 So now after introducing you to various magnetic interactions I am back to magnetic neutron diffraction. In many ways magnetic neutron diffraction is similar to X-ray diffraction. Let me start at the onset that often we do these experiments using powder crystals. Magnetism means a powder consists of small crystallites oriented in all possible directions. The magnetic neutron diffraction has an additional advantage that X-rays can give us structure, physical structure or chemical structure but magnetic neutron diffraction in addition gives us magnetic lattice. The only possible thing is the only technique with which magnetic crystallographic structures like ferromagnetism or antiferromagnetic structures have been determined. I will come to specific examples later. So magnetic scattering amplitude first the magnetic diffraction is due to the potential Vm. If you remember earlier when I was talking about nuclear scattering I talked about Vn which was a delta function and the nuclear potential. This is due to the unfilled orbitals and the magnetic moment which is there and due to unfilled shells that interacting with the applied magnetic village on the sample. So it has two parts actually. This mu n is magnetic moment of the neutron and it has got two parts as I wrote here spin an orbital. I will just give you the expressions because derivations will be out of scope at the moment. So ultimately I can define magnetic scattering length when there is a scattering length I must tell you that sigma of any scattering cross section is given by 4 pi into a scattering length square. So this V can be nuclear or V can be magnetic there I think I wrote as A magnetic. So I can define a scattering length associated with the magnetic scattering is given here which is actually in terms of centimeter of the order of 10 to the minus 12 centimeter. This is the mu n of the neutron and mu B is a Bohr magneton. This value you can see that 10 to the minus 12 centimeter so if I do 4 pi A square then it comes to about 10 to the minus 24 which is a one barn and square of this value and this is our in our experiment earlier I told you that I was looking at the Fourier transform of the nuclear density. Now I am looking at the Fourier transform of the unpaired electron spin density together with the Fourier transform of the nuclear density. So I get both of them when I do neutron diffraction for a magnetic sample while discussing monochromators I used this expression and I told you that if I do a vector diagram K is the momentum transfer and mu is the direction of the magnetic field then that I can define a vector Q which is basically the component of mu K sub mu dot K. So component of mu along K subtracted out from mu and this is Q if this magnetic moment is in plane or if the K vector is normal to the magnetic moment in the sample then this angle becomes 0 this angle becomes 90 degree and in that case your Q is equal to minus mu because mu dot K is 0 and magnitude of Q square is equal to 1. So this expression has got two components which I gave you earlier it has got a nuclear component and a magnetic component weighted by Q square often Q square is equal to 1 because the direction of the magnetic moment or the magnetic field is normal to the K vector or the magnetic momentum transfer. But most importantly when I talk about these components fn square is the square of the scattering amplitude which I gave you earlier if you remember I wrote down in case of nuclear scattering this V was delta r minus rj sum over j which was a nuclear potential so and this was e to the power i Q dot rj bj sum over j this the derivation was given by me for nuclear potentials present at the lattice sites. Now with that I also have an fm term fm square and fm square you can see apart from other term there is one f term what is this f actually this is the magnetic form factor now in case of nuclear potential the form factor is replaced by bj and I told you earlier also that bj if I consider this in Q space it is a continuous constant value if I consider Q versus plot of bj it is a constant value for all Q whereas in this case in the second case I have got magnetic form factor that means this is due to the unpaired electrons in the shells let us say it is a 3D electron in case of nickel, cobalt, iron the D group the D electron magnets which are known to us or 4F electrons in case of rare earth materials so that means in this case in this cases my spin is in a shell which is either 3D or 4F and it is a shell now the Fourier transform of this shell unlike bj which is constant for all Q this will tend to fall this we discussed earlier when we talked about extra diffraction because in extra diffraction which is due to the atomic charge cloud if you take a Fourier transform that also falls but the difference between these two is that here I consider the entire atomic charge cloud at a lattice side here I am considering only the shell which contains the unpaired electrons so this shell is in general larger average value of the R will be larger and so it will be falling faster with respect to Q values in case of magnetic materials so this is here and as I said if the sample is magnetized normal to the scattering mu dot k is 0 and Q is equal to minus mu and amplitude it is a unit vector and Q square is equal to 1 so now before I go forward there are two more terms e to the power minus 2w w if you remember that we talked about Debye-Waller factors this is the Debye-Waller factor because when I consider a crystallography structure whether magnetic or nuclear the entire electronic cloud and the whole object at a lattice side undergoing oscillations so this is the thermal oscillation and I showed you earlier that DeW is basically Q square and average value of R square so e to the power minus 2w is e to the power minus Q square and average value of this oscillation amplitude this gives us the thermal factor so this thermal factor again at any Q it will reduce the intensity and we can also evaluate from the reduced intensity what is the value of this so in case of x-rays it can be taken but this also part forms a part of the intensity so now the scattering for a magnetic crystal as I told you it has got two part the nuclear part and the magnetic part and the nuclear part is given by again I repeat that delta Q minus tau because for bright diffraction I discussed with you the evolved construction and this tau is equal to a reciprocal lattice vector so every time a diffraction appears depression peak appears then k minus k prime which is Q should be equal to a reciprocal lattice vector there I wrote G excuse my expression it is tau it is a reciprocal lattice vector and for the magnetic part all this remaining same there is a pre-factor and this structure factor form factor for the magnetic part which we have to accommodate so neutron diffraction for magnetic material as I showed you in this expression in this expression if the neutron beam is unpolarized then this this part will average out to zero and intensity for an