 Okay, good afternoon everybody and welcome back. So this afternoon, you have two lectures on applications of small angle neutron scattering. So I'll be giving the first one and I believe it's Judith Houston who's giving the second one. So my examples are going to focus on some aspects of what I touched on in my lecture yesterday on magnetic small angle neutron scattering. So, and what we're going to look at is we're going to spend some time talking about some examples relating to diffraction at small angles. And then we're going to look at using a field to control the signal and exploring the role of anisotropy and other factors in the diffuse small angle neutron scattering signal. So that's the plan for today. And as a part of that, I will also be raising some other issues that can come up during experiments. So we'll start off with some examples of diffraction at small angles. So I will start off with an illustration using vortex lattices in superconductors. I'll do that because that's something I do quite a bit. But I will also give some other examples, including some non-magnetic ones. And then we will use this as a way to illustrate the differences between pulse and continuous sources. And then I'll highlight some problems that may arise that one needs to think about a little bit. And as I said before, if you have any questions, please interrupt or type them into the chat. Okay. So there are a few to use this demonstration of vortex lattices and superconductors there are a small number of things that you need to know not too much about vortex lattices and superconductors basically a superconductor. And as you may know has zero resistance when it's inside the superconducting state and it expels magnetic field this is something called the Meissner effect. And this is what you may have seen various demonstrations of at various points. However, in some superconductors, in most actually above a certain critical field. You end up with the small regions of normal states, which are called flux lines, entering into the superconductor, and they lie parallel to the magnetic field. So this is supposed to be an illustration, you can think of them as tubes or pencils stacked parallel to the direction in which you apply the magnetic field. And so, as well as when you stack pencils together, you will normally get some sort of close packed lattice, which may be hexagonal in nature. And one of the useful things about these is that they have a fixed amount of magnetic flux in them. And so that means that as a function of the applied magnetic field, you can work out exactly how far apart they must be. And so it turns out that the distances between these objects fits perfectly into the small angle neutron diffraction range. And the objects are generally straight they are mono dispersed. And so they are an example for looking at the looking at diffraction in this in this range. So this is a sketch of the standard monochromatic type of small angle instrument. So we have our neutrons coming in. And then we have our detector here. And then we have our superconductor sample with these with these flux lines or water seas inside it. And basically the geometry that we're adopting here is that we have our lattices of water seas set up, and they are arranged in a particular pattern, and then we basically have them aligned parallel to the field. And then if we rotate, if we rotate the field by a very, very small angle, then we will end up in the brag diffraction the brag condition for scattering off the planes of the the arrangements of water seas. So that's the that's the geometry. So this is easiest, if the field is basically nearly parallel to the neutron beam. So you start off with the field parallel to the neutron beam, and you rotate slightly to meet the brag diffraction condition. And this is the usual geometry, the studies of vortex lattices, and also for magnetic skirmeons, which behave in a very similar way. And this is not the type of geometry you would use with the magnetic field necessarily, when looking at other magnetic materials. But in this case, it's the right one. So, yeah. So if you want to basically measure something, measure a brag peak, then just as you were done on an ordinary diffractometer, what you want to do is you want to try to capture that reflection completely. And the way that we usually do that is by a rocking curve. So effectively we have a rotation axis, as indicated here so we have the sample sitting here and we rotate and the beam would be coming into the into the screen. And then we rotate about that axis. And the video that you can see here is taken from niobium. So niobium is is is the favorite material for looking at these vortex lattices because it scatters extremely strongly. And indeed, it's actually used to check the alignment of fields in magnets to check that one knows exactly how the magnets how the axis of the field is oriented with respect to the external markers on the magnet because basically it's the vortices lie directly parallel to the field. So what you can see on here is we've got our detector are small and standard detector. And then as we rotate the sample we bring some of these spots into the diffraction condition and then they move out. So the center of the beam would be here. And you can see these spots coming in and out. And basically what we then do is we if you imagine if we concentrate on this spot here, we put a box around this particular region of the detector. And then we