AcademyNuclear Physics
Academy
Activity and Half-Life
Level 1 - Physics topic page in Nuclear Physics.
Principle
Activity, count rate, and half-life connect measured radiation rates to the number of unstable nuclei.
Notation
\(C\)
detected count rate
\(\mathrm{s^{-1}}\)
\(C_{\mathrm b}\)
background count rate
\(\mathrm{s^{-1}}\)
\(\epsilon\)
detection efficiency
1
\(A\)
source activity
Bq
\(\lambda\)
decay constant
\(\mathrm{s^{-1}}\)
\(m\)
sample mass
\(\mathrm{kg}\)
Method
Derivation 1: Correcting count rate
The detector records a fraction of source decays plus background.
Net count rate
\[C_{\mathrm{net}}=C-C_{\mathrm b}\]
Activity estimate
\[A=\frac{C_{\mathrm{net}}}{\epsilon}\]
Derivation 2: Atoms from mass
Number of radioactive atoms links mass to activity.
Number of nuclei
\[N=\frac{m}{M}N_A\]
Activity from mass
\[A=\lambda\frac{m}{M}N_A\]
Derivation 3: Dating ratio
For a remaining fraction, solve the exponential decay law.
Remaining fraction
\[f=\frac{N}{N_0}=e^{-\lambda t}\]
Age
\[t=-\frac{1}{\lambda}\ln f\]
Rules
Net count
\[C_{\mathrm{net}}=C-C_{\mathrm b}\]
Activity estimate
\[A=\frac{C_{\mathrm{net}}}{\epsilon}\]
Dating age
\[t=-\frac{1}{\lambda}\ln\left(\frac{N}{N_0}\right)\]
Examples
Question
A detector reads
\[90\,\mathrm{s^{-1}}\]
with background \[15\,\mathrm{s^{-1}}\]
Find net count rate.Answer
\[C_{\mathrm{net}}=90-15=75\,\mathrm{s^{-1}}\]
Checks
- Subtract background before inferring source activity.
- Count rate equals activity only for ideal geometry and efficiency.
- Use the same time units for \(t\), \(t_{1/2}\), and \(\lambda\).
- Dating assumes a closed system and known initial ratio.