Nuclear physics

At the heart of nuclear physics is one of the most famous equations in science: E = mc². Proposed by Albert Einstein, it reveals a fundamental relationship between energy (E) and mass (m). It states that mass is a form of energy and can be converted into other forms of energy, and vice versa. The constant c² (the speed of light squared) is a huge number, meaning a tiny amount of mass can be converted into a vast amount of energy. This is the principle behind the energy released in nuclear reactions. In nuclear physics, masses are often measured in unified atomic mass units (u), where 1 u is defined as 1/12th the mass of a carbon-12 atom. It's useful to know the energy equivalent of 1 u.

If you were to weigh a nucleus (like helium-4) and then weigh its individual components (2 protons and 2 neutrons) separately, you'd find something strange: the separate components weigh more than the assembled nucleus. This difference in mass is called the mass defect (Δm). So where does the 'missing' mass go? It is converted into energy when the nucleus is formed, according to E = mc². This energy is called the binding energy (E_B). The binding energy is the energy that holds the nucleus together. It is also, therefore, the minimum energy required to completely separate a nucleus into its constituent protons and neutrons.

While a large binding energy means a nucleus is tightly held together, it doesn't tell the whole story. A very large nucleus will naturally have a large total binding energy. A better measure of nuclear stability is the binding energy per nucleon, which is the total binding energy divided by the number of nucleons (protons + neutrons) in the nucleus. The higher the binding energy per nucleon, the more stable the nucleus. A graph of binding energy per nucleon against nucleon number (A) shows a distinct pattern. It rises sharply for light nuclei, peaks at a nucleon number of around 56 (Iron), and then slowly decreases for heavier nuclei. This curve is key to understanding energy release in nuclear reactions.

The binding energy curve explains why two types of nuclear reactions release enormous amounts of energy: fission and fusion.

Nuclear Fission: This is the process where a heavy, unstable nucleus (like Uranium-235) splits into two or more smaller, lighter nuclei. This happens because the heavy nucleus is on the right-hand side of the binding energy peak. When it splits, the resulting smaller nuclei are closer to the peak and thus have a higher binding energy per nucleon. This increase in binding energy per nucleon is released as energy. Fission is the process used in nuclear power plants.

Nuclear Fusion: This is the process where two light nuclei (like isotopes of hydrogen) combine to form a single, heavier nucleus. This happens because the light nuclei are on the left-hand, rising part of the binding energy curve. When they fuse, the resulting nucleus is further up the curve and has a higher binding energy per nucleon. Again, this increase is released as energy. Fusion is the process that powers the Sun and other stars.

Radioactive decay, the process by which an unstable nucleus transforms by emitting radiation, is governed by two key principles: it is spontaneous and random.

Spontaneous: This means the decay of a nucleus is not affected by any external factors. You cannot make a nucleus decay by heating it, crushing it, or putting it in a chemical reaction. The decay happens on its own, independent of its environment.

Random: This means it is impossible to predict when a particular nucleus will decay. If you have two identical unstable nuclei, one might decay in the next second, while the other might last for a million years. We have no way of knowing which will be which. However, for a very large number of nuclei (like in any macroscopic sample), the overall rate of decay is predictable. The evidence for this randomness comes from observing the count rate from a radioactive source with a Geiger counter. The clicks or readings will not be perfectly regular; they will fluctuate about an average value, showing that the decays are happening at random moments.

To describe the decay of a large number of nuclei, we use three key quantities.

Activity (A): This is the rate at which nuclei in a sample are decaying. It's measured in becquerels (Bq), where 1 Bq = 1 decay per second.

Decay Constant (λ): This is the probability that a single, individual nucleus will decay in a unit of time. It has units of s⁻¹, min⁻¹, or year⁻¹, etc. A large decay constant means a high probability of decay, so the substance decays quickly.

These two are related to the number of undecayed nuclei (N) in the sample by the equation A = λN.

Half-life (t₁/₂): This is a more intuitive measure. It is the average time taken for half of the undecayed nuclei in a sample to decay. It is also the time taken for the activity of the sample to fall to half its initial value. A short half-life means a substance is highly radioactive but will become safe quickly. A long half-life means it's less radioactive but will remain a hazard for a long time. Half-life and the decay constant are inversely related.

The random nature of decay at the individual level leads to a predictable, smooth, exponential decrease for a large population of nuclei. The number of undecayed nuclei, the activity, and the measured count rate all decrease exponentially with time. This behaviour is described by a single, powerful equation: x = x₀e⁻ˡᵗ. In this equation, x₀ is the initial value of the quantity (at time t=0), x is the value at time t, λ is the decay constant, and 'e' is the base of the natural logarithm (approximately 2.718). This equation is the cornerstone of all calculations involving radioactive decay over time, from medical imaging to carbon dating.