 Zelo, bojte. Zelo, mi je na bordelje, nekako izvajte. Zelo, bojte. Še začala, nekaj? Na našem kaj biti. Zelo, bojte, ki so svoju 30 minuči, nekaj svoje 25 minuči, nekaj svoje 5 minuči, nekaj biti. Mezaj, nekaj, nekaj, začala bojte. In... ... ... ... ... ... ... ... magnetic excitations and ground states in quantum magnets and frustrated magnets. And so, this lecture is going into depth, into neutron scattering, and then, that's the first half, and then the second half, there will be some examples. So, I think, go back, I told you about the energy of the neutron, in the kinetic energy related to the velocity, the velocity is, of course, related to the wave vector and the wavelength. That the neutron has two ways to interact with a sample. It can interact via the nuclear force and see the nuclei within a particular material, and it can also interact via its magnetic moment to see the electrons in the material. That the neutron can be produced in a nuclear reactor and also in an accelerator. Okay. So, now, I want to talk about the advantages of neutron scattering. Why one might choose that? The most obvious advantage for frustrated magnetism is the sensitivity of the neutron to magnetic order and fluctuations, to magnetic moments in a material. Besides that, it turns out that the neutron energy and wavelength are very well matched to condensed matter excitations and distances. So, neutrons, we can call them cold, thermal or hot, and we can create them with these types of energies. In temperature, we can convert that to milli-electron volts, and we have typical energies from 0.1 to 500 milli-electron volts, and that just about covers the range of magnetic excitations that you might find in a material. And also, for those energy ranges, for each energy, there is, of course, a particular wavelength, and these wavelengths are also well matched to the distances between magnetic ions in materials. So, if we take thermal neutrons with a wavelength of 2 angstrom, 2 angstrom is similar to interatomic distances in materials, and a 2 angstrom neutron will have kinetic energy of 20 MeV, and 20 MeV is typical of the type of magnetic excitations you might see in a material, that's energy scale. So, this is very well matched. It means you can measure the full B1 zone, the excitations over the full B1 zone as a function of wave vector and energy in a material. Now, for 2 angstrom electrons, the energy is actually of the order of electron volts, not milli-electron volts, and when you get to x-rays, it's kilo-electron volts. So, instrumentation is much more complicated with x-rays. Okay, another point is that neutrons are a weekly interacting probe. That means that most of the neutrons will pass straight through the sample, depending on how thick it is. And only a few neutrons or a small percentage of the neutrons are scattered. And this makes, so there are very few occasions where we have double scattering or multiple scattering of the neutrons. And this makes the equation simple, makes it simple to compare a scattering pattern that's been measured to a theoretical calculation. We don't have to deal with multiple scattering. So, in this graph, it's the nuclear scattering now of the neutron as a function of atomic number of the element. That's the red point. You can see it's of the order of centimeters. With x-rays, as the element gets heavier, the penetration depth, that the x-rays can go into the sample, gets smaller and is of the order of microns. And with electrons, it's even smaller. So, in those techniques, you have multiple scattering. OK. So, what happens when the neutron arrives at a sample? Well, it can do three things. Here we have a sample, and the neutrons arriving there, all, it's a collimated beam. The beam is all parallel. They're arriving in the same direction. So, the neutrons can either pass straight through the sample, and many of them will do that. And at the other side, we have the transmitted neutrons. Or they can be absorbed by the sample, they come to a stop. Or they can be scattered by the sample, in which case their direction changes. They're scattered, transmitted and absorbed. So, we can place our detector here or here. If we place our detector behind the sample, we measure the transmitted neutrons. And this is a real-space imaging technique. It's a form of microscopy, which you can do with neutrons. I won't talk about that. I'll talk about the scattered neutron beam. This is where the neutron changes direction after interacting with the sample. And this is an interference process that causes a change of direction. And it gives information about the density distribution of the positions of the atoms and their scattering power, and also the excitations. So, this is a typical way to draw the scattering experiment. We have a sample. We have instant neutrons with an instant wave vector Ki and an incident energy EI. Those two quantities are, of course, related. After scattering the sample, some of the neutrons are scattered at an angle 2 theta. And the scattered neutrons have a wave vector Kf and an energy Nef. And this angle is the scattering angle 2 theta. So, during the scattering process, the neutron can energy of the neutron may remain the same, or it may be changed. Elastic scattering is where the neutron