Nuclear Radius Calculator
Calculate the radius of any atomic nucleus using r = r₀ × A^(1/3), where A is the mass number.
Also computes nuclear volume, density, and surface area for any element.
Atomic nuclei are remarkably uniform in density. This leads to a simple empirical formula for nuclear radius:
r = r0 * A^(1/3)
A is the mass number (protons + neutrons). r0 = 1.2 femtometers (fm) = 1.2 x 10^-15 m is the empirical nuclear radius constant, determined from electron scattering experiments. Some sources use r0 = 1.25 fm for slightly different fits.
The cube-root dependence means nuclear volume scales linearly with A: V = (4/3)pir^3 = (4/3)pi(r0)^3 * A
Since volume is proportional to A and A is proportional to total nucleon mass, the nuclear matter density is approximately constant for all nuclei: rho ≈ m_nucleon / ((4/3)pi(r0)^3) ≈ 2.3 x 10^17 kg/m^3
That density is extraordinary: a teaspoon of nuclear matter would weigh about 2 billion tonnes. A neutron star is essentially a giant nucleus held together by gravity, with similar density throughout.
Worked examples:
- Hydrogen-1 (A=1): r = 1.2 fm (just a proton)
- Carbon-12 (A=12): r = 1.2 * 12^(1/3) = 1.2 * 2.289 = 2.75 fm
- Iron-56 (A=56): r = 1.2 * 56^(1/3) = 1.2 * 3.826 = 4.59 fm
- Uranium-238 (A=238): r = 1.2 * 238^(1/3) = 1.2 * 6.20 = 7.44 fm
The formula breaks down for very light nuclei (A < 4) where the liquid drop model underlying it does not apply well.