 So, in today's lecture, I will give you a brief recapitulation of what we did regarding the evaluating the scattering cross section for diffraction of neutrons. We have started, we have started with the Fermi-Golden rule and then we went ahead and calculated a fkk prime which is a scattering amplitude from either a single scattering object or an assembly of them. So, today I will go ahead and using the same formalism, I will show you that for neutrons we have something called coherent and incoherent scattering cross section which is not there in x-rays and I will explain to you why they are not there and then I will also go for a comparison between x-rays and neutrons because extra diffraction is the most commonly used tool by researchers in any field of condensed matter be it physics chemistry or material science. So, we will do a quick comparison and show you what are the similarities and what are the dissimilarities between neutron and x-ray diffraction. Please remember till now I am doing diffraction that means I am just trying to get the structure of a material using neutron diffraction. Now we will go to the lecture proper. So, as I indicated, so recap and calculating coherent and incoherent scattering cross section for neutrons will be a starting point for today's talk. So, earlier also I had written that for a rigid lattice rigid means physics concept wise this is a lattice which is 0 degree Kelvin. So, I have a regular arrangement of atoms but they are rigidly fixed at the sides. I will come later what happens when you come to finite temperature. So, this rigid lattice I can consider as a sum of delta function potentials at site RL and acting with a scattering amplitude BL with the neutron. Now, this delta function justification also I had given if you remember I said that the VR the scattering potential first for a single nucleus it is delta function because the nuclear potential is infinitely narrow or almost infinitely narrow with respect to the wavelength of neutron. So, lambda is much much greater than the nuclear radius and when I consider a lattice where the points are fixed in space like this they are fixed in space then this each point is a delta function and I can write now the VR is a sum of such delta functions and I wrote sum over all the lattice points. This was the potential offer by the entire lattice and if you remember my scattering amplitude f k k prime means the neutron when from an initial wave vector k the final wave vector k prime was given by by this and d sigma by d omega is equal to f k k prime square modulus square and you can see this the m by 2 pi h square in f k I have to cancel them in potential I have put as constant term 2 pi h square by m now with this so let me just repeat the last equation and we showed you I showed you for a rigid lattice because for a rigid lattice I will be replacing the p at the v by sum over delta l l prime and k k prime goes to k k minus k prime is equal to q so it goes to sum over l b l e to the power i q dot r l for the rigid lattice it becomes a sum over all those delta functions and the delta function gives you k prime k is equal to because e to the power i k dot r was the incident wave function and the outgoing wave function its complex conjugate was e to the power minus i k prime dot r so these two when I put here it gives me e to the power i q dot r delta r minus r l ultimately this along with the b it this integration becomes sum over l e to the power i q dot r l and v also has this b part so this is what I will do for the scattering amplitude called Goubert so this is what it was and d sigma by d omega is the square of that and v as I defined and k prime v k is 2 pi h square by b l e to the power i q dot r l where q is equal to k minus k prime as I derived for you so now the system is in some initial state which is not just its momentum but the scatterer here I have to consider the distribution of the isotopes and also distribution of the spins why because because one is that nuclear spin and neutron spin whether they're parallel or anti-parallel the value of b changes also it changes from one isotope i to another isotope j they are distributed all over the crystal and we assume that the nuclear spins are g-oriented it's a very reasonable assumption because it's difficult to align all the nuclear spins and so at any instant the nuclear spin can be either parallel or anti-parallel and the isotope is also randomly distributed and is signified by the assumption that isotope is randomly distributed of the sites independent one isotope position is independent of the other except statistically it is just given by the probability once I say that then now imagine your system is in a state lambda sigma spin an isotope and the final is one of the lambda prime sigma prime states so I can do a summation over all possible initial states when I do a squaring of it then what I have actually was I showed you d sigma by d omega was the modulus square of f k k prime so modulus means f k k prime and it's complex conjugate so it comes out like this so now I have to do a summation over the final states over all the initial states I need to take the probability of all the initial states then then you have complex conjugate this and I have to square this whole thing this is my scattering amplitude but you need wavelength you need solid angle I'm sorry you need solid angle in this summation I have summation over lambda prime sigma prime I have a summation over lambda sigma now you see I will just write briefly I have lambda prime sigma prime something lambda sigma sorry okay and then again there will be lambda sigma then then again one more bracket it will come from there and then lambda sigma then ultimately will have lambda sigma so now I can play with this and I'm slightly wrong here I'm sorry one second I have to slightly wrong say some over lambda sigma then here it will be lambda prime sigma prime lambda prime sigma prime then lambda sigma yes now here I have a summation I have a summation over lambda prime sigma prime and I have over the final states so this is a projection operator why let me just explain to you in more details suppose I have a state function psi in quantum mechanics now I want to know what are the components in these states lambda prime sigma prime component of this so I can write it like this this bracket gives me the component of psi along lambda prime sigma prime and this is a complex conjugate of that component of that if I call it the probability amplitude a that this sum over lambda prime sigma prime is nothing but is a summation over all values of a star a which is nothing but the sum of the probabilities over all the final states and that must be equal to one because if I sum all the components of a wave vector along this this is just like if I have a vector in three dimensional space if I sum over the thetas in all directions for that all the direction cosines if I add up that is then it will be cos theta sin theta sin