Video Transcript
A beam of electrons passes through
a crystal. A diffraction pattern of concentric
rings is formed on a screen behind the crystal that records the positions of
electrons that arrive at it, as shown in the diagram. The intensity of the rings is
plotted against the radial distance from the center of the pattern. The resulting intensity
distribution is shown three times, each time compared to another intensity
distribution that is shown below it. Which of the intensity
distributions would result from decreasing the velocity of the electrons in the
beam? (A) III, (B) I, (C) II, (D) none of
these distributions.
This question is asking us about
electrons which act like waves passing through a crystal lattice that essentially
acts as a line of slits in a screen. When waves pass through a slit,
they can be diffracted so that when they emerge, they spread out radially with the
most diffraction occurring when the width of the aperture is the same or similar to
the wavelength of the wave. Because electrons can act like
waves, electrons create similar diffraction patterns.
Waves emerging from neighboring
slits or a line of slits can interfere, either constructively, creating a greater
intensity at points of constructive interference, or destructively, reducing the
intensity.
For this question, we want to know
what would happen to a diffraction pattern created by electrons passing through a
crystal if the velocity of the electrons was reduced. For this, we need to understand how
electrons can behave like waves.
Remember that the de Broglie
wavelength for a particle can be found using the following equation. 𝜆 is equal to ℎ over 𝑝, where
lambda is the wavelength of the electron, ℎ is the Planck constant, and 𝑝 is the
momentum of the electron. Now then, if we decrease the
velocity of the electron beam, the momentum will decrease as well, since momentum is
dependent on velocity. So this means that decreasing the
velocity of the electrons will increase the associated wavelength of the
electrons.
If we think about how diffraction
patterns are affected by the wavelength of the wave, we know that the greater the
wavelength of the wave, the greater the distance between the consecutive bright
fringes of the diffraction pattern. A greater distance between bright
spots means that when we look at the ring patterns head on, we should expect the
concentric bright rings to spread out.
If we take a look at the diagrams,
we can see that the first graph shows a decrease in intensity, while the second
shows the electron intensity more tightly packed together. It’s difficult to spot, but only
the third graph shows the intensity of the electrons being more spread out, which
indicates that the concentric fringes would be spaced further apart.
So, the answer is graph III, which
corresponds to option (A). Lowering the velocity of the
electrons would result in a longer associated wavelength of the electrons and a more
spread out diffraction pattern.