Radioactive decay

Section: Nuclear Physics | Syllabus: Cambridge A Level Physics (9702)

Radioactive Decay

Nature of Radioactive Decay

Radioactive decay is the spontaneous transformation of an unstable nucleus into a more stable one.

Spontaneous: Not affected by external factors like temperature, pressure, or chemical bonding.
Random: It is impossible to predict when a specific nucleus will decay, but established probabilities apply to large numbers of nuclei.

Evidence for Random Decay

When measuring count rate from a radioactive source using a Geiger-Müller (GM) tube, the readings fluctuate even when conditions are constant. These fluctuations in count rate provide experimental evidence that radioactive decay is random.

Activity and Count Rate

Corrected Count Rate Corrected count rate = recorded count rate − background count rate

Background radiation arises from natural radioactive isotopes in the ground, building materials, cosmic rays, and radon gas in the atmosphere. The corrected count rate is proportional to the activity of the sample.

Activity and Decay Constant

Activity (A) The rate of decay of a radioactive sample (number of decays per unit time). SI Unit: Becquerel (Bq), where 1 Bq = 1 decay per second. Decay Constant (λ) The probability that an individual nucleus will decay per unit time. Unit: s^-1. A = λ N

A = -(dN/dt)

Half-Life

Half-Life (t_1/2) The time taken for the number of undecayed nuclei (or activity) to reduce to half its initial value. λ = ln 2t_1/2 ≈ 0.693t_1/2

Exponential Decay Equations

Since decay is proportional to the number of nuclei remaining, it follows an exponential law:

x = x_0 e^-λ t

where x can be Number of nuclei (N), Activity (A), or Count Rate (C). Figure 23.12: Decay Curve

Graph of Activity vs Time showing exponential decay. Mark points at t_1/2 (A_0/2) and 2t_1/2 (A_0/4).

Worked Examples

Worked Example: Calculating Activity

Question: A sample contains 2.5 × 10^20 atoms of an isotope with a half-life of 12 hours. Calculate the initial activity.

Solution

1. Convert half-life to seconds: t_1/2 = 12 × 3600 = 43200 s

2. Calculate λ: λ = (0.693/43200) = 1.60 × 10^-5 s^-1

3. Calculate Activity: A = λ N = (1.60 × 10^-5)(2.5 × 10^20) = 4.0 × 10^15 Bq

Worked Example: Exponential Decay Calculation

Question: The activity of a source falls from 400 Bq to 100 Bq in 24 minutes. Calculate the half-life.

Solution

400 Bq → 200 Bq → 100 Bq

This represents two half-lives.

2 t_1/2 = 24 min ⇒ t_1/2 = 12 min

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Radioactive decay