 Greetings, this is the last unit of this course and there are just a few classes in this. We already got the expression for the Breitwigner formula and today we will introduce the Fano parameters which describe the complete expression for the scattering cross section inclusive of the background and the resonance. So let me quickly recapitulate the result we obtained in the previous class. So this is the total cross section and we found that it had a three terms one coming from the background alone, one coming from the resonance part and a third part which came from the interference between the background and the resonance part right. So these are the three terms and then we have such terms for each partial wave L going from 0 through infinity and the expression for the Lth partial wave cross section is now written in terms of these three parts right. So let us look at this expression now so you have got a real part of this complex number so I look at this complex number and rewrite it in a slightly different way. So I take this complex number and I write it in terms of A and B so you have got in the complex number a real part here and an imaginary part here but then you also have got a real part here and an imaginary part in the numerator which is cosine psi and sine psi. So you multiply and divide by the complex conjugate of the denominator and then reconstruct this complex number so that you will have an effective real part and an effective imaginary part. So having done that you write this complex number in terms of this real parts which is now A cosine psi minus B sine psi divided by A square plus B square and the imaginary part will be B cosine psi plus A sine psi divided by A square plus B square. So we need the real part of this complex number the real part of Z so or Z as the young people call it my generation always called it as Z but now it is Z I think. So this is the real part of this complex number Z and what do we have here so this term will now go over here. So let us rewrite this scattering cross section for the lth partial wave in terms of this real part of the complex number Z and that is in this blue box. So this is a fairly straightforward substitution most of this is really very simple there is no involved mathematics it is just a matter of rearrangement of terms but done very cleverly by Fano and that is what we are going to appreciate today. So we essentially find that these two forms of writing the lth wave partial lth partial wave cross section in these two forms these are necessarily equivalent forms. What we will do now is to introduce the parameters which were defined by Fano in that famous paper and we rewrite this expression in terms of q and epsilon so that we express the effective cross section sigma l of the lth partial wave in terms of q and epsilon rather than the terms that you see over here. So this is just an effective rewriting of the same expression so there is no new result but the same expression when it is written in terms of these new parameters it gives us a very convenient and a very powerful tool to analyze the spectra the collision spectra. So here is the result now so let us rewrite it and we rewrite it by introducing two auxiliary numbers a and b and in terms of a and b we rewrite this cross section in terms of a and b so the middle terms becomes b square over a square plus b square b is half gamma so here you see you have got a quarter of a gamma square right and then you have got a square plus b square right so the middle term is here. So the only thing that is done is to rewrite it in terms of a and b over here and then you rearrange the terms because you have two terms coming from here one coming from the multiplication of twice psi sine psi b with a cosine psi and the other when you multiply this factor with b sine psi so you get these 1 2 3 4 terms and then you combine the terms in sine square psi because you have got sine square psi over here and the sine square psi also in the fourth term so you combine the two terms and rearrange these terms a little bit so that you get a square plus b square in the denominator in each of the three terms so that is the ideal. Now having done this you rearrange these terms one more time okay and I will not comment so much on these rearrangements they are quite straightforward but they are very effective they are extremely effective so you rearrange these terms and I am going to use a form so this expression for the lth partial wave cross section is very much the same as we got in the previous class except that it has been rewritten in terms of these auxiliary parameters a and b which are respectively just the energy difference from the resonance energy and the half width of the resonance okay so they are rewritten in terms of a and b so these are like intermediate auxiliary parameters. So here we are we have the a and b then we do know that the tangent of the phase shift at resonance is given by gamma over twice the difference of this energy and this is in fact how we had defined the resonance wave this was the definition of gamma as you will remember so now we have this expression and what we do is to define these two parameters epsilon and q following phano so this is how phano defined the two parameters epsilon and q so epsilon is nothing but the negative of the cotangent of the resonant phase shift and q is nothing but the negative of the cotangent of the background phase shift okay so these are two parameters and notice that this epsilon is nothing but a ratio of a over b with a negative sign okay so epsilon is just minus a over b and q is this q come comes from the background phase shift comes from the cotangent of the background phase shift