 now what is the relationship with the crystals now let's consider a cubic crystal again this photograph is from this figure is from introduction to solid state now you can see I have a lattice which is just the external structure plus I have to add atom bases which are atoms or molecules once I choose a lattice and bases then only I will get a crystal so now this is the most symmetric structure in crystallography a cubic structure please see that there is one plane here now the thing is that there are also other planes which I can draw like this there are other planes other planes but I can invoke those other planes by symmetry operation so I consider one which are inequivalent are these there is one one mirror plane along the body diagonal so if you join these opposite edges so there are actually there are three four four rotation axis one two three there is a rotation axis then there are four three four rotation axis which are actually if I go along the body diagonal so along the body diagonals there are actually three four four rotation axis so three four four rotation axis four three four because there are one two three four five six seven eight so the eight corners adding them two you get four so there are two fold axis sorry I am sorry there are three four axis three four axis four axis and there are three four axis four which are number is four then so for a cube let me just be little more clear about it this is a cubic figure so there are three four fold axis then when I add the body diagonals I noted around them then there are because there are eight corners so there are four diagonals four and the symmetry around the diagonals is three four three four also if you join the opposite center point of opposite edges then I will get two fold axis two fold axis and there are six of them two fold three fold four of those and four fold four fold there are three of them so these are the rotations that I have got here so there are three four fold rotations four three fold rotations six two fold axis but all of these are actually connected by symmetry operations with each other because I can combine let us say one this four fold axis if I rotate it around this I can come back to this four fold axis so I can generate the other axis from one axis then I can do the inversion by some symmetry operation also there is a center of inversion where all the diagonals meet across that I can do an inversion of operation an inversion operation means x y z will be going to minus x minus y minus z or in gestorography they write it x bar y bar z bar . So, here the cube so it is given as four four fold axis perpendicular mirror M a rotor inversion axis it is a rotation as well as inversion operation which is three bar and istinct మారొియతంధా 5 ఈఉలిత౿రికారి ఓితందరకూ. ఇంది న్తి ది నిందార౏ ఒాతి. So, this is actually the 410 gravel lattices, which gives all possible crystallographic structures in our known world of 1, 2, 3, 4, 6, 4 properties, symmetries. So, there are triclinic, monoclinic, triclinic, monoclinic, orthorhombic, trigonal, cubic, tetragonal, cubic, trigonal and hexagonal. And for example that most symmetric one is cubic, so and then cubic has got either simple cubic or BCC cubic or FCC cubic. So, here I have introduced one more thing, which is known as primitive primitive and non-primitive primitive crystal lattices. In a primitive crystal lattice, if you take one unit cell, there is one basis or one lattice point, one equivalent unit. Whatever it is, it can be iron, iron of course it is not simple cubic, but if you take a simple cubic structure, there will be one equivalent unit every in every unit cell. That means, how it comes because if you consider a cube, there are eight corners, each corner shared by each corner is shared by eight of these in itself. So, eight, one eighth and each one is shared by eight, so one eighth of each, so it gives me one unit cell. Now, if I put a body center, also an atom at the body center, which is often the case, then I have got one coming from this corner atoms and one coming from the central atom and there are two equivalent atoms. Now, when I come to FCC, then this cube has the corner atoms, the corner atoms, the corner atoms as well as one atom at the center of each face. Now, you can clearly see from the geometry that the one which is at the center of each face is shared by two of those, two of those. The one unit cell which is on the left hand side, the other one right hand side which I have not drawn. So, each one I have got half contribution and there are one, two, three, four, five, six faces. So, you get three atoms coming from the face centered positions and one atom coming from the corner positions. So, there are four atoms in an FCC unit cell. So, not only I have got primitive cubic unit cell, but you have so simple cubic face centered or body centered. So, the way we define a cubic cell is A equal to B equal to C and all of the angles are 90 degree. When it goes slightly off from here, A is equal to B not equal to C is a tetragonal. A is equal to B, but not equal to C, all the three angles are 90 degree. So, now in case of cubic symmetry and the translational symmetry, not only point book symmetry, allows me to have simple cubic body centered cubic or face centered cubic. In case of tetragonal, it is only body centered that is allowed when A is equal to B not equal to C, all three angles are 90 degree. So, accordingly I have written down here which actually have been derived from the symmetry and requirement of translational symmetry. Point book symmetry and the requirement of translational symmetry worked out by the mathematicians and the crystallographic. These are the possible unit cells which are known as Bravais lattices, 14 Bravais lattices in Karnes matter. Along with that, you have got 32 point group symmetries. I explained to you what are the point groups and here I have given the