 Good afternoon everybody. My name is Elizabeth Blackburn. I work at the physics department at Lund University. And this afternoon I'm going to be talking to you about magnetic aspects of small angle neutron scattering. So that will include some information on how we look at magnetic materials generally using neutrons and then very more specifically what we can learn from doing small angle neutron scattering experiments. About magnetic materials. Okay, so I should say please feel free to interrupt either verbally or through the chat if you have any questions or anything that you want to ask for more information about. Right. So to get started. Okay, so what we will what I will talk about today is the difference between nuclear and magnetic scattering of neutrons. What that means for our experiments, and here by experiments I am implicitly referring to small angle neutron scattering experiments, how we can optimize our experiments to take advantage of these differences to pick out magnetic features. And what are the things that we need to account for when measuring magnetic materials. So I think already you've learned quite a bit about the kinds of things that one can learn from small angle neutron scattering. And I'm going to all of those things remain true for magnetic materials with magnetic components. But there are some additional things that change some of the assumptions that that we make. Okay. So let you know the picture to the side here is is a commemoration of the discovery of the neutron by by James Chadwick way back when. Right. So, what do neutrons interact with so you've already you've already covered this to some extent or but as you know they couple to nuclei via the strong force, and this is a short range interaction. And so we can basically approximate that as as if a single nucleus exists at a single point so that the nuclei in the material appoint scatterers. And so we can then say that the associated scattering potential is this formula here so this is the scattering potential. This is the delta function indicating that the nucleus we can effectively treat it as a single point be here is our scattering length, which we will eventually convert into scattering length densities. And then we have some pre factors here relating to the mass of the neutron and the plank constant. So this is just due to the interaction between between nucleons because the neutron is a nucleon. However, as you can see up here if we look at the basic properties of the neutron. In addition to to its mass it also carries a spin. And that means it's got a spin one half, so it looks a bit like a little magnetic dipole and the magnetic dipole moments associated with that is minus 1.9 nuclear magnetons now. And we'll come to what that's what that actually means in a moment. Okay. And that moment is now, I hope. Right. So, what I've written here is that this is equal to 1000th of a mu B. So a mu B is a ball magneton, the ball magneton is the magnetic dipole moment associated with an electron. So when we are looking at most materials, any magnetic components or the strongest parts of the magnetic component usually come from the electron. So this ball magneton is our scale of reference. Okay, so in addition to interacting with the strong force the neutrons also couple to magnetic fields. And so, and so they will be able to diffract off periodic variations in magnetic fields and interact with changes, changes in the magnetic field distribution, much in the same way as they interact with the changes in the, in the atomic density. So, in a solid, these magnetic fields are generated by magnetic nuclei and electrons. However, because the moment associated with the magnetic nuclei is usually very small, the 1000th of that of the electron, the electron, the electron behavior usually dominates. So, for now we will basically ignore any effects from magnetic nuclei, they exist, they can be important sometimes. But for the purposes of this, of this course will pretend they don't exist. For now. And we mostly assume that we can model the magnetic fields that arise in materials by taking the atom and placing a magnetic moment on it so we're basically treating the atoms as carrying particular magnetic moments with little magnetic dipoles and then figuring out the associated magnetic fields from there. Okay, how does this help us if we want to look at magnetic structures. So we can see that we can interact with variations in the magnetic field. But there are, but then how do we actually get information out about our magnetic materials. So there are a couple of ways that you can think about this. I'm going to start with the, what is sometimes referred to in the color approach, where the idea is that the magnetization on a particular atom can be thought of as a color so that it's just some simple label. So I'm now going to use a magnet, a non magnetic example to illustrate this. And so far we're not really talking about small angle stuff this is rather more general diffraction. But if we take copper three gold, which is a particular alloy. And when it's at high temperatures three quarter, there's a 75% chance that this atom will be copper and a 25% chance that this will be gold. And in that case, if we were to do a diffraction experiment, an x-ray powder diffraction experiment, then we will see various Bragg peaks that correspond to the to the periodicities associated with the atomic planes. For example, our 111 this peak here corresponds to the difference between between these planes here. If we call this down, then we end up with the coppers going to particular places and the golds going to particular other places. This is why this is why this is called a color approach. We've basically given each of these sites, which were previously gray the color. So they look different. And if they look different, then that means that they have some different symmetry and we can no longer say that this atom here is the same as this one here. Instead, this one is now gold and this one is now copper. And so that means that we end up with some additional Bragg reflections that take into account the change in the change in the symmetry. And so we can see this we can see this change in the structure of copper three gold by looking for the appearance of additional of additional reflections. So this is not magnetic. If we take a magnetic example. This is manganese oxide. So this was the first material to be investigated by neutron scattering to show that it had a particular type of magnetic behavior called anti ferromagnetism. So we start out at high temperatures. And this is the, this