[MUSIC] We now have to begin and look very formally at the way in which we construct repeating units in three dimensions. These are the so called unit selves. By the end of this lecture you should be familiar with the seven lattice types which are found in three dimensions. In addition, you have to recognize the difference between the Bravais lattices. And in three dimensions there are 14 of these which are divided into primitive types and standard types. You will recall that in two dimensions, we spoke about five Bravais lattices. These included centered and primitive cells. The oblique lattice was primitive. For the rectangular lattice it could be primitive or centered. The square lattice was primitive. And the hexagonal lattice was also primitive. In two dimensions we could describe the cell edges in terms of a and b and their directions in terms of x and y. When we move to three dimensions, all we need do is add to this nomenclature. So now, we have directions XY plus zed. And we have edges to the sail AB plus C. In addition to a single angle between those directions, which was gamma, we can now add alpha and beta. With that additional nomenclature, we can create all of the Bravais lattices, that are found in three dimensions. Now we will look at the transformation between the seven Bravais lattices found in three dimensions. The general lattice, or so-called triclinic lattice, is one in which the three distances, a, b, and c, are all different, as are the three angles, which must be non-90. To explore this we set some arbitrary distances to a, b, and c 2.5, 4, and 2. We also set the angles alpha, beta, and gamma to be 80 degrees, 105 degrees, and 115 degrees. You'll notice that at the corners of the unit cell we have placed a green sphere. This is a lattice point, it should not be confused with an atom. This is really a mathematical representation of the nature of the symmetry in this cell. It is very useful to view these three-dimensional lattices looking down the cell edges. In other words, looking along x, looking along y or looking along zed. In this way we get a very direct representation of the relationship between the cell edges and the angles between them. Notice again, that the angle between A and B, will always be gamma. The angle between A and C, will always be beta, and the angle between B and C, must always be represented as alpha. The General Bravais Lattice has the lowest symmetry of any of the lattices. This is because all of the distances and angles are unique. As we work through this series of primitive lattices, we find that increasingly the cell edges become equal and the angles fix themselves at 90 degrees or 120 degrees. After triclinic we move to the monoclinic lattice. In this case, a, b, and c remain different. But alpha and gamma turn into 90 degree angles and beta is normally non-90. This relationship becomes more obvious when we look at the two dimensional projections. So only when we look down the b direction, do we see the unique angle, which in this case has been arbitrarily set. We now morph the cell further to create the Orthorhombic Bravais Lattice. Again, a, b, and c are all different. But now we force the angles alpha, beta, and gamma to be all right angles. When we do the two dimensional projections of the orthorhombic cell, looking along the principal directions, we see all of the right angles correctly reflected. After the Orthorhombic Bravais Lattice, we move towards a tetragonal cell Again increasing the symmetry. In the tetragonal cell, we find that we now have a equal to b, and c unique. All of the angles are still set at 90 degrees. We can fully represent the nature of this cell through two principal axial directions. Looking along c which is unique and looking along either a or b which are identical. In the Rhombohedral Lattice we set a and b and c all equal and we set all of the angles equal as well but they're not equal to 90 degrees. One way to think about the rhombohedral unit cell is it's a little like a squashed cube. If we project along the body diagonal of the rhombohedral unit cell, it looks like a hexagonal arrangement. We see all of the equally distant A, B and C. And the angle between these in projection become 60 degrees. In the Hexagonal Bravais Lattice, we set a equal to b and c unique. But now the angle between a and b, or gamma, is set towards 120 degrees. If we look along one of the short edges, a or b, we see a rectangular projection. If we look down the unique c axis, we see the 120 degrees between the two equally distant directions. Finally, we come to the highest symmetry Bravais lattice. This is where a, b, and c are all equal. And all the angles are equal to 90 degrees. Therefore, looking down any of the principal directions is identical and we can represent the two-dimensional projection just by one figure. So now, let's summarize what we have learned about the primitive Bravais Lattices in three dimensions. They come in seven varieties. The most general one is the triclinic form. In this case, all of the angles and all of the distances are different. As we move from triclinic through to the other forms, we gradually equalize either angles or distances or both. Therefore the Monoclinic Bravais Lattice has the second lowest symmetry followed by the orthorhombic, the tetragonal and then the rhombohedral and hexagonal. Finally we finish at the cubic lattice. This always has the highest symmetry because alpha, beta, and gamma are 90 degrees, and all the cell edges are equal. We've now seen the seven primitive lattices. But for some of these crystal families, there are also centered lattices and that's what we need to examine now. If we look at the triclinic system, it only appears as a primitive view in itself. But once we move to monoclinic, we find that centering takes place. For the monoclinic primitive cell, there are only lattice points at the corner. But for this system it's also possible to place lattice points in the base. As illustrated here we are showing the top and bottom faces as being centered. Notice the linkage between three-dimensional projection in the middle of the drawing and the positioning of the lattice points in the two dimensional projections shown at the right hand side. For the monoclinic system there is only base centering. There is no face centering nor is there bodies centering. The orthorhombic system shows the most extensive types of centering of all of the crystal families.We can have body centering where we place the additional lattice point in the center of the unit self. We can also create base centers. Those base centers can be in any opposing pairs of the faces of the unit cell. Here we are showing by centering in the AB plane. And you'll notice again how this additional lattice point is represented when we project along a and b and c. We can also have face centering and this is where every opposing pair of faces contains an additional lattice point. This is reflected again in the two dimensional projections shown at right. So in a case of the orthorhombic system, we can have Body centering, base centering, and face centering. The tetragonal system is simpler. It only displays body centering with the extra body centered lattice point shown at the center of the two dimensional projections. The cubic system shows two types of centering. There is body centering and also face centering. So let's summarize. We have now described seven Bravais lattices and these lattices are all primitive, which is to say their lattice points only appear at the corners of the unit cell. But it is also possible to create centered Bravais Lattices. And there are seven more of these. The Triclinic, the Hexagonal, and the Rhombohedral cells are never centered. But all of the other crystal systems show varying degrees of centering. In this lecture you have learned that there are 7 crystal systems. These seven systems are cubic, hexagonal, trigonal, tetragonal, orthorhombic, monoclinic, or triclinic. These definitions arise by looking at specific relationships between the edge distances, A, B, C, and the angles alpha, beta, gamma. Formerly we say that there are 14 Bravais lattices in three dimensions. And these Bravais lattices may be primitive or centered. Again the nature of the centering is dependent upon the underlying symmetry of the lattice.
