Band theory of solids

- [Instructor] In a previous video, we spoke about how electrons fill up different energy levels in a single atom. We saw that using the Pauli's Exclusion Rule, which says that no two electrons can ever have identical energy levels. Electrons fill up in the following manner, say for example, sodium, which has 11 electrons, they fill up this way. But the big question was what happens if you have two atoms very close to each other, almost forming a molecule like two sodium atoms very close to each other? We also saw that now each atom cannot have a configuration like this, electrons cannot fill up this way, because now if they do, then Pauli's Exclusion Rule gets violated, and we end up with lots of electrons with identical energy levels. So the big question now is what's going to happen with this system? Now, if you're not familiar with this stuff, or you need a refresher, it would be a great idea to go back, watch that video first, and then come back over here. But anyways, when atoms come very close to each other, what really happens is that the atomic orbitals of each atom overlap with each other and transform into what we now call as a molecular orbital. So if you look at the one S orbital of this odium, and the one S orbital of this sodium, they combine and form a one S molecular orbital. You see, atomic orbitals are something that is unique for each atom, but now a molecular orbital is a unique energy level for the entire molecule. And guess what, because two atomic orbitals are overlapping our molecular orbital will end up having two energy levels. So, if we are looking at one S, let me just try to draw that. So if you redraw the energy level, but now not for the atom, for this entire molecule, then the one S of this and one S of this combine and form a new one S molecular orbital. And that molecular orbital itself has two energy levels, two energy levels. These two energy levels belong to one S molecule orbital. We usually call them as one S, and one S star. And again, this is something that you may have already learned in chemistry, molecular orbital theory. This lower energy orbital is called as the bonding orbital and the higher energy orbital is called as the antibonding orbital. And again, if you need more clarity, it would be great to go and watch those videos in chemistry. But guess what, we don't need things so rigorously over here, all we need to understand is that the two atomic orbitals combine and become one molecular orbital, and they end up having two energy levels. And the way I like to think about this is I just like to think of this as so conveniently, solving Pauli's Exclusion Principle, because now the two electrons from this sodium can fill up this level of the molecular orbital, and the two electrons of this sodium can fill up this, this molecular orbital, energy of the orbital. And we are fine Pauli is now happy because now these two electrons are no longer identical. And the same thing is going to happen to our two S orbitals, they're also going to overlap and now you'll have a two S molecular orbital, which will also have two energy levels. And guess what, we don't have to remember this star. Star, I'm just gonna call this as one S molecular, two S molecular. And the same thing is going to happen over here as well, two electrons from this atom maybe can occupy the lower ones, and the two S electrons of the other can occupy the higher ones, and so on and so forth. And now notice, all electrons have different energy levels. So Pauli is fine now. And so now what do you think is going to happen if we add a third sodium atom to the mix? Well, now three atomic orbitals are overlapping. And as a result, our molecular orbital conveniently ends up having three energy levels, three energy levels like this. I'm not gonna draw them again. So you'll end up having a new draw somewhere over here. Now, each molecular orbital will have three energy levels. And I think now you can understand where we're going with this, the more atoms that we add, the more energy levels our molecular orbitals end up having. And eventually, if we have an entire solid, which is made of sodium, which has something like 10 to the 23 atoms packed together, then we'll, our new molecular orbital of this entire solid will have now 10 to the 23 levels. So if you're to draw this, this would be interesting. What would that look like? Again, this now is the energy level of all the electrons of this solid. And now the one S molecular orbital will have 10 to the 23 levels. And the way we can draw, I can draw that. I will draw the lowest one over here and the topmost one, and then all I have to do is fill 10 to the 23 levels. That's easy. Here's how I do that. I mean, think about it, if I were to draw 10 to the 23 levels inside, don't you think this is what it would look like? 10 to the 23 stacked up lines, that's what it would look like, right? And so notice is no longer can we identify individual energy levels, because the gaps between them would be extremely tiny. And as a result, we like to think of this energies as continuous energies. And so we like to think of this as an energy continuum. We can forget about the small gaps that, energy gaps that are present between them. And then we do think of it this way, we can now call this, we can now call this as, they're not quite as a molecular orbital anymore, but instead, we'll call this as an energy band. So this is called as an energy energy energy band. And the word band is signifying that we are ignoring the small spaces that exist between those 10 to the 23 levels, and we're just assuming that this whole thing is one big chunk of energy. All the energy levels from here all the way to here, are available, continuously available. And that energy continuum is what we call as an energy band. And similarly, we'll have a two S band. So we're gonna draw it the same way. I'll draw the lower one and the higher one, and then 10 to the 23, lines in between. So this we will call it as the two S band. So on and so forth. And there are other bands as well like three S and three P bands, I just ran out of space over here. And notice that since one single discrete level can hold two electrons, if there are N atoms, then there will be N levels over here. So let's write that down, if there N atoms over here, then there'll be N levels. And so the total number of electrons that it can fit would be two N. So this band can fit two N electrons, this band can also fit two N electrons. But notice that a single P level can fit six electrons. So the two P band can now fit six N electrons, and so on and so forth. Alright, to summarize, we saw that if you have a single atom, or if you have a gas, because even in a gas atoms are so far apart, we can assume they're infinitely far apart. And we can pretty much assume that they're single atoms. So this also works for a gas. So we can write that down, this works for a gas. So if you're dealing for gases, you're dealing with gases, then every electron has a discrete energy level. And if an electron wants to go from one level to another, it really has to jump, there are no continuous energies available, sort of like steps over here. But as atoms come close to each other and eventually form a solid, then they're forming energy continuum, and we call the continuum as bands, and within the bands, the energies which are available are continuous. Electrons can possess any continuous energy it wants within that band. And the name of this theory is no surprise, the band theory of solids. And now using this theory, we can understand how free electrons are generated and why certain materials have readily free electrons available making them a conductor and why some others don't.