 So, let me summarize what we targeted and did in this entire course quickly for your help in understanding various aspects of Neutron scattering. So, this is a brief course summary. I cannot discuss everything lecture by lecture, but very briefly. So, I started with the definition of thermal neutron because when I discuss Neutron scattering for condensed matter is a thermal neutron which are useful and I discussed with you that in the reactor after moderation after moderation we have a spectrum which is typically at the moderator temperature of around for drew ball at a set is around 330 Kelvin approximately 50 degree centigrade temperature of the moderator. Then you have a maxolian distribution of the energy of energy of neutrons and typically this has the peak around 30 millilefton volts and lower energy neutrons are known as cold neutrons typically below 5 millilefton volts and high energy neutrons are called hot neutrons. So, mostly we use for thermal neutron scattering neutrons from this range as well as from cold and hot regions, but mostly in the thermal region. These are typical distribution that we discussed and I justified thermal neutrons because they have wavelength close to inter atomic distance. They have energies typical with as we saw throughout our discussion on inelastic atoms and molecules of phonons diffusion and also vibrational spectrum. They are neutrons so they can get deep in the sample they have got very good contrast between isotopes and neighboring atoms unlike X-rays which depends on X-rays depend on the electron charge cloud and neighboring atoms their contrast is poor, but here not only neighboring atoms, but even between isotope the contrast is good because the neutron nucleus interaction which is the strong interaction in by nature when they have very good contrast between isotopes and notably between 1H1 and 1H2 hydrogen and deuterium. So we can replace hydrogen with deuterium and vice versa depending on the requirement and we can generate very good contrast which is very important because when you replace hydrogen with deuterium the chemistry remains same more or less dynamics become slower, but otherwise they are same and we can study selective parts of an object using neutron scattering. They have a magnetic moment of minus 1.91 nuclear magneton and possibly they are the unique tool so far as magnetic structures are concerned microscopic magnetic structures are concerned and it's a non-destructive characterization so always it is welcome for any samples. I started with the Kwame Golden Rule and describe how I can work out the cross sections using this expression here psi k and psi k prime are the incoming and the outgoing wave vectors I discussed in this course density of states as rho k and here I could use the density of states multiplied by the transition probability from k to k prime and I mentioned to you that for diffraction experiments the magnitude of k and k prime remain same and we talk about structure in terms of elastic diffraction and I derived the angle dependent diffraction law that is G sigma by d omega number of neutrons per unit solid angle and then I found I showed you that there are two parts one which actually takes care of the interference from atoms at site RL and RL prime which gives us the structure and that is given by a pre-factor which is a coherent scattering length and also we have got an incoherent scattering length which is b square average minus b average square and the total scattering cross section is an add of a coherent term and the incoherent term and similarly the total angle dependent scattering cross section has got a coherent term and an incoherent term but in case of diffraction the incoherent part is not interesting it's basically nuisance and it gives me background whereas this coherent part gives me the neutron diffraction pattern from which I try to solve for the structure but I showed you later that when it came to self correlation function in that case incoherent scattering plays an important role and we studied incoherent quasi elastic neutron scattering internally for almost always using hydrogenous material so this was a take and I showed you how to calculate the coherent and incoherent scattering part for that you need to find out b average and b square average so b average comes from the fact that if I use up spin and down spin neutrons then I get two scattering length b plus and b minus and that b plus and b minus the average is given because total spin of the neutron and the system is twice i plus 1 here i plus minus half plus 1 because i plus minus half is the spin of the system and neutron and this is either 2i or 2i plus 2 when it is plus it is 2i plus 2 when it is minus it is 2i is the total number of possible states with spin then then weighing according to the number of quantum projections b average is nothing but i plus 1 upon twice plus i plus 1 b plus when it is plus and twice i upon twice i plus 1 plus twice i which is i upon twice i plus 1 when it is the neutron spin is opposite to the nuclear spin and b square average simply with the same weightage we find out the b plus square and b minus square and if there are several isotopes like if I talk about hydrogen it has got a certain fraction of 1H1, 1H2, 1H3 and the relative concentrations if they are ck then I can use the same formula but now I have to do it for each and every isotope and it becomes ck ik plus 1 so one more suffix k comes and this ck is nothing but the concentration of the isotope so these are the ck for them I have to written right down and then I can calculate b average and b square average and b square average minus b average square will give me the incoherent part and b average will give me the diffraction part the structure factor I mentioned for high neutrons and x-rays the only difference is here in the form factor new x-rays have form factors which fall with q and so at high angle you get low intensity for x-rays in case of neutrons instead of form factor I have got a scattering length bj and this has doesn't have any q dependent so you can you can say the form factor is constant with respect to q for neutrons also I told you or that structure first we can find out absolute periodic structure at 0 degree Kelvin when we come to a finite temperature there are if these are the mean positions in a crystallographic lattice then there are vibrations around the mean positions as we go to higher and higher temperature higher and