Navigating 3D Structures - Crystallographic Points

Now that we have a good idea about what types of patterns can arise in crystal structures through Bravais lattices and motifs, we'll want to start navigating those structures in order to describe the positions of atoms, the vectors (or directions) between atoms, and the planes that atoms occupy. To do this, we have to develop a bit of mathematics in the navigation of crystal structures.

Crystallographic Points in a Unit Cell

We've conveniently described the basic unit of a crystal using a unit cell with lattice parameters $a$, $b$, $c$, $\alpha$, $\beta$, and $\gamma$ (Figure 5.6.1). This is great because we can develop a notation that allows us to concisely describe any point within the unit cell. This is crystallographic point notation and it is used to describe the positions of atoms in unit cells, as well as other sites of interest such as interstitial sites (more on that later).

To define any position in a unit cell with respect to the origin (which is conventionally defined as the furthest corner away from the viewer on the bottom plane - indicated with a black dot in Figure 5.6.1), we can define fractional positions $qrs$ (or alternatively $q.r.s$). This is best shown graphically in Figure 5.6.1.

• $qa$ is the absolute position of the point along the $\textbf{a}$ lattice direction.
• $rb$ is the absolute position of the point along the $\textbf{b}$ lattice direction.
• $sc$ is the absolute position of the point along the $\textbf{c}$ lattice direction.

This means that $qrs$ defines the position within the unit cell. For example, if I have a point at the origin, it is positioned as $0a$, $0b$, and $0c$. In our crystallographic notation, that is position $qrs = 000$. If we have a point at the body-centered position - the middle of the unit cell, it is at absolute position of $\frac{1}{2}a$, $\frac{1}{2}b$, and $\frac{1}{2}c$, or $qrs = \frac{1}{2}\frac{1}{2}\frac{1}{2}$. This point is shown explicitly in Figure 5.6.1.

This notation is great because it is concise and generally applicable to any crystal system. Let's practice with the same Bravais lattice we were working on in the previous page in Exercise 5.6.1.

We'll want to be able to work with any arbitrary point $qrs$, but there are some that are more important because the show up in the Bravais lattices themselves. These are the

Corner positions:

$$ \begin{array}{c c c c} 000 & 100 & 010 & 001\\ 101 & 110 & 110 & 101 \end{array} $$

Body-centered Position

$$ \begin{array}{c } \frac{1}{2}\frac{1}{2}\frac{1}{2}\\ \end{array} $$

Face-centered Positions

$$ \begin{array}{c c c} \frac{1}{2}\frac{1}{2}0 & \frac{1}{2}0\frac{1}{2} & 0\frac{1}{2}\frac{1}{2}\\ 1\frac{1}{2}\frac{1}{2} & \frac{1}{2}1\frac{1}{2} & \frac{1}{2}\frac{1}{2}1 \end{array} $$

Before we move on - you might be wondering if the $q$, $r$, and $s$ positions need to be between 0 and 1. The answer is no, not explicitly. Most of the time, we're working on the level of the unit cell, so we only need the fractional positions to describe locations of atoms. However, we'll sometimes want to visualize atoms in adjacent unit cells to help us understand a structure. As such, we might look at (e.g.) an atom positioned at $\frac{1}{2}\frac{3}{2}\frac{1}{2}$. More on this in Section 5.11.