We use miller indices to describe directions in unit cell structures when we’re talking about crystal structures when we’re dealing with a rectilinear shape. The cartesian coordinate system is sufficient for describing this and we can use just three coordinates when we start talking about hexagonal close-packed cell structures, though it becomes difficult to use integer values when describing those vector unit directions because of the shape of hexagons. So instead a four four axis coordinate system is used.
So let’s take a look at a hcp structure, so you can see here that we’ve got a vector direction drawn inside of our box and that’s what we’re trying to we’re going to try and calculate here. The the coordinate system that we’ve drawn here has three axes on: what’s called the basal plane or essentially, you could sort of think of it as the ground and they’re each 120 degrees to each other, and then there is the z axis, which is perpendicular to all three planes, all three axes: okay, so what we’re going to do is to determine this first, we’ll have a three coordinate indices that will create and then we’ll convert that into four four axes! To do this, we’ll ignore a three and describe the direction of this vector using only a 1, a 2 and z. So let’s go and do that?
Okay, so you can see the vector that’s been drawn in in this drawing and we have to travel only in the a1 and a2 directions and describe the direction from the tail to the tip. Okay- and there are two ways: i could go about this, but it makes sense to me at least to start going in the a one direction and then go in the a2 direction. Okay and to achieve this i’m going to go this way like this right out to the tip there and then i’ll go this way: okay, so i traveled in the negative a1 direction and then i traveled in the negative, a 2 direction and then looking over here. This is not quite accurately drawn, but this tip of this vector is supposed to be halfway up, see: okay, so notice that i’m talking about this in terms of its unit lengths where this is a the length of one of the sides- and this is c and with hexagonal close packed structure- a is not equal to c. Okay. So let’s write this in terms of its unit lengths first, so we went negative a and then we went negative a and then we went c over 2, so negative, a negative, a c over 2, okay and now we’ll convert this into unit values. So remember that a is one just like. If we were doing a unit circle, we consider r to be one! That’s the same principle applied here, so this is negative, one negative one and one half okay and now this is an important step before we move on to plugging this into these equations?
We need to reduce this to its simplest, integer form.
Okay. So this ends up being negative.
Two, because i have to multiply everything by two negative two and then this will become one okay, and this now we’re going to use this to convert to a four axis or a miller. Bravais indices, that’s how it’s pronounced, and so the final vector directions will be: u v, t and w. So we’ll equate this with: u prime v, prime and w prime and then we’ll plug into these equations! Also, don’t forget that when we’re dealing with a direction like a vector that we enclose our answers in brackets and typically these commas will not be in there either i’ll get rid of those at the end and if we’re doing a plane, then we would use regular old parentheses. Okay, so let’s just go through and start solving for each of the directions!
Now? u equals one-third oops one third of two times: u, prime, that’s 2 times negative 2, that’s negative: 4 minus v, prime minus negative, two: okay and that’s 4, plus 2 or minus 4, plus 2, that’s minus 2, and so this is negative? 2/3 v, prime or v equals 1/3 of 2 times v prime, which is exactly the same, and if you do this out, you’ll see that you get the exact same values? So this one will also equal negative 2/3. Okay t equals the negative of u plus v! Now that’s this u and this v, not these ones up here. So this is minus 2/3, minus 2/3 and that equals negative 4/3 in the parentheses times? The negative is positive, so this is 4/3 and the final one is the easiest, because w just equals w prime and so w equals 1. Okay!
Now, let’s collect this together in our square brackets negative 2/3, negative, 2/3, 4/3 and 1. Now we need to again reduce this into its lowest integer form and we’re lucky, because we have the same denominator across the board! So we’ll just multiply everything by 3.
This will become negative? 2 negative 2 3 4 12?
This is 4 and this is 3. But we’re not done yet because remember that to put this in its correct form, negative values just have a two bar over them bar over them, so 2 bar 2 bar for 3, and that’s our answer! Ok, so that process i should work for you every time. .