 In this lecture we will continue with the techniques of Neutral Reflectometry and before we get to any experimental data or examples, it is important that I discuss the way we handle the data, the way we fit them, the way we evaluate the reflectivity pattern for a given model structure. So with this, I continue and so earlier I showed you that if you consider the continuity of the wave function and its wave function and its derivative at an interface for a single infinite medium, it is an infinite medium, so then the reflectivity comes the way I derived earlier. I will just repeat it quickly. So if the incident beam is of unit amplitude and e to the power ikz, then the reflected beam since it is going opposite direction in qz it becomes reflectivity r e to the power minus i, sorry it is qz, qz, iqz. So this is the forward moving part, this is the reflected part which have the phase factor abroad back to the interface and I have got a transmitted part which is transmission coefficient, transmission amplitude into e to the power i q prime or q2z which is q2 is the wave vector transfer in the medium and then outside the medium I can call it q1. So these two equations they give me from the continuity of the wave function, you can see at z equal to 0, I consider the interface in z equal to 0, it is 1 plus r which is equal to t which is the continuity of psi gives me, t at z equal to 0 is a wave function in the medium and 1 plus r the incident beam and its reflected component is the psi of the wave. Similarly q1 if I differentiate it to get d psi by dz, q1 into 1 minus r will give me q2 t, just by dividing equation 1, 1 and 2 I can find out the reflection amplitude is q1 minus q2 upon q1 plus q2 and that gives me in terms of angle gives me sine theta because q1 equal to 4 pi by lambda sine theta and q2 is nothing but square root of n square minus cos square theta. So this gives me in terms of angle sine theta minus n square minus cos square theta under square root and sine theta plus square root of n square minus cos square theta in which q is the momentum transfer. So then we will always like all other diffraction experiment that we did, we will be getting the reflectivity pattern in q space actually we plot the reflectivity as a function of either angle or q momentum transfer and from there we have to obtain the structure in real space. So the task the general part of the task is this which we have done for all other diffraction patterns. So continuity of wave function and its amplitudes I will use it later again in this lecture and then. So now n equal to cos theta and I will take the two cases one is that when theta c is less than theta that means when theta c is less than theta then you can see n sorry not n reflection amplitude r equal to a I mean this will be theta theta minus square root of theta c square minus theta square when theta is less than theta c this is an imaginary number so r becomes in the form of a minus i beta upon a plus i beta and then the reflection intensity reflected intensity r into r star this is equal to 1 and this is one part this is basically it signifies the total external reflection external reflection from the interface depending on and because this theta c is dictated by the density in the medium so dictated by the medium properties this is up to critical angle intensity reflected is equal to 1 and then when theta is much much greater than theta c then I can write this as theta minus theta square root 1 minus theta c square by theta square and this I can write as theta minus 1 minus theta c square by theta to the power half and then this comes to if I expand it theta minus theta into 1 minus theta c square by 2 theta square theta square and this goes to theta c square by 2 theta square and similarly you can do the expansion and then when theta is much much greater than theta c I have got this region where I have got what is known as Fresnel reflectivity falling as q to the 1 by q to the power 4 that means if the q is doubled your intensity falls by 2 to the power 4 16 times so this it has written in terms of qc because this theta c upon 16 theta square theta theta square was r if I square it it will be theta c to the power 4 divided by this was 4 I am sorry this was 4 there 4 theta square it will be 16 theta to the power 4 and I can write qc is equal to 4 pi by lambda sin theta c under small angle of approximation that is justified because even up to up to q is equal to 0.3 angstrom inverse the angle theta will be somewhere around 3 degrees so I can use sin theta equal to theta approximation so it is 4 pi by lambda theta c and q is equal to 4 pi by lambda theta and then I get this equation in terms of q so this is important so I have when theta less than theta c I have reflectivity is equal to 1 this is what I showed you previous day and now it explains when theta is much much greater than theta c then this is the fall and it is q to the power minus 4 now at this point let me point it out to you then because reflectivity falls so rapidly with q and we want to fit the reflectivity data over a reasonably large q range to see this we plot the reflectivity in log scale and our fits are also in log scale because if I plot it in linear scale the intensity in this region will be all is sensible minus 4 compared to 1 and they will give very little weighty when I am trying to fit the data because they are almost in the background in that linear scale and that's why we don't need linear scale we use log scale for plot of reflectivity always all the data that I will be showing you you will find mostly it's in log scale or sometimes they multiply it by q to the power 4 to normalize with respect to final reflectivity so it will be q to the power 4 and then ln of intensity that is also done often but in general always we plot it in the log scale to bring the weightage of the points across the reflectivity profile to similar values so this is the final reflectivity for a single layer but the thing is this is this calculation has been done assuming that the interface is absolutely flat but now let us look at a real interface there is roughness at the interface and I wanted to show you this see this is the AFM picture of a nickel film so we can see that this is not flat and actually you can see if you see the height here is a three-dimensional picture it's shown in nanometers so you can see this height is a scale of four nanometers it can go up to 40 angstroms so the interface is not flat in general and rather it has got a structure like this which is for fluctuating height so in this fluctuating height now I can assume a mean height a mean height a mean height and a fluctuation around that a Gaussian fluctuation then you please see that I can