 In this lecture, I will cover diffraction at various length scale. I will introduce you to it and in the following lectures, I will discuss in details various important techniques under diffraction. And I will also introduce you to the fact that the diffraction experiment is done in reactors and in spallation neutron sources or pulse neutron sources in two different ways and I will introduce you to both of these techniques in reasonable details. Now going to the fact that we can do diffraction at various length scale, what I mean is that mostly we are introduced to diffraction either in light where we talk about diffraction or the bending of beams at an edge or we talk about diffraction to find out crystallographic structures. But actually diffraction is an elastic experiment where you determine structure at various length scales in various skew ranges because they are the conjugate of each other. So, with this starting let me tell you, I had shown this slide to you earlier. Again I have bring it to you notice that studies with neutrals in condensed matter can give us structure and dynamics as we said. In case of structure, we measure intensity versus angle or we will translate to intensity as a function of q because for structure we are not bothered about energy transfer and q is given by 4 pi by lambda sin theta which is the angle in this slide. And we measure intensity as a function of q from where we go back to g r which is a general correlation function in condensed matter. So, that means in diffraction we cover a wide range of length scales. So, for example, for crystal structure we work at angstrom and subangstrom level and for liquid and amorphous systems we can go up to fractions of an angstrom because in liquid and amorphous systems we have local structures, but no long range order. Then at mesoscopic length scale we can study in homogeneities typically 10 to 100 angstrom like micelles precipitates in metals or even pores in stones at 10000 angstrom and also under the same category of studies we have thin films where we can understand layered structures that means their thickness, interface roughness through reflectometry. Reflectometry can be done using neutrons and x-rays both, but importantly because neutrons are sensitive to magnetic moment for all these techniques which I mentioned here, all the techniques that I mentioned here and over playing feature is the magnetic scattering of neutrons. The other half is about measurement of intensity as a function of energy and angle both. So, there like for structure I said range of structures in case of dynamics we change what time scale of dynamics. So, you can study phonons, you can study rotational diffusions or you can even study slower dynamics at nanosecond time scale of relaxation of polymer backbone. These experiments per se are more difficult than the experiments where we do structures because here we have to analyze the energy of the scattered beam and then the question will come at what q range and in what e range. So, length scale of dynamics and energy scale of dynamics, length scale is given by q and energy gives us the time scale of the dynamics. If I do not have e resolution then q range gives us the length scale of the structure that we are trying to study. So, now in this half of my talks I will be in general addressing the elastic scattering that means no energy analysis and as I told earlier that in a reactor the incident beam is collimated as well as made a monochromatic beam using monochromators or velocity selectors and we define the energy and direction, direction is nothing but given by k it is a k vector which is magnitude wise 2 pi by lambda and then the scattered beam goes in this two dimension it looks as a circle, but actually it will go on a sphere and then we can use detectors usually detectors and inside the shielding way well above one ton. So, it is difficult to cover the entire 4 pi with detectors in a reactor in a monochromatic beam usually we will be in a planar geometry where either you have the end on detectors taking data serially angle by angle which was done earlier or today we have got position sensitive detector where the neutron detectors at a certain position gives the direction of the neutron scattering because knowing the distance from the sample from the sample to the detector and knowing the resolution of the detector we can tell it exactly what is the position and what is the position resolution in the experimental set up. So, with this typically this is the powder diffractometer at Dhruva I just show you the outside of the detector bank. So, here as I showed that there is a sample from which the neutron beam is scattered goes in various directions and we have position sensitive detectors the bank that I showed you. So, here if the distance is d and the position resolution is delta L and then one is that knowing the position average position this if this distance is d then from the distance the channel number I can see that channel number by d will give idea of theta and the delta L by d will give you the resolution in the delta theta of the setup. So, this is the detector bank you can see it from the back side of the detector bank the monochromatized drum is here at the center of which there is a monochromatizing crystal the sample position we can't say it is over somewhere over here below and then you have this detector bank and this is the counting electronics as well as data collection software. So, this is Dhruva I just want to show you typical data this is how it looks like after read weld analysis. So, I have to introduce you to read weld analysis for data specifically I will be deal with crystallography together with magnetic crystallography because neutron is extremely important not only for finding out crystallography structure but also to determine the magnetic structure actually that is a string of neutrons because crystallography structure can be easily detected by X-rays. X-rays till now by far the most important tool to find out structure of materials except for the fact that if our materials have low Z elements like you can see oxygen over here X-rays are insensitive to low Z material and that's why whenever my structure has got low Z materials neutron is a better