 So, in the last module, I talked to you about guinea approximation and showed you how you can get hydrodynamic radius or radius of gyration of one example of the pluronics. Next is porous law. Here I can write down the P of qr which is f square qr here in terms of pair correlation function but here is the pair correlation function and if you remember last earlier I had done again when I did the form factor if you remember sine qr by qr r square dr. I took a constant density and put rho outside but if I talk in terms of pair correlation function that means given a sphere if there is a particle here or some density here what is the probability of getting the density at another place which is at a distance r from here and that is given by gr and in common knowledge this is called the pair correlation function. So, now the haiku expansion gives the porous law and the porous law for smooth surfaces goes as 1 by q to the power 4. So, where sp by vp is the surface to volume ratio and here if I talk about log of you can see a is the it is some constant plus background that is the intensity here because p r gives me intensity p qr. So, i is equal to a background plus a upon q to the power n and if I do a log plot of it that means log of intensity minus the background I get log of a minus n log of q this n actually gives you the dimension of the surface. So, one is a dimensional surface I will explain to you see most of our objects that you see a straight line a circle or a pyramid let us say. So, this is one dimensional this is two dimensional this is three dimensional but the fact is that there are objects which have dimensions which between these how let me just use one simple example. Suppose I have got a scale of length l on which with which I measure this object. Now please see this is l l l l. So, if I measure it with a scale or with a ruler with a ruler which has got l as minimum count I get 1 2 3 4 l the length but now on this object let me try to put something else. So, now let me put at half a distance things like this. Now you see now in this if this object if I have a length which is l minimum I still have 4 l if I reduce my length to one-third the sensitive the minimum measuring length to one-third of l then I measure 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 I measure 16. Let us take a simple let us say one dimension like this if this length is l if I have a measuring ruler or measuring rod which can measure l by 2 I will it will linearly scale it will go to 2 if I make it one-third it will go to 3 but here you see you made it one-third you measure 16 so this is a fractal object. So, fractal objects so it doesn't go linearly or with any power of 3 integral power of 3 it is a non integral power of 3 when I reduce the scale to one-third this is not 3 it is 3 to the power I can say n it is not 3 square or 3 cube but this is some other number so this n is a fractal dimension so that means this concept comes the way things are embedded in space. So, for example one example is possibly if you take a tree and its leaves if you try to measure the dimension or let us say the coastline of India the coastline of India let us say if you see this is how it goes looks like I am being a little poor in drawing so if I look here if I measure it to the length scale l I will get some length of this coastline but if I magnify this coastline here it is actually somewhat like this and if I from l if I go to l by 2 or l by 3 the length I measure is not 3l or 2l but much larger or even if I take a part of this and again further I expand it even this is not a straight line it will be something like this and then again further expands so as you go to smaller and smaller length scales the length of the coastline of India changes and not linearly but because it is a fractal it is a fractal level now this fractal dimension we can measure because I have my porous law which is a by q to the power n plus b and in our experiments we can find out whether the object that I am trying to see or measure the surface to volume ratio whether it is the smooth surface it is the fractal surface so I have just taken an example of some of the fractal surfaces from this in a tool box this is a one dimensional two dimensional three dimensional object and the porous law will give you q to the power minus 1 minus 2 minus 4 but when you go to mass fractals or surface fractals you find these power law changes the power law changes the power law changes and from here we can always find out the fractal dimension of the object that you are going to measure using sun's experiment I will give you examples later so now I have introduced you to guinea region which is low q and porous region which gives me surface to volume ratio iq so with this let me get on to something called contrast factor excuse me as I told you earlier that my intensity has got several parts forget about n into v square volume of particle number of them you have got a contrast factor square of that the solvent and the particle then you have the form factor for the particle and then you have something called a structure factor what is the structure factor this I will borrow the concept from liquid and amorphous system what I taught you so if you have a very dilute system very dilute system very dilute system then one particle's location is not correlated with the location of another particle and what do you measure in this experiment is just as I described you so far the form factor for a particle but now if I keep increasing the density I keep increasing the density now their locations become correlated because one the particles cannot penetrate into each other and then as I argued earlier in for liquid and amorphous system that there is short train correlation and which will look like this that means you have a you have peaks in q value peaks in q value but these peaks are not well defined peaks as you find in drag diffraction but it basically signifies the correlation shells around a central particle and it's average over the entire ensemble so now we have iq multiplied by sq that gives me the intensity so so basically when you this is what this slide integrated the whole thing if you have a dilute system you have got a form factor here where you have a concentrated system you have got a peak in s of q and what you have is a multiplication which is p of q s of q and both of them acting together so then this is what the multiplication of the two is something what you are going to measure when you do a small angle neutron scattering please note that I am borrowing concepts from microscopy to mesoscopic world so that means now when I talked about microscopic world I talked about atoms or ions surrounded by atoms or ions here I'm talking about larger particles like I showed you the micelles from the surfactant particles this can be posed in a solid rock it can be precipitated in a matrix of metallurgical some alloys so and there the particles are large but the concepts are very similar you have got a structure factor which tells you the nearest neighbor distances and you have got a form factor which talks about the scattering length density and my experiments measure two of them so first there's a size dependence and size dependence means suppose I have the simplest example I take a spherical object so my p of q will fall faster if I take a dilute system for a larger radius particle I can the intensity will fall faster for a smaller radius particle the intensity will fall slower and from the log of q plot log of intensity plot I can get the radius of gyration