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)

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 (Ebind): 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 modeChange in (A,Z)Particle(s) emittedTypical penetrating powerCommon detector
α‑decay(A‑2, Z‑2)⁴He nucleus (α‑particle)Low – stopped by a sheet of paperScintillation or semiconductor detector (high α efficiency)
β⁻‑decay(A, Z+1)Electron + antineutrinoMedium – penetrates paper, stopped by ≈1 mm AlGeiger‑Müller tube, plastic scintillator
β⁺‑decay (positron)(A, Z‑1)Positron + neutrino (annihilation → 2 × 511 keV γ)Medium – γ‑rays are highly penetratingScintillator + coincidence with 511 keV γ
γ‑decay(A, Z) unchangedHigh‑energy photonHigh – requires lead or several cm of concreteNaI(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)

  1. Probability that a single nucleus decays in a short interval \(dt\):

    \$dP = \lambda\,dt\$

  2. For a large population \(N\), the expected change is

    \$dN = -\lambda N\,dt\$

  3. Rearrange to the differential equation

    \$\frac{dN}{dt} = -\lambda N\$

  4. 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\$

  5. Exponentiate:

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

  6. 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

QuantitySymbolDefinition / RelationUnits
Decay constant\(\lambda\)Probability per unit time that a single nucleus decayss\(^{-1}\) or yr\(^{-1}\)
Half‑life\(t_{1/2}\)Time for the quantity to fall to half its initial values, min, yr
Mean life\(\tau\)Average lifetime of a nucleuss, 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

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

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

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

  4. 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}}\$

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

  6. 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.)

  1. 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}

    \]

  2. Insert values into the decay law (using years for \(t\)):

    \[

    A = A_{0}\,e^{-\lambda t}=3.0\times10^{6}\,e^{-0.1315\times12}

    \]

  3. Evaluate the exponent:

    \[

    -\lambda t = -0.1315\times12 = -1.578

    \]

    \[

    e^{-1.578}=0.206

    \]

  4. 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)

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

  2. Recall the key relations:

    \(\displaystyle \lambda = \frac{\ln2}{t_{1/2}},\;

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

    A = \lambda N,\;

    C = \varepsilon A.\)

  3. Check that all quantities use consistent units before substitution.

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

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

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

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

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