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

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

Â it turns out that there are 14 of those.

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

Â and it represents a structure in which the lattice points

Â are distributed in space using a cubical description.

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

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

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

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

Â cells in three dimensions.

Â Each one of those points is then one-eighth.

Â And there are eight corners in the cube.

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

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

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

Â But for the time being,

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

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

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

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

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

Â However, what's different is the fact that

Â one of the axis is different than the other two.

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

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

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

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

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

Â Again, all the interaxial angles are 90 degrees.

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

Â So now, what we have to do is to specify

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

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

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

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

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

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

Â And you can think about the rhombohedral structure where, for

Â example, you look at a cube.

Â A cube is a very special case of a rhombohedron

Â in that each of the interaxial angles are 90 degrees.

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

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

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

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

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

Â interaxial angles, along with a, b and c.

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

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

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

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

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

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

Â is we actually have a total of three Bravais Lattices.

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

Â that is, that lattice contains one lattice point.

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

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

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

Â it's referred to as face center cubic,

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

Â we also have lattice points which sit at the face.

Â So the corner positions now count as, again,

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

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

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

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

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

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

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

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

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

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

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

Â now we have body center cubic.

Â And again we have a non-primitive unit cell.

Â We have a total of two lattice points.

Â The ones that are associated with the corners,

Â followed by the one that lies holy in the center.

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

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

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

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

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

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

Â Now, we look at the hexagonal and

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

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

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

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

Â we have the monoclinic, and we have the triclinic.

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

Â There are two Bravais Lattices associated with the monoclinic and

Â one with the triclinic.

Â That gives us a total of four in those three remaining

Â of the seven crystal systems.

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

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

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

Â and the 14 Bravais Lattices.

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

Â the 14 Bravais Lattices, thank you.

Â