Nuclear radiation (Topic 11)
On this page
An example of mass–energy equivalence is the annihilation of an electron–positron pair. When the two particles collide, they annihilate and produce two photons. The mass of the electron and positron is converted into two photons, i.e., energy.
When there is a change in the mass of an object or system, then there is an equivalent change in the amount of energy represented by Einstein’s equation:
A moving object possesses kinetic energy. Thus, the equation implies that the object’s mass when it is moving is greater than its mass while at rest (rest mass).
During radioactive decay, unstable nuclei decay by ejecting radiation in the form of particles such as alpha particles or high-energy photons in order to transition to a more stable state.
For example, during alpha decay, the initial nucleus (the parent nucleus) is unstable and emits an alpha particle (a helium nucleus) to become more stable, resulting in a more stable daughter nucleus:
The daughter nucleus recoils in the opposite direction to the alpha particle’s trajectory, so that mass, momentum and energy are conserved.
Energy is released during nuclear decay reactions. Therefore, there must also be an equivalent decrease in the total mass of the system.
This means that the total mass of the daughter nucleus and alpha particle combined must be less than the mass of the parent nucleus, i.e. the decrease in mass is equivalent to the amount of energy released from the reaction
This principle is also true for beta-minus and beta-plus decay:
In some nuclear reactions, energy may be absorbed. This occurs when there is an increase in the total mass of the system, i.e. the total mass of the products is greater than the total mass of the reactants.
This increase in mass, is equivalent to the amount of energy absorbed in the reaction, For example, this occurs in the fission of light elements and the fusion of heavy elements (heavier than iron).
Question walkthrough
Kinetic Energy Released in Alpha Decay
Calculates the total kinetic energy released when uranium-238 undergoes alpha decay, using the mass defect between the parent and daughter nuclei.
In order to split nuclei apart, we need to supply energy in order to overcome the strong attraction between the nucleons due to the strong nuclear force.
According to Einstein’s mass–energy equation, which states mass and energy are equivalent, the total mass of the split nucleons must be greater than the mass of the bound nucleus itself. This is because the energy is supplied in order to break the nucleus apart.
The definition of the mass defect of a nucleus is the difference between the mass of the separated nucleons and the mass of the bound nucleus.
The mass defect of a nucleus can be converted into a corresponding amount of energy using Einstein’s equation. This energy represents the binding energy of the nucleus.
The definition of the binding energy of a nucleus is the minimum energy required to completely separate a nucleus into its constituent protons and neutrons.
The more tightly bound a nucleus is, the harder it is to separate the nucleons and split them apart. Remember, the binding energy is what holds the nucleus together and not the entire atom.
We can measure how tightly bound a nucleus is and make comparisons with other nuclei by considering the average binding energy per nucleon.
The average binding energy per nucleon is defined as the total binding energy of a nucleus divided by the number of nucleons it contains. It represents the energy required to remove a single nucleon, on average, from the nucleus.
The greater the average binding energy per nucleon, the more tightly bound the nucleus. The more tightly bound the nucleus, the more stable it is.
Atomic mass units are used to express the mass of atoms and nuclei.
One atomic mass unit is equal to the mass of a neutral carbon-12 atom divided by 12:
- the small difference in mass between the proton and neutron has a significant impact on the binding energy of nuclei.
- it is essential to make accurate calculations of the average atomic mass of an element that exists in a mixture of isotopes.
The mass of a nucleus is approximately given by its nucleon number multiplied by the atomic mass unit, since the proton and neutron each have a mass of approximately
The electron has a mass of approximately
By plotting the average binding energy per nucleon against the atomic mass for all elements in the periodic table, we can visualise which elements have the most tightly bound nuclei and see how this changes as the size of the nucleus increases.
The binding energy per nucleon increases as the atomic mass increases up to the isotope iron-56. From iron-56 onwards, the binding energy per nucleon starts to decrease slightly. Since the curve peaks at iron-56, this means that this is the most stable (and tightly bound) nucleus in nature.
