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Nuclear Physics

Nuclear Physics
PHYS490: Nuclear Physics 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 1 PHYS490: Schedule 2017  Lectures : Thursday 11:00 – 13:00 (CHEMBRUN) Friday 12:00 – 13:00 (MATH106)  Lectures : weeks 1 6  Tutorials : weeks 3, 6  Text book : K.S. Krane (or LaTeX notes on VITAL)  Eddie Paul (E.S.Paulliv.ac.uk) Oliver Lodge Lab. 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 2 PHYS490: Nuclear Physics 1. NucleonNucleon Force 2. Nuclear Behaviour 3. Forms of Mean Potential 4. Nuclear Deformation 5. Hybrid Models 6. Nuclear Excitations 7. Rotating Systems 8. Nuclei at Extremes of Spin 9. Nuclei at Extremes of Isospin 10. Mesoscopic Systems 11. Nuclear Reactions 12. Nuclear Astrophysics 13. Neutrinoless Double Beta Decay 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 3 0. A Brief Introduction 2/24/2017 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 5 Atomic and Nuclear Sizes 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 6 More and More Isotopes Z Segre Chart N 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 7 Discovery History  Today around 3000 isotopes have been observed  Only 284 are stable 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 8 Limits of Stable Nuclei 2/24/2017 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 singleparticle (outofphase) and collective (inphase) 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 2/24/2017 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 manybody 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 11 Nuclear Physics in the Thirties: Splitting of the Atom CockcroftWilson accelerator – atom ‘split’ in 1932 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 12 Nuclear Physics in the Forties The first cyclotrons were built in Berkeley, California 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 13 Oliver Lodge Lab. Opening (1969) 2/24/2017 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 15 Nuclear Physics in the Eighties TESSA3: 16 (small) γray detectors at Daresbury, UK 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 16 Even Bigger Arrays  This picture shows ESSA30, an array of 30 (small) γray detectors at Daresbury, UK  It was a European collaboration  Again, spot the Liverpudlians 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 17 Nuclear Physics in the Nineties Gammasphere: 100 (big) γray detectors, USA 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 18 Nuclear Physics in the Noughties 2003: The Hulk destroys Gammasphere 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 19 Nuclear Physics Tomorrow The next generation of Radioactive Ion Beam (RIB) accelerators in Europe 2/24/2017 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 subatomic 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 21 1. NucleonNucleon Force 2/24/2017 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 2/24/2017 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 quark sea + 3 valence quarks 2/24/2017 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 25 Levels of Reality 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 26 Fundamental Forces 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 27 Fundamental Particles Forces  Quarks (fermions): Down (d) Up (u) Strange (s) Charmed (c) Bottom (b) Top (t)  Force Mediators (bosons): Photon (γ) Gluon (g) Z particle W particle 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 28 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 29 Properties of the NN Force  The force is spin dependent  The force is charge symmetric  The force is (nearly) charge independent  The force has a noncentral 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 Phys490latex.pdf for more details 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 30 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 31 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  Nucleonnucleon force dependent on both spin and isospin 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 32 OnePion Exchange Potential  At large distances the potential is constructed as arising from the exchange of one pion: OPEP  The form of the potential is: 2 2 V = g (1/3 σ .σ + S 1/3 + 1/μr + 1/(μr) ) OPEP s A B AB 2 μr x τ .τ 1/r μ e A B where: 2 μ = m c/ħ and: S = 3(σ .r)( σ .r)/r σ .σ π AB A B A B 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 33 Addition of (Iso)Spins  Spin σ and isospin τ are vectors  Cosine rule gives: 2 2 2 (σ + σ ) = σ + σ + 2 σ .σ A B A B A B  Parallel spins (triplet state): σ .σ = 1 A B  Antiparallel spins (singlet state): σ .σ = 3 A B  We need to know σ .σ in the description of the A B nucleonnucleon (NN) force 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 34 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) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 35 Calculations for Light Nuclei  In addition to twobody NN interactions, threebody NNN interactions must also be included in the theoretical description of light nuclei 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 36 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 37 The Deuteron  The deuteron consists of a bound proton neutron system  Its groundstate is the only state which is bound; the first excited state is unbound  The ground state has π + spin and parity I = 1  The deuteron is not a spherical nucleus 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 38 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 2 a time t ≈ ħ/mc 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 39 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 40 Hypernuclei Nuclei including excited nucleons including heavy quarks: e.g. Lambda particle Λ 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 41 2. Nuclear Behaviour 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 42 Mirror Nuclei  The force between two nucleons has the property of charge symmetry and charge independence 20  The two nuclei Na 20 and F are examples of mirror nuclei  The numbers of protons and neutrons are exchanged 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 43 Isospin Substates  By analogy with spin, an isospin T state has (2T+1) substates  The substates correspond to states in different nuclei 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 44 Isobaric Analogue States  Isodoublet states occur in oddA nuclei  Isotriplet states occur in evenA (eveneven and oddodd) nuclei 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 45 A=51 Mirror Nuclei 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 46 Mirror Nuclei: f Shell 7/2 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 47 Independent Particle Model  In principle, if the form of V ij the nucleonnucleon potential is known for bare nucleons, then the energy of a nucleon moving inside a nucleus can be calculated  This is a very difficult r = r r i j problem to solve as the nucleon interacts simultaneously with all the other nucleons Energy as a function  Use an average potential of separation 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 48 Independent Particle Model  The Hamiltonian is of the form: H = Σ (T + V ) i ij  It has 3A degrees of freedom and is too complicated to solve except for the lightest nuclei (A 12)  Instead we use an average “meanfield” potential: H = H + H mean field residual where H contains interactions between nucleons residual that are not accounted for by the average potential, especially interactions among valence nucleons 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 49 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 longer, then nuclei could have been crystalline 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 50 Particles in a (Potential) Box  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 Energy levels up to the (PEP) ‘Fermi level’ 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 51 Degenerate Fermi Gas Model  This is a simple model in which nucleons are placed in a 3 volume V = 4πR /3 and the interactions between them are ignored  A Fermi sea is formed, filled up to the energy corresponding to the Fermi momentum: 2 2 2 E = p /2m = ħ k /2m F F F  The binding energy per nucleon is: B = E/A = 3/5 T + 1/2 V F 0 where T is the kinetic energy at the Fermi surface F 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 52 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 E : S = B = 1/5 T F F  Wrong (S 0) – the nucleon has an effective mass (m m ) when moving in a nucleus n 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 53 Some Nuclear Quantities  Number density (A/V) measured: 3 18 3 ρ 0.17 fm ( 1.5 x 10 kg/m )  Fermi momentum: 1 k = p /ħ 1.4 fm F F  Fermi energy: E 10 MeV F  Kinetic energy of a nucleon in the nucleus: 3/5E 6 MeV F corresponding to a velocity v/c 0.14 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 54 Nuclear Potentials There are two approaches: 1. An empirical form of the potential is assumed, e.g. square well, harmonic oscillator, WoodsSaxon 2. The mean field is generated selfconsistently from the nucleonnucleon interaction 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 55 3. Forms of Mean Potential 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 56 Shell Model – Mean Field A nucleon