Undergrad Highest Order Diffraction Ring

[Willfrid Somogyi]

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?

 

[Simon Bridge]

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

 

[Charles Link]

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.

 

[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

 

[M Quack]

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.

 

[Charles Link]

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.

For a given UV wavelength as the source, there will be an upper limit for the "n", as described in post #3. Once you locate a Bragg peak for a given wavelength (with some integer "n"), you can always take shorter wavelengths of integer multiples "m"(with no limit on the integer multiple m) and those will also give a Bragg peak (as you correctly describe in your posting). Please correct me if it is not the case, but I think the structure factor that you get in the reciprocal lattice formulation is "q" dependent. When you get a structure factor that is nearly 0 (for some q that satisfies the Bragg condition), that usually would imply that there is a second plane of identical atoms equally spaced so that you would find a Bragg peak at the same angle for "2q" which is one-half the wavelength (as well as any integer multiples of 2q.). Although for more detailed and precise calculations the solid state experimentalists use the reciprocal lattice formulation, the atomic plane model gives a fairly accurate description of Bragg scattering.