 Today we are going to discuss about the high resolution electron microscopy or HRTM in details. In the last lecture I have shown you the microscope with different parts of the microscope as well as different operations of the normal electron microscope. And in this class we are going to discuss first what is the basic meaning of high listening microscope and how such a microscopic characterization technique can be used for different purposes. So first we will show you some of these examples which I have even shown in the class one before the last one. This one is basically an image taken on cobalt and gallium dot zinc oxide left inside of the image shows the normal bright fill image of zinc oxide doped with cobalt and gallium with a electron diffraction pattern taken along 00001 zone axis the right hand side of the image is basically taken from one such crystal showing the lattice fringes that is the columns of the atoms are clearly shown. In fact if we measure the distance between the two atomic layers it should come as a de spacing of 00001 plane of zinc gallium cobalt of zinc oxide. The next one is the little picture of same crystal shown as bright dots the last picture you have seen the lattice fringe that is the different lines showing the columns of atoms here you can see white and the black dots as the signature of columns of atoms. So but we do not know which atoms stands for what particular element but we can clearly see the atoms very nicely these images are taken in normal conventional electrolux electromicroscopes with the fake or filimission gun control machines. The next image is again taken from CDS crystal the cadmium sulfide where you can see a very small crystallites about approximately 4 to 5 nanometer diameter where each dots each white dots signifies a colobama atom similarly black dots which are not observed clearly also signifies a particular colobama atoms the revision hyaluronicroscopy has evolved quite a lot as we stand today we can actually have very high power aberration corrected hyaluronicoscope in which we can even see different atomic arrangements which are not possible even furious back the late picture shows the silicon crystals you can even see clearly the silicon dumbbells we know that silicon dumbbells forms inside the crystals the silicon atomic dumbbells can be easily deciphered from this picture right inside shows even the information limits a resolution level rather of 0.8 Armstrong the distance between the centers of the two atoms in the dumbbell structure the bottom picture shows the Stonson titanate or SRTI O2 where the this a shows the atomic positions along certain crystallographic directions B source conventional hyaluronicroscopic image where you cannot even see different atomic species properly because of the resolution problems on the other hand the C source the different atomic species even one can see Stonsium which are the big atoms like this a white color the titanium ok titanium is basically sitting like this that one can even see the oxygen atoms there residual contrast kind of oxygen atom can be seen because these are all aberration character in microscopy is COS detector is used we will all discuss this thing in detail in subsequent lecture if a one is very careful even one can see the vacancy in the atomic in the oxygen atom positions so therefore this is the state of part today of hyaluronicroscopy so if you want to understand the hyaluronicoscopy in detail we need to look at from a very simple picture of a imaging system but I would like to tell you something more of a hyaluronicoscopy in the sense that among all the structural character and techniques hyaluronicoscopy tells you the atomic structure along a particular incident in incident electron beam directions to the atom resolution level which is not possible in any of these characterization techniques not only that as we know that if we can take this atom musical images along different electron beam incident directions then we can build up a three dimensional structure and that three dimensional structure is what is exactly we develop from the XA diffraction or Newton diffraction studies so in the viewpoint we can say that it is possible to obtain the structural informations from different directions of these incident electron beam in electron microscope as well as also it is possible to build up a three dimensional structure to the atomic resolution level so normally electron microscope or conventional electron microscopes which we use nowadays has a resolution level of 1.2 to 1.6 Amstons but with the help of aversion guide electron microscope we can reach a resolution limit of 0.6 even better than that once in Amsterdam or even better than that so therefore it is possible to obtain the structural information to that level to the resolution levels of less than one Amstons is what is what makes hyaluronicoscopy so powerful technique obviously as I have shown you pictures where you can see even atoms actually we do not see the atoms you see the potential of an atom when a electron interacts with the material and it forms a particular image and we see basically potential of the atom in the imaging plane we do not see the atoms but many people write in the textbook and even the in the lectures that with quote and unquote we see atoms well fine so these are all very good features while so microscopy that we can see atoms virtually we can build up the structure we can get even composition information at the atomic energy level and we can do many other things but it has its own problems like any other techniques the potential of this technique is severely affected by the fact that we need to interpret the images so quantity interpretation of the images makes this technique rather I know difficult one for many users and it has been found that to quantitatively interpret an image we need to have a parody idea of the structure of the material and therefore hyaluronicoscopy is to some extent dependent on the many other techniques from where you can get a parody structure of the material which is very valid in case of complex structure like was a crystal or you know in commensurate material structures