 Welcome to this material characterization course. In the last class, we looked at the effect of crystallite size on the peak intensity of X-ray diffraction. Similarly, we also looked at the effect of stream on the X-ray intensity peak after diffraction. And before we get into the details, further details, I would like to show you a little bit about the intensity peak details, because we always say that whether it is the effect of crystal size as well as strain or instrumental broadening, everything we account for the breadth of the peak. So, how to understand this and how to correct this peak for the required intensity from which you either calculate the crystallite size or characterize the strain in the material. So, look at this slide. This is a typical X-ray diffraction peak where you have a peak intensity as well as the integrated intensity which is under area under the peak. And then you have the, typically we always measure the full width at half maximum and then this is the background, you have to subtract I background. And you have the intensity at full width of maximum is equal to I, that means intensity peak minus intensity background divided by 2. And we also discuss about this peak broadening is due to sample as well as the instrumentation reasons. So, that can be represented by this w square sample is equal to w square full width at half maximum minus w square instrumentation. So, this is a typical peak representation of X-ray diffraction. So, from there how to I mean separate all these effects that is exactly we want to see. Before that we would like to see some of the models, typical models for X-ray diffraction. It is very clear that if you want to quantify some of these effects you need to model them first. And then some of the models which are represented in this table for an X-ray diffraction you see that the first column is a function, second column is the intensity. So, you see the first the Gaussian, if your peak can be fitted into the Gaussian this model based upon this expression, then we can do certain type of calculations. And then secondly if your peak can be modeled with the Laurentian function like this, then that has got some advantage we will see. And then you also have a pseudo void model which is represented by this equation. I will not get into the details of this, but you should know the basic idea of the fitting the X-ray diffraction then how it can be utilized. So, let us look at the meaning of the details of the terms which is given in this equation. I2 theta i is the intensity at ith point on the peak. That means any point on the peak, Ip is the peak intensity, 2 theta i is the position along the profile, 2 theta p is the peak position, w fw hm is the full width half maximum of the peak and eta is the mixing fraction having a value between 0 to 1 and 1. So, if you have this kind of an expression then we have some advantage of using this mathematical model to separate some of the effects namely instrumentation or a crystal size effect or the strain in the lattice and so on. So, we will have a look at those aspects today and before we really look into the separation of these effects from an X-ray diffraction, I would like to go back to little more fundamental aspects of X-ray diffraction itself. And I would like to draw some of the typical X-ray diffraction pattern for a crystalline material as well as for a liquid and an amorphous material and then you will have a gases, how the X-ray diffraction peak will look like and then how do we understand them. So, let me draw some of the typical X-ray diffraction peak then we will continue our discussion. So, what I have drawn is a typical X-ray diffraction pattern from three different kind of material. One is crystal, the other is for the lipids and amorphous solid and then you have a monatomic gas. So, what do we understand from this? This is now well known to us. The intensity is 0, almost 0 except at particular 2 theta value typically known as a Bragg angle. So, one thing you have to understand from this very important you see you take a single atom then you assume that X-rays are interacting with this atom that atom scatters the X-rays in all directions. And in a periodic lattice you just assume that you get only with certain 2 theta angle this intensity is seen. That means rest of the all directions the intensity is cancelled by the destructive interference. So, which is also very important aspects of a diffraction as much as the constructive interference you have to appreciate that. So, the destructive interference plays a crucial role in appreciating this the constructive interference that happens only at particular angle of I mean Bragg law that is Bragg angle. Rest all the other directions is completely cancelled by the all the atoms which is arranged in a three dimensional periodic lattice. So, that is point number one. If you look at the intensity versus 2 theta peak for a lipids and an amorphous solid it is quite interesting to note that you do not have a very sharp peak for a given 2 theta. So, it is quite random you say it is something called you know it is a lack of order you see that there is no a concrete order or a three dimensional periodicity. What you see the peak is one or two peak statistically chosen by the system and then what you really observe is a very broad peak. So, you can so it is only a statistical preference of particular inter atomic distance which will give a two preferential angle peak and then eventually you see only a broad peak which is true for an amorphous solid and liquids which really lacks the order the crystal order orderliness. And if you look at the monotomic gas it is much more interesting you do not even see that any kind of a peak broadening because of the a complete randomness. So, you can write that the features no periodicity at all I would say no structural periodicity no structural periodicity and the curve is this is point number 1 point number 2 is it is featureless horizontal it is since it does not have any structural periodicity you will have the a featureless a curve or a peak like this you will obtain