 miracle. I think we are fine. About the poster session, contact me if you want to present a poster. If you have sent already to the secretary then don't contact me because I have the information. Thank you. Good morning everyone. In this talk I plan to talk about magnetism a dwi'n ddweud y gallu'n ddweud ddefnyddio'n ddweud. Mae'n fwyaf o gweld ar gyfer pethau ddweud yn y ddweud. Mae'r ddweud yn ystod y methu sydd y ddweud yn oed o'r mefnod megynysgwysbwyll a dwi'n ddweud yn'r ddweud yn y ddweud? Dwi'n ddweud o ddweud yr oedd o'r ddweud a ddweud o'r ddweud? Felly, dyma'r ddweud yn y ddweud. Felly mae'n ddweud yn ddweud o ddweud, i chi'n dod yr ysgrif Lingwlad amser yn gweithio mussteol. Cymru o'r fawr. Mae Llywodraeth, fy Llywodraeth, rwyf llwyddol yma'u magnes ar gwenddol. neu'r ysgrif Llywodraeth. Mae Llywodraeth. Aled yn ei siŵr. Mae Llywodraeth. Ydych chi'n ddiddorol, mae'r ysgrif Llywodraeth o'r dwyliad Oh, y magnet sefydliad, ryw i'n gwael, gweld y smelly nesaf iawn i'n gweld. Mae yma fyrwb.él y byddai. Sorry, sorry, sorry. That's it. Okay, so, yes. Wrong slide. Magnets sefydliad, ac rwyf wedi gweld gweithio am hynny gyda teimlo o gweld magnet. Mae gweld magnet ar mlyno gweld, y spinwave, y bryddof, hanfodd y mewn cynghlytau, strydiadau cyd-a dechrau a gwaith cyd-fyddfa chi yn y'r cyfriffedd yma, antall fydd yn ystyried gyda'r ddechrau gwasanaethau cyfrafff. Yr ystafell yma, mae yw tro i'r mewn cyfriffeddau yma, yw'r greu'r cyfriffeddau sefydliadau cyfriffeddau cyndog wahanol gairfaf, erbyn yr hwnnw erbyn y baith cyfriffeddau cyfriffeddau cyfriffeddau i'r traditional sense--" with static moments, you have fluctuations in the ground state, and you have different types of excitations from spin waves." Then, I shall talk briefly about how you can select materials for frustration, which look like good candidates for investigation, and can be used to test theories. Then, I'll talk about experiments. This is probably the next hour of the lunch. a'r tyfan yn ffordd o bethau o'r niwr gyda'r cyfnodd rhannu o'r cyfnodd rhannu cyfnodd rhannu, ac o'r ffordd yw'r cyfnodd rhannu, ac rwy'n rhoi'n gweld sut rwy'n meddwl cyfnodd rhannu. Yn cyfnodd rhannu cyfnodd rhannu, fel y bod yn ni wna, yw'r cyfnodd rhannu yn ymwylltol, a'r cyfnodd rhannu. Ac mae'n gael i'w ddwych yn gechydigol yng Nghwm Fyrir, mae'r ddwych yn eich ddwych hwn, mae'n mynd i'r mewn ffwrdd agnodol, bod hynny'n fyddai'n gweithio'r ddwych, ac mae'n gweithio'n gweithio'r ddwych yn gyfodol maen nhad. Ac mae'n ddwych yn gechydigol yng ngheil. Mae'n ddwych yn gweithio'n gweithio'n gweithio'r ddwych, The spin angular momentum is a quantum number, it's a quantum quantity and therefore it can only take certain predefined values with respect to a particular axis in the case of spin. Five halves like the manganese two plus iron, this is a light transition metal iron very typically found in multiferous materials. It has been five halves and therefore if you put it in a magnetic field you can actually split these states into six particular orientations with respect to an axis. Okay, so then we have magnetic ions, that's the first thing to decide upon in a material. Then you want to think about your interactions, let's take a very simple interaction, the Heisenberg interaction, this is isotropic. Of course we can have ferromagnetic or anti ferromagnetic depending on the sign of the interaction strength and it's given by this two particle or two magnetic iron dot product. And then in a typical lattice we have different types of interactions, so the blue dots here represent the magnetic ions. In a two dimensional material like is drawn here we can have different types of interactions along different directions and this would all be due to the crystal symmetry. For example, if the A axis was different from the B axis we'd expect the interactions along the A axis to be different from those along the B axis. So example here is Lancinum cuprate, but we can go to other systems, I think you just learnt about one dimensional systems where we can get one dimensional magnets. For example potassium copper trifluoride, here the interactions only go in one direction now and you can use field theories to study them. Or we can have alternating magnets, alternating interactions for example strong and weak or ferromagnetic and anti ferromagnetic going in a particular direction. We can also have anisotropic interactions and this can also be important when I talk later about frustration. So you have magnetic ions, they're coupled together by magnetic interactions into a lattice. At high temperatures the system is fluctuating, the spins are fluctuating, the system is effectively a paramagnet. You start cooling the material down and when the interaction strength becomes of the order of the temperature then they can start to settle into a fixed either short range ordered or long range ordered state. The long range ordered state would happen below the nail or curie temperature. That's the real space image, but we often measure in reciprocal space, so what we would see here is just very diffuse scattering. What we would see here at low temperatures are Bragg peaks due to this order, these are not structural Bragg peaks. They would be Bragg peaks due to magnetic order due to this regular arrangement. So you could have a system which has structural Bragg peaks given