Radioactive decay is a fascinating process where unstable atomic nuclei break down, emitting particles and energy. This natural phenomenon comes in different forms, each with unique characteristics and applications in science and technology.

Understanding radioactive decay is crucial for grasping nuclear physics and chemistry. It helps explain how elements change over time, enables radiometric dating, and forms the basis for nuclear energy production and medical treatments.

Radioactive Decay Modes and Particles

Types of radioactive decay

• Alpha ($\alpha$) decay involves the emission of an alpha particle (helium nucleus, $^4_2He$) consisting of 2 protons and 2 neutrons from the atomic nucleus resulting in a decrease of the atomic number by 2 and mass number by 4
• Beta ($\beta$) decay occurs in two forms:
  • Beta minus ($\beta^-$) decay involves the emission of an electron from the atomic nucleus as a neutron converts into a proton, an electron, and an antineutrino, increasing the atomic number by 1 while the mass number remains constant
  • Beta plus ($\beta^+$) decay involves the emission of a positron (antiparticle of an electron) from the atomic nucleus as a proton converts into a neutron, a positron, and a neutrino, decreasing the atomic number by 1 while the mass number remains constant
• Gamma ($\gamma$) decay involves the emission of high-energy electromagnetic radiation (photons) from the atomic nucleus as an excited nucleus transitions to a lower energy state without changing the atomic number or mass number

Particles in nuclear decay

• Alpha particles are helium nuclei ($^4_2He$) emitted during alpha decay carrying a positive charge of +2, having a mass of approximately 4 atomic mass units (amu), and typically possessing kinetic energy in the range of 4-9 MeV
• Beta particles are electrons ($e^-$) or positrons ($e^+$) emitted during beta decay carrying a negative charge of -1 (electrons) or a positive charge of +1 (positrons), having a mass of approximately 1/1836 amu, and possessing varying kinetic energy with a maximum value depending on the specific decay
• Gamma rays are high-energy photons emitted during gamma decay with no charge, no mass, and energy typically in the range of keV to MeV
• Radioactive decay releases energy due to the conversion of mass to energy (E=mc^2) carried away by the emitted particles and photons as the daughter nucleus achieves a lower mass and greater stability than the parent nucleus
• Radiation detectors are used to measure and analyze the particles and energy released during radioactive decay

Nuclear Decay Equations and Kinetics

Equations for nuclear decay

• Alpha decay equation: $^A_ZX \rightarrow ^{A-4}_{Z-2}Y + ^4_2He$ where X is the parent nucleus, Y is the daughter nucleus, A is the mass number, and Z is the atomic number
• Beta minus decay equation: $^A_ZX \rightarrow ^A_{Z+1}Y + e^- + \bar{\nu}_e$ where $\bar{\nu}_e$ represents an electron antineutrino
• Beta plus decay equation: $^A_ZX \rightarrow ^A_{Z-1}Y + e^+ + \nu_e$ where $\nu_e$ represents an electron neutrino
• Gamma decay equation: $^A_ZX^* \rightarrow ^A_ZX + \gamma$ where the asterisk (*) indicates an excited nuclear state

Half-life and decay kinetics

• Half-life ($t_{1/2}$) is the time required for half of a given quantity of a radioactive substance to decay, which is constant for a specific radionuclide and can be calculated using the equation: $t_{1/2} = \frac{ln(2)}{\lambda}$, where $\lambda$ is the decay constant
• Decay constant ($\lambda$) is the probability of decay per unit time and is related to half-life by: $\lambda = \frac{ln(2)}{t_{1/2}}$
• Exponential decay equation: $N(t) = N_0e^{-\lambda t}$ where $N(t)$ is the quantity of the radionuclide at time $t$, $N_0$ is the initial quantity of the radionuclide, $\lambda$ is the decay constant, and $t$ is the elapsed time

Radiometric Dating

Principles of radiometric dating

• Radiometric dating is based on the radioactive decay of unstable isotopes and compares the ratio of a radioactive isotope to its decay product in a sample, assuming that the initial ratio is known and that the sample has remained a closed system (no loss or gain of isotopes)
• Carbon-14 dating is used for dating organic materials up to ~50,000 years old based on the decay of carbon-14 to nitrogen-14 with a half-life of approximately 5,730 years
• Uranium-lead dating is used for dating rocks and minerals based on the decay of uranium-238 to lead-206 (half-life of 4.47 billion years) and uranium-235 to lead-207 (half-life of 704 million years)
• Potassium-argon dating is used for dating rocks and minerals based on the decay of potassium-40 to argon-40 with a half-life of 1.28 billion years
• Radiometric dating is applied to determine the age of fossils, rocks, and archaeological artifacts, reconstruct past climates and environments, study the evolution of life on Earth, and investigate the formation and history of the Earth and solar system

Nuclear Stability and Energy

Nuclear stability and isotopes

• Nuclear stability is determined by the ratio of neutrons to protons in the nucleus, with stable nuclei generally having a specific range of neutron-to-proton ratios
• Isotopes are atoms of the same element with different numbers of neutrons, which can affect their stability and radioactive properties
• Radioactive series describe the sequential decay of unstable isotopes, forming a chain of decay products until a stable isotope is reached

Nuclear reactions

• Nuclear fission is the splitting of heavy atomic nuclei into lighter nuclei, releasing energy and often neutrons that can sustain a chain reaction
• Nuclear fusion is the combining of light atomic nuclei to form heavier nuclei, releasing large amounts of energy in the process