Structure of Materials::Symmetry in Crystals

Symmetry in Crystals.

We looked at Translational symmetry, which is translating from one lattice point to another, which is present in crystals. The second was a mirror symmetry; the example of mirror symmetry can also be present in 3-D or 2-D. For example, in this case, you can see a mirror plane.

You have a mirror-like this, horizontal mirror, vertical mirror, diagonal mirrors, but in this case, you do not have a mirror plane there on the right. If you have a mirror plane like this, the number of options of the mirror planes have reduced. You have a mirror plane, but you do not have all the mirror planes, as you see on the left.

Similarly, rotational symmetry options have reduced because of the motif. So, what I wanted to emphasize at this point is, it is not the apparent shape that you look at; it is the consideration of the criteria, whether it has rotational symmetry, mirror symmetry, etc. These are important in defining a particular type of lattice.

 

Now, get back to the third class. So, this was again an example of Reflection symmetry. So, you can see that above the Taj Mahal was built in such a manner so that you have a mirror plane across the Taj Mahal. Also, there are a lot of other objects, which do show this kind of symmetry or our own human body.

For example, the human body has this symmetry. In the case of the human body, you can see that we are nature has made a fairly symmetric. So, that you can draw a vertical mirror plane across us and the left and right side unless we have any physical deformity, we are fairly symmetric. So, we have seen translational symmetry, reflection, and rotation symmetry. The fourth one is the Inversion symmetry.

Inversion is an operation; for example, I draw a cube here, AB is a cube diagonal. So, the center of the cube is the center of Inversion, and you are bringing this point across in such a fashion, so, that you bring it to B. So, basically your point x, y, z becomes minus x, minus y, minus z.

 

So, this operation is called Inversion, and this is an aspect that is found in 3-D crystals. So, if I now come back to symmetry 1-D crystal show Translation, Reflection at best. So, they may only show Translation and may not show reflection depends upon the motif. 2-D has Translation, Reflection, and Rotation. 3-D crystals have Translation, Reflection, Rotation, and Inversion. So, Translation is represented by T, and Rotation is represented by R. So, now let us get back to the crystals.

To summarize now, that particular point which I was trying to make. If I put a motif like this, so it has a Translation symmetry, it has a 4-fold, it has a 2-fold, it has a mirror plane like this. Similarly, it has a mirror plane in the other fashion. So, these are the three symmetries you can see, which are present. So, this is, obviously, in the case of 2-D. Moreover, if you draw in 3-D, you will have Inversion as well present. So, now what you can do at home is, find symmetry in alphabets, one of the simplest things you can do.

You can try both Hindi and English alphabets, and you will find that Roman alphabets are a little bit more symmetric as compared to the Hindi alphabets. You can use common car symbols around you like Honda, H, and Wolksvagen, W, and so on. So, when you walk, try to notice the symmetry, what are the symmetry elements that are present around you. We come back to the basis of classification of lattices into 7 Crystal systems and 14 Bravais lattices. we saw that we have 7 Crystal systems, and we have 14 Bravais lattices.

 What is the Defining Symmetry? So, crystal systems are Cubic, Tetragonal, Orthorhombic, Hexagonal, Rhombohedral, Monoclinic, and Triclinic. The cube has four 3-fold axes. I will come back to what you mean what we mean by this. Tetragonal must have one 4-fold, at least, which could be there because of the motif. So, for example, a cube if it does not have four 3-folds, even though it may look like a cube, it is not a cube.

Similarly, let us go to Orthorhombic. Orthorhombic must have three 2-fold rotations. If it does not have 3-fold rotations, Orthorhombic crystal, it is not Orthorhombic crystal. In the case of Hexagonal, you have one 6-fold mandatory, and in case of Rhombohedral, you have one 3-fold, and in case of Monoclinic, you have one, let us write one 2-fold, and in case of Triclinic, you have none. So, these are the Defining Symmetries of Cubic. However, there is a lot more to symmetry, we write things like space groups and because it is not only Rotational symmetry, which is taken into account for crystals, it is also Rotational symmetry, mirror planes something called a glide and screw which are defined by basically atomic arrangement in the crystals.

So, you write things like point groups and space groups for materials, but we do not have time for all that. So, the latices classified into seven crystal systems, Cubic must have four 3-fold, anything else is possible only beyond that, only when it has four 3-fold. So, you can have further final classifications in the Cubic system, but it must have four 3-fold axes. Tetragonal must have one 4-fold, Orthorhombic must have three 2-fold, and so on. So, these are the Defining Symmetry element for each of these. Now let us see, let us begin with Cubic.

