 So, now when I consider the whole system, there are there is a distribution of the initial states. So, I have to do it between k prime, k prime potential and k and have to square it. So, now the thing is that k prime minus k minus k prime gave me e to the power i q dot rl here right, but now I have a squaring to do and also I have to do a statistical averaging because not only that it goes from k to k prime if the initial state was lambda and the neutron spin was sigma is also going from lambda to lambda prime sigma prime. So, I have to do a summation over all the initial states with the respective probabilities and also summation over all the final states. So, let me say once again what I am trying to say is the following see now d sigma by d omega will be given by a squaring of b l e to the power i q dot rl by this is the summation, but you also have to do a here I have taken care of k minus k prime, but I am doing it because it is going from some final state from some initial state where this lambda includes energy spin everything and this also includes energy spin everything in the final state and now I have to square this square this because this is the square of this gives me d sigma by d omega is the square of the scattering amplitude. So, this is what I wrote here. So, it is the final state it is the initial state then p lambda p sigma sum over all probabilities of initial states of spin energy or anything else isotope. Now, you can ask why I have not put p lambda prime p sigma prime because this expression gives me probability of all initial states then the probability of transition from the initial state to the final state to all possible final states. So, that takes care of all the probabilities and that is why there is no p lambda prime p sigma prime just summation over all the final states and probability of all the initial state and summation over all the final states, but now this bracket I can move around. So, now lambda prime sigma prime I call it dot dot dot lambda sigma just treat this part square is equal to and then sum over lambda prime sigma prime sum over lambda sigma that is what I have to do. Now, here I can this square I can recap like this lambda prime sigma prime lambda sigma into lambda prime sigma prime one complex conjugate into lambda sigma because there the complex conjugate of one with another will give me the modulus. Now, you see these are separated wave functions. So, then sorry here it will be because complex conjugate will be lambda prime sigma prime. So, now I can move them around. So, I can write this equal to the summation and everything double summation remains and I can write it equal to lambda sigma lambda prime sigma prime I bring this bracket to this side then I have lambda prime sigma prime this is also complex conjugate I will show you how the complex conjugate comes then lambda sigma and then I have summation I have summation over lambda sigma and lambda prime sigma prime. So, interestingly I have a summation over lambda prime sigma prime lambda prime sigma prime lambda prime sigma prime what does it signify this is in quantum mechanics known as a projection operator you might have known it, but still I will quickly explain to what it means it means that these are the all possible final states. So, it is like a projection of the wave function into all the components in this space lambda sigma space that means for example, suppose I take a wave function psi. Now, psi if I can break it up into a basis vector then psi is nothing, but a i psi i where psi i forms a basis vector and the fact is that just like normal classical geometry if you add up three components of a vector then the respective components like if I call it a i then their sum over a i square should be equal to 1 because the overall this is the x possible this is the this is one component this is this is another component and this is another component and if you see that cos theta plus sin theta sin phi plus sin theta cos phi if you square up these components all these components you will get equal to 1 it is exactly same only it may not be three dimensional it can be higher dimensional, but the components will add up to 1 for the wave function here if I pre multiply and post multiply with a wave function then this is nothing, but the component of this wave function projected on this component of this wave function projected on this. So, I can call it a i or I might call it a lambda sigma and this two multiplication gives me a lambda sigma square and that is equal to 1 that is why in normal quantum mechanical parlance I call this sum of this projection operator equal to 1. So, what it helps me how it helps me is here because of that I can take the sum over all the final states and then I can put them equal to 1 and then what I am this expression equal to 1 and then what I am left with is sum over p lambda p sigma lambda sigma then the two compressed conduits of each other and lambda sigma. So, I need to add up over all the initial states with this bracket notation lambda sigma and then in between I will have BL complex conjugate BL and e to the power i q dot RL minus RL prime. Stay with me I will explain it to once again let us see. So, I had sum over L e to the power i q dot RL BL when I do this summation when I do this squaring what I need to do actually first I justify that I will have lambda sigma