 This lecture will have two modules. The first module will be concluding section for the neutral reflectometry part which is of specular reflectivity and then we will switch over to elastic scattering from elastic scattering. Till now all our discussions were on elastic scattering related to structure and now we will switch over to inelastic neutral scattering for dynamics in materials. So, this is how this lecture will have two modules. So, first module will have of specular neutral reflectivity. We had discussed specular neutral reflectometry for neutrons and x-rays and we found out that we can find out physical and magnetic structures, interface various interface properties. We can study growth kinetics and formation of alloys. One example was used on nickel aluminium material, nickel aluminium alloy in form AL3NI. We can also study in-situ growth when the source is strong and you have large number of neutrons you can do in-situ growth, in-situ studies and then the last part of this talk of this series is of specular neutral reflectometry which can quantify interface morphology physical and magnetic. So, what do you mean by of specular? First so far we have discussed reflectivity where the incident angle and the scattered angle they were same not scattered reflected angle theta incident and theta reflected they were same, but there can be cases or I can do studies where the incident angle is not equal to the reflected angle and in that case the q vector so far this was normal to the surface of the film. Now you have a q vector I will exaggerate it which is at an angle to the surface and I can have two components out of it I can split the q vector in two components which is one is qz let me draw it properly. So, this is the reflecting surface the angles are not same. So, still this is theta i this is theta r or I may be calling theta f also sometimes and this is the q value. Now this q value can be split into two components one is qz which we are familiar with, but we also have a component which is parallel to the plane we can call it qx here if xz is the reflecting plane. So, in this picture you can see I have shown the surface and I have shown it with respect to a one dimensional detector because in the instrument in drawer a one dimensional detector is used. So, in that you have one few channels because it cannot be in a perfect delta function peak to instrumental resolution. So, this channel a delta function reflection which follows snails law that means theta i equal to theta f on the detector will be broadened by instrumental resolution and give the specular peak, but beyond the specular peak we also have diffuse or off specular reflectivity to bring home the point what do you mean by off specular reflectivity. If you remember in olden days a movie projector used to project on a clock screen. Now that clock screen you could it is not a mirror for a mirror if we project the movie on a mirror you can only see it from certain angles, but when we projected it on a clock screen we could see the movie we are aware from any part of the movie theater because it was a diffuse scattering screen a rough screen which used to reflect diffusively. Here also because snails law is not valid we have reflected intensity beyond the specular peak or it is not a mirror like reflection and you have Q value which are got two components Qz and Qx which I explained just now. Now the fact is that in any scattering experiment we probe structure we probe structure along Q which was Qz for specular reflectivity. Now that we have a component which is along the surface plane then it will also be able to probe the surface structure in terms of I will show you just now high tide correlation function on the surface. So I will use one or two examples here this is a nickel film which was intentionally was taken through a corrosion process using sodium chloride and the image you see here one is before pitting corrosion and one is after pitting corrosion and we had done specular reflection from the surface and also of specular and I will show you what we get from these two since this pitting corrosion one is you can from the AFM picture you can see visibly the surface structure is modified and you can see granules appearing on the surface after the pitting corrosion. So how to quantify the surface morphology that is the question we can ask. So we can quantify the morphology with specular and non-specular reflectivity we know that surface roughness we have been characterizing with a Debye Waller like factor where this was this factor was put in front of RF to get the reflected intensity and this sigma square gave the average roughness of an interface not just the surface but of an interface in terms of this is the average fluctuation of the surface around a mean serve mean height. So this was sigma square this was done for specular reflectivity. Now let's see how we do the characterization of the corrosion for this nickel surface. So nickel film was around 800 angstrom thick so approximately 0.1 micron first passivated and then exposed to 0.05 molar sodium chloride we know that it causes corrosion first was that we did specular reflectivity and we know what we can get we can get the density profile interestingly here I show you two specular reflectivity patterns one is as deposited and one is after corrosion. Now one thing is clear you can see that the interface I mean the critical angle is slightly less because there is a loss of density because the part of the film has been eaten away and here the specular reflectivity profile was fitted with a six density six layer pattern the six layer model and you can see the layers each were of around 100 angstrom thick and with different scattering length densities because as we penetrated the film we find that the film goes to higher density and because the exposure to the corroding fluid was on the surface so it penetrated and you can see that this is the density profile so this is the air film interface so these are we have shown there as histograms so these models tell us that as we go inside how the corrosion has changed the scattering length density profile it has gone up and this is again the substrate side so there is a step over there also but from air to the air and the film interface the corrosion is there clear but now how to quantify how to quantify surface morphology one part is of course we can calculate the sigma for the surface and that is partial quantization but we can do a much finer work when we do a diffuse scattering