 In this video, we are going to see how we can use the effective number of atoms, the A effective, to calculate the packing efficiency of different unit cells. So what exactly do we mean by packing efficiency? To understand, let us say we had a bunch of shoes that I wanted to pack inside this bag. Now you know from experience that if I randomly throw these shoes inside the bag, I won't be able to pack in as many shoes compared to what I could if I had done it in a very neat and ordered way, right? So in this random scenario, I could pack in say, let me see 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16. So we had 16 shoes that I packed in out here. While in this scenario, I could pack 1, 2, 3, 4 times 5, 20 shoes. So clearly, I utilized the available space much more efficiently in this case compared to just randomly throwing shoes. So packing efficiency can now be thought of as how efficiently we can pack stuff in a given volume. Now that we know what packing efficiency means, how do I put a value to it? How do I describe this mathematically? Well, let us say that the bag that I had was of 50 liters. And let us assume that each of these shoes had a volume of 2 liters each, fairly big shoes if you ask me. So out of the available 50 liters in each scenario, in the first case, I had 16 shoes, each having a volume of 2 liters. So the total volume that is occupied out of this 50 is 32 liters, right? Similarly out here, there were 20 shoes, so the total volume that is occupied is 40 liters. Now if I try to calculate per unit volume, then the volume occupied will be 32 liters by 50 liters and 40 liters by 50 liters which will come out to be equal to 0.64 and 0.8 respectively. Now this ratio of the total volume occupied by the total available volume, this ratio actually gives the fraction of the total available volume that is being occupied, right? So the volume occupied by volume available will always be a dimensionless quantity, a fraction. And this fraction is called the pecking fraction. So greater the pecking fraction, more will be the volume occupied per unit available volume, so more will be the pecking efficiency. Now pecking efficiency is generally reported in percentages rather than in fractions. So the percent pecking efficiency is defined as pecking fraction multiplied by 100. So randomly throwing shoes has a pecking efficiency of 0.64 into 100, so it's 64%. While arranging shoes in a neat and ordered way resulted in an efficiency of 80%. Let us now turn our focus into calculating the pecking fraction of different unit cells and let's start with the simple cubic unit cell. A simple cubic unit cell has lattice points only at the corners and if I replace these lattice points with the motif which in case of metals is mostly a single atom, then the unit crystal is going to look something like this, right? Now do remember that the lattice points represents the center of the atom by convention. So the whole of this atom doesn't lie 100% within this unit cell but is instead shared between different unit cells. Now I have drawn unit cells only along this direction but there are actually unit cells in all the directions, right? If we now focus only on this unit cell, only on this cube, we will realize that these atoms do not fill up the whole of this cube but instead there are these gaps that are left behind. So what we are really interested in is in calculating the pecking fraction of this particular cube. Now to calculate the pecking fraction, we need to know the volume that is available and the volume that is occupied. So if this cube has an edge length A, then the volume available is the volume of this cube which is nothing but A cube, right? Now to calculate the volume occupied, we need to think about the volume of these atoms that lie within the cube. So how do we do that? How do we figure this out? Well, we know from a previous video that atoms at these corners penetrate into 8 different unit cells. So this atom is only 1 8 inside this cube. So the volume of this atom that is inside, the volume inside, will only be 1 8 the total volume of the atom, right? So atoms at each of these 8 corners are only 1 8 inside. So the volume of this cube that is occupied with these atoms will be equal to the volume of each of these atoms that lies inside the cube. So it's 8 times the volume of each atom that is inside. Now because the volume inside is nothing but 1 8 the total volume of the atom, so 8 and 8 will cancel out and ultimately I'll be just left with the volume of an atom. Now this makes sense because the A effective of a simple cubic unit cell as we have seen from the previous video comes out to be equal to 1. So we can think of this unit cell as having one atom that lies 100% inside. So the volume occupied will be equal to 1 times the volume of the atom, right? So we can now go ahead and say that to calculate the volume occupied, all we need to do is figure out the effective number of atoms within the unit cell and multiply it with the volume of the atom. So the pecking fraction of any unit cell can be written as A effective multiplied by the volume of the atom that makes up the unit cell divided by A cube. Now for a simple cubic unit cell the A effective is equal to 1 and if we consider these atoms as spheres having radius r then the volume of the atom can be considered as 4 by 3 pi r cube which is the volume of a sphere and so the pecking fraction will be equal to 1 into 4 by 3 pi r cube divided by A cube. Let us now try to calculate the pecking fraction of atoms arranged in the form of an FCC. An FCC arrangement has atoms not only at the corners but also at the center of each of the faces. Now I'd like to point out that all the atoms are actually identical to each other in all respects. I am just drawing these ones in green just to highlight the fact that they are face centered atoms. Let me also keep the unit lattice out here for reference. Now in an FCC the effective number of atoms inside the unit cell will be equal to 1 by 8 times of 8 as atoms at each of these corners are only 1 8 inside plus 1 by 2 into 6 as atoms at each of these face centers are only 50% inside. So the effective atomic number will come out to be equal to 4. So we can now think of this cube as effectively having 4 atoms inside it. And if the radius of these atoms was equal to r then the volume of this cube that is occupied by these spheres can be written as 4 into 4 by 3 pi r cube, right? So if this cube had an edge length of A then the pecking fraction can be written as 4 into 4 by 3 pi r cube divided by A cube. This 4 is the effective number of atoms that can be thought of as being 100% inside the cube the A effective. So what do you think would be the pecking fraction of a BCC unit cell? Pause the video and try to come up with your answer. So to calculate the pecking fraction we need to figure out the volume of these atoms that are inside the cube the volume occupied and the volume that is available. So if we say that this cube has an edge length of A then the volume available is going to be A cube, right? Now to figure out the volume occupied we need to figure out the effective number of atoms inside this unit cell and multiply it with the volume of the atom. So if we consider these atoms as spheres of radius r then the volume of each atom will be 4 by 3 pi r cube. Now A effective for BCC will be 1 by 8 into 8. This is the contribution from the corner atoms plus 1 as the body center atom lies 100% inside the cube. So this gives a value of 2. So the pecking fraction of a BCC unit cell will be 2 into 4 by 3 pi r cube by the volume of the cube which is A cube. So to summarize to calculate the pecking fraction we need to figure out the effective volume of the atoms. Set the unit cell and divide it with the volume of the cube. If we now look at these 3 unit cells which are made up of the same atom having some radius say r. So if we have the same r we might be tempted to say that because FCC has the highest number of atoms per unit cell. So because A effective of FCC is the highest so it has the highest pecking fraction. However this would be an incorrect assumption because this value of A is not the same in all the 3 cases. While atoms at the corners of a simple cubic unit cell touch each other, these atoms do not touch in FCC or in BCC. This is because if you have an arrangement in which these corner atoms touch each other and then if you put in a face centered atom then the resulting configuration cannot be a cube but it will instead be a cuboid. So if you touch these corner atoms it will be impossible for these two atoms to touch each other. So both these edges will have different lengths. So to have a cube with equal edges we need to spread out these atoms which ultimately results in them not touching each other. Similarly in BCC the atoms at these corners cannot touch each other because if they did they would result in a cuboid rather than a cube. If we put an extra atom at the center between these two layers it will be impossible for these two atoms to touch each other and so to make a cube we need to spread them apart and so none of the atoms at the corners touch each other. So just by knowing the A effective of a unit cell we will not be able to calculate the pecking fraction. We also need to know the value of the edge length A. So how do we calculate this edge length for each of these systems? Let us explore more in the next video.