 So, now I will continue with inelastic neutrons scattering for stochastic dynamics with neutrons. Now here by stochastic dynamics what I mean here this is a ball bouncing from the ground is a periodic motion whereas you can see this if we consider this let us consider hydrogen atom it is diffusing let us say H2O in water this motion is stochastic we know that Brownian motion and that the difference between the two is that this is periodic motion this is a stochastic motion now for this I need to give you an idea about the energy scales let us do an energy hierarchy high to low we know let us say a hydrogen atom its ionization energy is 13.6 electron volt the first excitation energy is around 3.4 electron volt we were talking about vibrational modes let us say vibrational modes of hydrogen they will be around say 200 to 300 milli electron volts that means 0.2 to 0.3 electron volts these will have a strong overlap with the phonons the phonons phonons will have typical energies let us say 10 milli electron volts to 100 200 100 should be the parliament possibly 100 milli electron volt where most of the instruments in the world they will go up at best of 100 milli electron volts so this is there is some overlap but still lower energy then I will go up to about the diffusion this is stochastic motion which I am going to discuss in this part here we may be talking about energy transfer of 10 microelectron volt to let us say 1000 microelectron volts so this is the range of energy transfers that we need to go ahead with if you want to if you want to talk about or the understand diffusion in solids so with this let me go ahead and introduce you to diffusion in solids so first what so far what you have discussed is this please see I have talked about diffraction where you had a neutron hitting a monochromatic coming to a sample and then we had a detector and this was actually we talked about sq and we here we were just bothered about in which angle the neutrons have gone and basically this was no energy analysis was done in case of phonons we had a monochromatic with the sample but we did energy analysis so here it was sq here it was sq omega but in both these cases they were coherent because in case of diffraction if you remember we talked about bi bj prime e to the power iq dot ri minus rj prime when we talked about the cut out intensity and in this case also we were talking about coherent motion of atoms and molecules in a crystalline soil so both these were coherent and we used b coherent to as to understand these structure or dynamics so we are we will use samples with large values of b coherent except for the fact that for density of state measurements I can use a sample with a large b square value because there was an incoherent approximation to coherent one-front structure but now when we go ahead and do stochastic motions then we will be talking about incoherent coherent cross-section because I will show you we will be talking about talking about self correlation of an atom with itself in time so now when we talked about vibrational energies as I showed you 200 to 300 millilektron volts then you there will be a so it will be a peak which you can see in Raman scattering or even in neutron scattering at an omega value corresponding to the energy of that particular vibrational mode periodic motion and then for harmonic oscillator this is the ideal and this is resolution product in case of stochastic motion it is the Doppler shift the way which I explained that the neutron comes and an atom is undergoing a random motion in a liquid or in a solid liquid a solid then this random motion this neutron hits it and goes back and this may be moving towards the neutron or away from the neutron or in some other direction and this Doppler shifts Doppler shifts shifts the energy Doppler shifts the neutron energy and neutron energy gets a broadening why so now the data will look somewhat like this so if I consider omega equal to 0 as a no energy transfer now this will get broadened the inelastic will come somewhere far away from the omega equal to 0 line so that's why it is called quasi elastic elastic neutron scattering or called q e n is here what I am looking at is the broadening of the elastic peak due to motion of the molecules the Doppler shift the atom and this will gives me information about the diffusion in the medium by medium I mean it can be a liquid or it can be a solid now let's first look at the double differential scattering cross section d2 sigma d omega d e that means neutrons per solid angle unit solid angle per unit energy interval there are two this is of course kf by k i is for normalizing the flux number of neutrons coming in by number of neutrons going out so this is the number of neutrons going out proportional to the velocity of at energy e f and there's a number of neutrons coming in with an energy e i the velocity given by the proportional to k i we have two parts one is coherent part another one is incoherent part this part gives me pair correlation correlations whether structure or dynamics it talks about two particles a particle at one point and the particle at another point if it is structure or a particle at time 0 at origin another particle at position or at time t so yes incoherent is the time and space account now when we talk about incoherent part then it is only the self correlation that means the particle at origin 0 at time t equal to 0 what is the probability it is at a position or at time to the same particle and here there are different particles when you talk about coherent that's why in phone or in structural work