in the N nucleons in Mean Field of a nucleus N1 nucleons  Assumption – ignore detailed twobody interactions  Each particle moves in a state independent of the other particles  The Mean Field is the average smoothedout interaction with all the other particles  An individual nucleon only experiences a central force 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 57 Square Well Potential  Simplest form of potential  Since we have a spherically symmetric potential we can separate the solutions into angular and radial parts  Radial solutions are Bessel functions which satisfy the boundary condition R (R) = 0 nℓ  The eigenenergies are labelled Infinite square by n and ℓ: well potential 2 2 2 E = (ħ /2mR )ξ U nℓ nℓ 0 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 58 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 59 Properties of the Solutions  Higher n : higher energy (more kinetic energy)  Higher ℓ : higher energy (larger radius, less bound)  The lowest state is : 1s (n = 1, ℓ = 0) explains 1/2 ground state of the deuteron: L = ℓ + ℓ = 0 1 2  Note that two orbits can have similar energies if one has larger n and smaller ℓ, or vice versa 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 60 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 61 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 62 Harmonic Oscillator potential  Easy to handle analytically  Form of potential: 2 2 V (r) = U + ½mr ω HO 0  Solutions are Laguerre polynomials  Eigenenergies may again be Simple harmonic labelled by n and ℓ : oscillator potential E = (2n + ℓ + ½) ħω – U nℓ 0 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 63 Harmonic Oscillator potential  Eigenenergies can also be labelled by the oscillator quantum number N: E = (N + 3/2) ħω – U N 0  For each N there are degenerate levels with n and ℓ that satisfy: 2(n1) + 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 N ℓ  The parity of each oscillator shell is: (1) = (1) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 64 WoodsSaxon Potential  Usually finite potential forms are used such that V(r)  0 if r » 0  The WoodsSaxon potential is considered to be the most realistic nuclear potential  For protons a Coulomb potential V (r) is added C 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 65 (Wrong) Magic Numbers 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 66 SpinOrbit Coupling  In order to account for the correct nucleon numbers at which the higher shell closures occur, a spinorbit term is added – Mayer, Haxel, Jensen, Suess (1948)  For the modified harmonic oscillator: 2 2 2 V (r) = U + ½mr ω – 2/ħ αℓ.s HO 0  Since: 2 ℓ.s = ½ħ 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) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 67 Predicted Shell Structure  The harmonic oscillator shells are shown to the left in this diagram 2  In the middle, an ℓ 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 68 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 69 Systematics Near Z(N) = 50 N = 50 Z = 50 100 132  Sn (Z=N=50) and Sn (N=82) are doubly magic nuclei 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 70 Neutron Separation Energies 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 71 Residual Interactions  The residual interaction ν between nucleons is the difference between the actual twonucleon potential V experienced by a nucleon in a state α and the α average potential  Matrix elements of ν, ανβ are only appreciable near the Fermi Surface  The interaction ν is a twobody operator because it changes the state of two nucleons. It can be treated in a number of ways: 1. from the free twonucleon potential (difficult) 2. as a free parameter (fit to experimental data) 3. parameterised using physical intuition 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 72 Quadrupole + Pairing Interaction  If we assume that the interaction takes place near the Fermi surface, i.e. near r = R then V (r ,r )  V (R) ℓ i j ℓ  The quadrupolequadrupole (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 Monopole pairing maximum spatial overlap π + I = 0 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 73 Hartree Fock Method  The philosophy here is that the nuclear potential is selfconsistent 1. We calculate the nucleon distribution (density) from the net potential 2. Then we evaluate the net potential from the nucleonnucleon interaction 3. Then we iterate 4. The potential is selfconsistent if the one with which we end up is the same as the one we start with 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 74 4. Nuclear Deformation 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 75 Evidence for Deformation 1. Large electric quadrupole moments Q 0 2. Lowlying rotational bands ( E  II+1 )  The origin of deformation lies in the long range component of the nucleonnucleon residual interaction: a quadrupolequadrupole 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 76 Simple Nuclear Shapes  The general shape of a nucleus can be expressed in terms of spherical harmonics Y (θ,φ) λμ  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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 77 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 78 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  ‘Prolate’ (rugby ball): β 0 laboratory frame axes  ‘Oblate’ (smartie): β 0 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 79 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 Triaxial shapes : x ≠ y ≠ z x = z y γ ≠ n 60° 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 80 Theoretical Deformations 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 81 Shape Coexistence 3 1 2 184  The nucleus Pb has three lowlying + 0 states 1. Spherical 2. Oblate 3. Prolate  This plot shows the calculated ‘potential energy surface’ 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 82 Deformation Systematics Theory Doubly Magic: Spherical Midshell: Deformed Oblate ‘Spherical’ Prolate Neutron Number N 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 83 Proton Number Z + First Excited 2 Energies + 1 E(2 )  Moment of Inertia Neutron Number N 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 84 Proton Number Z Deformation: Rotational Bands Neutron Number N 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 85 Proton Number Z Nilsson Model  In order to introduce nuclear deformation Nilsson modified the harmonic oscillator potential to become anisotropic: 2 2 2 2 2 2 V = ½mω x + ω y + ω z 1 2 3 with ω R = ω R and ω = ω ≠ ω k k 0 0 1 2 3  If axial symmetry is assumed (γ = 0) then the deformation is described by the parameter ε: ε = (ω – ω ) / ω 1,2 3 0  It can be shown that ε β 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 86 Nilsson Diagram (Energy vs. ε)  In order to reproduce the observed nuclear 2 behaviour Cℓ.s and Dℓ terms need to be added (C and D are constants)  The ℓ.s term is the spin orbit term 2  The ℓ term has the effect of flattening the potential to make it more realistic (like the shape of the Woods Saxon potential) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 87 Nilsson SingleParticle Diagrams N Z 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 88 Nilsson Labels  The energy levels are labelled by the asymptotic quantum numbers: π Ω N n Λ 3  ‘N’: N = n + n + n is the oscillator quantum number 1 2 3  ‘n ’: n is the zaxis (symmetry axis) component of N 3 3  ‘Λ’: Λ = ℓ is the projection of ℓ onto the zaxis z  ‘Ω’: Ω = Λ + Σ is the projection of j = ℓ + s onto the zaxis N ℓ  ‘π’: π = (1) = (1) is the parity of the state 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 89 The Λ, Σ, Ω Quantum Numbers  Spin projections: Ω = Λ + Σ = Λ ± ½ 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 90 Asymptotic Quantum Numbers 2  Because of the additional ℓ.s and ℓ terms the physical quantities labelled by n and Λ are not constants of 3 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 91 Large Deformations  This figure ignores the ℓ.s 2 and ℓ terms  Deformed shell gaps emerge when ω and ω are in the 3 1,2 ratio of small integers, i.e. ω / ω = p/q 3 1,2  A superdeformed shape has p/q = ½ or R :R = 2:1 3 1,2  A hyperdeformed shape has R :R = 3:1 3 1,2 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 92 5. Hybrid Models 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 93 Deformed Liquid Drop  Assuming that the nucleus behaves as a charged liquid drop, a semiempirical expression can be obtained for the total energy: 2/3 2 1/3 E(A,Z) = a A + a A + a Z A V S C  To correct for deformation the nuclear radius R is 0 replaced by: R = R (1 + δ) , R = R (1 ½δ) 3 0 1,2 0  The energy for small δ then becomes: 2/3 2 2 1/3 2 E(A,Z) = a A + a A (1 + 2/5 δ ) + a Z A (1 – 1/5 δ ) V S C 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 94 DeformedSpherical Energies  It is then predicted that the nucleus is always spherical (i.e. ΔE = 0 for 2 δ 0) unless Z /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 ΔE(δ) = E(δ) – E(δ=0) account shellmodel effects, i.e. effects from individual nucleon motion 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 95 Shell Correction  Additional terms arising from the symmetry energy (which prefers N = Z) and the pairing energy (Δ, 0, Δ for eveneven, oddeven and oddodd nuclei, respectively) can