Without having any evidence to support his claim on the periodic arrangements of atoms in a lattice, he further postulated that the crystalline structure can be used to diffract x-rays, much like a gradient in an infrared spectrometer can diffract infrared light.
His postulate was based on the following assumptions: the atomic lattice of a crystal is periodic, x- rays are electromagnetic radiation, and the interatomic distance of a crystal are on the same order of magnitude as x- ray light.
Laue's predictions were confirmed when two researchers: Friedrich and Knipping, successfully photographed the diffraction pattern associated with the x-ray radiation of crystalline \(Cu SO_4 \cdot 5H_2O\). The arrangement of the atoms needs to be in an ordered, periodic structure in order for them to diffract the x-ray beams.
A series of mathematical calculations is then used to produce a diffraction pattern that is characteristic to the particular arrangement of atoms in that crystal.
Diffraction and measurement of such small wavelengths would require a gradient with spacing on the same order of magnitude as the light.
In 1912, Max von Laue, at the University of Munich in Germany, postulated that atoms in a crystal lattice had a regular, periodic structure with interatomic distances on the order of 1 A.So we get: \[ BG = BC = d \sin \theta \label\] Thus, \[ 2d \sin \theta = n \lambda \label\] This equation is known as Bragg's Law, named after W. The x-rays that are diffracted off the crystal have to be in-phase in order to signal.Only certain angles that satisfy the following condition will register: \[ \sin \theta = \dfrac \label \] For historical reasons, the resulting diffraction spectrum is represented as intensity vs. The main components of an x-ray instrument are similar to those of many optical spectroscopic instruments.Applying some basic trigonometric properties, the following two equations can be shown about the lines: \[CD = x \cos(θ o)\] and \[HG = x \cos (θ) \] where \(x\) is the distance between the points where the diffraction repeats.Combining the two equations, Diffraction of an x-ray beam, occurs when the light interacts with the electron cloud surrounding the atoms of the crystalline solid.Calculation of the phase difference can be explained by examining Figure 1 below.In the figure below, two parallel waves, BD and AH are striking a gradient at an angle \(θ_o\).These include a source, a device to select and restrict the wavelengths used for measurement, a holder for the sample, a detector, and a signal converter and readout.However, for x-ray diffraction; only a source, sample holder, and signal converter/readout are required.Some of the light will be diffracted at an angle \(theta\), and the remainder will travel deeper into the solid.This process will repeat for the many planes in the crystal.