Diffraction on periodic Structures

[Gamdschiee]

Thank you, I think I get the principal, but what do you mean with the angle $\phi$?

And could you explain again please, when to use a + sign before the cos() and a - sign before it? I can't imagine that part how I should work out this.

 

[Charles Link]

Thank you, I think I get the principal, but what do you mean with the angle $\phi$?

The $\theta$ is a polar type angle. If we are just working basically in two dimensions e.g. with a diffraction grating, the $\phi$ is omitted. Similarly if we reflect from each plane so that $\theta_i=\theta_r$, we can essentially ignore the effect of the geometry of how the atoms are arranged on that plane. (That is only the case for the $m=0$ maximum). Then we can consider the planes as uniform, and only need to concern ourselves with the polar angle $\theta_i$ (spherical coordinates) relative to the z-axis. If we are considering the $m=1$ case that occurs for reflection off of a single plane, it only occurs for $\theta_i$ where $(1)(\lambda)=d(\sin(\theta_i)+\sin(\theta_r))$ (e.g. a rectangular array of atoms) if the azimuthal $\phi$ angle is zero. $\\$If we were to test this with a diffraction grating using a pinhole type (monochromatic=laser) source instead of a slit with a parabolic mirror to collimate the beam, and rotated the grating to some $\theta_i$, so that it has a plane wave incident on it at angle $\theta_i$, we could then test for the effect on the resulting $m=1$ bright spot that is found in a spectrometer where the far-field pattern is observed on the exit slit because a second parabolic mirror is used to focus the far-field pattern in the plane of the exit slit. If the grating is tilted backwards, it will make the focused $m=1$ spot occur slightly elevated on the upper part of the exit slit, but if that angle is tilted too much, I think the focused spot quality would be greatly reduced. It would start to become a big blob rather then a small focused spot. $\\$ Here, again is where I would really need to consult an x-ray crystallography textbook to see if they treat the case of a peak that doesn't satisfy the Bragg condition.

 

[Charles Link]

This last item, which really came about because of your question @Gamdschiee in post 29, could use some further research. I was unable to locate any mention of it in a google search. Normally, researchers in this field of x-ray scattering are quite thorough. I think it is likely that there will indeed be peaks of this nature occurring on a somewhat regular basis, and I expect they do take them into account. This finer feature doesn't appear to be presented in most textbooks discussions on Bragg scattering, but mathematically, the conditions for constructive interference appear to be met by all of the atoms in this case, so I would expect occasionally such peaks would occur and be observed experimentally.