where we do not have any idea where the exactly atoms sit so therefore describing the hyaluronicoscopic images quantitatively from those kind of crystals or those kinds of materials is a big challenge for us although advent of computers have made our life easy but it is not very simple still so far to run the simulations in the computer one needs to have lot of apparel knowledge about the structure of the material completion of the material many other things which needs to be obtained from different other characterizing techniques so by knowing all these aspects let us now look at what is actually a hyaluronicoscopy from the very basic point of view and how we can understand it from the image imaging tool of an electron microscopes let us start with a very simple thing that is the camera or other camera obscure in camera we know it is basically black box and in which you have a pinhole and the light passes through this pinhole and from the image so therefore if we have a image in image plane like this and we have a window or a small hole or pinhole through which the lights passes through and we form an image there this is what is basically a camera looks like the obscure camera means the old camera modern a camera has many other lenses so therefore if I describe if I have to describe this camera image formation in terms of you know mathematical aspects so we can always assume that this image to be one dimensional and the functions are very simple so let us do that if we assume that image has object has can be given by a function like fx how may object is given by a function like fx and this object is basically getting focused on the image plane while these rays are passing to the pinhole so therefore object the the rays are passing through a small aperture and if we assume this aperture to be at the window through which the light is passing through then we can interpret it the image basically the contribution of the point x in the image plane by this function called f image x which is nothing but the ax minus x prime into fx fx prime dx integral okay so where ax minus x prime is nothing but the aperture function so that means if I want to get the image information or the information in the image plane we need to use the aperture function aperture actually modifies the image during formation now this is very simple equation or very simple equation which is used in the imaging technique which can be available any books you can actually take convolution of this to ax and this equation can be written like this ax convolution of ax and fx giving the x to the image function so this function integration of the aperture function and the and this object can be simplified like this the convolution product of ax and fx and obviously any image whenever we form we need to to want to get the diffraction information one needs to do the Fourier transformation so in the Fourier space this expression can be written like this if I am in the G but G is the spectral frequency so AG in convolution of AG and HG so we know that convolution of the any function once we transfer into Fourier space remains the convolution of the Fourier components but G is basically now the spatial frequency so G is nothing but a spatial frequency because in the Fourier space everything is described in terms of frequency or inverse of lambda and AG or rather let us write like this to depict it the Fourier space AG is given as Fourier transform of ax can be given as known as modulation transfer function are known as also the modulation transfer function of the imaging device so this is a very important aspect of any imaging device because the whatever is getting image from the object plane to the image plane is getting modified by the aperture function so any simple camera if we see the image even though we are assuming that the whatever is there in the object is getting directly projected into the image plane is not true so this is what actually dictates many things like resolution of them of the imaging instruments and information limit of the imaging instruments and many other aspects so every imaging device is that is why I said is characterized by this function called modulation transfer function or simply called transfer function in the books it describes basically magnitudes of the this spectral frequency or G which is getting transferred to this device from the image object planes to the image plane so how much of this magnitude of the values of the spectral frequency which are present in the object is getting transferred to the image plane via this device is what tells you the transfer function so we will discuss more about the transfer function in the subsequent slides so let me just tell you something about also resolution which can be easily derived from this picture but once you go to the next slide it will be clear so let us suppose the mathematically this is my the image in the object function fx as a function of x in one dimensional so you can see lot of lot of spikes and lot of basically other things and if we just take this Fourier transformation of this fg it becomes a very discrete speaks in this Fourier space because this has transferred from the image space this Fourier space so therefore everything is a function of frequency and we can always say the aperture function is like this as shown in this picture so aperture function is one in the aperture and zero everywhere else because this is small hole through which light is passing through so therefore any places outside this small hole light is unable to pass through therefore you can assume this would be one and the aperture during other places and if I just Fourier transform this this again gives me lot of speaks okay now if I take the convolution of fx and ax it becomes a function like this with with the frequency with a busker distribution so which is summation nothing but is in this and this and most importantly if I take convolution of fg and ag it gives me 1 2 3 which are basically can be called as the signal in a basically any emerging device so therefore whatever there in the in the object space is can only be converted into the image space once you have very