2.7A Radius Ratio & Coordination Number Part 1

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From the course by Georgia Institute of Technology

Material Behavior

326 ratings

Georgia Institute of Technology

Material Behavior

326 ratings

Have you ever wondered why ceramics are hard and brittle while metals tend to be ductile? Why some materials conduct heat or electricity while others are insulators? Why adding just a small amount of carbon to iron results in an alloy that is so much stronger than the base metal? In this course, you will learn how a material’s properties are determined by the microstructure of the material, which is in turn determined by composition and the processing that the material has undergone. This is the first of three Coursera courses that mirror the Introduction to Materials Science class that is taken by most engineering undergrads at Georgia Tech. The aim of the course is to help students better understand the engineering materials that are used in the world around them. This first section covers the fundamentals of materials science including atomic structure and bonding, crystal structure, atomic and microscopic defects, and noncrystalline materials such as glasses, rubbers, and polymers.

From the lesson

Atomic Structure and Bonding [Difficulty: Easy || Student Effort: 2hrs]

In this module, we will discuss the structure of the atom, how atoms interact with each other, and how those interactions affect material properties. We will explore how the types of atoms present in a material determine what kind of bonding occurs, what differentiates the three types of primary bonds - metallic, ionic, and covalent, and the implications of the type of bonding on the material microstructure. You will learn how atoms arrange themselves as a natural result of their size and bonding. This knowledge will provide you with a foundation for understanding the relationship between a material's microstructure and its properties.

2.7A Radius Ratio & Coordination Number Part 14:30

2.7B Radius Ratio & Coordination Number Part 24:58

2.7C Radius Ratio & Coordination Number Part 33:46

Meet the Instructors

Thomas H. Sanders, Jr.
Regents' Professor
School of Materials Science and Engineering

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.

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