 Earlier I had shown this tree to you saying what we can see with neutrons in condensed matter as its structure and dynamics. Now I have finished this part of the talk which is longer and possibly more used by people. So measure intensity versus angle I mentioned this and when it comes to what range of crystal structures I should also include liquid and amorphous diffractometer these are all short in structures crystal structures at angstrom liquid and amorphous diffractometer also in the angstrom range then I talked about sands where you can see micelles precipitates and inhomogenities at typically about 10 nanometers 100 angstroms to even micron size and the last bit of it what we did was reflectometry neutron reflectometry which was unpolarized polarized specular and non-specular. So with this now with sands reflectometry crystallography I mean crystal structure crystal diffraction liquid and amorphous diffraction we have completed all the parts which are structure and I must mention here that because I was talking about structure which means spatial correlation either atom atom in case of crystal structure or inhomogenities like micelles and others. So far all these include coherent scattering length coherent scattering length this is a information that I want to share and now this part is over this part is over and now we go on to dynamics. So now over here you need to measure intensity of the scattered beam not only as a function of angle which you are doing here but also we have to find out the energy of scattered beam. So you have E f minus E i which are the energy differences K f minus K i which are the wave vector differences and we will be measuring what is known as scattering law which is this is equal to h cross omega and this is equal to Q wave vector transfer. This is what we will be measuring for dynamics not only Q but also energy of the scattered beam. So the question comes what is the range of dynamics that we can study using neutrons. So I just mentioned here phonons at 10 to the minus 50 second typical time scales rotational diffusion we can measure or we can measure very slow dynamics using polymer backbones but here I should have included I did not my apologies for that we can also measure dynamics of molecular vibration molecular vibration vibration and this will have even slower minor dynamics not slower I am sorry faster so these as we go higher in energy the dynamics is becomes faster because time scales are shorter which are inverse to the energy the energy changes are larger when I go to longer time scales that means delta t or whatever we are measuring a larger then we go to energy differences which are smaller so we are going to slower processes. So slower processes have smaller energy transfer faster processes have higher energy transfer this is true not only in this case it might extend it all the way up to nuclear physics where you do scattering experiments at very high energy transfer because we are looking at time scales which are much much shorter than what we are doing here. So now the discussion on inelastic neutrons scattering for dynamics will target the question what Q ray and what energy ray but before that let me just bring it to your notice that in case of neutrons the thing is that the wavelength one angstrom and the energies they are closely matching and neutrons penetrated very deeply so we can span a very large part of the Q omega space for example if I am doing Raman scattering Raman scattering Raman or infrared absorption spectroscopy I can find out the energy vibrational energy levels of a molecule but the fact remains that the wavelengths are typically around 3000 to when I would infrared 10000 angstrom of the impinging radiation look at the impinging radiation wavelength here so when I try to evaluate the Q value by 4 pi by lambda sin theta sin theta at base can be 1 and this lambda tells me that for this kind of experiment infrared or ion I am close to Q equal to zero whereas in case of neutrons I can span a very large part of Q ray so that is the advantage of neutrons that in case of x-rays wavelength is around 1 angstrom actually copper has got 1.54 angstrom copper k alpha but the energy is too large it's around 12 kilo electron volts so there the problem is we can't span the energy range so in case of neutrons I can span the Q values and I can also span the energy ranges and you can see energies from sub microelectron volt to tens of millileft on volts to hundreds of millileft on volts I can span using neutrons and also a large range of Q so using the here various spectrometers have been mentioned like backscattering time of flight triple axis pinnico I cannot describe all of them to you but I will target to explain to you the function of the major inelastic neutrons spectrometers at Dhruva and at other parts in the world so before I go into the discussion I want to take you back to the general theory that I discussed with you several lectures back at the beginning of the course please remember this was the scattering law d to sigma d omega d e that's why it will tend to measure you can see we wrote this in terms of an initial state k lambda specified by energy and momentum and the final state which is e lambda prime for diffraction experiment lambda equal to lambda prime and we put a delta function by hand stating that in any scattering process the