higher temperature and that is taken care of by a factor known as Debye-Waller factor where e square average is the average size or deviation of the atom with respect to its mean position at a finite temperature so this is due to thermal fluctuation so this is due to dynamics and this is taken care of by a Debye-Waller like factor in case of diffraction and most importantly what I mentioned was that we are seeking information in real space g of rt is basically the correlation function in real space but a Fourier transform over q that means grt d3r takes me to iqt which is an intermediate scattering law so function and then one more integration a Fourier transform over time takes me to sq omega which is the scattering law which we are measuring now it is desirable that I do a double Fourier transform and get all my information on g of rt from my experiment but usually that doesn't happen so I have to start with a model for example for quasi-nitro scattering we started with a model which is e to the power minus r square upon 4 dt of a Fickian diffusion model and then iqt became e to the power minus dq squared t and then sq omega became a Lorentzian of dq square whole square plus omega square so usually we have to figure out a model when you do only structure work then I can forget about omega I'm talking about sq and this comes from iq and they are same if I don't bother about omega part of it so as I said just now the sq omega comes as a Fourier transform over iqt and iqt can also come as a Fourier transform sq omega and from these two expressions you can see iq0 is nothing but d omega sq omega and this is what we do normally when you do our experiments we have the incoming let us say in a reactor monochromatic beam and we have either position sensitive detectors or end on detectors looking at the looking at the sample so we don't do energy analysis means we do an integration over energy and what we get is iq0 so this gives an instantaneous picture on the other hand if we do an experiment where we ensure that the energy transfer is zero then what I get is a time integration of the structure but both of them they are identical because if I consider an experiment in which I find out iq0 but then I collect the data over a finite time which is much larger than the time scales in the system and ultimately I get an averaging over time frame by frame and this is what we get inherently from an experiment at a zero energy transfer so this is the two ways we can do diffraction and most importantly we try to find out pair correlation functions so various experiments this is the proper way proper grammar of writing down the pair correlation function because in quantum mechanics they don't they don't commute with each other so we have to keep them separately and this is a definition and the grammar of writing down a pair correlation function if r i is not equal to r j then it is g distinct that means correlation that means a particle at origin at time zero at origin zero what is the problem of another particle at position r at time t that gives me g of r t classical iconite is a single correlation function and it's a pair correlation function if r zero is not equal to r j and if r r zero is not equal to if i sum over all the positions one a particle is certainly correlated with itself at zero time so at zero time the particle is only correlated with itself which is a delta and this is the time t equal to zero that means this gives me if i have let us say i just take the example of a linear chain so this is the zero particle it is connected with all the particles at zero Kelvin i'm saying so this gives you the pair correlation function this is what you obtain in a diffraction experiment they are since s q omega and g of r t are Fourier inverse of each other so we have to bring in uncertainty principle in our experiments and if i want certain delta r that is related to the delta q in my experiment and if i'm targeting certain timescale or delta t in my system that is related to the energy transfer in my experiment so we need to choose the experiment depending on what you want to see so basically my choice of momentum transfer gives me my resolution in space in structure my resolution in delta t or resolution delta omega in energy transfer gives me the band of dynamics that i can see of various timescales delta t this is just an example which i used that in a diffraction experiment the overall the delta of my experiment is given by 2 pi by q max because that is the range of momentum transfer that i'm having in my experiment and this whole data is giving me my structure in the system so delta r is dependent on 2 pi by q max and i have given an example for a wave vector transfer 10 angstrom inverse with an incident neutral energy of 1.2 angstrom we need to go to almost 140 degree for a typical product diffractometer and then quantum resolution comes out to be around 0.6 angstrom inverse typically in the range of inter particle distances similarly if i want to make the resolution poorer in real space that means here as an example is 30 angstrom then we can go to smaller q transfer as an example if we go to 0.2 angstrom inverse with a 4 angstrom neutron we do the experiment at an angle of 4 degrees only 140 degrees i said here 4 degrees and at this is a typical small angle neutron scattering angular range i discuss with you core consideration and how we have beam lines various beam lines in various places reaching the core then various various additions to the core like cold neutron source and hot neutron source which will shift the spectrum towards the desirable energies experiments where we want to use long wavelength or slower neutrons i need a cold neutron source or experiment where we need large energy transfer or smaller wave vector transfer we need hot neutron source i discuss the beam lines for neutron transportations as well as neutron guides that carry neutrons by the principle of total external reflection before we impinge the beam on a sample we also tailor the geometry of the beam using in coil following into collimeters we remove unwanted neutron energies using filters to give you compromise between size of the beam and the broadening of and the width of the beam rather what is the angular resolution we have what is known as solar collimeters where a large beam is split by use of neutron absorbing materials like cadmium in case of reactors we use monochromatic neutron beams and we need monochromators monochromators