measure all of the intensity in that box as a function of the angle that we've changed. And if we do that, then we will end up being able to construct peaks that look like this. So I have the box that I would be imaginary box I was drawing was here, and that would correspond to this green, this green peak. And then if I were to draw a box around say this one, then I would get this orange peak, for example. So I make my measurement. And then I can use this information to infer some of the behavior of the some of the properties of the niobium. So, if I were to extract quantitative data, then I take these rocking curves. If I want to make nice pictures, then I should sum over all of the rocking curves. And then I will get some picture that allows me to see all of the individual features so this is the sum of the video that you just saw. So we have the center of the beam here and then we can see our first order diffraction spots, second order diffraction spots, and etc. And this is obviously a picture in reciprocal space. So the real space orientation in this case is the same shape, but rotated by 90 degrees about this shouldn't say B axis rotated 90 degrees in the in the plane of the, I don't know about the B axis so that, so that basically this spot would be up here and this spot that this spot would be the here or rather the flex lines that would correspond to those basings would be in those in those positions. Okay, so one thing to note here, and we'll come back to this in a moment is to think about the effects of gravity. So, in this type of instruments you may often have a very long distance between the sample and the detector, and for that matter, between the velocity selector and the sample. And the neutrons, they're traveling very fast, but they are not traveling fast enough that we cannot see the effects of gravity so basically over this long path length. You will start to see that that some of the some of the spots they basically drew. And this, this drooping effect is due to the some of the neutrons. They require a certain they basically move down in the in the y direction so they move downwards because they're basically following a parabola and that parabola starts somewhere back here, and then they basically moving down a bit. So, as you move your detector further and further away, you may need to correct for this kind of thing. It's very obvious when you are looking at these spots. It's also obvious if you pay attention to the position of the beam center as you move further out. But obviously this would also happen if you're looking at something that's nominally cylindrically symmetric, and you may end up not quite seeing is it so it says cylindrically symmetric in this case. Okay. One of the very nice things about small angle scattering when you use it for diffraction purposes is that you really can. It makes it very easy to get your data out in absolute units, which can be which can be very helpful. So this is a key strength in neutrons gathering in general. And so I will use the example of these vortex lattices to illustrate to this. So basically we try to measure the direct beam. We use that to normalize our results so that we can then basically extract actual field but actual values for the magnetic fields that are causing the distortions inside these inside these materials. So for vortex lattices, this is done via something called the Kristen formula, which is in this box here. But one can do similar things for other features such as scumions for example, and in this formula. We've got the integrated intensity that we've measured at a particular spot. So that's taking the area under say this green spot, we've got the intensity of the beam so this is the, this is the direct beam. And one does that by basically removing any beam stops, and then measuring, measuring the total, the total intensity in the central in the central spot. And then we have things like the wavelengths of the neutrons the thickness of the sample gamma which is this 1.9 one factor that came up in the lecture yesterday I didn't highlight it particularly but it's relating to the gyromagnetic ratio of the neutron. We've got our flux quantum, which is basically the amount of flux each of these vortex lattices hold the associated wave vector. And then this quantity F which is actually the quantity that we can then compare that we want to extract. And this is basically the Fourier component of the spatial variation of field inside our vortex lattice. So you can think of this as being basically related to the, to the size of the change in the in the magnetic field in the flux lattice and in the bulk of the superconductor. And so typically, the value of this of this form factor will be of about the order of a chaos. So that is something that we can detect. And if we do our normalizations properly, then we can extract actual values of this field in reproducible as the reproducible quantity. So that's an example of what you can do if you're looking at vortex lattices. And what I've said there applies to other types of other types of material. And so the question then is what sort of periodic structures will exist on these length scales where we may have chemical structures with very large unit cells. We could have ordered arrays of large length scale objects. And we could also have magnetic superstructures. And this is just a picture that's prepared by prepared by I forgot her first name, Edela, the University of Bath. I apologize for