PHYS490: Nuclear Physics 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 1 PHYS490: Schedule 2015 Lectures in Brodie Tower (BROD-405): Wednesday 13:00 – 14:00 Thursday 11:00 – 13:00 Lectures: weeks 1 - 6 Tutorials: weeks 3, 6 Eddie Paul ([email protected]) Room 411 Oliver Lodge Lab. 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 2 PHYS490: Nuclear Physics 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 1/29/2015 Nucleon-Nucleon Force Nuclear Behaviour Forms of Mean Potential Nuclear Deformation Hybrid Models Nuclear Excitations Rotating Systems Nuclei at Extremes of Spin Nuclei at Extremes of Isospin Mesoscopic Systems Nuclear Reactions Nuclear Astrophysics Neutrinoless Double Beta Decay PHYS490 : Advanced Nuclear Physics : E.S. Paul 3 0. A Brief Introduction 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 4 Prehistory (400 BC) This chart of Plato and Aristotle shows the relation of the four elements and their four qualities A fifth element was ether or material of the heavens (dark matter in early cosmology !) The chart was used for over 1000 years 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 5 Atomic and Nuclear Sizes 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 6 Limits of Stable Nuclei 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 7 More and More Isotopes 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 8 Discovery History Today around 3000 isotopes have been observed Only 284 are stable 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 9 The Unique Nucleus The nucleus is a unique ensemble of strongly interacting fermions: nucleons Its large, yet finite, number of constituents controls the physics Both single-particle (out-of-phase) and collective (in-phase) effects occur Analogy to a herd of wild animals. Individual animals may break out of the herd but are rapidly drawn back to the safety of the collective 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 10 Nuclear Models Quantum mechanics governs basic nuclear behaviour The forces are complicated and cannot be written down explicitly It is a many-body problem of great complexity In the absence of a comprehensive nuclear theory we turn to models A model is simply a way of looking at the nucleus that gives a physical insight into a wide range of its properties 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 11 Nuclear Physics in the Thirties: Splitting of the Atom Cockcroft-Wilson accelerator – atom ‘split’ in 1932 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 12 Nuclear Physics in the Forties The first cyclotrons were built in Berkeley, California 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 13 Oliver Lodge Lab. Opening (1969) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 14 Nuclear Physics in the Seventies An Open University program from 1979, shot in the Liverpool Physics Department, showing the forefront of nuclear structure experimentation (and fashion) at the time! Also on youtube: http://youtu.be/s43rxUA8euY 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 15 Nuclear Physics in the Eighties TESSA3: 16 (small) γ-ray detectors at Daresbury, UK 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 16 Even Bigger Arrays This picture shows ESSA30, an array of 30 (small) γ-ray detectors at Daresbury, UK It was a European collaboration Again, spot the Liverpudlians ! 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 17 Nuclear Physics in the Nineties Gammasphere: 100 (big) γ-ray detectors, USA 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 18 Nuclear Physics in the Noughties 2003: The Hulk destroys Gammasphere ! 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 19 Nuclear Physics Tomorrow The next generation of Radioactive Ion Beam (RIB) accelerators in Europe 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 20 Nuclear Physics in context Nuclear Physics is ‘the study of the structure, properties, and interactions of the atomic nuclei’ Nuclear Physicists investigate nuclear matter on all scales, from sub-atomic particles to supernovae Research areas include the structure of the nucleus at different temperatures and pressures, the origin of elements, and the structure and evolution of stars 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 21 1. Nucleon-Nucleon Force 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 22 The Nucleon – a Spin ½ Fermion The nucleon is a hadron, i.e. it feels the strong force The Ford Nucleon (1957) nuclear powered car 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 23 The Nucleon – a spin ½ Fermion It consists basically of 3 quarks but gluons (force mediators) must also be considered Only 2% of the mass (Higgs mechanism) comes from quark masses. The other 98% arises from the kinetic energy of the constituents Only 30% of the intrinsic spin can be accounted for from the constituent quarks proton 1/29/2015 quark sea + 3 valence quarks PHYS490 : Advanced Nuclear Physics : E.S. Paul 24 Building Blocks and Energy Scales Depending on energy and length scales, different constituents may be considered as the building blocks of the atomic nucleus 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 25 Levels of Reality 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 26 Fundamental Particles & Forces Quarks: Down (d) Up (u) Strange (s) Charmed (c) Bottom (b) Top (t) Force Mediators: Photon (γ) Gluon (g) Z particle W particle 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 27 The Strong Force The strong force is fundamentally an interaction between quarks It is really a residual colour force mediated by the exchange of gluons 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 28 Properties of the N-N Force The force is spin dependent The force is charge symmetric The force is (nearly) charge independent The force has a non-central component The force depends on the relative velocity or momentum of the nucleons The force has a repulsive core ‘Exchange model’: force mediated by pion exchange See Phys490_latex.pdf for more details 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 29 One Pion Exchange The origin of the nuclear force arises at the fundamental level from the exchange of gluons between the constituent quarks of the nucleons At low energies (<1 GeV/nucleon; >1 fm) the interaction can be regarded as being mediated by the exchange of π mesons – pions 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 30 Spin σ and Isospin τ Matrix mechanics was formulated by Born, Heisenberg and Jordan (1925) Nucleon intrinsic spin takes only two values: up and down Introduction of Pauli 2x2 spin matrices Same formalism used to describe nucleon: isospin up (neutron), isospin down (proton) Introduction of Pauli 2x2 isospin matrices Nucleon-nucleon force dependent on both spin and isospin 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 31 One-Pion Exchange Potential At large distances the potential is constructed as arising from the exchange of one pion: OPEP The form of the potential is: VOPEP = gs2 (1/3 σA.σB + SAB [1/3 + 1/μr + 1/(μr)2]) x τA.τB 1/r μ2e-μr where: μ = mπc/ħ 1/29/2015 and: SAB = 3(σA.r)( σB.r)/r2 - σA.σB PHYS490 : Advanced Nuclear Physics : E.S. Paul 32 Addition of (Iso)Spins Spin σ and isospin τ are vectors Cosine rule gives: (σA + σB )2 = σA2 + σB2 + 2 σA.σB Parallel spins (triplet state): σA.σB = 1 Antiparallel spins (singlet state): σA.σB = -3 We need to know σA.σB in the description of the nucleon-nucleon (N-N) force 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 33 Quark Meson Coupling Model The Quark Meson Coupling (QMC) Model of the nucleus takes into account both the fundamental interactions among quarks within the neutrons and protons, and also the interactions between the neutrons and protons (meson exchange between pairs of quarks) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 34 Calculations for Light Nuclei In addition to two-body N-N interactions, three-body N-N-N interactions must also be included in the theoretical description of light nuclei 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 35 Repulsive Core (Pauli Principle) Radius of nucleon: ~ 1 fm Radius of hard core: ~ 0.2 fm Nucleon mean free path: ~ 7 fm Volume of hard cores is only ~ 2% of nuclear volume 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 36 The Deuteron The deuteron consists of a bound protonneutron system Its ground-state is the only state which is bound; the first excited state is unbound The ground state has spin and parity Iπ = 1+ The deuteron is not a spherical nucleus 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 37 Range of the Nuclear Force The range of an interaction is related to the mass of the exchanged particle The Heisenberg Uncertainty Principle gives: ΔE Δt ≈ ħ A particle can only create another particle of mass m for a time t ≈ ħ/mc2 during which interval the particle can travel at most ct Taking ct as an estimate of the range R gives: R ≈ ħ / mc This yields R ≈ 1.4 fm for pion exchange 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 38 Deuteron Wavefunction The maximum of the wavefunction is only just inside the potential well with a considerable exponential tail outside The RMS separation between the neutron and proton is 4.2 fm, larger than the range of the nuclear force (~ 1.4 fm) The deuteron is loosely bound ! The binding energy is only B/A ~ 1 MeV/A 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 39 Hypernuclei Nuclei including excited nucleons including heavy quarks: e.g. Lambda particle Λ 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 40 2. Nuclear Behaviour 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 41 Mirror Nuclei The force between two nucleons has the property of charge symmetry and charge independence The two nuclei 20Na and 20F are examples of mirror nuclei The numbers of protons and neutrons are exchanged 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 42 Isospin Substates By analogy with spin, an isospin T state has (2T+1) substates The substates correspond to states in different nuclei 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 43 Isobaric Analogue States Isodoublet states occur in odd-A nuclei Isotriplet states occur in even-A (even-even and odd-odd) nuclei 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 44 A=51 Mirror Nuclei 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 45 Mirror Nuclei: f7/2 Shell 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 46 Independent Particle Model Vij r = ri - rj Energy as a function of separation 1/29/2015 In principle, if the form of the nucleon-nucleon potential is known for bare nucleons, then the energy of a nucleon moving inside a nucleus can be calculated This is a very difficult problem to solve as the nucleon interacts simultaneously with all the other nucleons Use an average potential PHYS490 : Advanced Nuclear Physics : E.S. Paul 47 Independent Particle Model The Hamiltonian is of the form: H = Σ (Ti + Vij) It has 3A degrees of freedom and is too complicated to solve except for the lightest nuclei (A < 12) Instead we use an average “mean-field” potential: H = Hmean field + Hresidual where Hresidual contains interactions between nucleons that are not accounted for by the average potential, especially interactions among valence nucleons 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 48 Nuclear Mean Free Path Why is it that the Independent Particle picture of nuclear motion works ? The Pauli Exclusion Principle (PEP) gives nucleons essentially infinite mean free path However, if the range of the nuclear force was 2 to 3 times stronger, then nuclei could have been crystalline 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 49 Particles in a (Potential) Box Energy levels up to the ‘Fermi level’ 1/29/2015 The short range interaction between nucleons means that each nucleon moves in an average potential The average separation (~ 2.4 fm) is larger than the range of the nuclear force (1.4 ~ fm) Nuclei cannot easily change state unless close to the Fermi surface (PEP) PHYS490 : Advanced Nuclear Physics : E.S. Paul 50 Degenerate Fermi Gas Model This is a simple model in which nucleons are placed in a volume V = 4πR3/3 and the interactions between them are ignored A Fermi sea is formed, filled up to the energy corresponding to the Fermi momentum: EF = pF2/2m = ħ2kF2/2m The binding energy per