Question

The mean momentum $\bar{p}$ of a nucleon in a nucleus of mass number $A$ and atomic number $Z$ depends on $A, Z$ as 
 

• A. $\bar{p}\propto A^{1/3}$
• B. $\bar{p}\propto Z^{1/3}$
• C. $\bar{p}\propto A^{-1/3}$
• D. $\bar{p}\propto (AZ)^{-2/3}$

Hint:

In nuclear physics, many quantities scale with $A^{1/3}$ due to the radius–mass relation $R\propto A^{1/3}$.

Answer 1

The Correct Option is C

Step 1: Use the Fermi gas model.
In nuclei, nucleons behave approximately like a degenerate Fermi gas. The Fermi momentum is $\displaystyle p_F \propto n^{1/3},$ where $n$ is the number density of nucleons.

Step 2: Relate density to mass number $A$.
The nuclear radius is $\displaystyle R \propto A^{1/3}.$ Thus the volume $\displaystyle V \propto R^3 \propto A.$ So number density $\displaystyle n = \frac{A}{V} \propto \frac{A}{A} = \text{constant}.$

Step 3: But mean momentum depends on Fermi momentum.
Even though density is nearly constant, the average momentum scales weakly as $\displaystyle \bar{p}\propto A^{1/3}$ because the Fermi momentum is proportional to the cube root of nucleon number within a constant-density potential well.

Step 4: Conclusion.
Hence mean nucleon momentum increases as $A^{1/3}$, matching option (A).