21.3 Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei emit particles and energy to reach a more stable state. Understanding the different types of decay, how to write nuclear equations, and how decay rates work gives you the foundation for topics like nuclear energy, medical imaging, and radiometric dating.

Radioactive Decay Modes and Particles

Types of radioactive decay

There are three main types of radioactive decay you need to know, and each one changes the nucleus in a different way.

Alpha ($\alpha$) decay releases an alpha particle, which is a helium nucleus ($^4_2He$) containing 2 protons and 2 neutrons. Because the nucleus loses 2 protons and 2 neutrons, the atomic number drops by 2 and the mass number drops by 4. This produces a completely different element.

Beta ($\beta$) decay comes in two forms:

• Beta minus ($\beta^-$) decay: A neutron in the nucleus converts into a proton, releasing an electron and an antineutrino. The atomic number increases by 1 (new proton), but the mass number stays the same since the total number of nucleons hasn't changed.
• Beta plus ($\beta^+$) decay: A proton converts into a neutron, releasing a positron (the antimatter version of an electron) and a neutrino. The atomic number decreases by 1, and the mass number again stays the same.

Gamma ($\gamma$) decay is different from the other two. The nucleus doesn't emit a particle with mass. Instead, an excited nucleus releases energy as a high-energy photon (electromagnetic radiation). Neither the atomic number nor the mass number changes, so the element stays the same.

Particles in nuclear decay

Each type of radiation has distinct properties that affect how far it can travel and how much damage it can do:

• Alpha particles ($^4_2He$): charge of +2, mass of about 4 amu, kinetic energy typically 4–9 MeV. They're the heaviest and most ionizing, but they can be stopped by a sheet of paper or a few centimeters of air.
• Beta particles ($e^-$ or $e^+$): charge of -1 (electrons) or +1 (positrons), mass of about $\frac{1}{1836}$ amu. They're much lighter and more penetrating than alpha particles, but a sheet of aluminum can stop them.
• Gamma rays: no charge, no mass, energy typically in the keV to MeV range. They're the most penetrating and require dense shielding like lead or thick concrete.

All radioactive decay releases energy because the products have slightly less total mass than the parent nucleus. That "missing" mass is converted to energy according to $E = mc^2$. The daughter nucleus ends up at a lower energy state and is more stable than the parent.

Nuclear Decay Equations and Kinetics

Equations for nuclear decay

Writing nuclear equations follows one key rule: both the mass numbers (top) and atomic numbers (bottom) must balance on each side of the arrow.

• Alpha decay: $^A_ZX \rightarrow ^{A-4}_{Z-2}Y + ^4_2He$
• Beta minus decay: $^A_ZX \rightarrow ^A_{Z+1}Y + e^- + \bar{\nu}_e$
• Beta plus decay: $^A_ZX \rightarrow ^A_{Z-1}Y + e^+ + \nu_e$
• Gamma decay: $^A_ZX^* \rightarrow ^A_ZX + \gamma$ (the asterisk * means the nucleus starts in an excited state)

For example, the alpha decay of uranium-238 looks like this:

$^{238}_{92}U \rightarrow ^{234}_{90}Th + ^4_2He$

Check: mass numbers $238 = 234 + 4$ ✓ and atomic numbers $92 = 90 + 2$ ✓

Half-life and decay kinetics

Half-life ($t_{1/2}$) is the time it takes for half of a radioactive sample to decay. This value is constant for any given isotope and doesn't depend on how much material you start with or external conditions like temperature.

The key equations connecting half-life, the decay constant, and the amount remaining are:

• Decay constant: $\lambda = \frac{\ln(2)}{t_{1/2}}$ The decay constant $\lambda$ represents the probability that a given nucleus will decay per unit time. A larger $\lambda$ means faster decay and a shorter half-life.

• Exponential decay: $N(t) = N_0 e^{-\lambda t}$ Here $N_0$ is the initial quantity, $N(t)$ is the quantity remaining after time $t$, and $\lambda$ is the decay constant.

Quick approach for whole half-lives: If you just need to find how much remains after a whole number of half-lives, you can skip the exponential formula. After $n$ half-lives, the fraction remaining is $\left(\frac{1}{2}\right)^n$. So after 3 half-lives, you'd have $\frac{1}{8}$ of the original sample left.

Radiometric Dating

Principles of radiometric dating

Radiometric dating uses the predictable rate of radioactive decay to determine the age of a sample. The basic idea: measure the ratio of a radioactive parent isotope to its stable daughter product, and use the known half-life to calculate how much time has passed.

Two assumptions are critical for this to work:

1. The initial ratio of parent to daughter isotope is known (or can be reasonably estimated).
2. The sample has been a closed system, meaning no parent or daughter isotopes have been added or lost since the sample formed.

Different isotopes are useful for different timescales:

For much older samples (millions to billions of years), methods like uranium-lead or potassium-argon dating are used instead, since carbon-14 decays too quickly to be detectable after about 50,000 years.

Nuclear Stability and Energy

Nuclear stability and isotopes

Whether a nucleus is stable depends largely on its neutron-to-proton ratio. For lighter elements (up to about $Z = 20$), a roughly 1:1 ratio tends to be stable. Heavier elements need progressively more neutrons than protons to remain stable.

Nuclei that fall outside the "band of stability" (the range of stable neutron-to-proton ratios) are radioactive and will decay toward that stable zone. Nuclei with too many neutrons tend to undergo $\beta^-$ decay (converting a neutron to a proton), while nuclei with too many protons tend to undergo $\beta^+$ decay or electron capture.

Radioactive decay series: Some heavy isotopes don't become stable after just one decay. Instead, they go through a chain of successive decays, producing different elements along the way, until they finally reach a stable isotope. For example, $^{238}U$ goes through 14 decay steps before ending as stable $^{206}Pb$.

Nuclear reactions

Beyond radioactive decay, nuclei can also undergo two other important types of reactions:

• Nuclear fission is the splitting of a heavy nucleus (like $^{235}U$) into two lighter nuclei, releasing energy and additional neutrons. Those neutrons can trigger more fission events, creating a chain reaction. This is the process behind nuclear power plants and atomic weapons.
• Nuclear fusion is the combining of light nuclei (like hydrogen isotopes) to form a heavier nucleus, releasing even more energy per unit mass than fission. Fusion powers the sun and other stars, but achieving sustained fusion on Earth remains a major engineering challenge.

Both processes release energy because the products have a higher binding energy per nucleon than the starting materials, meaning the products are more tightly held together and more stable.