4.1 Crystallographic Planes and Directions

1. Given an illustration or description of a crystallographic direction in a cubic unit cell, correctly define the direction indices for that direction, or vice-versa (draw the vector)
2. Given an illustration or description of a crystallographic plane in a cubic unit cell, correctly define the Miller indices for that plane, or vice-versa (draw the plane)
3. Explain the concepts of families of planes and families of directions
4. Correctly describe any unique direction or plane, or any family of directions or planes, using the appropriate notation, including correct enclosures and all other conventions
5. Demonstrate how Bragg's Law is derived from the treatment of x-ray diffraction as simple reflection from adjacent parallel crystallographic planes
6. Utilize Bragg's Law to demonstrate the relationship between order of reflection, x-ray wavelength, interplanar spacing, and the angle of diffraction, theta

We've now spent a fair amount of time learning a number of important crystal structures.  In the process, we've hinted the idea of describing important directions and planes within crystals.  For example, we discussed the close-packed directions in BCC (cube diagonals) or in FCC (face diagonals).  We have even described the close-packed planes in FCC and HCP crystals.  It will become essential later when considering properties of materials, whether mechanical, electrical, optical, or other to be able to understand and describe specific crystallographic directions.  

Quickly, it becomes difficult and highly inefficient to describe directions using words, so we must develop more efficient scheme.  For example, we can describe the close-packed directions in FCC as the face diagonals, but how can we differentiate between two unique face diagonals, as illustrated by the red and blue vectors in Figure 1?  

The answer is also given in Figure 1, of course, with the specific numbers, or indices given to each vector.  

What about planes?  Take a look at the example in Figure 2.  

This is one of my all time favourite planes.  It is the close-packed plane in FCC.  How would we describe this plane and distinguish it from the planes in Figure 3?  

 Again, the answers are given in these figures with the specific indices, but let's look at the process of naming directions and planes systematically.

With crystallographic directions and planes I like to be as systematic as possible.  This is especially important when you are just learning this topic.  Later, you may find that you can take some shortcuts or identify and draw some vectors intuitively, but let's begin with a systematic approach.

1. A vector may be translated if necessary to clarify the problem.  Or, a new origin may be defined.  Note that these two statements are really identical in this context.
2. Determine the projection of the vector onto each of the x, y, and z axes.  Or, determine the point coordinates of the head of the vector starting at the origin.
3. "Housekeeping" - this is just some organization to ensure that the data is presented in a standard manner.

a) Negative signs are moved above the number and are referred to as "bar", as in $\lbrack11\overline1\rbrack$ 

b) Convert to lowest integer values

c) No commas

d) Indices are enclosed in square brackets

Let's try a few examples!  Consider the two vectors drawn in Figure 4. 

Let's take a look at the red vector.  

Following the steps that we have established we have:

1. There is no need to translate the vector or re-establish a new origin.  The conventional origin is very convenient in this example.
2. The projection onto the x, y, and z axes are 0, 1, and 0.5.  These are also the values of the point coordinates of the head of the vector.
3. We can't have that nasty fraction in our final indices, so we multiply all three numbers by 2, yielding 0, 2, 1.  To present this as a vector we must enclose in square brackets and drop the commas, so we have $\lbrack021\rbrack$.

Let's take a look at the blue vector.  Following the steps that we have established we have:

1. In this case it makes sense to either translate the vector or re-establish a new origin.  In Figure 4 both have been done.
2. The projection onto the x, y, and z axes are 1, 0, and 0.  This is the same, regardless of whether we translate the vector or not.  If we define the new origin at the start of the vector, labelled O' in the figure, we see that these are also the values of the point coordinates of the head of the vector.
3. We don't have any fractions to clear.  To present this as a vector we must enclose the numbers in square brackets and drop the commas, so we have $\lbrack100\rbrack$.

How about another example?  Take a look at Figure 5.

What's going on with this vector?  Where's it pointing?  To make this easier to visualize, let's translate it back into the unit cell.  This is illustrated in Figure 6. 

In addition to translating the vector back into the unit cell, I've also defined a new origin, called it $O'''$, and indicated the point coordinates of the head of the vector, relative to this new origin.  Now we can use the point coordinates directly as the projections of this vector onto each axis, giving us -0.5, -0.5, 1.  Finally, the housekeeping step requires that we multiply by 2 to clear the fractions and enclose in square brackets with no commas for a vector of $\lbrack\overline1\overline12\rbrack$.

Sometimes we may want to refer to a group of vectors, like the face-diagonals, for example, without the need to specify a specific one.  In this case, we refer to a so-called family of directions.  Vectors are in the same family of directions if they have the same atom types at the same positions along the vector.  For example, all of the face-diagonals in an FCC crystal will be close-packed.  That is, the atoms are all touching along the face diagonals.  To indicate that we are referring to a family of directions we use a different enclosure: instead of square brackets we use angle brackets.  For example, the face-diagonals would be referred to as the $<011>$ family of directions.  There isn't a clear convention for whether to put the zeroes on the left or the right.  You'll find both used, so it wouldn't be wrong to also say that the face-diagonals are the $<110>$ family of directions.  Personally, I prefer to put the indices in increasing order, so I prefer the former approach over the latter, but you can decide what you like.  