is the structure of the chemical structure and there is no magnetization associated at the moment. And then you will end up with some diffraction. This is now a neutron diffraction rather than x-ray diffraction. And you'll see that there are some peaks associated with this structure. We call it down and a certain type of magnetic order develops and we can treat this by putting arrows onto manganese atoms. And you'll notice that this arrow is pointing up, whereas all of the ones in this plane are pointing down and then all of the ones in this plane are pointing up, etc. And so that gives rise to the appearance of additional Bragg reflections just as with the copper three gold case. And in this case, this is a purely magnetic reflection that has appeared because it only exists because of the difference in the direction of the magnetic moment here and the magnetic moment here. So that means that if we have magnetic order in our material, so some periodic variation in the magnetic order, we will be able to see Bragg reflections from that. Okay, so what that points to is that our color approach of giving things a particular color, it can take us a certain distance. But we also have to take into account the fact that magnetic moments are vectors. And this may sound like an obvious statement, but it has a number of very important consequences. So the magnetic moment is not just a label, the moment direction matters. And the direction in which a particular moment may lie is really maybe restricted by the nature of the atom that it's sitting on or the orientation of the electron orbitals in the crystal lattice or various other effects. And we can detect the magnetic moment direction using neutrons because we have an interaction between two different dipoles the first dipole being the atomic magnetic moment. So in the previous example, the little arrow sitting on the manganese and also the neutron magnetic dipole, which may be pointing in any particular direction. So to get straight to the results, and we'll go through this in a moment very briefly, the only magnetization that we can be sensitive to is that which is perpendicular to our scattering vector. Okay, so our scattering vector capital Q. So if our capital if our scattering vector points this way, then we can see magnetic magnetic, we can see magnetic signal that is perpendicular to it, but we can't see anything that's parallel to it. And we can get extra information if we are able to control the direction of the neutron dipole so we refer to this as polarizing the neutron beam. And so there are a couple of examples in this cartoon. So this cartoon here represents most of the information that you need to know about how neutrons interact with magnetic moments for at least an initial interpretation of data. But we'll go into a little bit more detail, not too much detail, but a little bit more detail. So, in a given neutron scattering experiment what we're eventually trying to measure is the differential cross section here, the d sigma by the omega where the where the omega is the solid angle of the pixels or the detector that we're looking at. And so this is basically the quantity that we measure the number of neutrons that are scattered into that solid angle in a particular direction normalized by the neutron flux into that solid for the same solid angle. And we can go a step further and look at the energy resolution so most of the time in small angle neutron scattering we're not too bothered about this energy, this energy resolution aspect. So when you're looking at materials at low temperatures this can become a little bit more important because of some effects that has on the symmetry of the response for energy, neutron energy loss and neutron energy gain. So as it says here, I've given some links at the end there's a slide at the end with some references. So if you want to learn about getting to the final equation then you can look there. Okay, we end up with an expression for our cross section that depends upon this potential here V. Okay, and in the end it's actually the V squared after after taking into account the interaction between the neutron and the sample coming in and then coming out of the sample. We have this potential, and we already know that if we're looking at the structural case then this potential is just a delta function multiplied by the scattering length. And so then in the usual nuclear scattering case, we will end up with a contribution that goes as the square of the scattering length. Okay, right. So for magnetic scattering, as you may have guessed the interaction potential is much more complicated, as the forces involved are not central, are not central over long range longer ranges. And we have to take into account the fact that the magnetic moments of vectors and expression for the potential looks like this. So this is this be here represents the magnetic, the magnetic field associated with the sample and its interaction with the magnetic moment of the neutron. And then there are various terms here so I put this in, but I'm not really, I'm not going to derive this at this point. If you're interested in finding out about it contact me and I can give you further information about getting to this. So what we can do with this is, is that we can then use that to get to basically to motivate the cartoon that I showed you on the previous slide. And I just also wanted to highlight that here we're treating in the nuclear case with treating the nucleus as a point scatterer, and in the magnetic case the electron is distributed over a larger spatial area. So we can get some consequences that we'll also touch on in a moment. Okay, so we have this expression we have this potential, we take our potential from the previous slide, and we can eventually get an expression in terms of the for the for our cross section in terms of the total the magnetization or really the Fourier transform of the magnetization this is M of q. So if you work through the work through it properly what you find out is that the actual term that you get in your cross section is this sigma. So this is related to the to the neutron, the neutron dipole, taking the dot product with something that's written as mag and perpendicular so this is the, the the contribution of the magnetization that is perpendicular to q. So that's what my cartoon was showing and this is the way that this is mathematically expressed. So, you have this particular expression and this particular expression is basically the mathematical representation of this picture here. But so we will only be able to see magnetic signal in in