AcademyMatter Waves
Academy
Nuclear Atom
Level 1 - Physics topic page in Matter Waves.
Principle
Large-angle alpha scattering shows that atomic positive charge is concentrated in a tiny nucleus.
Notation
\(Z\)
nuclear charge number
\(e\)
elementary charge
\(\mathrm{C}\)
\(r\)
distance from nucleus
\(\mathrm{m}\)
\(K\)
alpha-particle kinetic energy
\(\mathrm{J}\)
\(k\)
Coulomb constant
\(\mathrm{N\,m^{2}\,C^{-2}}\)
\(b\)
impact parameter
\(\mathrm{m}\)
Method
Derivation 1: Model the alpha-nucleus force
An alpha particle has charge \(+2e\). A nucleus with charge \(+Ze\) repels it by Coulomb's law.
Coulomb force
\[F=\frac{k(2e)(Ze)}{r^2}\]
Potential energy
\[U=\frac{k(2e)(Ze)}{r}\]
Derivation 2: Estimate closest approach
For a head-on approach, the alpha particle momentarily stops at closest approach.
Energy conversion
\[K=\frac{2kZe^2}{r_{\min}}\]
Closest approach
\[r_{\min}=\frac{2kZe^2}{K}\]
Derivation 3: Interpret rare large deflections
Most alpha particles pass through because atoms are mostly empty space. Rare large deflections require a compact massive positive centre.
Rules
Alpha charge
\[q_\alpha=+2e\]
Nuclear charge
\[q_N=+Ze\]
Closest approach
\[r_{\min}=\frac{2kZe^2}{K}\]
Nuclear scale
\[R\sim10^{-15}\,\mathrm m\]
Examples
Question
An alpha particle scatters through a large angle. What does that imply about the atom?
Answer
It passed close to a concentrated positive nucleus. A diffuse positive charge would not produce the strong repulsive force needed for a large deflection.
Checks
- Large deflections are rare but decisive.
- The nucleus carries positive charge and most atomic mass.
- Electrons occupy the surrounding atomic region.
- Classical nuclear orbits alone cannot explain stable atoms.