Week 2 - Nuclear Model | SEMF | BE

Question 1

What is the liquid drop model? Limitations? What do the first three and final two terms discuss?

Answer 1

This is the formula, alongside empirical measurements, through which the SEMF relies.

It provides good approximation for atomic masses however, fails to explain the existence of lines of greater BE at certain numbers of protons and neutrons.

These numbers are known as ‘magic numbers’ and are the foundation of the nuclear shell model.

First three terms assumes the nucleus as a drop of incompressible fluid with each term describing a physical property. The final two are quantum mechanical considering the nucleus as a gas of fermions confined by the nuclear potential and the final occupying discrete energy levels.

Question 2

How is the radius of a nucleus estimated?

Answer 2

R = R0•A^1/3
R = nuclear radius (fm)
R0 = constant of proportionality = 1.25fm
A = nucleon / mass number

Volume proportional nucleon number
Volume proportional to radius^3
Radius proportional to mass number^1/3

Provides evidence for liquid drop model, more nucleons means size will increase

Question 3

What is the density of a nucleus? Why is it constant? What does this mean?

Answer 3

Rho = (3•m)/(4•pi•r0^3)
m = mass of nucleon (Kg)
All other elements are constants, hence density is constant. Therefore adding more nucleons must increase the size of the nucleus for density to remain
Rho (nucleus) = 2.7•10^17kg•m^-3
Rho (neutron star) = 4.24•10^17kg•m^-3

Question 4

What are the 5 main terms of the SEMF & what do they address?

Answer 4

1. Volume term: increase in volume means increase in mass number and thus binding energy (+ term)
2. Surface term: SNF weaker for nucleons on outer surface, hence counters the overestimation of the 1st term (- term). This effect has a smaller impact for larger sized nuclei.
3. Coulomb term: deals with the force required to overcome the repulsion between protons in the nucleus (- term). Can be Z^2 or Z(Z-1) (second technically more accurate as a single proton wouldn’t have anything to repel against).
4. Asymmetry term: relates to Paulis exclusion principle “ no two protons or neutrons can exist in the same state of the atom”. Without charge (coulomb energy) the SNF is the dominant contribution to nuclear potential and acts both equally on protons and neutrons. Hence equal number of protons and neutrons results in a lower energy state (-ve if Z & N are not equal)
   • spacing between states inversely proportional to the size of nucleus — hence larger nuclei have smaller BE loss for asymmetries in N&Z
5. Pairing term: because N & Z levels are slightly different, it is favourable to complete the Z or N level (even-even +, even-odd 0, odd-odd -). Maximum overlap for two identical nucleons in the same spatial state, hence higher BE for same paired nucleons. Hence lowest BE for single paired nucleons.

Question 5

Why is a nucleus more stable with both neutrons and protons instead of just one type?

Answer 5

A nucleus is a bound state where the nucleons are confined by nuclear potential. Particles that are confined have discrete energy levels (quantised)

The pairing term illustrates how a mass number comprised of both types of nucleons would be more energetically stable than the same mass number comprised of either Zs or Ns.

Question 6

What is the issue with the SEMF?

Answer 6

The SEMF is based on experimental data rather than theoretical predictions, hence it does not make accurate predictions.

It’s great for understanding general trends however, fails to explain the existence of lines of greater BE at certain proton neutron numbers (magic numbers).

Question 7

Describe what ‘Mass Defect’ is and what it means in practise?

Answer 7

The combined mass of an atom is less than the total mass of its constituents. The MD is the loss which occurs when the atom is formed from its constituents.

As per E=mc2, energy and mass are related; the atom is in a lower energy state than its constituents.

Question 8

What is binding energy and how can it be used as a tool?

Answer 8

Binding energy is the energy equivalent of the mass defect.

A minimum amount of energy is required to break apart a nucleus. Likewise to join together individual nucleons to form a nucleus.

For a nucleus to form it’s total energy must be less than it’s constituents and the left over energy is release as binding energy.

It’s difficult to measure nuclear masses and sometimes atomic masses are given, thus electrons must be taken into account.

BE/N is an indicator of stability with a higher number indicating greater stability.

Question 9

What is the difference between MeV/c^2 and MeV/c?

Answer 9

MeV/c2
> mass equivalence to energy
> derived from Einsteins E=mc2
> states kg = jm-2s2

MeV/c
> used to express momentum
> utilised in relativistic physics (particles moving close to speed of light)
> kgms-1 (SI unit for momentum) = Jm-1s