation of single-particle
        energy levels and wave functions, explanation of "magic numbers" due to
        shell gaps. Contour plots of single particle probability densities ("nuclear
        orbitals"). Electric multipole (EL) transitions and Weisskopf unit.
3.1c Notes: deformed shell model (PDF)        3.1d Slides (PDF)
        Deformed shell model ("Nilsson model"): qualitative description of deformed
        nuclear ground states, "shape isomers" and "shape coexistence".

3.2 Collective motion
3.2a Slides: collective motion; rotation, vibration, giant resonances (PDF)
3.2b Notes: surface vibrations, rotations (PDF)
3.2c Notes: rot/vib bands, superdeformation, giant resonances (PDF)
        Double- and triple-humped fission barriers and fission isomers of actinide nuclei:
        macroscopic-microscopic method (liquid drop + deformed shell model).
3.2d Slides: fission barriers and fission isomers in macroscopic-microscopic model (PDF)

4. Microscopic nuclear structure and reaction theories

4.1 The nucleon-nucleon (N-N) interaction
4.1a Notes: N-N interaction (PDF)
4.1b Notes: N-N interaction (PDF)
4.1c Slides: N-N interaction (PDF)
        Structure of N-N interaction potential based on invariance principles,
        dependence on spin and isospin, strong spin-orbit term, tensor
        interaction and one-pion exchange potential, multiple meson exchanges.
        The Argonne v-18 N-N potential: plots of radial functions. Angular momentum
        eigenstates for N-N scattering, phase shift analysis of measured p-p and n-p
        differential scattering cross sections (Nijmegen data set).

4.2 Nuclear many-body theory in "occupation number representation"
4.2a Notes (PDF)
        Creation and annihilation operators for fermions, anti-commutators.
        State vector corresponding to single Slater determinant. One-body operators
        and ground state expectation values: calculate ground state density and
        total kinetic energy. Two-body and three-body operators: nuclear many-body
        Hamiltonian and total ground state binding energy.

4.3 Brief discussion of "ab initio" calculations for light nuclei
4.3a Slides: "ab initio" calculations for light nuclei (PDF)
        a) Green's function Monte-Carlo with Argonne-v18 + NNN interaction.
        b) "No-core" shell model results.         c) Coupled cluster theory.

4.4 Ground state mean field theory: static and constrained Hartree-Fock (HF)
4.4a Slides: static Hartree-Fock formalism (PDF)
        Pair correlation function for nucleons (Pauli "exchange hole"), average distance
        between nucleons. Independent particle motion in mean-field potential.
        Derivation of HF equations from variational principle. Mean field "Hartree"
        potential and Fock exchange term, need for effective interactions (Bruckner G-matrix),
        Bethe-Goldstone equation for G-matrix. Phenomenological effective interactions:
        Skyrme interactions, Skyrme energy density functional and mean field.
4.4b Slides: static Hartree-Fock numerical results (PDF)
        Contour plots of mass density distributions for spherical and deformed nuclei,
        plots of mean field potentials for protons and neutrons, HF single-particle
        energies, total binding energies, and rms-radii. Constrained HF calculations:
        potential energy surface and fission barrier, cranked HF calculations: rotational bands.

4.5 Beyond the mean field: residual interactions
4.5a Slides: residual interactions and BCS pairing
        Pairing due to short-range residual interaction, structure of BCS ground state
        for nuclei, pair occupation probability, spectral distribution of pairing,
        HF + BCS pairing Hamiltonian, modification of ground state density.

4.6 Ground state mean field theory with pairing: Hartree-Fock-Bogoliubov (HFB)
4.6a Slides: static HFB theory and numerical results (PDF)
        HFB formalism: quasi-particle transformation, generalized variational principle,
        mean-field and pairing field Hamiltonian, HFB equations in coordinate space,
        normal density and pairing density. Mean field potential and Fermi energies
        for stable nuclei and near the 2n-dripline. Contour plots of normal density
        and pairing density for protons and neutrons, 2n-separation energies and location
        of 2n-dripline for Zr isotopes, pairing gaps, quadrupole moments.
        HFB nuclear mass table: S2n and ground state deformations for all nuclei.
        HFB with (Q20,Q30) constraints: potential energy surface for 238U fission,
        and single-particle energies for 224Ra.

4.7 Random Phase Approximation (RPA): theory of collective excitations
4.7a Slides: Nuclear RPA theory and numerical results (PDF)
        Microscopic theory of excited states: particle-hole excitations built on the HF
        ground state. Collective motion = coherent superposition of large number of
        p-h excitations. TDHF --> linear response theory --> RPA equations. Discussion
        of early RPA calculations for 208Pb: quadrupole and octupole surface vibrations
        and giant multipole resonances. Recent results: splitting of GDR in deformed nuclei
        (example 238U), and quasiparticle RPA (=QRPA) calculations for exotic neutron-rich nuclei.

4.8 Time-dependent Hartree-Fock (TDHF): dynamic theory of nuclear reactions
4.8a Slides: Unrestricted TDHF: fusion and deep-inelastic reactions (PDF)
4.8b TDHF animations: the reaction 48Ca + 132Sn at Ecm = 130 MeV for two different impact parameters
        Heavy-ion fusion at b = 4.45 fm (MPEG4 movie, 122 kb)
        Deep-inelastic reaction at b = 4.6 fm (MPEG4 movie, 74 kb)
4.8c Density-constrained TDHF theory (DC-TDHF)
        DC-TDHF: heavy-ion potentials and sub-barrier fusion
        Fusion of stable and neutron-rich nuclei, "hot" and "cold" fusion leading to superheavy elements.

4.9 Summary and Outlook
4.9a Slides: Glimpses of the Future (PDF)