be added  Alternatively the total energy can be calculated using meanfield potentials  This is not simply the sum of the individual eigenenergies e because the potential energy of each i nucleon would be counted twice 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 96 Shell Energy  The eigenvalue for each nucleon is: e = T  +  V  i i j≠i ij  The total energy is: T  + ½ V  = ½e + ½T  i j≠i ij i i  For the harmonic oscillator potential: T  = V  =  V so that E = ¾e i i j≠i ij i  This method has difficulty in producing the correct energy because errors in e give rise to large errors in i the summation e i 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 97 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 g (e) =g (e) F AV density g(e) in the two (b) and (c) show shell effects. models A change in nuclear binding  And calculated the arises from: ‘fluctuation’ energy g (e) – g (e) AV F 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 98 Level Density 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 99 Shell Correction Energies 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 100 Fission Isomers  If the increase in liquid drop energy for increasing deformation ΔE(δ) is small 2 enough (e.g. Z /A 35) then any secondary minimum in the total energy arising from the shell correction will become similar in energy to the Superdeformed band first head is isomeric.  This second minimum Its decay can penetrate corresponds to a barrier either way superdeformed nuclear state 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 101 240 Superdeformed Pu 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 102 6. Nuclear Excitations Singleparticle and collective motion 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 103 Collective Motion in Nuclei  Adiabatic approximation: identify fast and slow degrees of freedom 2  Molecules: electronic motion fastest, vibrations 10 6 times slower, rotations 10 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 singleparticle modes can perhaps be separated but they will interact strongly 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 104 Types of Nuclear Excitation π +  All eveneven nuclei have a ground state with I = 0 , a consequence of nuclear pairing  Closedshell nuclei are spherical and excited nuclear states can only be formed by breaking pairs of nucleons or by vibrations  For oddmass nuclei (near closed shells) the lowlying excited states map out the singleparticle spectrum of states around the Fermi level  ‘Deformed’ nuclei exhibit regular rotational bands: quadrupole or octupole shapes etc… 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 105 Excitations in Spherical Nuclei  All eveneven nuclei have π + I = 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 106 Noncollective Level Scheme  Complicated set of energy levels  No regular features, e.g. band structures  Some states are isomeric 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 107 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 108 Beta (Y ) Vibration 20 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 109 Gamma (Y ) Vibration 22 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 110 Octupole (Y ) Vibration 30 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 111 Octupole (Y ) Vibration 31 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 112 Octupole (Y ) Vibration 32 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 113 Octupole (Y ) Vibration 33 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 114 Realistic Vibrational Levels  For each given mode of nucleus vibrator vibration, each phonon has an associated angular n=3 momentum and parity, e.g: n=2 + quadrupole 2 n=1 octupole 3 n=0  For a pure vibrator there are groups of degenerate levels for two or more phonons 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 115 Multiphonon Vibrational States N = 3 (3 phonon) N = 2 (2 phonon) N = 1 (1 phonon) 124 Sn, spherical N = 0 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 116 Giant Resonances Monopole L = 0 Isovector Isoscalar Dipole L = 1 Quadrupole L = 2 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 117 Rotations of a Deformed System  The lowlying levels of deformed eveneven 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: K 2 2 2 2 H = (ħ /2) R = (ħ /2) (IJ) rot where  is the moment of inertia Nuclear spins and J is additional angular momentum generated by, e.g. the odd particle in an oddA nucleus or by vibrations 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 118 Coriolis Coupling  Note that rotation cannot take place about the symmetry (z) axis  The rotational Hamiltonian can be expanded: 2 2 2 2 R = (I – J) = I – 2I.J + J 2 2 2 = I + J – 2K (I J + I J ) + + where I = I ± i I , J = J ± i J and J = I = ±K ± x y ± x y z z  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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 119 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: 2 2 E = (ħ /2)I(I+1) + J(J+1) 2K rot with I = K, K+1, K+2… and K is a constant of the motion 2  Then: E = E + (ħ /2)I(I+1) rot K where E is the energy of the lowest band level K 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 120 232 Vibrational Bands in Th  Here the lowlying levels are all collective, i.e. rotational and vibrational π +  Ground State Band: K = 0 π +  β Band: K = 0 π +  γ Band: K = 2 π  Octupole Band: K = 0 π + Oct.  Note that if K = 0 then Beta Gamma the I values 1, 3, 5… are not present GSB π  For K = 0 the I values 0, Reflection symmetric shape, 2, 4… disappear 232 Th is a deformed nucleus 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 121 226 Alternating Parity Bands in U  This nucleus is reflection asymmetric (i.e. β ≠ 0) in its 3 ground state: it has octupole deformation  The nuclear wavefunction in its intrinsic frame is not an eigenvalue of parity: 2 2 Ψ (x ,y ,z) ≠ Ψ (x, y, z)  In the laboratory frame (i.e. averaged over all nuclear orientations) the levels have alternating parity 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 122 Reflection (A)symmetry 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 123 Electric Dipole Moment  In a nucleus with octupole deformation, the centre of mass and centre of charge tend to separate, creating a nonzero electric dipole moment 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 124 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 125 157 Rotational Bands in Ho  This nucleus shows three band structures built on different Nilsson states 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 126 7. Rotating Systems 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 127 Moment of Inertia  The energy of a rotating nucleus is given by: 2 E = (ħ /2) II+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) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 128 Rotational Bands: γray Spectra 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 129 Inglis Moment of Inertia  Inglis (1952) showed that the moment of inertia of a Fermi gas rotating about the xaxis is:  = 2 pÎ h/(e e ) x x p h where the summation is over all possible 1particle 1hole excitations in a deformed shell model  The rigidbody moment of inertia is: 2  = (2/5) m AR 1 + 0.3β n 0 which is higher than observed 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 130 Nuclear Moments of Inertia  Nuclear moments of inertia are lower than the rigid body value – a consequence of nuclear pairing 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 131 Pairing Gap  A rough estimate of the energy required to create a particlehole excitation is 2Δ, where Δ is the pairing gap  A typical value for Δ is 1 MeV 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 132 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 ωÎ x ω  The Hamiltonian is: H = H – ωÎ DSM x ω  Energy in the rotating frame: E = E ω Î  x 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 133 Routhian and Aligned Angular Momentum  The Routhian is simply the energy in the rotating frame of reference: ω E  The aligned angular momentum is just the expectation value of the operator Î : x Î  x 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 134 Spin and Rotational Frequency  There are two important relations which arise since E ω is independent of ω and E is independent of I: ω dE /dω = Î  x and dE/dI = ω dÎ /dI x 2 Nuclear spin I and its  Since: Î  = √I(I+1)K ħ x projections onto the then for K = 0: Î  I ħ x rotation axis I and x and hence: deformation axis K dE/dI ω ħ 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 135 Gamma Ray Energies and Rotational Frequency  The energy of a rotational band for K = 0 is: 2 E = E + (ħ /2) I(I + 1) , I = 0, 2, 4… 0  The energy difference between consecutive levels ΔE represents the gammaray energy Eγ  The spin difference between consecutive levels is Δ I = 2  The rotational frequency ω is defined as: ωħ = dE/dI = ΔE/ΔI = E /2 γ i.e. the frequency is just half the gammaray energy 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 136 Moments of Inertia  The energy of a rotational band for K = 0 is: 2 E = E + (ħ /2) I(I + 1) , I = 0, 2, 4… 0  Then: 2 dE/dI = (ħ /2) (2I + 1) and: 2 2 2 d E/dI = ħ / defines the ‘dynamic moment of inertia’ which is independent of spin  By using finite differences: 