 In this lecture we will continue with magnetic neutron diffraction, how we carry on magnetic neutron diffraction, the process flow I will explain to you, but any diffraction whether it is crystallography or magnetic needs pre knowledge of little bit of crystal symmetry and the space lattices, I will introduce you briefly to that because otherwise this is a full-fledged subject in itself, but I will introduce you to the extent that you can use it for neutron diffraction and then I will introduce you to some results that you obtained using neutron diffraction. With this I start today's lecture. So as I told you earlier the riddle refinement, the technique that I can use for finding out the crystallographic structure it is an optimization tool, very important to understand that, that means we are starting with a structure and both crystallographic as well as magnetic and then I keep refining it till we get a good fit between the experiment and the theory. So good fit also depends on the error values that we have on the experimental points and that means we have to collect data for sufficiently long time to support one model or other. So running through the parameter space the starting point that means the starting structure should be good. This good is a very loaded word because when I say good that means when I start from a point I should be able to first it should be physically reasonable that means for example a very bad example possibly I know that iron it can form BCC lattice. Now that means my starting guess should not start with iron in an hexagonal closed packing because then I am too far away from the physical reality that many iron has got a BCC lattice. So I will start with an assumption with a BCC lattice and possibly and I will show you some examples that and then I try to figure out the moment. So one needs to start with a crystallographic structure plus a magnetic structure when I am looking at magnetic neutron diffraction and microscopic magnetic structures like ferromagnets, antiferromagnets, perimagnets. To start with I need a propagation vector in the last lecture I explain to you with an example maybe I will repeat it I showed you that if I have a planar moment and if the planes are translated along the z axis calling this as A this is as B I define the magnetism in some unit as 0 1 0 and the translation is along 0 0 1 direction. So here what I discussed earlier that I said that once I translate it by one lattice plane it is an antiferromagnetic order and then the next it comes back to ferromagnetic order. This is a simple example but in this case it was very simple to show you that magnetism which I called m 0 1 0 it goes to 0 minus 1 0 if I use a propagation vector which was 0 0 half this is a magnetic propagation vector and how I did it I did e to the power minus it was m I can call it 0 1 0 e to the power twice pi i and k dot t t was the translation vector and k was the propagation vector and propagation the translation vector was 0 0 1 for everyone and then k dot t I can show you that sorry k is a yes so 0 0 half so when I translate it by 0 0 1 it became m 0 1 0 e to the power i twice pi i and it was it was 0 0 1 0 0 half so it was half others was 0 and then m 0 1 0 into e to the power i pi which was m 0 1 0 cos pi plus i sin pi this is 0 this is minus 1 it became minus m 0 1 0 so that is how I could choose a propagation vector because here the antiferromagnetism means the repeatability is 2 t so it is very easy to choose a propagation vector as 0 0 half because I know that every time I go one lattice plane here to here 1 d spacing my magnetic moment reverts so but in reality actually it is a simple picture there can be more difficult or more complicated magnetic propagation vector sometimes it may even require thousands of lattice planes and also there are cases where the structure is commensurate with a crystallographic structure that means I start with one crystallographic lattice plane with a certain magnetic moment in a certain direction after an integral number of lattice planes I come back to the same moment which will match with another lattice plane maybe several or thousands of lattice planes away in case of incommensurate structure which is a more complicated problem this structure the magnetic structure is not commensurate with the crystallographic lattice structure and there will be some gap so I will discuss those things in the time cups so it's an optimization tool and so one needs to start with the crystallographic structure and the magnetic structure that means a magnetic structure a propagation vector side moments etc now how the process flows first a neutron powder diffraction is above the ordering temperature will give me the crystallographic structure in the paramagnetic phase under the assumption this has an assumption that no phase transition phase transition structural phase transition I should mention specifically sorry no structural channel phase transition phase transition occurs between paramagnetic phase and ordered phase ferromagnetic and so antiferromagnetic ferrimagnetic so if that remains true then I can take the data in the paramagnetic phase by going above the ordering temperature or T greater than Tc or T nil temperature and get all structural parameters using full proof this is equivalent to carrying out an extra diffraction on a extra diffraction on a crystallographic powder sample only here the difference is that the scattering amplitudes are different otherwise you should get the same pattern and both of them in this case the nuclear powder diffraction pattern can be solved using the suite of programs called full proof in this regard I must again tell you that in case you are interested in a full-fledged tutorial on full proof to solve magnetic structures please look for the lectures by Professor Anil Jain in our course on the course was on neutrons as probe of condensed matter condensed matter taken between 2020 and 21 21 and I point out to the the lectures taken by Anil Jain and Professor Esam Yusuf where a full-fledged tutorial was done how to fit a magnetic structure using full proof so with this mention so after collecting the data above the powder the magnetic transition now we cool the sample and collect again neutron powder data below the ordering temperature that means when the sample is magnetically ordered so when it is magnetically ordered if the magnetic order is commensurate with the commensurate with the structure crystallographic structure like ferromagnetic ferromagnetic then we will have additional intensity at the ferromagnetic peaks so if there is a ferromagnetic peak above order temperature I will show you there will be added intensities because of the magnetic order if it is anti-ferromagnetic then a new peaks will appear in this intensity versus q the most general representation in that case in intensity versus q not only you will have the crystallographic