Nuclear Physics: Radioactivity

Radioactivity is the spontaneous emission of radiation from unstable atomic nuclei, a fundamental process that powers stars, provides tools for medicine and energy, and allows us to date ancient artifacts. Understanding its mechanisms is essential for harnessing its benefits and mitigating its risks, from radiotherapy to nuclear safety. At its core, radioactivity is governed by statistical laws that describe how unstable nuclei transform to achieve greater stability.

Types of Radioactive Decay and Their Properties

Unstable nuclei achieve stability through different decay processes, each characterized by distinct particles and properties. The four primary types are alpha, beta-minus, beta-plus, and gamma decay.

Alpha decay occurs primarily in very heavy nuclei (e.g., uranium, radium). The nucleus emits an alpha particle, which is identical to a helium-4 nucleus (two protons and two neutrons). This emission reduces the atomic number by 2 and the mass number by 4. For example, radium-226 decays to radon-222. Alpha particles are relatively large and carry a double positive charge, making them highly ionizing but poorly penetrating. They can be stopped by a sheet of paper or a few centimeters of air. Think of them as bulky, slow-moving marbles that lose energy quickly through collisions.

Beta decay comes in two forms. Beta-minus decay involves the transformation of a neutron into a proton, an electron (the beta-minus particle, ${\beta }^{-}$), and an antineutrino. This increases the atomic number by one while the mass number stays the same, as in the decay of carbon-14 to nitrogen-14. Conversely, beta-plus decay (or positron emission) occurs when a proton transforms into a neutron, a positron (the beta-plus particle, ${\beta }^{+}$), and a neutrino. This decreases the atomic number by one, as when fluorine-18 decays to oxygen-18. Beta particles are much smaller and faster than alpha particles, are less ionizing, and have greater penetrating power, requiring a few millimeters of aluminum to stop them.

Gamma decay is the emission of high-energy photons, called gamma rays, from a nucleus that is in an excited state. This process often follows alpha or beta decay. Unlike alpha and beta decay, gamma emission does not change the composition (proton or neutron count) of the nucleus; it only reduces its energy. Gamma rays are electromagnetic radiation with no mass or charge, making them the most penetrating form of nuclear radiation. They require several centimeters of lead or meters of concrete for significant attenuation. They are the invisible, high-energy "laser beams" of nuclear decay.

Writing Balanced Nuclear Decay Equations

A balanced nuclear equation must conserve both mass number (A, total nucleons) and atomic number (Z, total protons). The sum of these numbers on the left side (parent nucleus) must equal the sum on the right side (daughter nucleus + emitted particles).

For alpha decay of uranium-238: $${}_{92}^{238}U{\to }_{90}^{234}Th{+}_2^{4}He$$ Mass number: 238 = 234 + 4. Atomic number: 92 = 90 + 2.

For beta-minus decay of carbon-14: $${}_6^{14}C{\to }_7^{14}N{+}_{-1}^{0}e+{\bar{\nu }}_e$$ Here, ${}_{-1}^{0}e$ represents the emitted electron. The antineutrino (${\bar{\nu }}_e$) carries away excess energy and momentum but has negligible mass and no charge, so it doesn't affect the balancing of A and Z.

For beta-plus decay of sodium-22: $${}_{11}^{22}Na{\to }_{10}^{22}Ne{+}_{+1}^{0}e+{\nu }_e$$ The positron is denoted ${}_{+1}^{0}e$.

For gamma decay, the equation simply shows the parent nucleus in an excited state (denoted with an asterisk) relaxing to its ground state: $${}_{27}^{60}C{o}^{\ast }{\to }_{27}^{60}Co+\gamma$$

Background Radiation and Sources

We are constantly exposed to low-level background radiation from natural and artificial sources. Natural sources constitute the majority of exposure for most people and include:

1. Radon Gas (approx. 50% of background dose): A radioactive gas emitted from uranium decay in rocks and soil, which can accumulate in buildings.
2. Cosmic Rays: High-energy particles from space that interact with the atmosphere.
3. Terrestrial Radiation: Radioactive isotopes like potassium-40, uranium, and thorium present in the Earth's crust and building materials.
4. Internal Radiation: Radioactive isotopes (like carbon-14 and potassium-40) naturally present in our food and water.

Artificial sources include medical procedures (X-rays, radiotherapy), fallout from historical nuclear weapons testing, and the nuclear industry. Understanding background levels is crucial for accurate radiation measurements and for putting risks from other sources into perspective.

