Cambridge Notes, Past Papers, Revision Questions

Radioactive Decay – Cambridge AS & A‑Level Physics (9702)

1. Overview of Nuclear Physics (Syllabus 23 N)

1.1 Mass‑defect & Binding Energy

Mass‑defect (Δm): the difference between the summed masses of the separate nucleons and the actual mass of the nucleus.
\[
\Delta m = \bigl(Zmp + Nmn\bigr) - m_{\text{nucleus}}
\]

Binding energy (E_bind): the energy required to separate a nucleus into its constituent nucleons. From Einstein’s relation,
\[
E_{\text{bind}} = \Delta m\,c^{2}
\]

Binding‑energy per nucleon curve: rises rapidly for light nuclei, peaks around iron (A≈56) and then falls slowly for heavier nuclei.
- Peak → nuclei are most stable (e.g. ⁵⁶Fe).
- Left of the peak: fusion releases energy (light nuclei combine).
- Right of the peak: fission releases energy (heavy nuclei split).

Energy released in a nuclear reaction:
\[
Q = \bigl(m{\text{reactants}}-m{\text{products}}\bigr)c^{2}
\]
Example (fusion of deuterium):
\[
{}^{2}\!{\rm H}+{}^{3}\!{\rm H}\rightarrow{}^{4}\!{\rm He}+n+17.6\ \text{MeV}
\]

1.2 Radioactive Decay

A spontaneous, random process that transforms an unstable nucleus into a more stable configuration.

Each individual nucleus has a constant probability per unit time of decaying – the decay constant λ.

2. Types of Radioactive Decay

Decay mode	Change in (A,Z)	Particle(s) emitted	Typical penetrating power	Common detector
α‑decay	(A‑2, Z‑2)	⁴He nucleus (α‑particle)	Low – stopped by a sheet of paper	Scintillation or semiconductor detector (high α efficiency)
β⁻‑decay	(A, Z+1)	Electron + antineutrino	Medium – penetrates paper, stopped by ≈1 mm Al	Geiger‑Müller tube, plastic scintillator
β⁺‑decay (positron)	(A, Z‑1)	Positron + neutrino (annihilation → 2 × 511 keV γ)	Medium – γ‑rays are highly penetrating	Scintillator + coincidence with 511 keV γ
γ‑decay	(A, Z) unchanged	High‑energy photon	High – requires lead or several cm of concrete	NaI(Tl) crystal, HPGe detector

3. Conservation Laws in Decay

Charge (Z) – conserved.

Mass number (A) – conserved when the emitted particle’s mass is accounted for (e.g. α‑particle carries A = 4).

Energy & momentum – carried away by emitted particles and any accompanying γ‑rays.

Lepton number – conserved (β⁻ emits an electron (+1) and an antineutrino (‑1)).

4. Radioactive Decay Series (Syllabus 23 N)

Natural radionuclides often belong to long decay chains that terminate in a stable nucleus. The three principal series are:

Uranium‑238 series → ²⁰⁸Pb (t₁/₂ ≈ 4.5 × 10⁹ yr)

Thorium‑232 series → ²⁰⁸Pb (t₁/₂ ≈ 1.4 × 10¹⁰ yr)

Uranium‑235 series → ²⁰⁸Pb (t₁/₂ ≈ 7.0 × 10⁸ yr)

Each step follows the exponential law (see §5). The overall activity of a series is governed by the half‑life of the parent nuclide.

5. Exponential Decay Law

Fundamental equation

\$x = x_{0}\,e^{-\lambda t}\$

where

$x$ – quantity at time $t$ (number of undecayed nuclei $N$, activity $A$ or detector count rate $C$).

$x{0}$ – initial quantity at $t=0$ ($N{0}, A{0}, C{0}$).

$\lambda$ – decay constant (units s$^{-1}$ or yr$^{-1}$).

$t$ – elapsed time (consistent units with $\lambda$).

