A2 Physics 



Nuclear Physics
9 August 1945  Atom bomb over Nagasaki
Contents
Rutherford's experiment
The diameter of a nucleus
Activity and halflife
Radioactivity
Radioactivity series
Energymass equivalence
Binding energy
Mass defect
Binding energy per nucleon and nuclear stability
Fission and fusion
Thermal nuclear reactor
Safety features of a thermal nuclear reactor
Radioactive waste
Rutherford's Experiment
Debate about the structure of the atom was put to rest after Ernest Rutherford's famous gold leaf experiment in 1911. Rutherford fired a stream of positively charged alpha particles towards a thin film of gold that was only a few atoms thick. He foudn that most of the alpha particles went straight through, suggesting that the majority of the atom is empty space.
Some of the alpha particles deflected marginally and others bounced back towards the source of the alpha particles. This suggsted that there was a small concentration of positive charge in the centre of a mostly empty atom. The centre of the atom became known as the nucleus, where almost all of the mass of an atom resides (this is, of course, excluding the small amounts of mass from electrons).
The image below shows the paths that alpha particles have taken when passing by a nucleus. The closer to the nucleus the more the alpha particle is deflected. Moving towards the nucleus head on the alpha particle bounces directly back to the source.
Some of the alpha particles deflected marginally and others bounced back towards the source of the alpha particles. This suggsted that there was a small concentration of positive charge in the centre of a mostly empty atom. The centre of the atom became known as the nucleus, where almost all of the mass of an atom resides (this is, of course, excluding the small amounts of mass from electrons).
The image below shows the paths that alpha particles have taken when passing by a nucleus. The closer to the nucleus the more the alpha particle is deflected. Moving towards the nucleus head on the alpha particle bounces directly back to the source.
The Diameter of a Nucleus
There are a number of ways of estimating the size of the nucleus.
A high energy beam of electrons is directed at an element (p.d. ~100MV). The de Broglie wavelength of such a beam is around 10^15m (roughly the diameter of a nucleus). A detector can be used to measure the angle at which the electrons are diffracted. Diffraction pattern determines the space in between nuclei; thus the nuclear radius can be measured.
The nuclear radius, R, can be shown that it depends on the nucleon number, A, in the following relationship:
A high energy beam of electrons is directed at an element (p.d. ~100MV). The de Broglie wavelength of such a beam is around 10^15m (roughly the diameter of a nucleus). A detector can be used to measure the angle at which the electrons are diffracted. Diffraction pattern determines the space in between nuclei; thus the nuclear radius can be measured.
The nuclear radius, R, can be shown that it depends on the nucleon number, A, in the following relationship:
Where r0 = 1.05 fm.
It makes sense for R to be proportional to A^(1/3) because the volume of the nucleus depends on the amount of nucleons inside:
It makes sense for R to be proportional to A^(1/3) because the volume of the nucleus depends on the amount of nucleons inside:
The following 2 graphs can be drawn using the equation above.
Activity and HalfLife
When an atom decays via α or β the proton number changes, changing the element. The mass of a radioactive isotope decreases over time because of this decay.
Activity, A = the number of nuclei of the isotope that decay per second (measured in Becquerel, Bq).
A radioactive isotope may emit energy (photons) of a specific energy, E, each day. The power (energy transferred per second) from a radioactive source = AE. This is because A is the number of decays per second.
Half life, t½ = the time taken for the mass of the isotope to decrease to half the initial mass, or, time taken for the activity to half (measured in seconds/hours/years, etc.)
It is impossible to predict which individual nucleus will decay in a sample. Every nucleus has the same starting probability. This is to do with the random nature of decay. However, macroscopically (and with enough particles in our sample) we can say with assurance that half of the particles will decay over a period of one half life.
Consider a sample of a radioactive isotope, X, that initially contains N0 nuclei. In time Δt the number of radioactive nuclei decrease by ΔN. ΔN is proportional to:
• N, the number of nuclei X remaining at time t.
• The duration of the time interval Δt.
