 So, as I am just summing it up, the researchers are often interested in inhomogenities 1 nanometer to 1,000 nanometer, so precipitates installites, micelles in liquids, proteins, pores in a medium and I must mention here the small angle neutron scattering and small angle X-ray scattering in sands and sacks are important experimental tools across communities, but here the thing is that I am doing experiments at a very low angle, so that means you please consider the case that I have got a sample and then I have got a detector, I have got a direct beam with and I have got a diffracted beam somewhere here, the fact is that because I am going to very very small angles, very small angles I need to be careful that I do not go and intercept the direct beam, so I am almost in direct beam, so for that matter I have to then collimate the direct beam to a large extent that means I have to reduce the delta theta and this is possible by using collimeters in the beam path but at a cost, when you put collimeters in the beam path you get a smaller angular spray, but your number of neutrons go down because you are cutting down their numbers by putting collimeters, so that is why again that when you use a small angle neutron scattering instrument on a cold neutron source just start with a larger number of neutrons, that is where the importance of the cold neutron source comes into picture that, so you use a highly collimated beam in case of a small angle neutron scattering and then you are doing the experiment almost at the direct beam path, in the direct beam path, so quick look to the findings that you can see schematically, here I am doing a Fourier transform over a density, I have done it earlier actually if you remember when I talked about form factor for a spherical electronic charge distribution in an atom, I got exactly that is an expression except some constants like 1 by b, so this is the structure factor or sorry the form is the form factor for a charge distribution like this. Now instead of electronic charge distribution if I take a scattering length density distribution, so please look at this figure, so I have these objects, I have these objects that we can gradually larger in this becoming larger, they have their sizes, so now I can talk to, instead of scattering length density instead of electronic density I am talking about scattering length density of these objects, if I want to see that then I do a scattering experiment and I measure iq, now when the objects are smaller then iq falls lower, you know if this objects are delta functions I showed you earlier it is constant all over q because a delta function is a constant in q space a delta function real space, as the objects becomes larger and larger your iq falls faster and faster and the typical size is given, it is a typical size given by twice by t, so I can look at the size of these particles looking at the intensity profile in a scattering experiment, but please remember it is not just qualitative, very shortly I will come to the expressions that I can use to get size of such particles, so the small angle neutrons scattering you can see that it is really almost close to a direct beam the detector, sometimes we use a position sensitive detector I will show you the instrument in Dhruva which uses a position sensitive detector, the q range you can say typically 0.001 to 1 angstrom inverse, lambda is equal to 4 to 10 angstrom and the 2 theta the scattering angle is typically around 0.5 to 10 degrees, so let us see if I talk about a 4 angstrom beam 4 angstrom beam, so please note q is equal to I chose it so that I can cancel some of the constants quickly, lambda is 4 angstrom sine theta, so I call it at low angle I assume sine theta equal to theta, so it is pi theta, so typically if I talk about pi theta if q is 0.001 let us say then theta will be 0.001 by pi in radians, so it will be into 180 by pi if I go to theta it will be around 2 degrees, I request you to please do this simple calculation and check, so we have to go to very very low q even with a 4 angstrom neutron, if we use a neutron which is that is 3 angstrom or 2 angstrom, because if I do not have a pole source then I may not be able to choose 4 angstrom because the numbers will be less, if I go to smaller wavelength I need to go to say from 4 angstrom to 2 angstrom from 2 degree I need to measure at 1 degree and I have to restrict my direct beam much below 1 degree, so that I can do the experiments, so this is the small angle in cross scattering and now my intensity is restricted on the lower side in the q or in the small angle with the direct beam and I keep measuring as a function of angle and on the higher side my q max is restricted by the background, this is an important factor because wherever you do the experiment either inside a reactor hall or in a neutron guide hall, you have background neutrons and background neutrons when your intensity because it is falling with q, when it matches the background then I can stop the experiment, I have to stop the experiment because now I have gone into the background, so q minim and q maximum are dictated by the direct beam and the background and the sample the signal to background ratio, now I will quickly and briefly take you to the theory of science, you will be appreciated, if you see as I told you a few slides earlier that I have a matrix and I have these particles which I want to study, these are the particles, so now earlier also when I did the extra scattering or neutron if you remember this was