Radioactivity, Nuclear Reactions, and Decay Modes

Fundamentals of Atomic Structure and Radioactivity

• Elements with atomic numbers of $20$ or higher usually have one or more isotopes containing unstable nuclei.

• Radioactive decay (radioactivity) is defined as the spontaneous disintegration of a nucleus accompanied by the emission of particles, electromagnetic radiation, or both.

• Nuclei are inherently unstable due to the relative numbers and combinations of protons and neutrons within the nucleus; unstable nuclei break apart and undergo transformations to eventually form stable nuclei.

• Emission during radioactive decay involves the release of energy, physical particles, or both from the decaying atom.

• Transmutation refers to the transformation of an atom of one element into an atom of a different chemical element, which occurs whenever nuclear reactions alter the proton count in the nucleus.

• Decaying systems are inherently unpredictable prior to observation; regardless of the isotope, it cannot be automatically predicted beforehand what specific mode of decay an atom will undergo (e.g., whether a carbon atom will undergo alpha decay or beta decay).

Nuclear Terminology and Isotope Representation

• Nucleons are defined as the subatomic particles located within the atomic nucleus, specifically comprising protons and neutrons.

• A nuclide is a general term applied to any atom of a specific nuclear species, characterized by its atomic composition.

• Isotope representation methods:

  • Nuclear Symbol Notation:

  • Written in the general form ${}^{A}_{Z}\text{X}$, where $A$ represents the mass number, $Z$ represents the atomic number, and $\text{X}$ represents the chemical symbol of the element.

  • Mass number ($A$) is the sum of protons plus neutrons in the nucleus.

  • Atomic number ($Z$) is the total number of protons in the nucleus.

  • In standard nuclear symbols, the atomic number $Z$ is occasionally omitted because the chemical symbol uniquely identifies the proton count (e.g., ${}^{133}_{55}\text{Cs}$ or ${}^{133}\text{Cs}$).

  • Hyphen Notation:

  • Formatted by writing the element name or symbol followed by a hyphen and the mass number.

  • Examples include $\text{cesium-133}$, $\text{Carbon-14}$, or $\text{C-14}$.

Types of Radiation and Nuclear Decay Modes

• Radioactive Decay Notation Principles:

  • General symbolic expression: $\text{Radioactive nucleus} \rightarrow \text{New nucleus} + \text{Radiation}$.

  • Fundamental conservation law: The sum of the mass numbers ($A$) and the sum of the atomic numbers ($Z$) must be equal on both sides of the arrow in all nuclear reaction equations.

• Alpha ($\alpha$) Radiation and Alpha Decay:

  • Involves the emission of an alpha particle, which is identical to a helium-4 nucleus (${}^4_2\text{He}^{2+}$, ${}^4_2\text{He}$, or $\alpha$).

  • Physical composition: Contains $2$ protons and $2$ neutrons, giving it a mass number of $4$ and an atomic number of $2$.

  • Nuclear Effect: Reduces the mass number ($A$) of the parent nucleus by $4$ and reduces the atomic number ($Z$) by $2$.

  • Shielding and Penetration: Alpha particles have low penetrating power and are stopped or blocked by a single sheet of paper placed in front of them.

  • Nuclear Reaction Example: Uranium-238 decays into Thorium-234 via alpha emission:     ${}^{238}_{92}\text{U} \rightarrow {}^{234}_{90}\text{Th} + {}^4_2\text{He}$

  • Mass and atomic number balance verification:

  • Superscript sum: $238 = 234 + 4$

  • Subscript sum: $92 = 90 + 2$

• Beta ($\beta$) Radiation and Beta Decay:

  • Involves the emission of a beta particle, which is a high-energy electron (${}^0_{-1}e$, ${}^0_{-1}\beta$, or $\beta$) ejected directly from the nucleus.

  • Nuclear Mechanism: Results from the breakdown of a neutron within an unstable nucleus into a proton and an electron.

  • Nuclear Effect: The generated proton remains in the nucleus, increasing the atomic number ($Z$) by $1$ (add $1$) while decreasing the neutron count by $1$. The overall mass number ($A$) remains unchanged because an electron has a mass number of $0$.

  • Shielding and Penetration: Beta particles penetrate paper but are blocked or stopped by lead or thick glass.

  • Detection: Radioactivity and beta particles are detected using a Geiger counter.

  • Nuclear Reaction Examples:

  • Decay of Carbon-14 into Nitrogen-14:       ${}^{14}_{6}\text{C} \rightarrow {}^{14}_{7}\text{N} + {}^0_{-1}e$

  • Free neutron breakdown equation:       ${}^1_0n \rightarrow {}^1_1\text{H} + {}^0_{-1}e$

• Positron Emission:

  • Involves the emission of a positron (${}^0_{+1}e$ or $\beta^+$), which is an antimatter particle possessing the same mass as an electron (mass number $= 0$) but carrying a positive charge ($+1$).