nucleon is: B = -E/A = -3/5 TF + 1/2 V0 where TF is the kinetic energy at the Fermi surface 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 51 Nucleon Effective Mass The nuclear force has the property of saturation so that B(A,Z) is independent of A caused by the Pauli Exclusion Principle (PEP), its spin and isospin dependence, and (less importantly) the repulsive core The nuclear separation energy S is the difference between the energy of a nucleon outside the nucleus and the energy of the Fermi level EF: S = B = -1/5 TF Wrong ! (S > 0) – the nucleon has an effective mass (m* > mn) when moving in a nucleus 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 52 Some Nuclear Quantities Number density (A/V) measured: ρ ~ 0.17 fm-3 (~ 1.5 x 1018 kg/m3) Fermi momentum: kF = pF/ħ ~ 1.4 fm-1 Fermi energy: EF ~ 10 MeV Kinetic energy of a nucleon in the nucleus: 3/5EF ~ 6 MeV corresponding to a velocity v/c ~ 0.14 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 53 Nuclear Potentials There are two approaches: 1. An empirical form of the potential is assumed, e.g. square well, harmonic oscillator, Woods-Saxon 2. The mean field is generated self-consistently from the nucleon-nucleon interaction 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 54 3. Forms of Mean Potential 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 55 Shell Model – Mean Field A nucleon in the Mean Field of N-1 nucleons N nucleons in a nucleus Assumption – ignore detailed two-body interactions Each particle moves in a state independent of the other particles The Mean Field is the average smoothed-out interaction with all the other particles An individual nucleon only experiences a central force 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 56 Square Well Potential Simplest form of potential Since we have a spherically symmetric potential we can separate the solutions into angular and radial parts Infinite square well potential 1/29/2015 Radial solutions are Bessel functions which satisfy the boundary condition Rnℓ(R) = 0 The eigenenergies are labelled by n and ℓ: Enℓ = (ħ2/2mR2)ξnℓ2 - U0 PHYS490 : Advanced Nuclear Physics : E.S. Paul 57 Square Well Quantum Numbers ‘n’ is the principal quantum number (number of nodes in wavefunction) ‘ℓ’ is the orbital angular momentum ( j = ℓ±½ is the total particle angular momentum ) The energies depend simply and monotonically on n and ℓ 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 58 Properties of the Solutions Higher n : higher energy (more kinetic energy) Higher ℓ : higher energy (larger radius, less bound) The lowest state is : 1s1/2 (n = 1, ℓ = 0) - explains ground state of the deuteron: L = ℓ1 + ℓ2 = 0 Note that two orbits can have similar energies if one has larger n and smaller ℓ, or vice versa 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 59 Square Well Labels The levels are labelled by n and ℓ ( ‘s’ = 0, ‘p’ = 1, ‘d’ = 2, ‘f’ = 3, ‘g’ = 4, ‘h’ = 5, ‘i’ = 6, ‘j’ = 7, ‘k’ = 8 ) Each level has 2ℓ + 1 substates The first few levels (different from H atom): Level Occupation Total 1s 2 2 1p 6 8 1d 10 18 2s 2 20 1f 14 34 2p 6 40 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 60 Square Well Wavefunctions For ℓ ≠ 0 there is an effective centrifugal barrier which modifies the shape of the potential Low n high ℓ states are moved towards the nuclear surface e.g. compare 1s and 1f states 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 61 Harmonic Oscillator potential Easy to handle analytically Form of potential: VHO(r) = -U0 + ½mr2ω2 Solutions are Laguerre polynomials Simple harmonic oscillator potential 1/29/2015 Eigenenergies may again be labelled by n and ℓ : Enℓ = (2n + ℓ + ½) ħω – U0 PHYS490 : Advanced Nuclear Physics : E.S. Paul 62 Harmonic Oscillator potential Eigenenergies can also be labelled by the oscillator quantum number N: EN = (N + 3/2) ħω – U0 For each N there are degenerate levels with n and ℓ that satisfy: 2(n-1) + ℓ = N, N ≥ 0, 0 ≤ ℓ ≤ N Even N contains only ℓ even states; odd N, odd ℓ The degeneracy condition is: Δℓ = 2 and Δn = 1 (e.g. N = 4 3s, 2d, 1g orbits) It is the fundamental reason for shell structure, i.e. clustering of levels The parity of each oscillator shell is: (-1)N = (-1)ℓ 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 63 Woods-Saxon Potential Usually finite potential forms are used such that V(r) 0 if r » 0 The Woods-Saxon potential is considered to be the most realistic nuclear potential For protons a Coulomb potential VC(r) is added 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 64 (Wrong) Magic Numbers 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 65 Spin-Orbit Coupling In order to account for the correct nucleon numbers at which the higher shell closures occur, a spin-orbit term is added – Mayer, Haxel, Jensen, Suess (1948) For the modified harmonic oscillator: VHO(r) = -U0 + ½mr2ω2 – 2/ħ2αℓ.s Since: ℓ.s = ½ħ2[j(j+1) - ℓ(ℓ+1) – ¾] the energy is modified by -αℓ if j = ℓ + ½ and by +α(ℓ+1) if j = ℓ – ½ Note: j = ℓ + ½ levels are lowered in energy relative to j = ℓ - ½ levels (opposite to the atomic case!) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 66 Predicted Shell Structure The harmonic oscillator shells are shown to the left in this diagram In the middle, an ℓ2 term is added to make the potential more realistic (‘modified oscillator’) A spin orbit term ℓ.s is added to the right with its strength (fitted to experiment) adjusted to obtain the correct nuclear magic numbers 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 67 Experimental Shell Effects The energies of the first excited 2+ states in nuclei peak at the magic numbers of protons or neutrons ‘B(E2)’ values ( 1/τ where τ is the mean lifetime) of the 2+ states reach a minimum at the magic numbers ‘Magic’ nuclei are spherical and the least collective 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 68 Systematics Near Z(N) = 50 N = 50 100Sn 1/29/2015 Z = 50 (Z=N=50) and 132Sn (N=82) are doubly magic nuclei PHYS490 : Advanced Nuclear Physics : E.S. Paul 69 Neutron Separation Energies 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 70 Residual Interactions The residual interaction ν between nucleons is the difference between the actual two-nucleon potential Vα experienced by a nucleon in a state α and the average potential Matrix elements of ν, α|ν|β are only appreciable near the Fermi Surface The interaction ν is a two-body operator because it changes the state of two nucleons. It can be treated in a number of ways: 1. from the free two-nucleon potential (difficult!) 2. as a free parameter (fit to experimental data) 3. parameterised using physical intuition 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 71 Quadrupole + Pairing Interaction Monopole pairing Iπ = 0+ 1/29/2015 If we assume that the interaction takes place near the Fermi surface, i.e. near r = R then Vℓ(ri,rj) Vℓ(R) The quadrupole-quadrupole (ℓ = 2) interaction is the most important correction to a spherical field, and is relatively long range The pairing interaction (left) is the important short range component. It leads to greater binding between nucleons if their angular momenta are coupled to zero spin, with maximum spatial overlap PHYS490 : Advanced Nuclear Physics : E.S. Paul 72 Hartree Fock Method The philosophy here is that the nuclear potential is self-consistent 1. We calculate the nucleon distribution (density) from the net potential 2. Then we evaluate the net potential from the nucleon-nucleon interaction 3. Then we iterate 4. The potential is self-consistent if the one with which we end up is the same as the one we start with 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 73 4. Nuclear Deformation 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 74 Evidence for Deformation 1. Large electric quadrupole moments Q0 2. Low-lying rotational bands ( E I[I+1] ) The origin of deformation lies in the long range component of the nucleon-nucleon residual interaction: a quadrupole-quadrupole interaction gives increased binding energy for nuclei which lie between closed shells if the nucleus is deformed. In contrast, the short range (pairing) component favours sphericity 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 75 Simple Nuclear Shapes The general shape of a nucleus can be expressed in terms of spherical harmonics Yλμ(θ,φ) The λ = 1 term describes the displacement of the centre of mass and therefore cannot give rise to excitation of the nucleus – ignore ! The λ = 2 term is the most important term and describes quadrupole deformation 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 76 Nuclear Shapes The λ = 3 term describes octupole shapes which can look like pears (μ = 0), bananas (μ = 1) and peanuts (μ =2,3) The λ = 4 term describes hexadecapole shapes In general most nuclei are prolate with a small additional hexadecapole deformation 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 77 Principal Axes The description of the nuclear shape simplifies if we make the principal axes of our coordinate system (x, y, z) coincide with the nuclear axes (1, 2, 3) For quadrupole shapes we then need only two parameters (β, γ) to describe the shape Intrinsic (nuclear) and laboratory frame axes ‘Prolate’ (rugby ball): β > 0 ‘Oblate’ (smartie): β < 0 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 78 Quadrupole β and γ Parameters prolate oblate x=z>y x>y=z Axially symmetric shapes 60° γ = n 60° prolate x=y<z 0° oblate x=y>z -60° oblate x<y=z prolate x=z<y 1/29/2015 Triaxial shapes : x ≠ y ≠ z PHYS490 : Advanced Nuclear Physics : E.S. Paul γ ≠ n 60° 79 Theoretical Deformations 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 80 Shape Coexistence 1 2 3 The nucleus 184Pb has three low-lying 0+ states 1. Spherical 2. Oblate 3. Prolate 1/29/2015 This plot shows the calculated ‘potential energy surface’ PHYS490 : Advanced Nuclear Physics : E.S. Paul 81 Deformation Systematics Theory Proton Number Z Doubly Magic: Spherical Midshell: Deformed Oblate ‘Spherical’ Prolate Neutron Number N 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 82 Proton Number Z First Excited 2+ Energies E(2+) [Moment of Inertia]-1 Neutron Number N 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 83 Proton Number Z Deformation: Rotational Bands Neutron Number N 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 84 Nilsson Model In order to introduce nuclear deformation Nilsson modified the harmonic oscillator potential to become anisotropic: V = ½m[ω12x2 + ω22y2 + ω32z2] with ωk R k = ω0 R0 and ω1 = ω2 ≠ ω3 If axial symmetry is assumed (γ = 0) then the deformation is described by the parameter ε: ε = (ω1,2 – ω3) / ω0 It can be shown that ε ~ β 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 85 Nilsson Diagram (Energy vs. ε) In order to reproduce the observed nuclear behaviour Cℓ.s and Dℓ2 terms need to be added (C and D are constants) The ℓ.s term is the spinorbit term The ℓ2 term has the effect of flattening the potential to make it more realistic (like the shape of the WoodsSaxon potential) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 86 Nilsson Single-Particle Diagrams N Z 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 87 Nilsson Labels The energy levels are labelled by the asymptotic quantum numbers: Ωπ [N n3 Λ] ‘N’: N = n1 + n2 + n3 is the oscillator quantum number ‘n3’: n3 is the z-axis (symmetry axis) component of N ‘Λ’: Λ = ℓz is the projection of ℓ onto the z-axis ‘Ω’: Ω = Λ + Σ is the projection of j = ℓ + s onto the z-axis ‘π’: π = (-1)N = (-1)ℓ is the parity of the state 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 88 The Λ, Σ, Ω Quantum Numbers Spin projections: 1/29/2015 Ω=Λ+Σ=Λ±½ PHYS490 : Advanced Nuclear Physics : E.S. Paul 89 Asymptotic Quantum Numbers Because of the additional ℓ.s and ℓ2 terms the physical quantities labelled by n3 and Λ are not constants of the motion, but only approximately so These quantum numbers are called asymptotic as they only come good as ε ∞ However, the quantum numbers N, Ω and π are always good labels provided that: 1. the nucleus is not rotating and 2. there are no residual interactions 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 90 Large Deformations This figure ignores the ℓ.s and ℓ2 terms Deformed shell gaps emerge when ω3 and ω1,2 are in the ratio of small integers, i.e. ω3/ ω1,2 = p/q A superdeformed shape has p/q = ½ or R3:R1,2 = 