 Hello everyone, welcome to the last class on advanced characterization techniques. So, in the course of this series, we have gone through you know the basics of x-ray diffraction and scattering. We have studied small angle x-ray scattering as well as grazing incidence small angle x-ray scattering. Then we also studied electron diffraction and finally, we studied some spectroscopic technique using x-rays. In the last class, we had studied a few things about neutrons and in the final lecture today, we are going to extend what all we studied about neutrons to a technique which we had studied earlier that is small angle scattering. In the earlier classes, we had gone through small angle x-ray scattering and we had also realized what are the advantages of using neutrons. And therefore, in the present class, we are going to extend the same geometry or the same concept of small angle scattering to neutron scattering. So, as we had discussed earlier that interaction of photons with matter can lead to scattering or as well as diffraction. Now, scattering of x-rays is essentially from electrons and we had seen earlier in the last class that scattering of neutrons is from the nucleus. We had also seen now this slide is almost borrowed from what I had shown you during small angle x-ray scattering that this is what is your diffraction where your incident x-rays or for that matter neutrons interact with the lattice planes and which occurs at a very small level of the order of the wavelength of the incident x-rays or neutron. While when we talk about scattering, scattering occurs at a higher length scale and small angle scattering most of the times is from interfaces. Now, this is what we had seen earlier also and the same concept remains valid even for neutron scattering. However, I hope you remember that in the last class, we had seen that depending on the energy, we can get neutrons of very different wavelengths. And therefore, we can see a variety of objects theoretically. I hope you appreciate that we can see a lot of wide range of objects using neutron diffraction compared to x-ray diffraction. This is precisely because we have neutrons in the energy range in a wide energy range which in turn gives us a wide wavelength regime. Having said that generally most of the neutron scattering or small angle neutron scattering also has the same size regime as that of small angle x-ray scattering. As we had seen earlier that at the very small length scale, we have very large angle and this constitutes the normal diffraction. As we have for x-ray, we have similar condition for neutron diffraction also. However, at large length scales, we get smaller angle. However, I am now going to say something which may kind of surprise you. We are talking about small angle neutron scattering and in the last class or in the couple of classes before, we talked about small angle x-ray scattering. However, this term small angle within codes is actually a misnomer because if at all we are having a change in wavelength, the angle at which we are getting scattering is going to change and this is particularly valid for neutrons where we can get a wide variety of wavelengths. Therefore, the correct term should be actually low q or low k diffraction for what we term as small angle scattering and normal wide angle scattering can be referred to as high q diffraction or scattering. So, this is just the kind of terminology that has kind of stuck with the technique. But I hope you appreciate that in small angle scattering, we are mostly dealing with what is happening at very low q values while at higher q values or k values, we deal with wide angle x-ray scattering which is the conventional intensity versus 2 theta plot. So, let us move over from there and I would just like to remind you again that the small angle neutron scattering theory is exactly similar to that of small angle x-ray scattering theory and for most practical applications, we are having the same size regime as we have for small angle x-ray scattering. But the big advantage that small angle neutron scattering offers because of the use of neutrons is that we can penetrate deeper. We can penetrate deeper within containers which are made of metallic materials and containing various soft matter like your polymers and what all things are happening to it as a function of flow or applied stress and therefore, we can study the rheological as well as flow behavior of the soft matter using small angle neutron scattering. At the same time a big advantage that neutrons offer over x-rays is their ability to detect light elements. So, particularly this is very important when it comes to biological materials and therefore, small angle neutron scattering has extensive application in biological, chemical as well as ferrofluid materials. Here again for ferrofluids, you know that neutrons are sensitive to the magnetic moment and therefore, can give us information about the magnetic structure. So, all this information is obtained using small angle neutron scattering which is generally not obtained in small angle x-ray scattering. One more important thing that we had stressed in the last class was about the ability of neutrons to separate between isotopes. Now, this ability of neutrons can be used to get better contrast. We will talk about this in the couple of slides, but I hope you appreciate that the advantages of neutrons over x-rays can be exploited in this technique to get relevant information from our experiment. So, just