Q.Are the nucleons fundamental particles, or do they consist of still smaller parts? One way to find out is to probe a nucleon just as Rutherford probed an atom. What should be the kinetic energy of an electron for it to be able to probe a nucleon? Assume the diameter of a nucleon to be approximately $10^{-15}\ \text{m}$.

The key idea is the de Broglie wavelength criterion: to resolve a structure of size $d$, the probing particle must have a wavelength $\lambda \lesssim d$. For a nucleon of diameter $10^{-15}\ \text{m}$, we need $\lambda \approx 10^{-15}\ \text{m}$.

For an electron, the de Broglie wavelength is $\lambda = h/p$. At such small wavelengths, the electron is highly relativistic, so its kinetic energy $K \approx pc$ (since rest energy $m_e c^2 \approx 0.5\ \text{MeV}$ is negligible compared to the required energy). …

The key idea here is the same one Rutherford used to probe the atom: to “see” a structure, the probe’s wavelength must be smaller than the size of the target. For a nucleon of diameter $10^{-15}\ \text{m}$, we need an electron with a de Broglie wavelength $\lambda \lesssim 10^{-15}\ \text{m}$.

Let’s work through this step by step.

1. The probing condition In any scattering experiment, the resolving power of the probe is limited by its wavelength. To resolve details of size $d$, we need $\lambda \leq d$. For a nucleon, $d \approx 10^{-15}\ \text{m}$, so we require:

2. De Broglie wavelength For a particle of momentum $p$, the de Broglie wavelength is:

where $h = 6.626 \times 10^{-34}\ \text{J·s}$ is Planck’s constant.

So the required momentum is:

3. Check if relativistic The rest mass energy of an electron is $m_e c^2 = 0.511\ \text{MeV} = 8.187 \times 10^{-14}\ \text{J}$. The momentum we found corresponds to an energy scale we can estimate:

Converting to MeV:

Since $pc \approx 1.24\ \text{GeV}$ is much larger than $m_e c^2 = 0.511\ \text{MeV}$, the electron is highly relativistic. We must use relativistic energy-momentum relations.

4. Relativistic energy For a relativistic particle, the total energy $E$ is:

Since $pc \gg m_e c^2$, we can approximate: …

Method: Estimating the Probe Energy Needed to Resolve a Given Length Scale

This is the general technique behind every "what energy/particle do I need to see structure of size $d$" question — the same logic Rutherford used with alpha particles on the atom, extended here to probing the nucleon itself.

Steps

Step 1: Translate "probe a structure of size $d$" into a wavelength condition

A wave-like probe can only resolve features at least as large as its own wavelength. So the working condition is:

where $d$ is the size of the structure you're trying to see (here, the nucleon's diameter).

Step 2: Use the de Broglie relation to convert the wavelength requirement into a momentum requirement

Every particle used as a probe (electron, proton, etc.) has a de Broglie wavelength $\lambda = h/p$. Requiring $\lambda \lesssim d$ therefore fixes a minimum momentum:

Step 3: Check whether the probing particle will be relativistic …