  • Nuclear Mechanism: A proton inside an unstable nucleus converts into a neutron and a positron. The neutron stays in the nucleus, while the positron leaves the atom.

  • Nuclear Effect: Decreases the atomic number ($Z$) by $1$ while leaving the mass number ($A$) unchanged.

  • Nuclear Reaction Examples:

  • Conversion of Potassium-39 to Argon-39:       ${}^{39}_{19}\text{K} \rightarrow {}^{39}_{18}\text{Ar} + {}^0_{+1}e$

  • Conversion of Aluminum-24 to Magnesium-24:       ${}^{24}_{13}\text{Al} \rightarrow {}^{24}_{12}\text{Mg} + {}^0_{+1}e$

• Electron Capture:

  • Occurs when an inner-shell orbital electron is captured by the nucleus of its own atom (${}^0_{-1}e$ on the reactant side).

  • Nuclear Mechanism: The captured inner electron combines with a nuclear proton to form a new neutron.

  • Nuclear Effect: Decreases the atomic number ($Z$) by $1$ while leaving the mass number ($A$) unchanged.

  • Nuclear Reaction Example:     ${}^{106}_{47}\text{Ag} + {}^0_{-1}e \rightarrow {}^{106}_{46}\text{Pd}$

• Gamma ($\gamma$) Radiation and Gamma Emission:

  • High-energy electromagnetic radiation ($\gamma$ or ${}^0_0\gamma$) with energy levels exceeding those of X-rays.

  • Physical Nature: Represents pure energy emission without mass or electric charge (mass $= 0$, charge $= 0$).

  • Nuclear Mechanism: Released when an unstable nucleus undergoes structural particle rearrangement.

  • Associated Emission: Frequently accompanies alpha or beta decay processes.

  • Nuclear Effect: Does not change the mass number ($A$) or the atomic number ($Z$) of the nuclide.

  • Medical Application: Focused gamma radiation is utilized in medical therapies to destroy cancer cells.

  • Nuclear Reaction Examples:

  • Alpha decay of Thorium-230 with gamma emission:       ${}^{230}_{90}\text{Th} \rightarrow {}^{226}_{88}\text{Ra} + {}^4_2\text{He} + \gamma$

  • Beta decay of Thorium-234 with gamma emission:       ${}^{234}_{90}\text{Th} \rightarrow {}^{234}_{91}\text{Pa} + {}^0_{-1}e + \gamma$

Half-Life Dynamics and Quantitative Calculations

• Definition of Half-Life ($t_{1/2}$):

  • The amount of time required for exactly one-half of the radioactive nuclei present in a sample to undergo decay.

• Temporal Variation across Nuclides:

  • Half-lives range from tiny fractions of a second to billions of years depending on the nuclide.

  • Naturally occurring radioactive isotopes usually have long half-lives.

  • Isotopes created through nuclear processes generally have shorter half-lives.

  • Decay timelines for individual unstable atoms cannot be predicted.

• Sequential Half-Life Decay Progression:

  • Starting with a $100\,\text{g}$ sample:

  • After $1$ half-life ($1\,t_{1/2}$): $50\,\text{g}$ remains.

  • After $2$ half-lives ($2\,t_{1/2}$): $25\,\text{g}$ remains.

  • After $3$ half-lives ($3\,t_{1/2}$): $12.5\,\text{g}$ remains.

  • After $4$ half-lives ($4\,t_{1/2}$): $6.25\,\text{g}$ remains.

  • Starting with a $20\,\text{mg}$ sample:

  • $20\,\text{mg} \rightarrow 10\,\text{mg} \rightarrow 5\,\text{mg} \rightarrow 2.5\,\text{mg}$.

• Decay Curve:

  • A graphical diagram plotting the remaining quantity of a radioactive isotope against elapsed time, illustrating exponential decay.

• Quantitative Half-Life Worked Examples:

  • Sample Problem 1:

  • Question: A radioactive isotope has a half-life of $6\,\text{days}$. How much remains of a $50.0\,\text{g}$ sample after $18\,\text{days}$?

  • Calculation:

    • Calculate total half-life cycles elapsed:         $\text{Elapsed half-lives} = \frac{18\,\text{days}}{6\,\text{days}} = 3\,\text{half-lives}$

    • Halve the original sample three times sequentially:         $50.0\,\text{g} \xrightarrow{1} 25.0\,\text{g} \xrightarrow{2} 12.5\,\text{g} \xrightarrow{3} 6.25\,\text{g}$

  • Result: Exactly $6.25\,\text{g}$ remains after $18\,\text{days}$.

