 So far we have derived the scattering amplitude for zero Kelvin lattice. Then we went ahead and evaluated the scattering amplitude for finite temperature. Then we introduced the energy conservation through a delta function and derived the time correlation which is correlated to the double differential scattering cross section for a scattering system. Now I will introduce you to the correlation functions in space of real time and real space to q space and to q omega space which is important. Because many of our experimental results sometimes we will be talking about the real space, sometimes we may talk about q space and their correlations and sometimes in planning of the experiment sometimes will depend on this with this as start or lecture 5. So we have derived the scattering law of a system. By scattering law what I mean is this. I wrote d2 sigma d omega d prime sorry this will become a E prime is a Fourier transform over sum over jj prime vj vj prime q and then there is a statistical average and sum over all possible sides distinct and non-distinct. So we have derived the scattering over ensemble average with the where we have used the statistical average. Now as a simplistic assumption if I take away the side dependence of the potential if I say that all the potential all the sides are same then this part I can just take out of the summation and it is k prime by k because of the flux normalization and I can write as average of v square q I am sorry it is not it is slightly wrong. So I was writing d2 sigma d omega d E prime and this comes out as I have one v square q when you take away the side dependence average value of that of the potential it comes out of the summation series its average over that and also we have got a summation over jj prime of an ensemble average of v to the power minus iq dot rj0 a time correlation function which is averaged over all the possible values of energy with the statistical weightage which is p lambda given by p lambda beta is 1 by kT Maxwell Boltzmann distribution and partition function is z. So with that we can write d2 sigma by d omega d E prime as n into k prime by k m by 2 pi h square square v square q sorry that v square q multiplied by the factor sq omega sq omega is known as scattering law for the system its expression is this 1 by 2 pi hn Fourier transform over this statistical average of correlation function at time t. So this is actually q dependent and we take away the time dependence over a time Fourier transform I get the q and omega dependence. Similarly from this sq omega this correlation function if you go in the other direction from q space if I try to go to r space again I have to put a Fourier transform and I have to write and I can go to real space also from the q space but then I have to integrate out I will take away all the constant factors I have to do an integration over q space and then I have to write down is the ensemble average of that and you have to do a Fourier transformation over q now so that means I was in momentum transfer and time space if I go do a Fourier transformation over q I go to real space and time similarly if I go and do a Fourier transformation over omega where h cross omega is energy transfer I go to an omega space and if I do a Fourier transform back on time I go to iqt from here. So this is an interesting way of looking at our data we collect our data in sq omega space our data we collect in terms of not sq omega sq omega space in terms of scattering log for a system which is a function of q and omega. For example for a diffraction experiment we know q is equal to 4 pi by lambda sin theta so this is the momentum transfer and we know that we collect the intensity as a function of q we get this kind of Bragg pic which I showed you the Bragg pic so this is in q space and we get intensity as a function of q from here if you want to go to the real space I have to do a Fourier transform I can go to g of r correlation functions in r going one step ahead for inlustric experiments if I am in real space one Fourier transform will take me to on r will take me to q space and time space one more Fourier transform will take me to q space and energy transfer space. So these are the correlation functions in q omega space qt space real time and momentum transfer and this is the physical system whose expression we are seeking. So this is known as a greens function when you go to real space and time which is a Fourier transform over q d3q q to the power minus iq dot r of the correlation function summed over all the sides this is what is interesting in this derivation. So now the prescription for real space and time to q omega space is Fourier transform over energy and momentum. This is what I have written down here that sq omega is a double Fourier transform I am sorry it is d3 here also you can write so this is so I have one Fourier transform over space and another Fourier transform over time to go to scattering law space q in q and omega to n omega. So now I have written it down specifically here that I have sq omega iqt and g of rt and now I can tell you that if I just talk about a diffraction experiment in many ways I can do it. I can do it by setting omega equal to 0 then it will be sq omega equal to 0 and I can also do it as an integration over omega and these two I will take up in the next part of my lecture.