Thirty-two point groups, thirty-one diffraction groups, one table: the pattern's symmetry is the crystal's, read through reciprocity.
The Buxton tables, derived rather than typed: choose a point group and a zone axis and read the diffraction group with the whole-pattern, bright-field, dark-field and ±G symmetries a CBED pattern must show; or enter the symmetries you observed and list every point group and zone-axis class that could have produced them. The schematic shows what each symmetry looks like on the screen.
Sampled over all axes with indices up to 3 in the cell chosen on the right; the representative is the lowest-index axis of each class. Click a row to move the zone axis there.
Enter what you see in one zone-axis pattern: the symmetry of the whole pattern (the arrangement and inner detail of all discs, including HOLZ lines), the symmetry of the bright-field disc alone, and, if you have tilted to the Bragg condition of a pair of reflections, how the +G and −G discs are related. Leave a field at “any” when it was not measured.
The 31 diffraction groups with the observable symmetries each one produces. The list is generated by running every point group through every zone axis and collecting the distinct results, so it is a check on the derivation as much as a reference: 31 distinct groups appear, the number Buxton, Eades, Steeds and Rackham found in 1976.
The axial ratios only matter for the zone-axis sampling in the non-cubic systems: they decide which directions coincide with symmetry elements. Any value gives the same classes; a special value (c/a = 1 in a tetragonal cell) merges none.
Indices in the conventional cell of the system (three-index for hexagonal and trigonal: [u v w] with the third index t = −(u + v) implied).
Each disc carries an asymmetric motif replicated by the symmetry operations, so the pictured symmetry is exactly the one listed. Whole pattern: the motif in every disc of the zero layer, with the mirror lines and the rotation axis of the projected pattern. Bright field: the central disc alone, which can be more symmetric than the pattern because the horizontal mirror or 2-fold (the 1R operation) adds a 2-fold about the disc centre. ±G: the pair of dark-field discs tilted to their Bragg conditions; 2 means the two are related by the 2-fold about the zone axis, 2R means they are identical by translation (a centrosymmetric crystal), 21R both, 1 neither.
A zone-axis CBED pattern has a symmetry that is decided by the crystal point group and the orientation of the zone axis within it, and Buxton, Eades, Steeds and Rackham showed in 1976 that the symmetries a microscopist can observe (the whole pattern, the bright-field disc, the dark-field discs and the relation between a +G and −G pair) fall into 31 classes, the diffraction groups, each of which is produced by a small number of (point group, zone axis) combinations. Reading the tables backwards is the standard route to a point group from CBED, and it is the first step of a space-group determination, before the Gjønnes-Moodie lines are used to find the glide planes and screw axes.
This page does not store the tables. It derives them. For a chosen point group the 3 × 3 matrices of the group are generated from two or three generators, and every operation is sorted by what it does to the zone axis: an operation that leaves the axis unchanged acts on the pattern as an ordinary two-dimensional rotation or mirror, an operation that reverses it acts through the reciprocity theorem as the same in-plane operation combined with the R operation (a 180° rotation of each disc about its own centre), and every other operation plays no part. The set of in-plane operations, with or without their R tags, is the diffraction group; the observable symmetries follow from it by Buxton's rules, and the names follow the standard notation, with the subscript R marking the tagged operations. Running the derivation over all 32 point groups and all zone axes with indices up to 3 produces exactly 31 distinct groups, which is the check that the derivation is right.
From the crystal: pick the point group, set the axial ratios if the crystal is not cubic, and enter the zone axis. The readout gives the diffraction group and what to expect on the screen; the table below it lists every zone-axis class of the point group with a representative axis, which is the list to consult when choosing which zone axes to record for a point-group determination. The pattern needs to be recorded with the zone axis exactly parallel to the beam, with a disc size that shows the fine structure, and at a thickness at which the dynamical detail (the HOLZ lines, the fine structure in the discs) is visible, because the projection symmetry of a thin crystal is always higher than the true one.
From the pattern: record the bright-field symmetry (the symmetry of the fine structure within the central disc alone), the whole-pattern symmetry (the arrangement and inner detail of all discs, including the HOLZ ring), and, if the pair is available, the ±G relation, and the page lists every combination of point group and zone-axis class that would produce them. A second zone axis of the same crystal usually cuts the list to one point group; the crystal-system filter applies whatever is already known from the lattice.
The page concerns point groups. The diffraction group cannot distinguish a space group's translational elements: the Gjønnes-Moodie dynamic-absence lines that reveal glide planes and screw axes are a separate analysis, and the SAED simulator on this site shows the kinematically forbidden reflections that they run through. The trigonal, hexagonal and tetragonal groups that exist in two settings (32 and 312, 3m and 31m, −3m and −31m, −6m2 and −62m, −42m and −4m2) are handled by the setting switch, which rotates the zone axis by 30° or 45° about c; the monoclinic groups use the b-unique setting with the 2-fold axis along [010]. The zone-axis classes are sampled up to index 3; a higher-index axis always falls in one of the same classes, since it is either on a symmetry element or in a general position, but the representative shown for a class is the lowest-index member found in the sampling. The observable symmetries are those of Table 2 of Buxton et al., reproduced in chapter 21 of Williams and Carter: whole pattern, bright field, dark field of a general reflection, and the ±G relation; the special-position dark-field symmetries, which depend on where the reflection sits relative to the symmetry elements, are not tabulated here.
The two hands of a chiral crystal (the enantiomorphic space-group pairs, P31 and P32 for example) share a point group and so share every diffraction group; CBED separates them only through the dynamical intensities of the HOLZ reflections, which this page does not compute.
@misc{tripathy2026cbedsymmetrytables,
author = {Tripathy, Manisha},
title = {CBED Symmetry Tables},
year = {2026},
howpublished = {\url{https://untetheredatom.com/tem/cbed-symmetry-tables}},
note = {Interactive web tool}
}