2:1 A hyperdeformed shape has R3:R1,2 = 3:1 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 91 5. Hybrid Models 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 92 Deformed Liquid Drop Assuming that the nucleus behaves as a charged liquid drop, a semi-empirical expression can be obtained for the total energy: E(A,Z) = -aVA + aSA2/3 + aCZ2A-1/3 To correct for deformation the nuclear radius R0 is replaced by: R3 = R0(1 + δ) , R 1,2 = R0(1 - ½δ) The energy for small δ then becomes: E(A,Z) = -aVA + aSA2/3 (1 + 2/5 δ2) + aCZ2A-1/3 (1 – 1/5 δ2) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 93 Deformed-Spherical Energies ΔE(δ) = E(δ) – E(δ=0) 1/29/2015 It is then predicted that the nucleus is always spherical (i.e. ΔE = 0 for δ > 0) unless Z2/A > 49 in which case the nucleus prefers infinite deformation (i.e. it fissions) This is clearly wrong ! The liquid drop model must be extended to take into account shell-model effects, i.e. effects from individual nucleon motion PHYS490 : Advanced Nuclear Physics : E.S. Paul 94 Shell Correction Additional terms arising from the symmetry energy (which prefers N = Z) and the pairing energy (Δ, 0, -Δ for even-even, odd-even and odd-odd nuclei, respectively) can be added Alternatively the total energy can be calculated using mean-field potentials This is not simply the sum of the individual eigenenergies ei because the potential energy of each nucleon would be counted twice 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 95 Shell Energy The eigenvalue for each nucleon is: ei = Ti + j≠iVij The total energy is: Ti + ½j≠iVij = ½ei + ½Ti For the harmonic oscillator potential: Ti = Vi = j≠iVij so that E = ¾ei This method has difficulty in producing the correct energy because errors in ei give rise to large errors in the summation ei 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 96 Strutinsky Shell Correction To obtain both the global (liquid drop) and local (shell model) variations with δ, Z and A, Strutinsky developed a method to combine the best of both models (a) Liquid drop: He considered the behaviour of the level gF(e) =gAV(e) (b) and (c) show shell effects. density g(e) in the two models A change in nuclear binding And calculated the arises from: ‘fluctuation’ energy gAV(e) – gF(e) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 97 Level Density 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 98 Shell Correction Energies 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 99 Fission Isomers Superdeformed band head is isomeric. Its decay can penetrate barrier either way 1/29/2015 If the increase in liquid drop energy for increasing deformation ΔE(δ) is small enough (e.g. Z2/A > 35) then any secondary minimum in the total energy arising from the shell correction will become similar in energy to the first This second minimum corresponds to a superdeformed nuclear state PHYS490 : Advanced Nuclear Physics : E.S. Paul 100 Superdeformed 1/29/2015 240Pu PHYS490 : Advanced Nuclear Physics : E.S. Paul 101 6. Nuclear Excitations Single-particle and collective motion 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 102 Collective Motion in Nuclei Adiabatic approximation: identify fast and slow degrees of freedom Molecules: electronic motion fastest, vibrations 102 times slower, rotations 106 times slower These different motions have very different time scales, so the wavefunction separates into a product of terms In nuclei the timescales are much closer Collective and single-particle modes can perhaps be separated but they will interact strongly ! 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 103 Types of Nuclear Excitation All even-even nuclei have a ground state with a consequence of nuclear pairing Iπ = 0+ , Closed-shell nuclei are spherical and excited nuclear states can only be formed by breaking pairs of nucleons or by vibrations For odd-mass nuclei (near closed shells) the low-lying excited states map out the single-particle spectrum of states around the Fermi level ‘Deformed’ nuclei exhibit regular rotational bands: quadrupole or octupole shapes etc… 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 104 Excitations in Spherical Nuclei All even-even nuclei have Iπ = 0+ in their ground states Excitations can only occur by breaking of pairs or by vibrations The energy difference between the first excited and ground states is a rough Doubly magic (spherical) measure of the pairing energy nuclei 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 105 Noncollective Level Scheme Complicated set of energy levels No regular features, e.g. band structures Some states are isomeric 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 106 Vibrations From the liquid drop dependence on deformation we can estimate the restoring force if the nucleus is deformed from its equilibrium deformation A vibration can be any distortion in the nuclear shape Equally spaced energy levels for each phonon of vibration 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 107 Beta (Y20) Vibration 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 108 Gamma (Y22) Vibration 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 109 Octupole (Y30) Vibration 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 110 Octupole (Y31) Vibration 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 111 Octupole (Y32) Vibration 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 112 Octupole (Y33) Vibration 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 113 Realistic Vibrational Levels vibrator n=3 n=2 n=1 n=0 nucleus For each given mode of vibration, each phonon has an associated angular momentum and parity, e.g: quadrupole 2+ octupole 3 For a pure vibrator there are groups of degenerate levels for two or more phonons 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 114 Multiphonon Vibrational States N = 3 (3 phonon) N = 2 (2 phonon) N = 1 (1 phonon) 124Sn, spherical N=0 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 115 Giant Resonances Monopole L=0 Isovector Isoscalar Dipole L=1 Quadrupole L=2 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 116 Rotations of a Deformed System K Nuclear spins 1/29/2015 The low-lying levels of deformed even-even nuclei which lie far from closed shells form a regular sequence of levels that are much lower in energy than the pairing energy. This arises from rotation The Hamiltonian is: Hrot = (ħ2/2) R2 = (ħ2/2) (I-J)2 where is the moment of inertia and J is additional angular momentum generated by, e.g. the odd particle in an odd-A nucleus or by vibrations PHYS490 : Advanced Nuclear Physics : E.S. Paul 117 Coriolis Coupling Note that rotation cannot take place about the symmetry (z) axis The rotational Hamiltonian can be expanded: R2 = (I – J)2 = I2 – 2I.J + J2 = I2 + J2 – 2K2 - (I+J- + I-J+) where I± = Ix ± i Iy, J± = Jx ± i Jy and Jz = Iz = ±K The quantity K is the projection of I along the deformation axis The coupling term (I+J- + I-J+) corresponds to the Coriolis force and couples J to R 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 118 The K Quantum Number The operators I± link states with K differing by ±1 The term (I+J- + I-J+) can be ignored if: (1) rotational bands with ΔK = 1 lie far apart (2) the particular band does not have K = ½ The excitation energies then become: Erot = (ħ2/2)[I(I+1) + J(J+1) -2K2] with I = K, K+1, K+2… and K is a constant of the motion Then: Erot = EK + (ħ2/2)I(I+1) where EK is the energy of the lowest band level 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 119 Vibrational Bands in 232Th Here the low-lying levels are all collective, i.e. rotational and vibrational Ground State Band: Kπ = 0+ β Band: Kπ = 0+ γ Band: Kπ = 2+ Octupole Band: Kπ = 0π = 0+ then Oct. Note that if K Beta Gamma the I values 1, 3, 5… are not present GSB π = 0- the I values 0, For K Reflection symmetric shape, 2, 4… disappear 232Th is a deformed nucleus 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 120 Alternating Parity Bands in 226U This nucleus is reflection asymmetric (i.e. β3 ≠ 0) in its ground state: it has octupole deformation The nuclear wavefunction in its intrinsic frame is not an eigenvalue of parity: Ψ2 (x ,y ,z) ≠ Ψ2 (-x, -y, -z) In the laboratory frame (i.e. averaged over all nuclear orientations) the levels have alternating parity 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 121 Reflection (A)symmetry 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 122 Electric Dipole Moment In a nucleus with octupole deformation, the centre of mass and centre of charge tend to separate, creating a non-zero electric dipole moment 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 123 Octupole Magic Numbers Octupole correlations occur between orbitals which differ in both orbital (ℓ) and total (j) angular momenta by 3 Magic numbers occur at 34, 56, 88 and 134 Nuclei with both proton and neutron numbers close to these are the best candidates to show octupole effects 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 124 Rotational Bands in 157Ho This nucleus shows three band structures built on different Nilsson states 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 125 7. Rotating Systems 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 126 Moment of Inertia The energy of a rotating nucleus is given by: E = (ħ2/2) I[I+1] The nuclear moment of inertia (at low spin) is found to be one third to one half of the value expected for a rotating liquid drop Nuclear pairing introduces a degree of superfluidity Rotation counteracts pairing (cf strong magnetic field applied to superconductor) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 127 Rotational Bands: γ-ray Spectra 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 128 Inglis Moment of Inertia Inglis (1952) showed that the moment of inertia of a Fermi gas rotating about the x-axis is: x = 2 |p|Îx|h|/(ep-eh) where the summation is over all possible 1-particle 1-hole excitations in a deformed shell model The rigid-body moment of inertia is: = (2/5) mnAR02[1 + 0.3β] which is higher than observed 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 129 Nuclear Moments of Inertia Nuclear moments of inertia are lower than the rigidbody value – a consequence of nuclear pairing 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 130 Pairing Gap A rough estimate of the energy required to create a particle-hole excitation is 2Δ, where Δ is the pairing gap A typical value for Δ is 1 MeV 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 131 Cranking Model The deformed shell model (e.g. Nilsson Model) can be modified to include pairing To include rotation it is convenient to subtract the effect of rotational forces (Coriolis and centripetal) Classically the ‘potential’ energy of these forces is ω.I so the corresponding quantum operator is ωÎx The Hamiltonian is: Hω = HDSM – ωÎx Energy in the rotating frame: Eω = E - ω Îx 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 132 Routhian and Aligned Angular Momentum The Routhian is simply the energy in the rotating frame of reference: Eω The aligned angular momentum is just the expectation value of the operator Îx: Îx 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 133 Spin and Rotational Frequency There are two important relations which arise since E is independent of ω and Eω is independent of I: dEω/dω = - Îx and dE/dI = ω dÎx/dI Nuclear spin I and its projections onto the rotation axis Ix and deformation axis K 1/29/2015 Since: Îx = √[I(I+1)-K2] ħ then for K = 0: Îx ~ I ħ and hence: dE/dI ~ ω ħ PHYS490 : Advanced Nuclear Physics : E.S. Paul 134 Gamma Ray Energies and Rotational Frequency The energy of a rotational band for K = 0 is: E = E0 + (ħ2/2) I(I + 1) , I = 0, 2, 4… The energy difference between consecutive levels ΔE represents the gamma-ray energy Eγ The spin difference between consecutive levels is Δ I = 2 The rotational frequency ω is defined as: ωħ = dE/dI = ΔE/ΔI = Eγ/2 i.e. the frequency is just half the gamma-ray energy 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 135 Moments of Inertia The energy of a rotational band for K = 0 is: E = E0 + (ħ2/2) I(I + 1) , I = 0, 2, 4… Then: dE/dI = (ħ2/2) (2I + 1) and: d2E/dI2 = ħ2/ defines the ‘dynamic moment of inertia’ which is independent of spin By using finite differences: dE=ΔE=Eγ, dI=ΔI=2, d2E=ΔEγ, d2I=Δ2I=4 we can evaluate even if we do not know I ! 