by the blue dots and magnetic Bragg peaks would appear below the nail temperature. These could be represented by the red dots in this reciprocal space plane. Now we have a long range ordered ground state, the spins are pointing for a long time in a particular direction. But we can create excitations about that state and they are typically spin waves. Spin waves are a collective fluctuation of the spin moments about the ordered state. Spin waves on a ferromagnet or on a ferromagnet, this is a snapshot in time. In reciprocal space we would see a well defined trajectory like a sinusoidal type dispersion as a function of wave vector and energy. It's an excitation now so it costs energy and the energy trajectory through wave vector would follow a sinusoidal path. This is a ferromagnet, spin wave dispersion and this is the spin wave dispersion of an antiferromagnet. Okay so this was probably revision. Onconventional magnets, so here now we want to have on ground states where we don't have long range order or it is at least suppressed and different types of excitations. So where do we get unconventional magnetism? Well normally we can get it if we are able to get the spins to fluctuate at low temperatures instead of ordering they fluctuate. Quantum fluctuations for example can suppress long range magnetic order and if you don't have long range magnetic order then spin wave theory is going to fail because it assumes a long range ordered ground state. And what might you need to get this to happen? Well some of the ingredients are to have a low spin value like spin half rather than spin five halves. Antiferromagnetic exchange interactions, these are less stable towards disorder. Low dimensional interactions and frustrated interactions. And if you just take the simple two spin Hamiltonian shown here, convert it from coordinates to ladder operators. This term represents the long range order and this run represents the fluctuations and it's focusing on this term to disorder the ground state and prevent static magnetism. So the ingredients were first of all to have a low spin value. So I've mentioned spin five halves iron has six possible states with respect to a particular axis. Okay so if you operate now here you're going to change the spin quantum number by one unit. Let's say from five halves to three halves the general direction is quite similar it's still pointing generally up. If you were to do that on the spin half iron you would flip the spin this one from down to up. So if you only got two states you start changing the state you can only go down if you're up or up if you're down. So this is much more susceptible to fluctuations. The other ingredient was to have anti-ferromagnetic interactions rather than ferromagnetic interactions. This is of course not always true but it is true in simple systems. And here I've drawn the ferromagnetic ground state for a one-dimensional magnet. Of course a one-dimensional magnet does not have long range order but over a short distance it could have. So if we were to act with our Hamiltonian on just let's say two ferromagnetically aligned spins. So that state would be up up then the Hamiltonian would give us back the same state. So we would have an eigen state of the Hamiltonian the two spins being up. Now if we take a simple opposite system which is just a simple the nail state one spin up one spin down and act with the Hamiltonian on it. Now we go to this line the Hamiltonian acting on one up and two down starts mixing in not just one up and two down but one down and two up due to this term here. And so we don't get back the same state. So an anti-ferromagnetic ground state is actually not an eigen state of the Hamiltonian. Or simple nail state is not an eigen state of the Hamiltonian. The third ingredient was to have low-dimensional interactions. So this is quite simple to understand. It's just a simple magnet, a cubic system like this. It's actually anti-ferromagnetic along this direction and ferromagnetic in the other directions. But each spin is coupled by magnetic ions to the left, to the right, to the top, to the bottom, to the front, to the back. It has six neighbours. And this is enough to create a mean field that is able to stabilise long-range order or static magnetism. In one dimensions each spin only has two neighbours to the left and to the right. And this is not enough to stabilise long-range magnetic order. The fourth ingredient was frustration. And this I'll go on into more depth about. Frustration can be understood as competing interactions. So for some lattices it's just simply not possible for the spins to find a configuration that will satisfy all the interactions present. This is what geometrical frustration is. Now if you take the square lattice with anti-ferromagnetic interactions you can see nearest neighbour anti-ferromagnetic interactions. You can see that you can satisfy every bond. Every bond is anti-ferromagnetic and every bond has an up and a down spin at either end. But if you have a triangular system you cannot satisfy this. If you have anti-ferromagnetic interactions on a triangle you can satisfy one of the bonds. With anti-parallel spins. But the second bond would want this to be up but the third bond would want this to be down. And so it's not possible to satisfy all the interactions simultaneously. And it's the