# Van Laue Formulation: X-ray Diffraction from Crystal Structure

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In summary, the Van Laue formulation states that for a sharp intensity peak to be observed on the screen when shining an X-ray beam on a crystal, there must be a specific direction and path difference for all scattered rays. This condition can be expressed mathematically as the dot product of the Bravaise vectors and the difference between the incident and scattered wavevectors being equal to an integral multiple of 2π. This raises two questions: (1) the probability of all scattered waves being in the same direction is low, and (2) atomic separation in a crystal means that the incident wavevectors for each plane of lattice points would have different magnitudes, so why does the formulation assume a constant magnitude? It may be helpful
When we shine an X-ray beam on a crystal, according to Van Laue formulation, for a sharp intensity peak to be observed on the screen there is a specific direction ##\mathbf{\hat{n}}## , in which for all the X-rays with wavelength ##\lambda## and wavevector ##\mathbf{k}=\dfrac{2\pi}{\lambda}\mathbf{\hat{n}}## incident on the crystal, a scattered ray will be observed in a direction ##\mathbf{\hat{n}^{'}}## with wavelength ##\lambda## and wavevector ##\mathbf{k}^{'}=\dfrac{2\pi}{\lambda}\mathbf{\hat{n}^{'}}##. This must be occurred simultaneously for all the lattice points in the crystal and the path difference between the rays scattered by each of the lattice sites in the crystal must be an integral number of wavelength.
In other words, the condition that all scattered rays will interfere constructively is that:
$$\mathbf{R} \cdot(\mathbf{k}-\mathbf{k}^{'})=2\pi m$$
for integral ##m## and all Bravaise vectors ##\mathbf{R}##.

I have two questions related to this formulation:

(1) How all the wavevectors ##\mathbf{k}^{'}## can be scattered simultaneously from all the lattice points in the same direction ##\mathbf{\hat{n}}^{'}##? Is'nt the probability for such thing to happen very low?

(2) Von Laue assumed in his formulation that all the incident X-rays will hit the lattice points with the same wavevector ##\mathbf{k}##. But we know that the atomic separation in a crystalline solid is comparable in length to the wavelength of the X-ray. Therefore, when an X-ray comes across the first plane of lattice points, it will diffract, and I think hence that the X-ray incident on the second plane of lattice points would have a wavevector with magniude different from that incident on the first plane of lattice points. Is'nt this right? So why in the Von Laue formulation it is assumed that every incident X-ray hitting the lattice points has the same magnitude ##k##?

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You seem to think that interference takes place among independent multiple photons. How about rethinking it that interference talked is about a single photon ?

## 1. What is the Van Laue formulation?

The Van Laue formulation is a mathematical equation that describes the scattering of X-rays from a crystal structure. It was developed by German physicist Max von Laue in the early 20th century and is a fundamental tool in the field of X-ray crystallography.

## 2. How does the Van Laue formulation work?

The Van Laue formulation uses the principles of diffraction to explain how X-rays interact with a crystal structure. When X-rays are directed at a crystal, they are scattered in different directions by the atoms in the crystal lattice. This scattering pattern can then be analyzed to determine the arrangement of atoms in the crystal.

## 3. What information can be obtained from the Van Laue formulation?

The Van Laue formulation allows scientists to determine the positions of atoms within a crystal, as well as the distances between them. This information is crucial for understanding the structure and properties of crystals, which has important applications in various fields such as materials science, biology, and chemistry.

## 4. How is the Van Laue formulation used in research?

The Van Laue formulation is used extensively in research to study the structure of crystals. It is commonly used in X-ray crystallography experiments, where X-rays are directed at a crystal and the resulting diffraction pattern is analyzed to determine the crystal structure. This technique has been instrumental in advancing our understanding of the atomic and molecular world.

## 5. Are there any limitations to the Van Laue formulation?

While the Van Laue formulation is a powerful tool, it does have some limitations. It can only be used for crystals that have a regular and repeating atomic structure. It also requires high-quality crystals and specialized equipment, making it a time-consuming and expensive technique. Additionally, the Van Laue formulation cannot provide information about the chemical composition of a crystal, only its structure.

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