 Well after seeing this so now let me just quickly tell you that what is I am trying to So this is the self correlation function R at t plus tau this is equal to From R minus L with a jump it is coming to R in with a Probability of jump length L at the with a time scale tau which is the residence time At the side so physically it is easy to understand it is sitting at a site It can jump from here to that side So this jump length if it is L then from R minus L it comes to R in a time t plus tau When PL tau is the jump length for a residence time of tau so it sits here after sometime it jumps After sometime it again jumps after sometime it again jumps so there's a residence time and there's a jump length So jump length probability given by L Dependence also on the time tau and when these are small quantities because you're talking about a medium In which a particle is diffusing through short jumps and it's short time scales I can do a Taylor expansion of this in time and in space. So this is in time and The right hand side. This is in space now here GS of RT can be taken outside In this expression because There is no PL tau involved in this term. So once I take it out and we know that PL tau Once you integrate over all jump lengths It has to be equal to 1 at least the particle jumps with any length So PL tau DL is equal to 1 if it is only PL tau DL so this goes outside the so I can write down GS T Plus tau equal to GS. It's a self-always self and on a RT Plus tau DGS by DT and Then then it is VGS by DT and then DT and I can do So this Taylor expansion of the whole thing so you can see that one is this One is this in time tau. This is the next will be tau squared D to T by DT to GS RT and so on and so forth and the right hand side. I do it same on space the GS RT minus L into Delta GS plus half of L like LJ like LX LX LY the components of jump length in XY and Z direction the second order derivative first order derivative and these two I am equating Now when I equate one thing is there this L into DGS By DX this tau. Now L has equal chance of being positive Or negative So just L tau if I do this over the entire space Just for DX same true for dy and dz if I do this integration It should give me 0 because this is first order in L and then LX LY LZ can be positive and negative with equal probability when some of all of them They become 0. So that means this term L into Delta GS RT I wrote DGS by DX but now there will be LX DGX by DGS by DX LY DGS by dy so on and so forth and then there's a second order term This term goes to 0 because L is isotropic. So integral of PL tau space is unity and the first term in both the sides are equal So GS and GS they'll cancel and what I get is this equation here So after neglecting the higher-order terms we get this equation and then because as I said that PL tau is isotropic So li square means LX square LX square P LX tau DX dy DZ and then there will be LX square plus LY square plus LZ square and Then you can see because they are isotropic This gives me nothing but an average Li square which is LX square plus Li square plus LZ square average and One-third of that anytime this is one-third because all three components equally possible and then I get this equation this equation is a equation for Motion that is dictated by this Self-correlation function of a particle in If I write it actually in slightly simpler form in one dimension, then you will realize this is DGS XT DT Equal to a rate we write a constant as D Because fixed second law says Mixing of space and time so DC by DT Equal to D D to C by DX to for a one-dimensional Diffusion and this is for three-dimensional diffusion. So the C and GS they are identical So I can use the solution of fix diffusion equation and use it here for GS of RT So here so now the solution of the fix law also must satisfy some very basic Constancies one is that at T equal to zero the self-correlation function is Delta because we know that at T equal to zero it is starting from some point which are calling it to origin This origin is not fixed. It can be anywhere in the Medium, but this is delta function for this particular particle and GS of RT Which at any time if I integrate over the whole space? DR is basically D3R D3R so integration over the whole space is equal to one because the physical significance of this is that the particle has to be somewhere at that time T and That's why GS of RT integrate over all the space is equal to one. So these are the Things which the fix law must satisfy in its solution and the solution we are aware That's for a diffusing particle For a diffusing particle at any time T starting if the particle is here There's a Gaussian the Gaussian has a width at any time which is 4 DT So if the particle starts from the origin at time T equal to zero as time goes on This is T equal to zero It is it Opens up like a Gaussian Widens like a Gaussian with a width of 4 DT. So that means One is that if I can measure at some time the width then I can find out D and If I go to infinity Then this becomes broader and broader and actually ultimately the particle is uniformly distributed all over the space So this is the significance of this solution of Fix equation which also I'm using as GS of RT because by taking a Fourier