 First thing, the reverse refinement technique is the most widely used technique for magnetic structure refinement. I use the word refinement, I have masked it in red because this is not an ab initio solution for crystallographic and magnetic structure rather to start with a given structure, magnetic and crystallographic and then you keep refining it. So he said I wrote here we need to start with a crystallographic structure for both the structure and magnetic part. So there is a starting point and then you refine that structure. So that means while we are doing this fitting program, running this fitting program we must understand that we are undertaking optimization of the given structure. So this program is available online and as I told you anybody interested to learn the technique can look into the lectures given in that course and this is used for both for both x-rays and neutrons all over the world and tutorials are also available online. So similar to crystallographic units now let us come back to magnetic unit cell. Now crystallographic unit cell we understand where the it is a repetitive cell that through translation gives me the entire crystal. But in case of magnetic unit cell, the magnetic unit cell can be different from the crystallographic unit cell. For example if I consider an anti-ferromagnetic sample then you can see that with the simplest example if I consider crystallographic to a linear chain. So in case of ferromagnetic material the repeat distance is L. Now if it is anti-ferromagnetic then nearest neighbor it will be anti-ferromagnetic then repeating ferromagnetic then again anti-ferromagnetic. So repeat distance has gone to 2L. So L to 2L 2L the cell has doubled when it is going from ferro to anti-ferro. But this is a one of the simplest possible examples that I can give there are other kind of unit cells which is commensurate or incommensurate with respect to the crystallographic cell I will come to it. So there is something called a magnetic magnetic propagation vector. Now if from one crystallographic structure unit cell to next the magnetism changes slightly the magnetic unit cell may be thousands of atoms in there and may have a much larger length scale compared to the crystallographic unit cell. So in that case for the fitting of our data we will use the crystallography unit cell and the propagation vector which will use the properties of the crystallographic unit cell and tell me at what distance the magnetic cell is repeating. Like as I showed you in the linear chain when I go from ferromagnetic to anti-ferromagnetic it goes from L to 2L the repeat distance and actually if you go from magnetic from ferromagnetic to anti-ferromagnetic if you look at the intensity of q or 4 pi sin theta by lambda you will find that intensity because L has goes to 2L the reciprocal lattice vector has gone to half and you will find that if the ferromagnetic peak matches with the crystallographic peak the anti-ferromagnetic peak will be approximately if it is small angle if theta sin theta is equal to theta it will be at half the angle for an anti-ferromagnetic peak. So you do see a new peak appearing if an anti-ferromagnetic order takes place. Now that means I have to find out the magnetic propagation vector for the additional peaks. Now here the magnetic moment I consider a 0th cell as one of the cells crystallographic or structural cells and then the magnetic moment in this cell at any side I can have components in various directions. So then I have to write down the magnetic moment at a site so let me just geometrically show. So suppose this is my unit cell unit cell and my moment this is the ith point in my unit cell and this is the M. Now if I consider a crystallographic axis A, B and C then this M will have components along A, B, C and basically that means the it's a vector sum of the components components in three directions. So let me here I have written this L is nothing but direction cosine and then the vector sum of the components L can be 0, 0, 1, 1, 0, 0, 0, 1, 0. So a sum of component, three components in A, B, C or X, Y, Z whatever you want to say direction and that you multiply gives you the magnetic magnetic vector at a certain lattice site. Now if there is only 0, 0, 1 component then I can write it as Mi is nothing but Mi 0, 0, 1. So that means in this diagram in this diagram now it is in the the moments are in the 0, 0, 1 direction. So then I can write Mi equal to Mi 0, 0, 1. Now since this magnetism is a periodic property either it is repeating every unit cell, two unit cells or maybe thousand unit cells. Any periodic function we know from mathematics that you can do Fourier expansion in terms of vectors in the Brillo zone. What is a Brillo zone? A Brillo zone is let me just explain to you if these are the nearest neighbors, nearest neighbors at a distance A. Then in the k space I can draw a line to the nearest vector in the reciprocal space at 1 by A and then draw perpendicular vector to that and that is my first Brillo zone and this is at pi by A and minus pi by A if I take the origin here. So this distance is this is known as a Brillo zone in reciprocal lattice space. So so now in this Brillo zone there are various values of k vectors allowed k vectors in a crystal lattice and I can write down the magnetization vector magnet vector at a side in terms of a Fourier components Mi k to the power minus twice pi i k t k is that propagation vector and t is the translation. If there is only one propagation vector for the system then I can write it as Mi equal to Mi k this summation will have a single component twice pi i k t. Let me just explain it with an example. I just take an antiferromagnetic order in one direction there is a plane and the direction switches between plus and minus from one plane to another as we go in c direction. Now I am going in c direction so my translation t is in c direction it is zero zero one and my moment is along the b direction as I showed in the drawing along the b direction. So I can write in some units this is zero one zero so moment is zero one zero translation is zero zero one. Now let us see in this case when it's an antiferromagnetic alignment if I take a propagation vector of zero zero half what happens so propagation vector is zero zero half so m u after one translation what happens it is zero one zero in the base plane then exponential minus twice minus twice pi i k dot t. Now k dot t is how much let us see k is zero zero half the propagation vector for the antiferromagnet I have chosen and t is zero zero one so this term