energy is the same before and after the scattering event. It's like there, it recoils from the atom during the collision, but doesn't lose energy. But the neutron can also give energy to the sample or absorb energy from the sample. So, it can create a phonon or a magnon, in which case it will lose that amount of energy. Or it can absorb a firmly excited phonon or magnon, and then afterwards it will have more energy. So, elastic scattering is where the neutron is unchanged, and no excitation has been created in the sample. In elastic scattering is where the neutron gains or loses energy. So, Ei does not equal Ef. Also, we have to have conservation of energy in momentum, so whatever energy change happens to the neutron must be given to the sample. So, the difference in the neutron energy is the difference between the initial and final neutron velocity squared, or wave vector squared. And that equals h cross omega, which represents the energy of the excitation or the energy given to the sample. We also have to have conservation of momentum, which is, of course, a vector. The difference, the vector difference between the initial and final wave vector of the neutron equals the scattering vector q, which is the momentum given to the sample. Okay, so now we can think about elastic scattering. If we have elastic scattering Ei equals Ef, there is no change in the neutron energy. H cross omega equals zero, the modulus of Ki equals the modulus of Kf. So, elastic scattering can be represented by the scattering triangle, where we have Ki. Initial neutrons arriving at the sample, which is the green dot. They are scattered at the angle to theta. Because it's an elastic scattering event, the length of Ki and Kf are the same, but the directions are different. We can make a vector triangle by rearranging the arrows. The difference is the wave vector q. You can do small angle neutron scattering and go down to 0.1, but you have to be very columnated, yes. You can do much better with x-rays, actually. If you do small angle neutron scattering, then you're looking at large objects. You can see that from Bragg's law. So, you've got a small theta, you can have a large d. Oh, I'll show you in a minute. So, back to the scattering triangle. The wave vector difference is q, or half of q is given here. Go back to theta q, or half of q is... So, from trigonometry, we can say that sine theta equals 2 by q over the wave vector k, where k is the modulus of Ki, which equals the modulus of Kf. And actually, this is now Bragg's law. So, just to go back to the scattering triangle, we could also rearrange it like this. It was Ki, Kf, scattering from the sample at angle 2 theta, and this is the wave vector difference q. So, what we have to do in the experiment is we have to arrange whatever is interesting, an interesting direction in our experiment along the wave vector q, which can be varied by changing the angle 2 theta. So, let's say we have a reciprocal lattice. We want, for example, a reciprocal lattice point to land along this wave vector q. That means you have to rotate the sample until this orientation is achieved, and at that point, you'll see a Bragg peak, or the feature we're looking for in your detector, which is placed at an angle 2 theta from the original incident beam. So, inelastic scattering can also be represented by these triangles. This triangle is for the neutron loses energy, and an excitation is created. So, here we have Ki and Kf. Ki is longer than Kf, which means that the initial energy is greater than the final energy. So, the neutron lost energy and it gave that energy to the sample and created an excitation at the wave vector q. And this triangle represents the neutron gains energy by destroying an excitation in the sample. Here, now Ki, the initial wave vector of the neutron is shorter than the final wave vector, so the initial energy of the neutron is smaller than the final energy. So, the neutron gained energy in the process, and it gained energy by destroying something, an excitation, and it destroyed it at the wave vector q. Okay, so, the mathematics of this is, people normally talk about cross sections. This is the cross section, it's the double differential cross section, that's sigma. Double differential with respect to solid angle omega and energy E. Okay, so, the important part here is that there is a matrix element between the initial and final state of the neutron, the initial and the final state of the neutron in the sample. Initial state, final state. We respect to this potential V, which is the potential between the neutron and the sample. So, what goes into V? This interaction potential is, well, we're thinking magnetically now. The neutron has a magnetic moment, and that moment is going to have an energy depending on the magnetic field that it experiences in the side, the sample. And we have magnetic fields in the sample because we have net magnetic moments due to unpaired electrons and orbital moments as well due to the unpaired electrons. And the neutrons will see this because of their magnetic moment. So, the interaction looks like this. That is the product of the field with respect to the magnetic moment of the neutron. That's all it is, except that the field is a very complicated expression involving the