 Hello everyone, welcome to this material characterization course. In the last class, we just looked at the X-ray diffraction and its application to the crystal structure determination and we looked at the crystal size, how it is being determined by the X-ray diffraction and also we have seen that the effect of the strain and how to determine that through X-ray diffraction and importantly we just looked at how to separate these two effects that is the crystallite size effect as well as the strain effect through Williams and Hall plots. And in today's class, we would like to look at some of the other important applications such as the determination of the crystal structures and then phase identification, one of the most important applications of X-ray diffraction and most of us find it very useful in material science and engineering and finally we would look at most industrially important application of X-ray diffraction that is the stress measurements. So we will look at the very basic principles of all these applications in this general course and we will not get into the very detailed about each aspects because each one of them will have its own will be dealt with a very in a broad manner in a specialized course. So in the today's class, we will start discussing about the determination of the crystal structures using X-ray diffraction. So if you recall the earlier lectures, what we have talked about the intensities, X-rays intensities and how they contribute to the I mean the arrangement of atoms and so on when we look at the reciprocal lattice concepts and so on, just recall those things and then we can broadly I mean classify some of the basic ideas which behind the I mean which determines the total integrated intensity of the X-ray diffraction. If you look at it from the basic aspects of X-ray diffraction, the crystal I mean crystal structure determines the type of diffraction pattern and the shape and size of the crystal system determines the angular positions of this diffraction pattern and if you look at the number of atoms and its arrangement determines the total intensity of the X-ray diffraction pattern. So but broadly what we can do is we can just make a small table to illustrate this aspect. For example, if you can write like this a crystal structure at one end where you have the unit cell and atom position on the other hand you can write a diffraction pattern. So what you can just look at it is the unit cell determines the line position of the diffraction pattern, the atom position determines the integrated line intensity. So this is some of the basic aspects of X-ray diffraction and its intensity related to the crystal structure but commonly we are interested in determining the unknown crystal system. The most popular method of determining the crystal system is through a sine squared theta measurements and I would like to just caution this particular measurement of sine squared theta depending upon the type of the diffractometer one uses. So one has to take care of the correction of the sine squared theta and then it varies with methods to methods. We will not get into the details but I will just briefly discuss how this sine squared theta is going to be used to in determining the crystal structure especially the cubic system which is the simplest system you will understand and from there we can extend it to few more other crystal system and then you can realize how it is useful and in determining the particular crystal system. So we will take up one particular example, first we will talk about indexing the patterns of the cubic crystals and where you have the two important law or you can generally it is written like this you have Bragg's law n lambda is equal to 2 d sine theta it more generally written as lambda is equal to 2 d sine theta and you can also write d is equal to lambda dine 2 sine theta you can write like this and the plane spacing equation in cubic crystal that also you can use it that is d is equal to a divided by square root of h square plus k square plus l square. If you look at these two relations where you have d is equal to lambda by 2 sine theta and for a cubic crystal this relation is valid. Now we can have a common relationship between these two and then we can write a one expression for a sine square theta and then that sine square theta is the value which is going to be most useful to our calculations. So let us see what is that relation? So you write like this so you can write from this two equations sine square theta divided by h square plus k square plus l square which is written as sine square theta by s which is equal to lambda square by 4 a square. Here we can write this sum h square plus k square plus l square is always an integral and then the lambda square by 4 a square is a constant. Since this is a constant now we have to find the integers which will have you know the some of the relations with this sum, some of the integers we have to find which will satisfy this. Let us see how we can do that. So what I have written is the aim is to find out a set of integers s which will yield a constant quotient when divided one by one into the observed sine square theta value. So what you observe from an experimentation is a sine square theta value and then from there using this relation we can find out a set of integers which will yield the quotient. So what are those ints set of integers that is the quotient. So we will now find for a simple cubic, so look at these numbers for a simple cubic system these are the integers which will sum of this h square k square plus l square which will follow this quotients will be an allowed reflection or whatever it from the cubic system. So that is 1, 2, 3, 4, 5, 6, 8, 9 and this is for a body centered cubic system 2, 4, 6, 8, 10, 12, 14 and so on. And this is a face centered cubic system 3, 4, 8, 11, 12, 16 and so on. And finally this is for a diamond cubic system this is 3, 8, 11, 16 and so on. So you should notice that some of the numbers are missing, some numbers are missing in this series. For example you can write certain integers such as 7, 15, 23, 28, 31 etcetera will not find in this series because the sum will not yield these kind of numbers. So these are all rolled out from this series and then you can actually make a table for a observed sine square theta value you make a table and then put a s in one column and you have put all the system in the other column whichever it is allowed and then we can tabulate it and also you can tabulate the sine square theta value. And that is how the crystal structure is determined using sine square theta for a simple cubic system. Now we will look at another simple system called you know tetragonal system how it is done. Here the find out the sine square theta relation which is so you have sine square theta is equal to A into h square plus k square plus C l square this is the relation for tetragonal system. So how to find out the crystal structure from this relation? So let us look at we can write where A which is a constant again and C is a