Selected area electron diffraction
In a transmission electron microscope (TEM), a parallel beam of high-energy electrons passes through a thin crystalline specimen. Because the de Broglie wavelength of the electrons (~0.025 Å at 200 kV) is much smaller than typical interatomic spacings, the crystal acts as a diffraction grating. By inserting an aperture in the image plane, a specific region of the specimen can be isolated for diffraction analysis. The resulting pattern of spots on the viewing screen is a direct map of the reciprocal lattice of the crystal, projected along the beam direction.
The zone axis and Weiss zone law
The electron beam travels along a crystallographic direction called the zone axis [uvw]. Only lattice planes (hkl) whose normals are perpendicular to the beam contribute to the zero-order Laue zone (ZOLZ) pattern. Mathematically, this is expressed by the Weiss zone law: hu + kv + lw = 0. This constraint selects a 2D slice of reciprocal space, which is why the pattern appears as a regular 2D array of spots rather than a 3D distribution.
Structure factor and selection rules
Not every plane satisfying the zone law produces a visible spot. The structure factor F(hkl), which depends on the atomic positions within the unit cell, can cause systematic absences. For a simple cubic (SC) lattice, all reflections are allowed. For BCC, reflections require h+k+l to be even. For FCC, h, k, and l must be all odd or all even. Diamond cubic has an additional constraint: when h, k, l are all even, their sum must be divisible by 4. These rules arise from destructive interference between atoms at different positions in the unit cell.
Relativistic electron wavelength
At TEM accelerating voltages (100-300 kV), electrons travel at a significant fraction of the speed of light, so relativistic corrections are essential. The wavelength is given by λ = h / √(2meeV(1 + eV/2mec²)), where the term in the denominator accounts for the relativistic increase in electron mass. At 200 kV, this gives λ = 0.00251 nm, about 2% shorter than the non-relativistic prediction.
Reading the pattern
Each spot corresponds to a set of crystal planes (hkl). The distance R of a spot from the central (transmitted) beam is inversely proportional to the interplanar spacing d: R = λL/d, where L is the camera length. Low-index planes with large d-spacings appear close to the center; high-index planes appear further out. The symmetry of the spot pattern directly reveals the point group symmetry of the crystal when viewed along the chosen zone axis. Comparing patterns from BCC and FCC structures along the same zone axis shows characteristic differences in which spots appear and their arrangement.