 Till now I have been discussing with you diffraction from crystalline solids and now I will move on to diffraction from liquid and amorphous systems and very briefly I mentioned the special features of single crystal diffraction in this lecture. When we talked about crystalline solids before that in the general formalism of neutron diffraction if you remember I introduced you to the potential energy as seen by a neutron B of r twice pi h cross square m sum over L B L delta r minus r. So, this is a scattering amplitude for an arrangement of scatterers. So, there are scatterers at point L and their scattering potential total scattering potential is the sum of this delta function. In this expression I have not mentioned anywhere that this is crystalline crystalline. This is just a general arrangement of scatterers and I did not assume crystalline and I also showed you that under Born approximation the scattering amplitude of going from k to k prime wave vector is basically given by this expression where k prime vk is a Fourier transform of the potential. And then finally I wrote the scattering intensity per unit solid angle d sigma by d omega has a coherent part coherent part where the coherent scattering amplitude is multiplied by the structure factor and then it also has a coexisting incoherent background which is coming from the fluctuation. This is a recollection of what we did, but in these expressions no one have assumed that the material is crystalline, but these are solid and the atoms are fixed at some points. Now, another question comes when I am talking about liquids and amorphous systems then for liquids the atoms and molecules are moving along. So, for a liquid the atoms and molecules are continuously diffusing inside the liquid and a structure work what will it show? So, if you remember it was e to the power i omega t s q omega if I integrated it over omega I am supposed to get i q t. And in this integration if I put t equal to 0 then it is a integral over s q omega d omega which gives you i q 0. For most of our structure works I can call it i q or s of q which I will be measuring in a scattering experiment. So, that means this gives me i q 0 means time t equal to 0, but there is no defined origin of time. So, basically the experiment is done over a finite time and they keep adding snapshots over snapshots over snapshots and I get a time average picture which is same as what we get from a static particles at their sides. And so, liquids and I will show you experimental results for that that liquids, liquids and amorphous solids have similar structure. Similar structure only in case of liquids I will add in case of liquids it is a snapshot and in case of solids, solids it is a static picture, static picture. Otherwise I will show you structurally they are similar amorphous solids and liquids both of them are similar. Now with this I would like to say that so far I will add up we discussed the periodic array, but periodic array means they have long range order. Does it mean that liquids and amorphous solids they do not have any structure at all because they are disordered materials in our common parlance it is not true. All of these they also have local structure close r structure small r structure. I will give an example let us consider case of sodium chloride very simple to explain with respect to a molten sodium chloride salt. If I take a molten sodium chloride salt then the central sodium atom it will like to be surrounded by chlorine because it is ionic potential ionic potential ionic potential. So if I calculate the structure factor for this molten salt or G of r in the real space and time with distance then up to some distance your G of r is 0, pair correlation function is 0 because nothing can come closer than that it can be a hard sphere here it is ionic code, but after that the chlorine occupies will have a sharp peak and then it will fall then after that there will be most probably the chlorine number will go down there will be more sodium, but then again there will be a second range of correlation and then oscillate and finally go to the density of the liquid. So you have do have local structure and we talk about first sharp diffraction peaks in amorphous solids and liquids known as FSDP. So always these amorphous and so called liquid and dissolved materials they do have local structure and this is of great interest in many cases because we would like to know the molecular atomic arrangement in such liquids and I will elaborate it with a few examples. So in case of periodic lattice we do have some of delta functions if I take other temperature effect. So my correlation function is a sum of delta functions which are atom sitting at location and its Fourier transform is this which is an intensity pattern that I will obtain if I want diffraction experiment with a periodic lattice I have just picked up a random one on lanthanum, stoichiom and lithium data from our Drouwer diffraction machine. So here because of periodicity we also have lots of selection for example when I do the diffraction from a periodic lattice I told you that a scattering vector Q has to be equal to G one of the reciprocal lattice vector because I can define a reciprocal lattice in case of a periodic solid. In case of liquid and amorphous structures we do not have a reciprocal lattice. So this kind of selection rules are not applicable to liquid and amorphous material and because of this Q equal to G I discussed with you before I went to diffraction problems that there is something called Ewald construction where the scattering vector hits the Ewald sphere every time there is a reciprocal lattice point if there