the expression for the scattering amplitude, amplitude this a, but there it was there was an fj and a structure factor and q had to become equal to g because it is crystallographic material, but this fj is a Fourier transform of the electronic charge in case of x-ray, in case of neutrons I replace that with bj the coherence scattering which is give me a which gives you a delta function in q space, but this fj is a Fourier transform of the electronic charge density and in this case it is a Fourier transform over this is a form factor, so in lecture 3 I introduced you to this form factor for atoms for the electronic charge density, here I am introducing the same thing with respect to this kind of objects, I will use the simplest example for example I have let us say I have a liquid in which I put this surfactant molecules which has got a hydrophilic head group and a hydrophobic tail group, so up to a certain density they will be sitting on the surface the tails sticking out and the heads on the water, if you keep increasing the density now to hide from the water this is how the structure will form it will be a near spherical not always spherical, but here the head group looks out to the water because it is hydrophilic and the tail group is inside, so this is called a micelle in chemistry, now you see this is the object and I am looking at the form factor of this object instead of fj, so now it is this is now depending on the scattering length density which I discussed earlier, so I measured the scattering length density and also now I have got a scattering length density contrast because you have a matrix here the matrix is water, so water has got us, so I can say sld of h2o and sld scattering length density the object and there must be some contrast, if there is no contrast this is like putting glass inside water, if I put a piece of glass inside water the refractive indices match you cannot see the glass it is very similar to that you have put a object whose scattering length density is same with water you cannot see in this scattering script because it is as good as the entire matrix, but when there is a scattering length contrast that means this scattering length is not equal to the scattering length of density of matrix that they are different then you can see them and you can measure the reflected not reflected I am sorry the scattered intensity at small angle and you can find the form factor that means the shape and size of this particle you can study using this technique and it is very important, so now let us say the form factor let us say a spherical symmetric object, so there this also I had introduced earlier it is very simple if you consider a sphere of density rho then e to the power i q dot r then you have to integrate if it is a spherical particle then it is r square dr d theta d phi per a spherical symmetric object so it becomes e to the power i q r cos theta sin theta d theta 0 to pi then you have r square dr and the phi gives you 2 pi so now this I will give you hints I will I do not want to do the whole derivation for you but you can see 0 to pi e to the power i q r cos theta sin theta d theta the same as minus 1 to 1 to the power i q alpha let us call it d alpha if I consider cos theta equal to alpha this is very easy to the integration and ultimately what you get with this integration the form factor for a spherical symmetric object which is this and this under low q when q is going to 0 comes to I am sorry about this there is a mistake it is e to the power of minus q square by 3 r g square so instead here this r g is a radius of gyration this is 3 fifth r square per a sphere why let me just quickly show you see r square average that is r g square nothing but r square integrate over a sphere and then the r value probability is 4 pi r square dr for a shell do this r to the power 4 4 pi goes out it becomes r to the power pi by 5 and then it's 4 with outside so it comes to 3 5th g square please derive this now if this relationship known as guinea relationship is rho square which is a contrast not just a density density contrast these raise the volume of each particle into this is wrong e to the power minus q square by 3 r g square here you can see if I will put log of i q then plot it against q square that will be a straight line and that slope of the straight line would give me r g radius of gyration I have used an example from this thing is a periodic is a tri block copolymer which is soluble in water and dissolving d2o actually because this is a hydrogenous this polymer is a hydrogenous something if you put it in h2o you can't see so it is dissolved in d2o I'll talk about something called a contrast factor shortly if I you can see this is an experimental result so it's called a guinea plot for small angle or sounds data taken from 10 gram by gram p 85 pluronic in d2o at 20 degree centigrade the slope of the thing gives you the r g sometimes it's called hydrodynamic radius also here or you can see this is the average r value for a sphere so this is the finding now you can see that this experiment tells me that pluronic forms force the coils into a sphere of this size when I do a measurement in the low q region and please note the q square 0.0002 to 0.0016 so q is actually here square root of this square root of 0.0016 is equal to 16 10 to the power minus 4 so it is 4 into 10 to the power minus 2 so this is 0.004 and strum inverse so we have to go really really low q values for this experiment I'll stop here and go to the next module shortly