Abstract
Here we introduce the concept of space groups, Sohncke groups, crystal classes, Bravais classes, site symmetry, Wyckoff positions, and orbits.
The biggest conceptual change over the last 100 years in the way physicists think about the world is symmetry.
– Lawrence M. Krauss
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Notes
- 1.
This is the only example of symmetry nomenclature being different for two- and -three-dimensional notations.
- 2.
A Euclidean normalizer (also known as a Cheshire group) is the group of motions that maps the pattern of symmetry elements of a space group onto itself. These are not necessarily plane/space groups themselves.
- 3.
Previously called Bravais flocks.
- 4.
See p. 64.
- 5.
An affine group is defined by transformations in space that leave straight lines as straight lines.
- 6.
There are actually 24 possible orientations, but not all are unique. The handedness of the coordinate system doesn’t change when two axes are reversed; so, for example, settings abc, \( \boldsymbol{a}\overline{\boldsymbol{b}}\overline{\boldsymbol{c}}=\overline{\boldsymbol{a}}\boldsymbol{bc} \), \( \overline{\boldsymbol{a}}\boldsymbol{b}\overline{\boldsymbol{c}}=\boldsymbol{a}\overline{\boldsymbol{b}}\boldsymbol{c} \), and \( \overline{\boldsymbol{a}}\overline{\boldsymbol{b}}\boldsymbol{c}=\boldsymbol{ab}\overline{\boldsymbol{c}} \) are all equivalent.
- 7.
The unique axis a (third setting) is not actually used in the International Tables, and the ambiguous terms “first setting,” “second setting,” and “third setting” have now all largely been dropped in favor of the more specific unique axis c, unique axis b, and unique axis a terms.
- 8.
The Hermann-Mauguin notation for the other five monoclinic space groups, none of which contain either centring or glide planes, is independent of the cell choice.
- 9.
The first English collection of equivalent positions in space groups appeared in “The analytical expression of the results of the theory of space groups” by Ralph Walter Graystone Wyckoff (1897–1994), first published by the Carnegie Institution of Washington in 1922.
- 10.
In such case, we run out of letters a-z, and so the general position is the 8α.
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Review Questions
Review Questions
-
1.
Briefly define/explain the term symmorphic space group.
-
2.
For each space group listed below, identify the point group and Bravais lattice:
P6122
Cmc21
\( Fd\overline{3} \)
P21/c
I41md
\( R\overline{3}c \)
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3.
Many space group notations are dependent upon the orientation of the axes which one chooses to describe a structure. In such cases, there is a “standard” setting, but potentially many other “non-standard” ones as well. In the example of Pban (#50), if the axes were reoriented such that abc ➔ cab, use what you know about symmetry operations and space-group notation to determine what the new notation would be, and explain your answer.
-
4.
The standard setting for space group No. 62 is Pnma. If the coordinate system were reoriented such that abc ➔ bca, what would the new notation be?
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5.
According to the International Tables for Crystallography, the site symmetry of atoms on 1a sites of space group P3 should be “3…”; however, inserting just a single atom into the 1a site of this space group results in a site point symmetry which is not “3…”.
-
(a)
What is the resultant site symmetry (remember to think in three dimensions)? Conjecture why it is not the same as what is listed. Use a diagram to help explain if necessary.
-
(b)
Briefly explain why one would not describe a crystal in which only the 1a site is occupied in this space group. What space group would be more correct?
-
(a)
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6.
While the cubic lattice must have fourfold rotational symmetry, the only requirement for cubic symmetry in a crystal is the simultaneous existence of four triads along <111>. Fourfold symmetry is not a requirement. Using standard short international (Hermann-Mauguin) symbols and space-group numbers, list three examples of cubic space groups which lack fourfold rotational symmetry – and whose point groups also lack fourfold symmetry. Give at least two examples of actual cubic compounds which crystallize in each of these space groups at ambient conditions. The compounds should have no more than three components (atomic species).
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7.
Compare and contrast the three space groups P3 (No. 143), R3 (No. 146), and P6 (No. 168) in terms of their lattices and crystal systems. For each of the three cases, determine the number of equivalent general positions (x,y,z) (equal to the order of the group multiplied by the appropriate factor for centring) and specify the axis system used.
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Ubic, R. (2024). Space Groups. In: Crystallography and Crystal Chemistry. Springer, Cham. https://doi.org/10.1007/978-3-031-49752-0_8
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DOI: https://doi.org/10.1007/978-3-031-49752-0_8
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