 Greetings, so we have the second lecture of the unit 5 and this will be on the Born approximations. Notice that there is a plural S at the end because it is not just one approximation but a series of approximations which is why it is referred to as a Born approximations. Before I get into this I will like to remind you that I did you know rush through several portions in the previous class in the last class. For example, when we wanted to estimate the asymptotic distance and then I had a few slides with these blue arrows on the side and I just want to remind you that there is very simple reorganization of terms and there is no big physics or mathematics which is no trick which is involved in this. It is just rearrangement nothing else and as part of this technology enhanced learning NPTEL program one advantage that I can take off is that the PDF which has all of these details is available at the course web page. So you can download it and go through all the intermediate steps which I have rushed through but there is nothing very big about any of these steps but they are all available in the PDF file at the course web page. So we went through this to get an estimate of this e to the ikr by r which we have used in our previous class and then we put it into the Lippman-Schwinger equation and this is where we begin our discussion for today's class. So the Lippman-Schwinger equation had one difficulty which is that it was it offered us a formal solution but not a particularly useful one because of what I called as the catch-22 situation and I have explained the term catch-22 it has become commonly accepted in the English language in the colloquial way because of this famous novel and there was this movie which was made on this and some of you may have seen the movie that you I do not want to go into the story of the movie because that will take us much too away from the discussion of the bond approximations where it is a very nice story and it argues that there was somebody who said that he needed you know some waivers and he argued that he should get that waiver because he was not mentally alert enough to take care of those situations and then the administration argued that if he is in such a frame of mind that he can plan such a strategy then he cannot have any issue and he would actually be mentally alert. So that is the kind of situation that you try to pose a problem or solve a problem but then the solution and the problem become integral parts of each other and you there is no getting away from that. So that is what I call as a catch-22 situation. So you have any problem in physics you when you seek a solution and then you want to have an unknown left hand side which is to be described by an equation and on the right hand side it is necessary that everything on the right hand side is known and only in terms of that can you determine the left hand side ever right. Now how can you do that if the right hand side has the very same function which you handle on the left hand side. So this is the catch-22 situation we have to find some way of coming out of this. So this is what the bond approximations will help us to achieve. So let us see how to go about doing that. As a 0th order solution what we propose is that we just put in the incident plane wave which we know okay there is nothing unknown about this and we plug it in over here. This looks like ad hoc but then there is some sort of merit to it because what it does to the right hand side is that you have the incident plane wave over here but then it appears again over here but in the second term it incorporates various things. What does it incorporate? It has got the potential. So that is the quantity of interest. It also has got the Green's function with appropriate boundary conditions. So there is some information which has gone into the second term regardless of the fact that you have made an approximation to that. So this is just a start. This is not the end of it because you can always improve upon it. So you can take it as an early solution and then make an effort to improve upon it. This is the kind of thing we do also in the Hartree-Fock self-consistent field method when you don't have a solution you propose some sort of a solution and then iterate on it till you get convergence. So it is something of this kind. So you propose this as a zero-ethodis solution and now everything on the right hand side is known because this solution is now replaced by this incident plane wave and now everything on the right hand side is known. So in terms of which you can actually evaluate the left hand side. So that is the strategy of the Born approximation. Now having done this you can improve upon it because this is our zero-ethodis solution as we just discussed but then you can go further because you can go to the next higher-order solution if you take the first-order solution. So here the superscript is zero so watch it very carefully. So the notation is that this is the zero-ethodis solution which is nothing but the incident plane wave and now you go for the first-order solution. This is the superscript 1 in which you put the zero-ethodis term here and now you can go further because in terms of this zero-ethodis solution which you already know is the incident plane wave you can evaluate the first-order term and now you can put this first-order term over here in place of this. So the complete solution is psi superscript plus that is the correct solution. We are approximating it now by psi superscript plus but with the superscript 1 which was our first-order correction. So we pick the first-order correction from here put it over here and now again you have got a right hand side which is not perhaps the exact solution nevertheless it is better than what you started out with. And you can make subsequent