is a reciprocal lattice point cutting the Ewald sphere you will have a diffracted beam in that direction. We do not have this kind of peaks in case of liquid and amorphous systems but what we have is something like this. This is as I told you just now this is the it is a simulated pattern for an arrangement of hard spheres. So hard spheres means the spheres they can at best touch each other and if their radius is r then the minimum distance between the centers can be 2 r nothing can be below that so that is why it is 0. So this is the 2 r but then in this arrangement because they are touching each other you will have arrangement of other hard spheres around a central hard sphere and then I will have a sharp thing. So I must mention it here for your we should be cautious when I say center this center is also not fixed any atom can be taken as a center but and then this local arrangement. So when I talk about this distribution this is an it is an average over a whole arrangement of spheres and any any solid any sphere can be a center of our coordinate system and then you have this after that you have the second you can see what happens actually the second the second shell around the central atom central atom can be anywhere and it is an average over various ensembles ensemble of the hard spheres. So this average shows of nearest and very sharp first peak in the geophore this is geophore then there is a second peak that is the second so if I have a center here the first one is pretty well defined but the second one is diffused second one is diffused second one third one is even more diffused and when you go to infinity or very far away my infinity is a distance much much larger than the radius of the sphere much much larger then you get a constant geophore because then it goes to the normal density of the liquid. So this is you can see this is what I get when I talk about a crystalline solid this is what I get when I talk about an amorphous solid or liquid this is the pyrolyne and this is what we will be trying to understand and measure in our experiments using liquid and amorphous systems. So how the pair collision function looks I have chosen some results from this using common materials metals that you are aware copper aluminum nickel in solid form all of them they form an FCC lattice iron form VCC these are they are called FCC lattice so they are FCC or VCC and this is what if you measure the S of Q it will measure S of Q data taken from here you can see and you can simulate the same data you can see using a random arrangement of spheres and here the experimental data and the fitted curves are shown for the copper copper aluminum and nickel so you can see that when we melt when they melt still they retain the local structure and their geophore evaluated from the fit of this are given on the top panel exactly what I have talked to you about. So hard spheres are a good point to start in case of liquid and amorphous system simulation in case of crystallographic structure if you remember we talked about 32 point groups 14 Bravais lattices and 230 crystal space groups so using those I mean so we can input the crystal structure as one of those space groups and also we can define the magnetic structures in terms of the propagation vector as I initially in case of liquid and amorphous systems either we have to dissolve dissolve to detect Fourier transform what I mean is when I am talking about we are aware that G of R is a reverse Fourier transform from S of Q this is what we can measure experimentally experimentally and we can directly Fourier transform it to get G of R but when we say we can directly Fourier transform it that means this range of my experiment of Q has to be 0 to infinity so this is how it looks for the common metals in molten form as I told you that this is an example this is rubidium at 40 degree centigrade taken from this reference and the structure factor you can see the these triangles are the experimental points and the continuous curve is from a random dense packing of hard spheres and they match reasonably well I would say over the entire Q range reason being that just now I told you that I need to Fourier transform over a very large range of Q often that is not possible for example you see this experiment we could go to 20 angstrom inverse in the experimental set up so that infinity gets cut into 20 and when you have a sharp drop if you put S Q equal to 0 at that then that sharp drop we know in Q space will give rise to artificial oscillations in G of R where we have Fourier transform so here the authors they have done the other thing they have actually simulated a dense packing of hard spheres calculated the GR somewhere played with the GR and matched it with the experimental value so similar to what we did in case of crystallographic material here also we usually start from a model and then go ahead and create the S of Q and check it with respect to the experimental S of Q I will come to this in more details in the later parts so in crystalline solids we know FCC and HCP has a packing fraction of 0.74 actually you can see this but this is the FCC let me just remind you and FCC crystal phase centered crystal which is actually as I mentioned it is your nickel, copper and aluminum so you have got one at every corner one of every corner and one at every corner and also one at the center of every phase that is FCC that is FCC actually this drawing doesn't indicate the fact that you can consider the atom as a sphere so center there is a large sphere then there is a sphere here I am sorry that I have got sphere here sphere here so four there are four atoms atoms per unit cell in itself