4.4 Radioactivity

Nuclear Structure, Stability, and Fundamental Forces

The nucleus contains two types of subatomic particles, collectively called nucleons:

• Protons: Positively charged ($+1.602 \times 10^{-19}\text{ C}$), with a mass of $1.673 \times 10^{-27}\text{ kg}$.
• Neutrons: Electrically neutral ($0\text{ C}$), with a mass of $1.675 \times 10^{-27}\text{ kg}$.

A nucleus of any element is written using standard nuclear notation:

Where $Z$ is the atomic number (the number of protons, which determines the element), $A$ is the mass number (total number of nucleons, $A = Z + N$), and $X$ is the chemical symbol. The number of neutrons is $N = A - Z$.

Isotopes are atoms of the same element that have the same number of protons but different numbers of neutrons. For example, carbon-12 ($^{12}_6\text{C}$) and carbon-14 ($^{14}_6\text{C}$) share identical chemical behaviour because they have the same electron configuration, but carbon-14 has two extra neutrons and is radioactive.

Fundamental Forces in the Nucleus

Protons inside a nucleus experience powerful electrostatic repulsion. Nuclear stability depends on the interplay of fundamental forces:

1. Strong Nuclear Force: An attractive force that acts equally between all nucleons (proton-proton, proton-neutron, neutron-neutron) over a very short range (about $10^{-15}\text{ m}$). It holds the nucleus together against electrostatic repulsion.
2. Electromagnetic Force: An infinite-range repulsive force acting between positively charged protons.
3. Weak Nuclear Force: A very short-range force responsible for radioactive beta decay.
4. Gravitational Force: An attractive force between masses, but negligible on the nuclear scale.

A nucleus is stable when the strong nuclear force holds its nucleons together securely. Very large nuclei have so many protons that electrostatic repulsion makes them unstable; they often emit alpha particles to reduce size. Nuclei with too many neutrons for their number of protons are also unstable; a neutron transforms into a proton via beta decay, governed by the weak nuclear force. An unstable nucleus decays spontaneously toward a more stable state.

Mass Defect and Binding Energy

A nucleus always has less mass than the sum of the individual protons and neutrons that constitute it when separated. This missing mass is converted into energy that was released when the nucleus formed.

Mass defect ($\Delta m$): the difference between the total mass of the separate nucleons and the mass of the nucleus.

Binding energy: the energy needed to separate a nucleus completely into its individual protons and neutrons. According to Einstein's mass-energy equivalence, it equals the energy equivalent of the mass defect:

The stability of a nucleus is measured by its binding energy per nucleon (total binding energy divided by the mass number $A$). The greater the binding energy per nucleon, the more tightly bound the nucleons are and the more stable the nucleus. Nuclei near iron-56 have the highest binding energy per nucleon (roughly $8.8\text{ MeV}$ per nucleon), making them the most stable in the universe.

Radioactive Emissions and Nuclear Equations

Radioactivity is the spontaneous disintegration of an unstable nucleus accompanied by the emission of ionising radiation.

Properties of Alpha, Beta, and Gamma Radiation

In a magnetic field, a moving charge experiences a force ($F = qvB$) at right angles to both the particle's velocity and the magnetic field lines, making the particle travel in a curve. Alpha and beta particles deflect in opposite directions because they carry opposite charges. Beta particles deflect much more noticeably because their charge-to-mass ratio ($q/m$) is thousands of times larger (about 3,700 times) than that of an alpha particle.

Writing and Balancing Nuclear Equations

To write any nuclear reaction:

1. Write the parent nucleus with mass number $A$ on top and atomic number $Z$ on the bottom.
2. Write the symbol for the emitted particle ($^4_2\text{He}$, or $^0_{-1}\text{e} + \bar{\nu}$).
3. Ensure total mass numbers (top) balance on both sides of the arrow.
4. Ensure total atomic numbers (bottom) balance on both sides.
5. Use the new atomic number $Z$ to identify the daughter element from the periodic table.

