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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

Nuclear Phenomenology


Nuclear and Particle Physics

Tricia Vahle & Simon Dean

(based on Lecture Notes from Ruben Saakyan)


nuclear notation
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
masses and binding energies
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
binding energy
Binding energy

Binding energy per nucleon as function of A for stable nuclei

nuclear forces
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
nuclear forces1
Nuclear Forces


  • 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)



r = R d<<R


B/A ~ V0




Approximate description

of nuclear potential

nuclei 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 sizes1
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


shapes and sizes
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

shapes and sizes1
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

liquid drop model semi empirical mass formula
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
Semi-empirical mass formula
  • “0th“term
  • 1st correction, volume term
  • 2d correction, surface term
  • 3d correction, Coulomb term
semi empirical mass formula1
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
semi empirical mass formula constants
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
nuclear stability
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



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

Electron capture




even mass nuclei
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 dB/dA

28.3 ≈ 4(B/A – 7.7×10-3 A)

  • Above A=151a-decay becomes energetically possible
a decay1


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)

spontaneous fission
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
spontaneous fission1
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