now please look at this schematic so on the surface we have undulations like this and if I consider a mean or average surface below it then the height varies depending on the roughness of the interface now this is along the qx vector that we defined earlier for off-specular reflectometry here the reflected intensity is given as a Fourier transform over x and y e to the power minus i qx x qy y of a height height correlation function so this is qz is the average z value because your diffuse the broadening of the peak due to off-specular reflectivity happens at a certain value of z component of qz it is that this g x y is very interesting you see this g x y is given as two sigma square minus two z x y z zero zero so this is the height height correlation function in general for a two-dimensional surface height at x y with respect to an origin zero zero is known as height distance correlation function c x y given by this x y gives r which is x square plus y square if I have both the components so c of r at a distance from any point this origin has nothing very specific about it because we have an ensemble average so we can set the origin anywhere and look for this correlation function actually that is how the ensemble average is obtained so here it has got sigma square so this is basically two sigma square one minus c x y that's what this expression means and then this one minus c x y c x y is a correlation function you can see it decays with a distance r so it is sigma square into e to the power minus r by xi to the power two h this empirical formula we obtain for a fractal structure of the interface please know that here xi is a correlation length is a correlation length length on the surface and two h is a parameter where h is known as earth parameter earth parameter which actually gives the fractal dimension of the surface I will not repeat what I said earlier about fractal a fractal dimension basically where the surface if I try to measure the length it doesn't change linearly if we change the scale of measurement it follows other power laws and that is like India's coastline the actually the surface embedded by a tree and I took one example so basically if the surface roughness I will show you later in in actual figures the surface roughness is characterized by its fractal dimensionality and it can be quantified there the fractal dimension 3d equal to 3 minus h now it's a two-dimensional substrate if h equal to 1 then it that fractal dimension becomes it's not 3d it's d I'm sorry d so it becomes 2 that means it will be a flat two-dimensional surface if h is equal to 1 but often you find h is less than 1 the lesser than 1 if h goes to 0 then basically the embedded fractal surface though in it is supposed to be two-dimensional it reaches almost a three dimensional fractal dimensionality and as h is smaller and smaller it goes to more and more fractal dimensionality and as h becomes closer to 1 it is closer to two dimensionality so that means a smooth surface will have h equal to 1 and the rough surface is h has h has a value which is less than 1 with this much of introduction let me just mention to you that this is what we will be trying to fit to the experimental value using and by fitting a height-height correlation function in this Fourier transform so let us just I we did the experiment at Dhruva of this film which was taken through a pitting corrosion cycle and before corrosion after corrosion this is the qz peak around which we have collected the data this is actually because it's a one-dimensional PhD it is naturally we can collect the data over a range of qx we call it because we consider x qx as the deviation from specular reflectivity so in this data we have shown and of course you can see that the qx value is much smaller than the value of qz what we use there and that's why we can have a mean qz value and the tail which depends on the height-height correlation function by fitting these two before and after corrosion I will just give you the parameters that we obtained from the fits we fitted this for the height-height correlation function and what we found actually sigma equal to 20 angstrom and 14 angstrom 20 angstrom before and 22 angstrom after we had also compared it with the roughness or the height not the roughness actually the height-height correlation function that you obtained by using AFM this is AFM data so we also looked at the AFM data from the same film surface and because it is 2 sigma square into 1 minus cxy and which is an exponential function so as xy goes to infinity we get a saturation value it rises and then saturates please know this in log scale so this is what we expect to get when you measure the height-height correlation function from AFM data and this is what we get from the neutron of specular reflection from the surface the matching is excellent for example before corrosion the correlation length was 800 angstrom after corrosion this increased to 1200 angstrom the AFM says it was 650 and 950 comparable most importantly the fractality of the surface or the fractal dimension you can see that before corrosion it was 0.95 h has become smaller so because of pitting corrosion the fractality of the surface has increased or rather dimension has gone more towards three dimensionality because 0.95 is greater than 0.90 and this is and d is equal to 3 minus h so this one is closer to two dimension compared to this one and the matching between the two data are excellent but then the question might come if we can do the thing with AFM then why do of specular neutron reflectometry answer comes from the fact that neutrons penetrate deep with AFM in we can do the height-height correlation function determination only on the surface of a film whereas in neutron reflectometry we can carry out experiments for hidden interfaces also which is not possible by AFM that is what the advantage of this this study was more of a comparative nature and the comparisons show that neutron reflectometry data and AFM data had an excellent matching but now I will relate something which is possible only and only with neutrons and possibly this is a unique study which I am going to describe to you which is quite interesting for of specular neutron reflectometry so we will we attempt to measure magnetic roughness of an interface what do you mean by magnetic roughness basically if we have a magnetic interface magnetic film with an interface like this then a certain thickness of the surface this is a magnetically dead layer dead layer and this