2 2 2 dE=ΔE=E , dI=ΔI=2, d E=ΔE , d I=Δ I=4 γγ we can evaluate  even if we do not know I 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 137 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): 2  =  δ hydro rig  This estimate is much too low  We require shortrange pairing correlations to account for the experimental values 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 138 Kinematic and Dynamic MoI’s  Assuming maximum alignment on the xaxis (I I), the x kinematic moment of inertia is defined: (1) 2 1  = (ħ I) dE(I)/dI = ħ I/ω  The dynamic moment of inertia (response of the system to a force) is: (2) 2 2 2 1  = (ħ ) d E(I)/dI = ħ dI/dω (1) (2) Rigid body:  =  (2) (1) (1)  And  =  + ω d /dω (1) (2) High spin:  ≈  2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 139 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 158 ( Er)  A more gradual increase is 174 called an upbend ( Hf) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 140 Crossing Bands  A backbend corresponds to the crossing of two bands (‘g’ and ‘s’ configurations)  The states we observe are called yrast states (thick yrare line) which have the lowest energy for a given spin yrast  The sband, where s stands for ‘Stockholm’ or ‘super’, arises from the breaking of a Yrast and yrare states: pair of nucleons. Their dizziest and dizzier in angular momenta j and j 1 2 the Swedish language align with the rotation axis 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 141 Pair Breaking  For the ground state band: 2 E = (ħ /2 ) I(I + 1) g g  For the sband: 2 2 E = (ħ /2 ) (I – J ) + E s s J where J = j + j and E is the 1 2 J energy required to break a pair of nucleons: 1/2 E 2Δ 24 A MeV J  The aligned angular momentum of the sband increases by approximately: 158 j + j – 1 ( 12ħ for Er) 1 2 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 142 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 143 Pair Breaking and Rotational Alignment A Backbending movie follows showing pair breaking and rotational alignment 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 144 Backbending Movie 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 145 Backbending Demonstration This movie shows Mark Riley’s “backbending machine” 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 146 8. Nuclei at Extremes of Spin 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 147 HighSpin 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 148 Generation of Angular Momentum  There are two basic ways of generating highspin states in a nucleus 1. Collective (inphase) motions of the nucleons: vibrations, rotations etc 2. Singleparticle effects: pair breaking, particle hole excitations. The individual spins of a few nucleons j generate the i total nuclear spin 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 149 High I Bands x  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 xaxis, i.e. I = j + R x x  The quantity I is approximately a good quantum number x and hence a given nuclear state can be described by a single value of I x  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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 150 Aligned Particles  Alignment effects should be prominent for nuclei with a few nucleons outside a 158 closed shell, e.g. Er 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  Eventually all the particles outside the closed shell (spherical) core align I = Σj i R = 0  These move in equatorial orbits giving the nucleus an oblate appearance 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 151 Band Termination neutron backbend proton backbend Gamma Ray Energy 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 152 158 Band Termination in Er  When we align the np protons and nn neutrons outside the closed shell the total spin is: np nn I =  j (p) +  j (n) i i i i and the rotational band is said to ‘terminate’ 158  At termination Er can be thought of as a spherical 146 Gd core plus 4 protons and 8 neutrons generating a total spin 46ħ  The configuration is: 4 2 3 3 π(h )  ν(i ) (h ) (f ) 11/2 13/2 9/2 7/2  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) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 153 High K (I ) Bands z  If we have many unpaired nucleons outside the closed shell then alignment with the xaxis becomes difficult because the valence nucleons lie closer to the zaxis, 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 = I = Σj = ΣΩ z z 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 154 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. π 178  For example, the K = 8 band head in Hf is isomeric with a lifetime of 4 s. This is much longer than the lifetimes of the rotational states built on it. π  The K = 8 band head is formed by breaking a pair of protons and placing them in the ‘Nilsson configurations’: Ω N n Λ = 7/2 4 0 4 and 9/2 5 1 4 3 N(1) N(2)  In this case: K = 7/2 + 9/2 = 8 and π = (1) .(1) = 1 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 155 178 K Isomers in Hf  A low lying state with spin I = 16 and K = 16 in 178 Hf 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 156 174 High K bands in Hf  This nucleus has 347 known levels and 516 gamma rays 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 157 Superdeformation  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 liquiddrop contribution to the total nuclear energy includes the rotational energy, which can be substantially reduced at high spin by increasing the moment Nuclear potential at of inertia low and high spin  At sufficiently high spin a secondary minimum can become energetically favourable 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 158 152 Superdeformed Band in Dy  The experimental signature of these superdeformed (SD) shapes is a very regular sequence of equally spaced γ rays 152  In Dy the (first) SD band spans a spin range 20 – 60 ħ  Nowadays multiple SD bands are known in this and other nuclei β 0.6, 2:1 axis ratio 2 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 159 Some Big Numbers 152  The SD band of Dy emits 20 gamma rays in 13 10 s. The total energy released is: 19 E 20 MeV (1 eV = 1.6 x 10 J) γ 12 13  The power is: (3.2 x 10 J) / (10 s) = 32 W  The rotational frequency is: ħω 500 keV, so 20 20 7 ω 8 x 10 radians/sec  10 Hz or 10 rotations in 13 10 s  same as number of days in 30,000 years  The decay of the SD band passes through a longlived 12 isomeric level (86 ns)  5 x 10 rotations  same as number of days since the Big Bang 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 160 132 Superdeformation in Ce  SD bands exist in cerium (Z = 58) nuclei with a major/minor 132 axis ratio of 3:2. This band in Ce (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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 161 Superdeformed Systematics 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 162 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 152 in Dy along with the superdeformed shape  Each shape has a (local) ‘minimum’ in the nuclear ‘total energy surface’ 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 163 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 164 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 130150 can occurs for macroscopic accommodate the most spin objects 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 165 Jacobi Shape 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 166 9. Nuclei at Extremes of Isospin 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 167 Limits of Nuclear Existence Segre Chart Known Nuclei Stable Nuclei Proton Dripline Fission Limit Terra Incognita Neutron Dripline 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 168 Where Are The Driplines 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 169 Where Are The Driplines  In this experiment – fragmentation of 48 a beam of Ca – no counts were 26 observed for O  This defines the neutron dripline for oxygen isotopes 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 170 Where is the Neutron Dripline 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 171 Heavy N = Z Nuclei  Shell corrections give minima in the nuclear energy at nonzero values of deformation  Bigger effect if both proton and neutrons occur at these ‘magic numbers’  Also a big effect for N = Z 80  The N = Z = 40 nucleus Zr is an example  It is difficult to study this nucleus: it is 10 neutrons lighter than the lightest stable zirconium isotope 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 172 Exotic Nuclei 12  The nucleus C has six protons and six neutrons  It is stable and found in nature 22  The nucleus C 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 173 Isospin: T = (N Z) / 2 z A A = 21 21 C 6 15 N Z 21 21 21 21 21 21 21 21 C N O F Ne Na Mg Al 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 T Z Neutron rich Proton rich 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 174 Nuclei Far From Stability 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 175 ProtonRich Nuclei  The proton dripline is defined by the least massive bound nucleus of every isotopic chain (S drops to zero) p  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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 176 Radioactivity: Normal 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 