 We are now going to talk about crystalline solids. We can simply call them crystals from now on at a much greater detail. And when we do that, there is this particular term that keeps cropping up a particular phrase called the Bravais lettuces, the Bravais lettuces. So in this video, we are going to try and actually understand what lettuces are. What it means to call a lettuces a Bravais lettuces and how they are related to the study of crystals. This is a naturally occurring crystal of sodium chloride. Now if I like really look closely into the crystal, if I like say zoom into this part, then I will find out that this crystal is made up of these sodium and chloride ions that repeat throughout the crystal in a very regular ordered way. Right? Now if you could somehow miraculously stand on any of this particular NSL unit and if this crystal was like infinite in all directions, then if you looked around, the kind of pattern that you'll observe will be exactly the same as observed from any other unit. So you might like say take this particular unit and look around, you'll see that the kind of pattern that is getting formed is exactly the same. In other words, whatever way you try and look into this crystal, you'll find the exact same pattern and so we say that this crystal has long range order because the pattern is same everywhere so it is long range order. Similarly if you take any other crystal, say if you take a crystal of calcium fluoride and if you look into it, we will see that it is again made up of these calcium and fluoride ions that repeat throughout the crystal in a very regular ordered way. Right? And just like earlier, whatever way you look into the crystal, you'll find the exact same pattern that is getting formed everywhere. So a crystal can be defined as periodic repetition of atoms, repetition of atoms or we can say groups of atoms with long range order, with long range order. Now because all crystals have long range order, they have the same pattern everywhere so all crystals also have translational symmetry. So what do we mean by this translational symmetry? Let me explain. Let us take a look at this two dimensional crystal. This particular crystal is made of two atoms A and B and this AB unit is repeated throughout the crystal in a very ordered way. Now if this crystal was infinite in all directions, then the kind of pattern that you'd observe standing on this particular unit will be exactly the same as standing on any other unit. Right? Now because the environment around every repeating unit is exactly the same, so if I take this whole crystal and if I move it from one repeating unit to another repeating unit, if I do that, we can see that all the other atoms also line up. So if I translate the crystal from one repeating unit to another repeating unit, the crystal will remain exactly the same. So this is what we mean by translational symmetry of a crystal. Let us take another example. This is a different crystal made by the same unit AB. Now obviously if this was an infinite crystal, then the kind of pattern around this particular unit would have been exactly the same around say any other unit. So if I take this whole crystal and if I move from one repeating unit to another, there is definitely going to be translational symmetry. Let's talk about a few more examples. First let me like take these crystals and put them up. Okay, so now will you call this to be a crystal? Of course not. Right? There clearly is no order in this particular system. And obviously if we move from one repeating unit to another, the atoms are not going to coincide and so there is no translational symmetry. So clearly this is not a crystal. Let me take one final example. Do you think this is a crystal? Now just by looking at it, we can intuitively say that this is a crystal, right? And this is because we humans are very good at recognizing order, but we can go ahead and check for translational symmetry. And because there is translational symmetry, so definitely this is a crystal. Now we can look at these three crystals and see that even though they are made up of the same repeating unit AB, the way these units are arranged, the kind of pattern that these three crystals make is pretty different. Now we can see it, but how do we understand this mathematically? To do that, we are going to replace our repeating unit AB with a point. So I'm going to put a point out here and this point is going to represent this particular repeating unit. Now, similarly, I should again put a point out here and I should do that throughout my crystal. So now I can think of this particular crystal simply in terms of these points, right? Just like in the original crystal, these points will also have translational symmetry. Now I can also do the same thing with my other crystal. I can replace the repeating unit with points and in this way, we can reduce the pattern of the crystal into these points. And once more, just like in the original crystal, these points are also going to have translational symmetry. And in this crystal, these points has translational symmetry. Now of course, there won't be any translational symmetry in this particular case because this is not even a crystal. Now one more thing that I'd like to add is that it doesn't really matter where I exactly put my point. So to replace this unit, I could have put my point out here instead. So in this case, I should also put my point out here. And if I do it throughout the crystal, I will end up with the exact same pattern, right? So it actually doesn't matter where you exactly put your point. But by convention, we put it generally at the center of an atom. Now these points that I replace my repeating unit with these points are called lattice points. So these are different lattice points. And a collection of lattice points is called the overall lattice. So this is a different lattice made of these lattice points. And similarly, this is another lattice that is made of these lattice