Introduction to Materials Science and Engineering

Prof. Rajesh Prasad

Department of Applied Mechanics
Indian Institute of Technology, Delhi

Lecture - 10
Miller indices of directions

Hello, today we will start a new topic. Till now we have been discussing Bravais lattices,
14 Bravais lattices and the 7 crystal systems. We looked how symmetry helps in
classification of crystals into these schemes. We will now start a new topic: the Miller
indices of directions and planes. These are techniques or tools to specify various directions.
When we work with crystals, we need to specify or name different directions and planes
in a crystal. Miller indexing is a standard method which is being used for this purpose.

We will look at Miller indices of directions and in the next one we will take Miller indices
of planes.

(Refer Slide Time: 01:06)

Let us look at Miller indices of directions. Suppose we have a face centred cubic lattice.
All these large circle are the lattice points and we want to specify a certain direction. Let
us say this blue direction is the edge of the cube now. Of course, the cube has this edge. It
has another two horizontal edges and one vertical edge. I have picked out one of them-this
blue one and I want to give a name to it. Of course, in common language I can say that it
is the edge of the cube on it is bottom face coming out of the screen. This one is an edge
lying in the screen and this is a vertical edge and so on; however, Miller indexing will give
us a specific notation, a specific system to name this blue line.

Let us look at how we do that. We will go step by step: The first step in Miller in indexing
is to choose an origin on the direction. So, I have chosen this back corner as my origin,
pointed out in red. The origin always has to be on the direction or vice versa the direction
should pass through the origin.

This freedom of choice exists in crystallography in the crystallographic coordinate system;

we are free to choose the origin anywhere we wish. If I want to index this blue direction I
choose the origin on the blue direction and I took this point as the origin. The next step is
to choose a coordinate system, crystallographic coordinate system with axis parallel to the
unit cell edges. In this case I have chosen x, y and z with the x, y and z directions parallel
to the unit cell edges here, I have red for illustration purpose. I have taken a cube even in
a non-cubic crystal even if the angle between x and y is not 90° and even if z is not
perpendicular to x and y, we will always choose our x, y and z parallel to the unit cell
edges. This is what is called the crystallographic coordinate system.

We will be using the crystallographic coordinate system with unit cell edges as our axis.
We have done that for this direction. Now we have taken this red origin and red axis. The
next step is to find the coordinates of another point on the direction in terms of ‘a’, ‘b’ and
‘c’. These are the three lattice parameters. In this case, they are the edge lengths ‘a’ is the
edge length of the unit cell along the x axis, ‘b’ is the edge length of the unit cell along y
axis and ‘c’ is the edge length along the z axis.

So, in terms of these three vectors, the ‘a’, ‘b’ and ‘c’ vectors, I will now try to express the
blue vector, which is the vector of my choice direction of my choice as in terms of these
three vectors. Here it is very simple. It is 1 times ‘a’ because the direction is along the x
axis and it is of the length equal to ‘a’. It is 1 times ‘a’, 0 times ‘b’ and 0 times ‘c’. I just
take these coefficients 1, 0, 0 to represent this direction. So, I find the coordinates of
another vector in terms of ‘a’, ‘b’ and ‘c’. So, the first 1 means one times ‘a’, the second 0
means 0 times ‘b’, and then 0 times ‘c’. The next step in this case redundant, but we will
write it out because we will use it in the next example is to reduce the coordinates to
smallest integers and this can be done either by dividing by a common factor or multiplying
by common factor.
Suppose, we had fractions then, we will multiply by some common factor such that the
fractions get cancelled or suppose if we had a common factor in all these three, then we
will divide by that common factor to cancel out the common factor. So, this is a step of
reducing the coordinates to smallest integers. In this case and nothing is required. 1, 0, 0
are already the smallest integers. So, we carry on with that and then the final step is to just
put these three numbers in a square bracket. This is an important step, a square bracket is
not my choice in this presentation or this slide. It is an internationally agreed upon
convention that the directions will always be represented by numbers inside square
bracket. So, we will follow this convention. So, [100] is the direction which is represented
by this blue line.

There is a slight difference between vector terminology and the Miller indices of a
direction. Although I picked up this blue vector too along this line and use that to calculate
my Miller indices, once I have found the Miller indices [100] it is not representing just this
blue vector, but this entire x axis. So, the entire x axis as well as the negative x axis can be
represented. This full line is represented by the number [100]. Another peculiarity of
convention here is that when I wrote the components separately I am writing it with
commas, but in the Miller indices I am not using any commas. This is a useful convention
unless and until we have a 2 digit Miller indices for one of the components. If it is only 3
numbers we write them without any commas and it is understood that the first number is
with respect to the x axis, the second one with respect to the y axis and third one with
respect to z.