inability of the system to decide on a particular ground state. It has several ground states that could be up, could be down, could fluctuate between the two. It can't decide what to do. This is what suppress long range magnetic order to temperatures that are lower than would be expected given the strength of the exchange interactions. That would be Tn is very much smaller than the Curie Weiss temperature. That's a representative exchange interaction strength and this is the ordering temperature. And also even when you do order and you've gone to zero Kelvin if you could get there you'd still find that the size of the ordered moment was very much smaller or smaller than the available spin moment. That's the ground state. It's partially ordered and can be partially dynamic. The remaining spins would be in a state of fluctuation. And then the excitations are also changed. We don't have a simple long range ordered ground state. We can't really use a simple theory like spin wave theory which assumes this ground state. You can use it but it will only be an approximation. You'll find that the true excitations are renormalised. That means they are shifted downwards or upwards with respect to the excitations predicted by a spin wave calculation. And they are also much broader than the prediction which of course has been waved in predicts sharp excitations. That's magnetic frustration and if we take magnetic frustration to the extreme very strong magnetic frustration then we can get a spin liquid ground state. This is when even at the lowest temperature you can achieve experimentally or if you do your theory you put in the ground state you'll find that you cannot find long range magnetic order. Not just that, not the simple order of ferromagnet or anti-ferromagnet. You also can't find static magnetism. It's not even a short range order. The spins are in fact moving in the ground state. And also that the excitations cannot really be described as magnons or spin waves. Magnons and spin waves appear as sharp excitations from an ordered ground state and are characterised by the spin quantum number of one. In fact the true excitations are in many cases spin-ons. Spin-ons have a fractional quantum number of spin a half and are particularly interesting excitations because of this fractional quantum number and because experimentally you can't create them. You can't create a single spin-on because your scattering rule cannot give you a fractional excitation. Your experimental selection rules, any experimental selection rule is unable to give you a spin half excitation. So now classical spin liquid. So this can happen when you've got a large spin of course and then you can have many possible ground states. And the system can visit these ground states at a finite temperature. So we have a highly macroscopic, this macroscopically degenerate ground state manifold of excitation, of possible ground states. For a quantum spin liquid it's possible to take a superposition so the quantum fluctuations allow you to access the ground states and possibly also the low line excitations. And they may select a particular ground state which is a particular superposition of the classical manifold. And this would probably be a fluctuating system where the spins are dynamic and not static. The spin liquids have no long range magnetic order, no static magnetism considered to have a highly entangled ground state possibly with a topological order and spin-on excitations. So now practical considerations for getting frustrated magnets in real life. There are different types of frustration, if I go back. There are different types of frustration to be considered. Some just arise from the crystal symmetry with first neighbour interactions. Some require first and second neighbour interactions to be able to generate the competition between different states that destabilises the ground state. And then thirdly you can introduce anisotropy and this acts as an additional method of creating competition between interactions and the magnetic ions. So I will talk about each one. Geometrical frustration is the first one, the simplest one. Here we have geometries like triangles or tetrahedra. So we have three spins here. You can satisfy two of the bonds but not the third bond on the triangle. And if you choose a crystal with a structure that involves magnetic ions on triangles you are likely to see frustration. Tetrahedra can also appear in crystal structures. Here we have four spins making this sort of three-fold pyramid. Again it's not possible to, if all the bonds are the same and are anti-ferromagnetic, it's not possible to satisfy them all. You can satisfy this one but whatever we put here is not going to satisfy both of these. The same for that one. So this requires some type of motif like the triangle or the tetrahedron and it requires anti-ferromagnetic interactions in the simple case on the first neighbour bonds. So here are some examples of frustrated magnets. In two dimensions we can have corner sharing triangles like this. This system is quite frustrated but actually develops long range magnetic order at low temperatures. But if you make the edge, instead of edge sharing triangles you make the corner sharing as shown here you can actually generate more frustration and this system is thought to be a spin liquid. You can also have triangles