Nuclear Phenomenology 3 C 24 Nuclear and Particle

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Nuclear Phenomenology 3 C 24 Nuclear and Particle Physics Tricia Vahle & Simon Dean

Nuclear Phenomenology 3 C 24 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

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

Masses and binding energies • Something we know very well: – Mp = 938.

Masses and binding energies • Something we know very well: – Mp = 938. 272 Me. V/c 2, Mn = 939. 566 Me. V/c 2 • 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 – –DMc 2 – the binding energy B. – B/A – the binding energy per nucleon, the minimum energy required to remove a nucleon from the nucleus

Binding energy per nucleon as function of A for stable nuclei

Binding energy per nucleon as function of A for stable nuclei

Nuclear Forces • Existence of stable nuclei suggests attractive force between nucleons • But

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 force is short which does not correspond to the exchange of gluons

Nuclear Forces d +V 0 V(r) r=R d<<R Range~R B/A ~ V 0 0

Nuclear Forces d +V 0 V(r) r=R d<<R Range~R B/A ~ V 0 0 r • Charge symmetric pp=nn • Almost charge –independent pp=nn=pn – mirror nuclei, e. g. 11 B 11 C • 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) -V 0 Approximate description of nuclear potential

Nuclei. Shapes and sizes. • Scattering experiments to find out shapes and sizes •

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

Nuclei. Shapes and Sizes. • Nucleus is not an elementary particle • Spatial extension

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 inverse Fourier transform found from

In practice ds/d. W falls very rapidly with angle

In practice ds/d. W falls very rapidly with angle

Shapes and sizes • Parameterised form is chosen for charge distribution, form-factor is calculated

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 c = 1. 07 A 1/3 fm a = 0. 54 fm • Charge density approximately constant in the nuclear interior and falls rapidly to zero at the nuclear surface

Radial charge distribution of various nuclei

Radial charge distribution of various nuclei

Shapes and sizes • Mean square radius • Homogeneous charged sphere is a good

Shapes and sizes • Mean square radius • Homogeneous charged sphere is a good approximation Rcharge = 1. 21 A 1/3 fm • If instead of electrons we will use hadrons to bombard nuclei, we can probe the nuclear density of nuclei rnucl ≈ 0. 17 nucleons/fm 3 Rnuclear ≈ 1. 2 A 1/3 fm

Liquid drop model: semi-empirical mass formula • Semi-empirical formula: theoretical basis combined with fits

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

Semi-empirical mass formula • “ 0 th“term • 1 st correction, volume term •

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

Semi-empirical mass formula • 4 th correction, asymmetry term • Taking into account spins

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

Semi-empirical mass formula Constants • Commonly used notation a 1 = av, a 2

Semi-empirical mass formula Constants • Commonly used notation a 1 = av, a 2 = as, a 3 = ac, a 4 =aa, a 5 = 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 Me. V/c 2

Nuclear stability • n(p) unstable: b-(b+) decay • The maximum binding energy is around

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

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

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

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

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

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

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

a-decay • a-decay is energetically allowed if B(2, 4) > B(Z, A) – B(Z-2,

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 d. B/d. A 28. 3 ≈ 4(B/A – 7. 7× 10 -3 A) • Above A=151 a-decay becomes energetically possible

a-decay TUNELLING: T = exp(-2 G) G – Gamow factor G≈2 pa(Z-2)/b ~ Z/

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

Spontaneous fission • Two daughter nuclei are approximately equal mass (A > 100) •

Spontaneous fission • Two daughter nuclei are approximately equal mass (A > 100) • Example: 238 U 145 La + 90 Br + 3 n (156 Me. V 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

Deformed nuclei

Deformed nuclei

Spontaneous fission • The change in total energy due to deformation: DE = (1/5)

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