directions that is perpendicular to our scattering vector. And if we take this expression for M of q and we break it down into component parts, then we have individual moments. So these are the arrows on the manganese in my earlier example. Let's talk about the lattice, the structure factor. We have thermal factor in that thing representing the influence of thermal energy on how well we can average a structure. And then we also have a form factor f f of q here, and this is an illustration this is the representation of the fact that the magnetization distribution is not point like like that exists over a reasonable reasonable range in real space. And this sketch here this picture here is an example of the form factor for Holmium for a particular Holmium iron Holmium three plus iron. And so this line here the magenta line. That is the representation of the form factor and there are several contributions to it shown by the black and the red. So what this is what this really means is that your magnetic signal drops off pretty rapidly as a function of the scattering vector so the x axis here is basically q. And so this drops off relatively relatively rapidly. And so this is obviously helpful so this means that you will be normally get a strong strong magnetic signal or as strong a magnetic signal as you will ever get in the small angle regime, because you're normally here, where basically this is very very close to one for most materials. So if you can get a magnetic signal in a small angle scattering experiment it will usually be pretty strong, which is, which is helpful. Okay, so that was my little diversion as to on where this particular cartoon comes from but this cartoon is the is the kind of key takeaway I would say to remember moving forward. And so I'll now come to what does that actually mean for our experiments so I'll be sticking with a couple of simple structures to illustrate a couple of points. So if I take this this simple structure here, and I have given it a particular magnetic structure so this is the real space representation of a simple cubic structure, simple cubic nuclear structure. And then I have some magnetic moments on the atom so that in this plane they're pointing upwards and in this plane they're pointing downwards. And that means that symmetrically this atom is different to this one and that means that we will end up with with the nuclear brag reflections shown by the gray circles and then magnetic brag reflections shown by the blue circles. And so if we have this as our material this is what we would see in a diffraction pattern. And so we now rotate the moment so that they lie along this direction. So they, in some sense they possess the same symmetry if we just consider them as colors because this one is pointing one way, this one is pointing the same way as that one, but these two are pointing in opposite directions. However, the actual direction of this arrow makes a difference and what we will see from this material is that we won't see a reflection in these two places. So here, if we carry out an experiment and we were to measure here, we would be able to distinguish between the structure that you can see on the left, and this structure here. Okay. And this is, this is particularly relevant if one is doing small angle diffraction, because what matters in this case is the is the relationship between the scattering vector, and the magnetization so if we're talking about the magnetic reflection that is here figure out what our scattering vector capital Q is, then it's basically lying in a direction which is in in the way that I've drawn this in the qx direction. And as you can see the x direction is perpendicular to the direction of the moments. So this one is visible. And if we now go to this particular case these moments are lying along the x direction. And so therefore we end up with a cancellation, and we cannot see cannot see what's happening here. So by looking at magnetic peaks that are very very close that are that are kind of close to our zero position. And we know that the moment we know very well what the direction of the scattering vector is and we can use that to infer information about the direction of the moments. Okay, right, moving now to what may be more familiar to you as a small angle neutron scattering pattern. So this is an example of some data that has been collected from a magnetic material and what you can see here is that this data, these data are not on the detector are not radially symmetric so if this is the beam center covered by a beam stop. Then you can see that we do not have a cylindrically symmetric pattern instead we have some lobes here, and then less scattering in this in this region here. If I then were to put on to this sketch of the of the small this cartoon of the small angle scattering instrument. Then I have my neutron beam coming in and then if I go to a particular point on the detector, then I can work out what my queue is. The direction of this queue is slightly exaggerated because in my picture in my picture is not to scale. But this queue will basically be almost so if I'm looking at a point on the detector here that corresponds to the tip of this green arrow here, then the direction of my scattering vector is almost but not quite parallel to the green and in this in this cartoon, the difference is larger than it would be in most experiments. So if I look at this particular position on the detector. Then my queue my total scattering vector is basically parallel to this and so that means that that I will only be able to see magnetic signal that is pointing either in the x direction or the z direction. Similarly, if I look at a position on the detector at the end of the blue arrow, then in that case my queue if you imagine this rotating round will be basically parallel to the qx direction. And in that case if I'm looking along the qx direction, I will only be able to see magnetic scattering coming from the y direction and the z direction, which is out of the out of the plane of the sector. So, if we consider the data that's contained in this particular image. Then, what we can observe is that with the scattering vector parallel to qy so upcoming up here, there is no, we have less scattering so there's no additional magnetic signal scene. Whereas if we look along this direction, we have some additional scattering, and this is additional magnetic scattering, and so therefore what we can conclude from that is that the scattering system has a significant magnetization, significant magnetization or contributions that are parallel to qy. And then if this is say a cubic