 So, now let us come to this expression. So, as I said, so one is that g l and s of q, they are Fourier transform of each other. So, then that tells us that we need to go to larger q values, if I do a real Fourier transform and we want to see small r structure. Smaller r demands larger q. And in case of liquid and amorphous systems, typically we can think of going to 15 to 20 angstrom inverse. Now, more importantly, if I need smaller lambda or high energy, high energy neutrons, then let me mention to you that I discussed with you something called a hot source. A hot source what it does is this number of neutrons versus energy, sorry number of neutrons versus energy. If this is a maxwellian at room temperature and 25 to 30 milli electron volts peak, then smaller energy neutrons are less in number, higher energy neutrons are also less in number. So, for this part I discussed with you that we use what is known as cold neutron sources to enhance the number in the low energy range, low energy range to give gains. So, that we can use low energy neutrons and I will talk you, discuss with you the usage of a cold neutrons in experimental facilities. Similarly, for higher energy neutrons we can think of increasing the shifting the spectrum to the higher energy side by using something called a hot neutron source. So, higher energy of neutrons smaller, but at the moment there are not too many hot neutron sources available in ISIS durable. There is a hot neutron source where a graphite at around 15 to 1600 degree centigrade is maintained inside the reactor core and that shifts the basically neutrons enter the thermal neutrons from these regions, they enter the hot source neutrons from this region and get the spectrum get shifted and our liquid and amorphous instrument should be using neutrons of such energies. So, hot neutron sources are preferable for liquid and amorphous diffractors. This is one difference in experimental facility which studies liquid and amorphous material. In case of palatial neutron sources interestingly we get lot more number of hot neutrons inherently because in case of palatial neutron sources once the proton beam hits a target, it generates neutrons with very high actually it can work to 100 mega electron and these are actually these are brought down to thermal energies, low energies using moderators surrounding the target material, but if we use less moderated neutrons then inherently hot. So, in case of a palatial neutron source we have inherently we have access to more number of hot neutrons and liquid and amorphous experiments can be reasonably done better in a palatial neutron source. So, that is what I wrote here experiments at hot source in a reactor or hot neutrons in a palatial neutron source as the preferable choices. So, with that I will come to the experimental setup what we have in Dhruva. In Dhruva we have this looks very similar to the instrument that we use for powder and the powder material for crystallography. This is the detector bank surrounding the sample here certain the sample in this one, but we have five detectors here five position sensitive detectors and the largest angle we use is around 148 degrees. We can also go to smaller lambda one step process, but our monochromator we can set it for smaller lambda. So, at the same angles we can go to higher q 148 is the largest angle, but the available q value is approximately 15 angstrom inverse. So, 15 angstrom inverse I must compare this with respect to the sandals spectrometer sandals at isis isis palatial neutron source rather for depletion laboratory UK. Here the sandals the angle use in this is not large the largest angle is that is bank of detector is at 38 degrees, but because we have access to very small wavelength neutrons even up to 0.1 angstrom it is possible to go up to 50 angstrom inverse in q. I must mention here one thing when I say q actually what I want to say and this is normally always put in q in h cross q is a momentum transfer, because I am measuring my intensity in momentum space in through my scattering, but always we talk about q as if this is the momentum transfer which is slightly I should say miss understood or miscoting because the momentum transfer h cross q, but all the experiments we will be discussing about q. So, for I have talked to you about q and later also we will be talking to you about q when I talk about diffraction experiment. So, we can go up to 50 angstrom inverse in q in sandals and I will try to give some results from there also. So, now coming back to the experiments that is done at that has been done in Dhruva I will just go ahead. So, before I start again in case of crystalline material q is equal to g as I mentioned you and this expression gives me the structure factor bj 80 duper minus i twice pi i xj 8 this is from the crystallography, but in case of liquid amorphous material the structure factor can be written like this it is starting from the same source, but this gets modified to this I will avoid the derivation here, but this in this expression you can see this is the structure factor and this takes care of the incoherent part, but b in a two component system is a concentration of the ith average of b coherent is given by the concentration weighted concentration of the coherent scattering length. The sk is the total structure factor and you also discuss about partial structure factor. For example, if I talk about let us say here we discuss the s of q for say vitreous silica or silica or geominium selenium glasses resourcing this. So, in these cases actually we have got distances several distances. So, partial structure factor means there can be silica or silica distance in case of silicon oxygen, silicon oxygen distance and oxygen oxygen distance all of these will give us the partial structure factor and the sum of them will give us the total structure factor that you measure in an experiment and these can be stimulated and tested against the experimental result. Similarly, in case of geominium selenium glasses I will have geominium geominium distance and also geominium selenium distance. So, we can use both of and both of these comprise the partial structure factor and the sum of them uses total structure factor. And often we write about particle distribution, particle distribution in a radial particle distribution which is nr if I get it from s of q to g of r then the particle distribution n r equal to 4 by r square rho g