Question

Show that the density of the nuclear matter is the same for all nuclei.

Hint:

Because nuclear radius varies as $A^{1/3}$ and mass varies as $A$, the density of nuclear matter remains constant for all nuclei.

Answer 1

Step 1: Radius of a nucleus.
The radius of a nucleus is given by the empirical relation:
\[ R = R_0 A^{1/3} \] where $R_0$ is a constant ($R_0 \approx 1.3 \times 10^{-15}\ \text{m}$) and $A$ is the mass number of the nucleus.

Step 2: Volume of the nucleus.
The volume of a nucleus is given by the formula for the volume of a sphere:
\[ V = \frac{4}{3}\pi R^3 \] Substituting $R = R_0 A^{1/3}$:
\[ V = \frac{4}{3}\pi (R_0 A^{1/3})^3 \] \[ V = \frac{4}{3}\pi R_0^3 A \]

Step 3: Mass of the nucleus.
The mass of the nucleus is approximately proportional to the mass number:
\[ M = A m_n \] where $m_n$ is the mass of a nucleon (proton or neutron).

Step 4: Density of nuclear matter.
Density is defined as:
\[ \rho = \frac{M}{V} \] Substituting the values:
\[ \rho = \frac{A m_n}{\frac{4}{3}\pi R_0^3 A} \] \[ \rho = \frac{m_n}{\frac{4}{3}\pi R_0^3} \]

Step 5: Conclusion.
Since the mass number $A$ cancels out, the density does not depend on the size of the nucleus. Therefore, the density of nuclear matter is approximately the same for all nuclei.