Size of the nucleus

Geiger and Marsden's alpha-particles of 5.5 MeV came no closer than about 4.0 × 10⁻¹⁴ m to a gold nucleus. Pure Coulomb repulsion explained the scattering, so the nucleus must be smaller than that.

Faster alpha-particles come closer. At some energy the short-range nuclear force starts to act and the scattering departs from Rutherford's pure-Coulomb prediction; where that happens gives the nuclear size.

Nuclear sizes of many elements have been measured accurately by scattering fast electrons, instead of alpha-particles, off targets.

Radii from electron scattering differ slightly from those from alpha-particle scattering: electrons probe how the nuclear charge is spread, while alpha-particles and similar projectiles probe the nuclear matter itself.

A nucleus of mass number A has radius R = R₀A^(1/3) (Eq. 13.5), with R₀ = 1.2 × 10⁻¹⁵ m = 1.2 fm (1 fm = 10⁻¹⁵ m).

Volume goes as R³, so it is proportional to A: the density of nuclear matter is the same for all nuclei, like drops of one liquid. It is about 2.3 × 10¹⁷ kg m⁻³, against 10³ kg m⁻³ for water, because ordinary matter is mostly empty space.

Example 13.1: an iron nucleus of mass 55.85 u (9.27 × 10⁻²⁶ kg) and A = 56 has density 9.27 × 10⁻²⁶/[56 × (4π/3)(1.2 × 10⁻¹⁵)³] = 2.29 × 10¹⁷ kg m⁻³.

Matter in neutron stars has a density comparable to this: it is squeezed so hard that the star resembles one huge nucleus.