or the negative of the cotangent of the background phase shift so we will rewrite these terms so instead of a and b we will now rewrite it in terms of the phano parameters epsilon and q now how do we do that so we notice that if you take the reciprocal of q reciprocal of e you get b over a okay and then you have the partial wave cross section which is 12 plus 1 sin square psi so you combine these two terms using just the usual properties of the cotangent functions okay and notice that you already get the q parameter appearing over here but the epsilon is yet to arrive in our expression so we again reorganize these terms okay so this is a straightforward rearrangement of the terms so you have you rewrite this term with a square plus b square as common you have got three terms and these are the three terms so now you have b square you factor out from this so you get only q square in this term and then a square over b square and then since b square has been factored out you get 1 over b so this becomes minus twice a over b q and what is this this is nothing but the square of q minus a over b but a over b is nothing but the negative of epsilon okay so now we can write the lth partial wave cross section which contains all the three contributions one from the background then from the resonance part and then from the interference part so all the three together because that is what an experiment is going to see the experiment does not distinguish you know when it when you have a detector it is not going to sense which part is coming from which term so you will get a net effect and that net effect is now written in terms of these two parameters so the a over b is here the q is here and that allows us to write the scattering cross section for the lth partial wave in terms of the phono parameters q and epsilon so the numerator here is just the square of q plus epsilon and the denominator is just 1 plus epsilon square okay so this is effectively just the same expression there is nothing new in this that we have found today except that we have rewritten the scattering cross section for the lth partial wave in terms of two parameters one related to the phase shift from the background and the other coming from the phase shift at resonance okay thus net scattering phase shift we had written as a sum of two parts in our previous classes in the last unit what we did was to express the total scattering phase shift as a sum of two parts so we separated out the dynamics resulting from the background and the dynamics which was coming from the resonance structure so these are the three terms that we began with okay the background part the resonance part and the interference part okay labelled by b r and i respectively and now we have these phono parameters q and epsilon and in terms of this we have the same cross section the same scattering cross section which contains information about all of these three together but written in this single term but in terms of two parameters which are q and epsilon so these are the this is our effective expression for the scattering cross section in terms of the phono parameters notice that when the background phase shift is small when sigma goes to 0 then you have only the resonant part of the scattering cross section right and when q is minus epsilon the cross section can actually vanish okay so there is nothing no stopping from the scattering cross section to disappear and it is not always that when you have resonance effects the scattering cross section must only increase because it can also decrease it can even vanish it can even go to 0 so here is a figure from Branstad and Jochen's book which I would like to discuss and this is figure 12.4 in Branstad and Jochen's book and it has got four panels A, B, C and D and they represent different cases which are described in the in the caption for the figure so here you notice that the background cross section this is for S waves so which is why the quantum number L is equal to 0 which is the subscript of xi so this is for L equal to 0 S waves scattering so that is the example which is discussed over here and for this case here the background cross background phase shift which is xi is 0 so xi 0 is 0 and this would correspond to q being either plus infinity or minus infinity so you will see how it is related it is very straightforward then here the background cross section phase shift is higher than over here so it is about half way between 0 and pi by 2 so this background phase shift is about pi by 2 and then in panel C the background phase shift is about pi by 2 this is pi by 4 in panel B in panel C the background phase shift is pi by 2 and in panel D it is 3 pi by 4 so this is pi by 2 this is pi this is pi and this is about three fourth so this background phase shift is about three fourth of pi and these are the corresponding cross sections and they come in all kinds of shapes and that is the reason it becomes very fruitful to analyze them in terms of what we call as a shape parameter which is the q because they all have different q values so let us look at these figures carefully panel by panel so I will discuss each panel separately so in each of these you have got you have got a dashed curve which is just what the maximum cross section at that energy which would be possible but then because of resonances the cross section changes sometimes widely it also goes to 0 so the dash curve only represents the maximum cross section which can be which can occur at that particular energy or momentum so let us look at these parameters carefully so we are discussing L equal to 0 the S wave scattering in terms of the Fano shape parameters and notice that when this background cross the