in three dimensions. It gets a bit harder to visualise but you can make, it's called the hypercagome lattice you can make corner sharing triangles like this and they actually can fill up a whole three-dimensional lattice in this way. Also in three dimensions the famous lattice and other famous lattice is the pyroclore lattice. Here now we have corner sharing tetrahedra. So geometrical frustration in the simplest sense is that due to geometry it's not possible to satisfy all the interactions. Normally you need triangular or tetrahedral motifs and you need anti-ferromagnetic interactions between first neighbours. Now the second type is frustration due to competing interactions. I mean there's more than one interaction. There's not just the first neighbour, there's also the second or further neighbour interactions. And here the second neighbour and further neighbour interactions can compete with the first neighbour interactions to produce frustration. They don't all have to be anti-ferromagnetic now. Some can be ferromagnetic but at least one must be anti-ferromagnetic. We can go to the square lattice, square lattice. We have the magnetic ions here. Square lattice is unfrustrated normally with first neighbour interactions. Here we have both first neighbours, that would be the first neighbour but we also have the second neighbour across the diagonal, that's the J2. So now if we just satisfy the first neighbour interaction with assuming anti-ferromagnetic bonds we can have a simple up-down pattern. This satisfies the solid lines. I mean they are anti-ferromagnetic. So this could be the ground state. But now if we just have the second neighbour and we don't have the first neighbour we have to have a different ground state. Here now we must have anti-parallel spins along the dotted lines. Actually the system separates into two sub-lattices, the red one and the blue one and they don't connect to each other. The connection would be the first neighbour interaction. If I turn on the first neighbour interaction let's say it's ferromagnetic it's satisfied here but not satisfied here. Or if it's anti-ferromagnetic then it's satisfied here but it's not satisfied here. So the first neighbour interaction would compete with the second neighbour interaction. This is a plot showing J1 here. J1 is anti-ferromagnetic or ferromagnetic. J2 is zero along this line. J2 is anti-ferromagnetic or ferromagnetic. And J1 is zero along this line. So let's say we've got a J2 which is anti-ferromagnetic. If we start introducing a bit of J1 that means we start going round the circle this way. At some point the J1 and the J2 will compete with each other to the point that the system can't decide on a ground state. And we enter a possible spin liquid state in this region. The same happens if we introduced a ferromagnetic J1 by going this way round the circle. Again, if it becomes large enough it will compete with the second neighbour interaction. Okay, another example is the honeycomb. The honeycomb lattice, actually if you just consider first neighbour interactions is not frustrated. We have magnetic ions forming a honeycomb. The first neighbour are the black lines. If you have the first neighbour only we can have a ferromagnet or a ferromagnet or an anti-ferromagnet you can satisfy every single bond. Going back to the ferromagnet now if that's the first neighbour now we can introduce second neighbours these are the dotted red lines. So these couple this and this spin also this and this, this and this, this and this. Now if the second neighbour is anti-ferromagnetic you can see there's a problem there's a competition between what the two interactions would like. The same goes if the first neighbour is anti-ferromagnetic and we introduce an anti-ferromagnetic second neighbour again there is competition between the two interactions. This of course is parallel aligned spins whereas this bond is anti-ferromagnetic. In three dimensions this can also happen on the diamond lattice. This is the diamond lattice so here the blue and green are actually equivalent. In the diamond lattice we have tetrahedra shown here and there's also an iron in the centre of the tetrahedra. If you have anti-ferromagnetic interactions you could pattern first neighbour interactions you could pattern the lattice with spins like this simple up and down between nearest neighbours but if you were to introduce anti-ferromagnetic second neighbour interactions there would be a problem because it would then disagree with the first neighbour interaction. The third type of frustration that I will discuss is frustration arising from anisotropy. This is now competition between anisotropy and the interactions. So far I have been only discussing interactions which are isotropic where the spin can point in any direction with the same magnitude. But in some materials, especially materials of unquench orbital moment you can have preferred directions for the spins to point in. And then depending on the crystal symmetry these preferred directions of the different spins may actually be non-colinear and therefore may compete with the interactions which prefer parallel or anti-parallel spins. So the typical example is the pyrochlor lattice. Pyrochlor lattice is corner sharing tetrahedra here's the tetrahedra again and here's the pyrochlor lattice showing the corner