Nuclear Physics

Study of atomic nuclei, nuclear forces, radioactivity, and nuclear reactions.

Definition

Nuclear physics studies the constituents and interactions of atomic nuclei. It encompasses nuclear structure, radioactive decay, nuclear reactions, and applications in energy and medicine.

Key Concepts

  • Nuclear Composition — protons ($Z$) and neutrons ($N$), mass number $A = Z + N$
  • Isotopes — same $Z$, different $N$ (e.g., $^{12}6\text{C}$ vs $^{14}6\text{C}$; $^{235}{92}\text{U}$ vs $^{238}{92}\text{U}$)
  • Nuclear Forces — strong force binding nucleons, short range; competes with long-range Coulomb repulsion
  • Nuclear Radius — $R = R_0 A^{1/3}$ where $R_0 = 1.2\ \text{fm} = 1.2 \times 10^{-15}\ \text{m}$
  • Mass Defect — $\Delta m = Zm_p + Nm_n - m_N$; the nucleus mass is less than the sum of its constituent nucleons
  • Binding Energy — $E_B = (\Delta m)c^2 = [Zm_p + Nm_n - m_N]c^2$; energy required to break the nucleus into its constituent particles (or energy released when nucleons combine to form a nucleus)
  • Binding Energy per Nucleon — $\frac{E_B}{A}$; measure of nuclear stability. Higher value = more stable nucleus. Peaks at Fe-56 ($\approx 8.8\ \text{MeV/nucleon}$)
  • Stability Limits — atoms become unstable when $Z > 83$, $N > 126$, or $N/Z \gtrsim 1.5$
  • Nuclear Fission — heavy nucleus splitting, chain reactions, critical mass
  • Nuclear Fusion — light nuclei combining, thermonuclear reactions, proton-proton cycle
  • Carbon Dating — C-14 half-life (5,730 yr), uranium-lead dating
  • Nuclear Waste Management — LLW/ILW/HLW classification, storage, disposal, recycling

Radioactive Decay

Definition: A process where an unstable nucleus spontaneously decays or breaks down, emitting particles and rays to form a more stable nucleus. Discovered in 1897 by Henri Becquerel.

Decay Modes

Mode Particle $\Delta Z$ $\Delta A$ Trigger
Alpha ($\alpha$) $^{4}_{2}\text{He}$ $-2$ $-4$ Nucleus too heavy
Beta minus ($\beta^{-}$) $^{0}_{-1}e$ (electron) $+1$ $0$ Too many neutrons
Positron ($\beta^{+}$) $^{0}_{+1}e$ (positron) $-1$ $0$ Too many protons
Gamma ($\gamma$) Photon $0$ $0$ Nucleus in excited state

Key principle: Radioactive decay only occurs when the mass difference $\Delta m$ (or Q-value) is positive.

Examples:

  • Alpha: $^{238}{92}\text{U} \rightarrow ^{234}{90}\text{Th} + ^{4}_{2}\text{He}$
  • Beta minus: $^{14}{6}\text{C} \rightarrow ^{14}{7}\text{N} + ^{0}_{-1}e$
  • Positron: $^{18}{9}\text{F} \rightarrow ^{18}{8}\text{O} + ^{0}_{+1}e$
  • Gamma: $^{12}{6}\text{C}^{*} \rightarrow ^{12}{6}\text{C} + \gamma$
graph TB
    start((Unstable Nucleus))
    start --> check["Assess N/Z ratio<br/>and excitation state"]

    check --> heavy["Nucleus too heavy<br/>A > 200"]
    heavy --> alpha["Alpha Decay<br/>Emit He-4 nucleus<br/>Z decreases by 2<br/>A decreases by 4"]

    check --> neutron["Too many neutrons<br/>High N/Z"]
    neutron --> beta_minus["Beta-Minus Decay<br/>neutron to proton + electron<br/>Z increases by 1"]

    check --> proton["Too many protons<br/>Low N/Z"]
    proton --> beta_plus["Positron Emission<br/>proton to neutron + positron<br/>Z decreases by 1"]

    check --> excited["Nucleus in excited state"]
    excited --> gamma["Gamma Decay<br/>Emit high-energy photon<br/>Z and A unchanged"]

    alpha --> stable["More Stable Nucleus"]
    beta_minus --> stable
    beta_plus --> stable
    gamma --> stable

    style start fill:#ffe3e3,stroke:#c92a2a
    style stable fill:#d3f9d8,stroke:#2f9e44

Decay Law and Half-Life

Decay Law: The rate of decay is proportional to the number of radioactive nuclei: $$-\frac{dN}{dt} = \lambda N$$

Integrating gives the decay equation: $$N(t) = N_0 e^{-\lambda t}$$

where $\lambda$ is the decay constant (probability per unit time of decay).

Half-Life ($T_{1/2}$): Time for half the nuclei to decay: $$T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$$

Activity

Activity ($A$) is the rate of decay of a sample: $$A = \lambda N = -\frac{dN}{dt}$$

Activity also follows exponential decay: $$A = A_0 e^{-\lambda t}$$

Units:

  • Becquerel (Bq): SI unit, $1\ \text{Bq} = 1\ \text{decay s}^{-1}$
  • Curie (Ci): $1\ \text{Ci} = 3.70 \times 10^{10}\ \text{Bq} = 3.70 \times 10^{10}\ \text{decays s}^{-1}$

Nuclear Reactions

A physical process in which the identity of an atomic nucleus changes. Must obey:

  1. Conservation of charge (Z) — total atomic number conserved
  2. Conservation of mass number (A) — total nucleon number conserved
  3. Conservation of energy — total energy conserved

