KAHIBARO
Table of Contents
Uniformity of Nuclear Density
One striking fact about atomic nuclei is that their density is almost the same for very different nuclei. A light nucleus and a heavy nucleus do not differ very much in density, even though they differ greatly in mass and size. This is unusual compared with everyday objects, where larger objects can have very different densities from smaller ones.
This near constant nuclear density is one of the key clues that nuclear matter behaves in a special way. It suggests that nucleons, protons and neutrons, are packed together with roughly the same average spacing in most nuclei.
Density from Mass and Volume
Density is mass divided by volume. For a nucleus, we write
$$
\rho = \frac{M}{V}
$$
where $M$ is the nuclear mass and $V$ is the nuclear volume.
To estimate nuclear density, we use two simple ideas. First, the mass of a nucleus with mass number $A$ is approximately
$$
M \approx A m_N
$$
where $m_N$ is the mass of one nucleon, about
$$
m_N \approx 1.67 \times 10^{-27}\ \text{kg}
$$
Second, the nucleus is approximately spherical, so its volume is
$$
V = \frac{4}{3}\pi R^3
$$
Using the usual nuclear radius relation,
$$
R = R_0 A^{1/3}
$$
we get
$$
V = \frac{4}{3}\pi (R_0 A^{1/3})^3 = \frac{4}{3}\pi R_0^3 A
$$
Now substitute into the density formula:
$$
\rho = \frac{A m_N}{\frac{4}{3}\pi R_0^3 A}
$$
The factor $A$ cancels:
$$
\rho = \frac{m_N}{\frac{4}{3}\pi R_0^3}
$$
So the density does not depend on $A$. This is why nuclei of many different sizes have nearly the same density.
For nuclei, the relation $R = R_0 A^{1/3}$ leads directly to an almost constant density:
$$
\rho \approx \frac{m_N}{\frac{4}{3}\pi R_0^3}
$$
This means nuclear density is approximately independent of the mass number $A$.
Numerical Estimate
Take
$$
R_0 \approx 1.2 \times 10^{-15}\ \text{m}
$$
Then
$$
\rho \approx \frac{1.67 \times 10^{-27}}{\frac{4}{3}\pi (1.2 \times 10^{-15})^3}
$$
This gives a value of about
$$
\rho \approx 2.3 \times 10^{17}\ \text{kg/m}^3
$$
A commonly quoted range is
$$
\rho \sim 2 \times 10^{17}\ \text{kg/m}^3
\text{ to }
3 \times 10^{17}\ \text{kg/m}^3
$$
This is an enormous density. A tiny amount of nuclear matter has a very large mass.
Why the Density is So Large
The nucleus is extremely small, with a radius of only a few femtometers, but it contains nearly all the mass of the atom. Since the nucleons are confined to such a tiny volume, the density becomes extremely high.
The nearly constant density tells us that adding more nucleons usually increases the volume in proportion to the number of nucleons. In simple terms, heavier nuclei are bigger because they contain more nucleons, not because the nucleons are packed much more tightly.
Comparison with Ordinary Matter
The scale of nuclear density becomes clearer when compared with familiar materials.
| Material | Approximate Density, $\text{kg/m}^3$ |
|---|---|
| Air | $1.2$ |
| Water | $10^3$ |
| Iron | $7.9 \times 10^3$ |
| White dwarf matter, typical order | $10^9$ |
| Nuclear matter | $\sim 10^{17}$ |
Nuclear matter is vastly denser than ordinary solids and liquids. Even compared with very dense astrophysical matter, it is extreme.
Typical nuclear density is of order
$$
10^{17}\ \text{kg/m}^3
$$
This is many trillions of times greater than the density of ordinary materials.
Physical Meaning
The approximate constancy of nuclear density supports the idea that each nucleon occupies about the same effective volume inside the nucleus. This is one reason simple nuclear models often treat the nucleus as a drop of nearly incompressible matter.
This does not mean the density is perfectly uniform everywhere inside the nucleus. Real nuclei usually have a dense central region and a surface where the density falls off. But as an average property, the density remains nearly constant from one nucleus to another.
Visualizing Constant Density
If the radius grows like $A^{1/3}$, then volume grows like $A$, and mass also grows like $A$. So mass and volume increase together, keeping density nearly unchanged.
Important Summary
Nuclear density is the average mass per unit volume of a nucleus. Because nuclear radius follows $R = R_0 A^{1/3}$, the volume is proportional to $A$, just like the mass. As a result, the average nuclear density is nearly constant for all nuclei and has a typical value around $10^{17}\ \text{kg/m}^3$.
Key result:
$$
\rho_{\text{nucleus}} \approx 2 \times 10^{17}\ \text{kg/m}^3
$$
and it is approximately the same for both light and heavy nuclei.