 So in the first class of hydrolysis electron microscopy I have discussed to you the basic fundamentals of hydrolysis electron microscopy and we are going to continue from the class which where we ended in the last class. So as I say so I have shown you that the contest transfer function which is known as the transfer function in a hydrolysis electron microscope is basically a very complex function and it can be written like the one I showed you transfer function is equal to 2 where the first factor is as a function of G is basically the aperture function and this actually is a strong function of the G and Psi is basically the angular factor which comes into play. So from this function we have seen that how the Psi actually plays a very important role and finally we derive the optimum values of the Psi the one which I can write like this is depends on de-focus as well as depends on the CS or the aberration spherical aberration constant. So therefore the factors which control the constant transfer function basically at the de-focus and the spherical aberration constant of a microscope obviously ? goes into the picture because ? is the wavelength of the electron beam and G is the spatial frequency which is inverse of ? and so therefore the de-focus and the spherical aberration constant the markets the values of Psi and Psi can be as a function of G as we have discussed here square with little de-focus and the fourth power for the CS. So we know that for a particular microscope CS is a constant CS values never changes for a particular microscope but obviously it can vary from microscope to microscope also it can be changed if you change the objective lens of a microscope. So knowing that CS is fixed by particular microscope the free parameter parameter which can play the important role is the ? F or the de-focus and that is where we actually shown you how this can be determined using different values of CS and different values of de-focus how this function can change. So we discussed this factor in detail and find out what the optimum focus limit for highly seen microscopy. So this is the slide where I am showing you the actual ideal transfer function when the microscope lens is a perfect as you see that the no microscope is a basically a perfect one because there will be lens defects like spherical aberration and other factors astigmatism and as well as this chromatic aberration so chromatic aberrations can is very difficult to correct and because of the aspects that the beams which are falling on the sample may have different energies but astigmatism can be 100% corrected on the other hand the spherical aberration which is a important factor going into this equation is very difficult to correct but off late it has been found as spherical aberration constant can be spherical aberration can be easily corrected in the electron microscope and can be used. So this is the ideal spherical aberration the transfer function which is shown here. So obviously in ideal transfer functions the as a function of G the obvious the value of zero in the transfer function should be as small number as possible the number of zeros here is this and this is for an ideal microscope which is not possible to achieve at any cost so in a real microscope the transfer function so the actual the here I am showing you the ideal transfer function for any term is electron microscope. So the you can see that the transfer function varies starts from zero is goes down and finally it is basically a square function reaching zero again at certain value of G which is at G1 and so as you know that more 0 means more less number of information going into the going to the sample to the image so that is why the ideal transfer function will have only one zero here and all these information which are here are basically at a low spatial frequency of transferred to the image from the sample so which is this kind of situation is never really achieved because all these microscopes will have defects and the defects change this transfer function drastically and I am showing you one such picture where the actual transfer function is shown TG as a function of G here and TG values can actually can be varying from plus 2 to minus 2 but what you can see is that there are lot of variation not only number of zeros have increased in these transfer function but also the special frequencies depending special frequencies the values are changing so there are higher order values there are lower values of the transfer functions now one can actually see the effect of transfer function effect of the space spherical aberration constant this transfer function this is what is shown here which I have shown in the last class itself if I change the spherical aberration constant CS from 1 to 3 millimeter the transfer function changes drastically for one basically the transfer function is a very large width before the first zero comes here and again as a CS increases this zero is slowly moving to the left side the lower value of G and therefore less number of a space lesser the special frequency can be transferred in the actual microscope and this is done at a fixed value of defocus minus 60 manometers so once we optimize CS value okay here it is seen that optimizations here value is basically one and then we can look at what happens basically as a function of the defocus as a function of delta F so a delta F is varied here for 200 global microscope with CS one millimeter from point minus 30 nanometers to minus 70 nanometers as we vary from minus 30 to minus 70 the profile of the transfer function for files a function of G is changing drastically for very low value of delta F that is minus 30 the G1 comes at a frequency like