becomes e to the power exponential minus twice pi i zero zero one dot zero zero half into half so it becomes e to the power minus twice pi i pi i sorry half is a pi i exponential minus pi i so now so this is what I have written so now let me go back and calculate it out so my moment was zero one zero in the plane I showed e to the power minus i pi is zero one zero cos pi minus minus i sine pi zero minus one it is equal to zero minus one zero so after translation of t my magnetic moment has become zero minus one zero from zero one zero that's exactly what I have shown in the diagram that it is an antiferromagnet order so zero one zero this is zero one zero zero minus one zero so given this magnetic propagation vector there is only one propagation vector here which is k equal to zero zero half which I could for this I could straight away guess because the it's very clear l became two l so k becomes half so instead of t which is one because it is 2t now my k is half very easy to guess in this case I've taken it from this reference so but while you're fitting a magnetic sample data we need to start with a guess of the propagation vector I would say a physically reasonable guess which comes from knowledge in magnetic structures like I showed you here various kinds of alignments for example this is a ferromagnetic sample here actually the propagation vector will be zero zero one in the previous diagram what I showed as for an antiferromagnet it is zero zero half it is zero zero one here it will be zero zero half this is a ferromagnet ferrimagnet so then with every t I also have to add an alpha t that means the length of the vectors are also changing not just that they are changing in site and they are also changing in size similar this is a triangular kind of lattice it's a canted lattice it's a umbrella kind of lattice where all of these vectors we have to represent in terms of a b c and then the reciprocal lattice and in the reciprocal lattice in the Brillo zone I can choose a set of k vectors which can give this correctly I show a sine or cosine kind of variation because the length is a ferrimagnet but the length of the moment is changing similarly these are circular helix where as we go ahead or as we go in one direction here for simplicity I can consider it is a zero zero one direction the magnetic moment keeps rotating and finally comes back to its original direction after a certain number of lattice points and 2 pi divided by that will be the propagation vector and the direction of h a plus k b plus l c then these are elliptic orbit so here not only if it is rotating the size is also changing and there are various possible structures in magnetic crystallography that you can consider and are being considered and that's a part of the game of solving a magnetic structure but in the read well refinement about read well refinement what I want to say it's a least square fitting technique that means you start with a diffraction pattern in hand so I have got a experimental diffraction pattern if experimental diffraction pattern I can say I or Y experimental which you have measured at a certain temperature theta or q and on the other hand you have got a calculated pattern and I have to compare these two that means at every point my every point where I have measured the data every point I measure the data I also have model data I have considered a model and have generated this and I'm comparing these two patterns so I have got I have got why experimental this is very common I have got why model I have to minimize the values of them but then there are many points so some override if I want depending on physical reasoning I can put a weight factor over here so I have to calculate this and this is known as a weight like various names can be given but this is fundamentally this is the numerical comparison between two models that I am using for my fitting so as I said so here it is error I dot A and then it's a weighted sum of the difference between the observed or experimental which I said and the calculated value the intensity of diffraction at every angle again is a convolution of several parameters like you have the the convolution of r theta which is resolution then wavelength dispersion then a sample related part plus a background and a convolution is something I must mention that the convolution is if there is a function y1 theta I mean let me say in terms of x minus x prime y2 x prime dx integration of that is the convolution convoluted value at a point x so experimentally suppose I have got a peak like this now if my instrumental resolution is a Gaussian like this then each and every point will tend to spread out based on that Gaussian and then I will have a broader peak whereas this is the theoretical peak and this is the convoluted peak so apart from these two over there when I wrote I have got convolution resolution function I have got a wavelength dispersion because the wavelength of neutrons that I use is not unique I use a band of neutrons because I need intensity if I try to make it mono energetic I agree very few neutrons so I have got a band of then there's a sample there can be a preferred orientation in the sample that can be strained in the sample if I know them then I also they also will cause broadening I'll put as s theta and then there's a background because anywhere I do the experiment there will be a background neutrons and that at every angle this will have a value and whereas the calculated value has to be convoluted with all these then and then only I can compare the calculated value convoluted with all these with the experimental value and with this I come the end of this lecture in the next lecture I have given a very general expression for the read well refinement but now I will tell you specifically what are the inputs there are several tutorials available online which I will give you the references and I have also mentioned like what are the parameters important parameters that can be given that needs to be given in read well refinement program so let me emphasize once again this is actually making a given model more and more commensurate with the experimentally observed pattern known as read well refinement it has done by Hugo read well and give the program the program name is full prof and it's a very very important program suit which is used heavily and for the magnetism society this is possibly the most important package in use today so I'll come back to read well refinement in the next lecture I will briefly mention give you some examples and then we'll go ahead