 Today I will talk a little bit about my PhD project. So I'm a PhD student in the University of Luxembourg. And first of all, I will talk a little bit about, yeah. So about this nutrient scattering technique, how it basically, what this idea was, it's more than scattering. So basically, why do we use neutrons for probing the materials? The thing is that the net electric charge of neutrons is zero. So neutrons can penetrate materials with hope. And due to the magnetic moment of the neutrons, when the neutron beam is like coming to material, it interacts with the magnetic structure of the material. And so the neutrons are scattered in several directions. And the neutron scattering experiment, we can then detect the two-dimensional detector like the scattering image and get some information about the inside magnetic structure of such material. And so our interest in my PhD project, it's about the nanoparticles and the sense of neutron scattering. You can distinguish your nanoparticles as follows. So first of all, you can say, you have like nuclear scattering from the nuclear structure of the particle. The nuclear scattering, it's the interaction via the strong force of the neutron and the nuclei inside the material. On top of this, we have also the interaction of the neutron with the balance electrons in the atoms of the material. So it's due to the magnetization of the material. And additionally to this, we also have like nuclear-magnetic interference. So yeah, this plate is somehow with nuclear structure here and this kind of higher wall should visualize like a spin structure in such particle. So see these spin structures could be very complex depending on which kind of interactions lay the role of the game in such a material. So then I will come to the outline of my talk. First of all, I will talk about some simulation results and about simulation techniques we are using and I will present some of our results from the last years. And on the second step, this is the biggest part for the talk today. I will talk about some analytical calculations I did and these analytical calculations, they represent in the end like the simulation data. So that here is these simulations, they are like large-scale simulations and with these kind of analytical calculations, we would like to get a better understanding of which kind of effects contribute mainly to the scattering responses we may observe in some experience. And in the end, I'm also talking a bit about what I've done in my research day here in Lund at Linz and the University of Lund and I will give some short overview. So first of all, I could come to the simulations. So basically, there's some overview of how you can see these kind of simulations. So you can distinguish in two parts. The first part is, let's say, to really simulate the magnetism inside the materials. So first you start to define your material when you start such simulations. So you say, have this kind of crystal structure inside of the material and then I have well-defined the interactions between the spins in this material. So for example, you have like nearest neighbors exchanging the actions, magnetic dipole and the actions or magnetic crystalline anisotropies. And especially in these nanoparticles also the outer shape of the structure plays a role. So on the outer surface, for example, of a spherical particle, you also may take into account some surface anisotropy contributions and these contributions may lead to additional spin this order in these kinds of materials. So when you've defined your configuration of your material, next step is to decide which kind of algorithms or numerical methods are well suited to calculate like the magnetism inside this material. Once you've done this, you can talk about which kind of resorts we can calculate. So on the one hand, we can calculate like the critical spin structures. These are typically spin structures where that minimize the magnetic anatomium. You can simulate this material for different applied fields. So you can, for example, simulate your material over the hysteresis loop. You can also simulate phase diagrams where you're in temperature or you would say very some material parameters to carry out maybe a sperm young phase or these kinds of things. On the other hand, you can also like simulate spin dynamics. They really get time, can be pantomated, spin motion. So this part here above this, is let's say from simulation point of view, it's the biggest part, this costs the most time in this kind of calculations. So once you have like done this first part, second part is tend to transfer the results from the magnetic simulation to scattering cross sections. So how does this, how it's done? It's like you take your, for example, equilibrium spin structures. So it's a data set of mechanization vectors and position data. You have to perform the Fourier transform. And once you've calculated the Fourier transform, you can calculate like a, 2D science for section or you may be ultimately interested in some one heat scattering cross section or if you do again, the inverse Fourier transform to get some correlation function describes like the magnetic correlations in the material. So for these kind of simulations, there are some established software packages already. So they are widely used. For example, we have one pion. This is some software package from the University of York, UK. This is a more atomistic kind of software. So here it's really that you can say I have this kind of crystal structure in my material. You can define it. Other software packages like Remarks or Oomph, they are called more like micro-magnetic simulation packages. So there it is, like you don't