 Hello everyone, welcome to this material characterization course. In the last class we looked at how to use X-ray diffraction in determining the crystallite size effect and also the effect of strain in the lattice and also we have looked at the method of separating these effects by Williamson Hall's plot. And today's class we will look at some of the fundamental applications of X-ray diffraction. I will briefly discuss about the basic principles of each of these applications and beyond that the we have to look into much more details in the any specialized to course because in a general characterization course I will not get into the details of the methodology which we are adopting and I will discuss the basic principles of for example crystal structure determination as well as the phase identification and also stress measurements. So these things we will look at in the class probably in the later classes I will solve some problems using these techniques in one of the tutorials if it required. So first and foremost important thing is the crystal structure determination. So the most common method of application of this I mean X-ray diffraction is determining the crystal structures. The simplest structure being the cubic crystal structures which we will use for indexing them. The first step is to find out the relation between so this is the Bragg law you can rearrange this equation like this and if you look at the plane spacing equation for the cubic system it is d is equal to 8 divided by square root of h square plus k square plus l square and combining these two we can write something like this. So once you have this relation then we can use this relation to index this cubic pattern we will see how we will do it. So what I have written is the sum h square plus k square plus l square that is s is always an integral and lambda square by 4 a square is a constant for any one pattern then the problem of indexing the pattern of a cubic substance is one of the finding set of integers yes which will yield a constant coefficient. So we can write for a simple cubic. So you have with this rule we can form this a kind of selection rule for the cubic system and you cannot choose integers like this because this sum will not will add up to these numbers so you can exclude them and by this we can simply index this hkl plane from the sine squared theta in fact you can make a table out of this sine squared theta versus this kind of a possible hkl plane for a cubic system and similarly you can use some others set of values for tetragonal system so the sine squared theta must obey this this is a relation for a tetragonal system we will see how this is going to help us where so where you have a is equal to lambda square by 4 a square plus c is equal to lambda square by 4 c square of constants and then now we have to find out this values using this kind of method. So a and c will give you lattice parameters so what I have written here is in order to find the constants a and c which will give the lattice parameters the value of a is obtained in this manner from hk0 lines you put l equal to 0 then your sine squared theta should satisfy this condition and then permissible values are 1, 2, 4, 5, 8 that kind of a combination. So now you write so what I have written is these are the permissible values for this condition therefore the hk0 lines must have sine squared values in the ratio of this integers and a will be some number which is 1, 1 by 2, 1 by 4, 1 by 5, 1 by 8 times the sine squared theta values of this lines so that is how you find out the value of a and similarly you will be able to find the values of c also. So c is obtained from the other lines on the pattern and the use of sine squared equation in this form sine squared theta minus a into h square plus k square which is equal to cl square. So I will repeat c is obtained from the other lines that means other than hk0 lines on the pattern and use the sine squared equation in the form sine squared theta minus a into h square plus k square is equal to cl square and the differences represented by this equation that is the left hand side of this equation are set up for various assumed values of h and k in an attempt to find a consistent set of c l square in fact you can form a table and which must be in the ratio of 1, 4, 9, 16 and so on. So from there the c value can be found out once these values so once these values what we are mentioning here is found out c can be calculated so that is how it goes with the tetragonal system. Now we will look at other system so you have for a hexagonal system the sine squared theta relation is like this sine squared theta is equal to a into h square plus hk plus k square plus cl square where a and c are constants where a is equal to lambda square by 3a square and c is equal to lambda square by 4c square and similar to what we have seen in this tetragonal system we will have the permissible values in hexagonal system like for h square plus hk plus k square values are 1, 3, 4, 7, 9, etcetera again you can make a table and then find out this constant in a similar manner. For the other crystal systems the sine squared theta relations will have more number of constants for example monoclinic and triclinic system then the calculations procedure becomes little elaborate and we need some kind of a program is programming is required to find out all those things. So this is a basic idea behind using the sine squared theta values to indexing the simple crystal systems and one more thing which I have not mentioned how accurately we will be able to measure this sine squared theta is again depending upon