to revise and put things in perspective, I hope you remember that we had this sample on which whether neutrons or electrons are getting you know kind of their incident and we see that it scatters the incident beam. And you see if Ki is your incident beam wave vector and Kf is your scattered wave vector you know we had defined our scattered wave vector Q as Q is equal to 4 pi sin theta by lambda. And the overall intensity that we got on the screen or our detector is nothing but phi is nothing but the distribution of particles and PQ is nothing but the form factor. If you remember like this contribution comes from all the atoms that constitute our particle and the SQ is nothing but the structure factor right and the structure factor comes because of the arrangement of these particles. So, this theory is exactly similar to that of small angle x-ray scattering. So, again if you remember we had two regimes right we had the Koiner regime and the Porot regime. The Koiner regime was valid at Q max Rg less than or equal to 1 this product Q max is nothing but your wave vector right into Rg where Rg is the radius of gyration which was for sphere as square root of 5 by 3 times at the radius of the sphere. So, it gives you the slope of this line minus Rg square by 3 if you plot line of intensity versus Q square this is our Koiner regime. However at high Q now mind you this higher Q is not the high Q regime which corresponds to the wide angle scattering. But at high Q regime we see that line of i is proportional to Q power minus 4 and we see that from this information about what all slope we get we can get information about the shape of the particle right this is what we did in small angle x-ray scattering. Now the same concept can be borrowed and applied directly to small angle neutron scattering. Having said that let us try to see one particular thing or rather one particular concept that is not seen or not quite prominent in x-rays and that is the ability to separate between the isotopes. But before we go there let us kind of rebuild and revise how actually scattering occurs. So all of us are aware that any particle if we take the particles are characterized by a scattering density right which is rho of r. Now this rho of r or the density of scattering depends on what all atoms are there right so the particle which comprises of atoms. So the scattering tendency of the particle is nothing but the summation of scattering tendency or scattering ability of all these atoms right. So therefore you see that your scattering density rho of r is nothing but summation Bi. Now it has to be averaged over the volume of the particle and therefore the scattering density is nothing but summation Bi over rho. Now these particles are not standing alone instead most of the cases these particles are there in a solution right. So we also have to assume like what is the density of the solvent under consideration so let it be say rho 0. So if at all our neutrons were to pass through a solution comprising of a solvent of scattering density rho 0 and particle with scattering density rho of r where r is a function of the location of the particle the intensity that we get can be given by this formula. Here mind you look what we are having over here so this is the difference in the scattering density of the particle and the solvent and this is what gives us the contrast. So this is the resultant scattering density. Here again the value of Q which we had seen the wave vector is nothing but 4 pi sin theta by lambda right and you see this exponential term and you see it is proportional to d cube r right and the whole thing square. So this is again very similar to our structure factor right and this also accounts for the size part. So moving on the form factor is related as we had seen in case of small angle X-ray scattering also to the scattering density. Therefore our P of Q is nothing but f of Q square. Now this is very similar to what we got in case of diffraction right the intensity that we get is directly proportional to the square of the amplitude right. The same over here and this f of Q is nothing but the difference in the scattering density of the particle and the solution right or the solvent okay. So this is the f of Q term similarly if you remember the structure factor which we had calculated in small angle X-ray scattering has remains the same when it comes to small angle neutron scattering and it has to do with the spatial position of the particles. So see we have a form factor which depends on the size and shape of the particle while the structure factor is decides the or rather is dependent on the spatial position of the particles and here you can see that r alpha and r beta are nothing but you know nothing but the spatial coordinates of the two particles. So and there is a big summation sign and everything is averaged for the particle under consideration okay. So now let us move ahead so you see that we do have a excess amplitude or contrast length so this is very important. So this is what we want to maximize in small angle X-ray scattering in order to get a estimate of the shape as well as size of the particle in the solution we need to maximize the contrast right and this is what is the contrast right. So I would like to remind you that this is your scattering ability was nothing but summation bi over v okay. Isn't it rho is equal to summation bi over v into dq and r. So what we get over here is actually a contrast length and this is the