Atoms & Nuclei — Nucleus & Radioactivity (Medium)

Quick Recap — The Nucleus & Radioactivity

• Size & density: $R=R_0 A^{1/3}$ (so $R\propto A^{1/3}$); nuclear density is roughly the same for all nuclei (mass $\propto A$, volume $\propto A$).
• Binding energy: mass defect $\Delta m=[Zm_p+(A-Z)m_n]-M$; $BE=\Delta m\,c^2$ ($1$ u $=931.5$ MeV). The BE-per-nucleon curve peaks near $A\approx56$ (iron) — both fusion of light nuclei and fission of heavy nuclei release energy.
• Decays: $\alpha$: $A\to A-4$, $Z\to Z-2$; $\beta^-$: $Z\to Z+1$, $A$ same; $\gamma$: no change in $A,Z$.
• Decay law: $N=N_0 e^{-\lambda t}$; half-life $T_{1/2}=\dfrac{0.693}{\lambda}$; mean life $\tau=\dfrac{1}{\lambda}$; activity $A=\lambda N$. After $n$ half-lives a fraction $\left(\tfrac12\right)^n$ remains.

Worked mini-example. For radii, $\dfrac{R_1}{R_2}=\left(\dfrac{A_1}{A_2}\right)^{1/3}$; comparing $A=27$ and $A=64$ gives $\left(\dfrac{27}{64}\right)^{1/3}=\dfrac{3}{4}$.