A moments thought indicates that the electrostatic repulsion due to a collection of protons in close proximity would result in any potential nucleus flying apart. A new and fundamental force is thus required to explain awhat holds a nucleus together against this repulsive force. The new force is termed the Nuclear or Strong force. Although the details of this force are only crudely understood, a potential energy diagram for two nucleons reveals much information:

The potential well is much deeper than the electrostatic repulsion which is shown for comparison.Obviously this is only relevant for the proton-proton case. The strong force acts equally for protons and neutrons so is charge independent. Particles on which the strong force acts are termed
' Hadrons'. In the search for a theory of the strong interaction, Yukawa suggested the formula
 

 where R is the range of the force and m is the mass of the field quantum (vector boson).

Q1) (a)If force is defined as minus the rate of change of potential energy with distance or in symbols F = - dE / dr , then sketch a force-separation curve for two nucleons , explaining clearly your reasoning.

(b) If a deuteron is an ordinary hydrogen atom with an extra bound neutron, show that the binding energy of the deuteronis about 2.2 MeV. Hence determine the least frequency of gamma rays that could be used to split a deuteron into a free neutron and a proton.
 

(b)Pions have a baryon number of zero and may be neutral(p^o) , negative (p^-)or positively charged (p⁺).If mesons consist of a quark/antiquark pair , deduce the quark structure of pions, using quark data from the data page.

Q3) (a) If the nuclear radius R is given by the simple formula R = R_oA ^1/3 where A is the mass number and R_o is a constant then show that the density of any nucleus is constant.

(b) If the value of R_o is 1.2 x 10^-15 m then deduce the radius of a ¹²C atom and calculate its density.Comment on this figure(the density of gold is 1.96 x 10³ kgm^-3.)

Q4) If the uncertainty principle is stated in the form DE.Dt > h / 2p and we assume that the speed of pion travel is ~ c then deduce the Yukawa formula m = h / 2pRc . This will only apply for an event in which energy DE is NOT conserved if the duration of the event is less than h / 2pDE

Clearly the pions are field quanta for the strong force so we can draw Feynman diagrams for field interactions:
 
 

1. Gluons ,which hold quarks together to form nucleons - the quarks exchange gluons .They are massless and travel at light speed - the theory of gluons requires that quarks have another property called ' colour' as well as flavour, the exchange of gluons resulting in a change of quark 'colour'.
 
 

2. Pi - Mesons , which nucleons exchange to form bound atomic nucleii.These account for hadron-hadron interactions and are (as shown above) , massiveand hence travel at sub-light speed. The associated transfer of momentum as the pions are exchanged constitutes a force.

The strong interaction is unable to account for beta decay. As far as the structure of matter is concerned , another short range interaction is responsible for the decay of nuclei that have top-heavy neutron/proton ratios.This is termed the Weak interaction.

This force also affects non-nuclear particles, such as electrons and neutrinos.

Q5) Use the Yukawa formula to estimate the mass of the field quanta of the weak interaction if its range is 1 x 10^-17 m. Give your answer in electronvolts.

Q6) Name the three field quanta for the weak force . They are collectively termed ' Intermediate Vector Bosons'.

Q7) Draw a Feynman diagram for proton decay at the quark level if it is given by the process

We apply the conservation of energy and momentum to this process, and show that energy does not appear to be conserved:
 
 

Q7(a) If the relativistic increase in mass of a particle is given by
 
 

then write down formulae for (i) Total relativistic energy , (ii) Relativistic momentum. Hence show that the relativistic equation for the energy of any particle in the weak interaction is given by the equation:

.

where E is total energy, p is particle momentum and m_o is the rest mass of the particle.

If the nucleus is at rest before the disintegration then the momentum of each part afterwards will be equal and opposite.If we denote this as p, then by conservation of energy:

.

Q8) Explain the significance of each term in this equation.

Q9) Show that all electrons emitted by stationary A nuclei should have the same energy given by:

However , the electrons in particular decays have varying energies, ranging from a minimum of m_ec² which is the rest mass-energy to the value just calculated, as the following diagram of the beta decay spectrum shows:
 
 

Q11) If the missing energy is carried away by another particle, termed the Neutrino, what can you deduce about

Q12) Using E = mc² , deduce that energy and momentum are linked by p = Ev / c² where v is the velocity of a particle.

Q13) Using the result of Q12 and the relativistic formula for energy, deduce the speed at which neutrinos must travel.

Q14) If the quantised 'spin' of the elementary particles n,p,e is +1h , then suggest a further reason why n --> p + e^- is not a complete weak interaction. What conservation principle does it violate ? What spin must the extra particle have ? Why do you think the particle is termed the antineutrino ?

Q15) Calculate the minimum energy that an antineutrino must have to cause the reaction

The antiparticle of a given particle should have exactly the same mass but opposite electric charge. Also the antiparticle can be annihilated by the corresponding particle, the two rest masses being converted into energy. Positrons (e⁺) are easily produced when a gamma ray photon strikes a nucleus and produces an electron/positron pair. The positron will be annihilated by the first electron it meets.

Q16) What is the least energy and frequency gamma ray needed for pair production ?
 
 

The antiparticle of the proton is the antiproton p . According to theory, a proton with sufficient kinetic energy striking another proton could cause antiproton production with the reaction:

The kinetic energy of the incident proton is converted directly into rest-mass energy of the proton/antiproton pair. This was carried out successfully in 1955.

Q17) (Difficult) In the above reaction scheme , if a high velocity incident proton strikes a stationary proton, and the momentum of the incident proton is shared equally between the emergent particles, show that the kinetic energy of the incident proton is six times its rest mass. ie K = 6m_pc² . [ Hints : You will need C of M, C of E, and the relativistic total energy formula :

Q18) Explain the curvature of the particle tracks. Explain why the tracks for the neutrinos and neutral pion are straight lines. Give decay equations for them⁺particle and the p⁺particle.

The weak interaction is responsible for the decay of particles and usually involves an electron - n_e pair or a muon - n_mpair.The electron and its neutrino have lepton number equal to +1 as do the muon and its associated neutrino. Antiparticles have lepton number equal to zero. Lepton numbers are conserved in the weak interaction,as is baryon number.
 

The strong interaction acts on Hadrons (baryons and mesons) and holds them together. Nucleons have baryon number +1 and mesons have baryon number 0. Antiparticles have baryon number -1 (nucleons) and 0 (mesons).Charge,baryon number and strangeness are conserved in the strong interaction.
 
 

In total there are 14 conservation laws which dictate how elementary particles may interact . The ones we have dealt with above are:

• (1) Conservation of Total Energy (including rest-mass).
• (2) Conservation of Total Linear Momentum.
• (3) Conservation of Total Angular Momentum.

• (4) Conservation of Electric Charge.
• (5) Conservation of Baryon Number.
• (6) Conservation of Lepton Number.
• (7) Conservation of Strangeness.