(taken from H-J. Wolesheima, Models of atomic nuclei 1) Introduction 2) Drop model of nucleus 3) Nucleus as Fermi gas 4) Shell model of nucleus 5) Unified (collective) model of nuclei 6) Further models Octupole vibrations of nucleus. (taken from H-J. Wolesheima, GSI Darmstadt) Extreme superdeformed states were predicted on the base of models

Introduction Nucleus is quantum system of many nucleons interacting mainly by strong nuclear interaction. Theory of atomic nuclei must describe: 1) Structure of nucleus (distribution and properties of nuclear levels) 2) Mechanism of nuclear reactions (dynamical properties of nuclei) Development of theory of nucleus needs overcome of three main problems: 1) We do not know accurate form of forces acting between nucleons at nucleus. 2) Equations describing motion of nucleons at nucleus are very complicated – problem of mathematical description. 3) Nucleus has at the same time so many nucleons (description of motion of every its particle is not possible) and so little (statistical description as macroscopic continuous matter is not possible). Real theory of atomic nuclei is missing  only models exist. Models replace nucleus by model reduced physical system.  reliable and sufficiently simple description of some properties of nuclei.

Models of atomic nuclei can be divided: A) According to magnitude of nucleon interaction: Collective models (models with strong coupling) – description of properties of nucleus given by collective motion of nucleons Singleparticle models (models of independent particles) – describe properties of nucleus given by individual nucleon motion in potential field created by all nucleons at nucleus. Unified (collective) models – collective and singleparticle properties of nuclei together are reflected. B) According to, how they describe interaction between nucleons: Phenomenological models – mean potential of nucleus is used, its parameters are determined from experiment. Microscopic models – proceed from nucleon potential (phenomenological or microscopic) and calculate interaction between nucleons at nucleus. Semimicroscopic models – interaction between nucleons is separated to two parts: mean potential of nucleus and residual nucleon interaction.

Liquid drop model of atomic nuclei Let us analyze of properties similar to liquid. Think of nucleus as drop of incompressible liquid bonded together by saturated short range forces Description of binding energy: B = B(A,Z) We sum different contributions: B = B1 + B2 + B3 + B4 + B5 1) Volume (condensing) energy: released by fixed and saturated nucleon at nuclei: B1 = aVA 2) Surface energy: nucleons on surface → smaller number of partners → addition of negative member proportional to surface S = 4πR2 = 4πA2/3: B2 = -aSA2/3 3) Coulomb energy: repulsive force between protons decreases binding energy. Coulomb energy for uniformly charged sphere is E  Q2/R. For nucleus Q2 = Z2e2 a R = r0A1/3: B3 = -aCZ2A-1/3 4) Energy of asymmetry: neutron excess decreases binding energy 5) Pair energy: binding energy for paired nucleons increases: +  for even-even nuclei B5 = 0 for nuclei with odd A where   aPA-1/2 -  for odd-odd nuclei Volume energy Surface energy Coulomb energy Asymmetry energy Binding energy We sum single contributions and substitute to relation for mass: M(A,Z) = Zmp+(A-Z)mn – B(A,Z)/c2 M(A,Z) = Zmp+(A-Z)mn–aVA+aSA2/3 + aCZ2A-1/3 + aA(Z-A/2)2A-1±δ Weizsäcker semiempirical mass formula. Parameters are fitted using measured masses of nuclei. (aV = 15.85, aA = 92.9, aS = 18.34, aP = 11.5, aC = 0.71 all in MeV/c2)

Using Weizsäcker formula many regularities can be derived. Behavior of stability curve: During beta decay A is not changed. For nuclei with odd A masses is lying on parabola and only one stable isotope exists. For nuclei with even A such parabolas are two due to pair contribution ±δ → more stable isotopes exist. We find the most stable nucleus in the line of isobars: reminder: M(A,Z) = Zmp+(A-Z)mn–aVA+aSA2/3 + aCZ2A-1/3 + aA(Z-A/2)2A-1±δ We carry out derivation: mp – mn + 2Z0aCA-1/3+2aA(Z0-A/2)A-1 = 0 And we obtain: Separation energy of neutron, proton and alpha particle: Energy obtained by separation of nucleon α particle. Kinetic energy of produced α particle after decay is: Eα = [M(A,Z) – M(A-4,Z-2) – mα]c2 It is also negative value of separation energy.

Ekv = nkvħω Eokt = noktħω The energy of stability border for different decays and particle emissions or fission can be obtained by Weizsäcker formula . Particle must come through Coulomb barrier before realization of decay or fission . It is problem for nuclei from fission. Description of vibrations and rotations of nuclei can be obtained by drop model: Nuclear matter is practically incompressible but production of surface vibrations is simple. Quadruple – nucleus is changing from pressed to elongated ellipsoid. Octupole – deformation has pear-shaped form. Energy of vibration state depends on frequency: Ekv = nkvħω Eokt = noktħω Where nkv, nokt are numbers of appropriate quanta. Quadruple quantum has spin J=2 and octupole has spin J=3. Special type of vibrations are independent oscillations of proton and neutron drops – giant dipole resonances. Ground state Experiment Model γ vibrations For deformed drop possibility of rotation. Description of rotational states: where J – moment of inertia, spin I = 0,1,2, …

