1.1.1.3 Nuclear Properties and Scales

Atomic nuclei exhibit remarkable uniformity in their intrinsic properties, governed by the strong nuclear force. Despite vast differences in atomic mass, nuclei share consistent characteristics in size, density, and energy binding—revealing the saturation nature of nuclear forces.

Nuclear Radius

The radius of a nucleus is not fixed but scales with the cube root of its mass number $A$ , reflecting the near-incompressibility and constant density of nuclear matter. The empirical formula for the nuclear radius is:

$R \approx R_{0} A^{1 / 3}$ , where $R_{0} \approx 1.2 fm$ (1 fm = $10^{- 15}$ m).

This relation implies that nuclear volume is proportional to the number of nucleons ( $V \propto R^{3} \propto A$ ), confirming that nucleons are closely packed like incompressible droplets. For example:

Hydrogen-1 ( $A = 1$ ): $R \approx 1.2 fm$
Carbon-12 ( $A = 12$ ): $R \approx 1.2 \times 12^{1 / 3} \approx 2.7 fm$
Uranium-238 ( $A = 238$ ): $R \approx 1.2 \times 238^{1 / 3} \approx 7.4 fm$

This model underpins the **liquid drop model** of the nucleus and is validated by electron scattering experiments (e.g., Hofstadter’s work, Nobel Prize 1961).

Density of Nuclear Matter

Because nuclear volume scales linearly with $A$ , the **density of nuclear matter** is approximately constant across all nuclei:

$ρ = \frac{mass}{volume} \approx \frac{A \cdot m_{nucleon}}{\frac{4}{3} π R^{3}} = \frac{A \cdot (1.67 \times 10^{- 27} kg)}{\frac{4}{3} π (1.2 \times 10^{- 15})^{3} A} \approx 2.3 \times 10^{17} {kg/m}^{3}$ .

This extraordinary density—about $2.3 \times 10^{14}$ times that of water—means a sugar-cube-sized chunk of nuclear matter would weigh over 200 million metric tons. Such density arises because nucleons are separated by only ~1–2 fm, held together by the short-range strong force.

Mass Defect and Binding Energy

The mass of a stable nucleus is always **less** than the sum of the masses of its individual protons and neutrons. This difference, called the mass defect ( $Δ m$ ), arises because energy is released when nucleons bind together.

The mass defect is defined as:

$Δ m = [Z m_{p} + N m_{n}] - m_{nucleus}$ ,

where $m_{p}$ , $m_{n}$ , and $m_{nucleus}$ are the rest masses of the proton, neutron, and the neutral atom (accounting for electrons in practice), respectively.

According to Einstein’s mass–energy equivalence principle ( $E = m c^{2}$ ), this missing mass is converted into **binding energy** ( $B$ )—the energy required to disassemble the nucleus into its free nucleons:

$B = Δ m \cdot c^{2}$ .

Binding energy is typically expressed in **MeV** (mega-electronvolts). For example, the binding energy of $^{4} He$ (alpha particle) is about 28.3 MeV, meaning it is highly stable.

A more useful quantity is the binding energy per nucleon ( $B / A$ ), which peaks around iron-56 ( $^{56} Fe$ ) at ~8.8 MeV/nucleon. This curve explains why:

Fusion of light nuclei (e.g., H → He) releases energy (increasing $B / A$ ),
Fission of heavy nuclei (e.g., U → Ba + Kr) also releases energy (moving toward higher $B / A$ ).

Thus, mass defect and binding energy are central to understanding nuclear stability, stellar nucleosynthesis, and energy production in reactors and stars.