Random decay
Definition
Random process
A process in which it is impossible to predict when an individual event will happen.
- Radioactive decay is a random process.
- It is impossible to predict which nucleus in a sample will decay next or when any particular nucleus will decay.
- Every unstable nucleus of the same isotope has the same chance of decaying in any given second.
- This chance does not depend on how long the nucleus has already existed or on its surroundings, such as temperature or pressure.
- Although a single nucleus is unpredictable, a sample contains a huge number of nuclei, so the overall rate of decay is predictable.
- Popcorn behaves in a similar way, because you cannot say which kernel will pop next, but you can predict roughly how long it takes for half of them to pop.
Activity and count rate
Definition
Activity
The number of unstable nuclei in a radioactive source that decay each second, measured in becquerels (Bq).
Definition
Becquerel
The unit of activity, equal to one decay per second.
Definition
Count rate
The number of decays recorded by a detector each second or each minute.
- The activity of a radioactive source is the number of nuclei that decay each second.
- Activity is measured in becquerels, Bq\text{Bq}Bq, where 1 Bq1\ \text{Bq}1 Bq is one decay per second.
- A Geiger-Muller tube connected to a counter detects radiation and records each detection as a count.
- The number of counts recorded per second or per minute is the count rate.
- The count rate is lower than the activity, because the detector only catches the radiation that travels towards it.
- Because decay is random, repeated count-rate readings from the same source vary, so several readings are taken and a mean is calculated.
- A detector also picks up background radiation from the surroundings, so the background count rate is measured without the source and subtracted from each reading.
Half-life
Definition
Half-life
The time taken for the number of undecayed nuclei in a radioactive sample, or its activity, to fall to half of its starting value.
- The half-life of a radioactive isotope is the time taken for half of the unstable nuclei in a sample to decay.
- It is also the time taken for the activity, or the count rate, of the sample to fall to half of its value.
- The half-life is the same whatever amount of the isotope you start with, so 1 g1\ \text{g}1 g and 1 kg1\ \text{kg}1 kg of an isotope have the same half-life.
- The half-life is the same at every stage, so it takes the same time to fall from 800800800 to 400400400 undecayed nuclei as from 400400400 to 200200200.
- Half-lives vary hugely between isotopes.
- Radon-220 has a half-life of about 565656 seconds.
- Technetium-99m has a half-life of about 666 hours.
- Iodine-131 has a half-life of about 888 days.
- Carbon-14 has a half-life of about 570057005700 years.
- Uranium-238 has a half-life of about 4.54.54.5 billion years.
Why random decay gives a half-life
- Every nucleus has the same chance of decaying in each second, so the number that decay each second is a fixed fraction of the number still left.
- A large sample has many nuclei, so many decay each second and the activity is high.
- As nuclei decay, fewer are left, so fewer decay each second and the activity falls.
- Because the same fraction always decays in the same time, the time taken for the number to halve is always the same, and this time is the half-life.
- The half-life is a statistical quantity, so it describes large numbers of nuclei and cannot tell you when one particular nucleus will decay.
- With only a few nuclei the random variation is large, so the halving pattern only appears clearly when the number of nuclei is large.
Practical
Modelling random decay with dice
- Aim: to model the random decay of unstable nuclei using dice, and to see how random events produce a predictable half-life.
- Apparatus: 100100100 six-sided dice, a large tray with raised sides, a results table and a pencil.
- Method, setting up:
- Count out exactly 100100100 dice, each of which represents one unstable nucleus.
- Draw a results table with columns for the throw number and the number of dice remaining, and record 100100100 dice at throw 000.
- Method, throwing:
- Tip all of the dice into the tray together so that they roll freely.
- Remove every dice that lands showing a six, because these dice represent nuclei that have decayed.
- Count the dice that are left and record the number in the table.
- Gather the remaining dice and throw them again, removing the sixes each time.
- Continue until fewer than five dice are left.
- Method, repeating:
- Repeat the whole experiment at least three times, or combine results with other groups, and calculate the mean number remaining after each throw.
- Results: about one sixth of the remaining dice are removed on each throw, so the number falls from 100100100 to roughly 838383, 696969, 585858 and 484848, halving after about four throws.
- Analysis: the number of throws needed for the dice to halve is about the same whether you start counting from 100100100, from 505050 or from 252525, which models a constant half-life.
- Variables: the number of throws is the independent variable, the number of dice remaining is the dependent variable, and the type of dice and the face chosen to mean decay are control variables.
- Watch out: you cannot predict which dice will show a six, just as you cannot predict which nucleus will decay, and the pattern becomes less clear as the number of dice gets small.
- Safety: keep the dice in the tray and pick up any that fall on the floor so that nobody slips on them.
Example
Finding a half-life from count rates
- A Geiger-Muller tube measures a background count rate of 303030 counts per minute.
- With a radioactive source in place, the count rate is 830830830 counts per minute at the start and 430430430 counts per minute after 202020 minutes.
- The background is subtracted from each reading to find the count rate from the source alone.
- The corrected count rate has halved in 202020 minutes, so the half-life of the source is 202020 minutes.
Common Mistake
- Do not say that half-life is the time for half of a sample to disappear or for half of its mass to be lost, because the daughter nuclei remain.
- Do not say that half-life is half the time for the whole sample to decay, because the number left keeps halving and never quite reaches zero.
- Do not say that each nucleus has a half-life, because half-life describes a large group of nuclei, not a single one.
Exam technique
Explaining half-life and randomness
- Define half-life as the time for half the unstable nuclei to decay or for the activity to halve, because "half the atoms" alone may not be credited.
- To explain random decay, state that you cannot predict when a particular nucleus will decay, or which nucleus will decay next.
- When given count rates, subtract the background count before looking for the time taken to halve.
- Give a half-life with its unit, such as minutes, days or years.
Self review
- What does it mean to say that radioactive decay is random?
- What is meant by the half-life of a radioactive isotope?
- What is the unit of activity?
- Why is background radiation subtracted from count-rate readings?
- Why can the half-life of a sample be predicted even though the decay of each nucleus is random?
