In this lesson we're going to introduce the relationship between the radius ratio and the coordination number that exists when packing together ionic materials. The radius ratio is defined in terms of the ratio of the radius of the cation, to that of the radius of the anion. Generally we see that the radius of the cation is typically smaller than the radius of the anion. And hence the relationship of r/R will be a number that ranges between 0 and 1. As the radius ratio increases, what we see is that there is a corresponding change in the coordination number at certain specific radius ratios. Once we have identified the behavior of the coordination number, we can see how it leads directly into packing to form crystal structures in these ionic salts. The coordination number and the radius ratio go hand in hand with respect to one another. When we look at a coordination number of 2, for example, which is the simplest of all coordination numbers, what we have is on either side of the cation, the anion. And if we consider this like a daisy chain or a string of pearls, we see that we will have the alternating anion, cation, anion, cation. And what this winds up doing to the material is that it will establish an overall electrostatic balance on the short range as well as the long range with respect to the positively charged cation and the negatively charged anion. If we consider higher coordination numbers. For example if we draw a figure of three circles and we position those so that they are at the centers of the vertices of an equilateral triangle and we look at the figure on the right hand side, we have developed a critical radius ratio that for a cation and an anion, we have the anions just touching one another as you can see on the picture to the right and the cations are just touching the anions. And if we construct a triangle which is given there, which is a right triangle, and we know the relationship that exists between a 30, 60, 90 degree triangle, namely that of 1, 2, square root of 3. We can then determine what that radius ratio is and that critical radius ratio turns out to be a value of 0.155. And it's determined by using that triangular relationship. So the dimension that is on the side which measures the cation, anion distance, that's the hypotenuse of the right triangle. And so what we have then is the fact that we have little r plus big r for the hypotenuse. When we look at the one side, the base of that right triangle, the big R represents then the radius. So we now can use those relationships to describe that critical value of 0.155. Now, if we were to take and go below that critical value of 155 and that's what's indicated on the left hand side of this picture. We begin to see that there are crossover points. Because what we need to try to accomplish is that the cation is surrounded by the anions and they're touching. And what we've see here is, if we try to touch then what we find is, we get overlap. And that overlap is the consequence of Pauli exclusion principle and two electrons having all the same quantum numbers. So as a consequence, if the radius ratio drops below 0.155, then what happens is the coordination number then becomes the value of 2. Already its ration is below 155, then the coordination number is 2. Thank you.