Consider a beam of mono-energetic electrons incident on apowder of crystal fragments, each with theatoms arranged in a cubic array with d being the distance between nearestneighbors.
(a)if the transmitted electrons form a diffraction pattern, describe what will be observed.
(b) what is the lowest energy electrons can have, in order to produce any diffraction pattern.(c) what happens if the electron energy is lower than what you have found in part (b).

Concepts:

Bragg diffraction

Reasoning:

The Bragg condition for constructive interference is nλ = 2d sinθ, where θ is the angle between the incident wave vector and the scattering planes.

Details of the calculation:

(a)  Powder diffraction produces concentric rings for the maxima of different orders.  The maxima are circular because the crystal fragments are oriented randomly in space.  Some fragments are oriented such that they produce the "top" of the interference maximum in relation to the beam direction, while others are oriented to produce the left, right, bottom, or other parts of the interference maximum. The result is a circular ring at the angle 2θ.

image

(b) The largest angle θ for a first order maximum can occur is θ = 90o.  

The longest wavelength for a first-order maximum therefore is λmax = 2d.

This is the wavelength of the lowest energy electrons.

Using de Broglie formula we can find the corresponding energy of electrons.

Emin = p2/2m = (h/λ)2/2m = h2/(8d2m).

(c)  If E < Emin, we only have the zeroth order maximum, the electrons pass through the crystal without deflection.