Energy is released in radioactive decay, and this can be shown using the binding energy per nucleon plot. If a nucleus spontaneously alpha-decays, the binding energy of the parent nucleus is less than the binding energy of the daughter nucleus and the alpha particle.
For example, uranium-238 alpha-decays into thorium-234. The resulting thorium-234 has a lower mass number and, according to the binding energy per nucleon against nucleon number curve, it has greater binding energy per nucleon than uranium-238. The excess energy is released in the decay in the form of kinetic energy of the thorium nucleus and the alpha particle.
We can use Einstein’s mass–energy equation to calculate the binding energy of a nucleus:
We can get a measure of how tightly bound a nucleus is by looking at the average binding energy per nucleon. The greater the average binding energy per nucleon, the more tightly bound the nucleus and the more energy that is needed to completely split the nucleus apart.
Question walkthrough
Binding Energy of a Plutonium-239 Nucleus
Calculates the binding energy of a plutonium-239 nucleus from its mass defect, using the individual masses of its constituent protons and neutrons.
Uranium is the most common fuel used in nuclear power stations. The isotope uranium-235 readily undergoes fission when absorbing a low-energy neutron. These low-energy neutrons are known as thermal neutrons.
As uranium-235 undergoes fission when absorbing a low-energy neutron, there will be a mass defect between the parent nuclei and the daughter products. This mass defect is converted to useful energy for our power stations.
Whereas, the isotope uranium-238 requires high-energy, fast neutrons to fission, which are less common in nuclear reactors. U-238 nuclei are more likely to capture a neutron to form uranium-239 because the probability of fission with low-energy thermal neutrons is low. The unstable nuclei will then quickly decay into plutonium-239 and not fission.
Both uranium-235 and uranium-238 can spontaneously undergo nuclear fission without the need for neutrons, but this process is rare. When uranium-235 absorbs a neutron, it becomes uranium-236 and has a much greater chance at spontaneously fissioning.
When the uranium-236 fissions, it releases fission products and neutrons. These products have very high kinetic energy, which is utilised in a nuclear reactor. They transfer their kinetic energy via collisions to a coolant, which increases the coolant’s temperature. The coolant is then used to heat and boil a circuit of water to create steam to drive a generator turbine, resulting in electricity. This is the basic operating principle of a nuclear power station.
Below is an example of a typical neutron-induced fission reaction of uranium-235:
The uranium-235 nucleus absorbs a thermal neutron to become uranium-236. The highly unstable uranium-236 fissions almost immediately to produce the two daughter nuclei barium-141 and krypton-92. Three fast (high energy) neutrons are also released in the reaction.
The total mass of nuclear fission products is less than the total mass of the particles before fission due to the total binding energy of the products being greater than the total binding energy of the particles before fission.
The difference in mass corresponds to the difference in binding energy , and is the energy released in the reaction.
The energy released is in the form of kinetic energy of the fission products and energy of any other particles produced, such as photons.
The three neutrons produced in a uranium-235 fission reaction are fast neutrons. If they are slowed to lower energies to become thermal neutrons, then they may go on to cause further fission of other nearby uranium-235 nuclei. This is known as a chain reaction, as shown below.
Further fissions release more neutrons, which cause further fissions and so on. After fission events, there will be neutrons, so the growth of neutrons is exponential.
Nuclear reactors are able to control this reaction to produce a steady output of power.
Nuclear fusion is the process of combining small nuclei into larger nuclei, releasing several of energy per reaction. This process is what occurs in the cores of stars.
In order for nuclei to fuse, they have to be brought very close together within a few femtometres, At these short distances, the strong nuclear force takes effect and overcomes the electrostatic repulsion between the protons created by the Coulomb force.
Fusion requires very high temperatures in order for the nuclei to have enough kinetic energy to get close enough for the strong nuclear force to take effect. The conditions in stars are ideal for fusion due to the very high temperatures and their very high density.
Protons are positively charged. Like charges repel, so protons in the atom should repel each other.
