Structure of Materials:: Crystal Structure: Lattice and Basis

 Crystal Structure: Lattice and Basis

In the previous lecture, we learned about the fundamentals of bonding without getting into any quantitative treatment, and we will do some quantitative treatment when we learn about those solids and their structures a little later in the course. So, for the sake of simplicity, let me just brief that bond energy determines properties like melting point, coefficient of thermal expansion, and elastic modulus. Higher the bond energy higher the melting point, higher the modulus, and lower the coefficient of thermal expansion. So, now let us move on to the atomic structure of materials, and the objective is to learn how are atoms arranged in space.

Of course, there is whole mathematics that is behind the structures but will not get into those that mathematics right now.
So, how are these atoms arranged in space? What do we mean by a crystal structure? So, before we get into this, we know that various bonds connect the atoms, but the question is, how these atoms are spaced in the universe or how are they replaced in the space? So, before we talk of atoms, we start with random points and space.

So, there are various methods by which you can do that the points may be spaced like this, or the points may be space like that, these are just to illustrations, and there are various other possibilities. So, in one situation, you have a distribution which is random in this case what do you call it, periodic or regular; at least you can see a pattern there on the right side, but you cannot see a pattern here on the left side. So, here it is random, or without any periodicity, or lack of periodicity, most materials in nature except for few tend to have atoms arranged in a regular fashion. So, when you look at the atomic structure, you will see that the atoms are placed in space at regular patterns. Where did the motivation behind, which come from works of some previous scientist, for example, there was a scientist called steno.

Steno was a scientist who lived between 1638 to 1686. He made drawings like quartz, and he made drawings of hematite. He made these regular shapes, now these shapes there many of them he drew not only these, so I am going to draw only a few of them. So, he made the shapes because he observed that crystals tend to have certain shapes, and there is a constancy about phase angles. Some of these angles that you see here, they tended to have certain relations with each other, and you could feed these angles into a mathematical framework and get an order about these angles. There was a relationship between these angles. So, steno was the first guy to observe that crystals have certain geometrical patterns about them, and the angles of phases and edges can have correlations.

Then, later on, Huygens, who lived between 1629 to 1695, made drawings of calcite crystals. So, calcite crystal had a peculiar sort of geometry. From the calcite crystal geometry, if we see macroscopically, you can see a regular shape, and then that must be about the atom sitting inside the crystal. It also because the atoms in this structure sitting in an ordered fashion in something like that, I am not saying it is exactly ordered structure, but there must be some order about the atoms which are sitting inside. If the atoms are sitting inside in a regular fashion, the crystal itself will manifest in a regular shaped. So, this was the underlying basis of why you can say that first thought was the regular shape of crystals, which is probably due to the regular arrangement of atoms in crystals. So, these were some earlier indications of why atoms could be arranged in a periodic fashion in a space.

So, we replace the atoms with points, and then, of course, you have something like that. So, this is non-periodic, and this is periodic, and if I put atoms here instead of points, I make a crystal. So, in this crystal, we consider that the atoms have a spherical shape. If the material is having no longer in periodicity, such materials are called amorphous. And the materials in which periodicity is present are called as crystalline. So, amorphous materials are typically things like glasses, but all other materials almost I would not say all of them, but almost all of them. So, all others almost are crystalline in nature, which has a periodic arrangement of atoms.

In this case, you had a random arrangement of points without any periodicity in the structure, here you have a periodic or regular arrangement of points. So, in this case, every point has a different neighborhood. Because each point is randomly distributed in a space there are no correlations of lengths and angles and directions, as a result of the if I look around myself there are four guys, if a certain four guys are in certain distance certain angles, but if another guy close to me looks at his surroundings, he points at all there could be five, there could be six and at different angles and directions.
            The coordination number will be different, but the coordination number is something that is defined by fix distances. In this case, even the distance is not fixed. So, there is no fixed coordination number. So, that every point has a different neighborhood; in this case, when you standard point b, you see the same arrangement. So, each point has an identical neighborhood. So, that is the first thing that now let me make another structure that may look periodic by still not periodic. So, let me draw something here.

Now, if I let us say standard point A and standard point B, do they have identical neighborhoods? This is a hexagonal arrangement. For A, you have one neighbor on here another neighbor here, for B, you have one neighbor here, another neighbor here. So, the number of neighbors is the same, but is the arrangement of neighbors not the same. For A, you see two neighbors on the left at a certain angle in a certain direction and another neighbor on the right in a certain direction; the distances are the same. However, the directions are different for B, you see two neighbors on the right, and one on the left. It is a mirror image, but it is not identical.

Now, let me put a point here, so you have a point A, and you have point B do they identical neighborhood now?