really care now about the crystal structure of the material, but you say you have some more continuum like formulations. So you say, for example, you have two nanometer sized cubes and in such a two nanometer sized cube, your magnetization is constant. And then you have these interaction between these cubes and you would like to find energy minimals and how it's resulting spin configurations are located by. So this is the part for the magnetic simulations. There are also several other packages to do these kind of things. For my simulations on particles, I implemented something myself. And for the free transform, I implemented some CUBE code. There I can say from my experience, it's really helpful to do this kind of create transform calculations on a graphics card because that saves you a lot of time. Let's say when you calculate on a CPU, it takes maybe two hours on a GPU, you can be using it to 15 seconds. So this is a good thing you should do with this. And yeah, also for the free transform, you might think it would be a good idea to use like fast free transform algorithms, but for these fast algorithms, the thing is of course they are fast, but they have a limited resolution in fully spaced. So it's better to use some brute force method because then you get some better resolution, especially when you would like to investigate the larger structures. So yeah, some equations about the magnetic simulations. So this is what I did not equation I already discussed a bit. So here we have some example for such a magnetic hematoma. This is now discreet hematoma where we have some nearest neighbors exchanging the actions that is exchanging the action want to keep the spins more parallel. And this for example, the Zeeman interaction is an applied field. So you can apply some fields to your material. And so the energy minimization is performed by a system of Landau-Lipschitz equation. So you started some initial configuration must propose something and then you calculate like the dynamics of the spins and they will end up in some equilibrium configuration. So you find some stationary state. At the stationary state, we take them and we calculate that schedule in response. So this is the main equations for such a kind of atmospheric simulations. Then the equations for the heat transform. I mean, this is not just a basic creation set. So you have this data from the simulations, the magnetization backdoors and position data. You're summing this up and this discreet heat transform. And finally, this you can calculate like the scattering cross section or sort of quantity. You can see it like a Fourier energy density, magnetic Fourier energy density, something like this. So if you take the volume integral over this expression, you will get like the magnetic energy in the tube. And this quantity, so the scattering cross section, we can measure in such experiments, some small and scattered experiments. Okay, so far, so with this short introduction about the simulation techniques, now I will talk about some results from our group. So one simulation results, we have this for larger iron particles, let's say in the diameter size of 40 nanometers. In this kind of size range, we observe also like vortex type spin structures. So it's mean like this, that you don't have uniformly magnetized particles. So the spins are swirling around in some vortex. And this kind of vortex spin structure, it's mainly induced in this case by the magnetic dipole interaction. So this is the main contribution that this is relevant for such a spin structure. And here I've also listed which kind of interactions are included in the simulation. So we have dipole interaction, some uniaxial anisotropy, that's the exchange energies of it's formula in this micro-magnetic approach, because this is continuous formulation. And another question is how can we, or how does such a vortex types to look like in the sun's cross section or the related magnetic related functions that you can observe in a scattering experiment? And for this we have here these two plots. So let's first think about some separated particle where the spins are fully polarized. So they are all let's say closely parallel oriented. Then you see these kind of black line. Let's say this field of neutron scattering from particles. There's some black scattering curve that's typically called the spherical shape factor or the spherical form factor. So that's what we see here. And so basically here you see just the shape of the particle when it's fully saturated. And when you go then to some lower fields, so let's say in the zero field and run in state where these kind of vortex type structures appear, we'll see that like this scattering set on this points will decrease at low q's. And we also see these minimals, they are shifted to larger q values. And yeah, we have here some maximums. So these, let's say the main features will observe from these kind of spin structures. If we do now the inverse free transform vector field space to get some correlation functions, we see for these vortex type spin structures, this postulatory like curve. So this is a really prominent feature with these kind of let's say, similar piece of spin structures. So I can think about why do we get here negative correlations because that's the basic thing when we get this postulatory behavior. It's when you think about the correlation of two vortex type spin structures in a particle. So here's vortex type spin structure inside the apple, for example, now you take the ghost particle and you do the correlation. So they are shifted over each other. They add some overlap and you integrate over it. And if you think now about the vortex, what you have is in