spin and the orbital of the electrons. For comparison, you can look at the nuclear potential, which is given here, which is much, much simpler. It's simply a delta function. It's a very short range in the strong nuclear force is a very short range interaction. That's why it's represented by delta function and then a scattering power B. So, now we're going back to the magnetic cross section, which is shown here, now simplified in terms of the scattering function. So, this is now had some rearrangements done to it. The magnetic V has been put into the equation. Just I want to point out some things that might be interesting. So, here we have a magnetic form factor, which reduces the intensity as the, with increasing wave vectors. So, this is something to look out for when you're looking for magnetic signal. Divine boiler factor reduces the intensity with increasing temperature. There's a polarization factor, which is this part, which ensures that you only see spin components perpendicular to the wave vector transfer. And most important is this part, which is the spin-spin correlation function, which describes how two spins in the material, let's say at lattice site A and somewhere else in a different unit cell, how they are correlated as a function of distance and time. That's this. And basically, what neutron scattering observes is the Fourier transform of the spin-spin correlation function. And that's something that can be calculated, depending on your model. OK, so now actual measurements how to do actual measurements. Well, I'm going to focus on inelastic neutron scattering, because that way we can measure excitations. So, clearly you need to know your initial energy and the final energy of the neutron. So, you need to be able to select initial energy and measure a final energy. The two types of instruments, the triple axis spectrometer and the time of light spectrometer. So, I talked about the triple axis spectrometer, which is shown here. So, first of all, I should mention that all the sources of neutrons, the reactor, the accelerator, both of them produce a wide beam of neutrons. So, we have to select one wavelength or one energy, it's the same thing. We have to choose particular wavelength of the neutrons to do our experiment. And that's done using the monochromator. It's actually branch scattering at the monochromator. The monochromator will select one particular wavelength. So, that's a monochromator crystal doing branch scattering. You have to produce... And it's nuclear branch scattering at this point. It's just a material that's chosen to get scattered very strongly, so we have an intense beam of that, now one wavelength. So, it's a wide beam here, it's a single wavelength there. Okay, then we get to the sample, which is here. It's drawn in with a circle around it, because normally we have to cool the sample in a cryostat, because we want to measure at a particular temperature, or in a particular magnetic field, and that would be the equipment to do it. Then we have an analyzer. Analyzer is used to measure the energy of the scatterneutron. So, now I control the scattering triangle. We have Ki, it's just this direction, we have Kf, which, because we've chosen to put our analyzer at this 2-theta angle, which means our Q wave vector transfer is in this direction, because it's the vector difference between them. Okay, now the analyzer, so we know that the neutrons are going this direction because we've put our analyzer to collect them in that 2-theta direction. But what we don't know is the energy of these scattered neutrons, and that's the purpose of the analyzer, which is also a crystal of graphite, which will select one particular wavelength. So it'll only scatter one particular wavelength, depending on its brag angle to the instant beam, which can be altered, so you can scan that to measure to find out what is the energy of the scattered neutrons. Okay, so now a bit more about the monochromator. The monochromator I said was a crystal, the idea of the monochromator is to select one wavelength or one energy of neutrons. And it's just a crystal, which is used to do brag scattering. So the neutrons might have a white beam, which has some distribution as a function of wavelength or energy, that looks like this. And we're going to select the neutrons which have this wavelength only. For which we need to do brag scattering, nuclear brag scattering. So now I'm drawing the planes of our monochromator. It's a single crystal with a d-spacing, and we scatter the white beam of neutrons from that, and it will reflect only one wavelength, corresponding to its d-spacing and the angle theta. And if you want to select a different wavelength, we change the angle. That's what it looks like in reality. It's a series of blades of graphite, and these blades can actually be curved to focus the beam and gain more intensity. The other bit of equipment that's important is now we need to measure the final energy of the neutrons. After the scattering event, and again we use brag's law and the graphite crystal, and we select our angle two theta to choose just one wavelength that might be in the scattered beam, and then we can scan that, then to select a different wavelength. And again, this is what it looks like in reality. This one is