Question

Learning Goal: To understand how to find the wavelength and diffraction patterns of electrons. An electron beam is incident on a single slit of width aaa. The electron beam was generated using a potential difference of magnitude VVV. After passing through the slit, the diffracted electrons are collected on a screen that is a distance LLL away from the slit. Assume that VVV is small enough so that the electrons are nonrelativistic. Ultimately, you will find the width of the central maximum for the diffraction pattern.

Answer 1

Answer:

y = L h / a √√ (2q V m)

Explanation:

This is a diffraction exercise, so we must use the D'Broglie relation to encode the wavelength of the electron beam.

 

                  p = h / λ

                  λ= h / p

the moment is

                   p = m v

 

                   λ = h / mv

 

Let's use energy conservation

                  E = K

                  q ΔV = ½ m v²

                   v = √ 2qΔv / m

 

                  λ = h / (m √2q ΔV / m)

                  λ = h / √ (2q ΔV m)

Having the wavelength of the electrons we can use the diffraction ratio

                  a sin θ = m λ

First minimum occurs for m = 1

                   sin θ = λ / a

                   

let's use trigonometry for the angles

                   tan θ = y / x

as in these experiments the angles are very small

               tan θ = sin θ / cos θ = sin θ

               sin θ = y / x

we substitute

               y / x = λ / a

               y = x λ / a

we replace the terms

               y = L h / a √√ (2q V m)