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## Nuclear Phenomenology

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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)
- 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

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

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

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 found from

inverse Fourier

transform

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 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

- 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

- “0th“term
- 1st correction, volume term
- 2d correction, surface term
- 3d correction, Coulomb term

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 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

- 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

b-decay. Phenomenology

- Rearranging SEMF
- Odd-mass and even-mass nuclei lie on different parabolas

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-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)

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 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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