 Hello, I have completed the course on Neutron Scattering and I thought that those who are waiting for the examination, some discussion and also those who have completed the course, some discussion and some general information about neutrons and neutron scattering will be relevant. This is just to further your knowledge about the particle that you are using for neutron scattering experiments. The technical part of the course is over. This is more of general information which may be good for future applications also and for some general knowledge in the field. So we have been talking about scattering of thermal neutrons. So let me just say that again and again I said that neutron is a neutral particle of mass almost equal to that of proton. Actually neutron mass, mass of neutron is possibly slightly higher than mass of proton because we can consider neutron as consisting of a proton and an electron and a free neutron. That means the neutron which has come out of the nucleus is not stable and it actually decays into a proton and an electron plus an electron antineutrino. So this is what happens but the lifetime is quite long. Actually free neutron lifetime is around 15 minutes, approximately 15 minutes I mean say more accurately it is around 874 seconds. So this time is much longer compared to the time scale that we measure in neutron scattering. Typically I would say 10 to the power minus 9 to 10 to the power minus 13 seconds. So for all practical purposes all the experiments that we do using thermal neutrons, cold neutrons or epithermal neutrons we can consider the neutron as a stable particle for these experiments. Though neutron lifetime its measurement is still a major serious issue and in several reactors particle phases work on them because these have consequences on the lifetime of the universe but that's beyond the scope of the present time. Neutron mass as I said if you talk about energy its 935 mega electron volts is the mass of neutron in energies otherwise it is around 1.674 into 10 to the power minus 27 kg and this is some information on neutrons and we get neutrons from fission that means when a neutron is bombarded a thermal neutron that means almost zero energy bombards the uranium nuclei because it is neutral it's easy to penetrate the nucleus it forms a compound nucleus and then it breaks into usually two components not exactly always two or same mass but typically you can say barium and krypton are the fission products but there are ranges of fission product plus it produces 2.5 on an average more number of neutrons which are thermalized as I discussed earlier and used for thermal neutron scattering for condensed matter studies. You must remember that this is what happens in a nuclear reactor I talked to you about a spallation neutron sources where again there's a proton accelerated proton beam impinges a high Z target so it is not a fission reaction high Z target it can be tantalum it can be zirconium it can also be uranium but usually fissile uranium is not used and then the reaction is not a spallation which is boiling off so when it hits a high Z target it boils off and gives nearly 30 neutrons here and then these 30 neutrons are again similar to a fission reactor are brought down in energy and then used for using neutron scattering experiments but interestingly here we use a monochromatic beam usually in a reactor and we do the momentum transfer which I discussed with Egan again by variation of angle with a monochromatic beam maybe in different settings you can change but again change the theta to get to various Q values in a spallation neutron source it's a pulsed source so that neutrons are coming in pulses typically in pulses like 50 hours 20 hours in a second since it's a pulsed source it is also convenient to use the entire polychromatic beam for scattering experiments or in lambda versus lambda if I plot a neutron scattering experiment of beams we can use and since it is pulsed we can use what is known as time of flight time of flight spectroscopy that means we use non relativistic mechanics and when you use a polychromatic beam this lambda 4 pi by lambda sin theta this lambda in a polychromatic beam for a fixed theta this lambda is changing and when this lambda is changing your Q value is changing so I can do for a single setting without moving my detector or with the one setting of the detector I can get a range of lambda or range of Q because now Q and lambda they are tagged in case of a spallation neutron source and typically I can tell you a one angstrom neutron moves typically around 4000 meters per second so really speaking a distance of 70 80 100 meters if I consider from the source to the detector we can easily tag the time starting from the source and it's linear so that means four angstrom neutrons if I consider it will go around 1000 meters per second so in some of the problems that I've given you can use these things and you can calculate out various parameters the problems given so this is one part I wanted to share with you regarding neutron scattering I continuously use the term Born Approximation and I use terms like kf v ki to calculate out the scattering amplitudes but here one thing needs to be told to you that neutron is an S wave scattering S wave scattering I will just I did not discuss this part because this is what that means if I have a point target if I have a plane wave of neutrons then this plane wave I can break it up into spherical waves to spherical waves let me just show you in a spherical waves like this and these spherical waves have various l values l value means we know l is the momentum with respect to the center around which we do the breaking so I can find out the e to the power ikz if it's a