  • Sample Problem 2:

  • Question: In $27.5\,\text{seconds}$, a $72.0\,\text{g}$ sample of a radioactive isotope decays to leave $2.25\,\text{g}$. What is the half-life of this nuclide?

  • Calculation:

    • Count the number of half-life decay cycles from $72.0\,\text{g}$ to $2.25\,\text{g}$:         $72.0\,\text{g} \xrightarrow{1} 36.0\,\text{g} \xrightarrow{2} 18.0\,\text{g} \xrightarrow{3} 9.0\,\text{g} \xrightarrow{4} 4.5\,\text{g} \xrightarrow{5} 2.25\,\text{g}$

    • Total cycles elapsed = $5$

    • Calculate the duration of one half-life cycle:         $\text{Half-life } t_{1/2} = \frac{27.5\,\text{seconds}}{5} = 5.5\,\text{seconds}$

  • Result: The half-life is $5.5\,\text{seconds}$.

Nuclear Fission and Chain Reactions

• Definition of Nuclear Fission:

  • A reaction process in which heavy nuclei split into lighter, stable, medium-mass nuclei.

• Energy and Neutrons Released:

  • Fission releases massive amounts of energy (atomic energy) alongside fast-moving neutrons.

• Occurrence and Initiation:

  • Fission occurs naturally or can be artificially induced by bombarding heavy target nuclei with small particles such as neutrons.

• Primary Example — Fission of Uranium-235:

  • Neutron bombardment of Uranium-235 (${}^{235}_{92}\text{U}$):     ${}^1_0n + {}^{235}_{92}\text{U} \rightarrow {}^{236}_{92}\text{U}^* \rightarrow {}^{92}_{36}\text{Kr} + {}^{141}_{56}\text{Ba} + 3\,{}^1_0n + \text{Energy}$

  • An unstable Uranium-236 intermediate nucleus briefly forms before splitting into Krypton-92, Barium-141, three high-speed neutrons, and energy.

• Chain Reaction Dynamics:

  • The fast-moving neutrons released during initial fission hit adjacent heavy nuclei, causing them to split in turn.

  • Applications:

  • Uncontrolled Chain Reactions: Lead to nuclear weapon explosions (atomic bomb).

  • Controlled Chain Reactions: Utilized in nuclear reactors, which are equipped with containment and control systems designed to prevent runaway chain reactions.

Nuclear Fusion

• Definition of Nuclear Fusion:

  • A process where two or more low-mass nuclei (such as hydrogen or helium nuclei) combine or fuse together to form a heavier, stable nucleus.

• Energy Yield:

  • Releases extraordinarily large amounts of energy, surpassing the energy yield per unit mass of nuclear fission.

• Temperature Requirements:

  • Requires extremely high temperatures (around $100,000,000\,^\circ\text{C}$) to overcome electrostatic repulsion between positively charged hydrogen nuclei.

• Astrophysical Context:

  • Powers the Sun and other stars, supplying their thermal and light output.

• Stellar Multistep Fusion Chain Reactions:

  • Reaction 1:     ${}^1_1\text{H} + {}^1_1\text{H} \rightarrow {}^2_1\text{H} + {}^0_{+1}e$

  • Reaction 2:     ${}^1_1\text{H} + {}^2_1\text{H} \rightarrow {}^3_2\text{He}$

  • Reaction 3:     ${}^3_2\text{He} + {}^3_2\text{He} \rightarrow {}^4_2\text{He} + 2\,{}^1_1\text{H}$

  • Alternative Reaction Path:     ${}^3_2\text{He} + {}^1_1\text{H} \rightarrow {}^4_2\text{He} + {}^0_{+1}e$

Practical Applications and Summaries of Radioactivity

• Radiocarbon Dating & Age Determination:

  • Carbon dating uses the known half-life of Carbon-14 (${}^{14}\text{C}$) to determine the age of ancient organic remnants, fossils, and human artifacts based on remaining radiation levels.

• Medical Cancer Treatments:

  • Targeted radiation therapy utilizes gamma rays and beta decay to selectively destroy malignant cancer cells.

• Diagnostic Medical Radiotracers:

  • Short-lived radioactive isotopes serve as diagnostic tracers inside the human body for medical imaging procedures.

• Electrical Energy Production:

  • Nuclear power plants convert energy from controlled fission into electricity. Fission remnants have super long half-lives, providing continuous electrical generation over long spans.

• Medical Instrument Sterilization:

  • Radiation kills pathogens and sterilizes surgical instruments and medical supplies safely.

• Space Exploration Power:

  • Nuclear power units supply long-duration power for space travel and extraterrestrial exploration equipment, such as Mars rovers.