table. I don't have time to explain all of these, but let me just consider one. Let us say cubic, easy to see. You see in cubic the symmetry operations are given here and the symmetry operations international table, it gives me these are the symmetry operations. Symmetry elements are given here according to the elements on the left. So, this I have taken from this source you can and this is used heavily by us because we have to do for example tetragonal. We do say that when I input it to full crop, it is a 4 by mm structure. That is what the input, this indices actually. To mention that it is a tetragonal or for hexagonal it can write 6 by mm structure. So, there is a 6 fold axis and you can see that the mm, there are 3 mirror planes. Here there are 3 mirror planes and the 4 fold axis. In case of orthorhombic, you have got 3 to 4 fold axis, you also have 3 mirror planes. But we write this indices for any program that we use to solve the crystallographic structure. So, I have told you about the 14 Bravais lattices. These are the 14 Bravais lattices given here and the third row point groups. This together with 2 more things. 14 Bravais lattices and 32 point groups give us 230 space groups in crystallography. There is 2 more symmetry operations I need to tell you. There is with transitional symmetry, we have something called screw axis and glide planes. So, it is not just point groups, 14 Bravais lattices also we have to consider screw axis and glide planes. What are they? I will just quickly tell you. Let us see along with translation. If a rotation is there for example here you see take this atom A. Actually these are all atoms or units can be a molecule also which I am putting at this point. So, you see there is a translation of C by 4 and a rotation of 90 degree. Another translation of C by 4, another rotation of 90 degree position 3, another rotation of C by 4 and the rotation of 90 degree. It takes me from 1 to 4 identical positions in the unit cell or in the repetitive unit cell. So, this axis here not only I translated, but I also went in a screw that is why it is known as a screw axis. So, translation associated with rotation is known as a screw axis. And this together with 32 point groups and 14 Bravais lattices gives me a screw axis. Similarly, I have taken a simple example of a glide plane the sources are here. You can see instead of rotation if I move in a certain direction and I reflect and I reflect then you see it is a half a translation reflection. The unit is shown as a small engine. So, the engine reflected below the plane it can be a molecule actually it is available in case of molecule. Then again another half translation and another reflection in the glide plane the glide plane is shown here. So, that means you translate and rotate you get a screw axis to translate and reflect you get a glide plane. There is an inherent difference between these two. In case of screw axis if you put a molecule then it maintains the handedness of the molecule. But in case of glide plane this translation and reflection does not matter the handedness. And there are cases where in reality in crystallographic structures that such things happen. So, screw axis and glide planes are the last two symmetry operation that give us the all 230 space groups we know. But ultimately after talking about all these things all we need to give as an input is the space group classifications. Then we try to solve the crystallographic structure in the paramagnetic phase. Now in addition to the crystallographic space group there are magnetic space groups that have an extra dimension. Because we also add time and there is a time inversion symmetry and reflection around the time axis is allowed. Because I know that in the simplest explanation current gives you magnetic field. But current is dq by dt and dq by dt goes to minus dq by dt if I do a time inversion. So, like all other Newtonian motions the magnetic field does not have time inversion symmetry changes sign. And that has to be accounted for when I input my data for solution of a magnetic structure. So, in a magnetic space group each side has a spin colored black and white. And time inversion or time reversal will interchange them in a magnetic space group. So, now I come to the fourth step but this you have to actually learn through a tutorial. So, once the propagation vector is determined, crystallographic structure is determined. Then the program Basiref gives us the basis vectors of the irreducible representation. That means these are the basis vectors with which we can generate any translation along with the propagation vector. With the help of this program we can determine the Magnetic Symmetry Operators who first introduced the Magnetic Symmetry Operators and use them directly as the basis vectors in the irreducible representation of the crystallographic group. After talking to you about the procedure how you input the magnetic structure as well as the crystallographic structure for the for determination of magnetic structure. I will come to some of the results very interesting results unique to neutral diffraction. Okay, so we have discussed the genesis of symmetry operations, the space groups, the space groups coming from the point group operations and the 14 Bravais lattices in case of Karnes Matter crystallography. Now I will share with you some of the very interesting results so that you understand the importance of the technique and the capability of it to probe actual magnetic structures. So first I will talk to you about a compound which is known as Prussian blue analog done in our group, the author say here. Interestingly it is neutral diffraction. Now why I chose this because I have chosen iron compound because you know that iron is a ferromagnetic material as an element, it is well known but look at the structure that you obtain when you use a compound for iron and iron cyanide in