Formula Sheet — Nuclear Physics
Comprehensive formula reference extracted from Nuclear Physics. All formulas use standard SI units unless otherwise noted.
1. Nuclear Structure & Properties
Nuclear Composition
$$A = Z + N$$
| Variable | Meaning | Units |
|---|---|---|
| $A$ | Mass number (total nucleons) | dimensionless |
| $Z$ | Atomic number (protons) | dimensionless |
| $N$ | Neutron number | dimensionless |
Isotope notation: $^{A}_{Z}\text{X}$
Nuclear Radius
$$R = R_0 A^{1/3}$$
| Variable | Meaning | Units |
|---|---|---|
| $R$ | Nuclear radius | m (or fm) |
| $R_0$ | Empirical constant ($R_0 = 1.2\ \text{fm} = 1.2 \times 10^{-15}\ \text{m}$) | m |
| $A$ | Mass number | dimensionless |
Nuclear Density
$$\rho \approx 2.3 \times 10^{17}\ \text{kg/m}^3$$
2. Mass Defect & Binding Energy
Mass Defect
$$\Delta m = Zm_p + Nm_n - m_N$$
| Variable | Meaning | Units |
|---|---|---|
| $\Delta m$ | Mass defect | kg (or u) |
| $Z$ | Atomic number (number of protons) | dimensionless |
| $m_p$ | Proton mass | kg (or u) |
| $N$ | Neutron number | dimensionless |
| $m_n$ | Neutron mass | kg (or u) |
| $m_N$ | Measured nuclear mass | kg (or u) |
Binding Energy
$$E_B = (\Delta m)c^2 = [Zm_p + Nm_n - m_N]c^2$$
| Variable | Meaning | Units |
|---|---|---|
| $E_B$ | Binding energy | J (or MeV) |
| $\Delta m$ | Mass defect | kg (or u) |
| $c$ | Speed of light | m/s |
Binding Energy in MeV (Convenient Form)
$$E_B = \Delta m \times 931.5\ \text{MeV/u}$$
| Variable | Meaning | Units |
|---|---|---|
| $E_B$ | Binding energy | MeV |
| $\Delta m$ | Mass defect | u |
| $931.5$ | Energy equivalent of 1 atomic mass unit | MeV/u |
Binding Energy per Nucleon
$$\frac{E_B}{A}$$
| Variable | Meaning | Units |
|---|---|---|
| $E_B$ | Total binding energy | MeV (or J) |
| $A$ | Mass number (total nucleons) | dimensionless |
Peak stability: Fe-56 at $\approx 8.8\ \text{MeV/nucleon}$
3. Radioactive Decay
Decay Modes Summary
| Mode | Emitted Particle | $\Delta Z$ | $\Delta A$ | Condition |
|---|---|---|---|---|
| Alpha ($\alpha$) | $^{4}_{2}\text{He}$ | $-2$ | $-4$ | Nucleus too heavy ($A > 200$) |
| 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 |
Alpha Decay Example
$$^{238}{92}\text{U} \rightarrow {}^{234}{90}\text{Th} + {}^{4}_{2}\text{He}$$
Beta-Minus Decay Example
$$^{14}{6}\text{C} \rightarrow {}^{14}{7}\text{N} + {}^{0}_{-1}e$$
Positron Emission Example
$$^{18}{9}\text{F} \rightarrow {}^{18}{8}\text{O} + {}^{0}_{+1}e$$
Gamma Decay Example
$$^{12}{6}\text{C}^{*} \rightarrow {}^{12}{6}\text{C} + \gamma$$
4. Decay Law & Half-Life
Decay Law (Differential Form)
$$-\frac{dN}{dt} = \lambda N$$
| Variable | Meaning | Units |
|---|---|---|
| $N$ | Number of radioactive nuclei at time $t$ | dimensionless |
| $t$ | Time | s |
| $\lambda$ | Decay constant | s⁻¹ |
| $-\frac{dN}{dt}$ | Rate of decay (activity) | s⁻¹ |
Decay Equation (Integrated Form)
$$N(t) = N_0 e^{-\lambda t}$$