this and there are several zeros here after this G1 and as you go from minus 30 to minus 50 the G1 value shift to the right side that we more information for lower spectra frequency ranges can be transferred to the image plane but as we increase the value delta F to space minus 70 nanometers you can see that G1 actually never reaches the zero value or zero axis value and G2 rather so G1 actually is coming at a higher value but G2 is another kind of spikes coming to picture which is giving or restricting the transfer function to a much lower value of G which can be transferred to the actual image. So these are the two effects of CS and delta F then for a given microscope as CS values are known how do I optimize the delta F values and this is basically done by scientists called Otto Caesar in 1949 that is what we will do so as you can understand that the presence of zeros in the what is called in these transfer functions are basically tells you the gaps in the output signal correct because these zeros means they had giving no output signal as far as the transfer of the actual things in the sample to the image plane is concerned. So that means these frequencies can be thought of filter out so really best transfer function will be one in which there are least number of zeros the one I have shown you one slide back here is basically the idle transfer function where there are only one zero that is only one zero so therefore best one will be the one you where there will be least number of zeros in the transfer function so that the best resolution is possible. So in what Sergio has done Sergio Otto Sergio along back in 1949 he has basically described we have basically described the optimization of the focus or defocus in a particular microscope what is said that the transfer function can be optimized by balancing the effect of spherical aberration against a particular value of defocus so that means if I need to get a particular value of defocus where the effect of CS can be balanced this Delta F is known as Sergio defocus which is given by this formula minus 1.2 CS lambda to the power half and many times we write this equation as minus 1.2 SCZ where SCZ is known as given the Sergio focus defocus CS lambda to the power half. So what you can see clearly from this picture is that the Sergio focus is basically tells us that all the beams will have at the defocus all the beams will have nearly constant phase out to these first zero a fast crossover in the zero G axis that is what is shown here this is the first crossover at the zero defocus a zero G axis. So therefore all these beams all the electron beams will be nearly constant phase out to this and this is what basically required to obtain the best possible resolution or information in electron microscopes this performance the best performance obviously will be expected from microscope at values first value of first crossover value on the zero G of the G axis unless until obviously we do lot of image processing later on that is we improve the contrast by doing some other means but this is what is obtained from a particular for a particular machine with a given value of CS now how does this formula arrive that can be basically done by using simply several aspects. So what you can see from this from this picture is that from this what is called for the slide is that the closest we can get to the idle car idle car which I have shown you is basically corresponding to sigma corresponding to this value the Xi G to the equal to minus 120 degrees and that correspond to sign Xi G to be equal to minus one. So that means this is the obviously maximum value or whatever minimum value of sign possible because sign is a bounded function between 0 and 1 or minus 1 and 1 basically depends on the angle. So minus one is the minimum value of the six sign is possible and this is the gives you the best optimal possible the value of transfer function now let us see how this can be basically derived using this formula which I have already given you for Xi and then for Xi we have seen that the Xi is given depends on the defocus and of and also the spherical aversion constant so this is equal to ? ? F G to the power square ? G to the power square plus half of ? CS ? Q G to the power 4. So if I basically minimize this function at a particular value of G I can get basically the Sager focus so let us do that if I minimize this I get a simple equation twice ? ? F ? G plus again twice ? CS ? Q G to the power Q and this obviously has to be 0 so we can write down this as this plus CS ? square G square that can be easily seen so when ? will be – 120 so we can obtain from this equation that this is – 2i ? by 3 equal to ? ? F ? G square plus half of ? CS ? Q G to the power 4 and combining these two equations one can obviously obtained the simple mass simplified expression of Sager focus or the focus that is F CHS CH is equal to – 4 by 3 CS ? to the power – CS 4 by 3 CS ? to the power one by half so that becomes 1.155 CS ? to the power so that can be easily derived by this way just by putting the first derivative of ? respect to G to be 0 for a particular value of ? that is 120 degrees because that 120 ? ? is equal to – 1 so by using these equations we can derive that Sager optimum Sager focus become 1.155 CS ? to the power half which is very close to 1.2 and this by finding this calculation obviously one can arrive at the same Sager focus and which is basically nothing but that balancing act which you do of CS by putting the focus value at a particular number and this is routinely done in a microscope nowadays where the optimum image is obtained at a particular value of Sager focus. Now obviously one can go ahead and even go to the next