and lambda sigma, but what more I have got is actually the complex conjugate of this is this L is a dummy variable. So, another summation will be L prime BL prime star e to the power minus i q dot RL prime. So, now when I add these two in the middle of lambda sigma what I get is e to the power i q dot RL minus RL prime then I have lambda sigma lambda sigma and I also have here I have BL prime star BL in the prefactor. So, I have got let me write it once again lambda sigma BL prime star BL e to the power i q dot RL minus RL prime lambda sigma I have to sum up over all the sides I have to sum up over all the initial states of spin energy isotope everything. So, I get and of course, I have to take probability of the initial states. So, I have got p lambda p sigma and I explained to you why the summation over lambda prime sigma prime it went to 1 because there is a projection operator. So, I get d sigma by d omega equal to this expression I hope I have been able to bring it home that this is the differential scattering cross section per unit solid angle given that the atoms let me call them atoms or whatever units are sitting at these sides of corresponding scattering length of BL and BL prime and I have to do the summation over all the initial states. Now, the question is I have written it like this over here, but you please see this expression cannot have any spin and isotope dependence because this is the wave vector transfer and this is the position vector. So, I can take them outside. So, after taking them outside so that means I have a summation over l l prime before that I have a summation over lambda sigma I have got lambda sigma BL prime star BL e to the power i q dot rl minus rl prime lambda sigma. So, this part this part e to the power i q rl minus rl prime does not have any dependence on isotope or spin. So, this I can take out of this integral. So, that means now I have got sum over l l prime e to the power i q dot rl minus rl prime vectors vectors, but now I have got lambda sigma lambda sigma BL prime star BL lambda sigma that means this side at l prime and l there are scattering lengths which depend on the isotope it depends on energy if you talk about energy changing or energy in the of the neutron it depends on the spin of nucleus versus the neutron up or down and this averaging sorry again I missed here there should be p lambda p sigma. This averaging with respect to the initial probabilities is something which is unique for neutrons because this scattering length is dependent on isotope and spins it is varying from side to side and this averaging we need to do. So, that means we need to find out this part will till later p lambda p sigma lambda sigma star BL this is an averaging over the entire sample which we have to perform. So, now to please look at this expression. So, this is what I was trying to reach through this discussion. So, now this is unique about neutrons and this you will not see for x-rays which is the closest cousin of neutrons so far as diffraction is concerned and why not I will explain to you in my talks later and even today because in case of x-rays the scattering is from the charge cloud around scattering particle like a natum and this charge cloud then it does not vary from side to side it does not depend on let us say polarization of x-rays or it also does not depend on the isotope. I mean if I am talking about scattering from nickel so nickel and nickel 62 their charge cloud size is same and so there is no variation so far as x-ray scattering length is concerned. So, scattering cross section depends on the isotope nuclear spin and neutron spin and that is contained in this. So, now let me say we need to now define an averaging process in which I call it BL prime BL star this is and also it is an average of that I need to do the averaging over initial states. So, this value of BL prime star compressed conjugate and BL and average over that now there are two parts to it now let me come to what I call as coherent and incoherent scattering cross section. In this double summation which is there is a summation over L L prime so it is a double summation now you imagine I am talking about L H side and L prime side. So, there are pairs there are half N N minus 1 pairs now in that when I am talking of the same side that means L is equal to L prime then it is BL star BL same side and average of that and that is nothing but B square and average of that. So, that means for L equal to L prime it is B square average, but when L is not equal to L prime let us assume that the isotope and spin in one side has no correlation with the isotope and spin at the other side except for that they are statistical that means if two isotopes are if an isotope has one percent abundance then a site will have that particular isotope as a probability of 0.01 and other isotope will be 99.99. So, if it is a 10 percent abundance then the probability at a site will have that isotope is 0.1 or the probability will not have the isotope there is 0.9. So, this is not dependent only statistical averaging. So, then BL prime BL star average will be given by an L not equal to L prime 1 B average over isotopes and then square of that. So, I have got 1 B square average and when L is not equal to L prime because they are not correlated I will do the averaging separately. So, this is equal to let me just write one more