 In this module, we will discuss magnetism in solids. Now, this is a part of the module or the portion in which I will be teaching you how to do magnetic neutron diffraction. But before we get into the subject proper, we need to understand various kinds of magnetism that we find in solids and a brief introduction will be necessary before we get into neutron diffraction because you should know the kind of magnetic materials that will be handling and the examples that I will be choosing from. So we know that primarily there are three kinds of magnetic magnetisms in solids, diamagnetism, paramagnetism, ferromagnetism. So now I will start with diamagnetism, diamagnetism, diamagnetism is inherent property of all because diamagnetism comes from the fact that every electron orbit is a magnet. So this is inherent in every material and this opposes any applied magnetic field. That means whenever I try to apply a magnetic field on any material, diamagnetism will try to oppose it. So every material is inherently diamagnetic and this I will give you very briefly a classical theory which comes from Lenz's law regarding diamagnetism. So I show you here that every electron has a magnetic moment of E by 2 mc and angular momentum is E by 2 mc and m and that is the magnetism of an electron. Now I show here this is the magnetic field that you have applied on a material and this is the direction of the angular momentum for an electron in an orbit. So if I say that I have applied a magnetic field, there are many many electrons I choose one of them and the angular momentum is this direction. Now we know that this will tend to precise around this magnetic field and this precision is given by a Larmor frequency which is given by the rate of change of magnetic moment. You can see that it is coming as a cross product of the mu of the electron and the magnetic field H applied. So dm by dt for my system is equal to mu cross H of vector cross product. Now this comes this shows that if it is perpendicular to both of them then it is acting in this direction normal to both H and the magnetic moment of the or the magnet of the magnet of the electronic magnet, electron magnet which is basically due to rotation of electron in an orbit. Now I can add up this magnetic field's orbit by orbit and calculate the diamagnetism in the material and that is exactly what. So the force is normal to H as well as M as I said and the M preceses around the applied field and the alarmor frequency is EH by 2 MC. Then the electron orbit is precessing and producing a magnetic field to oppose this applied field H and then the mu induced is given by E by 2 MC M omega L rho square is the classical Lenz's law and all I can tell you that this rho square is the average radius of an orbit. So we can take average radius of all the orbits and this comes typically as average value of R square multiplied by Z value number of electrons in the orbits and Z value of the material and that multiplied by N and then E square by 6 MC square is the factor coming from the pre-factor coming from the expression. So now this gives the susceptibility of a diamagnet and that this if you evaluate it it has got a very low value about 10 to the power minus 7 Z which is around 10 to the power minus 6 if I multiply it with the value of Z. So that means every material has a part which is of the order 10 to the power minus 7 to 10 to the power minus 6 depending on the number of electrons in its orbits which opposites of magnetic field. But over and above I have atoms which have got unfilled electrons in their shells. So going one step ahead there are most popular or best known are the 3D transition metals iron, cobalt, nickel these have unpaired electrons in DNF shells. So here I have shown that nickel 2 plus it is a 3D aid that means in the third orbit in the D shell there are 8 electrons and before I try to evaluate the magnetic moment of this I can say that if there are unfilled shells then there will be magnetic moment. If all the shells are filled then you know every sub orbit as I shown here they will have 2 electrons each and overall the magnetic moment will be 0 but this is unfilled and for that we need Hund's rule. So Hund's rule says maximum value of S allowed by Pauli exclusion principle then maximum value of S allowed by Pauli exclusion principle consistent with that maximum value of L and if the shell is less than half filled J is equal to L minus S and J is equal to L plus S when more than half filled. So I will do it for nickel so nickel is 3D 8 so now iron is 3D 6 so because the D shell is unfilled now you can see D is SPD 012 so you have 2 to the 4 plus 1 5 10 electrons I can accommodate so let me just draw the shells here. So minus 2 minus 1 0 1 2 minus 2 minus 1 0 1 2 I have got 8 electrons first let me put all of them in one in each orbital because this is the best possible configuration coulombically but now I have to put three more and I have to maximize my L value so then I put three more in these three shells now I am left with 1 and 2 so that means now my L value will be 3 2 plus 1 3 S value is 2 into half which is 1 so J value this is more than half filled so it will be 3 plus 1 equal to 4 so now you see this is what it is we have drawn in the same way and only this I am sorry because this will be minus 1 minus 2 because the maximum value of S and maximum value of L. So this gives me and these orbitals which are actually empty it is T2G and EG orbitals T2G has got six of these and E2G has got two of these total eight and this gives me the ground state of nickel with two unfilled electrons so this orbitals I why I mention because these orbitals which are having unfilled electrons they are actual physical drawing is like this this is a 3D xy xz and zy which are you can see diagonal to x and y and 3D z square and 3D x square minus y square the plots are taken from the sources here so these these are actually degenerate if I take an independent neutral atom but