| Variable | Meaning | Units |
|---|---|---|
| $N(t)$ | Number of nuclei remaining at time $t$ | dimensionless |
| $N_0$ | Initial number of nuclei ($t = 0$) | dimensionless |
| $\lambda$ | Decay constant | s⁻¹ |
| $t$ | Elapsed time | s |
Half-Life
$$T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$$
| Variable | Meaning | Units |
|---|---|---|
| $T_{1/2}$ | Half-life (time for $N$ to halve) | s (or yr, etc.) |
| $\lambda$ | Decay constant | s⁻¹ |
| $\ln 2$ | Natural logarithm of 2 ($\approx 0.693$) | dimensionless |
Alternative decay form using half-life: $$N(t) = N_0 \left(\frac{1}{2}\right)^{t/T_{1/2}}$$
5. Activity
Activity Definition
$$A = \lambda N = -\frac{dN}{dt}$$
| Variable | Meaning | Units |
|---|---|---|
| $A$ | Activity (decays per unit time) | Bq (or Ci) |
| $\lambda$ | Decay constant | s⁻¹ |
| $N$ | Number of radioactive nuclei | dimensionless |
Activity Decay
$$A = A_0 e^{-\lambda t}$$
| Variable | Meaning | Units |
|---|---|---|
| $A$ | Activity at time $t$ | Bq (or Ci) |
| $A_0$ | Initial activity ($t = 0$) | Bq (or Ci) |
| $\lambda$ | Decay constant | s⁻¹ |
| $t$ | Elapsed time | s |
Activity Units
| Unit | Symbol | Definition |
|---|---|---|
| Becquerel | Bq | $1\ \text{Bq} = 1\ \text{decay s}^{-1}$ |
| Curie | Ci | $1\ \text{Ci} = 3.70 \times 10^{10}\ \text{Bq}$ |
6. Nuclear Reactions
Mass Difference (Q-value Calculation)
$$\Delta m = \sum m_{\text{reactants}} - \sum m_{\text{products}}$$
| Variable | Meaning | Units |
|---|---|---|
| $\Delta m$ | Mass difference | kg (or u) |
| $m_{\text{reactants}}$ | Total mass of reactants | kg (or u) |
| $m_{\text{products}}$ | Total mass of products | kg (or u) |
Reaction Energy (Q-value)
$$Q = (\Delta m)c^2$$
| Variable | Meaning | Units |
|---|---|---|
| $Q$ | Reaction energy | J (or MeV) |
| $\Delta m$ | Mass difference | kg (or u) |
| $c$ | Speed of light | m/s |
Classification:
- $Q > 0$: Exothermic (exoergic) — energy released
- $Q < 0$: Endothermic (endoergic) — energy absorbed
7. Stability & Decay Triggers
Stability Limits
| Condition | Consequence |
|---|---|
| $Z > 83$ | Unstable (radioactive) |
| $N > 126$ | Unstable |
| $N/Z \gtrsim 1.5$ | Unstable |
Decay only occurs when: $$\Delta m > 0 \quad \text{or} \quad Q > 0$$
8. Key Constants
| Constant | Symbol | 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 | $1\ \text{u}$ | $1.6606 \times 10^{-27}\ \text{kg}$ |
| Energy equivalent of 1 u | $c^2$ | $931.5\ \text{MeV/u}$ |
| Speed of light | $c$ | $3.00 \times 10^8\ \text{m/s}$ |
| Electron volt | $1\ \text{eV}$ | $1.602 \times 10^{-19}\ \text{J}$ |
| Mega-electron volt | $1\ \text{MeV}$ | $1.602 \times 10^{-13}\ \text{J}$ |
| Nuclear density | $\rho$ | $\approx 2.3 \times 10^{17}\ \text{kg/m}^3$ |
| Nuclear radius constant | $R_0$ | $1.2\ \text{fm} = 1.2 \times 10^{-15}\ \text{m}$ |
| Carbon-14 half-life | $T_{1/2}$ | $5730\ \text{yr}$ |