The electrostatic force of attraction between two like charges is inversely proportional to the square of the distance between them:
Therefore, as becomes small, such as the spacing between protons in the nucleus, the electrostatic force of repulsion becomes extremely large.
The size of the nucleus is approximately All protons in an atom are squeezed within this small space. This distance of separation corresponds to extremely large electrostatic forces between protons of about
The electrostatic repulsion between protons in the nucleus is balanced by the strong nuclear force. Colloquially referred to as the strong force:
- The strong nuclear force acts between all nucleons (protons and neutrons) equally.
- The strong force is exclusively a short-range force and becomes practically zero when nucleons are separated by distances of about where is a femtometre, or
- The strong force is repulsive at extremely short distances, meaning that the nucleus does not crush itself.
- The strong force is attractive between distances of approximately and
There are multiple fusion reactions that can occur and are often sequential reactions. For example, the production of helium-3 in stars results from the proton–proton chain:
The cycle is repeated again with the two protons produced at the end. Each reaction releases energy.
Currently, there are no power stations utilising nuclear fusion. There are challenges with maintaining the very high temperatures for sufficient periods of time and confining the hot fuel in the reactor. The first controlled nuclear fusion reaction that produced more energy than it took to sustain only occurred in 2022, but it was still far from being economical.
Future fusion devices, such as ITER in Europe, will use deuterium and tritium nuclei as fuel. The fuel will be heated to temperatures above 150 million kelvin, where it becomes a plasma. In ITER, powerful magnetic fields will be used to confine the plasma to a doughnut shape to prevent it from touching the vessel walls and losing energy.
The fusion of tritium and deuterium nuclei produces a helium nucleus and a free neutron.
This reaction releases of energy in the form of kinetic energy of the products. The neutron possesses 80% of the energy and will be absorbed by a lithium blanket surrounding the fusion reactor. This interaction of the neutron with lithium breeds more tritium to be used as fuel.
Ionising radiation includes three main types: alpha particles, beta particles and gamma rays, which have sufficient energy to remove electrons from atoms:
- The ionisation power of radiation is the ability to ionise atoms: a greater mass leads to a greater ionisation power.
- Penetration power of radiation is the ability to penetrate through matter: a smaller mass leads to a greater penetration power.
The table below summarises some of the properties of each type of radiation.
Alpha particles
- Made of two neutrons and two protons: a helium nucleus
- Emitted from nuclei that are too large to be stable
Beta particles
- Beta-minus particles are high-energy electrons, emitted from nuclei with too many neutrons to be stable.
- Beta-plus particles are high-energy positrons, emitted from nuclei with too many protons.
Gamma rays
- High energy electromagnetic waves
- Emitted from a nucleus that has excess energy after alpha or beta decay
This tip details an experiment to investigate the penetration of different forms of radiation using a radioactive source and Geiger–Müller tube (the sensing element of a Geiger counter):
- Independent variable: specimen absorber material.
- Dependent variable: count rate.
- Controls: distance between source and Geiger counter, background radiation and the sealed source.
Method:
- Without the source, measure the background count rate three times for one minute and take the average.
- Place the sealed source a fixed distance of from the Geiger–Müller tube and take another reading. Note that the source should be sealed in a lead box and only moved with tongs.
- Now place a specimen material between the source and Geiger–Müller tube. The specimen materials used are: paper, aluminium sheets and lead.
- Using one specimen material at a time, take three count rate readings over one minute and calculate an average.
Question walkthrough
Identifying Radiation Type by Absorption
Identifies the types of radiation emitted from a source by interpreting count rate drops as paper, aluminium, and lead absorbers are introduced.
X-rays are a form of ionising radiation: as they pass through matter they will lose energy due to attenuation. This causes a gradual decrease in the intensity of the X-ray, where intensity is defined as the power per unit of cross-sectional area and has units of
The penetrating power of the X-ray is dependent on the material it is passing through. An incoming X-ray will be attenuated based on:
- the thickness of the material,
- the density of the material, and
- the atomic number of the material.