177 Radioactivity: Exotic 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 178 Alpha and Proton Emitters 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 179 Jyväskylä, Finland (Feb 2006) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 180 Jyväskylä (midnight June 21 2009) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 181 Recoil Decay Tagging 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 182 JUROGAM + RITU + GREAT 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 183 Deformed Proton Emitter  The halflives of proton radioactivity are sensitive to both specific orbitals and nuclear deformation 131  Measured halflives in Eu 141 and Ho could only be understood if these nuclei were deformed  This was later confirmed by the observation of 141 rotational bands in Ho 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 184 Proton Decay Half Lives ℓ = 0 Coulomb Barrier ℓ = 5 Coulomb Barrier plus Centrifugal Barrier  The halflives 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 185 Fine Structure in Proton Decay 131  In Eu proton decay has been observed both to the ground state and the first 130 excited state of Sm  This establishes the first + 130 2 state in Sm at an energy of 121 keV  This low energy implies a large moment of inertia and large quadrupole (prolate) deformation for 130 the exotic Sm nucleus 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 186 Fine Structure in Alpha Decay 109  In Xe alpha decay has been observed both to the ground state and the first excited state of 105 Te  This establishes the relative energies of the neutron d and 5/2 105 g orbitals in Te 7/2 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 187 Superdeformed Proton Emitter SD 58 57 An SD band in Cu decays by proton emission into Ni in 58 competition with γ decay to the lowspin Cu states 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 188 Direct TwoProton Decay  A new mode of decay, direct twoproton decay, had been predicted long ago, but until recently, experimental efforts had only found sequential emission through an intermediate state  To prove diproton emission, specific nuclei are needed where the sequential Are the two protons correlated emission is energetically (diproton emission) or uncorrelated 18 forbidden e.g. Ne (sequential proton emission) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 189 45 Direct TwoProton Decay of Fe Decay energy spectrum correlated 45 with Fe implantation 45 from Fe  The experimental Qvalue implies diproton emission with a barrierpenetration halflife of 0.024 ms 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 190 NeutronRich 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 neutronrich nuclei  These nuclei are increasingly the focus of present and future experimental (and theoretical) effort 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 191 Light NeutronDripline Nuclei  The neutron dripline has only been reached for light nuclei 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 192 11 Level Inversion in Be  The ordering of the neutron 1s and 1p orbitals 1/2 1/2 appears to be inverted in the 11 nucleus Be and lighter N = 7 isotones unbound 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 193 Nuclear (Neutron) Haloes 11  The spatial extent of Li with 3 protons is similar to 208 that of Pb with 82 protons 11 9  Li is modelled as a core of Li plus two valence neutrons 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 194 Size of Lithium Isotopes  Interaction cross sections give a measure of the nuclear matter distribution (radius)  A sudden jump is seen in 9 11 going from Li to Li  However, the electric quadrupole moments are similar (charge distribution)  Hence, excess neutron Root mean square radii tail or halo 1/3 Textbook: R = r A 0 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 195 Nuclear Sizes 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 196 Halo Systematics  Neutron haloes have now been seen in nuclei 19 as heavy as C (Z = 6, N = 13)  Note proton haloes also predicted 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 197 Borromean System  Halo nuclei have provided insight into a new topology with a Borromean property  The twobody subsystems of the stable threebody 11 9 system Li ( Li + n + n) are themselves unstable bound unbound 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 198 Neutron Skins 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 199 Proton Skins Theory 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 200 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 201 New Shell Structure Z=8 N=20 N=16 N=8 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 202 Unexpected Things Happen Far From Stability  The heaviest known tellurium isotope is 136 Te  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’ 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 203 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 204 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 208  For example, the nucleus Pb 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) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 205 Uranium  “You know what uranium is, right, it's a thing called nuclear weapons and other things, like, lots of things are done with uranium, doing some bad things” Donald Trump 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 206 Super Heavy Elements (SHE) Flerovium Livermorium Copernicium Roentgenium Darmstadtium Meitnerium Hassium Bohrium SHE Protons  Quantal shell effects stabilise energy  Up to Z = 116 results confirmed  Dubna: Z = 114, 116, 118  Berkeley: Z = 118 ‘discovered’ then Neutrons retracted 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 207 Element 115  Well known for antigravity properties  The fuel of UFO’s 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 208 2016: Four New Elements nihonium (Nh), moscovium (Mc), tennessine (Ts), oganesson (Og) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 209 The Island(s) of Stability 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 210 Alpha Decay Chains  The heaviest nuclei are unstable against  decay  The decay half life (in s) is given empirically by: 1/2 log t = 1.61 Z E 10 1/2 α 2/3 – 1.61 Z – 28.9  Here E (in MeV) is the  decay α energy, related to the mass difference of the parent (Z, A) and daughter (Z2, A4) nuclei  decay provides the  The lifetimes are very long 3 technique to identify (10 s) on the nuclear time heavy elements scale 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 211 SHE Synthesis at GSI 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 212 Elements 116 and 118 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 213 Element 117 (2010)  Dubna (Russia)  Phys. Rev. Lett. 104, 142502 (2010) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 214 Superheavies at High Spin  The groundstate rotational band of 254 No (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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 215 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 leptonlike 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 half life 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. 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 216 10. Mesoscopic Systems 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 217 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 218 Mesoscopic Systems  ‘Mesoscopic’ systems contain large, yet finite, numbers of constituents, e.g. atomic nuclei, metallic clusters 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 219 Nanostructures and Femtostructures  ‘Nanostructures’: intense research is ongoing for quantum systems that confine a number of electrons 9 within a nanometresize scale (10 m), e.g. grains, droplets, quantum dots 15  Nuclei are femtostructures (10 m)  All these systems share common phenomena but on very different energy scales: nuclear MeV; molecular eV; solidstate meV 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 220 Mesoscopic systems N complexity Nuclei Hedroplets Metal clusters Emergent phenomena: Liquidgas surface, droplet features superconductivity / superfluidity Nanoparticles thermal phase transitions shell structure, quantal shapes (liquid) spatial orientation, rotational bands rotational/magnetic response quantum phase transitions E Quantum dots 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 221 macroscopic Quantality Parameter  The ‘quantality’ parameter (Mottelson 1999), 2 2 Λ = ħ / M a V , measures the strength of the twobody 0 attraction V expressed in units of the quantal kinetic 0 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) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 222 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 lowenergy excitations have ‘infinite’ meanfree path  Constituents T = 0 matter 3 He Λ = 0.21 ‘liquid’ 4 He Λ = 0.16 ‘liquid’ H Λ = 0.07 ‘solid’ 2 Ne Λ = 0.007 ‘solid’ Nuclei Λ = 0.4 ‘liquid’ 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 223 Fermi Liquid Droplets  ‘Clusters’ are aggregates of atoms or molecules with a welldefined 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 224 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 1/3 A (i.e. radius) where A is the number of constituents 