 Now, I go to the most important quantity that important quantity is the two particle distribution functions which is defined here. So, now I want to know that what is the probability if I have one molecule here and another molecule there. In other words, I also want to know later in a homogeneous system how many molecules are there I have been discussing. So, now I go by the following important definitions that radial distribution function is joint probability of having R1 and R2 one at R1 another R2 and that is divided by rho square because if this probability is independent of each other Pn is independent of each other then Pn R1 R2 would be it is P2 then R1 R2 then I will have P1 R1 P1 R2 and that I just showed in homogeneous system this is nothing but density. So, then P R1 R2 if there is no correlation then will be just by rho square. So, it makes sense now to define a quantity where the joint point distribution which is Pn R1 R2 is divided by rho square because in the limit in a homogeneous no correlation and also in the limit when R1 R2 are widely separated then in a when R1 R2 are widely separated in this notation R is R1 minus R2 R1 minus R2. So, you show it please make these corrections in a second edition we will fix all these things. So, the radial distribution function now we have a this GR is the radial distribution function again I motivate it that it is in a it is number of molecules and in a shell at a distance R this gives the GR and it is a very pivotal quantity in theory of liquids now go back. So, we have showed this is rho R now we want to do two particle distribution just I motivated it that that then will be in the from n particle distribution which is defined here n factorial n minus n factorial this quantity then I get two particle P2 R1 R2 I am little bit uncomfortable with both R1 and R2 because at the end of the day in homogeneous liquid it is only the distance between them it is both R1 R2 not negate it is I can select my co-ordinated one and then I can only call consider R2 as a distance R. So, basically it is a distance a scalar that becomes which is far simpler quantity R1 R2 if I keep it as a vector then it is 6 dimensional but what really matters in the homogeneous liquid is one dimension. So, which the tremendous simplification and something people should be aware of. So, just from the last one I get this thing I integrate over R3 R4 R5 R2 Rn and then between then the configuration integral may have configuration integral and n factorial n minus 2 now that is of course I can now divide I can start seeing how rho square will come out I can then this quantity will be rho square n going to infinity this quantity will be rho square and then if they are not interacting then Zn is V to the power n and this V square will another V to the power n minus 2 will come here. So, I will get back these conditions that. So, this is a very cleverly written by multiplying and divided by this square. So, the motivation of gr is now clear motivation of gr is that is very is clear and so in n particle gr is V2 R1 R2 by rho square but R1 R2 what we need actually is a R1 minus R2 this quantity. So, this is the radial distribution function which is the probability of getting 2 that they are 2 particles at separation R that is the central quantity of our discussion here. So, we now go generalize to n particle distribution many times we need the 3 particle distribution G3. So, from the discussions I have done this is Gn and then rho R I give a this delta function is kind of a simple but a straightforward thing that we now want to be talking of n particle this is not quite correct Rn equal to so I that means that n is missing this delta R minus R I. So, they are n particles and this is the complete distribution of n particles that means what is saying that it is essentially Pn but delta function representation of that but the quantity is that I can go to Gn n particle. So, they are what one is saying that I have n particles in a capital N particle system I have tagged small n particles and I want to know what is the probability of this in a huge system there are many molecules which I am not which I am going to integrate over but I have. So, the green plus red at the total n so but my small n is 3. So, I want this 3 particle positions and the R1 R2 and R3 and that what is the probability of having 3 particles at R1 R2 and R3 when n equal to 3 then I have that description and I define that as the 3 particle the power distribution then normalized by rho cube it is done for inhomogeneous system that I give otherwise I will just make it Pn n R2 d power n divided by rho to the power n that would be this quantity. As I said delta function means you can say I can write this thing now as sum over delta R minus R I and I equal to 1, 2, 3. So, I have the position of this 3 and that is rho then will be rho 3. So, this is the definition of delta function that as an advantage we will see little later. So, homogenous system is averaging that we are talking this comes out like that is not very important. So, now however even 2 particle distribution it becomes kind of interesting because then I can now write 2 particles at R1 and R2 then I can write one particle at I have to sum over if is this in just constant particles the same constant identical particles then I can say my rho R at R1 and R2 is just integrate over all the other molecules n number of integration that the R to the power n but before all our definitions we are making this integration to n minus 2. But now because of delta function representation I can put it in and say ok one particle at R another particle is R prime actually if I have to look at that then that should be R1 and R prime is R2 then this is becomes the delta R1 minus R i delta R2 minus Rj I sum over all i and j and if I make them double some then I have to take over counting I have avoid that I have to take half which is half. So, this is rho R1 R2. So, if I can do that now I the number of ways I can pick it up is n into n minus 1 by 2 particles then this is the same thing as that we have done in the last page also and same thing again repeated here. So, nothing we are doing here. So, we have now developed a definition of gr. So, we define gr in terms of e to the power minus beta ur. So, starting from an partition function starting from partition function we get this quantity and a definition of gr we have not evaluated here yet but we have this structure. Next what happens? So, now we take a little bit back and say ok what do you know about gr before you evaluate gr which is difficult but has been done with great accuracy in many large number of systems but there is two ways one can get it directly one is radial one is the computer simulation and this is neutron scattering and also as I told you these days in X-ray in random system. So, what we are doing essentially are random systems. So, extra factor of random systems and liquid one of them but of course the what I am doing is also goes for glasses that goes for very much glasses which is more and materials surface things like in mobiles the surface of the mobiles we need great need for the structure of the liquid crystals. So, all those things exactly follows this terminology that is why the