Nuclear and particle physics explore the deepest structures of microscopic matter: from radioactive decay, strong nuclear binding, and nuclear fission/fusion in atomic nuclei, to subatomic quarks, leptons, gauge bosons, and electroweak symmetry breaking in the Standard Model.
This cheat sheet organizes 15 foundational physical laws and governing equations in nuclear and particle physics, categorized into three core domains: Radioactivity & Decay, Nuclear Reactions & Energy, and Particles & Fundamental Interactions, complete with mathematical formulations, quantum numbers, and high-energy physics applications.
🔗 Physics Equations Suite: Classical Mechanics · Electromagnetism · Fluid & Thermal · Optics & Acoustics · Relativity & Cosmology · Astro & Geophysics · Quantum Mechanics
1. Radioactivity & Decay (5 Items)
01 · Radioactive Decay Law
Exponential Decay Law
$$N(t) = N_0 e^{-\lambda t}$$
- ①Physical Essence: The number of unstable nuclei decaying per unit time is strictly proportional to the current remaining number of parent nuclei; decay is a memoryless Poisson quantum process.
- ②Mathematical Notation: $N(t)$ is remaining nuclei at time $t$, $N_0$ is initial count at $t=0$, $\lambda$ is decay constant ($\text{s}^{-1}$), $t$ is elapsed time.
- ③Theoretical Significance: Foundation of nuclear chronometry; $\lambda$ depends purely on nuclear internal structure and is invariant to temperature, pressure, gravity, or chemical bonding.
- ④Boundary Conditions & Applications: Valid for macroscopic statistical ensembles ($N \gg 1$). Applied in radiometric geochronology, carbon-14 archaeological dating, and spent fuel storage calculations.
02 · Half-Life
Decay Half-Life Period
$$T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.69315}{\lambda}$$
- ①Physical Essence: The time required for half of the radioactive parent nuclei in a sample to undergo decay.
- ②Mathematical Notation: $T_{1/2}$ is half-life, $\ln 2 \approx 0.693$, $\lambda$ is decay constant. Remaining fraction after $n$ half-lives is $(1/2)^n$.
- ③Theoretical Significance: An immutable intrinsic characteristic parameter defining each specific radioisotope.
- ④Boundary Conditions & Applications: Spans extreme timescales: bismuth-209 has $T_{1/2} \approx 2 \times 10^{19}\text{ years}$, whereas beryllium-8 decays in $10^{-16}\text{ seconds}$. Critical for nuclear medicine radiotracer selection (e.g. Technetium-99m, $T_{1/2} \approx 6\text{ h}$).
03 · Radioactivity & Activity
Disintegration Rate
$$A(t) = -\frac{\mathrm{d}N}{\mathrm{d}t} = \lambda N(t) = A_0 e^{-\lambda t}$$
- ①Physical Essence: The total rate of nuclear disintegrations occurring in a radioactive source per second.
- ②Mathematical Notation: $A$ is activity measured in Becquerels ($\text{Bq} \equiv \text{disintegrations/s}$), historical unit is Curie ($1\text{ Ci} \equiv 3.7 \times 10^{10}\text{ Bq}$).
- ③Theoretical Significance: The core metric defining radiation intensity and biological exposure risk of radioisotopes.
- ④Boundary Conditions & Applications: Direct metric measured by Geiger-Müller counters and liquid scintillation counters; benchmark for radiotherapy dose planning.
04 · Alpha Decay
Helium Nucleus Emission & Quantum Tunneling
$$^A_Z\text{X} \longrightarrow {}^{A-4}_{Z-2}\text{Y} + {}^4_2\alpha\ (^4_2\text{He}) + Q_\alpha$$
- ①Physical Essence: Heavy nuclei discharge an extremely stable helium-4 nucleus ($^4_2\text{He}$) via quantum mechanical tunneling through the Coulomb barrier.
- ②Mathematical Notation: $A$ decreases by 4, $Z$ decreases by 2; $Q_\alpha$ is shared kinetic energy ($Q_\alpha = (m_X - m_Y - m_\alpha)c^2$).
- ③Theoretical Significance: First real-world application of quantum tunneling (Gamow's theory); explains the Geiger-Nuttall law where slight kinetic energy differences cause astronomical differences in half-life.
- ④Boundary Conditions & Applications: Alpha particles carry high ionizing power but low penetration (stopped by paper); used in Americium-241 smoke detectors and targeted alpha radiotherapy (TAT).
05 · Beta Decay
Electroweak Weak Interaction Decay
$$\beta^-: n \to p + e^- + \bar{\nu}_e,\qquad \beta^+: p \to n + e^+ + \nu_e$$
- ①Physical Essence: Mediated by the weak interaction $W^\pm$ gauge bosons; changes a down quark into an up quark (or vice versa), converting a nucleon while emitting a lepton pair.