choice but more importantly if it is a magnetic structure for example iron ferromagnetic iron nickel cobalt or their compounds like nickel zinc Fe2O4. So, these structures and their magnetic structures are sometimes commensurate sometimes incommensurate and neutron is possible with the only tool to determine the structure of it. Here the data you see this is as per angle versus intensity fitted and the data is from Dhruva. So, this is a typical data and you can see the how the fit looks like and from here we can at least in this case of this sample magnetic structure was determined. Now, this is the photograph I have given the source here of the one of the most popular and possibly most used diffractometers at the reactor in ILL Grenoble. You can see the similarity with the previous photograph this is at Dhruva this is at ILL Grenoble. Here also we have a position sensitive detector bank the beam comes on to the sample at the center of the sample table and of course in this case we have much higher resolution much better intensity but typical structures are very similar in reactors where usually you have a sample position surrounded by position sensitive detectors and things like resolution here intensity because I if you may not remember I showed you that there is a vertical detector bank for this particular there is a detector bank which is vertically focused. So, this is vertical not horizontal. So, you have detectors vertically focused detectors raising the intensity at the sample position being vertically focused and each each face has got a detector number of detector strips. So, this is vertical and it is bent like this in the vertical direction it is bent actually it is bent like this as I showed you and horizontally also it gives a large beam by bending the crystal in that direction. So, these are the ways one can enhance intensity and make a compromise between the intensity and resolution in various spectrometers but in general the powder diffractometers in most of the reactor sources look like this as I showed you they look like this and then let us go to how it will be in a spallation neutron source. In a spallation neutron source we have a polychromatic beam and you measure time of flight. How do you measure time of flight? So, primarily there is a proton beam which comes and hits a target hits a target which can be uranium and a high-z material like titanium. Now when it hits a target we start a clock we start the clock it is like starting a stopwatch the neutron goes actually neutron does not go directly this is wrong let me correct myself from the dump it goes to a moderator moderator it gets moderated then it goes to the sample and gets scattered and once it comes to the detector I stop the clock. So, start to stop I measure the time of flight this will be typically either I don't know whether I mentioned it will be typically around say four angstrom neutron four angstrom neutron it covers something like thousand meters per microsecond so you can imagine if it is one angstrom neutron it will be covering around four thousand meters per microsecond these length scales are typically tens of meters it can vary depending on the resolution that you demand from your system so it is easy to measure the time of flight in these polychromatic pulse beams if you want to do a similar experiment in a reactor then I have to chop the beam if I chop the beam I throw away a lot of neutrons and suddenly I can do time of flight spectroscopy in a reactor but at the cost of lots of neutrons whereas in case of spallation neutron sources because it is typically a pulse source the source is suitable for time of flight spectroscopy for example the Rutherford Appleton Laboratory in RL they have got a 50 hertz source 50 hertz source that means 50 pulses per second that means there are 20 millisecond between the pulses so I can count neutrons pulse by pulse one first pulse start the clock neutron detected stop second pulse start the clock neutron reaches detector stop the pulse so that is how pulse by pulse we can do the spectroscopy so in this case 2d sin theta equal to lambda now I will quantify 2d sin theta equal to lambda is a Bragg law but in we know lambda is equal to h by mv for a neutron or for any particle is the Broglie wavelength and v for a particular wavelength of neutron is given by v is equal to l by t which is a time of flight so 2d sin theta will be equal to h by ml by t this is the Bragg's law for a pulse source for a particular time of flight in a polychromatic beam so now in case of pulse sources we don't try to monochromatize the source we choose neutrons in a pulse but of course we cannot take all the neutrons so there is something called choppers combination of choppers will actually allow a certain ring of wavelength for our application and after that we do time of flight spectroscopy using this formula so now my resolution in case of Bragg's law for a monochromatic beam with a mosaics straight it reads like delta d by d square is one is a wavelength resolution other part is the angle resolution here by using this formula I get delta d by d square depends on delta t by t the time of flight resolution delta l by l the time of flight resolution because time of flight will have some uncertainties like detector thickness and the angular resolution so both of all three of these can be improved if we use a detector bank at far away because if l is large then for a given delta l for any kind of length determination delta l by l will be fractionally small similarly for a large t by delta t by t will be small if I go further because further length larger t longer l longer the l longer is t and delta t be smaller and similarly if I use a detector bank if I use a detector bank with detectors like matrix matrices like this matrices like this if the if I consider the neutron detector in a in one of these strips has got some delta x delta y that angular resolution goes as delta x by l and larger the l given the size of a detector the angular resolution will be better delta theta will be better and a large angle it will be even better so larger flight path gives larger time and a detector bank with with matrices of detectors with better angular resolution and for that and in backscattering