r, nr gives me the probability of particle numbers between r and r plus dr. So, nr dr will give us the particle number of particles in r and r plus dr 4 by r square rho g r, rho is the average particle density. So, this gives the particles with within r and r plus dr. So, let me just show you a result and how we did the data analysis. In this case the data analysis was by us using a model known as Monte Carlo g of r I will briefly tell you what it is. So, the solid lines are the data of germanium selenium glass g e s selenium 1 minus 6 and you can see various values of x that were used and the solid line is the data discrete points are the I am sorry the solid lines are the fit the discrete points of the data and there is excellent fit between the data and the solid. What we did actually in this case we started we did not do direct Fourier transform because you can see that our results stopped approximately 14 Armstrong inverse. So, direct Fourier transform is certainly not a good idea, but we what we did actually what we assumed a g of r which is a percussier weak heart sphere. What it is I will just briefly explain to you in case of percussier weak heart sphere we locate heart spheres in a certain volume a number of heart spheres in this case there will be selenium and germanium atom sizes and then we packed them the in a certain volume, but there are if the if in this case heart sphere if the heart spheres intercept then the inter atomic potential is infinity because two heart spheres cannot penetrate each other forbidden and if they do not penetrate then v r of r is 0. So, v of r is infinity r less than equal to r 0 size of an atom. So, with this assumption this percussier weak heart sphere we create a linear one-dimensional g of r by using the percussier weak heart sphere model for our sample. So, this g of r is our input to start with for a given configuration for a given configuration and we generate the configuration using an inter atomic potential which is meant for this heart sphere. That means, we just put the heart spheres if they intercept each other we will immediately say this configuration is not possible put it somewhere else and at some point we just after creating for some time we will let it equilibrate and then you find out g of r one-dimensional g of r. Now, once you have created the g of r we can always do a Monte Carlo result we Fourier invert and we have a Monte Carlo result for the structure factor in Q and we also have the experimental results. So, now what we do here do actually try to minimize the difference between the experimental result and the Monte Carlo S of Q. So, I have put an S of Q. So, one S of Q is experimental experimental then another S of Q is model Monte Carlo model which has come from g of r which has come from g of r that we start and then we have model and we have to minimize the difference between these two and that is what we did in this experiment in this data. So, this is and sigma i square is the error bar which is square root of n at every point. So, now every time we change the g of r the S of i Monte Carlo it changes and then there is a chain associated with chi square which you call it delta chi square and this we accept the chain with the probability e to the power minus delta chi square by t. I will explain to you that will be the details because this Monte Carlo technique has been heavily used in the Ising model it is very similar, but I would like to tell you how it is done with the taking the help of Ising model. So, in case of Ising model spin half I will just consider spin half particle. So, I take a lattice this is an Ising model Ising model lattice. So, here I can flip a spin at a side when I flip it then we know that the interaction energy is j i j S i dot S j in this case let us assume nearest neighbor. So, then for an for one atom it is only 1, 2, 3 and 4 these four atoms are concerned these four atoms are concerned as nearest neighbor. If I flip this there is an increase in energy delta e from this expression increase of energy delta e. So, in one state I randomly flip one spin this is same as what I am doing for the liquid and amorphous system I randomly shift one sphere when I randomly move as give a spin there is an energy difference of delta e. This I will be comparing with delta chi square a virtual energy. Now, delta e if there is a change in energy of delta e we if delta is less than 0 then we accept them because that is reducing the energy. If it is increasing the energy do I reject it? No. I accept it with a probability to the power minus delta e by d this is a Boltzmann factor that is what a Monte Carlo simulation Monte Carlo simulation of ising model. So, that means it accepts even if the energy is increased with a probability to the power minus delta e by k t. How does it do? Now, you can evaluate the value we know that if delta e goes to infinity this becomes 0 if this that means we cannot turn but this is the finite number here it will be because I do not know whether the nearest neighbor was already flipped or not. So, if all of them are flipped then there will be 4 into some basic unit if 2 of them are already flipped it will be 2e. So, what we do actually in the computers today there are facilities to generate random numbers random numbers between 0 and 1. Now, this one if I scale it to 0 and then after I have got this change in energy if it is less than energy I accept the move it is greater than energy it to the power minus delta f t probability. So, I choose a random number and compare this with the random number if this is if the random number is greater than equality to the minus delta by k t that means it allows me the move I move that means it is flipped in this case there is no internal energy involved but a structure is involved and the chi square with that involved now I compare the delta chi square with the energy change in case of spin glass that I showed you here delta e when I do that then I move the probability the move I accept the move with the probability of e to the power minus delta chi square by t now this is a this is just an increase in a parameter space and error bar. So, there are no elements t is a fictitious temperature but I can fit a fictitious temperature what will happen actually if I use a very large t you can see e to the power minus delta chi square by t it will be a deliver minus a very small number. So, that means it will be almost equal to 