phase shift is 0 look at this when would when xi 0 is 0 what do you get for cotangent of xi it will be either plus infinity or minus infinity right so that is how the parameters are related so the back whenever the background phase shift is 0 the Fano q parameter would be plus or minus infinity so the value of q is essentially determined by the phase shift of the background scattering when the background phase shift is pi by 4 which is what we see saw in the second panel right when the background phase shift is pi by 4 what is the negative cotangent of pi by 4 it is minus 1. So, what when the background phase shift is pi by 2 with the background phase shift is pi by 2 the cotangent would go to 0 and q is equal to 0 and in the fourth panel the background phase shift was 3 pi by 4 and the Fano q parameter would be plus 1 so these are simply coming from the values of the background phase shift but they give you a range of q values which can be anything from plus infinity to minus infinity and the background phase shift can be anything from 0 to pi or modular pi right. So, the background phase shift can be anything which means that the q parameter can be just about anything and as you see from these figures they determine the shape of the resonance. So, here is the case which was in panel A and in this case the background phase shift is 0 q is plus infinity or minus infinity and what do you get you get a cross section if you put these values q equal to plus infinity or minus infinity in the expression for the scattering cross section you get essentially the pure bright Wigner resonance which we have discussed at length in our previous unit in our previous classes. So, you get the pure bright Wigner resonance in this case when q is either plus infinity or minus infinity. So, that can always be a special case that you have just a pure resonance but then you may have as in case B a background phase shift which is pi by 4 and in this case q is minus 1 and notice that this would give you a net cross section which is given by this formula here and what does it turns out to have an asymmetric line shape. So, typically the resonance profiles have asymmetric line shapes, but not necessarily they may also be symmetric like in the bright Wigner pure resonance it does have a symmetric line shape in this case it is asymmetric in case C the background phase shift is pi by 2 which begins over here and q is 0 and when q is 0 you have got only the square of epsilon in the numerator here in this formula, but what is epsilon it is related to the resonance phase shift and the resonance phase shift is pi by 2 we have seen that in our previous discussion. So, what does it mean you get a scattering cross section which actually goes to 0 and this resonance is often called as a window resonance because as you can see you get a window like behavior a window like shape in the scattering cross section. So, this is often referred to as a window resonance and then we also had the other case which is the background phase shift being three fourth of pi and then you have got q equal to plus 1 and as was the case for q equal to minus 1 you have an asymmetric line profile. So, most of the profiles for various different values of q will be asymmetric in certain cases you may get some kind of symmetry that depends and the actual shape the information about the shape of the profile is then contained in the q parameter. So, this is the famous Fanot's q parameter and you can see that how specially if you are an experimentalist you would like to look at the shape and analyze it in some simple terms. So, Fanot parameters let you analyze the shape of the resonances in a very neat form. So, here is our expression for the scattering cross section of the lth partial wave. So, this formula is no different from this one there is nothing in this formula which is not there in the first formula the two are completely equivalent. But the second one is in terms of parameters which have got rather simple interpretation when you are looking at a scattering spectrum. When you look at the scattering cross section and study it as a function of energy or momentum then you can analyze the shapes of the resonances very neatly using the Fanot shape parameters. And here I have given some reference, but a plural reference says because there are so many of them and I am borrowing some discussion from the book by Macy and Burhup on electronic and ionic impact phenomena this is a very classic book on scattering theory. And then of course there are large number of papers by Fanot and other reviews and so on. So, I am not going to attempt to review the entire literature extensively except point out some of the key references. So, these are the three terms which are of importance to us the background term, the resonance term and the interference term. And these play this formula in terms of the Fanot parameters becomes extremely useful when you study the configuration interaction between bound to bound and bound to continuum transition. So, that is what you get in a phenomenon like auto ionization. In auto ionization you have got alternate channels which are open for the photo absorption process. The photo absorption can result in either a bound to bound transition from an inner electron of the atom or from a bound to continuum transition from an outer electron because both the processes can be degenerate and it will be a resonant process because the bound to bound transition is always between discrete levels which are quantized. So, this will be a resonant process and essentially you have these two possibilities. So, you can have an electron in the bound state in a two electron system you can have one electron in