Structure of Materials :: Primitive and Non-primitive Lattices

Primitive and Non-primitive Lattices

In this lecture, we will discuss crystallography. So, what we will talk about, and this is the distinction between so, primitive and non-primitive lattices.

Now, this understanding is important to understand the crystallography to a reasonable level. So, let me give you a recap of lecture four. In the previous lecture, we learned about what is a lattice.

So, the lattice, a 3D arrangement of a regular arrangement of points in space with the condition that each point must have identical surroundings. So, this is point lattice, as long as you have a point, it is point lattice. Moreover, if you put some objects such as atoms to it makes as a crystal lattice.

So, the lattice is a 3D arrangement of a periodic arrangement of atoms in a space. So, instead of points, you have atoms or molecules a group of atoms. Moreover, the second thing that we considered was, a lattice can be represented by vectors such as cartesian vectors.

So, let us say these are x, y, z, and vector R is equal to n1a1 +n2a2+n3a3. So, n1, n2, n3 are integers, which determine the number of translational steps that you take along a1, a2, and a3 axis. The vector R will be n1a1 +n2a2+n3a3. So, this is what this vector will be. So, and lattice parameters of this can be determined as a, b, c, α, β, γ. a, b, c are the lengths or unit cell lattice translations, and α, β, γ are the angles. Now let me now move to the next topic of this series called primitive versus non-primitive lattice.

So, earlier, we were talking about point lattice. Now I was saying object or atom, in the technical language we call it as a Motif or basis. So, when you combine these two , what you get is a crystal lattice. So, what is this Motif?

So, either you can say a Motif or basis. So, this is an atom or group of atoms. So now, let me just give you a simple example. So, you had a periodic lattice like this. So, translations are not exactly equal, but I hope you understand what I mean here. So now, let me replace each of these points by atom A. So, this becomes a periodic lattice of A. Now, for the sake of simplicity, we have taken A as a circular or a spherical atom in 3D.

        Now, it does not need to be like this, for the sake of change or modification, I can convert this as a molecule. So, for example, if it was a molecule like this. So, B atom, which is attached to A. So, the B atom, which is the smaller one. So now, in this case, this whole thing is a molecule. So now, this is put on the same point you can make it in a different manner.

From the point lattice, what I do here is, I put one atom here, another atom here, and so on. The unit cell is the repeatable unit cell. So, in the previous case, the repeatable unit cell is still is this one, or you can have this another parallelogram. Let us say this one, but for the sake of simplest and most symmetric, we take the square one or rectangular one. So, these are the two smallest unit cells, two examples of the smallest unit cells.
        In this case, what is the scenario? Now is the smallest unit cell the same as before, or is it this one. No, it is not. You can see that it violates the definition of lattice since each lattice point does not have the same neighborhood. So, the smallest lattice, in this case, happens to be this on, drawn with a different color. So, what is the motif or basis in this case?
        The basis is combined where two atoms put together, the unit cell has gotten bigger than what we have been going so far, you can just make it smaller and blue remain the same size. However, you can now construct a smaller unit cell, if you have a group of atoms positioned. So, if you look at this unit cell, it is like a dumbbell right. You can move the corner of the unit cell anywhere.

So, what it will look like a dumbbell-shaped representation, so let us say this is my red atom, and this is my purple atom somewhere here. So, this is a group of atoms or a molecule. This is a simple example of A B kind of molecule. You can have a much more complicated one. So, you can have AB2, A2B3, etc.
        So, the moment you have more than one atoms. You need to look at the repeatable unit cell very carefully. So, as to ensure that at least one formula unit lies within the unit cell, so, how many formula units are there in a one-unit cell? You have one formula unit in the unit cell, and you can see that there is one A and one B. You can have more than one of more than one as well, and we will see that later on, but at least one must be there. For a that is the bare minimum requirement. Now, so, based on this, let me define this. So, you have a lattice-like this.

I just draw a few of them, and when I have one atom of this, of same kind, then I draw a repeatable unit cell. This unit cell is the smallest, and how many atoms it consists of? Only one atom. So, this is called a primitive unit cell.