Alpha decay: The parent nucleus ejects an alpha particle, reducing mass number by 4 and atomic number by 2:

Example (finding the daughter for Polonium-212 decay):

Beta ($\beta^-$) decay: A neutron inside the nucleus transforms into a proton, releasing an electron and an antineutrino ($\bar{\nu}$):

The atomic number increases by 1 while the mass number stays the same:

Example: Carbon-14 decay:

Wolfgang Pauli predicted the neutrino in 1930 to explain why emitted beta particles have a continuous range of kinetic energies rather than a fixed value. Without an unseen third particle sharing the available energy and momentum, conservation of energy and momentum would fail. The antineutrino is exceptionally difficult to detect because it carries zero charge and almost no mass, allowing it to pass through light years of matter without interacting.

Gamma decay: An excited nucleus releases excess energy as a high-energy photon without changing its atomic or mass numbers.

Radiation Detection, Background Radiation, and Safety

Detectors identify ionising radiation by the charged ions or charge carriers it creates when passing through matter.

The Geiger-Müller (GM) Tube

A GM tube consists of a cylindrical metal cathode enclosing argon gas at low pressure, with a thin mica entrance window at one end and a thin wire anode running down the centre. Radiation enters through the window and ionises argon atoms. A high voltage applied between the cathode and anode accelerates the freed electrons toward the wire anode. As they gain speed, these electrons collide with more argon atoms, triggering an avalanche of secondary ionisations. This produces a measurable pulse of electrical current that registers on a ratemeter or scaler.

A solid-state detector is an alternative modern sensor: a semiconductor device in which incoming radiation produces electron-hole pairs, creating a current pulse proportional to the particle's energy.

Measuring Background Radiation

Background radiation is measured by connecting a GM tube to a counter or scaler in the absence of any artificial radioactive sources:

1. Record the count over a measured duration (such as 300 seconds) and divide by the time to find the background count rate in counts per second ($\text{s}^{-1}$).
2. When measuring an experimental source, subtract this background count rate from the total measured count rate to obtain the corrected count rate.

In terms of experimental error:

• Radioactive decay is random, so counts fluctuate from second to second. Counting over a long time and repeating readings reduces this random error.
• Failing to subtract the background count rate is a systematic error that makes every measured activity too high.

Sources of Background Radiation and Safety

Background radiation is present everywhere in the environment. In Ireland, natural sources contribute the vast majority, with radon gas accounting for more than half of average public exposure. Radon is a radioactive alpha emitter formed by the natural decay of uranium in granitic bedrock, which can seep up into unventilated basements and homes. Other sources include cosmic rays from space, radioisotopes in rocks and soil, food (such as potassium-40), and medical procedures like diagnostic X-rays.

Ionising radiation damages biological cells, breaks chemical bonds, and mutates DNA, increasing the risk of cancer. Practical radiation safety relies on three rules: minimise exposure time, maximise distance from the source (using tongs), and employ appropriate shielding (such as lead blocks).

The Law of Radioactive Decay and Modelling Decay

Nuclear decay is spontaneous and completely random: we cannot predict when a specific nucleus will decay, and the rate cannot be altered by temperature, chemical reactions, or pressure.

The Law of Radioactive Decay

For a large sample, the activity ($A$) is directly proportional to the number of undecayed nuclei ($N$) present:

Where:

• $A$ is activity measured in becquerels (Bq) ($1\text{ Bq} = 1\text{ decay per second} = 1\text{ s}^{-1}$).
• $N$ is the number of undecayed nuclei remaining.
• $\lambda$ is the decay constant (unit: $\text{s}^{-1}$), the constant of proportionality representing the probability per second that an individual nucleus will decay.
• In calculus notation, activity is the rate of loss of undecayed nuclei: $A = -\frac{\text{d}N}{\text{d}t}$.