transform over space of this function I can get the Intermediate scattering law and which will be actually I'm telling you it goes as e to the power minus D Q squared T. So So now that I mentioned you it is easy to do a Fourier transform of this and Then the Fourier transform of this over R gives me e to the power minus D Q squared T and once more if I do the Fourier transform of this Exponential function. This is an exponential function And what I will get is a Lorentzian. So this is the Lorentzian which I mentioned again and again that Here when I showed that the Due to diffusion the particle has a Lorentzian distribution in omega. This is a Lorentzian. I'm talking about This is a Lorentzian. I'm talking about so this Lorentzian you can see the width D Q square Can be used to determine the D values Given that we know the Q value for the experiment So this is the Lorentzian which will be measuring in our experiments And if the particle is diffusing in an infinite medium, then we only have this Lorentzian but if it is Diffusing in a finite medium, then I will also have a plus one delta omega term and Elastic in current structure factor which tells me the geometry of the space in which the particle is Diffusing so this is for an infinite medium This is for a finite medium and I will show you how to derive a Q for a finite medium so now this is what I Stated just now the scattering exhibits the shape of a Lorentzian function Who's half width or full width whatever you want to say with half width at half maxima Increases with the momentum transfer Q according to a D Q square law This also often you will find I'll write it as 1 by tau or where tau is the residence time. I can quickly do a Dimension analysis because a D is given in centimeters square per second and Q is centimeter Inverse square so what I get D here D Q square is centimeter square per second the centimeter inverse square is per second which is one upon time and Here I can also write the whole thing as 1 by tau squared Where 1 by tau tau is given as a residence time. So we will be Talking interchangeably about D Q square where and 1 by tau and these four equivalent unit wise and physically also because D Q square dictates how fast it is diffusing which will also depend if there is a residence time at every site so it's a Lorentzian but Take away from this expression is that if I do the experiment if I can measure the width of the energy transfer peak Which is a Lorentzian then from there I can find out the diffusion constant which is a very important factor or an important physical quantity for understanding dynamics Especially self dynamics in various media so let me just Give you an example now. I just now I talked about Diffusion Fickian diffusion, but now let me just take an example which is a two-fold jump rotation Let me just explain to you Suppose there are two sides Two sides. So it can be an H2 molecule. It can be an H2 molecule and which may be rotating In a medium not able to move. So it's just rotation. I'm talking about So this hydrogen goes from side one to side two side two to side one So basically in brief I am talking about a two-fold Jump rotation So it is changing site as I showed you the particle goes from one side H2 Of course, it is not right because there are two particles I'm talking about a single particle and there are two sides And the particle will go from this side to This side and this side to this side is the jumping So if I write down the residence time as tau at a site Then I can write a very simple loss and gain equation. Let us say it started With the site r1. Let me call it. This is the site r1. There's a site r2 1 and 2 so The change in probability of being at site r1 One is that this is like radioactive decay if the residence time is tau. It's then it's a statistical fact fact that probability minus P r1 tau by tau will tell me after tau time What is the probability that it has leaked out or gone to the other side? And this is a loss term and the gain term is if the particle was at second side What is the probability That it has leaked into this side. So this is the gain. So basically it is gain minus loss gives me the Particle the probability that particle at r1 t and its rate of change With time the probability similarly. I can write the same thing for the second side r2 t dt Where the loss term now it is that it is the particle is lost from second side to first side and there is a gain term Which is coming from the first side to second side if I add up these two equations We get a very obvious result that p r1 t Plus p r2 t equal to a constant actually this constant should be one because It should be either here or there at a time t So together they should be one But here from these two equations when I add up dp r1 t dt Plus dp r2 dt equal to zero gives me a constant I can solve it But this is a very simple First order equation to solve this But I just give you the solutions So this is Basically two constants a and b have to be determined for this solution and this is the Time variable and how the probabilities are changing with that. So this I can straight away convert into First solutions because at t equal to zero p r1 t is a plus b and this is a minus b So a plus b It can be equal to one if