a horizontally focused analyzer. OK, so triple axis spectromes. This is a picture of the one we have in Berlin. The monochromator rests inside this drum. Everything has to be shielded for safety reasons. The sample would sit on this table. The analyzer is inside this box, and the detector is at the end. A scatter and triangle would look like this. So now you can do measurements by, you can do different types of measurements. So one measurement that's useful is to measure, to scan the energy at a particular wave vector. Does my system have an energy gap and what is the energy of the gap, for example? So to do that, this is now a reciprocal space drawing. We have the initial wave vector and the final wave vector and q. This is the scattering triangle. And we want to keep q fixed because we're going to measure at a particular wave vector and change the energy, which means change of difference between Ki and Kf. So you can change it this way, keeping the wave vector fixed. And then this is used, for example, if you have flat modes, as in this material, you can scan in energy through the flat mode, so there's one mode here and one mode here, and then you see two peaks. Alternatively, you might want to scan as a function of wave vector, keeping the energy fixed, in which case the length of Ki and Kf must be fixed, although the angle between them can change to change q. And this is a way to scan through, for example, double sides of a dispersion. If I did a scan this way, I would see two peaks as it passes through here and here. Okay, the second type of instrument is the time of flight spectrometer. So now the point is to use time and distance as a way to find your neutron energies. You don't use a monochromator crystal anymore. You use time and distance. So, and this is done by cutting the beam into pulses. Normally the beam is a continuous beam coming out of the reactor, or it can be a pulse beam, in which case you don't need to cut it in pulses. You have that already. So here we would have the instant neutrons coming in this direction. We would have choppers. The sample would be here. And now we have a detector bank. These are little green dots inside a detector chamber. Each dot has a different 2-theta angle. So the method is to have choppers in the beam. We have a continuous beam of neutrons. We need to pulse the beam of neutrons by putting a chopper in. A chopper is a disk with a hole in it and the disk goes around. And once per revolution it lets the neutrons through. So that's the chopper, actually it's a double disk chopper with a hole here. They counter rotate and at one point the both holes are open and it lets the pulse of neutrons through and then it closes again until the next revolution. OK, so that's the first chopper but we still have a wide beam of neutrons. All we know is that pulse started at time t0. So you need a second chopper placed a distance away which opens at a certain time later and you can phase it to open at a selected time later and you have a distance between these two choppers. You know the time difference between this one opening and this one opening. So you know the velocity of the neutrons. So having two choppers allows you to only allow through the neutrons which have a particular velocity and you can select that velocity by arranging this t2 time with respect to t0. When you know velocity of course you know the kinetic energy then you know the wavelength the wave vector, etc. OK, so we've monochromated the neutrons we have one particular wavelength for energy of neutrons. Go back. So here we have two choppers so we know the wavelength of the neutrons when they arrive at the sample. The next step is to measure the scattered neutrons not just their scattering angle which depends on which detector they go into but also their energy. But again these detectors don't just measure the angle they also measure the time and we know all the distances. So we know how long it took to arrive at this detector we know the distance from this detector to the sample so we know the velocity of the neutrons that was scattered. So we also have the final energy all can measure it. So this diagram is time versus distance this is actually an instrument and it's distance along the instrument. Instrument has choppers chopper 1, chopper 2, chopper 3, chopper 4 then the sample then the detector bank. At the function of time we have chopper 1 opens it opens regularly and lets through neutrons but it's a white beam we have to use chopper 2 to select neutrons only arriving at a particular time later to get through therefore they have a particular velocity so now we have one particular wavelength by the time it gets to the sample now the sample is going to scatter in whatever way the sample chooses to scatter and it's going to be scatter elastically but also inelastically by taking some of the neutron energy therefore some of the neutrons will be slower some might also be faster but the detectors measure the time of arrival and therefore also the energy of the scattered neutrons. So this is a diagram of the Merlin time of flight spectrometer here we have a chopper so we have the initial wave vector Ki we have the sample here at this point so we have the choppers down here we have the sample and then we have the