step that is the first crossover is by this the next crossover can also be calculated and next crossover as you basically comes at a particular G value the next crossover can come at a particular G value given by this this is 1.51 CS – 1 by 4 ? – 3 by 4 okay and this crossover is very important in the sense that at this defocus at this basically Sager defocus which corresponding to this value of G will give us the resolution limit of the microscopes and this resolution limits can be obtained by sticking inverse of this which is equal to 0.66 CS ? Q to the power 1 by 4. So this gives us the optimum resolution limit or the actual resolution limit for any time is electro microscopes. So by knowing this number CS of the object in lens and ? we can basically get that that is why you understand now why people use million volt electron microscopes because ? can be extremely reduced by using higher and higher accelerating voltage from 100 to 200 300 people have gone up to 1.51500 actually kilo volts. So by an obviously another way of improving resolution is by correcting the CS which you are going to see today even the end of the lecture how we can do that. So this actually puts us the limits of resolution this but this is not the limits of the information information limit is much much actually higher value than the resolution limits in microscopes okay information sorry much much lower value than the resolution of the microscope information can be achieved even much lower in fact there are report that 1970s a detail of 0.66 was actually done or achieved and when the interpreted resolution was about 3.3 Armstrong. So you can see that that actually it details an image doesn't mean that you can gain any useful information about the microstructure. So there is a distinct difference between the resolution limit and the information limits as far as the microscope is concerned. So therefore these two function one is as said here another one is that ?F said here which is equal to 1.2 CS lambda to the power half these two are the master equations in the high resolution electron microscopy they actually gives us give us the limits for any uses in the high electron microscope and this was due to auto said here and other one the first equation is basically developed by a Glaser. So therefore and these two people are these two scientists are called the pioneers of the highly selective microscopy for developing this concept in the highly selective microscopes. So that's actually sets the tone of the highly selective microscopes. So many students or many users have miss concepts and that highly selective microscopy means just getting a lattice fringe or just getting some information on the on the computer screen all on the imaging screen where you can see the atomic planes okay but that's not the case there are a lot of many interesting aspects one is to know. Now as we have discussed here or we have shown you that the transfer functions we normally do not take any of the things which are beyond this first 0 something which is basically coming beyond this number or beyond the G1 value okay which is basically of no use to us but why it is so or whether this is really to or not that's what is gives us the new concept called envelope damping functions. So plots on this TG as a PSYG whatever PSYG is also related to TG PSYG versus G as you can see here can they be extended from this far 0 to the other values that is the question we need to ask us because as you understand if we can extend this value to the higher G numbers and G is basically spectral frequency therefore we may be able to achieve much better information in the real space because G is in the reciprocal space. So if G higher value of G means lower value of X, Y or Z real space variables. So if we can extend this G to the higher values and include them into transfer functions in a electron microscope we will be able to achieve much higher resolution and much higher information limits also can be achieved. So if that is the case is it possible answer is no we are we will and we do not use this higher you know crossover or we do not extend the values of the G beyond this first crossover normally why this is mainly because of the damping effects because of the other issues like this spatial coherency of the electron beams and also chromatic aberrations. So which normally comes into play when spherical aberration is corrected and these are basically due to spatial coherency means as the electrons falls on a sample or basically if you look at the electron beam how small it wherever it is even it is 1 nanometer or 0.5 nanometer whether the spatial coherency of the electron beam exist or not is very difficult to ensure even using the the Fegg guns. So because of these resolution is hampered or which because of these these higher G values can never be accessed second one is the chromatic aberrations although we can use FEGs or the field emission guns to reduce the beam energy spread to even very low values still it is very difficult to get 0 chromatic aberrations or it is actually not yet possible to correct the chromatic aberration to that level. So because of presence of both this chromatic aberration and the specific spatial coherency problems this kind of these frequencies can never be actually used to get information from the sample now obviously the exactly mathematically this can be derived and mathematical envelope form of this envelope functions is very complex and one can actually simply write that T effective function as a function of G equal to T G EC EA EC is basically due to chromatic aberration and EA is due to spatial coherence of electron beam and once you put this envelope function into the transfer