material we can relate that directly to whatever the relevant axis is in our in our sample. So, we could say therefore say that if we make this measurement, we can make some statement about the magnetization that it has components. I've drawn this where they're all pointing in the same in, they're all pointing upwards but there could be some up and down components as long as they're pointing in the y direction either up or down. So, if we did a different measurement, and we obtained basically this pattern rotated, and that would tell us that what we were looking at with the magnetization would be rotated. So that the magnetization that we're probing as a component, it has its components along the x direction. Okay, so what we can get from that is that we can get information. We can get information of the magnetization, and we can also use that to differentiate between magnetic contributions and non magnetic contributions so in this in this in this particular example we might, we might be able to assume that there were no magnetic contributions here, and only magnetic contributions here, and therefore the behavior in this direction represents the from things like the particle shape, the particle size, the polydispersity that kind of thing, and that those will probably exist in this direction too, but then we also have some additional magnetic some magnetic information. And I'll give another example of that right now so some samples some materials may already have some inbuilt magnetization that is forced to point in a particular direction. However, it's helpful for us as a when designing an experiment if there is some way that we can control this ourselves. So, I'm going to use an example using a magnetite particles so F e three or F e three or four particles. So these are found in many places this is just a picture of little magnetite particles inside magneto tactic by bacteria. So these are bacteria that use this like spine of magnetite particles to orient themselves, and so that they can move in response to magnetic fields. If you are interested there's some some nice videos you can find by searching for little bacteria building permits. And in the case of the of these bacteria these these particles are arranged in a line. So, if I take, if I take a, like a big, a big amount of these magnetite particles in a close packed array. Then I will end up with a set of the particles in a sort of amorphous arrangements, a bit like this example with little metal little metal balls, where there is some characteristic separation distance but there's no long range order. So, what we would see from scattering in that case is a ring like this on our on our detector, where the position of the ring in Q can be used to figure out the size of the average size of the particles and I think you've already covered how to do how to do things like that. So, we will, we will have this ring of scattering that is associated with the separation of the particles, but we can also apply a magnetic field. And if we apply that magnetic field, then our magnetite particles will respond. If we apply a large enough field, we will get all of those particles to be at what we call saturation magnetization so that they have a uniform magnetization that points in the direction of the applied magnetic field. So if in this case this is meant to be a field that is applied in the x direction of the detector plane so in the qx direction. Using the axes that are shown here. And so we expect that the magnetization will all lie in this particular direction. And so, if we consider what's happening along the y direction. If we forget about our magnetic field for the moment we would expect a nuclear contribution denoted by n here, n squared because in the end our cross section is always taking the square of the contributions. So the magnetic magnetization in the x direction magnetization is in the z direction. And then if we look along the x direction. It's the same except that instead of this mx here we have an m y here. Okay, so if the magnetic field that we apply is large enough. As I said the magnetization of the particles all points in the same direction which is parallel to the field. So what does that mean it means all of the magnetization is in x squared. And we don't have any or we will assume that we don't have any in these other parts so that we should expect to see a contribution only in the y direction. And if we were to look at a direction in between so at 45 degrees, then we would expect to basically get a vector sum of the contributions along while x so we will get a smoothly we will have a maximal maximal magnetic signal here. And that will smoothly decay until it gets to zero here following a trigonometric function. So, if we take our observation here from, and we basically split it up to consider the two different the two different directions so these lines here represent what we what we call sectors that have been looked at so we have a we have two vertical sectors and two horizontal sectors. And we expect to see a magnetic signal in the vertical sectors and no magnetic signal in the horizontal sectors. And if you look at the data this is shown here. And so we can see so this is taking this is basically taking the intensity as a function of the magnitude of q, as we as we move through the sectors. And what you find is that the red line is the is from the horizontal sectors, and the black line is from the, from the vertical sectors. And what you can see is that there is a very small difference between the two with the black line being slightly slightly higher. And so one can use that to basically figure out what the contribution from the magnetic scattering is and although this is a very small difference. It's measurable. And it corresponds to what is expected for this particular material which is that there is a basically a fraction of you can calculate the fraction of the magnetic scattering that is associated with this with this small increase in the black line with respect to the red line. And one other thing to note here is that this you can think about this in terms of the scattering like density so we have the contribution from the nuclear components. We have the magnetic components, and this is the same order of magnitude as the nuclear component, so that the magnetic signal is not necessarily a lot weaker. In this case, in this case the overall contribution is, is small, but the, it's not invisibly small, let's say. Okay, right. So, we can use an external magnetic field to control our, we can use an external magnetic field to help us to help us figure out what