detector bank shown here and the choppers select a particular wave vector of neutrons and then the neutrons are scattered by the sample into the detector bank and depending on which detector they go into they will have different two theta angles you can draw the scattering triangle with Ki Kf Kf is the now we have to choose one particular detector so at this detector detector scattering angle is two theta this is an elastic scattering event but sometime later neutrons that created excitations will arrive these will be slower because they gave up some of their energy to create the excitation so they will arrive later in time they will have less energy therefore the modulus of their wave vector will be smaller so now we have some inelastic neutrons arriving they have the same two theta angle because I'm only looking at one detector here but it's a shorter wave vector so the wave vector transfer Q has a different direction and so on as time goes on can put this into a reciprocal space diagram as the wave vector as the neutrons slow down or the ones that created excitations arrive later they will be smaller wave vector modulus of their wave vectors and when you do the scattering triangle you'll find that the wave vector transfer has a different direction now depending on how you arrange your sample you'll produce some trajectory through your sample okay that was just I just want to say one more thing I was of course only looking at one detector as a function of time and there's a large detector bank in each detector we'll be doing this so that's the background to neutron scattering or inelastic neutron scattering I want to talk about some examples now so the first example real measurements on the two pages oh yes um yes okay so currently going back here if you imagine that every detector is measuring as a function of time so it's measuring a whole spectrum and that there are lots of detectors this many you can imagine then you get a lot of data and if you want to measure if you want to have an overview of your material this is the best technique as you see you don't have a great deal of control as to go back to here you don't have a great deal of control as to where you are measuring so you have to do the whole thing you have to measure everything and then you can look at the data and work out and redistribute in terms of that structure so if you want to measure at a very specific point and you want to measure as a function of energy or as a function of magnetic field of temperature it's better to use the triple axis because you can go to a very specific point and then change whatever parameter you want to the time of flight is gives a bigger data bigger data sets yes it gives the overview in general you have to count longer to get this and therefore it's not always an advantage but there are new sources being made now which hopefully will be stronger so the first example is going to be a zero-dimensional magnet so i'm talking about the simplest thing you can imagine it's just two spins coupled together by an antiferromagnetic exchange interaction and assuming this is Heisenberg 2 and then you can work through this and realize that your ground state is a singlet represented by this wave function that's down and up plus minus up and down and then that's in the ground state and the excitations all have the same energy in a gap at a gap j and they are a triplet there are three of them and they can be assigned to spin quantum number one this is a singlet spine spin quantum number zero if you measure s of this or s z of this you'll find at zero okay well that can actually be realized in materials but more likely is that you have a dimerized material where you have pairs of spins like this this pair and this pair but there is a bit of weak coupling between them and then the simple picture gets modified because this coupling between them let's say j dash produces some sort of dispersion but so long as this one is smaller than this one then the gap remains okay then you put this in a magnetic field and if you get your field right as you increase the field you'll split the triplet you can even drive one of the triplet states into the ground state and so we have a singlet ground state non-magnetic give nothing in neutron scattering now we've driven one of the states into the ground state so we expect now to see magnetic drag peaks at the field where this goes into the ground state these things have been measured this is maps onto Bose-Einstein condensation so at a particular field the gap closes we get a condensation of the magnons into the ground state we get this type of phase diagram at very high fields we force all the spins to point along the field direction so we get a ferromagnet so these systems can be quite interesting they're very simple they have a singlet ground state gapped one magnon excitations the gap it can be dispersive depending on the interdimer coupling we can also have extra excitation like two magnon states bound states and Bose-Einstein condensation so a real material this is strontium chromate chromium this is one of the other valences of chromium it's chromium 5 plus chromium 5 plus so we said that chromium had six extra electrons five went into the D-shell one went into the S, four S so now we're going to