functions they are coming as a multipliers as you can clearly see and this basically gives us limitation of using this higher values of G into the transfer function this effectively means the envelope function is basically acting as a virtual aperture in the back focal plane of the objective lens regardless is whatever the value of the defocus that is what is the actual meaning of that so that means physically if we at all put an aperture in the in the skull in the back focal plane to remove this unwanted noise this must be less than size or let the virtual aperture present because of this enable envelope functions so that is why you know presence of this virtual aperture means higher order pass bands higher order pass band means this pass bands higher order values are never accessible it cannot be accessible because of presence of these envelopes that is that is why we I say that this is nothing but envelope damping functions are these the these envelopes are actually these functions frequency is getting damped out because of presence of the envelopes due to chromatic aberration and the spherical coherency and this is what can be shown there so as you see here this is the sign psi as a function of G so there are many such higher order frequencies or higher order G values which gets damped out because of the presence of this envelopes due to EC and EA so effect of EC and EA is to damp out this higher order frequency so we cannot access them so whatever may be the situation as you can clearly see that there are all the zeros more than higher than the G1 but this beyond that the values of the TG is very small and whether this can actually affect the resolution or the information on the microscope is needs to be analyzed and it has been seen that it can be it can affect so therefore although resolution limits will be given by the first crossover the information limits can be higher information limit can be given by this point G2 which is at a higher frequency higher spatial frequency so that is why in many microscopic images highly some microscope is nowadays whatever you see all the resolution is about say 0.8 Armstrong for the Titan microscope or point 5 Armstrong for the chromatic aberration microscopes but information can be actually picometer levels so we can achieve information much higher to quite a much higher resolution levels are much higher value better values than the resolution level given the microscopes that is what is possible in the microscope because of this at the day these corporations of the small you know higher order G values and in the in the transfer functions so these all give us some extra information now my knowing all these aspects how actually in a real microscopes change or operate the microscopes because as you see the focus of a strong role to play in getting the actual high-reson images because Delta F what you have seen is a very important aspects and we have to go to the sedger de-focus limit to obtain the best resolution image so how do I achieve the Delta F sedger in a real microscope that can be done in many ways obviously first thing one can do is that I know if for example vast task case of time constant transfer function is that when the transfer is minimized that is you just for the start for defocusing the orbital lens in such way that you do not get see any contrast on the screen or on the computer skin or on the basically fluorescent skin so if you do not see that that becomes your best from that you start actually changing the focus value and reach the sedger de-focus value so this minimum value here as you can see big is corresponding to this number of sign psi and this number of sign size happens to be 0.3 so therefore minimum contrast the minimum contrast for the the orbital and focus lens is given by minus 0.44 CS lambda to the for half this can be calculated in an actual microscope knowing the lambda and the CS but in real microscopic sense you do not need to do you can simply change the objective focus in such way that you do not see on the screen anything that becomes your the minimum contrast in the microscope and from there you start changing the focal length and as you change the focal length of the routine microscope you will be you should be able to reach the sedger focus otherwise what can people do is that in some special cases transfer function k actually settings special setting of transfer function are also used one such is basically aspect is to use something known as pass band or use a larger window in the transfer function okay so that we can allow this higher special frequencies to contribute the image how it is done obviously you as you see from this figure I clearly from the next figure which I am showing here from this figure very clearly you can see is that that this requires that psi to be constant obviously psi means this value of psi psi is depending on the figure G and also obviously depends on the CS and delta F this requires psi to be constant and also D psi to be constant and D psi DG to be very small one such example is shown here for silicon as you can see the sign size varying from minus one minus one to plus one as a function of G here G is the special frequency and what you can see is that many pass bands and if I select a pass band which correspond to the 111 reflection of silicon what you can see is that I can clearly get to see that side remains almost constant for this from this part to this part of G and D psi by D G is also very small that means it is all not zero but is very small value so because this side there is also very small and size the many constant so that we can use this higher order passing bands and normally this kind of higher order passing