1.1. The Unstable Nucleus: A State of Disproportion

Radioactivity is a fundamental process in nuclear physics where unstable atomic nuclei, also known as radioisotopes, undergo spontaneous decay to become more stable.¹ The stability of a nucleus is primarily determined by the ratio of its protons and neutrons. When this ratio is unbalanced, the nucleus possesses excess energy and is compelled to release it in the form of particles or electromagnetic waves, a process collectively referred to as radioactive decay.²

1.2. A Detailed Look at the Decay Processes

The three primary types of nuclear decay are alpha (α) decay, beta (β) decay, and gamma (γ) decay. Each process transforms the unstable nucleus in a specific way, altering its composition and energy state to achieve stability.³

Alpha (α) Decay

In alpha decay, the nucleus ejects a high-energy alpha particle, which is identical to the nucleus of a helium atom, consisting of two protons and two neutrons.² This form of decay is typically observed in very heavy elements, such as uranium, that need to reduce both their mass and proton count to become more stable.¹ The emission of an alpha particle causes the parent nucleus's mass number (A) to decrease by four, and its atomic number (Z) to decrease by two, resulting in a new element.¹ For example, the alpha decay of Uranium-238 transforms it into Thorium-234.¹

Beta (β) Decay

Beta decay is a process governed by the weak nuclear force, in which a nucleon within the nucleus changes its identity to achieve a more stable proton-to-neutron ratio.⁵ This type of decay does not change the total number of nucleons (the mass number, A) but fundamentally changes the element's identity by altering its atomic number (Z).⁵ There are two primary forms of beta decay.

Beta-minus ($\beta^{-}$) Decay: This process occurs in nuclei with an excess of neutrons.⁵ A neutron is converted into a proton, and in this transformation, a high-energy electron (known as a beta particle) and an electron antineutrino are emitted from the nucleus.⁹ The emission of the electron and antineutrino is necessary to adhere to fundamental conservation laws, specifically the conservation of electric charge and lepton number.⁶ The electron is a lepton with a lepton number of +1, and the antineutrino has a lepton number of -1. Since the parent nucleus has a total lepton number of zero, the sum of the decay products' lepton numbers must also be zero to maintain this conservation. This process results in the mass number (A) remaining unchanged, while the atomic number (Z) increases by one, creating a new element with a higher proton count.⁵ A common example is the decay of Carbon-14 into Nitrogen-14 ¹⁵ or Sodium-24 into Magnesium-24.¹⁷

Beta-plus ($\beta^{+}$) Decay: In nuclei with too many protons, a proton is converted into a neutron, and a positron (the antimatter equivalent of an electron) and an electron neutrino are emitted.⁶ This process also conserves lepton number, as the positron has a lepton number of -1 and the neutrino has a lepton number of +1.¹⁰ As a result, the mass number (A) remains constant, but the atomic number (Z) decreases by one.⁵

Gamma (γ) Decay

Gamma decay is a process where an excited nucleus releases surplus energy without changing its mass number or atomic number.³ This occurs when a nucleus that has undergone alpha or beta decay is left in a high-energy, or "excited," state.¹⁷ To reach its most stable, or "ground," state, it emits a high-energy photon, known as a gamma ray.² Since gamma rays are pure electromagnetic energy and have no charge or mass, they do not alter the composition of the nucleus itself.¹

1.3. A Comparative Summary of Radiation Properties

The three types of radiation have distinct properties that determine their effects and applications. These properties are summarized in the table below.

This table, compiled from multiple sources ², provides a quick reference for students to differentiate between the types of radiation. The properties of penetrating power and ionising ability are inversely related: highly ionising particles like alpha particles lose their energy quickly, traveling only a short distance, while the weakly ionising gamma rays can pass through most materials.²

2.1. The Official Definition (Cambridge IGCSE Syllabus)

The half-life of a radioactive isotope is precisely defined as the time taken for half the nuclei of that isotope in any sample to decay.²¹ This definition is a cornerstone of the IGCSE Co-ordinated Science syllabus. This period of time is also the time it takes for the mass of a radioactive sample to halve, or for its activity (the rate of decay, measured in Becquerels, Bq) to fall to half of its original level.²³

2.2. The Unchanging Nature of Half-Life

A crucial aspect of half-life is that it is a constant value for a given isotope, unaffected by external physical conditions such as temperature, pressure, or chemical state.²⁴ This constancy stems from the fact that radioactive decay is a nuclear process, driven by the strong and weak nuclear forces, which operate on a scale far removed from the everyday forces that govern chemical reactions and the kinetic energy of atoms. Consequently, the random, probabilistic nature of individual nuclear decay events remains independent of environmental factors. This distinguishes nuclear processes, such as radioactive decay, from chemical reactions, where reaction rates are highly sensitive to temperature and concentration.

