Introduction to Materials Science and Engineering

Prof. Rajesh Prasad

Department of Applied Mechanics

Indian Institute of Technology, Delhi

Lecture - 04

Classification of lattices

Hello. We have defined till now crystal as periodic arrangement of atoms and lattice as periodic set
of points. In this video, we will look at the classification of lattices.

(Refer Slide Time: 00:24)

Lattices are periodic set of points and the periodicity can vary in the 3 directions of space. So, the
axis and the angles of the crystal can have many different values.

So, it is important to have a system of classification of these lattices; 2 important systems are
common: 1 of them classifies the crystal into crystal systems-7 crystal systems and another one into
14 Bravais lattices.

So, let us let us look at them. Here is the list of 7 crystal systems.

(Refer Slide Time: 01:12)

We have cubic, tetragonal, orthorhombic, hexagonal, trigonal or rhombohedral (there are 2
alternative names for the same system), monoclinic and triclinic.

Each of these systems have a conventional unit cell which we have shown here. So, in the cubic
system you have a unit cell in which all sides are equal a=b=c.

All angles are equal: α=β=Ƴ and all these equal angles are equal to 90°.

So, basically you have a cube as your unit cell, and that unit cell is repeated in space to generate the
entire lattice.

Tetragonal crystal system has a conventional unit cell which is almost like cube. All angles are equal
to 90°. Two sides are equal, but third side is not equal.

In orthorhombic also all axes are mutually orthogonal, all angles are 90°, but none of the 3 sides are
equal.

Similar relationships are given here for all other systems. You don’t really have to worry about
memorizing all this in one go, but gradually you will become familiar and in case if you need and you
do not remember, you can always look it up in some book. Let us try to understand the meaning of
the crystal system and Bravais lattices.

So, we have 7 crystal systems and 14 Bravais lattices. In each crystal system, there are 1 or more
Bravais lattices. P, which stands for primitive or simple, is present in all of them.

So, we have 7 primitive or simple lattices- one each in the 7 systems. So, P stands for primitive or
simple and which means lattice points are only at the corners of the unit cell.

The next type is I, or body-centred, which means lattice points are at corners as well as the body
centre.

This Bravais lattice is present in cubic, tetragonal and orthorhombic.

We now have 7+3 = 10 Bravais lattices. 4 more Bravais lattices are there.

F stands for Face-centred which means lattice points are at corners;

Corners will always have lattice points whether in simple or body-centred or face-centred. Corners
plus all face centres. You have cubic F and you have orthorhombic F.

So, two F lattices are there. Finally, C is called either end-centred or it has an alternative name also:
base-centred. Here the lattice points are at corners and not on all faces as in face-centred, but only
on one pair of parallel faces.

The C-centred or end-centred lattices are there in orthorhombic and in monoclinic.

So, we have now 7 P, 3 I, 2 F and 2 C. So, this constitutes the 14 Bravais lattices. Let us familiarize
ourselves a little bit more with the Bravais lattices.

(Refer Slide Time: 06:39)

So, let us look at the 3 cubic Bravais lattices. In the cubic system, we said we have P, we have I and
we have F. So, what do they look like?

This is my cubic unit cell and when I say cubic P or primitive cubic or simple cubic, more common
name is simple cubic.

So, the Bravais lattice name includes the crystal system name as well as either the symbol- cubic P or
in language, we can say simple cubic and the simple cubic lattice will have, the simple means the
lattice points are only at the corners.

So, only corners are identified as lattice points. Don’t think of these as atoms. They are only points. I
am highlighting them to emphasize that the lattice points are only at the corners.

Let us draw the next one. Let us draw cubic I. This will be body-centred cubic. Both names are
synonymous or equivalent. So, you can say cubic I or you can say body-centred cubic.

Here the lattice points will be at the corners. As I said, corners will always be lattice points, but there
will be an additional lattice point right in the centre of the cube.

So, this will be the body-centred cubic lattice and finally, let us try to draw the face-centred one.

