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Nuclear Phenomenology

Nuclear Phenomenology. 3C24 Nuclear and Particle Physics Tricia Vahle & Simon Dean (based on Lecture Notes from Ruben Saakyan) UCL. Nuclear Notation. Z – atomic number = number of protons N – neutron number = number of neutrons A – mass number = number of nucleons (Z+N)

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Nuclear Phenomenology

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  1. Nuclear Phenomenology 3C24 Nuclear and Particle Physics Tricia Vahle & Simon Dean (based on Lecture Notes from Ruben Saakyan) UCL

  2. Nuclear Notation • Z – atomic number = number of protons N – neutron number = number of neutrons A – mass number = number of nucleons (Z+N) • Nuclides AX (16O, 40Ca, 55Fe etc…) • Nuclides with the same A – isobars • Nuclides with the same Z – isotopes • Nuclides with the same N – isotones

  3. Masses and binding energies • Something we know very well: • Mp = 938.272 MeV/c2, Mn = 939.566 MeV/c2 • One might think that • M(Z,A) = Z Mp + N Mn - not the case !!! • In real life • M(Z,A) < Z Mp + N Mn • The mass deficit • DM(Z,A) = M(Z,A) - Z Mp - N Mn • –DMc2 – the binding energy B. • B/A – the binding energy per nucleon, the minimum energy required to remove a nucleon from the nucleus

  4. Binding energy Binding energy per nucleon as function of A for stable nuclei

  5. Nuclear Forces • Existence of stable nuclei suggests attractive force between nucleons • But they do not collapse  there must be a repulsive core at very short ranges • From pp-scattering, the range of nucleon-nucleon force is short which does not correspond to the exchange of gluons

  6. Nuclear Forces d • Charge symmetric pp=nn • Almost charge –independent pp=nn=pn • mirror nuclei, e.g. 11B 11C • Strongly spin-dependent • Deutron exists: pn with spin-1 • pn with spin-0 does not • Nuclear forces saturate (B/A is not proportional to A) +V0 V(r) r = R d<<R Range~R B/A ~ V0 0 r -V0 Approximate description of nuclear potential

  7. Nuclei. Shapes and sizes. • Scattering experiments to find out shapes and sizes • Rutherford cross-section: • Taking into account spin: Mott cross-section

  8. Nuclei. Shapes and Sizes. • Nucleus is not an elementary particle • Spatial extension must be taken into account • If – spatial charge distribution, then we define form factor as the Fourier transform of can be extracted experimentally, then found from inverse Fourier transform

  9. In practice ds/dW falls very rapidly with angle

  10. Shapes and sizes • Parameterised form is chosen for charge distribution, form-factor is calculated from Fourier transform • A fit made to the data • Resulting charge distributions can be fitted by • Charge density approximately constant in the nuclear interior and falls rapidly to zero at the nuclear surface c = 1.07A1/3 fm a = 0.54 fm

  11. Radial charge distribution of various nuclei

  12. Shapes and sizes • Mean square radius • Homogeneous charged sphere is a good approximation Rcharge = 1.21 A1/3 fm • If instead of electrons we will use hadrons to bombard nuclei, we can probe the nuclear density of nuclei rnucl ≈ 0.17 nucleons/fm3 Rnuclear ≈ 1.2 A1/3 fm

  13. Liquid drop model: semi-empirical mass formula • Semi-empirical formula: theoretical basis combined with fits to experimental data • Assumptions • The interior mass densities are approximately equal • Total binding energies approximately proportional their masses

  14. Semi-empirical mass formula • “0th“term • 1st correction, volume term • 2d correction, surface term • 3d correction, Coulomb term

  15. Semi-empirical mass formula • 4th correction, asymmetry term • Taking into account spins and Pauli principle gives 5th correction, pairing term • Pairing term maximises the binding when both Z and N are even

  16. Semi-empirical mass formulaConstants • Commonly used notation a1 = av, a2 = as, a3 = ac, a4 =aa, a5 = ap • The constants are obtained by fitting binding energy data • Numerical values av = 15.67, as = 17.23, ac = 0.714, aa = 93.15, ap= 11.2 • All in MeV/c2

  17. Nuclear stability • n(p) unstable: b-(b+) decay • The maximum binding energy is around Fe and Ni • Fission possible for heavy nuclei • One of decay product – a-particle (4He nucleus) • Spontaneous fission possible for very heavy nuclei with Z  110 • Two daughters with similar masses p-unstable n-unstable

  18. b-decay. Phenomenology • Rearranging SEMF • Odd-mass and even-mass nuclei lie on different parabolas

  19. Odd-mass nuclei Electron capture 1) 2) 3)

  20. Even-mass nuclei b emitters lifetimes vary from ms to 1016 yrs

  21. a-decay • a-decay is energetically allowed if B(2,4) > B(Z,A) – B(Z-2,A-4) • Using SEMF and assuming that along stability line Z = N B(2,4) > B(Z,A) – B(Z-2,A-4) ≈ 4 dB/dA 28.3 ≈ 4(B/A – 7.7×10-3 A) • Above A=151a-decay becomes energetically possible

  22. a-decay TUNELLING: T = exp(-2G) G – Gamow factor G≈2pa(Z-2)/b ~ Z/Ea Small differences in Ea, strong effect on lifetime Lifetimes vary from 10ns to 1017 yrs (tunneling effect)

  23. Spontaneous fission • Two daughter nuclei are approximately equal mass (A > 100) • Example: 238U  145La + 90Br + 3n (156 MeV energy release) • Spontaneous fission becomes dominant only for very heavy elements A  270 • SEMF: if shape is not spherical it will increase surface term and decrease Coulomb term

  24. Deformed nuclei

  25. Spontaneous fission • The change in total energy due to deformation: DE = (1/5) e 2 (2as A2/3 – ac Z2 A-1/3) • If DE < 0, the deformation is energetically favourable and fission can occur • This happens if Z2/A  2as/ac ≈ 48 which happens for nuclei with Z > 114 and A  270

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