remove five of them and we're left with one only in the D-shell one electron only will have a spin half this is the actual structure of strontium chromate it consists of bilings of triangles and these are shifted with respect to each other and actually the strongest interaction was believed to be this interaction this bilayer interaction so here are pictures of actual crystals these are two crystals that's the black objects and these two were grown with different orientations and we have two of them in our neutron beam just to get more intensity but because they have different orientations that's why they have different angles so they have they point in different directions so that their C axis will be the same and their A axis will be the same this is a powder measurement of the data so now powder is this function of wave vector and energy and there is scattering intensity shown here so the colors bright colors are the scattering intensity so scattering intensity around 5 MeV and no scattering intensity at lower energies so this suggests yes it is dimerized these excitations do appear to be gapped we can take a cut over here so integrate over this wave vector region and plot as a function of energy you get this data take a lower energy data set and show really there is nothing at lower energies so from this you can say well my dimer interaction corresponds to the middle of this band so my J dimer this was one is going to be 5.5 MeV but my band is not sharp that means there must be dispersion so there must have interdimer interactions and the general strength of the interdimer interactions is the width of this band here so this material looks very exciting because we had these triangular layers and then we had this sort of dimerized coupling and this is an investigation of the single crystal now and for this the Merlin spectrometer was used so in this spectrometer the sample goes here the incident beam is coming in here and the detector bank is on the edges of this block and for one fixed position of the sample you can get a data set that looks like this projected onto two reciprocal lattice planes two directions you change the angle position bit you get a different picture and you can continue in this manner rotating the sample and lap out a large region of reciprocal space and then you can recombine all that data in terms of the reciprocal lattice of the material itself and get actually a four dimensional data set with the three wave vector directions and energy these are planes through the data set and then pull out certain planes like this slice as a function of wave vector and energy where we see a single dispersion and this other direction where we see three dispersions so that could be fitted now we know how dispersion energies and their trajectories and they can be fitted and then plotted on a wave vector and energy diagram this and then we need a model and unfortunately although the system appeared to have perfect triangular layers it's clear because we have three dispersions that we must have three domains the only reason why we would have three domains is if we have distorted triangular layers because there's three ways the triangle can distort so in fact our Hamiltonian is much more complicated and involves more interactions and our triangle as you see there's there are three possible values around the triangle they're not equal so it's not an equal lateral triangle and it's unfortunately not frustrated anyway that it can be modeled using a random phase approximation and then the intensities can be modeled using a single mode approximation this is the data two directions in the crystal and powdered data compared to the model so the point I was making about you can really do a quantitative comparison between neutral data and theory this is an example of it okay the next example so I have two dimensional systems they're one dimensional so one dimensional magnets the next example so actually I'm going to now just talk about the chain this is the one dimensional spin half Heisenberg antiferomagnet system is not frustrated of course it has simple Heisenberg interactions which are antiferomagnetic between the spins first tackled by Hans beta in 1931 so he realized that there was no long-range order in the ground state and actually you can write this nail state which is of course incorrect and 50 percent of the spins are actually flipped from that state and this was a long-standing problem in condensed matter since that time but some insight was made 50 years later by Fidevan Taktajan who realized that the fundamental excitations were spin-ons and not magnons so I go back to the chain of course it shouldn't show long-range order as I've drawn but this is over a short distance it might show some short-range order so I say this is the approximate ground state we can create an excitation so I want to talk about spin-ons now we can create an excitation by flipping a spin like this one and it's cost energy here and here because we have antiferomagnetic interactions but we have parallel aligned spins so it's cost two j, one here and one here but if I flip three spins it would have cost the same amount of energy one j here and j here because this is where we have the ferromagnetically aligned spins we do five spins gain