in structural work at temperature t equal to 0 the position of one particle in a crystallizing that is tells me the position of all other particles in case of phonons the position and velocity of a particle at one position tells me the position and velocity of all other particles they are related to phonons so this is the coherent part this is the incoherent part and S incoherent and S coherent they depend only on the sample and it's coherent and incoherent cross section as I discussed earlier that incoherent one is b square average minus b average square and coherent part is average value of b and similar average of b so this is the average part which comes here and the incoherent part comes here and in because of hydrogen sigma incoherent 80 burns nearly 80 burns it is much much greater than sigma so when we talk about self correlation functions and try to do an experiment unlike all other experiments which we discussed so far which were actually pure correlation function based hydrogen is desirable particle desirable atom because this has got a very large cross section between this now I discussed this earlier when I introduced the scattering law if you remember that in the intermediate scattering function I had an ensemble average not ensemble average is a statistical average between the initial state of energy lambda what I calculated was actually j particle at origin and j prime particle at time t so this is the correlation function in intermediate space because here these are constants q is the variable so this is iq t now Fourier transform over time takes me to scattering law this is what I measure in a neutron scattering experiment and the other way around if I take a Fourier transform over over momentum I go to g of RT which is a real space g of RT so if I go from here over time integral I go test q omega which we measure if I go Fourier transform over q as I wrote here I go to the correlation function in the real space and time now g of RT if I consider the same particle classically it means delta of r minus r0 plus r at time t for the same particle average over all the atoms and molecules in the system so this is basically this gives tells me that given a particle at origin at time t equal to 0 what is its probability being at arm at time t this is g of RT which we will get if we can get it from sq omega and now so this is what I mentioned it specifically that sq omega Fourier transform over time it is the double Fourier transform over time gives me space and time gives me sq omega and sq omega is single so this one Fourier transform over space g RT into the iq dot r d3 r this gives me iq t and one more Fourier transform takes me to sq omega just to remind you so if I have s of q omega or if I know by some model what is the g of RT I can estimate what will be the analytical form of s of q omega and this gives me s of q omega or s of q omega I can model into g of RT and get informations about the system specifically on the diffusion constant so for Fickian diffusion this solution is known gs means self correlation function that means this is easy to explain suppose a particle is at origin at time t equal to 0 there is a delta function delta 0 now as time goes the particle diffuses and there is a Gaussian distribution of it the particle with time and then the width of the Gaussian function is 4 dt that means as time increases this becomes broader and broader and ultimately t equal to infinity this width becomes uniform so this is the relationship that we went from Fickian diffusion now let me come to the Fourier transform because I need iqt and s of q omega please note that this expression e to the power minus r square 4 dt g of RT I can do a Fourier transform over space e to the power minus r square by 4 dt e to the power iq dot r d 3 i is r square dr d theta d phi r goes from 0 to infinity theta goes from 0 to pi and 0 to 2 pi over pi I leave this integration to you but at the end of it because you can write iq r iq dot r equal to iq r cos theta and then do it sin theta d theta 0 to pi I leave it to you to show that it is sin qr if I remember by q this will give and then you have to do a integration over r r square dr and ultimately the intermediate intermediate scattering law scattering function will be e to the power minus d q square t now the d q square t you need one more Fourier transform and you get s in coherent q omega this is a Lorentzian so the scattering law for the self correlation function starting from here shows a Lorentzian and whose full width at half maxima is dictated by d q square now if I can do this experiment that means measure s in coherent q omega at various q wells then you can see that the width of the Lorentzian will vary with d q square and from there I can try to get information on the diffusion constant very interesting and this Lorentzian its width is actually typically tens of microelectron volts to hundreds of microelectron volts so this is a very low energy transfer experiment in which if we measure the Lorentzian actually Lorentzian plus it is convoluted with a Gaussian resolution function will be a pseudo void function which we have to from which we have to find out the value of d q square and then that d q square further plotting at various values of q will give me the diffusion constant in the system if the temperature of the experiment is t then at the temperature t what is the value of the diffusion constant so now there are two issues one is that my s in coherent of course it has got a Lorentzian as I wrote here because of the time correlation or diffusion but there is another part where