in this article i'm going to give you a basic introduction into unit cells now there's three structures you need to be familiar with the first one on the left is the simple cubic structure the second one in the middle is the body centered cubic structure and the last one is the face centered cubic structure so each of these cubes represent a single unit cell and for the simple cubic structure within one unit cell there's only one atom for the body centered structure you have two atoms per unit cell and for the face centered cubic structure there are four atoms per unit cell the coordination number of the simple cubic structure is six i'm going to run out of space so i'm just going to write c equals six that's the coordination number the coordination number is the number of atoms that's attached to a single atom so every atom in the cubic structure is attached or adjacent to six other atoms for the body centered cubic structure the coordination is eight if you look at the atom in the middle you can see that it's attached to eight other atoms this is one two three four five six seven and the eighth one is in this other corner that you can't see based on the way it's drawn the coordination number for the face centered cubic structure is 12. now the next thing we need to talk about is the volume about 52 of the volume of the cube actually consists of the volume of the atoms in this cube so 48 is empty space for the body centered cubic structure it's 68 efficient so what that means is that out of the entire cube the atoms are arranged in such a way that they occupy 68 of the total space in the cube so that means 32 percent is unused space now the face centered cubic structure is the most efficient cubic structure the percent by volume is 74 percent so the way the atoms are arranged they're arranged in such a way that they take up the most space they're maximizing the use of the volume of the cube so only 26 is empty space in that structure so that's why it's also known as the cubic closest packing because it it's the most efficient use of space when the atoms are arranged and the face centered cubic structure now the next thing we need to talk about is the edge length the edge length I like to call x now the textbooks might use different letters but x is basically the side length of a cube so that's the edge length r is the radius so this is the radius and for this structure this is the radius here now for this uh sphere inside this is the diameter which is twice the radius and here this is r this is 2r and this is r as well now we need to know is the relationship between the edge length and the radius that is the atomic radius of the atoms in each of these structures for the simple cubic structure x is equal to 2r so what that means is that the edge length is twice the value of the atomic radius of the atoms in that structure now if you want to visually see it here's another r so therefore this whole thing is x so x is 2r now for the body centered cubic structure the edge length x is equal to 4 over root 3 times the atomic radius now this information is useful if let's say you're given the density of an element and you want to find the edge length or the atomic radius of that element and they'll have to specify which structure you're dealing with sometimes you might be given the atomic radius and you need to calculate the edge length in a certain cubic structure as well as the density of the material so i'm going to go over some example problems on that in a different video now for the face centered cubic structure the edge length is equal to the square root of 8 times r so you may want to write this information down if you have a test on this stuff now let's go over the simple cubic structure let's focus on this structure and the details that we mentioned now we said that there's one atom per unit cell so here's how you can get that answer we need to realize is that in a simple cubic structure there's eight atoms that's at the edge of the cube now notice that we don't have a whole atom but each atom at the edge is one-eighth of an atom and so one-eighth times eight is equal to one and that's why there's only one atom per unit cell in the simple cubic structure now why is the coordination six well if you focus on one atom there's going to be another atom above it and another atom below it there's going to be another atom to the right and another atom to the left or you can say this is uh east and this is wes or basically just the x direction and let's call this the z direction now there's also another atom in the y direction as well so therefore every atom is surrounded by six other atoms and that's why the coordination number is equal to six for the simple cubic structure now as we already mentioned the edge length is 2 times the radius of the atom so let's call this r and this is r and the edge left is equal to x so this side is x and this side is x the volume of a cube is left times width times height so it's going to be x times x times x so therefore the volume of a cubic structure or the entire cube is x cubed is the edge length raised to the third power but each edge length is equal to 2r it's twice the atomic radius so make sure you remember this equation when dealing with simple cubic structures now the last thing we need to talk about is the volume we said that the atoms in the simple cubic structure only use 52 percent of the volume of the entire cube now to get this ratio or this decimal value which is 0.52 you need to take the volume of the atoms in the unit cell and divide it by the volume of the cube so there's only one atom per unit cell and the volume of a spherical atom is four over three pi r cubed the volume of the cube is always going to be x cubed so we need to do is replace x with 2r so we're going to have 4 over 3 pi r cubed divided by 2r raised to the third power so that's four over three pi r cubed two to the third is eight and then this is gonna be r cubed so you can cancel r cube now four thirds pi that's four pi over 3 as a decimal if you want to write it it's 4.188 and then divide that by 8. if you do that this will give you 0.5236 so 52.30 excuse me 52.36 of percent volume of the cube is basically the volume of the atoms that are in the unit cell so that's how you can get this value now let's talk about the body centered cubic structure so we said that there's two atoms per unit cell now you need to realize that every atom at the corner is one eighth of an atom now we know that there's going to be eight corners to deal with this is number one we use a different color that's the first corner this is the second third fourth fifth sixth seventh and the eighth one is in the bottom left if you were to extend this picture so it's going to be 1 8 times 8 and then don't forget the atom in the middle so that's one whole atom so one eighth times a is one plus one it's two and that's why we say that there's two atoms per unit cell in the body centered cubic structure now the coordination number is eight and as you mentioned before it's easy to see that this atom at the middle is attached to eight other atoms which is basically the eight corners one two three four five six seven and the other one that you don't see which is supposed to be there so the coordination number is eight now let's calculate the volume of the body centered cubic structure at least the ratio of the volume used by the atoms relative to the volume of the entire cube however before we should do that we need to talk about the edge length now we said that the edge left is 4 over square root 3 times r so let's see how we can come up with that figure now for the body centered cubic structure we need to realize is that this is r and from here to here is 2r and this is another r so let me just redraw this so you could see what i'm talking about exactly so here is the sphere in the middle and then here's the sphere at the this edge of this cube and here's the other a sphere so this sphere that's twice the radius that's the diameter and then this is r and this is r so the distance between the bottom edge of the cube and the top edge that's 4r that's we need to realize for the body center cubic structure so the longest diagonal of the cube is 4r now let's see how we can relate that to the edge length of the cube now we know that every side of the cube is equal to x now let's call the diagonal between these two points let's call it l that's different from the diagonal that we want to find which is here that's d so notice that this is a right triangle hopefully you can see the right triangle the right triangle is at the bottom face of the cube and so therefore we could say that l squared is equal to x squared plus x squared so now we need to draw another right triangle which has l as the base this is the next hypotenuse one of the legs is x and the hypotenuse of this triangle is the longest diagonal d so focusing on the triangle in green we can see that l squared plus x squared is equal to the hypotenuse of the triangle d squared so in this equation we need to replace al squared with x squared plus x squared and so we're going to get this equation l squared which is x squared plus x squared and then plus this other x squared that's equal to d squared so 3x squared is equal to d squared now if we take the square root of both sides the square root of 3 times the edge of x is equal to the length of the diagonal and the length of the log is diagonal we can see that it's 4r so root three x is equal to four r so solving for x we need to divide both sides by the square root of three and so the edge length not sure what happened there the edge length is 4 over root 3 times r so that's how you can relate the edge length to the atomic radius of the atoms in the body centered cubic structure so now we can calculate the volume that's used up by the atoms relative to the volume of the cube and we said it was 68 for the body center cubic structure so