you know approximations following essentially the same logic. So you get the first-order correction, the second-order correction, you can get a third and fourth-order correction and in general the n-eth-order solution which is indicated by the superscript n will have the n minus one-eth solution of the previous step which will be plugged in over here. So that is the plan of this series of approximation which is why it is called as the bond approximations as a plural so that you have got a series of approximation and you can go to n-eth-order you can go to infinite order in principle. So the exact solution will require infinite order but then you can go as far as you need to and also as far as you possibly can depending on a given situation. So let us have a look at the second-order solution. So you have got the superscript 2 over here the superscript 1 which is n minus 1 in this case right 2 minus 1 and this is your second-order solution. Now let us write it a little more fully because the first-order solution itself had the zero-eth-order term plus this correction right. So the first-order term consisted of these two terms which I have now put in place of this explicitly. So now you have got one term over here a second term which is coming from this integration and then you have got this term and then a third term which will have two integrations one over r prime and the other over r double prime. So each is a triple integral but then there will be 3 plus 3 6 integrations in the third term. So in general if you look at the n-eth-order term the potential itself like this is the second-order term sorry let me go back. So in the second-order term this is the second-order term the potential appears here and it also appears here. So it appears twice and for a weak potential the powers as you have more and more powers of the potential the subsequent terms in the bond approximation series will hopefully become smaller and smaller and that is what makes it possible to use this approximation in a very constructive manner. So this is the general structure of the second-order term so let us write these terms explicitly. So these are the three terms the first is nothing but the incident plane wave, second has got the potential once and the third term will have the potential twice one is here and the second is here. Make sure that you have the appropriate dummy label of integration. An integration variable it is a dummy label it gets integrated out but then it has to be appropriate for each integrand. So make sure that you do not make any careless mistake about it so there are these three terms and let us now focus our attention on the integrants. What is it that you are integrating out? So let us look at them and in this integrand so let us look at the second term this we know this is the plane wave incident plane wave and then the other part or the main essential part of the integrand is this GU. So that is like the heart of the integral because you have this term of course it is a part of the integrand the integrand you can think of the integrand as factored into this piece and then into this piece. So this is what is in the blue box is then like the heart of the integral and it is often referred to as the kernel of the integral. So it is not the integrand itself but it is the main part of the integrand. So you have identified the kernel in this term and then there is likewise in the third term this is the one if you agree that this is something that is known we already know what it is and then the remaining part of the integration which really has got the heart of the integral or the kernel of the integral that is what is in this red box. So let us write these terms clearly now so these are the three terms we have identified the three kernels and these are written as k1 and k2. So k1 kernel is this one GU k2 is this GU GU but then the argument of this U is r prime the argument of this U is r double prime so you have to be very careful about that. So this is the recognition of the essential core of the integrand which is the kernel and we now write this kernel which is the one with subscript 1 has got the potential once and the one with subscript 2 has got the potential twice once here and again over here. The arguments of the kernel again you have to be careful because this one will depend on r and r prime this one will depend on r and r double prime r single prime gets integrated out. So that is the kernel that we will be using. So this is the second order term and you can now write the second order term as a sum of three parts one is here second is this integral and the third is this integral. So these three pieces can be written as phi 0, phi 1 and phi 2 and then you can write the second order solution as a sum of these three terms phi 0 plus phi 1 plus phi 2. So you can in the nth order solution in the nth order solution you will have n terms and you can go to n plus 1th order or even infinite order if you like. So the nth order solution will be a sum of similar terms and now you know how to come up with these terms how to come up with the next term. So the n plus the nth term will have all of these terms n plus 1 terms phi m going from 0 to n. So there will be n plus 1 terms over there and your solution is the one with superscript plus which will actually require you to go all the way to infinity only then you will have the complete solution. So that is the this procedure of getting an approximate solution to the scattering problem using the Lyman Schrodinger equation and making an approximation appropriately depending on the powers of the potential. So this is this whole scheme is called as a Born approximation and that will give you an approximate expression of the scattering amplitude. So the scattering amplitude which we know is this integral