the bound state and the other also in a excited bound state in a. So, this will be a bound to bound transition from discrete bound to discrete bound and this channel is degenerate with an outer bound into the continuum and these two represent two different configurations. The occupation numbers of the single electron states are obviously different in these two. So, you essentially have a configuration interaction between the continuum and the discrete and that is what Fano parameters help us analyze. So, this is the classic paper which I mentioned earlier also most atomic physicists have got this reference by heart in no volume 124 number 6 everything they almost know it by heart it is a classic paper. So, we analyze this in terms of the square well potential right the expressions we got were in terms of the scattering by a square well that was our prototype of a potential. But then our interest is in realistic potentials and the realistic potentials would be somewhat different from square well potential. The square well potential is just an attractive model potential and we will therefore, now like to ask how will this analysis go if you worked with a more realistic potential then some of the parameters may end up having either a different interpretation or they if you are to use the same kind of expression then you must find out what would be the meanings of these parameters and how to interpret them. So, let us discuss that. So, let us consider a pure Breitwigner resonance which has its maximum at E equal to E R this is a symmetric you know classic Lorenzian shape. But most of the resonance profiles are often asymmetric and then we can deal with potentials which are much different than the square well potentials and the typical resonance profile as you see in this figure you have got the background cross section which goes almost flat over this small energy region over a wider energy region background cross section of course, would change. But over the narrow region it would be relatively flat and over this the actual cross section has got a completely different symmetry. It is an asymmetric cross section and obviously, it has some value of q which is somewhere between minus infinity and plus infinity and we can specify exactly what is the value of q for this particular shape resonance. So, the maximum of the cross section as you see is not necessarily at the value of E R it can be somewhere else. So, the criterion of the resonance energy is that it is the energy at which the resonant part of the phase shift goes through pi by 2 that is the criterion and not the value of the cross section itself the cross section can hit the roof and it can be a maximum it can also disappear in a window resonance it can even go to 0 or it can have any other value in the intermediate range anywhere between 0 and whatever is the maximum value that it can get. So, you have got the background part which is relatively flat over this narrow energy region and to understand these terms we will very quickly recapitulate some of the terms that we have discussed at length for the square well potential which we have discussed at length in our previous classes. So, I will only summarize the essential results just to relate so that we can relate to those terms and we you will remember that we had solutions for the inner region we had solutions for the outer region we had the wave functions the exact radial functions for the outer region and for the inner region we have the expression for the continuity of the logarithmic derivative coming from the continuous nature of the radial function and also its derivative and what happens as a result of this is that when you exploit the continuity of the radial function and its derivative you have the corresponding expression for the outer region outside you have got the total phase shift which is the sum of the background part and the resonance part and near the resonance if you rewrite this expression what it gives you for the internal wave function is a function of an amplitude which is the most amplitude. So, what the result what this whole analysis is resulting in is that the particle tends to get bound in the well for this particular case so that is what happens at a resonance. So, there is a bound state which and the particle tends to get mind it tends to get bound in that in the well. So, you can have for l not equal to 0 you can have the centrifugal barrier you can have metastable states in this and then for l equal to 0 we know that there is no centrifugal barrier because it is l into l plus 1 by r square whereas. So, S wave resonances are coming because of the virtual bound states which we have discussed at some length especially in the context of the Levenson's theorem we have discussed this feature to us in considerable detail. So, now you have this potential scattering so this is no longer just a square well it can have some other shape and this figure over here in the lower right corner over here represents an alternative potential shape it can be anything else in general it is quite complex it will not even have an analytical form. And in most situations you end up getting numerical forms so you cannot really sketch them so easily like this. But by and large the only thing that is conveyed in this figure is that this is how you will deal with potential scattering from those potentials which are different from the square well potential and this is a prototype of all of them. So, you can have discrete bound states over here you can have metastable resonances over here which will give you shape resonances and then you can analyze this in terms of the phase shifts at resonance and the phase shift in the background