phi and sin theta sin theta cos phi if I square the add them up it will be cos square theta plus sin square theta will be equal to one it is the same thing in the wave vector space and that's why we put lambda prime sigma prime lambda sigma if I have if I consider transition probability to all the final states they will go to one so now I remain within so I have put this probability equal to sum equal to one I showed you why so now I have the scattering cross section per unit solid angle is given by excuse me sum over all the initial states e to the power i q dot rl minus rl prime that came from the potential which is delta r minus rl there are two parts of it there are two dummy variables l and l prime so it comes over sum over l l prime and that is averaging over sigma lambda sigma lambda that means I have to average the value of bl prime bl bl prime star bl over all possible initial states and states of target and neutron so in this case this brings a difference between neutrons and x-rays because scattering cross section here it depends on the spin and the isotope whereas if you let us consider you are looking at diffraction pattern of x-rays from aluminum just as an example aluminum or nickel or type or any other method now aluminum has got a charge cloud which scatters so aluminum has a charge cloud which scatters and has a nucleus in this case in the diffraction experiment my interaction of the neutron is with this one the nucleus and the x-rays they are getting scattered from the charge cloud and you see the charge cloud if I consider a if I consider a lattice of aluminum their same charge cloud everywhere if does not depend on the isotope does not depend on the isotope or spin so that's why in case of x-rays we don't have any averaging process whereas in case of neutrons we do have an averaging process which we will be using to calculate coherent cross section and incoherent cross section I will do the mathematics now but before that let me tell you the cause of this fluctuation is that in case of neutrons first the nuclear spins are not oriented and the neutron spin can be either parallel or anti-parallel to the nuclear spin and then the scattering cross section is different so is the cause of the isotopes the scattering cross section varies from isotope to isotope and you have to take an averaging on that also and that gives me an average scattering cross section and the fluctuation around them before I do the mathematics that is the biggest difference between neutron and x-rays diffraction now let me go ahead and calculate it so as I said that there is an averaging involved between the the initial states for bl bl prime so so here this is the average value so the average value if I take sum over all possible spin orientation as one then is all possible initial states p lambda lambda bl prime star bl lambda so I have to do the averaging over all possible random isotope and nuclear spin distribution so now let me try to write down this average value so now I have this at the l prime side the scattering cross section is bl prime and its complex conjugate is bl bl prime star and this and I am doing an averaging over all the possible in states that which is also have to put the this I'll come to later so now in this averaging let me show you write down this as this is the average value let us say now when l is equal to l prime then these two are same and what I get is that l equal to l prime so both of them are same so for l equal to prime I get b square and I have to average over that average over that when l is not equal to l prime then you have to do the averaging separately because our assumption is that two sites are uncorrelated so far as their isotopes and spins are concerned so I can do the averaging separately two independent values can be averaged independently that's clear mathematics so we have what is known as this is also give me an average value b average star so ultimately I will have b average average and square of that so I have two values when l is not equal to l prime and l equal to l prime and with this I find I get b square average and b average square so now what I can write actually excuse me so the sites are not correlated so the averaging is just separately this thing so now this summation was this averaging was there now let me write it down in an interesting way so I will write equal to these are tronica delta I have written this averaging as b average square plus delta l l prime b square average minus b average square I want to show you what I wrote actually this is equal to b average square b square average when l is not equal to l prime and when l is not equal to l prime it is b average square these two statements I have absorbed in one expression so now you see when l is not equal to l prime this term is zero and I have b average square so that means distinct sites when they come into picture this averaging I have b average square and this part is not there and when l is equal to l prime then this is one so b square average minus b average square these two cancel and I'm left with b square average so there are two parts in this averaging b average square and b square average so now please know that now my scattering cross section is sum over l l prime l l prime b average square when is l is not equal to l prime then I get a term which is b square because l is not equal to l prime so in this summation l is not equal to l prime e to the power i q dot rl minus rl prime so my d sigma by d omega scattering cross section per unit solid angle has two parts when is when l in that summation when l is not equal to l prime I have e to the power i l is not equal to l prime to b average square and another part when l is equal to l prime I have b square because l equal to l prime in this term then it is b square average minus b average square and sum over l equal to l prime so it will just give me n number of that is sides in the crystal now please note that q dot rl minus rl prime so this is the distance between the l and l prime side and this q is the transfer vectors this has got an angle dependence angle dependence and look at this term this does not have any angle dependence because there is no q which is 4 pi by lambda sin theta q is equal to it is magnitude and the direction is dictating what is the value of q dot rl minus rl prime this has angle dependence and this does not have any angle dependence so this is the term b average which dictates diffraction and this term it does not have any angle dependence this adds the background to this thing so now after deriving this now you see so that's what I wrote here so this part is the incoherent part because it does not have any angle dependence and it depends on the fluctuation around the mean b square average minus b average square is the fluctuation around the mean value of the scattering length at the side and the average value of the scattering length at the average over all the isotopes and spins gives me the diffraction term