Crystal Symmetry

At this point, some of you may have noticed something about the crystals that we're navigating. Namely, that these crystals always possess some sort of symmetry. For example, let's look at the crystal structure for iron (Figure 5.9.1(a)), which has atoms (i.e. a motif) on the lattice points of a cubic-I Bravais lattice.

Now, if I take away the coordinate system and orient this iron structure structure in a variety of different ways, such as those shown in Figure 5.9.1(b.)-(c.), we observe something interesting... we could not uniquely apply any set of coordinates to those structures. That is, is the direction pointing up $[100]$? $[010]$ $[00\bar{1}]$? Is the exposed crystallographic plane the $(100)$? The $(010)$? The $00\bar{1}$. It is impossible to say, because the crystal itself has some symmetry - or invariance to various operations in space. For example, for this iron crystal we can rotate it $90^{\circ}$ about the [001] direction projecting from the origin, and we reproduce the structure precisely! (This is exactly what I did between Figure 5.9.1(b) and (c)). Or, as it turns out, I could rotate the stucture $120^{\circ}$ about the $[111]$ direction projecting from the origin.

Indeed, for this iron crystal structure there are 96 different symmetry operations that, when performed on the crystals, yield atoms in the exact same positions. Some of these rotations are easier to see ($90^{\circ}$ about the [001] projecting from the origin). Some are more difficult ($120^{\circ}$ about the [111] direction projecting from the origin). And this is one of the highly symmetry and intuitive crystal to navigate. Crystal symmetry gets pretty complicated, and one can spend a lifetime studying it.

For our purposes we'll limit ourselves to identifying whether certain crystallographic directions or planes ($[uvw]$ and $(hk\ell)$) belong to the same crystallographic families of directions or planes, respectively. That is, we could ask something like: for the iron crystal in Figure 5.9.1(a), are the [100] and [010] directions equivalent? Or: do the [100] and [010] directions belong to the same family of directions for that crystal?

There is, of course a very systematic way of doing this. In some classes (upper division undergraduate and graduate-level courses in MSE, chemistry, or mathematics), we spend a few weeks constructing the foundations of group theory as applied to symmetry groups of crystal structures. That is, of course, not within the scope of an introductory MSE course. So, we'll introduce a simpler (and more limited) set of rules that works for the crystals we'll encounter in this class that will allow us to identify whether two crystallographic planes or directions belong to the same family - that is, whether these directions and planes are symmetry-equivalent.

The Importance of Symmetry-equivalence

Before we show the rules, let's briefly consider why this is important.

Material Anisotropy: Perfect crystals, like the ones we've been describing, are important in all sorts of fields: semiconductors, high-temperature structure materials, turbine blades, catalysis, optics, etc. For example, in turbine blades used for energy production (88% of energy we use is produced using a high-temperature turbine of some sort) we use special materials called nickel superalloys, which have the crystal structure shown in Figure 5.9.1. As such, the materials have different properties along different directions: they can be much stiffer along the $[111]$ or $[\bar{1}11]$ compared to the $[100]$ This is because the higher degree of atomic packing in the $[111]$ or $[\bar{1}11]$ directions (Figure 5.9.2)

Crystal Growth and Microstructure: Different surfaces have different reactive/energy. As such, when materials are growing/crystallizing, they prefer specific crystallographic directions. This can be seen from the morphology/microstructure of snow flakes (Figure 5.9.3)(a), which grow specifically along certain crystallographic directions. It's also important in engineering materials like aluminum, which will have dendritic structures that nucleate and grow based on the crystal symmetry (Figure 5.9.3)(b).

Catalysis, Corrosion, and Surface Reactivity: Different surfaces - which are very important features of materials - have properties that can be highly dependent on the family of planes that are exposed. For example, in catalytic converters in gas-powered vehicles, which help reduce the amount of nasty nitrogen oxides and carbon monoxide that's released into the environment. To maximize the rate of this reaction, materials scientists and chemical engineers work to control the surfaces exposed to the exhaust gas

Characterization: When we shoot X-rays or electrons at a material, we can often get a diffraction pattern due to the crystal structure. The diffraction patterns that appear are due to the crystal structure, and the symmetries we observe in these patterns are due to the symmetries within the crystal. Understanding these symmetries and crystallographic families are indispensable - without understanding these symmetries we would not be able to quantify crystals, and without understanding crystals, we can not engineer things made out of crystals. We wouldn't have semiconductors, modern medicine and vaccines, light-emitting diodes... really any modern technology.