nice signal coming out of there this is what actually seen or observed mathematically when we transfer any points in the object planes to the image plane through and the aperture function or through any imaging device other way and TM or HAT image consisting of many many imaging plates or imaging you know imaging lenses so therefore for each of the lens there is an imaging function a transfer function so and they are all cumulatively added so therefore the total image quite a quality of image which you observe obtained after the object passes the light passes through or the electron may passes through all these lenses is basically can be obtained if we have convolution of all these the aperture functions in this Fourier space so this allows us to basically define the resolution so resolution is basically of an image instrument is defined as a cutoff or 1 by cutoff means 1 by row cutoff in the 6 that means this is the 1 by row cutoff which tells you a distance difference between the signal and the noise so if I take the ratio of signal noise there is a cutoff below which nothing can be transferred by the the imaging device so any other frequencies a special frequencies here above this 1 by row values which are marked here can no longer be transferred by the imaging device so this sets us the resolution this solution actually you know is basically it can be easily compared with the real age definition of solution which I have discussed in the class 2 3 class before so this is basically in a physical sense definition of resolution of any machine now Fourier transformation of this transfer function basically or this AG or rather whatever we have said is basically known as the in plus transfer function which we will discuss later this is nothing but generalization of the aperture function of the camera it is basically you have seen here you can see this there here also you can see that this is basically sharply picked plot of the as a function of FGA convoluted AG as a function of the 1 by wavelength of the reciprocal space so therefore the width of this function basically can relate to the real age resolution criteria. Now this is very simple and as I said the solution can be easily plotted like this so therefore if I plot the this is as a resolution in the in the x axis or the signal actually not x axis so if you clearly see that this is a noise level so therefore any frequency or any special frequency beyond 1 by row can no longer be transferred by that by these imaging device into the image plane so therefore we tell this is the resolution of the imaging device so all the other frequencies all the other values of G special frequencies can be easily transferred by the machine or by the upper these the imaging device is a camera or a electron microscope it can be transferred into the image plane and that is why this can be defined as a resolution of the machine. So now theoretically these are all possible practically no microscope is a perfect one or no camera is also perfect one there are a lot of defects inside it and these defects do play a serious do play a serious role in basically reducing the resolution level from these two even low much higher. So let us now discuss by knowing this you know so these aspects very well let us now discuss about the electron microscopes this is basically very simple thought I just gave you at the beginning of this lecture let us now talk about the image formation in a electron microscope we know the image formation electron microscope is a coherent process what does it mean object as well as transfer functions in the in the electron microscopes are very complex functions and they are complex functions of both or they are the complex function with the amplitude and also phase component because they are all waves so like any other waves there will be amplitude component there will be basically a phase component so as we know that wave function so whenever a basically whenever if you think of an object in electron microscopes or sample electron microscopes we know that electron beams falls on the on the sample and passes through the sample also because it is a very thin one see if I think of that this is an sample in the electron microscope and electron will falls like this and interacts and then either get diffracted or not get diffracted through this exit plane okay so by knowing this this is what happens in electron microscope I discussed with you in the next class before by knowing this now if I think that wave function or electron wave function basically is at the exit plane is given by this we can always write this where basically the R is the exit plane and Psi is the function or electron wave functions so if I have you know this kind of consideration we can always think this wave function of this plane as a planner you know source or the spherical waves okay these are the planner source of the spherical waves here at the exit plane huge in types we can always do that and obviously then in that case if they are all planner source of the source of light or electron as they exist plane so we can think of that they did the sample is acting as a grating or diffraction grating and then according to Farnofer's diffraction theory Farnofer was one of the scientists who basically worked tremendously on the diffraction theory had to use theory the complex magnitude of the diffracted wave is in the direction of the wherever the electron beam is going this the action can be the diffracted interaction can be given by certain value of the reciprocal vector so therefore we can always tell from the corner for diffraction theory that complex aperture diffracted wave and the action given by the any reciprocal value that is vector is can be written given by the Fourier transformation of the object function. So this is the object function if I take a Fourier transformation of this object function that is this is nothing but the diffracted beam I are the what is called the diffraction diffracted beam intensity this is what is the what is called the Farnofer diffraction theory tells us from the very basic knowledge now in an electron microscopes we know that objective lens so let me just