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 136 Nuclear Rotation The assumption of the ideal flow of an incompressible nonviscous fluid (Liquid Drop Model) leads to a hydrodynamic moment of inertia (surface waves): hydro = rig δ2 This estimate is much too low ! We require short-range pairing correlations to account for the experimental values 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 137 Kinematic and Dynamic MoI’s Rigid body: (1) = (2) High spin: (1) ≈ (2) 1/29/2015 Assuming maximum alignment on the x-axis (Ix ~ I), the kinematic moment of inertia is defined: (1) = (ħ2 I) [dE(I)/dI]-1 = ħ I/ω The dynamic moment of inertia (response of the system to a force) is: (2) = (ħ2) [d2E(I)/dI2]-1 = ħ dI/dω And (2) = (1) + ω d(1)/dω PHYS490 : Advanced Nuclear Physics : E.S. Paul 138 Backbending The moment of inertia increases with increasing rotational frequency Around spin 10ħ a dramatic rise occurs The characteristic ‘S’ shape is called a backbend (158Er) A more gradual increase is called an upbend (174Hf) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 139 Crossing Bands A backbend corresponds to the crossing of two bands (‘g’ and ‘s’ configurations) yrare yrast Yrast and yrare states: dizziest and dizzier in the Swedish language 1/29/2015 The states we observe are called yrast states (thick line) which have the lowest energy for a given spin The s-band, where s stands for ‘Stockholm’ or ‘super’, arises from the breaking of a pair of nucleons. Their angular momenta j1 and j2 align with the rotation axis PHYS490 : Advanced Nuclear Physics : E.S. Paul 140 Pair Breaking For the ground state band: Eg = (ħ2/2g) I(I + 1) For the s-band: Es = (ħ2/2s) (I – J )2 + EJ where J = j1 + j2 and EJ is the energy required to break a pair of nucleons: EJ ~ 2Δ ~ 24 A-1/2 MeV The aligned angular momentum of the s-band increases by approximately: j1 + j2 – 1 (~ 12ħ for 158Er) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 141 Destruction of Pairing Strong external influences may destroy the superfluid nature of the nucleus In the case of a superconductor, a strong magnetic field can destroy the superconductivity: the ‘Meissner Effect’ For the nucleus, the analogous role of the magnetic field is played by the Coriolis force, which at high spin, tends to decouple pairs from spin zero and thus destroy the superfluid pairing correlations 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 142 Pair Breaking and Rotational Alignment A Backbending movie follows showing pair breaking and rotational alignment 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 143 Backbending Movie 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 144 Backbending Demonstration This movie shows Mark Riley’s “backbending machine” 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 145 8. Nuclei at Extremes of Spin 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 146 High-Spin States As the nucleus is rotated to states of higher and higher angular momentum, or spin I, it tries to assume the configuration which has the lowest rotational energy The spin I is made up of a collective part R and a contribution J arising from single particles The energy can be minimised by reducing R or by increasing the nuclear moment of inertia The pairing is broken by the effect of rapid rotation 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 147 Generation of Angular Momentum There are two basic ways of generating high-spin states in a nucleus 1. Collective (in-phase) motions of the nucleons: vibrations, rotations etc 2. Single-particle effects: pair breaking, particlehole excitations. The individual spins of a few nucleons ji generate the total nuclear spin 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 148 High Ix Bands In backbending the value of R (collective spin) is reduced by breaking a single pair of nucleons and aligning their individual angular momenta j with the x-axis, i.e. Ix = jx + R The quantity Ix is approximately a good quantum number and hence a given nuclear state can be described by a single value of Ix The alignment of broken pairs becomes easier if 1. the particle j is high but its projection Ω small 2. the Coriolis force is large: small and high ω 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 149 Aligned Particles Alignment effects should be prominent for nuclei with a few nucleons outside a closed shell, e.g. 158Er with 8 neutrons above the N = 82 closed shell If we continue to rotate faster and faster then more of the valence pairs break and align I = Σji R=0 1/29/2015 Eventually all the particles outside the closed shell (spherical) core align These move in equatorial orbits giving the nucleus an oblate appearance PHYS490 : Advanced Nuclear Physics : E.S. Paul 150 Band Termination neutron backbend proton backbend Gamma Ray Energy 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 151 Band Termination in 158Er When we align the np protons and nn neutrons outside the closed shell the total spin is: I = inp ji(p) + inn ji(n) and the rotational band is said to ‘terminate’ At termination 158Er can be thought of as a spherical 146Gd core plus 4 protons and 8 neutrons generating a total spin 46ħ The configuration is: π(h11/2)4 ν(i13/2)2(h9/2)3(f7/2)3 The terminating spin value of 46 is generated as: (11/2+9/2+7/2+5/2) + (13/2+11/2) + (9/2+7/2+5/2) + (7/2+5/2+3/2) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 152 High K (Iz) Bands If we have many unpaired nucleons outside the closed shell then alignment with the x-axis becomes difficult because the valence nucleons lie closer to the z-axis, i.e. they have high Ω values The sum K of these projections onto the deformation (z) axis is now a good quantum number K = Iz = Σjz = ΣΩ 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 153 K Forbidden Transitions It is difficult for rotational bands with high K values to decay to bands with smaller K since the nucleus has to change the orientation of its angular momentum. For example, the Kπ = 8- band head in 178Hf is isomeric with a lifetime of 4 s. This is much longer than the lifetimes of the rotational states built on it. The Kπ = 8- band head is formed by breaking a pair of protons and placing them in the ‘Nilsson configurations’: Ω [N n3 Λ] = 7/2 [4 0 4] and 9/2 [5 1 4] In this case: K = 7/2 + 9/2 = 8 and π = (-1)N(1).(-1)N(2) = -1 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 154 K Isomers in 178Hf A low lying state with spin I = 16 and K = 16 in 178Hf is isomeric with a half life of 31 years ! It is yrast (lowest state for a given spin) and is ‘trapped’ since it must change K by 8 units in its decay 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 155 High K bands in 174Hf This nucleus has 347 known levels and 516 gamma rays ! 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 156 Superdeformation Nuclear potential at low and high spin 1/29/2015 Shell effects can give large energy corrections for large values of prolate deformation, e.g. when the major/minor axis ratio is 2:1 The smooth liquid-drop contribution to the total nuclear energy includes the rotational energy, which can be substantially reduced at high spin by increasing the moment of inertia At sufficiently high spin a secondary minimum can become energetically favourable PHYS490 : Advanced Nuclear Physics : E.S. Paul 157 Superdeformed Band in β2 ~ 0.6, 2:1 axis ratio 1/29/2015 152Dy The experimental signature of these superdeformed (SD) shapes is a very regular sequence of equally spaced γ rays In 152Dy the (first) SD band spans a spin range 20 – 60 ħ Nowadays multiple SD bands are known in this and other nuclei PHYS490 : Advanced Nuclear Physics : E.S. Paul 158 Some Big Numbers The SD band of 152Dy emits ~ 20 gamma rays in ~ 10-13 s. The total energy released is: Eγ ~ 20 MeV (1 eV = 1.6 x 10-19 J) The power is: (3.2 x 10-12 J) / (10-13 s) = 32 W ! The rotational frequency is: ħω ~ 500 keV, so ω ~ 8 x 1020 radians/sec 1020 Hz or 107 rotations in 10-13 s same as number of days in 30,000 years ! The decay of the SD band passes through a long-lived isomeric level (86 ns) ~ 5 x 1012 rotations same as number of days since the Big Bang ! 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 159 Superdeformation in 132Ce SD bands exist in cerium (Z = 58) nuclei with a major/minor axis ratio of 3:2. This band in 132Ce (THE original SD band – discovered by the Liverpool Nuclear Physics Group) is now seen up to spin approaching 70ħ – one of the highest spins ever seen in the atomic nucleus ! 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 160 Superdeformed Systematics 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 161 Shape Coexistence For a given nuclear system at a given value of spin, a number of configurations can exist These configurations may have different shapes Weakly deformed triaxial and oblate shapes coexist in 152Dy along with the superdeformed shape Each shape has a (local) ‘minimum’ in the nuclear ‘total energy surface’ 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 162 Hyperdeformation Superdeformation represents a secondary minimum in the nuclear potential energy, with typically a 2:1 axis ratio Hyperdeformation represents a third minimum, with an axis ratio 3:1 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 163 Critical Angular Momenta Nuclei can only attain a finite amount of spin before they fly apart (fission) Just before this fission is a predicted region of extended triaxial (x ≠ y ≠ z) shapes This is known as the Jacobi regime Such behaviour also Nuclei with mass 130-150 can occurs for macroscopic accommodate the most spin objects 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 164 Jacobi Shape 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 165 9. Nuclei at Extremes of Isospin 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 166 Limits of Nuclear Existence Segre Chart Known Nuclei Stable Nuclei Proton Dripline Fission Limit Terra Incognita Neutron Dripline 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 167 Where Are The Driplines? 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 168 Where Are The Driplines? In this experiment – fragmentation of a beam of 48Ca – no counts were observed for 26O This defines the neutron dripline for oxygen isotopes 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 169 Where is the Neutron Dripline? 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 170 Heavy N = Z Nuclei Shell corrections give minima in the nuclear energy at non-zero values of deformation Bigger effect if both proton and neutrons occur at these ‘magic numbers’ Also a big effect for N = Z The N = Z = 40 nucleus 80Zr is an example It is difficult to study this nucleus: it is 10 neutrons lighter than the lightest stable zirconium isotope ! 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 171 Exotic Nuclei The nucleus 12C has six protons and six neutrons It is stable and found in nature The nucleus 22C has six protons and sixteen neutrons ! It is radioactive and at the limit of nuclear binding Characteristics of exotic nuclei: excess of neutrons or protons, short half life, neutron or proton dominated surface, low binding 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 172 Isospin: Tz = (N - Z) / 2 A = 21 A 21 C 6 15 Z TZ N 21 C 21 N 21 O 21 F 21 Ne 21 Na 21 Mg 21Al 6 15 7 14 8 13 9 12 10 11 11 10 12 9 13 8 +9/2 +7/2 +5/2 +3/2 +1/2 –1/2 –3/2 –5/2 Neutron rich 1/29/2015 Proton rich PHYS490 : Advanced Nuclear Physics : E.S. Paul 173 Nuclei Far From Stability 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 174 Proton-Rich Nuclei The proton dripline is defined by the least massive bound nucleus of every isotopic chain (Sp drops to zero) For nuclei beyond the dripline the last proton has a positive energy and is unbound This proton does not escape instantaneously as it must overcome the Coulomb Barrier 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 175 Radioactivity: Normal 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 176 Radioactivity: Exotic 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 177 Alpha and Proton Emitters 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 178 Jyväskylä, Finland (Feb 2006) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 179 Jyväskylä (midnight June 21 2009) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 180 Recoil Decay Tagging 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 181 JUROGAM + RITU + GREAT 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 182 Deformed Proton Emitter The half-lives of proton radioactivity are sensitive to both specific orbitals and nuclear deformation Measured half-lives in 131Eu and 141Ho could only be understood if these nuclei were deformed This was later confirmed by the observation of rotational bands in 141Ho 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 183 Proton Decay Half Lives ℓ = 0 Coulomb Barrier ℓ = 5 Coulomb Barrier plus Centrifugal Barrier The half-lives of proton radioactivity are sensitive to the orbital angular momentum of specific states A centrifugal barrier occurs in the potential proportional to the orbital angular momentum 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 184 Fine Structure in Proton Decay In 131Eu proton decay has been observed both to the ground state and the first excited state of 130Sm This establishes the first 2+ state in 130Sm at an energy of 121 keV This low energy implies a large moment of inertia and large quadrupole (prolate) deformation for the exotic 130Sm nucleus ! 