for a monotomic type of a gas. So, now let us look at the let us come back to the X-ray diffraction peak where you have the contribution from a crystal size as well as the strain effect of the lattice. So, how to separate them that we will see now. So, the first point to note down here is when you talk about the crystal size which you obtain from the X-ray diffraction experiments if you are the crystal sizes obtained from a peak width measures give volume average sizes whereas, from the peak shape analysis give the number averages. So, these two subtle things you have to appreciate whether it is peak shape analysis or from the peak width analysis. So, now let us look at the contribution from. So, the contributions from crystal size micro strain and an instrumental effects can be separated if the peaks are Laurentian or Gaussian shaped. So, just like just before we have seen in the slides where we have how this peaks are modeled and if your peak is close to this models then separating this effects are straight forward. So, we will see how suppose your B experiment is the experimentally measured B experiment that is full width half maximum B size. So, you have different contribution to the peak width B experiment is the experimentally measured full width half maximum B size is the FWHM due to the crystallite size B strain is the FWHM due to micro strain and B instrument is the FWHM due to the instrument effect. Now if it is for Laurentian peaks B experiments equal to B size plus B strain plus B instrument and for Gaussian peaks. So, for a Gaussian peaks it is a square of these terms that is B square experiments equal to B square size plus B square strain plus B square instrument. So, now correction for the instrument is done you can write that also as a point. So, how to get this instrumental width contribution the correction for the instrumental width in either case that is both these cases can be obtained by recording a diffraction pattern under identical conditions of the same substance, but in a well unhealed large grained condition. So, that you do not have a contribution from the crystallite size itself that is why it is it is emphasized a large grained condition that means you obtain that by complete annealing treatment you relieve all the strain and then record this peak that is considered as the instrumentation contribution and then we will have these two alone size and strain. So, now let us write so what I have drawn here is a plot of B cos theta by lambda versus sin theta by lambda that is B cos theta versus sin theta that is what I have written here separation of size and strain can be done by plotting B cos theta as a function of sin theta. So, let us write so this type of plot that is B cos theta versus sin theta is known as Williams and Hall plot Williams and Hall plot and implicitly assumes that the peak shape are Lorentzian. So, let us now try to understand this how this separates the size and strain terms of what is the crystallite size equation I am saying here rearranging the terms of earlier equation for crystallite size that is Schroer formula we can write B cos theta is equal to 0.9 lambda divided by t and this one equation we know and we also then the other equation that is strain equation minus 2 delta d by d. So, these two equations are quite familiar to us this one is a Schroer formula I have just rearranged and this one is again rearranged if you differentiate the Bragg's law with respect to theta and d you get this expression that we have seen in the last class. So, combine these two we can now explain this plot how this separates the first remark is if so let us look at what is that details of this using these two equations if the size broadening is the only significant contribution to the peak width then B cos theta is the constant for all peaks that means its Williams and Hall plot is a horizontal line. So, like this you have a straight line and if the strain broadening is the important contribution then B cos theta is a linear function of sin theta. So, like this. So, what you have the data points here plotted for magnesium oxide as well as aluminum oxide, magnesium and then this is for alumina. So, now we can see that this portion here is this. So, now let us complete that discussion this plot as I mentioned belong to two materials magnesium and alumina after two conditions after ball milling and ball milling followed by an annealing at 1350 degree centigrade for two hours. So, you have the when you say ball milling that means you are basically trying to make the powders more fine that means you are straining the system or straining the lattice. So, you have the strain accumulation is there then after this annealing the complete strain is relieved. So, you have only the size contribution not the strain contribution and that is how this two effects are separated in these two examples. So, if you look at the complete procedure it is first you have to measure the instrumental broadening, subtract them and then use these two equations one for strain one for sorry one for size and one for strain and then plot this because theta versus sin theta then you will be able to separate these two effects. So, this is one typical examples of application of x-ray diffraction where we have demonstrated how this size effect as well as strain effect can be measured and then other typical applications of x-ray diffraction is a crystal structure determination which everybody commonly use this technique and also the x-ray diffraction is primarily used for the phase identification. Suppose if you have a mixture of two or three phases in a substance and then how to identify each phase that is again a primary application of this technique and then finally people use this technique quite often for a stress measurements that is residual stress measurements of all the engineering compounds that is one of the vital industrial application and also x-ray diffraction is used for crystal orientation determination namely a texture analysis. So, we will touch upon all this application part with small case studies in the coming classes. Thank you.