energy here and here so each of these excited states or approximate excited states actually have the same energy and so some combination of these has to be considered to true excitations and it turns out that the excitations really can be sided as these end points of this flipped over region but now you see the issue because we had it was a spin half system the spin was pointing down and the neutron forced it to point up so the change in spin was one but that quickly disintegrates into two spin half objects at each end of this flipped over region so the true excitation of actually these spin half objects or spin-ons which are flipped or which are at these end points and you can see we have to create two of them from the initial spin flip so spin-ons are fractional spin half particles you cannot create them with the neutron scattering the neutron scattering rule is delta s is one so we can't create us one spin-on we have to create two or multiple pairs the individual spin-on might have a dispersion in wave vector and energy that looks like this but we never see that because we can't create one said we have to create two and that's what leads to what's known as the spin-on continuum if you create a spin-on here and a spin-on here these two or these two we have wave vector one wave vector two energy one, energy two so if you add to give the wave vectors and energies the actual energy wave vector the neutron has to give up is something over here I think some of the wave vectors and the sum of the energies if it's this and this or you can take a different pair of wave vector and energy and you'll get a and you'll observe that at a different place and for each pair of spin-ons you'll observe this at a different point and so if you look at all these combinations you find out that actually the scattering observed in an experiment lies between an upper boundary and a lower boundary and is known as the multispin-on continuum so this broad continuous scattering shown here is actually a consequence of creating two particles in the scattering event okay so back to the theories there were various other theories used to tackle the molar ansatz also it can be tackled by field theory this problem but then in 2006 John Sebastian Koh and his colleagues tackled the problem and were able to solve it to very high order and this was then 75 years after it was originally proposed and their prediction I mean this is their prediction for the spin-on continuum as a functional wave vector energy which can be directly compared to the neutron scattering data okay so now what will we compare it to we needed a sample and the sample could be potassium copper trifluoride which has copper ions 2 plus copper 2 plus yes I think it's a what is the kind of scale well I can show you some line plots later but the red is intense and blue is weak and dark blue is background but I'll also show some line plots later so compound is potassium copper trifluoride with spin half ions in copper has a strong interaction due to orbital ordering just in one direction of the crystal it's actually it's a traginal traginal crystal system and the chains lie along the C axis that's a unique axis and the A and B the interactions along the A and B plane are much weaker than they are along the C axis it has interchain coupling though just a few percent but this is enough to give rise to long range magnetic order so as usual the real material cannot reproduce the model ideally so Hamiltonian is coupling along the chain and then weak interchain coupling experiments were performed on the time of flight spectrometer called maps this is the actual raw data without any background subtraction this is the wave vector along the chain direction as and and this is the energy and the excitations are shown as now we have the color scale the excitations are shown by the colors with strong excitations at the pi point and weak at zero and then again pi and the scattering actually extends a little bit upwards into this region so I've drawn over the two spinon continuum where it would be expected just to show you that we have interchain coupling this is the dispersion perpendicular to the chain so we see a weak dispersion this is due to the interchain coupling but here the energy is about 12 MeV compared to 50 MeV so off the background subtraction this is the actual data from the sample to be compared with the theory from John Sebastian Koh obviously these are just color plots they don't really tell you any details so you can take actual cuts through the data at constant wave vectors going upwards in energy and these are red points shown here so this one would be this one and then this would be this and then this and then the line through the data the solid blue line is the theory of John Sebastian Koh and then there are other theories that's the pink line is the molar ansatz and the dotted line is also the theory of John Sebastian Koh but not taking account of all scattering events we can also do the cuts at constant energy see down here we see two peaks they disperse apart at the top we just see one broad feature and again is compared to the theories and the the accurate theory is the solid blue line okay so now five minutes left I will talk about some two-dimensional magnets first on