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 185 Fine Structure in Alpha Decay In 109Xe alpha decay has been observed both to the ground state and the first excited state of 105Te This establishes the relative energies of the neutron d5/2 and g7/2 orbitals in 105Te 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 186 Superdeformed Proton Emitter SD An SD band in 58Cu decays by proton emission into 57Ni in competition with γ decay to the low-spin 58Cu states 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 187 Direct Two-Proton Decay A new mode of decay, direct two-proton decay, had been predicted long ago, but until recently, experimental efforts had only found sequential emission through an intermediate state Are the two protons correlated (di-proton emission) or uncorrelated (sequential proton emission)? 1/29/2015 To prove di-proton emission, specific nuclei are needed where the sequential emission is energetically forbidden e.g. 18Ne PHYS490 : Advanced Nuclear Physics : E.S. Paul 188 Direct Two-Proton Decay of 45Fe Decay energy spectrum correlated with 45Fe implantation from 45Fe The experimental Q-value implies di-proton emission with a barrier-penetration half-life of 0.024 ms 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 189 Neutron-Rich Nuclei Physics of weak binding The question of which combinations of protons and neutrons form bound systems has not been answered for most of the nuclear chart because of a lack of experimental access to neutron-rich nuclei These nuclei are increasingly the focus of present and future experimental (and theoretical) effort 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 190 Light Neutron-Dripline Nuclei The neutron dripline has only been reached for light nuclei 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 191 Level Inversion in 11Be The ordering of the neutron 1s1/2 and 1p1/2 orbitals appears to be inverted in the nucleus 11Be and lighter N = 7 isotones unbound 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 192 Nuclear (Neutron) Haloes The spatial extent of 11Li with 3 protons is similar to that of 208Pb with 82 protons ! 11Li is modelled as a core of 9Li plus two valence neutrons 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 193 Size of Lithium Isotopes Root mean square radii Interaction cross sections give a measure of the nuclear matter distribution (radius) A sudden jump is seen in going from 9Li to 11Li However, the electric quadrupole moments are similar (charge distribution) Hence, excess neutron tail or halo Textbook: R = r0A1/3 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 194 Nuclear Sizes 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 195 Halo Systematics Neutron haloes have now been seen in nuclei as heavy as 19C (Z = 6, N = 13) Note proton haloes also predicted 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 196 Borromean System Halo nuclei have provided insight into a new topology with a Borromean property bound The two-body subsystems of the stable three-body system 11Li (9Li + n + n) are themselves unstable ! unbound 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 197 Neutron Skins 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 198 Proton Skins? Theory 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 199 Quenching of Shell Structure Adding more and more neutrons to a nucleus may change the shell structure It has been predicted that the shell gaps (magic numbers) are washed out far from the stability line The ℓ.s term is diminished 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 200 New Shell Structure? N=20 Z=8 N=16 ? N=8 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 201 Unexpected Things Happen Far From Stability The heaviest known tellurium isotope is 136Te It is two neutrons outside the N = 82 shell closure However its measured B(E2) value is much lower than expected ! The 2+ energy is also too ‘low’ 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 202 Extremes of Mass and Charge Investigations of the heaviest nuclei probe the role of the Coulomb force and its interplay with quantal shell effects in determining the nuclear landscape Without shell effects nuclei with more than 100 protons would fission instantaneously However, ‘superheavies’ with Z up to 118 have been identified ! 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 203 Deformed Superheavy Nuclei Modern theory not only predicts which combinations of N and Z can be made into heavy nuclei but also indicates that stability arises in specific cases from the ability of the nucleus to deform For example, the nucleus 208Pb at the shell closures Z = 82, N = 126 is spherical but nuclei with substantially deformed ground states are predicted around the next shell closures at Z ~ 114, N = 184 Different theories suggest the next proton shell closure at Z = 114, 120, 126 (note N = 126 occurs for neutrons) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 204 Super Heavy Elements (SHE) Copernicium Roentgenium Darmstadtium Meitnerium Hassium Bohrium Protons Neutrons 1/29/2015 Flerovium Livermorium SHE Quantal shell effects stabilise energy Up to Z = 112 results confirmed Dubna: Z = 114, 116, 118 Berkeley: Z = 118 ‘discovered’ then retracted PHYS490 : Advanced Nuclear Physics : E.S. Paul 205 Element 115 Well known for antigravity properties The fuel of UFO’s 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 206 The Island(s) of Stability 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 207 Alpha Decay Chains decay provides the technique to identify heavy elements 1/29/2015 The heaviest nuclei are unstable against decay The -decay half life (in s) is given empirically by: log10 t1/2 = 1.61 Z Eα-1/2 – 1.61 Z2/3 – 28.9 Here Eα (in MeV) is the decay energy, related to the mass difference of the parent (Z, A) and daughter (Z-2, A-4) nuclei The lifetimes are very long (>10-3 s) on the nuclear time scale PHYS490 : Advanced Nuclear Physics : E.S. Paul 208 SHE Synthesis at GSI 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 209 Elements 116 and 118 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 210 Element 117 (2010) Dubna (Russia) Phys. Rev. Lett. 104, 142502 (2010) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 211 Superheavies at High Spin The ground-state rotational band of 254No (Z=102) has been identified up to spin 20+ (at least!) The energy spacing of the levels is consistent with a sizeable prolate deformation with an axis ratio 4:3 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 212 The Heaviest Element University researchers have discovered the heaviest element yet known to science. The new element, Governmentium (Gv), has one neutron, 25 assistant neutrons, 88 deputy neutrons and 198 assistant deputy neutrons, giving it an atomic mass of 312. These 312 particles are held together by forces called morons, which are surrounded by vast quantities of lepton-like particles called pillocks. Since Governmentium has no electrons, it is inert. However, it can be detected, because it impedes every reaction with which it comes into contact. A tiny amount of Governmentium can cause a reaction that would normally take less than a second, to take from 4 days to 4 years to complete. Governmentium has a normal halflife of 2 to 6 years. It does not decay, but instead undergoes a reorganisation in which a portion of the assistant neutrons and deputy neutrons exchange places. In fact, Governmentium's mass will actually increase over time, since each reorganisation will cause more morons to become neutrons, forming isodopes. This characteristic of moron promotion leads some scientists to believe that Governmentium is formed whenever morons reach a critical concentration. This hypothetical quantity is referred to as a critical morass. When catalysed with money, Governmentium turns into Administratium (Ad), an element that radiates just as much energy as Governmentium, since it has half as many pillocks but twice as many morons. 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 213 10. Mesoscopic Systems 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 214 Finite Fermionic Systems The behaviour of micro particles (atoms, electrons, nuclei, nucleons and other elementary particles) can be described by quantum theory Macroscopic bodies obey the laws of classical mechanics These two ‘worlds’ largely differ from each other In nature there is no sharp border between the micro and macro world and there are objects that exist in the intermediate range The atomic nucleus, a finite fermionic system, is an example of such a mesoscopic system 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 215 Mesoscopic Systems ‘Mesoscopic’ systems contain large, yet finite, numbers of constituents, e.g. atomic nuclei, metallic clusters 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 216 Nanostructures and Femtostructures ‘Nanostructures’: intense research is ongoing for quantum systems that confine a number of electrons within a nanometre-size scale (10-9 m), e.g. grains, droplets, quantum dots Nuclei are femtostructures (10-15 m) All these systems share common phenomena but on very different energy scales: nuclear MeV; molecular eV; solid-state meV 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 217 Mesoscopic systems complexity Nuclei He-droplets Metal clusters Emergent phenomena: Nanoparticles E -Liquid-gas surface, droplet features -superconductivity / superfluidity -thermal phase transitions -shell structure, quantal shapes (liquid) -spatial orientation, rotational bands -rotational/magnetic response -quantum phase transitions macroscopic N Quantum dots 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 218 Quantality Parameter The ‘quantality’ parameter (Mottelson 1999), Λ = ħ2 / M a2 V0, measures the strength of the two-body attraction V0 expressed in units of the quantal kinetic energy associated with a localisation of a constituent particle of mass M within the distance a corresponding to the radius of the force at maximum attraction For small Λ the quantal effect is small and the ground state of the many body system will be a configuration in which each particle finds a static optimal position with respect to its nearest neighbours (crystalline) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 219 Nuclei as Quantum Liquids If Λ is large enough the ground state may be a quantum liquid in which the individual particles are delocalised and the low-energy excitations have ‘infinite’ mean-free path 1/29/2015 Constituents 3He 4He H2 Ne Nuclei T = 0 matter Λ = 0.21 ‘liquid’ Λ = 0.16 ‘liquid’ Λ = 0.07 ‘solid’ Λ = 0.007 ‘solid’ Λ = 0.4 ‘liquid’ PHYS490 : Advanced Nuclear Physics : E.S. Paul 220 Fermi Liquid Droplets ‘Clusters’ are aggregates of atoms or molecules with a well-defined size varying from a few constituents to several tens of thousands Conduction electrons in clusters are approximately independent and free Nucleons in nuclei also behave as delocalised and independent fermions Hence analogies exist between these two systems 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 221 The Spherical Droplet Both clusters and nuclei are characterised by a constant density in the interior and a relatively thin surface layer