While we're on the topic of decisions, you'll need to decide on how you want to speak these indices.  For example, how do you speak the $\lbrack1\overline11\rbrack$ vector?  Well, you have two options; one is to say, "one, one-bar, one," while the other way is, "one, bar-one, one."  You see, you may choose to speak the bar either just before or just after the number that it is associated with.  The important thing is that you say the bar and its number quickly, one after another, and leave a longer space between them and the other indices.  I prefer to say, "one-bar," but I won't judge you if you want to say, "bar-one."  I know some really smart people who say, "bar-one."  The other decision you'll need to make is whether you want to say, "zero," or "oh," as in, "one, one-bar, oh," ($\lbrack1\overline10\rbrack$).  You'll hear both.  Saying "oh," has a certain hipster vibe, doesn't it?

So, now we need to discuss the naming of planes.  Let's get right into things with the steps that we take for naming crystallographic planes.  The numbers that we use to name planes are known as Miller Indices.  Here's the process:

1. The plane must either be translated so that the origin does not reside on the plane, \emph{or} the origin must be redefined somewhere that is not on the plane.
2. Starting at the origin, determine the distance along each axis that must be taken to intercept the plane.  
3. Take the reciprocal of all of the indices from step 2.
4. Housekeeping
1. Negative signs are moved above the number and are referred to as bar
2. No commas
3. Indices are enclosed in parentheses

Let's consider an example.  How about the plane in Figure 7?

Let's follow the steps:

1. The conventional origin (back-bottom-left) is on the plane so we need to define a new origin or move the plane so that the origin is not on the plane.  In this case I think it is easiest to just define a new origin, $O'$ at the back-bottom-right corner.
2. From $O'$ we can try travelling either in the positive or negative x direction but we never touch the plane.  This is because the plane is parallel to the x axis.  We could say that the intercept is at infinity.  In the y-direction, we need to travel one lattice parameter in the negative y-direction, starting from $O'$ before we hit the plane.  Finally, in the z-direction we need only travel half of a lattice parameter in the positive direction before we hit the plane.  So, we have intercepts of $\infty,-1,0.5$ 
3. We take the reciprocals of the numbers from step 2, giving us $0,-1,2.$  Notice how that nasty infinity symbol has disappeared?
4. We drop the commas and enclose in parentheses, giving $(0\overline12)$. 

Just for kicks, let's see what would happen if we started with a different origin, as in Figure 8.

From this origin we must travel $\infty,1,and-0.5.$  We take the reciprocal of these numbers and enclose them in parentheses, giving $(01\overline2)$.  You'll likely notice that the Miller Indices determined this time are the negative multiple of the ones determined previously.  This is fine.  In fact, both planes are identical.  There is no significance to sign on a plane.  As a convention, if multiplying the indices of a plane by negative one will decrease the number of negative signs, we do so, to simplify things.  For example, instead of writing $(\overline1\overline12)$ we would write $(11\overline2)$.

Just like we had families of directions, we have families of planes.  To indicate that we are referring to a family of planes, we enclose the Miller indices in curly brackets.  For example, the family of planes defining the faces of the cubic unit cell are the $\left\{001\right\}$ family of planes, and the close-packed planes in the FCC crystal structure are the $\left\{111\right\}$ family of planes.

An important experimental technique that we can explore now that we have some knowledge of crystallographic planes is x-ray diffraction, or XRD.  In XRD, an x-ray source shines radiation on a sample and an x-ray detector detects the intensity, or brightness of the radiation that comes back off the sample, as shown in Figure 9.  

An interesting (read:frustrating when you are learning this) artifact of history is that in the early days of XRD the machines would read out twice the angle theta and so, to this day, XRD patterns are always plotted against $2\theta$.  

You'll notice that at a few specific angles, there are peaks in the intensity of the radiation coming off the sample.  The x-ray waves are actually diffracted by the atomic spacing as both the atomic scale and the wavelength are close to $10^{-10}$ m, however, we can model the interaction as if it was simple reflection.  

Take a look at Figure 10.  X-ray radiation comes at the sample and some is reflected off the top crystallographic plane of atoms, while some is reflected off the next parallel plane down.  The spacing between these planes is called the interplanar spacing and we denote it with $d$, since it is a distance and use the letters h, k, and l, to refer to the Miller indices in the x, y, and z positions, giving the interplanar spacing the name $d_{hkl}$.  Geometrically, we see that the distance traveled by beam 2 before it hits the detector is equal to $\overline{AB}+\overline{BC}$ and since these two line segments are equal, the extra distance travelled by beam 2 is equal to $2\overline{AB}=2d_{hkl}\sin\theta$.  In order to get peaks in the XRD pattern we need beam 1 and beam 2 to be in-phase with each other, that is, beam 2 needs to be pulled back by a distance equal to the wavelength, or some integer multiple of it, $n\lambda$.  If we put these two together we end up with the equation that governs XRD, known as Bragg's Law:

 $n\lambda=2d_{hkl}\sin\theta$