phi kf this is the final state momentum in the scattered state and this is the initial momentum and you need to evaluate this matrix element. So here the superscript is plus but you will be making an approximation to this. So you can evaluate this matrix element and the corresponding scattering amplitude at various levels of approximation the first order the second order and so on. So this is your complete expansion and if you put this term over here your scattering amplitude will be a sum of a number of these matrix elements as you can see directly right because in this vector you will have n number of vectors for the nth order solution. So you will have the first order solution so this is when you get a solution at this level where you have only the first power of U you have a solution which you call as a first order Born approximation to the scattering problem. So this is your first order Born approximation then you have the second order Born approximation and then you have a sum of all these terms. So the nth order scattering amplitude will be a sum of these n matrix elements and these this is what you call as a Born series. So this is after max Born and here I have written this F with a bar on top of it but you can immediately see that so far as the first Born approximation is concerned it is already this right. So here in this if you look at the jth term in the scattering amplitude then the potential U will appear j times and g0 the green's function will appear j-1 time. So as you can see very clearly from this so this is what we have got and if we look at the nth order solution so this is the main story but essentially the Born series is then a sum of these all these terms and you can represent them by diagrams notice that in the second order term the potential will appear twice so it is as if you have got a particle which is incident in this state finally it gets out in this momentum kf which is the same as in over here so the final state momentum is always kf but intermittently it can get multiply scattered so that is represented pictorially. So as if the particle gets scattered in the scattering region gets scattered by some other part in the source region again and then again and depending on how many times the potential term appears in the Born series it gets multiply scattered. So essentially it is a multiple scattering series and you can represent this pictorially as well so these are the g0 is of course the green's function or the propagator and let us now focus our attention on the first order Born approximation. So here this delta is the momentum transfer it is the difference between these two vectors the initial state momentum and the final state momentum you can always write it along with the h cross if you like so these are the k vectors and h cross k is the momentum right so delta which is ki-kf is the momentum transfer and this is the scattering amplitude in the first Born approximation. So you will need to evaluate this integral and you can manipulate these terms because this is nothing but a plane wave with momentum kf this is the complex conjugate so this comes with a minus sign here and this one on the right comes with a plus sign and then you have the difference ki-kf which is the momentum transfer so that this is the integral that you have to determine now this looks like a very familiar expression okay you can see what it is and it will already occur to you how to evaluate this you can also write it in terms of the real potential because the difference between the reduced potential and the real potential was only in terms of this m and h cross and so on so you can scale it appropriately write it in terms of this what is your conclusion that the scattering amplitude in the first Born approximation is essentially the Fourier transform of the potential right so that is a very useful and a very powerful result all you have to do is to you if you have some form of the potential in mind put it over there and get its Fourier transform and that will give you the scattering amplitude. So this is the result in the first Born approximation in the first Born approximation the scattering amplitude is proportional to the Fourier transform of the potential now having done this let us go ahead and see what it gives us for the scattering cross section the differential cross section is nothing but the modulus square of the scattering amplitude so you can go ahead and get that okay and notice that because it goes as the modulus square if there was any sign involved in v like an attractive potential would come with a minus sign and a repulsive potential will come with a plus sign if there was any sign which you needed to worry about the information about that sign would become irrelevant when you take the modulus square right the result is that the differential cross section remains the same regardless of the potential being attractive or negative it does not matter whether it is plus or minus you get essentially the same result. So this is a very interesting feature of the first Born approximation so here you have the Fourier transform and this appears as a volume integral you can simplify this integration because you know that so far as the total cross section is concerned it is a double integral not a triple integral it is the differential cross section is over all the angles and you can carry out this angle integration over the azimuthal angle and over the polar angle theta going from 0 to pi azimuthal angle going from 0 to 2 pi and this is the differential cross section which is the modulus square of the scattering amplitude. So this integration will give you the total cross section you can always choose one axis of symmetry which will give you a factor of 2 pi when you integrate over the azimuthal angle right from symmetry