break peak but below ordering temperature you will also have magnetic break peak so this is when your order is anti-ferromagnetic that because this happens because for anti-ferromagnetic order you can see in the simplest picture the unit cell has doubled when the unit cell has doubled 2d sin theta equal to lambda sin theta has to go to lower value if it is in the low angle where I can say theta equal to sin theta then it will be theta by 2 which is not the case always but in the lowest possible assumption in the 0th order assumption you will also get peaks at half the intensity so after this thing when you start the program of course the first thing is that determine the propagation vectors for the magnetic structure petrial and error or okay and once the propagation vector has been determined use the program basi erupts in order to get the basis vector of the irreducible representation lot of jumbo mambo let me tell you basically the symmetry operations of a crystal they form a group group means any combination of any two symmetry operation combination of any two symmetry operations combination of any two symmetry operations is symmetry operation they carry after b gives rise to another symmetry operation c which is also in the group so that's why they form a group and there's an unit main operator which does not make any change in the system so now this is what so the irreducible representation because I can always represent the symmetry operations by a matrix for example a rotation we know can be denoted by Euler matrix Euler n matrix a rotation around an axis by theta so now if I consider a 90 degree rotation this I can write this and you can see that this this will be a matrix representation of a symmetry operation which will I'm sorry no I'm very sorry so it will be a symmetry operation which is represented as a matrix so the irreducible representation is one which can be broken down into further representations the matrix representation which can be broken down into further or you may say that is the smallest size representation this flow I have taken it from a tutorial on magnetic structure determination and refinement using neutron powder diffraction and full prof by jr carvayan and this is the site if some of you are interested you can also take help from this tutorial so how this is how the process flows but now it is important that before I go further even if briefly I introduce you to to the symmetry operations so here I have taken the example from our well done book kittel solid state introduction to solid state physics by kittel this just shows that this has got just one full symmetry one so it leaves apart invariant but come imagine now in this same diagram I have got an object here and I have got a mirror plane when I have a mirror plane perpendicular to this then this object is reflected onto this side and then I call this symmetry operation as one m now I can also invoke a rotational symmetry a point group operation basically means that the operation leaves the points invariant in space so here if I have let us say two oxygen molecules by atoms at these two points instead of this circle in some structure then they have got one in symmetry similarly I show you a rotational symmetry this is actually like this rotational symmetry and this is a two-fold rotation because if I rotate this if I take this point around this axis axis passing through the center of this circle 180 degree the point goes here from here it goes here and now if these start with if these are the two atoms sitting here then this system or this structure remains unchanged through a rotation of pi and this is pi means 360 by 180 degree it's a two-fold rotation next I have taken example another example here there is a two-fold rotational symmetry you can see plus I have used two mirrors one mirrors two mirrors normal to the plane now interestingly you can see if I use one mirror after rotation we have got these four points and this automatically gives rise to the second mirror so this justifies by argument that the symmetry operations in a group two operations invoke brings in a third operation so here two m invokes another mirror plane and mirror plane is represented by m rotations are invoked by one two three four and six because this we have learned that the possible rotations of one two three four five is not possible six because we can fill space with these rotational symmetries a rectangle a triangle a square a hexagon can fill space but a pentagon can't and that's why we say that the possible rotations are one two three four and six five fold symmetry in solids is not possible this book the cover page also shows why pentagon can't fill space but today we know even as an information for you that yes even if if we consider pentagons we can fill space not with rotational symmetry but still we can fill space because that is the that is the target that using a crystallographic unit cell but repeating the unit cell we should be able to fill up space that's why you can create a crystal which is which I can see with millions of unit cells so today this is also accepted that five fold symmetry is possible the group of crystals are known as quasi crystals but in this lecture I will retain I will retain the old rules that one two three four and six fold symmetries are allowed and then what are the 32 point groups of symmetry point groups in crystallography these are combinations of symmetry operations given the symmetry of a unit cell in a crystal let me just do it for a cuboid so let me take a cuboid now let me take a cuboid cuboid in this cuboid please know that this is a two fold axis that means I can rotate it by 180 degree around this I'm sorry I should draw passing through the intersection of the diagonals of the diagonals the diagonals which is clear there's a two fold axis around this because 180 degree rotation will leave the parallel pipe and unchain I have one more axis at the intersection of these two diagonals in this phase and one more here so there are three two fold axis but interestingly normal to these axes I also have mirror planes this will be one mirror play over there I have drawn it here you see this is one mirror plane normal to the axis on this phase this one is one mirror plane which is normal to the vertical axis and this is one more mirror plane normal to this axis so there are three mirror planes so that means I have got three two fold axes and three mirror planes and the mirror planes are normal so this is the international convention and if you are planning to do structural analysis using grid field fitting you should know that so for this it is 2 by m 2 by m 2 by m so and also you can see that combination of any two of the symmetry operations here will give back to get back us to one more symmetry operation so 2 by m 2 by m 2 by m is the international convention to present there are several conventions one of them is this