Radioactive decay
A source counted with a Geiger tube, decay by decay, with the √N scatter of real counting statistics and a background that does not decay.
Nuclear & quantum — Radioactive decay, the absorption of radiation, photons, and energy levels in the atom.
Experiments already set up on it
Measuring a half-life
Wait, count, repeat. ln R against t is straight, with gradient −λ.
- A = A₀e^{−λt}, so ln A against t is straight with gradient −λ
- t½ = ln2/λ
- Repeats scatter because counting is random, not because the apparatus is poor
- The background does not decay, so it bends the log graph at the end
Why count for longer?
The same source, counted for different times: √N/N falls as 1/√N.
- A count of N carries an uncertainty of √N
- So the *fractional* uncertainty is 1/√N, and falls as you count for longer
- Nothing about the apparatus changed — only how long you watched
What is the background?
Take the source away and the counter still clicks. Measure what is left.
- There is a count rate with no source at all
- It does not decay, so it must be subtracted from every other reading
- Measuring it well means counting for a long time, because it is small
What you can measure
Each of these can be recorded over time, aggregated over a run, or read at the moment something happens — through an instrument that carries its own uncertainty.
- Timet (s)
- Counts so farN
- Count rateR (1/s)
- Counting time so fart_c (s)
- Activity of the sourceA (1/s)
- Nuclei leftN_n
- Fraction leftf
What you can change
Any of these can be the thing you vary, and every combination of the ones you choose is run as its own trial.
- Half-lifet_h (s)
- Activity when preparedA₀ (1/s)
- Efficiency of the tubeε
- Background rateR_b (1/s)
- Wait before countingt_d (s)
- Count fort_g (s)
The equations behind it
What radioactive decay is built on. Every one of these is in the lab's formula reference too, where it copies straight into a write-up.
Radioactive decay
- Number of nuclei remaining
- Activity
- Activity with time
- Half-life
Also in Nuclear & quantum
Absorption of radiation
Alpha, beta and gamma through absorbers of your choosing: only one of the three is absorbed exponentially, and the counter still does not subtract the background.
Photoelectric effect
Light on a metal plate and a retarding voltage turned up until the current stops: the stopping voltage is found rather than calculated, and brightness changes nothing about it.
Muon decay & time dilation
Cosmic-ray muons counted at any altitude you like: far more arrive than a muon’s lifetime allows, and only a slow-running clock accounts for it.
Electron diffraction
Electrons accelerated through a voltage and diffracted by graphite: the rings move as 1/√V, which is de Broglie’s λ = h/p made visible.
Atomic spectra
Hydrogen through a grating spectrometer: each line is an electron falling between two levels, and a graph of them gives the Rydberg constant.
Rutherford scattering
Alpha particles onto a thin foil, counted at an angle: the 1/sin⁴(θ/2) law is what a point nucleus gives, and a head-on alpha says how small it is.