Therefore:
Consider a sample of a radioactive isotope, X, that initially contains N0 nuclei. In time Δt the number of radioactive nuclei decrease by ΔN. ΔN is proportional to:
• N, the number of nuclei X remaining at time t.
• The duration of the time interval Δt.
Therefore:
Where λ is known as the decay constant (it is negative to show that the number of nuclei decreases).
Rearrange the equation above:
Rearrange the equation above:
The activity, A, is the rate of disintegration:
The solution to these equation is attained through integration (see capacitor decay derivation, its almost identical in this respect):
Where e^λt is an exponential function.
The following equation can be changed by taking the natural logs (ln) of each side.
The log of two multiplied numbers is the log of one plus the log of the other. In this case the above equation becomes:
This is the equation for a straight line graph. When we plot a graph of ln(N) against time, t, the gradient = λ, and the yintercept is ln(N0)
Where N0 is the number of undecayed particles in the initial sample. 
The decay constant, λ (s^(1)) = the probability of an individual nucleus decaying per second.
e.g. if there are 10000 nuclei present and 300 decay in 20 seconds, the decay constant is (300/10000)/20 = 0.0015s^1.
HalfLife Equation
At time t=0, N=N0. At time T = t½ , N = 0.5N0. Substitute this into N=N0e^(λt):
This leads to the general form of the equation for half life:
Radioactivity
Unstable isotopes will decay into more stable forms. The 3 modes of decay are alpha, beta and gamma.
In a magnetic field charged particles will be deflected due to the Lorentz force. Alpha will deflect less than beta if they are travelling at the same speed (see magnetism). Gamma is an EM wave and therefore carries no charge; it is unaffected by a magnetic field.
In a magnetic field charged particles will be deflected due to the Lorentz force. Alpha will deflect less than beta if they are travelling at the same speed (see magnetism). Gamma is an EM wave and therefore carries no charge; it is unaffected by a magnetic field.
The intensity of gamma rays decreases by the inverse square of the distance from the source. Meaning that if you double your distance from a gamma ray source (assuming it emits uniformly over a sphere) the intensity drops to by a quarter. Below is a useful diagram that demonstrates this principle:
This rule is true for almost all point sources of light/EM waves. Lasers are the exception, they can retain their intensity over large distances.
The table below gives a few more common particle interactions:
The table below gives a few more common particle interactions:

Perhaps unintuatively, atoms do not have the same number of protons as neutrons. In fact, as atoms get larger they tend to increase the number of neutrons at a greater rate than protons. This is because adding protons to an atom gets more and more difficult due to electromagnetic repulsion. If the atom is large enough (greater than the range of the strong nuclear force) it becomes more and more difficult to add protons as the atom gets bigger.
The graph opposite gives an idea of what types of atoms will decay by which means. Blue line = neutron rich Red line = proton rich At the top of the blue line heavy atoms with a large proportion of neutrons to protons will best get rid of their neutrons by emission of an alpha particle (this only happens in heavier elements). Lower down the blue line beta occurs whereby neutrons decay into (the slightly less massive) protons. For atoms on the red line to become more stable they need to get rid of their protons. This is achieved through beta+ decay whereby protons turn into neutrons (see table above). 
Radioactivity Series
The chart opposite is a pictoral representation of an atom undergoing several decays in series.

EnergyMass Equivalence
Einstein's famous equation describes how energy, E, and mass, m are related, where c is the speed of light. The equation shows how a small amount of mass can contain large amounts of energy within it due to this c^2 'conversion factor' being so large. If one was able to fully redeem the amount of energy equivalent to 100kg (as given by the equation) it would satisfy the world's consumption of energy per year! Thats the mass of one 6ft man!
Binding Energy
The binding energy of a nucleus is the “work that must be done to separate a nucleus into its constituent neutrons and protons”.
When a nucleus forms from nucleons, energy is released as the SNF does work pulling the nucleons together. The energy released is equal to the binding energy of the nucleus. Because energy is released when a nucleus forms from separate nucleons, the mass of a nucleus is less than the mass of the separate nucleons.
