 Now that we know what unit cells are and the role symmetry plays while determining a unit cell, in this video we are going to talk at a much greater detail about different Bravais lattices in two dimension and try and classify these lattices into different categories based on the kind of unit cells that they can have. Doing this will actually help us understand and better appreciate the different unit cells that are possible in the more complex three dimensional lattices. So let's jump right into the video. Now before we can even begin to think about classification, we should ask ourselves a question. How many different Bravais lattices are actually out there in two dimension? Do remember that all Bravais lattices needs to have translational symmetry at every lattice point. You can pause the video and think about this for a moment. Well the answer to this question is infinite. There are actually infinite different Bravais lattices that are possible in two dimension. Now to really understand this point, we need to think about how we can create Bravais lattices in the first place. Well we can start by placing points at equal intervals in a straight line. May I keep placing points at an interval of a units? Now if I do that till infinity, I will end up with a lattice that has translational symmetry at every lattice point. So this is nothing but a Bravais lattice in one dimension. Now to do it in two dimension, I can simply take this set of points and place them at a distance b exactly above the first line. Now if I kept repeating this at equal interval, I'd have created a lattice in two dimension that has translational symmetry at every lattice point. So this is how we can create a Bravais lattice in two dimension. Now to create this lattice, I had placed my second line at a distance b which was equal to a. But I could have done it otherwise, right? I could have placed the second line at a distance b which is not equal to a. Again if I kept doing that at regular intervals, I would have created a different two dimensional Bravais lattice. Now instead of placing this second line b exactly above a, I could have placed it at an angle theta and in this way if I had kept doing it, I would have created another different lattice. Now because all these lattices are built by translation, so there definitely is translational symmetry. So by changing the values of a, b and theta, I can create different Bravais lattices. And because there are like infinite different possibilities, so clearly there will be infinite different Bravais lattices in two dimension. So now that we have established that there are infinite different lattices, is there any way of classifying these different lattices into some categories? Well, we know that a lattice is named after its unit cell and different lattices might have different unit cells. So one way of classifying these lattices could be on the basis of the unit cell that makes up the lattice. So how many different unit cells are possible in two dimension? Let us try and explore. The most common unit cell that comes to mind is a square, right? Now if we try to build up a lattice using a square as my unit cell, then the lattice is going to look like this, right? We can now go ahead and call all such lattices that has a square as its unit cell as the square lattices, the square lattice. So what other unit cells can you now think of? We started with a square which is actually the most symmetric structure in two dimension. All the sides and all the angles in a square are equal, right? So all the sides are equal in a square and all the angles are equal. We can now change some of these parameters to get different unit cells, right? For example, we can have a rectangle in which all the angles are equal but the sides are not and using a rectangle I'll get a different lattice that is going to look like this. So we can again go ahead and call all such lattices as the rectangular lattices. So what other unit cells can you now think of? Well, we can come up with a rhombus, right? In a rhombus, all the sides are equal but all the angles are not equal and a lattice formed by using the rhombus as the unit cell is going to look like this. So we can go ahead and call this our rhombus lattices. Well, is there any other shape that we are missing out? Well, yeah, right. We can have a parallelogram in which neither the sides are equal nor the angles are equal and the lattice made by such oblique shaped parallelograms is going to look like this. And we can call all such lattices as the oblique lattices. We can now take a step back and say that hey, all Bravais lattices can be classified either as a square or rectangular or as rhombuses or as obliques. However, it turns out that this picture is neither a hundred percent correct nor is it hundred percent complete. For starters, all rhombus lattices actually have a different non-primitive unit cell, this one which is more symmetric. Because a rectangle has more symmetry elements compared to a rhombus, the overall global symmetry of this lattice will be better represented by a rectangle rather than the rhombus. So the more appropriate unit cell for this lattice is actually this non-primitive rectangle centered because there's an extra lattice point at the center rather than this rhombus. So all rhombus lattices should actually be called the rectangular centered rather than the rhombus. Now if you are new to this series on solids, if this is your first video, I highly recommend you to watch the previous video titled what is a unit cell to better understand the role of symmetry while determining a unit cell. Okay, now besides the fact that rhombus lattices are mislebeled, some of the rhombuses in which the opposite angles are 60 degrees, let me call this the rhombus 60. In such lattices, the most symmetrical unit cell is in fact a hexagon rather than a rectangle. A hexagon in fact is a highly symmetrical structure. If I try to rotate a hexagon along this axis, then the hexagon will actually repeat itself six times. So this is in fact a C6 axis of symmetry. So because the most appropriate unit cell out here is a hexagon, so these kind of rhombuses are not rectangular centered, rather they should be called the hexagonal centered lattice. Hexagonal centered. A quick note out here, the hexagonal centered lattices are simply called the hexagonal lattices most of the time. Now if you are stuck around till now, I am sure you might be scratching your head and asking yourself, well how many unit lattices are actually out there? Well it turns out that there are only these five different unit cells that are possible in two dimension, yes only five that can be categorized into these four systems. Now this actually makes a lot of sense because unit cells by definition should cover up the whole lattice via translation and if you think about it, a square, a rectangle, a hexagon and an oblique are the only fundamental shapes that can fully cover up an area, a 2D area, only via translation. Other shapes like triangles and pentagons cannot and so a triangle and a pentagon can never be a unit cell. So clearly there can only be these four different systems of unit cells in two dimension. We can now also dive a bit deeper and ask ourselves why we can have a rectangle centered unit cell but we don't seem to have a squared centered or an oblique centered. Well if we tried making a lattice using a square center, a square center, the lattice is going to look like this, right? Now if you look closely we will see that there is actually a square, a primitive square which has exactly the same symmetry as this square center. So this non-primitive square center is not more symmetric than this primitive square. So this will actually be called a square lattice rather than a square centered lattice, right? So all square centered lattices will actually be square lattices. However, this is not the same case when we talk about rectangle centers. A lattice made of rectangle centers will not have a more symmetric unit cell. In fact the primitive unit cell out here is actually a rhombus. So rectangular centered lattices have to be called rectangular centered and nothing else. Now a lattice made up of simple hexagons is actually a honeycomb lattice which as we know from previous videos is not a Bravais lattice as it does not have translational symmetry at every lattice point. For it to have translational symmetry there has to be a lattice point at the center. So hexagonal lattices are actually hexagonal centered lattices by default. Now finally if we talk about the oblique centers, the non-primitive oblique center is not more symmetric than the primitive oblique. So even these lattices like the square center ones will be called the oblique lattices. Now besides lattice points at the center, we cannot have a lattice point at some other position say out here because a lattice made by such unit cells will not be translationally symmetric at every lattice point, right? So we cannot have non-centered unit cells in two dimensions. So to conclude all Bravais lattice cells in two dimensions can actually be categorized into these four systems depending on the kind of unit lattice that they have.