Since the force between nucleons is short range, the size of a nucleus is reasonably well-defined. Experiment shows that nuclei are roughly spherical with a radius
Thus, the density of a nucleus is approximately constant, and the volume of a nucleus is proportional to the number of nucleons contained in it.
The radius parameter can vary with the method used to measure it and is generally in the region 1.1 - 1.4 fm (with larger values corresponding to the electromagnetic radius).
Rutherford interpreted the initial scattering experiments that showed the existence of a massive, positively charged, compact nucleus in the atom. The Rutherford cross-section describes the scattering resulting from electrostatic forces:
This was derived by classical considerations and associated a scattering angle with a unique impact parameter, in turn associated with a unique closest approach to the nucleus.
The potential due to the nuclear force is strongly attractive and short range. Adding it to the Coulomb potential results in deviations from Rutherford scattering at small distances of approach (large scattering angles). This leads to an approximate size for the nucleus.
Other methods exist for determining this. An atom can capture a meson ( Mev) which behaves as a heavy electron. X-rays are emitted as it cascades down to the 1s atomic orbital. The energy can be predicted by the usual Bohr formula, but in heavy atoms there are perturbations as the Bohr radius approaches the nuclear size.
Today, electron scattering is usually used to measure nuclear size. Energetic electrons ( MeV) are usually used for this. These are highly relativistic; the electron invariant mass is 0.511 MeV, so the Lorentz factor . The momentum p of the electron is related to its de Broglie wavelength :
Since MeV and MeV, resolutions of 1 fm or better require .
From many such electron scattering measurements, the nuclear density (nucleons per unit volume) can be parameterised as
where the surface diffusivity fm and the radius parameter fm. The central density . See Figure .
Note that this indicates that