The Liquid Drop Model can be used to calculate the binding energy of a charged droplet The binding energy can be expanded in powers of A1/3 (i.e. radius) where A is the number of constituents 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 222 Spherical Droplet Energy The energy of a droplet may be expressed as: ELD(N,Z) = fA + 4πσR2 + WZ + C Z2e2/R = fA + bsurfA2/3 + WZ + bcoulZ2A-1/3 Here R = r0A1/3 is the radius of the droplet, A the number of atoms and Z is the net charge The first term (fA) is the ‘volume energy’ which contains the binding energy per particle f of the bulk material The second term (4πσR2)is the ‘surface energy’ where σ is the coefficient of surface tension 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 223 Spherical Droplet Energy (cont) The third term (WZ) contains the ‘work function’ W which is the energy required to remove one electron from the bulk metal The fourth term (C Z2e2/R) represents the ‘Coulomb energy’ of the charged constituents In nuclei the charge is evenly distributed because the symmetry energy (quantal effect) keeps the ratio of neutron to protons roughly constant: thus C=3/5 For a cluster charge tends to accumulate at the surface and C tends to 1/2 for a large cluster 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 224 Shell Structures A bunching together of the energy levels of a particle in a two- or three-dimensional potential represents a shell structure Metallic clusters show shell structures similar to nuclei Clusters can contain more constituents than stable nuclei 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 225 Supershell Structures Metallic clusters also exhibit a supershell structure The basic shell structure is enveloped by a long wavelength oscillation (beat pattern) Nuclei become unstable well before the first half-period of the long wavelength oscillation is seen 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 226 Periodic Orbit Theory Supershell structure from interfering periodic orbits 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 227 Loss of Spherical Symmetry Deformation occurs in subatomic and mesoscopic systems with many degrees of freedom, e.g. nuclei, molecules, metallic clusters The microscopic mechanism of ‘spontaneous symmetry breaking’ was first proposed by Jahn and Teller (1937) – for molecules Nuclei with incomplete shells tend to deform because the level density near the Fermi surface is high (unstable) for a spherical shape When the shape of the nucleus changes, nucleonic levels rearrange such that the level density is reduced (stable) – ‘nuclear Jahn-Teller effect’ 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 228 Shapes Of Clusters Nuclei can easily deform because they consist of delocalised nucleons (liquid) The presence of heavy discrete ions leads to a more varied response of clusters Nevertheless, similar shapes are predicted for nuclei and clusters despite the very different nature of the interactions between the constituents 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 229 Differences Between Atomic Nuclei and Metallic Clusters There is only one kind of nuclear matter It has a single ‘equation of state’ However, all materials have their own equation of state In a cluster, as in bulk matter, it is the constituents that determine the density and binding energy 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 230 Nuclear Phase Diagram At sufficient temperature or density nucleons are expected to dissolve into a quark-gluon plasma (QGP) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 231 Nuclear Molecules Speculation about the existence of clusters in nuclei, such as alpha particles, has existed for a long time Ikeda Diagram 1/29/2015 Initially stimulated by the observation of alpha particle decay PHYS490 : Advanced Nuclear Physics : E.S. Paul 232 Beryllium-12 A beryllium nucleus containing 8 neutrons and 4 protons has been found to arrange itself into a molecular-like structure, rather than a spherical shape that some naïve theories might suggest M Freer et al. Phys. Rev. Lett. 82 (1999) 1383 1/29/2015 Beryllium-12 can be thought of as two alpha particles and four neutrons PHYS490 : Advanced Nuclear Physics : E.S. Paul 233 Chain States: Nuclear Sausages Cluster Model calculations for 12C show evidence for a ‘chain state’ consisting of three α particles in a row – axis ratio 3:1 (i.e. ‘hyperdeformed’) Similarly calculations for 24Mg show evidence for a chain state consisting of six α particles in a row – axis ratio of 6:1 ! 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 234 Bloch-Brink Cluster Model Brink presented the light alpha conjugate nuclei as almost crystalline structures These nuclei contain specific arrangements of the alpha clusters Narrow resonances in 12C + 12C scattering data suggested larger clusters may occur ‘Nuclear Molecules’ 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 235 Binary Cluster Model It has been observed that measured quadrupole moments of many superdeformed bands follow: Qo ~ 2 Ro2[ Z A2/3 – Z1 A12/3 – Z2 A22/3] This expression results from considering the states of the nucleus (Z, A) to be composed of two clusters (Zi, Ai) in relative motion For example, a strongly deformed band has recently been found in 108Cd (Z = 48) The predicted fragmentation for 108Cd is: 58Fe (Z = 26) + 50Ti (Z = 22) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 236 11. Nuclear Reactions 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 237 Examples of Nuclear Reactions 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 238 Introduction In a typical nuclear reaction a (light) projectile a “hits” a (heavy) target A producing fragments b (light) and B (heavy) Schematically this can be written a+A b+B In this nuclear “transmutation” we need to consider both kinetic energy and binding energy (E = mc2) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 239 The Impact Parameter Reactions can be classified by the impact parameter b Central collisions occur for small b, e.g. fusion Peripheral collisions occur at large b, e.g. elastic and inelastic scattering, transfer reactions Deep inelastic or massive transfer reactions occur at intermediate values of b 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 240 Collision Kinematics The Q value is: [ (MA + Ma) - (MB + Mb) ] c2 Exothermic (Q > 0) reactions give off energy – kinetic energy of reaction products Endothermic (Q < 0) reactions require an input of energy to occur. By considering the kinetic energy available in the centre-of-mass frame, the threshold energy is: Ta > |Q| [ (Ma + MA) / MA ] 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 241 The Compound Nucleus Consider the reactions: a + A C* a + A* b + B* γ + C* The incident particle a enters the nucleus A and suffers collisions with the constituent nucleons, until it has lost its incident energy, and becomes an indistinguishable part of the excited compound nucleus C* The compound nucleus ‘forgets’ how it was formed and its subsequent decay is independent of its formation: “Bohr’s Hypothesis of Independence” 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 242 Compound Nucleus Example Consider a beam of alpha particles of energy 5 MeV/A (or MeV per nucleon) impinging on 60Ni: + 60Ni 64Zn* At this (kinetic) energy, the incident particle is nonrelativistic, β = v/c = 0.1, and it will take the alpha particle ~10-22 s to travel across the target nucleus In a compound nucleus, the first emission of a nucleon or gamma ray takes > 10-20 s Hence the alpha particle traverses the compound nucleus hundreds of times and loses its identity ! 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 243 Geometric Cross-Section In the classical picture, the projectile and target nuclei will fuse if the impact parameter b is less than the sum of their radii A disk of area π(R1 + R2)2 is swept out This area defines the geometric cross-section Remember: units of cross-section are area (1 barn = 100 fm2; 1 fm = 10-15 m) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 244 Coulomb Excitation Coulomb Excitation (Coulex) is the excitation of a target nucleus by the long-range electromagnetic (EM) field of the projectile nucleus, or vice versa The biggest effect occurs for deformed nuclei with high Z: In these nuclei, rotational bands can be excited to more than 20 ħ In pure Coulex, the charge distributions of the two nuclei do not overlap at any time during the collision. 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 245 Coulex Example 234U bombarded by 5.3 MeV/A 208Pb The beam energy is kept low – below the Coulomb Barrier – so that other reactions, e.g. fusion, do not compete In this example: Beam energy = 5.3 x 208 MeV = 1.1 GeV The Coulomb Barrier (in the lab frame) is: { Z1Z2e2 / [4πε0(R1 + R2)] } x {(A1 + A2) / A2} C-o-M barrier 1/29/2015 = 1.3 GeV PHYS490 : Advanced Nuclear Physics : E.S. Paul 246 Intermediate Energy Coulex At higher beam energies (> 30 MeV/A), well above the Coulomb Barrier, Coulex can still take place but in competition with other violent reactions The process is now so fast that only the first excited states (2+ for even-even nuclei) are populated Intermediate energy Coulex is characterised by straight line trajectories with impact parameters larger than the sum of the radii of the two colliding nuclei 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 247 IE Coulex Example (GSI RISING Experiment) A gold target (179Au) bombarded by a 140 MeV/A radioactive 108Sn beam The beam energy is: 140 x 108 MeV = 15.1 GeV At this energy, β = v/c =0.48 – the projectile is travelling at half the speed of light ! Note: 108Sn is not stable – cannot make a target, but can generate a short-lived Radioactive Ion Beam (RIB) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 248 Intermediate Energy Coulex Ideally suited for use with fragmentation beams (Ebeam > 30 MeV/u) Large cross sections (~100 mb) Can use thick targets (~100 mg/cm2) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 249 Coulex Cross Sections For Intermediate Energy Heavy Ions, the Coulomb excitation cross section can be approximated as: σπλ = [Z1e2/ħc] B(πλ;0λ) [πR2/e2R2λ(λ-1)] for λ ≥ 2 Here Z1 is the charge of the projectile and R is the sum of the radii of target and projectile The cross section is peaked at forward angles within the angular range Δθ ≈ 2Z1Z2e2/RE where E is the beam energy 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 250 Neutron-Rich Sulphur Isotopes Energy spectra in target (top) and projectile (bottom) frames of reference for 40S + 197Au at MSU β = v/c = 27% H. Scheit et al. Phys. Rev. Lett. 77, 3967 (1996) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 251 Neutron Capture Low-energy neutron-capture cross-sections exhibit peaks or resonances corresponding to a compound system An example is the capture of a 1.46 eV neutron by 115In to form a highly excited state (6.8 MeV !) in 116In The high excitation energy in 116In arises due to the binding energy of the neutron 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 252 Neutron Capture Cross-Sections At 1.46 eV, the measured total cross-section for neutron capture by 115In is σ ≈ 2.8 x 104 barn However the geometrical cross-section (πR2) is only ≈ 1.1 barn Quantum effect: we need to consider the de Broglie wavelength (λ/2π) instead of the nuclear radius – slow neutrons have a large wavelength and hence a long-range influence The cross-section becomes: σ = πR2 σ = π(λ/2π)2 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 253 De Broglie Wavelength The momentum of the neutron is: pn = √{2mnE} = √{2 x 939 x 1.46 x 10-6} = 0.052 MeV/c The de Broglie wavelength is then: (λ/2π) = ħc/pnc = 197/0.052 = 3.7 x 103 fm The cross-section then becomes 4.3 x 105 barn The measured value is only 6% of this estimate ! We must also consider other effects such as the spins of the neutron, target and compound systems 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 254 Decay of 116In* n + 115In γ + 116In* 116In* 4% 96% For this neutron energy of only 1.46 eV Γn/Γγ = 0.04, also Γn/Γ ≈ 0.04 (Γ = Γn + Γγ) This decay fraction can be related to the formation cross-section: σ = π(λ/2π)2 x Γn/Γ (Γn/Γ ≈ 4%) Recall the measured formation cross-section was only 6% of the estimate using the de Broglie wavelength ! 