9.3 Radioactivity and Decay Processes

Radioactivity is the spontaneous emission of radiation from unstable atomic nuclei, a natural process that transforms one element into another. Henri Becquerel discovered it in 1896 while studying uranium salts. This topic covers the three main types of radioactive decay (alpha, beta, and gamma), how to balance nuclear equations, and the concept of decay series.

Radioactivity and its origins

Discovery and fundamental concepts

Radioactive nuclei are unstable, and they release energy by emitting particles or photons until they reach a more stable configuration. Becquerel discovered this phenomenon with uranium, and Marie and Pierre Curie expanded the field by identifying additional radioactive elements like polonium and radium.

• Natural radioactivity occurs in elements with atomic number greater than 83 (e.g., uranium, thorium, radium)
• Artificial radioactivity is induced in otherwise stable nuclei through nuclear reactions, producing isotopes like technetium-99m and cobalt-60 used in medicine and industry

Causes and characteristics of radioactivity

A nucleus becomes unstable when it has an unfavorable proton-to-neutron ratio or carries excess energy. The nucleus "wants" to reach a lower energy state, and radioactive decay is how it gets there.

• Half-life ($t_{1/2}$) is the time it takes for half of a sample's radioactive nuclei to decay. Each radioisotope has its own characteristic half-life, and it doesn't change with temperature, pressure, or chemical environment.
• Radioactive decay follows first-order kinetics, meaning the decay rate is proportional to the number of radioactive nuclei present at any moment. The number of undecayed nuclei as a function of time is:

$N(t) = N_0 \, e^{-\lambda t}$

where $N_0$ is the initial number of nuclei and $\lambda$ is the decay constant.

• The decay constant ($\lambda$) quantifies the probability of decay per unit time and relates to half-life by:

$t_{1/2} = \frac{\ln(2)}{\lambda}$

• The activity ($A$) of a sample is the number of decays per second: $A = \lambda N$. Activity is measured in becquerels (1 Bq = 1 decay/s) or curies (1 Ci = $3.7 \times 10^{10}$ decays/s).

Types of radioactive decay

Alpha decay

An alpha particle consists of two protons and two neutrons, identical to a helium-4 nucleus. When a heavy nucleus emits one, its atomic number drops by 2 and its mass number drops by 4.

• Alpha particles are highly ionizing because of their +2 charge and large mass, but they're the least penetrating form of radiation. A single sheet of paper or a few centimeters of air will stop them.
• The energy spectrum of alpha particles shows discrete peaks because the nuclear energy levels are quantized.
• Alpha decay is most common in heavy nuclei ($Z > 83$) where the strong nuclear force can no longer fully overcome the Coulomb repulsion among protons.
• Example: radium-226 decaying into radon-222:

$_{88}^{226}Ra \rightarrow _{86}^{222}Rn + _{2}^{4}He$

Check the balance: $A$: 226 = 222 + 4. $Z$: 88 = 86 + 2. Both sides match.

Beta decay

Beta decay comes in three forms, all involving the conversion of one type of nucleon into another inside the nucleus.

• $\beta^-$ decay: A neutron converts into a proton, emitting an electron and an electron antineutrino. The atomic number increases by 1; mass number stays the same. This occurs in nuclei that have too many neutrons relative to protons.
• $\beta^+$ decay: A proton converts into a neutron, emitting a positron and an electron neutrino. The atomic number decreases by 1; mass number stays the same. This occurs in proton-rich nuclei.
• Electron capture: An inner orbital electron is absorbed by the nucleus, converting a proton into a neutron and emitting a neutrino. The changes in $Z$ and $A$ are the same as $\beta^+$ decay, but no positron is emitted. Instead, characteristic X-rays are produced as outer electrons fill the vacancy.

Beta particles are moderately ionizing and moderately penetrating, typically stopped by a thin sheet of aluminum (a few millimeters thick). Unlike alpha particles, beta particles have a continuous energy spectrum because the emitted energy is shared between the electron (or positron) and the neutrino. This continuous spectrum was actually the key evidence that neutrinos existed; without them, energy and momentum wouldn't be conserved.

Example of $\beta^-$ decay (carbon-14 to nitrogen-14):

$_{6}^{14}C \rightarrow _{7}^{14}N + _{-1}^{0}e + \bar{\nu}_{e}$

Gamma decay

Gamma rays are high-energy photons emitted when an excited nucleus drops to a lower energy state. No particles are ejected, so there's no change in atomic number or mass number.

• Gamma rays are the least ionizing but the most penetrating form of radiation. You need thick lead or several centimeters of concrete to attenuate them significantly.
• The energy spectrum is discrete, with peaks corresponding to specific nuclear energy level transitions.
• Gamma emission often accompanies alpha or beta decay, since the daughter nucleus is frequently left in an excited state.