5.1 Derivation (boxed for quick reference)

Probability that a single nucleus decays in a short interval $dt$:
\$dP = \lambda\,dt\$

For a large population $N$, the expected change is
\$dN = -\lambda N\,dt\$

Rearrange to the differential equation
\$\frac{dN}{dt} = -\lambda N\$

Integrate from $N_{0}$ at $t=0$ to $N$ at time $t$:
\$\int{N{0}}^{N}\frac{dN}{N}= -\lambda\int_{0}^{t}dt\$
\$\ln\!\left(\frac{N}{N_{0}}\right) = -\lambda t\$

Exponentiate:
\$N = N_{0}e^{-\lambda t}\$

Because activity $A$ and count rate $C$ are directly proportional to $N$, the same form holds:
\$A = A{0}e^{-\lambda t},\qquad C = C{0}e^{-\lambda t}\$

5.2 Key Related Quantities

Quantity	Symbol	Definition / Relation	Units
Decay constant	$\lambda$	Probability per unit time that a single nucleus decays	s$^{-1}$ or yr$^{-1}$
Half‑life	$t_{1/2}$	Time for the quantity to fall to half its initial value	s, min, yr
Mean life	$\tau$	Average lifetime of a nucleus	s, yr
Activity	$A$	Decays per second; $A = \lambda N$	Bq (s$^{-1}$)
Count rate	$C$	Detected events per second; $C = \varepsilon A$	counts s$^{-1}$

Useful relationships:

t_{1/2} = \frac{\ln 2}{\lambda},\qquad

\tau = \frac{1}{\lambda},\qquad

A = \lambda N,\qquad

C = \varepsilon A

6. Graphical Representation

Linear plot (x vs t) – shows a decreasing exponential curve; half‑life points can be marked.

Semi‑log plot (log x vs t) – yields a straight line. Gradient = $-\lambda$; intercept = $\log x_{0}$. This is the standard method for extracting $\lambda$ from experimental data.

Typical sketches (not to scale):

– Left: $x$ vs $t$ on linear axes, with half‑life markers.

– Right: $\log_{10}x$ vs $t$ on semi‑log axes, straight‑line fit (gradient = $-\lambda$).

7. Practical / Experimental Context (AO3)

7.1 Designing a Decay‑Measurement Experiment

Select a radionuclide whose half‑life allows measurable change during the lab (e.g. $^{60}$Co, $^{137}$Cs).

Use a detector with known efficiency $\varepsilon$ (Geiger‑Müller tube, scintillation counter, or semiconductor detector).

Measure background count rate $C_{\text{bkg}}$ over a long interval; subtract from all subsequent readings.

Record counts for a fixed counting time $\Delta t$ (commonly 60 s) at regular intervals $t_i$. Convert to count rate:
\$Ci = \frac{Ni}{\Delta t} - C_{\text{bkg}}\$

Plot $\log{10} Ci$ against $t_i$; fit a straight line (least‑squares). The slope $-m$ gives $\lambda = m\ln 10$.

Calculate the half‑life:
\$t_{1/2} = \frac{\ln 2}{\lambda}\$
and compare with the accepted value.

7.2 Error Analysis

Statistical (Poisson) uncertainty:
\[
\sigma_{N}= \sqrt{N}\;\;\Longrightarrow\;\;
\sigma_{C}= \frac{\sqrt{N}}{\Delta t}
\]

Propagation through the logarithm:
\[
\sigma{\log C}= \frac{1}{\ln 10}\,\frac{\sigma{C}}{C}
\]

Include systematic contributions (detector efficiency, timing, background subtraction) when quoting the final uncertainty on $\lambda$ or $t_{1/2}$.

8. Safety, Shielding & Applications

Radiation protection – the three “T’s”:
Time – minimise exposure; Distance – increase separation; Shielding – use material appropriate to the radiation type (paper for α, aluminium for β, lead or concrete for γ).

