noc19-cy16-Lecture 41 - XRD - Analysis of Pattern

Now, I will start the 9th week of this course. In this week, I will continue the discussion on X-ray diffraction that we started in the last week and we will complete the discussion with on X-ray diffraction, X-ray crystallography and electron microscopy. So, I will first continue where we left off with X-ray diffraction and we will see how to analyze the diffraction pattern. So, in this first lecture of week 9, we are going to do analysis of XRD pattern. So, what are the general features of an XRD pattern? So, suppose you look at a typical XRD pattern, then it looks something like this. So, there is an intensity on one axis and there is 2 theta, which is the angle on the other axis and typically, this looks like there are lot of peaks in this intensity and so on. Typically, the range of any powder X-ray diffraction pattern is from 10 to 80. The theta scale is linear and the range of theta is about typically from 10 to 80 and intensity shows up as such peaks. So, what is important about these peaks? So, what is important is the location of this peak. So, what is the 2 theta value at which this peak is located? That is one important property of this peak. In fact, the most important property is where the peak is located, where the maximum that is the peak is located. The other properties relate to the intensity of the peak. So, what is the intensity of the peak? What are the relative intensities of the different peaks? So, that is the other property of interest and finally, the third property is actually related to the width of the peaks. So, you have the location, intensity and width of the peaks. Now, we will be analyzing a lot about the location of the peaks. The very fact that it is a peak means that the intensity is high but then there are also the relative intensities of the different peaks and these are generally slightly harder to analyze but we will analyze that also and the width we would not be talking much about. So, the general way to do is that there is a huge database of XRD patterns, different compounds. This is called the international center for diffraction data. This is ICDD. Now, there is also the international crystal structures database ICSD. This is the international crystal structures database. So, why I am mentioning these two databases is that every compound that has been crystallized, its powder diffraction data is available on this international center for diffraction data. So, if you want to identify your compound, you can compare with this database. Identification can be obtained by comparing with database. So, this can be done. In fact, the number of databases, number of compounds right now is greater than 800,000 compounds whose powder diffraction data is available and you can compare with them and you can identify your compound but sometimes this does not work. Sometimes, comparison does not work and what would be the cases? So, one case would be that you have a new compound which is not in the database or your compound might be impure, defected. So, the diffraction data is actually not exactly the same as the pure compound or it might be a mixture of two compounds. So, what this means is that it is useful to know how to analyze. So, what we are saying is that yes, you can directly compare with the database but still it is useful to know how to analyze and also this knowledge can also help you verify that the analysis that is obtained by comparison is actually correct. Now, one of the ways to analyze is, I mean the whole analysis is based on Bragg's law. And so Bragg's law and peak intensities, so what the basic idea is that the peak of intensities corresponds to constructive interference and so we have n lambda equal to 2 d sin theta. So, I can rewrite this as lambda equal to 2 d by n sin theta. So, now what you have at the peaks is that you know the wavelength of light used and you know the theta. So, you can determine the distance between planes up to this n factor. Now, what is done is usually we take n equal to 1 and this does not change anything. I mean if you can think n equal to 2 will just mean d spacing of half of n equal to 1. So, we can just take n equal to 1 and that will give you the relation lambda equal to 2 d sin theta. So, we can use this to analyze the patterns. So, what we will get is we will obtain. So, we know lambda and we measure theta where the intensity is maximum and from that you will get values of d. So, we obtain d for and now what next? So, now the d spacings if you this d is in fact one way to get accurate d is an experimental way to get accurate d values in experiment. So, these are obtained using a standard compound. So, KCL is used. So, for example, KCL can be used. So, what is done is that you take potassium chloride whose d values are very accurately known and you mix it with this compound. So, that you can actually check that d values of KCL are accurately reproduced in your XRD pattern and if there are any shifts in the d values, then you accordingly have to shift the d values for your compound. So, let us say that we have all these d values. Let us say that we have all these d values accurately. Then what can you do? So, the point is we would like to determine. So, goal is to determine crystal structure and for a start, we will determine A, B, C and alpha, beta, gamma, but what is the problem? The problem is that your d of HKL is a function of A, B, C, alpha, beta, gamma, H, K, L. So, this distance between lattice planes that we get depends not only on A, B, C, alpha, beta, gamma, but it also depends on HKL and so we have to determine all these quantities. All these quantities have to be determined. So, if you do not know what HKL corresponds to or what A, B, C, alpha, beta, gamma corresponds to, then how can you actually get any of these? How can you get all these values? So, clearly this becomes very complicated. So, this process of assigning HKL values peaks is called indexing of peaks. That means you are assigning Miller index to the peaks. So, I mean once you know A, B, C, alpha, beta, gamma, then you can easily assign this HKL values to the peaks, but if you do not know A, B, C, alpha, beta, gamma, then in general it is very difficult. We will see some simple cases, where this can be done. So, obviously the simple case let us say if you have a cubic or let us say you have an orthogonal system, that means alpha equal to beta equal to gamma equal to 90 degrees, then you have 1 over DHKL square is equal to H square over A square plus K square over B square plus L square over C square. Now, you have to determine A, B and C and by looking at all the peaks, you can imagine that you could be able to do this. In the simplest case for the cubic case, it is really simple. For cubic, you just have the relation that 1 over DHKL square is equal to H square plus K square plus L square