Rotating Crystal 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). The rotating crystal method provides for the incident x-ray beam to keep the wavelength fixed but make the crystal turn, 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 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 clearance between parallel lattice planes can be obtained through , where is the displacement from the origin to the point on the circle within the reciprocal lattice [1]. (The subscript refers to Miller indices, which denote a family of planes orthogonal to the reciprocal lattice vector: ).

Use the "lattice" setter bar to select from the five different types of 2D lattices. The wavelengths that satisfy Bragg's law are associated with specific radii of the Ewald's circle, which can be selected through the "Ewald radius" and "θ (rad)" setter bars.

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

Two boxes show the vectors that generate the direct lattice and the reciprocal lattice; to highlight the crystal rotation, the fixed vectors are shown in red and the rotating vectors are shown in black. Further, the rotation is highlighted by orange dashed arcs.

Snapshot 1: no diffraction; on the direct lattice this becomes clear through the representation of a single vertical line

Snapshot 2: diffraction obtained with a centered rectangular lattice with axial distance and axial angle

Snapshot 3: Second-order diffraction obtained by means of a rectangular lattice with an axial distance and axial angle . As you can see, this takes place when one more point lies on the clearance vector in the reciprocal lattice. In this case, Bragg's law is valid with resulting in where , generating planes that are located halfway through the lattice planes, as shown by the side corresponding to the direct lattice.

Reference

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