The Fermi-gas model dVp = d3p = dpxdpydpz = 4π p2 dp Nucleons are fermions (have spin 1/2). Because of Pauli exclusion principle only one fermion can be in one state. States characterized by strictly given discrete values of energy and momentum exist at potential of nucleus. In the ground state all the lowest states allowed by Pauli principle are filled by nucleons. Such system of fermions is named as degenerated fermion gas → nucleons can not change their state (all near states are filled) → they can not collide and behave as noninteracting independent particles. System of N fermions in volume V and with temperature T: Probability of fermion occurrence in state with energy E: where k is Boltzman constant and EF – Fermi energy. We determine Fermi momentum pF ( nonrelativistic approximation EF = pF2/2m ): We introduce phase space: extension of coordinate space by momentum space (6 – dimensional space). Space element is: dV = dx·dy·dz → dV = d3r = r2sindr·d·d If angular direction is not important, we integrate through all angles: dV = 4π r2 dr Analogously for momentum space element: dVp = d3p = dpxdpydpz = 4π p2 dp Phase space element: dVTOT = dV·dVp From Heisenberg uncertainty principle: ΔpxΔx ≥ ħ ΔpyΔy ≥ ħ ΔpzΔz ≥ ħ

Volume dVTOT of elementary cell in phase space is h3 Volume dVTOT of elementary cell in phase space is h3. number d of elementary cells with one particle with momentum p  p+Δp is for volume V: Nucleons have s = 1/2  in every cell gs = (2s+1) = 2. For T = 0: p < pF  2 particles in cell p > pF  0 particles in cell. and then: Fermi gas is degenerated for EF >> kT. For EF << kT → classical gas and Maxwell distribution. Nucleus is mixture of two degenerated fermion gases: Z protons and N neutrons closed in volume V = (4/3)R3 = (4/3)r03A. Fermi energy for neutrons and protons at nucleus: in first approximation: mn  mp = m, Z  N  A/2: Deepness of potential well (binding of last nucleon is B/A): V0  EF + B/A  37 MeV + 8 MeV  45 MeV Further we calculate whole kinetic energy: Hence for A = Z+N nucleons: Average kinetic energy per A (for Z A):

Shell model of nucleus Assumption: primary interaction of single nucleon with force field created by all nucleons. Nucleons are fermions  one in every state (filled gradually from the lowest energy). Experimental evidence: 1) Nuclei with value Z or N equal to 2, 8, 20, 28, 50, 82, 126 (magic numbers) are more stable (isotope occurrence, number of stable isotopes, behavior of separation energy magnitude). 2) Nuclei with magic Z and N have zero quadrupol electric moments  zero deformation. 3) The biggest number of stable isotopes is for even-even combination (156), the smallest for odd-odd (5). 4) Shell model explains spin of nuclei. Even-even nucleus  protons and neutrons are paired. Spin and orbital angular momenta for pair are zeroed. Either proton or neutron is left over in odd nuclei. Half-integral spin of this nucleon is summed with integral angular momentum of rest of nucleus  half-integral spin of nucleus. Proton and neutron are left over at odd-odd nuclei  integral spin of nucleus. Shell model: 1) All nucleons create potential, which we describe by 50 MeV deep square potential well with rounded edges, potential of harmonic oscillator or even more realistic Woods-Saxon potential. 2) We solve Schrödinger equation for nucleon in this potential well. We obtain stationary states characterized by quantum numbers n, l, ml. Group of states with near energies creates shell. Transition between shells  high energy. Transition inside shell  law energy. 3) Coulomb interaction must be included  difference between proton and neutron states. 4) Spin-orbital interaction must be included. Weak LS coupling for the lightest nuclei. Strong jj coupling for heavy nuclei. Spin is oriented same as orbital angular momentum → nucleon is attracted to nucleus stronger. Strong split of level with  orbital angular momentum l.

without spin-orbital coupling with spin-orbital Energy Number per level per shell Total number Sequence of energy levels of nucleons given by shell model (not real scale) – source A. Beiser

Unified (collective) model of nucleus The most part of nuclei has both single-particle and collective properties. Common description is possible by unified (collective) model: Nucleus is divided to even-even hard core and some number of outer valence nucleons. Hard core need not be spherical (given by influence of valence nucleons) → possibility of rotation. We assume adiabatic approximation – ΔErot<<ΔEpart → collective motion proceed more slowly than single-particle → rotation does not disturb single-particle state → rotation bands over single-particle states: Erot = (ħ2/2J)[I(I+1)]. Often nonspherical deformed potential – Nilsson potential – is used. Deformed potential → energy of single-particle level depends on magnitude of deformation. Interaction between collective and single-particle states. → Mixture of different states. Creation of complex system of single-particle, vibrational and rotational states (levels). Unified model makes possible relatively good description of complicated system of energy levels at nucleus, state characteristics, transition probabilities, quadrupole moments of nuclei. Further models of nuclei Very strong influence of pairing. Nucleon pair can be so strongly bound, that we can look on it as on single particle (spins of two fermions are summed → boson). This is base of interacting boson model (IBA). Strong pairing is also base of independent quasiparticle models. Microscopic models use nucleon potential and solve equations of motion. Big problems with mathematical solution.