Nuclear Physics & Radioactivity Cheat Sheet

Key Facts

• A nuclide is written as $^{A}_{Z}\mathrm{X}$, where $A$ is the mass number, $Z$ is the atomic number, and the number of neutrons is $N = A - Z$.
• In every nuclear reaction, total mass number and total atomic number are conserved, so $\sum A_{\text{reactants}} = \sum A_{\text{products}}$ and $\sum Z_{\text{reactants}} = \sum Z_{\text{products}}$.
• Alpha decay emits $^{4}_{2}\mathrm{He}$, so a parent nucleus changes according to $^{A}_{Z}\mathrm{X} \rightarrow ^{A-4}_{Z-2}\mathrm{Y} + ^{4}_{2}\mathrm{He}$.
• Beta-minus decay emits an electron and an antineutrino, so $^{A}_{Z}\mathrm{X} \rightarrow ^{A}_{Z+1}\mathrm{Y} + ^{0}_{-1}e + \bar{\nu}$.
• Beta-plus decay emits a positron and a neutrino, so $^{A}_{Z}\mathrm{X} \rightarrow ^{A}_{Z-1}\mathrm{Y} + ^{0}_{+1}e + \nu$.
• Radioactive decay follows $N = N_0 e^{-\lambda t}$, where $N$ is the number of undecayed nuclei, $N_0$ is the initial number, and $\lambda$ is the decay constant.
• Half-life and decay constant are related by $T_{1/2} = \frac{\ln 2}{\lambda}$, and after $n$ half-lives the remaining amount is $N = N_0\left(\frac{1}{2}\right)^n$.
• Nuclear binding energy is found from $E_b = \Delta m c^2$, where the mass defect is $\Delta m = Zm_p + Nm_n - m_{\text{nucleus}}$.

Vocabulary

Nuclide

A specific nucleus identified by its number of protons $Z$ and total nucleons $A$.

Isotope

Atoms of the same element with the same $Z$ but different neutron numbers $N$.

Half-life

The time $T_{1/2}$ required for half of the radioactive nuclei in a sample to decay.

Activity

The decay rate of a radioactive sample, given by $A_{\text{act}} = \lambda N$.

Mass defect

The missing mass $\Delta m$ between separate nucleons and the actual nucleus, converted into binding energy.

Binding energy

The energy $E_b$ required to completely separate a nucleus into individual protons and neutrons.