you get that result directly and then you have got an angle dependent scattering amplitude which you will have to evaluate in this theta integral. So if you have a spherically symmetric potential then this integration which is a triple integral right the scattering amplitude is a triple integral. So you have one integration over the axis of symmetry which gives you a factor of 2 pi so you have got this minus 1 over 4 pi here one integration over the azimuthal angle around the axis of symmetry gives you a factor of 2 pi then you have integration over the remaining 2 degrees of freedom are going from 0 through infinity and theta going from 0 to pi. So those are the 2 integrals that you now have to determine one of which is already determined in this 2 pi. So this 2 pi and this 4 pi will give you a factor of 1 over 2 and with that 1 over 2 we write the scattering amplitude as minus half and then you have got a radial integral and a integration over the polar angle theta. Now this looks like a integration that you have done what you call it like a jar times right. So might as well put a new variable there like carry out the integration over cosine theta instead of theta and that makes it easy. So I will not spend much time discussing this. So you have got the integration for this new variable which is cosine theta going from minus 1 to plus 1 and then you carry out this integration so you get put the limits you get this familiar form. So this is the usual integration that you would have done so many times and now what do you get? You are left with a radial integral from 0 through infinity the integration over theta has now been carried out. There is an r square here there is a 1 over r here so you get only one power of r right and everything else is now taken care of your instead of using the exponential form you have written it as a sinusoidal function. So this is the integration that you have to carry out in the first bond approximation the momentum transfer is h cross delta this diagram you would have drawn a number of times in a number of different situations I think most commonly in x-ray diffraction when you use this Ewald sphere kind of right. So you have got the net momentum transfer which is Ki minus Kf so this is the difference vector and then you have got an isosceles triangle over here. So here you have got a right angle triangle of which this side has got a magnitude of delta over 2 this side has got a magnitude of K which is the angle opposite to the 90 degrees to the right angle right and this the third side will be K times sin of half theta right. So from this Pythagoras theorem you can use that and what it gives you for delta or is this 2 K sin theta by 2 and you have the Pythagoras theorem which relates the squares of these sides. So you get you get a relation for delta square or half delta square but then if you differentiate that you will get 2 delta d delta. So using that you will get sin theta d theta equal to delta d delta over K square what is the range of theta? Theta goes from 0 to pi and theta equal to 0 is forward scattering, theta equal to pi is backwards scattering. So the difference vector K will go from 0 to twice K right. So the range of delta itself will be 0 to 2 K. So if you carry out the integration instead of theta you carry it over K you can transfer the integration. So this is the evolved sphere this is in three dimensions okay we have drawn the figure on a plane and this is now your total cross section in the bond approximation the first bond approximation and you need to evaluate this integral instead of integration over theta you can integrate over the momentum transfer delta which will have a minimum value of 0 corresponding to forward scattering and twice K corresponding to backwards scattering right. So you integrate from 0 to 2 K alright. So these are our main results and now let us look at the high energy limit which is where the bond approximation is very commonly used and everybody believes and rightly so for most situations although there are some exceptions on which I will comment later. If you now look at the high energy limit now energy goes as quadratically with K right h cross square K square by 2 m is the energy. So the high energy limit will be obtained by carrying out this integration with this limit K going to infinity. So here you have this 2 pi over K square integration going from 0 through infinity and what do you see there is a 1 over K square here and 1 over K square as K tends to infinity will give you a 0. So that is your result that the bond approximation first bond approximation cross section will in the limiting case if you go to high enough energies it will go to 0 and the rated which it will go to 0 is 1 over K square or 1 over energy. So it will fall as 1 over E. So this is your first bond approximations scattering amplitude. Now let us take the case of coulomb potential but in particular let us take the case of the screen coulomb potential and when you adjust when you tune the screening parameter in the yukawa potential or in the screen coulomb potential you can always take the limit and find what would be the value for the coulomb potential itself. So let us determine this for the screened coulomb potential so this is the yukawa type screened coulomb potential and you can write it in terms of this parameter alpha or it is inverse it is the same. So you have the screened coulomb potential and now you should immediately recognize that when you put this potential over here this will give you a real number right there is nothing over here which has a chance of giving you anything imaginary. Now what happens to the optical theorem because in the optical theorem we discussed earlier in the previous set of classes that the scattering cross section goes as the imaginary part of