sharing tetrahedra in three dimensions. Now let's pretend we now have isotropic interactions just simple isotropic interactions for example from a chromium ion and we put these on the pyrochlor. If the first neighbour interaction is anti-ferromagnetic then the system is highly geometrically frustrated possibly it's been liquid. But of course if the interactions first neighbour interactions are ferromagnetic all spins can just point in the same direction and every bond is satisfied and there's no frustration. But if we introduce anisotropy and here the anisotropy is different for each ion we have a magnetic ion here four on each tetrahedra and the anisotropy direction points towards the centre of the tetrahedra and of course it's a different direction for this ion because the centre is a different direction for this ion than it is for this ion. The anisotropy directions are non-colinear. Now if you have anti-ferromagnetic interactions the spins can only point parallel or anti-paral to this anisotropy direction and in fact with anti-ferromagnetic interactions the system can find a unique ground state and develops a type of ordering where all the spins point into the tetrahedra this one and then out of the neighbouring one this one and then out it's an alternating pattern but if we have ferromagnetic interactions on the pyrochlor lattice and the same anisotropy now the order is not so simple two spins must point in and two spins must point out this is something that can be worked out just by minimising the energy on a single tetrahedra you'll find that two spins want to point in and that's the minimises the energy The problem is however that it can be any two spins pointing in and any two spins pointing out and there's a very large number of ways you can do this if you would take this pattern this is just one option we could reverse the spins around this loop for example or probably this loop and we'd still satisfy the two in two out rule so there's many different ground states and in fact ferromagnetic interactions between highly anisotropic ising spins on the pyrochlor lattice gives rice the famous spin liquid known as spin ice in the presence of classical spins which has monopole excitations I believe so but maybe this is one yes I believe so but I should leave that one for the experts I believe so yes I can't hear you I'm talking about classical spins just now okay let me have an expert that gave you an idea of types of frustration now I want to talk about real materials and how you can actually achieve this so real materials can actually approximate these ideal models and can be used to test the theoretical concepts they can actually also instead of just testing theories they can also provide inspiration to theories sometimes a real material may show unusual excitations and on ground state and it wasn't a model that had been considered before and then it provided inspiration for new calculations so what do we need to consider we need to consider magnetic ions which magnetic ions should we have on our lattice and for this we must consider not just a magnetic ion or it's valence because the valence will determine the number of unpaired electrons and therefore the magnetic moment by a huns rule and also some magnetic ions have more susceptible to being anisotropic than other magnetic ions for example rarer of ions tend to be anisotropic whereas light transition metals tend to be isotropic so these are all things to consider then there's the symmetry of the crystal so this crystal symmetry will place these magnetic ions in certain positions due to the crystal symmetry so satisfied a crystal symmetry and if you choose carefully you can get lattices of magnetic ions which show triangular or tetrahedral arrangements and can be frustrated symmetry also can determine the anisotropy directions so here I'm going to talk about light transition metals light transition metals, ions this would be the 3D shell in the periodic table the 3D shell is shown here light transition metal ions normally have quenched orbitals due to the strong crystal field because they are form an outermost shell within the ion and so when you put them in a crystal the surrounding the surrounding for example oxygen ions produce a quenching field then the magnetic moment is due to the spin only or mostly and the magnetic moment is isotropic many ions have several valences so many of these transition metals can take several valences and each valence will have a different value of spin and one needs to be careful about that of course spin half is particularly interesting if you want to have a quantum magnet so the light transition metals this is the 3D shell the 3D shell has five orbital states each can be filled by two electrons spin up and spin down so in total there are ten states also the 4S shell also tends to get filled the 4S is here filled or partially filled and this has two states so let's look at copper copper is a very important ion for magnetic materials copper has bare copper metal has looks like argon argon is here it has argon plus an additional 11 electrons to get to here 11 electrons 10 of them go into the 3D shell and one of them goes into the 4S shell so the 4S shell can hold two electrons but it just holds one so you can remove an electron from copper the question is which electron well it just so happens