Now, if I can draw a bigger unit cell like this, how many atoms does it have? It has two atoms. So, this is called a non-primitive unit cell. You can go higher and higher, things are simple when you have just one atom, and things become a little complicated when you have more than one atom. So, I gave you one example last time. So, if you take the example of more than one atoms. You can see a non-primitive unit cell that is next order non-primitive unit cell contains twice the number of atoms. As the primitive unit cell contains the area, the volume of the non-primitive unit cell is a number of atoms multiplied by the area or volume of the primitive unit cell. So, you can see that, VNP =2*VP.

In the first example, and here I draw little green dots. This is the same as before, and it again has a primitive unit cell, which is like this. This is the primitive unit cell, and it contains one molecule of AB. Again, you can draw the non-primitive one, and this will contain 2 of AB. Now let us make things slightly more tricky, with this what I do is that I again draw a smaller simple lattice.

Furthermore, I put the green atom here. So, which is the repeatable unit cell now? Is this the repeatable unit cell? Yes, this is not, right? What about this? So, this is not. So, you will know the straightness, it is not very straight here, what about this? This is the smallest unit cell, which means this is the primitive lattice. So, this is the primitive lattice. So, the primitive unit cell does not necessarily have to contain one formula unit in the case of molecules. It can contain more than one formula unit, and it depends upon the relative orientation of molecules to each other. And then you have to figure out which is the smallest repeatable unit cell. So, again, you can make it a little bit more complicated; there are several examples of that. So, these are certain illustrations that you can do away with.

Now, let me give you a few examples of these primitive and non-primitive lattices in a little quick manner.

We said that the motif or basis is an atom or a group of atoms that are associated with each lattice point. And I gave you a demonstration of when you put a molecule replaces an atom that things could be quite different. The definition of primitive a non-primitive unit cell does not remain the same.

 We said that you could have a square lattice, like what you see on the left, and then this could be a motif. So, this is a much more complicated motif, and there you have three atoms instead of having just two atoms motif. Can does not need to be a very symmetric object? It can be asymmetric objects. For example, it can be something like that you can have a pattern like a trolley or other types also. So, this is sort of a trolley lattice, and the only condition is that all these atoms or groups of atoms or objects should be arranged in such a manner. So, they make up the periodic lattice. So, that there is a distinction about the neighborhood of each of the lattice points that you see.

 So, for example, this periodic arrangement of hearts is like a lattice. If you replace each of the points by a heart, it becomes a love pattern, something like that.

And you can have a monkey pattern, and you can have a dog pattern you have provided all the dogs or monkeys are arranged in such a fashion so that you make a periodic one.

So, this is the illustration of how you can have a primitive lattice with a different arrangement of the heart is arranged in such a fashion. So, each alternative heart is in the same orientation. So, as a result, what you have is when you now represent it using the lattice, the motif is this. So, as a result, the periodic lattice does not remain the same as what you had earlier, you need to define it differently.

So, I will show you how, you do that now.

So, this is, for example, the set of dogs who are standing, you can make a periodic lattice like this. Moreover, this is the unit cell that you draw the green one. So, these are all primitive unit cells; however, in this image on the right, I have changed the orientation of
each alternating animal.

So, for example, let me just bring two arrows. So, this dog is standing up, and every alternate one is standing upside down. So, here now, if you want to draw a periodic lattice, on the left cannot be the periodic lattice here. So now, here when you want to make a lattice now is bigger, which contains one upside facing the upstanding animal and one downside facing the animal.

So, basically, the formula unit in the previous case is this animal one animal. In this case, a formula unit is a group of 2 animals. One is standing up, and one is standing down. This is what happens with atoms. So, you can consider an animal that is standing up like one type of atom, and the animal which is standing upside down is another type of atom. So, basically what you have to do is that you replace the point in each point lattice with the motif for basis. Moreover, in the case of crystal structures, we do that with atoms or groups of atoms or molecules.