the particle starts from site one And then a minus b Should be equal to zero Putting t equal to zero because it was so now If I at time t equal to zero if I solve these two I get straight away This solution p11 And p12 that is the two probabilities One is that it was at site one and what is the probability it continues at Site one up. So site one At time t equal to zero. What is the probability it continues same for the other side Now using these probabilities. I can write down this intermediate scattering function Which has got a time dependent part and a time independent part. This is the eisf part because here the Jump from the between the sites are stochastic stochastic And the time variation Is given by this part But the fact is that because it's a finite size We also have something called a elastic incoherent structure factor Which has got a q dependence like this. I have specifically write down the q dependence for a zero q and a one q q dependent This is the pre factor Of the time dependent part And this is the pre factor if I want I can put a delta omega here So when I take a Fourier transform, this will give me the elastic line elastic line broadened by the instrumental resolution This will give me the Lorentzian line Broadened by the instrumental resolution and the pre factor a one q So this is true for a two side two side diffusion Jump diffusion between two sides and often that happens In many of our problems. So as I said that so iqt gives us A zero q which is this the q dependence for the elastic incoherent structure factor And the other part gives me a Lorentzian Or if you see the previous expression I wrote dq square whole square And this is omega square tau square So there's very very similar In nature. So this is a Lorentzian distribution So this Lorentzian time distribution can give me the jump times tau But this part a zero q its variation with q tells me What is the nature of the jump or the geometry not nature really What is the geometry of this jump here? It is two fold it can be even three points It can be even three points. So the particle may be jumping between these points Or it can be even a six fold Jump where the particle is jumping between Six sides or the particle may be diffusing over a circle and all these can be We can make similar equations This is very simple here I took half of this in case of Three points I will have three terms And two will be the gain terms one will be the loss term when the particle starts from here and here and here and One can easily solve I have taken one example. I'll give the other results. So now let us Just write down it for a powder sample in case of so far I have kept calculating sq omega Now only difference is that in case of powder sample we have to take all possible orientations of q And that is you can see that it is Sine theta d theta d phi Actually here this d phi won't be there it is here Sine theta d theta d phi will give you the The orientational average of sq omega And then what do we get is this This is the Elastic incoherent structure factor and this is the natural Lorenzian Where the pre-factor the q dependent pre-factor of EISF which comes from the geometry of rotation Is given by these expressions these are Bessel functions Of 0th order which are basically sine x by x 0th order spherical Bessel functions. So with this Once we do an experiment We can measure the Lorenzian and we can also measure a0q or a0 prime q For a powder sample as a function of q and get the kind of rotation Geometry that is there in the system. So just A pictorial Demonstration that we can have two fold jump rotation We can have uniaxial continuous rotational diffusion We can have three fold jump diffusion And so an isotropic rotational diffusion means it is this this axis is not fixed This axis can be anything and then it can have a rotational diffusion over a sphere here It will be over a circle here it will be over a sphere. So these are the possible situations, but the possible physical Constraints over which a molecule Can diffuse So now let me just give you the expression for elastic incoherent structure factors So for jump rotation as I told you earlier Two fold three fold six fold These are the expressions. So this is for a two fold like Actually derived it for you For three fold it will be extended to this expression and for six fold For isotropic rotational diffusion It is j0 square q r where r is the radius of rotation And j0 is the spherical Bessel function of 0th order and so This schematic shows you that what will be the variation of EISF with respect to q for a two fold three fold and six fold and continuous jump diffusion Rotational diffusion and by measurement of EISF we can Select or we can see which one of these models are valid for a molecule Like say some organic molecule which is undergoing a rotational diffusion So we are not only talking about translational diffusion We are not only talking about translation But we are also talking about rotation and We are introducing a term for rotational diffusion because if there are three sides Just like translational diffusion the particle can come from here to here go back there Or from here to here or from here to here. So they can undergo jump diffusion between these three sides