Families of Directions and Planes

When we talk about crystallographic directions or planes that belong to the same family, we mean that they are crystallographically symmetric. For a crystal structure that has a unit cell with orthogonal bases vectors (cubic, orthorhombic, or tetragonal), we can come up with a set of simple rules to identify whether two directions or planes are within the same family. We'll show the rules first, and then show some examples.

Below, we'll introduce a flow chart to identify whether two directions are of the same family for orthorhombic crystals. At the end, we invite you to always confirm with a drawing. Understand that in a more advanced analysis, we identify a set of symmetry operations for a crystal and produce invariant directions via those operations - but that's really too much for an introductory class. This will help us understand if the structure (and therefore properties) of a pair or group of directions or planes are the same.

Let's start with the flowchart in Figure 5.9.6.

When you are done applying the flowchart, choose one of the indices and put angled brackets around it to define the family of directions, e.g. $\langle 111\rangle$. If you are finding family of planes, put curly brackets around one of the indices: $\{111\}$.

Cubic-P: $a = b = c$, $\alpha = \beta = \gamma$

1. Let's test the pair of directions $[110]$ and $[111]$:

   • Question 1: Is the crystal system orthogonal?: Yes, $\alpha = \beta = \gamma$
   • Question 2: Are the indices $[u_1 v_1 w_1]$ and $[u_2 v_2 w_2]$ permutations of each other, or their negatives? No: you cannot permute $[110]$ (or its negatives) to get $[111]$ $\rightarrow$ Not in the same family. See Figure 5.9.7(a.).

2. Let's test the pair of directions $[121]$ and $[11\bar{2}]$:

   • Question 1: Is the crystal system orthogonal?: Yes, $\alpha = \beta = \gamma$
   • Question 2: Are the indices $[u_1 v_1 w_1]$ and $[u_2 v_2 w_2]$ permutations of each other, or their negatives? Yes: you can permute $[121]$ (or its negatives) to get $[11\bar{2}]$.
   • Question 3: Are the "exchanged" values $[u_1 v_1 w_1]$ and $[u_2 v_2 w_2]$ associated with identical unit cell lengths? Yes, $v_1 \rightarrow \bar{w}_2$ and $w_1 \rightarrow v_2$, respectively. Note, the addition of the overbar does not matter. $\rightarrow$ They are in the same family! See Figure 5.9.7(b.).

Families of Crystallographic Planes

Families for crystallographic planes are similar to those of crystallographic directions, but now instead of thinking of the journey along a direction, you consider the pattern on the plane. Also, we label a family of planes as ${hk\ell}$ instead of $\langle uvw \rangle$.

Strictly, families of planes are related by the symmetry of the lattice, but here we can simply check if they have some permutation of the index of the family and that the pattern of atoms or lattice points in the plane is identical to the others in the same family.

For example, let's look at the two planes in Figure 5.9.8. First, let's identify the planes using the axial intercepts and Eq. 5.8.1.

• The plane in (a.) intercepts the axes at $qrs = 1\infty1$. The Miller indices are therefore

\begin{align} h &= \frac{1}{n}\frac{1}{1}\\ k &= \frac{1}{n}\frac{1}{\infty}\\ \ell &= \frac{1}{n}\frac{1}{1}\\ \end{align}

Where $n = 1$. This is the $(101)$ plane.

• The plane in (b.) intercepts the axes $qrs = 11\infty$. The Miller indices are therefore

\begin{align} h &= \frac{1}{n}\frac{1}{1}\\ k &= \frac{1}{n}\frac{1}{1}\\ \ell &= \frac{1}{n}\frac{1}{\infty}\\ \end{align}

Where $n = 1$. This is the $(110)$ plane.

Clearly, the $(110)$ plane will belong to the ${110}$ family, but does the $(101)$ plane belong to the ${110}$ family? The possible members of the ${110}$ family are the permutations of the $hkl$ indices:

\begin{array}{ccc} \hline\hline (110) & (101) & (011)\\ (\bar{1}10) & (\bar{1}01) & (0\bar{1}1)\\ (1\bar{1}0) & (10\bar{1}) & (01\bar{1})\\ (\bar{1}\bar{1}0) & (\bar{1}0\bar{1}) & (0\bar{1}\bar{1})\\ \hline \end{array}

Since the system is cubic, all of these permutations are of the same family. Try drawing lattice points on any of them. You will see that they are all the same. For our two planes in Figure 5.9.8(a.) and (b.), the arrangement of lattice points looks like shown in Figure 5.9.8(c) and (d). Now try Exercise 5.9.2 to try a different Miller plane.