divided by A square and you can easily just by trial, but just by trying different values of HKL, you can determine the lattice parameter A and what you have to do is you have to take each of the peaks and for each of the peaks, you try different values of HKL and using that, you can identify both the type of crystal and the lattice parameter. So, this is easy to do and this I would expect you to be able to do. We will do some exercises on this. So, we will get some practice. Now, there are few more things that affect the X-ray diffraction pattern and I will just mention these here. The first is something called an atomic form factor. Basically, your X-ray XRD depends on scattering of X-rays by atoms and different atoms scatter differently. For example, if you have NACL, then NA and CL atoms scatter differently. That means the intensity of scattering, the scattering intensity of NA and the scattering intensity of CL will be different. In general, the scattering intensity F p is proportional to Z times 1 plus cos square theta. So, there is a factor of Z, which is the atomic number and then, there is also a factor of theta and this is approximately proportional to. So, what that means is that since it is approximately linearly proportional to Z, what you can say is that lighter atoms scatter less. Hence, they are hard to detect. In fact, for this reason, you may have heard that hydrogen atoms are very hard to detect and almost impossible to detect. So, hydrogen atoms cannot be detected. The other point is that high angles, high theta implies low scattering. So, if theta is large, then the scattering is very low because theta is close to 90 degrees. Then, the scattering intensity will be very low. So, usually we do not go to very high theta values. So, the other point is that it is difficult to distinguish light atoms such as carbon, nitrogen and oxygen, which have very similar atomic masses. They are light atoms with very similar atomic masses and they are hard to distinguish. So, if you have an organic compound, carbon, nitrogen and oxygen, it is usually quite hard to determine the structure. So, the atomic form factor will also affect the scattering intensity, which will also affect the intensity of the XRD peaks. There are additional factors that affect the intensity of XRD. One is called the geometric structure factor and this refers to the phenomena of interference between atoms, the basis of crystals. Here, what I mean is that we have a unit cell. You have this unit cell and so, these spacings are with respect to some unit cell. The unit cells basically have only 7 types corresponding to 7 crystal systems. Now, within these 7 types, there are 14 Bravais lattices. So, there are 14 Bravais lattices and then, you have all the space groups and so on. Now, you can also have bases. So, this geometric structure factor is a factor. So, this is a factor. It is denoted by capital F of h k l. So, we say intensity is proportional to capital F of h k l square. This is called the geometric structure factor. This arises because the intensity is due to different atoms within the unit cell. So, this is due to interference of scattering by different atoms in the unit cell. So, the different atoms of the same unit cell, they each of them scatter the x rays and it is the interference due to scattering of these atoms that leads to something called the geometric structure factor. We will look at this in more detail, but I will just mention that this leads to something called systematic absences in x r d. So, this is a very crucial factor that is used to identify the compounds or identify the structure. So, in certain values of h k l, you would expect a peak, but that would be missing because of this geometric structure factor. We will see this in more detail in the next lecture. Now, I will just mention two more factors that affect the intensity of x r d patterns. One is called the temperature factor and this temperature factor basically says that the atomic scattering factor changes with temperature. In particular, at higher temperatures, it is less and the reason for this is that when you increase the temperature, then the atoms vibrate more. So, due to increased vibration of atoms, what we mean is the following that if you have the atoms of a solid, now these atoms are not actually exactly fixed at their lattice position. They have some vibration and the amplitude of this vibration increases with temperature and this leads to a lowering of the structure factor and this is given by this temperature factor. So, what is typically an expression is written that the atomic form factor at some high temperature is the atomic form factor at 0 temperature multiplied by exponential of minus b sin square theta by lambda. Now, this factor of b is equal to 8 pi square and u bar square. u bar square is the mean square displacement. So, the mean square displacement of the atom is how much it displaces. What is the typical displacement and this mean square displacement is proportional to the temperature. So, the scattering atomic form factor decreases with temperature. Now, in addition to this, there is something called a refinement factor, which is now routinely used in all x ray diffractometers. This is a factor that is used to actually, as the name suggests, it compares calculated structure with observed structure. So, it compares the calculated structure with the observed structure and I would not spend too much time on this, but basically, the idea is that there is a very well known procedure called Rietveld refinement. So, Rietveld refinement is an automated program that is there in most of the modern x ray diffractometers. This is named after the scientist Hugo Rietveld. So, the idea is that you compare the structure factor that you observe with some model calculated structure factor and you try to minimize this difference. So, this difference is used to actually improve the structure. So, if the difference is too large, then you sort of modify the structure a bit till this difference is very less. So, it is used to refine the structure. Now, it is actually quite a complicated program and I would not be getting into the details of how this program works, but it is good to know that there is such a procedure that is there in all the x rays. So, I will just mention here that it is automated in most instruments in most modern x rays. So, we would not discuss about it anymore, but it is good to know that all these factors are there. So, we saw that you have a temperature factor, refinement factor and this is in addition to the atomic form factor and the structure factor and all these factors affect the intensity of the of the x ray diffraction pattern. So, with this, I will conclude this lecture, the first lecture of week 9 and in the next lecture, I will talk in more detail about the geometric structure factor. Thank you.