Example (excited cobalt-60 de-exciting):

$_{27}^{60}Co^* \rightarrow _{27}^{60}Co + \gamma$

The asterisk ($*$) denotes the excited nuclear state.

Balancing nuclear decay equations

General principles

Every nuclear equation must conserve two quantities:

1. Mass number (A): the total number of nucleons on the left must equal the total on the right
2. Atomic number (Z): the total charge (proton number) must balance on both sides

Use standard notation with mass number as the superscript and atomic number as the subscript. For beta decay, don't forget to include the neutrino or antineutrino.

Steps for balancing a nuclear equation

1. Write down the parent nuclide with its $A$ and $Z$ values.
2. Identify the type of decay and write the emitted particle with its $A$ and $Z$ (e.g., $_{2}^{4}He$ for alpha, $_{-1}^{0}e$ for $\beta^-$).
3. Subtract the emitted particle's $A$ and $Z$ from the parent's values to find the daughter's $A$ and $Z$.
4. Use a periodic table to identify the daughter element from its $Z$ value.
5. Verify: confirm that $A$ and $Z$ both balance on each side.

Specific decay equations

These are the general forms you should know:

• Alpha decay: $_{Z}^{A}X \rightarrow _{Z-2}^{A-4}Y + _{2}^{4}He$
• Beta-minus decay: $_{Z}^{A}X \rightarrow _{Z+1}^{A}Y + _{-1}^{0}e + \bar{\nu}_e$
• Beta-plus decay: $_{Z}^{A}X \rightarrow _{Z-1}^{A}Y + _{+1}^{0}e + \nu_e$
• Electron capture: $_{Z}^{A}X + _{-1}^{0}e \rightarrow _{Z-1}^{A}Y + \nu_e$
• Gamma decay: $_{Z}^{A}X^* \rightarrow _{Z}^{A}X + \gamma$

Practice example: If $_{92}^{238}U$ undergoes alpha decay, the daughter has $Z = 92 - 2 = 90$ and $A = 238 - 4 = 234$. Element 90 is thorium, so the daughter is $_{90}^{234}Th$.

Decay series and daughter nuclides

Decay series concepts

Most heavy radioactive nuclei don't become stable after a single decay. Instead, they go through a decay series, a chain of successive decays until a stable nuclide is finally reached.

• The three naturally occurring series are the uranium series (starts at $_{92}^{238}U$, ends at $_{82}^{206}Pb$), the thorium series (starts at $_{90}^{232}Th$, ends at $_{82}^{208}Pb$), and the actinium series (starts at $_{92}^{235}U$, ends at $_{82}^{207}Pb$). Each terminates at a stable isotope of lead.
• The uranium-238 series involves 14 decay steps (8 alpha decays and 6 beta decays) before reaching stable $_{82}^{206}Pb$. You can verify this: 8 alpha decays reduce $Z$ by 16 and $A$ by 32, while 6 $\beta^-$ decays increase $Z$ by 6. Net change: $Z$ goes from 92 to $92 - 16 + 6 = 82$, and $A$ goes from 238 to $238 - 32 = 206$.
• Branching decay occurs when a nuclide can decay by more than one mode. Each branch has a specific branching ratio that gives the probability of that path.

Daughter nuclides and equilibrium

The product of any radioactive decay is called a daughter nuclide. It may itself be radioactive (and decay further) or it may be stable.

When a parent has a much longer half-life than its daughter, the system eventually reaches secular equilibrium. At that point, the daughter decays at the same rate it's being produced, so the activity of parent and daughter become equal:

$\lambda_1 N_1 = \lambda_2 N_2$

Here $\lambda_1$ and $N_1$ are the decay constant and number of atoms of the parent, while $\lambda_2$ and $N_2$ are for the daughter. Since $\lambda_2 \gg \lambda_1$ in secular equilibrium, the daughter is present in much smaller quantities ($N_2 \ll N_1$) but decays fast enough to match the parent's activity.

Applications of decay series

• Radiometric dating: Carbon-14 dating works for organic materials up to ~50,000 years old ($t_{1/2} \approx 5,730$ years). Uranium-lead dating handles geological timescales of billions of years ($t_{1/2}$ of $^{238}U \approx 4.47 \times 10^9$ years). In both cases, you measure the ratio of parent to daughter isotopes and use the decay equation to calculate the sample's age.
• Nuclear forensics: Analyzing the ratios of isotopes in a sample can reveal its origin and processing history.
• Radon mitigation: Radon-222 is a gaseous daughter in the uranium-238 series. Because it's a noble gas, it seeps out of rock and soil into buildings. Understanding the decay chain helps engineers design ventilation systems to reduce radon buildup.
• Medical imaging: Technetium-99m, the most widely used medical radioisotope, is produced from the decay of molybdenum-99. Its short half-life (~6 hours) and gamma emission make it ideal for diagnostic scans, since it provides a clear image without delivering excessive radiation dose to the patient.