They now have identical neighborhoods. So, I would say merely having a periodic arrangement there, but it is not fine; the identical neighborhood has to be obeyed. So, I would say this structure is not a lattice. So, by definition, when point arranged themselves in a space in such a fashion, so that the arrangement is periodic and every point has an identical neighborhood, these two conditions qualify that arrangement as a point lattice. So, this is called as point lattice. So, that two definitions are periodicity and identical neighbors. So, these are two distinct in scenarios, so once we defined a periodic
lattice in this fashion, let me draw a periodic lattice.

 

This is a periodic lattice, and in this periodic lattice, I can draw the smallest repeating unit, this is the smallest repeating unit, called a unit cell. The sides of this unit cell are called as unit cell parameters. Here, a and b are lengths, and γ is the angle between the two edges; these are called unit cell parameters.
        Now, my question is the choice of this unit cell unique? I can also make a unit cell when you have to have the smallest repeating unit. If you make this one here and if I put a point somewhere here, this is also a valid unit cell. So, the choice of the unit cell is not unique. So, which one you prefer to choose? You choose the one with the highest symmetry.         So, that is why the concept of symmetry comes into the picture.
So, one chooses a unit cell with the highest symmetry, and we will see what the meaning of this highest symmetry is, and we will come to the definition of the symmetry in the next lecture. So, you can have a similar arrangement in 3D.

So, it will make a unit cell, which is a 3D unit cell, and in 3D, you will have lattice parameters as a, b, c, and α, β, γ. So, it will look something like that a parallelogram, here a, b, c, and angles are α, β, γ. So, between a and b, you will have γ, and between b and c, you will have α, and between a and c, you will have the β. So, this is the periodic structure that will be a 3D unit cell of material, so basically, the unit cell requires few points for description.The first thing you need to measure is size and shape, which are determined by α, β, γ, and a, b, c. What is the other thing that you require now if you replace the points by atoms? Because atoms can make it a little complicated, you may not have just one type of atom, you may have different types of atoms. So, that is why we begin with the point, now let us say we replace the point by an identical atom. What we required is the kind of atom and the fractional coordinates of the atoms. So, these few things are required to specify. So, if I replace these points by atoms here.

So, these are all atoms, so to describe the unit cell, you require these positions to be known now I have we have seen. So, a, b, c are the lattice parameters, and α, β, γ are the interaction angles or angle between the edges the unit cell can also be defined in a little bit more quantitative manner.

 

So, if you have an arrangement of points in space, you have to define the first origin. Then chose a lattice factor R, and this R can be defined as n1a1+n2a2+n3a3 in the 3D, or here you can see just in 2D. So, if you have a lattice factor, R., now, you have these two vectors you can define. These two vectors will make the unit cell; alternatively, you could have chosen your vector, and this will be the periodic vector that will construct the lattice. So, depending upon the choice of lattice vector, you can create these arbitrary unit cells, you can have this one as well. So, there are various choices by which you can make the where these unit cells, but as we discussed as we just talked about earlier, it is the symmetry of that unit cell, which will be determined which has to be taken as an equilibrium unit itself. So, now, what is the difference between the lattice and crystal?

Can you tell me, what is the difference between lattice and crystal? Lattice is only about the points in space, or you can say periodic arrangement of points in space. Then, what is the crystal? Crystal is a 3D arrangement of atoms in space. So, based on this particular aspect of lattice now within this lattice, I have a unit cell, and if these are all atoms, this will be called a crystal lattice.

Now, we said that lattice plus atoms make a crystal lattice. Moreover, these atoms basically can be put in a more specific term, which is called a motive or basis, it could be atom or group of atoms, or multiple random types of atoms arranges various places. So, for example, let me draw a simple lattice.

Now, I can just put a simple one atom, and this is the simplest 2D lattice; I can make some change here, I can add a molecule. It is not just one atom that goes to a point, and it is this molecule that goes to a point. So, you have replaced the points with these molecules does this make a lattice? Does it still retain the definition of lattice in the previous case? I did in the previous case I did now that you have an atom, which is that you can consider this molecule as an asymmetric atom.

 So, the question is, does this modified motive retain the definition of lattice? Let me give you another scenario instead of having this. So, I will draw it again a little smaller now again I draw these atoms, now instead of putting them in this fashion, I let us say I will put them in this fashion. So, these atomic arrangements have modified from the previous one. So, the question arises; rather, they still maintain the validity of lattice; they seem as if they are periodically arranged in the lattice, But the question is do they maintain the validity of the definition of the lattice. So, we will discuss these aspects related to placing a different kind of objects or motifs in space and how do they alter the definition of lattice and then invoke the symmetry to understand this in the next few lectures.

For Next Lecture Click below

 Structure of Materials : Bonding in Materials

Structure of Materials : Correlation between bond and physical properties

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