This is a very famous picture by Escher, which depicts air, water, and earth. So, you can see that there are three kinds of species here. There is fish, there is a bat, and what you have is a lizard. So, these three animals represent the three constituents of the universe air, water, and earth. Now, in this pattern, can you find out the primitive lattice, first of all? So, I put one point here, the one in the green here. And then I put the green points all over the place. So, you can see that there is a certain periodicity with it. Each green point has these three inward looking fishes, three bats, and three lizards. So now, three lizards which are not visible. So, what you can do is that you can move this point somewhere else. So, you can say that every periodic pattern as a result of crystal has a unique lattice associated with it. So, when you now make a lattice out of it, if you go to the previous one, what is the lattice in this case?

However, as I said, at the point of lattice does not need to be there. So, what is the motif in this case now? Can you determine what the motif is? The motif is how many bats does it contain three bats. How many lizards do you have? Three, and how many fishes you have? Do you have three fishes? There are some which are being cut here, but there are some which are entering from here. So, overall, they make up three fishes. So, overall, within the one-unit cell, you have three fishes, three lizards, and three bats. This is the motif in this case.

You can consider each lizard is one atom, each bat is one atom and each fish as one atom. So, three atoms. So, each lattice now has three atoms A B and C. Why 3 of them? Because all 3 of them are differently oriented with respect to each other. So, as a result, each has three species in the primitive lattice. So, you can move this lattice wherever you want, and still, it remains a periodic lattice. I can put the center here, and it still remains the same. If I now draw the lattice, the lattice would be this. So, it does not matter where you put the unit cell corner. It still contains three fishes, three bats, and three lizards. So, this is just an illustration of what the lattice could be.

So, this is a kind of pattern if I remove the pattern from there. This is how it will look like.

I was talking about the primitive and non-primitive unit cell. The primitive unit cell consists of one lattice point. Technically speaking, each atom, in this case, each animal is a lattice point.

However, and for each of them, the lattice point is identical. On the other hand, if you look at the non-primitive unit cell, for example, this is the non-primitive unit cell. So, go to the pointer. So, this will be the non-primitive unit cell, and it contains consists of 2 lattice points. Here in this case again, a primitive unit cell consists of one lattice point; however, one lattice point is now consisting of 2 atoms; A and B of 2 animals, upside down and downside up. Moreover, the non-primitive one will contain four animals, two upside and two downside. That is the difference between the primitive and non-primitive unit cells. So, the volume of a non-primitive unit cell will be equal to the number of motifs, or the number of lattice points, multiplied by the volume of the primitive unit cell. The number of lattice points is more precise definitions because lattice points can be one atom; it could be a group of atoms.

So, this is what the definition of the lattice can be summarized as. You can have written your primitive lattices in a periodic pattern.

So, you can see that in the first 2 cases, you have chosen to keep the corner of the unit cell at the lattice points themselves. However, it does not matter you can keep the corner of the unit cell anywhere in that within the pattern as long as you have one lattice point within the unit cell. Here the lattice points are shared, and you have one lattice point within the unit cell. This is a non-primitive unit cell. Likewise, you make different possible options of primitive unit cells you can make, several options of non-primitive unit cells as well.

These are some patterns that are provided and try to draw the primitive unit cells and non-primitive unit cells and compare their areas compare the lattice points. What is the lattice point made of determining the motif in each of these cases? So, these are just two examples that I am giving here, but you can Google for patterns.

So, let me now summarize at this point, you have a point lattice, and when you replace points and the point lattice by atoms or group of atoms or molecules, you make a crystal structure. Now depending upon you can associate in case of one atom, it is easy, and each smallest unit cell will consist of one atom, and it will be called a primitive unit cell. So, in this case, the lattice point is associated with one atom. The things will change when you replace one atom by multiple atoms, multiple different types of atoms, or different molecules. That is where the relative orientation of a relative position of atoms with respect to each other will determine what kind of primitive lattice you will have? And how many lattice points? How many atoms or molecules will be associated with one lattice point? But it is also possible that one lattice point may contain more than one or more molecules. There are several examples of that. So, essentially, it will be governed by the relative positioning of molecules; they make a periodic pattern. That is the ultimate takeaway. So now, we will pause here. Furthermore, we will now go to the next lecture, which is on unit cells and crystal structures.

For Next Lecture Click below

Structure of Materials  

 Structure of Materials : Bonding in Materials

Structure of Materials : Correlation between bond and physical properties

 Crystal Structure: Lattice and Basis 

 

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