constant for any one pattern and then we can find out therefore h k 0. So you should also know why are we interested in this constants apart from the finding out the integers which will satisfy this sine square theta relation these constants will disclose the cell parameters ultimately our interest is to identify the crystal system. So the finding out the constant is also very important we will now see how to find out the constants first and then we will move on to the selection rules. In order to find the A let us see that the value of A that means if you put L is equal to 0 here then your equation is modified like this now you see that the permissible so once you have this then we can say that the h k 0 lines must have a sine square theta values in the ratio of these integers then we will be able to find this value of the constant. Now the C is the value of C is obtained from other lines which will use this relation so the permissible values which must be in the ratio 1 4 16 etc. So once these values are found out and your C can be calculated similar to what we have done it for A. So like that we can go ahead with the other systems also for example hexagonal system and monoclinic and triclinic system the only problem is the more the crystal system becomes asymmetric for example triclinic and monoclinic the number of constants that is whatever we are seeing A and C it will become many if you have many constants then manual calculations becomes tedious and you require a special computer programming and these are available in fact when I show the laboratory demonstration I will show how these calculations are done using a software automatically. So I will not go into the other systems now I will go to the other applications which I talked about is the phase identification by x-ray diffraction. See this is again one of the important application of x-ray diffraction especially when you have the phase mixtures suppose if you have 2-3 phases in a mixture how to find out the phase what is the phase and how to identify them that is the challenge. So one of the easiest method of finding out this by x-ray diffraction and if you recall the earlier discussion on x-ray I mean x-ray diffraction and it is integrated intensity you have to think about a qualitative I mean if you want to do it a quantitative rather than a qualitative assessment you have to accurately calculate the integrated intensity and then the integrated intensity it depends upon several factors if you have to recall those aspects and then we will discuss what is that the starting point of the problem how to solve this problem. So you have to understand the 3 important points before we look into this quantitative analysis of multiphase materials what is the basis the quantitative analysis by x-ray diffraction is based on the intensity of a diffraction of a particular phase of the interest in a mixture and that intensity of diffraction in turn depends upon the concentration of the particular phase in the mixture that is the second point important point the problem is the relation between the intensity and the concentration is not linear always and then the diffracted intensity depends markedly on the absorption coefficient which in turn will vary with the concentration. So these 3 points you have to keep in mind so the in order to identify the phase from the next ray diffraction the starting point is the intensity equation if you recall we have written a very long intensity equation I will just write it for the reference you know the meaning of all this term which we have already looked at in the earlier class just for the sake of completion I will rewrite those things and you will appreciate what are the terms which we need to really bother about. So this is the an expression for an integrated intensity for a given hkl plane we have already looked into what is the meaning of all these terms and you are much familiar with that now how do we this is the starting point the problem with this expression is this is applicable for only the pure substance and then we have to convert that into an easier form where we can simply adopt this intensity for the a 2 phase or a multi phase mixture or a component. So how do we do that so we can write everything for example I will write it for a 2 phase a mixture or a multi phase mixture where I am interested in the phase alpha a alpha phase in a mixture then I will write this expression like this I alpha is equal to k 1 c alpha divided by this is absorption coefficient. So the multiplication of c alpha is required here because the concentration of the alpha which is responsible for the diffraction intensity which is much lower if it were to be completely from alpha alone but now it is from the mixture so this is has to be written like this and then we can now write k 1 is a constant and which is unknown k 1 is a constant which is unknown because generally I0 is also unknown but we do not have to worry about this it becomes unimportant if you take a ratio of two intensities one from the standard one is from the unknown if you take the ratio then we do not have to worry about this a constant so that is the idea. So now let us see how we can rewrite this okay before I get into the example there are three methods so before we take up this kind of expression and then look at the ratio in identification of the unknown we would look at the what are the general methods available for doing this quantitative analysis one is external standard method where a line from a pure alpha is being used and you have direct comparison method where a line from another phase in a mixture is used. Standard method a line from a foreign material mixed with this specimen so these are the three standard methods which is used in the quantitative analysis of a multi phase materials or a mixture we will take up this particular technique that is direct comparison method and then we will take up a two phase mixture and try to identify how to calculate the concentration of an unknown phase from this intensity expression. See I will just make it more general normally in a metallurgical system or material science system a steel is taken as an example where you have martensite plus austenite or austenite plus ferrite kind of a mixture where people are interested in finding out the austenite fraction which is industrially more important. The people who right now I will not insist on that I will make it more general because the people do not have the materials background you can assume that in a two phase mixture like alpha and beta where you can consider r a gamma and alpha whatever it is where one of the fraction phase fraction is which is less but still we want to quantify them through x-ray diffraction and this method can be adopted. So now we will rewrite this expression in order to get this two phase mixture calculation. So let us take a two phase mixture gamma and alpha and also assume that gamma and alpha have two different crystal structure then in that case we can write this intensity expression in this form. You