region which is xi. And our question that we are addressing is how would the above relations all of these relations how will they get modified if we deal not with an ideal square well potential, but with some other potential the prototype of which is what our figure suggested. So, our first conclusion is that if our potential is no longer the square well potential what we had for gamma and gamma plays a big role over here notice that all the rapid fluctuations are coming because of gamma r and s and xi they change relatively smoothly across the resonance region it is the gamma which changes widely. And that is of importance and because gamma is changing widely for a potential which is other than the square well potential you will not be able to use the same expression for gamma as you did for the square well potential for which we had the exact solutions for the inner region and the outer region. For some arbitrary potential we can hope to expand gamma about the resonance energy and we carry out a power series expansion and consider only the first term. We already know that compared to the energy differences between adjacent resonances the width of a resonance is rather small which is why they appear as spikes. So, it is sufficient to take the first order term in the power series expansion for gamma. So, you have a power series expansion for gamma this is the first order term and this is what we will use for gamma. So, let us look at the expression for the resonant phase shift and this is just simply determined by the properties of the tangent functions which we have discussed earlier also. And notice that at resonance when E is equal to E r this denominator goes to 0. So, the tangent shoots up and the phase shift is n pi by 2. So, that is the criterion mind you for a resonance it is not in terms of what is the value of the cross section it is in terms of what is the value of the scattering phase shift of the resonance part of the scattering. The net phase shift being separated in the background part psi and the resonance part delta. So, this phase shift rho which is the part of the phase shift other than the background when it is specifically at the resonance we use a specific symbol for it which is delta with a superscript r. So, this is nothing, but rho at resonance and this will be n pi by 2. And essentially if you look at this alternative expression for the tangent of the phase shift when this tangent blows up as one would expect at resonance effectively you have gamma becoming nearly equal to r. So, that is denominator would go to 0. So, gamma being equal to r is the same condition as the resonance condition E being equal to r E r and the resonance phase shift going to pi by 2 or n pi by 2 where n is an odd number. So, gamma equal to r is now an equivalent criterion to recognize the resonance condition for a square well scattering potential. So, at resonance gamma is nearly equal to r. So, that is our conclusion what we will do now is that if we now carry forward the same interpretation that if gamma being associated or being set equal to r works for a scattering potential. And if we were to extend this correspondence to other cases also when you have a potential which is different from a square well potential then if we were to write the tangent of the phase shift using a similar expression what would be the meaning of gamma it could be different. Because you are reinterpreting this in terms of the square well parameters, but for a potential case presuming that gamma being equal to r works even for other cases. This will give us an effective interpretation of the width of the resonance width. So, let us look at other cases of potential scattering when I use the term potential scattering I am specifically looking at those potentials which are other than the square well potential. What I have for a square well potential is what I indicate explicitly by writing a square well scattering as a superscript over here. So, anything which is a potential scattering this is of course also a potential, but it is a specific potential that we have worked with and for which we have got the exact solutions. Whereas, this is in general some other potential which is a more realistic potential prototype of which we have the figure for and in general it would be some other potential other than the square well deep square well potential. And then we will ask the question that if this correspondence holds good even for potential scattering then what would be the interpretation of S and what would be the interpretation of gamma over here the interpretation of r is what we pinned down as a reference level. So, r would be the same as gamma, but gamma for the potential scattering is not the same as gamma for a square well scattering we have seen that you can expand it about the energy and you then include at least the first order correction term. So, we will now ask what would be the interpretation of the parameters S and gamma in this case for potential scattering. So, this is the expansion of gamma. So, you have got this is the first order change in gamma that you will need to consider when you deal with potentials other than the square well potential. And you do know that the representation of the remaining part of the phase shift which is other than the background phase shift which is rho has got a specific notation in our analysis which is delta superscript r. And we know that this expression holds good for the square well potential and we know that for the square well potential the tangent of the this phase shift you can write in terms of S, r and gamma and equivalently in terms of this gamma over the energy difference this is how we defined gamma. So, that