the real particle spins up point in this direction or this and in the ghost particle the spins up point in this direction. So you get some anti-parallel correlation. And from this anti-parallel correlation, we get then these negative values in this correlation function. That's why you observe this. So this is really the dominant feature that's really related to these vortex type structures. And if you would like to investigate these structures, this is what you should look for. Yeah, so these vortex type structures, you can see them as a big feature in the particles. And if you go to, let's say, small amount of particles, the thing is that the dipolar interaction becomes less dominant because the total magnetization is not that large anymore. So we have smaller particles. And like the even more GEDs and spin structures are then more induced from surface effects of the particles outside here. So one example here was this simulation with the L surface anapotropy. So what we do is this tree, teletomium, then take into account the nearest neighbor's exchange interaction. We have the Zeeman interaction for the applied field and some anapotropy contribution. And this anapotropy contribution would distinguish between the core and the surface of the particle. So in core, here we take the uniaxial anapotropy, this kind of term. And on the surface, we take now this, it's called the L model. So this is the L surface anapotropy, which depends also on which kind of outer shape of your particle you take. So if you take a cubic particle, well, sorry, the particle, then you get, of course, different results because they have different, yeah, different order in the surface. So, and now you see this example spin structure that this is in comparison to the ball-text-tap structure, less dominant or a smaller feature in the innermost genity. So then the idea is, of course, also here that the features we will observe in the Sun's cross section are also not that dominant anymore. And what you see is now in this Sun's cross section, so the orange curve, it's like the fully polarized state. So here is the uniform metatars case. This is the, you see these typical ball factor oscillations, really sharpie. And when you increase now the surface anapotropy, so then it starts that you get some inhomogeneities in your spit structure. And you see that these sharp oscillatory peaks, they are smeared out and also shifted to some, yeah, here in this case to higher q-values, if you take a different surface anapotropy model, could also be that they are shifted to smaller q-values. So it's not clear from the beginning to say why is it now shifting to higher q-values or shifting to lower q-values. It's not that clear to say. Of course, it's not a simulation result. And the next steps where I do this analytical calculation yes, I have got to get more insights about why it comes like this. So yeah, so these are two applications with it in the last year. This was one side the simulation paper, the other side I also tried some analytical calculations here where I used some sterical harmonics approach to solve them like the magnetization distribution analytical. So this was some kind of perturbation theory to solve this micro-magnetic equations. Yeah, this was also some research from the simulations. So this is now simulation of the historias loop and we see here now one article. So one of these articles, so the simulation take into several articles the average to be size. And here you see one dimensional schedule response. Our change is now with different applied fields. So you see the applied field and the average methodization. And this is now the 2D spin flip science perception. So in the three color right set, you see this four fold like shape now if you come to the state where we have the magnetization reversal, we observe quite different scattering patterns. So this is one idea for our next project and there's a collaboration of a subpoena dish group from the University of Newspork because they can prepare some samples that we have time to make some measurements at PSIY. So you have some proposal PSI. So to display maybe on the cause of one dimensional function it's sometimes better to understand. So this is very similar like this. So basically we can distinguish let's say between the core of the particle so this cyan color, you can say this is some of the core of the particle then you see you don't have that much spin deviations from the average optimization. So this is like a deviation angle of the local spin in some slice of the particle. And on the surface of the particle we see that we then have more spin deviations. This is also how it is currently done. So in this more analytical, easy simple analytical models to fit some scattering data from nanoparticles. It's like this that you take a thorough magnetic core and you say that you have a shell where the magnetization is reduced and often depending on your sample it's already fits the data quite good. So this was now about the simulation part. So also if you have some questions in between can just interrupt me, it's no problem. And the next step is now I will describe is also some work I carried out here in between. Some analytical calculations the idea was some analytical models so now the fun starts with some more equations. Yeah, so basically again the structure of the simulations you can see the kind of like this where these three steps, micro-magnetic simulation you get a static magnetization vector field you perform for heat transform you can calculate the science perception. The idea is now to split up these steps and say just have a look now at this step here and see what we can learn from it. Yeah, so