 Last time I stopped before I could introduce you to various magnetic interactions in solids and I promised you that before we get into neutron diffraction for magnetic structures, I will give a brief introduction to at least some of the major magnetic interactions in solids and direct exchange I discussed with you which comes from the we know that I discussed it with respect to hydrogen like molecules. So here if there is an atom A and atom B so it will be H2 but what I wanted to point out to you the direct exchange comes we need exchange electron one with electron two based on Pauli exclusion principle and I do get a G exchange term which is given by the wave function of A with electron one or R1 I can say then wave function of B with electron two VAB which is the interaction potential then Psi A R2 I have exchanged now R1 and R2 Psi B R1 this whole thing to be integrated over double volume integral and I mentioned to you earlier VAB is given by E square one upon sorry let me just clear it a little bit VAB is equal to E square A2 one by RAB that the distance between the nuclei plus one by R12 between the two electrons minus between the nucleus and the electron minus this is the interaction potential this is purely columbic in nature but this exchange term comes due to Pauli exclusion principle where you can see that this thing in this integration I have kept VAB as I wrote here but I take it from a state where the things are exchanged and where the things are not exchanged and this comes because two electrons are identical particles this is the integral part making it a little confusing this is the integral part G exchange so the direct exchange it comes from overlap of electronic orbitals and we know that I talked to our earlier like it like e2g orbital which looks like this if this is the z direction xy plane then e2g looks somewhat like this e2g one of the e2g orbitals so this comes from overlap integral of these orbitals not exactly atomic orbitals I mean not exactly s electron like orbital that I showed you so that is the direct exchange but there are several other exchange interactions like the next one is a super exchange interaction now in this case it is not the direct overlap of the wave functions of two neighboring atoms but here you can see this is needs an intermediate atoms so it is the next two nearest neighbors two manganese dz square orbitals they are interacting through an oxygen pz orbital so this is typically always manganese for example manganese is a 3d4 element and 3d4 means I will just quickly remind you that 3d4 how to calculate the ground state of manganese so 3d4 is d is spd spd012 so there are five orbitals possible two three four five excuse my unequal size minus two minus one zero one two these are the orbitals now I have to put electrons in them four electrons so it goes to one here and I can put them parallely which is favorable they will be far apart there is a four so now g l is equal to two plus one plus zero plus minus one so l is equal to two two plus one plus zero plus minus one two s is equal to there are four parallel electrons so four into half equal to two the orbitals is less than half filled so j is equal to l minus s equal to zero so now you can see because s equal to two there are five possible states but this is for the whole atom now not for one individual electron and then l equal to two that means spd note this capital letter and write because this is for the whole atom and g is equal to zero so it's a 5d0 for manganese and for this 5d0 manganese interacting through so is a dz square orbitals and oxygen is a 2p oxygen is eight electrons so one is two 2p6 so it's interacting via 2p electron and you can see that if this is up spin then it will force it to become down this is by Pauli exclusion principle and this will force it down see mostly it favors antiferromagnetic order but that is in this case so for this 5d0 ground state so it is a through intermediate this gives an antiferromagnetic interaction between two next to nearest neighbor manganese magnetic atoms now the but there are actually a set of rules given by good enough conglomerate rules this problem was first targeted by PW Anderson and later more successfully by good enough conglomerate so it says that between two magnetic ions manganese in the previous slide which coupled through an intermediate non-magnetic ion which was oxygen 2 minus here so it was 2s 2 2p6 two oxygens are going out two electrons have gone out so 2 minus the super exchange will be strongly antiferromagnetic which I showed you but there are rules regarding others the coupling between ion with field orbital and one with the half field orbital will be ferromagnetic the coupling between ion with either a half field orbital and one with a vacant orbital can be either ferromagnetic or antiferromagnetic but generally favors ferromagnetism these are known as good enough conglomerate rules the derivations are beyond the scope of this course but what I wanted to say that so we discussed about direct exchange where two atomic orbitals overlap then we talked about super exchange where two magnetic ions are interacting through an intermediate non-magnetic ion which is oxygen so manganese oxide is a known antiferromagnet antiferromagnet means neighbors are parallel anti-parallel aligned then there's something known as double exchange now double exchange is similar to super exchange another form of super exchange very similar but I request you to note the fact that here I am putting mn 3 plus that means three electrons have been taken out from the d4 so orbital so we have these t2g orbitals filled and one in the eg orbital energy wise they are split now in a crystal field next is that the other ion it has a different vacancy valence valence state which has got four electrons taken out so there's a vacancy here now this electron can hop from here to oxygen 2p state and hop to this vacant side and keep on hopping back and forth and now by using quantum mechanical techniques I can calculate this hopping parameter or the transfer integral and that will give rise to a double exchange this was proposed by Zener in 1950s and this works specially in compounds not in elemental magnets compounds with mixed valence compound because you can see that I need two interacting ions which will have different valence valency now there is one more very interesting this interaction known as Rudermann-Kittel-Kassuer-Yoshida interaction RKKY more commonly here the interaction between two magnetic sides is by polarizing conduction electrons so it is the technique actually this interaction was discovered when people discussed about or did not understand the nuclear lines and it is basically the one nuclear spin can polarize the nearby conduction electrons and that can further force another nuclear spin to align in some way here instead of nuclear spin we are talking about inner d electrons so I will just try to give you a pictorial representation of this interaction let's say this is the spin of a d layer this is the spin magnetic moment of combined d electrons now this will force the nearby conduction electrons to be oppositely aligned because of Pauli exclusion principle and then another one magnetic ion since this is oppositely aligned to