Types of Radioactive | AP Physics 2 Unit 7 Study Guide

Getting Started

Radioactive decay is a process that occurs at the subatomic scale, within the dense, positively charged nucleus of an atom. Many atomic nuclei are inherently unstable due to an imbalance in the forces holding them together or an unfavorable ratio of neutrons to protons. The core question we will explore is: How do these unstable nuclei spontaneously transform into more stable configurations, and what fundamental physical laws govern these transformations?

What You Should Be able to Do

After working through this section, you should be able to:

Write and balance nuclear decay equations for alpha, beta-minus, beta-plus, and gamma decay.
Identify the parent nucleus, daughter nucleus, and emitted particles in a decay reaction.
Apply the principles of conservation of nucleon number, charge, and lepton number to analyze nuclear decays.
Compare the effects of different decay types on the composition of a nucleus.
Explain why gamma decay is often a secondary process that follows an initial alpha or beta decay.

Key Concepts & Mechanisms

System & Preconditions

Our system of interest is a single, unstable parent nucleus, which we denote as $_{Z}^{A} X$ . Here, X is the chemical symbol for the element, Z is the atomic number (the number of protons), and A is the nucleon number (or mass number), which is the total number of protons and neutrons. The precondition for any radioactive decay is that the nucleus is in an unstable, higher-energy state. A more stable, lower-energy configuration is accessible, and the decay is the process by which the nucleus transitions to that state. In our analysis, we assume the nucleus is an isolated system at the moment of decay and that three fundamental conservation laws are always obeyed.

Key Steps / Relations

The transformation from an unstable parent nucleus to a more stable daughter nucleus is governed by strict conservation rules. For any decay, the total value of these conserved quantities must be the same before and after the event.

Conservation of Nucleon Number (A): The total number of nucleons (protons + neutrons) must remain constant. Nucleons can change their identity (e.g., a neutron to a proton), but they are not created or destroyed.
- Sum of A (reactants) = Sum of A (products)
Conservation of Charge (Z): The net electric charge must remain constant. Since protons and electrons/positrons are the primary charge carriers in these reactions, this is closely related to the conservation of the atomic number.
- Sum of Z (reactants) = Sum of Z (products)
Conservation of Lepton Number (L): Leptons are a class of fundamental particles that includes electrons, positrons, and neutrinos. To account for their appearance in beta decay, we assign a lepton number: L = +1 for electrons and neutrinos, and L = -1 for their antimatter counterparts (positrons and antineutrinos). All other particles, like protons and neutrons, have L = 0. The total lepton number must remain constant.
- Sum of L (reactants) = Sum of L (products)

Outputs & Effects

The primary effect of radioactive decay is the transformation of the parent nucleus into a different nuclide (the daughter) and the emission of one or more particles. While nucleon number, charge, and lepton number are conserved, the identity of the nucleus (its Z) and its internal energy state are altered. Furthermore, a small amount of mass is converted into kinetic energy for the products, following the principle of mass-energy equivalence ( $E = m c^{2}$ ).

1. Alpha (α) Decay

This process occurs primarily in heavy nuclei that have too many protons and neutrons. The nucleus ejects an alpha particle (α), which is a helium nucleus ( $^4_2$He). - **General Equation:**$ ^A_Z X \rightarrow ^{A-4}_{Z-2} Y + ^4_2 \alpha$

Effect: The daughter nucleus (Y) has 4 fewer nucleons and 2 fewer protons than the parent (X). The element changes.

2. Beta-Minus (β⁻) Decay

This occurs in nuclei with an excess of neutrons. A neutron within the nucleus transforms into a proton, and to conserve charge and lepton number, an electron (the beta-minus particle, $_{- 1}^{0} e$ or β⁻) and an electron antineutrino ( $\overset{ν}{ˉ}_{e}$ ) are created and emitted.

General Equation: $_{Z}^{A} X \to_{Z + 1}^{A} Y +_{- 1}^{0} e + \overset{ν}{ˉ}_{e}$
Effect: The nucleon number A is unchanged, but the atomic number Z increases by one. The element changes.

3. Beta-Plus (β⁺) Decay

This occurs in nuclei with an excess of protons. A proton transforms into a neutron, emitting a positron ( $_{+ 1}^{0} e$ or β⁺, the electron's antiparticle) and an electron neutrino ( $ν_{e}$ ).

General Equation: $_{Z}^{A} X \to_{Z - 1}^{A} Y +_{+ 1}^{0} e + ν_{e}$
Effect: The nucleon number A is unchanged, but the atomic number Z decreases by one. The element changes.

4. Gamma (γ) Decay

This process does not change the composition of the nucleus. It occurs when a nucleus is in a high-energy, or "excited," state, often immediately following an alpha or beta decay. The nucleus transitions to a lower energy state by emitting a high-energy photon called a gamma ray (γ).

General Equation: $_{Z}^{A} X^{*} \to_{Z}^{A} X + γ$ (The asterisk denotes an excited state).
Effect: The nucleon and atomic numbers are unchanged. The nucleus simply loses energy.

Regulation & Limits

These models describe the how of a decay, not the when. The probability of a decay occurring is statistical in nature. For a decay to happen spontaneously, the total mass of the parent nucleus must be greater than the total mass of all the products; this "missing" mass is converted into the kinetic energy of the emitted particles.

Key Models & Diagrams

The different types of radioactive decay can be summarized by their effect on the nucleus and the particles they emit. The following table links the decay type to its standard representation and its observable consequences.

