Nuclear Size Shape and Charge Distribution Simple Notes

The atomic nucleus is astonishingly small — about a hundred thousand times smaller than the atom — yet it holds essentially all the atom's mass. Its radius follows a beautifully simple rule: R = R₀·A^(1/3), where A is the mass number and R₀ ≈ 1.2 femtometres (1 fm = 10⁻¹⁵ m). This cube-root law means nuclear volume is proportional to A, so every nucleus has nearly the same density, about 2×10¹⁷ kg/m³.

Nuclear size matters because it sets the scale of all nuclear physics: reaction cross-sections, the Coulomb repulsion that fission must overcome, and the charge distribution mapped by electron scattering. The fact that density is constant was the first clue that nucleons pack like marbles in a bag — the insight behind the liquid-drop model.

Key Concepts

• Femtometre (fermi): 1 fm = 10⁻¹⁵ m, the natural unit of nuclear sizes. A proton's radius is about 0.84 fm.
• Radius formula: R = 1.2 × A^(1/3) fm. Doubling the volume needs eight times the nucleons, so heavy nuclei grow slowly.
• Charge distribution: measured by firing electrons at nuclei (Hofstadter's experiments); most nuclei show a roughly uniform charge density with a fuzzy edge about 2 fm thick.
• Form factor: the Fourier transform of the charge distribution; its fall-off with scattering angle reveals the size.
• Nuclear shapes: many nuclei are spherical, but some are deformed — prolate (rugby-ball) like many rare earths, or oblate (discus) like some light nuclei — measured via quadrupole moments.

Formulas

• Nuclear radius: R = R₀·A^(1/3), R₀ = 1.2 fm. For lead-208: R = 1.2 × 208^(1/3) = 1.2 × 5.93 = 7.1 fm.
• Nuclear density: ρ = A·m_nucleon / (4/3·π·R³) ≈ 2.3×10¹⁷ kg/m³ for all nuclei — denser than anything else in the universe except neutron stars.
• Rutherford scattering baseline: at low energies the nucleus looks point-like; deviations from the Rutherford formula at high energy reveal the finite size.

Worked Example

Compute the radius and average density of the lead-208 nucleus (A = 208).

1. Cube root: 208^(1/3) ≈ 5.93 (since 5.93³ ≈ 208.5).
2. Radius R = 1.2 fm × 5.93 = 7.11 fm = 7.11×10⁻¹⁵ m.
3. Volume = 4/3·π·R³ = 4.189 × (7.11×10⁻¹⁵)³ = 4.189 × 3.594×10⁻⁴³ = 1.505×10⁻⁴² m³.
4. Mass = 208 × 1.66×10⁻²⁷ kg = 3.45×10⁻²⁵ kg. Density = 3.45×10⁻²⁵ / 1.505×10⁻⁴² = 2.29×10¹⁷ kg/m³.
5. A teaspoon of this matter would weigh about a billion tonnes — the famous illustration of nuclear density.

Common Misconceptions

• "The nucleus has a sharp surface." Electron scattering shows a skin about 2 fm thick over which the density fades; there is no hard edge.
• "Bigger atoms have proportionally bigger nuclei." Atomic size is set by electron shells and varies wildly; nuclear radius grows only as A^(1/3), so uranium's nucleus is merely ~3 times wider than helium's.
• "All nuclei are spheres." Quadrupole moment measurements prove many are permanently deformed footballs or discs.

Real-World Applications

• Nuclear reactors: fission and capture cross-sections scale with nuclear size, fixing fuel and moderator design.
• Medical isotopes: cyclotron target yields are calculated from measured charge distributions and radii.
• Electron microscopy calibration: nuclear form factors from scattering calibrate the highest-resolution probes of matter.

Nuclei follow R = 1.2·A^(1/3) fm, giving a universal density near 2×10¹⁷ kg/m³. Electron scattering maps a diffuse-edged charge distribution, and quadrupole moments reveal prolate or oblate deformations. Size and shape together anchor every calculation in nuclear science.