Laue's Method for 2D Lattices Using Ewald's Circle

This Demonstration shows possible types of 2D lattices, the corresponding reciprocal lattices and Ewald's circle for the reciprocal lattice (right side). These determine the parallel lattice planes for which Bragg's law is satisfied (left side). Laue's method determines the positions of both the crystal and the incident x-ray beam, kept fixed by changing the wavelength, which leads to the observed diffraction pattern. The upper-right corner shows a real representation of the x-ray beam wavelength, which is related to the diameter of the Ewald's circle, being inversely proportional to the wavelength. The origin of the reciprocal lattice is always located on the edge of the Ewald's circle. The distance between parallel lattice planes is given by , where is the displacement from the origin to the point on the circle within the reciprocal lattice [1].

Use the "lattice" SetterBar to select from the five different 2D lattices. The wavelengths that satisfy Bragg's law are associated with specific radii of the Ewald's circle, which can be selected with the "beam" SetterBar.

In order to take all possible geometric cases into consideration, different options have been chosen for each crystal lattice, which can be selected with the "cases" SetterBar.

Snapshot 1: Backscattering of the incident x-ray beam. The angle of reflection is 180 degrees. The lattice considered is oblique with an axial distance and axial angle .

Snapshot 2: Scattering from a centered rectangular lattice with an axial distance and axial angle , . Note that, by symmetry, two points (in addition to the origin) are located on the circle and both of them generate a diffracted beam; just one of them is shown in the actual lattice, together with the associated parallel lattice planes.

Snapshot 3: Scattering from a hexagonal lattice with an axial distance and axial angle . Note that, by symmetry, five points (in addition to the origin) are located on the circle and all of them generate a diffracted beam; just one of them is shown in the actual lattice, together with the associated parallel lattice planes.

Reference

[1] C. Kittel, Introduction to Solid State Physics, 3rd ed., New York: Wiley, 1966.