that, where it shows different different scales of objects and also give some information indication of different techniques that are well suited. So we're basically focusing on things that may occur in this particular region. And sometimes some of these other objects will fall into fall into these into these areas. So to give an example of something like small angle diffraction in a different context. Karen, that's it. Karen, Edela. Then what we can, we can consider basically things like liquid crystal that are behaving in a little bit like liquid crystal material. So these are actually micelles. And basically they're disordered so you see nothing. And then at some point, we get some orientational order on the application of sheer in this particular case. There is some some order. They're basically all oriented in this particular direction. And we have some characteristic length between them. Then we see a pattern like this. And depending on how ordered this is this may get closer to a true brag peak, but we can still extract information on the separation from that. And then eventually as you apply more sheer in this case, you get it going to an isotopic case. So if you want to examine these features here, you can do the same type of rocking as I just described. Yes. And so other examples of this type of behavior would be in certain types of liquid crystals, as you move through the transitions in liquid crystals. Okay. In terms of the long range chemical structures and long range magnetic superstructures. It depends on the specifics of the particular material. So I've picked an example here. So this is the, this is the chemical structure for what this is an example from the family of hexa ferrite. So basically, as you can see these are these are materials containing iron and oxygen, hence the ferrite. And then, then you have something out the front here, which could just be the original one of these is barium with nothing else and then the iron and oxygen. And these, these materials basically have a very, very long repeat length along the C direction. So what, what this, this region here represents one unit cell of this material and as you can see, there are a lot of things stacked up stacked up the C axis. And the thing to focus on to try to make sense of this particular picture is safe use concentrate on the blue atoms which represent the irons or the magnesiums in this particular case. And so you can see here that we have a diagonal component, a vertical component, a diagonal component, a vertical component, etc. repeating up until we get to the point. So this has a large, a pretty large unit cell, and it contains iron, and the iron has a carries a magnetic moment. And in fact, it can develop a very complicated magnetic structure, which is supposed to be illustrated by the bits here to the right. So basically, the irons carry the magnetic moment, and they are split into different types of behavior. So you can see here that it's written L three S three, etc. L three is referring to, to this diagonal component, and it has a large moment on this kind of larger cone here pointing upwards, and then the vertical parts classified as S three and then it points downwards in this case. And so over the whole of this unit cell we build up a rather complicated magnetic structure. And that leads with a with an even longer repeat distance than the chemical structure. And this, this particular example will fall into the range of what we can see using small angle neutron scattering. And so this is an example from a material like that. And this is a screenshot of the detector at D 33 at the Institute of Langiova. So this is a small angle neutron scattering instrument where you can take the detector to be about 30 meters away from the sample. And it is equipped with a central detector, but also with four side detectors. And basically, the magnetic reflection the large magnetic superstructure in this case is is shown here. Basically, the software that was used to prepare this particular picture is a software called grasp which is what they can do it's it's it's doesn't do the same kinds of things is not optimized for the same kinds of things as SAS view. But it is optimized for basically looking at, looking at magnetic reflections and considering how well at reflections brag reflections, and considering how they change as a function of applied angle. In this case, you can see that there are several little boxes, and these are the regions that are being integrated over to construct the rocking curves to extract integrated intensity, and in the way that I was telling you about earlier for the, for the vortex lattice. One other thing to notice here is that you can see some features. The circle sort of associated around here. And so I believe that in this case this circle here represents the detector window. So you have your sample before the window to the detector tank, and that has a particular size, and in this case, our detectors were placed in such a way that some parts of the device would never receive any neutrons, because, because the neutron beam have to go through a whole load of the tank is through the window to get in. So, that's another thing to be aware of when setting up, setting up your particular experiment. Okay. I also wanted to use this to give an illustration of the difference between time of flight and monochromatic measurements. So the layout of the instrument that I showed you before. This one here is for the