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 225 Spherical Droplet Energy  The energy of a droplet may be expressed as: 2 2 2 E (N,Z) = fA + 4πσR + WZ + C Z e /R LD 2/3 2 1/3 = fA + b A + WZ + b Z A surf coul 1/3  Here R = r A is the radius of the droplet, A the 0 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 2  The second term (4πσR )is the ‘surface energy’ where σ is the coefficient of surface tension 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 226 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 2 2  The fourth term (C Z e /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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 227 Shell Structures  A bunching together of the energy levels of a particle in a two or threedimensional potential represents a shell structure  Metallic clusters show shell structures similar to nuclei  Clusters can contain more constituents than stable nuclei 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 228 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 halfperiod of the long wavelength oscillation is seen 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 229 Periodic Orbit Theory Supershell structure from interfering periodic orbits 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 230 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 JahnTeller effect’ 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 231 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 232 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 233 Nuclear Phase Diagram At sufficient temperature or density nucleons are expected to dissolve into a quarkgluon plasma (QGP) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 234 Extra Slides: Start 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 235 Nuclear Molecules  Speculation about the existence of clusters in nuclei, such as alpha particles, has existed for a long time  Initially stimulated by the observation of alpha particle decay Ikeda Diagram 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 236 Beryllium12  A beryllium nucleus containing 8 neutrons and 4 protons has been found to arrange itself into a molecularlike structure, rather than a spherical shape that some naïve theories might suggest  Beryllium12 can be thought of as two alpha M Freer et al. Phys. Rev. Lett. 82 (1999) 1383 particles and four neutrons 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 237 Chain States: Nuclear Sausages 12  Cluster Model calculations for C show evidence for a ‘chain state’ consisting of three α particles in a row – axis ratio 3:1 (i.e. ‘hyperdeformed’) 24  Similarly calculations for Mg show evidence for a chain state consisting of six α particles in a row – axis ratio of 6:1 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 238 BlochBrink 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 12 12 C + C scattering data suggested larger clusters may occur  ‘Nuclear Molecules’ 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 239 Binary Cluster Model  It has been observed that measured quadrupole moments of many superdeformed bands follow: 2 2/3 2/3 2/3 Q 2 R Z A – Z A – Z A o o 1 1 2 2  This expression results from considering the states of the nucleus (Z, A) to be composed of two clusters (Z , A ) i i in relative motion  For example, a strongly deformed band has recently been 108 found in Cd (Z = 48) 108  The predicted fragmentation for Cd is: 58 50 Fe (Z = 26) + Ti (Z = 22) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 240 Extra Slides: End 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 241 11. Nuclear Reactions 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 242 Examples of Nuclear Reactions 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 243 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 2 kinetic energy and binding energy (E = mc ) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 244 The Impact Parameter Reactions can be classified  Central collisions occur for by the impact parameter b 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 245 Collision Kinematics  The Q value is: 2 (M + M ) (M + M ) c A a B b  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 centreofmass frame, the threshold energy is: T Q (M + M ) / M a a A A 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 246 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” 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 247 Compound Nucleus Example  Consider a beam of alpha particles of energy 5 MeV/A 60 (or MeV per nucleon) impinging on Ni: 60 64  + Ni  Zn  At this (kinetic) energy, the incident particle is non relativistic, β = v/c = 0.1, and it will take the alpha 22 particle 10 s to travel across the target nucleus  In a compound nucleus, the first emission of a nucleon or 20 gamma ray takes 10 s  Hence the alpha particle traverses the compound nucleus hundreds of times and loses its identity 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 248 Geometric CrossSection  In the classical picture, the projectile and target nuclei will fuse if the impact parameter b is less than the sum of their radii 2  A disk of area π(R + R ) is swept out 1 2  This area defines the geometric crosssection  Remember: units of crosssection are area 2 (1 barn = 100 fm ; 15 1 fm = 10 m) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 249 Extra Slides: Start 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 250 Coulomb Excitation  Coulomb Excitation (Coulex) is the excitation of a target nucleus by the longrange 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. 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 251 Coulex Example 234 208  U bombarded by 5.3 MeV/A Pb  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: 2 Z Z e / 4πε (R + R ) x (A + A ) / A 1 2 0 1 2 1 2 2 CoM barrier = 1.3 GeV 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 252 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 eveneven 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 253 IE Coulex Example (GSI RISING Experiment) 179  A gold target ( Au) bombarded by a 140 MeV/A 108 radioactive Sn 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 108  Note: Sn is not stable – cannot make a target, but can generate a shortlived Radioactive Ion Beam (RIB) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 254 Intermediate Energy Coulex  Ideally suited for use with fragmentation beams (E 30 MeV/u) beam  Large cross sections (100 mb) 2  Can use thick targets (100 mg/cm ) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 255 Coulex Cross Sections  For Intermediate Energy Heavy Ions, the Coulomb excitation cross section can be approximated as: 2 2 2 2λ σ = Z e /ħc B(πλ;0λ) πR /e R (λ1) πλ 1 for λ ≥ 2  Here Z is the charge of the projectile and R is the sum 1 of the radii of target and projectile  The cross section is peaked at forward angles within the angular range 2 Δθ ≈ 2Z Z e /RE 1 2 where E is the beam energy 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 256 NeutronRich Sulphur Isotopes  Energy spectra in target (top) and projectile (bottom) frames of reference for 40 197 S + Au at MSU  β = v/c = 27  H. Scheit et al. Phys. Rev. Lett. 77, 3967 (1996) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 257 Neutron Capture  Lowenergy neutroncapture crosssections exhibit peaks or resonances corresponding to a compound system  An example is the capture of 115 a 1.46 eV neutron by In to form a highly excited state 116 (6.8 MeV ) in In  The high excitation energy 116 in In arises due to the binding energy of the neutron 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 258 Neutron Capture CrossSections  At 1.46 eV, the measured total crosssection for neutron 115 4 capture by In is σ ≈ 2.8 x 10 barn 2  However the geometrical crosssection (πR ) 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 longrange influence  The crosssection becomes: 2 2 σ = πR  σ = π(λ/2π) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 259 De Broglie Wavelength  The momentum of the neutron is: 6 p = √2m E = √2 x 939 x 1.46 x 10 n n = 0.052 MeV/c  The de Broglie wavelength is then: 3 (λ/2π) = ħc/p c = 197/0.052 = 3.7 x 10 fm n 5  The crosssection then becomes 4.3 x 10 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 260 116 Decay of In 116 115 In  n + In 4 116  γ + In 96  For this neutron energy of only 1.46 eV Γ /Γ = 0.04, also Γ /Γ ≈ 0.04 (Γ = Γ + Γ ) n γ n n γ  This decay fraction can be related to the formation crosssection: 2 σ = π(λ/2π) x Γ /Γ (Γ /Γ ≈ 4) n n  Recall the measured formation crosssection was only 6 of the estimate using the de Broglie wavelength 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 261 Proton Capture  For chargedparticle 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, protoncapture crosssections are 1 barn at maximum 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 262 Extra Slides: End 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 263 Heavy Ion Fusion Reactions 12 58  For heavy projectile ions, e.g. C or Ni, 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: 1080 MeV  Since the neutron binding energy is only 8 MeV, several neutrons are emitted before gammaray emission dominates  These Heavy Ion Fusion Evaporation reactions bring large amounts of angular momentum into the