go there we know that objective lens actually placed behind the sample that is what you see here the this is the objective lens here in the microscope here also so sample is basically immersed in the object lens as I told you in the class I have been shown you in the while showing the microscopes so obitulium is basically behind the object therefore it focuses all these waves into a plane called a back focal plane okay and that is also shown here this is the black focal plane you can see here it basically all the whatever waves coming out of the existing planes they are focused by the objective lens on the back focal plane and then actually the this positions of this back the this you know focus beams gives you basically diffracted spots or diffraction beams rather and when these diffraction beams as then focused on the other form the image on the imaging plane we get a diffraction pattern therefore back focal plane of the objective lens contains the Fourier transformations of the object basically if you think of it that way in the electron microscope we can change actually that is what we have shown you we can change the settings of the electron microscope rather we can change actually the focal length of these different lenses like intermediate or the first intermediate lens to either focus this back focal plane on to the image plane and see to obtain these to see these diffraction patterns and obviously the intensity of it of this diffraction beams will be intensities will be proportional to square of this Fourier transformation of this the exit wave in the Fourier plane and in the object is periodic there is a crystal the diffraction pattern will be ready the object is continuous diffraction model will be continuous these are all understandable this is the basically classic sense of diffraction pattern formations in a classical TM so that is normally routinely done in the microscopes whenever we insert a sample first thing we are going to crystals and then put the aperture in for the select a particular aperture in and get the diffraction pattern that any any electron microscope users who are seriously in a microscopy can immediately recognize this whenever go to the whenever he or she goes to the microscope there is a second kind of stage of image formation this is the first kind of image formations and in factually one can use this the diffraction patterns to do diffraction contrast imaging which I will talk in a within moments time so in a second set of image formations the back focal plane can be basically can be thought of acting as a set of huge in sources of the spherical waves we can always think of their acting actually as a set of huge in sources or the sources of the electron beam as a spherical waves which can be then allowed to interfere so if this diffraction spots are allowed to interfere and then with it can form the interference patterns on the image plane so whenever that happens we get something what is called lattice fringe images which I shown you in the beginning of the lecture so this basically is called whatever was forming this you know the by interfering this set of huge in spherical waves and then forming a lattice fringe wave if lattice fringe is nothing but inverse Fourier transfer machine which basically reconstruct the object function psi r in the image plane so therefore by just looking at the camera picture and comparing you with the electron microscope one can simply understand how this basic things actually happens in electron microscope let me show you the how the electron diffraction pattern form as I said if I have a object here the objective lens basically focuses all these they are diffracted beams into the back focal plane and that is what we get diffraction spots when he basically image this the diffraction spots a focus refraction for basically on the image plane this is with simple like a formation of nothing but Fourier transformation of the the electron beams or the exit waves at the back or the exit surface of the crystal so now once we can use as I said we can use this diffraction pattern to image or get diffraction contrast which is normally done in the electron microscope standard electron microscope so this is my diffraction pattern on the back focal plane okay as you can see focused by the and if the objective lens this is the objective lens here and this is the diffraction pattern on the back focal plane so if I take the the the bright spot or the the undefracted spot or the transmitted spot or the fourth scatter spot whatever way you can explain then whatever we form is nothing but a bad fill image in the field is bright and the image is looks black on the other hand if I take any of the diffracted beam here like this and use this to form the diffracted the image then we what you get is a dark fill image in the field of use dark image is basically bright and this is what is shown here in this way so by using in the aperture of the aperture we can either form the back right fill image or you can form a dark fill image and this is routinely done in electron microscopes anyone who is doing normal series microscopy are very well conversant with this on the other hand what we are saying is the second sort of imaging process just a few minutes back I discuss is nothing but if I use all this diffracted and the transmitted beams and allow them to interfere and then I form what is known as a lattice fringe or lattice image this is what is shown here you could see that I have marked a larger area all I put a bigger aperture in the in the diffraction pattern and selected at least you know 9 the 8 diffracted spot along with the transmitted spot and we can form this kind of the phase contrast images this is the second set of the image formation which is done routinely in a electron microscopes nowadays because of the advent of different you know high quality microscopes this is now also can be done routinely so as you can see that that in a real practice by putting in aperture in the focal plane of objective lens this possible to form either a face contours or interfere face contest image which is basically nothing but the interference of these diffracted