this. So this is a FeCN64H2, this is a diffraction pattern. Now as I told you earlier when discussing the tutorial with Karvel here the data is taken at 50 K above the ordering temperature and then below the ordering temperature at 1.5 K. Now interestingly you see the peak intensity, the intensity is appearing in these places. So here this is a diffraction pattern which immediately tells you that certainly it is not just a ferromagnetic material because in case of ferromagnetic material the intensity is only boost the already existing bright peaks but you have got extra peaks here and this has been fitted by using the full prop suite by the authors, I have mentioned given the reference and you see the structure that you obtain, the microscopic structure. Here in this structure the magnetic unit cell is double of the normal unit cell because here in this compound there are two irons Fe1 and Fe2 and their magnetic moments are absolutely different. One has got a five Bohr magneton which is at 000 but the other one at half half half has got a magnetic moment of 0.8 plus minus 0.02 Bohr magneton. It is very much lower and but they are aligned ferromagnetically rather I should say that they are aligned parallelly with each other but the magnetic moments are different from one side to another and that is why the magnetic unit cells are double. So at the edge of this magnetic unit cells you can see these are the irons which have got high moment and this is the oxygen, sorry iron moment which is low. They are the low moments. So high moments and low moments these two together comprise the iron lattice and this is a unique result and this you can get by no other technique because here when I started with an iron lattice I started possibly I don't know the I don't know their starting point of the calculation or the optimization but ultimately at the end the results that fits best this diffraction pattern gives me two magnetic moments at two iron sides aligned parallelly. So magnetic unit cells are double and they have got a parallel structure that means just as we have in ferromagnetic structure but the local moments are different. I will take another example for you. This is a copper nickel MNSB. So here in this pattern this pattern has been fitted. So interestingly this is a pattern which gives us ferromagnetic as well as antiferromagnetic peaks. So this compound again starts with a very simple magnetic moment of nickel. Nickel we know that it's again a ferromagnetic but here with 0.54 magnet and for a nickel element but here the ordering in this compound is different. Here we have got the coexistence of ferromagnetic and antiferromagnetic order in this case. I just show you the fitted results. So here the antiferromagnetic spin structure comes from the orientation here. You can see that the crystal structure here and they are actually this octahedra FeO5 octahedra here but at the center you have the atoms here and you can see the spins which have been fitted and there is a 120 degree antiferromagnetic spin structure. When it is 120 degree oriented you always have a component which is antiferromagnetic and a component which is ferromagnetic. And in this case interestingly when you fit the crystal structure then we find that there is a distortion. This is you can see this is the magnetic moment versus temperature. So this is the phase transition point where it goes from order to disorder structure. This matches with the change in A and C parameters of the crystal lattice. So this magnetic phase transition is associated with the change in the volume of the unit cell. That means if you remember I was calculating Jij which is the exchange integral. So the exchange integral changes in such a manner that you get an antiferromagnetic order together with a change in the volume and A and C parameter of the cell. So basically the cell expands you can see at this point suddenly expands. A and C both have shown a tendency to expand and this is a very interesting result. So you have got a needle temperature of 130 degree for this compound. So I just discussed with you two examples because if one goes through the literature there will be thousands of papers where actually all the major neutron sources like ILL, Dronable or Oakree, SNS Oakree and also our own reactor Dhruva a very large number of research is going on magnetic materials and their microscopic structure not only because of our interest in the fundamentals of them because their macroscopic physical properties are of interest for magnetic memories and many other applications. So I will just briefly mention here to complete the magnetization measurements there is one more tool known as neutron depolarization technique. So neutron depolarization technique depends on precision of a polarized neutron beam around a field in the sample or rather the field in a domain. I will try to explain simply this but this is a transmission measurement so here I do use a polarized neutron beam let me just emphasize for all these experiments that I present to you we use unpolarized neutron beam but for depolarization experiments we need a polarized neutron beam we apply magnetic field in a sample there are domains inside the sample magnetic domains and they are aligned with an external applied field H and in the neutron path the domains force the neutron movement to undergo precision and this precision causes depolarization depolarization a loss of polarization of the neutron beam which we measure in this experiment and this is an excellent technique to get an estimate of the domain size this was introduced by Theoregg Welt at TELT to find out the domain size it's in 1970s and this has also been very successfully utilized in Dhruva time permits I will try to show you some of the results taken at Dhruva with this I come to an end for the lecture on magnetic neutron studies for understanding Karnes-Meter structure