Schrödinger's Cormorant

5.3 Something to Bragg about

Evidence from electron diffraction to support the de Broglie hypothesis.

When x-rays (i.e. electromagnetic radiation) are directed at a crystal lattice, they are reflected most strongly in the direction whereby the angle of incidence equals the angle of reflection. However the intensity of the reflected radiation depends strongly on what this angle actually is. An explanation for this was put forward by Lawrence Bragg and his Dad, who realised that this angular dependence was a function of the distance between “reflecting planes” in the lattice – in effect the distance between the centres of parallel sets of atoms. They proposed a simple law which embodies this idea. Let’s have a go at deriving this law, because we’re going to need it to help Louis out.

Figure V-i is a diagram which shows how Bragg diffraction works. Although it’s not entirely realistic - the atoms are modelled by points on parallel lattice planes – this model effectively illustrates how a regularly spaced array of reflecting particles can lead to the characteristic intensity variation.

Fig. V-i

The incident wave front hits the parallel planes of atoms at an angle θ. Along path A, the x-ray has to travel further to hit an atom than it would along line B. Nevertheless, in this example, the emerging rays are still in phase, so a high intensity reflection will be observed. You can see, though, that if we changed the wavelength or the separation of the layers, this might not be so. Time for a little maths:

Question 5-1

You should have found Bragg’s law (or, possibly, Braggs' law), which may be written down as:

nλ = 2d sinθ (V-2)

This means that if you know the x-ray wavelength and you measure the angles, between the x-ray beam and the surface, where the reflection intensities are maximal, you can work out the distances between the centres of atoms in adjacent layers. This was the starting point for the development of x-ray diffraction as a method for determining molecular structure: measurements of the positions of atoms in the crystal are made by measuring the angles of scattering of x-rays of known wavelength.

For us, though, the importance of all this is something else: if Prince Louis is right about wave-particle duality, we might also be able to use this phenomenon to try to measure the wavelength of an electron and hence test whether his equation has any validity. “Yeah, we did that already” chime in a couple of American voices.

Within a couple of years of Louis first making his proposal, Clinton Davisson and Lester Germer found themselves firing a beam of electrons at a nickel surface, hoping to find out something interesting about the nickel. Instead they found out something remarkable about the electrons: they had the information needed to test Louis’ prediction. “Just because we did it by accident doesn’t mean we’re not great,” Clinton and Lester point out, brandishing their Nobel prize medals. Alexander Fleming hurries over to back them up.

What they noticed was that - just as happens with x-rays - there was a strong dependence of the intensity of the scattered electron beam on the angle between the incident electron beam and the nickel target. In other words it looked like they were seeing diffraction. This was the first direct evidence of electrons behaving in a way that seemed more like a wave than a particle. Our next job is to go further and find out whether their data are quantitatively consistent with the de Broglie relationship.

Question 5-2

The remarkable agreement between the result of the Davisson-Germer experiment lends a lot of credibility to Louis de Broglie’s vision of wave-particle duality and supports the validity of his equation. It’s such a simple equation that you can easily apply it to any moving object. Like a cormorant:

Question 5-3

The wavelength for a macroscopic body, like Cormorant, is ridiculously short. His calculated wavelength is around one septillionth of the size of a hydrogen atom. And this is not a seabird-specific outcome; it happens because – as Louis’ equation makes plain - as the particle’s mass increases, its wavelength decreases. As a result, the general picture is the same for any such object travelling at anything but an immeasurably slow velocity: its de Broglie wavelength is so small that any wave-like properties would be entirely negligible. Only when we get down to something approaching the atomic scale can we expect to see interesting wavy effects like diffraction and interference. Going back to chapter one, remember that Cormorant was persuaded to try flying at a pair of narrow slits to see if he could be diffracted. We can now see why the experiment was doomed to failure:

Question 5-4

We started, in Chapter 1, with the idea that the seemingly unlikely proposition that particles have wave-like properties might explain their quantised energy levels. The de Broglie equation and the experimental data that support it are therefore very exciting. However the equation raises more questions than it answers. Time to think a bit more deeply about what it means.

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