and transmitted electron beams from the the back focal plane or we can select one of these basically transmitted beam or the diffracted beam and then this allow this interference to happen and from the images but as you as I told you electron microscope like any other objects in this world which man makes is not a perfect device so therefore anything anyway which is passing through this objective lens any electron wave he undergoes phase shift and also amplitude reduction and amplitude reduction nothing but a damping therefore we need to consider the transfer functions of it to objective lens to really describe actual image and this is what we will do it a transfer function in objective lens or objective electron microscope in objective lens can be written or can be given in a very simplified way I would not discuss the detail derivation of that but transfer function can be written very easily so I if I write transfer function in a electron lens like this Tg Tg is nothing but to a G G is special frequency and sin Xi G so this is what is basically simplifies transfer function in a like in a orbit or electron microscope of the objective lens where a G is basically tells of the effect of the aperture function effect of beam selecting aperture and size E which is very important parameter here is basically phase shift it tells are the phase shift and a G is the aperture function so this is a very important functions of the objective lens in electron microscope this will tell us not only the resolution but also the information limit which you can obtain in the high listening to microscope that is what I am telling so wave function at the image plane obviously can be written like a very simplified wave function can be written like this this is wave function in image plane is image plane is if it is R then can be written as inverse wear transformation so inverse wear transformation is like this in my transformation of the transfer function and the size E and intensity obviously will be proportional to square of the wave function that we know so that is therefore and this is very simplified theory we okay we assume in this theory that the crystal is a very very thin so there is no in I know interaction between these diffracted beams inside the crystals once they come out then you know the interfere from the image so that is not all this is what is called as a coherent approximation so if you assume the coherent approximation then only these formula is correct otherwise the formula gets complicated and those complicated formula needs to be derived based on the concept of transmission cross coefficients okay how this the beams are actually interacting and because of that there is a cross coefficients so if you know a transmission cross coefficient then we can actually see this function then will be much more complex which I can actually give you so Psi R can be written as integration of G E plus G prime into multiplied by transfer function G E plus G prime G into Psi G thus basically tells you the interaction between these two different special frequencies G and G prime so forget about it we are not going to use this we are going to use the coherent approximation and for that the what is called wave function at the image plane can be calculated using inverse wave transformation of the transfer function and this is the size E well by knowing this now actually one can seriously look at the exact nature of this phase shift or the size G function which you are going to do in the today's lecture very seriously and that is basically the purpose of because this will tell us many important aspects of the image formation. So as you see that size E as I said these goes into the transfer function very importantly this is the aperture function so aperture function can be derived as I showed you in any any electron micro any device but let us first look at what is the nature of this function the studies indicate that size E is basically depends on three factors or it actually has a combined effect of CS the spherical average constant of the lens as I said any lens is not a perfect lens so like the camera also we know the lenses of spherical aberrations mostly spherical aberrations chromatic aberrations also comes into picture because chromatic aberrations comes from the difference in the energy levels of the incident electron beam but let us for the sake of understand or the simplicity assume that spherical aberration is the most important defect and then second factor is the delta F or the focus and third factor is what is known as this is a delta F is the focus which is basically the depth of the tilt lens defocus and third factor is obviously the wavelength we will see that and they all CS and wavelength are related we will see also that so if we can consider we can find out a point in the at the specimen okay basically what does it mean the effect of this on the size you means that if I have a point in the image plane that point will be focused as a disc because of the effect of spherical aberration and the focal and are the defocus of the objective so therefore we can basically tell that that you know with the image whatever has been formed on the image plane as a disc can be given by this radius delta theta equal to CS theta Q plus delta F theta this function tells us the disc which is formed so if you want this to be small then you want CS and delta F to be optimized and that is can be done by analyzing the whole problem mathematically which you will do the next 10 minutes or so finally so we are discussing about this effect of the different parameters on size E so let me just describe it in detail all that as I said that size E this function which is going to the transfer function very as a factor depends on three aspects the CS the spherical aberration constant defocus of the orbital lens and the astigmatism of the of the tiller so astigmatism can be easily corrected as you know even a normal camera also astigmatism can be corrected but the these two parameters CS and delta F they need to be optimized to basically have the best value of transfer function so that we can obtain very good resolution in the images now we