Crystallography · Lattices & Point Groups series · Part 3 of 3
Crystallography III: Space Groups
Screw axes and glide planes: symmetry, but make it move.
A space group adds translation to a point group: screw axes and glide planes, not just rotations and mirrors. Explore them in four real structures, then use the same symmetry machinery to work out how many equivalent orientations a precipitate can form inside a matrix.
Part 3 of 3 in the Crystallography series. Previous: Crystallography II, Stereographic Projections · see also Crystallography I, Lattices & Point Groups and Interfaces & the Coincidence Site Lattice.
Chapter 1's point groups describe symmetry that leaves at least one point fixed: rotations, mirrors, inversion. A space group adds the lattice's own translations back in, and two new kinds of operation appear as a result: a screw axis is a rotation combined with a translation along the rotation axis itself (a 21 screw rotates 180° and shifts by half a lattice repeat), and a glide plane is a mirror combined with a translation within the mirror plane. Mode 1 shows both in four real structures: diamond, zinc blende, cuprite, and cementite, the last two of which are directly relevant to steels and their oxides.
Mode 2 turns the same machinery toward a practical question: given a precipitate's point group and a specific orientation relationship to its parent matrix, how many distinguishable ways can that precipitate sit inside a grain? The answer is a subgroup-index calculation, not a lookup table, verified here against two textbook results (cubic-to-tetragonal gives exactly 3 variants, cubic-to-trigonal gives exactly 4) before being opened up to any point-group pair.
Space groups & Wyckoff positions
Drag to orbit · scroll/pinch to zoom.
Why only two of the four get precisely placed elements
Diamond and cuprite are both what International Tables for Crystallography calls dual-origin-choice groups: the same structure can be legitimately described from two different, equally valid origins, and the exact screw/glide generator you'd write down depends on which one a source uses. Cementite's generators were checked directly against an independent listing before being used here (an earlier hand-derived attempt was caught failing its own internal consistency check, see the build notes), and zinc blende's are mechanically safe to construct outright, since it's a symmorphic group: its full operator list is exactly its point-group operations combined with its centering translations, no origin judgment call involved. Diamond's and cuprite's screw axes and glide planes are named here from their verified space-group symbol, honestly, without individually placing a glyph this page can't fully verify.
Precipitate shape from symmetry
Orientation relationship: direction pair 1 (matrix [UVW] ∥ precipitate [uvw])
Direction pair 2 (pins rotation about pair 1)
Two related but different numbers
The readout below reports two counts side by side because they answer slightly different questions. Variants is the number of crystallographically distinguishable orientations the precipitate itself can adopt: it accounts for the precipitate's own symmetry, so an axis with no preferred end (a 4-fold or a mirror-related pair) doesn't get double-counted. The pole multiplicity below it is a simpler, more literal quantity: how many images the precipitate's primary axis lands on under the matrix's symmetry alone, treating it as a directed arrow rather than an unsigned axis. The two agree exactly when the precipitate has no symmetry of its own that reverses that particular axis, and can differ by a clean small factor (often exactly 2) when it does, which is itself worth noticing rather than a discrepancy to explain away.
Simplifications made in this lab
Wyckoff atom positions for all four structures are drawn from published structure determinations, not re-derived from scratch here; screw axes and glide planes are individually placed in 3D only for cementite and zinc blende, for the reason explained in Mode 1. The precipitate-variant calculator assumes a fully rigid orientation relationship pinned by two direction pairs; real precipitates can also form partially-constrained ORs (only one direction fixed, the rotation about it left free), which this simpler fully-pinned case doesn't cover.
References
- Hahn, T. (ed.) International Tables for Crystallography, Volume A. International Union of Crystallography, 5th ed., 2005. (Space group formalism, Wyckoff positions, origin choices.)
- Fasiska, E. J. & Jeffrey, G. A. "On the cementite structure." Acta Crystallographica 19, 463–471 (1965); Wood, I. G. et al. "Thermal expansion and crystal structure of cementite." J. Appl. Cryst. 37, 82 (2004). (Cementite Wyckoff coordinates.)
- Kirfel, A. & Eichhorn, K. "Accurate structure analysis with synchrotron radiation: the electron density in Al₂O₃ and Cu₂O." Acta Crystallographica A46, 271–284 (1990). (Cuprite structure.)
- Bhadeshia, H. K. D. H. Worked Examples in the Geometry of Crystals, 2nd ed. Institute of Materials, 2001. (Topic outline for chapter 4 of this series; equations and structure data above are independently derived and sourced, not reproduced from this text.)
BibTeX
@misc{tripathy2026spacegroupslab,
author = {Tripathy, Manisha},
title = {Space Groups in 3D: Wyckoff Positions, Screw Axes & Glide Planes, and Precipitate Variants},
year = {2026},
howpublished = {\url{https://untetheredatom.com/crystallography/space-groups-lab}},
note = {Interactive teaching resource}
}