this becomes equal to this at resonance right. So, this is how this width gamma was defined and now to write it in terms of S and r we write this gamma in terms of this expansion on the right hand side. So, you have got this expansion here and then you have got S and r over here, but these may have a different interpretation than what we had for the square well case. So, now our criterion to identify the interpretation of S and r and then also the resonance width gamma is that we presume that just as in the case of the deep square well potential this is our reference for comparison that we interpret r to be the same as gamma. So, this terms becomes equal to this right. So, these two terms would be equal and they will cancel each other and you get an effective expression for the tangent of phase shift in which we do not have S and gamma anymore you do have the derivative of gamma with respect to energy coming from this term and this is valid for the potential scattering which is for the case when the scattering potential is something different from the square well potential. So, this is the expression that we have presuming that gamma and r are the same, but this is no longer the gamma of the square well potential it is one corresponding to the actual potential and this is the leading term and then you have got a correction which is the first derivative of gamma with respect to energy. So, here I have rewritten this I have got E minus E r in the denominator which I now write in terms of E r minus E. So, I have a minus sign over here. So, these two terms are essentially the same except that I have absorbed the negative sign in the denominator and attach it to the numerator and what this allows me to do is to determine what would be how to interpret the resonance width. The resonance width has now got a new meaning for the potential scattering because if the resonance width is to be so, defined that half of that divided by the energy difference will still be related to the tangent of the phase shift and resonance then it we now have a correspondence between this part which is in this blue box and this part which is in this blue box. So, these two parts must correspond to each other and that gives us an interpretation of the resonance width gamma for the potential scattering right because these two must correspond to each other. So, that you can write gamma or you can interpret gamma once again in the same fashion that half of that divided by the energy difference will be the tangent of the phase shift at resonance. So, this is the correspondence which is emerging from this comparison. So, here this particular correspondence it allows us to write the expression for the tangent of the resonance phase shift in the same form as we did for the square well scattering. And now because gamma has been interpreted in this particular form we cannot presume any longer that this gamma will always be a positive quantity because you know that the dynamics is governed by all these cotangent curves and they can take all kinds of values. So, you cannot guarantee that gamma is a positive quantity. So, it does not have to be a positive quantity and then the if you look at the properties of the tangent function then if gamma is positive then what happens to the resonance phase shift that it increases through an odd multiple of pi by 2, but if it is negative it actually decreases through an odd multiple of pi by 2. So, that is an upshot of this particular analysis. So, this is for a typical resonance coming from some other potential. So, this is something which is of great importance in the analysis of autorization resonances and in particular the configuration interactions between the bound to bound and bound to continuum states. And a classic example of this is two electron system which I will discuss in some details. And a two electron system like the helium atom or it could be the hydrogen negative ion. So, these are the simplest examples of two electron systems, but then a two electron system can be a subset of any many electron system and you can analyze the atomic properties of many electron atoms in terms of the analysis for the two electron systems. So, we will describe this in terms of zero order approximation and in the zero order approximation you pretend that the interaction between the two electrons is switched off which means that you can continue to use the hydrogenic quantum numbers. Now, you will remember that the hydrogenic quantum numbers like N, L, M and so on they essentially come from symmetry properties they come from those Eigen values of those operators which commute with the Hamiltonian. But once you break the symmetry of the Hamiltonian because once you have you do have the spherical symmetry for the hydrogen atom. So, N, L, M are good quantum numbers for the hydrogen atom, but they would not be really good quantum numbers for any other atom any atom other than the hydrogen atom. So, those quantum numbers will not be the so called good quantum numbers, but we will continue to use the hydrogenic quantum numbers, but then you will now have configuration interactions between these different hydrogenic states. So, we will have one particle hydrogen like states which will define different configurations. So, I will give examples of some of these and the actual state actually really will be a configuration interaction which means that if you have a superposition if you have the atomic system described by a superposition of two configurations one in which you have got one set of hydrogenic quantum numbers, other in which you have a different set of quantum numbers, hydrogenic quantum numbers that obviously you are conceding that the