 So, let me remind you, when we did without polarization analysis, we had two component potential v plus and v minus. And I also showed you two Schrodinger wave equations for the two polarization plus and minus. Now, with one step ahead, now I have got a four component potential. That means there is a potential for non-spin flip, that means plus plus minus minus and two potential for spin flip, plus minus and minus plus. Now, just compare this with the two component potential, now I have got four components. Why four components? Now, let me quickly justify to physics wise. If you imagine there is a sample in plane, magnetization direction is this, now suppose the in plane magnetization is not parallel with the magnetic field, but at an angle. Let us say this is the, so at an angle gamma, this is the magnetization. Now, let us consider this is the neutron spin, this is the angle gamma and this is the angle m in plane that it makes with the applied field, this is the applied field direction. Now, this m I can resolve into two components. One is m cos gamma and one is m sin gamma and this m, this m comes from magnetic scattering length Bm. So, Vm can be written in terms of that we know, but now because it is at an angle there are two components, one is m cos gamma, one is m sin gamma, now take this m cos gamma, this is of magnetic origin, it does get added with the nuclear potential B coherent, but see this is the spin of the neutron either up or another experiment either down and this is either parallel or anti-parallel to this direction of this component of the magnetic moment and because they are parallel, this magnetic field will not cause any spin flip for the spin. So, that means m cos gamma cos gamma will not give any, it is a non-spin flip component and consider m sin gamma, now this is the spin direction for the neutron and this is m sin gamma, this allows precision of the magnetic moment around m sin gamma and this causes spin flip for the reflected beam, so m sin gamma is the one which will cause spin flip. So, now in this four component potential, now please check that in this four component potential, I have got V plus plus is rho B coherent plus B y that means m cos gamma from this plot and this is twice pi h square by m rho B coherent plus B m cos gamma, V minus minus is just this sin goes negative as I explained just now, but now we have got two components of potential which is either plus minus that means up goes to down or down goes to up given by the normal component of the spin and this is this spin flip component is purely magnetic of origin, spin flip component is purely magnetic in origin, why I mentioned this? Because you can see the non-spin flip part has nuclear as well as magnetic either plus or minus, so this is similar to what I showed you for V plus and V minus, but here either spin flip potential it has its origin only in the normal component of the magnetic moment and it's purely magnetic in origin, so now I can write down the Fresnel reflectivity earlier if you remember I wrote q1 minus q2 upon q1 plus q2 going from one medium to another it is just q1 I have written qz and q plus minus is given by when qz is in air it has got two parts nuclear part and the magnetic part and qm square again it is given by I can talk about ky square kx square and kx square plus ky square equal to I can write km square and qm square is equal to 2km qm equal to 2km, so I can write it in terms of qz qn qz is the in air the value of qz without any refraction effects this was qz for the beam in air, but this is q plus and q minus of the propagation vector in the medium which depends on nuclear as well as magnetic part, so now still we will write down the Paris formalism as I told you for psi plus and psi minus their continuity and the continuity of their differentiation and now the there will be components of psi plus, so we have to add in this continuity of psi plus psi plus we will have psi plus as well as r minus plus into psi in the medium because now we also have to consider in a parrot formalism if you remember I consider continuity at n minus 1 and nth layer and n plus 1th layer came in the recursion relation, but here when I talk about psi plus here I also have a reflected component of psi in the medium, so now our parrot formalism the continuity equation will have one extra term which I am not writing over here, but that is straightforward, so thus again let me highlight the observed non-spin flip intensity gives nuclear scattering length density and m cos gamma and the spin film component gives only m sin gamma and from these two components we can reveal the magnetic moment as m equal to vector sum of m cos gamma and m sin gamma, so I can say m is equal to mx plus my which is basically m square equal to m square cos square gamma plus m square sin square gamma, so we can get the value of m if I can determine m cos gamma and sin gamma basically we have to determine the angle at which the magnetic moment is oriented from the spin flip intensity, so I will use only one example over here which is a cobalt gadolinium multilayer, so this is the typical structure or model structure for this we have gadolinium and cobalt of 140 angstrom and 80 angstrom thickness 8 bilayers on silicon, both of them are ferromagnetic cobalt of course magnetic at room temperature it has got a QD temperature around 1400 Kelvin, gadolinium is ferromagnetic just below room temperature around 290 Kelvin. Here in this example that I have chosen what we observed actually is a very interesting structure of structure in cobalt gadolinium multilayer here first again the additional informations from other measurements one is XRD which says that it is a polycrystalline sample XRR gives me the scattering length density or the physical density as a function of depth and magnetic hysteresis loop hysteresis loop. Now this magnetic hysteresis loop interestingly I just show you the hysteresis loop at 5 Kelvin under a field cooling field of 500 hot state plus and 500 hot state minus you can see the shift in the hysteresis loop we know that the hysteresis loop should look like this this is m versus h so initially when you start applying the field the thing gets magnetized the thing gets magnetized to saturation magnetization when I reduce it it goes to remnant