# Highest Order Diffraction Ring

## Main Question or Discussion Point

So we have the standard diffraction equation for crystals, being: nλ = 2dsinθ. If we keep the wavelength and the lattice parameter the same then n simply depends on θ. So there must be a maximum order diffraction ring that you could obtain as once you get to 90° you're essentially going in the opposite direction with the same angles you've already covered.

Intuitively you'd think to just rearrange to find the maximum n as n = (2dsin90°)/ λ or n = 2d/λ. But my question is, would you begin to get weird effects from the "slits" in planes below that would change this result?

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Simon Bridge
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You basically want to know if the Bragg's Law holds for large angles.

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So we have the standard diffraction equation for crystals, being: nλ = 2dsinθ. If we keep the wavelength and the lattice parameter the same then n simply depends on θ. So there must be a maximum order diffraction ring that you could obtain as once you get to 90° you're essentially going in the opposite direction with the same angles you've already covered.

Intuitively you'd think to just rearrange to find the maximum n as n = (2dsin90°)/ λ or n = 2d/λ. But my question is, would you begin to get weird effects from the "slits" in planes below that would change this result?
The maximum order n is the maximum integer that is less than $2d/\lambda$. This will normally occur at some angle close to 90 degrees, i.e. perpendicular to the crystal planes. If you choose $\lambda>2d$ then you don't get any Bragg maxima. The angles only go from 0 to 90 degrees. The incident angle measured from the normal to crystal planes is $\theta_i=90-\theta$. I believe in Bragg scattering, you need to locate the peaks by rotating the crystal w.r.t. the source. Once you find an orientation of the crystal that gives a maximum, you then find the location where the scattered UV is going. Two conditions are satisfied for Bragg scattering: 1) The angle of incidence=angle off reflection off of the individual parallel planes 2) The spacing between the crystal planes must satisfy the Bragg condition, i.e. $n \lambda=2d \sin \theta$ where $n$ is an integer so that the reflected waves off of all of parallel (and equally spaced) crystal planes constructively interferes.

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• Willfrid Somogyi
You basically want to know if the Bragg's Law holds for large angles.
Yeah, that's a much simpler way to put it but I was also wondering if anyone could tell me what happens after Bragg's law breaks down

Bragg's law written as nλ = 2dsinθ is really a quite crude approximation. For a more systematic approach to the problem you have to understand the concept of the reciprocal lattice. The "d-spacing" in Bragg's law is the inverse of the length of a reciprocal lattice vector Q, usually written with its Miller indices (H,K,L). A basic rule of the reciprocal lattice is that multiples of a reciprocal lattice vector (n*Q) are again reciprocal lattice vectors. Hence the "n" in Bragg's law.
It is much more systematic to identify, say, a (333) reflection as (333) instead of 3*(111), because the strength of a reflection depends on the structure factor. In some cases the structure factor can be zero, then the reflection does not exist. For example in the diamond structure (111) exists and is a nice, strong reflection, but (222) is forbidden, whereas (333) and (444) exist again. No in principle there is no fundamental limit to "n". In practice, however, there is a lower limit for λ, and with λ = 2dsinθ/n this gives an upper limit for n - or more precisely the length of the reciprocal space vector Q=d/n.