until it starts with some equations sorry for that. So the idea from my side was now to make some kind of power series approach to describe like spin structures in these particles and you can see it like this. Here we have like the magnetization vector field components so an X and Y and Z and now I'm summing up over different particles. So it's like this, this index new refers to let's say we have here some particle, here particle there's a particle, here's a particle so think about this room it's filled with different particles on different positions and the positions where the particles are placed this is basically this vector A here and this S function it's like the form factor of the particle so this is basically just the describes the outer shell for example if you take this apple it defines inside the apple this function gives you one outside the apple it's zero. So this form factor function I might apply with some power series expansion and in this power series I can take into account in homogeneities in the spin structure. Now I'm handling all these summation symbols and so on it's a bit messy therefore I prefer some index notation and some combination of Einstein and multi index notation so basically this expression here it's just the same idea like this just shorter form to describe so here we have this expansion coefficient of the power series, these coefficients these are the powers so this is like this property here this product of these three different coordinates and we have like the particles form factor so it's the main idea to start and now from there, from this point we can describe what I will show in the next step the main features that will arise in the sun's construction so okay now we have the real space multiplication vector field and now comes the funny part we have to free transform this expression and here's the good thing we have these kind of power like expressions because if you're familiar with free theory you immediately see okay here we can apply that differentiation theory and also the shift theory so basically we don't really have to compute to free transform we can just say these powers are transformed to some differential operator this exponential comes from the shift theory and the complex unit comes also from the differentiation theory so basically that's already free transform so that's it so this step was quite easy and here we have now the derivatives of the Fourier transform of this shape factor function so if we take higher order in homogeneity for the spin structure into account we also get higher order derivatives that contribute to our expiry so now we have this multiplication free components so this is free space already and to describe now the Zansper section as you know in the Zansper section you use the information of the phase so what it contributes to the Zansper section are like the correlation terms of the multiplication vector field so then I defined like these correlation matrix and so it's like the products of the multiplication vector components in Fourier space and one of them is complex conjugated and so from this expression so this is really the general form so with this expression you can choose particle forms as you like and yeah you get some these but in the next steps I will specify and we will go to some spherical shape so okay now once we have these correlation functions we can depth them into the Zansper section so they are now in here these are these correlation components and that's basically it from the general side so now for these studies we've done so far one assumption was typically that we assume an ensemble of dilute particles so here we do some let's say simplification we neglect this exponential here and we say that the particle forms they are all the same so we have some monodisperse ensemble of particles in the sense of scattering this means that you don't have any inter-particle scattering correlations so these kind of correlations in your scattering response are connected and this is basically also then you have to take care about when you're preparing some sample that your sample also satisfies these kind of approximations it's widely used in this way okay so now we will make it more specific now we'll see from this general description what we can learn about vortex times the structures in particles so now we have to start with the spherical form factor and we have to go through all of these kind of calculations but we simplified immediately at the first step and we say just take into account first of all the first order terms so if the zero order term, the first order term and you will see what you will get from this calculation so now if you're familiar with non-particles and scattering you know the first zero order term because this contribution is nothing else than like the form factor of the particle let's say if you're uniformly necrotized particle you only have this contribution and now this contribution comes from the first order inhomogeneities so you can calculate all these kind of functions you can separate them in angular contributions and all the radial functions so these small g's they only carry the information the radial behavior and these capital G functions they take into account like the unanswered topic part for these contributions okay so now you continue with these kind of things putting on these kind of things again together going to the sunsport section if you also take a capital X and finally we get some expression for the i of q so this is now an absolutely average sunsport section and here again we see this is the zero order contribution from uniformly necrotized particles this is the first order contribution