talk about this as a Gaussian whose full width at half maxima sigma is a signature of this fluctuation or sigma is a roughness we call it the roughness but this length scale has to be much less than the linear resolution cube or the coherence length of the incident radiation because the coherence dictates up to what distance two rays will interfere and this length scale of fluctuation has to be smaller than that to invoke this picture and I call it divide wall or like I will remind you that we got a similar parameter when it talked about thermal agitation if you remember I talked about thermal agitation of a crystal lattice they grow larger as the thermal agitation takes place and we wrote down I is equal to I 0 0 degree Kelvin by minus q square u square by 3 u square by 3 because it was actually it was cos square theta for a sphere is equal to one third the average gives me one third actually I got so this expression q square u square by 3 also we used it in the guinea region in small angle in sands if you remember I wrote I is equal to again some I 0 into the power minus q square Rg square by 3 where Rg was the radius of gyration of the object and its form factor gives me this so whenever we have fluctuation around a mean here Rg basically is the length scale over which the object is there you can find out the distance and then the Rg gives the radius of gyration in case of actual dynamic devourer factor the time average picture gives me u square by 3 because the atom is now oscillating in all possible directions in all possible directions and that oscillation has an amplitude of u square by 3 and now here we have this roughness parameter for so this is a static case where the there is a fluctuation around a mean value and that's why I call it Debye Waller like it is not Debye Waller it's a Debye Waller like factor and we put it by hand we don't derive it I have derived the Fresnel reflectivity and this is a Debye Waller like factor given there is a roughness of the surface so this roughness parameter also one obtains from the reflectometry data and I will show you so let me go quickly tell you so I talked to you about a medium whose potential offered to the neutron beam passing is this given by twice by h square by m b coherent row this has come from the coherent scattering length density let me write down coherent scattering length density but together with it so I have a coherent scattering length density b row if the system is non-magnetic if I have a magnetic or magnetized medium then there is a magnetic scattering length density magnetic scattering length density city call it bm so now my potential will have twice by h square by m you have I will call it b coherent as b nuclear which is offered by the nuclear potential which are actually sum over all the scatterers in a unit volume that's why row comes into picture and we are talking terms of density but we also have a magnetic scattering length which is either added or subtracted from the nuclear scattering length depending on the magnetization in the medium and the magnetization of the neutron whether they're parallel or anti-parallel so now this medium the magnetized medium offers a potential which is either more or less depending on the polarization of the neutron and polarization of the medium this is the beauty of neutron scattering you don't see this in x-rays so that's why now I show here this was earlier what I stated this was from unpolarized medium unpolarized neutron reflectometry this is the correct picture but when you have a magnetized medium and a polarized neutron you can see that if this is the nuclear potential either you have plus v magnetic or you have minus v magnetic and then we have v equal to v nuclear plus v magnetic for spin up neutrons and v minus v nuclear minus v magnetic for the spin down neutrons and the potential energy is given as a sum of the two so this gets modified to this when your magnetized medium so now and I have discussed earlier the specular reflectivity when incident beam makes the same angle as the reflected beam that the snails law and you have a transmitted media so I have got a specular reflectivity and then the specular reflectivity critical angle since the v is different for up and down neutrons so now for polarized beams my critical angles will be different so if I try to plot let's say this is if it is unpolarized neutron intensity versus q plot then I can say that for polarized beam for up spin we will have one reflected intensity for down spin we will have another reflected density and this difference depends on the magnetic moment density in the medium so let me just show some examples so now consolidating roughness and magnetic potential I have shown you a nickel film on glass it's the data taken in our instrumented drover you can see that it has got a flat edge normally for any reflectivity experiment this intensity is taken as one and then we have the fall in the reflectivity profile and this follows q to the power minus four plot so this is typical is an experimental data which matches nicely with the frontal reflectivity profile and here I have plotted or simulated reflectivity profile from a nickel fire the same nickel film but it is simulated so first there are two different polarizations r plus and r minus you can clearly see the difference here in the critical angle of the two beams the the way the experiment is done you have a polarized beam so the way the experiments are done this is your sample surface this is a sample surface it is magnetized in some direction you have a neutron beam incident on it and reflected now this neutron beam incident neutron beam has a spin either parallel or anti-parallel with respect to the sample magnetization and that gives me what I showed you just now that they are the two r plus and r minus over and above the difference in critical angle this has been simulated for two roughnesses interface roughness of one angstrom and interface roughness of 15 angstrom now you can see the effect of this uh Debye Waller factor here you can see even for the up up up polarized neutron as you go to larger and larger q the reflectivity falls much faster because of e to the power minus q square sigma square sigma is a interface roughness and q square is a q value in this measurement same is true for the down's neutrons up to some point they follow the same intensity but then the roughness of one angstrom falls much less rapidly than roughness of 15 angstrom so these show us that from our experiments I can find out the roughness parameters from the critical angle I can find out the density of the film and from the so from and now comes if the film is of finite thickness so this I will attempt in the next part of this talk