 So, we are in the last leg of inelastic neutron scattering I was discussing stochastic motion the possible instruments and the experiments that have been done. I was talking about quasi-elastic neutron scattering I will give a few more examples in that then I will explain to you spectrometer known as pin-eco spectrometer which can be used for understanding dynamics at possibly that is the slowest dynamics of the longest time scale of 10 to the power minus 9 second that is the nanosecond range and in the second part of the talk today I will summarize whatever I did so far in this course because it is important to tell you exactly how to go about your experiments and what analysis we have been doing regarding neutron scattering technique. So, coming back to stochastic motions using quasi-elastic neutron scattering if you remember last time I discussed the dynamics of propylene this molecule in sodium ZSM zero light which has got this kind of channels in it and as I mentioned to you earlier also that often you have competing time scales or time scales which are quite different from each other in the same system in this system there are translational motions translation of the molecules and rotations the rotational motions are much faster almost a factor of 10 smaller time scale. So, I should say 10 to the power minus 1 factor of 10 smaller time scale compared to translation and translation is slower. So, the experiment was done in two different instruments one is the Marx mode quasi-elastic neutron scattering here the resolution is around 200 micro electron volt. So, around 0.2 milli electron volts and here this was around 3 milli electron volts. So, the resolution here is narrower by a factor of approximately 10 3 milli electron volts and 0.2 milli electron volts. So, the translation which is a which is slower of the two translation was observed or was detected the diffusion constant using the instrument known as Marx QVNS because when I talk about two different time scales as I mentioned to earlier that it is possible that both of them are quasi-elastic, but one is a narrower component and one is a broader component. The narrower component is associated with slower dynamics slower dynamics slower dynamics slower dynamics makes it the time scales longer and so you see it in a narrow one because the if you remember that the Lorentzian has a 1 by tau square width. So, where tau is the time scale associated with the dynamics. So, for translation since it is slower the Lorentzian is narrower and here in this case because the rotation is much faster. So, it is broader part it may be taken as a background for the experiment in which I measure the slower translational dynamics and the I this this quasi elastic broadening represents the translational dynamics. On the other hand when I go to a instrument which has got a poorer resolution that is 3 milli electron volt compared to 0.2 milli electron volts in case of the slower dynamics in that case this part possibly will become like a delta function and I can measure the width over here that in the present case will you know talk involve rotation. So, what I wanted to emphasize or bring home that to do or to measure various time scales we need instrument with various different resolutions and I discuss this with respect to two instruments triple axis triple axis and q in s instrument in Dhruva, but I will just use an example where you use other bigger instruments for example this is the iris quasi elastic neutron scattering instruments at the Rutherford-Eppelton laboratory this works on time of flight principle. So, there is a long flight path from the source to the sample after the sample interestingly you will find that there are mica analyzers and graphite analyzers, but the beam goes to the mica analyzer and comes back to the detector. So, this is almost a 180 degree reflection. So, theta is around 90 degree and we know cot theta gives the best resolution when theta is around 90 degree 90 degree or 2 theta close to 180 degree. So, both the analyzers are at back scattering geometry and these are the analyzers curved on a circle and the detector bank is at the center. So, that means this is a vertical geometry. So, the sample is here the reflected beam comes back vertically to the detector. So, this the displacement between the sample and the detector bank is vertical. So, the detector is here and the analyzer is I am sorry the analyzer is there the sample is over here. So, it goes to the analyzer and comes back to the detector. So, it is not only that there is a horizontal angle which is measured on this detector bank and it gives you the q value, but also there is a vertical displacement. So, it is neutron which is reflected back I mean scattered back not reflected scattered which to a vertically displaced the analyzer which sends it back to the detector. And here so, in this kind of sources because we need low energy neutrons it is acting on a hydrogen cold source. So, there is a nearly 20 Kelvin hydrogen cold source which shifts the spectrum as I explained to you earlier that your thermal spectrum is shifted in energy if you take a cold source and you have better intensity at long wavelength neutrons and you get more neutrons over here. So, the geometry is such that you have better resolution because the analyzer is in a back scattering geometry you have higher intensity of long wavelength or low energy or cold neutrons. So, this is a high flux time of flight instrument which is there in a spallation neutron source. So, I will just quickly show you the result, but interestingly this result comes from experiment at three places. So, it has quasi-elastic neutrons scattering was done at ion 6 ILL Grenoble with an energy resolution of 85 microelectron volt. It has been done at ISIS where the result half width and half maxima was around 17.5 microelectron volt a factor of 5 down it was done at ion 6 in Grenoble where the resolution was around 60 microelectron volt and also it was done at TOF TOF at FRM 2 with a similar 85 microelectron volt resolution. This is the quasi-elastic scattering where the experimental data has been fitted with a quasi-elastic part and an elastic part because we know that if it is if it is a diffusion over infinite length then there is a delta omega term and which is possible only at q equal to 0 because it has go with the the hydrogen that you are discussing is diffusing to infinity. If it is a finite geometry diffusion then always we have a delta omega with a q dependent form factor like term which tells us the geometry of the diffusing