The Laue equations can be written as Δ k = k o u t − k i n = G. X-ray diffraction in real space Braggs Law. They are named after physicist Max von Laue (1879–1960). A is the amplitude, k is the wave vector, and 2f is the angular frequency. In crystallography and solid state physics, the Laue equations relate incoming waves to outgoing waves in the process of elastic scattering, where the photon energy or light temporal frequency does not change upon scattering by a crystal lattice. To understand how this works, consider an incoming wave with wavevector k that scatters off two atoms separated by a lattice vector R into an outgoing wave with wavevector k ′ (see figure).Equations describing diffraction in a crystal lattice Laue equation As a result of interference, the scattered radiation pattern reveals the reciprocal lattice of the crystal. X-rays, neutrons, or electrons) is directed at a material of interest. In a diffraction experiment, a beam of high-energy waves or particles (e.g. Such experiments are some of our most powerful tools for determining the crystal structure of materials. However, here is a reminder of such a cell:ĭetermining crystal structures using diffraction experiments ¶ The Laue condition ¶Īnother reason to understand the reciprocal lattice is that it manifests directly in diffraction experiments. In the previous lecture we already discussed how to construct Wigner-Seitz cells. This means that any wavevector k ′ outside the 1st Brillouin zone is related to a wavevector k inside the first Brillouin Zone by shifting it by a reciprocal lattice vector: k ′ = k + G. But what unit cell to choose? We learned that the choice of a primitive unit cell is not unique.Ī general convention in reciprocal space is to use the Wigner-Seitz cell, which is also called the 1st Brillouin zone.īecause the Wigner-Seitz cell is primitive, the 1st Brillouin zone (1BZ) contains a set of unique k-vectors. And because waves with wavevectors differing by a reciprocal lattice vector G are identical, we only need to understand the dispersion in a single primitive unit cell of the reciprocal lattice. An important reason to study the reciprocal lattice is that we are often interested in understanding the dispersion relation of electronic or vibrational modes in a material. We have now seen how the structure of the reciprocal lattice is directly determined by the structure of the real-space lattice. The importance of the 1st Brillouin zone ¶ The reciprocal lattice in three dimensions ¶ Let us now generalize this idea to describe reciprocal lattices in three dimensions. The set of points G = 2 π m / a forms the reciprocal lattice in k-space. We then observed that waves with wavevectors k and k + G, where G = 2 π m / a with integer m, are exactly the same:Į i ( k + G ) n a = e i k n a + i m 2 π n = e i k n a , To obtain the dispersion relation, we considered waves of the form In lecture 7 we discussed the reciprocal space of a simple 1D lattice with lattice points x n = n a, where n is an integer and a is the spacing between the lattice points. Recap: the reciprocal lattice in one dimension ¶ In this lecture, we will 1) study how real-space lattices give rise to lattices in reciprocal space (with the goal of understanding dispersion relations) and 2) consider how to probe crystal structures using X-ray diffraction experiments. In the last lecture, we introduced crystallographic terminology in order to be able to discuss and analyze crystal structures. Interpret X-ray powder diffraction data.Compute the intensity of X-ray diffraction of a given crystal.Construct a reciprocal lattice from a given real space lattice.This usually means that the atomic beam crosses the light wave exactly. Define the reciprocal space, and explain its relevance Bragg diffraction of atoms at thick standing light waves requires that the wave-matching condition is fulfilled.The Laue condition and structure factor for non-primitive unit cellsĮxercise 1*: The reciprocal lattice of the bcc and fcc latticesĮxercise 2: Miller planes and reciprocal lattice vectorsĮxercise 4: Analyzing a 3D power diffraction spectrum The reciprocal lattice as a Fourier transformĭetermining crystal structures using diffraction experiments The reciprocal lattice in three dimensionsĮxample: the reciprocal lattice of a 2D triangular lattice Recap: the reciprocal lattice in one dimension
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