that it's normally the lone electron in the 4S shell that means that the 3D shell is completely filled with electrons and a completely filled shell has spin zero because it's got five up spins and five down spins so net spin is zero so now let's remove another electron we have copper 2 plus so we have to remove one now from the 3D shell so now we have nine electrons and each orbital can have one up and one down that means one orbital will only have one electron giving a net moment of spin half so copper can take these two balances so Herbert Smithite Herbert Smithite thought to be a quantum cagamate antiferromagnate actually this is its chemical formula it's a natural mineral and it looks like this but you can also grow it in the lab so we have to think about what would the valence of our copper be it can be one plus or two plus one plus would not be interesting because it would be non-magnetic well zinc always takes two plus oxygen always takes two minus hydrogen takes minus one chlorine takes plus one and they have this ratio one, three, six, six, two and of course the total sum of the charges must be zero so we can work through the maths two sinks, three coppers which are valence are unknown six oxygens, six hydrogens two chlorines their valences of course means that the valence of the copper must be two to balance the charge which is good because that means we have copper two plus ions which have spin half right so this is the crystal structure of Herbert Smithite and every ion is shown here and one thing to note is that it has a space group R three bar M the three bar suggests three fold symmetry M would be a reflection plane R would mean a trigonal group ok so what we're interested in is the copper ions these are the magnetic ones and they are blue so here the bright blue ones show the top layer and then below there's another layer and there's some faint blue ones so we just look at the bright blue ones of course we also have zinc and chlorine and oxygen but what you can see here is copper copper copper and in between you have an oxygen the bond is not straight but the direct distance from this copper to this copper if you drew the direct distances around them you would get a triangle even though the bonds themselves don't form triangles this is actually the AB plane of Herbert Smithite and rotate the crystal around and now look in the AC plane so that means you've turned this plane and turned it now horizontal so now the copper ions occupy a layer and this layer is separated from this layer and in between the two layers we have the grey ions which are zinc and zinc are non-magnetic so all this seems to suggest we have triangles of copper another triangle here another triangle here if we draw just the direct distances between them we will get a cagome lattice of copper ions that's in the AB plane this cagome lattice occupies a confined distance along C just this it forms a layer and this cagome plane is separated from the layer below by zinc and chlorine and all this suggests is that this distance is quite large with non-magnetic ions between and probably there is no connection between these layers no magnetic connection but these distances are shorter with oxygen bonds and probably have strong anti-ferromagnetic interactions in fact it is known that when you have copper, oxygen, copper so here we just have one of the triangles you have copper, oxygen, copper and the bond between them is 180 degrees it's not 180 degrees maybe it's 160 degrees but it's close to 180 degrees then according to the good enough Canymorri-Anderson rules we should have anti-ferromagnetic interactions ok so this is a well-studied material it's a candidate for a quantum cagome it's been used to test theories it has one drawback and all real materials have drawbacks and this is that it has copper zinc disorder a small amount of copper zinc disorder that means there's coppers on the zinc side and zinx on the copper side that means there may be some vacancies on the cagome layers some non-magnetic ions and this can be a bit of a problem so it only approximates an ideal cagome of course next example is copper chromate so we actually have two magnetic two possible magnetic ions here copper and chromium copper can be have valence one or two chromium can have many valences so oxygen always has minus one so that the sum of the valence of the copper and chromium must be four plus four that's all we know for now chromium has five electrons sorry six electrons it's positioned here so it's argon plus six electrons argon is a stable unit plus six electrons one goes into the 4s shell and five of them go into the 3d shell so I just tell you actually now that chromium will like the valence three plus because it has many options for valences but actually in an octahedral environment as happens in this ion so the chromium is surrounded by six oxygens and in an octahedral environment it will normally take the valence three plus so if that's three plus we're going to have to remove three electrons we're going to have three electrons in the 3d shell none in the 4s shell now we have to use hunts rule hunts rule first rule says you need to maximise the spin you fill up with spin up first and after that with spin down we've only got three electrons they're all going to be spin up so the total spin is three halves now if chromium