Legal limits (UK/International guidance):
Public exposure < 1 mSv yr$^{-1}$; occupational exposure ≈ 20 mSv yr$^{-1}$ (averaged over 5 yr).

Medical applications: PET and SPECT (β⁺ emitters), radiotherapy (γ‑rays from $^{60}$Co, β⁻ emitters such as $^{90}$Y).

Industrial applications: radiography, tracer studies, nuclear power (fission of $^{235}$U, $^{239}$Pu).

Environmental monitoring: radon surveys, fallout measurement, waste management.

9. Worked Example (Unit‑Consistent)

Problem: A sample of $^{60}$Co has an initial activity $A{0}=3.0\times10^{6}\,\text{Bq}$. Its half‑life is $t{1/2}=5.27$ years. Calculate the activity after $t=12$ years. (Answer to two significant figures.)

Calculate the decay constant (yr$^{-1}$):
\[
\lambda = \frac{\ln 2}{t_{1/2}} = \frac{0.693}{5.27\ \text{yr}} = 0.1315\ \text{yr}^{-1}
\]

Insert values into the decay law (using years for $t$):
\[
A = A_{0}\,e^{-\lambda t}=3.0\times10^{6}\,e^{-0.1315\times12}
\]

Evaluate the exponent:
\[
-\lambda t = -0.1315\times12 = -1.578
\]
\[
e^{-1.578}=0.206
\]

Final activity:
\[
A = 3.0\times10^{6}\times0.206 = 6.2\times10^{5}\ \text{Bq}
\]
Rounded to two significant figures: $6.2\times10^{5}\,\text{Bq}$.

10. Common Mistakes

10.1 Conceptual

Confusing decay constant $\lambda$ with half‑life $t{1/2}$. They are related by $\lambda = \ln 2 / t{1/2}$, not equal.

Assuming the measured count rate $C$ equals the activity $A$; detector efficiency $\varepsilon$ must be accounted for: $C=\varepsilon A$.

Neglecting background radiation when analysing count‑rate data.

10.2 Procedural

Mixing time units (e.g. using seconds for $t$ but a decay constant expressed per year).

Using a linear plot for exponential data – a semi‑log plot is required to obtain a straight line.

Omitting error propagation when extracting $\lambda$ from experimental data.

11. Summary Checklist (AO1 + AO2)

Write the exponential law $x = x_{0}e^{-\lambda t}$ and label each symbol.

Recall the key relations:
$\displaystyle \lambda = \frac{\ln2}{t_{1/2}},\;
\tau = \frac{1}{\lambda},\;
A = \lambda N,\;
C = \varepsilon A.$

Check that all quantities use consistent units before substitution.

Re‑arrange the equation to solve for the required unknown (usually $\lambda$ or $t$).

For experiments, plot $\log_{10} C$ against $t$; the gradient gives $-\lambda$.

Subtract background and correct for detector efficiency when converting count rate to activity.

Perform a full uncertainty analysis (statistical + systematic) and quote results with appropriate limits.

Apply the three “T’s” of radiation protection: minimise time, maximise distance, use suitable shielding.

Quantity	Symbol	Definition / Relation	Units
Decay constant	\(\lambda\)	Probability per unit time that a single nucleus decays	s\(^{-1}\) or yr\(^{-1}\)
Half‑life	\(t_{1/2}\)	Time for the quantity to fall to half its initial value	s, min, yr
Mean life	\(\tau\)	Average lifetime of a nucleus	s, yr
Activity	\(A\)	Decays per second; \(A = \lambda N\)	Bq (s\(^{-1}\))
Count rate	\(C\)	Detected events per second; \(C = \varepsilon A\)	counts s\(^{-1}\)

understand the exponential nature of radioactive decay, and sketch and use the relationship x = x0e–λt, where x could represent activity, number of undecayed nuclei or received count rate

Radioactive Decay – Cambridge AS & A‑Level Physics (9702)

1. Overview of Nuclear Physics (Syllabus 23 N)