bands occur periodically and one can actually obtain such equation like this delta Fpn to be equal to 8n plus 3 divided by 2 CS lambda all half so when equal to 0 that become the first this value here okay so and that becomes nothing but 3 by 2 CS to the power half 3 by 2 is 1.5 and if you take root of that that becomes 1.15 so that is because that becomes actually the same as such a focus for n equal to 0 so this technique gives you the access to the higher special frequencies the here you can see here higher physical frequencies thus the finer details in the real space as I said higher value of G means finals in the real space and this price is basically is that only price we paste that now there are many zeros 1 2 1 2 3 4 5 extra zeros other than the first zero so because of this more number of zeros and zeros corresponding to no output signal and therefore there will be transfer the more zeros means lower partial frequencies will give you at more zeros and less information transfer transfer function will be heavily affected that is the only problem but this is used widely to basically select the focus in a hydrolysis electron images in many microscopes. So therefore an inertial I could say that in the in the lecture for the last lecture in this lecture I have given you very what is called brief picture of the transfer function how the transfer function actually varies as a function of ? F CS and obviously facial frequency G and how we can use it and different ways to obtain the optimum focus values are a by balancing the CS and this all are done routinely nowadays by computer okay so we do not need to do yourselves many times what people many of the users to is that that instead of trying to obtain the optimum say the focus we take through focus images that is we take images at different level of focus and then we see which is the optimum focus or which is the basically correspond to say their focus that can be easily done routinely nowadays using these the digital cameras where we do not need to spend money to gather grab images like earlier days we used to spend the what is called camera plates where each camera prejudice to God's about hundreds rupees so in a normal imaging technique where you use the digital microse digital imaging process you do not rather actually need to spend money for more images or less images you take so that's why nowadays one can easily take three focus images where at least 20 images as a function of focus can be taken and then from there one can actually decide which is the optimum focus image so that is actually trial and error method which normally people use but for the beginners actually this is the best method possible in fact when I started using HRTM I also did the same way taken through focus images from the armors of thin flames and then find out which is the optimum focus well that's basically the state of art till 1999 or 2000 from there a lot of changes happened in electron microscopy especially people have started correcting the CS as we have seen CS plays a major role in the in deciding the resolution as well also the focus so if we can able to correct the CS then obviously we can get better resolution better the information in the hallucinatory images and that is what has been done by a scientist called heroes in age rose in 1990 he proposed that is possible to correct the CS value in an electron microscope okay by using a set of spherical lenses and hexapols and this was a major discovery major-major discovery in a field of electron microscopy as you know in a normal camera we can basically correct the spherical aberrations by using a set of lenses okay by basically diverging lens type we can basically do this change over so in a nutshell this is what I would like to tell you about hallucinatory microscopy and I have basically started from a normal camera and how the transfer function can change in electron microscopes what are the factors controlling the transfer function and how to actually optimize them now as I have says to show you several lectures several slides actually in the last lecture and in the today's lecture also that CS the spherical aberration constant plays a very important role in deciding the resolution of the term is electron microscope rather to break the resolution limit of the term is electron microscope requires the CS to be corrected and this started in 1990 by the classic paper published by H rose from an optic as you know that in a normal camera we can correct the spherical aberration by putting a set of converging diverging lens now in electron microscope this can be done by using set of spherical and hexapols that is what is shown in the slide so as you can see this is the standard optic objective lens this one the standard objective lens and the electron beams from there is passes through a set of spherical lens and except all lens and by this when it passes through all this actually except all lens and do not affect the paraxial path of the rex okay but it corrects the aberration in fact which has been reported that the it is possible to actually have spherical aberration constant negative and by getting that one can actually break the resolution barrier of one Armstrong and this is what has been done later on by many scientists okay and by professor noturban and others in in the answers cross-center of electron microscope in in Germany they have been successful in making such a character possible and by putting characters below the orbital lens and other orbital lens where both of the the was called probe and the image characters can be inserted and such insertion of the characters leads to tremendous change of resolution so as you can see if the CS is corrected