- ②Mathematical Notation: $\beta^-$ emits electron $e^-$ and electron antineutrino $\bar{\nu}_e$; $\beta^+$ emits positron $e^+$ and electron neutrino $\nu_e$.
- ③Theoretical Significance: Wolfgang Pauli predicted the neutrino (1930) to preserve conservation of energy and angular momentum; Chien-Shiung Wu confirmed parity violation (1956).
- ④Boundary Conditions & Applications: Generates a continuous beta energy spectrum; positrons emitted in $\beta^+$ decay are the foundation of Positron Emission Tomography (PET scans) in oncology.
2. Nuclear Reactions & Energy (5 Items)
06 · Mass Defect & Binding Energy
Nuclear Binding Energy Formula
$$B = \Delta m \cdot c^2 = \left[Z m_p + (A - Z)m_n - m(A, Z)\right]c^2$$
- ①Physical Essence: When free nucleons bind into an atomic nucleus, strong nuclear attraction releases binding energy, resulting in a total nuclear mass strictly less than the sum of its constituents.
- ②Mathematical Notation: $B$ is binding energy, $\Delta m$ is mass defect, $Z$ is proton count, $A-Z$ is neutron count, $1\text{ u} \approx 931.494\text{ MeV}/c^2$.
- ③Theoretical Significance: Quantifies the total work required to completely disassemble an atomic nucleus into free individual protons and neutrons.
- ④Boundary Conditions & Applications: Essential calculation baseline for nuclear reaction Q-values ($Q = \sum m_{\text{reactants}}c^2 - \sum m_{\text{products}}c^2$).
07 · Binding Energy per Nucleon
The Curve of Binding Energy
$$\frac{B}{A} \quad \left(\text{Peaks at }{}^{56}\text{Fe} \approx 8.8\text{ MeV/nucleon}\right)$$
- ①Physical Essence: The average binding energy per nucleon peaks around Iron-56 ($^{56}\text{Fe}$) and Nickel-62 ($^{62}\text{Ni}$). Nuclei lighter than iron release energy via fusion; nuclei heavier than iron release energy via fission.
- ②Mathematical Notation: $B/A$ is binding energy per nucleon; jumps from $1.1\text{ MeV}$ (deuterium) to $7.1\text{ MeV}$ (helium-4), peaking at $8.79\text{ MeV}$ ($^{56}\text{Fe}$), and descending to $7.6\text{ MeV}$ ($^{238}\text{U}$).
- ③Theoretical Significance: Dictates the thermal energy limits of the universe, explaining why iron is the thermodynamic graveyard of stellar nucleosynthesis.
- ④Boundary Conditions & Applications: Drives core-collapse supernovae when massive stars exhaust silicon burning into iron cores.
08 · Heavy Nucleus Fission
Induced Nuclear Fission
$$^{235}_{\ 92}\text{U} + {}^1_0\text{n} \longrightarrow {}^{141}_{\ 56}\text{Ba} + {}^{92}_{36}\text{Kr} + 3\,{}^1_0\text{n} + 200\text{ MeV}$$
- ①Physical Essence: A heavy fissile nucleus absorbs a thermal neutron, forming an unstable compound nucleus that splits into two medium-mass fragments and multiple secondary neutrons.
- ②Mathematical Notation: Releases $\approx 200\text{ MeV}$ per fission event (primarily as fragment kinetic energy $\sim 168\text{ MeV}$ and prompt gamma rays $\sim 8\text{ MeV}$).
- ③Theoretical Significance: Explained by the liquid-drop model; secondary neutrons enable self-sustaining nuclear chain reactions.
- ④Boundary Conditions & Applications: Powers commercial Pressurized Water Reactors (PWR), Boiling Water Reactors (BWR), and thorium molten salt reactors.
09 · Light Nucleus Fusion
Thermonuclear Fusion
$$^2_1\text{H} + {}^3_1\text{H} \longrightarrow {}^4_2\text{He} + {}^1_0\text{n} + 17.6\text{ MeV}$$
- ①Physical Essence: Light hydrogen isotopes overcome the electrostatic Coulomb repulsion under extreme temperatures and pressures, merging into helium and releasing massive kinetic energy.
- ②Mathematical Notation: $17.6\text{ MeV}$ total yield: $3.5\text{ MeV}$ carried by alpha particle ($^4\text{He}$), $14.1\text{ MeV}$ carried by energetic neutron.
- ③Theoretical Significance: The primary energy engine of the cosmos (solar p-p chain and CNO cycle in stars).