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 255 Proton Capture For charged-particle capture (and decay) we must consider the Coulomb Barrier which inhibits the formation or decay of a compound system The proton needs sufficient energy to overcome the Coulomb Barrier (several MeV) and hence its de Broglie wavelength is smaller (than in the case of neutron capture Consequently, proton-capture cross-sections are ~ 1 barn at maximum 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 256 Heavy Ion Fusion Reactions For heavy projectile ions, e.g. 12C or 58Ni, the Coulomb Barrier is high and the particle enters a continuum of high level densities and overlapping resonances The excitation of the compound nucleus is also high: 10-80 MeV Since the neutron binding energy is only ~ 8 MeV, several neutrons are emitted before gamma-ray emission dominates These Heavy Ion Fusion Evaporation reactions bring large amounts of angular momentum into the compound system 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 257 Fusion Evaporation Reactions 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 258 David Campbell Florida State University 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 259 Fusion Cross-Section The angular momentum brought into the compound system depends on the impact parameter b: ℓ =bp The partial fusion crosssection is proportional to the angular momentum: d σfus(ℓ ) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul ℓ 260 Compound Formation And Decay 100Mo(36S,4n)132Ce Beam energy: 4.31 MeV/A 1/29/2015 Compound nucleus formation: 10-20 s Neutron emission: 10-19 s Statistical (cooling) dipole gamma-ray emission: 10-15 s Quadrupole (slowing down) gamma-ray emission: 10-12 s After 10-9 s the nuclear ground state is reached after 1011 rotations PHYS490 : Advanced Nuclear Physics : E.S. Paul 261 Cold Fusion Superheavy elements (SHE’s) can be formed by lowenergy fusion-evaporation reactions in which only one neutron is emitted 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 262 Transfer Reactions Transfer reactions occur within a timescale comparable with the transit time of the projectile across the nucleus Cross sections are a fraction of the nuclear area The de Broglie wavelength of a 20 MeV incident nucleon is 1 fm and it interacts with individual nucleons at the nuclear surface The projectile may lose a nucleon (stripping reaction) or gain a nucleon (pick up reaction) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 263 Transfer Reactions 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 264 Neutron-Induced Fission 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 265 Direct Reactions Proceed in a single step, timescale comparable to the time for the projectile to traverse the target (10-22 s) Usually only a few bodies involved in the reaction Excite simple degrees of freedom in nuclei Mostly surface dominated (peripheral collisions) Primarily used to study single-particle structure Examples: elastic and inelastic scattering 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 266 Elastic Scattering Both target and projectile remain in their ground state a+A a+A Nuclei can be treated as structureless particles 100 Example: Investigation of nuclear matter density distributions in exotic nuclei by elastic p-scattering (inverse kinematics) -3 rm(r) [fm ] 10-1 10-2 11 Li matter 10-3 9 Li -4 10 11 Li core 10-5 0 2 4 6 8 10 r [fm] 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 267 Inelastic Scattering Both target and projectile nuclei retain their integrity, they are only brought to bound excited states a + A a* + A* Can excite both single-particle or collective modes of excitation Example: investigate the GMR by (,’) inelastic scattering, gives access to nuclear incompressibility, key parameter of nuclear EOS Knm (Z,N) ~ r02 d2(E/A) / dr2 | r0 Example: safe and unsafe Coulomb excitation (below and above Coulomb barrier) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 268 Transfer Reactions One or a few nucleons are transferred between the projectile and target nuclei Probes single-particle orbitals to which nucleon(s) is (are) transferred Characteristics of the entrance channel determines selectivity of the reaction, i.e. alpha particle with T=0 leads to states with the same isospin as the ground state, but proton with T=1/2 leads to states with T=T±1 Examples numerous: (d,p) , (p,d) , (t,p) , (t,3He) … 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 269 Charge Exchange Reactions Reactions that exchange a proton for a neutron, or vice versa Net effect is the same as β+ or β- decay But not limited by Qβ – can reach higher excited states and giant resonances Many different probes: (p,n) , (d,2He) , (t,3He) , but also with heavy ions, e.g. (7Li,7Be) or exotic particles, e.g. (π+,π0) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 270 Knockout Reactions One or a few nucleons are ejected from either the target and/or the projectile nuclei, the rest of the nucleons being spectators Exit channel is a 3-body state Becomes dominant at high incident energies Populates single-hole states, from which spectroscopic information can be derived Examples: (p,2p) , (p,pn) , (e,e’p) , heavy-ion induced knockout, e.g. (9Be,X) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 271 Compound Nucleus Reactions The two nuclei coalesce, forming a fused system that lasts for a relatively long time (10-20 to 10-16 s) De-excitation follows by a combination of particle and/or gamma decay Compound system has no memory of entrance channel, the cross section of the exit channel is independent Occurs for central collisions around Coulomb barrier energies 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 272 Types of Nuclear Reaction 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 273 12. Nuclear Astrophysics Linking Femtophysics with the Cosmos 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 274 Evolution of the Universe Link to Nuclear Physics… Nuclear reactions are the only way to transmute one element into another Nuclear reactions account for ‘recent’ synthesis of elements in stars ‘Astrophysical’ origin of the elements 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 275 Nuclei Power Stars Stars are luminous, hot, massive, self-gravitating collections of nuclei (and electrons) To generate sufficient light via release of gravitational potential energy, a star would only live for ~107 years Stars must have an internal energy source to prevent gravitational collapse faster than their observed lifetimes ~108 – 109 years Chemical energy too small… Nuclear Fusion Reactions 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 276 Stellar Evolution Nuclear Reactions are responsible for both preserving and evolving the collection of nuclei Preserving: nuclear reactions generate energy which balances the self-gravitation of ~1030 kg star Evolving: nuclear reactions change the chemical composition and therefore the star’s inner structure and energy generation rate Stars are gravitationally confined thermonuclear reactors 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 277 Periodic Table Of Elements 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 278 Elemental Abundances Similar distribution everywhere Spans twelve orders of magnitude Hydrogen: 75% Helium: 23% C to U (‘metals’): 2% D, Li, B and Be underabundant Exponential decrease up to Fe A peak occurs near Fe Almost flat distribution beyond Fe 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 279 Elemental Signatures Galactic distribution of the 1809 keV gamma ray in 26Mg 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 280 Origin of the Elements Big Bang: 1H, 2H, 3He, 4He, 7Li (Z = 3) Thermonuclear fusion in a rapidly expanding mixture of protons and neutrons Interstellar Gas: Li, Be, B (Z = 5) Spallation and fusion reactions between cosmic rays and ambient nuclei Stars: Successive energy-releasing fusion or ‘burning’ of light elements Low (< 8 M): Li, C, N, F (Z = 9) Massive (> 8 M): Li, B, C, to Fe (Z = 26) (maximum BE) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 281 Big Bang Nucleosynthesis Big Bang Theory states that the Universe began 13.7 billion years ago in a hot and dense state After 1 s only protons, neutrons and lighter stable particles were present At this time there existed 1 neutron for every 6 protons For the next 5 minutes nuclear reactions occurred… For 1 proton: 0.08 4He, 10-5 2H, 10-5 3He, 10-10 7Li nuclei 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 282 Elemental Abundances: Timeline 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 283 Stellar Evolution Low-mass stars (< 2.3 M): Ignition of H, but He core becomes ‘degenerate’ before ignition Intermediate-mass stars (3 M < M < 8 M): Ignition of H, He, C, O white dwarf remnant High-mass (‘massive’) stars (M > 8 M): Ignition of H, … Si core collapse supernova 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 284 Massive Stars Stars are gravitationally confined thermonuclear reactors Each time one kind of ‘fuel’ runs out, contraction and heating ensue, unless degeneracy is encountered For a star over 8 M contraction and heating continue until an iron (Fe) core is made Gravitational collapse ensues, after no energy-providing fuel is left 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 285 Nuclear Burning Stages Massive Star Fuel Main Product H He He C, O C Ne O Si Ne, Mg O, Mg Si, S Fe 1/29/2015 Secondary Product 14N 18O, 22Ne s-process Na Al, P Cl, Ar, K, Ca Ti, V, Cr, Mn, Co, Ni Temp. (GK) Time (yr) 0.02 0.2 107 106 0.8 1.5 2.0 3.5 103 3 0.8 1 week PHYS490 : Advanced Nuclear Physics : E.S. Paul 286 Death Of A Star Heavier elements sink to the centre of the star Fusion of elements beyond Fe requires an input of energy Energy from nuclear reactions can no longer oppose gravitational collapse 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 287 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 288 Turning Hydrogen into Helium The fusion of four protons into helium is the only way to produce enough energy over the timescale of the Solar System. The main reaction is: 4 1H 4He + 2 e+ + 2 ν It is unlikely that 4 protons just happen to come together to form the He nucleus ! Instead the 4 protons are processed into the He via a series of simpler reactions: The ‘pp chain’ or the ‘CNO cycle’ 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 289 The Proton-Proton (pp) Chain 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 290 The pp Chain Reactions 1H + 1H 2H + e+ + ν 2H + 1H 3He + γ 3He + 3He 4He + 2 1H pp1, Q = 26.20 MeV 3He + 4He 7Be + γ 7Be + e- 7Li + ν 7Li + 1H 2 4He pp2, Q = 25.66 MeV 7Be + 1H 8B + γ 8B 8Be + e+ + ν 8Be 2 4He pp3, Q = 19.17 MeV 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 291 The pp Chain Reactions 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 292 Helium Burning: The Triple Chain 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 293 The Triple Chain To produce nuclei beyond 8Be, three alpha particles can combine to produce a 12C nucleus (Q = 7.275 MeV): + + 12C or 4He + 4He + 4He 12C Since the probability for a 3-body reaction is extremely low, the reaction is expected to take place in two steps (1) + + 8Be, Q = -0.092 MeV, but 8Be is unstable (τ ~ 10-16 s) and decays back into + (this explains the A = 8 mass gap) (2) + 8Be 12C, Q = 7.367 MeV 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 294 12C E(0+) Abundance: The Hoyle State = 7.654 MeV Triple alpha: Q = 7.275 MeV The triple alpha process does not account for the full abundance of 12C – the fourth most abundant element in the universe Hoyle (1954) predicted the existence of a resonant 0+ state in 12C at E ~ 7.7 MeV This was later confirmed in experiment ! 