the forward scattering amplitude right and the imaginary part no matter what angle in this case actually goes to 0. So obviously you know that the optical theorem is not going to be valid at least in its form in the form in which we have established it in the case of the first bond approximation. So we already expect that we will have difficulty with the optical theorem and now let us get rid of this R now you have this expression for the first bond approximation scattering amplitude so you have to determine this integral now which is again you would have carried out integrations of functions this is a product of two functions of this kind quite routinely it is easiest to do it if you put this in the exponential form and take the sum of two integrals that is much easier. So these are usual tricks and if you carry out this integration which you can work out very easily the result is that the first bond approximation scattering amplitude for the screened Coulomb potential goes as u0 over delta square plus alpha square. Now this is a very important result the differential cross section will go as a modulus square so you get u0 square and the square of this denominator okay it is the same for the attractive and for the repulsive potential as we commented earlier okay and now if you use these forms you plug in the explicit form of the scattering potential so instead of the reduced potential you use the real potential and if this is some sort of a screened Coulomb interaction between so this is scattering between a charge z1e times which is being scattered or interacting with another charge z2e then you will have this z1z2e square and then you will have the other terms in which we now find this h cross and m because we are using the real potential not just the reduced potential u. So we already using that evolved construction we have shown that this delta is twice k sin theta by 2 right from that previous diagram for the evolved construction and delta square which comes over here in the denominator we can find it in terms of k square which we can write in terms of e and h cross square. So we can put all the terms together and write it in a form and you probably begin to recognize this result because it is very familiar it is something that you would have seen earlier and it comes as a surprise that you are you seem to be heading toward a classical result as you can see that you are heading toward that okay. So if you see this this is a very interesting situation and I have just rewritten this for this this alpha is coming from the Yocawa potential this is the one which scales the Coulomb by the exponential term and instead of putting it in these m e and h cross and so on I write it in terms of these atomic unit for length which is a 0. So in terms of a 0 it is 4 z square for an electron proton scattering so z 1 and z 2 are both equal in this case. So you get a factor of z square instead of z 1 z 2 okay and now you have this a 0 square and you have this term here. Now this is the differential cross section you can integrate it to get the total cross section integrate over all the angles right. So integration over theta and phi and the range of integration will be for z for theta going from 0 to pi corresponding to the momentum transfer going from 0 to twice k. So this is the total cross section in the bond approximation and here is how it will behave okay. So first thing to note is that the total cross section at this energy is not 0 whereas the imaginary part of the forward scattering amplitude is 0. So the optical theorem is let down in the first bond approximation. Now if you plot this it gives us some interesting characteristic features of the bond approximation. So here is 1. So this is the differential cross section plotted in units of u 0 square a 4 so that is just some scaling so do not worry too much about it. And here it is plotted as a function of the momentum transfer okay and this is measured in terms of 1 over a so this is some sort of a range which is involved in scaling down the Coulomb potential by the Yukawa factor okay. So this is how the differential cross section behaves in general. Now let me show you another plot which is really very interesting which is here okay. Now this is a similar plot but it has been here you see curves for different values of k so this is for k equal to 1 this is for k equal to 2 k equal to 3 k equal to 5 and so on but what are you plotting you are plotting a ratio okay. This is a ratio of the differential cross section in the first bond approximation at an arbitrary angle theta okay that is in the numerator and what is in the denominator is the differential cross section in the first bond approximation not at an arbitrary angle but in the forward scattering direction at theta equal to 0 okay. So this tells us to what extent scattering in other angles is important relative to scattering in the forward direction. So this ratio what you are plotting is a measure of the importance of scattering in different directions compared to the scattering in the forward direction. Notice that here if you go to high energy you know this is k equal to 1 k equal to 2 k equal to 3 so as you go inward energy is increasing as you go in this direction and as you go to the high energy limit which is what we considered almost and this is plotted as a function of the angle theta right. So what you find is that almost all the scattering is in the forward direction because this ratio is equal to unity over here okay. So at the high energy limit almost all the scattering is in the forward direction. So forward direction means it will go in some small cone in the forward direction right. So you will have a direction of incidence and you will have a small measure of cone and that cone will become smaller and smaller as the energy