- ④Boundary Conditions & Applications: Requires meeting the Lawson criterion ($n \cdot T \cdot \tau_E \ge 3 \times 10^{21}\text{ keV}\cdot\text{s/m}^3$) for net ignition. Active research in magnetic confinement (ITER, Tokamaks) and inertial confinement (NIF lasers).
10 · Criticality Condition
Neutron Multiplication Factor
$$k_{\text{eff}} = \frac{\text{Neutrons produced in generation } n+1}{\text{Neutrons absorbed or lost in generation } n}$$
- ①Physical Essence: Ratio measuring the reproduction rate of fission neutrons between successive generations inside a nuclear reactor or assembly.
- ②Mathematical Notation: $k_{\text{eff}} < 1$ is subcritical (chain reaction terminates); $k_{\text{eff}} = 1$ is critical (stable steady power); $k_{\text{eff}} > 1$ is supercritical (power escalates exponentially).
- ③Theoretical Significance: Fundamental stability criterion governing nuclear engineering; controlled using delayed neutron precursors ($0.65\%$ in $^{235}\text{U}$) to grant control rods mechanical response time.
- ④Boundary Conditions & Applications: Core operational setpoint for reactor control systems, emergency SCRAM systems, and nuclear fuel enrichment safety limits.
3. Particles & Fundamental Interactions (5 Items)
11 · Four Fundamental Interactions
Fundamental Forces of Nature
$$\text{Strong} (1) > \text{EM} (10^{-2}) > \text{Weak} (10^{-6}) \gg \text{Gravity} (10^{-38})$$
- ①Physical Essence: All physical processes in the universe are mediated by four distinct gauge interactions: Strong (gluons), Electromagnetic (photons), Weak ($W^\pm, Z^0$), and Gravity (gravitons).
- ②Mathematical Notation: Relative coupling strengths at subatomic distances: Strong ($\sim 1$, range $\sim 10^{-15}\text{ m}$), Electromagnetic ($\sim 1/137$, infinite), Weak ($\sim 10^{-6}$, range $\sim 10^{-18}\text{ m}$), Gravity ($\sim 10^{-38}$, infinite).
- ③Theoretical Significance: Governs atomic stability, radioactive beta decay, chemical bonds, and macroscopic planetary dynamics.
- ④Boundary Conditions & Applications: Electroweak unification proved electromagnetism and the weak force merge at $\sim 100\text{ GeV}$; Grand Unified Theories (GUT) and Quantum Gravity seek ultimate unification.
12 · Fermi Coupling Constant
Effective Weak Interaction Strength
$$G_F = \frac{\sqrt{2}}{8}\frac{g^2}{M_W^2 c^4} \approx 1.1663788 \times 10^{-5}\ \text{GeV}^{-2}$$
- ①Physical Essence: Low-energy effective coupling constant for four-fermion contact weak interactions, suppressed inversely by the massive $W$ boson mass squared ($M_W \approx 80.4\text{ GeV}$).
- ②Mathematical Notation: $G_F \approx 1.166 \times 10^{-5}\text{ GeV}^{-2}$, $g$ is $SU(2)_L$ gauge coupling, $M_W$ is $W$ boson mass.
- ③Theoretical Significance: Reconciled why the weak interaction appears extremely weak at low energies despite its gauge coupling $g$ being comparable to electromagnetism.
- ④Boundary Conditions & Applications: Essential benchmark constant extracted precisely from muon lifetime decay measurements ($\tau_\mu$).
13 · Neutrino Oscillation
Pontecorvo-Maki-Nakagawa-Sakata (PMNS)
$$P(\nu_\alpha \to \nu_\beta) = \sin^2(2\theta) \sin^2\left(\frac{\Delta m^2 c^3 L}{4\hbar E}\right)$$
- ①Physical Essence: Neutrinos created in a specific lepton flavor state ($\nu_e, \nu_\mu, \nu_\tau$) periodically transform into different flavors as they propagate across space.
- ②Mathematical Notation: $\theta$ is mixing angle in PMNS matrix, $\Delta m^2 = m_2^2 - m_1^2$ is mass-squared difference ($\text{eV}^2$), $L$ is propagation baseline, $E$ is neutrino energy.
- ③Theoretical Significance: Conclusively proves neutrinos possess non-zero mass ($\Delta m^2 \neq 0$), providing the first verified physics beyond the minimal Standard Model (2015 Nobel Prize).
- ④Boundary Conditions & Applications: Resolved the historical Solar Neutrino Problem; investigated by underground detectors (Super-K, SNO, KamLAND, Jiangmen JUNO).