compound system 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 264 Fusion Evaporation Reactions 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 265 David Campbell Florida State University 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 266 Fusion CrossSection  The angular momentum brought into the compound system depends on the impact parameter b: ℓ = b  p  The partial fusion cross section is proportional to the angular momentum: d σ (ℓ ) ℓ fus 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 267 Compound Formation And Decay  Compound nucleus 20 formation: 10 s  Neutron emission: 19 10 s  Statistical (cooling) dipole gammaray 15 emission: 10 s  Quadrupole (slowing down) gammaray 12 emission: 10 s 9  After 10 s the nuclear ground state 100 36 132 Mo( S,4n) Ce 11 is reached after 10 Beam energy: 4.31 MeV/A rotations 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 268 Cold Fusion  Superheavy elements (SHE’s) can be formed by low energy fusionevaporation reactions in which only one neutron is emitted 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 269 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) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 270 Transfer Reactions 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 271 NeutronInduced Fission 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 272 Extra Slides: Start 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 273 Direct Reactions  Proceed in a single step, timescale comparable to the 22 time for the projectile to traverse the target (10 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 singleparticle structure  Examples: elastic and inelastic scattering 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 274 Elastic Scattering  Both target and projectile remain in their ground state a + A  a + A  Nuclei can be treated as structureless particles 0 10  Example: 1 10 Investigation of nuclear matter 2 10 11 Li matter density distributions 3 10 in exotic nuclei by 9 Li elastic pscattering 4 10 11 (inverse kinematics) Li core 5 10 0 2 4 6 8 10 r fm 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 275 3 (r) fm r mInelastic Scattering  Both target and projectile nuclei retain their integrity, they are only brought to bound excited states a + A  a + A  Can excite both singleparticle or collective modes of excitation  Example: investigate the GMR by (,’) inelastic scattering, gives access to nuclear incompressibility, key parameter of nuclear EOS 2 2 2 K (Z,N) r d (E/A) / dr r nm 0 0  Example: safe and unsafe Coulomb excitation (below and above Coulomb barrier) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 276 Transfer Reactions  One or a few nucleons are transferred between the projectile and target nuclei  Probes singleparticle 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 3  Examples numerous: (d,p) , (p,d) , (t,p) , (t, He) … 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 277 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 Q – can reach higher excited states β and giant resonances 2 3  Many different probes: (p,n) , (d, He) , (t, He) , but also 7 7 with heavy ions, e.g. ( Li, Be) or exotic particles, e.g. + 0 (π ,π ) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 278 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 3body state  Becomes dominant at high incident energies  Populates singlehole states, from which spectroscopic information can be derived  Examples: (p,2p) , (p,pn) , (e,e’p) , heavyion induced 9 knockout, e.g. ( Be,X) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 279 Extra Slides: End 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 280 Compound Nucleus Reactions  The two nuclei coalesce, forming a fused system that 20 16 lasts for a relatively long time (10 to 10 s)  Deexcitation 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 281 Types of Nuclear Reaction 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 282 12. Nuclear Astrophysics Linking Femtophysics with the Cosmos 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 283 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 284 Nuclei Power Stars  Stars are luminous, hot, massive, selfgravitating collections of nuclei (and electrons)  To generate sufficient light via release of gravitational 7 potential energy, a star would only live for 10 years  Stars must have an internal energy source to prevent gravitational collapse faster than their observed 8 9 lifetimes 10 – 10 years  Chemical energy too small…  Nuclear Fusion Reactions 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 285 Stellar Evolution  Nuclear Reactions are responsible for both preserving and evolving the collection of nuclei  Preserving: nuclear reactions generate energy which 30 balances the selfgravitation of 10 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 286 Periodic Table Of Elements nihonium (Nh), moscovium (Mc), tennessine (Ts), oganesson (Og) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 287 Elemental Abundances  Similar distribution everywhere  Spans twelve orders of magnitude  Hydrogen: 75  Helium: 23  C to U (‘metals’): 2  D, Li, B and Be under abundant  Exponential decrease up to Fe  A peak occurs near Fe  Almost flat distribution beyond Fe 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 288 Elemental Signatures Galactic distribution of the 26 1809 keV gamma ray in Mg 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 289 Origin of the Elements 1 2 3 4 7  Big Bang: H, H, He, He, Li (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 energyreleasing 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)  2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 290 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… 4 5 2 5 3 10 7  For 1 proton: 0.08 He, 10 H, 10 He, 10 Li nuclei 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 291 Elemental Abundances: Timeline 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 292 Stellar Evolution  Lowmass stars ( 2.3 M ):  Ignition of H, but He core becomes ‘degenerate’ before ignition  Intermediatemass stars (3 M M 8 M ):   Ignition of H, He, C, O white dwarf remnant  Highmass (‘massive’) stars (M 8 M ):  Ignition of H, … Si core collapse supernova 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 293 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 energyproviding fuel is left 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 294 Nuclear Burning Stages Massive Star Fuel Main Secondary Temp. Time Product Product (GK) (yr) 14 7 H He N 0.02 10 18 22 6 He C, O O, Ne 0.2 10 sprocess 3 C Ne, Mg Na 0.8 10 Ne O, Mg Al, P 1.5 3 O Si, S Cl, Ar, K, Ca 2.0 0.8 Si Fe Ti, V, Cr, 3.5 1 week Mn, Co, Ni 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 295 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 296 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 297 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: 1 4 + 4 H  He + 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’ 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 298 The ProtonProton (pp) Chain 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 299 The pp Chain Reactions 1 1 2 +  H + H  H + e + ν 2 1 3  H + H  He + γ 3 3 4 1  He + He  He + 2 H pp1, Q = 26.20 MeV 3 4 7  He + He  Be + γ 7 7  Be + e  Li + ν 7 1 4  Li + H  2 He pp2, Q = 25.66 MeV 7 1 8  Be + H  B + γ 8 8 +  B  Be + e + ν 8 4  Be  2 He pp3, Q = 19.17 MeV 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 300 The pp Chain Reactions 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 301 Helium Burning: The Triple  Chain 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 302 The Triple  Chain 8  To produce nuclei beyond Be, three alpha particles can 12 combine to produce a C nucleus (Q = 7.275 MeV): 12 4 4 4 12  +  +   C or He + He + He  C  Since the probability for a 3body reaction is extremely low, the reaction is expected to take place in two steps 8 8  (1)  +  +  Be, Q = 0.092 MeV, but Be is 16 unstable (τ 10 s) and decays back into  +  (this explains the A = 8 mass gap) 8 12  (2)  + Be  C, Q = 7.367 MeV 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 303 12 C Abundance: The Hoyle State  The triple alpha process + E(0 ) = 7.654 MeV does not account for 12 the full abundance of C – the fourth most abundant element in the universe  Hoyle (1954) predicted the existence of a + 12 resonant 0 state in C Triple alpha: at E 7.7 MeV Q = 7.275 MeV  This was later confirmed in experiment 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 304 The CNO Cycle  The CNO (carbonnitrogenoxygen) 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: 12 13 + 13 14 15 + 15 12 C(p,γ) N(e ,ν) C(p,γ) N(p,γ) O(e ,ν) N(p,α) C  The net result is: 1 4 + 4 H  He + 2 e + 2 ν, Q = 26.73 MeV 13  The cycle is limited by β decay of N (τ 10 min) and 15 O (τ 2 min) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 305 CNO Reactions 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 306 CNO and PP Chain: Temperature Dependence 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 307 Breakout into the Hot CNO Cycle  At higher temperatures, proton capture on 13 N can begin to compete with the β decay and the cycle can break out into the hot CNO cycle 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 308 CNO and Hot