In this lesson, we're going to introduce the concept of these 7 crystal systems,

and a term which we refer to as the Bravais Lattice, and

it turns out that there are 14 of those.

The first and the simplest of all for us to understand is the cubic lattice,

and it represents a structure in which the lattice points

are distributed in space using a cubical description.

And one point that I want to bring out here is that,

if we look at each one of those points in space, they are representing here

a total of one-eighth for each one of those points in space.

So recognizing that each point is surrounded by a total of eight unit

cells in three dimensions.

Each one of those points is then one-eighth.

And there are eight corners in the cube.

And therefore, there is one lattice point associated with this cube.

Now, just remember around that lattice point, we can put a bases

which could be represented by a collection of atoms on around that point.

But for the time being,

what we're doing is we're talking about the basic seven crystal systems.

So the first is cubic, a b and c are all equal to 1 another and we know that alpha,

beta and gamma, the three interaxial angles, are all equal to 90 degrees.

Now, when we look at the next of the crystal systems, the tetragonal system, in

the case of the tetragonal system, we have all three interaxial angles at 90 degrees.

However, what's different is the fact that

one of the axis is different than the other two.

So in this particular case, a is equal to b but it's not equal to c.

So when describing this structure we have to provide two pieces of information

in addition to the fact that we're in the tetragonal system.

We need to give the values for a and we also need to give the value for c.

If we now look at the third in the seven crystal systems, the orthorhombic.

Again, all the interaxial angles are 90 degrees.

However, a, b, and c are no longer equal to one another.

So now, what we have to do is to specify

the magnitude of each of those a, b, and c components.

Another type of lattice that we can describe is something that's referred to

as a hexagonal lattice and a hexagonal lattice develops in two

dimensional space or three dimensional space in a hexagonal structure.

And in the case of a hexagonal structure, what we find is that a and

b which lie in the plane of the hexagon is perpendicular to

the c axis which is a different value than a or b.

And the interaxial angle, of course, in the plane of the hexagon is a 120 degrees.

So now what we have to do here is we have to specify that interaxial angle and

we need to describe the fact that a and b are equal.

However, they're not equal to c.

Now, the next in our seven crystal systems is the rhombohedral.

And in the case of a rhombohedral each faces a rhombus, that is each face

the lengths of the sides are all equal to one another and

therefore what we have is a, b, and c are all equal.

However, the interaxial angles alpha, beta, gamma are not equal to 90 degrees.

And you can think about the rhombohedral structure where, for

example, you look at a cube.

A cube is a very special case of a rhombohedron

in that each of the interaxial angles are 90 degrees.

But again, each face, being a square, is also a rhombus.

Now, when we look at the monoclinic system, what we see here is that

we have 2 angles that are equal to 1 another, and they're equal to 90 degrees.

And however, the third angle, beta, is not.

And we also have the additional requirement that we have to specify a,

b, and c because they're not equal.

Now, it turns out the last of the seven crystal systems is triclinic.

In the case of the triclinic lattice system, a, b, and c are not equal.

Nor are the interaxial angles equal to the 90 degrees, or equal to 1 another.

So as a consequence of that we have to specify all those

interaxial angles, along with a, b and c.

So this then describes for us what we mean by the seven crystal systems.

Now, we're going to depart a little bit, and

we're going to talk about the 14 Bravais Lattices.

And what we see here are, again, the representations of the cube,

the tetragonal lattice, the octahedral lattice, and the hexagonal lattice.

And this time, what we see for the cubic lattice,

is we actually have a total of three Bravais Lattices.

One of them, where we have what we refer to as a primitive cell,

that is, that lattice contains one lattice point.

Remember, each one of those corner positions represent one-eighth.

They're eight corners and therefore we have a total of one lattice point.

When we look at the second figure and if you look at that figure carefully,

it's referred to as face center cubic,

that is not only do we have lattice points that sit at the corners at the cube but

we also have lattice points which sit at the face.

So the corner positions now count as, again,

one-eighth times eight of those so that's now one lattice point.

And if you look at the faces, there are total of six faces that are associated

with a cube, and what we'll see then is each of those points are shared

by either the unit cell beside them, above them or in front of them.

So consequently, those six faces then become three lattice points.

So the three lattice points on the faces along with the lattice points

at the corner gives us a total number of four lattice points.

So we refer to this type of cell, since we have more than one lattice point,

a non-primitive cell whereas in the case of the simple cube, it turns out that that

happens to be winding up being a primitive lattice with one lattice point.

We turn our attention now to the third of the cubic structures and

now we have body center cubic.

And again we have a non-primitive unit cell.

We have a total of two lattice points.

The ones that are associated with the corners,

followed by the one that lies holy in the center.

Now, if we go down and we take a look at the tetragonal units, we now

have two of those, a simple tetragonal, and we have a body centered tetragonal.

If we look at the orthorhombic, what we see is a total of four of those cells.

One of which is primitive, and the other two are non-primitive,

a body, a base, and a face centered tetragonal unit cell.

And what that does is it provides us some additional units to our Bravais Lattices.

Now, we look at the hexagonal and

it turns out that there is just simply one of those.

So now if we count up all the number of lattices that we have in here, or

the Bravais Lattices, we have a total of ten so far.

If we look at the remaining three crystal systems, we have the rhombohedral,

we have the monoclinic, and we have the triclinic.

And what we see is that there are one lattice associated with rhombohedral.

There are two Bravais Lattices associated with the monoclinic and

one with the triclinic.

That gives us a total of four in those three remaining

of the seven crystal systems.

And then, consequently, what we have is the 10 from the previous slide,

and the 4 on this 1, is a total of 14 Bravais Lattices.

In this lesson, what we describe were the seven crystal system,

and the 14 Bravais Lattices.

In the next lesson, what we'll be describing is why is there a need for

the 14 Bravais Lattices, thank you.

3.5 Crystal Systems and Bravais Lattices

Material Behavior

Meet the Instructors