2.3. Half-Life as a Statistical Measure

While it is impossible to predict precisely when a single, unstable nucleus will decay, the half-life provides a reliable, statistical measure for the decay of a large population of nuclei.²³ This can be understood through an analogy: if one were to flip a large number of coins, the outcome of a single flip is random, but the overall result for the entire population is statistically predictable. In a similar way, for a sample containing billions of radioactive nuclei, the time it takes for half of them to decay is a predictable and measurable quantity. This is why half-life is a practical metric for characterising the decay rate of a radioisotope.

This section provides a practical guide to the types of calculations required for the IGCSE Co-ordinated Science syllabus, focusing on interpreting numerical data and decay curves. As per the syllabus, all calculations provided herein do not account for background radiation.²¹

3.1. Calculations from Numerical Data

The fundamental principle of half-life calculation is that for every half-life that passes, the quantity of the radioactive substance (whether measured by mass, number of nuclei, or activity) is reduced by a factor of two.²³

The relationship between the number of half-lives and the remaining substance can be easily visualized in a table.

The amount of a substance remaining after a specific number of half-lives can be calculated using the formula:

where $N_t$ is the remaining quantity and $N_0$ is the initial quantity.²⁷

Worked Example 1: Calculating Half-Life from Activity

A radioactive sample has an initial activity of 880 Bq. After one year, the activity has dropped to 220 Bq. What is the half-life of the material? ²³

• Step 1: Determine how many times the activity has halved.
• Initial activity: 880 Bq
• After one halving: 880 Bq / 2 = 440 Bq
• After two halvings: 440 Bq / 2 = 220 Bq
• The activity has halved two times to reach 220 Bq.
• Step 2: Conclude that two half-lives have passed in one year.
• Step 3: Calculate the half-life.
• Half-life = Total time / Number of half-lives
• Half-life = 1 year / 2 = 6 months
• The half-life of the material is 6 months.

Worked Example 2: Calculating Remaining Mass

A 144 g sample of Carbon-14 has a half-life of 5,730 years. How much of the sample will remain after 17,190 years?

• Step 1: Calculate the number of half-lives that have passed.
• Number of half-lives = Total time / Half-life
• Number of half-lives = 17,190 y / 5,730 y = 3 half-lives
• Step 2: Determine the fraction of the sample remaining after 3 half-lives.
• After 1 half-life: 1/2 of the original mass.
• After 2 half-lives: (1/2) × (1/2) = 1/4 of the original mass.
• After 3 half-lives: (1/4) × (1/2) = 1/8 of the original mass.
• Step 3: Calculate the final mass remaining.
• Final mass = Initial mass × Fraction remaining
• Final mass = 144 g × (1/8) = 18 g
• After 17,190 years, 18 g of the Carbon-14 sample will remain.²⁹

3.2. Interpreting Decay Curves (Graphs)

A decay curve is a graph that plots the activity or count rate of a radioactive sample against time. The curve is exponential, starting with a steep slope and gradually becoming less steep as the activity decreases.²³ To determine the half-life from a decay curve, a straightforward graphical method can be employed:

1. Step 1: Identify the initial activity ($A_0$) on the vertical axis of the graph.²³
2. Step 2: Calculate and locate the value that is exactly half of the initial activity ($A_0$/2). For example, if the initial activity is 80 counts per minute, find the 40 counts per minute mark on the vertical axis.³²
3. Step 3: Draw a horizontal line from the $A_0$/2 value on the vertical axis to the decay curve.
4. Step 4: From the point where the horizontal line intersects the curve, draw a vertical line straight down to the time axis. The time at this point is the half-life of the isotope.²³

It is important to note that in real-world measurements, a low, constant level of background radiation is always present.³² This radiation, which comes from natural sources in the environment, must be subtracted from the total measured count rate to obtain the true activity of the radioactive source. However, for the purpose of the IGCSE syllabus, all provided data for half-life calculations is assumed to be already corrected for background radiation.²¹

The concept of half-life is not merely an academic exercise; it has vital applications in various fields, from archaeology and geology to medicine and industry.

