AcademyNuclear Physics
Academy
Radioactivity
Level 1 - Physics topic page in Nuclear Physics.
Principle
Radioactive decay is random for each nucleus but exponential for a large population.
Notation
\(N(t)\)
number of undecayed nuclei
1
\(N_0\)
initial number of nuclei
1
\(\lambda\)
decay constant
\(\mathrm{s^{-1}}\)
\(A(t)\)
activity
Bq
\(t_{1/2}\)
half-life
\(\mathrm{s}\)
\(\tau\)
mean lifetime
\(\mathrm{s}\)
Method
Derivation 1: Decay law
The number decaying per unit time is proportional to the number still present.
Rate equation
\[\frac{dN}{dt}=-\lambda N\]
Population
\[N(t)=N_0e^{-\lambda t}\]
Derivation 2: Activity
Activity is the decay rate magnitude.
Activity definition
\[A=-\frac{dN}{dt}\]
Activity law
\[A(t)=\lambda N(t)\]
Derivation 3: Half-life
Half-life is the time required for the population to halve.
Half condition
\[\frac{N_0}{2}=N_0e^{-\lambda t_{1/2}}\]
Half-life
\[t_{1/2}=\frac{\ln2}{\lambda}\]
Rules
Decay law
\[N=N_0e^{-\lambda t}\]
Activity
\[A=\lambda N\]
Half-life
\[t_{1/2}=\frac{\ln2}{\lambda}\]
Examples
Question
How much of a sample remains after three half-lives?
Answer
\[(1/2)^3=1/8\]
remains.Checks
- Activity is measured in becquerels: \(1\\,\\mathrm{Bq}=1\\,\\mathrm{s^{-1}}\).
- Half-life is independent of the initial sample size.
- Exponential decay never reaches exactly zero in the model.
- Decay probability per nucleus per unit time is \(\lambda\).