# Exploring Bragg/von Lau Scattering: A Discussion of X-Ray Experiments

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• sterilemeep
In summary: In both cases, light is scattered. The only difference between the two is the geometry in which the scattering happens.
sterilemeep
TL;DR 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?
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.

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Last edited:
Lord Jestocost
Thank you for your reply (and sorry it's taken me so long to respond.) I wasn't able to connect your reply to my question, unfortunately. But...

I have been thinking about this on an off for a while though, but the answer is actually fairly simple: Bragg scattering is the geometric optics approximation to Fraunhofer diffraction.

It's snell's law with phase constraints that come from periodic media. It is a good approximation, as most X-ray wavelengths are orders of magnitude smaller than what they are scattering off.

My question above was the result of trying to equivocate snell's law with diffraction.

## What is Bragg/von Lau scattering?

Bragg/von Lau scattering is a phenomenon in which X-rays are diffracted by a crystal lattice, producing a pattern of constructive and destructive interference. This can be used to study the structure and properties of materials.

## How is Bragg/von Lau scattering used in X-ray experiments?

In X-ray experiments, Bragg/von Lau scattering is used to analyze the diffraction patterns produced by X-rays passing through a crystal. This can provide information about the arrangement of atoms in the crystal lattice and the properties of the material.

## What are the key differences between Bragg and von Lau scattering?

Bragg and von Lau scattering are both types of X-ray diffraction, but they differ in the orientation of the crystal and the angle at which the X-rays are diffracted. Bragg scattering occurs when the crystal is perpendicular to the X-rays, while von Lau scattering occurs at an oblique angle.

## What types of materials can be studied using Bragg/von Lau scattering?

Bragg/von Lau scattering can be used to study a wide range of materials, including crystals, powders, and thin films. It is particularly useful for analyzing the structure and properties of crystalline materials.

## What are some practical applications of Bragg/von Lau scattering?

Bragg/von Lau scattering has many practical applications, including in materials science, chemistry, and engineering. It can be used to determine the structure of new materials, analyze the quality of crystals, and study the properties of materials under different conditions.

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