of course takes three plus valence that must mean that the copper takes one plus valence and one plus has spin zero so we're just dealing with one magnetic ion this is the structure of copper chromate these are the copper layers and then these are the chromium oxygen layers actually the chromium are inside this octahedra this is the little shadow inside and each one is surrounded by six oxygens that's called an octahedral environment that's a crystal field so I tell you with a crystal octahedral environment you'll take the three plus valence normally also you notice that these are edge-sharing now actually we should probably look at this layer here so you've got a chromium inside here or chromium oxygen octahedra here and here and here and here and actually they form a triangular lattice that's to be expected because the space group is trigonal it's called a three-bar M three-bar would suggest that you've got three-fold symmetry so the chromium ions form a triangular lattice as you see this triangle layer of chromium ions is well separated from the next layer by a non-magnetic layer of non-magnetic copper ions one other thing to bear in mind is that of course these chromium ions are going to interact with each other via exchange interactions and the question is what is the strength of what is the sign of those exchange interactions a ferromagnetic or anti-ferromagnetic well with three electrons in an octahedral environment you tend to fill the T2G levels these are the T2G orbitals and the T2G orbitals point along the edges of the octahedra and so each chromium ion will have its orbitals pointing towards each other and of course each orbital is filled by just one electron and that favours an anti-ferromagnetic interaction so this is actually a good example of a triangle lattice anti-ferromagnet but it's a real material and it has additional terms in its Hamiltonian in this case it has a second neighbour coupling it also has some other issues which make it quite interesting it has very strong coupling between the magnetic system and the lattice so it has magnon phonon coupling and this also leads to multiferous behaviour in this system ok so now I'm going to next topic is to talk about experimental methods so I just briefly outlined some methods here and then I will talk about neutron scattering so what do you want to know you've got a material you think it's interesting you've grown it in your lab what do you do next well question one do we have long range magnetic order because if we don't or it is suppressed this is a good sign susceptibility can show you the transition very simply you put the material in a magnetic field and you measure its magnetisation its response to that field and you do this as a function of temperature and most materials will show a transition which will be a peak or some sort of feature in the susceptibility at a particular temperature indicating the transition the susceptibility can also be used to determine the curie vice temperature which is an indication of the strength of the interactions so we're looking at materials which have no nail temperature at all no ordering or where the ordering temperature is very much smaller than the curie vice temperature in particular materials which have enhanced correlations well above the nail temperature temperatures higher than the nail temperature are probably showing a response some collective response to an applied magnetic field suggesting a collective magnetic behaviour but which is unable to stabilise long range order and so that's some feature to look out for you can also look for the transition temperature by doing heat capacity measurements as a function of temperature again you can see your transition should be a lambda and normally a very sharp peak at a particular temperature indicating the transition but you also, I mean heat capacity is basically a measure of the density of states, a measure of the entropy and so your measurement of the heat capacity as a function of temperature can show you if you have a low line density of states at low energies which would suggest you have competing ground states or alternative ground states to the one that was selected and also if not all the entropy is released exactly at the transition temperature but continues to be released at higher temperatures this is another sign that your system shows unconventional magnetism now if you actually want to investigate that ground state move on spin rotation is a technique to use it's very sensitive to magnetic order very weak magnetic order can show up in move on spin rotation it can determine whether you have static magnetic order or even if you don't have long range magnetic order so it can determine a spin glass and alternatively it can also show you if your ground state is not static if the spins are moving so this is a very important way to investigate the ground state and then the last method is neutrons and I will talk more about that in the next part so I have 5 minutes so I will just start on that so I'm going to talk quite in detail about neutron scattering because it's an important technique for quantum magnets it's a way to measure the excitations and gives a very direct comparison direct and quantitative comparison between a theoretical model and a real material so you can do a quantitative comparison of theories