shall go through a very simple exercise to derive this the value of this size E let us do that and as I said that the basically the exercise is nothing but the combining this effect of CS and the defocus of the outer lens so if I do that we find that any object any point in the object normally will be focused on to the image plane like a disc because of these presence of the CS and the delta F and so therefore we can write down the diameter of the disc as delta theta to be equal to CS theta Q plus delta F into theta and so basically the rays which are passing to the objective lens they at an angle theta are not focused to the same point okay because of the spherical aberrations and finite value of the of the delta F that is why they are basically coming at that but we know that that are not only one value of theta we need to consider but we need to consider all the values of theta theta is nothing but the diffraction angle so therefore we need to consider all the diffraction angles and that case we need to average out the value the this value as a function of the diffraction angle and which can be done by simply doing the integration of this equation form 0 to theta CS theta Q plus delta F theta D theta and as you can see if you once integrate this equation one can get this as a CS theta to the power 4 by 4 plus F delta F by 2 into theta square this is what is the basically the disc average disc diameter if we integrate this we can get now we know that from Bragg's law that we can very easily write from Bragg's law that 2 D sin theta B equal to L lambda so therefore we can write down for TM basically for normal microscope we know theta is very small if theta is very small then we can write down 2 theta B sin theta equal to almost like theta equal to we can write down very simply that lambda into G where G is basically 1 by 2 D you can see that G is 1 by 2 D which is the reciprocal space these 1 by D actually is the reciprocal lattice vector so if this is the case the 2 theta P is basically very specific value because this is the diffraction scattering angle for the diffraction to happen now we can inside this if this was called value into this function and then we can get the whole function like this the Psi G Psi G is the phase that is equal to D 2 lambda y 2 pi by lambda into D G and this is nothing but then 2 pi by lambda into C s theta to the power 4 theta to the power C s theta to the power 4 by 4 plus delta f theta square by 2 and this can be easily written in a mass simplified form like this so delta f into delta f by delta f into lambda square now we can convert this theta into lambda as you can see from there so therefore we can finally get this to be equal to twice pi by lambda into C s lambda 4 G 4 by 4 plus delta f into lambda G by 2 theta V that is equal to lambda square G square by 2 so that is basically gives me the value of Psi G and therefore that means Psi basically this Psi is basically function of pi or Psi committed as a pi delta f lambda pi delta lambda into G square plus plus C s half pi into C s lambda to the power C and G to the power 4 so what I can see is a very complex nature of this function Psi if I take a sign of this that becomes very simple complex function and it is very difficult to even calculate this functions normal simple mathematical way we need to use certain tools to calculate that but whatever you see that the Psi depends on not only the delta f lambda and G okay depends on delta f lambda G but also C s very strongly so that is why one needs to optimize the delta f and this C s in electron microscope very clearly and then the next and basically ideally the transfer function should be like this form 0 to G1 so the specific value and beyond that it will be all 0 this is the ideal transfer function in electron microscope what this is never possible see if I plot this transfer function as a function of G here generalized functions okay obviously depends on the values of delta f the lambda and C s if I consider C s to be you know 1 millimeter and delta I have to be 50-58 nanometers and the easy row that is the electron beam by the radiation the voltage by the electron accelerator is to be 200 k L electron volts then this is the nature of Tg so you can see that it is not doesn't look like the ideal nature rather it is basically from 0 it goes down the increases there are lot of crossovers that makes the life very difficult now let us look at the nature of this graph as different functions so the next one is basically as a function of C s so if you can see if we increase the C s from 1 to 2 to 3 the function that is totally changing as you can see here the for the value of 1 which is very low for C s with a particular value of the accelerating voltage and the de-focus we can see that there are different crossover crossover here is basically crossover there is coming a different values of G even one can even see that crossover is basically changing basically coming at a lower value of G as you increase the value of C s now if we keep the C s to be 1 millimeter because that is why the best possible you have seen and then vary the focus by the de-focus from 30 to 50 to 70 minus a nanometer and one can see that how these things are changing in fact one can see as you go from minus 30 to minus 50 basically this width is changing the G1 is also changing the crossover point at which this transformation is 0 is also changing but once you go to very high de-focus values we can see there is a two spikes coming into picture one at a G1 other at a G2 and both of them are basically not meeting this the line at 0 so that makes the life very complicated therefore what could have seen that these two parameters the C s and the delta f has a strong role in basically determine the transfer functions and any electron microscopes C s is obviously given by the orbital length configuration C s can never be you know changed so given a microscopes you can what you can change is the objective focal length or the delta have value the refocus so in the next lecture we are going to see the effect of de-focus and how we can use this to obtain various informations in the highly center micrographs.