Reaction Energy (Q-value): $$\Delta m = \sum m_{\text{reactants}} - \sum m_{\text{products}}$$ $$Q = (\Delta m)c^2$$

  • $Q > 0$: exothermic (exoergic) — energy released
  • $Q < 0$: endothermic (endoergic) — energy absorbed
graph LR
    subgraph fission["Nuclear Fission"]
        direction TB
        f1["Input: Heavy nucleus<br/>U-235, Pu-239"]
        f2["Process: Neutron capture<br/>Nucleus splits"]
        f3["Output: Lighter nuclei<br/>2-3 neutrons + Energy"]
        f4["Condition: Critical mass<br/>Chain reaction"]
        f1 --> f2 --> f3 --> f4
    end

    subgraph fusion["Nuclear Fusion"]
        direction TB
        fs1["Input: Light nuclei<br/>Deuterium + Tritium"]
        fs2["Process: High T and P<br/>Nuclei combine"]
        fs3["Output: Heavier nucleus<br/>Neutron + Energy"]
        fs4["Condition: T > 10^8 K<br/>Plasma confinement"]
        fs1 --> fs2 --> fs3 --> fs4
    end

Key Formulas

Formula Description
$R = R_0 A^{1/3}$ Nuclear radius ($R_0 = 1.2\ \text{fm}$)
$\Delta m = Zm_p + Nm_n - m_N$ Mass defect
$E_B = [Zm_p + Nm_n - m_N]c^2$ Binding energy
$E_B = \Delta m \times 931.5\ \text{MeV/u}$ Binding energy in MeV (using $c^2 = 931.5\ \text{MeV/u}$)
$\frac{E_B}{A}$ Binding energy per nucleon
$N(t) = N_0 e^{-\lambda t}$ Radioactive decay
$T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$ Half-life
$A = \lambda N = -\frac{dN}{dt}$ Activity
$A = A_0 e^{-\lambda t}$ Activity decay
$Q = (\Delta m)c^2$ Reaction energy
$\Delta m = \sum m_{\text{reactants}} - \sum m_{\text{products}}$ Mass difference

Key Constants (Lecture Values)

Constant Value
Proton mass ($m_p$) $1.00728\ \text{u}$ ($1.67262 \times 10^{-27}\ \text{kg}$)
Neutron mass ($m_n$) $1.00867\ \text{u}$ ($1.67492 \times 10^{-27}\ \text{kg}$)
Electron mass ($m_e$) $0.000549\ \text{u}$ ($9.10938 \times 10^{-31}\ \text{kg}$)
Atomic mass unit ($1\ \text{u}$) $1.6606 \times 10^{-27}\ \text{kg}$
$1\ \text{u}$ in energy $931.5\ \text{MeV}/c^2$
Speed of light ($c$) $3.00 \times 10^8\ \text{m/s}$
$1\ \text{eV}$ $1.602 \times 10^{-19}\ \text{J}$
$1\ \text{MeV}$ $1.602 \times 10^{-13}\ \text{J}$
Nuclear density ($\rho$) $\approx 2.3 \times 10^{17}\ \text{kg/m}^3$

Binding Energy Curve (Stability Curve)

The graph of binding energy per nucleon ($E_B/A$) versus mass number ($A$) reveals four distinct regions:

1. Steep Rise — Light Nuclei ($A < 50$)

  • $E_B/A$ increases rapidly with $A$
  • Small nuclei (e.g., hydrogen, helium, lithium) are relatively unstable
  • To gain stability, they undergo nuclear fusion, combining into heavier nuclei and releasing massive energy

2. Peak of Stability — The Iron Group ($A \approx 56$)

  • Maximum at Fe-56 with $E_B/A \approx 8.8\ \text{MeV}$
  • The most stable nucleus in the universe
  • Nucleons are packed as tightly as physics allows
  • Neither fusion nor fission occurs naturally here

3. Gradual Decline — Heavy Nuclei ($A > 62$)

  • Curve slopes downward as $A$ increases
  • The short-range strong nuclear force struggles to overcome the long-range Coulomb repulsion between many protons

4. Radioactive Zone — Very Heavy Nuclei ($A > 200$)

  • $E_B/A$ drops to $\sim 7.5$–$8.0\ \text{MeV}$ (e.g., Uranium-235)
  • Nuclei are unstable and "top-heavy"
  • Undergo nuclear fission (splitting) or radioactive decay to move toward iron, increasing $E_B/A$ and releasing energy

Nuclear Waste Management

Nuclear waste management deals with the safe handling, storage, and disposal of radioactive byproducts from nuclear power generation and medical procedures.

Classification of Radioactive Waste

Type Description Examples
Low-Level Waste (LLW) Low radioactivity Clothing, filters
Intermediate-Level Waste (ILW) Moderate radioactivity Resins, chemical sludges
High-Level Waste (HLW) Highly radioactive Used nuclear fuel

Management Methods

Storage Temporary containment in shielded facilities, allowing for decay over time or awaiting final disposal.

  • Example: Sellafield, UK — stores high-level waste in stainless steel tanks.

Disposal Long-term disposal by burying radioactive waste deep underground in stable rock formations, designed to isolate waste for thousands of years.

  • Example: Onkalo Repository, Finland — world's first deep geological repository for spent nuclear fuel; located in stable bedrock at 400–450 meters deep.

Recycling Treating spent nuclear fuel to separate out materials that can be reused in new fuel.

  • Example: La Hague Plant, France — reprocesses spent fuel from France and other countries.

Safety and Environmental Concerns

  • Long-term monitoring of disposal sites is essential
  • Potential risks of leakage or contamination must be managed
  • Public awareness and international cooperation are key

Related Concepts

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