So, let us begin with Cubic first, and we can put the motif like this. This is the simplest motif. So, we have options of P, I, and F. P is primitive, I is BCC, and F is FCC. We can see that there is no end centered Cubic, we will discuss that later. So, cube typically has three 4-fold axes along the body diagonal. So, all these will have a 3-fold rotation around that axis. So, it has six 2-fold axes along with equivalent to face diagonals, so, 6 of these will provide you six 2-fold rotations. So, this is how cubic symmetry will be. In the case of Tetragonal, I will give you a few examples.

 In the case of Tetragonal, we know that there is a primitive Tetragonal, body-centered Tetragonal. So, Tetragonal will have one of 4-fold, and if you have one or 4-fold, it will also have two of 2-fold. So, that also you can see that, if you when you draw a Tetragonal crystal, so, this is your Tetragonal crystal. So, if you draw a line like that, this is a, a and c, this will give you a 4-fold rotation, and this is and Defining criteria in the case of Tetragonal. Similarly, you can see in the case of Orthorhombic and Hexagonal.

Now, I will come to the next point, why do we not have 28 Bravais lattices? What is the minimum symmetry that is required? So, you may have a 4-fold, you may have a 3-fold, but If you lose a 3-fold, it does not remain a Cube. So, Crystallographically speaking, a Cube is a Cube, only when it has four 3-fold rotations possible. Otherwise, it is not a Cube. The cube must be brought into the position of self coincidence by performing the minimum symmetry operation.

Although 4-fold and 2-fold can bring it into a cube shape back into it, 3-fold will not be able to. So, which means it has lost one symmetry element. So, that is the minimum defining criteria. So, if you can perform four 3-fold operations on a cube, then 4-fold, 2-folds are automatic, but having 4-folds and 2- folds do not necessarily mean a 3-fold is automatic. So, that is why we choose the minimum Defining Symmetry.

Why do we not have 28 provides lattices? Moreover, we have only half of this, only 14. So, what are the reasons? The reasons are the first reason is that it is based on symmetry, and the second reason is based on size. That is, the other possibilities convert into something else because of symmetry because they fulfill the symmetry criteria of other lattices. Similarly, as far as possible, we must be choosing the smallest size with the best possible symmetry. So, the smallest size and best possible symmetry lead to other combinations. So, the possibilities convert into something else. So, we have a Crystal system table, and we have Bravais lattices.

 We have Cubic, Trigonal, Orthorhombic, Rhombohedral, Hexagonal, Monoclinic, and Triclinic. So, we define these into and the classes or let me write here P, I, F and C. In case of Cubic I have these two, Tetragonal I have only these, Orthorhombic I have all of them, Rhombohedral only P, Hexagonal only P, only monoclinic has P and C and Triclinic does not have any of them. It has the only P nothing else.

Why is C-Centred Cubic missing? So, let us draw the C-Centred Cubic lattice. Now, the question arises is; does it have the Defining Symmetry? Four 3-folds. If I draw a 3-fold from here to here, does it have a 3-fold? Will I be able to bring it into self coincidence by performing a 3-fold rotation here? We will not be. So, what have we done here? We have lost the 3-fold symmetry criteria. For 3-fold symmetry criteria, as a result, although it looks like a cube, it is not a cubic system, but then what it is? Is it a lattice to begin with? See, what was the definition of a lattice? This is point A, and this is point B; both must have the same neighborhood.

So, we can see that B has four neighbors, here A also has four neighbors, because one will be here; another will be here; another will be here. So, it is a Lattice. So, what is it then? What can we reconstruct out of it? So, it must be something. So, what is it? We can now draw two unit cells.

If I construct a unit cell like this, which is an orange-colored unit cell, what you get here is a Tetragonal. So, we can form a simple Tetragonal cell, which has a smaller size. Endcentered Cubic is nothing but a Simple tetragonal cell. So, we will see the other opportunity for other possibilities in the next class.

To summarize this class, we have seen that there are few Defining Symmetries in Crystals, Translation Symmetry, Reflection Symmetry, Rotation Symmetry, and Inversion symmetry. These are followed in 3-D cases, and as we have seen that Bravais lattices and Crystal systems are certain Defining Symmetries, then Bravais lattices are chosen out of those Crystal systems, based on their size and symmetry. We have seen one example, and we will see more in the next class.

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 

Bravais Lattices Symmetry in Crystals 

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