Half-life turns random decay into something you can predict

1. No individual nucleus can be predicted, but with billions of them the fraction that decays in a given time is remarkably steady.
2. The words any sample matter: the half-life is the same whether you start with a gram or a tonne, because it is a fraction rather than an amount.
3. Half-life is fixed for a given isotope and cannot be altered by heating, cooling or chemically reacting the sample.
4. Values range enormously, from fractions of a second for some isotopes to billions of years for uranium-238.
5. Because the count rate is proportional to how many undecayed nuclei remain, the count rate halves over the same interval, which is what makes half-life measurable in a lab.

Counting in half-lives: halve, halve, halve again

1. After one half-life, half the original nuclei are left; after two, a quarter; after three, an eighth.

2. The pattern generalises to any number of half-lives $n$.

   $$fraction\; remaining={\left(\frac{1}{2}\right)}^{n}wheren=\frac{time\; elapsed}{half-life}$$

3. Each halving takes the same length of time, so the sample never quite runs out: half of whatever is left always survives the next interval.

4. The nuclei that have decayed have not vanished, of course; they have become nuclei of the daughter, so the number of daughter nuclei rises as the number of parent nuclei falls.

Reading a half-life off a decay curve

1. A decay curve plots count rate on the vertical axis against time on the horizontal axis, and it always falls steeply at first and then flattens.
2. To read the half-life off the graph, work through three steps.
   1. Take the count rate at the start and halve it.
   2. Draw a line across from that halved value to the curve, then straight down to the time axis.
   3. The time you read off is the half-life, because that is how long the count rate took to fall by half.
3. Repeat the construction starting from a different point on the curve and average your answers, since the halving interval should come out the same wherever you begin.

Subtract the background before you halve anything

1. Real readings from a detector always include background radiation as well as the source.
2. The giveaway on a graph is the tail: an uncorrected curve levels off at the background count rate rather than heading towards zero.
3. Halving an uncorrected reading gives a half-life that is too long, because part of what you halved was never going to decay away.
4. So subtract the background count rate from every reading first, and only then halve and read across.
5. If the background figure is not given directly, look for a reading taken with no source present, or read the value the tail of the curve settles at.