Why are diffraction rings closer together when electrons travel at a greater speed in electron diffraction?

Question

I know that at higher speeds the de Broglie wavelength decreases so the electrons diffract less, but does the fact electrons repel affect it in any other way?

What I was thinking was that since electrons now reach the screen in less time because of the greater speed, they have less time to repel and thus repel less so rings are closer together. Is this a correct reason and does this relate in any way to the wavelength reason?

These images are from Electron diffraction.

A beam of electrons is accelerated in an electron gun to a potential of between 3500 V and 5000 V and then allowed to fall on a very thin sheet of graphite (see diagram above). The electrons diffract from the carbon atoms and the resulting circular pattern on the screen (see diagrams below) is very good evidence for the wave nature of the electrons.

The diffraction pattern observed on the screen is a series of concentric rings. This is due to the regular spacing of the carbon atoms in different layers in the graphite. However since the graphite layers overlay each other in an irregular way the resulting diffraction pattern is circular. It is an example of Bragg scattering.

Answer 1

What I was thinking was that since electrons now reach the screen in less time because of the greater speed, they have less time to repel and thus repel less so rings are closer together. Is this a correct reason and does this relate in any way to the wavelength reason?

Short Answer is no. You would get the exact same pattern if you shot the electrons one by one and not in a beam where they could theoretically interact with each other via coulomb repulsion. Note that the pattern tells you something about the distribution of the "impact positions of the particles".

Stating in the graph here it is purely particle-like and here it is purely wave-like is oversimplifying. You should think of the particles as something different, which happens to behave like waves in one limit and like particles in the other, and how it behaves in between is more complex and described by shrödinger equation.

Answer 2

Assuming that we are dealing with parallel electron beams, this can be explained through Bragg's Law, $n\lambda = 2d sin\theta $, with $2\theta$ the angle between the incident and diffracted ray. As $\lambda$ is decreased, $sin\theta$ should decrease for the same value of d, hence the rings are now closer to the central spot.

Answer 3

The momentum of an electron $p$ increases with energy.

Let's say the lattice spacing of a material is $a$. Then we can assign a de Broglie momentum to the lattice (of sorts anyways) that is $q=\hbar \frac{2\pi}{a}$.

We can think of the crystal/material as giving a kick off momentum to the electron that is of order $q$, so that the electron exits with momentum $\mathbf{p}+\mathbf{q}$. If $q$ is perpendicular to $p$, then we have the exiting angle is approximately $\theta\sim q/p$. Thus, for larger electron energy $p$ goes down and so does the scattering angle $\theta$.

This behavior is ompletely identical to light passing through a diffraction grating. As you decrease the wavelength (larger momentum), the angular spacing between the diffracted beams becomes smaller.