Help with Xray Scattering: Reconciling Bragg Scattering with Fraunhofer Diffraction

  • #1

Summary:

Most (if not all) XRay scattering discussions center around Bragg's Law of reflection. However, most experiments seem to be better described by Fraunhofer diffraction. Is there a way of connecting these phenomena? Or could someone with more Xray experience help me reconcile these approaches?

Main Question or Discussion Point

Overview of thread:
1. Quick derivation of bragg scattering
2. Discussion of modern xray experiments as they relate to bragg/fraunhofer
3. Summary of points.

Bragg/von Lau Scattering:
(I will be following Ashcroft if you want to sing along, pg 98-99)
Imagine you have light incident on some crystal structure with wavevector ##k=2π\hat{n}/λ##. You make the following assumption-- the light is scattered elastically (its wavelength doesn't change). For constructive interference, the path difference between any two scattered rays must be an integer number of wavelengths, which gives the von Lau condition:
$$
R \cdot{} (k-k') = 2 \pi m
$$
with R being a lattice vector, k being the incident, and k' being the outcident. This is equivalent to saying that the difference between the incident and outcident vectors must be a reciprocal lattice vector (2πmR).

Because the scattering is elastic |k|=|k′|, and we can use this to derive the following:
$$
\vec{k} \cdot{} \hat{K} = \frac{1}{2} K
$$
(Graphically, you can see this in the following geometric construction: you have two vectors of the same length. Subtracting them gives a third vector. Because the two original vectors are the same length, this makes an isosceles triangle. You can verify that each of the two equal lines in an isoceles triangle, when projected onto the third line, each compose 1/2 of the third line. See Ashcroft pg 99 for a picture of this (or my badly drawn Figure 1.)

Problem
Imagine you have a beam of light incident on a 1D lattice (see Figure 2), where the scattering vector is incident along ##\hat{x}## and the crystal bravais lattice is along ##\hat{z}.## In this geometry k⋅K=0, so bragg's law predicts that no scattering will occur (as far as I can see, see Figure 3). The issue then is that this is the geometry that is used for a lot of xray experiments! This is the geometry for instance of a syncrotron, where a beam of light is incident on a sample, and the detector measures transmission. If you look at a lot of Xray literature, the plots will often be in terms of the bragg angle ( 2θ).

To sum up, most theoretical descriptions of xray scattering use Bragg scattering (all the ones I've seen), when it appears that Bragg scattering gives nonsensical results in a very common experimental geometry. I can think of two solutions.

1. I'm an idiot and completely misunderstood Bragg scattering, or modern Xray science (in which case, could you point me in the direction of some resources that tackle this issue??!)
2. It doesn't matter/is an experimental approximation. Most crystal lattices are on the order of 10's of angstroms, so you'd only need a deviation from a pure ## \hat{x}## incident by arctan(.5∗K/k) to meet the bragg condition, which would be small.

I'd love to hear anyones thoughts about this!
 

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Charles Link
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