if I have a correlation if it is going to zero that is one kind of correlation but suppose an atom a hydrogen atom is inside a box of finite size then if I consider at time t equal to infinity this particle due to its stochastic motion will get uniformly distributed inside this box and so now this i q at infinity if I consider it does not go to zero but goes to a finite value now this finite value is because the particle is in a box or in a cage if I say I don't know the dimension of the cage but studying this we can see that since it is distributed inside the cage the correlation function doesn't go to zero and then I can write it as i incoherent q omega q infinity because at time t equal to infinity it gets distributed and there is a factor called elastic incoherent structure factor we know from our dealing so far whatever we discuss that that elastic factor is always coherent structure factor is always due to coherent but here a single particle but as it goes to t equal to infinity get distributed inside a box then this is like a form factor of an atom which we will be seeing in terms of an intensity a delta function is a delta delta function omega equal to zero as a function of q which is known as elastic incoherent structure factor this plus we have a Lorentzian with half width at half maxima I wrote d q square but if the temporal relationship is e to the power minus t by tau plus a constant so one is that the probability of of the same particle as t goes to infinity is a uniform value depending on the case size and there is a e to the power minus t by tau part now e to the power t by tau earlier I wrote e to the power minus d q square t the incoherent so t by tau d q square and t by tau the time scale of motion they are related and then I have an aq term which is eisf elastic incoherent structure factor because this is incoherent scattering safe correlation but the particle if it is in a cage then it has got a structure factor centered at zero energy transfer and a Lorentzian with a pre factor of course there are as I wrote here there is a this is there are what should I say this instrumental resolution factor which will be coming so this is the resolution factor this is the Lorentzian and this is the elastic incoherent structure factor and the Fourier transfer of this one will give me a Lorentzian whose width will give me 1 by tau or d q square both are correlated and also one elastic factor the elastic factor should have been ideally speaking a delta function but now because of resolution it is broadened and this experiment has a particular q value it is done at a certain angle the certain momentum vector transfer and the intensity of this peak because we know when it did form factor it goes like this form factor falls as a function of q the intensity of this delta function peak will depend on the q value and from there that means if I know this fall then I can talk about the cage in which the particle is moving and its dimensions so interestingly I can find out the diffusion constant from the width of the Lorentzian from the width of the Lorentzian but also from the intensity of the elastic peak which is elastic incoherent structure factor as I told you that there is an i incoherent q as t goes to infinity its Fourier transform gives me a delta omega part so it is energy equal to 0 and the q dependent elastic constant elastic part which gives me the geometry of the cage I will come back to it once again if I consider that the particle goes out to infinity for example a water molecule in a glass of water I can consider the glass of the dimension of the glass in our length scales infinity so the water molecule can go anywhere and then my probability of finding the particle with time t goes to 0 because it has gone to infinity and then in that case we will not have a delta function peak we will have so this part will not be there what we will have one second yes so when I have the probability going to 0 exponentially up to infinity if it goes in that case this part won't be there what we will get is only the Lorentzian only the Lorentzian of width 1 by tau or dq square and measurement at various angles at various q values will give me the width of the peak and from there we can find out the diffusion constant for the system so now that means my instrument resolution broadened S incoherent q omega has got an elastic part has got a Lorentzian part and broadened by the resolution so that means it is a convolution and we know that convolution if I write it as hx hx is given by resolution x minus x prime into data at x prime dx prime so every point of the actual true width is broadened by the resolution function and this will look like this and from this we have to deconvolve the resolution and then get the width from width you get the dq square so this is the elastic part and this is the quasi elastic part if it is a finite geometrical confinement for the particle then we will have an elastic part as I showed you elastic incoherent structure factor so please look at that term it is elastic incoherent structure factor this is because the same particle as we go to t infinity get distributed if it is a box then it's at a finite volume or if it is an unbound movement then it can go to infinity and when it is a bound movement then we get a elastic term now I will explain to you with respect to the instrument that we can use at Dhruva how the quasi elastic scattering is done so here this is quasi elastic neutral discriminated marks mode at Dhruva so first we have a double crystal monochromator this is very interesting because you have two