cage or medium and plus you have also the Lorentzian in energy which gives you the poise elastic broadening referring and getting back to which takes us back to the diffusion constant that we can find. So, in this case the elastic elastic peak is shown and also the quasi-elastic fitted peak and then the full width at half maxima are plotted you can see here, but the highlight of this actually for one sample that is which is the attempt to find out the dynamics of water in onidensis bacteria experiments were done at three different sources with three different resolutions. In addition the experimentalists used a hydrogen deuterium contrast in case of quasi-elastic neutron scattering we know that hydrogen has largest incoherent incoherent scattering cross section which is nearly 80 burns we are aware of that part. So, if we replace hydrogen with deuterium in parts of the bacterial cell then we can highlight the diffusion of hydrogen because for incoherent q omega it is the self correlation it is the self correlation and for measurement of this we need something which has got large incoherent incoherent scattering in this case hydrogen. So, similar to small angle neutron scattering was sans here also they had used contrast by selectively deuterating a part of the bacteria and this data is basically for everything which is inside the bacterial cell, but it was most importantly this experiment was done to look at the dynamics of hydrogens or water in the cycloplasm of the cycloplasm of the cell. So, such kind of very slow dynamics in biological systems I mentioned to you earlier can be studied using quasi-elastic neutron scattering at various sources I had given examples earlier from our experience at droga and this is a combination of three experiments one at iris the which is at Rutherford-Aperton laboratory ILL I am sorry in RL iron 6 quasi-elastic neutron spectrometer at ILL granable with slightly higher resolution. So, which will measure the broader part of the spectrum of the broader part of the translational spectrum and also at top top at FRM 2 and this is a combination of data data is about full width at half maxima, but this data refer to different kind of motions the narrower resolution or the better resolution data will involve will rather will will describe the slower motion the and the faster motions will describe the full width at half maxima at faster rate and this is what I wanted to show you regarding quasi-elastic neutron scattering from various samples. So, this was an example taken from biology which is quite common these days and people come to understand dynamics possibly neutron quasi-elastic neutron scattering is the only tool that can give you such slow dynamics in biological systems and its role is immensely important to understand structure I showed you earlier the structure at mesoscopic length scale using sands and in this case dynamics inside a cell the cytoplasmic dynamics I want to bring you to another spectrometer which is a very interesting spectrometer known as spin eco spectrometer. So, how do you measure inelastic neutron scattering using a spin eco spectrometer here I am just showing you the neutron spin first you polarize a neutron beam possibly by reflecting with a super mirror and then you have a pi by 2 flipper so because this is reflected from super mirror a super mirror maybe polarization in this direction the reflected beam has this is the polarization direction which is parallel to the polarization in the super mirror plane so that means if I have the beam coming it has got a polarization which is along the beam now there is a pi by 2 flipper here this is flippers I have discussed with you earlier they are basically combination of magnetic fields so this pi by 2 flipper flips it to 90 degree because I have got a coil this is a coil actually block diagram of a coil through which the neutron passes the field is in this direction see if the field is in this direction and the neutron spin is normal to it then as the neutron travels through the coil it preceses it preceses around the field it's called Larmor precision all of you are aware that this magnetic field forces a spin to precese if it is normal to it after that I have got a pi flipper the pi flipper I will come to it shortly pi flipper basically flips the polarization of the neutron by an angle pi and then there is a precision coil in which the neutron preceses opposite direction and after that again there is a pi by 2 flipper and energy analyzer and the detector so what it does so basically what it does is this now first if I the neutron with a single spin then it preceses preceses reaches the sample gets scattered by the sample and then if I precese it in the opposite direction then if there is no energy exchange then this neutron spin will come out with the same spin over here so after that so it was a pi by 2 flipper spin takes it to a spin in this direction then again a pi by 2 flipper takes the spin in this direction and in case of spin echo that means if there is no energy transfer it comes out with the same polarization now let us consider the case that there is an energy exchange between this neutron and the sample so now if this is e1 this is e2 if the velocity is v1 and the velocity is v2 then v1 is not equal to v2 in case of energy exchange when such an energy exchange takes place these two parts the coil before the sample and the coil after the sample they are identical so that means in case of no energy exchange there is an eco condition it comes out with the same polarization but now in this case if there is an energy exchange then the time spent time spent in this coil is different because the velocity is different if it is a slower velocity it will spend longer time if it is a higher velocity or higher energy then it will spend shorter time and this precision will in the I mean the precision PRE CE precision angle will not be same after the sample and it will come out with a direction which is different from this different from spin echo condition and this can tag the energy transfer now what is done in reality that first you use a velocity selector velocity selector is a monochromator which can give large energies broad energy spectrum so poorer resolution typically around 15 percent broadening so we have got a neutron beam which is passing through the first coil I will show it as a block diagram so now after pi by two flipper after there is a neutron coming in this direction it has been flipped by pi by two and now it is going through the first coil first coil before the sample since there is a