already from from this function we really can describe the main effects that we also were observing in our simulations so this really worked yeah so here's another comparison again this where the results from these micro magnetic simulations and here you see now some plots from these analytical functions what we see is basically if we increase now this i1 coefficient to some certain magnitude then it starts that we also see oscillations in the correlation function yeah from some analytical reasoning if we increase it even more and we see that really gets these kind of expandable also in the up to the average sunsport section you also see that for for example the ratio of these coefficients of 12 you see that for small key values the intensity is really decreasing so it's really like this that this first order model already carries the main effect the main effects of these kind of cortex structures and here again it's the question maybe now from this side why is it like this already the first order contribution carries main information and the thing is that it's all about symmetry so if you think about the straight line so let's say you think about a coordinate cross you have a linear function you see in this this this area your linear function may be negative and here it's positive and now you can think about in a negative part spins are pointing this direction in the positive part spins are pointing opposite so you already have with this linear function really you can describe this like vortex type of behavior that's circulating around if you think about the first semester about classical mechanics just like that you have some rotating solid body and there you know from the center of this body to the outside your tangential velocity is of course increasing so there you also have this kind of behavior so it's really like this kind of picture that you can have in your mind okay so now at this point we have described the one dimensional functions it's not a next interesting thing is can you also describe two dimensionless level responses there's a theory and the answer is yes but we have to push some more effort and more thinking into it so from some simulation we have seen in these vortex structures that the two-dimensional scattering has this kind of angular anisotropy so this was from the sample of nanoparticles this like the average scattering response to the offset from this this simulation and now the idea is how can we describe this by using our theory now again we start with this first order model so here we now specify like the basic optimization vector field we say we have this uniformly contribution this homogeneous part and we have a first order contribution it's here described as this yeah this is vortex vector field so basically this v function it's nothing else than this picture we see here it's vortex and with the ratio of these two coefficients and zero and one you can now switch between the vortex state and the uniform magnetized state now if you just take take like this and calculate the sunspot section you will not see the image like in simulation so what you have to do is so it's also the idea from my side we take now this and say this is basically one particle and in a real sample you have not have it not like this that all the particles or all the vortices are oriented in this direction but they may be also oriented in this direction or that direction that or that that so you have some distribution of vortex rotation axis configurations so what i do is i take some rotation matrices i transform the space vector field with two angles so i can rotate it around in space i derive again the magnetization coefficients that i that we need in our expansion and this is now how we can see it in this linear idea so this the constant part and this is the linear part of this non-generalized rotate that magnetization vector field so now we have to work with this take let the equations form and again for the formulas so we can do this coefficients correlation coefficients into the two-dimensional sounds cross-section it is also having functions and now we can have a picture in your mind as i explained with this apple you have for example some critical conical open angle and only above this angle you say your vortices are oriented somehow so there's some some point where you stop and and this angle the quality alpha c it is also field dependent so if you apply a really strong field you might have a small opening angle and all the particles are more closely oriented through your main axis if you choose a smaller applied field or go to zero field you have a more you have more variations so you take into account more angles and yeah what we need to do we need to do now what we need to do is to calculate average all of this kind of thing and here i take into account some or model it with some uniform distribution in this area and happily we can solve all the integrations and finally we get really a quite nice expression it's always good when you get some associated results of polynomial that's always nice to have and we finally see really our theory can describe what we also see in simulations so now we have this nice analytical expression it gives you like this picture in the zero field so in the remnant state and this is the comparison to the result from the micrometric simulation so really with this first order model you can let's say explain the main effects really we'll observe in such a sample yeah with these two parameters and zero and one so of course you can vary these parameters now you can say alpha is also field dependent so if you go through a high field let's say to this long scaled applied field variable so