is the magnetic interface this is a physical interface so we attempted to do an experiment where we tried to get the height-height correlation function for the magnetic interface also obtained the height-height correlation function for the physical interface and compare these two and the nature of these two interfaces so here so here because we have a physical interface and the magnetic interface buried under it so we have now the scattering amplitude in terms of two height-height correlation functions please know just like earlier I wrote b coherent plus minus b magnetic when you describe the potential for a neutron specular neutron reflectometry here you please see the scattering amplitude is equal to a part which is due to structure structural interface and we have a part which depends on the magnetic scattering length otherwise same it depends on the nuclear scattering length or the physical scattering length we have here we have a magnetic height-height correlation function and this is a structural height-height correlation function square of this gives me d sigma by d omega where now squaring of this this gives me s of structure structure that means height-height correlation function here it will be z s of x y and z s of x prime and y prime the two heights I will be correlating using a height-height correlation function and a part which is magnetic that means the this is a z m x y z m x prime y prime so it will be z m x y z m x prime y prime so this will give height height so this is in terms of height this is in terms of height height correlation function because you will have when you square this you will have terms like z s x y minus z s x prime y prime and z m x y z m x prime y prime and an interference term which gives me b n b m and the height height correlation function between these two interfaces is the so it will be z m x y minus z s x prime y prime so how the height of the magnetic interface over here over here is correlated to the height of the physical interface at x prime y prime so that means we have three terms s s s m and z m minus z m x y minus z s x prime y prime s s s s m s surface structure and magnetic pre-multiplied by b n and b m it is quite straightforward from here to here so I can if there is no correlation between these two surfaces that means the magnetic surface and the physical surface there no there is no correlation the correlation is zero then this term will go to zero if we assume that so then you will get the height height correlation function of the structural surface and the magnetic surface or interface so now let's see what we obtained so this was done because you can see the magnetic part can be added or subtracted depending on the polarization of the neutron beam whether it is parallel or anti-parallel to the magnetization of the film so we what we did was p d nr means polarized polarized diffuse neutron reflectivity so in this case we have gone one step ahead and not only we are doing off-specular reflection we are doing off-specular reflection in the polarized mode that means my probing beam is up polarized and down polarized and this nickel film I had earlier also shown you it had was fitted with specular reflectivity data and we had a low density layer high density layer and a substrate and what we are looking at actually the interface here interface here actually there is not one interface but there is a physical interface and there is a magnetic interface below this and we try to fit that now this is the data log scale so that in log scale you can clearly see the distinct difference between the r plus and r minus data indicating that there is indeed off-specular reflection from this expression you can see that all the three three terms are present and there is a difference otherwise if this was not there then only for both of them I love s s s and s mm and then they will be identical since they are not identical that indicates that there is a difference and now we obtained the parameters of the interface as I said a b and c from the fits to the pd nr data so I will talk about the a interface please look at the interface which is of interest to us now here first what we fitted at the sigma value because we had i tight correlation function and if you remember it is 2 sigma square into 1 minus c s x y for 1 and 2 sigma so I will call it sigma s square to sigma m square 1 minus c m x y there are two correlation functions that I fitted now structural roughness was larger than magnetic roughness the average roughness the correlation length for this interface for the two surfaces this and this structural and magnetic the magnetic softener interface has a longer correlation length and the harsh parameter this most interesting for the structural surface the harsh parameter is much smaller compared to the harsh parameter for the magnetic interface that means the magnetic interface is much more smooth and closer to two dimension than the structural roughness it is much correlated over a much longer distance compared to the physical interface and the average roughness is also much smaller compared to the magnetic the magnetic interface has much smaller roughness compared to the physical interface so this actually justifies because we know that high tight correlation for a magnetic interface should be dependent on the dipolar inter dipole dip magnetic dipole dipole interaction and this is supposed to be long range and long range and the magnetic interface is supposed to be much smoother because of long range interaction and weak interaction between the two high tight parts unlike the air and the film interface so what we did actually from this we use a mid term displacement formula mid term displacement formula not formula displacement mid term displacement protocol to generate these two fractal surfaces and this is the physical interface that that to generate the two dimensional physical interface with these parameters these parameters and this is the magnetic interface with these parameters you can visibly see the distinction between the two interfaces the which one is which is physical one which is hidden behind it which is magnetic and so we can do polarized diffuse neutral reflectometry to obtain the difference between a magnetic interface and a physical interface possibly this is a unique result which has not been done in many other places with this I close this module the complete module on structural studies using neutrons and in the next part I will start inelastic neutron scattering for dynamics