 We will continue with our discussion on self-correlation function and stochastic motion. If you remember that we are continuously, I have continuously mentioned to you that for stochastic motion we are studying self-correlation function. This is I need to mention you that in the structure work structure work structure and phonons phonons we like diffraction liquid and amorphous systems, sands polarized Newton reflectometry phonons dispersion relation then density of states these are all are coherent scattering cross sections are used because for structure at any length scale that is short length scale in liquid and amorphous crystallography structure or in mesoscopic structure we are talking about how one particular atom or molecule or a scattering unit is position with respect to another one. So, we are talking about g of r t when they are distinct particles that is also true for phonons because these are collective oscillation of atoms. So, the there is coherence in their motion and knowing the motion of one particle we can always derive the position and motion of other particles and density of states are simply integration over q the phonon wave vector and it is also a coherent phenomena. So, for all this we use g of r t and we use coherent scattering cross section coherent scattering cross section cross section. In these experiments the incoherent scattering cross section is a nuisance it is a background. But now when you talk about self correlation function g s of r t which we do in our stochastic motion experiments it is the same particle which is at an origin at time t equal to 0 its location at a time t at a place r what is the probability. So, this is a self correlation function now we are discussing about self correlation function correlation function and we know that we have i q t which is a Fourier transfer over r and q if i come from r to q in this direction or q to r in this direction q to r in this direction it is g of r t and then once more over omega and time i go to s of q omega. And in experiments we measure s of q omega and we try to find out what is the g of r t. So, in the present case that I am discussing stochastic motion this g of r t is specifically it is self correlation function. So, this is also related to self correlation function and what we get is a scattering law for the dynamics of a single particle. But then this single particle is particle by particle we take average. So, we take ensemble averages and this is what we are looking for and in quantum mechanical terms I can write the pair correlation function in this manner. Please see here I have written down the position vectors in terms of operators these are operators in quantum mechanics. And that is why I have taken extra pin in writing that the two delta functions where r minus r prime plus r i 0 and delta r prime minus r i t at the position of the same particular time 0. And time t, but because in general these operators do not commute I have taken care to write them as the average of two delta functions. And then the integration over space for d r prime, but this one if I convert it to classical expression it becomes simpler to understand. And it is basically delta r of r i minus r i t minus r i 0. So, it is the position vector at t equal to 0 and t equal to t sometime t for the distance. And this is the delta function r equal to r i t minus r i 0 this is one. So, this is classical incoherent r t for self correlation. I think I should also mention this is GS self correlation function. And the incoherent intermediate scattering function which I wrote just now for you the incoherent function. This is nothing but a Fourier transform over d 3 r for this of this G incoherent classical I am writing classical to make it more simple of e to the power i q dot r d 3 r G incoherent r t. And this is nothing but once I put this delta function in this as G incoherent you can see it is e to the power minus i q r i t r i 0 sum over all the particles and also average over the ensemble. So, if I have number of particles in the system at least formalism wise I should add up particle by particle this particle its position at 0 0 and its position at t. And I do it by particle by particle and I also do an ensemble averaging for the function e to the power minus i q r i t minus r i 0. So, now when we talk about t equal to 0 and t equal to infinity basically when this averaging is done r i t minus r i 0 interestingly if I talk about a particle at time t equal to 0. And the particle at time t equal to infinity then it is not correlated the time t equal to infinity the position is no way correlated with the position at time t equal to 0. And then this averaging I can do it separately. So, I have broken it up into minus i q dot r i infinity and ensemble average of that and ensemble average of the other part r i 0. So, please know I started from this pair correlation function in real space its Fourier transform gives me this and here this there is an averaging over ensemble that ensemble average of these two variables or here these two positions I am talking about I am dealing them as classical variables and not as operators then I can do it independently. And this is nothing but average value of e to the power i q dot r i infinity at infinite time and at 0th time the particle position can be anywhere at 0th time. And the particle position can be anywhere at after infinite time as elapsed and they are not correlated. So, this averaging is done separately and this is nothing but position of the particle of the ith particle ensemble average of that and mod of that and square this comes up as a mod of its square. So, basically what I am looking for is the position of a particle in a in a in space. Now, if this particle is diffusing over entire space then it is any position can be chosen or if it is chosen in this infinite size space dimensional space or if it is a finite dimensional space then I need to find out the average position of a particle r i at a time at at any time t or t equal to 0 and it is mod square. So, it is basically I need to find out r i given a particle what is the probability of its position vector r i is the position vector being r i and then basically ensemble average of that and square of that. So, now if the particle is restricted inside a sphere let us take a simple case. If it is a particle is restricted in a sphere what is the probability that this is in a small volume d3r. What is the probability that it is in a small volume d3r it is nothing but 1 by v into this fundamental volume d3r that is the probability that the particle is in a volume d3r. And this is equal to 1 by v then r square dr sin theta d theta d phi in