Structure of Materials :: Bravais Lattices Symmetry in Crystals

Bravais Lattices and Symmetry in Crystals. 

In this lecture, we are going to discuss Bravais lattices and the introduction of symmetry in crystals. So, let me give you a brief recap. We discussed the primitive, non-primitive lattices in the last class. What is the motif or basis? And how the relative orientation of atoms, molecules, or motif determines the type of primitive units that you will have. This must follow the definition of the primitive lattice, that is, within the primitive unit cell, it should be a repeatable unit, there should be no gaps or discontinuities, and it should be repeatable. So, if you choose the smallest possible cell, which must take into account the orientation of molecules with respect to each other, it should be such that so that it is repeatable. It has an identical neighborhood for all of the species associated.

Seven crystal systems with different symmetry elements and characteristics

So, let me now go to the next topic. There are 7 crystal systems and14 Bravais lattices in 3-D. Moreover, we saw that each non-primitive lattice, such as a face-centered cubic or body-centered cubic lattice, in the case of a cubic system, is composed of the number of primitive lattices depending upon the number of lattice points. So, for example, a bodycentered cubic has two lattice points, which means it is equivalent to two primitive cubic lattices. Similarly, the face-centered cubic lattice has four lattice points, and it is equivalent to have it is equivalent to four primitive lattices. So, one should be able to draw the primitive lattices within the non-primitive lattices easily.

For example, a primitive lattice, it is in 2D. In this, what we have is an array of atoms. We have drawn the first primitive unit cell, a1 is a primitive lattice vector, a2 is a primitive lattice. But, the choice of the primitive cell is not unique, essentially you can choose any primitive vector which can give rise to a primitive unit cell. So, you have primitive lattice vectors as a1’, a2’, however, is different it is not same as a2, a2’ is from this atom to that atom, but it still gives you a primitive unit cell the area of these two cells is going to be equal to each other. You can see in the third one, and you say a1”, and a2”. So, the choice of primitive lattice vectors, as you can have multiple choices, it is not a fixed choice as long as you can make a primitive unit cell out of those two vectors or those three vectors in 3-D. Similarly, in this case, you have a1’’’, you can see that this is the unit cell that you are drawing is a non-primitive unit cell, which is bigger.

Similarly, there are multiple choices for non-primitive unit cells, as well. So, in this case, you can have one non-primitive unit cell, and this could be a lattice vector, or that could be a lattice vector. So, what I am trying to emphasize is when you choose a particular primitive unit cell, the choice of primitive unit cell vectors is multiple. Why did those vectors always give you a primitive unit cell of the same type of the same area?

In BCC, the first set is,

 

This set of vectors in a still constructed primitive lattice or you can alternatively have rather set of vectors, which seems more convenient in BCC, what you choose depends upon the symmetry, but there are multiple possibilities. This is a BCC unit cell. So, we are checking the atoms which are down there. This is the one in the center, this is on the right side, this is the atom below, and this is the atom which is somewhere on the downside. So, for example, you could have chosen from here to here, this could be one lattice vector. So, in this case, we are taking this point as an origin, that is why we have chosen the atom which is down there. So, you can see that this is y, this is x, and this is z. So, this vector is half of y, in this direction; half of z, which is this direction and then half of x. So, this is facing you right. So, x is in this direction this atom is within the cell, and this is outside the cell in front of you, this is to the right of the central atom in the unit cell, this is the bottom of the central atom in the unit cell. So, you can see that set is,

And by correcting these vectors, you can make a unit cell something like this. So, you have a lattice, and you have the lattice translation. Now you connect them, and you should end up with something like that what you have. So, this is a primitive cell, and the volume is half of the one primitive unit cell volume.

This is in the case of FCC, where you can have vectors. So, choose this as an origin; this is a1, a2, and this is a3. So, corner atoms connecting to three face center atoms as a result,

If you choose your origin differently, your vectors and signs will change. So, if you connect from using these three vectors, you get this parallelogram or parallelepiped in within the cube. This is the primitive cell. The non-primitive unit cell does consist of primitive? What is the shortest lattice translation vector? That is what we look at, so, that is the primitive lattice vector, which is the primitive lattice vector because one primitive cell is made up of two primitive cells. So, you can always choose a primitive lattice vector within the non-primitive cell.