terminology though we are doing in the context of liquids this is a terminology and the nomenclature that goes into perfect whole of physics and chemistry. So, the radial distribution function at the radial distribution function same as gr is RTL. So, how do they look now we have plotted here radial distribution function of say around this blue one this little bit ceiling drawing that because there is too much of a structure really not too much of a structure but what it does this one that this is the central one the red ones at the first layer and the blue ones at the second layer and I actually blue one I should I should draw some more some more maybe ok. So, now you see near in the first layer this one there is a very sharp peak almost reminds you of a crystal because liquid has a considerable amount of short range order all the liquids what a short range order it is of the kind of thing we discussed in Ising model the same short range order molecules are placed quite like a crystal in the first layer. So, this is the radial distribution function structure there is a sharp first peak but the first structure in the first layer is due to the excluded volume interaction because the molecules are packed together they are touching the surface then comes the second layer because the first layer is structure second layer is also structure that structure is like you are packing people see in a bus like in places like Calcutta and Bombay bus are very crowded. So, if you are in a center of a crowded bus then the people around you will be finite number and they will be you will find four or five people around you then if it is a big bus then another four or five or a ten around the second layer. So, the hard sphere interaction or the that you cannot penetrate two molecules cannot penetrate that it has a volume and size that dictates this arrangement that is what I meant that is the hard sphere part of the interaction repulsive part of the interaction that determine the structure of a liquid is very very important to know the structure of a liquid is determined by this repulsive hard sphere interaction dynamics also to a great extent by that but phase transition and many properties is a competition between this repulsive part of the interaction and the attractive part of the interaction needs to together drive the properties of a liquid and so this is what now GR if I now do this calculation that just it is exactly what Bernal did in his he was genius this way he did. So, if you do that then this is the Bernal kind of GR that we get radial distribution function in a homogeneous system of that gives you how many molecules are placed at what distance. Now, the important thing to know the structure factor where do you get the GR? So, how do you get GR? Experimentally so long I have been talking of partition function experimentally it does not care about your partition function it can he wants to measure and indeed he can directly measure this was shown first time and done on the first time by Vanno a great physicist in around 1950 it is I think 47 to 50 around that time lady who did not up the second world war when it comes scattering becomes he could get the scattering cross section from liquid and that give you the structure factor is and that from that these are the equations we briefly derived but we will not go very detail into that that the structure factor is K which is measured from neutrons scattering cross section is exactly by the radial distribution function. So, SK is essentially given by with certain p factor this is a equation what it is weight in gold. So, it is essentially Fourier transform of so this is my GR this is a Fourier transform other factor like 4 pi here and so this this is the one that I get from neutrons scattering. So, from differential cross section we describe little by the vice sigma this is this total scattering cross section differential because what is the amount that goes into the incoming beam coming then gets a scattered then what goes into an angle located at sigma that is the cross section is connected to total number because exchange of momentum exchange of momentum this K is the difference of momentum between incoming and outgoing and that is because of the positions the length scale and that will show the derivation but this is as I told you which is what is this equation v is a golden equation. So, static structure factor is K by neutrons scattering is gives you the radial distribution for it can directly to an extent that is the most important take home message up to this lecture right okay. So, now this is now going to do now so that study the geometry that incident beam comes in gets scattered. So, the Ki is incident beam K2 is the scattered this goes to forward scattering the representation of neutrons scattering if we can do that then with the plane wave the coming neutron is a plane wave around that is with K Ki going out is also a plane wave going K2 then one can write the wave function blast total wave function that is going out is a linear combination of the two but most important part is that that is a comes from Fermi golden rule those scattering cross section square of the interaction between them and once you do that this is the Fermi golden rule that the this is the square of the interaction between the so the particle coming and here is my sample the neutron interact with my molecules here and get scattered off and that case is given by the scattering here which is nothing but interaction of the neutron at particles at position Ri and incoming neutron and that is R minus Ri some of the delta function the kind of thing we told before. So, this one now can be transformed into by fairly simple fairly simple algebra and using the delta function representations one can show that the scattering cross section is connected to this K prime is written as Ki K2 minus Ki that we will do just K later. So, this is the structure factor and one can also show that the structure factor is K is same as rho K rho minus K that is done in the book and that is the same as this quantity and so static structure factor which is measured by forget about this K by this this is experimental quantity this is experiment and this relation is the theory and this SK is nothing but the radial distribution function as we have yeah this is the equation. So, you know SK from neutron scattering and then we can invert this Fourier transform and get the GR and SK is getting from neutron scattering by the following fairly trivial manipulation that this is measured experimentally that directly gives SK once you know this parameter BRD that we are done now is known as the static structure factor and this gives the correlation and we just show that this gives GR most important quantity not just in the quits but in glasses in materials and everywhere else that is what the huge number of people are measuring in neutron scattering okay we are done. So, thermodynamic one can get the thermodynamic function for radial distribution functions and we will continue that in the next class little bit of this radial distribution functions and that is what today we stop and we will take up from here and then we will go tomorrow to a very related thing which is theory of polymers okay.