this this will force it either depending on the oscillation of the so now this spin density if I may say will oscillate between plus and minus it will either give it a positive alignment or if this conduction electrons they go back to up spin near the next side then this will force them force it to antiferromagnetic ordering this is actually many times it is used between magnetic defects in an otherwise crystalline lattice and this interaction is known as r k k y interaction where the intermediate intermediate conduction electrons conduction electrons they are responsible for this I just show you drawing here the indirect r k k y magnetic interaction this has been taken from this paper where it's the long range you can see so electron density of a system is given to 10 to the power 11 per centimeter square and one impurity is fixed at y 1 equal to 15 nanometer and the change in location of other impurity you can see this is nanometers so 12 25 45 15 so basically first close to the first impurity the electrons are polarized now this polarization causes as we go further down opposite polarization of the conduction electrons then this opposite polarization in the next distance will force the electron to be again polarized up this is simply because of poly exclusion principle and now we can see the polarization of the polarization of the electrons they have an oscillatory behavior depending on the polarization here if a magnetic defect is situated at this point then this will be polarized opposite to the polarization of the electron and ultimately this defect or atomic or molecular side is forcing the alignment of another side which is quite far away you can see this is the distances of the rough of 12 25 45 nanometers to 120 to 250 angstrom so there's a long range magnetic interaction known as RK co-interaction where the intermediaries are the electrons the conduction electrons so or one may say that these are dynamic interaction using electron spins close and away from one spin and then forcing the other spin to align ferromagnetically or antiferromagnetically with respect to the first spin and often used for defects or impurities magnetic impurities in solids dipolar exchange interaction is another type of interaction which I mentioned in my transparency earlier so it's basically the exchange interaction between two rotation or two pairs of rotation states of a molecule here this is what I've shown here schematically is a potassium rubidium molecule so rubidium that is so here you can see that the one state is zero zero that means zero means one is the total spin of the electrons in this molecule it is zero and the other state spin one so there's zero zero state and also there is a if the principal quantum number is one then there can be three possible values minus then one it can be minus one zero one this depends on the energy hierarchy of the spin one system so primarily I am talking about exchange between the zero zero and one zero one one one minus one states of a molecule and again I can calculate the transfer integral and I can give you the energetic value of this double exchange integral so here this is a dipolar spin exchange interaction of molecules in a little so spin zero to spin one and spin one to spin zero this is the interaction in this case the most interesting one is I have this figure has been taken from this source in case anybody is interested you can look up the article because it has been determined using some technique known as ramsey scattering optical scattering because the difference of energy there is a difference of energy involved in in this interaction which is 2.2 gigahertz unlike other exchange interactions where we don't mention any energy down between the two states we are considering for exchange this is the last one I am using this is interesting because here so far in the exchange interaction I was writing si dot sj that is si cross hj so here that means there's a cross product term and cross product term means you can see there component between the two which is normal to this plane of this figure and most interestingly it use a weakly ferromagnetic behavior in an antiferromagnetic I will try to give a vectorial diagram for this consider these an ideal end to antiferromagnetic to antiferromagnetic now if I can have some canteen that means if I add an antiferromagnetic term to these two ferromagnetic term to these two a small ferromagnetic component coming from the cross product then it looks like this now as I showed you see it goes this way it goes this way so this is one spin the other spin so that's why what is mentioned here that in this exchange interaction it gives a weak ferromagnetism due to the cross product of the term and it's a source of weak ferromagnetism in an otherwise antiferromagnetic but this was very difficult to determine experimentally but now without talking about this interaction I cannot really bring you up to date with the magnetic interaction listing because this this interaction is important for generation of magnetic skirmeons now some of you may be aware but this is the latest order magnetic structures but that macroscopic length scale or mesoscopic length scale I just show you the schematic because I don't have the chance to give you the complete theory of it here the fact is that this cross product you see there is a this this dimension is almost a micron size I am giving you a TEM photograph TEM image of this skirmeons you can see in this one micron length so typically this size can be fractions of a micron it can go to 1000 angstroms also so this is a macroscopic to mesoscopic size objects and look at the spin canting so here the spin canting because of the cross terms you can see they keep turning and finally coming to parallel position at the boundary of this skirmeon this is one skirmeon now I must mention to you that this is beyond neutron diffraction or crystallographic neutron diffraction of a solid because the sizes are much larger but this can be investigated using small angle neutron scattering which will be a part of this module of diffraction but small angle takes place at a much larger or mesoscopic length scale and skirmeons have been studied using small angle neutron scattering if I have time along with other problems in small angle neutron scattering I will try to introduce you to the studies on skirmeons so this completes my promise of introducing you to the major magnetic exchange interactions so I talked to about direct exchange super exchange through an intermediary double I did not talk about indirect exchange interaction double exchange where you take about talk about two different balances of the same magnetic ion then RKKO interaction where interaction between two magnetic atoms possibly defects in a solid are caused due to polarization of conduction electrons and then the alofini ski moria interaction which is a cross term of si and sj to interacting spins and this gives rise to skirmeons mostly due to loss of inversion symmetry at the interfaces of thin films you find skirmeons in thin films I cannot give too much of theoretical details for all of them but I hope I have introduced you to most of the major magnetic exchange interaction after this I will go over to magnetic neutron diffraction