Crystallography (I) – Basis and Lattice

Finally, I can make a note for myself as a continuation from last post. I hope you will find this useful.

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Have you ever wondered how atom arrangement in solid? Then welcome! If you interest with this, I recommend you to study further in crystallography, a branch of physics investigating the arrangement of atoms in the crystalline solids. With the aid of transmission electron miscoscopy (TEM), the possibility of having resolution up to nanometer scale, hence seeing the real atom is realized, and even down to Angstrom scale with the help of aberration-corrected technique in the TEM. An example of how atoms look like:

The crystal structure of the mineral cordierite taken from approximately 200 Angstrom thick using high resolution TEM. Source: A. Putnis, Introduction to Mineral Sciences, Cambridge University Press, 1992 - frontispiece, available from http://www.doitpoms.ac.uk/tlplib/crystallography3/intro.php
Figure 1. The crystal structure of the mineral cordierite taken from approximately 200 Angstrom thick using high resolution TEM. Source: A. Putnis, Introduction to Mineral Sciences, Cambridge University Press, 1992 – frontispiece, available from http://www.doitpoms.ac.uk/tlplib/crystallography3/intro.php

The atoms are represented as a white spots arranged in 6-fold rings, while the black area corresponds to hollow channels through the structure. The distance between the black spots is approximately 9.7 Angstrom. If we take only one 6-fold rings, compare it with its vicinity first, and observe throughly the whole figure, do you notice something? Yes, It looks like all the atoms have the same size and they repeat themselves in regular manner, where, in this case in x-and y-direction. Of course, when it comes to bulk structure, the arrangement of atoms will have the same behaviour in x-, y- and z.

Here we deal with two important concept. First concept is what we call translational symmetry, defined as the invariance of a system under any translation, as it is illustrated in Figure 2. With this characteristics, the size of the atoms will not change as it translates from one place to another.

The shape of the circle is maintained as it experiences translation symmetry from point A to point B
Figure 2. The shape of the circle is maintained as it experiences translation symmetry from point A to point B

Generally, the number of atom which exist in here is not limited by how many atoms and whether it is identical or different atoms, as it is depicted in Figure 3. It can be only one atom (Cu), two identical atoms (Si), two different atoms (GaAs or GaN) or up to \textit{n} number of atoms (protein crystals). No matter how many atoms exist in here, as long as it follows the translational symmetry, this/these what composes/compose basis.

The number of atom and its size is maintained as it experiences translation symmetry from point A to point B
Figure 3. The number of atom and its size is maintained as it experiences translation symmetry from point A to point B

The second part is what we called as lattice. In mathematical point of view, lattice is actually specific coordinates in space. The reason why these atoms are arranged regularly is because of they sit on the lattice. By observing the lattice points, we can identify what pattern these atoms are forming. Once more, the arrangement of atoms in lattice is required to be a repeatable. So, how to recognize the lattice then? This flash animations of Lattice Point Game is good starting point. Here I put my own result (selected lattice is highlighted with yellow color) in Figure 4

Identifying brick pattern
Figure 4. Identifying brick pattern, taken from http://www.doitpoms.ac.uk/tlplib/crystallography3/lattice.php

In the same way, we can identify the lattice pattern of Figure 1, and in the Figure 5a we recognize it has triangular pattern. Furthermore, as we observe other than recognized lattice pattern carefully, we notice that there is another lattice pattern. Figure 5b shows two different arrangements of lattice, one with triangular with black dots on it (red) and one without the black dots (green). Each of them forms different orientation of triangle.

Identifying lattice pattern
Figure 5a. Identifying lattice pattern, taken from Figure 1. Source: A. Putnis, Introduction to Mineral Sciences, Cambridge University Press, 1992 – frontispiece, available from from http://www.doitpoms.ac.uk/tlplib/crystallography3/lattice.php
Two lattice patterns
Figure 5b. Two lattice patterns, taken from Figure 1. Source: A. Putnis, Introduction to Mineral Sciences, Cambridge University Press, 1992 – frontispiece, available from from http://www.doitpoms.ac.uk/tlplib/crystallography3/lattice.php

Now, by putting basis (atom/atoms) on lattice, the crystal structure is formed. As we can see from Figure 6, the crystal structure will have same size of atoms everywhere (fulfilling criteria of the translational symmetry) and they are arranged periodicaly (lattice).

Lattice + Basis=Crystal structure

Figure 6. Combination of basis and lattice forms crystal structure
Figure 6. Combination of basis and lattice forms crystal structure

That is all for the first part. The concept introduced in here is heavily influenced by these resources:

http://www.doitpoms.ac.uk/tlplib/crystallography3/lattice.php

http://www.physics.iisc.ernet.in/~aveek_bid/PH208/Lecture%202%20Crystal%20lattice.pdf

http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_1/basics/b1_3_1.html

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