CNO Cycles 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 309 Breakout of the Hot CNO Cycle 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 310 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 311 p Reactions 14  Starting with O: 14 17  O +   F + p 17 18  F + p  Ne 18 21  Ne +   Na + 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 312 The p Process 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 313 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 314 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 315 Neutron and Proton Capture  Neutron capture reactions are responsible for the production of elements above Fe  The relative ncapture / βdecay efficiencies lead to two extreme cases: sprocess (slow) and rprocess (rapid)  Nuclear structure details determine the rprocess: connection between Astrophysics and Nuclear Physics  Extreme and transient conditions near compact remnant stars can yield nuclei on the protonrich side of the stability region: rpprocess 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 316 Rapid Proton/Neutron Capture 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 317 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 318 Neutron Capture Reactions  Stellar abundances of the elements imply two different processes: The sprocess (slow): 8 3 Low neutron flux, N(n)  0 (10 n/cm ) The rprocess (rapid): 20 3 High neutron flux, N(n)  ∞ (10 n/cm )  Rapid neutron capture (rprocess), and also rapid proton capture (rpprocess), produce exotic nuclei far away from the valley of stable nuclei 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 319 Neutron Capture Reactions Very small (n,γ) cross sections at N magic numbers Evidence for nuclear processes governing nucleosynthesis 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 320 Astrophysical Sites 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 321 Creation of Heavy Elements 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 322 Influence of Shell Structure on Elemental Abundances 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 323 Creation of ProtonRich Nuclei  The rpprocess lasts 101000 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 324 Endpoint of rp Process 106,107 Te very recently studied at Jyväskylä Small island of alpha decay just above protonrich tin (Z=50) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 325 Astrophysical rpprocess Sites  Novae  Xray bursters  Shock waves passing through the envelope of supernova progenitors  ThorneZytkow objects, where a neutron star sinks to the centre of a supergiant 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 326 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) 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 327 Thermonuclear Energy Generation 1 4  4 H  He 6.7 MeV/u 4 12  3 He  C 0.6 MeV/u ‘triple ’ 4 1 104  5 He + 84 H  Pd 6.9 MeV/u ‘rp process’  Gravitational potential energy: 200 MeV/u 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 328 Astrophysical Reactions  The elemental abundances depend crucially on the reaction rates (crosssections), i.e. proton/neutron capture vs. β decay  These important crosssections can now be measured using accelerated beams of radioactive beams 21 22  An example is the Na + H  Mg + γ reaction 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 329 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 330 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 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 331 The Role of Nuclear Physics 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 332 Cosmophysics: A New Field  The fields of Cosmology and Astrophysics can be combined in two ways: 1. Cosmophysics 2. Astrology The End 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 333 13. Double Beta Decay 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 334 Introduction  Doublebeta 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 eveneven nuclei  More than sixty naturally occurring isotopes are capable of undergoing doublebeta decay (energetically)  Ten such isotopes have been experimentally observed: 48 76 82 96 100 116 128 130 150 238 Ca Ge Se Zr Mo Cd Te Te Nd U 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 335 Double Beta Decay  Twoneutrino double beta decay ββ(2ν): (Z,A)  (Z+2,A) + 2 electrons + 2 antineutrinos conserves not only electric charge but also lepton number 19 Halflife (measured) 10 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 26 Halflife (predicted) 10 years 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 336 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 doublebeta decay proceeds only when neutrinos are massive Majorana particles  Recent neutrino oscillation experiments suggest that neutrinos have a nonzero mass of the order 50 meV  The Standard Electroweak Model postulates that neutrinos are massless and lepton number is conserved  New Physics 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 337 The Signal of ββ(0ν) Decay 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 338 Neutrino Mass  Upper limits of neutrino mass are shown to the left from twoneutrino doublebeta decay measurements  Neutrinooscillation experiments suggest a mass scale of the order 50 meV 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 339 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 340 Experiment 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 341 Theory 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 342 Nuclear Chocolate 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 343 Fundamental Forces 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 344 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 345 The Nuclear ManyBody Problem: Energy, Distance, Complexity heavy nuclei few body quarks gluons vacuum nucleon quarkgluon few body systems many body systems QCD soup QCD free NN force effective NN force 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 346 Life, The Universe Everything 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 347 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 348 Marielle Slides… 2/24/2017 PHYS490 : Advanced Nuclear Physics : E.S. Paul 349 Isospin Dependence of Mean Field and Residual Interactions Shell structure predicted to change Mean field modifications in exotic nuclei (particularly neutron  surface composed of diffuse rich). neutron matter Weak binding, impact of the particle continuum, collective skin modes and  derivative of mean field potential clustering in the skin… weaker and spinorbit interaction reduced Residual interaction modifications  partly occupied orbits  V monopole interaction: coupling  of protonneutron spinorbit partners  deformed intruder configurations Quenching of ‘old’ shells and emergence of new magic numbers in exotic neutronrich nuclei Neutron shell Gap Neutron Shell Gap 4 1 10 Ca (Z=20) 8000 GAP = M(Z,N2) M(Z,N) 6000 N + M(Z,N+2) M(Z,N) 4000 = S (N) S (N+2) 2N 2N 2000 20 28 0 16 18 20 22 24 26 28 30 Neutron Number 8000 CaGAP FGAP KGAP 7000 8000 NeGAP ClGAP NaGAP SGAP 6000 MgGAP SiGAP 6000 5000 4000 20 4000 3000 2000 2000 16 20 1000 28 0 0 18 20 22 24 26 28 12 14 16 18 20 22 Neutron Number Neutron Number GAP (keV) N GAP (keV) N Gap = S (N)S (N+2) keV N 2N 2NSingleNeutron Removal in the psd shell ns 1/2 ns intruder 1/2 pshell sdshell E.Sauvan et al., Phys. Lett. B 491 (2000) 1, Phys. Rev. C 69 (2004) 044603. New Magic Number at N=16 Present in stable nuclei but missing V monopole interaction : coupling of  in nrich nuclei where the spin protonneutron spinorbit partners orbit partner of the valence neutrons are not occupied by protons T. Otsuka et al. Phys. Rev. Lett. 87 (2001) 082502. Examples of experimental evidence:  Twoneutron separation energies  Inbeam fragmentation gamma spectroscopy  1nremoval crosssections and longitudinal momentum distributions (direct reactions) + Systematics of the 3/2 in the N=15 isotones 23 25 27 Mg O Ne 4.5 4.0 1f 7/2 3.5 3.0 2.5 2.0 1d 5/2 1.5 1.0 1d 3/2 0.5 2s 0.0 1/2 6 8 10 12 atomic number The energy of the 1d neutron orbital rises when protons 3/2 are removed from its spinorbit partner, the 1d orbital. 5/2 23 25 27 O Ne Mg 8 10 12 1d 3/2 1d 3/2 1d 3/2 2s 2s 2s 1/2 1/2 1/2 1d 1d 5/2 5/2 1d 5/2 1s, 1p 1s, 1p 1s, 1p 1s, 1p 1s, 1p 1s, 1p n p n p n p excitation energy (MeV) Transfer Reaction Example 46 47 Ar(d,p) Ar at 10.7 A.MeV Modification of residual in inverse kinematics interactions at N=28 N=28 gap : 4.47(8)MeV Excitation energy 47 spectrum for Ar p 3/2 47 Ar p 1/2 f 5/2 f 7/2 MUST at GANIL/SPIRAL L. Gaudefroy et al, PRL 97, 092501 (2006). Knockout Reactions Example Systematics of (e,ep) on Stable Nuclei Departures of measured spectroscopic factors from the independent singleparticle model predictions Electron induced proton knockout reactions: A,Z (e,ep) A1,Z1 similar proton separation energies See only 6070 of nucleons expected Effect of longrange and shortrange correlations W. Dickhoff and C. Barbieri, Prog. Nucl. Part. Sci., 52 (2004) 377. HighEnergy SingleNucleon Removal A New Spectroscopic Tool dominant core+1N core g  = 2 p 0 d/ dp  = 2 and  = 0 mixture Target  = 2 x g  E d/dp   core n p 2  (J )  C S 1n core REVIEW:Hansen Tostevin, Ann. Rev. Nucl. Part. Sci. (2003)
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