The half-life ($T_{1/2}$) is the time taken for half of the undecayed nuclei in a radioactive sample to decay (or the time for its activity to halve):

After $n$ half-lives, the fraction remaining is $(1/2)^n$ ($1/2, 1/4, 1/8, \dots$).

Worked Example: Activity and Decay Over Half-Lives Problem: Iodine-131 has a half-life of 8 days. A sample contains $2 \times 10^{15}$ nuclei. Calculate: (i) the decay constant and activity, and (ii) how long it takes for $1.75 \times 10^{15}$ nuclei to decay.

Working:

1. $T_{1/2} = 8 \times 24 \times 3600 = 6.91 \times 10^5\text{ s}$.
2. $\lambda = \frac{0.693}{6.91 \times 10^5\text{ s}} = 1.00 \times 10^{-6}\text{ s}^{-1}$.
3. $A = \lambda N = (1.00 \times 10^{-6}\text{ s}^{-1})(2 \times 10^{15}) = 2.0 \times 10^9\text{ Bq}$.
4. If $1.75 \times 10^{15}$ nuclei decay, then $2.5 \times 10^{14}$ remain. This is $1/8$ of the original, which is 3 half-lives, so $t = 3 \times 8 = 24\text{ days}$.

Modelling Random Decay

Although individual decays are unpredictable, every nucleus of a particular isotope has the same probability of decaying in any given second. In a large collection of nuclei, a constant fraction decays each second, generating a smooth mathematical curve.

The dice model: If you roll 600 ordinary six-sided dice, each die has a 1 in 6 chance of landing on a six. If you remove all dice that land on a six (representing decayed nuclei) and roll the survivors again, roughly $1/6$ of the remaining dice disappear on each roll. Plotting remaining dice against the roll number produces an exponential decay curve that drops by half approximately every 3.8 rolls. That roll count is the half-life of the dice model.

Reading half-life from a decay curve:

1. Plot corrected count rate or undecayed nuclei on the y-axis against elapsed time on the x-axis, drawing a smooth curve through the points.
2. Choose any initial activity value on the curve (e.g. $800\text{ counts s}^{-1}$).
3. Read the time value where the activity drops to half that initial value ($400\text{ counts s}^{-1}$). The difference in time is one half-life.
4. Repeat from $400$ to $200\text{ counts s}^{-1}$ and average the time intervals to reduce reading errors.

Radioisotopes in Practice

Selecting an isotope for an engineering or medical task requires choosing the correct radiation type (determining how far it penetrates and its ionising strength) and an appropriate half-life (how long it remains active).

• Smoke detectors (Americium-241): Americium-241 emits alpha particles that ionise air inside a small chamber, allowing a tiny electric current to pass between two electrodes. Smoke particles entering the chamber absorb the alpha particles and neutralise ions, dropping the current and triggering the alarm. Alpha particles cannot escape the detector's casing, so the risk from outside is negligible (the source must not be opened, swallowed or inhaled). Its half-life of 432 years ensures the detector operates reliably for years without needing replacement.
• Medical tracers and cancer therapy (Iodine-131): The human thyroid gland naturally concentrates iodine. Administering Iodine-131 delivers targeted beta particles and gamma rays directly to diseased thyroid tissue. Its short half-life of 8 days ensures the patient receives an effective therapeutic dose without remaining radioactive for extended periods.
• Carbon dating (Carbon-14): Cosmic rays continuously produce carbon-14 in the upper atmosphere, which living organisms absorb through photosynthesis and the food chain. When an organism dies, intake stops and carbon-14 decays via beta emission with a half-life of approximately 5,730 years. Measuring the remaining activity per gram of carbon reveals how long ago the organism died.
• Space exploration power sources (Plutonium-238): Deep-space probes traveling far from the Sun cannot rely on solar panels. Radioisotope thermoelectric generators (RTGs) capture the heat released by alpha-decaying Plutonium-238 and convert it into electricity using thermocouples. A half-life of 87.8 years supplies steady power for decades.