frustrated and then frustrated so square lattice is on frustrated it's a two-dimensional magnet this is a spin half square lattice and spin five half square lattice rubidium manganese fluoride manganese has been five halves this is a theoretical prediction for the excitations using spin wave theory we don't expect spin wave theory to be accurate if we take a cut at constant energy through the theory you would expect to see rings of dispersion you see the dispersion cones coming out of the Bragg peaks and indeed that's what you see this is actually now the data this is the data at a low energy we see the dots if you increase the energy you see the rings and at the very top the rings merge these are sharp rings resolution limited what you would expect triangular lattice is a triangle lattice does actually develop long range order with 120 degree type ordering theoretical predictions are that you spin wave theory works to a degree but for spin half it becomes increasingly problematic at high energies where you see continuum scattering there's also a renormalization of the energy scale but spin wave theory works to a degree and you see sharp excitations and sharp rings now to the kagme I mentioned this a bit earlier the kagmes corner sharing triangles in the classical case spin wave theory tends to work adequately but if you go to spin half and this is now the example of Herbert Smithite that I was mentioning earlier this is what you see in the constant energy slices you see rings but the rings are broad and merged together you never see sharp rings and this is a sign of continuum scattering rather than sharp magnum excitations and continuum scattering says you've got multiple particles and it could be an indication of having spin on excitations and it comes at final example which is this material calcium chromate this is material that didn't start from a theory it was a material that looked interesting just because it didn't have any long range magnetic order the structure is very complicated and consists of corner sharing triangles but each triangle it is in fact a kagme system of kagme bilets but the triangles have different sizes and the two layers the triangles and two layers also have different sizes from each other so this system seems to behave like a spin liquid with broad excitations these are the actual crystals this is the data now it's going up in energy the data is always broad never see sharp rings and if you take a cut as a function of energy in wave vector look at this one first we see this broad features again this is the magnetic scattering this is a different direction we also see this broad magnetic scattering and then this sharp scattering is actually a phonon and that shows you the resolution compared to the magnetic signal so it turned out that this material oh sorry just this is more data so it turned out that this material has a low energy scale and so it's possible to put it in a magnetic field strong enough to force all the spins point along the fields direction so this is the magnetization the magnetization saturates at 12 tesla so although it appears to have diffuse scattering and behave like a spin liquid if you force it to be a ferromagnet by putting in a magnetic field you force our ferromagnetic ground state this simplifies the theory it gets rid of the quantum fluctuations and you should see spin wave excitations and then these spin wave excitations can be fitted to get the exchange interaction so that we can know our Hamiltonian so that's what was done it was put in a strong magnetic fields to force all the spins to point along the field direction excitations were measured compared to spin wave theory to extract the exchange interactions and this was the final model with the color coding for the exchange interactions matching the colors of the bonds shown on here green is ferromagnetic and blue is anti-ferromagnetic and there's two green triangles and two blue triangles and then there's an interlayer coupling five interactions in total and that's just the simple model so having got a Hamiltonian the question is why at least in zero magnetic fields why does this system appear not to have any long range order at a broad excitations what is special about this Hamiltonian well then it was compared to the method of pseudo-fermian functional renormalization group which was able to show that with this set of interactions no long range order would be expected and it was also this technique was also able to calculate the expected scattering and showed diffuse ring like features with sort of qualitative analogy to what we observe okay so this was an example of where in fact the material was found before the model was found so I will stop now and just summarize talked about conventional magnets with long range order and spin wave excitations unconventional magnets in particular frustrated magnets do not have long range order in the ground state and they have spin on excitations and they can arise from geometric frustration competing interactions or anizotropy and they can be measured using various techniques in particular neutron scattering which triple axis spectrometer or the time of light spectrometer and there are many examples of quantitative comparison theory and experiment shown here thank you