14 · Standard Model Gauge Group
Quantum Electroweak & Chromodynamic Symmetries
$$SU(3)_C \times SU(2)_L \times U(1)_Y \xrightarrow{\text{SSB}} SU(3)_C \times U(1)_{\text{em}}$$
- ①Physical Essence: The fundamental local gauge symmetry group of the Standard Model: $SU(3)_C$ governs color charge in QCD; $SU(2)_L \times U(1)_Y$ governs electroweak interactions.
- ②Mathematical Notation: Broken via the Brout-Englert-Higgs mechanism (Spontaneous Symmetry Breaking, SSB) to $U(1)_{\text{em}}$, endowing $W^\pm, Z^0$ bosons and fermions with mass while leaving the photon strictly massless.
- ③Theoretical Significance: The pinnacle theoretical framework of subatomic particle physics; successfully predicted neutral currents, the top quark, and the Higgs boson ($125\text{ GeV}$, confirmed at CERN in 2012).
- ④Boundary Conditions & Applications: Excludes gravitation; does not account for dark matter, dark energy, or the cosmic matter-antimatter baryon asymmetry.
15 · Quark Composition of Nucleons
Valence Quark Model
$$p = |uud\rangle,\qquad n = |udd\rangle$$
- ①Physical Essence: Protons and neutrons are composite hadrons (baryons) composed of three valence quarks bound tightly by gluon color fields in Quantum Chromodynamics (QCD).
- ②Mathematical Notation: Up quark $u$ has charge $+2/3e$; down quark $d$ has charge $-1/3e$. Proton $p = (+\frac{2}{3}) + (+\frac{2}{3}) + (-\frac{1}{3}) = +1e$; Neutron $n = (+\frac{2}{3}) + (-\frac{1}{3}) + (-\frac{1}{3}) = 0$. Bare quark masses account for only $\approx 1\%$ of nucleon mass; $99\%$ arises from gluon binding energy via $E=mc^2$.
- ③Theoretical Significance: Explains deep inelastic scattering experiments at SLAC, hadron multiplets, and color confinement.
- ④Boundary Conditions & Applications: Essential framework for high-energy hadron colliders (LHC), quark-gluon plasma (QGP) research, and neutron star equation of state models.
4. Physical Constants, Symbols & SI Units
| Physical Quantity | Symbol | SI Base or Derived Unit | Common High-Energy / Nuclear Unit | Equivalence / Value |
|---|---|---|---|---|
| Atomic Mass Unit | $u, \text{Da}$ | Kilogram ($\text{kg}$) | Dalton ($\text{Da}$) | $1\text{ u} \approx 1.6605 \times 10^{-27}\ \text{kg} \approx 931.494\ \text{MeV}/c^2$ |
| Electron Volt | $\text{eV}$ | Joule ($\text{J}$) | $\text{keV}, \text{MeV}, \text{GeV}, \text{TeV}$ | $1\text{ eV} \approx 1.6021766 \times 10^{-19}\ \text{J}$; Fission $\approx 200\text{ MeV}$ |
| Radioactive Activity | $A$ | Becquerel ($\text{Bq}$) | Curie ($\text{Ci}$) | $1\text{ Bq} \equiv 1\ \text{s}^{-1}$; $1\text{ Ci} \equiv 3.7 \times 10^{10}\ \text{Bq}$ |
| Absorbed Radiation Dose | $D$ | Gray ($\text{Gy}$) | Rad ($\text{rad}$) | $1\text{ Gy} \equiv 1\ \text{J/kg}$; $1\text{ rad} = 0.01\ \text{Gy}$ |
| Equivalent Dose | $H$ | Sievert ($\text{Sv}$) | Rem ($\text{rem}$) | $1\text{ Sv} \equiv 1\ \text{J/kg}$ (Weighted); $1\text{ rem} = 0.01\ \text{Sv}$ |
| Reaction Cross-Section | $\sigma$ | Square meter ($\text{m}^2$) | Barn ($\text{b}$) | $1\text{ b} \equiv 10^{-28}\ \text{m}^2 = 100\ \text{fm}^2$ |
| Decay Constant | $\lambda$ | Per second ($\text{s}^{-1}$) | Per year ($\text{yr}^{-1}$) | $\lambda = \ln 2 / T_{1/2}$ |
| Nuclear Length Scale | $R$ | Meter ($\text{m}$) | Femtometer / Fermi ($\text{fm}$) | $1\text{ fm} \equiv 10^{-15}\ \text{m}$ (Nuclear diameter scale) |
| Fermi Constant | $G_F$ | $\text{J}\cdot\text{m}^3$ | $\text{GeV}^{-2}$ | $G_F \approx 1.1663788 \times 10^{-5}\ \text{GeV}^{-2}$ |