6.4NuclearStretch

Rutherford Scattering and Nuclear Radius

Before 1911, the atom was pictured as a uniform ball of charge. Firing alpha particles at gold foil thinner than this page shattered that picture entirely, revealing a nucleus ten thousand times smaller than the atom, holding almost all of its mass.

40 min Video by Science Shorts Module 6.4 Nuclear and particle physics

Nuclear Radius & Alpha Scattering — A-level PhysicsWatch the full walkthrough before the notes below.

Open on YouTube

What you'll be able to do

Describe the alpha scattering experiment and its key observations
Explain how the results are evidence for a small, dense nucleus
Estimate nuclear radius using the closest-approach method
Use $R = R_{0} A^{1/3}$ and explain constant nuclear density

The alpha scattering experiment

Most alpha particles fired at gold foil pass straight through undeflected; a small fraction deflect through large angles; a rare few bounce back almost the way they came. This is only explicable if the atom’s positive charge and mass are concentrated in a tiny, dense nucleus.

Closest approach

An alpha particle heading directly at a nucleus decelerates, converting kinetic energy into electric potential energy, momentarily stopping at the point of closest approach.

E_{k} = \frac{Qq}{4 π ε _{0} r _{m i n}}

Closest-approach estimate of nuclear radius.

E_{k} = 5.0 \times 1.60 \times 1 0^{- 13} = 8.0 \times 1 0^{- 13}

r_{m i n} = \frac{Qq}{4 π ε _{0} E _{k}} = \frac{( 3.2 \times 1 0 ^{- 19} ) ( 1.264 \times 1 0 ^{- 17} )}{4 π \times 8.85 \times 1 0 ^{- 12} \times 8.0 \times 1 0 ^{- 13}}

Answer

r_{m i n} \approx 4.6 \times 1 0^{- 14}

Nuclear radius and density

Nuclear radius follows $R = R_{0} A^{1/3}$ , $R_{0} \approx 1.2$ fm. Since volume ( $\propto R^{3}$ ) is proportional to $A$ , nuclear density is approximately constant (~ $1 0^{17}$ kg m⁻³) for all nuclei.

1Ratio:

(\frac{125}{8})^{1/3}

Answer

\approx 2.5

times larger.

Equation recap

R = R_{0} A^{1/3}

Nuclear radius (R₀ ≈ 1.2 fm).

Common mistakes to avoid

Thinking most alpha particles were deflected.

The vast majority passed straight through — only rare deflections gave evidence for a small nucleus.

Using R = R₀A² or R₀A instead of a cube root.

Nuclear radius follows R=R₀A^(1/3), consistent with nuclear volume (not radius) being proportional to A.

Key takeaways

Rutherford scattering (mostly undeflected, rare large-angle) shows a small, dense, positive nucleus.
Closest-approach method estimates nuclear radius from alpha particle energy.
R = R₀A^(1/3); nuclear density is approximately constant for all nuclei.

Test yourself

Ready to lock in Rutherford Scattering and Nuclear Radius? Pick a mode and earn XP & Dobloons.

Rutherford Scattering and Nuclear Radius

40 min Video by Science Shorts Module 6.4 Nuclear and particle physics