That is the cube as usual lattice points at the corners, but to qualify as cubic F or face-centred cubic,
we have to give additional lattice points at the centres of all the faces.
So, at the centre of left and right face, at the centre of bottom and top face, as well as at the centre
of the front and the back faces. So, all 6 faces will be centred.

So, then we will have a cubic F lattice. So, that is the meaning of these symbols and the location of
the lattice point.

Cubic system does not have end-centred or as we said can be called base-centred also, but
orthorhombic has.

(Refer Slide Time: 11:19)

So, let us look at the example of base-centred in an orthorhombic. Orthorhombic has all 3 sides
unequal, but axes are still at 90°.

So, that is my unit cell box and the location of lattice points at corners and to make it base-centred
or end-centred, I have to centre one pair of opposite faces, one pair of parallel faces not all faces.

If I centre all faces, I get the face-centred lattice, but if I centre let us say, only the bottom and the
top faces, lattice points, additional lattice points only on one pair of faces, then I end up with end-
centred or base-centred orthorhombic lattice.

It is called orthorhombic C, because if I choose my axis if I choose my axis x, y and z in this way. Note
that in crystallography the a is along the x-axis b is along the y-axis and c is along the z-axis.

So, the face which we have centred is the face containing a and b. This face is called the C face.

So, when I say orthorhombic C, I mean orthorhombic C this means C face is centred. Of course, one
can have, if I had chosen to centre the other faces, I can have orthorhombic A, orthorhombic B, or B
also possible; however, since only one pair of face is centred.

You can always choose your z-axis or the c-axis perpendicular to that face and make it orthorhombic
C.

So, we have we have seen, we have simple

(Refer Slide Time: 14:29)

where only corners are there. Corners as lattice point. Then we had face-centred F, then we have
body-centred and we also had a base-centred or end-centred.

So, the question can be asked why don’t we have any lattice which we can call edge-centred by
edge-centred, we will mean points at the centres of each edge.

So, let us consider edge-centred cube. So, points at corners and at centres of each edge centres of
each edge. Let me construct one unit cell like that.

Again I have my cube. I put points at the corners and I also put additional point at the centres of
each edges.

The centres of each edge. Why no lattice is is listed like this? An edge-centred cubic lattice or an
edge-centred orthorhombic lattice.

The answer to that can be seen by examining the surroundings of the point.

So, let us look at, let us look at point A. Let me call this point A and let me call this point B and let me
choose this direction as my x direction.

Let me call the edge length of the cube as a. So, if I move from point A in the direction x, at a
distance a/2, I find another neighbour, this one but if I move from point B in the same direction that
is in the x direction, the same distance a/2, I do not find any additional point.

So, we conclude that the points A and B are not equivalent (not translational equivalent). So, the set
of points do not form a lattice.

So, let us again look at the complete list of Bravais lattices which we had.

(Refer Slide Time: 19:41)

So, we were we were having cubic P (simple cubic), cubic I and cubic F. We had tetragonal P and
tetragonal I.

We had orthorhombic P, orthorhombic I, orthorhombic F, and orthorhombic C. So, orthorhombic is

quite rich. It has all the 4 varieties.

We have hexagonal P and that is all. We have trigonal P, we have monoclinic P and monoclinic C and
we have triclinic P.

So, we can see that 7 different crystal systems are there and in each crystal system, like in
orthorhombic, 4 possible Bravais lattices could have been there, P, I, F and C.

But only orthorhombic has all 4. So, 7 systems, 4 types, 7×4=28 lattices were possible, but we have
only 14 Bravais lattices.

Why so many other lattices which were possible are not there? In particular, why don’t we have
cubic C? Why cubic C is absent from the Bravais list?

Of course, similar question can be asked for all other empty boxes. Why is tetragonal F not there,
why is tetragonal C not there and so on.

So, all these empty boxes is a question mark and it is this question which we will take up in the next
video.