0. So, all the moves will be accepted with high probability. This will mean when you have a high temperature for a medium then the dynamics takes over and the energy increase is accepted because there is lots of energy in the medium to randomize the material and when if t equal to 0 then no moves will be accepted because this will be 0 and my random number space is 0 to 1. So, this you can see that if t is large this goes to almost 1 and almost all the moves are taken. So, even I can start with a large t and keep coming down with simulation there are various techniques which I do not want to discuss here but this is how we generate the model change the model Monte Carlo s of q compared with this experimental and finally accept the s of q which gives you minimum error and this is what is shown and this is experimentally overtake s of q and the model and the from that model we can get various parameters of liquid and amorphous systems in case of germanium selenium glass this is t of r this is pair distribution function and this was the one we published in journal Monte Carlo s of q the data and the figures are taken from this journal. Now, the most used technique for liquid and amorphous systems is a reverse Monte Carlo by R L McGree v. So, we had used a one dimensional g of r to simulate our s of q in case of germanium selenium glass, but the reverse Monte Carlo takes a three dimensional element of atoms and molecules. So, this is much closer to reality, but in a normal Monte Carlo like as I showed you in this ising model one tries to minimize the energy of the whole free energy because it can fluid higher energy configurations can also be accepted, but generally we are trying to find out the minimum not in energy space, but free energy space, but in that case if you see here in normal Monte Carlo we need to define an interaction energy. So, here also when you are talking about arrangement of atoms and molecules I need to talk about internal energy if I want to do a simulation physical simulation, but I would say this is a mathematical simulation, but physical, but what we do actually what I talk just now that I play with the arrangement of the atoms in three dimension and create the structure factor Monte Carlo comparing structure factor experimenter and try to minimize I will just try to show the result for vitreous silicon this was a paper by McRuby. So, you see this is the neutron the structure factor simulated as well as experimenter and this is the atomic arrangement. This actually this is for SiO4, SiO4 forms a tetrahedra we talk about SiO4 units. So, it is a tetrahedra SiO4. So, if I draw tetrahedra so, this is the tetrahedra tetrahedra that you are talking about tetrahedra now we can play with the orientation of this tetrahedra even length of the bonds and then try to generate the experimental value. So, this the arrangement so, it says the projections of atomic positions in a 7-anestromatic section from a representative configuration large atoms are silicon small atoms are oxygen, silicon oxygen bonds from all atoms within the section tron. So, the some bonds terminate and atoms outside of the section we should know. So, basically it is played with the silicon oxygen tetrahedra which is the basic unit for vitreous silicon and most interestingly this is a corner sharing tetrahedra. So, that means if I consider another tetrahedra these corners so, with respect to one tetrahedra the other tetrahedra can share corner with it and they can orient with respect to one another they can some bonds they will get elongated, but they can orient. So, the tetrahedra remains because these bonds are strong. So, the silicon oxygen tetrahedra remains more or less undisturbed, but their orientation changes their corner sharing tetrahedra not edge sharing, but these corners they are shared and these were the findings of this work by McRuby and using reverse Monte Carlo. So, at the moment full RMC package is available online is anybody can use it and it is simple. If you are interested to do neutral or x-ray studies of liquid and amorphous material this is the best possible package which you can use to do the Monte Carlo simulation reverse Monte Carlo simulation and get a good fit to the experimental results. I talked about molecular clusters this is a work done in a in Dhruva in our group I will the reference I missed here this is actually done by PSR Krishna and group I will show you the reference in the next lecture. Here actually total structure factor which has been measured it is a sum of two parts one is that there are molecules propanal molecules age bonded simple alcohols. So, one is the distinct that means this is the structure these are correlation between two molecules and this is the structure inside the intramolecular structure and intermolecular structure and here through the simulation our reverse Monte Carlo simulation one finds that there are clusters of molecules here. So, that means in the liquid while we assume that the molecules are able to move freely actually there are clusters but I must mention this clusters must be having some time scale. So, these clusters are forming and breaking and again forming and breaking but the thing is that at any instant of time these clusters they exist in this alcohol and that is the finding of this work on liquid and amorphous material. As I told you that up to 50 angstrom universe is possible at sandals I just attempt to show one result from sandals this is a nature paper I will again show the reference I missed it my apologies for that I will bring out the reference in the next lecture. So, most interesting finding in this because it is nature you see in this solution of magnesium perchlorate they observe from the partial structure factors that there is a drastic defect on water structure there is a high pressure they get a water structure which is commensurate with higher pressure on oxygen because we know that oxygen has got a structure like this. So, every H2 in the solution has this structure so and there is also hydrogen bonding between two hydrogens. So, there is a signature that the oxygen that the water is under higher pressure in this solution. So, this is a result from thing sandals and you can see that the data are taken up to very high Q values 20 angstrom universe. So, with this I will like to stop today on the liquid and amorphous systems a brief introduction to single crystal will be done in the next part.