it's now a high field then you say okay then this conical open angle goes to zero and then we see the sine square anisotropy now we see from uniform the magnetized particles here zero field and a larger conical open angle so you have a larger distribution and then you will really see this kind of image yeah these two dimensional things i carried them out in one time so it's everything is not published about this so but paper as well let's say 85 to 90 percent so to publish this here i hope so yeah so this was about the vortex particles and now you can go one step further because now we are at the step where we have a linear approximation and if you want to describe particles the surface anisotropy i also take into account the second part of approximation so equations getting more and more messy again but you can also calculate this kind of things and here we'll just show the final results so yeah the black dots these are some simulated scattering curves so they are from some simulations on different particles the surface anisotropy and the red curves are then shifted to be similar to that you see already the second order approximation was just an integral approach can describe it and in the second order approach you have like again seven parameters so if you sit back in the first order approximation you have only two parameters but then if you take the second order model you gain five more parameters the thing is why is it like this it's because you all also get like correlations from the second order contribution and zero order contribution so typically it's like this that you get correlations from uh yeah even integers and and and odd integer contributions so we don't have correlation between first order zero order but we have correlations between second order and zero order and these correlations are in in the fifth and the sixth coefficients and so this is the particle radius all right so this was tough a lot of equations so finally summary and workflow based on what what have you done we've described like the nanopart magnetic nanoparticle by shape function times some non-uniform vector field we have approximated it with some polynomial expansion you also use some linear transform to rotate the particles around you can calculate the initial free transform and in the end also get this averaged sans cross section we can describe these kind of effects all right yeah this was mainly the part about the research topics and now we're just trying to talk something about my research day and moon so also that some started some collaborations include a list of blackburn this was on some yx and hexaflowert material was some preparation i started here and also we had tried some ideas or we started to try some ideas how to get some informations about about some let's say magnetic ordering and material and yeah just keep it like this briefly questions on it later just ask then another contribution i'm working on a slide to this sassu software so the idea is there's a software package and we developed in our group last year and the idea is to merge our software package to sassu environment so this is a software package for scattering data analysis or fitting of scattering curves there was a white check that's the worst mean and then i also had several talks now in this time so i had some talk in the beginning in the virtual university it was also called Hayden in the cool world came left one it's not today also i attended several workshops and talks and there was also a funny thing and because it's a really nice environment that we liked here at links and the university was booked and there's a read some funny funny story there was some journalist that contacted about the like mechanism plays on your fridge the question was does these magnetic field rising from them harm your foot and so it gains you some cancer and that was the question and then Andrew came to me because i was talking to him that i can relate some fields and so on and then asked me can you calculate when at which distance from magnet the magnetic flux density is below the earth's magnetic field so and these kind of things with it and yeah Andrew answered this journalist and then they published like this in some Romanian journals so i cannot read it but yeah if you're interested you can yeah it's also a good thing i started doing some sports so i need to get again some 10 kilometer run that's good to do this kind of thing because last year it was too much of sitting around and yeah so things is a good place i think it's good to get some new tea cup coffee cup and it's good to meet new people here and to discuss science and that's also with with Andrew this idea then that i will come in the next year for three months to Oxford and visit there some people yeah we like it from the side some outlook yeah can assimilation you can always make it more complicated so my colleagues is currently simulating some skirmish structures and nanoparticles so there's VMI interactions included this kind of hot discussed topic and we have done until now so many calculations now and what we also need to do in future more is also to find more collaborations to do some experiments to find people that can manufacture samples so that you get some real data that's how we set that room next year all right then there's some acknowledgement i would like to like to thank links all of you and i'd like to thank my supervisor i'm fiasz michels our postdocs fiasz evan my colleague evelyn she's also a PhD student laxenberg my second supervisor in sami kakafi from university of kakafi go on france and finally i would also like to thank Elizabeth thank you very much and thank you all for your kind attention