velocity spread so now if I consider looking at the neutron from this direction now it will fan out so it will fan out because neutrons of different velocity will undergo different precision this precision can be as well as 20,000 10,000 precision inside this length of the coil now it has fanned out there is a pi flipper there is a pi flipper in my schematic there is a pi flipper so it flips by pi so here now it has fanned out let me just change the color a little bit the faster possibly is here and the slower possibly is here because of faning out the faster neutron has this angle and the slower neutron has this angle in an average this is the angle of precision now when I flip it by pi then they interchange now this faster by pi flipping will go there if you consider this fastest and the slower will go to the other direction in the other direction and then I have once I flipped it now if I took it through the second coil through the second coil without any energy exchange I should have gone back to the same velocity distribution but now for inelastic this will be different and what I find is a counts versus phase current because you keep changing the phase current so this is the counts in up and in down neutron you can see depending on the intensity and precision you have got an oscillatory input in average is basically this dot line and this in average is up and down this is for the incoming and this is for the outgoing now that means this distribution if I can measure it in a detector using an analyzer then I can find out what is the intensity of these neutrons and ultimately I can transfer it into s of q omega d omega how it is here I will just quickly this is a spin nico instrument at nist this is the arms on which the first term and the second arm and this angle dictates what is the q value of this instrument so now let me just quickly tell you the algebra of this process so phi one is a precision angle if the velocity is v then l by v is a time it spins and gamma is a gyromagnetic ratio for neutrons then gamma v l by v this gamma b t basically is the precision that undergoes a neutron for spending time t in the field path in the field v into l now this velocity has a white distribution and now if the outgoing velocity is v2 then that two neutrons they get defaced with respect to each other because of different precision angle and the defacing I can write down as gamma v l one by v one minus one by v two for this phi difference they're exactly same in resonance and for that I can write in terms of energy exchange h cross omega so I multiply that the numerator by h cross omega and the denominator by half m v1 square which is the energy of the neutron so e equal to h cross omega in elastic condition equal to half m v1 square so I get this expression the the phase phi due to precision in radiance is given by this expression this is nothing but t omega now what we do actually after the after the instrument after the instrument we have got a pi by 2 flipper so that flipper flips it to the direction or the plane in which the neutron spin came in after you have got an analyzer this analyzer analyzes the spin of this outgoing neutron and this spin depends on basically the after the analyzer it depends on cos of omega t because omega is the energy exchange and t is a time it has spin in the field and cos omega t average value of that over all the velocities will give me average value of cos omega t which is actually measured in the detector because analyzer does not do it spin by spin but it integrates over all the omega after pi by 2 flipper and then sends it to the detector that means what I measure is an average value of cos omega t cos omega t and that average value of cos omega t is nothing but s q omega is the probability of scattering with an energy transfer omega with a momentum transfer q into cos omega t omega t is the precision angle in the two arms and the defacing caused by that and we integrate over the all the energies that so d omega but now we know that e to the power i omega t s q omega d omega is equal to i qt and if I do it over dt bring it on this side I get s q omega now this expression and this is a real part of that expression gives me i qt so the takeaway is that in case of spin echo what I am measuring is not s q omega but because this integral is integration is done by the instrument it is what I measured is i qt the intermediate scattering law and which for a fiki and diffusion we know it is given by e to the power minus d q square p so I can directly fit an exponential function and measure the d values from the intermediate scattering law and this is a result so this is about a proton dynamics in oxide materials it's a berium zirconium indian oxide in which we have changed this this x value or content of indium in this to between 0.1 and 0.5 it's a perovskite structure but the experimental result which is normalized to i q 0 you can see that this fall this is basically the intermediate scattering function that I'm measuring directly so in this case for the two x values I find that the x equal to 0.1 value 0.1 here x equal to 0.1 in zirconium 0.9 in oxide and 3 minus x by 2.05 so 2.95 we have exponential fall and this can be fitted with the this thing with an exponential fall and a constant but this time constant or residence time of a proton is given by 60 picosecond and but when we go to x equal to 0.5 this becomes a stretched exponential and the authors they mentioned that the at this is both at 500 kelvin that in this case the stretched exponential comes in because there are many exponential functions or there are many time scales associated with the residence time of the protons and all of them added together gives us a stretch exponential this is a typical result from a spin echo instrument I just would like to point out to you the time scale is in nanoseconds so possibly this is the slowest dynamics that one can see in a neutrons catholic experiment using spin echo spin echo instruments are available in several major reactor sources like ill granable and nyst at ncnr at nyst usa where they're available and very very very slow dynamics in the range of nanoseconds if you compare this with respect to let us say phonons which are actually in the range of 10 to minus 14 to minus 15 second is a much slower dynamics even slower than bulk diffusion that we discussed earlier so we can use the spin echo instrument for observing such slow dynamics so with this my discussions on inelastic neutrons scattering ends in the next part I'll quickly summarize what we did in the course till now