total energy neutron plus system which you are studying should be conserved so the conservation comes from this delta function next step we wrote the delta function in this form as an integral over time this is a delta function energy is a Fourier transform of integration over time of e to the power minus i t h cross omega which is the energy transfer and the difference this is this is for neutron and this is for the energy difference in the system from between e lambda and lambda prime states and this should be giving us a delta function and the interaction potential if I had imposed a delta function potential here otherwise in general I can write the potential also as a Fourier transform over cube space of the spatial potential vjr j is the site and vjr is the potential at the jth side now here from there by putting this expression for the delta function I could write this expression equal to in terms of a Fourier transform over time to go to sq omega so the Fourier transform of time of a function which is a sum over the initial states and the final states lambda lambda prime p lambda being the probability of the initial state which is e to the power e lambda by kt for experiments done at temperature kt the scattering experiment and e lambda by kt is a Boltzmann factor that is p lambda but interestingly here because there's a square of that I can break open I'll I'll just try to tell you there's a square I'm not writing everything so when I break it this becomes lambda lambda prime and then one more bracket for d2 sigma d omega d and here I have let us say lambda prime vj prime e to the power i q dot rj prime lambda this kind of expression we know that for a stationary state earlier also I mentioned to you lambda is given by e to the power i e lambda t by h cross lambda which is time independent if I do that then please know so I had lambda prime vj prime q anyway I got from this expression this expression but I have put in this expression that lambda prime the time dependence has gone on to e to the power i e lambda prime by h cross e to the power i q dot rj prime and from here e to the power minus i t e lambda by h cross into lambda but this here I will use this expression use this expression is given by because h lambda for an eigen function gives me e lambda lambda so I can write it and then I can write the expression you can see so I have got vj prime q e to the power i because it's vj dagger it was so e to the power i t e lambda by h cross e to the power i q dot rj minus i e e lambda by h cross lambda now I can further write this as so now this part e to the power i h t by h cross e to the power i q dot rj for any operator a in quantum mechanics the time dependence in this picture is given by and this is time dependence so here the operator is a position operator rj prime hence this expression allows me to write to write rj which was time dependent so far it's time dependent rj prime rj prime rj prime t so now this part gives me the time dependence of rj prime and I can write it write the expression as the whole expression ultimately boils down to e to the power minus i q dot rj 0 e to the power i q dot rj prime t so this expression here to get the scattering law d to sigma d omega d e I need the time correlation of the position operators that means given a point at time zero a molecule is at a point r at time zero where is it at time t I can do a modeling for example for a simple let us say this is it will be obvious if a particle is moving with a velocity v is equal to u plus f t simple school level problem given the time I can find out the velocity and then I can find out the position if it is moving in a linear path so given r0 I can calculate all the r prime t so the correlation function is known but for many problems we have to take recourse to various kinds of modeling to get this correlation function because this correlation function and its Fourier transform will directly give me the s of q omega so now I write it in terms of scattering line q omega space and this scattering law is given by the time Fourier transform of this correlation function and its ensemble average this angular brackets give me the ensemble average and this contains the dynamics of the system so we can study all dynamics in the system provided one my energy range allows that secondly I know some way of finding out rj0 and rj prime t and my next part will be explaining how do we calculate such correlations and also I mentioned it earlier let me repeat it again that if I consider this is in q space then this function is also Fourier transform of a correlation function g in real space so now a real a function in the real space g of rt goes to an intermediate scattering function which is Fourier transform over q and r and this goes to s of q omega over omega and time so it is a time Fourier transform if I try to come to the other side it will be a energy Fourier transform and this is a double Fourier transform of the pair correlation function so ultimately we have landed up at g of rt which we need to know and g of rt means the pair correlation function if a particle is at origin at time t equal to zero what is the probability of finding the particle the same particle or another particle if it is a self if it is the same particle then it is a self correlation function if it is another particle then it is a pair correlation