Objectives:

• To have an understanding of electron diffraction, including experimental evidence of this effect (you should understand that electron diffraction provides evidence for wave-like  behaviour of particles)
• To understand that the diffraction of electrons travelling through a thin slice of polycrystalline graphite is due to the atoms of graphite and the spacing between the atoms
• To be able to select an use the de Broglie equation  $\lambda = \frac{h}{p}$

Your understanding of light so far involves it behaving as a wave or as a particle, the photon; this has been seen by number of different experiments including reflection, refraction, diffraction, interference (Young’s slit experiment and diffraction grating), polarisation and the photoelectric effect. In 1924 a scientist name Louis de Broglie published his thesis for his PhD and wrote:

“The fundamental idea of [my 1924 thesis] was the following: The fact that, following Einstein‘s introduction of photons in light waves, one knew that light contains particles which are concentrations of energy incorporated into the wave, suggests that all particles, like the electron, must be transported by a wave into which it is incorporated… My essential idea was to extend to all particles the coexistence of waves and particles discovered by Einstein in 1905 in the case of light and photons.” [1]

So in a bold move, Louis de Broglie predicted that matter could behave as a wave and predicted that it could therefore interfere with other pieces of matter. If interference could occur that this suggests that one could obtain some form of constructive interference or destructive interference (could a piece of matter really interact with another piece of matter to obtain some form of ‘bright fringe’ or ‘dark fringe’?)

Louis de Broglie justified his theory using theoretical physics originating from Einsteins mass-energy equation $E = mc^{2}$this is a shortened version of the equation and a very brief derivation:

Einsteins equation can really be written as:

$E = mcv$

where v is the speed of the particle in question. Since this particle is usually a photon it is denoted with the letter $c$ (hence $E = mcv = mcc = mc^{2}$

If matter can behave as a wave it makes sense that it should have a wavelength, so the equation $E = \frac{hc}{\lambda}$ is substituted in for $E$:

$\frac{hc}{\lambda} = mcv$

The $c$‘s cancel leaving:

$\frac{h}{\lambda} = mv$

This equation now links matter (anything with a mass) to something having a wavelength. Rearranging for $\lambda$ gives:

$\lambda = \frac{h}{mv}$

Since momentum is the product of mass and velocity, $p = mv$:

$\lambda = \frac{h}{p}$

where
$\lambda$ is the wavelength
$h$ is the Planck constant
$p$ is the momentum of the matter

This equation links the wavelength of a particle to its momentum. This is the de Broglie equation.

If we take an example of a cricket ball being thrown at a set of stumps (and lets just say the stumps are separated by the width of the cricket ball). A typical cricket ball has a mass of $160 \ g$ and can be thrown by a bowler at a speed of up to $160 \ kmh^{-1} = 44 \ ms^{-1}$. We can now work out the wavelength that this cricket ball could have:

$p = mv = 160 \times 10^{-3} \times 44 = 7,040 kgms{-1}$

$\lambda = \frac{h}{p} = = \frac{6.63 \times 10^{-34}}{7,040}$

$\lambda = 9.42 \times 10^{-38} \ m$

This suggests that the wavelength is exceptionally small, much smaller than the width of the stumps, and much much smaller than the distance between two adjacent atoms ($\approx 10^{-10} \ m$). This explains in the first instance why diffraction cannot be observed if the ball passes through the stumps – after all maximum diffraction occurs when the wavelength matched the width of the gap the wave passes through.

Using the de Broglie equation, $\lambda = \frac{h}{p}$ we can see that in order for the wavelength to increase (and hopefully then diffraction or some other phenomena can then be observed) the momentum, $p$ needs to decrease. Since the wavelength of the cricket ball above was shown to be of the order $10^{-38}$, the momentum would need to decrease by a factor of $10^{31}$ if it were then to be of a similar order to visible light (of which we have previously witnessed wave phenomena).

So by changing the cricket ball to an electron and lets say we diffract it through two atoms (if we had such control), then we can use the knowledge that the mass of an electron, $m_{e} = 9.11 \times 10^{-31} \ kg$ and the idea that we accelerate it to a high speed, say $v = 3 \times 10^{6} \ ms^{-1}$ (a hundredth of the speed of light) to determine the wavelength and hence see if diffraction would occur and if it could be observed:

$p = mv = 9.11 \times 10{-31} \times 3 \times 10{6} = 2.733 \times 10^{-24} kgms{-1}$

$\lambda = \frac{h}{p} = = \frac{6.63 \times 10^{-34}}{2.733 \times 10^{-24}}$

$\lambda = 2.43\times 10^{-10} \ m$

This is now of the correct order of magnitude and as such could theoretically diffract electrons. These electrons may then cause interference with one another…

… so we may get areas of bright electrons and dim electrons???

Watch the following video clip, this shows how an experiment can be performed to determine whether electrons can actually diffract.

This video shows us that electrons can be diffracted, which in turn tells us that matter can behave like a wave. It has particle and wave like properties – wave-particle duality!

Physicists went one step further and wanted to observe what was happening when the electrons went through the gaps and hence the behaviour causing these interference patterns.

This is where the world of quantum physics really begins and where it shapes our future in terms of technology:

This is what the Schrödinger’s cat thought experiment really refers to. Is the cat dead, alive, or is it it both? Well in reality it is not both! – but on a quantum physics level, the cat refers to the electrons and whether the electrons are behaving as a wave or as a particle or both. This video from minutephysics will explain it in more detail:

Physics is an incredible subject to study and helps us learn about the world around us. There are however some areas that are more difficult to fathom because of how dissimilar we are to that idea, a bit like how we live in a 3-dimension plane (if we exclude time), is there a fourth dimension and if there is what would it look like? If a photon travels at $3 \times 10^{8} \ ms^{-1}$ to the right and another photon travels at  $3 \times 10^{8} \ ms^{-1}$ to the left, how fast is the second photon going with respect to the first? In the first instance you would probably say $6 \times 10^{8} \ ms^{-1}$, this is however incorrect, the relativistic speed to one another is $3 \times 10^{8} \ ms^{-1}$. This again is another area of physics known as special relativity – it, like quantum physics are different areas available for study.