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 295 The CNO Cycle The CNO (carbon-nitrogen-oxygen) cycle converts H (hydrogen) into He (helium) by a sequence of reactions involving C, N and O isotopes and releasing energy in the process. It occurs in stars with masses › 1.5 M The main reaction scheme is: 12C(p,γ)13N(e+,ν)13C(p,γ)14N(p,γ)15O(e+,ν)15N(p,α)12C The net result is: 4 1H 4He + 2 e+ + 2 ν, Q = 26.73 MeV The cycle is limited by β decay of 13N (τ ~ 10 min) and 15O (τ ~ 2 min) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 296 CNO Reactions 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 297 CNO and PP Chain: Temperature Dependence 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 298 Breakout into the Hot CNO Cycle At higher temperatures, proton capture on 13N can begin to compete with the β decay and the cycle can break out into the hot CNO cycle 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 299 CNO and Hot CNO Cycles 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 300 Breakout of the Hot CNO Cycle 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 301 Nucleosynthesis At still higher stellar temperatures, reactions begin to compete that can break out of the hot CNO cycle and ignite a runaway sequence of nuclear burning: nucleosynthesis p reactions r (‘rapid’ neutron) rp (‘rapid’ proton) processes 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 302 p Reactions Starting with 14O: 14O + 17F + p 17F + p 18Ne 18Ne + 21Na + p etc Elements from oxygen (Z = 8) up to scandium (Z = 21) are produced Heavier elements cannot be formed in this manner since the Coulomb Barrier between the particle and the target nucleus becomes too large and prevents their fusion 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 303 The p Process 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 304 Creation of Elements Beyond Fe Fusion of elements up to Fe (Z=26) releases energy, the nuclear binding energy The nuclear binding energy is a maximum for Fe To produce elements heavier than Fe via nuclear fusion requires an input of energy – the binding energy decreases for heavy nuclei So, how are elements heavier than iron formed ? 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 305 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 306 Neutron and Proton Capture Neutron capture reactions are responsible for the production of elements above Fe The relative n-capture / β-decay efficiencies lead to two extreme cases: s-process (slow) and r-process (rapid) Nuclear structure details determine the r-process: connection between Astrophysics and Nuclear Physics Extreme and transient conditions near compact remnant stars can yield nuclei on the proton-rich side of the stability region: rp-process 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 307 Rapid Proton/Neutron Capture 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 308 Explosive Nucleosynthesis Evidence: Technetium (Tc: Z = 43) has no ‘stable’ isotopes but atomic Tc lines have been identified in red giants with strong lines of Y, Zr, Ba, La (Z = 57) Elements beyond Fe: Nuclear fusion is ruled out since the binding energy (B/A) is maximal at iron Neutron Capture: Can occur at ‘low’ temperatures but we need ‘high’ temperatures to activate sources of neutrons 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 309 Neutron Capture Reactions Stellar abundances of the elements imply two different processes: The s-process (slow): Low neutron flux, N(n) 0 (108 n/cm3) The r-process (rapid): High neutron flux, N(n) ∞ (1020 n/cm3) Rapid neutron capture (r-process), and also rapid proton capture (rp-process), produce exotic nuclei far away from the valley of stable nuclei 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 310 Neutron Capture Reactions Very small (n,γ) cross sections at N magic numbers Evidence for nuclear processes governing nucleosynthesis 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 311 Astrophysical Sites 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 312 Creation of Heavy Elements 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 313 Influence of Shell Structure on Elemental Abundances 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 314 Creation of Proton-Rich Nuclei The rp-process lasts 10-1000 s It is a series of radiative proton capture reactions and nuclear β+ decays that processes the lower mass nuclei into higher mass radioactive nuclei 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 315 Endpoint of rp Process 106,107Te very recently studied at Jyväskylä 1/29/2015 Small island of alpha decay just above proton-rich tin (Z=50) PHYS490 : Advanced Nuclear Physics : E.S. Paul 316 Astrophysical rp-process Sites Novae X-ray bursters Shock waves passing through the envelope of supernova progenitors Thorne-Zytkow objects, where a neutron star sinks to the centre of a supergiant 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 317 Creation of the Elements Movies X ray burster: The rp process converts hydrogen and helium into heavier elements up to tin (Z=50) Supernova explosion: The r process is responsible for the origin of about half the elements heavier than iron found in nature, including elements such as gold (Z=79) or uranium (Z=92) 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 318 Thermonuclear Energy Generation 4 1H 4He 6.7 MeV/u 3 4He 12C 0.6 MeV/u ‘triple ’ 5 4He + 84 1H 104Pd 6.9 MeV/u ‘rp process’ Gravitational potential energy: 200 MeV/u 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 319 Astrophysical Reactions The elemental abundances depend crucially on the reaction rates (cross-sections), i.e. proton/neutron capture vs. β decay These important cross-sections can now be measured using accelerated beams of radioactive beams An example is the 21Na + H 22Mg + γ reaction 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 320 Nuclear Reactions Nuclear reactions play a crucial role in the Universe They provide energy for life on Earth They produced all the elements we depend on 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 321 The Nuclear Landscape There are ~ 280 stable nuclei By studying reactions between them we have produced ~ 3000 more (unstable) nuclei, which have profoundly influenced many research areas: Big Bang, neutrino physics, diagnostic and therapeutic medicine, geophysics, archeology, climate studies etc There are ~ 4000 more nuclei which we know nothing about and which may hold many surprises. Their study will generate further practical applications of Nuclear Physics 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 322 The Role of Nuclear Physics 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 323 Cosmophysics: A New Field The fields of Cosmology and Astrophysics can be combined in two ways: 1. Cosmophysics 2. Astrology The End 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 324 13. Double Beta Decay 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 325 Introduction Double-beta decay is a rare transition between two nuclei with the same mass number A involving change of the nuclear charge Z by ±2 units Two beta decays occur simultaneously in a nucleus It is a rare second order weak interaction event The decay can only proceed if the initial nucleus is less bound than the final one, and both must be more bound than the intermediate nucleus These conditions are only fulfilled for even-even nuclei More than sixty naturally occurring isotopes are capable of undergoing double-beta decay (energetically) Ten such isotopes have been experimentally observed: 48Ca 76Ge 82Se 96Zr 100Mo 116Cd 128Te 130Te 150Nd 238U 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 326 Double Beta Decay Two-neutrino double beta decay ββ(2ν): (Z,A) (Z+2,A) + 2 electrons + 2 antineutrinos conserves not only electric charge but also lepton number Half-life (measured) ~1019 years Neutrinoless double beta decay ββ(0ν): (Z,A) (Z+2,A) + 2 electrons violates lepton number conservation and is forbidden in standard Electroweak Theory Half-life (predicted) ~1026 years 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 327 Massive Majorana Particles? Charged leptons (electrons, muons) are Dirac particles, distinct from their antiparticles (charge conjugation) Neutrinos may be the ultimate neutral particles, as envisioned by Majorana, identical with their antiparticles This fundamental distinction becomes important only for massive particles Neutrinoless double-beta decay proceeds only when neutrinos are massive Majorana particles Recent neutrino oscillation experiments suggest that neutrinos have a non-zero mass of the order 50 meV The Standard Electroweak Model postulates that neutrinos are massless and lepton number is conserved New Physics??? 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 328 The Signal of ββ(0ν) Decay 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 329 Neutrino Mass Upper limits of neutrino mass are shown to the left from two-neutrino double-beta decay measurements Neutrino-oscillation experiments suggest a mass scale of the order 50 meV 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 330 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 331 Experiment 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 332 Theory 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 333 Nuclear Chocolate 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 334 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 335 The Nuclear Many-Body Problem: Energy, Distance, Complexity few body heavy nuclei quarks gluons vacuum quark-gluon soup QCD 1/29/2015 nucleon QCD few body systems free NN force PHYS490 : Advanced Nuclear Physics : E.S. Paul many body systems effective NN force 336 Life, The Universe & Everything 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 337 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 338 Marielle Slides… 1/29/2015 PHYS490 : Advanced Nuclear Physics : E.S. Paul 339 Isospin Dependence of Mean Field and Residual Interactions Shell structure predicted to change in exotic nuclei (particularly neutron rich). Weak binding, impact of the particle continuum, collective skin modes and clustering in the skin… Mean field modifications surface composed of diffuse neutron matter derivative of mean field potential weaker and spin-orbit interaction reduced Residual interaction modifications partly occupied orbits V monopole interaction: coupling of proton-neutron spin-orbit partners deformed intruder configurations Quenching of ‘old’ shells and emergence of new magic numbers in exotic neutron-rich nuclei Gap = S (N)-S (N+2) [keV] Neutron Shell Gap 2N 2N GAPN = M(Z,N-2) - M(Z,N) + M(Z,N+2) - M(Z,N) N = S2N(N) - S2N(N+2) Neutron shell Gap 1 104 Ca (Z=20) 8000 6000 4000 2000 0 16 20 18 20 28 22 24 26 28 30 Neutron Number 8000 F-GAP Ne-GAP Na-GAP Mg-GAP 5000 4000 20 N N GAP (keV) 6000 3000 2000 0 12 14 6000 4000 2000 16 1000 8000 GAP (keV) 7000 Ca-GAP K-GAP Cl-GAP S-GAP Si-GAP 20 16 18 Neutron Number 20 22 0 18 20 28 22 24 Neutron Number 26 28 Single-Neutron Removal in the p-sd shell ns1/2 ns1/2 intruder p-shell sd-shell E.Sauvan et al., Phys. Lett. B 491 (2000) 1, Phys. Rev. C 69 (2004) 044603. New Magic Number at N=16 V monopole interaction : coupling of proton-neutron spin-orbit partners T. Otsuka et al. Phys. Rev. Lett. 87 (2001) 082502. Examples of experimental evidence: Two-neutron separation energies In-beam fragmentation gamma spectroscopy 1n-removal cross-sections and longitudinal momentum distributions (direct reactions) Present in stable nuclei but missing in n-rich nuclei where the spinorbit partner of the valence neutrons are not occupied by protons excitation energy (MeV) Systematics of the 3/2+ in the N=15 isotones 23O 4.5 27Mg 25Ne 4.0 1f7/2 3.5 3.0 2.5 2.0 1.5 1d5/2 1.0 1d3/2 0.5 0.0 8 6 10 12 2s1/2 atomic number The energy of the 1d3/2 neutron orbital rises when protons are removed from its spin-orbit partner, the 1d5/2 orbital. 23O 25Ne 10 8 1d3/2 2s1/2 1d5/2 1s, 1p 1s, 1p p n 27Mg 1d3/2 2s1/2 1d5/2 1s, 1p 1s, 1p p n 12 1d3/2 2s1/2 1d5/2 1s, 1p 1s, 1p p n Transfer Reaction Example Modification of residual interactions at N=28 46Ar(d,p)47Ar at 10.7 A.MeV in inverse kinematics N=28 gap : 4.47(8)MeV Excitation energy spectrum for 47Ar p3/2 p1/2 f5/2 47Ar f7/2 MUST at GANIL/SPIRAL L. Gaudefroy et al, PRL 97, 092501 (2006). Knockout Reactions Example Systematics of (e,ep) on Stable Nuclei Departures of measured spectroscopic factors from the independent single-particle model predictions Electron induced proton knockout reactions: [A,Z] (e,ep) [A-1,Z-1] See only 60-70% of nucleons expected! Effect of long-range and short-range correlations similar proton separation energies W. Dickhoff and C. Barbieri, Prog. Nucl. 377. Part. Sci., 52 (2004) High-Energy Single-Nucleon Removal A New Spectroscopic Tool core+1N p0 core g d/dp dominant =2 = 2 and =0 mixture Target g Excore -1n(Jpcore) =2 d/dp n C2S REVIEW:Hansen & Tostevin, Ann. Rev. Nucl. Part. Sci. (2003)

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