term the term on your right hand side which we want to maximize. Now in order to get sufficient signal sufficient number of particles must be there right like if the particle size is very small you should have sufficient number of these particles very small particles in your solution. So that you get good enough signal and from this signal you can decipher the size shape as well as distribution of these particles in the solvent and in order to do this you have to make certain approximations and we assume that all particles should be identical right their orientation is random. We have lower concentration so that there is no interference of scattering cause from one particle with that of cause by the other particle. Another important thing is we ideally like a two component system so you have a solvent and your particle right which is which may be a macro molecule you don't like you know any interaction taking between these two and having the presence of third phase. So in this case the situation is very clean and you can do very easy analysis of your sample and get information about the size shape as well as distribution of the particle under consideration. So before I proceed you see here the contrast term the contrast length that I have shown over here we have just noted down for different nucleus mind you this contrast term is for nucleus I had mentioned that you know it is for different atoms but mind you our neutrons we should remember that neutrons do not interact with the means they do interact with the atom but they do not interact with the electron cloud of the atom but with the nucleus of the atom and you see for light elements we do have different values and very good values you know light as well as heavy elements. Another important thing is look what we are having here hydrogen and deuterium and as I had mentioned that you know they have different scattering cross section here again we see that they have different scattering length not only the magnitude is different but also their sign is different and this is where lies the biggest you know advantage of neutron diffraction you know that most of the biological systems comprises of hydrogen. So many a times we can play and replace hydrogen with deuterium and try to get or improve the contrast now this is something that is very difficult to do using small angle or rather it should be not possible it is not possible in small angle X-ray scattering okay. So again let us go back and try to work on this you know macromolecule solvent stuff so scattering due to solvent and macromolecule needs to be separated right like at the end of the day we are getting signal from both the solvent as well as the macromolecule that we are studying. So we need the overall scattering is essentially rho r rho of r is rho v plus rho f r where rho v is your average scattering density of the particle now this comes from the particle while rho f of r comes from the fluctuation about this mean now this is caused essentially because of the interaction of the particle okay with the solvent and therefore the scattered intensity I hope you appreciate can be given as nothing but rho v which we have over here minus rho 0 square i v q plus rho v minus rho 0 i v f q and plus i f q term so you know this fluctuation term also plays a very important role in deciding the intensity right and this is where it comes the overall intensity so if at all we have to see contrast we have to change this fluctuation right and this will happen how this will happen again this will happen by changing the interaction between the solvent and the macromolecule so that is what I mean that you know we can modify rho 0 by mixing H2O or what or D2O deuterium oxide right so if you have a particular solvent you can add water in it right or you add heavy water that is D2O in it now this will ensure that our i v q as well as i f q can be determined from the contrast series right like once all one in one case you have only your solvent and macromolecule in the other case you add water and in the other case you add D2O and then you compare right so your i v let us go back and see what is your i v i v is the scattering density of the means the intensity due to scattering by the particle while i f is actually due to the fluctuation part right and the fluctuation is going to change as we change the solvent so it is a bit mathematically involved but I hope you appreciate by changing different you know solvents we can get some information having said that there is however you know there are however two problems the first problem is that there is perturbation in the solvent at the interface like if you change the solvent there is almost always some perturbation so the perturbation level may change at the same time what may happen like if you are changing something with say hydrogen right if you are just in a solvent if you are putting water what may happen is the hydrogen iron which is in your macromolecule may get exchanged with the water the hydrogen iron in the water that you are adding it will things will get even more problematic if your hydrogen is replaced with deuterium but having said that it is still the aforementioned technique of you know what you know changing the contrast by adding hydrogen and hydrogen water and d2o is quite valid and as we move ahead we again come across this intensity versus intensity as function of q term which is nothing but i0 plus we see here radius of gyration q square by three term now this is