Mass Defect
Mass defect of nucleus, Δm = “difference between the mass of separated nucleons and the mass of the nucleus”.
The binding energy of a nucleus, Ebind, is calculated using the mass defect:
Binding Energy per Nucleon and Nuclear Stability
The binding energy per nucleon is the “average work done per nucleon to remove all the nucleons from a nucleus”. It is a measure of the stability of a nucleus.
The range of the strong nuclear force (SNF) is relatively small (~0.5 to 5fm). It is the strongest of the 4 fundamental forces. Small atoms are held together by the SNF with the effects electromagnetic (EM) repulsion being negligible. As the atom gets bigger than the range of the SNF then EM forces begin to take hold i.e. repulsion between protons. This is why smaller atoms are more 'bound' i.e. more difficult to pull apart. Pulling the nucleons apart requires a lot of energy, and likewise fusing nucleons to small atoms requires a lot of energy. This is why the curve is much steeper at the 'fusion' part of the binding energy per nucleon curve. Fusing a proton to a hydrogen atom releases a large amount of energy. Conversely, the decay of a heavy element (e.g. uranium) only releases a small amount of energy i.e. the energy change from U238 to U235 is very small compared to the energy change from hydrogen to helium. Currently there is no energy efficient way to harness this fusion energy because it takes too much energy to bring nucleons together in the first place!
Fission and Fusion
FissionThe splitting of (heavy) atomic nuclei. This further releases neutrons which are absorbed by neighbouring atoms to make the process happen again in a chain reaction. This process releases energy. In a nuclear fission reactor the chain reaction is often started using a neutron gun. In order for the chain reaction not to get out of control 'control rods' are used to moderate the number of reactions by absorbing some neutrons. 
Fusion
Fusion is the joining of atomic (usually hydrogen) nuclei, releasing energy. The process by which this happens is quite complicated. One of the ways in which helium is formed is via the following process:
The energy released at each stage can be calculated using the mass defect, Δm i.e. the mass lost as energy (E=Δmc^2) in the reaction. The final stage releases ~12.9MeV of energy.
Thermal Nuclear Reactor
 Fuel rods are made of enriched uranium. U235 is fissionable where as U238 isn't. Natural uranium contains 99% of U238 whereas enriched uranium contains about 98% U235.
 Neutrons are injected into the reactor to begin the fission reactions in the fuel rods (made of enriched uranium). These neutrons need to be slowed down in water (the 'moderator') in order that they are the correct speed to be absorbed by the fuel rods. These neutrons are called 'thermal neutrons' because they are in thermal equilibrium with the water surrounding them, its not to do with heat.
 A chain reaction in the fuel rods releases many neutrons. Some of these are absorbed by the fuel rods. The fuel rods can be raised and lowered in order to moderate the chain reaction and to ensure that the chain reaction doesn't get dangerously out of control.
Safety Features of a Thermal Nuclear Reactor
 Steel  withstand high pressures and temperatures and can absorb alpha, beta and neutrons.
 Concrete  absorbs neutrons and gamma.
 Control rods  can be raised and lowered to control rate of neutron production (and thus fission reactions).
 Fuel rods  when removed they are radioactive and are therefore only handled remotely.
Radioactive Waste
Radioactive waste most commonly comes from nuclear power stations, specialist research, industry or hospitals. Radioactive waste can be categorized in the following 3 ways according to their activity:
1) Highlevel radioactive waste: Spent fuel rods from nuclear power stations. Must be removed by remote control and placed in ponds to cool for up to a year. Then stored in large steel containers where and processed for remaining useful materials. The rest is stored deep underground in sealed containers, potentially for centuries. There are many long term problems with this... 2) Intermediatelevel waste: Specialist radioactive materials with low activity. Sealed in drums in encased concrete, stored in buildings of reinforced concrete. 3) Lowerlevel waste: Lab equipment and contaminated clothing. Sealed and stored in trenches. 