monochromatic measurement. And here you set the wavelengths for one measurement, given that you have a spread of wavelengths but you have a central you have a central wavelength and a relatively small spread. So if you look at a particular momentum transfer, then you just move the detector backwards and forwards, and you may play with the collimation to alter the beam divergence, as is most optimal for your experiment. If you're using a time of flight sound spectrometer, then in this case typically the detector is fixed. So the example that I've taken here is Taycan at Jay Park. The detector is fixed and you have multiple detectors actually to cover different angles. And then you basically in your experiment you are measuring multiple wavelengths at the same time. And it is the spread of the wavelengths that sets the momentum transfer range that you can access. In the end we're trying to get to the same quantities, but here we changed the move the detector distance to set the key range and here we play with the wavelengths to set the key range. Okay. And that's just a photo of D33 the instruments I mentioned, and it's neighbor D11 where this tank is I think 40 meters long. Very large. So if I'm looking at Bragg diffraction peaks, then I already showed you this video for the measuring niobium measuring the flux line lattice in niobium. And so we rotated the sample to get into the Bragg reflection condition for various peaks. And if we're doing our time of flight measurement, then in principle we can set our sample at a particular angle at a particular angle so not necessarily straight on but slightly off. And then we're using the different wavelengths to construct information about the peak. So here, this is basically one measurement. But what is changing between the different frames in this video is that we are looking at different wavelength bins. So the spot that you can see here is a spot from a skirmy on lattice. And you can see that it's moving across the detector. And that is not because the queue is changing but because the wavelength is changing so the spot always appears at the same queue. But we are able to pick up the spot of different wavelength values. And so if we're careful, then we can convert, we can end up with the same information as we would get from this. So eventually the information that we want to know is the relative intensity or the absolute intensity of a given reflection. And in this case, we can look at it in the way that I showed you before, we change the angles and then we sum over the intensities. And in this case, we are looking at the intensity at different wavelengths, and using that in a similar way to extract information. Now typically you should probably measure at a couple of different angles to get sufficient information here. But you can get quite a lot of information from one angle position. And you can also see that early on at small, there was a spot appearing here, so some kind of spurious spot that only appears at a certain point. And that is probably something, some scattering, some Bragg scattering from the window, I would have guessed. If the windows are made of silicon, then you can get, then you can get sharp Bragg peaks appearing at certain times. Okay, so that is a kind of illustration of the difference. And hopefully it gives you an idea of the kind of things you need to be aware of when considering the differences between what you're measuring in the monochromatic case and in the time of light case. Okay, so I also said that I would say something about unexpected problems. So I showed you this picture before, and I commented on the effect of gravity. And so, although this is something that one just has to correct for in our experiment in the kind of experiments that I do. And so being able to see that you have this effect actually motivated some experiments to try to see if, if, if when you created a gravitational potential well for the neutrons to exist in that you would also be able to get that you might also be able to see quantized states. And so an experiment was set up, this has nothing to do with the small angle neutron scattering parts, but it was motivated, I would say, in part by the original observation of the effects of gravity. So the idea is that the neutrons come in, they reflect off a mirror, there's an absorber. And then you end up basically with the neutron sitting at particular positions which are quantized quantized heights, let's say, so that we can indeed see bound quantum states in a gravitational potential well. This is the kind of key plot where you see that you're getting neutrons, like so, in these kinds of steps. So it's just something a little bit different. Right, I'm going to move on to the next section, but if you have any questions. Now would be a good time to ask. Okay. So, we're now going to look at the case of these two cases using a field to control the signal and exploring the role of anisotropy and other factors. So, I'm just going to start by thinking about the effects of alignment in a ferromagnet. So if we have a real ferromagnet, then we have these magnetic domains, as we talked about a little bit yesterday, and they form these closure circles to reduce stray fields in general. As illustrated here, we go from having many stray fields to a little bit of stray field to hardly any stray field. And so, and the boundaries between these, between these domains, we have the domain walls, which in this picture are drawn as straight line