line this is equal to BL prime average and BL average now because this statistical quantity the sites there is nothing to choose between two sites. So, these two averages must be same if I if I do over the entire sample the summation these two averages should be same and they should be B average and this is give me this will give me B average square. So, I have got a B square average and I have got a B average square. So, now please know that this sigma by d omega this expression will come to now that I have to do this averaging what I get is this one is summation over L L prime we know this averaging is over average over lambda sigma please note that this summation I have not done yet this averaging is over lambda sigma. So, now this average when L is equal to L prime then it gives me B square average and when L is not equal to L prime then it gives me B average square. So, I can write this average first like this please know this is just consolidating this expression that when L equal to L prime it is B square average and when L is not equal to L prime it is B average square look at this expression when L is equal to L prime I wrote it as equal to B average square and delta L L prime B square average minus B average square. So, when L equal to L prime then these two will cancel out and I will have B square average that is what I said when L equal to L prime it should be B square average and when L is not equal to L prime then then this term is 0 I have B average square which is what I said that when L is not equal to L prime you have got a B average square because two sides are independent of each other and averaging is being done over isotope and spin. So, it will be B average and square of that, but now with this expression I can write d sigma by d omega into two parts one is d sigma by d omega I call it coherent I will tell you why and d sigma by d omega in coherent because you please note that d sigma by d omega coherent is I am defining as B average square e to the power i q dot r minus r L prime sum over L L prime that is when L is not equal to L prime this expression gives me i q dot r minus r L prime for a lattice this is the expression which gives me diffraction and look at the other expression because when L equal to L prime then e to the power i q r L minus r L prime it goes to 1 and I have got this summation gives me summation over L L prime it is gives me just because L not equal to L prime will not be there this n into this is what I wrote B square average minus B average square and see this has no dependence on q dot r L minus r L prime it has no angular dependence. So, this is that is why I call it incoherent because it has no this expression has got angle dependence this gives me diffraction this does not have any angle dependence this is give me a background. So, in case of neutron diffraction B square average minus B average square this term gives me incoherent scattering incoherent means the sites the scattering from one side is not correlated with the scattering from another side and it gives me a background and when I talk about coherence then I talk about B average. So, basically when I go from side to side there is a mean scattering length when I go from one side to another side to another side to another side when I do the averaging over all the spins and initial states I have got a B average sitting at every side. So, and also there is a fluctuation. So, there is a mean value and there are fluctuations around this there is fluctuation around this fluctuation is given by this which gives me an incoherent background. And this expression the B average gives me diffraction this is completely different from what you find in x-rays because in x-rays the charge cloud does not change from side to side. So, there is nothing like B square average minus B average square. So, this will be 0 for x-rays we and we have this side which gives me diffraction only this is not just a scattering length in case of x-rays I will explain to you later and derive for you that it comes this B average is comparable to form factor if you are familiar with extra diffraction you know what is the form factor this will be familiar it will be compared with the form factor. So, now neutron has a coherent and incoherent part in its cross section and it will have a lot of implications which we will discuss later. So, basically excuse my drawing basically what you find as intensity versus Q in case of neutrons you get the Bragg peaks this is what I mean and this comes from B average this coherent peaks and also you have an incoherent background lying below it if you have incoherent scattering cross section non-zero for that particular sample for example one of them is hydrogen which has got a very large incoherent cross section and why I will explain to you later. So, now let me consolidate I started with Fermi golden rule I derive scattering amplitude then I derived the scattering cross section d sigma d omega angular angle dependent scattering cross section then I did an averaging over the initial states of spin and isotope and I showed you that we have a coherent part which gives me diffraction and an incoherent part which gives me background and this is unique about x-rays. So, with this I come to the end of this lecture I will more discussions will follow regarding neutron diffraction in the later lectures. Thank you.