points. So a lattice is nothing but a periodic arrangement of points. Periodic arrangement of points. So a crystal is a periodic arrangement of atoms while a lattice is a periodic arrangement of points. Now this lattice is purely a geometrical construct. And only when I replace my lattice points with my repeating unit, I get the crystal. So this repeating unit that I have to put at each of the lattice points, this is called the motif or the basis. So a crystal is nothing but equal to the lattice plus the basis. The lattice gives the underlying translational symmetry of the crystal. And only when I add the basis to it, I'll get my actual physical crystal. Now mathematically, there are many different kinds of lattices. There are many different ways of arranging points so that there is periodicity. However, out here when we are studying crystals, the kind of lattice that we are going to encounter is always going to have translational symmetry when I go from one lattice point to any other lattice point. So these lattices that we get by reducing crystals, these are actually a special kind of lattice and they are called the Bravais lattice named after the French mathematician Auguste Bravais who kind of built the maths to describe this particular lattice system. For example, if we look at this particular 2D crystal, if you look at it closely, you'll realize that this is made of this particular repeating unit. So this is the basis or the motif. And if I replace my motif with points, if I do that, then the kind of lattice that I'll get is going to be translationally symmetric at every lattice point. So it's going to be a Bravais lattice. So what are non-Bravais lattices? So we have learned that a lattice is nothing but a periodic arrangement of points. So say I take one point, this one, and say I put place another point out here, and then I keep repeating this at equal intervals. So this is a periodic arrangement of points, right, all the points come after equal intervals. So right now I've just created a 1D lattice. And hey, if this was an infinite 1D lattice, then it would have been translationally symmetric at every lattice point. So this would have been a Bravais lattice, right? But we are talking about non-Bravais lattices. Okay. So now what if I place points like this? What if I place two points out here, and then I put a gap of two units, and then I again place two points, then I again take a gap of two units, two points and so on. So even this is a periodic arrangement of points, right? This set of points keeps repeating at regular intervals. So there is definite periodicity. So this is also a lattice. However, is this a Bravais lattice? If we move from one lattice point to another lattice point, all the other points do not coincide. So it is not a Bravais lattice. However, because it is repetitive, so there is translational symmetry, but we need to go till here, right? So this is a lattice, but this is not a Bravais lattice. And we'll come to that again. Now what if I place a point out here, and then I place a point after one unit, and then I place a point say after two units, and then I place a point say after three units, and then after four units and so on. Now this is not even a lattice. Well, this is because the points are not getting repeated at regular intervals. There is no periodicity. So this is actually a periodic. So therefore, this is not a lattice. Now if we come back to our second lattice, this one, if I replace these two points with this particular point, only this one say. In other words, if we consider this to be the lattice point, and these two points as the basis, I'll be left with this particular lattice. And if I add more points on both sides, then you might have guessed this becomes a Bravais lattice, right? In other words, a periodic pattern can always be converted into a Bravais lattice. Let us now take a look at this arrangement of points. Now this is also a periodic arrangement, and to help you better visualize it, let me show you these guidelines. This looks like a honeycomb. So let's call it the honeycomb lattice. Now is this a Bravais lattice? Now if you take the whole lattice and move from one lattice point to another lattice point, if I do that, we can clearly see that all the atoms do not line up, right? So this does not have translational symmetry to every lattice point. So this is not a Bravais lattice. However, if we move a bit further, we can see that now the points line up, right? So this definitely is a repeating pattern. And of course, it has to be repeating, that is periodic for it to be called a lattice, just that the periodicity does not repeat at every lattice point. However, as I have already told you, any repeating pattern can always be converted into a Bravais lattice. So even out here, there are many ways of making a Bravais lattice out of it. And one of the ways would be to consider these two atoms as my basis and replace it with a lattice point out here. So even out here, this will be the lattice point. And the other two atoms can simply be thought of as the basis. So in this way, if I put it throughout the crystal, I'll ultimately end up with a lattice that looks like this, right? And this particular lattice is translationally symmetric to every other lattice point. So all the lattice points are identical out here. And all the atoms will coincide when I go from one lattice point to any other lattice point. So clearly this is a Bravais lattice. In other words, any repeating pattern, because it repeats, so it is going to have translational symmetry. And this underlying translational symmetry can always be expressed using the Bravais lattice. Now because crystals are made by repeating the basis or the motive, so all crystals will have an underlying translational symmetry. And because all translational symmetry can be fundamentally expressed in terms of the Bravais lattices, so therefore any crystal structure can always be explained in the form of a Bravais lattice.