Crystallography (II) – Unit Cell

So we have talked about lattice in the previous part. One should know that lattice is actually composed of the smallest possible regular (repetitive) array of unit cells, which is defined as the smallest building blocks of the lattice. First, let us inspect what the unit cell of the crystal structure from Figure 6 is:

Figure 1. Identification of unit cell from crystal structure in Figure 6 from https://andreaslm.wordpress.com/2015/06/21/crystallography-i-basis-and-lattice/
Figure 1. Identification of unit cell from crystal structure in Figure 6 from https://andreaslm.wordpress.com/2015/06/21/crystallography-i-basis-and-lattice/

Based on the repetition pattern of the arranged atoms in the lattice, we find that the unit cell in this crystal structure is hexagonal unit cell. Remember that repetition of the smallest possible pattern plays role here.

To elaborate more, how the unit cell build the lattice, let us assume we have unit cell formed as it is illustrated in the Figure 2a As we stack this unit cell, in the sense of arranging the unit cell with the closest neighboring unit cell and so on, eventually the lattice is produced, as we can see from Figure 2b.

Figure 2a. Unit cell
Figure 2a. Unit cell
Figure 2b. Lattice
Figure 2b. Lattice

Ok, now let us dive to the real example, a NaCl crystal structure, Figure 3. The gold atoms are Na and green atoms are Cl. There are three unit cells identified in this crystal structure and from our understanding of the relation between between unit cell and lattice, there must be only one unit cell composing the lattice. It means that we need to examine the possible options out of three unit cells and eliminate the other two.

Figure 3. NaCl crystal structure with its possible identified unit cells. Source: http://minerva.mlib.cnr.it/mod/book/view.php?id=269&chapterid=77
Figure 3. NaCl crystal structure with its possible identified unit cells. Source: http://minerva.mlib.cnr.it/mod/book/view.php?id=269&chapterid=77

All the three unit cells have the same possibility to arrange the NaCl crystal structure by put the respective unit cell one to each other. To determine which is the real unit cell, we need to consider:

  1. The symmetry of unit cell –> having higher symmetry is more likely to be real unit cell.
  2. The volume of unit cell –> having smaller volume is more likely to be real unit cell.

Basically, there are two different shapes of unit cell in here: A being parallelogram, B and C being square. The unit cell A has 2-fold axis of rotation, meaning that if you pick one corner and rotate it 360° (either clockwise or counterclockwise), it only fit every 180° rotation to its corresponding shape (Figure 4a). The unit cell B and C has 4-fold axis of rotation, as it firs perfectly when it is rotated 90° (Figure 4b and 4c).

Figure 4a. 2-fold rotation of parallelogram, unit cell A
Figure 4a. 2-fold rotation of parallelogram, unit cell A
Figure 4b. 4-fold rotation of square, unit cell B
Figure 4b. 4-fold rotation of square, unit cell B
Figure 4c. 4-fold rotation of square, unit cell C
Figure 4c. 4-fold rotation of square, unit cell C

We also discover that according the volume of the unit cell (remember, lattice is composed of the smallest possible pattern), the rank of compactness is unit cell C being the most dense with minimum space between atoms followed by unit cell A and unit cell B for having the widest empty space among the atoms.

Based on the two established criteria, we can conclude that unit cell C is the real unit cell composing lattice and thus the crystal structure of NaCl.

The figures and concept I made in here are based on these sources:

http://departments.kings.edu/chemlab/animation/untolat.html

http://minerva.mlib.cnr.it/mod/book/view.php?id=269&chapterid=77

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