of these particles of these particles now shape dependence many times the organic assemblies like I told you myself they need not be spherical so there can be rod like they can be like a disc like a disc or they can be rod like and for all these you can find out the average hydrodynamic radius or radius of gyration which is r g square for a rod I have written here where r is the radius and l is the length l is the length but the nature of fall is different in small angle nitro scattering so this Allah helps you to figure out the not only the size of the particle but also shape of the particle and similarly now if it is a solution mostly this is true for solution when you increase the concentration of such particles your s of q will change for example when the concentration is 5 percent to when the concentration is 10 percent of a radius 25 angstrom that is 2.5 nanometer and the charge of 25 coulomb you can see the s of q the peak position in the pair correlation function changes so similarly when the charge increases but the radius remains same the when the charge increases this exactly similar to what I discussed earlier the in case of sodium chloride it was the ionic liquid large mortal material where the ionic interaction stabilizes the distances here also same thing if there is charge then most likely the positively charged core will like to be like to repel and maintain a certain distance which is for nearest neighbor as we increase the charge this distance because better defined and s of q peak goes up with charge because higher charge means more coulombic repulsion and they define the nearest shell more accurately or more with more what should I say more definition better definition less fluctuation so this is the size dependence and considerable difference dependence of the data in case of small angle neutrons scattering for various particles but these are not microscopic particles again and again I am hoping these are mesoscopic size particles this is a very interesting part of this top and of this technique I wrote intensity depends on contrast now this is where we can do a very interesting thing please see that this is the particle and the solvent and p q s now I can play with rho p and rho s which are the scattering length densities rho is sum over scattering length density of a particle divided by volume you can using various contrast actually make different parts of a particle visible an example here the particle is invisible here you can play with the matrix scattering length density and you can make the particle visible how I will tell you shortly because this is a very important technique used by organic chemists and biologists for science let's say you have a particle which is a core and shell structure as I show here there's a core there's a shell I can play with the matrix with the matrix I can manipulate the contrast of this particle with the matrix and when I match the core with the matrix you see the shell or if I match the shell with the matrix you see the core so using sands in two different experiments I can find out the core structure I can find out the shell structure how scattering is different for neutrons on x-rays for different isotopes so for a hydrogen especially for hydrogen hydrogen 1h1 1h2 deuterium 1h3 they have different coherent scattering lengths for neutrons and that gives us a very interesting technique in our hand which is contrast matching of a mixture of mixture of h2o and d2o let me just work it out for you so this for example I have noted down hydrogen has got a negative scattering length why negative I will explain later but now accept the fact that hydrogen has got a negative scattering length of minus 3.74 hydrogen has minus 3.74 femtometer this is scattering length deuterium has 6.664 femtometer oxygen has 5.8 femtometer this is interesting let me just work out so that means h2o h2o has got 2h so minus 3.74 into 2 plus 5.8 femtometer as it's scattering length d2o has 6.664 into 2 plus 5.8 femtometer so this goes to d2o has no so h2o and d2o now let's consider the density of the queen so you know that is 1cc is 1 gram we know that and you also know that 18 grams of h2o will have 6.023 into 10 to the power 23 h2o molecules each molecule have worked out each molecule have worked out here I have worked out here these two values so now 18 grams of h2o has got 18 because 8 o16 h2o come right now 18 grams 16 plus 2 18 s and for d2o it is so this 18 has 6.023 into the power 23 so in 1cc in 1 gram it is 1 by 6.023 into 10 to the power 23 number of molecules and each molecule has got each molecule has got sorry and has got this is the scattering length so then that gives me the scattering length density for h2o similarly for d2o it is 1h2 so the it is 1h2 so 2d2o will be 2d2o so the molecular weight of d2o is molecular weight of d2o is 2 into 2 because there's a mass is plus 16 so it is 20 so 20 grams of d2o will have 6.023 into 10 to the power 23 d2o molecules now per cc we just evaluate 1 by 20 of this that is the number of molecules multiply it with the scattering length that I evaluated for 1h2o and 1d2o and what you get is a very interesting thing scattering length of d2o is 6.38 10 to the power 10 because 10 to the power 15 meters 10 to the power minus 13 centimeters this 10 to the power 23 and it is length divided by length cube so it comes out to be centimeter to the power minus 3 most interesting thing is that in the simple calculation you will find that the scattering length density of d2o is 6.3 10 to the power 10 which is a positive number and the scattering length of h2o is minus 0.56 into 10 to the power 10 centimeter to the power minus 2 now you see I can mix d2o and h2o they are the same chemistry same so when I mix these two then I can even make the scattering length density of the mixture as 0 but most importantly if I know the chemical formula of the core and the shell in our examples whatever experiments you are doing I can match we can make a solution so presumption is that I am putting these large objects in a solution of d2o and h2o excuse me when I put them in a solution I can match the scattering length density of the solution either with the shell or with the core and in the process I can sort of illuminate different parts of the object in my experiment and these are I make samples for contrast variation by matching it for example if it is only hydrogen based I can put it in pure d2o if it is hydrogen but with different scattering length densities I can play with the mixture and tube for example if I map put one cc of water plus one cc of d2o you can see it becomes 6.3a minus can I just see the number minus 0.56 0.56 now the thing is that into 10 to the power 10 centimeter inverse so you can see that if I put one cc of d2o d2o with 10 cc of water it is a factor of 10 then it will be 5.6 sorry one cd2o plus 10 cc of water what is the scattering length density for this that is 6.38 minus 5.6 into 10 to the power 10 centimeter inverse if I increase it slightly more maybe 11 cc or 12 cc I can make this equal to 0 so I can make the matrix 0 but that is not the done thing always what is done is basically in this kind of core shell structure to match various parts of the object even if and then you can find out the geometry of the object because you can find out the size relationship from that experiment so you can find out the shape you can find out the volume by illuminating various parts of the object it is an extremely strong technique available only with neutrons because h21 d2o has got a huge contrast between the two things