understand that radioactive decay is both spontaneous and random

Radioactive Decay – Spontaneous and Random

Learning Objective

Explain why radioactive decay is spontaneous and random. Use the quantitative relationships – decay constant (λ), activity (A), half‑life (T½) and the exponential law – to predict the behaviour of a large sample and to interpret experimental data.

1. What is Radioactive Decay?

• Unstable nuclei lose energy by emitting radiation (α, β, γ) and transform into a different nucleus (or the same nucleus in an excited state).
• No external influence is required; the decay is an intrinsic property of the nucleus.

2. Spontaneity (Syllabus 23.2.2)

• The decay rate is independent of temperature, pressure, chemical state, magnetic or electric fields.
• Experimental evidence: half‑life of a radionuclide measured at room temperature, at 500 °C and in a strong magnetic field are identical within experimental error.
• Therefore the process is spontaneous – it occurs without any external trigger.

3. Randomness (Syllabus 23.2.1)

• For a single nucleus the exact instant of decay cannot be predicted.
• Each nucleus has a constant *mean* probability per unit time of decaying – the decay constant λ (s⁻¹).
• The number of decays in a short time interval Δt follows a Poisson distribution:

  σ = √(A Δt)

  where A is the activity (mean count‑rate) and σ is the standard deviation of the counts.

• When many nuclei are observed the count‑rate fluctuates about a mean value; the fluctuations become Gaussian only for very large counts.

Evidence from a Geiger‑Müller Counter

Record the number of counts each second for a weak β‑source. Typical readings might be 12, 9, 15, 11 counts s⁻¹. The spread of values follows the Poisson law, confirming that each decay event is independent and random. Over longer periods the average count‑rate stabilises, illustrating the statistical nature of the process.

4. Decay Constant λ

• λ is a property of the radionuclide only; it does not depend on the amount of material.
• It is the mean probability per unit time that a single nucleus will decay.
• Relation to half‑life (see Section 6): λ = 0.693 / T½

5. Activity (Syllabus 23.2.4)

The activity A is the number of decays per unit time (s⁻¹). For a sample containing N undecayed nuclei:

A = λ N

• Unit: becquerel (Bq) = 1 decay s⁻¹.
• In a detector the measured count‑rate (C) is related to the true activity by the efficiency ε:

  C = ε A  (0 < ε ≤ 1)

6. Exponential Law (Syllabus 23.2.5)

For a large ensemble the number of undecayed nuclei after a time t is:

N(t) = N₀ e^‑λt

Because A = λN, the activity follows the same form:

A(t) = A₀ e^‑λt

This equation also describes the count‑rate when the detector efficiency is constant (C(t)=ε A(t)).

7. Half‑Life (Syllabus 23.2.6)

The half‑life T½ is the time required for half of the original nuclei to decay.

λ = 0.693 / T½  or  T½ = 0.693 / λ

Example rearrangement (exam style): If λ = 2.0 × 10⁻³ s⁻¹, then

T½ = 0.693 / (2.0 × 10⁻³) ≈ 3.5 × 10² s ≈ 5.8 min.

8. Log‑Linear Plot (Syllabus 23.2.7)

• Taking natural logarithms of the exponential law gives:

ln A = ln A₀ – λt

• A plot of ln A (or ln C) against time is a straight line with slope –λ and intercept ln A₀.
• This method is frequently used in Paper 5 to determine λ or T½ from experimental data.

9. Sketching the Exponential Decay (Syllabus 23.2.7)

• Vertical axis: quantity (N, A or count‑rate). Horizontal axis: time.
• Mark the initial value (N₀ or A₀) at t = 0.
• Draw vertical dashed lines at T½, 2T½, 3T½ …; the curve should pass through (T½, ½ N₀), (2T½, ¼ N₀), (3T½, ⅛ N₀).
• Indicate that the curve is smooth for a large sample, whereas individual decay events are discrete and irregular.
• Include a small inset showing a log‑linear plot (ln A vs t) as a straight line.