Non-primitive lattice vector would be a cube. So, non-primitive lattice vector would be this, that and that, but these are shortest lattice translation vector which has primitive lattice vectors.

So, I think in one of the earlier lectures, I asked how to draw 2-D lattices which are possible. So, there are few possibilities you can see the first, a is not equal to b and θ is not equal to 90o. The other two possibilities are a is not equal to b, but θ is equal to 90o, and the third one, a is not equal to b, and θ is equal to 90o, but you have atom a in the center. So, this is a rectangular centered lattice. So, this is an oblique lattice, this is a rectangular and centered one, this is hexagonal in which a is equal to b, θ is equal to 120o, and then you have a square lattice where a is equal to b and θ is equal to 90o.

So, these are the possibilities that exist in 2D, five possibilities of Bravais lattices. So, now we have been talking about the primitive and non-primitive unit cell, and we have also said key that there are multiple possibilities of primitive unit cells. One can have a square depending on the type of arrangements, and you can have a parallelogram. So, multiple possibilities are provided; they have only one lattice point per unit cell. The question was, how do you define a criterion? So, that you do not end up with multiple possibilities. How you fit them into certain criteria, and that is where the system of this crystal system came into being. The classification according to the crystal system based on lattice parameters and their correlations.

 So, how do you get this criterion? This is as you can see this based on symmetry. So, you can be intuitive that cube is more symmetric as compared to tetragon because a cube has three equal sides, it has all 90o angles, and tetragon has all the 90o angle, but it has one side which is different as compared to other two. Does the question arise what this criterion is? There are certain crystallography symmetric considerations which have to be followed, to evolve this criterion. We will now take up that symmetry criterion in the next few minutes.

So, what we now start with this called Symmetry in Crystals, and why do we need to understand is? So, that we can understand the logic behind the basis of crystal system classification and choice of Bravais Lattices. This is a very complicated topic. So, unfortunately, in this course, we do not have enough time to get around complete aspects of crystallographic, but we will try to establish a simple basis on how to deal with it. So, what is symmetry?.

That is the first question. So, the answer to this question is, symmetry is an operation, which brings an object into it is the original state. So, for example, if I take this square, what is the symmetry operation I can perform on it so that it remains it looks the same. One possible option is if I choose this as a center of the square, and I turn it 90o rotation around this axis. So, the axis is perpendicular to the plane of the paper. So, if I apply 90o rotation, then this again looks the same right, it comes back to a square shape. So, this is a 90o rotation. So, this is called a Rotation symmetry. Similarly, if you take a triangle, equilateral triangle, what operation do you need to perform on it? So, this is a center of the triangle, and I provide a 120o rotation. So, it appears in the same shape. So, these are just examples of operations that you can perform to bring the object into the same shape. So, why we need to understand is because their symmetry classifies lattices.

 

So, it is not only this rotation, which is one symmetry element. There are multiple symmetry elements. So, what are these symmetry elements? So, as I said, symmetry is an operation, when you perform on an object, you bring into the position of selfcoincidence. So, let us now look at what are these symmetry operations types of symmetry operations?.

So, types of symmetry operations, the first is translational symmetry because if you start just from 1-D Lattice. So, let us say, if you have this case of 1-D lattice, and you just put an atom here. So, you can see that, if you move from this point to that point by a vector T, in an infinite array of points in 1-D, then this lattice translation vector T, brings into the position of self-coincidence because this point is identical to that point, then this is a translation. So, this is a case of what we call translational symmetry, and this is a defining symmetry in 1-D. So in 1-D, you must have Translational symmetry.

 

Now, if I change the motif around it, so, this is again in 1-D. Instead of keeping motif as one atom there, I keep motif like this. So, what I have here? I have Translation T, but I also have mirror symmetry. You can make this little worst. You can make the mirror disappear if you make this. So, let us say this becomes dark. So, the mirror is disappeared right, but it still has because the motif now is. So, motif initially was A, and now it is AA, now the motif is AB. In 1-D, you can have operations like Translation and Mirror or Reflection. They apply to 1-D, 2-D, 3-D, but the only two cases which are possible in 1-D are these 2. So, let us move to a little bit more complicated.