the magnetic contributions are. We can also play around with directing the polarization of the neutrons so that means that we then try to try to make all of the neutrons that are in the beam and that are hitting the sample, have their dipole moments pointing in the in the same direction. There are several methods as to how one can do this I'm not going to go into how that's done. So, once you are able to do that, then there are several, several directions that may be more or less interesting to place that polarization. So for example, if I place that polarization of the neutron beam, such that it is parallel to the total to the scattering vector. So this is this particular case here. Then, when the neutron interacts with a with a magnetic moment. So, when we have the interaction between the moment on on an atom and the neutron dipole moment, if the two are perpendicular, then the neutrons moment will be flipped. So that means that the neutron has a moment that's pointing in this direction before and after the scattering event, the neutron moment will be pointing in exactly the opposite direction. And so this is illustrated on the right hand side. So if we send up one of these send the neutron in with this green arrow, then if the neutron direction is not flipped, then the green arrow will be pointing in the same direction coming out. So if it is flipped, then instead it will have turned into this blue arrow pointing down. So, so if we have our neutron polarization in this particular direction, then all of the magnetization will induce this spin flip scattering. Basically, spin flip scattering is in general caused only by the magnetic contribution to the scattering. So if I am measuring this particular peak, if I were to send in a polarized beam and I measured the polarization before and afterwards, I would be able to get rid of the I would be able to isolate rather the magnetic contribution alone instead of having to look at it on top of the large nuclear contribution. So this is an alternate way to get essentially the same information. And sometimes it's helpful to do that by using something like an external field, and sometimes it's helpful to do that using polarization analysis. The other side of using polarization analysis is that the ways that you polarize the neutrons, almost all well they all involve throwing away at least 50% of your neutrons. And if you're, this can be a very heavy cost to pay in certain experiments so one has to decide whether the, which is more important, the precision in the measurement or the intensity. This is the most common type of way of using the polarization of the neutrons because it allows you to rapidly detect what is magnetic and what is not magnetic. However, you can do other things. So if the neutron polarization is parallel to a magnetic contribution, this is what's shown by this green arrow, this dashed green arrow here. The magnetic, there will be scattering by the magnetized by the magnetic contribution but it will not give you spin flip scattering it will give you non spin flip scattering. So, if you already know that your scattering is magnetic and only magnetic, then you can place your neutron polarization in this direction, and you'll be able to work out what contribution the magnetization has in this direction and in this direction. Basically, if we so we can we can try to get this type of information. There are some costs associated in terms of lots of neutrons so there are several different experimental profiles that are commonly that are commonly used. You can polarize before the sample so that means that if this is the sample you set the polarization here, and then you just measure the other side, what happens you measure the intensity that you observe. If you send in the neutrons with the spin up the green arrow here, and then you measure them with the spins pointing in the opposite direction and this gives you something that's called the flipping ratio. And, and you can use that to extract some information. You get more complete information by polarizing both before the sample and as in this cartoon after the sample, and then you basically at four different types of measurement that you can make you get non spin flip. So you get the up spin, going to an up spin, you get the up spin going to a down spin, a down spin going to an up, and a down spin going to a down. Again, you lose some more of your neutrons and taking these these additional polarization analysis steps, but then you can be a lot more precise about what's actually happening with your magnetization. And if you really want to you can also change the direction of this neutron spin so this is pointing upwards but we could put it in either of the two orthogonal directions. And then for each of those orthogonal, each of those three orthogonal directions, we would be able to measure four different make four different measurements. So, if, if we really need to we can make 12 different sets of measurements to really pin down what's happening with the magnetization. Okay, so, and I should say that within the, within the literature and the larger magnetic sand community. There is quite some debate about the best ways to proceed with particular measurements so in some cases, applying a really large magnetic field to force your magnetization to point in one direction. That is a perfectly, perfectly good way to do to avoid having to use any of this polarization. But sometimes what one finds if one does the polarization is that the field is not large enough to always saturate the material so it can be material dependent, and you don't necessarily know that until you've done the experiment. So this is a kind of active area of research at the moment figuring out the best experimental protocols to be sure of what you are seeing in a given experiment. Right. And so in the. Okay. Right, I'll just take a quick pause there and ask if anyone has any questions that they'd like to ask at this point. Okay. So now we've gone through. So we've gone through how the neutrons interact with magnetic with the magnetic field distribution that is set up inside a sample. And so, and the different ways in which we can probe that and the different types of information that we can get out from the scattering experiment. So the next question is what can we actually study with small angle neutron scattering if we are looking at magnetic materials. So, there are two main categories of things that are looked at for magnetic materials, and they break down into diffraction at small angles. So this is just, let's say standard diffraction. It's just that the