4.1. Radiometric Dating: The Case of Carbon-14

One of the most well-known applications of half-life is radiocarbon dating, used to determine the age of organic artifacts.²⁶ This method relies on the known half-life of Carbon-14, which is approximately 5,730 years.³⁴ While an organism is alive, it continuously exchanges carbon with its environment, maintaining a stable ratio of the radioactive Carbon-14 isotope to the stable Carbon-12 isotope. Once the organism dies, it stops this exchange, and the Carbon-14 begins to decay into Nitrogen-14 at a constant, predictable rate.³⁴ By measuring the remaining ratio of Carbon-14 to Carbon-12 in the fossilized or preserved remains, scientists can calculate how many half-lives have passed since the organism's death and, consequently, its age.³⁵

While this method is highly effective, it has limitations. The accuracy of carbon dating is predicated on the assumption that the atmospheric concentration of Carbon-14 has remained constant over time.³⁵ However, human activities such as atmospheric nuclear bomb tests in the 1950s and the burning of fossil fuels (which releases ancient, Carbon-14-depleted carbon) have introduced fluctuations.³⁵ For this reason, scientists often use calibration curves to adjust their dating results, showcasing the intricate and evolving nature of scientific measurement.

4.2. Medical and Industrial Applications

The half-life of a radioisotope is a critical factor in selecting it for specific medical and industrial applications. A careful balance must be struck between the time needed for the isotope to perform its function and the need to minimize the radiation dose to the patient or the environment.

Medical Applications

The choice of a radioisotope for medical use depends on its intended purpose—diagnosis or therapy. For diagnostic imaging, such as a PET scan or bone scan, a radioisotope with a short half-life is preferred.³⁶ An isotope like Technetium-99m, with a half-life of just six hours, is ideal because it provides a sufficient period for imaging while ensuring that the radiation quickly diminishes to a safe level once the procedure is complete, thereby minimizing the patient's exposure.²⁴ In contrast, therapeutic applications, such as the treatment of cancer, require radioisotopes with a longer half-life to deliver a sustained and destructive dose of radiation to malignant cells over time.³⁶ Iodine-131, with an 8-day half-life, is a classic example used for treating thyroid cancer, as the thyroid naturally absorbs iodine.³⁶

Industrial Applications

In industry, radioisotopes are selected based on the specific requirements of the task and the desired longevity of the source. For example, radioisotopes with long half-lives, such as Caesium-137 (30.08 years), are used in thickness gauging to provide a stable, long-lasting source of radiation.³⁶ This is a cost-effective solution as it does not require frequent replacement. For tasks such as tracing leaks in pipelines, short-lived radioisotopes like Sodium-24 (half-life of 14.9590 h) are used.³⁸ The short half-life ensures that the isotope will decay to a non-radioactive state relatively quickly, preventing long-term environmental contamination.³⁶

5.1. Key Takeaways and Summary

Radioactive decay is a probabilistic nuclear process by which unstable nuclei transform into more stable configurations. The rate of this decay is characterized by the half-life, which is the constant time required for half of the nuclei in any sample to decay. This principle allows for predictable calculations of remaining mass, activity, or the age of a sample. The half-life is a fundamental property of a radioisotope, independent of all external factors. It is a critical concept with extensive real-world applications in scientific dating, medicine, and industry.

5.2. Exam-Ready Checklist

• Definition: Recall the precise definition: "the time taken for half the nuclei of that isotope in any sample to decay."
• Decay Effects: Remember the effects of each decay type on the nucleus:
• Alpha decay: Mass number (A) decreases by 4, Atomic number (Z) decreases by 2.
• Beta decay: Mass number (A) is unchanged, Atomic number (Z) increases by 1 ($\beta^{-}$) or decreases by 1 ($\beta^{+}$).
• Gamma decay: Both mass and atomic numbers are unchanged.
• Calculation Methods: Be prepared to perform calculations from both numerical data and graphical decay curves.
• Background Radiation: Note that exam calculations will not include background radiation, which simplifies the process to reflect ideal conditions.²¹
• Applications: Understand the distinction between radioisotopes with short and long half-lives and their uses in areas such as medical diagnosis and treatment, and scientific dating.