increases or the range increases. So you can expect it to go as 1 over k a some sort of an order of magnitude estimate for this angle and if you look at that so this is the result that you get that at high energies almost all the scattering is in the forward direction. Now this angle is about is of the order of 1 over k a and this is the kind of cone in which most of the high energy scattering takes place. Now this is a good result very useful one and you can see because delta is twice k sin theta by 2 this we know from the previous analysis of the Ewell diagram then delta over 2k will be of the order of sin theta over 2 or right and theta over 2 you know when this angle is small this sin of this angle is nearly equal to the angle itself which is of the order of 1 over k a. So you get a factor of 1 over twice k a for delta over 2k right. So you get 1 over twice k a over here and in the high energy a bond approximation you get your this result we have seen earlier we have written it in terms of this Bohr radius a0 also and what we get is if you look at the limit alpha going to 0 or a goes to infinity which is when the Yuccava potential will go to the Coulomb potential right. So as the Yuccava potential goes to the Coulomb potential this term alpha square vanishes and then you get the sin square theta by 2 but then you have got another power of 2 over here and that is essentially the classical result. In classical Rutherford scattering that is exactly the result that you get and this is rather exciting that the bond approximation or the Coulomb potential obtained from the Yuccava potential by taking the limit of this alpha going to 0 you get you recover the classical Rutherford result. So this is the problem with the optical theorem that we had noticed that the imaginary part of the forward scattering amplitude actually goes to 0 but then if you determine the scattering amplitude using this integration of the differential cross section right you do get a non-zero scattering amplitude but then if you determine the bond approximation scattering amplitude in the second bond approximation it turns out and I will not work that out in any detail over here but then you do get the total cross section to be given by 4 pi over k times the imaginary part of the forward scattering amplitude but in the second bond approximation. So there is some sort of a nonlinearity over here and this is sometimes referred to as a nonlinear nature of the bond approximation or the bond series of approximations and it is only with reference to what you expect from the optical theorem and you can still write an expression which is somewhat similar to the bond approximation and you can expand the scattering amplitude and the scattering cross section in different powers of the potential and then look at the corresponding powers of if you make a comparison of corresponding powers then you see that there is this nonlinearity pops out of that. So this is a straightforward extension which I will not discuss any further. The only thing I will like to remind you or make a comment on that one expects the bond approximation from all this discussion that it will work at high energy and at high energy then the particle comes at such immense energy that it sees the target potential interacts with it gets scattered off and there will be no effect of the electron correlations because if the target consists of many electrons which it almost always does when you do atomic scattering then you know very well that the single particle approximation does not describe the atom fully correctly because it leaves out what does it leave out? It leaves out the coulomb correlations. So these coulomb correlations are left out and you do not expect these correlations to play any important role in high energy scattering because you think that it is going to come so fast that it will not be disturbed by the correlations which will be so weak and so insignificant that they won't matter. In other words one would be tempted to conclude that in high energy the independent particle approximation and the bond approximation will always be valid and it is not a bad approximation as such it is not a bad conclusion in most situations it holds. However there are certain considerations due to which you cannot really get rid of the consequences of correlations and as a matter of fact it turns out that has been discussed in this reference here that the independent particle approximation in X-ray photoemission so that is high energy photoemission. In that it turns out that it is all means if it all the bond approximation works it is almost always an exception and not quite a rule and this is because of certain correlations that you really cannot completely get rid of so the independent particle approximation actually breaks down. So one has to be concerned about some of these things I can show you one of the results like if you were to look at the bond approximation result you would expect this line over here but this line is not what explains the experimental result which is well away from this bond approximation or independent particle approximation result and you have these additional features. So this is just a comment for further studies and something that you might want to keep at the back of your mind but other than that for most applications the bond approximation and the independent particle approximation is an excellent approximation in the high energy range and with that I will conclude this class over here. In the next class I will discuss Coulomb's category you will remember that the methods that we discussed earlier were not directly applicable for the Coulomb case and it needed a different approach so that is the one I will discuss in the next class. If there is any question for today I will be happy to take.