In 2-D, there is an addition of rotation element, for example, if I take this lattice Z, what is the rotation I need to provide on it to make it bring it in self- coincidence? I need to rotate it by 180o. So, if I rotate around this point by 180o, it will become the same shape. In the case of rotational symmetry, we define this as fold n-fold symmetry.

So, n is the number of folds of symmetry, and what is this n? n is equal to 360o divided by theta, or angle of rotation. So, this is the angle of rotation. So, in this case, what will n will what will n be? It will be 2. Now, how can you make a 2-D Lattice out of this? In the case of an equilateral triangle, θ will be equal to 120o, n will be equal to 3 if θ is equal to 90o, n is equal to 4.

Moreover, if you look at some flowers, let me it is not very symmetric, but. So, some flowers have 5 petals better. So, you have 5 petals here. So, here you need to provide a rotation of 72o, 5-fold. If you look at the ice flakes or if you look at things like this, they are 6-fold symmetry. So, here you need to provide a rotation of 60o, and this n will be equal to 6, and you can also have things like eightfold symmetry if you have a 45o rotation in case of certain objects.

So, there is no 7-fold symmetry, 13-fold; 11-fold, all those are absent here. So, and there is a mathematical basis that why I cannot get into details of that, but 7, 11 you can see that here, 9 is missing, 9-fold is not there; 13-fold is not there. Even 5-fold is not permitted in Crystallography because it does not fill the space.

See the point is, you can have a rotation of that degree, but if an object does not fill the space. In crystallography, the important thing is, in crystal crystalline materials, that operation must fill the space. So, a 5-fold object does not fill the space. So, as a result, crystalline materials do not show 5-fold symmetry. There is another class of material, which shows 5-fold symmetry are called as Quasi Crystalline Materials, but they are non-equilibrium materials.

So, similarly, other symmetries are also shown by those materials 10-fold symmetry or 9-fold symmetry, some materials could show them, but there are seen in crystalline materials normally. So, in the case of crystalline materials, what we are interested mostly in is n-fold 2-fold, 3-fold, 4-fold, and 6-fold and 1-fold symmetry. So, now, let us come back to this lattice, which I have drawn. So, you can see that in this case in this lattice.

So, if I provide a rotation around this point then even a 2-fold rotation is possible, is there 3-fold possible? There is no possibility of 3-fold. 4-fold is possible. 6-fold, 5-folds is not possible. So, this has 2 and 4. So, of course, around this point, it will have 4-fold, but 2-fold you can also have at these points. So, you define each of the points by maximum possible symmetry. So, this center here, this can provide you 4-fold. So, although it can also provide you 2-fold, you depicted by 4-fold, because 4-fold is the higher symmetry that you can achieve by rotating around this point. So, similarly around these points, these are depicted as 2 points because they cannot give you 4-fold. They can only give you 2-fold. So, you depict this symmetry points rotational symmetry points in the lattice in this manner.

Now, you can see that if you have a square lattice and if I choose a motif which is symmetric enough or which is circular, you get 2-fold and 4-fold, but now let us say, the lattice is a square, but I replace the motif by these triangles. So, I have changed the motif now. Does it have 4-fold or 2-fold symmetry?.

It does not have 2-fold, no it has no 4-fold. So, what I mean to emphasize here is, we cannot go by the conventional definition of what looks symmetric. We have to go by these definitions symmetry, which makes it very specific. So, although it looks like a square grid, it is actually not a square lattice because it does not follow 4-fold, it does not have a 4-fold symmetry, it does not even have 3-fold symmetry, because if you perform 3-fold symmetry operation, it does not remain the same the only operation, so it has only 1-fold symmetry. You can see that it has only 1-fold symmetry, rotational symmetry. So, this is why, in crystallography, a cube may not be a cube; if it does not have elements of symmetry that are specific to cube, which I will come in a short while. So, let us wind up here, and we can now take to the next lecture.

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 

Primitive and Non-primitive Lattices  

Crystal Systems and Bravais Lattice 

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