spacings the periodicities are very large. And so the scattering angles are very small. This allows us to look at long length scale periodic structures. And the most famous examples of these are things like scourmions. And also vortex lattices in superconductors. This obviously can also be done for non magnetic non magnetic examples as well. And there are there are plenty of those. And in some cases there are very long length scale magnetic structures that also that also appear within the within the diffraction range of sands. And then we have what we might call the true small angle scattering the diffuse small angle neutron scattering, which is in the non magnetic context is what you've been learning about over the last day and a half. And so I've already shown you this picture. And this lower picture is an example that actually combines contributions of both of these terms. So this is looking at the iron. This is a picture of a picture of the detector view from a self assembled iron oxide nano particles. So we can see that we have, we have some preferential order we can see some, we can see some cylindrically symmetric contribution here in the middle and then we can see that there is also some sort of periodicity, giving rise to these to these spots here and presumably also to these higher order spots here. So these are the main two things that are studied. So we also have to think about what is going to influence our observed magnetic small angle scattering so here concentrating on the diffuse components. So everything that you've already heard about still applies. So we could think about particle shape we could think about packing of particles, we could think about polydispersity we could think about inter particle interactions. And I've written this as particle shape here but obviously we could have a matrix that has some precipitates inside it for example and we can still use this type of model to describe the behavior so everything you've already learned applies. But then we have some additional things that we have to consider. And this is, this is a list of the main contributions. So we have magnetic anisotropy, we have domain walls, demagnetization factors, dead layers, and magnetic inter particle interactions, I'll say a few words about each of these. So, if we are looking at a magnetic material. So, most magnetic materials they are. They're not the same in all directions and are strong links between the crystal structure and the observed magnetization so this is just taking an example of the three three ferromagnetic elemental metals iron, nickel and cobalt. Iron and nickel both have cubic structures, slightly different cubic structures. And if you look at the magnetization applied. When you apply the field sorry I seem to have chopped off the axes here the applied magnetic field on the x axis. And if it's applied along a particular crystal, a particular high symmetry crystal direction, then you get different responses. So, we can see that if you apply the field along the 100 to iron, then you get this higher magnetization value and it kicks in earlier. And if you go to the 110 or the along the 111 direction. Then you get slightly different responses, and it takes a higher field to force all of the moments to lie parallel to that field, which is what reaching this saturation value implies. What we can say is that the iron has its easy access along the 100 directions and there are three 100 directions so I'm cubic iron has three has three easy axes. If we look at nickel, then it's cubic, but I think it's. Body centered cubic. And then we basically see something similar except that we have the easy access along the 111 type direction in which there are four. And if we go to cobalt cobalt is actually hexagonal material. In that case, we can see a strong difference between if we have the magnetic field applied along the c axis so perpendicular to the hexagonal planes, as opposed to in the hexagonal plane. So we can identify that we'll get a different magnetic response along different directions. Okay, so what that means is that if we are then looking at say a cobalt system. If we imagine that we have a whole set of cobalt nanoparticles, for example, or something like that. Then, then in addition to any shape anisotropy that we have in the particles, there will also be a magnetic anisotropy that we have to average over, if we want to figure out what the overall magnetization if we want to extract information on the magnetization from our small angle neutron scattering data. So that so that we have to we have to know this information about the anisotropy and also be able to incorporate that into our simulation of the small angle neutron scattering data. It's an additional complication. And this is one of the things where one can often use an applied external fields to help with this by effectively trying to ensure that everything is oriented in the same in the same direction because what will happen if the particles are able to move, if they can rotate so for example the obvious example be there in solution but this can actually also happen when you have particles embedded in a matrix the particles may rotate to to minimize the overall energy with respect to the field. And in that case you can that can be helpful in that then you know your only you know which direction, which it's easier to take care of the anisotropy in those cases so that's something one has to consider. Okay, I then move on to domain walls. So in a magnetic material the magnet will usually minimize its energy by forming domains so what that means is that, rather than have a single domain it's usually energetically favorable to have multiple domains. And then you're reducing the overall stray fields. So you can kind of see that in this cartoon with these arrows showing these are thin arrows showing the stray fields. So, there is a lot of interesting physics in that if you make the material small enough, you reach a point where the energy cost of forming the domain wall is too large. And those types of particles are called super paramagnetic particles. And there's quite a lot of interesting experiments on those, but most of the time you'll get the formation of domains. And the, the energy associated with these domain walls is based on the anisotropy that we just talked about, but also the strength of the exchange interaction that is what determines what the magnetic structure is material. So, we have the formation of these domain walls and bluntly these domains of these domain walls they are defects. So just like all of the defects, they are something that we are very