# Whats the difference between unit cell and motif?

The definition in my material is that motif and unit cells are both a repeating unit, but whereas motif is with respect to 2D lattice the unit cell is with respect to 3D lattice. But I can't find anything online with respect to that.

Conceptually, the unit cell is the larger object, the (if relating to 3D crystals), the parallelepiped containing the crystallographic motif. It is the unit cell which, by mere translation along $$\vec{a}$$, $$\vec{b}$$, and $$\vec{c}$$ builds the crystal. In an analogy, unit cells stash like shoe boxes next to / on top of each other.

In addition to the outer shape and apparent symmetry of the unit cell (the Bravais classes; i.e., triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, or cubic) the unit cell contains symmetry operators in its inner (the proper rotation axes, mirror planes, centres of inversion, screw axes, rotoreflection axes, glide planes) which act on the crystallographic motif. In an analogy, these symmetry operators are instructions how to copy and paste the motif within the unit cell. The motif itself is a smallest representative design pattern within the elected unit cell.

As an example, the illustration below shows how the $$c$$-glide plane relates two molecules of maleic acid in respect of their position within the unit cell, as well as in respect to their orientation relative to each other:

(credit: Hanson, R. M.; Jmol – a paradigm shift in crystallographic visualization in J. Appl. Cryst. 43, 2010, 1250-1260; doi: 10.1107/S0021889810030256 (open access))

Depending on the the space group symmetry within the unit cell, the position and the symmetry of the molecules within the unit cell, you may encounter motifs which on first inspection might look like incomplete. For example, if a centre of inversion (molecular level) coincides with a centre of inversion (unit cell level). The molecule then still is fully described in its geometry, because of the exact completion by the symmetry operators. As an example, see the following triptych:

(entry COD 2102215)

Here, the left image depicts the crystallographic motif alone. In the centre, you recognize the triazine molecule, «complete» as if you would draw it in a reaction scheme; but it has been «completed» by applying the symmetry operator of the unit cell. In the right image, using the same perspective (i.e., again the same direction of observation) you see the complete unit cell when all symmetry operators are applied. It depends on the software used how (if possible all three of) these levels are visualized with ease.

Primary reference for the crystal structure: Fridman, N.; Kapon, M.; Sheynin, Y.; Kaftory, M.; Different packing in three polymorphs of 2,4,6-trimethoxy-1,3,5-triazine in Acta Cryst. B60, 2004, 97-102; doi: 10.1107/S0108768103026284.

• Thanks for the answer. I love how you went in detail to explain the relation between the two (in a depth much better than my textbook too), but could you add a short definition of the two as well in the beginning? Jun 2, 2021 at 11:39
• I can't seem to find the definition of a motif on the net that I can compare with the definition of a unit cell. Thanks once again. Jun 2, 2021 at 11:41
• Unlike unit cell, motif is not a thing that can be defined rigorously. Jun 2, 2021 at 11:43
• Unit cell is a volume that can build the entire crystal if only the lattice translations are applied. Asymmetric unit cell is a set of atoms that can build the entire crystal if all space group symmetries are applied. Motif usually means, as far as I have seen, a set of atoms that can build the entire crystal if only the lattice translations are applied. So I do not quite agree with what @Buttonwood has written. Maybe motif is not always used with the same meaning. It is not so common as the other two concepts. Jun 2, 2021 at 12:03
• Here is a reference that is consistent with the meaning of motif that I understand: doitpoms.ac.uk/tlplib/crystallography3/structure.php Jun 2, 2021 at 12:07