10. Decay Series and Daughter Products (Syllabus 23.2.8)

Many radionuclides decay through a series of successive transformations until a stable nucleus is reached. Example (uranium‑238 series):

⁽²³⁸⁾U →[α] ⁽²³⁴⁾Th →[β⁻] ⁽²³⁴⁾Pa →[β⁻] ⁽²³⁴⁾U →[α] ⁽²³⁰⁾Th → …
→ … → ⁽²⁰⁶⁾Pb (stable)

• Each step has its own λ and T½; the overall activity of a secular equilibrium mixture is governed by the longest‑lived parent.
• Understanding decay series is useful for interpreting background radiation and for age‑dating techniques.

11. Worked Example – From Nuclei to Activity

Sample: 1.0 × 10⁶ atoms of a radionuclide with T½ = 30 min.

1. Decay constant: λ = 0.693 / (30 × 60 s) = 3.85 × 10⁻⁴ s⁻¹.
2. Initial activity: A₀ = λ N₀ = (3.85 × 10⁻⁴)(1.0 × 10⁶) = 3.85 × 10² s⁻¹ = 385 Bq.
3. Number of nuclei after 90 min (3 half‑lives): N = N₀(½)³ = 1.25 × 10⁵.
4. Activity after 90 min: A = λ N = 3.85 × 10⁻⁴ × 1.25 × 10⁵ ≈ 48 Bq.
5. Count‑rate expected in a detector of efficiency ε = 0.25: C = ε A ≈ 12 counts s⁻¹.

This illustrates that while the exponential law predicts the average activity accurately, the exact moment at which each of the remaining 1.25 × 10⁵ nuclei will decay remains unpredictable.

12. Common Misconceptions (Syllabus‑Focused)

1. “Heating a sample speeds up decay.” – Nuclear decay rates are independent of temperature or pressure.
2. “All atoms decay at the same moment.” – Decay times are random; only the ensemble follows the exponential law.
3. “We can predict when a particular nucleus will decay.” – The random nature is confirmed by Poisson fluctuations in short‑interval count‑rates.
4. “Radioactive decay is a chemical reaction.” – It involves changes in the nucleus, not electron rearrangements.
5. “Activity equals the raw count‑rate.” – The measured count‑rate must be corrected for detector efficiency (C = ε A).

13. Summary

• Radioactive decay is an intrinsic, spontaneous nuclear process.
• The moment of decay for any single nucleus is random; λ is the mean probability per unit time.
• Activity (A = λN) links the microscopic randomness to a macroscopic measurable quantity (Bq).
• Large ensembles obey the exponential law N(t)=N₀e⁻λt, and the half‑life relation λ = 0.693 / T½.
• Count‑rate fluctuations follow a Poisson distribution (σ = √A t); for large counts they approach a Gaussian shape.
• Log‑linear plots give a straight line of slope –λ, a common exam technique.
• Decay series consist of successive parent‑daughter transformations, each with its own λ and T½.

14. Self‑Check Questions

1. Define the decay constant λ and state its physical significance.
2. A radionuclide has λ = 2.0 × 10⁻³ s⁻¹. Calculate its half‑life.
3. A sample contains 5.0 × 10⁴ undecayed nuclei and λ = 1.2 × 10⁻⁴ s⁻¹. What is its activity in becquerels?
4. Describe an experiment using a Geiger‑Müller counter that demonstrates the random nature of decay.
5. Sketch an exponential decay curve for activity, label two half‑life intervals, and show the corresponding ln A vs t straight line.

15. Suggested Practical Activity

Objective: Observe Poisson fluctuations and verify the exponential decay law.

1. Place a weak β‑emitting source (e.g., ^90Sr) 5 cm from a Geiger‑Müller counter.
2. Record the number of counts each second for 2 minutes (120 data points).
3. Calculate the mean count‑rate ( C̄ ) and the standard deviation σ.
4. Check whether σ ≈ √C̄, as expected for a Poisson distribution.
5. Repeat the measurement after 30 minutes (≈ one half‑life for the source). Plot both data sets on the same graph and show that the mean count‑rate has roughly halved while the relative spread (σ / C̄) remains similar.

Discussion points: why the points do not lie on a straight line, how the spread confirms randomness, and how detector efficiency would affect the measured counts.