spherical coordinates and that is nothing but a small volume which is between r and r plus dr theta and theta plus d theta phi n phi plus d phi. This is a small elemental volume it is a trapezoid spherical trapezoid and then this is a fundamental volume in spherical polar coordinates. So, now if I take 1 by v out this has to be so e to the power iq dot r so I have to write e to the power iq dot r then 1 by v then the probability is r square dr sin theta r square dr sin theta d theta d phi. This is a probability of this small volume and now I need to integrate it over the entire sphere of radius r let us say r. So, in that case this integration becomes 1 by v e to the power iq dot r r square dr sin theta d theta d phi. So, r goes from 0 to r theta goes from 0 to pi and phi goes from 0 to 2 pi this covers the sphere and adds up for all the probabilities in all the small elemental volumes which I have taken all over the sphere. And now this integral we have done many times we have done it many times basically I will just give you hints that iq dot r is iq or cos theta so we need to integrate 0 to pi e to the power iq or cos theta cos theta sin theta d theta. And then I will get from here twice I mean it is e to the power iq or z iq or z d z cos theta writing cos theta z minus 1 to plus 1 and then next integral will be integration of 0 to r for the result of this integration multiplied by r square dr. And this we have done also when we did the form factor for a sphere it is the same integral and then I integrate q omega will have a form like this. So, for a finite sphere the form is like this and if it is an infinite I am sorry this is for a if it is a sorry it is iq r if radius is size r of the sphere if radius of the sphere is r. But if I talk about i incoherent q and this sphere going to infinity that means the particle is diffusing in an infinite medium then you can say these goes to 0. So, there is no time dependence for this i incoherent q and then that means iq r equal to for a finite sphere this is a size but this does not have any time dependence. And so if there is no time dependence then if there is no time dependence we have an aq let us say intermediate scattering function which does not have a time dependence. So, if I do a time Fourier term sum to go to s of q omega q omega this has no time dependence it is just e to the power i omega t d omega aq value of q. So, now this becomes a delta function. So, this is barring a few constant values this should be a q dependent part and a delta omega. So, that means in my scattered intensity I have a pure delta function with a q dependence in my experimental data and then this is known as elastic incoherent structure factor. I mentioned it earlier also that this is because of the finite size diffusion of the particle. Now if I go to a sphere which is infinite radius then this i incoherent goes to 0 goes to 0. So, then the aq which I wrote as a pre factor of delta omega this is 0. So, you do not have a pure. So, for diffusion in finite medium no EISF no inelastic part, but if the medium in which the particle is diffusing is finite. Actually this finite also when I say finite it is it is the r depends on q q r should be. So, if q is large then r has a small we know and when it is a finite size then depending on the angle or the q value that I can prove the finite size is quantified. But in general in an infinite medium there is a elastic incoherent structure factor, but in a finite medium we have a delta function when I say delta omega what I mean is this. To consider that there is an energy transfer in the medium in the interaction between the neutron and the diffusing particle. So, if delta omega is there that means this delta function is that omega equal to 0. So, this is no energy transfer. Now what we are discussing now is quasi elastic neutron scattering this is because this elastic line it gets broadened that is why it is called quasi elastic neutron scattering. It is almost elastic it is almost elastic this is basically as I mentioned to you earlier it is a broadening of this elastic line due to Doppler shifting of the neutron by the diffusing particle. So, that is why it is quasi elastic it is broadening of the line. You have in elastic intensity is far away from the omega equal to 0 part it can be omega 1, omega 2 and then others. So, phonons and others you know phonons we talked about energy transfer of the order of 10 to 80 milli electron volts. But now for incoherent scattering coming from self diffusion of a particle just broadens out the elastic peak and we measure this broadening. And this broadening can be 10 typically 10 microelectron volt to let us say 200 microelectron volt resolution. Depending on how tight energy resolution that you can have and this is broadening of the elastic line that is why it is called quasi elastic neutron scattering. And we look at this broadening in these experiments. Now thing is that one is that broadening of the elastic line due to diffusion. But as I discussed with you just now that there is a pre factor an aq and a term which has got no energy transfer. So, now we have not only broadening of the elastic line but we also have an elastic term. So, let me now draw it a little differently. So, that means now the time correlation in the systems gives me a Lorentzian. But that aq gives me an elastic elastic term riding over this Lorentzian. Why it is Lorentzian I will compute just now. This is the broadening generally gives a Lorentzian. I have mentioned it earlier again I will show you how it comes. So, I have got a Lorentzian plus an elastic line elastic line a delta omega line or omega equal to 0 line. If it is a particle which is diffusing in a finite medium if it is diffusing in a new finite medium I do have the Lorentzian. I do not have the overriding elastic omega equal to 0 line. Why I show it as broadening? Because this is the Lorentzian convoluted with the instrumental resolution instrumental resolution. And this is a delta omega, but this is also convoluted with the instrumental resolution. And that is why the delta function gets broadened. So, now what we have as follows that in my experiment let me see. So, I have a I have an elastic line and I have an inelastic Lorentzian excuse my poor drawing. It is a Lorentzian function riding on to it Lorentzian function. And this comes when the particle is diffusing in a finite medium I will use examples to show you what sort of finite mediums we are talking about. But this Lorentzian comes from the diffusion of the particle in that medium. And it is actually indicates the dynamics and the diffusion constant most importantly that you can get from this experiments.