again similar to what we had got using small angle x-ray scattering the only difference that we have over here is that in this case again your scattering ability or scattering density is a function of you know you see for your particle okay as well as but you have to subtract it from that of the solvent I am invariably using particle and macromolecule you know as as meaning the same though you should keep in mind that you know these macromolecules are pretty large and they can agglomerate and form a small particle okay so this is what we talked about was actually what is known as contrast variation so you know that compounds with close scattering length give low contrast that is because if the two atoms are having the same two elements are having the same same scattering ability we cannot differentiate between the scattering due to the individual element however small angle neutron scattering offers improvement in contrast by isotope substitution right so instead of if you are having hydrogen why do not you put deuterium right if you want to look for say a particular isotope of nitrogen right instead of n14 you substitute it with n15 right so this gives us the possibility of erasing the scattering from solvent by using normal and deuterated solvent this means you take a solvent which is normal right and then in one which contains water and in the other case you put heavy water and see try to match how the contrast is changing and if you do it that way what we can do is see the whole idea is if we have more than one type of particle right or more than one type of macro molecule what we can do is we can play with the elements and we can glow so it is just like if the entire wall is painted in white you paint the region of interest in black so there is a lot of contrast and if you want to see the outer part what you do is the black region is painted in white and the rest of the region covers into black this is what has been shown over here so you see here what we do is when you are you have two entities right our row one and row two so these are two particles right so now what we do is we do match contrast matching so in contrast matching what actually happens is we look at row two right the particle two and make its contrast similar to that with that of solvent so what actually happens we are not able to differentiate between particle two and the solvent but this gives us the maximum contrast for particle one right so we can now very clearly see the particle one over here fine now what about particle two well we play the same trick now instead of matching the you know scattering ability of the solvent with that of particle two we match it with that of particle one so now particle one is invisible and we see the contrast only of particle two that is great because you know using these two techniques one after the other we have got information not only about particle one but also about particle two which means we have been able to differentiate between two species which have very very close scattering length or scattering ability so you see how by intelligently doing contrast matching we can get information and separate out two species which are having very close scattering tendency let me remind you that this can be done only in small angle neutron scattering because of the sensitivity of neutrons to differentiate between the isotopes of the same elements so this is one of the biggest advantage of going for small angle neutron scattering just to give you some you know kind of feel about where all we can do use small angle neutron scattering I have cited a few examples I would not go back and you know kind of repeat the same things that we did using small angle x-ray scattering because with small angle neutron scattering we can do all the things that we can do using small angle x-ray scattering and therefore in the next couple of slides I am going to show you certain things which you can do only with small angle neutron scattering right so if you remember small angle x-ray scattering or for that matter any x-ray scattering or diffraction experiments is always are always limited by the penetration depth of x-rays while if you want to study flow behavior right of a polymeric hydrogels polymers or polymer composites polymer nano clay composites well there is no way like we have to apply stresses on it right or to make the polymer flow now how do we do that well we need containers and that is what has been shown over here the common coitre flow cell geometry as well as the cross section of common flow which are used in small angle neutron scattering environment now keep this thing in mind like you know like all these containers contain our polymer or hydro gel of interest and we are going to apply stresses on it right shear stress or to see how they are flow we are going to actually squeezing the material into these cavities now how do we get signal and how do we see how the structure is evolving as a function of say stress or flow remember in order to actually get some information our probe has to have the ability to pass through these containers and gather important information from the polymer or hydro gel under consideration now this is absolutely impossible with small angle x-ray scattering right even if you are using a synchrotron the synchrotron will not the synchrotron radiation will not penetrate through these containers and we will not get good signal however small angle neutron scattering due to its high penetration depth