as thin lines, but as you may recall from yesterday, they have a particular length. And that length is set by the exchange interaction between the magnetic moments in this material, amongst other considerations. Okay. And so as we increase our magnetic field, then we will end up getting closer and closer to a saturation magnetization. And so supposing we're applying a field in a particular direction, we would start off with our domain pattern like so. We have this closure loop. And then as we apply the field in this particular direction, we will end up preferring certain directions of the magnetic moments. And eventually, some of these domains will win out. And then eventually the moments, those domains will start, the magnetic moments in them will all start to rotate to get closer and closer to the applied field. Until at last, we end up with all of the, all of the domain walls gone, and the magnetic moments applied parallel to the field, or the magnetic moment lines parallel to the applied field. So this picture here would be a point that is exactly on this dotted line, which is meant to represent the saturated magnetization. So as we move along this particular curve, we move through these different domain states. Okay, so if you actually do small angle neutron scattering experiments of what is happening in a material. So this is the magnetization curve like I just showed you for nano crystalline cobalt. The cobalt has is a furrow magnet. And you can make, you can make nano crystalline samples that to whether the average grain sizes is reasonably well controlled. So in this case it's about 10 nanometers for the grain size. And then if you do that. And you compare what you see in the small angle neutron scattering experiment as you move around this. As you move along these fields, then what you'll see is that here, basically, there should be very little difference between the orange star and the pink triangle. In fact, what you can see is that as you're moving up, you do see changes. So these changes here are expected to be due to the to the domain scattering. And the so due to the scattering from the domain walls, but also from the different contributions from the overall moment orientation of the different domains. And when you get in principle to this point here, then you might expect that in this case, all of the magnetic moments are pointing in the same direction. That means that there is no more magnetic contrast, because everything is doing the same thing. And therefore you would expect to have a diminished magnetic scattering. And here you have to think about the direction in which you are applying your given magnetic field. So if I'm applying my magnetic field parallel to the to the to the neutron beam. Then I will then everything in my detector should basically behaving be behaving in the same way. If I apply my magnetic field in a different direction, then I may might expect to get some differences between the vertical and the horizontal directions as we discussed. As we discussed yesterday. So in this particular case you can see small change it you can see changes smoothly as you as you move up in field. But then the difference between the 243 mini Tesla and the 1800 mini Tesla is perhaps larger than we might have expected if we just relied on the magnetization information because it would seem that we would expect only a very small difference. So this is a common issue when carrying out when looking at the behavior of when using small angle neutron scattering to study to study the behavior of this we basically see see a lot more changes than we're sensitive to in the in the magnetization. And that brings into question the idea of whether we can really use this as a background so what when I said that if we're looking at this there is no magnetic contrast what I am implicitly interested in doing is using using this as a means as something that I can then subtract off all of the data to to get information about the magnetic components only because I want to try to make the assumption that what I see here is just the nuclear is just the nuclear behavior the nuclear small angle neutron scattering. And then ideally I would subtract a couple of this subtract this from all of the others and then I would have my magnetic scattering. But because I see these changes here it does and indeed if you keep on going you'll still see some small changes. It does bring into question how high in field you have to go to actually be achieving to be actually be able to use this as a background. So, yes, so as it says here, the neutron can see transverse deviations in the magnetization long after the magnetic magnetization measurements are giving you no further information. So, one way to get around this there are there are several ways to get around this. One of those ways is with polarization analysis because that gives you a very clear indication of what you are seeing. But as I mentioned yesterday this can cause problems with the reduces your flux. I'm going to give an example of applying as large a field as you can with the field parallel to the beam and using this to give you a contrast free background. So, why put the field parallel to the beam so the idea is that then qx and qy should behave in the same the same way. And therefore you can carry out a radial averaging of the data and that will improve your statistics on that data and you don't have to worry about including any angular corrections, which you might