The Exponential Decay Law and Half-Life Calculations

Radioactive decay is a random process for an individual nucleus, but for a large sample, it follows a predictable statistical pattern described by the exponential decay law. The number of undecayed nuclei, $N$, at time $t$ is given by: $$N=N_0{e}^{-\lambda t}$$ where $N_0$ is the initial number of nuclei and $\lambda$ is the decay constant, unique to each isotope, representing the probability of decay per unit time.

A more intuitive concept is the half-life, $T_{1/2}$, defined as the time taken for half of the radioactive nuclei in a sample to decay. It is inversely related to the decay constant: $$T_{1/2}=\frac{\ln 2}{\lambda }\approx \frac{0.693}{\lambda }$$

To calculate half-life from a decay curve, you plot the remaining activity (or count rate) against time on a linear graph. The time taken for the activity to fall from any value to half of that value is the half-life. On a graph where the y-axis is logarithmic, the decay curve becomes a straight line, and the half-life can be easily found. For example, if an isotope's activity drops from 800 Bq to 400 Bq in 6 days, its half-life is 6 days. The decay constant would be $\lambda =\ln 2/T_{1/2}\approx 0.693/6\approx 0.116{\text{day}}^{-1}$.

Activity, $A$ (measured in Becquerels, Bq), is the rate of decay: $A=\lambda N$. It also decays exponentially: $A=A_0{e}^{-\lambda t}$.

Applications: Radioactive Dating

The predictable nature of half-life makes radioactivity a powerful clock. Radiocarbon dating is the most famous application. Carbon-14 ($T_{1/2}\approx 5730$ years) is produced in the atmosphere and absorbed by living organisms. When an organism dies, it stops exchanging carbon, and the C-14 content decays. By measuring the remaining ratio of C-14 to stable C-12 in a sample and comparing it to the atmospheric ratio, the time since death can be calculated using the exponential decay law.

For dating older geological samples, isotopes with much longer half-lives are used. Potassium-argon dating relies on potassium-40 ($T_{1/2}\approx 1.25$ billion years) decaying to argon-40 in rocks. By measuring the ratio of argon-40 to remaining potassium-40, scientists can date rock formations billions of years old. These techniques rely on knowing the initial conditions and assuming a closed system where no parent or daughter isotopes have been lost or added except through radioactive decay.

Common Pitfalls

1. Confusing Mass Number and Atomic Number in Equations: A common error is to balance the atomic number but forget the mass number, or vice versa. Always check both sums independently. For example, in beta-minus decay, remember the mass number of the electron is 0, not 1.
2. Misunderstanding Half-Life: The half-life is constant and does not depend on the amount of material or its physical/chemical state. A common misconception is believing that a more active sample has a shorter half-life. Half-life is an intrinsic property of the isotope itself.
3. Incorrectly Applying the Decay Law: Forgetting that $N$ and $A$ follow the same exponential form $e^{-\lambda t}$, or misplacing the negative sign in the exponent, is a frequent algebraic mistake. Ensure you use the correct form: $N=N_0{e}^{-\lambda t}$, not $N=N_0{e}^{\lambda t}$.
4. Overlooking Background Radiation: When performing practical decay investigations, failing to subtract the background count rate from all measurements will lead to inaccurate calculations of half-life and activity. Always measure background radiation separately.

Summary

• Alpha, beta, and gamma radiation are distinguished by their composition, charge, ionizing power, and penetration. Alpha is a helium nucleus (high ionization, low penetration), beta is a fast electron or positron (moderate), and gamma is a high-energy photon (low ionization, very high penetration).
• Nuclear equations must conserve both mass number (top number) and atomic number (bottom number). Alpha decay reduces A by 4 and Z by 2; beta-minus increases Z by 1; beta-plus decreases Z by 1.
• Background radiation is an ever-present low-level exposure from natural (radon, cosmic rays, terrestrial) and artificial sources, which must be accounted for in experiments.
• Radioactive decay follows an exponential law ($N=N_0{e}^{-\lambda t}$). The half-life ($T_{1/2}=\ln 2/\lambda$) is the constant time for half the nuclei in a sample to decay and is used to analyze decay curves.
• Radioactive dating, like radiocarbon dating, applies the exponential decay law by comparing the current ratio of a radioactive isotope to its stable decay product with the assumed initial ratio to determine the age of a sample.