are usually crystals that single crystals with some mosaic spray we use like we have discussed copper 111 i think we use pyrolytic graphite 004 so these are the typical crystal crystallographic set of planes which i use depending on what lambda we need at what angle and it works on the principle of Bragg reflection but you also have monochromators based on based on super mirrors where you literally reflect a neutron beam from a mirror optically or you also use mechanical assemblies like velocity selectors where we have cylindrical bodies with helical slots on them they allow a neutron beam of certain wavelength with a broad wavelength distribution to go through and last but not the least and very important i discussed with you about neutron detectors because neutron detectors are very important with respect to neutron intensity measurement in reactors and also installation neutron sources next i discussed with you that with after these descriptions that what can we see with neutrons so there are two parts one is structure part and one was dynamics part and in the structure part we measure intensity versus angles we don't do energy analysis and on the dynamics part we do energy as well as angle analysis in the structure part we went from crystal structure all the way to small angle neutron scattering and reflectometry in our discussion when we discussed inelastic neutron scattering we started with phonon dynamics and ultimately i discussed with you dynamics at nanosecond length scale using spin echo and all these experiments are also having an additional advantage the neutrons having a magnetic moment of minus 1.91 nuclear magneton that all of these have an additional parameter which can be magnetic magnetic mesoscopic structure magnetic dynamics like magnols or magnetic structure crystallographic structure and neutron is the most used tool for understanding magnetic structure and dynamics in condensed matter only which cube and which e range we need to use for a certain range of certain range of structural length scales or certain range of dynamical time scales that's what our choice has to be there when diffraction i went from powder diffraction to single crystal then local structure in liquid and amorphous systems small angle neutrons scattering to neutron reflectometry and we discussed neutron detectors in one lecture where we had discussed with you various types of neutron detector that i can use specifically at the moment position sensitive detectors that we use in reactors and also a scintillation detectors that can be used at spallation neutron sources i discussed with you about neutron polarizers and spin flippers because so i need to often polarize neutron beams and that can be done using branch scattering because if there are some materials like hoistler alloy where you can match the nuclear scattering length and the magnetic scattering length is almost similar amplitude so fn plus fm for one polarization is much larger than fn minus fm which is fn minus fn is almost equal to zero and we after bragg reflection from such an alloy like cu2 ml i get a polarized neutron beam with one spin direction it is also possible to have polarized neutron beam by use of super mirrors where we have critical angles much different for two spins and i have a can have a polarized beam by choosing an incident angle on the super mirror which is between these two values between these two values of critical angle this is the this is theta c for minus this is theta c for plus so i can use an ordinary monochromator first to get a monochromatic neutron beam and then i can reflect it from a super mirror to get a monochromatic polarized neutron beam so we choose the direction by collimeters we choose the energy by monochromators velocity selectors filters we also do polarization using bragg diffraction super mirror reflection of course we can also do using helium 3 transmission not available everywhere in some of the sources by transmission through a polarized helium gas we can get a polarized beam of neutrons in transmission this is a typical powder diffractometer which is used for magnetic neutron diffraction at through work this typical data which has been taken on this instrument and these are fits are by ridvent analysis of the data this is a data from the hrpd high resolution powder diffractometer at icis rather for dappleton laboratory please note the way the peaks go here and here here because it is a dispacing so this is the low q part here and this part is the high q part in the low d spacing range because 2d sine theta equal to n lambda if you see as the d spacing becomes longer or the time of flight becomes longer you go into the lower range of theta in the other diagram uh ridvel refinement is a very commonly used tool and actually most popular tool possibly with x-rays and neutrons which is basically uh chi-square minimization technique like all other techniques the y observed and y calculated we try to reduce the difference between them but it is refinement i told you earlier let me just remind you that you start with a structure known structure and refine it to get a good fit and that way we can get structure magnetic as well as physical structure of materials i just showed you one magnetic structure where you can see we can not only position the atoms by ridvel fitting but i can also give give magnetic moment values at various sides for example in this example the iron there are two irons having two different moments in this very very cyanide sample taken from this reference next i discuss with you the local structures in liquid and amorphous systems just as an example please see the difference here this is what a long range order crystallographic material these are product crystal we look like in a diffraction experiment you have sharp peaks all over q but when we talk about a liquid and amorphous system you can see that s of q it gives first sharp diffraction peaks or second sharp diffraction peaks but it goes to a flat line because in case of liquid and amorphous material you have only local structure and we don't have a long range order and the consequence of that and the s of q you get this kind of few peaks in low q region and then going into flat background and from here we can find out the local structure in glassy materials here we have found out the structure you can see the fit using a reverse Monte Carlo technique which actually minimizes the again the chi-square minimization process but