magnetic moment then cohesive field and then the loop from the positive to negative loop it goes to the same one and goes but if there is a antiferromagnetic coupling at the interface then the coupling needs to be broken in the hysteresis loop and that shows as known as this antiferromagnetic coupling this indicates a shift in the hysteresis loop either this way or the other way when the cohesive field in the positive and negative directions are not same because in one direction you have to apply higher field to break open and that's what is shown here that that means there is an antiferromagnetic coupling at the interfaces that we can try to guess from the magnetic hysteresis loop in a sample in our gadolinium cobalt sample another interesting thing is that gadolinium and cobalt both of them are strong neutron absorbers so this experiment was really difficult to make now this experiment was done at off-spec spectrometer at icis I will just show the results so at 300 Kelvin first the sample was ferromagnetic cobalt gadolinium was non-magnetic so at room temperature cobalt show should show a ferromagnetic nature and gadolinium is non-magnetic so when you plot the room temperature magnetic scattering length data along with the nuclear scattering data it looks like this so nsld is from the left side and magnetic scattering length density has been plotted on the right side so here it is there is no analysis of the reflected beam just the magnetic moment density profile we tried to find out and the red curve you can see the red curve here for gadolinium wherever it comes to gadolinium layer the magnetic moment density is zero and whenever we go to cobalt layer I can see a positive magnetic moment scattering length density so this is at 300 Kelvin now let us go down to 200 Kelvin when you go to 200 Kelvin you can I can assure you that there was a shift in the hysteresis loop and then what we find here that at 200 Kelvin the gadolinium and the cobalt the antiferromagnetically coupled at the interface understood but in this plot I have done r plus r minus and here it is so what I plotted is r plus plus r minus minus non spin flips for up and down non spin flip plots for up and down neutrons and this is how they looked at 300 k this is how they looked at 200 k but please pay attention to this part this is the spin flip what we have done actually I have plotted r plus minus r minus plus average spin flip here still the spin flip intensity with the given model of gadolinium and cobalt one antiferromagnetic to the other is very small there is not much of spin flip that means if this is the magnetic moment direction which is 500 gauss in this case the cobalt is aligned almost along it maybe few degrees this way that way and the gadolinium is aligned opposite to it so I have a cobalt layer a cobalt layer and a gadolinium layer so at the interface the antiferromagnetically coupled which makes it the hysteresis loop shift here this 200 k structure is between two ferromagnets which are antiferromagnetically coupled and the spin flip intensity spin flip intensity at 200 k is almost zero that means there is no normal company now let's go down further we have gone down to 125 k here you please know these are the non-spin flip intensities there is a spin flip intensity now the spin flip intensity is increasing and increasing because now your magnetic moment if this is the applied field is having an angle with respect to the applied field which can be broken up into two components two components so magnetic field has magnetic moment not the magnetic field magnetic moment has two components one along the field which is the cause for this non-spin flip reflectivity and it has got a normal component which causes the spin flip and we can fit these two and now please look at this structure which you obtain at 125 k we get something like a two-dimensional domain wall structure that means if I consider this is a gadolinium layer and this is a cobalt layer so one is that this is rotating actually it is I am not able to show it three dimensionally it keeps rotating and ultimately rotates and does a two-pie rotation inside the layer that's why I call it this layer behaves like a domain wall 2d domain wall domain domain wall and in case of cobalt it is again antiferous magnetic couple then it goes down rotates around the tree and then goes down sorry I am sorry because it has to go to two pi I have by mistake I have made it pi so it rotates and finally it comes back to the same orientation here here also that means I made a mistake if it's if I start from here it should be down then it keeps rotating two pi and then finally comes back as down so these are two pi domain wall structure and so this is what we show here and actually you can see in this plot that the magnetic moment direction has been plotted as a function of the thickness as you go from the gadolinium cobalt to gadolinium layer this is the in plane direction of the magnetic field which is 500 oz and you can see the rotation a helix getting created in this cobalt gadolinium structure this is a very interesting result in which by polarization analysis you can see below 125 you have got this component and I have fitted this and after fitting the spin flip and the non-spin flip components we decipher this structure a helix which is forming in this cobalt gadolinium multilayer it's a very interesting structure and possibly this is the only technique which can determine this structure without destroying the sample experimented so this is a two-dimensional domain wall and in case of 5k data we see the structure only in the gadolinium and the entire cobalt layer remains anti ferromagnetic coupled to the gadolinium layer so with this helix we see only in the gadolinium layer and we don't see the helix in the cobalt layer so that means 300 k cobalt is ferromagnetic 200 k both of them are ferromagnetic but coupled anti ferromagnetically at the interface that's why in both these cases we have only non-spin flip data then at 125 k we have got the helix in the cobalt layer as well as in the gadolinium layer in 5k we have the helix in the gadolinium layer and not in the cobalt layer this result shows the strength of neutron reflectometry polarized neutron reflectometry with polarization analysis and with this I stop here