the system of diffractometer we operate and there are set of procedures in the literature how to correct the sine theta value and after that these kind of calculations are carried out. In fact I will show you in the laboratory how some of these calculations are done with the interface of a software you directly get all the indexed pattern readily available and these are some of the basic aspects of indexing the crystal systems. So I will just move on to the next application called a phase identification which is another very important application of X-ray diffraction for example if you have a material which has a mixture of 2, 3 phases so how this phase is identified through X-ray diffraction that we will see with a simple case study. So what I have written here the quantitative analysis by diffraction is based on the intensity of diffraction pattern of a particular phase in the mixture of phases because we are interested in identifying the phase in a multi phase system more than one phase system. So the basic idea is the quantitative analysis is based on intensity of diffraction pattern from the particular phase in a mixture of phases. So what is the next step what is this intensity it depends on that is what we have to write now. So the intensity in turn depends upon the concentration of the phase of the interest in the mixture. So these three points are important so the relationship between the intensity and the concentration is not generally linear. So then it depends upon what it depends upon the mass absorption coefficient of that mixture which itself will vary as a function of concentration. So it is not straight forward here so in order to look at the possibility of identifying the phase using the X-ray diffraction we have to keep these three points and also the mass absorption coefficient of the mixture and its variation as a function of concentration also to be considered. So to find out the relation between the intensity and the phase concentration of the phase the starting point is the intensity expression. So this is a familiar equation to us because this is what the intensity the final intensity of X-ray diffraction is expressed we have gone through the meaning of each term in the earlier class but this is the standard pattern for the powder specimen we can write intensity. So this is a general expression for an intensity diffracted by a powder specimen you have this but what you have to remember is the expression what we have written is for a pure substance we will write this expression is applied only to a pure substance but our interest is to find out the phase in a mixture for example if you take alpha beta mixture so then this expression need to be written in the other form. So this expression has to be rewritten in this form for a particular phase of interest for example intensity alpha is equal to k1 multiplied by C alpha by mu m. The intensity must be multiplied by C alpha that is a volume fraction of that phase of interest to allow for the fact that the diffracting volume of alpha in the mixture is less than it would be if the specimen were pure alpha. So in order to take care of that you have to multiply that expression by C alpha and then we have to calculate this. So now let us write what is the meaning of this expression. So with respect to this expression k is constant the value of k1 is unknown because I0 that is incident intensity of x-ray is generally unknown but this is unimportant because if the ratio of I alpha to the intensity of the some standard reference line is formed if you can use the ratio between I alpha and a standard reference line then this things can be ignored. So there are three methods in general one is external standard method a line from pure alpha is used where you have direct comparison method a line from another phase in the mixture is used the third internal standard method where a foreign material is mixed with the specimen. So there are three methods by which you can do the phase identification in x-ray diffraction using this concept and we will just look at one of the important and very frequently used method to find out the phase fraction and identification that is called direct comparison method we will take up and then we will look at it okay. What I have written here is suppose if you have a two-fix mixture where for example you take a gamma and alpha in a mixture in general. So the you can write this intensity equation in two forms one depend on I mean theta hkl and the kind of substance that is R which is completely a substance dependent and this is a substance independent term and then you can write this general expression for these two individual phases in a mixture for in this case I gamma is equal to K2 R gamma C gamma divided by 2 mu m and I alpha is equal to K2 R alpha C alpha divided by 2 mu m they take the ratio of this which is R gamma C gamma by R alpha C alpha. Now the C gamma by C alpha can therefore be obtained from the measurement of I gamma by I alpha and then calculation of R gamma and R alpha can be obtained by the lattice constants and crystal structures and so on once we find that C gamma by C alpha ratio the value of C gamma can be found out from this relation C gamma plus C alpha is equal to 1. So like that even if you have one more phase you can find out this from this direct comparison method. So like that you can make use of this X-ray diffraction technique for identifying the phase and calculating the phase fraction and another important application is stress measurements. I will again discuss the very basic principles of stress measurements in the next class. Thank you.