(Refer Slide Time: 08:30)

Let us look at some more examples now, before looking at those examples, there is one
more point. Miller indices of a direction represents only the orientation of the line, not its
particular position in space or also its sense, I already told that not only the positive x axis,
but the negative x axis will also be represented by [100] if we do not really want to
distinguish positive and negative that is to say, if we are only interested in the line not in
the sense and in Miller indices that is usually the case then [100] represents the entire x
axis. Not only that because what we said about freedom of choosing the origin, if we had
parallel lines somewhere here then, I can again choose my origin here and this will become
my x axis or I can choose my origin there and this will become my x axis. So, all parallel
directions have the same Miller indices.

(Refer Slide Time: 09:42)

Let us now look at some more examples. Again we take this face centred cubic unit cell
and we want to indices a direction which is starting from this corner and passing through
the top face centre. This direction is the direction of my choice. So, the first step is to
choose a coordinate origin and a coordinate system. So, I chose this point as my origin and
axis along the unit cell edges. With respect to this coordinate system I now write the vector
OA. So, you can see, to reach OA, I have to go a/2 steps along x, a/2 steps along y and
then 1c step along z. So OA, the vector OA is (½)a + (½)b + 1c. So, the coordinates which
we will use in the Miller indices in terms of a, b and c are ½ , ½ and 1.

Now I will use the cancelling of fractions step which we did not require in the previous
one. In the case of [100] we did not require that, but now in the case of ½, ½, 1 we will not
call this direction [½ ½ 1] but will simply multiply by 2 to get [112] and of course, I put
them in the square bracket which we have agreed upon to use as a convention for
directions. So, the OA direction not just the OA vector, but the entire OA direction will be
represented by the Miller indices [112]. One more example, let us look at this black
direction now of course, I have to choose the origin on the black line and that freedom is
there.

So, I shift the origin to this point P on the black line. You can note that when I have shifted
the origin, I have kept the axis parallel. So, we have the freedom to choose our origin
anywhere, but in a given problem once we have specified the orientation of the axis, that
orientation cannot change. So, the x, y, z with the new origin is exactly parallel to the x, y,
z before. The black x, y, z is parallel to the blue x, y, z. Now, let us try to index this
direction along PQ which is one of the body diagonals of this cube. So, if we want to look
at this PQ, we will start with P and you can see that now I have to take a −1 step along x,
−1 step along y and 1 step along z to reach Q. So, the PQ vector is −1a − 1b + 1c.

So, the components are −1, −1, 1 in terms of a, b and c. Now I write this in a square bracket
with an additional convention that the negatives are written as bars over the numbers
instead of on the side as in usual mathematics. In the Miller indexing notation, a bar above
the number represents negative quantity. So, and it is read also as bar instead of minus 1.
So, we will call this direction PQ as bar 1 bar 1 1. So, the negative steps are shown as bar
over the number.

(Refer Slide Time: 14:04)

Let us now look at another convention which is used many times, we are not interested in
just one direction because we have talked about the symmetry in the crystal and crystals
can have symmetry, and symmetry relates many directions. So many directions become
equivalent because of the existing symmetry of the crystal. For example, if you take a
cubic crystal, all the edges of the cube are equivalent by the cubic symmetry. If we index
the edge along x axis it will be [100], if we index along the y axis [010] and indexed along
z axis, [001]. But suppose I am not interested in a specific direction, I just want to talk
about the cube edges for all directions along the cube edges which are equivalent by
symmetry.
Then there is a new notation that you can put the Miller index of any one of them in an
angular bracket. <uvw> means the specific direction [uvw] and all other direction related
to [uvw] by the symmetry of the crystal. And it is important that when we are using this
notation we have to know which crystal system we are talking about because different
crystals will have different symmetries and the symbol will mean different things. We will
show you this with the help of cubic and tetragonal examples.

(Refer Slide Time: 15:49)

So, let us look at first the cubic. In the Miller indices of cubic crystal the [100] direction is
̅ . So,
equivalent to [010], [001] as well as if you take the negatives, [1̅00], [01̅0] and [001]
all 6 direction are equivalent by the cubic symmetry. So, if we simply write <100>, if I
pick any one of them, I have picked up [100] you could have picked up [010] or [001] or
any of these 6. This is a family of 6 members, this is a family of symmetry related
directions and I pick up any member of the family to represent the entire family. So, when
I say <100> and I know that it is for cubic then I will mean all these 6 directions.

(Refer Slide Time: 16:56)

In tetragonal you know x and y are equivalent by symmetry, but not the z. So, if I say
<100> for tetragonal, it will only mean these four directions: [100], [010] and their
negatives. The third direction, [001] is absent from here in this list because tetragonal
symmetry does not make [001] equivalent to [100].

With this, we will end this video. In the next video, we will take the discussion on Miller
indices of planes.