A very thick, dense, high atomic number material will attenuate the X-ray a lot more than a thin, less dense, low atomic number material.
Different materials will attenuate X-rays to a different extent; this can be used to distinguish different materials inside the body such as bone and soft tissue.
On the periodic table, all elements are represented by nuclide notation:
Where:
- is the nucleon number: the total number of protons and neutrons. is also known as the mass number, since it gives the mass of the nucleus in units of the proton (or neutron) mass.
- is the proton number: the number of protons. is also known as the atomic number, since it determines the type of atom.
Radioactive decay of a nucleus alters the nucleon number and/or the proton number.
When balancing nuclear transformation equations, it is important to remember to conserve the following quantities on either side of the equation:
- Mass number,
- Atomic number,
Alpha decay occurs when a nucleus has too much mass, which makes the nucleus unstable.
In large nuclei with more protons, the larger electromagnetic repulsion between the protons becomes larger compared to the strong nuclear force holding the nucleus together and makes the nucleus unstable.
The daughter nucleus will have a lower mass number by four and a lower proton number by two.
Beta minus decay releases a high speed, high energy electron.
Beta minus decay occurs in nuclei with too many neutrons: a neutron in the nucleus is converted to a proton, an electron, and an electron antineutrino.
The daughter nucleus will have an increased proton number since the beta minus particle has a −1 proton number. The lepton number must be conserved, hence an antineutrino is also emitted.
Beta plus decay releases a positron – the antiparticle of an electron – which has a positive charge.
Beta plus decay occurs in nuclei with too many protons; a proton in the nucleus is converted to a neutron, a positron and an electron neutrino.
The daughter nucleus will have a decreased proton number since the beta plus particle has a +1 proton number. The lepton number must be conserved, hence a neutrino is also emitted.
Gamma decay occurs following decay from alpha and/or beta decay, where the daughter nucleus still has too much energy to be stable. No particles are released.
The daughter nucleus will have a preserved nucleon and proton number. Only energy is expelled through gamma decay.
Decays involving beta-minus or beta-plus can be confused. Remember that lepton number is conserved: in beta-minus decay an electron, is released with an electron antineutrino, whereas in beta-plus decay a positron is released with an electron neutrino,
Question walkthrough
Balancing a Deuterium-Tritium Fusion Reaction
Shows that a deuterium-tritium fusion reaction is balanced by checking that mass number and atomic number are conserved on both sides.
Unstable atomic nuclei can break down by radioactive decay. Radioactive decay releases radiation from the nucleus in the form of photons, matter or antimatter particles.
After the decay, the remaining nucleus, called the daughter nucleus, is more stable than the original nucleus.
Radioactive decay is a completely random process: in a sample of radioactive nuclei, it cannot be known which nucleus will decay first.
Radioactive decay is spontaneous: the decay of the nuclei in a radioactive sample is not affected by the presence of other nuclei or any external factors. For example, heating up a sample or increasing its pressure will not make the nuclei decay faster.
Radioactive decay is spontaneous and random. Therefore, radioactivity is described through an average decay rate.
Although it is impossible to know when a specific radioactive isotope will decay, radioactive samples have many, many nuclei, which makes the decay behaviour follow statistical patterns. For example, one gram of hydrogen contains nuclei.
The activity of a sample is the number of decays per unit time, and is measured in becquerels . An activity of of a beta radiation source means there are 1000 beta particles released per second.
Activity depends on the decay constant and the number of nuclei in the sample:
The decay constant, is the probability that a given nucleus will decay per second. It has units of inverse seconds , and a larger value results in a higher level of activity.
Question walkthrough
Activity of a Beta-Minus Source from Power
Calculates the activity of a beta-minus emitting sample from its total emitted power and the energy carried by each beta particle.
The half-life of a radioactive sample is the time taken for half of the nuclei to decay: this can be measured from experimental trials.
