 In the last video, we spent a fair bit of time talking about different Bravais lattices in two dimension and we found out that we could in fact classify these different lattices into five categories based on the kind of unit cell that they have. Now in this video, we are going to dive into the more complex three dimensional Bravais lattices and try and figure out if you can also classify these lattices into some categories. Just like we did with these two dimensional ones. Now before we can think about classification, we should again ask ourselves how many different Bravais lattices are actually possible in three dimensions. Now just like in two dimension, the answer to this question is infinite. There are infinite different Bravais lattices that are possible in three dimensions. Now you simply don't have to just take my word for it, I can also show you how we can reach to this conclusion. Well, we can start with a two dimensional Bravais lattice and take the whole lattice and translate it in the third dimension by a distance say C. Now if we keep doing that at regular intervals, I will ultimately end up with a lattice that has translational symmetry at every lattice point. All the lattice points coincide on translation, so this is how we create a Bravais lattice in three dimension. Now instead of putting the second layer exactly above the first layer, I could have placed it at an angle along the x-axis or maybe I could have instead done it along the y-axis. So in this way just like in 2D, by changing the values of A, B and C and the angles between them, I can create infinite different Bravais lattices in three dimensions. So now that we are fully established that there can indeed be infinite different lattices, we can now turn our focus into classifying these different lattices. Well, from our previous videos we know that a lattice is defined by its symmetry and the best possible way to figure out the overall symmetry of a lattice is by thinking about its unit cell. A unit cell in three dimension is the smallest, most symmetrical cell made by joining the lattice points that on translation can fully cover up the whole lattice. So to figure out the different types of Bravais lattices in 3D, we need to think about the different kinds of unit cells that are possible in three dimension. Now, before we do that, let us take a moment and retrospect about 2D lattices. In two dimensional lattices, we figured out that a triangle cannot be a valid unit cell because a triangle cannot fully cover up a two dimensional area only via translation. The smallest possible shape was in fact a parallelogram and by tinkering with the symmetry of this parallelogram, we found out that there could be these four fundamentally different shapes having different symmetries that could cover up a 2D space only via translation. Any other shape that you can think of besides these will not be able to cover up a two dimensional area only via translation and so it cannot be a valid unit cell. Now similarly, in three dimensional lattices, the smallest possible unit cell is in fact a parallelopiped and by changing and tinkering with the symmetry of this parallelopiped, it turns out that we can have these seven fundamentally different shapes having different symmetries that could fully cover up a 3D volume only via translation. Now based on symmetry, these shapes have been given some fancy names. This one is clearly a cube so it's called cubic. This one is like a cuboid but we call this the tetragonal and we can also have the orthorhombic, hexagonal, rhombohedral, monoclinic and the triclinic systems. Now it's not really important for us to go into the details of these different shapes but what's really important is that we can't have any other shape besides this that can cover a three dimensional space only via translation. Now of course you can have bigger cubes and smaller cubes and thicker and thinner tetragonals. What I really mean is that shapes having different symmetries besides these like say a tetrahedron or an octahedron will not be able to cover up such 3D lattices only via translation. In other words, whatever be the kind of three dimensional lattice that you choose because Bravais lattices themselves are translationally symmetric, they can be thought of to be built via translation and because these are the only possible shapes that can fully cover up a 3D space only via translation. So it follows that this lattice has to be made up of one of these shapes as its unit cell. So if we analyze this lattice we'll realize that this in fact is made up of these tetragons so this is nothing but a tetragonal lattice. We can now look at this whole business of classifying lattices differently and say that because these are the only possible shapes that can fully cover up a 3D space only via translation so only those lattices that are built using these shapes as the unit cell. Only those lattices can classify as Bravais lattices. However, it turns out that besides these primitive unit cells in which there are lattice points only at the corners, lattices built using unit cells that have an extra lattice point at the center, all such lattices also have translational symmetry when I go from one lattice point to another lattice point. Even if I move from the corner to the body center, there will be translational symmetry because when we move from the corner to the body center, the body center itself will move to this corner and so the translational symmetry will remain intact. So even these kind of non-primitive unit cells in which there's an extra lattice point at the center and we call these the body centered unit cells, even these can generate translational symmetric Bravais lattices. Now besides these body centered unit cells, even lattices built by face centered unit cells in which there are extra lattice points at the center of each phase as well as those built by these end centered unit cells in which there are extra lattice points at the center of any two opposing phases. Even these also generate lattices that are translational symmetric and so they are valid Bravais lattices. However, any other unit cell in which there's a lattice point at some other position, maybe say out here, lattices built by such unit cells will not have translational symmetry at every lattice point. So if I go from this corner to this lattice point, if I do that, all these other lattice points do not coincide. So all such non-centered unit cells do not generate valid Bravais lattices. To summarize, besides the lattices built by simply using these shapes and we call this the primitive unit cells, lattices built by the body centered, face centered and end centered unit cells also generate valid Bravais lattices. We should now be able to conclude and say that based on symmetry alone, there are these seven fundamentally different shapes that can qualify as a unit cell. But besides the primitive unit cell, we can also have body centered, face centered and end centered unit cell because even they generate valid Bravais lattices. So if we do the math, we can have 28 different types of Bravais lattices in three-dimension. However, it turns out that this is actually not right. And the actual answer is in fact 14 as some of these unit cells actually generate the exact same lattice. For example, a lattice made of end centered cubes can also be thought of to be made up of these simple tetragons. In other words, a lattice generated by end centered cubic unit cells is actually the exact same lattice as generated using simple tetragons. So what do we call such lattices? Do we call them end centered cubic lattices or do we call them tetragonal lattices? Now a cube is a highly symmetrical structure and if I try to rotate this cube along this axis, the cube will repeat itself after every 90 degrees. So this is a C4 axis of symmetry. However, if we take end centered cubes in which there are lattice points out here and now if I try to rotate it along this axis, these faces will not repeat after 90 degrees. In fact, they will repeat after 180 degrees. So this is no more a C4 axis of symmetry, but it is instead a C2 axis of symmetry. Now a cube is much more symmetric than a tetragon. However, an end centered cube isn't. And so when it comes to describing this lattice, we use the primitive tetragon as the unit cell rather than the non-primitive end centered cubic. So an end centered cubic unit cell is the same thing as a simple tetragonal. And based on symmetry, we call it the tetragonal and not the end centered cubic. And similarly, if we check for all these other combinations, we'll find out that there are a lot of redundancies. Many of these lattices are actually exactly the same thing. So the total number is not 28, but instead it comes out to be 14. Now all this hard work was first done by the French mathematician Auguste Bravais. So all these lattices are called the Bravais lattices.