Introduction - Prof. Sandeep Sangal

So, I am Sandeep Sangal, Professor in the Department of Material Science and Engineering at the Indian Institute of Technology, Kanpur. The course title Crystal Symmetry, X-ray Diffraction and Physical Properties suggests that this course has 3 components. I have taught similar courses over the last over 2 decades with varying emphasis on each of these components the crystal symmetry, X-ray diffraction and physical properties. In my experience what I have found is crystal symmetry or better known as crystallography is a poorly understood concept. Even though the subject itself transcends several disciplines in sciences which also include biology and engineering. And there are also gaps and misunderstandings in the concepts of crystallography. Therefore, this course is central theme would be crystallography and it would take up about half of the course. The importance of the subject can be noted from the fact that the year 2014 was a 100th year of crystallography and the United Nations declared this year 2014 as the International Year of Crystallography. The underlying principle of crystallography is symmetry. The kind of symmetries that we are interested in are like reflection, rotation, inversion, translation etcetera. And crystallography is about the interplay of these symmetries to generate patterns in the arrangement of atoms or molecules in a crystal. And this is important from the point of view of when we analyze crystals with X-rays or also when we look at properties of crystals. So, first the course will take up crystallography in the two dimensional space which is somewhat easier to understand and it will be done with some rigor. This will include introduction to point groups and space groups. It then becomes easy to extend to the third dimension and to be able to visualize the three dimensional space. The two dimensional patterns that we will look at are what are known in mathematics as wallpaper groups. These are repetitive patterns based on symmetry. As an example consider the following pattern. So, here we see little butterflies that have some kind of an orderly arrangement in this pattern and these butterflies could be replaced by atoms and molecules then it will become a crystal. So, if I look at this I try to explore what are the various symmetries that would be present in this pattern. So, let us take a look at this. I have drawn some vertical and horizontal red lines here. These are mirror lines and in three dimensions this will become mirror planes. Basically, if you take any of these mirrors reflect the pattern from one side to the other side the pattern will remain unchanged or what we call the pattern will come into self coincidence through any of these mirrors and these are not this pattern is infinitely large. So, you will have even the mirror lines will be repeated and I have just drawn a few of them. Are these the only symmetries? No, we can find some more symmetries in this pattern. For instance, I have put some green ellipses in this figure. The center of these ellipses are rotation axes. So, I take any of the centers rotate this pattern about 180 degrees the pattern will come into self coincidence or the pattern will look unchanged. So, it has a rotational symmetry of 180 degrees. These symmetries are also called as two fold rotational symmetries that we will understand in the due course through the lectures. There are some more symmetries present here. I have put some more of these rotational symmetries here, here, here and here and then you see some dashed lines. These dashed lines are kind of a special mirror. This is a mirror where reflection takes place and then there is also a translation part of it and again through this and such an operation is called a glide operation. So, these are glide lines in three dimensions there would be glide planes. So, through a glide operation also the pattern will remain unchanged. So, these are all the symmetries or what we call as crystallographic symmetries that are present in this and this small unit is enough to describe the complete pattern of symmetries and such a diagram would be called as a plane group that we will see in the due course through the lectures. Now, consider another pattern. Now, this pattern looks different from the previous one as you can see. Now, what are the symmetries that may be present here? Well, let me put all the symmetry elements at once. In this case we have only vertical mirrors. There are no horizontal mirrors as was the case in the previous situation, but there are horizontal glide lines that are present and there are the rotational symmetries that are present. So, this pattern has a different set of symmetry as a comparing with the previous one, but let us take a look at a third pattern now and this is the small unit that describes this pattern. Looking at the this pattern or this pattern appears to be very different from the previous pattern as you can see. There are some more motifs here they are not just butterflies, but there are some other motifs that have been added to this, but let us look at what are the symmetries that are present here. Now, what we will find in this case is that the symmetry underlying symmetry of this pattern is exactly the same as the previous pattern. Now, if this was a crystal where I had put different atoms in the 2 crystals, different molecules in the 2 crystals, but having the same symmetry pattern the 2 crystals may end up having some similar physical properties in terms of anisotropy or isotropy. So, that is why symmetry becomes important. Now, let me take one example piezoelectric materials. These are crystals which show what is called as a piezoelectric effect when you apply a stress to such a crystal to either compress it or extend it a potential difference develops in this crystal or the inverse piezoelectric effect is when I apply a potential difference these crystals change dimensions either they get compressed or extended. Such materials have a lot of applications to in fact, a common one is a domestic gas lighter. A gas lighter has piezoelectric crystals in it and when they are get compressed when you push the plunger of the gas lighter there is a spark. There is another application ultrasound used in medical imaging there also piezoelectric crystals become important and there are host of other applications. Now, piezoelectric materials what kind of materials will show piezoelectric effect? Any material which has a center of symmetry or what is called as an inversion symmetry that kind of a material will not show piezoelectric effect. So, rather a material which can show piezoelectric effect must have the absence of center of symmetry. So, this is where again symmetry becomes important towards physical properties. Now, X-ray diffraction is such an integral part of crystallography that there are courses in the universities curriculum which are simply named as X-ray crystallography. Here the basic principles of X-ray diffraction will be discussed and their applications to structure analysis. The third component which deals with tensor representation of physical properties of crystals will be discussed along with crystal symmetry what role does the crystal symmetry play in physical properties. Some of the important properties are conductivity, pyroelectricity, piezoelectricity, elasticity and many others. I invite you to join this course and explore with me the wonders of symmetry diffraction and properties. Thank you.