can give us some signal and we can really study in situ the flow behavior how the rearrangement of the particles right of the particles is taking place or the nano clay polymer composite how the flow behavior as well as the reorientation can be easily studied using small angle neutron scattering this is what has been shown over here and you see here we have a biopolymer that was subjected to a shear strain of 145 percent and this is how the pattern goes you see we do see some kind of an elongation over here right and what all information we have this is the experimental part we can incorporate again go back and you know kind of simulated and get this value which has been shown over here now this is done using again commercially available softwares or which are available at the neutron sources various neutron sources in the world and are very similar to the softwares that we use for small angle x-ray scattering but having said that the input that we are taking one is your structure factor and the other one and more most importantly is our form factor so you see we can get very nice information you know from the simulations and we can predict how the particles have sheared as a function of strain not only that we can also get the annular intensity average and this is what we plotted over here and also the sector average 1D scattering which clearly shows that there is a difference in the scattering tendency of the particles perpendicular and parallel to the direction of straining right and here again we see what is very important that there is indeed some reorientation of phases or particles in the bio polymer right so you see like how we can use this and get in information during in situ experiments right as a function of stress or flow behavior in polymers using small angle neutron scattering now as I had mentioned and as we also appreciated in the last class that small angle x-ray means x-ray scattering and neutron scattering or for that matter x-ray and neutron can be used as complementary sources so here I am going to present a small example from a very old paper which is on silicon carbon nitrogen ceramics and here both small angle x-ray scattering and small angle neutron scattering were used to actually characterize the material so this composites were prepared using pyrolysis of poly-cellul carbodyl my precursor at 1373k to obtain silicon 24.4 carbon 42.4 nitrogen 33.2 ceramic now these ceramics were amorphous up to 1673 that is 1400 degree centigrade and here you see after running at 1400 degree centigrade we do get some peaks indicating the crystalline nature okay but below this temperature it is absolutely amorphous but this is about wide angle x-ray scattering now let us go and look at what happens in small angle scattering so on my left hand side I have the small angle x-ray scattering data and on my right hand side I have my small angle neutron scattering data here again you see what we have you see here 15 n so here we have substituted right the nitrogen with its isotope while here we have the native nitrogen right the native nitrogen so now let us compare the 1100 one sample with 15n and the native n now you see the 1100 one is comprising of this style over here and you see where it goes so it goes over here somewhere right so it is here and the 1100 with the native nitrogen is a triangle and you see where it goes oh it goes over here so oh my god we do have a difference right theoretically we were not supposed to have this difference isn't it because as we know x-ray scattering is not sensitive to isotopes however this may be an exception and this can be attributed to some problem during the processing however if you go back and look at the silicon carbide SIC with n native centered at you know rather annealed at 1300 degree centigrade you see that it goes somewhere over here right this one right so we see that here in this region there is almost a bit of mass so let us assume that you know there is not much of a difference right and here again one point that I would like to emphasize is that though we talked about the difference between 1100 15n and native n you see the almost behavior right in this region it is almost parallel right so you remember at very low k if you are getting a parallel region this corresponds to what well this corresponds to in this region well let us not talk about it but it corresponds to the porod regime find you mind you this is in negative right so now if you go to the small angle neutron scattering so let us look at 1300 degree centigrade curve so you have these squares which are over here and here once you go you see with the native nitrogen right so this you see there is a huge bit of contrast right you see huge bit of contrast where very here is you see lot of contrast right in fact it is characteristically different right once you have a native nitrogen and here where you have the isotope right so you see how much contrast we get using small angle neutron scattering we saw that you know increase in temperature like let to the signal shift at lower edge values now this edge is similar to q or k that we had used now even then at low k right your low edge we saw a linear behavior which is similar to the porod regime now here you see here we do not see that right so you see the contrast we are getting different information right from these two scans though the range of the q is same so see with small angle x-ray scattering this is more of a linear log log straight line this is the