Decay Type	Emitted Particle(s)	General Nuclear Equation	Change in Z	Change in A
Alpha (α)	Helium nucleus ( $_{2}^{4} α$ )	$_{Z}^{A} X \to_{Z - 2}^{A - 4} Y +_{2}^{4} α$	Decreases by 2	Decreases by 4
Beta-Minus (β⁻)	Electron ( $_{- 1}^{0} e$ ) & Antineutrino ( $\overset{ν}{ˉ}_{e}$ )	$_{Z}^{A} X \to_{Z + 1}^{A} Y +_{- 1}^{0} e + \overset{ν}{ˉ}_{e}$	Increases by 1	No change
Beta-Plus (β⁺)	Positron ( $_{+ 1}^{0} e$ ) & Neutrino ( $ν_{e}$ )	$_{Z}^{A} X \to_{Z - 1}^{A} Y +_{+ 1}^{0} e + ν_{e}$	Decreases by 1	No change
Gamma (γ)	Photon ( $γ$ )	$_{Z}^{A} X^{*} \to_{Z}^{A} X + γ$	No change	No change

Key Components & Evidence

Nucleon Number (A): The total count of protons and neutrons. Its conservation is a fundamental rule in all nuclear reactions.
Atomic Number (Z): The number of protons, which defines the element. Changes in Z are direct evidence of elemental transmutation.
Alpha Particle (α): A helium nucleus ( $^4_2$He). Its detection is the definitive sign of alpha decay. - **Beta Particle (β⁻):** An electron ($ ^0_{-1}e$) emitted from a nucleus. Its charge and mass are evidence for the n → p + e⁻ + ν̅ process.
Positron (β⁺): The antimatter electron ( $_{+ 1}^{0} e$ ). Its detection is the key signature of beta-plus decay.
Gamma Ray (γ): A high-energy photon. Observing gamma rays with discrete energies is evidence of nuclei de-exciting from one quantum energy state to another.
Neutrino (ν) / Antineutrino (ν̅): Nearly massless, neutral particles. Their existence was first inferred from analysis of the energy spectrum of beta decay products, which showed that a third, undetected particle must be carrying away some energy.
Parent Nucleus: The initial, unstable nucleus before it undergoes decay.
Daughter Nucleus: The nucleus that remains after the decay process has occurred.
Conservation Laws: The strict rules (conservation of A, Z, and L) that govern all decays, allowing us to predict the products.

Skill Snapshots

Causation:
1. An excess of neutrons relative to protons causes a nucleus to be unstable, often leading to beta-minus decay to convert a neutron into a proton.
2. The emission of an alpha particle causes the strong nuclear force to have less influence relative to the electrostatic repulsion between protons, often resulting in a more stable daughter nucleus.
3. A nucleus being left in a high-energy excited state after a primary decay causes the subsequent emission of a gamma ray to reach its ground state.
Comparison:
1. Alpha decay is a process of particle ejection that changes the element and mass, whereas gamma decay is a process of energy release that changes neither.
2. Beta-minus decay increases the number of protons in the nucleus, while beta-plus decay decreases the number of protons.
3. Both alpha and beta decay result in transmutation (a change of element), while gamma decay does not.
Change Over Time:
- Baseline: An unstable parent nucleus, such as Uranium-238 ( $_{92}^{238} U$ ), exists with a specific number of protons and neutrons.
- Change 1: Over time, it undergoes alpha decay, ejecting a helium nucleus and transforming into a new element, Thorium-234 ( $_{90}^{234} T h$ ).
- Change 2: The resulting Thorium-234 is also unstable and subsequently undergoes beta-minus decay, transforming into Protactinium-234 ( $_{91}^{234} P a$ ).
- Continuity: Throughout this decay chain, the total nucleon number, charge, and lepton number are conserved at every single step.

Common Misconceptions & Clarifications

Misconception: The electron emitted in beta-minus decay comes from the atom's orbital electron shells.
- Clarification: The electron is created inside the nucleus at the instant of decay. It is a direct result of a neutron transforming into a proton (n → p + e⁻ + ν̅) and is immediately ejected.
Misconception: Mass is conserved in nuclear decays.
- Clarification: Mass-energy is conserved, but mass itself is not. In any spontaneous decay, the total mass of the products is slightly less than the mass of the parent nucleus. This "lost" mass has been converted into the kinetic energy of the daughter nucleus and emitted particles, as described by $E = m c^{2}$ .
Misconception: Neutrinos and antineutrinos are not important because they have almost no mass and no charge.
- Clarification: They are essential. Without the emission of a neutrino or antineutrino in beta decay, both lepton number and energy-momentum would not be conserved. Their existence is required to make the physics work.
Misconception: Any nucleus can undergo any type of decay.
- Clarification: The type of decay is determined by the specific instability of the nucleus. Nuclei that are too heavy undergo alpha decay. Nuclei with too many neutrons undergo beta-minus decay. Nuclei with too many protons undergo beta-plus decay.

One-Paragraph Summary

Radioactive decay is the fundamental process by which an unstable atomic nucleus transforms into a more stable configuration by emitting particles and energy. The primary types of decay are alpha, beta (minus and plus), and gamma decay, each altering the parent nucleus in a distinct way. These transformations are not random but are strictly governed by the conservation of nucleon number, charge, and lepton number. Alpha decay ejects a helium nucleus, reducing both mass and atomic number. Beta decay changes a neutron to a proton (or vice versa), altering the atomic number but not the mass number, and requires the emission of a neutrino or antineutrino. Gamma decay releases excess energy as a photon without changing the nucleus's composition. By applying these conservation laws, we can write precise nuclear equations that predict the products of any radioactive decay.

Types of Radioactive Decay - AP Physics 2: Algebra-Based Study Guide