Nuclear Radius (r = r₀A^⅓)

The cube root is the physics. If radius grows as A^⅓, then volume grows as A itself — add a nucleon, add one nucleon's worth of space — which means every nucleus from deuterium to uranium is packed to the same density, about 2.3 × 10¹⁷ kg/m³, a teaspoon of which would weigh a billion tonnes. Nucleons behave like drops of an incompressible liquid, and that liquid-drop picture is the foundation of the semi-empirical mass formula and of Bohr and Wheeler's 1939 theory of fission. Contrast the atom: electron clouds do not scale this way, so a gold atom is barely larger than helium's while its nucleus is four times wider.

The coefficient r₀ ≈ 1.2 fm is an empirical fit, not a defined constant. It descends from Rutherford's scattering analysis and was pinned down by firing electrons at nuclei and mapping where the charge sits — Hofstadter's Nobel-winning programme at Stanford in the 1950s. Fits to charge radii give 1.2 fm; fits to matter radii from hadron scattering drift toward 1.25 fm, and the lightest nuclei stray from the trend entirely, so treat the formula as good to a few per cent from carbon upward. Worked through: aluminium-27 gives r = 1.2 × ∛27 = 3.6 fm exactly, and lead-208 gives 1.2 × 5.925 ≈ 7.11 fm — a mass number 208 times hydrogen's buys a radius under six times larger.

Setting A = 1 returns r = r₀ = 1.2 fm for the proton, which is instructively wrong: the measured proton rms charge radius is 0.841 fm (the value that emerged from the decade-long "proton radius puzzle"). A single nucleon is not a droplet of nuclear liquid, and the mismatch marks where the model's jurisdiction ends. Inside it, though, the formula earns its keep — Coulomb barriers, alpha-decay tunnelling rates and the geometric cross-sections behind the barn all start from r₀A^⅓.