hydrogenic quantum numbers are not good quantum numbers because the state is described by a superposition of these two. So, you have the two electron system like the helium atom or the negative hydrogen ion and let us consider some configurations you can have 1s2 that is what would come to your mind. You may also have 1s2s for the two electrons right you may also have 1s3s and you may also have 1s, ECs, EC is a continuum energy. So, it can go into the continuum and then you may have infinite number of different possibilities. You may also have 1s2 and then 1s2p why not you may have 1s3p. So, you have got very many different possibilities. So, you have a large number of actually infinite number of possibilities different configurations in which the two electron system can reside and in general the two electron system if you want to describe it in a complete set of bases it will then have a superposition of all of these bases states. But then this is not the only one you may also have 2s2. So, you may have another series which is built on 2s2 configuration. So, 2s2, 2s3, 2s4, 2s continuum s you may also have 2s2 then 2s2p, 2s3p and 2s continuum p and so on right. And both the possibilities are considered over here you may have like each of these you know stipulates a bound to bound excitation from a discrete bound state to another discrete bound state. But it also accommodates transition from a bound state to a continuum state and it includes the configuration interactions between bound to bound and bound to continuum states. You may have other possibilities and here notice that there are 2 principal quantum numbers which are involved. So, you write one of them within the with an upper case n and the other with a lower case n. So, which one is written with an upper case and which one with a lower case. So, there is there is no you know means sometimes the notations differ, but you just have to be careful that one of them represents both are principal quantum numbers one for one of the electrons and the other for the other electron in any case the electrons are indistinguishable you are not putting a tag on any one of them, but it only represents the configuration of that particular state. So, you may have other possibilities this is built on 2s2 this is built on 2p2. So, you can have a 2p2 then 2p3p, 2p4p then 2p2, 2p3s, 2p4s and so on and how many can you think of like this let us pack them all in etc that is one of the very nice words in the English language it includes everything specially your ignorance. So, everything is included in that. So, here you are. So, if you look at the helium atom the 2 electron system you have got the 1s2 configuration then you can have a 1s2s then you can have a 1s3s 1s4s or 1sns in general till you can knock out 1s into the continuum s and that would be the series limit and this series limit is at 24.6 electron volts for the helium atom. Then you can have another one which is built on this 2s2 which is what I mentioned in the previous slide and then you can have the notation is this that 2s3s corresponds to one electron having a principal quantum number 2 and the other having 3 it does not matter which one is which. Then you can have another one and you can actually have an infinite series and notice that the discrete bound to bound of one series can be embedded in the continuum of another. So, those are degenerate and you can expect you know resonances coming from this possibility that whenever you have 2 alternative channels like in a Young's double slit experiment the amplitudes of the wave functions would superpose not the intensities. So, here the labeling is in terms of the independent electron picture which we know is not correct, but what we are doing is correct because we are using configuration interaction which is a superposition of the one electronic states. So, we have not shown in this figure the single electron continuum states above each ion limit for the different quantum numbers n and we have also not shown in this figure the double electron continuum. But then there are other figures in literature and I will show you this classic figure from Fano and Cooper's paper. So, here you have the series limits over here which is 24.6 over here then 65.4 for this then 79 for this and so on and there will be so many which go through the roof of this picture. So, you have this classic figure from Fano and Cooper's review in modern physics in 1968 many of you would have seen this is this is one of the most famous figures in literature on resonances. Notice that you have the only difference between this figure and the previous figure is that energy is increasing from left to right whereas in the previous figure the energy was increasing from bottom to the top. So, it is the same as a previous figure, but it is tilted through rotated through 90 degrees and what is also shown are the continuum over here. So, you have got the first ionization threshold at 24.6 which we have seen the second ionization threshold at 65.4 then you have at 72.9 and then you have many more. So, all these other ones I had not shown in the previous one, but then you do know that there are infinite numbers of them and notice that the bound to bound transitions some of these discrete lines they are embedded in the in this continuum and that is true for a large number of different cases. So, what you will then have is a resonance between a bound to bound transition and a bound to continuum transition you will get a resonance and you can analyze in terms of the Fano shape parameter. So, I will proceed from this point in the next class if there is any question for today I will be happy to take otherwise we conclude the class at this point.