the resolution of the microscope will be dependent on only on the the the chromatic aberration so therefore resolution is basically can be written as when CS is corrected is like this ? e by e and ? cc this is the chromatic aberration to the power half so when CS is corrected and ? a is basically 0.3 v a 200 kV microscope and you can we can clearly see DCS will be approximately 0.8 Armstrong in fact if CC is corrected or the chromatic aberration character which is going on now and in the theme project in US it is possible to reach when CC is corrected also DCS and CC is possible to reach 0.28 Armstrong so you can imagine that by using all kind of characters we can actually reach a limit of 0.3 approximately 0.3 Armstrong which is very very close to the smallest atomic size the the atomic diameter of carbon atom is 0.89 Armstrong so atomic diameter of hydrogen atom is 0.5 Armstrong so we will be able to reach a resolution limit of less than the size of an hydrogen atom that is the dream which is once needs to cherish but obviously such a dream will come at a very high cost maybe once the I know that the CS character microscope cost about several million dollars actually about 10 to 12 million dollars so once CS and CC characters boats are actually inserted in a time select a microscope length of the microscope will be three storey building and such a microscope operating such a microscope requires help of computers without computers one cannot actually operate such a microscope well this is all are getting done and in fact CS character microscopes are now available in India ID can please also going to get a CS character microscope very soon item address and many other places like TIFR Mumbai and the GN CSR Bangalore has also got CS character microscopes and soon there will be many users using the CS character microscopes and maybe in future in India will have CC character microscopes where we will be able to break the solution barrier to 0.3 Armstrong then we can see whatever you want to see for the atoms of any sizes can be easily matched and the as I have shown you that in the lot of lectures in the first class that if you have abrasion car CS character here the image shows that change as you can see if we correct CS we can clearly see the dumbbells silicon dumbbells so that means we can clearly achieve the solution of 0.8 Armstrong which is sub-Amsterdam resolution and from 0.8 to 0.3 will take us to a different domain of world or picture so by using simply CS character microscope as I shown you in the first in the first lecture HRTM even we can image the oxygen atoms in the Stonseam titanate it has been reported by GI at all in science 2003 and now it is a routine things people can actually image oxygen nitrogen carbon atoms even which are very small using because resolution limits are rich that is value 0.8 is very easy to achieve now I do not want to go into all kinds of numbers but let me show you how even 1970s people used to do electromicroscope and get hallucinomicroscope this was taken from the van dyke at all of the group of amylings and the Pantandalu in Belgium long back it is done in 1992 even before the characters have come into picture this is the 80 m dark fill image of AU4 MN what you can see here all the black and white dots of atoms which you do not know which one is what but you can see the anti-phase domain boundaries marked by this white circles also you can see lot of distortions huge number of distortion in the picture next one is taken from a set of pictures by many of us one of some of us is our so we can actually image gain bound with germanium you can clearly see it consists of the dislocations at different intervals each please one has one is dislocations low angle gain boundary characteristic of parallel set of dislocations you can see that gain boundary of SN3 Si3 and 4 silicon nitride as amorphous layer on the very small thickness about say 0.5 Ampster 0.5 nanometer sorry and then phase boundaries between NiO Ni N2O4 nickel aluminide which is spindle is very flat and very sharp last one D is basically profile image all along 0.001 surface of hematite where you can see the black and white dots of iron this was used to be the case in 1990s till 1990s was advent of CS character microscope has made a lot of changes and it was possible now to pinpoint which is what and to the resolution of the label less than amsterm so therefore basically this all nice and fine you can easily routinely get this in CS character microscopes but the real problem comes how to interpret the images or how to really you know they get the information from the images that is where the real challenge as I said the real handy cup solution to microscopy is to interpret the images or interpretation of images requires simulations simulation needs lot of you know apparently knowledge as per the crystal structure of the material is concerned at wing positions are concerned also interaction of the electrons with the sample how this interaction interaction is dynamic as we have seen so in the next class for I will just describe this all the simulation techniques and then I will move on to much advanced techniques like stem scanning transmissive electron microscopy where we can obtain even in normal routinely microscopes convincing my electron microscope like the one which I have shown in you in the beginning of this course it is possible to attain achieve different other kinds of information like jet contrast or the atomic number contrast information or even you can actually take highlights images using these stem features to obtain or to see the heavy atoms presents in a particular material which these are all discussions we will do in the next class.