the minimization is done through a Monte Carlo technique we also discuss very briefly single crystal diffractional material because it is difficult to get single crystals but if you have organic single crystals first the difference is that you have a large guanyometer on the sample table because the single crystal needs to be oriented so that the reflection the bright reflection comes onto the horizontal plane in which the detector is moving this is the detector I am here you can see the detector in this photograph and you can find out the structures have been issued using single crystal diffraction followed by small angle neutron scattering so in a small angle neutron scattering you can see the length scale drastically changes in case of crystallographic material we talked about structures of one angstrom resolution now we are talking about sizes which are large 1 to 100 nanometers 1000 angstrom to close to crystallographic structure this entire range covers the range of what do you call as small angle neutron scattering in this cases we don't look at the crystallographic structure of a medium but rather inhomogeneities like this that you can see as shown here in the schematic inhomogeneities and the size of these inhomogeneities in a medium which are often of lot of interest in industries for chemists for biologists and small angle neutron scattering is an important tool if you want to look at mesoscopic lengths such as mesoscopic length scales so then in the same mesoscopic length scale we can use neutral reflectometry for understanding structure and magnetism in thin films and we discussed the techniques of neutron reflectometry there are two kinds of reflectometry specular and off specular in case of specular reflectometry as you see here the results are shown you follow snails law the incident and the outgoing angle or same incident angle is equal to reflection angle of reflection and here by solving the for the structure or you can extract data regarding thickness roughness magnetic moment density in thin films and magnetic materials this is a very important tool for understanding magnetism in thin films there is a technique called parrots formalism where using which we can actually fit a given structure to the experimental data and these fits are obtained actually from PNR the polarized neutron reflectometry for a sample is shown here this is a nickel aluminum alloy and this is the extra reflectometry data the two techniques are very similar only fact is that in case of polarized neutron reflectometry we have an added advantage of finding out the magnetism at mega mesoscopic length scale in case of xrl we can find out the physical structure like density and thickness of thin films using extra reflectometry but advantage is also there that xrl is a technique which has much higher intensity often compared to PNR so any thin film sample before we carry out any PNR on that it is desirable that we do an xrl measurement to find out the structure of the film and we also discussed obstacle and neutron scattering where the angle of incidence and an angle of reflection they are not same and when they are not same in that case when they are same let me just remind you when these angles are same then my wave vector transfer is normal to the film surface but when they are not same when they are not same when theta i and theta f are not same then your wave vector transfer is at an angle to the normal to the film surface and we have got a component which is along the film surface we have a normal component and a component which is along the film surface which can be called qx and this study tells me height-height correlation on a surface so here we did polaroid I talked to you about obstacle and neutron reflectometry especially for magnetic interfaces and I showed you how magnetic interface is smoother than a physical interface at this point I switched over to inelastic neutron scattering and I discussed with you how you can do inelastic neutron scattering so here similar in a reactor you have a monochromatic neutron being prepared by a monochromator after that there is a sample followed by an energy analyzer which is important then inelastic neutron scattering because you need to know not only momentum transfer but also the energy transfer in the experiment and that energy transfer is measured using an analyzer system followed by a detector that this is known as a triple axis spectrometer very commonly used in almost all reactor sources for studying inelastic neutron scattering by using inelastic neutron scattering we can find out dispersion relation or in crystals and also phonon density of states I discussed with you this is an example of measurement of dispersion relation in zircon so basically dispersion relation means we have omega versus wave vector inside a brillo zone and we talk about acoustic phonons as well as optic phonons and transverse and longitudinal phonons and this is all about the phonon dynamics in zircon this is important because this using I mean understanding the phonon dynamics you can understand many thermal and thermodynamic processes in the solid and this is an example of measurement of density of states you can see here we have found out e as a function of q and this is as I mentioned you earlier while discussing it in details that we need to take care of the crystal symmetry and from the crystal symmetry and from group theoretical calculations we find out at which reciprocal lattice space I will do the measurement for which particular class of movements in a phonon dispersion relation and these are mentioned here that these are the various directions at which these not I mean various class of group group theoretical class of motions that have been measured and this is a combination of data from various sources in case of phonon density of states we since we use a powder sample so the there is an averaging over q range and this is how the data looks like we followed it by quasi elastic neutron scattering which I discussed in last two lectures and this is for understanding diffusion in material and this I discussed with you with respect to various organic materials in zero rate cages for when I chose the example for drova and also I showed you how such slow dynamics can be measured using spin eco spectrometer in the beginning of the talk today so this is a quick summary of all the things that we did for neutron in neutron scattering during these lectures delivered by me thank you