carefully observe here what we are doing. See you have now separated the integrated intensity into two forms one is k2 and another is r where k2 is a constant which is independent of the substance but r is independent on theta, hkl and kind of substance and so on. So we have separated the terms into two different constants and then we have written the diffracted intensity in this form i is equal to k2 times r divided by 2 mu. So we can write that for reference r depends on theta, hkl and kind of substance. So now we will rewrite this or adopt this expression for the two phase mixture gamma plus alpha. So you write i alpha i gamma is equal to k2 is a constant anyway r depends on theta and hkl so we will say r gamma and we are interested in finding out concentration of this phase so c gamma. So divided by your mass I mean absorption coefficient this is for one phase and this is for then another phase we have i alpha. So similarly you write k2 r alpha and c alpha which is in the mixture and then you can write like this. So division of this two equation yields so you write i gamma by i alpha which is equal to r gamma c gamma divided by r alpha c and r gamma and r alpha can be found out from the crystal structures, knowledge of crystal structure and lattice parameters. And once you know this r gamma and r alpha we will be able to find out the concentration of the unknown from this relation because it is a two phase mixture and once you know the r then you know this ratio so you know the c alpha you will be able to find out c gamma from this relation. So that is how the direct comparison method is used to identify the one of the phase in the phase mixture namely gamma and alpha. Suppose if you have one more unknown component then the constants becomes suppose r gamma r alpha and r beta and if you find out that you will also be able to calculate from this relation for example is equal to 1. Suppose if you have one more phase which requires then those things also can be identified through this kind of relation with the ratio method. So this is very simple and straightforward method and very effective and mostly widely used relation in the material science. So the final application I would like to show is stress measurements. Stress measurements is another very important area in terms of engineering components and it has got more relevance in the industry and we will discuss the basic principle behind measuring the stress in a component by x-ray diffraction. So we will just draw some few schematic quickly and then we will take it from there. So this is the coordinate x is towards the perpendicular to the blackboard this is z and this is y and suppose you assume that this is x-ray which comes and we are trying to identify assume that this is a cylindrical sample which is being pulled in the uniaxial direction and you have the dimensions of d and l and this is the x-ray which comes and impinges on the surface of the sample and then it get diffracted. So how are we going to use this geometry for identifying the relation. So we will write a basic expression for stress is equal to sigma y here because it is a y direction. So we write like this force by area in the y direction and you assume that there is no force in the x and z direction. So for this we can write some relation the stress sigma y produces a strain epsilon y and which is given by epsilon y delta l by l which is nothing but l final minus l initial divided by l not. This is a original length this is after you stretch it and then you get the strain and this strain is related to the stress y this is very famous Hooke's law. So the stress is related by strain by this relation and we can also relate this epsilon x and epsilon y or z g is equal to d final minus d initial divided by d not. So this is the strain in the x direction strain in the z direction is measuring by this change d final minus d initial by d not which is a diameter of the sample also can be represented by this. If we assume that material is if we assume that the material is isotropic isotropic then we can write epsilon x is equal to epsilon z is equal to mu times epsilon y this is also true where mu is the Poisson's reaction. So now how this relations are related to our x-ray diffraction that is the idea I have to write one more schematic quickly loading direction the d spacing is a smaller compared to the planes which are perpendicular to the loading axis you can see that the distance is different because of the stress. So this is the bottom line this is the bottom line for using the x-ray diffraction so we can quickly look at epsilon z is equal to d n minus d not divided by d not. So this is your initial d spacing this is normal the after the stress then what is the d spacing so that is the difference so we can write sigma y is equal to minus e divided by mu where d n minus d not by d not. So this is this is the relation which is forms the basis for this using x-ray diffraction you find the d spacing and I will write the details d n is the spacing of the plane parallel under stress. So d n is under stress and d not is without stress so you see that difference and I think this is the basic idea behind the measurement of stress using x-ray diffraction you will be able to identify the d spacing and you have to remember that you have to have the plane always perpendicular to the this is a normal plane where n p is the x-ray diffraction takes place where the plane which are perpendicular to this incidence so you have you have to make sure that this kind of information are always obtained only from the plane which are parallel to this load or perpendicular to this load axis or I would say parallel to the x-ray diffraction and so on. So you have to keep that in mind and in fact this is what we are talking about here a simple case actual cases a biaxial and triaxial stresses are measured with an elaborate procedures involved and people calculate the residual stress which is a very important component in an engineering application and we can readily quantify these stresses using x-ray diffraction similar based on the fundamental principles like this. So we will not get into the details because each one will come under a special course in itself but as a beginner you should know what is the basis of x-ray diffraction in applying for all this parameters like you know crystal structure determination phase identification stress measurements and so on. So in the next class I will take you to the lab and then show actual how we measure this in a material in a polycrystalline material how do we obtain an x-ray data and then how do we analyze with the interface software today and everything is automated. So you just get the final result in your desktop but you have to understand unless you get into this fundamentals here then only you will be able to appreciate what the software is doing in your interface. So that is the intention of this particular I mean illustrations what we have shown today's class. So we will see you in the next class with the demonstrations. Thank you.