so neutron scattering start at the beginning with the neutron neutron scattering is literally scattering neutrons from a sample you fire neutrons at your sample your sample scatters the neutrons you collect the scattered neutrons and you measure where they are scattered in which direction and you measure the energy that they have and the energy difference between the initial and final neutrons so it's all about how the neutron interacts with a sample now the neutron of course is a nuclear particle it has a mass similar to the proton it has no electrical charge and this is important because it can penetrate deep inside a material and it has a spin angular momentum of a half and this is important because it makes it sensitive to magnetic fields inside the material because it has a spin angular momentum it has a magnetic moment and actually it has a finite lifetime of 15 minutes so neutron scattering is in many ways like X-ray scattering but has some major differences of course in X-ray scattering you're always dealing with the fact that the speed of light is the main constant you don't have that with neutron scattering the neutron can take a range of energies so here the mass for the neutron the neutron has a momentum if it is moving it's always created with a motion its momentum is its mass times its velocity which of course is also a quantum mechanical particle it's also h cross times its wave vector that means of course that its velocity is related to its wave vector and its wave vector is related to its velocity now it has a wave vector it's a quantum mechanical particle so it has a wave vector so it has a wavelength of course a wavelength and a wave vector and the wave vector and wave length are related like this to pi over the wave vector so actually you can also write the wave vector in terms of the velocity remember the velocity can take a range of values depends on how you create it and then it has an energy and its energy is its kinetic energy it's half mv squared but velocity can be replaced by wave vector this so we have energy we have wave vector we have velocity and we have wavelength these are four quantities and they are all related and if you know one of them you know the other all the others so that was the point here to show you the maths for a neutron two minutes so the neutron, this is the important part how does the neutron interact with a material well first of all it's a nuclear particle and therefore it interacts by the strong nuclear force with other nuclei that it sees inside a material and this is a very short range force but it has a magnetic moment and therefore interacts with magnetic fields that it sees in a material why might there be magnetic fields? Well we have on-paired electrons going around in current loops in orbitals generating magnetic fields as they go or we have spin moments and so the on-paired electrons in a material can scatter the neutrons so if our neutrons come in this is our material we have primitive nuclei here with one electron going around and they form a lattice and we have our neutrons going in the first method is the neutron-nuclear interaction it will be deflected by its interaction with the nucleus but it can also interact by the on-paired electron and be scattered that way this is for x-rays and electrons and sources of neutrons how do we make neutrons we'll get them the two ways this is complicated or can be so the simplest way to do it is to use polarized neutrons where you know the direction of the spin of your neutron and you see how that changes after the scattering process so the phonons or the nuclei will not change the direction of the spin of the neutron but the magnetic scattering can do so that there are other things you can do like phonons tend to get stronger and wave vectors tend to get stronger of temperatures so you can do temperature dependence and wave vector dependence also to check with on-polarised neutrons so two ways to make the neutron the first is actually in a nuclear reactor is the fission process so here we have uranium 235 and uranium 235 can capture a spare neutron and going to excited state uranium 236 which is unstable and splits into barium and krypton releasing three neutrons in the process and a lot of energy we started off with one neutron and now we have three and they have a lot of energy so they're moving and they can move to the next uranium 235 nucleus get captured excited, split it release more neutrons so this is the chain reaction what happens inside the nuclear reactor and of course the goal in nuclear reactor is to utilise this energy but as a by-product you can also get neutrons the second method is spallation this is actually an accelerator so here we have now protons you start with protons and you accelerate them to very high speeds this is now within a few percent of the speed of light around an accelerator so they have a lot of energy and then you fire them at a target a heavy metal target and because they have so much energy they cause the targets to splinter and give off neutrons and protons so this is a very high energy technique compared to this and for each initial proton that you fire you might get 15-20 neutrons they're very high energy neutrons and you have to slow them down to make them usable I think I've gone past my time so I will continue after lunch thank you