plane wave I can break it up into l equal to zero to infinity or a radial part and a theta part for the spherical wave and then you can see that there is because neutron nucleus interaction is extremely short range you can I can say that if I break the plane wave into spherical wave of various l values 0 1 1 2 only the l equal to zero part of this plane wave contributes which is S wave so that's why neutron takes S wave scattering scattering scattering and this is scattering means that neutron nucleus interaction is extremely short range and out of the plane wave I can only choose that one which is going for a head on collision with the sample others other neutrons will not interact because it is a charge less particle and it is only S wave that means when it is going and hitting the target that neutron will only be considered others in the plane wave are not considered so it's a radial wave and then if that is the case I mean I have shown it in a diagram here and then the this part that we can write down the scattered wave amplitude at r going to infinity as consisting of a incident plane wave a scattering amplitude and a radial wave function which is going out if the sample is hit by a neutron then the scattered wave goes out spherically and this is given by spherical wave it is equal i k r by r so scattered flux if you consider f as a scattering amplitude is given by this and the scattering cross section is 4 pi modulus of f square where f is given by a real part and then imaginary part so now with this I will not discuss all these things all I can tell you that there is something called an optical theorem where the total scattering cross section is given by the 4 pi by k imaginary part of f where f is equal to f0 plus f1 it can be taken from the imaginary part of f and this is known as optical theorem in neutron scattering I did not discuss this earlier this comes more under the purview of nuclear physics but this is for your knowledge that this is the possibility that is this part of imaginary the imaginary part of component of the f gives us sigma total in terms of the expression given here and next part is the question paper that I have said now as I told you that questions are there using various techniques numericals are there but I would tell you that there are many questions where if you look at the answers by process of elimination without doing any calculations you can reach the right answer for example I can give a very macroscopic example for example if I consider that distance between Mumbai and Delhi I can give if I give the answer that one is 800 kilometers one is eight lakh kilometers and one is two centimeters you need not do any calculations just from the figure of Mary I mean from the distances listed you can see possibly 800 kilometers will be the closest to the answer and I have always said closest to so please you can do the numerical analysis or you can do the process of elimination by your logistics and arrive at the answer and my demand is that you give the closest answer because sometimes the numerical values may slightly differ from one another this is one and many of the questions are recalled type that means things which I discussed or gave you the information during my lectures you can use them to answer the question some questions they do need numerical calculations as I told you some of them you can arrive by process of elimination but many of them you need to do the calculations and I think the calculations will be simple enough for you if you have followed the lectures regarding scattering experiments please remember I have used two terms Bragg angle angle angle of scattering please note if I use the term Bragg angle you will realize that usually you are aware that 2d sin theta because the lambda is this angle theta but if you consider the angle of scattering then it is 2 theta so the Bragg angle Bragg angle is half of the scattering angle for the same setup maybe I use Bragg angle and scattering angle interchangeably I want you to accept this part that the Bragg angle and the scattering angle they are half of each other in this in these lectures and sometimes in the questions maybe and what's from there another part is that for neutron scattering cross-section which you might need one is that scattering cross-section as a function of energy of function neutron energy these are available as tables published by IEEA International Atomic Energy Agency for various elements you can access them from there because when you go to higher energies you can have resonance absorption also we are in the thermal energy range where these are not there and we consider the scattering cross-section cross-section as constant constant and I am refraining from mentioning the complete thing that you can use bright Wigner formula in nuclear physics to get scattering cross-section which also takes care of the inherent resonance scattering and you can write it as a radius part and then there are resonance energies plus half width for nuclear absorption and scattering and square of the whole thing so if E is much much less than e gamma the er sorry resonance then only 4 pi xi squared term remains which is same as what we were writing that 4 pi b square we wrote earlier so it is only this part but when you go close to resonance then you also need to consider the half width of the resonance at epithermal energies and this is known as the bright Wigner formula this I did not discuss earlier because for us the only the weak scattering limit and the scattering cross-section using coherent and incoherent links were enough to do all our theoretical and experimental data analysis thank you this with this I wish you all the best for the examination thank you