likely to see in a small angle neutron scattering experiment. So, if the defects if the domain walls are all aligned in a particular direction that will obviously give us some sort of additional additional periodic informational orientational information. But in most materials we may assume that there's a lot more randomization going on. So, when we have a magnetic material one of the ways in which it's the most often characterized is with a hysteresis loop, which is shown here, so that we increase them as we increase the field from a case where we assume everything is aligned altogether, and then we end up with the total magnetization dropping going through zero and then reaching saturation on the other side. And this can be modeled in terms of a domain picture so that when you're in the saturation case. So, there are no domain walls you've got rid of them all and everything is pointing in the same direction. And then as you move around the hysteresis loop you get the formation of domain what you get the formation of domains that eventually these the flipping so that when I get to this particular point, the arrow points in the other way. And if you have a material that's been sitting outside of a magnetic field for a long time, you will end up with the formation of domains through through thermal effects typically. You can destroy those that way you can alter those domains by applying the magnetic field for the first time which is what this this line here represents and we can see this is basically an artistic representation of what might be happening in a particular set of domains. And so we have all of these defects or domain walls. And so we can. We may get scattering off the domain walls we also may have some characteristic domain size so this is just like our particle. If we can think of these domains as particles where the difference between the particles is the direction at the moment and then you can use all of the things that you've been learning about to characterize that behavior. So that's this color picture of the way to think about magnetization all I'm saying is that this has a different magnetic this has a different scattering length density to this portion purely because of the direction of the moment. And so this is something that we should be able to should be able to look at. However, things get a bit more complicated when we actually think about what's happening inside the domain walls. So there are typically two types of domain wall that we can consider these are called the nail wall and the block wall. And hopefully, you can get an idea of what the difference is by from different these pictures so here we have a rotation in the plane of the wall and here the rotation is out of the plane of the spins. So you can perhaps see that from the side views of what's happening. The important point here is first of all that we have changes in the magnetization that we will be able to pick up in our scattering because, as you can see if we've aligned all the moments in this direction we have some transverse component here and we will be very sensitive to these transverse components because of the way because of this only picking up the magnetization perpendicular to the scattering vector. So for example if our scattering vector is set up so that it's parallel to this, then we will be very sensitive to what's happening here. So let me show you what you've been learning about looking at small angle scattering from particles. Unfortunately domain walls are not sharp interfaces, they typically have a, they have can have variable length but it's never one single going from one atom to the next, that you have the shift or that's very rare. The magnetization changes over a relatively long distance. And what this means is that the assumptions that are built into, for example, porods law regarding sharp interfaces, they're not always appropriate. And so that means that one has to modify those descriptions and to be able to take it take into account the actual nature of the interfaces that you are looking at. And so there are a variety of ways in which this can be done. But they obviously all add some additional complication into the process. And so that means if you're looking at a magnetic material, you will not expect to get a, you would not normally expect to get a q to the minus four dependence. If you're if you're extracting the intensity as a function of q. Okay. Right, so that was domain walls if I now move on to the demagnetization factor. So if I have any sample that is uniformly magnetized. It will have a particular shape. And there will be an associated demagnetization field with that particular shape. And this is just, this is just a natural consequence of thinking about what happens electromagnetically, if we have a particle in a magnetic field that has a magnetization induced by that field. And if we have a thin film, this is kind of our optimal case, then we would have no demagnetization factor. Basically, if you think about the directions of the magnetic field lines they just go straight through the film film. If we have a sphere, a perfect sphere, then we can calculate this demagnetization factor and it is one third. If we have an ellipse, sorry, an ellipsoid, then we can extract a formula for this for this demagnetization factor if we know enough about the properties of the ellipse of the ellipsoid. And if we have an irregular shape, then this is very, very difficult to calculate. And in fact, you would normally have to do this, you'd have to calculate it numerically. And the reason that this is important is that if the information you want to get out is related to the magnetization, then you need to be able to figure out what the effective field is for those particular particles. So this is something that one has to think about. Okay, so I've gone through the first three examples here. And then there are a couple of other factors here at the end. So magnetic dead layers, it's quite common that if I have a magnetic material at the surface or at some interface with another layer, there will be a small region where I will not get the magnetization I might expect. So I might have a cobalt atom, but because it's only interacting with a small number of other cobalt atoms and then something else on the other side of the interface or surface that I will not get magnetization on that cobalt. So that's known as a dead layer. And this is, this basically just means that you have to add in some additional information about the scattering length density. So it means if you're looking at a particle, a magnetic nanoparticle, you