when you put the same electronic orbitals inside a crystal then depending on the crystal field which is highly directional this degeneracy of energy is lifted in the crystal and then you have this once this degeneracy is lifted then you have to put electrons accordingly in this orbitals so again as I wrote here that so LSJ I have written so S is 1 so that means and J is 4 so you can write down the ground state you can see I have written here the magneton number for iron groups iron group means iron 2 plus cobalt 2 plus nickel 2 plus this is the most well known magnetic materials towards there is 3d6 3d7 3d8 and just now I calculated for nickel I have calculated the ground state which has got ground state J L and S values it is 3f4y go back there so now you see l equal to 3 so 0 1 2 3 s p df so it's a f state now you see why 3 this is 3 because this is 2s plus 1 into 2 so 2 into 1 into 2 it is 3 and J value is 4 go there so J value is 4 so this is how we write the ground state for an atom especially for a magnetic atom using moon's rule and this is used for ferromagnetic materials and also for the rarer group materials the rarer group is here the same technique I know I have used I can calculate out the ground state for all of these but I have taken it from a book eternal fifth edition so it is just for your knowledge but the thing is that everywhere you have got a calculated value of the magnetic moment and that is normally written in terms of this value mu B Bohr magneton which is e h by 2 mc let me just mention here that you can also write a magnetic moment for a nuclear particle like neutron which will be using later but because this mass is almost 2000 times higher the magnetic moment is almost 2000 times lower compared to an electron but here it is e h cross by 2 mc which is a spin magnetic moment of a free electron coming back to it so we can evaluate the ground states and what are the Bohr magnetons in these ground states and now we will be talking about materials which have got inherent magnetic moment so that allows us to write down so this is a crystallographic structure which you are familiar and we can find out the crystallographic structure using diffraction experiments extra diffraction experiments so these are regular structure but now consider this is a magnetic material let me say iron nickel cobalt 3d element then each and every side also has a magnetic spin which I calculated out just now using hoon's rule there's a magnetic spin associated with it so there is a magnetic spin but please note that I'm writing all of them parallel because you are familiar with the fact that these are inherently ferromagnetic but this is not true for all temperatures this is not true for all temperatures because if I raise the temperature then there is something called an exchange energy and temperature the exchange energy's rule is to align them this is exchange energy is working between the two magnetic moments at two sides and the temperatures rule is to misalign them so if I raise the temperature then it will be a structure like what you see on the left hand side so that means when j by kt kbt is a thermal energy it will always try to dynamically disorder the thing and j ij j ij ij is the exchange energy between two sides which tries to align but when I say align I must mention that though I have taken examples from only ferromagnets but there are others like antiferromagnets ferrimagnets they are all ordered structure joe antiferromagnet does not have a bulk value of magnetism but it is ordered so far as if I consider that these are oppositely aligned you must understand that this is also an ordered magnetic structure so each and every nearest side they are not aligned to each other not aligned to each other sorry I may be out now so well I created a frustration here but they are not aligned to each other but these are ordered moment so here you see this is an ordered moment where the overall s value that means s averaged over the entire ensemble of spins that's non-zero and this you achieve when you come down in temperature above a certain temperature which is known as curie temperature for ferromagnet this is disordered like this so now I have introduced you to sides with local moments but one is a paramagnet paramagnet which is disordered disordered and as you lower the temperature you go to ferromagnet antiferromagnet ferrimagnet depending on the material and their h i j and they are all ordered so this is a phase transition from disordered to order state and I must mention here I will get back to it later that whenever we do neutron diffraction we do experiments above the transition temperature that means at that time your magnetic order is lost but crystallographic order is present so you can get the physical crystallographic structure then you come down in temperature below the ordering temperature and then I have a ordered magnetic moments and they will aid the the diffraction pattern and then add up the intensities what you find from in the disordered magnetic state so you add up intensities in those peaks so there are ordered magnetic states like as I showed you just now that this is a ferromagnet which is s is not equal to zero there is antiferromagnet which is averages will be equal to zero but this is also ordered and also there are ferrimagnets where the magnetic moment often it happens and I will see I will use examples for these kind of materials where the local sites have different magnetic moments so one large one small and there's an order state of that but these order states are different from the disordered state when s is equal to zero again but this is what I call dynamically disordered because of temperature at higher temperature so these are the ordered magnetic states which you will be studying using neutrons but for that you also seek information from the disordered magnetic states which is above the curie temperature I will stop now before I go to the next module