Nuclear Phenomenology 3 C 24 Nuclear and Particle
- Slides: 25
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 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. 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
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 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 • Rutherford cross-section: • Taking into account spin: Mott cross-section
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
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
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 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 • 2 d correction, surface term • 3 d correction, Coulomb term
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 = 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 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
Odd-mass nuclei 1) 2) 3) Electron capture
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-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/ 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) • 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
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
- Strength and weaknesses of phenomenology
- Phenomenology and ethnography
- Ethnomethodology and phenomenology
- Lesson 15 nuclear quest nuclear reactions
- Fisión nuclear vs fision nuclear
- Nenl
- Phenomenology vs ethnography
- Phenomenology vs case study
- What is grounded theory in simple terms
- Phenomenology vs ethnography
- Types of phenomenology
- Philosophical assumption
- Phenomenology
- Strength of grounded theory
- Max van manen
- Rigid body vs particle
- Particle equilibrium in 2d and 3d engineering mechanics
- Kinetics of a particle: force and acceleration
- Kinetics of a particle: impulse and momentum
- Youtube https //www.youtube.com/watch v=vnp84pn0mjq
- Particle movement in solids liquids and gases
- Particle theory dissolving
- Solid+heat=
- Particle removal wet etch filters
- Wave particle duality questions
- Wave-particle duality