because we have eight at corners and each shared by 8 unit cell so each one contributes 1 8 so from the corners we get one atom and there are one two three four five six six phases and in the six phases we have got one atom at the center of each phase I am now drawing the size of the atom such that they touch each other then we have got six phases each is shared each sphere is shared by two neighboring unit cells so there are three so three there are six phases and each one is shared by half in one unit cell so half so it comes to three atoms from phase centers and total there are four atoms atoms per unit cell now I can show you that this is the diagram taken from Wikipedia so this is how they are packed so each one is here it is one night you can see this is half these are half this is one night and there are four atoms per unit cell so when you have four atoms per unit cell then you can calculate out if I look at the top phase and try to fit in then you can see this is one here one touching and the corner of the touching so here in this FCC packing each atom is touching the neighboring atom and this FCC so you can see that if this is the side a of the unit cell and r is the radius of it this comes to a and 2 a square this square is this this is r 2 r r that means 2 r r 4 r so a square plus a square is 2 a square by this is r 2 r r so 4 r square 2 r 2 16 r square or a square equal to 8 r square and a equal to 2 root 2 r so in the under the geometrical configuration that the atomic spheres are touching each other each side is 2 root 2 r and I request you to find out that in this sphere there are 4 r square 2 r square 2 r square 2 r square 2 r square 2 r square 1 root 2 r so in the under the geometrical configuration that the atomic spheres are touching each other each side is 2 root 2 r and I request you to find out that in this sphere there are 4 atoms so show that the packing fraction is 0.74 you can check it yourself and then this is same for HCP also same for HCP that is hexagonal cross-pack and so that is a but when into random packing of spheres in case of amorphous materials I would like to mention that this is a very interesting problem one is that the maximum packing you can get up to 0.68 and not only that packing arrangement depending on the packing fraction of the number of atoms that you have got in a certain volume you can get various phases of the material before I get into the experience proper is a very interesting problem is basically packing of heart spheres there have been cases where people are talking about packing of cylindrical materials there are cases where one discusses packing of flat biscuit like objects and chai kin is one experimentalist who has done a lot of this work this is one of his work this here you can see that it goes from liquid to glassy phase depending on the density of the heart spheres in between you have also can see FCC crystal interestingly at certain densities between liquid and glass there is a liquid solid coexistence and that means there are clusters of molecules which are actually these clusters are solid like but in between you have got gaps between them so there is a liquid solid coexisting phase and here they have given a line where the depending on the volume fraction we can see that starting from 0.494 you get random close packing and crystal close packing so random close packing gives me somewhere around 0.6364 now the liquid and glassy state of Jamina as I told you earlier I promised you that the snapshot picture of a liquid is very very similar to the amorphous structure in a solid so this is an experimental data where the Jaminium J E Jaminium selenium glass if the liquid and glassy state you can see the S of Q they are very very similar except the liquid which is a dashed line it is a slightly less intense but the structure is very similar if I talk about shells around the central atom they are very similar and also there are pre peaks reveal a peak at around one angstrom inverse which is the first sharp diffraction peak if this is also observed in many glassy and complex liquids and almost similar values of Q around one angstrom inverse this is actually is the characteristic of anomalous behavior in several properties this I will stop discussing here so now what is the difference in the experimental setup that's a very important thing so what I want to say here that how a crystal diffractometer will be different from a liquid and amorphous diffractometer let us just start with the most basic quantity that we are aware of invoking quantum mechanics we know that if we are going to see smaller and smaller are smaller and smaller are because the Fourier convert from Q space I need to go to larger and larger Q values so typically in our crystal diffractometers the Q value is well of the order of 10 angstrom inverse but now I am looking for local structure and I need to go to large Q so large Q is desirable but how do I go to large Q that is because this is this reason is because I am looking at local structure smaller R values also if I want to do a direct Fourier transform from Q space to R space I need to go to large Q values now how do I go to large Q values one is that by increasing the theta so my angular rate has to increase another possibility is that if I can go to smaller lambda values smaller lambda values smaller lambda as you can see from this expression at the same angle a smaller lambda or higher energy will give larger Q so either I mean so I need handles to go to larger angle that is possible I cover a larger angle using detectors but I can also use for the same angles another set of lambda which will give me larger Q values