I Questions About Bragg's Law of X-Ray Diffraction

[Dario56]

You guys provided many new interesting sources to read and expand upon my question. I think that this is a bit too detailed for what I currently need, but who knows. Maybe I'll want to get back to them later.

One new question popped to me. Basically, we take that diffraction centres are different in the XRD and Young's experiment. In the former case, we take that atoms are centres. In the latter, slits are centres which would be equivalent to the interatomic spacing in the context of XRD.

Is it justified to apply Huygens principle to atoms rather than spacing between them? Or in this case, we look at the diffraction from the perspective of Thompson scattering? I think it is the second case, though.

 

[Charles Link]

Interesting question that you are asking. With the XRD the atoms are considered to be the sources that basically send out the $E$ field as a sinusoid in all directions. In general each one scatters just a small portion of the incident amplitude. With the scattering, we are working with an E amplitude and the sinusoids (in a given direction) need to be summed with their phases. Once the sum is computed, the energy/intensity is then computed, (taking $E_{total}^2$), as I think you have got it figured out by now.

One might expect that one could compute the energy scattered in a given direction by each scatterer and sum them all, but that is, perhaps somewhat surprisingly, not how it works. I'm just adding a little extra detail here, even though I think you have it figured out. :)

Edit: Presently you are just computing the intensity pattern with an unknown proportionality factor. In more detailed calculations, the actual power can be computed for the various diffraction patterns, and they can be shown to conserve energy. For an example of the energy conservation, see https://www.physicsforums.com/insights/diffraction-limited-spot-size-perfect-focusing/

 

[Steve4Physics]

Is it justified to apply Huygens principle to atoms rather than spacing between them? Or in this case, we look at the diffraction from the perspective of Thompson scattering? I think it is the second case, though.

It doesn’t matter what the sources of the waves are. They can be atoms, or gaps, or even loud-speakers.

The essential requirements are that:
- the waves radiated by the sources overlap;
- the phase-difference between neighbouring sources is appropriate (e.g. zero for the Young’s double-slit experiment and some constant non-zero value for Bragg diffraction).

 

[sophiecentaur]

Therefore, how can we only consider parallel scattered waves in Bragg's law as non-parallel waves can also create diffraction pattern?

What are you suggesting about "non parallel (incident?) waves"? Diffraction still works - as in the production of holograms but this is not really relevant to Xray diffraction patterns which involve a pretty short coherence length of the incident waves. I have never come across Xray holography - but someone here may know differently.
The basics of Xray diffraction in crystals (Bragg scattering) assume a point source and a plane wave arriving at the crystal. If you could stack layers of optical diffraction gratings you would get a similar effect on the pattern with many gaps in it. Very often, there is not a single, large crystal available to look at and the simple idea of distinct single rays emerging fails. What does remain, though, tends to be cones of scattered rays with particular angles between them. Spacings between atoms can be calculated from the pattern.

 

[sophiecentaur]

Is it justified to apply Huygens principle to atoms rather than spacing between them? Or in this case, we look at the diffraction from the perspective of Thompson scattering? I think it is the second case, though.

I re-read this and I don't understand where you're going with this. Huygens construction is basically the diffraction formula and would be treating a large aperture across the whole crystal and would give you the whole of the pattern of emerging rays. This would produce an intense beam in the straight through direction and also the Bragg maxima when the angles are suitable - i.e. when both i and r are equal, giving an effective reflection. For other angles there is no such 'reflection'.
The basic Bragg formula ignores the spaces between and deals with point sources, which is an interference formula and a lot simpler.
I remember struggling with this in my brief exposure to Crystallography. Life was a lot easier when I started looking at RF transmitting or receiving arrays where there is only an 'incident wave' or an 'output wave'. (the 'other ray' is just in your head)