Half-life

1. Overview

Radioactive decay is a random process, meaning we cannot predict when a single nucleus will decay. However, because samples contain billions of atoms, we can use the concept of half-life to predict how the overall activity of a sample decreases over time. Understanding half-life is crucial for using radiation safely in medicine, industry, and the home.

Key Definitions

• Half-life ($t_{1/2}$): The time taken for half the nuclei of a specific isotope in a sample to decay. Alternatively: the time taken for the activity (or count rate) of a sample to decrease to half its initial value.
• Activity: The rate at which a source of unstable nuclei decays, measured in Becquerels (Bq). 1 Bq = 1 decay per second.
• Isotope: Atoms of the same element with the same number of protons but different numbers of neutrons.
• Count-rate: The number of decays detected per second by a device (like a Geiger-Müller tube).
• Background Radiation: The low-level radiation present at all times from natural (e.g., rocks, cosmic rays) and man-made sources.

Core Content

Understanding Half-life Since decay is exponential, a substance never "runs out" of radioactivity linearly. Instead, it halves every set period of time.

• After 1 half-life: 50% remains (1/2)
• After 2 half-lives: 25% remains (1/4)
• After 3 half-lives: 12.5% remains (1/8)

Using Decay Curves A decay curve is a graph showing Activity (y-axis) against Time (x-axis).

To find half-life from a graph:

1. Pick a starting activity on the y-axis.
2. Find the time value for that activity.
3. Divide the starting activity by 2.
4. Find the new time value for this halved activity.
5. The difference between these two times is the half-life.

Worked Example (Core) Question: A sample has an initial activity of 1200 Bq. Its half-life is 5 minutes. What is the activity after 15 minutes?

1. Calculate the number of half-lives: $15 \div 5 = 3$ half-lives.
2. Halve the activity three times:
   • 1200 $\rightarrow$ 600 (1st half-life)
   • 600 $\rightarrow$ 300 (2nd half-life)
   • 300 $\rightarrow$ 150 Bq (3rd half-life)

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Extended Content (Extended Curriculum Only)

Calculating Half-life with Background Radiation In real experiments, a Geiger-Müller tube detects both the source and the background radiation. To find the true half-life, you must subtract the background radiation first.

Formula: $Corrected Activity = Measured Activity - Background Radiation$

Worked Example (Extended) Question: The background radiation is 20 counts/min. A source measures 180 counts/min initially. After 4 hours, the source measures 60 counts/min. Calculate the half-life.

1. Find initial corrected activity: $180 - 20 = 160$ counts/min.
2. Find final corrected activity: $60 - 20 = 40$ counts/min.
3. Determine half-lives: $160 \rightarrow 80 \rightarrow 40$ (This is 2 half-lives).
4. Calculate duration: 2 half-lives = 4 hours, so 1 half-life = 2 hours.

Applications of Isotopes The choice of isotope depends on its penetrating power and its half-life.

Application	Radiation Type	Preferred Half-life	Reasoning
Smoke Alarms	Alpha	Long (years)	Alpha is easily blocked by smoke; long half-life means you don't need to replace the source often.
Irradiating Food	Gamma	Long	Gamma kills bacteria/mold throughout the food; long half-life provides a consistent dose over time.
Sterilising Equipment	Gamma	Long	Gamma penetrates packaging to kill bacteria on medical tools; long half-life ensures the machine stays effective.
Thickness Control	Beta	Long	Alpha is too weak (blocked by paper); Gamma is too strong. Beta absorption changes based on thickness; long half-life ensures changes in count-rate are due to metal thickness, not source decay.
Cancer Treatment	Gamma	Short/Long	Focused beams kill tumors. If injected, a short half-life is needed to limit the dose to the patient.
Medical Diagnosis	Gamma	Very Short (hours)	Must be penetrating to leave the body; short half-life ensures the patient isn't radioactive for long.

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Key Equations

1. Number of half-lives ($n$): $n = \frac{\text{Total Time}}{\text{Half-life duration}}$
2. Remaining Activity ($A$ ): $A = A_0 \times (\frac{1}{2})^n$ (where $A_0$ is initial activity)
3. Corrected Activity: $A_{corrected} = A_{measured} - A_{background}$

• Units: Activity = Becquerels (Bq); Time = Seconds, minutes, hours, or years.

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Common Mistakes to Avoid

• ❌ Wrong: Assuming a substance decays at a constant linear rate (e.g., if 100g decays to 50g in 10 mins, it will all be gone in 20 mins).
• ✓ Right: Decay is exponential. After 20 minutes, 25g would remain (half of 50g).
• ❌ Wrong: Calculating half-life by simply dividing the total time by the starting activity.
• ✓ Right: Use the total time to find how many "halving cycles" occurred.
• ❌ Wrong: Forgetting to subtract background radiation before halving the numbers in extended questions.
• ✓ Right: Subtract background radiation from all readings first, then perform the half-life calculation.

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Exam Tips

1. Read the Graph Axes: Ensure you are looking at "Activity" or "Count-rate" and check the units for time (hours vs. days).
2. The "Half-way" Check: If a question asks for the activity after a certain time, always draw a quick arrow diagram ($100 \rightarrow 50 \rightarrow 25$) to keep track of how many half-lives have passed.
3. Application Logic: When asked why a specific isotope is used, always mention two things: its penetrating power (can it get through the material?) and its half-life (safety vs. convenience).