may find one particular particle size from your structural, structural small angle neutron scattering. But then if you were to look just at the magnetic contribution, you would find that the particle size was slightly smaller. And that difference would be this dead layer around the outside. And then finally, magnetic inter particle interactions. So these definitely take place over long ranges because the magnetic field is a long range force. And they can be, they can be quite complicated. So this is, so we know that these happen, dealing with these at the moment is very much on a case by case basis. And this is figuring out how to deal with this in a much more systematic way as an active area of research at the moment. So to summarize, we've looked at the difference between nuclear and magnetic scattering of neutrons. I've tried to show what that means for our experiments, and how we can optimize our experiments to take advantage of this. And what we need to do to account for when we're measuring magnetic materials. Okay, so I mentioned that there will be some references at the end. So these are three good textbooks that go through the details. So this first one talks quite a lot about magnetization, and particularly, considering the effects of polarization. This book here goes through that in a lot more detail. If you're not already aware this, if you go to this particular website, you can get a copy of this which has a lot of practical information that's very helpful. And there's an excellent review from 2019 that runs through Magnetic Small Angle Neutron Scattering, and it runs through a lot of basically some of the examples that I've talked about here, and a little bit more about some of the experimental details that one has to go through. And I will be giving a talk tomorrow on applications of some of these aspects. So we'll be looking in a bit more detail at some examples of what you can extract from the small angle diffraction case. And also some examples, looking at nanoparticles using a field to determine the direction and then extracting information from that, and also using polarization analysis to extract information. But at that point I'll stop. Thank you for your attention. Any questions. Hi Professor Blackburn, I wrote a question in the chat. Oh right okay sorry I didn't see it. No worries, no worries. Would you like me to read it out? No I can see it now, I can see it. Looking at magnetic layers of amorphous iron, is it possible to use magnetic stands with external magnetic fields to see the response of the magnetization direction, and also find it. In this case it would be best to in the laboratory first of all figure out what field you need, so in a squid or a magnetometer of some description. Yes, quite to figure out what the magnetic saturation is there. And then you would be able to do an experiment to look at the behavior with respect to the film, the plane of the film and out of the plane and possibly other directions there would be a potential. So, to be honest, if the thin films like this then grazing incidents small angle neutron scattering which I think you'll hear about a bit in the next talk this afternoon will be the best tool to do that and you can certainly do that using this. You can do magnetic grazing incidents small angle neutron scattering, following the principles that I've outlined here, because to look at the thin film are kind of appropriately a reflection type geometry is most efficient. Okay. Okay. And the other thing I would just add that is that, as I said you want to get some idea of what the saturation is before you come to the neutron facility. So, what, what one often finds is that the neutron scattering can pick up, can pick up traces of traces of deep of spins that have not yet saturated well beyond where the magnetometer indicates everything is saturated. So, I didn't include a reference to that here but I think in the review of modern physics article there is there are some references to the literature on that or I can send you some information if you're interested. I have a, there's a nice paper that discusses it. That would be great. So if I understood it correctly, I first try with squids or the SM or some other magnetometer and then then do magnetic case sensor. Yeah, so that yes, that would be that would be one way to look at it. I guess that you could also get some information from reflectometry from polarized neutron reflectometry. So, they give so the reflectometry polarized neutron reflectometry tells you about what is happening in the direction. And the grazing incidents small angle neutron scattering can tell you also about what's happening away from in the plane a little bit. Okay, actually, samples has been sent for PNR measurements. Right. I was just trying to incorporate something in some way. Okay, great. Then I know. Thank you so much. Okay, and then surrender has a question. If we have course shell nanoparticles, and you get from the XMCD that the gold is magnetic, can we get some more information on the magnetic moments of the gold. Okay, so, so with the neutron scattering, you get the interaction with the magnetic field as a whole. And so if. So it can be difficult to identify specifically that your contribution is coming from the gold unless, unless this the I'm just going to show this other slide that I have with them something. So if the if the size of the shell. The shell is very thin. This would basically be impossible because the probably the errors would be larger than the thickness larger than the thickness of the shell. But if, if one is able to get enough resolution, then one would be able to say that there was a difference in the magnetic magnetic diameter and then the structural diameter then you could argue that the gold is magnetic that you could argue that you could get some information on that. I do think that probably I would have thought that resonant x-ray techniques would be a better way to explore this than to kind of confirm that you've definitely got the signal on the gold rather than the neutron scattering in this case, because I'm assuming that that the that the that the thickness is a small, but maybe you have some more information on that. So in principle, with x-rays, obviously you can do small angle x-ray scattering which is not sensitive to the magnetization in principle one could do resonance small angle x-ray scattering. And then you would be able to get to look directly at the gold contribution or the iron contribution. You can't then get absolute numbers out, but it can still be helpful for confirming that you're seeing something.