monochromators and one is a focusing monochromator vertical event and then the advantage of this is that I can so this is one monochrome first let me just drop plain monochrome straight away so this gives me a lambda and this gives me lambda after reflection same monochromator in this geometry if I change this angle if I change this angle if I change this angle then this will go somewhere else so what we do actually when you change this angle this this monochromator can move back and forth so then I can change this angle I can go there and I can also move this one so by keeping I mean the angle is changing but but if I change this angle then this angle also changes and I can move this to catch this beam and this is rotated both of them are rotated and translated to keep the outgoing beam direction same for two wavelengths I hope I have been able to answer you make you understand that if you see the neutron from the reactor falls on the first monochromator it gets reflected from it goes to the second monochromator goes out along this line now if I rotate the first monochromator and the second monochromator and move them then I can keep this outgoing direction fixed but I can change the wavelength of the neutron going on after that the sample is put so here the monochromator you cannot see you can see the sample position and the analyzer now here the analyzer is an extended crystal and you can see from here that is the angle on the analyzers are different and because they are different the neutrons of different energies will be reflected by this part and this part of the analyzer so that means the detector which is a one-dimensional detector here this detector captures neutrons of different energy at two different ends and ultimately I get the distribution of energy reflected from the sample in one scan of the detector or maybe I can extend the scan by rotating the detector around the sum around the analyzer so that's why you can see the analyzer arm is there and this is the one-dimensional detector arm which is there and the detector can rotate around the analyzer to change the energy range so this in one setting we get a certain energy range and for this setting the average Q value is given so I can have of energy resolution of 200 microelectron volts and the typical Q range is 0.6 to 1.8 angstrom inverse and time scales are 0.1 nanosecond to 10 to the power minus 12 or picosecond so this is then energy resolution is around 200 microelectron volts and I can study diffusion or stochastic motions using this spectrometer let me tell once again the neutron comes on this monochromator and this is reflected by two monochromators and I can by rotating and shifting the monochromator on a rail I can keep the keep the direction of the outgoing beam fixed also the second monochromator is not flat but it is vertically focused vertically focused means the beams is being focused onto the horizontal plane so that doesn't make my angle resolution poorer but it gives me some gain in intensity so it is vertically focused and it is in the Bragg reflection geometry in parallel with the first monochromator and there as I told you this is they remain parallel now if I change the angle then I have to change the angle here and I also have to move it and that outgoing direction remains same in this setup this is a double crystal monochromator this is a drover I will discuss some of the results in the next lecture because now I will discuss movement of various organometallic molecules inside a zeolite cage it is of deep technological interest that how these particles they move in such cage materials and I will come to these results later and I will also show you in the same go this is the iris spectrometer at at RA rather for Dappelton laboratory Isis source this is palatial neutron source so I just show you the main part of the instrument of course there is a triester for the sample the neutron comes here here is the sample and the analyzers are there are two sets of analyzers one is a graphite analyzer bank other one is silica analyzer bank and then then the detector bank basically the analyzer banks are mica and graphite so they give two different energy ranges here and interesting the detector is at the lower radius so here if you consider vertically if this is the this is the analyzer the sample reflects and goes to the so this is the analyzer and this is the detector so it is vertically then neutron first goes to the analyzer and reflected back to the detector so this is almost back scattering geometry and in the back scattering geometry we know that the black peak gets the best resolution because in the angle it is called theta delta so when you go to large angle that means theta equal to 180 degree if I go then this gives me the best resolution in terms of energy or delta D by D or energy in this case delta D by D ultimately will translate into energy of the outgoing beam so there in our max mode detector we had one position sensitive detector around a graphite analyzer here you have detector banks a large number of detectors most probably more than 50 and this is the details of it is available at this site and I will discuss results from both of these and in this case in this instrument they also have kept a diffraction detector at the back scattering geometry so this can also give you long B information about the sample structural information whereas at the same time you can do the study of inelastic or quasi-elastic neutron scattering using this detector so I showed you two detectors I showed you two instruments and I will discuss results in the next lecture