Radioactivity (6.4.3)
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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.
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.
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.
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,
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:
Carbon exists as three naturally occurring isotopes: C-12, C-13 and C-14. C-14 is radioactive and decays with a half-life of via beta decay.
The ratio between C-14 and C-12 in the atmosphere is approximately constant:
Living organisms absorb carbon through photosynthesis or by eating other carbon-rich organisms. The ratio of C-14 to C-12 will match atmospheric ratios. After an organism dies, it will no longer absorb carbon but the C-14 it contains will continue to decay.
Dead tissue samples can be tested to measure the C-12 to C-14 ratio and by comparison to atmospheric levels, the time since death can be estimated. This is known as carbon-dating.
There are negatives to using carbon-dating to find the age of a dead organism:
- The method assumes a constant C-14 to C-12 atmospheric ratio, ignoring possible fluctuations with time.
- Small organisms might have undetectable traces of C-14 compared to background radiation.
- For organisms around 40,000 years old, less than 1% of their C-14 remains, which is often too small to detect.
Question walkthrough
Carbon Dating an Ancient Burial Site
Calculates the age of an ancient linen sample from its C-14 to C-12 ratio compared to the atmospheric ratio, using the half-life of carbon-14.