geometry cot theta is the best should be nearly zero at 180 degree it is equal to zero so in backscattering geometry at a large distance with a long the flight time we get the best resolution and this is so a long flight path and the large flight time at large angle will give the best resolution and delta theta can be decreased as I told you using strip detectors usually in most of the detectors most of the experiments the primary flight path is the source to sample l1 and secondary flight path is sample to the detector l2 l1 plus l2 dictates the l usually l1 is kept long and l2 is few meters except for the cases when we want to have very high resolution so I just show you the schematic of one of the best powder diffractometers in rutherford appleton laboratory I have given the source here so this is called HRPD high resolution powder diffractometer in circle and shown here so there are detectors at shorter distances but the high resolution detector bank is nearly 90 meters from the sample 90 meters I will for comparison I can tell you to please compare it with the 100 meter rest dash in an olympic so it is that long so the neutron has to fly through 90 meter path before it reaches the detector so there are other technical difficulties which I am not discussing at the moment you have to understand that the neutron if it travels through air then it will be highly absorbed in air and then we lose intensity of neutron so basically it is traveling to evacuated tubes 90 meters long and then the detected at the high resolution detector bank in HRPD so in general this is like the this is the guide tube which brings the neutron the thermal neutron to the sample you have two sample positions and low angle detector banks and of course you have the wax scattering detectors in this particular experiment so now I show you the data from HRPD the first thing that you can notice is that because the resolution is extremely good you can see the peaks are extremely narrow but the other part is that if you see this x-axis it is de spacing so if I see the intensity as a function of theta in a monochromatic beam this is intensity then 2d sin theta is lambda so as I go to larger and larger theta sin theta goes up de spacing goes down so we see peaks are as I showed you there so small de spacing peaks come at larger theta on the other hand if I see the time of flight spectrum the de spacing you can see the de as the de spacing increases the formula that I wrote 2d sin theta equal to lambda which is h by m l t so you can see as the de spacing increases the time of flight increases for the given fixed flight path and another given fixed angle so here when we plot as a function of de spacing or as a function of time of flight in case of spellation neutron sources the diffraction pattern it looks like the mirror image of what we see in a diffraction pattern from a reactor source that's why the same pattern which I showed you earlier I just want to show you here again this is the pattern which I showed you from Dhruva I have kept it as an inset to this data from HRPD so this is the time of flight or de spacing axis there you can see that there are mirror image of each other if I put a mirror in the middle this and this data they're reflecting or reflection of each other but most importantly here you see lot sharper peaks in case of time of flight spectrum diffraction spectrum in a pulse source so but there are there's a very serious issue which is known as frame overlap problem in time of flight at pulse sources so this is like this now I have started the clock when the proton has hit the target and that is my t equal to zero after that with all uncertainties regarding moderation and transport the neutron beam is traveling now consider that I have a neutron beam of spectrum with fast and slow see this is time this is distance so we know distance is given by velocity into time so it's for smaller velocity it goes slower so it reaches the same distance at a later time at a later time whereas it reaches a faster time and possibly this is the band of neutrons which have allowed to pass through my choppers so this is the band which I'll be using for my diffraction and I need to measure time of flight but interestingly here you consider the next pulse again I have the same band it goes like this but now imagine this one this one and this one so the slower neutron of the previous pulse is taken over by the fastest new faster neutron of the next pulse so now this is like you might have seen sometimes that in a race some competitors they are lagging by one full lap and because it's going in circles you don't know whether he's first or he's the last because you see the other competitors going moving with him and it's very difficult so this is known as frame overlap this fact that the slower neutron of the previous pulse are caught up by the fastest neutrons of the next pulse so this is what I tried to show you pulse one pulse two pulse three through the chopper the same range of wavelength passes but if I go to and if I go to very long distances if I go to long distances there's a good chance that the pulses will be over lagging the beam is and the beam is polychromatic in case of pulse neutron sources so for an example consider icis as I told it's a 50 hertz source the time between the pulses 20 microsecond now a one angstrom neutron is traveling nearly 4000 meters per second if the detector is at 20 meters it reaches the detector at in 5 milliseconds that we can see so a 5 angstrom neutron is 5 times slower so it reaches the same detector in 25 millisecond then the one angstrom neutron from the next pulse reaches the detector it takes 20 meters it gets 20 plus 5 millisecond because it takes 5 milliseconds to reach and it started 20 milliseconds later so it reaches 25 milliseconds so the 5 angstrom neutron of the previous pulse has been caught up by the one angstrom neutron of the next pulse and so we can't determine the time of flight if I allow this frame overlap so that's why we not only use polychromatic beams but you also need to use frame overlap choppers that is restrict the wavelength band depending on how far we want to put the detector and that depends on what is the resolution that we are demanding in these experiments