unpolarized beam will be f n square plus q square f m square q square equal to 1 means f m square plus f n square so in case of unpolarized neutrons also we have got two parts in the intensity one is due to the nuclear part one is due to the magnetic part so we can determine the magnetic structure by fitting such a such a pattern so really speaking to find out magnetic structure in solids we need not polarize the neutron beam because one might think that since I am talking about spins which are aligned then I should also have a dressed or curtailed neutron beam where I take only one spin with respect to the spin which is there in the lattice but it is not necessary unpolarized beam as you can see from this expression I have got a nuclear part and also I have got a magnetic part and both can be fitted or taken out from a measured intensity only thing is that the magnetic part has this structure factor or form factor and of course I discussed with you already the q vector need not repeat that this is q so if the neutron beam is unpolarized basically the neutron beam polarization under for an aligned sample is basically a sum of i plus that means neutron beam and the magnetic spin they are parallel or neutron beam and the magnetic spin they are anti-parallel so it is half of that average of that i plus and i minus gives the unpolarized intensity which is nuclear scattering amplitude square and magnetic scattering amplitude square so we can find out the magnetic structure by fitting the intensity from both the sides so neutron beam is unpolarized of course there are instances where we use polarized neutron beam but in general for powder neutron diffraction for magnetic structure we don't need to polarize the neutron beam the polarization unpolarized beam here the interaction between nuclear and magnetic part the two f n dot f m part p dot q part that gets averaged out I'll just show the expression in this expression f m f n p dot q this averages out to zero for an unpolarized beam and we do have addition of these intensities separately coming from the magnetic part and the nuclear part let me just show you how well this can be done all over the world possibly in neutron magnetic diffraction is one of the major activities in the research reactors so but I show you the two instruments which are available at baba atomic research center which have been used extensively by my colleagues so both of them are based on position sensitive detectors the one instrument which is run by solid state physics division of baba atomic research center the q range is typically around because a point five to ten angstrom inverse we do need a resolution to resolve peaks which are close by this instrument has got a resolution of around point eight percent the other one is run by u gc d acsr which is an u gc center at baba atomic research center that runs this instrument which is a again a powder diffractometer with using position sensitive detector the q range is marginally different but has got slightly better resolving capability of nearby peaks because delta d by d which dictates the nearby peak is slightly better point three percent and we can you can see that the from the q max values quantum mechanics allows us to study historiographic structure because delta r dictates the value of this the distances the order of distances that we can measure using these instruments so I will discuss some pattern with some specific specific properties let me just bring to you a diffraction pattern of lanthanum calcium manganese oxide now two things please note that this pattern has been taken at 15 kelvin and 300 kelvin so this is a high temperature room temperature pattern this is the low temperature pattern the historiographic peaks which are possible are listed out here not that all of them you can see but many of them you can see here I just refer to because this is a fundamental reflection of ferromagnet so in a ferromagnet means the crystallographic structure ferromagnet is where a crystallographic structure I am just taking a very simple square lattice and the magnetic structures are same they have the same repeatability same repeatability so in this case that means that means here I will find crystallographic peaks and the magnetic peaks coinciding but please note that at 300 k you have this crystallographic peak which is also in magnet magnetic contribution nearly very small as we go to low temperature the magnetization of the sample improves and then the magnetic density increases and you can see it's from the rise in the peak intensity and this is a bright diffraction peak crystallographic peak but with the addition it is 0 to 2 peak and 2 to 0 they are degenerate peaks but with the addition of the magnetic intensity you can see the from 300 k to 15 k the intensity has increased so this increase is due to increase in the magnetic intensity reason being for a magnetic material if you consider the moment versus temperature it undergoes a second order phase that means as you go to lower and lower temperature the moment increases with lower temperature and that's what is indicative of the pattern here taken at 15 k and 300 k another interesting thing I want you to note that the magnetic peaks they come at low values of q this is being due to the fact that there is a magnetic form factor working as a multiplicative unit with respect to the bright peak intensity so things added to this is if your bright peak is a delta function then there's instrumental resolution which has been added on to this then convoluted with this then there is a background which has added on to this plus this magnetic form factor also one needs to take care of and that's why the magnetic peaks in magnetic neutron diffraction appear at relatively lower angles compared to the high angle peaks and high angle peaks we are not fine we will not find much of magnetic intensity now now I'll come to the actually the refinement technique we use for magnetic structures in this regard I must tell you that we had a course done in 2020-21 on neutrons for goodness matter which has been recorded and kept on site and regarding magnetic neutron diffraction professor SM Yusuf Yusuf and professor Anil Jain they did a detailed description of the technique and specially tutorials taken by professor Anil Jain on fitting of magnetic structure using Reedwell fitting program because this is a detailed program and it needs a large number of tutorials in this case I am just I request you in case some of you are keen so lecture in this course on neutrons for goodness matter lecture 13 to 21 it deals with magnetic neutron diffraction and there are a large number of tutorials anyone of you interested to learn this technique in greater details for your samples I will request you to use this this lectures are available in youtube youtube under hbni hbni and also can be downloaded from hbni homepage hbni homepage here in this course I will briefly introduce you to the full proof or the fitting refinement techniques for magnetic neutron diffraction