For any radioactive sample, the product of the half-life and the decay constant is :
This equation shows that half-life and the decay constant are inversely proportional, since the above equation can be rearranged to:
Therefore, the shorter the half-life, the larger the decay constant and the faster the decay.
The half-life of a radioactive sample, protactinium-234, can be experimentally determined. Protactinium-234 will decay via beta radiation into uranium-234.
Method:
- A bottle contains a uranium-234 salt solution, protactinium-234 and an organic solvent.
- Shaking the bottle dissolves the protactinium-234 in the solvent, and the uranium-234 remains in an aqueous solution: the two elements settle into layers as shown in the diagram.
- Measure the background radiation levels with a Geiger–Müller tube.
- Protactinium-234 will decay; the beta particle count rate is measured via the Geiger–Müller tube connected to a data logger.
- The uranium-234 will further decay via alpha radiation but this is absorbed by the glass bottle.
- Measuring the activity versus time, the half-life can be determined. Always subtract the background radiation from readings.
The nuclei in a radioactive sample decay at a rate proportional to the total number of nuclei, Mathematically, this is exponential decay: the number of undecayed nuclei decreases exponentially with time.
For an initial number of unstable nuclei the number of unstable nuclei remaining after a time can be found through the following equation:
Where is the decay constant. Substituting this into the equation for activity,
where the initial activity has been written as:
Radioactive decay is a random and spontaneous process:
- It cannot be known when a particular nucleus will decay.
- The rate of decay is unaffected by external conditions.
Radioactivity can be modelled by rolling a large number of dice:
- Every die represents an unstable undecayed nucleus in a sample.
- Roll each dice and remove those that land on one.
- The dice with a one have decayed into a stable daughter nucleus (of a different element)
- Remove the ‘decayed’ dice from the sample
- Repeat this rolling procedure several times.
- Record the number of ‘undecayed’ dice remaining after each roll.
The example plot above of the number of dice remaining against roll number for the method outlined above follows the same exponential decay curve as the activity of a radioactive source.
If the roll number corresponds to the time in seconds, then the probability of rolling a one is , which corresponds to the decay constant: the probability of decay of a given nucleus per second.
Question walkthrough
Deriving the Exponential Decay Equation
Derives the exponential equation of radioactive decay, N = N₀e^(-λt), starting from the fact that decay rate is proportional to the number of nuclei present.
The activity of a radioactive sample is:
Activity is defined as the number of decays per unit time, which can be written as:
Where:
- is the change in the number of nuclei in a short time period, and
- represents the time inverval. It must be short in comparison to the half-life so that the activity can be assumed to be constant. There is a minus sign because the number of nuclei remaining decreases.
Combining these two equations gives:
By selecting a small time interval the decrease in the number of undecayed nuclei in a radioactive sample can be modelled by:
Where is the number of undecayed nuclei at the start of the time interval. Note that the negative sign is not included as is defined as the decrease in the number of undecayed nuclei.
The equation can be used over successive time intervals to find the approximate number of undecayed nuclei over time. This method is referred to as iterative modelling. The number of undecayed nuclei is given exactly by which can be compared to the values obtained by the iterative modelling method.
The table below shows the results of the iterative modelling method compared to the true values for and which corresponds to
The two models show good agreement, even with the large time interval of (compared to the half-life). Using a smaller time interval would make the iterative model even more accurate.
In a radioactive sample, the number of unstable nuclei decays exponentially with time
Exponential decay shows a characteristic downwards curve, which can be used to calculate the half-life.
The half-life can be found by finding the time at which the initial number of unstable nuclei falls to ; shown in red as 20 seconds for the curve above.
After two half-lives, the number of unstable nuclei remaining will be . Therefore, multiple values of the half-life can be calculated and averaged to obtain a more accurate value.
An exponential graph can be plotted on a logarithmic scale to produce a straight line graph, which is useful for calculating the half-life. Rearranging
1) Take logarithms of both sides, remembering that
2) Rearrange into the equation of a straight line,
Plotting against gives a straight line with equal to the negative of the gradient.
The half-life can be calculated from: