Navigating 3D Structures - Crystallographic Planes

We've done crystallographic points (0-dimentional geometric entities) and directions (1D geometric entities). We'll also do crystallographic (or lattice) planes: 2D geometric entities. While similar to points and lines in that we'll describe them using a series of indices (here, we give them the term Miller indices, $h$, $k$, and $\ell$), planes are a bit more complex than points or lines. Miller indices actually represent a stack of parallel planes that are separated by some distance $d_{hk\ell}$, as shown in Figure 5.8.1. This representation is extremely useful to materials scientists and other folks who study and use crystals because it helps communicates a number of important things such as:

1. The periodicity of the atoms in the crystal in different directions.
2. How atoms are positioned in planes within the 3D crystal.
3. How light (especially X-rays) scatter from layers of atoms in the crystal.
4. How chemicals adsorb or react on a surface.
5. How planes of atoms may cleave during fracture.
6. How atoms slide across each other during deformation of materials.
7. How a complicated thing called the reciprocal lattice is constructed. This reciprocal lattice is the Fourier transform of the real-space crystal.

All of these topics are important in materials science and engineering - but the last is critical. The reciprocal lattice is a powerful construction that is fundamental to the way that waves travel through crystalline solids - and is therefore central to our understanding of electronic behavior - semiconductor behavior in general.

Construction of the reciprocal lattice is a topic for another time, but it is important to know a bit about it as we learn about lattice planes because there is a step in the mathematical derivation of the Miller indices that necessitates taking a reciprocal. While we won't be exploring reciprocal space in this class - the process for determining the Miller indices for lattice planes often confuses students due to this step. We'll use Miller indices because they are a cornerstone of crystallography, and it's just important to know why.

While Miller indices strictly defines a set of parallel planes in space, we're going to simply this by one step. We'll develop the algorithm - much like we did for lattice directions - to derive a lattice plane, but we'll only look at one plane. We won't be concerned with the periodicity of the set of planes like we show in Figure 5.8.1, we'll be more interested in which atoms the plane passes through in the unit cell. Please understand that this is strictly the wrong formulation to describe lattice planes. This formation will, however, allow us to spend less time on mathematical construction of Miller planes and more time on their application. For a more accurate approach necessary for X-ray or electron diffraction and solid-state physics, please see (e.g.) Hammond, The Basics of Crystallography and Diffraction, Ch. 5 and 6.

Let's say we want to describe the plane in Figure 5.8.1 that passes through the atoms located at $100$, $110$, $101$, and $111$. In materials science we use Miller indices to describe this plane. The process goes as:

1. Read off the intercepts of the plane with the axes in terms of fractional coordinates $qrs$.
2. Take the reciprocals of the intercepts.
3. Reduce to smallest integer values. (This will work for our purpose in this course, but in precise crystallography practice this isn't acceptable).
4. Enclose in parentheses.
5. Put a bar above any negative values and remove the minus sign

For the example we describe:

1. The intercepts with the $x$, $y$, and $z$ axis are $q = 1$, $r = \infty$ and $s = \infty$, respectively. What does it mean to have an intercept at $\infty$? Imagine that the plane we're describing were instead tilted very slightly towards the $c$ axis. In that case, we would intercept the $c$-axis at some astronomically large value of $s$. In the limit that the tilt is infinitesimal, the intersect would essentially be at an "infinite" position along the $c$-axis. So, we say that the intercept is at $s = \infty$ - or it never intercepts.
2. Next, we take the reciprocals: $$ \begin{align} q &\rightarrow \frac{1}{q} = \frac{1}{1} = 1\\ r &\rightarrow \frac{1}{r} = \frac{1}{\infty} = 0\\ s &\rightarrow \frac{1}{s} = \frac{1}{\infty} = 0\\ \end{align} $$
3. We then reduce to the smallest integer value ( no action needed in this example).
4. And enclose in parentheses: $(100)$

In short:

Where $n$ is equal to 1 and the subscript ($x$, $y$, $z$) denotes the axial intercept.

Caveats

• Sometimes it is necessary to draw an additional unit cell in order to see where the intercept is. For example, our algorithm reveals that the plane $(11\bar{1})$ intercepts the $x$, $y$, and $z$ axes at $q = 1$, $r=1$ and $s = -1$. To see this easily, you can draw another unit cell translated by $\mathbf{T} = 1\mathbf{b}$.
• Because we are taking the reciprocals of the $x$, $y$, and $z$ intercepts, we do not try to derive Miller indices for a plane that passes through the origin (indeed all planes can pass through the origin). Instead, move the origin to an identical lattice site on another corner of the unit cell! Since Miller indices simply represent a infinite set of parallel planes, this does not affect your analysis. Your solution will be the same (try it!) and you won't get confused by trying to work with an intercept at the origin.