do. In the case of yesterday I showed you an example of magnetite, and there you have to include a trigonometric correction to be able to handle the data correctly. If you're looking away from the strict true vertical and strict true horizontal. So, if you do that so this is taking this is basically taking these nano crystalline cobalt systems and applying as larger field as is as is possible. If you're wondering about the field values here. These might seem like strange values to have set. And the reason that these are particularly odd values is due to the application of the de magnetization factor. So you may recall I mentioned that yesterday you have to take into account the de magnetization factor associated with the shape of your particles. In this case the field applied in for this year was 16 Tesla. And but after application of the de magnetization factor that is an effective field experienced by the particles of 14.71 Tesla and similarly for all of these others. They correspond to. They correspond to more intuitive numbers that you would have applied the field that before making the correction. So what we can see here is that we're going through a particular we're looking at the, so on the right you can see the magnetization as a function of field with the various points measured. And as you can see 15 Tesla 14.7 Tesla is an order of magnitude well there's significantly larger than the other fields apply. So the hope is that this really is getting rid of everything and that this can be used as a true measurement, and so that you can extract the, the magnetic information from what's being seen from what's being seen here. So we have all of the individual fields illustrated here. And then the idea is to subtract is to subtract this information off. In these pictures this has all been normalized so that can, so it has all been corrected so and reduced so that one has absolute units for the small angle cross section. And this is what the subtractions look like if we subtract off the orange high field data. So you can now see that the basically we are reproduced we are getting a similar pattern, a similar type of shape of pattern qualitatively with some changes in the apparent intensity as we increase the field. Okay, so there are a number of things you can do this this particular experiment. The aim was to try to explore what was happening in the low q region with the aim of trying to see if there is some something equivalent to the Guinea approximation that we can think about quantitatively when looking at magnetic materials. So I know you've already learned about the beginning approximation. In terms of thinking about the radius of gyration associated with a polymer or a particle a particle size or shape. And that is based on certain assumptions about sharp edges. And as I mentioned yesterday and tried to mention just before now, in the case of our domains or magnetic structures we don't really have sharp interfaces. We have relatively, relatively diffuse interfaces that take place over a particular length scale. And that length scale is set by the strength of the magnetic interactions between between the individual magnetic moments. And so that we can we can construct what's called a micro magnetic or magnetic exchange length, which is basically this term here so a here is the exchange constant for our particular in this case cobalt for the exchange interaction. And so this is a length scale. And basically we have to find a way to incorporate that length scale in and see if we can in fact, in a perfect case, or in a, in a, how do you call it, a simple case, let's say, and see if we can actually get something that looks a bit like the Guinea law. So the that's the so that these nano crystalline cobalt systems are good for that because they have this kind of well controlled and particulate size and cobalt is pretty well understood as magnetic materials go. And so here, what has been done is so this is the data originally showed you and then one subtracts off the high field, and you end up with the curves I showed you just before so this is the this is the this is the cost section against the wave vector. And so if we now plot the logarithm of that against q squared, then you can see we get these nice straight lines. There are. So they're very well behaved here and then we start to see very small deviations up here as we get to lower fields but basically, we can we can basically get a straight line in this straight lines here when we apply that when we apply these changes to the axes. And that means that indeed we can think about this as behaving in a way similar to the Guinea approximation. But what we have to do is instead of just taking a radius of duration based on the particle shape we have to introduce an additional term relating to the magnetic exchange link so this contribution here. And so you can see. So the, the slope of these lines is used to get this this quantity this r squared. And then this is plotted against the applied field. And to give the to give a straight line to show that indeed in this case we can in fact develop and show that we are seeing something like this. And to show that indeed in this case we can in fact develop and show that we are seeing something like the Guinea law, but for magnetic materials if we make the appropriate additions to the to incorporate the information about the magnetic interfaces. Okay, so that is basically all that I have to say for the examples here. I'll stop.