Half-life
Section: Nuclear Physics | Syllabus: Cambridge IGCSE Physics (0625)
What is Half-life?
The half-life of a particular isotope is the time taken for half the nuclei of that isotope, in any sample, to decay.
- Half-life can equally be described as the time taken for the count rate (activity) to fall to half its original value, or the time taken for the mass of the isotope remaining to halve
- Half-life is constant for a given isotope - it cannot be changed by temperature, pressure or chemical reactions
| Isotope | Half-life |
|---|---|
| Iodine-131 | 8 days |
| Cobalt-60 | 5.3 years |
| Carbon-14 | 5730 years |
| Uranium-238 | 4.5 billion years |
The Half-life Pattern
After each half-life, half of the remaining radioactive nuclei decay.
After each half-life, the number of undecayed nuclei halves
| Number of Half-lives | Fraction Remaining |
|---|---|
| 1 | 1/2 |
| 2 | 1/4 |
| 3 | 1/8 |
| 4 | 1/16 |
- After n half-lives, the fraction of the original sample remaining = (1/2)ⁿ
Half-life Calculations
The definition of half-life can be used in simple calculations, which might involve information given in tables or decay curves. These calculations will not include background radiation.
Worked Example: Basic Half-life Calculation
Museli has a radioactive sample with an initial count rate of 800 counts/s and a half-life of 10 minutes. She wants to find the count rate after 30 minutes.
- Number of half-lives elapsed = 30 ÷ 10 = 3
- After 1 half-life: 800 ÷ 2 = 400 counts/s
- After 2 half-lives: 400 ÷ 2 = 200 counts/s
- After 3 half-lives: 200 ÷ 2 = 100 counts/s
Worked Example: Finding the Half-life
Muchindu has a sample with an initial activity of 1200 counts/min. After 24 hours, the activity is 150 counts/min. He wants to find the half-life.
- 1200 → 600 → 300 → 150 counts/min is a fall to 1/8 of the original activity, i.e. 3 half-lives
- 3 half-lives = 24 hours, so 1 half-life = 24 ÷ 3 = 8 hours
Decay Curves
Plotting the activity (count rate) of a sample against time produces a characteristic decay curve.
A radioactive decay curve - the time to halve is always the same, wherever you start on the curve
- The curve decreases exponentially and gets closer and closer to zero, but never quite reaches it
- The half-life can be read from the curve by choosing any point, finding half of that count rate, and measuring the time between the two points - this time is always the same, wherever on the curve you start
Worked Example: Reading Half-life from a Graph
Muumbe reads two points from a decay curve: at t = 0 hours, count rate = 800 counts/s; at t = 5 hours, count rate = 400 counts/s.
- The count rate has halved (800 → 400) between t = 0 and t = 5 hours
- Half-life = 5 hours
Calculating Half-life When Background Has Not Been Subtracted
Extended Only
In a real experiment, a detector measures the source's radiation together with background radiation, so a decay curve plotted from raw readings levels off at the background count rate, not at zero. Half-life must be calculated using the count rate due to the source alone, so background radiation must first be subtracted from each reading.
Worked Example: Half-life with Background Radiation
Museli records the following data, with a background count rate of 40 counts/min: at t = 0, 520 counts/min; at t = 10 min, 280 counts/min; at t = 20 min, 160 counts/min.
- Corrected count rate at t = 0: 520 − 40 = 480 counts/min
- Corrected count rate at t = 10 min: 280 − 40 = 240 counts/min
- Corrected count rate at t = 20 min: 160 − 40 = 120 counts/min
- The corrected count rate halves (480 → 240) between t = 0 and t = 10 min, so the half-life = 10 minutes
Applications of Radioactive Isotopes
Extended Only
The isotope chosen for a particular application depends on both the type of radiation it emits (linked to penetration and absorption) and its half-life.
| Application | Radiation Type | Half-life Needed | Reasons |
|---|---|---|---|
| (a) Household fire (smoke) alarms | Alpha | Long (hundreds of years) | Alpha ionises air in the detector to produce a small current, which smoke particles interrupt, triggering the alarm; alpha cannot penetrate the casing, so it is safe; a long half-life means the source does not need replacing for the lifetime of the alarm |
| (b) Irradiating food to kill bacteria | Gamma | Several years | Gamma penetrates deep into food and its packaging to kill bacteria throughout, without making the food radioactive; a half-life of several years makes the source economical to use over a long period |
| (c) Sterilising equipment | Gamma | Several years | Gamma penetrates sealed packaging, sterilising equipment inside without it needing to be opened; a half-life of several years allows continuous industrial use without frequent source replacement |
| (d) Measuring and controlling thickness | Beta (gamma for thicker materials) | Long (years) | Beta is partially absorbed by the material - if it becomes too thick, less radiation reaches the detector on the other side, and rollers can adjust the thickness accordingly; a long half-life means the count rate (and so the calibration) does not change noticeably over the equipment's working life |
| (e) Diagnosis of disease (medical tracers) | Gamma | Short (hours) | Gamma penetrates body tissue and can be detected outside the body using a gamma camera, and is less ionising so causes less damage to tissue; a short half-life keeps the radiation dose to the patient low, as the source decays away quickly once the scan is complete |
| (e) Treatment of cancer (radiotherapy) | Gamma | Several years | Gamma penetrates deep into the body to reach tumours, and can be focused from several angles to concentrate the dose on the cancer cells while minimising damage to surrounding tissue; a half-life of several years allows the same source to be used for many treatments |
Using beta radiation to monitor and control the thickness of a material
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