Introduction to Materials Science and Engineering
Prof. Rajesh Prasad
Department of Applied Mechanics
Indian Institute of Technology, Delhi
Lecture - 03
Unit Cell
Keywords: primitive and non-primitive unit cells; crystal coordinate system; lattice parameters
We are on the topic of geometrical crystallography. Last class we discussed
(Refer Slide Time: 00:18)
this important relationship: crystal = lattice + motif. We defined crystal, lattice, and motif.
Now, in this class we will look at more details about some of the properties of a lattice. So, lattice will
be the focus of this session.
(Refer Slide Time: 00:46).
We begin with lattice translation. We already have a periodic set of points which is a lattice, and I
have drawn an example here, a nice simple is a square lattice.
Lattice translation can be defined as any vector from one lattice point to another lattice point, I will
draw some examples here.
So, for example, I take this this as my lattice point, and I draw a vector, horizontal vector up to the
next nearest neighbour, in the horizontal direction.
So, this red vector is an example of a lattice translation. Or if I wish, I can draw another vector like
this. This is also a lattice translation. Or yet another, I can start from here, and draw this vector.
This is a lattice translation. So, all these vectors are lattice translations. Of course, a lattice will have
many such lattice translations. We will now define a unit cell.
(Refer Slide Time: 02:41)
Let me first give a little abstract definition. So, I will define a unit cell as a region of space, which can
generate the entire lattice or crystal by repetition through lattice translations.
So, connected region of space which can generate the entire lattice by repetition, through lattice
translation will be called the unit cell.
The unit cell can be either of the lattice or of the crystal. Let us see some examples.
(Refer Slide Time: 04:27)
Although it is an abstract definition, commonly, a unit cell is a parallelogram in 2D or a parallelepiped
in 3D with corners at the lattice points, different shapes can be selected, but the common shape
which is selected is a parallelogram in 2D or a parallelepiped in 3D.
So, again I have an example of a 2D lattice here, and if we wish, we can select a square in this lattice.
This square will be the unit cell.
This square has corners at the lattice point, and if I start repeating this square, I see that I can get the
entire lattice.
All the points can be obtained by such repetition. So, I have to periodically translate these unit cells,
and I have to recognize that lattice points are at the corners. However, a given lattice has not just
unit cell, but many different unit cells. So, the unit cells are not unique.
(Refer Slide Time: 07:01).
A given lattice can have many, in fact, infinitely many unit cells. We will see here for example, in the
square lattice, initially we had selected a nice square as our unit cell, but in the same lattice if I so
wish, I can select this red unit cell which is no more a square, but a parallelogram. And if I repeat this
parallelogram, I can again generate the entire lattice. Also we can select larger unit cells for example,
if I so wish, I can select a larger square, in which now lattice points are not only at the corners, but
also in the centre of the square. This is also possible. So, we have many different unit cells for the
same lattice.
The unit cells which I showed you can be classified as primitive and non-primitive unit cell.
(Refer Slide Time: 09:29)
Primitive unit cell will have lattice points only at the corners of the cell. So, we have already seen for
example, the square unit cell which we selected will qualify as a primitive unit cell.
The red unit cell, the parallelogram unit cell, is also a primitive unit cell, because by this definition
this also has lattice points only at the corners, but the larger blue unit cell which we had selected last
time, will be an example of a unit cell which is non-primitive.
So, the non-primitive unit cell, will also have a lattice point at its corner, but it will have some lattice
points other than the corner points as well as at some other point.
So, this will be an example of a non-primitive unit cell. You can have many different primitive unit
cells and you can have many different non-primitive unit cells as well.
For example, if I choose a still larger square, let us say like this, this also is a parallelogram with
lattice points at its corners.
So, it qualifies as a unit cell, but apart from the corners it has lattice points at its edges, and it has a
lattice point at its centre.
So, this will be another example of a non-primitive unit cell.
We now define what is called a crystal coordinate system.
(Refer Slide Time: 12:55)
So, unit cell itself, parallelogram or parallelepiped unit cell are defined by the vectors along its edges.
So, select a corner of the unit cell as the origin.
The vectors from the corner along the edges of the unit cell define the basis vectors of the lattice.
So, we have selected this as the origin, along the edges we have these two vectors a and b.
The angle between a and b is Ƴ. We can also define these as the axis. This our x-axis, this is our y-
axis, this was origin O.
So, Oxy forms the crystal coordinate system.
So, of course, the crystal coordinate system is associated with the selection of a unit cell.
So, if we select a different unit cell, for example, if we select this parallelogram unit cell, then my x
and y are no more orthogonal, my a and b are also not equal and the angle Ƴ is not 90°.
So, in this case a was equal to b, and Ƴ was 90°, but here for the same lattice, I have now selected a
different coordinate system where a is not equal to b, and Ƴ is not equal to 90°.
So, crystal coordinate system can be non-Cartesian. We will now define the lattice parameters.
(Refer Slide Time: 17:39).
So, the lengths of 3 edges of the unit cell (or the 3 basis vectors), and the 3 inter-axial angles
between them are called the lattice parameters.
Now I am talking in 3D. So, that is why we are talking of 3 basis vectors, 3 edges and so on. Of
course, in 2D there will be 2 basis vectors, and 2 edges and there will only be 1 angle.
So, let me try to draw a parallelepiped unit cell to define these. So, suppose I have a parallelepiped
unit cell and I choose this as my origin, and choose this as my x-axis, y-axis and z-axis.
So, they are not necessarily orthogonal, and the corresponding lengths a, b and c. These are also not
necessarily equal, and we have 3 angles- angle between b and c, α, angle between a and c, β, and
angle between a and b, Ƴ. This is the convention of naming. α is always taken to be angle between b
and c and so on.
So, once these 6 numbers are given, we have the complete description of the so called lattice
parameters of the crystal which means if we are given the lattice parameters we can construct the
unit cell of the crystal.
(Refer Slide Time: 22:29).
If we know the unit cell, by repetition of the unit cell, we know the lattice. And if we know the lattice,
then we have the information about crystal, or am I making a mistake here?
Lattice parameter gives me the unit cell, unit cell gives me the lattice, but lattice gives, if you have
been following this video till now, you will be worrying about this lattice from lattice going to crystal,
because we have said the lattice only gives me how to repeat, but it is not giving me what to repeat,
and unless and until I know what is being repeated, that is unless and until I know the motif, I do not
know the crystal.
So, this is not right. I need the information about the motif. So, if the motif information is given, then
only I know about the crystal.
(Refer Slide Time: 24:16)
So, we will continue this in the next video. Classification of lattices into 7 crystal systems and 14
Bravais lattices will be our next topic. Thank you.