function but the time part also includes the dynamics and g of rt gives me the dynamics of it if I take out the time then it becomes only a structural work and if I include time then I have to find out how to find g of rt for the system so then with this much of brief introduction to how we included time and dynamics in our formalism I will just take you to various time scales of dynamics which I also discussed briefly earlier so molecular vibrations few to tens of electron volts and other techniques like fti Raman can be used but not for all q values or not for all kinds of spatial correlation because q range gives me spatial correlation whereas these techniques they study at close to q equal to zero phonons are so these molecular vibrations can also be studied using neutrons next let us talk about phonons phonons are few millilectron volts to hundreds of millilectron volts Raman scattering can be used but again at q equal to zero and because for phonons we use the brilloise zone for plotting the finding out the phonons in them so we can only do Raman scattering for zone center phonons whereas for phonons over the entire brilloise zone I will come to it shortly we need to go to neutrons then there is something called stochastic dynamics what is stochastic dynamics the best example in a liquid let us say in water one H2O molecule how it moves around so it's it's a stochastic dynamics so stochastic dynamics like diffusion it can be studied using quasi elastic neutron scattering which I'll be discussing so these two techniques are discussing initially then I'll briefly touch upon molecular vibration and also a technique known as pin echo which is I should say one of the novel techniques not use too much it's a novel technique to understand very slow dynamics tenths of nanosecond time scales like polymer chains moving in a melt and that kind of dynamics using spin echo so this will be my discussion and the list over through which I will go now first let me discuss with two the phonons you are familiar from your master's degree that phonons are collective quantized oscillation of atoms in a crystalline material so I have borrowed this picture it's a lithium fluoride crystal you can here we assume a spring between the atoms of the constituent crystal so it has to be a crystal crystallographic structure when you talk about phonons and we consider that the atoms are bound by strings with each other now then I can write down the equation of motion for the s-th atom in the system in terms of the elastic constant because the displacement is us plus p minus us so displacement of us plus pth atom minus us atom gives the relative displacement multiplied by the spring constant and summed over all the neighbors gives me the equation of motion for the s-th atom so if it is a monatomic atom then of course cp is a spring and cp is a spring constant for particles moved from for atoms move away from each other by p units now in this I assume two things one the it's a time dependent is given by e to the power i omega t that means this gives me that there are vibrations which are quantized with energy h cross omega if I consider time dependence of e to the power i omega t then this equation gives me the equation in terms of displacement and omega square and also I assume a wave-like solution for the displacement of the atoms in the lattice so this displacement wave-like nature of this displacement is a fundamental under the cornerstone of phonons in solids so the displacement of s plus pth atom us plus p in given in terms of a constant and e to the power i s plus p and this is a wave vector and these are lattice spacing wave vector is the wave vector of the spatial variation of this displacement these we have to find out so with this when I substitute it in this equation I get cp e to the power i s plus p k a minus e to the power i s k a and I have substituted u is u s equal to u e to the power i s k everywhere and I lambda m omega square equal to e to the power i s k a minus one it should have been i pk I am sorry pk now this is summing over all neighbors now I know that if I was starting atom is here whether I go left or whether I go right there is a transmission of symmetry from this that means for p equal to plus and p equal to minus cp if the same distance away the force constant remains same so I can break it up into instead of summing over all atoms I can do it over p greater than 0 that means and I break for the neighbor maybe right and neighbor left so distance either plus a or plus p a or minus p a I can write it as sum over e to the power i p k a plus e to the power minus i p k a and minus 2 here this gives me if I add it up it omega square is given by this very simple we come from there to here because these two will give me 2 cos p k negative goes out so it gives 2 cos p k a minus 2 so taking 2 common I get 1 minus cos p k a now for nearest neighbor only my sum gives me instead of cp I have got only c1 and that gives me a solution which is 4 c1 by m sin k a by 2 so it's a sinusoidal curve and this is given by this is the force constant I know this omega and k are related by sin so this is the derivation for a monatomic nearest neighbor atom and you can see that we have a wave like solution or rather there are there are dispersion relations just sin like now question is that what sort of displacement we are talking about