Determination of Space Groups
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The space group notation combines the Bravais lattice type with the symmetry elements of the point group along with the proper symbols for the translational symmetries (screw axes and glide planes) if necessary.

It is necessary to mention that in many cases some easier techniques such as HRTEM imaging, EDS, EELS, chemical, mechanical and electrical measurements can provide enough data to extract the information about the crystal structure unambiguously, or understanding the space group of the crystal doesn’t tell us anything more about its properties than we may gain from understanding its point group. We do not need to determine the space group because such process doesn’t provide any additional value. Therefore, the space group determination is only necessary if the other techniques (including point group determination) still leaves some uncertainty on understanding the nature of the crystal.

In general, the determination of the space group of a crystal should be done by analyzing systematic absent reflections or groups of reflections. The analysis of the reflections can be performed by hand but is usually done automatically by programs for time-efficiency.

For a given Laue symmetry or point group, there may be several possible space groups. The choice may be reduced through use of "systematic extinctions" in diffractions indicating glide planes or screw axes. An experienced microscopist uses “shortcuts” (known-information and knowledge about the crystal) to isolate the possible space groups.

In practice, CBED technique is very useful to determine the space groups of crystals. In this case, we often record a CBED pattern along a proper zone axis and then observe the symmetry of the whole pattern, including HOLZ details. Rotation axes are observed directly with ZOLZ and HOLZ lines in CBED patterns when the beam is aligned with the rotation axis. For instance, the space group of β-pyrochlore oxide superconductor KOs2O6 was determined by CBED technique (see page1903 for details).

Precession electron diffraction (PED) patterns not only display more diffraction spots in the zero-order Laue zone (ZOLZ) than selected-area electron diffraction (SAED) and nano-beam electron diffraction (NBED) patterns but also show some diffraction spots in the high-order Laue zones (HOLZ) which normally do not show in SAED and NBED patterns. PED also can show more diffraction spots than the conventional diffraction techniques. Therefore, PED is a very useful for the determination of the space groups of a crystal.

Even though some techniques above can be used to determine the space groups of crystals, the determination procedure is usally not simple. The diffuculties can be significantly reduced if we know some relavent characteristics of the materials such as listed in Table 3900.

Table 3900. Materials’ characteristics related to their space groups.


Impossible Possible
L-amino acid Centrosymmetric space groups Non-centrosymmetric space groups
Centrosymmetric molecule Centrosymmetric space groups Non-centrosymmetric space groups
Racemic mixture Centrosymmetric space groups Non-centrosymmetric space groups
Piezoelectric crystals Centrosymmetric space groups, 432 point groups Non-centrosymmetric space groups
Pyroelectric crystals Centrosymmetric space groups Non-centrosymmetric space groups, only the presence of a polar axi
Ferroelectric crystals Centrosymmetric space groups Non-centrosymmetric space groups
Crystals with second-harmonic generation in laser irradiation Centrosymmetric space groups, 422 point group, 622 point group Non-centrosymmetric space groups