slope will be about minus 2 to the power minus 4 right so around minus 4 so actually this indicates the porod regime see here you see we see a nice porod regime right while here no porod regime is valid so you see depending on the wavelength and the character whether you are using x-ray or neutron we can get different regime so I just wanted to emphasize that you know we cannot just blindly take the entire theory of small angle x-ray scattering and extended to neutron scattering and there indeed are differences right and here again we saw that isotope substitution gave very much it gave quite a bit of contrast or characteristic difference in small angle neutron scattering right so this is how our study should aim right like you start with normal wide angle x-ray diffraction and then you can do small angle x-ray as well as small angle neutron scattering I am not going to go in details of this particular observation but let us go ahead and look at one of the best advantage of neutrons that is its ability to separate to get information about the magnetic structure so this example from a paper by Copitsa et al what we have done is there is what they have done is there is iron manganese alloy right which has shape member effect right so that they remember once you change you it as a function of stress and go from you know austenite one phase to other phase they remain remember its shape so we used polarized neutrons in transmission mode right and I have not gone given the details but in this case these are nice metallic materials right so we instead of having a suspension these were rolled plates of about 3 millimeter thickness right now you cannot imagine even synchrotron light to pass through 3 millimeters of iron manganese alloys right a sheet of iron manganese alloys which is about 3 millimeter thick even if the synchrotron radiation can penetrate the signal will be very very weak however neutrons can easily penetrate and by monitoring the transmission signal you can see that there was very small they were able to see that there is very small catering which essentially indicated that there is low magnetic heterogeneity not only that they also added carbon and nitrogen again you can add directly or you can add that respective isotopes you can add carbon 12 or carbon 14 or nitrogen 14 and nitrogen 15 however this study they have not done but you always have this option to maximize the contrast and this is what has been shown over here so you see that the momentum dependence on intensity so this is our momentum transfer or q of sands for silica doped iron manganese alloys you see that the difference in intensities of scattering is due to because of carbon and nitrogen so this way we can study what happens due to addition of trace elements or interstitial elements on the overall shape memory effect which has tremendous metallurgical as well as mechanical applications right so again using only and this can be done only using small angle neutron scattering because we cannot get any information about the magnetic structure using x-ray scattering so to summarize I hope you appreciate that small angle neutron scattering offers us a very powerful tool to study objects between 1 to 100 nanometers it can be used or rather it is used widely for several applications in the field of biology chemistry polymers as well as ferrofluids the most important advantage is the contrast variation method for analysis of density in samples containing more than one compound so we can get different materials right different kind of phases which are present right in the same material we are able to separate them in a much better way using small angle neutron scattering and this aspect which I have not touched about but as we already know there are very good softwares right which are available for analyzing small angle x-ray scattering as well as small angle neutron scattering data and we can reconstruct the size shape as well as distribution of the particle under consideration with small angle neutron scattering we can also use to get information about the size shape and distribution of the mic or rather the macro molecule you know which are of the order of which fall in the same size regime so with this we come to the end of this module having said that I hope I have been able to show you or rather give you a glimpse of the entire spectrum of advanced characterization technique that use diffraction scattering as well as spectroscopy principles the major aim of this module was to just expose you to all these techniques so that you are you become aware I have not really gone into greater details and there are plenty of text available on these techniques I therefore request you to go through the text which will be shown which is on the website where you can see what all textbooks can be offered having said that let me remind you that all these techniques what we have used are very much dependent on the instruments that we are having and therefore the best information can be obtained at the facility which offers all these techniques so the main point is if at all you are going to carry out a synchrotron diffraction or scattering experiment or neutron diffraction or neutron scattering experiments it is always suggested to spend some time at that particular facility and carry out your experiments with this I will wish you all the best for your advanced characterization techniques which we will probably find use at some time in your academic career thank you.