text: RHEED users construct Ewalds spheres to find the crystallographic properties of the sample surface. Ewalds spheres show the allowed diffraction conditions for kinematically scattered electrons in a given RHEED setup. The diffraction pattern at the screen relates to the Ewalds sphere geometry, so RHEED users can directly calculate the reciprocal lattice of the sample with a RHEED pattern, the energy of the incident electrons and the distance from the detector to the sample. The user must relate the geometry and spacing of the spots of a perfect pattern to the Ewalds sphere in order to determine the reciprocal lattice of the sample surface.
  The Ewalds sphere analysis is similar to that for bulk crystals, however the reciprocal lattice for the sample differs from that for a 3D material due to the surface sensitivity of the RHEED process. The reciprocal lattices of bulk crystals consist of a set of points in 3D space. However, only the first few layers of the material contribute to the diffraction in RHEED, so there are no diffraction conditions in the dimension perpendicular to the sample surface. Due to the lack of a third diffracting condition, the reciprocal lattice of a crystal surface is a series of infinite rods extending perpendicular to the samples surface. These rods originate at the conventional 2D reciprocal lattice points of the sample's surface.
  The Ewald's sphere is centered on the sample surface with a radius equal to the magnitude of the wavevector of the incident electrons,
  where λ is the electrons' de Broglie wavelength.
  Diffraction conditions are satisfied where the rods of reciprocal lattice intersect the Ewalds sphere. Therefore, the magnitude of a vector from the origin of the Ewalds sphere to the intersection of any reciprocal lattice rods is equal in magnitude to that of the incident beam. This is expressed as
  Here, k is the wave vector of the elastically diffracted electrons of the order (hl) at any intersection of reciprocal lattice rods with Ewald's sphere
  The projections of the two vectors onto the plane of the sample's surface differ by a reciprocal lattice vector G,
  Figure 3 shows the construction of the Ewald's sphere and provides examples of the G, k and k vectors.
  Many of the reciprocal lattice rods meet the diffraction condition, however the RHEED system is designed such that only the low orders of diffraction are incident on the detector. The RHEED pattern at the detector is a projection only of the k vectors that are within the angular range that contains the detector. The size and position of the detector determine which of the diffracted electrons are within the angular range that reaches the detector, so the geometry of the RHEED pattern can be related back to the geometry of the reciprocal lattice of the sample surface through use of trigonometric relations and the distance from the sample to detector.
  The k vectors are labeled such that the vector k00 that forms the smallest angle with the sample surface is called the 0th order beam. The 0th order beam is also known as the specular beam. Each successive intersection of a rod and the sphere further from the sample surface is labeled as a higher order reflection. 
  Because of the way the center of the Ewald's sphere is positioned, the specular beam forms the same angle with the substrate as the incident electron beam. The specular point has the greatest intensity on a RHEED pattern and is labeled as the (00) point by convention. The other points on the RHEED pattern are indexed according to the reflection order they project.
  The radius of the Ewalds sphere is much larger than the spacing between reciprocal lattice rods because the incident beam has a very short wavelength due to its high-energy electrons. Rows of reciprocal lattice rods actually intersect the Ewalds sphere as an approximate plane because identical rows of parallel reciprocal lattice rods sit directly in front and behind the single row shown. Figure 3 shows a cross sectional view of a single row of reciprocal lattice rods filling of the diffraction conditions. The reciprocal lattice rods in Figure 3 show the end on view of these planes, which are perpendicular to the computer screen in the figure.
  The intersections of these effective planes with the Ewalds sphere forms circles, called Laue circles. The RHEED pattern is a collection of points on the perimeters of concentric Laue circles around the center point. However, interference effects between the diffracted electrons still yield strong intensities at single points on each Laue circle. Figure 4 shows the intersection of one of these planes with the Ewalds Sphere.
  The azimuthal angle affects the geometry and intensity of RHEED patterns. The azimuthal angle is the angle at which the incident electrons intersect the ordered crystal lattice on the surface of the sample. Most RHEED systems are equipped with a sample holder that can rotate the crystal around an axis perpendicular to the sample surface. RHEED users rotate the sample to optimize the intensity profiles of patterns. Users generally index at least 2 RHEED scans at different azimuth angles for reliable characterization of the crystal's surface structure. Figure 5 shows a schematic diagram of an electron beam incident on the sample at different azimuth angles.
  Users sometimes rotate the sample around an axis perpendicular to the sampling surface during RHEED experiments to create a RHEED pattern called the azimuthal plot. Rotating the sample changes the intensity of the diffracted beams due to their dependence on the azimuth angle. RHEED specialists characterize film morphologies by measuring the changes in beam intensity and comparing these changes to theoretical calculations, which can effectively model the dependence of the intensity of diffracted beams on the azimuth angle.